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Electron localization function
Electron localization function
Electron localization function
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Electron localization function of the krypton atom at the Hartree–Fock / cc-pV5Z level of theory. Also shown is the radial density, 4πr2ρ(r), scaled by a factor of 0.0375.

In quantum chemistry, the electron localization function (ELF) is a measure of the likelihood of finding an electron in the neighborhood space of a reference electron located at a given point and with the same spin. Physically, this measures the extent of spatial localization of the reference electron and provides a method for the mapping of electron pair 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 shell structure of heavy atoms is obvious when plotting ELF against the radial distance from the nucleus; the ELF for radon has six clear maxima, whereas the electron density decreases monotonically 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 core and valence electron, and also shows covalent bonds and lone pairs, in what has been called "a faithful visualization of VSEPR theory in action".[1] Another feature of the ELF is that it is invariant concerning the transformation of the molecular orbitals.

Image of the ELF of water at level 0.8, generated using PyMOL

The ELF was originally defined by Becke and Edgecombe in 1990.[1] They first argued that a measure of the electron localization is provided by

where ρ is the electron spin density and τ the kinetic energy density. The second term (negative term) is the bosonic 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 uniform electron gas with spin density equal to ρ(r), which is given by

The ratio,

is a dimensionless 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

ELF = 1 corresponding to perfect localization and ELF = ⁠1/2⁠ corresponding to the electron gas.

The original derivation was based on Hartree–Fock theory. For density functional theory, the approach was generalized by Andreas Savin in 1992,[2] who also have applied the formulation to examining various chemical and materials systems.[3] In 1994, Bernard Silvi and Andreas Savin developed a method for explaining ELFs using differential topology.[4]

The approach of electron localization, in the form of atoms in molecules (AIM), was pioneered by Richard Bader.[5] Bader's analysis partitions the charge density in a molecule to "atoms" according to zero-flux surfaces (surfaces across which no electron flow is taking place).[6] 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.

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Electron localization function

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The electron localization function (ELF) is a fundamental tool in quantum chemistry that measures the local degree of electron localization in atomic and molecular systems by analyzing the Pauli exclusion principle's effects on the kinetic energy density of same-spin electrons, originally introduced by A. D. Becke and K. E. Edgecombe in 1990. Defined as a dimensionless function ranging from 0 (delocalized electrons) to 1 (fully localized), ELF is derived from the spin-resolved one-particle electron density and the associated kinetic energy density, providing a physically meaningful way to identify regions of electron pairing without relying on orbital transformations. In practice, ELF enables the topological partitioning of molecular space into basins—non-overlapping regions associated with atomic cores, bonding pairs, and lone pairs—facilitating a chemically intuitive visualization of electronic structure that aligns with concepts like the valence shell electron pair repulsion (VSEPR) theory. This partitioning is particularly valuable in density functional theory (DFT) and post-Hartree-Fock calculations, where ELF contours reveal shell structures in atoms (e.g., clear maxima for inner shells in heavy elements like radon) and bonding patterns in molecules, such as covalent bonds in N₂, lone pairs in NH₃, and even multicenter bonds in species like diborane. Unlike traditional electron density maps, which may not distinguish localized features, ELF's sensitivity to fermionic exchange effects offers deeper insights into chemical reactivity, aromaticity, and electron delocalization in transition states or conjugated systems. Historically, while the original formulation by Becke and Edgecombe focused on Hartree-Fock wavefunctions, subsequent developments have extended ELF to correlated methods and DFT, enhancing its applicability across computational chemistry software like PWmat and VESTA for visualization. Its robustness under molecular symmetry and independence from basis set choices make it a preferred metric for studying ionic, polarized, and metallic bonding, as demonstrated in examples ranging from LiF to Cu₄Sn₄ clusters. Overall, ELF bridges quantum mechanical calculations with classical chemical intuition, aiding in the interpretation of electronic properties and predicting molecular behavior.

