6.10 Conclusions

Large Coulomb finite size errors arise in many-body calculations when using the Ewald form of the electron-electron interaction. These errors are different from those encountered in single-particle theories such as DFT, and corrections derived only from single-particle theories are unsatisfactory.

By modifying the interaction terms in the Hamiltonian, the finite size errors may be considerably reduced. A modified interaction, the ``Modified Periodic Coulomb'' (MPC) interaction, in which the inter-particle interaction is exactly equal to $1/r$ at short distances and the long range interactions are replaced by a mean-field-like one-electron potential, is found in actual HF, VMC and DMC calculations to substantially reduce the finite-size errors in total energies.

The Ewald and MPC interactions may be used in tandem as an efficient diagnostic of Coulomb finite size errors. If the Ewald and MPC results agree then the Coulomb finite size error should be small. If the simulation cell is too small then the confinement of the XC hole makes the XC energy more negative. This source of error is intrinsic to using a finite simulation cell. However, even when the XC hole is artificially confined by a small simulation cell the MPC interaction still gives a better estimate of the XC energy than the Ewald interaction.

Excitation energies calculated within fixed-LDA-orbital HF theory show significant finite size effects. However, in correlated calculations the finite size effects are smaller and accurate excitation energies can be obtained using quite small simulation cells. In silicon the finite size errors in VMC electron-promotion (``optical absorption'') and electron addition/subtraction (``photoemission'') calculations are similar, and the optical promotion method has the greater statistical efficiency. The finite size errors for low lying excitations in silicon are small, and quite accurate results may be obtained from 16 atom cells. It appears that the limiting factor in obtaining accurate results for excitation energies is the accuracy of the excited state wavefunctions and not the size of supercell.

In every test the energy calculated with the MPC interaction is closer than the Ewald energy to the value for a very large system. Use of the MPC interaction therefore permits the use of smaller simulation cells for a specified accuracy than use of the Ewald interaction. The MPC is therefore an attractive and practical alternative to the Ewald interaction, suitable for use in all quantum many-body calculations of total energies in systems with periodic boundary conditions.