A Revival of Molecular Surface Electrostatic Potential Statistical Quantities: Ionic Solids and Liquids

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} div.type-section h2 { font-size: 20px; line-height: 26px; font-weight: 300; } div.type-section h3 { margin-left: 15px; margin-bottom: 0px; font-weight: 300; } .journal-tabs .tab-title.active a { } </style> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/slick.css?f38b2db10e01b157?1732615622"> <meta name="title" content="A Revival of Molecular Surface Electrostatic Potential Statistical Quantities: Ionic Solids and Liquids"> <meta name="description" content="In this paper, we focus on surface electrostatic potentials and a variety of statistically derived quantities defined in terms of the surface potentials. These have been shown earlier to be meaningful in describing features of these potentials and have been utilized to understand the interactive tendencies of molecules in condensed phases. Our current emphasis is on ionic salts and liquids instead of neutral molecules. Earlier work on ionic salts has been reviewed. Presently, our results are for a variety of singly charged cations and anions that can combine to form ionic solids or liquids. Our approach is computational, using the density functional B3PW91/6-31G(d,p) procedure for all calculations. We find consistently that the average positive and negative surface electrostatic potentials of the cations and anions decrease with the size of the ion, as has been noted earlier. A model using computed statistical quantities has allowed us to put the melting points of both ionic solids and liquids together, covering a range from 993 °C to 11 °C." > <link rel="image_src" href="https://pub.mdpi-res.com/img/journals/crystals-logo.png?a2c22e8538083c0b" > <meta name="dc.title" content="A Revival of Molecular Surface Electrostatic Potential Statistical Quantities: Ionic Solids and Liquids"> <meta name="dc.creator" content="Jane S. Murray"> <meta name="dc.creator" content="Kevin E. Riley"> <meta name="dc.creator" content="Tore Brinck"> <meta name="dc.type" content="Article"> <meta name="dc.source" content="Crystals 2024, Vol. 14, Page 995"> <meta name="dc.date" content="2024-11-17"> <meta name ="dc.identifier" content="10.3390/cryst14110995"> <meta name="dc.publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="dc.rights" content="http://creativecommons.org/licenses/by/3.0/"> <meta name="dc.format" content="application/pdf" > <meta name="dc.language" content="en" > <meta name="dc.description" content="In this paper, we focus on surface electrostatic potentials and a variety of statistically derived quantities defined in terms of the surface potentials. These have been shown earlier to be meaningful in describing features of these potentials and have been utilized to understand the interactive tendencies of molecules in condensed phases. Our current emphasis is on ionic salts and liquids instead of neutral molecules. Earlier work on ionic salts has been reviewed. Presently, our results are for a variety of singly charged cations and anions that can combine to form ionic solids or liquids. Our approach is computational, using the density functional B3PW91/6-31G(d,p) procedure for all calculations. We find consistently that the average positive and negative surface electrostatic potentials of the cations and anions decrease with the size of the ion, as has been noted earlier. A model using computed statistical quantities has allowed us to put the melting points of both ionic solids and liquids together, covering a range from 993 °C to 11 °C." > <meta name="dc.subject" content="surface electrostatic potentials" > <meta name="dc.subject" content="surface areas" > <meta name="dc.subject" content="volumes" > <meta name="dc.subject" content="statistical quantities" > <meta name="dc.subject" content="ionic salts" > <meta name="dc.subject" content="ionic liquids" > <meta name ="prism.issn" content="2073-4352"> <meta name ="prism.publicationName" content="Crystals"> <meta name ="prism.publicationDate" content="2024-11-17"> <meta name ="prism.volume" content="14"> <meta name ="prism.number" content="11"> <meta name ="prism.section" content="Article" > <meta name ="prism.startingPage" content="995" > <meta name="citation_issn" content="2073-4352"> <meta name="citation_journal_title" content="Crystals"> <meta name="citation_publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="citation_title" content="A Revival of Molecular Surface Electrostatic Potential Statistical Quantities: Ionic Solids and Liquids"> <meta name="citation_publication_date" content="2024/11"> <meta name="citation_online_date" content="2024/11/17"> <meta name="citation_volume" content="14"> <meta name="citation_issue" content="11"> <meta name="citation_firstpage" content="995"> <meta name="citation_author" content="Murray, Jane S."> <meta name="citation_author" content="Riley, Kevin E."> <meta name="citation_author" content="Brinck, Tore"> <meta name="citation_doi" content="10.3390/cryst14110995"> <meta name="citation_id" content="mdpi-cryst14110995"> <meta name="citation_abstract_html_url" content="https://www.mdpi.com/2073-4352/14/11/995"> <meta name="citation_pdf_url" content="https://www.mdpi.com/2073-4352/14/11/995/pdf?version=1732001402"> <link rel="alternate" type="application/pdf" title="PDF Full-Text" href="https://www.mdpi.com/2073-4352/14/11/995/pdf?version=1732001402"> <meta name="fulltext_pdf" content="https://www.mdpi.com/2073-4352/14/11/995/pdf?version=1732001402"> <meta name="citation_fulltext_html_url" content="https://www.mdpi.com/2073-4352/14/11/995/htm"> <link rel="alternate" type="text/html" title="HTML Full-Text" href="https://www.mdpi.com/2073-4352/14/11/995/htm"> <meta name="fulltext_html" content="https://www.mdpi.com/2073-4352/14/11/995/htm"> <link rel="alternate" type="text/xml" title="XML Full-Text" href="https://www.mdpi.com/2073-4352/14/11/995/xml"> <meta name="fulltext_xml" content="https://www.mdpi.com/2073-4352/14/11/995/xml"> <meta name="citation_xml_url" content="https://www.mdpi.com/2073-4352/14/11/995/xml"> <meta name="twitter:card" content="summary" /> <meta name="twitter:site" content="@MDPIOpenAccess" /> <meta name="twitter:image" content="https://pub.mdpi-res.com/img/journals/crystals-logo-social.png?a2c22e8538083c0b" /> <meta property="fb:app_id" content="131189377574"/> <meta property="og:site_name" content="MDPI"/> <meta property="og:type" content="article"/> <meta property="og:url" content="https://www.mdpi.com/2073-4352/14/11/995" /> <meta property="og:title" content="A Revival of Molecular Surface Electrostatic Potential Statistical Quantities: Ionic Solids and Liquids" /> <meta property="og:description" content="In this paper, we focus on surface electrostatic potentials and a variety of statistically derived quantities defined in terms of the surface potentials. These have been shown earlier to be meaningful in describing features of these potentials and have been utilized to understand the interactive tendencies of molecules in condensed phases. Our current emphasis is on ionic salts and liquids instead of neutral molecules. Earlier work on ionic salts has been reviewed. Presently, our results are for a variety of singly charged cations and anions that can combine to form ionic solids or liquids. Our approach is computational, using the density functional B3PW91/6-31G(d,p) procedure for all calculations. We find consistently that the average positive and negative surface electrostatic potentials of the cations and anions decrease with the size of the ion, as has been noted earlier. A model using computed statistical quantities has allowed us to put the melting points of both ionic solids and liquids together, covering a range from 993 °C to 11 °C." /> <meta property="og:image" content="https://pub.mdpi-res.com/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-ag-550.jpg?1732009038" /> <link rel="alternate" type="application/rss+xml" title="MDPI Publishing - Latest articles" href="https://www.mdpi.com/rss"> <meta name="google-site-verification" content="PxTlsg7z2S00aHroktQd57fxygEjMiNHydKn3txhvwY"> <meta name="facebook-domain-verification" content="mcoq8dtq6sb2hf7z29j8w515jjoof7" /> <script id="Cookiebot" data-cfasync="false" src="https://consent.cookiebot.com/uc.js" data-cbid="51491ddd-fe7a-4425-ab39-69c78c55829f" type="text/javascript" async></script>  <script type="text/plain" data-cookieconsent="statistics"> (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); 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Riley</div><div id="profile-card-drop13409518" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/profiles/289920/thumb/Kevin_Riley.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Kevin E. 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width: 13px; margin-left: 3px; max-width: 13px !important; height: auto; top: -5px;"></a></span> </div> <div class="nrm"></div> <span style="display:block; height:6px;"></span> <div></div> <div style="margin: 5px 0 15px 0;" class="hypothesis_container"> <div class="art-affiliations"> <div class="affiliation "> <div class="affiliation-item"><sup>1</sup></div> <div class="affiliation-name ">Department of Chemistry, University of New Orleans, New Orleans, LA 70148, USA</div> </div> <div class="affiliation "> <div class="affiliation-item"><sup>2</sup></div> <div class="affiliation-name ">Department of Chemistry and Biology, Xavier University of New Orleans, New Orleans, LA 70125, USA</div> </div> <div class="affiliation "> <div class="affiliation-item"><sup>3</sup></div> <div class="affiliation-name ">Department of Chemistry, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden</div> </div> <div class="affiliation"> <div class="affiliation-item"><sup>*</sup></div> <div class="affiliation-name ">Author to whom correspondence should be addressed. </div> </div> </div> </div> <div class="bib-identity" style="margin-bottom: 10px;"> <em>Crystals</em> <b>2024</b>, <em>14</em>(11), 995; <a href="https://doi.org/10.3390/cryst14110995">https://doi.org/10.3390/cryst14110995</a> </div> <div class="pubhistory" style="font-weight: bold; padding-bottom: 10px;"> <span style="display: inline-block">Submission received: 25 October 2024</span> / <span style="display: inline-block">Revised: 9 November 2024</span> / <span style="display: inline-block">Accepted: 13 November 2024</span> / <span style="display: inline-block">Published: 17 November 2024</span> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Section <a href="/journal/crystals/sections/inorganic_crystalline_materials">Inorganic Crystalline Materials</a>)<br/> </div> <div class="highlight-box1"> <div class="download"> <a class="button button--color-inversed button--drop-down" data-dropdown="drop-download-1522964" aria-controls="drop-supplementary-1522964" aria-expanded="false"> Download <i class="material-icons">keyboard_arrow_down</i> </a> <div id="drop-download-1522964" class="f-dropdown label__btn__dropdown label__btn__dropdown--button" data-dropdown-content aria-hidden="true" tabindex="-1"> <a class="UD_ArticlePDF" href="/2073-4352/14/11/995/pdf?version=1732001402" data-name="A Revival of Molecular Surface Electrostatic Potential Statistical Quantities: Ionic Solids and Liquids" data-journal="crystals">Download PDF</a> <br/> <a id="js-pdf-with-cover-access-captcha" href="#" data-target="/2073-4352/14/11/995/pdf-with-cover" class="accessCaptcha">Download PDF with Cover</a> <br/> <a id="js-xml-access-captcha" href="#" data-target="/2073-4352/14/11/995/xml" class="accessCaptcha">Download XML</a> <br/> <a href="/2073-4352/14/11/995/epub" id="epub_link">Download Epub</a> <br/> </div> <div class="js-browse-figures" style="display: inline-block;"> <a href="#" class="button button--color-inversed margin-bottom-10 openpopupgallery UI_BrowseArticleFigures" data-target='article-popup' data-counterslink = "https://www.mdpi.com/2073-4352/14/11/995/browse" >Browse Figures</a> </div> <div id="article-popup" class="popupgallery" style="display: inline; line-height: 200%"> <a href="https://pub.mdpi-res.com/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-ag.png?1732009038" title=" <strong>Graphical abstract</strong><br/> "> </a> <a href="https://pub.mdpi-res.com/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g001.png?1732001505" title=" <strong>Figure 1</strong><br/> <p>Computed electrostatic potentials on the 0.001 au iso-density contours of (<b>a</b>) hydrazine, (<b>b</b>) the hydrazinium cation, and (<b>c</b>) NH<sub>2</sub>NH<sup>−</sup>. The frameworks are shown in gray within the surfaces. The surface of (<b>a</b>) has both positive and negative values of V<sub>S</sub>(<b>r</b>); those of (<b>b</b>,<b>c</b>) are completely positive and negative, respectively. The color ranges, in kcal/mol, are therefore necessarily different for each. The color ranges for (<b>a</b>) are red, greater than 15; yellow, from 15 to 0; green, from 0 to −15; blue, more negative than −15. The color ranges for (<b>b</b>) are red, greater than 150; yellow, from 150 to 125; green, from 125 to 100; blue, less than 100. The color ranges for (<b>c</b>) are red, less negative than −110; yellow, from −110 to −130; green, from −130 to −150; blue, more negative than −150.