History and Development

Introduction by Becke and Edgecombe

In the late 1980s, the field of quantum chemistry was experiencing significant advances in density functional theory (DFT), driven by the development of more accurate exchange-correlation functionals, which highlighted the need for improved tools to analyze chemical bonding beyond basic electron density visualizations. Axel D. Becke, a prominent figure in these DFT developments, collaborated with Kenneth E. Edgecombe to address this gap by proposing a new measure for electron localization. Their work emerged in the context of growing interest in understanding electron pairing and delocalization in molecular systems using accessible, density-based quantities rather than complex wavefunction analyses. The seminal 1990 publication, titled "A Simple Measure of Electron Localization in Atomic and Molecular Systems," appeared in The Journal of Chemical Physics (volume 92, page 5397). In this paper, Becke and Edgecombe introduced the electron localization function (ELF) as a way to quantify the local likelihood of finding paired electrons, drawing directly from the Pauli exclusion principle and the concept of the Fermi hole—the region around a reference electron depleted of same-spin electrons due to exchange effects. Their motivation was to create a practical, unitary-invariant tool based on the parallel-spin pair probability from Hartree-Fock theory, enabling visualization of electron localization without reliance on specific orbital choices or full pair density calculations. This approach aimed to bridge the gap between theoretical electron correlation and intuitive chemical concepts like bonds and lone pairs. Initial applications in the paper demonstrated ELF's utility on simple systems, such as the hydrogen molecule (H₂) and helium atom (He), where the function exhibited maxima at bond midpoints in H₂, highlighting localized bonding pairs, and revealed clear shell structures in He. These examples underscored ELF's ability to delineate core electrons, bonding regions, and lone pairs in atomic and diatomic systems, providing a foundation for its broader use in interpreting chemical bonding.

Subsequent Refinements and Extensions

Following the introduction of the electron localization function (ELF) by Becke and Edgecombe in 1990, subsequent refinements in the mid-1990s focused on enhancing its utility for spatial partitioning of electron density. In 1997, Andreas Savin and collaborators advanced the ELF framework by introducing a topological partitioning approach that leverages the gradient field of the ELF to identify and delineate electron localization basins. This method defines attractors as local maxima of the ELF, with basins formed by tracing paths of steepest ascent along the ELF gradient from every point in space, assigning regions to the nearest attractor; these basins correspond to core, bonding, or lone pair domains, providing a rigorous position-space representation of chemical bonds. The refinements also incorporated f-localization domains, where the ELF value exceeds a threshold f, allowing quantitative analysis of bond strength through the interval between attractor maxima and saddle points. These developments, detailed in Savin's 1997 publication, enabled more precise identification of electron pairs in complex systems, bridging ELF with dynamical systems theory for basin analysis. During the 1990s, extensions of ELF were developed to handle spin-restricted and multireference wave functions, particularly for open-shell systems, with contributions from researchers like E. V. Ludena who explored density-based variants to visualize shell structures in atoms and molecules under approximations like the weighted density approximation (WDA) within Thomas-Fermi-Dirac theory. These adaptations separated spin contributions (α and β) to extend ELF's applicability to systems with unpaired electrons, such as radicals and transition states, ensuring the function captured localization in non-closed-shell configurations without violating the Pauli principle. Ludena's work in 1995, for instance, demonstrated how a density-based ELF (DELF) complemented traditional ELF by highlighting incipient shell structures in open-shell atomic systems, paving the way for broader use in correlated methods. By the early 2000s, ELF became widely integrated into density functional theory (DFT) frameworks, facilitating its routine computation in popular quantum chemistry software packages. Implementations in codes like deMon, which employs linear combinations of Gaussian-type orbitals for DFT calculations, allowed ELF analysis alongside Kohn-Sham densities for molecular systems. Similarly, Gaussian software supported ELF evaluation through output of wave functions suitable for post-processing, enabling topological analysis in DFT and hybrid functional calculations. VASP, a plane-wave DFT code, incorporated ELF computations by the early 2000s to study electron localization in periodic structures, leveraging its efficient handling of large systems. This widespread adoption, as evidenced in applications from 2000 onward, democratized ELF's use for bonding insights in both molecular and solid-state DFT simulations. Key milestones in the 2000s included extensions of ELF to solid-state systems by Bernard Silvi and colleagues, who adapted the topological analysis to periodic boundary conditions for studying metallic and ionic bonding in crystals. In their 2000 work, Silvi's group demonstrated how ELF attractors in interstitial regions of metals like beryllium reveal direct-space representations of delocalized bonding, with basins defined under periodic constraints to account for translational symmetry. These developments enabled ELF's application to high-pressure crystal chemistry and intermetallic phases, using custom codes for topological analysis in periodic systems. By incorporating periodic boundary conditions, Silvi's refinements in the 2000s extended ELF's scope from molecules to extended solids, providing tools for quantifying multicenter bonding and electron delocalization in materials.

