6.7 Coulomb finite size effects

In this section the CFSE is considered and the theory of the MPC interaction is described. These effects do not occur in single-particle theories and are more difficult to compensate for than the IPFSE.

The origin of the CFSE was described in section 6.4 as arising from the XC energy and the dependence of the Ewald interaction, $v_{\rm E}$, on the size and shape of the simulation cell. Expanding $v_{\rm E}$ around zero separation gives [86,87]

$$v_{\rm E}({\bf r}) = {\rm constant} + \frac{1}{r} + \frac{2 \pi}{3\Omega} \; {\bf r}^{T}\cdot{\bf D}\cdot{\bf r} + {\cal O}\left(\frac{r^4}{\Omega^{5/3}}\right) \;\;,$$ (6.7)

where $\Omega$ is the volume of the simulation cell, and the tensor D depends on the shape of the simulation cell. (For a cubic cell D= I.) The constant term arises from the condition that the average of $v_{\rm E}$ over the simulation cell is zero. The term of order $r^2$ and the higher order deviations from $1/r$ make the Ewald interaction periodic and ensure that the sum of interactions between the particles in the simulation cell gives the potential energy per cell of an infinite periodic lattice. These terms are responsible for the spurious contribution to the XC energy, which is the source of the large finite size effect in many-body calculations using the Ewald interaction. [86,87] For cubic cells the interaction at short distances is larger than $1/r$ and therefore the XC energy is more negative than it should be, and because the leading order correction is proportional to the inverse of the simulation cell volume the error per electron is inversely proportional to the number of electrons in the cell.

Clearly it is desirable to remove this spurious contribution to the XC energy, but the Hartree energy is correctly evaluated using the Ewald interaction. The key requirements for a model Coulomb interaction giving small CFSEs in simulations with periodic boundary conditions are therefore: (i) it should give the Ewald interaction for the Hartree terms, and (ii) it should be exactly $1/r$ for the interaction with the XC hole. Unfortunately, the only periodic solution of Poisson's equation for a periodic array of charges is the Ewald interaction, which obeys criterion (ii) only in the limit of an infinitely large simulation cell. The ``Model Periodic Coulomb'' (MPC) interaction satisfies both criteria, but does not satisfy Poisson's equation.

The MPC interaction can be written as [87]

$${ \hat{H}_{\rm e-e} = \sum_{i>j}f({\bf r}_i-{\bf r}_j)}$$

$$+ \sum_{i}\int_{\rm WS} \left[ v_{{\rm E}}({\bf r}_i-{\bf r})- f({\bf r}_i-{\bf r})\right] {\rho({\bf r})} \, {\rm d}{\bf r} \;\;,$$ (6.8)

where $\rho$ is the electronic charge density and

$$f({\bf r}) = \frac{1}{r_{\rm m}} \;\;.$$ (6.9)

The definition of the cutoff Coulomb function, $f$, involves a minimum image convention whereby the inter-electron distance, ${\bf r}$, is reduced into the Wigner-Seitz (WS) cell of the simulation cell lattice by removal of simulation cell lattice translation vectors, leaving a vector ${\bf r}_{\rm m}$. This ensures that $\hat{H}_{\rm e-e}$ has the correct translational and point group symmetry. The first term in equation 6.8 is a direct Coulomb interaction between electrons within the simulation cell and the second term is a sum of potentials due to electrons ``outside the simulation cell''. Note that the second term is a one-body potential similar to the Hartree potential. It depends on the electronic charge density, $\rho$, but is not a function of inter-particle separation.

The electron-electron contribution to the total energy is evaluated as the expectation value of $\hat{H}_{\rm e-e}$ with the many-electron wave function, $\phi$, minus a double counting term:

$$E_{\rm e-e} = \langle \phi \vert \hat{H}_{\rm e-e} \vert \ph... ...r}-{\bf r}')\right] \, {\rm d}{\bf r} \, {\rm d}{\bf r}' \;\;.$$ (6.10)

Evaluating the expectation value gives

$$E_{\rm e-e} = \frac{1}{2}\int_{\rm WS} \rho({\bf r})\rho({\bf r}') v_{\rm E}({\bf r}-{\bf r}') \, {\rm d}{\bf r} \, {\rm d}{\bf r}'$$

$$+ \left( \int_{\rm WS} \vert\phi\vert^2\sum_{i>j}f({\bf r}_i-{\bf r... ... r}') {f}({\bf r}-{\bf r}') \, {\rm d}{\bf r} \, {\rm d}{\bf r}' \right) \;\; ,$$ (6.11)

where the first term on the right hand side is the Hartree energy and the term in brackets is the XC energy. We can see immediately that the Hartree energy is calculated with the Ewald interaction while the XC energy (expressed as the difference between a full Coulomb term and a Hartree term) is calculated with the cutoff interaction $f$.

