Tight binding - Wikipedia

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class="vector-toc-numb">2</span> <span>Historical background</span> </div> </a> <ul id="toc-Historical_background-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematical_formulation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathematical_formulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Mathematical formulation</span> </div> </a> <button aria-controls="toc-Mathematical_formulation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Mathematical formulation subsection</span> </button> <ul id="toc-Mathematical_formulation-sublist" class="vector-toc-list"> <li id="toc-Translational_symmetry_and_normalization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Translational_symmetry_and_normalization"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Translational symmetry and normalization</span> </div> </a> <ul id="toc-Translational_symmetry_and_normalization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_tight_binding_Hamiltonian" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_tight_binding_Hamiltonian"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>The tight binding Hamiltonian</span> </div> </a> <ul id="toc-The_tight_binding_Hamiltonian-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_tight_binding_matrix_elements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_tight_binding_matrix_elements"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>The tight binding matrix elements</span> </div> </a> <ul id="toc-The_tight_binding_matrix_elements-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Evaluation_of_the_matrix_elements" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Evaluation_of_the_matrix_elements"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Evaluation of the matrix elements</span> </div> </a> <ul id="toc-Evaluation_of_the_matrix_elements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_to_Wannier_functions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Connection_to_Wannier_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Connection to Wannier functions</span> </div> </a> <ul id="toc-Connection_to_Wannier_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Second_quantization" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Second_quantization"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Second quantization</span> </div> </a> <ul id="toc-Second_quantization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example:_one-dimensional_s-band" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Example:_one-dimensional_s-band"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Example: one-dimensional s-band</span> </div> </a> <ul id="toc-Example:_one-dimensional_s-band-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Table_of_interatomic_matrix_elements" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Table_of_interatomic_matrix_elements"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Table of interatomic matrix elements</span> </div> </a> <ul id="toc-Table_of_interatomic_matrix_elements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" 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href="https://de.wikipedia.org/wiki/Tight-Binding-Methode" title="Tight-Binding-Methode – German" lang="de" hreflang="de" data-title="Tight-Binding-Methode" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Enlace_fuerte" title="Enlace fuerte – Spanish" lang="es" hreflang="es" data-title="Enlace fuerte" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AF%D9%84_%D8%AA%D9%86%DA%AF%E2%80%8C%D8%A8%D8%B3%D8%AA" title="مدل تنگ‌بست – Persian" lang="fa" hreflang="fa" data-title="مدل تنگ‌بست" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B0%80%EC%A0%91_%EA%B2%B0%ED%95%A9_%EA%B7%BC%EC%82%AC" title="밀접 결합 근사 – Korean" lang="ko" hreflang="ko" data-title="밀접 결합 근사" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Metodo_Tight_Binding" title="Metodo Tight Binding – Italian" lang="it" hreflang="it" data-title="Metodo Tight Binding" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%95%D7%93%D7%9C_%D7%94%D7%A7%D7%A9%D7%99%D7%A8%D7%94_%D7%94%D7%94%D7%93%D7%95%D7%A7%D7%94" title="מודל הקשירה ההדוקה – Hebrew" lang="he" hreflang="he" data-title="מודל הקשירה ההדוקה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Tight_Binding_Model" title="Tight Binding Model – Dutch" lang="nl" hreflang="nl" data-title="Tight Binding Model" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%BC%B7%E7%B5%90%E5%90%88%E8%BF%91%E4%BC%BC" title="強結合近似 – Japanese" lang="ja" hreflang="ja" data-title="強結合近似" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D0%B1%D0%BB%D0%B8%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5_%D1%81%D0%B8%D0%BB%D1%8C%D0%BD%D0%BE_%D1%81%D0%B2%D1%8F%D0%B7%D0%B0%D0%BD%D0%BD%D1%8B%D1%85_%D1%8D%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BE%D0%BD%D0%BE%D0%B2" title="Приближение сильно связанных электронов – Russian" lang="ru" hreflang="ru" data-title="Приближение сильно связанных электронов" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%B0%D0%B1%D0%BB%D0%B8%D0%B6%D0%B5%D0%BD%D0%BD%D1%8F_%D1%81%D0%B8%D0%BB%D1%8C%D0%BD%D0%BE%D0%B3%D0%BE_%D0%B7%D0%B2%27%D1%8F%D0%B7%D0%BA%D1%83" title="Наближення сильного зв'язку – Ukrainian" lang="uk" hreflang="uk" data-title="Наближення сильного зв'язку" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="/wiki/Electronic_structure" class="mw-redirect" title="Electronic structure">Electronic structure</a> methods</th></tr><tr><th class="sidebar-heading" style="background-color: #ddd"> <a href="/wiki/Valence_bond_theory" title="Valence bond theory">Valence bond theory</a></th></tr><tr><td class="sidebar-content"> <a href="/wiki/Coulson%E2%80%93Fischer_theory" title="Coulson–Fischer theory">Coulson–Fischer theory</a><br /><a href="/wiki/Generalized_valence_bond" title="Generalized valence bond">Generalized valence bond</a><br /><a href="/wiki/Modern_valence_bond_theory" title="Modern valence bond theory">Modern valence bond theory</a></td> </tr><tr><th class="sidebar-heading" style="background-color: #ddd"> <a href="/wiki/Molecular_orbital_theory" title="Molecular orbital theory">Molecular orbital theory</a></th></tr><tr><td class="sidebar-content"> <a href="/wiki/Hartree%E2%80%93Fock_method" title="Hartree–Fock method">Hartree–Fock method</a><br /><a href="/wiki/Semi-empirical_quantum_chemistry_method" title="Semi-empirical quantum chemistry method">Semi-empirical quantum chemistry methods</a><br /><a href="/wiki/M%C3%B8ller%E2%80%93Plesset_perturbation_theory" title="Møller–Plesset perturbation theory">Møller–Plesset perturbation theory</a><br /><a href="/wiki/Configuration_interaction" title="Configuration interaction">Configuration interaction</a><br /><a href="/wiki/Coupled_cluster" title="Coupled cluster">Coupled cluster</a><br /><a href="/wiki/Multi-configurational_self-consistent_field" title="Multi-configurational self-consistent field">Multi-configurational self-consistent field</a><br /><a href="/wiki/Quantum_chemistry_composite_methods" title="Quantum chemistry composite methods">Quantum chemistry composite methods</a><br /><a href="/wiki/Quantum_Monte_Carlo" title="Quantum Monte Carlo">Quantum Monte Carlo</a></td> </tr><tr><th class="sidebar-heading" style="background-color: #ddd"> <a href="/wiki/Density_functional_theory" title="Density functional theory">Density functional theory</a></th></tr><tr><td class="sidebar-content"> <a href="/wiki/Time-dependent_density_functional_theory" title="Time-dependent density functional theory">Time-dependent density functional theory</a><br /><a href="/wiki/Thomas%E2%80%93Fermi_model" title="Thomas–Fermi model">Thomas–Fermi model</a><br /><a href="/wiki/Orbital-free_density_functional_theory" title="Orbital-free density functional theory">Orbital-free density functional theory</a><br /><a href="/wiki/Linearized_augmented-plane-wave_method" title="Linearized augmented-plane-wave method">Linearized augmented-plane-wave method</a><br /><a href="/wiki/Projector_augmented_wave_method" title="Projector augmented wave method">Projector augmented wave method</a></td> </tr><tr><th class="sidebar-heading" style="background-color: #ddd"> <a href="/wiki/Electronic_band_structure" title="Electronic band structure">Electronic band structure</a></th></tr><tr><td class="sidebar-content"> <a href="/wiki/Nearly_free_electron_model" title="Nearly free electron model">Nearly free electron model</a><br /><a class="mw-selflink selflink">Tight binding</a><br /><a href="/wiki/Muffin-tin_approximation" title="Muffin-tin approximation">Muffin-tin approximation</a><br /><a href="/wiki/K%C2%B7p_perturbation_theory" title="K·p perturbation theory">k·p perturbation theory</a><br /><a href="/wiki/Empty_lattice_approximation" title="Empty lattice approximation">Empty lattice approximation</a><br /><a href="/wiki/GW_approximation" title="GW approximation">GW approximation</a><br /><a href="/wiki/Korringa%E2%80%93Kohn%E2%80%93Rostoker_method" title="Korringa–Kohn–Rostoker method">Korringa–Kohn–Rostoker method</a></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Electronic_structure_methods" title="Template:Electronic structure methods"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Electronic_structure_methods" title="Template talk:Electronic structure methods"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Electronic_structure_methods" title="Special:EditPage/Template:Electronic structure methods"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Solid-state_physics" title="Solid-state physics">solid-state physics</a>, the <b>tight-binding model</b> (or <b>TB model</b>) is an approach to the calculation of <a href="/wiki/Electronic_band_structure" title="Electronic band structure">electronic band structure</a> using an approximate set of <a href="/wiki/Wave_function" title="Wave function">wave functions</a> based upon <a href="/wiki/Quantum_superposition" title="Quantum superposition">superposition</a> of wave functions for isolated <a href="/wiki/Atom" title="Atom">atoms</a> located at each atomic site. The method is closely related to the <a href="/wiki/Linear_combination_of_atomic_orbitals" title="Linear combination of atomic orbitals">LCAO method</a> (linear combination of atomic orbitals method) used in chemistry. Tight-binding models are applied to a wide variety of solids. The model gives good qualitative results in many cases and can be combined with other models that give better results where the tight-binding model fails. Though the tight-binding model is a one-electron model, the model also provides a basis for more advanced calculations like the calculation of <a href="/wiki/Surface_states" title="Surface states">surface states</a> and application to various kinds of <a href="/wiki/Many-body_problem" title="Many-body problem">many-body problem</a> and <a href="/wiki/Quasiparticle" title="Quasiparticle">quasiparticle</a> calculations. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tight_binding&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The name "tight binding" of this <a href="/wiki/Electronic_band_structure" title="Electronic band structure">electronic band structure model</a> suggests that this <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanical model</a> describes the properties of tightly bound electrons in solids. The <a href="/wiki/Electron" title="Electron">electrons</a> in this model should be tightly bound to the <a href="/wiki/Atom" title="Atom">atom</a> to which they belong and they should have limited interaction with <a href="/wiki/Quantum_state" title="Quantum state">states</a> and potentials on surrounding atoms of the solid. As a result, the <a href="/wiki/Wave_function" title="Wave function">wave function</a> of the electron will be rather similar to the <a href="/wiki/Atomic_orbital" title="Atomic orbital">atomic orbital</a> of the free atom to which it belongs. The energy of the electron will also be rather close to the <a href="/wiki/Ionization_energy" title="Ionization energy">ionization energy</a> of the electron in the free atom or ion because the interaction with potentials and states on neighboring atoms is limited. </p><p>Though the mathematical formulation<sup id="cite_ref-SlaterKoster_1-0" class="reference"><a href="#cite_note-SlaterKoster-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> of the one-particle tight-binding <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> may look complicated at first glance, the model is not complicated at all and can be understood intuitively quite easily. There are only <a href="#The_tight_binding_matrix_elements">three kinds of matrix elements</a> that play a significant role in the theory. Two of those three kinds of elements should be close to zero and can often be neglected. The most important elements in the model are the interatomic matrix elements, which would simply be called the <a href="/wiki/Bond_energy" title="Bond energy">bond energies</a> by a chemist. </p><p>In general there are a number of <a href="/wiki/Energy_level" title="Energy level">atomic energy levels</a> and atomic orbitals involved in the model. This can lead to complicated band structures because the orbitals belong to different <a href="/wiki/Point_groups_in_three_dimensions" title="Point groups in three dimensions">point-group</a> representations. The <a href="/wiki/Reciprocal_lattice" title="Reciprocal lattice">reciprocal lattice</a> and the <a href="/wiki/Brillouin_zone" title="Brillouin zone">Brillouin zone</a> often belong to a different <a href="/wiki/Space_group" title="Space group">space group</a> than the <a href="/wiki/Crystal_structure" title="Crystal structure">crystal</a> of the solid. High-symmetry points in the Brillouin zone belong to different point-group representations. When simple systems like the lattices of elements or simple compounds are studied it is often not very difficult to calculate eigenstates in high-symmetry points analytically. So the tight-binding model can provide nice examples for those who want to learn more about <a href="/wiki/Group_theory" title="Group theory">group theory</a>. </p><p>The tight-binding model has a long history and has been applied in many ways and with many different purposes and different outcomes. The model doesn't stand on its own. Parts of the model can be filled in or extended by other kinds of calculations and models like the <a href="/wiki/Nearly-free_electron_model" class="mw-redirect" title="Nearly-free electron model">nearly-free electron model</a>. The model itself, or parts of it, can serve as the basis for other calculations.