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Electron localization function

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In <a href="Quantum_chemistry" title="Quantum chemistry">quantum chemistry</a>, the electron localization function (ELF) is a measure of the likelihood of finding an <a href="Electron" title="Electron">electron</a> in the neighborhood space of a reference electron located at a given point and with the same <a href="Spin_(physics)" title="Spin (physics)">spin</a>. Physically, this measures the extent of spatial localization of the reference electron and provides a method for the mapping of <a href="Electron_pair" title="Electron pair">electron pair</a> probability in multielectronic systems.

ELF's usefulness stems from the observation that it allows electron localization to be analyzed in a chemically intuitive way. For example, the <a href="Electron_shell" title="Electron shell">shell</a> structure of heavy atoms is obvious when plotting ELF against the radial distance from the nucleus; the ELF for radon has six clear <a href="Maxima_and_minima" class="mw-redirect" title="Maxima and minima">maxima</a>, whereas the <a href="Electron_density" title="Electron density">electron density</a> decreases <a href="Monotonic_function" title="Monotonic function">monotonically</a> and the radially weighted density fails to show all shells. When applied to molecules, an analysis of the ELF shows a clear separation between the <a href="Core_electron" title="Core electron">core</a> and <a href="Valence_electron" title="Valence electron">valence electron</a>, and also shows <a href="Covalent_bond" title="Covalent bond">covalent bonds</a> and <a href="Lone_pair" title="Lone pair">lone pairs</a>, in what has been called "a faithful visualization of <a href="VSEPR_theory" title="VSEPR theory">VSEPR theory</a> in action".<a href="#cite_note-Becke1990-1">[1]</a> Another feature of the ELF is that it is invariant concerning the transformation of the <a href="Molecular_orbital" title="Molecular orbital">molecular orbitals</a>.

The ELF was originally defined by Becke and Edgecombe in 1990.<a href="#cite_note-Becke1990-1">[1]</a> They first argued that a measure of the electron localization is provided by

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<annotation encoding="application/x-tex">{\displaystyle D_{\sigma }(\mathbf {r} )=\tau _{\sigma }(\mathbf {r} )-{\tfrac {1}{4}}{\frac {(\nabla \rho _{\sigma }(\mathbf {r} ))^{2}}{\rho _{\sigma }(\mathbf {r} )}},}</annotation>

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</math><img src="./035f82c76fb2d9e47ed1ef3eb62ea85e12522db0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.273ex; height:6.676ex;" alt="{\displaystyle D_{\sigma }(\mathbf {r} )=\tau _{\sigma }(\mathbf {r} )-{\tfrac {1}{4}}{\frac {(\nabla \rho _{\sigma }(\mathbf {r} ))^{2}}{\rho _{\sigma }(\mathbf {r} )}},}" loading="lazy"></dd></dl>

where ρ is the electron <a href="Spin_density" class="mw-redirect" title="Spin density">spin density</a> and τ the <a href="Kinetic_energy" title="Kinetic energy">kinetic energy</a> density. The second term (negative term) is the <a href="Boson" title="Boson">bosonic</a> kinetic energy density, so D is the contribution due to fermions. D is expected to be small in those regions of space where localized electrons are to be found. Given the arbitrariness of the magnitude of the localization measure provided by D, it is compared to the corresponding value for a <a href="Uniform_electron_gas" class="mw-redirect" title="Uniform electron gas">uniform electron gas</a> with spin density equal to ρ(r), which is given by

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<annotation encoding="application/x-tex">{\displaystyle D_{\sigma }^{0}(\mathbf {r} )={\tfrac {3}{5}}(6\pi ^{2})^{2/3}\rho _{\sigma }^{5/3}(\mathbf {r} ).}</annotation>

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</math><img src="./0cea0d61db2df3c765caa9d32a4a89617f3ef461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:26.281ex; height:4.009ex;" alt="{\displaystyle D_{\sigma }^{0}(\mathbf {r} )={\tfrac {3}{5}}(6\pi ^{2})^{2/3}\rho _{\sigma }^{5/3}(\mathbf {r} ).}" loading="lazy"></dd></dl>

The ratio,

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<annotation encoding="application/x-tex">{\displaystyle \chi _{\sigma }(\mathbf {r} )={\frac {D_{\sigma }(\mathbf {r} )}{D_{\sigma }^{0}(\mathbf {r} )}},}</annotation>

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</math><img src="./298aa39574b817ea1888dbfcd35fef3bda2a3fcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:16.129ex; height:6.676ex;" alt="{\displaystyle \chi _{\sigma }(\mathbf {r} )={\frac {D_{\sigma }(\mathbf {r} )}{D_{\sigma }^{0}(\mathbf {r} )}},}" loading="lazy"></dd></dl>

is a <a href="Dimensionless" class="mw-redirect" title="Dimensionless">dimensionless</a> localization index that expresses electron localization for the uniform electron gas. In the final step, the ELF is defined in terms of χ by mapping its values on to the range 0 ≤ ELF ≤ 1 by defining the electron localization function as

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {ELF} (\mathbf {r} )={\frac {1}{1+\chi _{\sigma }^{2}(\mathbf {r} )}}.}</annotation>

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</math><img src="./76c7c2b204503beb16322ed6861f68dc2d1fd2be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.588ex; height:6.176ex;" alt="{\displaystyle \mathrm {ELF} (\mathbf {r} )={\frac {1}{1+\chi _{\sigma }^{2}(\mathbf {r} )}}.}" loading="lazy"></dd></dl>

ELF = 1 corresponding to perfect localization and ELF = <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠1/2⁠ corresponding to the electron gas.

