Electron Localisation Function Calculation

• Electron localisation function (ELF) is a scalar measure that quantifies electron distribution and bonding by comparing local kinetic energy density with a homogeneous electron gas reference.
• Computational approaches involve orbital-based and semi-local parameterizations that utilize density gradients and Laplacians to map electron localization efficiently.
• Extensions such as mELF and gauge-invariant formulations incorporate dynamical correlations, spin non-collinearity, and many-body effects to refine localization analysis in complex systems.

The electron localisation function (ELF) is a scalar field constructed to provide a universal, dimensionless indicator of real-space electron localisation in atoms, molecules, and solids. ELF encodes Pauli exclusion effects and chemical bonding features, achieving wide adoption in solid-state theory, quantum chemistry, and electronic structure visualization. Originating from the work of Becke and Edgecombe, ELF is defined in terms of the one-particle density and kinetic energy density, with numerous extensions addressing dynamical correlations, spin non-collinearity, and computational efficiency in orbital-free implementations. The function ranges from 0 (fully delocalized) to 1 (fully localized), with 1/2 corresponding to the homogeneous electron gas reference.

1. Fundamental Theory and Definitions

ELF is grounded in the probabilistic exclusion of parallel-spin electrons imposed by the Pauli principle. For a single-determinant system, the conditional probability of finding a like-spin electron at a given position is reflected in the curvature of the parallel-spin pair density. Formally, for each spin channel $\sigma$:

$$D_\sigma(\mathbf{r}) = \tau_\sigma(\mathbf{r}) - \frac{1}{4} \frac{|\nabla \rho_\sigma(\mathbf{r})|^2}{\rho_\sigma(\mathbf{r})}$$

where $$\tau_\sigma(\mathbf{r}) = \sum_i |\nabla \psi_{i\sigma}(\mathbf{r})|^2$$ is the single-particle (Kohn–Sham or Hartree–Fock) kinetic energy density. The reference homogeneous electron gas (HEG) value is: $$D^{\rm unif}_\sigma(\mathbf{r}) = \frac{3}{5}(6\pi^2)^{2/3} [\rho_\sigma(\mathbf{r})]^{5/3}$$ The dimensionless localization index is $$\chi_\sigma(\mathbf{r}) = D_\sigma(\mathbf{r}) / D^{\rm unif}_\sigma(\mathbf{r})$$. The original ELF is then

$${\rm ELF}(\mathbf{r}) = \frac{1}{1 + [\chi_\sigma(\mathbf{r})]^2}$$

This formalism leads to ELF$\to 1$ for perfectly localized (e.g., core) regions or bonds, ELF$\to 1/2$ for the uniform gas, and ELF$\to 0$ for maximal delocalization or shell overlap (Pittalis et al., 2015).

2. Computational Approaches and Semi-Local Parameterizations

The direct evaluation of ELF from orbitals requires $\rho$, $$D_\sigma(\mathbf{r}) = \tau_\sigma(\mathbf{r}) - \frac{1}{4} \frac{|\nabla \rho_\sigma(\mathbf{r})|^2}{\rho_\sigma(\mathbf{r})}$$0, and $$D_\sigma(\mathbf{r}) = \tau_\sigma(\mathbf{r}) - \frac{1}{4} \frac{|\nabla \rho_\sigma(\mathbf{r})|^2}{\rho_\sigma(\mathbf{r})}$$1 on a grid. In contexts lacking orbitals—such as orbital-free DFT, energetic database post-processing, or reduced data—semi-local parameterizations are critical. Lindmaa et al. introduced a fully semi-local ELF depending only on the density and its first and second derivatives (Lindmaa et al., 2019). Two reduced variables are defined: $$D_\sigma(\mathbf{r}) = \tau_\sigma(\mathbf{r}) - \frac{1}{4} \frac{|\nabla \rho_\sigma(\mathbf{r})|^2}{\rho_\sigma(\mathbf{r})}$$2 Their semi-local Pauli ratio fit is

$$D_\sigma(\mathbf{r}) = \tau_\sigma(\mathbf{r}) - \frac{1}{4} \frac{|\nabla \rho_\sigma(\mathbf{r})|^2}{\rho_\sigma(\mathbf{r})}$$3

with $$D_\sigma(\mathbf{r}) = \tau_\sigma(\mathbf{r}) - \frac{1}{4} \frac{|\nabla \rho_\sigma(\mathbf{r})|^2}{\rho_\sigma(\mathbf{r})}$$4, $$D_\sigma(\mathbf{r}) = \tau_\sigma(\mathbf{r}) - \frac{1}{4} \frac{|\nabla \rho_\sigma(\mathbf{r})|^2}{\rho_\sigma(\mathbf{r})}$$5, $$D_\sigma(\mathbf{r}) = \tau_\sigma(\mathbf{r}) - \frac{1}{4} \frac{|\nabla \rho_\sigma(\mathbf{r})|^2}{\rho_\sigma(\mathbf{r})}$$6, $$D_\sigma(\mathbf{r}) = \tau_\sigma(\mathbf{r}) - \frac{1}{4} \frac{|\nabla \rho_\sigma(\mathbf{r})|^2}{\rho_\sigma(\mathbf{r})}$$7, $$D_\sigma(\mathbf{r}) = \tau_\sigma(\mathbf{r}) - \frac{1}{4} \frac{|\nabla \rho_\sigma(\mathbf{r})|^2}{\rho_\sigma(\mathbf{r})}$$8.