</p> "> </a> <a href="https://pub.mdpi-res.com/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g002.png?1732001505" title=" <strong>Figure 2</strong><br/> <p>The quantum chemically computed electrostatic potential (V<sub>QC</sub>(r)) compared with the electrostatic potential of a point charge (V<sub><span class="html-italic">q</span></sub>(r)) as functions of the distance (r) from the nucleus for the ions Na<sup>+</sup> and F<sup>−</sup>. The distance of the iso-density contour (<span class="html-italic">ρ</span>(r) = 0.001 au) that is used to compute V<sub>S</sub>(<b>r</b>) is marked in both plots.</p> "> </a> <a href="https://pub.mdpi-res.com/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g003.png?1732001507" title=" <strong>Figure 3</strong><br/> <p>Computed electrostatic potentials on the 0.001 au iso-density contours of (<b>a</b>) EA<sup>+</sup> and (<b>b</b>) [BMIM]<sup>+</sup>. The frameworks are shown in gray within the surfaces. The color ranges, in kcal/mol, are for (<b>a</b>) red, greater than 150; yellow, from 150 to 125; green, from 125 to 100; blue, less than 100. The color ranges for (<b>b</b>) are red, greater than 100; yellow, from 100 to 80; green, from 80 to 60; blue, less than 60.</p> "> </a> <a href="https://pub.mdpi-res.com/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g004.png?1732001508" title=" <strong>Figure 4</strong><br/> <p>Computed electrostatic potentials on the 0.001 au iso-density contours of (<b>a</b>) BF<sub>4</sub><sup>−</sup> and (<b>b</b>) PF<sub>6</sub><sup>−</sup>. The color ranges, in kcal/mol, are for (<b>a</b>) red, less negative than −129; yellow, from −129 to −133; green, from −133 to −137; blue, more negative than −137. The color ranges for (<b>b</b>) are red, less negative than −113; yellow, from −113 to −117; green, from −117 to −121; blue, more negative than −12.</p> "> </a> <a href="https://pub.mdpi-res.com/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g005.png?1732001509" title=" <strong>Figure 5</strong><br/> <p>Plot of predicted melting points using Equation (8) vs. experimentally determined melting points for the seventeen ionic systems in <a href="#crystals-14-00995-t003" class="html-table">Table 3</a>. R = 0.997.</p> "> </a> </div> <a class="button button--color-inversed" href="/2073-4352/14/11/995/notes">Versions Notes</a> </div> </div> <div class="responsive-moving-container small hidden" data-id="article-counters" style="margin-top: 15px;"></div> <div class="html-dynamic"> <section> <div class="art-abstract art-abstract-new in-tab hypothesis_container"> <p> <div><section class="html-abstract" id="html-abstract"> <h2 id="html-abstract-title">Abstract</h2><b>:</b> <div class="html-p">In this paper, we focus on surface electrostatic potentials and a variety of statistically derived quantities defined in terms of the surface potentials. These have been shown earlier to be meaningful in describing features of these potentials and have been utilized to understand the interactive tendencies of molecules in condensed phases. Our current emphasis is on ionic salts and liquids instead of neutral molecules. Earlier work on ionic salts has been reviewed. Presently, our results are for a variety of singly charged cations and anions that can combine to form ionic solids or liquids. Our approach is computational, using the density functional B3PW91/6-31G(d,p) procedure for all calculations. We find consistently that the average positive and negative surface electrostatic potentials of the cations and anions decrease with the size of the ion, as has been noted earlier. A model using computed statistical quantities has allowed us to put the melting points of both ionic solids and liquids together, covering a range from 993 °C to 11 °C.</div> </section> <div id="html-keywords"> <div class="html-gwd-group"><div id="html-keywords-title">Keywords: </div><a href="/search?q=surface+electrostatic+potentials">surface electrostatic potentials</a>; <a href="/search?q=surface+areas">surface areas</a>; <a href="/search?q=volumes">volumes</a>; <a href="/search?q=statistical+quantities">statistical quantities</a>; <a href="/search?q=ionic+salts">ionic salts</a>; <a href="/search?q=ionic+liquids">ionic liquids</a></div> <div> </div> </div> </div> </p> </div> <div class="row"> <div class="columns large-12 text-center"> <div class="abstract-image-preview open js-browse-figures"> <a href="#" class="openpopupgallery" data-target='article-popup-ga'> <img src="https://pub.mdpi-res.com/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-ag-550.jpg?1732009038" style="max-width: 100%; max-height: 280px; padding: 10px;"> </a> <div id="article-popup-ga" class="popupgallery" style="display: inline; line-height: 200%"> <a href="https://pub.mdpi-res.com/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-ag.png?1732009038" title="<strong>Graphical Abstract</strong>"></a> </div> <p>Graphical Abstract</p> </div> </div> </div> </section> </div> <div class="hypothesis_container"> <ul class="menu html-nav" data-prev-node="#html-quick-links-title"> </ul> <div class="html-body"> <section id='sec1-crystals-14-00995' type='intro'><h2 data-nested='1'> 1. Introduction to Electrostatic Potentials</h2><div class='html-p'>The nuclei and electrons of an atom or molecule or ion create an electrostatic potential V(<b>r</b>) at every point <b>r</b> in the surrounding space, given rigorously by the following equation:<div class='html-disp-formula-info' id='FD1-crystals-14-00995'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <mi mathvariant="normal">V</mi> <mo>(</mo> </mrow> <mstyle mathvariant="bold" mathsize="normal"> <mi>r</mi> </mstyle> <mo stretchy="false">)</mo> <mo>=</mo> <mstyle displaystyle="true"> <munder> <mo>∑</mo> <mi mathvariant="normal">A</mi> </munder> <mrow> <mfrac> <mrow> <msub> <mi mathvariant="normal">Z</mi> <mi mathvariant="normal">A</mi> </msub> </mrow> <mrow> <mo stretchy="true">|</mo> <msub> <mstyle mathvariant="bold" mathsize="normal"> <mi>R</mi> </mstyle> <mi mathvariant="normal">A</mi> </msub> <mo>−</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>r</mi> </mstyle> <mo stretchy="true">|</mo> </mrow> </mfrac> </mrow> </mstyle> <mo>−</mo> <mstyle displaystyle="true"> <mrow> <mo>∫</mo> <mrow> <mfrac> <mrow> <mrow> <mi mathvariant="sans-serif">ρ</mi> <mo>(</mo> </mrow> <mstyle mathvariant="bold" mathsize="normal"> <msup> <mi>r</mi> <mo>′</mo> </msup> </mstyle> <mrow> <mo>)</mo> <mi mathvariant="normal">d</mi> </mrow> <mstyle mathvariant="bold" mathsize="normal"> <msup> <mi>r</mi> <mo>′</mo> </msup> </mstyle> </mrow> <mrow> <mo stretchy="true">|</mo> <mrow> <mstyle mathvariant="bold" mathsize="normal"> <msup> <mi>r</mi> <mo>′</mo> </msup> </mstyle> <mo>−</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>r</mi> </mstyle> </mrow> <mo stretchy="true">|</mo> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </semantics></math> </div> <div class='l'> <label >(1)</label> </div> </div></div><div class='html-p'>In Equation (1), Z<sub>A</sub> is the charge on nucleus A, located at <b>R</b><sub>A</sub>, and ρ(<b>r</b>) is the system’s electronic density. The sign of V(<b>r</b>) in any region depends upon whether the positive contribution of the nuclei or the negative one of the electrons is dominant there.</div><div class='html-p'>Molecular electrostatic potentials were introduced as a tool in chemistry by Scrocco and Tomasi in the 1970s [<a href="#B1-crystals-14-00995" class="html-bibr">1</a>,<a href="#B2-crystals-14-00995" class="html-bibr">2</a>] and have been used extensively since then [<a href="#B3-crystals-14-00995" class="html-bibr">3</a>,<a href="#B4-crystals-14-00995" class="html-bibr">4</a>,<a href="#B5-crystals-14-00995" class="html-bibr">5</a>], and with considerable success, to interpret noncovalent interactions [<a href="#B6-crystals-14-00995" class="html-bibr">6</a>,<a href="#B7-crystals-14-00995" class="html-bibr">7</a>,<a href="#B8-crystals-14-00995" class="html-bibr">8</a>,<a href="#B9-crystals-14-00995" class="html-bibr">9</a>]; regions of positive and negative potential on one molecule will tend to interact favorably with, respectively, nucleophilic and electrophilic portions of another molecule. Of course, polarization, an intrinsic part of any Coulombic interaction [<a href="#B1-crystals-14-00995" class="html-bibr">1</a>,<a href="#B10-crystals-14-00995" class="html-bibr">10</a>], explains what are considered “counter-intuitive” interactions [<a href="#B11-crystals-14-00995" class="html-bibr">11</a>].</div><div class='html-p'>An important feature of the electrostatic potential is that it is a real physical property, an observable. It can be determined experimentally by diffraction methods [<a href="#B4-crystals-14-00995" class="html-bibr">4</a>,<a href="#B12-crystals-14-00995" class="html-bibr">12</a>,<a href="#B13-crystals-14-00995" class="html-bibr">13</a>] as well as computationally. The electrostatic potential should not be confused with partial atomic charges in molecules, which are arbitrarily defined (in many different ways) [<a href="#B11-crystals-14-00995" class="html-bibr">11</a>,<a href="#B14-crystals-14-00995" class="html-bibr">14</a>,<a href="#B15-crystals-14-00995" class="html-bibr">15</a>,<a href="#B16-crystals-14-00995" class="html-bibr">16</a>,<a href="#B17-crystals-14-00995" class="html-bibr">17</a>,<a href="#B18-crystals-14-00995" class="html-bibr">18</a>,<a href="#B19-crystals-14-00995" class="html-bibr">19</a>] because they are not physical observables. It should be noted, however, that some partial atomic charges have been designed to produce molecular dipole moments [<a href="#B18-crystals-14-00995" class="html-bibr">18</a>,<a href="#B19-crystals-14-00995" class="html-bibr">19</a>], which are physical observables and which dominate the long-range part of a neutral molecule’s electrostatic potential.</div><div class='html-p'>The electrostatic potential of a spherically symmetric neutral atom is positive everywhere (except, of course, at the actual position of the nucleus where it is undefined) and is monotonically decreasing as V(r) approaches infinity [<a href="#B3-crystals-14-00995" class="html-bibr">3</a>]; it is when atoms combine to form neutral molecules that negative regions of potential develop, often associated with lone pairs, π electrons and strained bonds [<a href="#B1-crystals-14-00995" class="html-bibr">1</a>,<a href="#B2-crystals-14-00995" class="html-bibr">2</a>,<a href="#B3-crystals-14-00995" class="html-bibr">3</a>,<a href="#B4-crystals-14-00995" class="html-bibr">4</a>,<a href="#B5-crystals-14-00995" class="html-bibr">5</a>], with associated global minima V<sub>min</sub>. Pathak and Gadre showed that there are no global maxima [<a href="#B20-crystals-14-00995" class="html-bibr">20</a>], such that the minima must be linked by saddle points.