6.5 General approach

A large part of the finite size error in the interaction energy arises from the use of the Ewald interaction. [86,87] The general approach to the problem of the finite size errors taken here differs from that of Ceperley et al. in an important respect. Here the finite size effects are reduced within the many-body calculation by modifying the interaction terms in the many-body Hamiltonian.

The Ewald interaction is a periodic function which differs from $1/r$ in such a way that the sum of interactions between pairs of particles within one cell gives the exact Coulomb energy per cell of a periodic lattice of identical cells. This ensures that the Ewald Hamiltonian gives the correct Hartree energy, but the deviations from $1/r$ give rise to a spurious contribution to the XC energy, corresponding to the periodic repetition of the XC hole. [86,87] A solution is to change the many-body Hamiltonian so that the interaction with the XC hole is exactly $1/r$ (see section 6.7) for any size and shape of simulation cell, without altering the form of the Hartree energy. This source of finite size error is therefore eliminated and equation 6.3, with the correction term calculated within a the LDA, gives a much better description of the remaining finite size errors. Greater accuracy can be obtained by adding an extrapolation term, but this term is much smaller than if the Ewald interaction is used.

The utility of changing the Hamiltonian to reduce the finite size errors is readily seen by considering an example. Suppose that a periodic boundary condition QMC technique is used to study an isolated molecule. If the molecule is placed at the center of the periodically repeated simulation cell, the calculated energy is that of an array of identical molecules, including unwanted inter-molecular interactions. The results improve as the simulation cell is made larger, but the convergence is slow, especially for molecules with permanent dipole moments. A better solution is to cut off all Coulomb interactions between charges on different molecules, i.e., to replace the Ewald interactions by truncated Coulomb $1/r$ interactions acting only within the simulation cell. As long as the molecular wave function has decayed to zero before the simulation cell boundary is reached, this procedure should give essentially exact results.

An alternative procedure is to evaluate the correction term in equation 6.3 using HF data. HF theory is an approximate method for solving the many-body Hamiltonian, and if the Ewald formula is used for the electron-electron interaction terms in both the many-body and HF theories, the finite size error in the HF exchange energy will tend to cancel the finite size error in the many-body interaction energy. However, this procedure gives too large a correction, presumably because the HF exchange hole is significantly different from the screened XC hole of the many-body system.