The resulting functional,

$$D_\sigma(\mathbf{r}) = \tau_\sigma(\mathbf{r}) - \frac{1}{4} \frac{|\nabla \rho_\sigma(\mathbf{r})|^2}{\rho_\sigma(\mathbf{r})}$$9

delivers a close qualitative match to orbital-based ELF, capturing shell structure and bonding topologies but with residual absolute deviations in regions of high localization ($$\tau_\sigma(\mathbf{r}) = \sum_i |\nabla \psi_{i\sigma}(\mathbf{r})|^2$$0–$$\tau_\sigma(\mathbf{r}) = \sum_i |\nabla \psi_{i\sigma}(\mathbf{r})|^2$$1) (Lindmaa et al., 2019).

3. Extensions: Correlation, Spin, and Many-Body Effects

Dynamical Correlation and Modified ELF: The original ELF neglects same-spin dynamical correlation and multi-reference static correlation. Pittalis et al. addressed dynamical correlation effects by incorporating a local renormalization: $$\tau_\sigma(\mathbf{r}) = \sum_i |\nabla \psi_{i\sigma}(\mathbf{r})|^2$$2 where $$\tau_\sigma(\mathbf{r}) = \sum_i |\nabla \psi_{i\sigma}(\mathbf{r})|^2$$3 with $$\tau_\sigma(\mathbf{r}) = \sum_i |\nabla \psi_{i\sigma}(\mathbf{r})|^2$$4 and $$\tau_\sigma(\mathbf{r}) = \sum_i |\nabla \psi_{i\sigma}(\mathbf{r})|^2$$5 the mean exchange-hole radius (related to the Slater potential $$\tau_\sigma(\mathbf{r}) = \sum_i |\nabla \psi_{i\sigma}(\mathbf{r})|^2$$6) (Pittalis et al., 2015). The modified ELF (mELF) is

$$\tau_\sigma(\mathbf{r}) = \sum_i |\nabla \psi_{i\sigma}(\mathbf{r})|^2$$7

mELF systematically enhances localization in atomic shells and heavy-atom bonds, especially in regions where dynamical correlation is pronounced.

Antiparallel-Spin Pairing and Complementary Localization: The standard ELF measures only parallel-spin localization. A coalescent pair locator (CPL) models on-top antiparallel pairing via exchange-correlation hole models, yielding a complementary function

$$\tau_\sigma(\mathbf{r}) = \sum_i |\nabla \psi_{i\sigma}(\mathbf{r})|^2$$8

with $$\tau_\sigma(\mathbf{r}) = \sum_i |\nabla \psi_{i\sigma}(\mathbf{r})|^2$$9, $$D^{\rm unif}_\sigma(\mathbf{r}) = \frac{3}{5}(6\pi^2)^{2/3} [\rho_\sigma(\mathbf{r})]^{5/3}$$0. Their combination,

$$D^{\rm unif}_\sigma(\mathbf{r}) = \frac{3}{5}(6\pi^2)^{2/3} [\rho_\sigma(\mathbf{r})]^{5/3}$$1

addresses chemical cases where the standard ELF fails (e.g., bonding in H$$D^{\rm unif}_\sigma(\mathbf{r}) = \frac{3}{5}(6\pi^2)^{2/3} [\rho_\sigma(\mathbf{r})]^{5/3}$$2), and improves trends for bond order, lone pairs, and electron shells (Pittalis et al., 2017).

Many-Body and Time-Dependent Systems: The exact ELF can be defined via the Laplacian of the same-spin reduced pair density, but in practice, single-particle approximations are typically used. In time-dependent contexts, a current term $$D^{\rm unif}_\sigma(\mathbf{r}) = \frac{3}{5}(6\pi^2)^{2/3} [\rho_\sigma(\mathbf{r})]^{5/3}$$3 is required for gauge invariance (TDELF). However, in strong-field dynamics and collisions, the approximate ELF loses accuracy, underestimating delocalization; improved functionals remain an open necessity (Durrant et al., 2015).