</div><div class='html-p'>What is the situation with monoatomic ions and molecular ions? Spherically averaged monoatomic cations have positive potentials everywhere in space; the values of V(r) decrease monotonically as r → ∞, as do their neutral counterparts [<a href="#B20-crystals-14-00995" class="html-bibr">20</a>,<a href="#B21-crystals-14-00995" class="html-bibr">21</a>]. Sen and Politzer proved in 1989 that the potential of monoatomic anions is positive close to the nucleus and then goes through a negative minimum, since V(r) → 0 as r → ∞ [<a href="#B22-crystals-14-00995" class="html-bibr">22</a>,<a href="#B23-crystals-14-00995" class="html-bibr">23</a>]. The positive monoatomic cations approach zero from the positive side, while the monoatomic anions approach zero from the negative side. The radial behavior of monoatomic ions has recently been revisited for both isotropic atoms and ions and the anisotropic halonium cations [<a href="#B24-crystals-14-00995" class="html-bibr">24</a>].</div><div class='html-p'>In using electrostatic potentials to analyze noncovalent interactions, V(<b>r</b>) is now generally computed on a molecular surface defined, following Bader et al., as an outer contour of the molecule’s electronic density [<a href="#B25-crystals-14-00995" class="html-bibr">25</a>]. Defining the surface in this manner has the advantage that it reflects features specific to the particular molecule, such as lone pairs, π electrons and atomic anisotropy [<a href="#B6-crystals-14-00995" class="html-bibr">6</a>,<a href="#B9-crystals-14-00995" class="html-bibr">9</a>,<a href="#B26-crystals-14-00995" class="html-bibr">26</a>]. The 0.001 au contour is commonly chosen for this purpose, and the electrostatic potential on this contour is labeled V<sub>S</sub>(<b>r</b>). Its locally most positive and most negative values, of which there may be several, are designated by V<sub>S,max</sub> and V<sub>S,min</sub>, respectively [<a href="#B26-crystals-14-00995" class="html-bibr">26</a>].</div><div class='html-p'>The useful information provided by a molecular surface electrostatic potential goes well beyond simply identifying positive and negative regions and utilizing the surface extrema. In the 1990s, Brinck et al. defined certain statistical quantities using the values of points on the surface electrostatic potential [<a href="#B27-crystals-14-00995" class="html-bibr">27</a>,<a href="#B28-crystals-14-00995" class="html-bibr">28</a>,<a href="#B29-crystals-14-00995" class="html-bibr">29</a>]; these have been found to be significant in quantifying important features of the surface electrostatic potentials of molecules [<a href="#B17-crystals-14-00995" class="html-bibr">17</a>,<a href="#B28-crystals-14-00995" class="html-bibr">28</a>,<a href="#B29-crystals-14-00995" class="html-bibr">29</a>,<a href="#B30-crystals-14-00995" class="html-bibr">30</a>].</div><div class='html-p'>The first such quantity defined was the average deviation of V<sub>S</sub>(<b>r</b>) [<a href="#B27-crystals-14-00995" class="html-bibr">27</a>]; it is a measure of the degree of internal charge separation that is present even in a molecule having a zero dipole moment, and is labeled Π [Equation (2)]:<div class='html-disp-formula-info' id='FD2-crystals-14-00995'> <div class='f'> <math display='block'><semantics> <mrow> <mo>Π</mo> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mi mathvariant="normal">n</mi> </mfrac> </mstyle> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi mathvariant="normal">n</mi> </munderover> <mrow> <mfenced close="|" open="|"> <mrow> <mi mathvariant="normal">V</mi> <mo stretchy="false">(</mo> <msub> <mstyle mathvariant="bold" mathsize="normal"> <mi>r</mi> </mstyle> <mi>i</mi> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> </msub> </mrow> </mfenced> </mrow> </mstyle> </mrow> </semantics></math> </div> <div class='l'> <label >(2)</label> </div> </div> where <math display='inline'><semantics> <msub> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> </msub> </semantics></math> is the average value of V<sub>S</sub>(<b>r</b>), while n is the number of points on the surface for which V<sub>S</sub>(<b>r</b>) is computed. This quantity has been shown to correlate with the Kamlet–Taft polarizability–polarity parameter and dielectric constants [<a href="#B27-crystals-14-00995" class="html-bibr">27</a>].</div><div class='html-p'>The next statistical quantities, the positive, negative and total variances of V<sub>S</sub>(<b>r</b>), were conceived to explain a trend seen in the solubilities of naphthalene and a series of substituted indoles in supercritical fluids [<a href="#B28-crystals-14-00995" class="html-bibr">28</a>]. These are defined in Equations (3)–(5) and reflect the ranges and variabilities of its positive and negative values, and of their sum. <div class='html-disp-formula-info' id='FD3-crystals-14-00995'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>+</mo> <mn>2</mn> </msubsup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mi mathvariant="normal">m</mi> </mfrac> </mstyle> <msup> <mrow> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi mathvariant="normal">m</mi> </munderover> <mrow> <mfenced close="]" open="["> <mrow> <msup> <mi mathvariant="normal">V</mi> <mo>+</mo> </msup> <mo stretchy="false">(</mo> <msub> <mstyle mathvariant="bold" mathsize="normal"> <mi>r</mi> </mstyle> <mi>i</mi> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> </mfenced> </mrow> </mstyle> </mrow> <mn>2</mn> </msup> </mrow> </semantics></math> </div> <div class='l'> <label >(3)</label> </div> </div><div class='html-disp-formula-info' id='FD4-crystals-14-00995'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>−</mo> <mn>2</mn> </msubsup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mi mathvariant="normal">n</mi> </mfrac> </mstyle> <msup> <mrow> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi mathvariant="normal">n</mi> </munderover> <mrow> <mfenced close="]" open="["> <mrow> <msup> <mi mathvariant="normal">V</mi> <mo>−</mo> </msup> <mo stretchy="false">(</mo> <msub> <mstyle mathvariant="bold" mathsize="normal"> <mi>r</mi> </mstyle> <mi>j</mi> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>−</mo> </msubsup> </mrow> </mfenced> </mrow> </mstyle> </mrow> <mn>2</mn> </msup> </mrow> </semantics></math> </div> <div class='l'> <label >(4)</label> </div> </div><div class='html-disp-formula-info' id='FD5-crystals-14-00995'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mrow> <mi>tot</mi> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>+</mo> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mrow> <mrow> <mo> </mo> <mi mathvariant="sans-serif">σ</mi> </mrow> </mrow> <mo>−</mo> <mn>2</mn> </msubsup> </mrow> </semantics></math> </div> <div class='l'> <label >(5)</label> </div> </div> In Equations (3) and (4), the summations are over the surface points with positive and negative electrostatic potentials, respectively; <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>−</mo> </msubsup> </mrow> </semantics></math> are the averages of the positive and negative values.</div><div class='html-p'>Finally, a balance parameter was defined to be a measure of the similarity between the positive and negative variances on the molecular surface [<a href="#B29-crystals-14-00995" class="html-bibr">29</a>], given by Equation (6). <div class='html-disp-formula-info' id='FD6-crystals-14-00995'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <mo mathvariant="sans-serif">ν</mo> <mo>=</mo> </mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>+</mo> <mn>2</mn> </msubsup> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>−</mo> <mn>2</mn> </msubsup> </mrow> <mrow> <msup> <mrow> <mfenced close="]" open="["> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mrow> <mi>tot</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </mfenced> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> </mstyle> </mrow> </semantics></math> </div> <div class='l'> <label >(6)</label> </div> </div> Such a balance parameter was needed for obtaining correlations for properties that involve self-interaction, such as boiling points [<a href="#B29-crystals-14-00995" class="html-bibr">29</a>,<a href="#B30-crystals-14-00995" class="html-bibr">30</a>,<a href="#B31-crystals-14-00995" class="html-bibr">31</a>]. ν simply indicates how <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>+</mo> <mn>2</mn> </msubsup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>−</mo> <mn>2</mn> </msubsup> </mrow> </semantics></math> compare. The more similar they are (whether large or small), the closer will ν be to its maximum possible value, 0.25, which corresponds to <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>+</mo> <mn>2</mn> </msubsup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>−</mo> <mn>2</mn> </msubsup> </mrow> </semantics></math> being equal. The product of the total variance and the balance parameter, ν<math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mrow> <mi>tot</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math>, has been found to be particularly effective for representing noncovalent molecular interactions involving self-interaction [<a href="#B17-crystals-14-00995" class="html-bibr">17</a>,<a href="#B30-crystals-14-00995" class="html-bibr">30</a>,<a href="#B31-crystals-14-00995" class="html-bibr">31</a>].</div><div class='html-p'>To give some perspective on these statistical quantities, <a href="#crystals-14-00995-t001" class="html-table">Table 1</a> lists these quantities for water, ammonia, N<sub>2</sub> and a few small organic molecules, computed at the B3PW91/6-31G(d,p) level on 0.001 au iso-density surfaces. The molecules are listed in order of increasing size. It is noteworthy that water is the smallest molecule in <a href="#crystals-14-00995-t001" class="html-table">Table 1</a> but has the largest Π, total variance, as well as ν and the product <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mrow> <mi mathvariant="sans-serif">ν</mi> <mi mathvariant="sans-serif">σ</mi> </mrow> </mrow> <mrow> <mi>tot</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math>. This has been noted earlier [<a href="#B27-crystals-14-00995" class="html-bibr">27</a>,<a href="#B30-crystals-14-00995" class="html-bibr">30</a>,<a href="#B31-crystals-14-00995" class="html-bibr">31</a>]. The least balanced molecule in <a href="#crystals-14-00995-t001" class="html-table">Table 1</a> is dimethyl ether, which helps to explain why its boiling point is similar to that of propane [<a href="#B27-crystals-14-00995" class="html-bibr">27</a>]. There are three molecules in <a href="#crystals-14-00995-t001" class="html-table">Table 1</a> with zero dipole moments: methane, N<sub>2</sub> and acetylene. Their respective Π values of 3.0, 4.4 and 12.4 kcal/mol indicate that acetylene has the greatest internal charge separation of these three molecules, even though all three have zero dipole moments [<a href="#B27-crystals-14-00995" class="html-bibr">27</a>]. In addition, acetylene’s value of <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mrow> <mi mathvariant="sans-serif">ν</mi> <mi mathvariant="sans-serif">σ</mi> </mrow> </mrow> <mrow> <mi>tot</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math> is indicative of it having greater interactive tendencies than do either methane or N<sub>2</sub>.