Charged systems with density functional theory

On this page, we describe technical issues with computing the energies of charged systems with periodic density functional theory (DFT) calculations. We first introduce the problem of divergence of the electrostatic energy in DFT calculations and illustrate how this divergence is removed for charge neutral computations. We then discuss why this divergence is present for charged systems. Finally, we present methods which have been implemented in VASP that allow for calculations of charged bulk, surface and molecular systems.

Treating divergence in charge neutral calculations

VASP makes use of efficient fast Fourier transforms (FFT) to compute the electrostatic potential from the charge density using the Poisson equation,

$$V(\mathbf{r}) = 4\pi \int \frac{\rho(\mathbf{r}^\prime)}{\left | \mathbf{r} - \mathbf{r}^\prime \right|} d\mathbf{r}^\prime,$$

where ${\textstyle \mathbf{r}}$ and ${\textstyle \mathbf{r}^\prime}$ are all points in real space. Fourier transforming the Poisson equation to reciprocal space,

$$V(\mathbf{G}) = \frac{4\pi}{\mathrm{G}^2} \rho(\mathbf{G}),$$

where ${\textstyle \mathbf{G}}$ is the reciprocal lattice vector and ${\textstyle \mathrm{G}}$ is its norm.

An obvious issue with computing ${\textstyle V(\mathbf{G})}$ is that it diverges for ${\textstyle \mathrm{G}\to 0}$. This divergence is handled in charge neutral density DFT calculations by cancelling out individual divergences for the total electrostatic energy, which is the sum of electron-electron, ion-electron and ion-ion energies

$$E_{\mathrm{electrostatic}} = E_{\mathrm{electron-electron}} + E_{\mathrm{ion-electron}} + E_{\mathrm{ion-ion}}.$$

which have the following functional forms,

$$E_{\mathrm{electron-electron}} = \frac{1}{2} \sum_{\mathrm{G}} \rho_{\mathrm{elec}}(\mathbf{G}) V_{\mathrm{elec}}(\mathbf{G}),$$

where ${\textstyle \rho_{\mathrm{elec}}}$ and ${\textstyle V_{\mathrm{elec}}}$ are the electronic charge density and potential respectively.

$$E_{\mathrm{ion-electron}} = -\sum_{\mathrm{G}} \rho_{\mathrm{ion}}(\mathbf{G}) V_{\mathrm{elec}}(\mathbf{G}),$$

where ${\textstyle \rho_{\mathrm{ion}}}$ is the ion charge density. VASP does not explicitly compute ${\textstyle E_{\mathrm{ion-electron}}}$, but the potential of the ion is reflected through the eigenvalues. The ion-ion interactions are treated by Ewald summation

$$E_{\mathrm{ion-ion}} = E_{\mathrm{long-range}} + E_{\mathrm{short-range}} + E_{\mathrm{self}} + E_{\mathrm{homogeneous}},$$

where ${\textstyle E_{\mathrm{long-range}}}$ is the only component which sums over the ${\textstyle \mathrm{G}}$ vectors and has the form,

$$E_{\mathrm{ion-ion}} = \frac{1}{2} \sum_{\mathrm{G}} \rho_{\mathrm{ion}}(\mathbf{G}) V_{\mathrm{ion}}(\mathbf{G}).$$

The summed ${\textstyle \mathrm{G}=0}$ terms of ${\textstyle E_{\mathrm{electron-electron}}}$, ${\textstyle E_{\mathrm{ion-electron}}}$ and ${\textstyle E_{\mathrm{ion-ion}}}$ are given by,

$$E_{\mathrm{electrostatic}}(\mathbf{G}=0) = \frac{1}{2}\rho_{\mathrm{elec}}(\mathbf{G}=0) V_{\mathrm{elec}}(\mathbf{G}=0) - \rho_{\mathrm{ion}}(\mathbf{G}=0) V_{\mathrm{elec}}(\mathbf{G}=0) + \frac{1}{2}\rho_{\mathrm{ion}}(\mathbf{G}=0) V_{\mathrm{ion}}(\mathbf{G}=0),$$

which, can be expressed through the reciprocal space Poisson equation as,

$$E_{\mathrm{electrostatic}}(\mathbf{G}=0) = \frac{1}{\mathrm{G}^2} \left [\frac{1}{2}\rho_{\mathrm{elec}}(\mathbf{G}=0)^2 - \rho_{\mathrm{ion}}(\mathbf{G}=0)^2 + \frac{1}{2}\rho_{\mathrm{ion}}(\mathbf{G}=0)^2 \right].$$