Non-Collinear Spin and Gauge Invariance: For non-collinear spin systems and relativistic open-shell cases, the standard ELF is not gauge-invariant. Desmarais et al. constructed a strictly U(1)$$D^{\rm unif}_\sigma(\mathbf{r}) = \frac{3}{5}(6\pi^2)^{2/3} [\rho_\sigma(\mathbf{r})]^{5/3}$$4SU(2)–gauge-invariant ELF: $$D^{\rm unif}_\sigma(\mathbf{r}) = \frac{3}{5}(6\pi^2)^{2/3} [\rho_\sigma(\mathbf{r})]^{5/3}$$5 where $$D^{\rm unif}_\sigma(\mathbf{r}) = \frac{3}{5}(6\pi^2)^{2/3} [\rho_\sigma(\mathbf{r})]^{5/3}$$6 incorporates the kinetic energy density, paramagnetic current, spin-magnetization gradients, and spin-current corrections, ensuring invariance under local phase and spin rotations. This extended ELF reveals marked quantitative and qualitative corrections, particularly in molecular magnets, spin-orbit-coupled solids, and even collinear open-shell systems (Desmarais et al., 2024).

4. Practical Recipes and Implementation

Standard (Orbital-Based) ELF

• Extract spin densities $$D^{\rm unif}_\sigma(\mathbf{r}) = \frac{3}{5}(6\pi^2)^{2/3} [\rho_\sigma(\mathbf{r})]^{5/3}$$7, kinetic energy densities $$D^{\rm unif}_\sigma(\mathbf{r}) = \frac{3}{5}(6\pi^2)^{2/3} [\rho_\sigma(\mathbf{r})]^{5/3}$$8, and density gradients from KS or HF orbitals.
• Compute $$D^{\rm unif}_\sigma(\mathbf{r}) = \frac{3}{5}(6\pi^2)^{2/3} [\rho_\sigma(\mathbf{r})]^{5/3}$$9, $$\chi_\sigma(\mathbf{r}) = D_\sigma(\mathbf{r}) / D^{\rm unif}_\sigma(\mathbf{r})$$0, and thus ELF as above.
• Visualization: Map ELF as isosurfaces or 2D slices to identify covalent, metallic, and shell regions.

Semi-Local ELF (Lindmaa et al., 2019)

1. Input: $$\chi_\sigma(\mathbf{r}) = D_\sigma(\mathbf{r}) / D^{\rm unif}_\sigma(\mathbf{r})$$1 on a real-space grid.
2. Compute derivatives $$\chi_\sigma(\mathbf{r}) = D_\sigma(\mathbf{r}) / D^{\rm unif}_\sigma(\mathbf{r})$$2, $$\chi_\sigma(\mathbf{r}) = D_\sigma(\mathbf{r}) / D^{\rm unif}_\sigma(\mathbf{r})$$3 using finite-difference or FFT techniques.
3. Calculate $$\chi_\sigma(\mathbf{r}) = D_\sigma(\mathbf{r}) / D^{\rm unif}_\sigma(\mathbf{r})$$4, $$\chi_\sigma(\mathbf{r}) = D_\sigma(\mathbf{r}) / D^{\rm unif}_\sigma(\mathbf{r})$$5 at each grid point.
4. Apply the parametrized functional for $$\chi_\sigma(\mathbf{r}) = D_\sigma(\mathbf{r}) / D^{\rm unif}_\sigma(\mathbf{r})$$6 and thus ELF$$\chi_\sigma(\mathbf{r}) = D_\sigma(\mathbf{r}) / D^{\rm unif}_\sigma(\mathbf{r})$$7.
5. Use in orbital-free DFT or large-scale datasets.

mELF and Antiparallel Indicators

• For mELF: compute the Slater potential using OEP/KLI, Becke–Roussel model, or LDA as practical.
• For CPL/ELFC: combine with a pair-dominated localization indicator as detailed (Pittalis et al., 2017).

Gauge-Invariant ELF for Non-Collinear Spin

• Evaluate spinor orbitals $$\chi_\sigma(\mathbf{r}) = D_\sigma(\mathbf{r}) / D^{\rm unif}_\sigma(\mathbf{r})$$8 and compute the full suite of spin densities, kinetic terms, paramagnetic currents, and their gradients.
• Assemble the gauge-invariant kinetic expression and curvature.
• Construct $$\chi_\sigma(\mathbf{r}) = D_\sigma(\mathbf{r}) / D^{\rm unif}_\sigma(\mathbf{r})$$9 as per (Desmarais et al., 2024).

ELF in Correlated and Many-Body Frameworks

• In DFT+DMFT, reconstruct effective densities and kinetic energy using correlated wavefunctions and occupations (James et al., 2020).
• For full many-body systems, obtain the pair density from the N-electron wavefunction; otherwise, apply the best available Kohn–Sham or post-KS approximation.

