In an attempt to understand the origin of these CFSE, it is instructive to write the electron-electron interaction in terms of a Hartree interaction, which describes the electrostatic energy of the system and an Exchange-Correlation interaction, which describes the interaction of each electron with its exchange-correlation hole, via an interaction ;
Note the exchange-correlation energy in Eq.() is the full many-body exchange-correlation energy, not the exchange-correlation energy as defined in density functional theory (see chapter ). It has been established [3, 74] that both the charge density, and the shape of the exchange-correlation hole converge rapidly with system size and therefore exhibit a very small finite size effect. An extreme example is jellium (see section ), where the charge density is exact for all simulation cell sizes but the CFSE is still present. It therefore appears that the likely source of the CFSE present in the VMC and HF calculations in figures and is due to the choice of the interaction, , used in Eq.(). In the calculations shown in figures and the Ewald interaction has been used to represent the Coulomb interaction between each electron and all the periodic images of all the other electrons produced by the periodic boundary conditions. The Ewald interaction has already been described in detail in chapter . Expanding the Ewald interaction for small r yields
where is the volume of the simulation cell and the tensor, D, depends on the geometry of the simulation cell (for a cubic cell D is the identity matrix) and the constant is defined in chapter so that the average value of the potential is zero. The deviations from 1/r model the effects of charges ``outside'' the simulation cell. In the XC integral, however, the interaction between each electron and its XC hole should be exactly 1/r, independent of the size of the simulation cell. For very large simulation cells the 1/r term in the expansion of the Ewald interaction dominates, but for typical cell sizes such as those used in figures and the second term is significant and produces a finite size error proportional to in the XC energy per electron. The XC energy is negative and the extra unphysical interaction makes the XC energy more negative. These observations explain why the HF energies in figure converge with increasing simulation cell size (i) from below, and (ii) with an error which is roughly inversely proportional to the number of electrons in the simulation cell as originally proposed in section .