</div><div class='html-p'>Most applications utilizing the statistical quantities defined in Equations (2)–(6) as well as surface areas and volumes have been applied to the physical properties of neutral molecules [<a href="#B17-crystals-14-00995" class="html-bibr">17</a>,<a href="#B26-crystals-14-00995" class="html-bibr">26</a>,<a href="#B27-crystals-14-00995" class="html-bibr">27</a>,<a href="#B28-crystals-14-00995" class="html-bibr">28</a>,<a href="#B29-crystals-14-00995" class="html-bibr">29</a>,<a href="#B30-crystals-14-00995" class="html-bibr">30</a>,<a href="#B31-crystals-14-00995" class="html-bibr">31</a>,<a href="#B32-crystals-14-00995" class="html-bibr">32</a>]. In contrast, our focus in this paper will be upon the surface electrostatic potentials of atomic and molecular cations and anions, highlighting previous work from the Politzer group [<a href="#B33-crystals-14-00995" class="html-bibr">33</a>,<a href="#B34-crystals-14-00995" class="html-bibr">34</a>,<a href="#B35-crystals-14-00995" class="html-bibr">35</a>,<a href="#B36-crystals-14-00995" class="html-bibr">36</a>,<a href="#B37-crystals-14-00995" class="html-bibr">37</a>,<a href="#B38-crystals-14-00995" class="html-bibr">38</a>] and by others [<a href="#B39-crystals-14-00995" class="html-bibr">39</a>,<a href="#B40-crystals-14-00995" class="html-bibr">40</a>,<a href="#B41-crystals-14-00995" class="html-bibr">41</a>,<a href="#B42-crystals-14-00995" class="html-bibr">42</a>,<a href="#B43-crystals-14-00995" class="html-bibr">43</a>,<a href="#B44-crystals-14-00995" class="html-bibr">44</a>,<a href="#B45-crystals-14-00995" class="html-bibr">45</a>,<a href="#B46-crystals-14-00995" class="html-bibr">46</a>,<a href="#B47-crystals-14-00995" class="html-bibr">47</a>,<a href="#B48-crystals-14-00995" class="html-bibr">48</a>,<a href="#B49-crystals-14-00995" class="html-bibr">49</a>,<a href="#B50-crystals-14-00995" class="html-bibr">50</a>,<a href="#B51-crystals-14-00995" class="html-bibr">51</a>]. Finally, we present data bridging concepts relevant to both ionic solids and liquids.</div></section><section id='sec2-crystals-14-00995' type='methods'><h2 data-nested='1'> 2. Methods</h2><div class='html-p'>The structures and surface electrostatic potentials of all molecules and ions have been computed at the B3PW91/6-31G(d,p) level using G16 [<a href="#B52-crystals-14-00995" class="html-bibr">52</a>] and the WFA-SAS code [<a href="#B26-crystals-14-00995" class="html-bibr">26</a>]. This method/basis set combination has been shown to be reliable for our purposes [<a href="#B53-crystals-14-00995" class="html-bibr">53</a>].</div></section><section id='sec3-crystals-14-00995' type=''><h2 data-nested='1'> 3. Surface Electrostatic Potentials of Molecular Ions</h2><div class='html-p'>The best way to introduce the electrostatic potentials of molecular ions is to compare them to those of neutral molecules. Take, for example, the series: NH<sub>2</sub>NH<sub>2</sub>, NH<sub>2</sub>NH<sub>3</sub><sup>+</sup> and NH<sub>2</sub>NH<sup>−</sup>; their surface electrostatic potentials plotted on the 0.001 au contour of the electronic density are shown in <a href="#crystals-14-00995-f001" class="html-fig">Figure 1</a>. Hydrazine has negative regions of V<sub>S</sub>(<b>r</b>) associated with the lone pairs on its nitrogens and positive regions associated with the hydrogens, shown in <a href="#crystals-14-00995-f001" class="html-fig">Figure 1</a>a. The most positive regions on the surface of the hydrazinium cation are associated with the three hydrogens bonded to the nitrogen at the left in <a href="#crystals-14-00995-f001" class="html-fig">Figure 1</a>b; the least positive region in blue is associated with the nitrogen at the right. Finally, the most negative region on the surface of NH<sub>2</sub>NH<sup>−</sup> is at the top right of <a href="#crystals-14-00995-f001" class="html-fig">Figure 1</a>c, with the least negative sites (in red) on the hydrogens to left.</div><div class='html-p'>Neutral molecules such as those listed in <a href="#crystals-14-00995-t001" class="html-table">Table 1</a> and hydrazine in <a href="#crystals-14-00995-f001" class="html-fig">Figure 1</a>a typically have regions of both positive and negative electrostatic potential on their 0.001 au surfaces [<a href="#B1-crystals-14-00995" class="html-bibr">1</a>,<a href="#B2-crystals-14-00995" class="html-bibr">2</a>,<a href="#B3-crystals-14-00995" class="html-bibr">3</a>,<a href="#B4-crystals-14-00995" class="html-bibr">4</a>,<a href="#B5-crystals-14-00995" class="html-bibr">5</a>,<a href="#B6-crystals-14-00995" class="html-bibr">6</a>,<a href="#B7-crystals-14-00995" class="html-bibr">7</a>,<a href="#B8-crystals-14-00995" class="html-bibr">8</a>,<a href="#B9-crystals-14-00995" class="html-bibr">9</a>,<a href="#B26-crystals-14-00995" class="html-bibr">26</a>,<a href="#B27-crystals-14-00995" class="html-bibr">27</a>,<a href="#B28-crystals-14-00995" class="html-bibr">28</a>,<a href="#B29-crystals-14-00995" class="html-bibr">29</a>,<a href="#B30-crystals-14-00995" class="html-bibr">30</a>,<a href="#B31-crystals-14-00995" class="html-bibr">31</a>,<a href="#B32-crystals-14-00995" class="html-bibr">32</a>], while cations have only positive values of V<sub>S</sub>(<b>r</b>) and anions only negative values of V<sub>S</sub>(<b>r</b>) on these surfaces [<a href="#B33-crystals-14-00995" class="html-bibr">33</a>,<a href="#B35-crystals-14-00995" class="html-bibr">35</a>,<a href="#B36-crystals-14-00995" class="html-bibr">36</a>,<a href="#B37-crystals-14-00995" class="html-bibr">37</a>,<a href="#B38-crystals-14-00995" class="html-bibr">38</a>,<a href="#B39-crystals-14-00995" class="html-bibr">39</a>,<a href="#B41-crystals-14-00995" class="html-bibr">41</a>,<a href="#B42-crystals-14-00995" class="html-bibr">42</a>,<a href="#B43-crystals-14-00995" class="html-bibr">43</a>,<a href="#B44-crystals-14-00995" class="html-bibr">44</a>,<a href="#B45-crystals-14-00995" class="html-bibr">45</a>,<a href="#B46-crystals-14-00995" class="html-bibr">46</a>,<a href="#B47-crystals-14-00995" class="html-bibr">47</a>,<a href="#B48-crystals-14-00995" class="html-bibr">48</a>,<a href="#B49-crystals-14-00995" class="html-bibr">49</a>,<a href="#B50-crystals-14-00995" class="html-bibr">50</a>,<a href="#B51-crystals-14-00995" class="html-bibr">51</a>]. Monoatomic anions and molecular anions will have negative potentials when plotted on outer contours of the electronic density; however, at contours very close to the nuclei, the potentials will be positive, as they are there dominated by the first term in Equation (1) [<a href="#B22-crystals-14-00995" class="html-bibr">22</a>,<a href="#B23-crystals-14-00995" class="html-bibr">23</a>,<a href="#B24-crystals-14-00995" class="html-bibr">24</a>].</div><div class='html-p'>This is apparent from <a href="#crystals-14-00995-f002" class="html-fig">Figure 2</a>, which shows V(<b>r</b>) as a function of the distance r from the nucleus, labeled V<sub>QC</sub>(r) in the figure, for both Na<sup>+</sup> and F<sup>−</sup>. The quantum chemically computed V<sub>QC</sub>(r) is compared with the electrostatic potential computed from a positive or negative unitary point charge, labeled V<sub><span class='html-italic'>q</span></sub>(r) in the figure. V<span class='html-italic'><sub>q</sub></span>(r) is obtained from Coulomb’s law as V<span class='html-italic'><sub>q</sub></span>(r) = q/r. Note that for all values of r, V<sub>QC</sub>(r) > V<sub><span class='html-italic'>q</span></sub>(r), since a part of the electronic charge is located outside <span class='html-italic'>r.</span> However, at larger distances this charge is very small and V<sub>QC</sub>(r) is nearly identical to V<sub><span class='html-italic'>q</span></sub>(r), and the two curves are indistinguishable from each other. This is true already at the distance where ρ(r) = 0.001 au, which is the contour that we usually use for computing V<sub>S</sub>(<b>r</b>).</div><div class='html-p'>Sen and Politzer proved that the V(r) of monoatomic singly negative spherically averaged ions must go through a minimum before approaching zero from the negative direction [<a href="#B22-crystals-14-00995" class="html-bibr">22</a>,<a href="#B23-crystals-14-00995" class="html-bibr">23</a>]. They also showed that these minima correspond to reasonable anionic radii for these ions and to lattice energies in a series of monoatomic cations (Li<sup>+</sup> to Fr<sup>+</sup>) [<a href="#B22-crystals-14-00995" class="html-bibr">22</a>], and that the quantity of electronic charge encompassed within the radial V<sub>min</sub> exactly equals the nuclear charge [<a href="#B22-crystals-14-00995" class="html-bibr">22</a>]. Ramasami and Murray recently showed that monoatomic spherically averaged ions with charges greater than −1 follow the same pattern [<a href="#B24-crystals-14-00995" class="html-bibr">24</a>]. As expected, F<sup>−</sup> also has a radial V<sub>min</sub>, and, as shown in <a href="#crystals-14-00995-f002" class="html-fig">Figure 2</a>, it appears at a much shorter distance than the 0.001 density contour.</div><div class='html-p'>Our first venture into looking at the statistical quantities of charged species larger than monoatomic ions was looking at lattice energies of a series of 17 anions, ranging in size from F<sup>−</sup> to SiF<sub>6</sub><sup>−2</sup> and including both inorganic and organic ions with charges primarily of −1 but a few with charges of −2 [<a href="#B33-crystals-14-00995" class="html-bibr">33</a>]. What we found is that for NH<sub>4</sub><sup>+</sup>, Na<sup>+</sup> and K<sup>+</sup> salts, there were good correlations between the charge Q of the anion, the V<sub>S,min</sub> on the surface of the anion and the product of the V<sub>S,min</sub> and the surface area of the anion [<a href="#B33-crystals-14-00995" class="html-bibr">33</a>].</div><div class='html-p'>Zwitterions are neutral but locally have stronger positive and negative regions of electrostatic potential in the vicinities of their positive- and negative-charged regions than do their nonionic counterparts [<a href="#B34-crystals-14-00995" class="html-bibr">34</a>]. This was explored for glycine, histidine and tetracycline. The average deviations of the surface electrostatic potentials of the zwitterions, the Π values, were in each instance greater for the zwitterionic forms than for the nonionic forms [<a href="#B34-crystals-14-00995" class="html-bibr">34</a>].</div><div class='html-p'>Moving on to molecular cations, a study was planned and carried out to assess the stability of gas phase carbocations [<a href="#B35-crystals-14-00995" class="html-bibr">35</a>]. What was found is that the stabilities of the carbocations generally follow the decrease of their V<sub>S,max</sub> values. This follows intuitively, as the +1 charge is more delocalized as the size of carbocation increases, and the V<sub>S,max</sub> value accordingly decreases [<a href="#B35-crystals-14-00995" class="html-bibr">35</a>].</div><div class='html-p'>In 2009, Politzer et al. published a paper entitled “An electrostatic interaction correction for improved crystal density prediction” [<a href="#B54-crystals-14-00995" class="html-bibr">54</a>]. The objective of this paper was to improve crystal density predictions for energetic molecules. In this paper, the authors added an electrostatic correction term to the already widely used term M/V<sub>m</sub> [<a href="#B55-crystals-14-00995" class="html-bibr">55</a>,<a href="#B56-crystals-14-00995" class="html-bibr">56</a>], where M is the molecular mass and V<sub>m</sub> is the volume of the 0.001 au iso-density surface envelope encompassing the molecule. Both statistical quantities Π and ν<math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mrow> <mi>tot</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math> were tested and yielded improved predictions [<a href="#B54-crystals-14-00995" class="html-bibr">54</a>]. A reviewer for this paper wondered if this could somehow be extended to energetic ionic salts.