Under the condition that ${\textstyle \rho_{\mathrm{elec}}(\mathbf{G}=0)=\rho_{\mathrm{ion}}(\mathbf{G}=0)}$, the individual divergences would cancel out and the total energy is convergent. Since the ${\textstyle \mathrm{G}=0}$ term of a Fourier transformed quantity is its average, the above condition is satisfied when the average electron and ion charge density are the same, i.e. the system is charge neutral.

Divergences in charged calculations

The total energy does not converge when ${\textstyle \rho_{\mathrm{elec}}(\mathbf{G}=0) \neq \rho_{\mathrm{ion}}(\mathbf{G}=0)}$, i.e. when the system is not charge neutral. Alternatively, we can think of this divergence as coming a compensating uniformly smeared out charge density, ${\textstyle \delta \rho}$ with energy ${\textstyle E_{\mathrm{compensating}}}$

$$E_{\mathrm{compensating}} = \frac{1}{\mathrm{G}^2} \left [ \delta \rho^2 \right]$$

which cancels out the divergent terms in ${\textstyle E_{\mathrm{electrostatic}}(\mathbf{G}=0)}$.

The consequences of adding a uniform compensating charge is system specific. For highly screened systems, the potential coming from a moderate compensating charge presents an acceptable amount of error. Conversely, systems containing vacuum (such as 2D materials, surfaces and molecules) have little screening and hence are more susceptible to artifacts caused by this compensating charge.

Treating charged systems in practice

VASP implements a few methods to deal with charged systems. These methods are specific to the dimensionality of the system that is being computed.

Bulk charged systems in orthorhombic cells can be treated by the applying analytical monopole-monolpole corrections (see LMONO for further practical details).

Surfaces and other two dimensional materials are treated with Coulomb kernel truncation methods. These methods replace the Coulomb kernel ${\textstyle v(\mathbf{G})}$, i.e., the ${\textstyle \frac{4\pi}{\mathrm{G}^2}}$ in ${\textstyle V(\mathbf{G})}$ with a 2D kernel (${\textstyle v_{\text{2D}}(\mathbf{G})}$) that removes electrostatic interactions between non-periodic replicas along the surface normal,

$$v_{\text{2D}}(\mathbf{G}) = \begin{cases} 4\pi / \mathrm{G}^2 \left[ 1 - e^{-\mathrm{G}_{\parallel}R} \left ( \cos(\mathrm{G}_\perp R_\mathrm{c}) + \mathrm{G}_\perp / \mathrm{g}_{\parallel} \sin(\mathrm{G}_{\perp}R_\mathrm{c}) \right ) \right] \quad & \mathrm{G} \neq 0 \\ 4\pi / \mathrm{G}_\perp^2\left[ 1 - \cos(\mathrm{G}_\perp R_\mathrm{c}) - \mathrm{G}_{\perp}R_\mathrm{c} \sin(\mathrm{G}_\perp R_\mathrm{c}) \right] \quad & \mathrm{G}_{\parallel} = 0 \\ -2\pi R_\mathrm{c}^2 \quad & \mathrm{G}=0 \end{cases}$$

where ${\textstyle \mathrm{G}_{\parallel}}$ is the norm of the ${\textstyle \mathbf{G}}$ vector parallel to the surface and ${\textstyle \mathrm{G}_{\perp}}$ is the ${\textstyle \mathbf{G}}$ vector along the surface normal. A similar change can be made to the Coulomb kernel to treat charged molecules (so-called 0D systems with ${\textstyle v_{\text{0D}}(\mathbf{G}}$)) to remove interactions with periodic replicas in all directions,

$$v_{\text{0D}}(\mathbf{G}) = \begin{cases} 4\pi/\mathrm{G}^2 \left( 1 - \cos\left(\mathrm{G}R_\mathrm{c}\right) \right) & \quad \mathrm{G}\neq 0 \\ 2\pi R_\mathrm{c}^2 & \quad \mathrm{G} = 0 \end{cases}$$