Mathematical Formulation

Definition of ELF

The electron localization function (ELF) is a dimensionless quantity that measures the local degree of electron localization in atomic and molecular systems, defined as a function of position r\mathbf{r} in space. It was introduced to provide a physically meaningful way to analyze electron pairing and localization based on the one-particle electron density. Formally, ELF is given by ELF(r)=11+χ2(r),\text{ELF}(\mathbf{r}) = \frac{1}{1 + \chi^2(\mathbf{r})}, where χ(r)\chi(\mathbf{r}) is the ratio of the local Pauli kinetic energy density tP(r)=T(r)−TW(r)t_P(\mathbf{r}) = T(\mathbf{r}) - T_W(\mathbf{r}) to the Thomas-Fermi kinetic energy density TTF(r)T_{TF}(\mathbf{r}) of a homogeneous electron gas, such that χ(r)=T(r)−TW(r)TTF(r)\chi(\mathbf{r}) = \frac{T(\mathbf{r}) - T_W(\mathbf{r})}{T_{TF}(\mathbf{r})}, with TTF(r)=cFρ(r)5/3T_{TF}(\mathbf{r}) = c_F \rho(\mathbf{r})^{5/3} and cF=310(3π2)2/3c_F = \frac{3}{10}(3\pi^2)^{2/3}. This formulation arises from considerations of the Pauli exclusion principle and the Thomas-Fermi model, emphasizing regions where electrons behave as if paired or localized similarly to a homogeneous electron gas. Equivalently, ELF can be expressed directly as ELF(r)=[1+(T(r)−TW(r)cFρ(r)5/3)2]−1,\text{ELF}(\mathbf{r}) = \left[1 + \left(\frac{T(\mathbf{r}) - T_W(\mathbf{r})}{c_F \rho(\mathbf{r})^{5/3}}\right)^2 \right]^{-1}, with T(r)T(\mathbf{r}) representing the positive-definite kinetic energy density from the quantum mechanical wavefunction, and TW(r)=∣∇ρ(r)∣28ρ(r)T_W(\mathbf{r}) = \frac{|\nabla \rho(\mathbf{r})|^2}{8\rho(\mathbf{r})} being the von Weizsäcker term that models the kinetic energy of a single orbital or bosonic system, where ρ(r)\rho(\mathbf{r}) is the electron density. The function ELF(r\mathbf{r}) ranges from 0 to 1, where values approaching 1 indicate high electron localization, such as in covalent bonds, lone pairs, or atomic cores; values near 0.5 correspond to the behavior of a homogeneous electron gas with moderate delocalization; and values below 0.5 signify regions of low localization or strong delocalization. For open-shell systems, spin-polarized versions ELF↑(r)^\uparrow(\mathbf{r}) and ELF↓(r)^\downarrow(\mathbf{r}) are defined separately for α\alpha and β\beta spin electrons using the respective spin densities ρ↑(r)\rho^\uparrow(\mathbf{r}) and ρ↓(r)\rho^\downarrow(\mathbf{r}), with the spin-averaged ELF given by ELF(r)=ELF↑(r)+ELF↓(r)2\text{ELF}(\mathbf{r}) = \frac{\text{ELF}^\uparrow(\mathbf{r}) + \text{ELF}^\downarrow(\mathbf{r})}{2}. This extension allows ELF to handle systems with unpaired electrons while maintaining the core interpretive framework of localization.