The original derivation was based on <a href="Hartree%E2%80%93Fock" class="mw-redirect" title="Hartree–Fock">Hartree–Fock</a> theory. For <a href="Density_functional_theory" title="Density functional theory">density functional theory</a>, the approach was generalized by Andreas Savin in 1992,<a href="#cite_note-2">[2]</a> who also have applied the formulation to examining various chemical and materials systems.<a href="#cite_note-3">[3]</a> In 1994, Bernard Silvi and Andreas Savin developed a method for explaining ELFs using <a href="Differential_topology" title="Differential topology">differential topology</a>.<a href="#cite_note-4">[4]</a>

The approach of electron localization, in the form of <a href="Atoms_in_molecules" title="Atoms in molecules">atoms in molecules</a> (AIM), was pioneered by <a href="Richard_Bader" title="Richard Bader">Richard Bader</a>.<a href="#cite_note-5">[5]</a> Bader's analysis partitions the <a href="Charge_density" title="Charge density">charge density</a> in a molecule to "atoms" according to zero-flux surfaces (surfaces across which no electron flow is taking place).<a href="#cite_note-6">[6]</a> Bader's analysis allows many properties such as multipole moments, energies and forces, to be partitioned in a defensible and consistent manner to individual atoms within molecules.

Both the Bader approach and the ELF approach to partitioning of molecular properties have gained popularity in recent years because the fastest, accurate ab-initio calculations of molecular properties are now mostly made using density functional theory (DFT), which directly calculates the electron density. This electron density is then analyzed using the Bader charge analysis of ELFs. One of the most popular functionals in DFT was first proposed by Becke, who also originated ELFs.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFA._D._Becke_and_K._E._Edgecombe1990" class="citation journal cs1">A. D. Becke and K. E. Edgecombe (1990). "A simple measure of electron localization in atomic and molecular systems". J. Chem. Phys. 92 (9): 5397–5403. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1990JChPh..92.5397B">1990JChPh..92.5397B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.458517">10.1063/1.458517</a>.</cite>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFSavinJepsen,_O.Flad,_J.Andersen,_O._K.1992" class="citation journal cs1">Savin, A.; Jepsen, O.; Flad, J.; Andersen, O. K.; Preuss, H.; von Schnering, H. G. (1992). "Electron localization in solid-state structures of the elements – the diamond structure". Angewandte Chemie International Edition in English. 31 (2): 187–188. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fanie.199201871">10.1002/anie.199201871</a>.</cite>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFSavinNesperWengertFässler1997" class="citation journal cs1">Savin, Andreas; Nesper, Reinhard; Wengert, Steffen; Fässler, Thomas F. (1997-09-17). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/10.1002/anie.199718081">"ELF: The Electron Localization Function"</a>. Angewandte Chemie International Edition in English. 36 (17): 1808–1832. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fanie.199718081">10.1002/anie.199718081</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0570-0833">0570-0833</a>.</cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFSilviSavin1994" class="citation journal cs1">Silvi, B.; Savin, A. (1994). <a rel="nofollow" class="external text" href="http://www.nature.com/articles/371683a0">"Classification of chemical bonds based on topological analysis of electron localization functions"</a>. Nature. 371 (6499): 683–686. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994Natur.371..683S">1994Natur.371..683S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F371683a0">10.1038/371683a0</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0028-0836">0028-0836</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4362878">4362878</a>.</cite>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFBader1994" class="citation book cs1">Bader, R. W. F. (1994). Atoms in Molecules: A Quantum Theory. Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-855865-1</bdi>.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFBader2001" class="citation journal cs1">Bader, Richard F. W. (2001-04-04). <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/s002140000233">"The zero-flux surface and the topological and quantum definitions of an atom in a molecule"</a>. Theoretical Chemistry Accounts: Theory, Computation, and Modeling. 105 (4–5): 276–283. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs002140000233">10.1007/s002140000233</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1432-881X">1432-881X</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120944734">120944734</a>.</cite>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li>Frank R. Wagner (ed.) <a rel="nofollow" class="external text" href="http://www.cpfs.mpg.de/ELF/index.php">Electron localizability: chemical bonding analysis in direct and momentum space</a>. Max-Planck-Institut für Chemische Physik fester Stoffe, 2002. (accessed 2008-09-02).</li></ul></div><div><div class="zim-footer">

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