<sup id="cite_ref-Harrison_2-0" class="reference"><a href="#cite_note-Harrison-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In the study of <a href="/wiki/Conductive_polymer" title="Conductive polymer">conductive polymers</a>, <a href="/wiki/Organic_semiconductor" title="Organic semiconductor">organic semiconductors</a> and <a href="/wiki/Molecular_electronics" title="Molecular electronics">molecular electronics</a>, for example, tight-binding-like models are applied in which the role of the atoms in the original concept is replaced by the <a href="/wiki/Molecular_orbital" title="Molecular orbital">molecular orbitals</a> of <a href="/wiki/Conjugated_system" title="Conjugated system">conjugated systems</a> and where the interatomic matrix elements are replaced by inter- or intramolecular hopping and <a href="/wiki/Quantum_tunneling" class="mw-redirect" title="Quantum tunneling">tunneling</a> parameters. These conductors nearly all have very anisotropic properties and sometimes are almost perfectly one-dimensional. </p> <div class="mw-heading mw-heading2"><h2 id="Historical_background">Historical background</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tight_binding&action=edit&section=2" title="Edit section: Historical background"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By 1928, the idea of a molecular orbital had been advanced by <a href="/wiki/Robert_S._Mulliken" title="Robert S. Mulliken">Robert Mulliken</a>, who was influenced considerably by the work of <a href="/wiki/Friedrich_Hund" title="Friedrich Hund">Friedrich Hund</a>. The LCAO method for approximating molecular orbitals was introduced in 1928 by B. N. Finklestein and G. E. Horowitz, while the LCAO method for solids was developed by <a href="/wiki/Felix_Bloch" title="Felix Bloch">Felix Bloch</a>, as part of his doctoral dissertation in 1928, concurrently with and independent of the LCAO-MO approach. A much simpler interpolation scheme for approximating the electronic band structure, especially for the d-bands of <a href="/wiki/Transition_metal" title="Transition metal">transition metals</a>, is the parameterized tight-binding method conceived in 1954 by <a href="/wiki/John_C._Slater" title="John C. Slater">John Clarke Slater</a> and <a href="/wiki/George_Fred_Koster" class="mw-redirect" title="George Fred Koster">George Fred Koster</a>,<sup id="cite_ref-SlaterKoster_1-1" class="reference"><a href="#cite_note-SlaterKoster-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> sometimes referred to as the <a href="#Table_of_interatomic_matrix_elements">SK tight-binding method</a>. With the SK tight-binding method, electronic band structure calculations on a solid need not be carried out with full rigor as in the original <a href="/wiki/Bloch%27s_theorem" title="Bloch's theorem">Bloch's theorem</a> but, rather, first-principles calculations are carried out only at high-symmetry points and the band structure is interpolated over the remainder of the <a href="/wiki/Brillouin_zone" title="Brillouin zone">Brillouin zone</a> between these points. </p><p>In this approach, interactions between different atomic sites are considered as <a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbations</a>. There exist several kinds of interactions we must consider. The crystal <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> is only approximately a sum of atomic Hamiltonians located at different sites and atomic wave functions overlap adjacent atomic sites in the crystal, and so are not accurate representations of the exact wave function. There are further explanations in the next section with some mathematical expressions. </p><p>In the recent research about <a href="/wiki/Strongly_correlated_material" title="Strongly correlated material">strongly correlated material</a> the tight binding approach is basic approximation because highly localized electrons like 3-d <a href="/wiki/Transition_metal" title="Transition metal">transition metal</a> electrons sometimes display strongly correlated behaviors. In this case, the role of electron-electron interaction must be considered using the <a href="/wiki/Many-body_theory" class="mw-redirect" title="Many-body theory">many-body physics</a> description. </p><p>The tight-binding model is typically used for calculations of <a href="/wiki/Electronic_band_structure" title="Electronic band structure">electronic band structure</a> and <a href="/wiki/Band_gap" title="Band gap">band gaps</a> in the static regime. However, in combination with other methods such as the <a href="/wiki/Random_phase_approximation" title="Random phase approximation">random phase approximation</a> (RPA) model, the dynamic response of systems may also be studied. In 2019, Bannwarth et al. introduced the GFN2-xTB method, primarily for the calculation of structures and non-covalent interaction energies.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_formulation">Mathematical formulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tight_binding&action=edit&section=3" title="Edit section: Mathematical formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We introduce the <a href="/wiki/Atomic_orbital" title="Atomic orbital">atomic orbitals</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{m}(\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{m}(\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b656f016ae081ed1bc240d7f37218e22673ec0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.107ex; height:2.843ex;" alt="{\displaystyle \varphi _{m}(\mathbf {r} )}"></span>, which are <a href="/wiki/Eigenfunction" title="Eigenfunction">eigenfunctions</a> of the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\rm {at}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\rm {at}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d802985cd4660448a482cbecc7997acc2e003997" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.625ex; height:2.509ex;" alt="{\displaystyle H_{\rm {at}}}"></span> of a single isolated atom. When the atom is placed in a crystal, this atomic wave function overlaps adjacent atomic sites, and so are not true eigenfunctions of the crystal Hamiltonian. The overlap is less when electrons are tightly bound, which is the source of the descriptor "tight-binding". Any corrections to the atomic potential <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4716a2c49bbbe155e8b399117ca78342e802cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.718ex; height:2.176ex;" alt="{\displaystyle \Delta U}"></span> required to obtain the true Hamiltonian <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> of the system, are assumed small: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(\mathbf {r} )=H_{\mathrm {at} }(\mathbf {r} )+\sum _{\mathbf {R} _{n}\neq \mathbf {0} }V(\mathbf {r} -\mathbf {R} _{n})=H_{\mathrm {at} }(\mathbf {r} )+\Delta U(\mathbf {r} )\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mrow> </munder> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(\mathbf {r} )=H_{\mathrm {at} }(\mathbf {r} )+\sum _{\mathbf {R} _{n}\neq \mathbf {0} }V(\mathbf {r} -\mathbf {R} _{n})=H_{\mathrm {at} }(\mathbf {r} )+\Delta U(\mathbf {r} )\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e94d9ced2884456ec643da29b8bc7dc7aa8f1a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:53.536ex; height:6.009ex;" alt="{\displaystyle H(\mathbf {r} )=H_{\mathrm {at} }(\mathbf {r} )+\sum _{\mathbf {R} _{n}\neq \mathbf {0} }V(\mathbf {r} -\mathbf {R} _{n})=H_{\mathrm {at} }(\mathbf {r} )+\Delta U(\mathbf {r} )\ ,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\mathbf {r} -\mathbf {R} _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(\mathbf {r} -\mathbf {R} _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d914ae640303458863b323e4ee48b3babebb19e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.761ex; height:2.843ex;" alt="{\displaystyle V(\mathbf {r} -\mathbf {R} _{n})}"></span> denotes the atomic potential of one atom located at site <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/997e29b5d369a92ec54ebbbadb9a21cff537e336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.222ex; height:2.509ex;" alt="{\displaystyle \mathbf {R} _{n}}"></span> in the <a href="/wiki/Crystal_structure" title="Crystal structure">crystal lattice</a>. A solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed229b721adfd0e48daa75f714788c1c7e37903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.188ex; height:2.509ex;" alt="{\displaystyle \psi _{m}}"></span> to the time-independent single electron <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> is then approximated as a <a href="/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_method" class="mw-redirect" title="Linear combination of atomic orbitals molecular orbital method">linear combination of atomic orbitals</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{m}(\mathbf {r-R_{n}} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> <mo mathvariant="bold">−</mo> <msub> <mi mathvariant="bold">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{m}(\mathbf {r-R_{n}} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fc3e902200625913df6eed61144aac7ab585cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.502ex; height:2.843ex;" alt="{\displaystyle \varphi _{m}(\mathbf {r-R_{n}} )}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{m}(\mathbf {r} )=\sum _{\mathbf {R} _{n}}b_{m}(\mathbf {R} _{n})\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{m}(\mathbf {r} )=\sum _{\mathbf {R} _{n}}b_{m}(\mathbf {R} _{n})\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51d0f7a4595e6ce4f3020f48e138fb1185a5fae4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.393ex; height:5.843ex;" alt="{\displaystyle \psi _{m}(\mathbf {r} )=\sum _{\mathbf {R} _{n}}b_{m}(\mathbf {R} _{n})\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n})}"></span>,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> refers to the m-th atomic energy level. </p> <div class="mw-heading mw-heading3"><h3 id="Translational_symmetry_and_normalization">Translational symmetry and normalization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tight_binding&action=edit&section=4" title="Edit section: Translational symmetry and normalization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Bloch_theorem" class="mw-redirect" title="Bloch theorem">Bloch theorem</a> states that the wave function in a crystal can change under translation only by a phase factor: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mathbf {r+R_{\ell }} )=e^{i\mathbf {k\cdot R_{\ell }} }\psi (\mathbf {r} )\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> <mo mathvariant="bold">+</mo> <msub> <mi mathvariant="bold">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> <mo>⋅</mo> <msub> <mi mathvariant="bold">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r+R_{\ell }} )=e^{i\mathbf {k\cdot R_{\ell }} }\psi (\mathbf {r} )\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13cb1d23ab46bf25edbdee76746ba27118b9fe32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.682ex; height:3.176ex;" alt="{\displaystyle \psi (\mathbf {r+R_{\ell }} )=e^{i\mathbf {k\cdot R_{\ell }} }\psi (\mathbf {r} )\ ,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {k} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea699cbc1f843f2e855577d57529430ec33a1ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.176ex;" alt="{\displaystyle \mathbf {k} }"></span> is the <a href="/wiki/Wave_vector" title="Wave vector">wave vector</a> of the wave function. Consequently, the coefficients satisfy </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\mathbf {R} _{n}}b_{m}(\mathbf {R} _{n})\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n}+\mathbf {R} _{\ell })=e^{i\mathbf {k} \cdot \mathbf {R} _{\ell }}\sum _{\mathbf {R} _{n}}b_{m}(\mathbf {R} _{n})\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n})\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\mathbf {R} _{n}}b_{m}(\mathbf {R} _{n})\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n}+\mathbf {R} _{\ell })=e^{i\mathbf {k} \cdot \mathbf {R} _{\ell }}\sum _{\mathbf {R} _{n}}b_{m}(\mathbf {R} _{n})\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n})\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43a00500ef45145a8ba5d48ae2f0b62bd41dcd5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:64.34ex; height:5.843ex;" alt="{\displaystyle \sum _{\mathbf {R} _{n}}b_{m}(\mathbf {R} _{n})\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n}+\mathbf {R} _{\ell })=e^{i\mathbf {k} \cdot \mathbf {R} _{\ell }}\sum _{\mathbf {R} _{n}}b_{m}(\mathbf {R} _{n})\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n})\ .}"></span></dd></dl> <p>By substituting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} _{p}=\mathbf {R} _{n}-\mathbf {R_{\ell }} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} _{p}=\mathbf {R} _{n}-\mathbf {R_{\ell }} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/300538361fdee2c0e58eac0e63283465261a8cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.144ex; height:2.843ex;" alt="{\displaystyle \mathbf {R} _{p}=\mathbf {R} _{n}-\mathbf {R_{\ell }} }"></span>, we find </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{m}(\mathbf {R} _{p}+\mathbf {R} _{\ell })=e^{i\mathbf {k\cdot R_{\ell }} }b_{m}(\mathbf {R} _{p})\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> <mo>⋅</mo> <msub> <mi mathvariant="bold">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> </mrow> </msup> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{m}(\mathbf {R} _{p}+\mathbf {R} _{\ell })=e^{i\mathbf {k\cdot R_{\ell }} }b_{m}(\mathbf {R} _{p})\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/286dbef1245b0f7324dc21122f3b95e21e872c18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.652ex; height:3.343ex;" alt="{\displaystyle b_{m}(\mathbf {R} _{p}+\mathbf {R} _{\ell })=e^{i\mathbf {k\cdot R_{\ell }} }b_{m}(\mathbf {R} _{p})\ ,}"></span> (where in RHS we have replaced the dummy index <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/997e29b5d369a92ec54ebbbadb9a21cff537e336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.222ex; height:2.509ex;" alt="{\displaystyle \mathbf {R} _{n}}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/831239c1a8dc373ca12d618a52f589155dfd6a79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.062ex; height:2.843ex;" alt="{\displaystyle \mathbf {R} _{p}}"></span>)</dd></dl> <p>or </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{m}(\mathbf {R} _{\ell })=e^{i\mathbf {k} \cdot \mathbf {R} _{\ell }}b_{m}(\mathbf {0} )\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> </msup> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{m}(\mathbf {R} _{\ell })=e^{i\mathbf {k} \cdot \mathbf {R} _{\ell }}b_{m}(\mathbf {0} )\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54cd80010e1bb2dc6f87b024a40ea675fa15fef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.023ex; height:3.176ex;" alt="{\displaystyle b_{m}(\mathbf {R} _{\ell })=e^{i\mathbf {k} \cdot \mathbf {R} _{\ell }}b_{m}(\mathbf {0} )\ .}"></span></dd></dl> <p><a href="/wiki/Normalizable_wave_function" class="mw-redirect" title="Normalizable wave function">Normalizing</a> the wave function to unity: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int d^{3}r\ \psi _{m}^{*}(\mathbf {r} )\psi _{m}(\mathbf {r} )=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mtext> </mtext> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int d^{3}r\ \psi _{m}^{*}(\mathbf {r} )\psi _{m}(\mathbf {r} )=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/304aadc4da45ee5fd26fd9d8384f03be832171e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.942ex; height:5.676ex;" alt="{\displaystyle \int d^{3}r\ \psi _{m}^{*}(\mathbf {r} )\psi _{m}(\mathbf {r} )=1}"></span></dd></dl> <dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\sum _{\mathbf {R_{\ell }} }b_{m}(\mathbf {R_{\ell }} )\int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\varphi _{m}(\mathbf {r} -\mathbf {R} _{\ell })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mtext> </mtext> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\sum _{\mathbf {R_{\ell }} }b_{m}(\mathbf {R_{\ell }} )\int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\varphi _{m}(\mathbf {r} -\mathbf {R} _{\ell })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe71b4c81a45a31e8ee03fb3c33c5e31de4dfcc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:56.337ex; height:6.843ex;" alt="{\displaystyle =\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\sum _{\mathbf {R_{\ell }} }b_{m}(\mathbf {R_{\ell }} )\int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\varphi _{m}(\mathbf {r} -\mathbf {R} _{\ell })}"></span></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =b_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k\cdot R_{n}} }\sum _{\mathbf {R_{\ell }} }e^{i\mathbf {k\cdot R_{\ell }} }\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\varphi _{m}(\mathbf {r} -\mathbf {R} _{\ell })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> <mo>⋅</mo> <msub> <mi mathvariant="bold">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </msub> </mrow> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> <mo>⋅</mo> <msub> <mi mathvariant="bold">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> </mrow> </msup> <mtext> </mtext> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mtext> </mtext> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =b_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k\cdot R_{n}} }\sum _{\mathbf {R_{\ell }} }e^{i\mathbf {k\cdot R_{\ell }} }\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\varphi _{m}(\mathbf {r} -\mathbf {R} _{\ell })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96f23f58dadf20405d332c45241e9cbe359fe7a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:66.013ex; height:6.843ex;" alt="{\displaystyle =b_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k\cdot R_{n}} }\sum _{\mathbf {R_{\ell }} }e^{i\mathbf {k\cdot R_{\ell }} }\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\varphi _{m}(\mathbf {r} -\mathbf {R} _{\ell })}"></span></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =Nb_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R} _{p}}e^{-i\mathbf {k} \cdot \mathbf {R} _{p}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{p})\varphi _{m}(\mathbf {r} )\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>N</mi> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </msup> <mtext> </mtext> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mtext> </mtext> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =Nb_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R} _{p}}e^{-i\mathbf {k} \cdot \mathbf {R} _{p}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{p})\varphi _{m}(\mathbf {r} )\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79be6bbe1542fa4d74c5797ad060a69709927939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:52.95ex; height:6.843ex;" alt="{\displaystyle =Nb_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R} _{p}}e^{-i\mathbf {k} \cdot \mathbf {R} _{p}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{p})\varphi _{m}(\mathbf {r} )\ }"></span></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =Nb_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R} _{p}}e^{i\mathbf {k} \cdot \mathbf {R} _{p}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} )\varphi _{m}(\mathbf {r} -\mathbf {R} _{p})\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>N</mi> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </msup> <mtext> </mtext> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mtext> </mtext> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =Nb_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R} _{p}}e^{i\mathbf {k} \cdot \mathbf {R} _{p}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} )\varphi _{m}(\mathbf {r} -\mathbf {R} _{p})\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eabc84da6d5b23caced01e32a30ced2df4009f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:52.319ex; height:6.843ex;" alt="{\displaystyle =Nb_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R} _{p}}e^{i\mathbf {k} \cdot \mathbf {R} _{p}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} )\varphi _{m}(\mathbf {r} -\mathbf {R} _{p})\ ,}"></span></dd></dl></dd></dl></dd></dl> <p>so the normalization sets <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{m}(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{m}(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3da710756281614bc30dcb9e41f1d21ff0b95198" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.644ex; height:2.843ex;" alt="{\displaystyle b_{m}(0)}"></span></i> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{m}^{*}(0)b_{m}(0)={\frac {1}{N}}\ \cdot \ {\frac {1}{1+\sum _{\mathbf {R} _{p}\neq 0}e^{i\mathbf {k} \cdot \mathbf {R} _{p}}\alpha _{m}(\mathbf {R} _{p})}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <mtext> </mtext> <mo>⋅</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </msup> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{m}^{*}(0)b_{m}(0)={\frac {1}{N}}\ \cdot \ {\frac {1}{1+\sum _{\mathbf {R} _{p}\neq 0}e^{i\mathbf {k} \cdot \mathbf {R} _{p}}\alpha _{m}(\mathbf {R} _{p})}}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42558df89ad5bcab07e9f231646eeea105d48203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:47.244ex; height:6.676ex;" alt="{\displaystyle b_{m}^{*}(0)b_{m}(0)={\frac {1}{N}}\ \cdot \ {\frac {1}{1+\sum _{\mathbf {R} _{p}\neq 0}e^{i\mathbf {k} \cdot \mathbf {R} _{p}}\alpha _{m}(\mathbf {R} _{p})}}\ ,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\alpha _{m}(\mathbf {R} _{p})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\alpha _{m}(\mathbf {R} _{p})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0816cfa0777e82efe96a0fcb12af87cf0469fba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.034ex; height:3.009ex;" alt="{\displaystyle {\alpha _{m}(\mathbf {R} _{p})}}"></span> are the atomic overlap integrals, which frequently are neglected resulting in<sup id="cite_ref-Lowdin_4-0" class="reference"><a href="#cite_note-Lowdin-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{m}(0)\approx {\frac {1}{\sqrt {N}}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{m}(0)\approx {\frac {1}{\sqrt {N}}}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb43dce18cc67aef9f3b6b14e5440e3962ea81c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.806ex; height:6.176ex;" alt="{\displaystyle b_{m}(0)\approx {\frac {1}{\sqrt {N}}}\ ,}"></span></dd></dl> <p>and </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{m}(\mathbf {r} )\approx {\frac {1}{\sqrt {N}}}\sum _{\mathbf {R} _{n}}e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n})\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mtext> </mtext> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{m}(\mathbf {r} )\approx {\frac {1}{\sqrt {N}}}\sum _{\mathbf {R} _{n}}e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n})\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df3638c044a32f891af46a73152365a3734cfb8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:37.859ex; height:6.676ex;" alt="{\displaystyle \psi _{m}(\mathbf {r} )\approx {\frac {1}{\sqrt {N}}}\sum _{\mathbf {R} _{n}}e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n})\ .}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="The_tight_binding_Hamiltonian">The tight binding Hamiltonian</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tight_binding&action=edit&section=5" title="Edit section: The tight binding Hamiltonian"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using the tight binding form for the wave function, and assuming only the <i>m-th</i> atomic <a href="/wiki/Energy_level" title="Energy level">energy level</a> is important for the <i>m-th</i> energy band, the Bloch energies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b725dd48d650bf1193424e68469b31f4e0336562" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.759ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{m}}"></span> are of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{m}=\int d^{3}r\ \psi _{m}^{*}(\mathbf {r} )H(\mathbf {r} )\psi _{m}(\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mtext> </mtext> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{m}=\int d^{3}r\ \psi _{m}^{*}(\mathbf {r} )H(\mathbf {r} )\psi _{m}(\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e4d7a2cfdf1b0096693e51482672c363ccdb8c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.514ex; height:5.676ex;" alt="{\displaystyle \varepsilon _{m}=\int d^{3}r\ \psi _{m}^{*}(\mathbf {r} )H(\mathbf {r} )\psi _{m}(\mathbf {r} )}"></span></dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})H(\mathbf {r} )\psi _{m}(\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mtext> </mtext> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})H(\mathbf {r} )\psi _{m}(\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3526456b9416785eaca1426020b19f16101a13d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:44.592ex; height:6.676ex;" alt="{\displaystyle =\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})H(\mathbf {r} )\psi _{m}(\mathbf {r} )}"></span></dd></dl></dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})H_{\mathrm {at} }(\mathbf {r} )\psi _{m}(\mathbf {r} )+\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mtext> </mtext> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mtext> </mtext> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})H_{\mathrm {at} }(\mathbf {r} )\psi _{m}(\mathbf {r} )+\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae390f3498a130ef0c5a07929511a2575c0461b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:92.788ex; height:6.676ex;" alt="{\displaystyle =\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})H_{\mathrm {at} }(\mathbf {r} )\psi _{m}(\mathbf {r} )+\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )}"></span></dd></dl></dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\sum _{\mathbf {R} _{n},\mathbf {R} _{l}}b_{m}^{*}(\mathbf {R} _{n})b_{m}(\mathbf {R} _{l})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})H_{\mathrm {at} }(\mathbf {r} )\varphi _{m}(\mathbf {r} -\mathbf {R} _{l})+b_{m}^{*}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k} \cdot \mathbf {R} _{n}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> </munder> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mtext> </mtext> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mtext> </mtext> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mtext> </mtext> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\sum _{\mathbf {R} _{n},\mathbf {R} _{l}}b_{m}^{*}(\mathbf {R} _{n})b_{m}(\mathbf {R} _{l})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})H_{\mathrm {at} }(\mathbf {r} )\varphi _{m}(\mathbf {r} -\mathbf {R} _{l})+b_{m}^{*}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k} \cdot \mathbf {R} _{n}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f14001d4f2648c0e9cbc1689655ceac9d4781648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:112.357ex; height:6.843ex;" alt="{\displaystyle =\sum _{\mathbf {R} _{n},\mathbf {R} _{l}}b_{m}^{*}(\mathbf {R} _{n})b_{m}(\mathbf {R} _{l})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})H_{\mathrm {at} }(\mathbf {r} )\varphi _{m}(\mathbf {r} -\mathbf {R} _{l})+b_{m}^{*}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k} \cdot \mathbf {R} _{n}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )}"></span></dd></dl></dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =b_{m}^{*}(\mathbf {0} )b_{m}(\mathbf {0} )\ N\int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} )H_{\mathrm {at} }(\mathbf {r} )\varphi _{m}(\mathbf {r} )+b_{m}^{*}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k} \cdot \mathbf {R} _{n}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo stretchy="false">)</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>N</mi> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mtext> </mtext> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mtext> </mtext> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mtext> </mtext> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =b_{m}^{*}(\mathbf {0} )b_{m}(\mathbf {0} )\ N\int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} )H_{\mathrm {at} }(\mathbf {r} )\varphi _{m}(\mathbf {r} )+b_{m}^{*}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k} \cdot \mathbf {R} _{n}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9579d70958bb989eba9ce4c96969b6e992f4e24b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:94.314ex; height:6.676ex;" alt="{\displaystyle =b_{m}^{*}(\mathbf {0} )b_{m}(\mathbf {0} )\ N\int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} )H_{\mathrm {at} }(\mathbf {r} )\varphi _{m}(\mathbf {r} )+b_{m}^{*}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k} \cdot \mathbf {R} _{n}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )}"></span></dd></dl></dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx E_{m}+b_{m}^{*}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k} \cdot \mathbf {R} _{n}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mtext> </mtext> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mtext> </mtext> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx E_{m}+b_{m}^{*}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k} \cdot \mathbf {R} _{n}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63a3d5b062f4af985b383ca1cfe7e3a6ffeef25e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:59.031ex; height:6.676ex;" alt="{\displaystyle \approx E_{m}+b_{m}^{*}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k} \cdot \mathbf {R} _{n}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )\ .}"></span></dd></dl></dd></dl> <p>Here in the last step it was assumed that the overlap integral is zero and thus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{m}^{*}(\mathbf {0} )b_{m}(\mathbf {0} )={\frac {1}{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo stretchy="false">)</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{m}^{*}(\mathbf {0} )b_{m}(\mathbf {0} )={\frac {1}{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00d007d699d7bda3e041e267e860e42e66f6bdfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.635ex; height:5.176ex;" alt="{\displaystyle b_{m}^{*}(\mathbf {0} )b_{m}(\mathbf {0} )={\frac {1}{N}}}"></span>. The energy then becomes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{m}(\mathbf {k} )=E_{m}-N\ |b_{m}(0)|^{2}\left(\beta _{m}+\sum _{\mathbf {R} _{n}\neq 0}\sum _{l}\gamma _{m,l}(\mathbf {R} _{n})e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\right)\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <mi>N</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{m}(\mathbf {k} )=E_{m}-N\ |b_{m}(0)|^{2}\left(\beta _{m}+\sum _{\mathbf {R} _{n}\neq 0}\sum _{l}\gamma _{m,l}(\mathbf {R} _{n})e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\right)\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14c84002bc575d1e425e560972fad9b6bc4006ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:60.647ex; height:7.843ex;" alt="{\displaystyle \varepsilon _{m}(\mathbf {k} )=E_{m}-N\ |b_{m}(0)|^{2}\left(\beta _{m}+\sum _{\mathbf {R} _{n}\neq 0}\sum _{l}\gamma _{m,l}(\mathbf {R} _{n})e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\right)\ ,}"></span> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =E_{m}-\ {\frac {\beta _{m}+\sum _{\mathbf {R} _{n}\neq 0}\sum _{l}e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\gamma _{m,l}(\mathbf {R} _{n})}{\ \ 1+\sum _{\mathbf {R} _{n}\neq 0}\sum _{l}e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\alpha _{m,l}(\mathbf {R} _{n})}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mtext> </mtext> <mtext> </mtext> <mn>1</mn> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =E_{m}-\ {\frac {\beta _{m}+\sum _{\mathbf {R} _{n}\neq 0}\sum _{l}e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\gamma _{m,l}(\mathbf {R} _{n})}{\ \ 1+\sum _{\mathbf {R} _{n}\neq 0}\sum _{l}e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\alpha _{m,l}(\mathbf {R} _{n})}}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d02f771de370be5010d7526285ddef14e4fcf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.855ex; height:7.676ex;" alt="{\displaystyle =E_{m}-\ {\frac {\beta _{m}+\sum _{\mathbf {R} _{n}\neq 0}\sum _{l}e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\gamma _{m,l}(\mathbf {R} _{n})}{\ \ 1+\sum _{\mathbf {R} _{n}\neq 0}\sum _{l}e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\alpha _{m,l}(\mathbf {R} _{n})}}\ ,}"></span></dd></dl></dd></dl></dd></dl> <p>where <i>E</i><sub>m</sub> is the energy of the <i>m</i>-th atomic level, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{m,l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{m,l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d722524d28a9caaade2384cd77899d9b22378d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.11ex; height:2.343ex;" alt="{\displaystyle \alpha _{m,l}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e066616c9ed741f3d70ef4ce962bcef909c4d62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.991ex; height:2.509ex;" alt="{\displaystyle \beta _{m}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{m,l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{m,l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16851d79ec949984f2f943569485ecd4284e2749" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.827ex; height:2.343ex;" alt="{\displaystyle \gamma _{m,l}}"></span> are the tight binding matrix elements discussed below. </p> <div class="mw-heading mw-heading3"><h3 id="The_tight_binding_matrix_elements">The tight binding matrix elements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tight_binding&action=edit&section=6" title="Edit section: The tight binding matrix elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The elements <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{m}=-\int {\varphi _{m}^{*}(\mathbf {r} )\Delta U(\mathbf {r} )\varphi _{m}(\mathbf {r} )\,d^{3}r}{\text{,}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>,</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{m}=-\int {\varphi _{m}^{*}(\mathbf {r} )\Delta U(\mathbf {r} )\varphi _{m}(\mathbf {r} )\,d^{3}r}{\text{,}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e98f7a24d3abd54f1154e0ef4608aa436dcbbd3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.063ex; height:5.676ex;" alt="{\displaystyle \beta _{m}=-\int {\varphi _{m}^{*}(\mathbf {r} )\Delta U(\mathbf {r} )\varphi _{m}(\mathbf {r} )\,d^{3}r}{\text{,}}}"></span> are the atomic energy shift due to the potential on neighboring atoms. This term is relatively small in most cases. If it is large it means that potentials on neighboring atoms have a large influence on the energy of the central atom. </p><p>The next class of terms <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{m,l}(\mathbf {R} _{n})=-\int {\varphi _{m}^{*}(\mathbf {r} )\Delta U(\mathbf {r} )\varphi _{l}(\mathbf {r} -\mathbf {R} _{n})\,d^{3}r}{\text{,}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>,</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{m,l}(\mathbf {R} _{n})=-\int {\varphi _{m}^{*}(\mathbf {r} )\Delta U(\mathbf {r} )\varphi _{l}(\mathbf {r} -\mathbf {R} _{n})\,d^{3}r}{\text{,}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84114b1baa1cf98ea93920535a3f7b8383e14e21" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.04ex; height:5.676ex;" alt="{\displaystyle \gamma _{m,l}(\mathbf {R} _{n})=-\int {\varphi _{m}^{*}(\mathbf {r} )\Delta U(\mathbf {r} )\varphi _{l}(\mathbf {r} -\mathbf {R} _{n})\,d^{3}r}{\text{,}}}"></span> is the <a href="#Table_of_interatomic_matrix_elements">interatomic matrix element</a> between the atomic orbitals <i>m</i> and <i>l</i> on adjacent atoms. It is also called the bond energy or two center integral and it is the dominant term in the tight binding model. </p><p>The last class of terms <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{m,l}(\mathbf {R} _{n})=\int {\varphi _{m}^{*}(\mathbf {r} )\varphi _{l}(\mathbf {r} -\mathbf {R} _{n})\,d^{3}r}{\text{,}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>,</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{m,l}(\mathbf {R} _{n})=\int {\varphi _{m}^{*}(\mathbf {r} )\varphi _{l}(\mathbf {r} -\mathbf {R} _{n})\,d^{3}r}{\text{,}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b3cb10b18d8ff62d74d95a1823a0c223a605b6f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.498ex; height:5.676ex;" alt="{\displaystyle \alpha _{m,l}(\mathbf {R} _{n})=\int {\varphi _{m}^{*}(\mathbf {r} )\varphi _{l}(\mathbf {r} -\mathbf {R} _{n})\,d^{3}r}{\text{,}}}"></span> denote the <a href="/wiki/Overlap_matrix" class="mw-redirect" title="Overlap matrix">overlap integrals</a> between the atomic orbitals <i>m</i> and <i>l</i> on adjacent atoms. These, too, are typically small; if not, then <a href="/wiki/Pauli_repulsion" class="mw-redirect" title="Pauli repulsion">Pauli repulsion</a> has a non-negligible influence on the energy of the central atom. </p> <div class="mw-heading mw-heading2"><h2 id="Evaluation_of_the_matrix_elements">Evaluation of the matrix elements</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tight_binding&action=edit&section=7" title="Edit section: Evaluation of the matrix elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As mentioned before the values of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e066616c9ed741f3d70ef4ce962bcef909c4d62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.991ex; height:2.509ex;" alt="{\displaystyle \beta _{m}}"></span>-matrix elements are not so large in comparison with the ionization energy because the potentials of neighboring atoms on the central atom are limited. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e066616c9ed741f3d70ef4ce962bcef909c4d62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.991ex; height:2.509ex;" alt="{\displaystyle \beta _{m}}"></span> is not relatively small it means that the potential of the neighboring atom on the central atom is not small either. In that case it is an indication that the tight binding model is not a very good model for the description of the band structure for some reason. The interatomic distances can be too small or the charges on the atoms or ions in the lattice is wrong for example. </p><p>The interatomic matrix elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{m,l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{m,l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16851d79ec949984f2f943569485ecd4284e2749" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.827ex; height:2.343ex;" alt="{\displaystyle \gamma _{m,l}}"></span> can be calculated directly if the atomic wave functions and the potentials are known in detail. Most often this is not the case. There are numerous ways to get parameters for these matrix elements. Parameters can be obtained from <a href="/wiki/Bond_energy#External_links" title="Bond energy">chemical bond energy data</a>. Energies and eigenstates on some high symmetry points in the <a href="/wiki/Brillouin_zone" title="Brillouin zone">Brillouin zone</a> can be evaluated and values integrals in the matrix elements can be matched with band structure data from other sources. </p><p>The interatomic overlap matrix elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{m,l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{m,l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d722524d28a9caaade2384cd77899d9b22378d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.11ex; height:2.343ex;" alt="{\displaystyle \alpha _{m,l}}"></span> should be rather small or neglectable. If they are large it is again an indication that the tight binding model is of limited value for some purposes. Large overlap is an indication for too short interatomic distance for example. In metals and transition metals the broad s-band or sp-band can be fitted better to an existing band structure calculation by the introduction of next-nearest-neighbor matrix elements and overlap integrals but fits like that don't yield a very useful model for the electronic wave function of a metal. Broad bands in dense materials are better described by a <a href="/wiki/Nearly_free_electron_model" title="Nearly free electron model">nearly free electron model</a>. </p><p>The tight binding model works particularly well in cases where the band width is small and the electrons are strongly localized, like in the case of d-bands and f-bands. The model also gives good results in the case of open crystal structures, like diamond or silicon, where the number of neighbors is small. The model can easily be combined with a nearly free electron model in a hybrid NFE-TB model.<sup id="cite_ref-Harrison_2-1" class="reference"><a href="#cite_note-Harrison-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Connection_to_Wannier_functions">Connection to Wannier functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tight_binding&action=edit&section=8" title="Edit section: Connection to Wannier functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Bloch%27s_theorem" title="Bloch's theorem">Bloch functions</a> describe the electronic states in a periodic <a href="/wiki/Crystal_structure" title="Crystal structure">crystal lattice</a>. Bloch functions can be represented as a <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{m}(\mathbf {k} ,\mathbf {r} )={\frac {1}{\sqrt {N}}}\sum _{n}{a_{m}(\mathbf {R} _{n},\mathbf {r} )}e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{m}(\mathbf {k} ,\mathbf {r} )={\frac {1}{\sqrt {N}}}\sum _{n}{a_{m}(\mathbf {R} _{n},\mathbf {r} )}e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dde41634c0385d90936c17d395ea0ea80bd3225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.627ex; height:6.343ex;" alt="{\displaystyle \psi _{m}(\mathbf {k} ,\mathbf {r} )={\frac {1}{\sqrt {N}}}\sum _{n}{a_{m}(\mathbf {R} _{n},\mathbf {r} )}e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\ ,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/997e29b5d369a92ec54ebbbadb9a21cff537e336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.222ex; height:2.509ex;" alt="{\displaystyle \mathbf {R} _{n}}"></span> denotes an atomic site in a periodic crystal lattice, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {k} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea699cbc1f843f2e855577d57529430ec33a1ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.176ex;" alt="{\displaystyle \mathbf {k} }"></span> is the <a href="/wiki/Wave_vector" title="Wave vector">wave vector</a> of the Bloch's function, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span> is the electron position, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is the band index, and the sum is over all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> atomic sites. The Bloch's function is an exact eigensolution for the wave function of an electron in a periodic crystal potential corresponding to an energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{m}(\mathbf {k} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{m}(\mathbf {k} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0b8ae7e3077cfc576a3814eed78668e69c6670d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.611ex; height:2.843ex;" alt="{\displaystyle E_{m}(\mathbf {k} )}"></span>, and is spread over the entire crystal volume. </p><p>Using the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> analysis, a spatially localized wave function for the <i>m</i>-th energy band can be constructed from multiple Bloch's functions: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{m}(\mathbf {R} _{n},\mathbf {r} )={\frac {1}{\sqrt {N}}}\sum _{\mathbf {k} }{e^{-i\mathbf {k} \cdot \mathbf {R} _{n}}\psi _{m}(\mathbf {k} ,\mathbf {r} )}={\frac {1}{\sqrt {N}}}\sum _{\mathbf {k} }{e^{i\mathbf {k} \cdot (\mathbf {r} -\mathbf {R} _{n})}u_{m}(\mathbf {k} ,\mathbf {r} )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{m}(\mathbf {R} _{n},\mathbf {r} )={\frac {1}{\sqrt {N}}}\sum _{\mathbf {k} }{e^{-i\mathbf {k} \cdot \mathbf {R} _{n}}\psi _{m}(\mathbf {k} ,\mathbf {r} )}={\frac {1}{\sqrt {N}}}\sum _{\mathbf {k} }{e^{i\mathbf {k} \cdot (\mathbf {r} -\mathbf {R} _{n})}u_{m}(\mathbf {k} ,\mathbf {r} )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0993e5f35905b3a9eceea2d84ab37620cc6d5d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:67.806ex; height:6.343ex;" alt="{\displaystyle a_{m}(\mathbf {R} _{n},\mathbf {r} )={\frac {1}{\sqrt {N}}}\sum _{\mathbf {k} }{e^{-i\mathbf {k} \cdot \mathbf {R} _{n}}\psi _{m}(\mathbf {k} ,\mathbf {r} )}={\frac {1}{\sqrt {N}}}\sum _{\mathbf {k} }{e^{i\mathbf {k} \cdot (\mathbf {r} -\mathbf {R} _{n})}u_{m}(\mathbf {k} ,\mathbf {r} )}.}"></span></dd></dl> <p>These real space wave functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {a_{m}(\mathbf {R} _{n},\mathbf {r} )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a_{m}(\mathbf {R} _{n},\mathbf {r} )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5925ceed9140c8eed82d176e22697143428ad2d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.072ex; height:2.843ex;" alt="{\displaystyle {a_{m}(\mathbf {R} _{n},\mathbf {r} )}}"></span> are called <a href="/wiki/Wannier_function" title="Wannier function">Wannier functions</a>, and are fairly closely localized to the atomic site <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/997e29b5d369a92ec54ebbbadb9a21cff537e336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.222ex; height:2.509ex;" alt="{\displaystyle \mathbf {R} _{n}}"></span>. Of course, if we have exact <a href="/wiki/Wannier_function" title="Wannier function">Wannier functions</a>, the exact Bloch functions can be derived using the inverse Fourier transform. </p><p>However it is not easy to calculate directly either <a href="/wiki/Bloch%27s_theorem" title="Bloch's theorem">Bloch functions</a> or <a href="/wiki/Wannier_function" title="Wannier function">Wannier functions</a>. An approximate approach is necessary in the calculation of <a href="/wiki/Electronic_structure" class="mw-redirect" title="Electronic structure">electronic structures</a> of solids. If we consider the extreme case of isolated atoms, the Wannier function would become an isolated atomic orbital. That limit suggests the choice of an atomic wave function as an approximate form for the Wannier function, the so-called tight binding approximation. </p> <div class="mw-heading mw-heading2"><h2 id="Second_quantization">Second quantization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tight_binding&action=edit&section=9" title="Edit section: Second quantization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Modern explanations of electronic structure like <a href="/wiki/T-J_model" title="T-J model">t-J model</a> and <a href="/wiki/Hubbard_model" title="Hubbard model">Hubbard model</a> are based on tight binding model.<sup id="cite_ref-Altland_6-0" class="reference"><a href="#cite_note-Altland-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Tight binding can be understood by working under a <a href="/wiki/Second_quantization" title="Second quantization">second quantization</a> formalism. </p><p>Using the atomic orbital as a basis state, the second quantization Hamiltonian operator in the tight binding framework can be written as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=-t\sum _{\langle i,j\rangle ,\sigma }(c_{i,\sigma }^{\dagger }c_{j,\sigma }^{}+h.c.)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mo>−</mo> <mi>t</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mi>σ</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo>+</mo> <mi>h</mi> <mo>.</mo> <mi>c</mi> <mo>.</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=-t\sum _{\langle i,j\rangle ,\sigma }(c_{i,\sigma }^{\dagger }c_{j,\sigma }^{}+h.c.)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45a5236d7c7dabc38ccce1069a4edf9464cffbab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:28.158ex; height:6.009ex;" alt="{\displaystyle H=-t\sum _{\langle i,j\rangle ,\sigma }(c_{i,\sigma }^{\dagger }c_{j,\sigma }^{}+h.c.)}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i\sigma }^{\dagger },c_{j\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>σ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i\sigma }^{\dagger },c_{j\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec5886b088eab5c8f260ac7926e719a52da28f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.637ex; height:3.509ex;" alt="{\displaystyle c_{i\sigma }^{\dagger },c_{j\sigma }}"></span> - creation and annihilation operators</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/521ce77b21714ede6aac99d4c01aadc7bf551c2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \displaystyle \sigma }"></span> - spin polarization</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef230e08afa41ab9bf7a8b92dca95c5f230a9bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle \displaystyle t}"></span> - hopping integral</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle \langle i,j\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle \langle i,j\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f61b830547088774f746718d6c32662cbf245c7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle \displaystyle \langle i,j\rangle }"></span> - nearest neighbor index</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle h.c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>.</mo> <mi>c</mi> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle h.c.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7c8ec160c843711a1e8e0639493283214868686" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.027ex; height:2.176ex;" alt="{\displaystyle \displaystyle h.c.