</div><div class='html-p'>This led to exploratory computational work involving energetic ionic salts; it seemed reasonable and necessary to find a correction term for the cation and the anion in each salt. For a dataset of 25 energetic ionic salts, we found the following equation to be an effective option [<a href="#B36-crystals-14-00995" class="html-bibr">36</a>]:<div class='html-disp-formula-info' id='FD11-crystals-14-00995'> <div class='f'> <math display='block'><semantics> <mrow> <mi>density</mi> <mo>=</mo> <mi>α</mi> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi mathvariant="normal">M</mi> <mrow> <msub> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">m</mi> </msub> </mrow> </mfrac> </mstyle> <mo>+</mo> <mi>β</mi> <mfenced> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> <mrow> <msubsup> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> </mfrac> </mstyle> </mrow> </mfenced> <mo>+</mo> <mi>γ</mi> <mfenced> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>−</mo> </msubsup> </mrow> <mrow> <msubsup> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">S</mi> <mo>−</mo> </msubsup> </mrow> </mfrac> </mstyle> </mrow> </mfenced> <mo>+</mo> <mi>δ</mi> </mrow> </semantics></math> </div> <div class='l'> <label >(7)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>−</mo> </msubsup> </mrow> </semantics></math> are the average values of the potential on the surfaces of the cations and anions, respectively, and <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">S</mi> <mo>−</mo> </msubsup> </mrow> </semantics></math> are the surface areas of the positive cations and negative anions. M is the molecular mass of the salt, and V<sub>m</sub> is the sum of the 0.001 au volumes of the cation (s) and anion (s) for each salt. This improvement over M/V<sub>m</sub> alone could be further explored, as has been pointed out [<a href="#B36-crystals-14-00995" class="html-bibr">36</a>]. Statistical quantities derived from the surface electrostatic potential together with other descriptors have also been used to develop machine learning models for prediction of solvation and partition properties of ionic liquids [<a href="#B57-crystals-14-00995" class="html-bibr">57</a>,<a href="#B58-crystals-14-00995" class="html-bibr">58</a>].</div><div class='html-p'>Finally, some perspectives on the sensitivities of ionic energetic materials were addressed by Politzer et al. [<a href="#B37-crystals-14-00995" class="html-bibr">37</a>,<a href="#B38-crystals-14-00995" class="html-bibr">38</a>] after some interest toward this endeavor was expressed. This is a challenge that continues to this day. One of the findings found computationally is that the impact sensitivities for a series of ammonium salts showed a tendency to decrease as the absolute value of the differences of the most negative surface potentials, the V<sub>S,min</sub>, and the least negative surface potentials, the V<sub>S,max</sub>, of the anions increased [<a href="#B38-crystals-14-00995" class="html-bibr">38</a>]. Further investigations of this complex phenomenon are warranted.</div></section><section id='sec4-crystals-14-00995' type=''><h2 data-nested='1'> 4. Statistical Quantities of Some Cations and Anions in Ionic Salts and Liquids</h2><div class='html-p'><a href="#crystals-14-00995-t002" class="html-table">Table 2</a> lists surface areas, volumes and some statistical quantities defined in terms of the electrostatic potential on the surfaces of a number of cations and anions that can be components of ionic salts and liquids. Ionic salts are solids at room temperature, while ionic liquids are generally defined as having melting points below 100 °C [<a href="#B59-crystals-14-00995" class="html-bibr">59</a>,<a href="#B60-crystals-14-00995" class="html-bibr">60</a>,<a href="#B61-crystals-14-00995" class="html-bibr">61</a>], with room-temperature ionic liquids having melting points below 25 °C. The ethylammonium, trimethylammonium, 1-ethyl-3-methylimidazolium and 1-butyl-3-methylimidazolium cations are given the common abbreviations EA<sup>+</sup>, TMA<sup>+</sup>, EMIM<sup>+</sup> and BMIM<sup>+</sup> in <a href="#crystals-14-00995-t002" class="html-table">Table 2</a>. All of the ions in <a href="#crystals-14-00995-t002" class="html-table">Table 2</a> have a charge of ±1.</div><div class='html-p'>To provide some perspective and to allow the reader to visualize the potentials, <a href="#crystals-14-00995-f003" class="html-fig">Figure 3</a> and <a href="#crystals-14-00995-f004" class="html-fig">Figure 4</a> show the V<sub>S</sub>(<b>r</b>) of the cations EA<sup>+</sup> and BMIM<sup>+</sup> and the anions BF<sub>4</sub><sup>−</sup> and PF<sub>6</sub><sup>−</sup>, respectively. The surfaces of the cations shown in <a href="#crystals-14-00995-f003" class="html-fig">Figure 3</a> are totally positive, showing variation in their surface electrostatic potentials, as has been noted recently for EMIM+ [<a href="#B49-crystals-14-00995" class="html-bibr">49</a>]. The larger color range cutoffs for EA<sup>+</sup> compared to those of BMIM<sup>+</sup> are consistent with their respective average positive surface potentials <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> </semantics></math> listed in <a href="#crystals-14-00995-t002" class="html-table">Table 2</a>, 124.5 vs. 86.1 kcal/mol, and their sizes. The most positive regions of EA<sup>+</sup> are associated with the hydrogens bonded to the nitrogen on the left in <a href="#crystals-14-00995-f003" class="html-fig">Figure 3</a>a and can be compared to those of the hydrazinium cation in <a href="#crystals-14-00995-f001" class="html-fig">Figure 1</a>b. The most positive region in <a href="#crystals-14-00995-f003" class="html-fig">Figure 3</a>b is to the bottom left in the region, between the C-2 hydrogen and the closest methyl hydrogen of the C-3 methyl group, where the positive regions overlap on the surface.</div><div class='html-p'>What is interesting to observe in <a href="#crystals-14-00995-f004" class="html-fig">Figure 4</a> is the fact that the color range cutoffs for both anions cover only a narrow range, compared to those of the cations shown in <a href="#crystals-14-00995-f003" class="html-fig">Figure 3</a>. Those of BF<sub>4</sub><sup>−</sup> are only slightly more negative than those of PF<sub>6</sub><sup>−</sup>, and are consistent with their sizes, their <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>−</mo> </msubsup> </mrow> </semantics></math> values and the ranges of their negative potentials (<a href="#crystals-14-00995-t002" class="html-table">Table 2</a>).</div><div class='html-p'>The cations are listed first in <a href="#crystals-14-00995-t002" class="html-table">Table 2</a>, from smallest to largest, as indicated by their surface areas and volumes. These are followed by the anions. Note that the average surface potentials <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>−</mo> </msubsup> </mrow> </semantics></math> for the cations and anions, respectively, decrease in magnitude in the same order that size increases. This has been observed in earlier studies [<a href="#B33-crystals-14-00995" class="html-bibr">33</a>,<a href="#B35-crystals-14-00995" class="html-bibr">35</a>,<a href="#B36-crystals-14-00995" class="html-bibr">36</a>,<a href="#B37-crystals-14-00995" class="html-bibr">37</a>,<a href="#B38-crystals-14-00995" class="html-bibr">38</a>] and can be attributed to the overall charge of each ion, either +1 or −1, being delocalized over a larger space as the size of the ion increases. For each monoatomic cation and anion, the V<sub>S,max</sub> and V<sub>S,min</sub> values are the same, as their potentials are isotropic [<a href="#B20-crystals-14-00995" class="html-bibr">20</a>,<a href="#B21-crystals-14-00995" class="html-bibr">21</a>,<a href="#B22-crystals-14-00995" class="html-bibr">22</a>,<a href="#B23-crystals-14-00995" class="html-bibr">23</a>,<a href="#B24-crystals-14-00995" class="html-bibr">24</a>], explaining why their Π and <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>+</mo> <mn>2</mn> </msubsup> </mrow> </semantics></math> or <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>−</mo> <mn>2</mn> </msubsup> </mrow> </semantics></math> values are zero. This is not the case for the molecular cations and anions, which show variation in their electrostatic potentials, as shown by their Π, <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>+</mo> <mn>2</mn> </msubsup> </mrow> </semantics></math> or <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>−</mo> <mn>2</mn> </msubsup> </mrow> </semantics></math> values as well as their V<sub>S,max</sub> and V<sub>S,min</sub> values.</div><div class='html-p'>Do the data in <a href="#crystals-14-00995-t002" class="html-table">Table 2</a> reflect features of the cations and anions that relate to the melting points of some ionic salts and ionic liquids? To explore this possibility, in <a href="#crystals-14-00995-t003" class="html-table">Table 3</a> are listed some melting points [<a href="#B62-crystals-14-00995" class="html-bibr">62</a>,<a href="#B63-crystals-14-00995" class="html-bibr">63</a>] for seventeen ionic salts and liquids. Note that the ionic salts have melting points often an order of magnitude greater, or more, than those of the ionic liquids. It can be seen in <a href="#crystals-14-00995-t002" class="html-table">Table 2</a> that Na<sup>+</sup> and K<sup>+</sup> are the two smallest cations listed and that they have the two largest average values (<math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> </semantics></math>), while the halide ions are the three smallest anions and have the three most negative <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>−</mo> </msubsup> </mrow> </semantics></math>, with the fluoride ion’s value being significantly more negative than those of the chloride and bromide ions. Another observation is that the surface areas and volumes of the cations cover a wider range of values than those of the anions.</div><div class='html-p'>With some insight from earlier work with ionic systems [<a href="#B31-crystals-14-00995" class="html-bibr">31</a>,<a href="#B33-crystals-14-00995" class="html-bibr">33</a>,<a href="#B34-crystals-14-00995" class="html-bibr">34</a>,<a href="#B35-crystals-14-00995" class="html-bibr">35</a>,<a href="#B36-crystals-14-00995" class="html-bibr">36</a>], the following multivariable relationship was found for the seventeen ionic salts and liquids in <a href="#crystals-14-00995-t003" class="html-table">Table 3</a> using the NCSS software [<a href="#B64-crystals-14-00995" class="html-bibr">64</a>]:<div class='html-disp-formula-info' id='FD12-crystals-14-00995'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <mi>melting</mi> <mo> </mo> <mi>point</mi> <mo>=</mo> <mn>182.0</mn> <mo> </mo> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> <mrow> <mo>−</mo> <mn>0.01445</mn> <mo> </mo> <mo>(</mo> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>−</mo> </msubsup> </mrow> <mrow> <mo>)</mo> <mo> </mo> <mo>+</mo> <mo> </mo> <mn>2.491</mn> <mo> </mo> <msubsup> <mrow> <mrow> <mo>(</mo> <mi mathvariant="normal">A</mi> </mrow> </mrow> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>−</mo> <mn>1174</mn> <mo> </mo> <mfenced> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> <mrow> <msubsup> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> </mfrac> </mstyle> </mrow> </mfenced> </mrow> <mrow> <mo>+</mo> <mspace linebreak="newline"/> <mo>−</mo> <mn>72.83</mn> <mo> </mo> <mo>(</mo> <msup> <mi>Vol</mi> <mo>+</mo> </msup> <mo>)</mo> <mo> </mo> <mo>−</mo> <mo> </mo> <mn>45267</mn> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(8)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>−</mo> </msubsup> </mrow> </semantics></math> are the average values of the potential on the surfaces of the cations and anions, respectively, and <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> </semantics></math> and Vol<sup>+</sup> are the surface areas and volumes of the positive cations. Note that the second term in Equation (8) is a cross-term, (<math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> </semantics></math><math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>−</mo> </msubsup> </mrow> </semantics></math>), fitting in with the Coulombic nature of ionic interactions. The first three terms in Equation (8) lead to an increase in the value of the melting point, with the fourth and fifth terms serving likely as correction terms. The R value for the correlation is 0.997 and the F-ratio is 328, showing improvement over any three- or four-variable correlations. Each term in Equation (8) has a <span class='html-italic'>p</span>-value of 0.000, indicating the validity of each variable.