In an attempt to gain further understanding on the nature of these CFSE, the VMC calculations shown in figure , were repeated using the Local Density Approximation (LDA) to Density Functional Theory (DFT) and Hartree-Fock (HF) theory. The k-point sampling in the LDA and HF calculations was chosen to be consistent with choosing simulation cell sizes of n= 2,3,4 and 5 as in the VMC calculations. The results of these LDA and HF calculations are plotted in figures and . Note the scale of the y-axis is the same in both graphs.
Figure: Total energy per atom calculated using LDA as a function of system size.
Figure: Total energy per atom calculated using HF, as a function of system size.
To facilitate comparison between the LDA and HF results the LDA orbitals were used to calculate the HF energies, so that the energy differences arise solely from the difference between the LDA exchange-correlation (XC) energy and the HF exchange energy. Figure shows that the LDA energy converges very rapidly with simulation cell size, whereas in figure , the HF exchange energy converges very slowly with simulation cell size as did the VMC in figure . For n=3 the finite size error in the LDA energy (IPFSE) is 0.012 eV per atom, which is much smaller than the HF finite size error of -0.211 eV per atom. The slow convergence of the HF exchange energy with the density of BZ sampling (which is equivalent to the size of the simulation cell) is well known[71, 72, 73] and is usually solved by increasing the quality of the BZ integration.
The rapid convergence of the LDA energy with simulation cell size is easily understood. In the LDA, the total energy can be written as a functional of the charge density
The LDA charge density has been shown  to converge rapidly with simulation cell size, hence the total energy also converges rapidly. As the LDA orbitals were used to calculate the HF energies, the kinetic and external potential energy in the LDA and HF calculations in figures and are identical for each system size. If one considers the Hamiltonian for the system as in Eq.(), the expectation value of the first and second terms of the Hamiltonian with respect to the trial wavefunction are the same in the LDA and HF and only differ in the final electron-electron interaction term;
Hence, the original interpretation of the the residual finite size effect after the subtraction of the IPFSE as a Coulomb FSE is consistent with these results.