}"></span> - the hermitian conjugate of the other term(s)</dd></dl> <p>Here, hopping integral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef230e08afa41ab9bf7a8b92dca95c5f230a9bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle \displaystyle t}"></span> corresponds to the transfer integral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25bf50ec2825620634e2c62f01af10965cd40bef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \displaystyle \gamma }"></span> in tight binding model. Considering extreme cases of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\rightarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\rightarrow 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a7c351847219a0d062a2966bcdd1987b74fa5b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle t\rightarrow 0}"></span>, it is impossible for an electron to hop into neighboring sites. This case is the isolated atomic system. If the hopping term is turned on (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle t>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>></mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle t>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa8ffb393df377e8dd11a6c9205e96551c5481d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle \displaystyle t>0}"></span>) electrons can stay in both sites lowering their <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a>. </p><p>In the strongly correlated electron system, it is necessary to consider the electron-electron interaction. This term can be written in </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle H_{ee}={\frac {1}{2}}\sum _{n,m,\sigma }\langle n_{1}m_{1},n_{2}m_{2}|{\frac {e^{2}}{|r_{1}-r_{2}|}}|n_{3}m_{3},n_{4}m_{4}\rangle c_{n_{1}m_{1}\sigma _{1}}^{\dagger }c_{n_{2}m_{2}\sigma _{2}}^{\dagger }c_{n_{4}m_{4}\sigma _{2}}c_{n_{3}m_{3}\sigma _{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>σ</mi> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle H_{ee}={\frac {1}{2}}\sum _{n,m,\sigma }\langle n_{1}m_{1},n_{2}m_{2}|{\frac {e^{2}}{|r_{1}-r_{2}|}}|n_{3}m_{3},n_{4}m_{4}\rangle c_{n_{1}m_{1}\sigma _{1}}^{\dagger }c_{n_{2}m_{2}\sigma _{2}}^{\dagger }c_{n_{4}m_{4}\sigma _{2}}c_{n_{3}m_{3}\sigma _{1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed3bd7387765cf103777f1959489bd7a6181f71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:78.391ex; height:7.009ex;" alt="{\displaystyle \displaystyle H_{ee}={\frac {1}{2}}\sum _{n,m,\sigma }\langle n_{1}m_{1},n_{2}m_{2}|{\frac {e^{2}}{|r_{1}-r_{2}|}}|n_{3}m_{3},n_{4}m_{4}\rangle c_{n_{1}m_{1}\sigma _{1}}^{\dagger }c_{n_{2}m_{2}\sigma _{2}}^{\dagger }c_{n_{4}m_{4}\sigma _{2}}c_{n_{3}m_{3}\sigma _{1}}}"></span></dd></dl> <p>This interaction Hamiltonian includes direct <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb</a> interaction energy and exchange interaction energy between electrons. There are several novel physics induced from this electron-electron interaction energy, such as <a href="/wiki/Metal-insulator_transition" class="mw-redirect" title="Metal-insulator transition">metal-insulator transitions</a> (MIT), <a href="/wiki/High-temperature_superconductivity" title="High-temperature superconductivity">high-temperature superconductivity</a>, and several <a href="/wiki/Quantum_phase_transition" title="Quantum phase transition">quantum phase transitions</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Example:_one-dimensional_s-band">Example: one-dimensional s-band</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tight_binding&action=edit&section=10" title="Edit section: Example: one-dimensional s-band"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Here the tight binding model is illustrated with a <b>s-band model</b> for a string of atoms with a single <a href="/wiki/Cubic_harmonic#The_s-orbitals" title="Cubic harmonic">s-orbital</a> in a straight line with spacing <i>a</i> and <a href="/wiki/Sigma_bond" title="Sigma bond">σ bonds</a> between atomic sites. </p><p>To find approximate eigenstates of the Hamiltonian, we can use a linear combination of the atomic orbitals </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |k\rangle ={\frac {1}{\sqrt {N}}}\sum _{n=1}^{N}e^{inka}|n\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>k</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>k</mi> <mi>a</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |k\rangle ={\frac {1}{\sqrt {N}}}\sum _{n=1}^{N}e^{inka}|n\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44182f3feab8f937f64570dd9b1c608cdb9b0154" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.368ex; height:7.343ex;" alt="{\displaystyle |k\rangle ={\frac {1}{\sqrt {N}}}\sum _{n=1}^{N}e^{inka}|n\rangle }"></span></dd></dl> <p>where <i>N</i> = total number of sites and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is a real parameter with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\pi }{a}}\leqq k\leqq {\frac {\pi }{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>a</mi> </mfrac> </mrow> <mo>≦</mo> <mi>k</mi> <mo>≦</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\pi }{a}}\leqq k\leqq {\frac {\pi }{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fdfa0a63f096b4848dd51015d534e7dbb0d3d35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.553ex; height:4.676ex;" alt="{\displaystyle -{\frac {\pi }{a}}\leqq k\leqq {\frac {\pi }{a}}}"></span>. (This wave function is normalized to unity by the leading factor 1/√N provided overlap of atomic wave functions is ignored.) Assuming only nearest neighbor overlap, the only non-zero matrix elements of the Hamiltonian can be expressed as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle n|H|n\rangle =E_{0}=E_{i}-U\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>U</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle n|H|n\rangle =E_{0}=E_{i}-U\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da394c371e002f01cfa0251e18922b2ba1256341" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.288ex; height:2.843ex;" alt="{\displaystyle \langle n|H|n\rangle =E_{0}=E_{i}-U\ .}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle n\pm 1|H|n\rangle =-\Delta \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mo>±</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle n\pm 1|H|n\rangle =-\Delta \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fac06669a5dcfbb4f8d964ec17e77b992b1ea68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.382ex; height:2.843ex;" alt="{\displaystyle \langle n\pm 1|H|n\rangle =-\Delta \ }"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle n|n\rangle =1\ ;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>1</mn> <mtext> </mtext> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle n|n\rangle =1\ ;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a1af2ba6d10f397300facfe7f58825727c572b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.734ex; height:2.843ex;" alt="{\displaystyle \langle n|n\rangle =1\ ;}"></span>   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle n\pm 1|n\rangle =S\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mo>±</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>S</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle n\pm 1|n\rangle =S\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e308ba6428e8192b07a322ac0f0c44ea0b29336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.074ex; height:2.843ex;" alt="{\displaystyle \langle n\pm 1|n\rangle =S\ .}"></span></dd></dl> <p>The energy <i>E</i><sub>i</sub> is the ionization energy corresponding to the chosen atomic orbital and <i>U</i> is the energy shift of the orbital as a result of the potential of neighboring atoms. The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle n\pm 1|H|n\rangle =-\Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mo>±</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle n\pm 1|H|n\rangle =-\Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e037f9af463e78261739fd7998b60dee7108c1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.801ex; height:2.843ex;" alt="{\displaystyle \langle n\pm 1|H|n\rangle =-\Delta }"></span> elements, which are the <a href="#Table_of_interatomic_matrix_elements">Slater and Koster interatomic matrix elements</a>, are the <a href="/wiki/Chemical_bond" title="Chemical bond">bond energies</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i,j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7814421833a1b6fae2769bb795158a0c02ecda0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.65ex; height:2.843ex;" alt="{\displaystyle E_{i,j}}"></span>. In this one dimensional s-band model we only have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span>-bonds between the s-orbitals with bond energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{s,s}=V_{ss\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>s</mi> <mi>σ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{s,s}=V_{ss\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe858e9be19cf013027474a66d73afbe1d42bc69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.115ex; height:2.843ex;" alt="{\displaystyle E_{s,s}=V_{ss\sigma }}"></span>. The overlap between states on neighboring atoms is <i>S</i>. We can derive the energy of the state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |k\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>k</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |k\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/063585a1e5a9a0b0fb63d1684d071fcf70f87d75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.763ex; height:2.843ex;" alt="{\displaystyle |k\rangle }"></span> using the above equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H|k\rangle ={\frac {1}{\sqrt {N}}}\sum _{n}e^{inka}H|n\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>k</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>k</mi> <mi>a</mi> </mrow> </msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H|k\rangle ={\frac {1}{\sqrt {N}}}\sum _{n}e^{inka}H|n\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af720092d4d94c2e4d94b70ea18106511b9c847" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.495ex; height:6.343ex;" alt="{\displaystyle H|k\rangle ={\frac {1}{\sqrt {N}}}\sum _{n}e^{inka}H|n\rangle }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle k|H|k\rangle ={\frac {1}{N}}\sum _{n,\ m}e^{i(n-m)ka}\langle m|H|n\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>k</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mtext> </mtext> <mi>m</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mi>k</mi> <mi>a</mi> </mrow> </msup> <mo fence="false" stretchy="false">⟨</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle k|H|k\rangle ={\frac {1}{N}}\sum _{n,\ m}e^{i(n-m)ka}\langle m|H|n\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d7d9cec7c0d0758e5d13a33450391031f64f68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.026ex; height:6.509ex;" alt="{\displaystyle \langle k|H|k\rangle ={\frac {1}{N}}\sum _{n,\ m}e^{i(n-m)ka}\langle m|H|n\rangle }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {1}{N}}\sum _{n}\langle n|H|n\rangle +{\frac {1}{N}}\sum _{n}\langle n-1|H|n\rangle e^{+ika}+{\frac {1}{N}}\sum _{n}\langle n+1|H|n\rangle e^{-ika}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi>i</mi> <mi>k</mi> <mi>a</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>k</mi> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{N}}\sum _{n}\langle n|H|n\rangle +{\frac {1}{N}}\sum _{n}\langle n-1|H|n\rangle e^{+ika}+{\frac {1}{N}}\sum _{n}\langle n+1|H|n\rangle e^{-ika}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a50a53fcbf73f38ef09ddeda09211fa244b4bc73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:69.709ex; height:6.343ex;" alt="{\displaystyle ={\frac {1}{N}}\sum _{n}\langle n|H|n\rangle +{\frac {1}{N}}\sum _{n}\langle n-1|H|n\rangle e^{+ika}+{\frac {1}{N}}\sum _{n}\langle n+1|H|n\rangle e^{-ika}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =E_{0}-2\Delta \,\cos(ka)\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mi mathvariant="normal">Δ</mi> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =E_{0}-2\Delta \,\cos(ka)\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99285c318ceca12c7d6d727d4191731754f51492" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.525ex; height:2.843ex;" alt="{\displaystyle =E_{0}-2\Delta \,\cos(ka)\ ,}"></span></dd></dl> <p>where, for example, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{N}}\sum _{n}\langle n|H|n\rangle =E_{0}{\frac {1}{N}}\sum _{n}1=E_{0}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mn>1</mn> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{N}}\sum _{n}\langle n|H|n\rangle =E_{0}{\frac {1}{N}}\sum _{n}1=E_{0}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0460670e08503ae8324fe4566d0c55fb90565ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.753ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{N}}\sum _{n}\langle n|H|n\rangle =E_{0}{\frac {1}{N}}\sum _{n}1=E_{0}\ ,}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{N}}\sum _{n}\langle n-1|H|n\rangle e^{+ika}=-\Delta e^{ika}{\frac {1}{N}}\sum _{n}1=-\Delta e^{ika}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi>i</mi> <mi>k</mi> <mi>a</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>a</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mn>1</mn> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>a</mi> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{N}}\sum _{n}\langle n-1|H|n\rangle e^{+ika}=-\Delta e^{ika}{\frac {1}{N}}\sum _{n}1=-\Delta e^{ika}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9de06a7d47cd6324c88aa1a6e195cea462e09da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:53.811ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{N}}\sum _{n}\langle n-1|H|n\rangle e^{+ika}=-\Delta e^{ika}{\frac {1}{N}}\sum _{n}1=-\Delta e^{ika}\ .}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{N}}\sum _{n}\langle n-1|n\rangle e^{+ika}=Se^{ika}{\frac {1}{N}}\sum _{n}1=Se^{ika}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi>i</mi> <mi>k</mi> <mi>a</mi> </mrow> </msup> <mo>=</mo> <mi>S</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>a</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mn>1</mn> <mo>=</mo> <mi>S</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>a</mi> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{N}}\sum _{n}\langle n-1|n\rangle e^{+ika}=Se^{ika}{\frac {1}{N}}\sum _{n}1=Se^{ika}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d7667706e11880dd07b541d9f135a691ef79bad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.611ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{N}}\sum _{n}\langle n-1|n\rangle e^{+ika}=Se^{ika}{\frac {1}{N}}\sum _{n}1=Se^{ika}\ .}"></span></dd></dl> <p>Thus the energy of this state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |k\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>k</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |k\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/063585a1e5a9a0b0fb63d1684d071fcf70f87d75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.763ex; height:2.843ex;" alt="{\displaystyle |k\rangle }"></span> can be represented in the familiar form of the energy dispersion: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(k)={\frac {E_{0}-2\Delta \,\cos(ka)}{1+2S\,\cos(ka)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mi mathvariant="normal">Δ</mi> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>S</mi> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(k)={\frac {E_{0}-2\Delta \,\cos(ka)}{1+2S\,\cos(ka)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47a379dc9169d5f8e2c51cdc96a3e0d107a0482f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.575ex; height:6.509ex;" alt="{\displaystyle E(k)={\frac {E_{0}-2\Delta \,\cos(ka)}{1+2S\,\cos(ka)}}}"></span>.</dd></dl> <ul><li>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6307c8a99dad7d0bcb712352ae0a748bd99a038b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=0}"></span> the energy is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=(E_{0}-2\Delta )/(1+2S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=(E_{0}-2\Delta )/(1+2S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b92f444f35356f0ddf33dd472adeb63fc3b351f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.028ex; height:2.843ex;" alt="{\displaystyle E=(E_{0}-2\Delta )/(1+2S)}"></span> and the state consists of a sum of all atomic orbitals. This state can be viewed as a chain of <a href="/wiki/Molecular_orbital" title="Molecular orbital">bonding orbitals</a>.</li> <li>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=\pi /(2a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=\pi /(2a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a77b90fb5cd4a8a2b993e59cda95cce446de8232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.006ex; height:2.843ex;" alt="{\displaystyle k=\pi /(2a)}"></span> the energy is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=E_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=E_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb31b83e6a518f3c01b491199cad9a21c17a5829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.644ex; height:2.509ex;" alt="{\displaystyle E=E_{0}}"></span> and the state consists of a sum of atomic orbitals which are a factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\pi /2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\pi /2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d334f4a4ed70eb3be2e39ba25b941de92cd3ae2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.469ex; height:2.843ex;" alt="{\displaystyle e^{i\pi /2}}"></span> out of phase. This state can be viewed as a chain of <a href="/wiki/Non-bonding_orbital" title="Non-bonding orbital">non-bonding orbitals</a>.</li> <li>Finally for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=\pi /a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=\pi /a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1591ac7475a01b7fcc1c299ca92e8e84a0003cf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.034ex; height:2.843ex;" alt="{\displaystyle k=\pi /a}"></span> the energy is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=(E_{0}+2\Delta )/(1-2S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=(E_{0}+2\Delta )/(1-2S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f782bb8fc02818258fad4c00b9c4538c2255aaf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.028ex; height:2.843ex;" alt="{\displaystyle E=(E_{0}+2\Delta )/(1-2S)}"></span> and the state consists of an alternating sum of atomic orbitals. This state can be viewed as a chain of <a href="/wiki/Antibonding" class="mw-redirect" title="Antibonding">anti-bonding orbitals</a>.