</div><div class='html-p'>A plot of predicted vs. experimentally determined melting points is shown as <a href="#crystals-14-00995-f005" class="html-fig">Figure 5</a>. As can be clearly seen the points for the ionic salts are at the top right of the plot, while those of the ionic liquids are at the bottom left, with the ammonium salts in between.</div><div class='html-p'>In the correlation shown in Equation (8) and in the corresponding plot in <a href="#crystals-14-00995-f005" class="html-fig">Figure 5</a>, the variables relating to the cations emerge as appearing more relevant than those for the anions. In fact, the only inclusion of the average negative surface electrostatic potential <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>−</mo> </msubsup> </mrow> </semantics></math> is in the cross-term (<math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>+</mo> </msubsup> </mrow> </semantics></math><math display='inline'><semantics> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="normal">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="normal">S</mi> <mo>−</mo> </msubsup> </mrow> </semantics></math>). This suggests that perhaps other contours of the electronic density might be better for characterizing the anions. As was shown recently, the radial V<sub>min</sub> for sphericallysymmetric monoatomic anions is within their 0.001 au iso-density surfaces [<a href="#B22-crystals-14-00995" class="html-bibr">22</a>]. Another possibility is that the characteristics of the cations are simply more important.</div></section><section id='sec5-crystals-14-00995' type=''><h2 data-nested='1'> 5. Concluding Remarks</h2><div class='html-p'>In this paper we have reviewed statistical quantities defined in terms of the surface electrostatic potential; these were originally defined to quantify the wealth of information beyond surface extrema that surface electrostatic potential plots provide. The original applications of these quantities were primarily for neutral molecules.</div><div class='html-p'>This study surveys earlier uses of the statistical quantities for ionic solids and is to be viewed as a starting point for future explorations in the growing field of ionic liquids. The properties of ionic liquids are complex; it may be that further studies involving surface electrostatic potential statistical quantities, as well as surface areas and volumes, will prove fruitful in this area.</div></section> </div> <div class="html-back"> <section class='html-notes'><h2 >Author Contributions</h2><div class='html-p'>Conceptualization, J.S.M., K.E.R. and T.B.; methodology, J.S.M. and T.B.; software, T.B.; validation and formal analysis, J.S.M., K.E.R. and T.B.; writing—original draft preparation, J.S.M.; writing—review and editing, K.E.R. and T.B.; funding acquisition, K.E.R. and T.B. All authors have read and agreed to the published version of the manuscript.</div></section><section class='html-notes'><h2>Funding</h2><div class='html-p'>This research was funded by the National Science Foundation (CHE230047, CHE2401967, DMR2425025) for K.E.R. and by the Swedish Research Council (DNR 2021-05881) for T.B.</div></section><section class='html-notes'><h2 >Data Availability Statement</h2><div class='html-p'>All data are included in the manuscript.</div></section><section id='html-ack' class='html-ack'><h2 >Acknowledgments</h2><div class='html-p'>J.S.M., K.E.R. and T.B. are grateful for the guidance and continuing inspiration provided by the late Peter Politzer (1937–2022).</div></section><section class='html-notes'><h2 >Conflicts of Interest</h2><div class='html-p'>The authors declare no conflicts of interest. 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[<a href="https://scholar.google.com/scholar_lookup?title=Handbook+of+Chemistry+and+Physics&author=Rumble,+J.R.,+Jr.&author=Bruno,+T.J.&author=Doa,+M.J.&publication_year=2021" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B63-crystals-14-00995' class='html-xx' data-content='63.'>Available online: <a href="https://en.wikipedia.org/wiki" target="_blank" rel="noopener noreferrer">https://en.wikipedia.org/wiki</a> (accessed on 23 October 2024).</li><li id='B64-crystals-14-00995' class='html-xx' data-content='64.'><span class='html-italic'>NCSS 2024 Statistical Software</span>; NCSS, LLC: Kaysville, UT, USA, 2024; Available online: <a href="https://www.ncss.com/software/ncss" target="_blank" rel="noopener noreferrer">https://www.ncss.com/software/ncss</a> (accessed on 23 October 2024).</li></ol></section><section id='FiguresandTables' type='display-objects'><div class="html-fig-wrap" id="crystals-14-00995-f001"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href="#fig_body_display_crystals-14-00995-f001"> <img data-large="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g001.png" data-original="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g001.png" alt="Crystals 14 00995 g001" data-lsrc="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g001-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href="#fig_body_display_crystals-14-00995-f001"></a> </div> </div> <div class="html-fig_description"> <b>Figure 1.</b> Computed electrostatic potentials on the 0.001 au iso-density contours of (<b>a</b>) hydrazine, (<b>b</b>) the hydrazinium cation, and (<b>c</b>) NH<sub>2</sub>NH<sup>−</sup>. The frameworks are shown in gray within the surfaces. The surface of (<b>a</b>) has both positive and negative values of V<sub>S</sub>(<b>r</b>); those of (<b>b</b>,<b>c</b>) are completely positive and negative, respectively. The color ranges, in kcal/mol, are therefore necessarily different for each. The color ranges for (<b>a</b>) are red, greater than 15; yellow, from 15 to 0; green, from 0 to −15; blue, more negative than −15. The color ranges for (<b>b</b>) are red, greater than 150; yellow, from 150 to 125; green, from 125 to 100; blue, less than 100. The color ranges for (<b>c</b>) are red, less negative than −110; yellow, from −110 to −130; green, from −130 to −150; blue, more negative than −150.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_crystals-14-00995-f001"> <div class="html-caption"> <b>Figure 1.</b> Computed electrostatic potentials on the 0.001 au iso-density contours of (<b>a</b>) hydrazine, (<b>b</b>) the hydrazinium cation, and (<b>c</b>) NH<sub>2</sub>NH<sup>−</sup>. The frameworks are shown in gray within the surfaces. The surface of (<b>a</b>) has both positive and negative values of V<sub>S</sub>(<b>r</b>); those of (<b>b</b>,<b>c</b>) are completely positive and negative, respectively. The color ranges, in kcal/mol, are therefore necessarily different for each. The color ranges for (<b>a</b>) are red, greater than 15; yellow, from 15 to 0; green, from 0 to −15; blue, more negative than −15. The color ranges for (<b>b</b>) are red, greater than 150; yellow, from 150 to 125; green, from 125 to 100; blue, less than 100. The color ranges for (<b>c</b>) are red, less negative than −110; yellow, from −110 to −130; green, from −130 to −150; blue, more negative than −150.</div> <div class="html-img"><img data-large="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g001.png" data-original="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g001.png" alt="Crystals 14 00995 g001" data-lsrc="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g001.png" /></div> </div> <div class="html-fig-wrap" id="crystals-14-00995-f002"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href="#fig_body_display_crystals-14-00995-f002"> <img data-large="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g002.png" data-original="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g002.png" alt="Crystals 14 00995 g002" data-lsrc="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g002-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href="#fig_body_display_crystals-14-00995-f002"></a> </div> </div> <div class="html-fig_description"> <b>Figure 2.</b> The quantum chemically computed electrostatic potential (V<sub>QC</sub>(r)) compared with the electrostatic potential of a point charge (V<sub><span class='html-italic'>q</span></sub>(r)) as functions of the distance (r) from the nucleus for the ions Na<sup>+</sup> and F<sup>−</sup>. The distance of the iso-density contour (<span class='html-italic'>ρ</span>(r) = 0.001 au) that is used to compute V<sub>S</sub>(<b>r</b>) is marked in both plots.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_crystals-14-00995-f002"> <div class="html-caption"> <b>Figure 2.</b> The quantum chemically computed electrostatic potential (V<sub>QC</sub>(r)) compared with the electrostatic potential of a point charge (V<sub><span class='html-italic'>q</span></sub>(r)) as functions of the distance (r) from the nucleus for the ions Na<sup>+</sup> and F<sup>−</sup>. The distance of the iso-density contour (<span class='html-italic'>ρ</span>(r) = 0.001 au) that is used to compute V<sub>S</sub>(<b>r</b>) is marked in both plots.</div> <div class="html-img"><img data-large="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g002.png" data-original="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g002.png" alt="Crystals 14 00995 g002" data-lsrc="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g002.png" /></div> </div> <div class="html-fig-wrap" id="crystals-14-00995-f003"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href="#fig_body_display_crystals-14-00995-f003"> <img data-large="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g003.png" data-original="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g003.png" alt="Crystals 14 00995 g003" data-lsrc="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g003-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href="#fig_body_display_crystals-14-00995-f003"></a> </div> </div> <div class="html-fig_description"> <b>Figure 3.</b> Computed electrostatic potentials on the 0.001 au iso-density contours of (<b>a</b>) EA<sup>+</sup> and (<b>b</b>) [BMIM]<sup>+</sup>. The frameworks are shown in gray within the surfaces. The color ranges, in kcal/mol, are for (<b>a</b>) red, greater than 150; yellow, from 150 to 125; green, from 125 to 100; blue, less than 100. The color ranges for (<b>b</b>) are red, greater than 100; yellow, from 100 to 80; green, from 80 to 60; blue, less than 60.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_crystals-14-00995-f003"> <div class="html-caption"> <b>Figure 3.</b> Computed electrostatic potentials on the 0.001 au iso-density contours of (<b>a</b>) EA<sup>+</sup> and (<b>b</b>) [BMIM]<sup>+</sup>. The frameworks are shown in gray within the surfaces. The color ranges, in kcal/mol, are for (<b>a</b>) red, greater than 150; yellow, from 150 to 125; green, from 125 to 100; blue, less than 100. The color ranges for (<b>b</b>) are red, greater than 100; yellow, from 100 to 80; green, from 80 to 60; blue, less than 60.