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 $\rho$, $\nabla \rho$, and $\tau_\sigma$ 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: $$s(\mathbf{r}) = \frac{|\nabla \rho|}{2 (3\pi^2)^{1/3} \rho^{4/3}}, \qquad q(\mathbf{r}) = \frac{\nabla^2 \rho}{4 (3\pi^2)^{2/3} \rho^{5/3}}$$ Their semi-local Pauli ratio fit is

$$\eta(s, q) = \frac{5}{9 \sqrt{2\pi}} \frac{ \exp\left[ -\frac{1}{2} \frac{\beta s^2}{f_{\epsilon,\delta}(s,q)} \right] }{ \left[ f_{\epsilon,\delta}(s,q) \right]^{\gamma/2} }$$

with $$f_{\epsilon,\delta}(s,q) = \frac{\epsilon}{\delta} \ln\left[ 1 + \exp\left( \frac{(e^\delta-1)\delta}{\epsilon}(s^2-q) \right) \right]$$, $\beta=1.122$, $\gamma=1.420$, $\epsilon=0.10$, $\delta=10.0$.

The resulting functional,

$${\rm ELF_{approx}}(\mathbf{r}) = \frac{1}{1 + \eta(s(\mathbf{r}), q(\mathbf{r}))^2}$$

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 ($\approx 0.1$–$0.2$) (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: $$\chi'_\sigma(\mathbf{r}) = \frac{\chi_\sigma(\mathbf{r})}{1 + z_\sigma(\mathbf{r})/2}$$ where $z_\sigma(\mathbf{r}) = 2c R_X^\sigma(\mathbf{r})$ with $c \approx 0.88$ and $R_X^\sigma$ the mean exchange-hole radius (related to the Slater potential $U_X^\sigma$) (Pittalis et al., 2015). The modified ELF (mELF) is

$${\rm mELF}(\mathbf{r}) = \frac{1}{1 + [\chi'_\sigma(\mathbf{r})]^2}$$

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

$${\rm CPL}(\mathbf{r}) = \frac{1}{1 + z_{\alpha\beta}(\mathbf{r})}$$

with $$z_{\alpha\beta}(\mathbf{r}) = c [|U_{x,\alpha}(\mathbf{r})|^{-1} + |U_{x,\beta}(\mathbf{r})|^{-1}]$$, $c\approx 0.29$. Their combination,

$${\rm ELFC}(\mathbf{r}) = \frac{1}{2} ({\rm ELF}(\mathbf{r}) + {\rm CPL}(\mathbf{r}))$$

addresses chemical cases where the standard ELF fails (e.g., bonding in H$_2$), 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 $$[\mathbf{j}_\sigma(\mathbf{r}, t)]^2/\rho_\sigma(\mathbf{r}, t)$$ 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)$\times$SU(2)–gauge-invariant ELF: $$\widetilde{\rm ELF}(\mathbf{r}) = \frac{1}{1 + \left(\frac{\widetilde D_{\rm nc}(\mathbf{r})}{D_H(\mathbf{r})}\right)^2}$$ where $\widetilde D_{\rm nc}(\mathbf{r})$ 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., 24 May 2024).

4. Practical Recipes and Implementation

Standard (Orbital-Based) ELF

• Extract spin densities $\rho_\sigma$, kinetic energy densities $\tau_\sigma$, and density gradients from KS or HF orbitals.
• Compute $D_\sigma(\mathbf{r})$, $D^{\rm unif}_\sigma(\mathbf{r})$, 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: $\rho(\mathbf{r})$ on a real-space grid.
2. Compute derivatives $\nabla \rho$, $\nabla^2 \rho$ using finite-difference or FFT techniques.
3. Calculate $s(\mathbf{r})$, $q(\mathbf{r})$ at each grid point.
4. Apply the parametrized functional for $\eta(s,q)$ and thus ELF$_{\rm approx}(\mathbf{r})$.
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 $\Phi_k(\mathbf{r})$ and compute the full suite of spin densities, kinetic terms, paramagnetic currents, and their gradients.
• Assemble the gauge-invariant kinetic expression and curvature.
• Construct $\widetilde{\rm ELF}$ as per (Desmarais et al., 24 May 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 $\sim$0.05–0.1 in bonding regions for heavy atoms; in uniform jellium, it interpolates between ELF=1/2 and ELF=1 as density decreases (Pittalis et al., 2015).
• CPL identifies spatially local bond character in H$_2$ 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., 24 May 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$\approx$0) 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)	$\rho$, $\nabla \rho$, $\tau_\sigma$	$1/[1+(\chi_\sigma)^2]$ (Pittalis et al., 2015)
Semi-local (Lindmaa et al.)	$\rho$, $\nabla \rho$, $\nabla^2 \rho$	$1/[1+\eta(s, q)^2]$ (Lindmaa et al., 2019)
Modified (mELF)	ELF + Slater potential $U_X^\sigma$	$1/[1+(\chi'_\sigma)^2]$ (Pittalis et al., 2015)
Antiparallel (CPL, ELFC)	ELF + $U_{x,\alpha}, U_{x,\beta}$	$1/(1+z)$, ELFC$=(\rm{ELF+CPL})/2$ (Pittalis et al., 2017)
Gauge-invariant (non-collinear)	Spinors, current, spin densities	$\tilde{\rm ELF}$ (Desmarais et al., 24 May 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.