ELF in Terms of Density and Its Gradient

The electron localization function (ELF) can be expressed explicitly in terms of the one-particle electron density ρ(r) and its gradient ∇ρ(r) by approximating the local kinetic energy density using the Thomas-Fermi-von Weizsäcker model, which incorporates both the homogeneous electron gas contribution and gradient corrections. This approximation allows for the computation of ELF without explicit knowledge of the wavefunction or orbitals, relying instead on density and its spatial derivatives. The approach is particularly useful in density functional theory calculations where the electron density is the primary output. The total kinetic energy density T(r) in this framework is approximated using a second-order gradient expansion, combining the Thomas-Fermi term for the uniform gas, the von Weizsäcker gradient correction, and a Laplacian term to account for higher-order effects: T(r)=310(3π2)2/3ρ5/3(r)+18∣∇ρ(r)∣2ρ(r)−13∇2ρ(r)T(\mathbf{r}) = \frac{3}{10} (3\pi^2)^{2/3} \rho^{5/3}(\mathbf{r}) + \frac{1}{8} \frac{|\nabla \rho(\mathbf{r})|^2}{\rho(\mathbf{r})} - \frac{1}{3} \nabla^2 \rho(\mathbf{r}) This expression captures the leading contributions to the kinetic energy per unit volume at position r, with the Laplacian term ∇²ρ(r) arising from the second-order gradient expansion of the kinetic energy functional. The von Weizsäcker term T_W(r), which represents the kinetic energy density of a bosonic system with the same density distribution (i.e., without Pauli exclusion effects), is defined as: TW(r)=18∣∇ρ(r)∣2ρ(r)T_W(\mathbf{r}) = \frac{1}{8} \frac{|\nabla \rho(\mathbf{r})|^2}{\rho(\mathbf{r})} This term quantifies the contribution from the inhomogeneity of the density, emphasizing regions where the density varies rapidly, such as near atomic cores or bond midpoints. To measure the local extent of electron localization, the function χ(r) is introduced as the ratio of the Pauli-excluded kinetic energy excess to the Thomas-Fermi kinetic energy density T_{TF}(r): χ(r)=T(r)−TW(r)TTF(r)\chi(\mathbf{r}) = \frac{T(\mathbf{r}) - T_W(\mathbf{r})}{T_{TF}(\mathbf{r})} where TTF(r)=310(3π2)2/3ρ5/3(r)T_{TF}(\mathbf{r}) = \frac{3}{10} (3\pi^2)^{2/3} \rho^{5/3}(\mathbf{r}). This definition highlights the impact of the Pauli exclusion principle, as T(r) - T_W(r) represents the additional kinetic energy due to fermionic statistics and antisymmetry requirements, normalized relative to the uniform electron gas. In regions where Pauli effects are minimal (e.g., within paired electron domains), χ(r) is small, leading to high ELF values; conversely, in delocalized regions like a uniform gas, χ(r) ≈ 1, yielding ELF ≈ 0.5, while stronger delocalization can make χ > 1, suppressing ELF further. The ELF is then obtained as ELF(r) = 1 / [1 + χ²(r)], ensuring values between 0 and 1. Substituting the expressions for T(r), T_W(r), and T_{TF}(r) yields the explicit form for χ²(r): χ2(r)=[310(3π2)2/3ρ5/3(r)+18∣∇ρ(r)∣2ρ(r)−13∇2ρ(r)−18∣∇ρ(r)∣2ρ(r)310(3π2)2/3ρ5/3(r)]2=[1+−13∇2ρ(r)310(3π2)2/3ρ5/3(r)]2\chi^2(\mathbf{r}) = \left[ \frac{ \frac{3}{10} (3\pi^2)^{2/3} \rho^{5/3}(\mathbf{r}) + \frac{1}{8} \frac{|\nabla \rho(\mathbf{r})|^2}{\rho(\mathbf{r})} - \frac{1}{3} \nabla^2 \rho(\mathbf{r}) - \frac{1}{8} \frac{|\nabla \rho(\mathbf{r})|^2}{\rho(\mathbf{r})} }{ \frac{3}{10} (3\pi^2)^{2/3} \rho^{5/3}(\mathbf{r}) } \right]^2 = \left[ 1 + \frac{ - \frac{1}{3} \nabla^2 \rho(\mathbf{r}) }{ \frac{3}{10} (3\pi^2)^{2/3} \rho^{5/3}(\mathbf{r}) } \right]^2
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