---
title: Electron Localisation Function Calculation
url: https://www.emergentmind.com/topics/electron-localisation-function-calculation
type: topic
---

# Electron Localisation Function Calculation

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 [1509.04089].

## 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 [1908.06947]. 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 ($\approx 0.1$–$0.2$) [1908.06947].

## 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 $U_X^\sigma$) [1509.04089]. 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 [1704.05005].

**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 [1505.07687].

**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 [2405.15530].

## 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 [1908.06947]

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 [1704.05005].

### 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 [2405.15530].

### ELF in Correlated and Many-Body Frameworks

- In DFT+DMFT, reconstruct effective densities and kinetic energy using correlated wavefunctions and occupations [2010.04694].
- 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 [1908.06947].
- mELF enhances localization metrics for inner shells and strong bonds, with typical increases of $\sim$0.05–0.1 in bonding regions for heavy atoms; in uniform jellium, it interpolates between ELF=1/2 and ELF=1 as density decreases [1509.04089].
- CPL identifies spatially local bond character in H$_2$ and better tracks shell contraction than ELF, and ELFC synthesizes both features [1704.05005].
- 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 [2405.15530].

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 [2010.04694].
- 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 [1505.07687].
- 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 [1505.07687].
- DFT+DMFT and other correlated band-structure codes now routinely compute ELF on charge/self-consistent correlated densities [2010.04694].

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) | $\rho$, $\nabla \rho$, $\tau_\sigma$ | $1/[1+(\chi_\sigma)^2]$ [1509.04089] |
| Semi-local (Lindmaa et al.) | $\rho$, $\nabla \rho$, $\nabla^2 \rho$ | $1/[1+\eta(s, q)^2]$ [1908.06947]    |
| Modified (mELF)   | ELF + Slater potential $U_X^\sigma$ | $1/[1+(\chi'_\sigma)^2]$ [1509.04089]|
| Antiparallel (CPL, ELFC) | ELF + $U_{x,\alpha}, U_{x,\beta}$ | $1/(1+z)$, ELFC$=(\rm{ELF+CPL})/2$ [1704.05005] |
| Gauge-invariant (non-collinear) | Spinors, current, spin densities | $\tilde{\rm ELF}$ [2405.15530] |
| DFT+DMFT-ELF      | Correlated occupations and orbitals | Same structure as standard, evaluated on DFT+DMFT wavefunctions [2010.04694] |


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.

Source: https://www.emergentmind.com/topics/electron-localisation-function-calculation