GitHub

Ewald Onsite Correction

An Ewald-based onsite correction is enabled when an Ewald block is present in the configuration file. The correction is only applied if effective charges are defined for all atomic species via the qeff parameter. For periodic systems, the total charge must be neutral, i.e., the sum over all qeff values must be zero.

Overview

The Ewald onsite correction introduces a physically motivated, long-range electrostatic contribution to tight-binding onsite energies.

It accounts for the electrostatic potential generated by all ions in the system and maps it onto onsite Hamiltonian terms.

The correction is of the form:

\[\Delta \varepsilon_i = -\gamma_{s(i)} \, \phi_i\]

where:

  • $\phi_i$ is the electrostatic potential at atomic site $i$ (computed via Ewald, PME, Wolf, or Zahn)
  • $\gamma_{s(i)}$ is a species-dependent coupling parameter
  • $s(i)$ denotes the atomic species of atom $i$

Physical Motivation

In a full electronic structure description, the electrostatic (Hartree) potential contributes to the Hamiltonian as:

\[\hat{H} = \hat{H}_0 - e \, \Phi(\mathbf{r})\]

Projecting onto localized orbitals $\chi_i$ gives:

\[\Delta \varepsilon_i = -e \, \langle \chi_i \mid \Phi(\mathbf{r}) \mid \chi_i \rangle\]

In this model, we approximate:

\[\langle \chi_i \mid \Phi(\mathbf{r}) \mid \chi_i \rangle \approx \Phi(\mathbf{R}_i)\]

which leads to a linear relationship between onsite energy shifts and site potentials.

The parameter $\gamma$ absorbs:

  • orbital averaging effects
  • screening
  • differences between model charges and true charge density

Electrostatic Potential

The potential $\phi_i$ is computed from effective charges $q_i$:

\[\phi_i = \sum_j q_j K(r_{ij})\]

where K(r) is the chosen kernel.

Supported Methods

Ewald / PME (method = "ewald")

  • Exact electrostatics under periodic boundary conditions
  • Includes:
    • real-space contribution
    • reciprocal-space contribution
    • self-interaction correction

Wolf / Zahn (method = "wolf" / "zahn")

  • Approximate, short-ranged electrostatics
  • Uses screened Coulomb kernel:

\[ K(r) = \mathrm{erfc}(α r) / r - \mathrm{shift}\]

  • No reciprocal-space contribution
  • Faster $(O(N))$, but approximate

Mean Centering

The electrostatic potential is only defined up to a constant.

To ensure consistency with onsite fitting, we subtract a per-species mean:

\[\phi_i \rightarrow \phi_i - \langle \phi \rangle _ {s(i)}\]

This ensures:

  • no arbitrary offset in onsite energies
  • compatibility across different structures