</li></ul> <p>This example is readily extended to three dimensions, for example, to a body-centered cubic or face-centered cubic lattice by introducing the nearest neighbor vector locations in place of simply <i>n a</i>.<sup id="cite_ref-Mott_7-0" class="reference"><a href="#cite_note-Mott-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Likewise, the method can be extended to multiple bands using multiple different atomic orbitals at each site. The general formulation above shows how these extensions can be accomplished. </p> <div class="mw-heading mw-heading2"><h2 id="Table_of_interatomic_matrix_elements">Table of interatomic matrix elements</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tight_binding&action=edit&section=11" title="Edit section: Table of interatomic matrix elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1954 J.C. Slater and G.F. Koster published, mainly for the calculation of <a href="/wiki/Transition_metal" title="Transition metal">transition metal</a> d-bands, a table of interatomic matrix elements<sup id="cite_ref-SlaterKoster_1-2" class="reference"><a href="#cite_note-SlaterKoster-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i,j}({\vec {\mathbf {r} }}_{n,n'})=\langle n,i|H|n',j\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <msup> <mi>n</mi> <mo>′</mo> </msup> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mo>,</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>n</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>j</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i,j}({\vec {\mathbf {r} }}_{n,n'})=\langle n,i|H|n',j\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa36ed8dcb99ab4d3afb3280898e259f9360817d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.322ex; height:3.176ex;" alt="{\displaystyle E_{i,j}({\vec {\mathbf {r} }}_{n,n'})=\langle n,i|H|n',j\rangle }"></span></dd></dl> <p>which can also be derived from the <a href="/wiki/Cubic_harmonic" title="Cubic harmonic">cubic harmonic orbitals</a> straightforwardly. The table expresses the matrix elements as functions of <a href="/wiki/LCAO" class="mw-redirect" title="LCAO">LCAO</a> two-centre <a href="/wiki/Chemical_bond" title="Chemical bond">bond integrals</a> between two <a href="/wiki/Cubic_harmonic" title="Cubic harmonic">cubic harmonic</a> orbitals, <i>i</i> and <i>j</i>, on adjacent atoms. The bond integrals are for example the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{ss\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>s</mi> <mi>σ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{ss\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05bbe9a1df5c5b2e205f544fd0e34607f8e1ad54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.07ex; height:2.509ex;" alt="{\displaystyle V_{ss\sigma }}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{pp\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>p</mi> <mi>π</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{pp\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb30021afc80ceaff032254e96e2eef0e4fa523e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.183ex; height:2.843ex;" alt="{\displaystyle V_{pp\pi }}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{dd\delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>δ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{dd\delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7118cb4493615decec68913c7b829fe9b3b3e24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.049ex; height:2.509ex;" alt="{\displaystyle V_{dd\delta }}"></span> for <a href="/wiki/Sigma_bond" title="Sigma bond">sigma</a>, <a href="/wiki/Pi_bond" title="Pi bond">pi</a> and <a href="/wiki/Delta_bond" title="Delta bond">delta</a> bonds (Notice that these integrals should also depend on the distance between the atoms, i.e. are a function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (l,m,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>l</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (l,m,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4eccfea9dabb29f936e20dc901af15641b77329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.006ex; height:2.843ex;" alt="{\displaystyle (l,m,n)}"></span>, even though it is not explicitly stated every time.). </p><p>The interatomic vector is expressed as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\mathbf {r} }}_{n,n'}=(r_{x},r_{y},r_{z})=d(l,m,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <msup> <mi>n</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\mathbf {r} }}_{n,n'}=(r_{x},r_{y},r_{z})=d(l,m,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5105d764b3eb3e8de07453b0dd063e2a3178fdae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.961ex; height:3.009ex;" alt="{\displaystyle {\vec {\mathbf {r} }}_{n,n'}=(r_{x},r_{y},r_{z})=d(l,m,n)}"></span></dd></dl> <p>where <i>d</i> is the distance between the atoms and <i>l</i>, <i>m</i> and <i>n</i> are the <a href="/wiki/Direction_cosine" title="Direction cosine">direction cosines</a> to the neighboring atom. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{s,s}=V_{ss\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>s</mi> <mi>σ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{s,s}=V_{ss\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe858e9be19cf013027474a66d73afbe1d42bc69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.115ex; height:2.843ex;" alt="{\displaystyle E_{s,s}=V_{ss\sigma }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{s,x}=lV_{sp\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>,</mo> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mi>l</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>p</mi> <mi>σ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{s,x}=lV_{sp\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b998ce8160eeaca791e8c944d1affa344cd9b91f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.034ex; height:2.843ex;" alt="{\displaystyle E_{s,x}=lV_{sp\sigma }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{x,x}=l^{2}V_{pp\sigma }+(1-l^{2})V_{pp\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>p</mi> <mi>σ</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>p</mi> <mi>π</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{x,x}=l^{2}V_{pp\sigma }+(1-l^{2})V_{pp\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90dbd9a667c7409974d0ef36c16985620c2ea38c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.896ex; height:3.343ex;" alt="{\displaystyle E_{x,x}=l^{2}V_{pp\sigma }+(1-l^{2})V_{pp\pi }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{x,y}=lmV_{pp\sigma }-lmV_{pp\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>l</mi> <mi>m</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>p</mi> <mi>σ</mi> </mrow> </msub> <mo>−</mo> <mi>l</mi> <mi>m</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>p</mi> <mi>π</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{x,y}=lmV_{pp\sigma }-lmV_{pp\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e1e17eadb478d3edfdf8886b751979c9941adb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.933ex; height:2.843ex;" alt="{\displaystyle E_{x,y}=lmV_{pp\sigma }-lmV_{pp\pi }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{x,z}=lnV_{pp\sigma }-lnV_{pp\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mi>l</mi> <mi>n</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>p</mi> <mi>σ</mi> </mrow> </msub> <mo>−</mo> <mi>l</mi> <mi>n</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>p</mi> <mi>π</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{x,z}=lnV_{pp\sigma }-lnV_{pp\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d5a0c6dac5314c243a02ec7f933a7d11e46ba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.594ex; height:2.843ex;" alt="{\displaystyle E_{x,z}=lnV_{pp\sigma }-lnV_{pp\pi }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{s,xy}={\sqrt {3}}lmV_{sd\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>,</mo> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mi>l</mi> <mi>m</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{s,xy}={\sqrt {3}}lmV_{sd\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52dd473775788adbf4e3f0db06ab013ebee4c70e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.022ex; height:3.176ex;" alt="{\displaystyle E_{s,xy}={\sqrt {3}}lmV_{sd\sigma }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{s,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}(l^{2}-m^{2})V_{sd\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{s,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}(l^{2}-m^{2})V_{sd\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4baafc92d9732d5fe51924f641eb8f44ea67930c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.562ex; height:5.843ex;" alt="{\displaystyle E_{s,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}(l^{2}-m^{2})V_{sd\sigma }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{s,3z^{2}-r^{2}}=[n^{2}-(l^{2}+m^{2})/2]V_{sd\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>,</mo> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{s,3z^{2}-r^{2}}=[n^{2}-(l^{2}+m^{2})/2]V_{sd\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/634217924cd42d49cc9dc2fac7339b1d4e8868d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:34.109ex; height:3.509ex;" alt="{\displaystyle E_{s,3z^{2}-r^{2}}=[n^{2}-(l^{2}+m^{2})/2]V_{sd\sigma }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{x,xy}={\sqrt {3}}l^{2}mV_{pd\sigma }+m(1-2l^{2})V_{pd\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>+</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{x,xy}={\sqrt {3}}l^{2}mV_{pd\sigma }+m(1-2l^{2})V_{pd\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/444cdee8cbc60e9c640a6800045a1142f695831c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.12ex; height:3.343ex;" alt="{\displaystyle E_{x,xy}={\sqrt {3}}l^{2}mV_{pd\sigma }+m(1-2l^{2})V_{pd\pi }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{x,yz}={\sqrt {3}}lmnV_{pd\sigma }-2lmnV_{pd\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mi>l</mi> <mi>m</mi> <mi>n</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mi>l</mi> <mi>m</mi> <mi>n</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{x,yz}={\sqrt {3}}lmnV_{pd\sigma }-2lmnV_{pd\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da9e3c5f3a1d0c1e1f63b759c023c3c0d678f9a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.818ex; height:3.176ex;" alt="{\displaystyle E_{x,yz}={\sqrt {3}}lmnV_{pd\sigma }-2lmnV_{pd\pi }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{x,zx}={\sqrt {3}}l^{2}nV_{pd\sigma }+n(1-2l^{2})V_{pd\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>+</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{x,zx}={\sqrt {3}}l^{2}nV_{pd\sigma }+n(1-2l^{2})V_{pd\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd48788f87589bf90b4b1ec30f46f0787f14722" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.781ex; height:3.343ex;" alt="{\displaystyle E_{x,zx}={\sqrt {3}}l^{2}nV_{pd\sigma }+n(1-2l^{2})V_{pd\pi }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{x,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}l(l^{2}-m^{2})V_{pd\sigma }+l(1-l^{2}+m^{2})V_{pd\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mi>l</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>+</mo> <mi>l</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{x,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}l(l^{2}-m^{2})V_{pd\sigma }+l(1-l^{2}+m^{2})V_{pd\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6cf7ac47edf72dfacab4c3a6bcd8104d7802b2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.725ex; height:5.843ex;" alt="{\displaystyle E_{x,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}l(l^{2}-m^{2})V_{pd\sigma }+l(1-l^{2}+m^{2})V_{pd\pi }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{y,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}m(l^{2}-m^{2})V_{pd\sigma }-m(1+l^{2}-m^{2})V_{pd\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{y,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}m(l^{2}-m^{2})V_{pd\sigma }-m(1+l^{2}-m^{2})V_{pd\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e1acf854c03927306796e839d0cdc22f3dcef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:53.296ex; height:5.843ex;" alt="{\displaystyle E_{y,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}m(l^{2}-m^{2})V_{pd\sigma }-m(1+l^{2}-m^{2})V_{pd\pi }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{z,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}n(l^{2}-m^{2})V_{pd\sigma }-n(l^{2}-m^{2})V_{pd\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>−</mo> <mi>n</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{z,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}n(l^{2}-m^{2})V_{pd\sigma }-n(l^{2}-m^{2})V_{pd\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c14594c20f620fa5a54c497d32669c30e64a05d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.954ex; height:5.843ex;" alt="{\displaystyle E_{z,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}n(l^{2}-m^{2})V_{pd\sigma }-n(l^{2}-m^{2})V_{pd\pi }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{x,3z^{2}-r^{2}}=l[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma }-{\sqrt {3}}ln^{2}V_{pd\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mi>l</mi> <mo stretchy="false">[</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mi>l</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{x,3z^{2}-r^{2}}=l[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma }-{\sqrt {3}}ln^{2}V_{pd\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25bd2504f6ab99b0e93be013e27443e6a4f2fd08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:48.324ex; height:3.509ex;" alt="{\displaystyle E_{x,3z^{2}-r^{2}}=l[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma }-{\sqrt {3}}ln^{2}V_{pd\pi }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{y,3z^{2}-r^{2}}=m[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma }-{\sqrt {3}}mn^{2}V_{pd\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>,</mo> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mo stretchy="false">[</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mi>m</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{y,3z^{2}-r^{2}}=m[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma }-{\sqrt {3}}mn^{2}V_{pd\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/494abd00e0412fc6d53b9b8542a3dfa803bbd41b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:50.895ex; height:3.509ex;" alt="{\displaystyle E_{y,3z^{2}-r^{2}}=m[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma }-{\sqrt {3}}mn^{2}V_{pd\pi }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{z,3z^{2}-r^{2}}=n[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma }+{\sqrt {3}}n(l^{2}+m^{2})V_{pd\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>,</mo> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mo stretchy="false">[</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{z,3z^{2}-r^{2}}=n[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma }+{\sqrt {3}}n(l^{2}+m^{2})V_{pd\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbfc0061911518a77f4fd2b8a3cacbfc25118d88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:56.599ex; height:3.509ex;" alt="{\displaystyle E_{z,3z^{2}-r^{2}}=n[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma }+{\sqrt {3}}n(l^{2}+m^{2})V_{pd\pi }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{xy,xy}=3l^{2}m^{2}V_{dd\sigma }+(l^{2}+m^{2}-4l^{2}m^{2})V_{dd\pi }+(n^{2}+l^{2}m^{2})V_{dd\delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>δ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{xy,xy}=3l^{2}m^{2}V_{dd\sigma }+(l^{2}+m^{2}-4l^{2}m^{2})V_{dd\pi }+(n^{2}+l^{2}m^{2})V_{dd\delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ade6f48c1d8acf798c05148758e565bd63c10e52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:63.526ex; height:3.343ex;" alt="{\displaystyle E_{xy,xy}=3l^{2}m^{2}V_{dd\sigma }+(l^{2}+m^{2}-4l^{2}m^{2})V_{dd\pi }+(n^{2}+l^{2}m^{2})V_{dd\delta }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{xy,yz}=3lm^{2}nV_{dd\sigma }+ln(1-4m^{2})V_{dd\pi }+ln(m^{2}-1)V_{dd\delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> <mo>,</mo> <mi>y</mi> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mi>l</mi> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>+</mo> <mi>l</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>4</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> <mo>+</mo> <mi>l</mi> <mi>n</mi> <mo