</div> <div class="html-img"><img data-large="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g003.png" data-original="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g003.png" alt="Crystals 14 00995 g003" data-lsrc="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g003.png" /></div> </div> <div class="html-fig-wrap" id="crystals-14-00995-f004"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href="#fig_body_display_crystals-14-00995-f004"> <img data-large="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g004.png" data-original="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g004.png" alt="Crystals 14 00995 g004" data-lsrc="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g004-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href="#fig_body_display_crystals-14-00995-f004"></a> </div> </div> <div class="html-fig_description"> <b>Figure 4.</b> Computed electrostatic potentials on the 0.001 au iso-density contours of (<b>a</b>) BF<sub>4</sub><sup>−</sup> and (<b>b</b>) PF<sub>6</sub><sup>−</sup>. The color ranges, in kcal/mol, are for (<b>a</b>) red, less negative than −129; yellow, from −129 to −133; green, from −133 to −137; blue, more negative than −137. The color ranges for (<b>b</b>) are red, less negative than −113; yellow, from −113 to −117; green, from −117 to −121; blue, more negative than −12.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_crystals-14-00995-f004"> <div class="html-caption"> <b>Figure 4.</b> Computed electrostatic potentials on the 0.001 au iso-density contours of (<b>a</b>) BF<sub>4</sub><sup>−</sup> and (<b>b</b>) PF<sub>6</sub><sup>−</sup>. The color ranges, in kcal/mol, are for (<b>a</b>) red, less negative than −129; yellow, from −129 to −133; green, from −133 to −137; blue, more negative than −137. The color ranges for (<b>b</b>) are red, less negative than −113; yellow, from −113 to −117; green, from −117 to −121; blue, more negative than −12.</div> <div class="html-img"><img data-large="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g004.png" data-original="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g004.png" alt="Crystals 14 00995 g004" data-lsrc="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g004.png" /></div> </div> <div class="html-fig-wrap" id="crystals-14-00995-f005"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href="#fig_body_display_crystals-14-00995-f005"> <img data-large="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g005.png" data-original="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g005.png" alt="Crystals 14 00995 g005" data-lsrc="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g005-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href="#fig_body_display_crystals-14-00995-f005"></a> </div> </div> <div class="html-fig_description"> <b>Figure 5.</b> Plot of predicted melting points using Equation (8) vs. experimentally determined melting points for the seventeen ionic systems in <a href="#crystals-14-00995-t003" class="html-table">Table 3</a>. R = 0.997.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_crystals-14-00995-f005"> <div class="html-caption"> <b>Figure 5.</b> Plot of predicted melting points using Equation (8) vs. experimentally determined melting points for the seventeen ionic systems in <a href="#crystals-14-00995-t003" class="html-table">Table 3</a>. R = 0.997.</div> <div class="html-img"><img data-large="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g005.png" data-original="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g005.png" alt="Crystals 14 00995 g005" data-lsrc="/crystals/crystals-14-00995/article_deploy/html/images/crystals-14-00995-g005.png" /></div> </div> <div class="html-table-wrap" id="crystals-14-00995-t001"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href='#table_body_display_crystals-14-00995-t001'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href="#table_body_display_crystals-14-00995-t001"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 1.</b> Computed values of molecular volumes (Vol), the average deviation of V<sub>S</sub>(<b>r</b>) (Π), positive, negative and total variances (<math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>+</mo> <mn>2</mn> </msubsup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>−</mo> <mn>2</mn> </msubsup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mrow> <mi>tot</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math>), balance parameters ν and products <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mrow> <mi mathvariant="sans-serif">ν</mi> <mi mathvariant="sans-serif">σ</mi> </mrow> </mrow> <mrow> <mi>tot</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math> <sup>a</sup>. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_crystals-14-00995-t001"> <div class="html-caption"><b>Table 1.</b> Computed values of molecular volumes (Vol), the average deviation of V<sub>S</sub>(<b>r</b>) (Π), positive, negative and total variances (<math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>+</mo> <mn>2</mn> </msubsup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>−</mo> <mn>2</mn> </msubsup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mrow> <mi>tot</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math>), balance parameters ν and products <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mrow> <mi mathvariant="sans-serif">ν</mi> <mi mathvariant="sans-serif">σ</mi> </mrow> </mrow> <mrow> <mi>tot</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math> <sup>a</sup>.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Molecule</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Vol</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Π</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <msubsup> <mi mathvariant="bold-sans-serif">σ</mi> <mo>+</mo> <mn>2</mn> </msubsup> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <msubsup> <mi mathvariant="bold-sans-serif">σ</mi> <mo>−</mo> <mn>2</mn> </msubsup> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <msubsup> <mi mathvariant="bold-sans-serif">σ</mi> <mrow> <mi mathvariant="bold">tot</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >ν</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <msubsup> <mrow> <mrow> <mi mathvariant="bold-sans-serif">ν</mi> <mi mathvariant="bold-sans-serif">σ</mi> </mrow> </mrow> <mrow> <mi mathvariant="bold">tot</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </mstyle> </semantics> </math></th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >water</td><td align='center' valign='middle' class='html-align-center' >26.0</td><td align='center' valign='middle' class='html-align-center' >23.9</td><td align='center' valign='middle' class='html-align-center' >152.7</td><td align='center' valign='middle' class='html-align-center' >134.7</td><td align='center' valign='middle' class='html-align-center' >287.4</td><td align='center' valign='middle' class='html-align-center' >0.249</td><td align='center' valign='middle' class='html-align-center' >71.6</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >ammonia</td><td align='center' valign='middle' class='html-align-center' >33.6</td><td align='center' valign='middle' class='html-align-center' >17.9</td><td align='center' valign='middle' class='html-align-center' >38.8</td><td align='center' valign='middle' class='html-align-center' >185.4</td><td align='center' valign='middle' class='html-align-center' >224.2</td><td align='center' valign='middle' class='html-align-center' >0.143</td><td align='center' valign='middle' class='html-align-center' >32.1</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >N<sub>2</sub></td><td align='center' valign='middle' class='html-align-center' >35.5</td><td align='center' valign='middle' class='html-align-center' >4.4</td><td align='center' valign='middle' class='html-align-center' >4.9</td><td align='center' valign='middle' class='html-align-center' >7.2</td><td align='center' valign='middle' class='html-align-center' >12.1</td><td align='center' valign='middle' class='html-align-center' >0.241</td><td align='center' valign='middle' class='html-align-center' >2.9</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >methane</td><td align='center' valign='middle' class='html-align-center' >41.5</td><td align='center' valign='middle' class='html-align-center' >3.0</td><td align='center' valign='middle' class='html-align-center' >5.8</td><td align='center' valign='middle' class='html-align-center' >0.9</td><td align='center' valign='middle' class='html-align-center' >6.7</td><td align='center' valign='middle' class='html-align-center' >0.113</td><td align='center' valign='middle' class='html-align-center' >0.7</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >acetylene</td><td align='center' valign='middle' class='html-align-center' >48.1</td><td align='center' valign='middle' class='html-align-center' >12.4</td><td align='center' valign='middle' class='html-align-center' >77.3</td><td align='center' valign='middle' class='html-align-center' >27.7</td><td align='center' valign='middle' class='html-align-center' >105.0</td><td align='center' valign='middle' class='html-align-center' >0.194</td><td align='center' valign='middle' class='html-align-center' >20.4</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >methanol</td><td align='center' valign='middle' class='html-align-center' >50.9</td><td align='center' valign='middle' class='html-align-center' >13.5</td><td align='center' valign='middle' class='html-align-center' >89.7</td><td align='center' valign='middle' class='html-align-center' >139.3</td><td align='center' valign='middle' class='html-align-center' >229.0</td><td align='center' valign='middle' class='html-align-center' >0.238</td><td align='center' valign='middle' class='html-align-center' >54.5</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >dimethyl ether</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >75.7</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >8.7</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6.2</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >118.3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >124.5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.047</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5.8</td></tr></tbody> </table> <div class='html-table_foot html-p'><div class='html-p' style='text-indent:0em;'><span class='html-fn-content'><sup>a</sup> Units: Vol is in Å<sup>3</sup>; Π is in kcal/mol; <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>+</mo> <mn>2</mn> </msubsup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mo>−</mo> <mn>2</mn> </msubsup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="sans-serif">σ</mi> <mrow> <mi>tot</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msubsup> <mrow> <mrow> <mi mathvariant="sans-serif">ν</mi> <mi mathvariant="sans-serif">σ</mi> </mrow> </mrow> <mrow> <mi>tot</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math> are in (kcal/mol)<sup>2</sup>; ν is dimensionless.</span></div><div style='clear:both;'></div></div> </div> <div class="html-table-wrap" id="crystals-14-00995-t002"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href='#table_body_display_crystals-14-00995-t002'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href="#table_body_display_crystals-14-00995-t002"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 2.</b> Computed surface quantities for a number of cations and anions. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_crystals-14-00995-t002"> <div class="html-caption"><b>Table 2.</b> Computed surface quantities for a number of cations and anions.