The charge density $\rho$ appears in equations 6.8, 6.10, and 6.11. In QMC methods, the charge density is known with greatest statistical accuracy at the end of the calculation. This is not a serious complication for VMC simulations as the interaction energy may be evaluated at the end of the simulation using the accumulated charge density. In DMC this is not possible because the local energy is required at every step. This point is investigated in section 6.8.3.

6.7.1 Systems of electrons and nuclei

In the previous sections only the electron-electron interaction was considered, not the electron-nucleus and nucleus-nucleus interactions. If the MPC interaction is physically reasonable it should be possible to apply it to all the interactions in the problem, not just the electron-electron terms. For systems of electrons and nuclei, under quite general conditions, the expressions for the MPC interaction simplify so that the electron-nucleus and nucleus-nucleus terms reduce to the Ewald form.

Consider a simulation cell with periodic boundary conditions containing $N$ electrons at positions ${\bf r}_i$ and $M$ nuclei of charge $Z_{\alpha}$ at positions ${\bf x}_{\alpha}$. The wave function of the electrons and nuclei is $\Psi(\{{\bf r}_i\},\{{\bf x}_{\alpha}\})$, and the total charge density (electrons and nuclei) is $\rho_{\rm T}({\bf r})$. The interaction energy calculated with the MPC interaction is

$$E_{\rm int} = \frac{1}{2}\int_{\rm WS} \rho_{\rm T}({\bf r})\rho_{\rm T}({\bf r... ...}({\bf r}-{\bf r}') - f({\bf r}-{\bf r}')] \, {\rm d}{\bf r} \, {\rm d}{\bf r}'$$

$$+ \int_{\rm WS} \vert\Psi\vert^2 \left[ \sum_{i>j}f({\bf r}_i-{\bf r}_j) \right. - \sum_{i} \sum_{\alpha} Z_{\alpha}f({\bf r}_i-{\bf x}_{\alpha})$$

$$+ \left. \sum_{\alpha>\beta}Z_{\alpha}Z_{\beta}f({\bf x}_{\alpha}-{... ..., \Pi_k \, {\rm d}{\bf r}_k \, \Pi_{\gamma} \, {\rm d}{\bf x}_{\gamma} \;\;\; .$$ (6.12)

We now employ the adiabatic approximation to separate the electronic and nuclear dynamical variables:

$$\Psi (\{{\bf r}_i\},\{{\bf x}_{\alpha}\}) = \phi (\{{\bf r}_i\};\{{\bf x}_{\alpha}\}) \Phi (\{{\bf x}_{\alpha}\}) \;\;\; ,$$ (6.13)

where the $\{{\bf x}_{\alpha}\}$ appear only as parameters in $\phi$. To make further progress a form for the nuclear part of the wave function, $\Phi$, must be assumed. The simplest assumption is that $\Phi$ can be written as an appropriately symmetrized product of very strongly localized non-overlapping single-nucleus functions. Regardless of whether the product is antisymmetrized, symmetrized, or not symmetrized, equation 6.12 reduces to

$$E_{\rm int} = \frac{1}{2}\int_{\rm WS} {\rho} ({\bf r}){\rho} ({\bf r}') [v_{{\... ...}({\bf r}-{\bf r}') - f({\bf r}-{\bf r}')] \, {\rm d}{\bf r} \, {\rm d}{\bf r}'$$

$$+ \int_{\rm WS} \vert\phi\vert^2 \sum_{i>j}f({\bf r}_i-{\bf r}_j) \, \Pi_k \, {\rm d}{\bf r}_k$$

$$- \int_{\rm WS} \vert\phi\vert^2 \sum_{i}\sum_{\alpha} Z_{\alpha} \, v_{{\rm E}}({\bf r}_i-\overline{{\bf x}}_{\alpha}) \, \Pi_k \, {\rm d}{\bf r}_k$$

$$+ \sum_{\alpha>\beta}Z_{\alpha}Z_{\beta} \, v_{{\rm E}}(\overline{{\bf x}}_{\alpha}-\overline{{\bf x}}_{\beta}) \;\;\;,$$ (6.14)

where the $\overline{{\bf x}}_{\alpha}$ denote the centers of the single-nucleus functions and ${\rho}$ is the electron density. Note that the electron-nucleus and nucleus-nucleus terms involve only the Ewald interaction and that the first two terms of equation 6.14 correspond precisely to the electron-electron interaction of equation 6.11. This result justifies the use of equation 6.8 for the electron-electron interactions while retaining the Ewald interaction for the electron-nucleus and nucleus-nucleus terms.