stretchy="false">(</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>δ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{xy,yz}=3lm^{2}nV_{dd\sigma }+ln(1-4m^{2})V_{dd\pi }+ln(m^{2}-1)V_{dd\delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fd3ab0b435ef59de049a4e5c0307860a6d50ae7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:56.57ex; height:3.343ex;" alt="{\displaystyle E_{xy,yz}=3lm^{2}nV_{dd\sigma }+ln(1-4m^{2})V_{dd\pi }+ln(m^{2}-1)V_{dd\delta }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{xy,zx}=3l^{2}mnV_{dd\sigma }+mn(1-4l^{2})V_{dd\pi }+mn(l^{2}-1)V_{dd\delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> <mi>n</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>+</mo> <mi>m</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>4</mn> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> <mo>+</mo> <mi>m</mi> <mi>n</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>δ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{xy,zx}=3l^{2}mnV_{dd\sigma }+mn(1-4l^{2})V_{dd\pi }+mn(l^{2}-1)V_{dd\delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e766f5c59b79ebb7d1a6a1e189d15809356e2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:56.693ex; height:3.343ex;" alt="{\displaystyle E_{xy,zx}=3l^{2}mnV_{dd\sigma }+mn(1-4l^{2})V_{dd\pi }+mn(l^{2}-1)V_{dd\delta }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{xy,x^{2}-y^{2}}={\frac {3}{2}}lm(l^{2}-m^{2})V_{dd\sigma }+2lm(m^{2}-l^{2})V_{dd\pi }+[lm(l^{2}-m^{2})/2]V_{dd\delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi>l</mi> <mi>m</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>l</mi> <mi>m</mi> <mo stretchy="false">(</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">[</mo> <mi>l</mi> <mi>m</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>δ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{xy,x^{2}-y^{2}}={\frac {3}{2}}lm(l^{2}-m^{2})V_{dd\sigma }+2lm(m^{2}-l^{2})V_{dd\pi }+[lm(l^{2}-m^{2})/2]V_{dd\delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/414f509dc16064238598749115e1fdc2ca07e2f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:73.645ex; height:5.176ex;" alt="{\displaystyle E_{xy,x^{2}-y^{2}}={\frac {3}{2}}lm(l^{2}-m^{2})V_{dd\sigma }+2lm(m^{2}-l^{2})V_{dd\pi }+[lm(l^{2}-m^{2})/2]V_{dd\delta }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{yz,x^{2}-y^{2}}={\frac {3}{2}}mn(l^{2}-m^{2})V_{dd\sigma }-mn[1+2(l^{2}-m^{2})]V_{dd\pi }+mn[1+(l^{2}-m^{2})/2]V_{dd\delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mi>n</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mi>n</mi> <mo stretchy="false">[</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> <mo>+</mo> <mi>m</mi> <mi>n</mi> <mo stretchy="false">[</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>δ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{yz,x^{2}-y^{2}}={\frac {3}{2}}mn(l^{2}-m^{2})V_{dd\sigma }-mn[1+2(l^{2}-m^{2})]V_{dd\pi }+mn[1+(l^{2}-m^{2})/2]V_{dd\delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8a8a0e68e491b8ce4ebe1529f6d746cacd8e2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:84.878ex; height:5.176ex;" alt="{\displaystyle E_{yz,x^{2}-y^{2}}={\frac {3}{2}}mn(l^{2}-m^{2})V_{dd\sigma }-mn[1+2(l^{2}-m^{2})]V_{dd\pi }+mn[1+(l^{2}-m^{2})/2]V_{dd\delta }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{zx,x^{2}-y^{2}}={\frac {3}{2}}nl(l^{2}-m^{2})V_{dd\sigma }+nl[1-2(l^{2}-m^{2})]V_{dd\pi }-nl[1-(l^{2}-m^{2})/2]V_{dd\delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> <mi>l</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>+</mo> <mi>n</mi> <mi>l</mi> <mo stretchy="false">[</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> <mo>−</mo> <mi>n</mi> <mi>l</mi> <mo stretchy="false">[</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>δ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{zx,x^{2}-y^{2}}={\frac {3}{2}}nl(l^{2}-m^{2})V_{dd\sigma }+nl[1-2(l^{2}-m^{2})]V_{dd\pi }-nl[1-(l^{2}-m^{2})/2]V_{dd\delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4fe1210f83794c80f97d831622e1f8b398ec835" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:80.96ex; height:5.176ex;" alt="{\displaystyle E_{zx,x^{2}-y^{2}}={\frac {3}{2}}nl(l^{2}-m^{2})V_{dd\sigma }+nl[1-2(l^{2}-m^{2})]V_{dd\pi }-nl[1-(l^{2}-m^{2})/2]V_{dd\delta }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{xy,3z^{2}-r^{2}}={\sqrt {3}}\left[lm(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma }-2lmn^{2}V_{dd\pi }+[lm(1+n^{2})/2]V_{dd\delta }\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> <mo>,</mo> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mrow> <mo>[</mo> <mrow> <mi>l</mi> <mi>m</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mi>l</mi> <mi>m</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">[</mo> <mi>l</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>δ</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{xy,3z^{2}-r^{2}}={\sqrt {3}}\left[lm(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma }-2lmn^{2}V_{dd\pi }+[lm(1+n^{2})/2]V_{dd\delta }\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50e7f65d8edb561c35c4bd39ea63042ddfcabe76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:78.795ex; height:3.509ex;" alt="{\displaystyle E_{xy,3z^{2}-r^{2}}={\sqrt {3}}\left[lm(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma }-2lmn^{2}V_{dd\pi }+[lm(1+n^{2})/2]V_{dd\delta }\right]}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{yz,3z^{2}-r^{2}}={\sqrt {3}}\left[mn(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma }+mn(l^{2}+m^{2}-n^{2})V_{dd\pi }-[mn(l^{2}+m^{2})/2]V_{dd\delta }\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> <mo>,</mo> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mrow> <mo>[</mo> <mrow> <mi>m</mi> <mi>n</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>+</mo> <mi>m</mi> <mi>n</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> <mo>−</mo> <mo stretchy="false">[</mo> <mi>m</mi> <mi>n</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>δ</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{yz,3z^{2}-r^{2}}={\sqrt {3}}\left[mn(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma }+mn(l^{2}+m^{2}-n^{2})V_{dd\pi }-[mn(l^{2}+m^{2})/2]V_{dd\delta }\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/841e7888da970b88d7505a352db178447b1b27b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:93.129ex; height:3.509ex;" alt="{\displaystyle E_{yz,3z^{2}-r^{2}}={\sqrt {3}}\left[mn(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma }+mn(l^{2}+m^{2}-n^{2})V_{dd\pi }-[mn(l^{2}+m^{2})/2]V_{dd\delta }\right]}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{zx,3z^{2}-r^{2}}={\sqrt {3}}\left[ln(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma }+ln(l^{2}+m^{2}-n^{2})V_{dd\pi }-[ln(l^{2}+m^{2})/2]V_{dd\delta }\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> <mo>,</mo> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mrow> <mo>[</mo> <mrow> <mi>l</mi> <mi>n</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>+</mo> <mi>l</mi> <mi>n</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> <mo>−</mo> <mo stretchy="false">[</mo> <mi>l</mi> <mi>n</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>δ</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{zx,3z^{2}-r^{2}}={\sqrt {3}}\left[ln(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma }+ln(l^{2}+m^{2}-n^{2})V_{dd\pi }-[ln(l^{2}+m^{2})/2]V_{dd\delta }\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d37b9c0dbb4e733532d3f54eee9a4148c50c6441" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:89.21ex; height:3.509ex;" alt="{\displaystyle E_{zx,3z^{2}-r^{2}}={\sqrt {3}}\left[ln(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma }+ln(l^{2}+m^{2}-n^{2})V_{dd\pi }-[ln(l^{2}+m^{2})/2]V_{dd\delta }\right]}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{x^{2}-y^{2},x^{2}-y^{2}}={\frac {3}{4}}(l^{2}-m^{2})^{2}V_{dd\sigma }+[l^{2}+m^{2}-(l^{2}-m^{2})^{2}]V_{dd\pi }+[n^{2}+(l^{2}-m^{2})^{2}/4]V_{dd\delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">[</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">[</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>δ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{x^{2}-y^{2},x^{2}-y^{2}}={\frac {3}{4}}(l^{2}-m^{2})^{2}V_{dd\sigma }+[l^{2}+m^{2}-(l^{2}-m^{2})^{2}]V_{dd\pi }+[n^{2}+(l^{2}-m^{2})^{2}/4]V_{dd\delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86503383acc3ab37b12e142128b73d7fbddff4ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:87.495ex; height:5.176ex;" alt="{\displaystyle E_{x^{2}-y^{2},x^{2}-y^{2}}={\frac {3}{4}}(l^{2}-m^{2})^{2}V_{dd\sigma }+[l^{2}+m^{2}-(l^{2}-m^{2})^{2}]V_{dd\pi }+[n^{2}+(l^{2}-m^{2})^{2}/4]V_{dd\delta }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{x^{2}-y^{2},3z^{2}-r^{2}}={\sqrt {3}}\left[(l^{2}-m^{2})[n^{2}-(l^{2}+m^{2})/2]V_{dd\sigma }/2+n^{2}(m^{2}-l^{2})V_{dd\pi }+[(1+n^{2})(l^{2}-m^{2})/4]V_{dd\delta }\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo stretchy="false">]</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>δ</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{x^{2}-y^{2},3z^{2}-r^{2}}={\sqrt {3}}\left[(l^{2}-m^{2})[n^{2}-(l^{2}+m^{2})/2]V_{dd\sigma }/2+n^{2}(m^{2}-l^{2})V_{dd\pi }+[(1+n^{2})(l^{2}-m^{2})/4]V_{dd\delta }\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46a9e6f0c6819483d95f177d9600245e80e57984" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:102.661ex; height:3.509ex;" alt="{\displaystyle E_{x^{2}-y^{2},3z^{2}-r^{2}}={\sqrt {3}}\left[(l^{2}-m^{2})[n^{2}-(l^{2}+m^{2})/2]V_{dd\sigma }/2+n^{2}(m^{2}-l^{2})V_{dd\pi }+[(1+n^{2})(l^{2}-m^{2})/4]V_{dd\delta }\right]}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{3z^{2}-r^{2},3z^{2}-r^{2}}=[n^{2}-(l^{2}+m^{2})/2]^{2}V_{dd\sigma }+3n^{2}(l^{2}+m^{2})V_{dd\pi }+{\frac {3}{4}}(l^{2}+m^{2})^{2}V_{dd\delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>σ</mi> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>π</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> <mi>δ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{3z^{2}-r^{2},3z^{2}-r^{2}}=[n^{2}-(l^{2}+m^{2})/2]^{2}V_{dd\sigma }+3n^{2}(l^{2}+m^{2})V_{dd\pi }+{\frac {3}{4}}(l^{2}+m^{2})^{2}V_{dd\delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f987d902bc15a2332d37a79825b5b584da12f675" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:79.383ex; height:5.176ex;" alt="{\displaystyle E_{3z^{2}-r^{2},3z^{2}-r^{2}}=[n^{2}-(l^{2}+m^{2})/2]^{2}V_{dd\sigma }+3n^{2}(l^{2}+m^{2})V_{dd\pi }+{\frac {3}{4}}(l^{2}+m^{2})^{2}V_{dd\delta }}"></span></dd></dl> <p>Not all interatomic matrix elements are listed explicitly. Matrix elements that are not listed in this table can be constructed by permutation of indices and cosine directions of other matrix elements in the table. Note that swapping orbital indices amounts to taking <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (l,m,n)\rightarrow (-l,-m,-n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>l</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>l</mi> <mo>,</mo> <mo>−</mo> <mi>m</mi> <mo>,</mo> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (l,m,n)\rightarrow (-l,-m,-n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34c1dba5833468e123c38fda4382ada0235ceff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.05ex; height:2.843ex;" alt="{\displaystyle (l,m,n)\rightarrow (-l,-m,-n)}"></span>, i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\alpha ,\beta }(l,m,n)=E_{\beta ,\alpha }(-l,-m,-n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>l</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>,</mo> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mi>l</mi> <mo>,</mo> <mo>−</mo> <mi>m</mi> <mo>,</mo> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\alpha ,\beta }(l,m,n)=E_{\beta ,\alpha }(-l,-m,-n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b582bc0108b9789b1c5f8985b59848592e81149d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.331ex; height:3.009ex;" alt="{\displaystyle E_{\alpha ,\beta }(l,m,n)=E_{\beta ,\alpha }(-l,-m,-n)}"></span>. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{x,s}=-lV_{sp\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>l</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>p</mi> <mi>σ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{x,s}=-lV_{sp\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9568bbf51b69d4fc073103a8739d96ab612ac95f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.842ex; height:2.843ex;" alt="{\displaystyle E_{x,s}=-lV_{sp\sigma }}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tight_binding&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Electronic_band_structure" title="Electronic band structure">Electronic band structure</a></li> <li><a href="/wiki/Nearly-free_electron_model" class="mw-redirect" title="Nearly-free electron model">Nearly-free electron model</a></li> <li><a href="/wiki/Bloch%27s_theorem" title="Bloch's theorem">Bloch's theorems</a></li> <li><a href="/wiki/Kronig-Penney_model" class="mw-redirect" title="Kronig-Penney model">Kronig-Penney model</a></li> <li><a href="/wiki/Fermi_surface" title="Fermi surface">Fermi surface</a></li> <li><a href="/wiki/Wannier_function" title="Wannier function">Wannier function</a></li> <li><a href="/wiki/Hubbard_model" title="Hubbard model">Hubbard model</a></li> <li><a href="/wiki/T-J_model" title="T-J model">t-J model</a></li> <li><a href="/wiki/Effective_mass_(solid-state_physics)" title="Effective mass (solid-state physics)">Effective mass</a></li> <li><a href="/wiki/Anderson%27s_rule" title="Anderson's rule">Anderson's rule</a></li> <li><a href="/wiki/Dynamical_theory_of_diffraction" title="Dynamical theory of diffraction">Dynamical theory of diffraction</a></li> <li><a href="/wiki/Solid_state_physics" class="mw-redirect" title="Solid state physics">Solid state physics</a></li> <li><a href="/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_method" class="mw-redirect" title="Linear combination of atomic orbitals molecular orbital method">Linear combination of atomic orbitals molecular orbital method</a> (LCAO)</li> <li><a href="/wiki/Holstein%E2%80%93Herring_method" title="Holstein–Herring method">Holstein–Herring method</a></li> <li><a href="/wiki/Peierls_substitution" title="Peierls substitution">Peierls substitution</a></li> <li><a href="/wiki/H%C3%BCckel_method" title="Hückel method">Hückel method</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tight_binding&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox 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ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-SlaterKoster-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-SlaterKoster_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-SlaterKoster_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-SlaterKoster_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFJ._C._SlaterG._F._Koster1954" class="citation journal cs1">J. C. Slater; <a href="/wiki/George_F._Koster" title="George F. Koster">G. F. Koster</a> (1954). "Simplified LCAO method for the Periodic Potential Problem". <i><a href="/wiki/Physical_Review" title="Physical Review">Physical Review</a></i>. <b>94</b> (6): 1498–1524. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1954PhRv...94.1498S">1954PhRv...94.1498S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.94.1498">10.1103/PhysRev.94.1498</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=Simplified+LCAO+method+for+the+Periodic+Potential+Problem&rft.volume=94&rft.issue=6&rft.pages=1498-1524&rft.date=1954&rft_id=info%3Adoi%2F10.1103%2FPhysRev.94.1498&rft_id=info%3Abibcode%2F1954PhRv...94.1498S&rft.au=J.+C.+Slater&rft.au=G.+F.+Koster&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATight+binding" class="Z3988"></span></span> </li> <li id="cite_note-Harrison-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Harrison_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Harrison_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWalter_Ashley_Harrison1989" class="citation book cs1">Walter Ashley Harrison (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=R2VqQgAACAAJ"><i>Electronic Structure and the Properties of Solids</i></a>. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-66021-4" title="Special:BookSources/0-486-66021-4"><bdi>0-486-66021-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electronic+Structure+and+the+Properties+of+Solids&rft.pub=Dover+Publications&rft.date=1989&rft.isbn=0-486-66021-4&rft.au=Walter+Ashley+Harrison&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DR2VqQgAACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATight+binding" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBannwarthEhlertGrimme2019" class="citation journal cs1">Bannwarth, Christoph; Ehlert, Sebastian; Grimme, Stefan (2019-03-12). <a rel="nofollow" class="external text" href="https://pubs.acs.org/doi/10.1021/acs.jctc.8b01176">"GFN2-xTB—An Accurate and Broadly Parametrized Self-Consistent Tight-Binding Quantum Chemical Method with Multipole Electrostatics and Density-Dependent Dispersion Contributions"</a>. <i>Journal of Chemical Theory and Computation</i>. <b>15</b> (3): 1652–1671. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Facs.jctc.8b01176">10.1021/acs.jctc.8b01176</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1549-9618">1549-9618</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Chemical+Theory+and+Computation&rft.atitle=GFN2-xTB%E2%80%94An+Accurate+and+Broadly+Parametrized+Self-Consistent+Tight-Binding+Quantum+Chemical+Method+with+Multipole+Electrostatics+and+Density-Dependent+Dispersion+Contributions&rft.volume=15&rft.issue=3&rft.pages=1652-1671&rft.date=2019-03-12&rft_id=info%3Adoi%2F10.1021%2Facs.jctc.8b01176&rft.issn=1549-9618&rft.aulast=Bannwarth&rft.aufirst=Christoph&rft.au=Ehlert%2C+Sebastian&rft.au=Grimme%2C+Stefan&rft_id=https%3A%2F%2Fpubs.acs.org%2Fdoi%2F10.1021%2Facs.jctc.8b01176&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATight+binding" class="Z3988"></span></span> </li> <li id="cite_note-Lowdin-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lowdin_4-0">^</a></b></span> <span class="reference-text">As an alternative to neglecting overlap, one may choose as a basis instead of atomic orbitals a set of orbitals based upon atomic orbitals but arranged to be orthogonal to orbitals on other atomic sites, the so-called <a href="/w/index.php?title=L%C3%B6wdin_orbitals&action=edit&redlink=1" class="new" title="Löwdin orbitals (page does not exist)">Löwdin orbitals</a>. See <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPY_Yu_&_M_Cardona2005" class="citation book cs1">PY Yu & M Cardona (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=W9pdJZoAeyEC&pg=PA87">"Tight-binding or LCAO approach to the band structure of semiconductors"</a>. <i>Fundamentals of Semiconductors</i> (3 ed.). Springrer. p. 87. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-25470-6" title="Special:BookSources/3-540-25470-6"><bdi>3-540-25470-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Tight-binding+or+LCAO+approach+to+the+band+structure+of+semiconductors&rft.btitle=Fundamentals+of+Semiconductors&rft.pages=87&rft.edition=3&rft.pub=Springrer&rft.date=2005&rft.isbn=3-540-25470-6&rft.au=PY+Yu+%26+M+Cardona&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DW9pdJZoAeyEC%26pg%3DPA87&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATight+binding" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Orfried Madelung, <i>Introduction to Solid-State Theory</i> (Springer-Verlag, Berlin Heidelberg, 1978).</span> </li> <li id="cite_note-Altland-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Altland_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexander_Altland_and_Ben_Simons2006" class="citation book cs1">Alexander Altland and Ben Simons (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0KMkfAMe3JkC&pg=RA4-PA58">"Interaction effects in the tight-binding system"</a>. <i>Condensed Matter Field Theory</i>. Cambridge University Press. pp. 58 <i>ff</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-84508-3" title="Special:BookSources/978-0-521-84508-3"><bdi>978-0-521-84508-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Interaction+effects+in+the+tight-binding+system&rft.btitle=Condensed+Matter+Field+Theory&rft.pages=58+%27%27ff%27%27&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=978-0-521-84508-3&rft.au=Alexander+Altland+and+Ben+Simons&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0KMkfAMe3JkC%26pg%3DRA4-PA58&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATight+binding" class="Z3988"></span></span> </li> <li id="cite_note-Mott-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Mott_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSir_Nevill_F_Mott_&_H_Jones1958" class="citation book cs1">Sir Nevill F Mott & H Jones (1958). "II §4 Motion of electrons in a periodic field". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LIPsUaTqUXUC"><i>The theory of the properties of metals and alloys</i></a> (Reprint of Clarendon Press (1936) ed.). Courier Dover Publications. pp. 56 <i>ff</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-60456-X" title="Special:BookSources/0-486-60456-X"><bdi>0-486-60456-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=II+%C2%A74+Motion+of+electrons+in+a+periodic+field&rft.btitle=The+theory+of+the+properties+of+metals+and+alloys&rft.pages=56+%27%27ff%27%27&rft.edition=Reprint+of+Clarendon+Press+%281936%29&rft.pub=Courier+Dover+Publications&rft.date=1958&rft.isbn=0-486-60456-X&rft.au=Sir+Nevill+F+Mott+%26+H+Jones&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLIPsUaTqUXUC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATight+binding" class="Z3988"></span></span> </li> </ol></div></div> <ul><li>N. W. Ashcroft and N. D. Mermin, <i>Solid State Physics</i> (Thomson Learning, Toronto, 1976).</li> <li>Stephen Blundell <i>Magnetism in Condensed Matter</i>(Oxford, 2001).</li> <li>S.Maekawa <i>et al.</i> <i>Physics of Transition Metal Oxides</i> (Springer-Verlag Berlin Heidelberg, 2004).</li> <li>John Singleton <i>Band Theory and Electronic Properties of Solids</i> (Oxford, 2001).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tight_binding&action=edit&section=14" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWalter_Ashley_Harrison1989" class="citation book cs1">Walter Ashley Harrison (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=R2VqQgAACAAJ"><i>Electronic Structure and the Properties of Solids</i></a>. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-66021-4" title="Special:BookSources/0-486-66021-4"><bdi>0-486-66021-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electronic+Structure+and+the+Properties+of+Solids&rft.pub=Dover+Publications&rft.date=1989&rft.isbn=0-486-66021-4&rft.au=Walter+Ashley+Harrison&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DR2VqQgAACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATight+binding" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFN._W._Ashcroft_and_N._D._Mermin1976" class="citation book cs1">N. W. Ashcroft and N. D. Mermin (1976). <i>Solid State Physics</i>. Toronto: Thomson Learning.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solid+State+Physics&rft.place=Toronto&rft.pub=Thomson+Learning&rft.date=1976&rft.au=N.+W.+Ashcroft+and+N.+D.+Mermin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATight+binding" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavies1998" class="citation book cs1">Davies, John H. (1998). <i>The physics of low-dimensional semiconductors: An introduction</i>. Cambridge, United Kingdom: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-48491-X" title="Special:BookSources/0-521-48491-X"><bdi>0-521-48491-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+physics+of+low-dimensional+semiconductors%3A+An+introduction&rft.place=Cambridge%2C+United+Kingdom&rft.pub=Cambridge+University+Press&rft.date=1998&rft.isbn=0-521-48491-X&rft.aulast=Davies&rft.aufirst=John+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATight+binding" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoringeBowlerHernández1997" class="citation journal cs1">Goringe, C M; Bowler, D R; Hernández, E (1997). "Tight-binding modelling of materials". <i>Reports on Progress in Physics</i>. <b>60</b> (12): 1447–1512. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997RPPh...60.1447G">1997RPPh...60.1447G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0034-4885%2F60%2F12%2F001">10.1088/0034-4885/60/12/001</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:250846071">250846071</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reports+on+Progress+in+Physics&rft.atitle=Tight-binding+modelling+of+materials&rft.volume=60&rft.issue=12&rft.pages=1447-1512&rft.date=1997&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250846071%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0034-4885%2F60%2F12%2F001&rft_id=info%3Abibcode%2F1997RPPh...60.1447G&rft.aulast=Goringe&rft.aufirst=C+M&rft.au=Bowler%2C+D+R&rft.au=Hern%C3%A1ndez%2C+E&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATight+binding" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSlaterKoster1954" class="citation journal cs1">Slater, J. C.; Koster, G. F. (1954). "Simplified LCAO Method for the Periodic Potential Problem". <i>Physical Review</i>. <b>94</b> (6): 1498–1524. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1954PhRv...94.1498S">1954PhRv...94.1498S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.94.1498">10.1103/PhysRev.94.1498</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=Simplified+LCAO+Method+for+the+Periodic+Potential+Problem&rft.volume=94&rft.issue=6&rft.pages=1498-1524&rft.date=1954&rft_id=info%3Adoi%2F10.1103%2FPhysRev.94.1498&rft_id=info%3Abibcode%2F1954PhRv...94.1498S&rft.aulast=Slater&rft.aufirst=J.+C.&rft.au=Koster%2C+G.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATight+binding" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tight_binding&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.cond-mat.de/events/correl12/manuscripts/pavarini.pdf">Crystal-field Theory, Tight-binding Method, and Jahn-Teller Effect</a> in E. Pavarini, E. Koch, F. Anders, and M. Jarrell (eds.): Correlated Electrons: From Models to Materials, Jülich 2012, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-89336-796-2" title="Special:BookSources/978-3-89336-796-2">978-3-89336-796-2</a></li> <li><a rel="nofollow" class="external text" href="https://tight-binding.com">Tight-Binding Studio</a>: A Technical Software Package to Find the Parameters of Tight-Binding Hamiltonian</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output 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autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Condensed_matter_physics_topics" title="Template:Condensed matter physics topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Condensed_matter_physics_topics" title="Template talk:Condensed matter physics topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Condensed_matter_physics_topics" title="Special:EditPage/Template:Condensed matter physics topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Condensed_matter_physics" style="font-size:114%;margin:0 4em"><a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/State_of_matter" title="State of matter">States of matter</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Solid" title="Solid">Solid</a></li> <li><a href="/wiki/Liquid" title="Liquid">Liquid</a></li> <li><a href="/wiki/Gas" title="Gas">Gas</a></li> <li><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></li> <li><a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a></li> <li><a href="/wiki/Fermionic_condensate" title="Fermionic condensate">Fermionic condensate</a></li> <li><a href="/wiki/Fermi_gas" title="Fermi gas">Fermi gas</a></li> <li><a href="/wiki/Supersolid" title="Supersolid">Supersolid</a></li> <li><a href="/wiki/Superfluidity" title="Superfluidity">Superfluid</a></li> <li><a href="/wiki/Luttinger_liquid" title="Luttinger liquid">Luttinger liquid</a></li> <li><a href="/wiki/Time_crystal" title="Time crystal">Time crystal</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:QuantumPhaseTransition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/220px-QuantumPhaseTransition.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/330px-QuantumPhaseTransition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/440px-QuantumPhaseTransition.svg.png 2x" data-file-width="512" data-file-height="369" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Phase phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Order_parameter" class="mw-redirect" title="Order parameter">Order parameter</a></li> <li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li> <li><a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li> <li><a href="/wiki/Critical_phenomena" title="Critical phenomena">Critical phenomena</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Electrons in solids</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hall_effect" title="Hall effect">Hall effect</a></li> <li><a href="/wiki/Quantum_Hall_effect" title="Quantum Hall effect">Quantum Hall effect</a></li> <li><a href="/wiki/Spin_Hall_effect" title="Spin Hall effect">Spin Hall effect</a></li> <li><a href="/wiki/Quantum_spin_Hall_effect" title="Quantum spin Hall effect">Quantum spin Hall effect</a></li> <li><a href="/wiki/Berry_phase" class="mw-redirect" title="Berry phase">Berry phase</a></li> <li><a href="/wiki/Aharonov%E2%80%93Bohm_effect" title="Aharonov–Bohm effect">Aharonov–Bohm effect</a></li> <li><a href="/wiki/Josephson_effect" title="Josephson effect">Josephson effect</a></li> <li><a href="/wiki/Kondo_effect" title="Kondo effect">Kondo effect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Drude_model" title="Drude model">Drude model</a></li> <li><a href="/wiki/Free_electron_model" title="Free electron model">Free electron model</a></li> <li><a href="/wiki/Nearly_free_electron_model" title="Nearly free electron model">Nearly free electron model</a></li> <li><a href="/wiki/Bloch%27s_theorem" title="Bloch's theorem">Bloch's theorem</a></li> <li><a href="/wiki/Fermi_liquid_theory" title="Fermi liquid theory">Fermi liquid theory</a></li> <li><a href="/wiki/Electronic_band_structure" title="Electronic band structure">electronic band structure</a></li> <li><a href="/wiki/Anderson_localization" title="Anderson localization">Anderson localization</a></li> <li><a href="/wiki/BCS_theory" title="BCS theory">BCS theory</a></li> <li><a href="/wiki/Tight_binding_model" class="mw-redirect" title="Tight binding model">tight binding model</a></li> <li><a href="/wiki/Hubbard_model" title="Hubbard model">Hubbard model</a></li> <li><a href="/wiki/Density_functional_theory" title="Density functional theory">Density functional theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Conduction</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li> <li><a href="/wiki/Mott_insulator" title="Mott insulator">Mott insulator</a></li> <li><a href="/wiki/Semiconductor" title="Semiconductor">Semiconductor</a></li> <li><a href="/wiki/Semimetal" title="Semimetal">Semimetal</a></li> <li><a href="/wiki/Electrical_conductor" title="Electrical conductor">Conductor</a></li> <li><a href="/wiki/Superconductivity" title="Superconductivity">Superconductor</a></li> <li><a href="/wiki/Topological_insulator" title="Topological insulator">Topological insulator</a></li> <li><a href="/wiki/Spin_gapless_semiconductor" title="Spin gapless semiconductor">Spin gapless semiconductor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Couplings</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Thermoelectric_effect" title="Thermoelectric effect">Thermoelectricity</a></li> <li><a href="/wiki/Piezoelectricity" title="Piezoelectricity">Piezoelectricity</a></li> <li><a href="/wiki/Ferroelectricity" title="Ferroelectricity">Ferroelectricity</a></li> <li><a href="/wiki/Flexoelectricity" title="Flexoelectricity">Flexoelectricity</a></li> <li><a href="/wiki/Electrostriction" title="Electrostriction">Electrostriction</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Magnetic phases</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amorphous_magnet" title="Amorphous magnet">Amorphous magnet</a></li> <li><a href="/wiki/Diamagnetism" title="Diamagnetism">Diamagnet</a></li> <li><a href="/wiki/Superdiamagnetism" title="Superdiamagnetism">Superdiamagnet</a></li> <li><a href="/wiki/Paramagnetism" title="Paramagnetism">Paramagnet</a></li> <li><a href="/wiki/Superparamagnetism" title="Superparamagnetism">Superparamagnet</a></li> <li><a href="/wiki/Ferromagnetism" title="Ferromagnetism">Ferromagnet</a></li> <li><a href="/wiki/Antiferromagnetism" title="Antiferromagnetism">Antiferromagnet</a></li> <li><a href="/wiki/Metamagnetism" title="Metamagnetism">Metamagnet</a></li> <li><a href="/wiki/Spin_glass" title="Spin glass">Spin glass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quasiparticle" 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Tight binding - Wikipedia