5. Validation, Limitations, and Use Cases

Validation of ELF and its extensions spans atomic, molecular, and extended condensed matter systems:

• The semi-local ELF reproduces shell and bonding topologies for fcc Al, diamond Si, and adsorbed graphene/Ni, with absolute deviations up to 0.1–0.2 in localized or weakly confining regions (Lindmaa et al., 2019).
• mELF enhances localization metrics for inner shells and strong bonds, with typical increases of $${\rm ELF}(\mathbf{r}) = \frac{1}{1 + [\chi_\sigma(\mathbf{r})]^2}$$00.05–0.1 in bonding regions for heavy atoms; in uniform jellium, it interpolates between ELF=1/2 and ELF=1 as density decreases (Pittalis et al., 2015).
• CPL identifies spatially local bond character in H$${\rm ELF}(\mathbf{r}) = \frac{1}{1 + [\chi_\sigma(\mathbf{r})]^2}$$1 and better tracks shell contraction than ELF, and ELFC synthesizes both features (Pittalis et al., 2017).
• The gauge-invariant ELF is critical for the correct identification of localization in non-collinear or relativistic systems, demonstrating large corrections—up to 40% in certain anti-bonding regions—over standard implementations (Desmarais et al., 2024).

Typical applications include:

• Bonding topology in solids and molecules.
• Delineation of shell structure in atoms.
• Visual discrimination of metallic, covalent, and dispersive bonding.
• Analysis of electron correlations and localization transitions in, e.g., Mott insulators (James et al., 2020).
• Systems lacking orbitals or in data-mining/big-data regimes.

Limitations:

• All single-determinant-based descriptors, including standard ELF and its DFT extensions, miss strong multireference and long-range correlation effects.
• Absolute ELF values differ between models; only relative spatial features (local maxima, basin topology) are robust.
• Semi-local parameterizations may fail (ELF$${\rm ELF}(\mathbf{r}) = \frac{1}{1 + [\chi_\sigma(\mathbf{r})]^2}$$20) in weakly-confined or low-density regions.
• Gauge-invariant functionals are essential outside collinear, non-relativistic regimes.

6. Extensions and Emerging Directions

Research continues on improved localization metrics:

• Time-dependent generalizations (TDELF) to account for current-driven delocalization, with explicit current corrections but limited accuracy in strong non-adiabatic processes (Durrant et al., 2015).
• Multi-reference and correlated wavefunction-based ELF for quantum Monte Carlo and post-DFT methods.
• Direct many-body measures such as the regional electron localization measure (RELM), quantifying global localization in exact wavefunctions (Durrant et al., 2015).
• DFT+DMFT and other correlated band-structure codes now routinely compute ELF on charge/self-consistent correlated densities (James et al., 2020).

A plausible implication is that further unification of ELF with modern quantum embedding and dynamical correlation frameworks will be central to next-generation descriptors of localization in the presence of both strong correlations and topological band effects.

7. Summary Table: ELF Expressions and Regimes

ELF Variant	Input Quantities	Key Formula / Reference
Standard (Becke–Edgecombe)	$${\rm ELF}(\mathbf{r}) = \frac{1}{1 + [\chi_\sigma(\mathbf{r})]^2}$$3, $${\rm ELF}(\mathbf{r}) = \frac{1}{1 + [\chi_\sigma(\mathbf{r})]^2}$$4, $${\rm ELF}(\mathbf{r}) = \frac{1}{1 + [\chi_\sigma(\mathbf{r})]^2}$$5	$${\rm ELF}(\mathbf{r}) = \frac{1}{1 + [\chi_\sigma(\mathbf{r})]^2}$$6 (Pittalis et al., 2015)
Semi-local (Lindmaa et al.)	$${\rm ELF}(\mathbf{r}) = \frac{1}{1 + [\chi_\sigma(\mathbf{r})]^2}$$7, $${\rm ELF}(\mathbf{r}) = \frac{1}{1 + [\chi_\sigma(\mathbf{r})]^2}$$8, $${\rm ELF}(\mathbf{r}) = \frac{1}{1 + [\chi_\sigma(\mathbf{r})]^2}$$9	$\to 1$0 (Lindmaa et al., 2019)
Modified (mELF)	ELF + Slater potential $\to 1$1	$\to 1$2 (Pittalis et al., 2015)
Antiparallel (CPL, ELFC)	ELF + $\to 1$3	$\to 1$4, ELFC$\to 1$5 (Pittalis et al., 2017)
Gauge-invariant (non-collinear)	Spinors, current, spin densities	$\to 1$6 (Desmarais et al., 2024)
DFT+DMFT-ELF	Correlated occupations and orbitals	Same structure as standard, evaluated on DFT+DMFT wavefunctions (James et al., 2020)

These formulaic, practical, and conceptual advances continue to solidify the ELF as a foundational construct for real-space analysis of electronic localization phenomena in a broad array of quantum systems.