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Cation or Anion</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Surface Area (Å<sup>2</sup>)</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Volume (Å<sup>3</sup>)</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'><semantics> <mstyle mathvariant="bold"> <mrow> <msubsup> <mrow> <mover> <mi mathvariant="bold">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="bold">S</mi> <mo>+</mo> </msubsup> </mrow> </mstyle> </semantics></math> <math display='inline'><semantics> <mstyle mathvariant="bold"> <mrow> <mi mathvariant="bold">or</mi> <mo> </mo> <msubsup> <mrow> <mover> <mi mathvariant="bold">V</mi> <mo>¯</mo> </mover> </mrow> <mi mathvariant="bold">S</mi> <mo>−</mo> </msubsup> </mrow> </mstyle> </semantics></math> (kcal/mol)</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Π (kcal/mol)</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'><semantics> <mstyle mathvariant="bold"> <mrow> <msubsup> <mi mathvariant="bold-sans-serif">σ</mi> <mo>+</mo> <mn>2</mn> </msubsup> </mrow> </mstyle> </semantics></math><math display='inline'><semantics> <mstyle mathvariant="bold"> <mrow> <mo> </mo> <mi mathvariant="bold">or</mi> <mo> </mo> <msubsup> <mi mathvariant="bold-sans-serif">σ</mi> <mo>−</mo> <mn>2</mn> </msubsup> </mrow> </mstyle> </semantics></math> (kcal/mol)<sup>2</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >V<sub>S,max</sub> (kcal/mol)</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >V<sub>S,min</sub> (kcal/mol)</th></tr></thead><tbody ><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Na<sup>+</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >22.3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >9.88</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >250.0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >250.0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >250.0</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >K<sup>+</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >38.4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >22.35</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >191.1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >191.1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >191.1</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >NH<sub>4</sub><sup>+</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >47.5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >30.13</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >171.8</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3.6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >17.7</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >180.6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >164.9</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >EA<sup>+</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >95.2</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >77.13</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >124.5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >23.1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >655</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >164.0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >87.8</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >TMA<sup>+</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >131.1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >125.9</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >107.6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >10.7</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >176</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >146.7</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >90.3</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >EMIM<sup>+</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >163.8</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >153.4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >96.6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >8.6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >121</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >122.8</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >72.2</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >BMIM<sup>+</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >208.1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >201.5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >86.1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >15.3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >330</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >120.7</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >49.1</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >F<sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >33.6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >18.31</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−202.6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−202.6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−202.6</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Cl<sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >60.9</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >44.72</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−149.1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−149.1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−149.1</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Br<sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >66.9</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >51.44</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−142.4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−142.4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−142.4</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >NO<sub>3</sub><sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >77.6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >58.25</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−134.1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4.8</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >37.0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−117.9</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−150.6</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >BF<sub>4</sub><sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >82.4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >62.47</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−131.5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4.2</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >23.9</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−121.6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−142.6</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >PF<sub>6</sub><sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >104.6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >87.69</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−118.0</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4.3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >24.5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−108.5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−125.5</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >MeSO<sub>4</sub><sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >119.6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >106.05</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−109.4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >24.3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >754</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−52.1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−139.0</td></tr></tbody> </table> </div> <div class="html-table-wrap" id="crystals-14-00995-t003"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href='#table_body_display_crystals-14-00995-t003'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2073-4352/14/11/995/display" href="#table_body_display_crystals-14-00995-t003"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 3.</b> List of experimentally determined and predicted melting points (mps) for some ionic salts and liquids. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_crystals-14-00995-t003"> <div class="html-caption"><b>Table 3.</b> List of experimentally determined and predicted melting points (mps) for some ionic salts and liquids.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Ionic Salt or Liquid</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Mp (°C) <sup>a</sup> [Exp]</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Mp (°C) [Pred]</th></tr></thead><tbody ><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >NaF</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >993</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >984</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >KF</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >858</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >891</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >NaCl</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >800</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >790</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >KCl</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >770</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >743</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >NaBr</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >747</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >766</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >KBr</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >730</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >724</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >NH<sub>4</sub><sup>+</sup>Cl<sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >338</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >266</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >NH<sub>4</sub><sup>+</sup>Br<sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >235</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >249</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >NH<sub>4</sub><sup>+</sup>NO<sub>3</sub><sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >169.6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >228</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >[BMIM]<sup>+</sup>Br<sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >78.2</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >56</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >[EMIM]<sup>+</sup>Br<sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >76.8</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >67</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >[EMIM]<sup>+</sup>PF<sub>6</sub><sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >60.1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >33</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >[BMIM]<sup>+</sup>Cl<sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >67.85</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >65</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >[BMIM]<sup>+</sup>NO<sub>3</sub><sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >36.01</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >46</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >[EMIM]<sup>+</sup>BF<sub>4</sub><sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >14</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >52</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >[EA]<sup>+</sup>NO<sub>3</sub><sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >12</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >11</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >[BMIM]<sup>+</sup>PF<sub>6</sub><sup>−</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >11.4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >26</td></tr></tbody> </table> <div class='html-table_foot html-p'><div class='html-p' style='text-indent:0em;'><span class='html-fn-content'><sup>a</sup> Taken from references [<a href="#B58-crystals-14-00995" class="html-bibr">58</a>,<a href="#B59-crystals-14-00995" class="html-bibr">59</a>].</span></div><div style='clear:both;'></div></div> </div> </section><section class='html-fn_group'><table><tr id=''><td></td><td><div class='html-p'><b>Disclaimer/Publisher’s Note:</b> The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). 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A Revival of Molecular Surface Electrostatic Potential Statistical Quantities: Ionic Solids and Liquids