Recent results [3, 74, 68, 69, 75] have provided strong numerical evidence that not only does the shape of the exchange-correlation hole converge rapidly with simulation cell size, it is also a relatively short ranged quantity. This is illustrated in figure , which shows the exchange-correlation hole calculated using VMC , for diamond-structure silicon, using a simulation cell containing 54 atoms. One electron is placed at the centre of a silicon-silicon covalent bond and the other electron position is within the (110) plane. Figure shows a slice through the QMC charge density in the same (110) plane. The position of the central electron from figure has been marked with a large white circle.
Figure: Exchange-Correlation hole in diamond-structure silicon from Ref., with at the bond centre and ranging over the (110) plane. The black circles represent the positions of the nuclei.
Figure: VMC Charge density calculated for 3x3x3 diamond structure silicon plotted in the (110) plane through the centre of a silicon-silicon covalent bond.
The short range nature of the exchange-correlation hole ensures that the exchange and correlation energy associated with each electron as written in Eq.() is well described purely within the simulation cell surrounding each electron. The use of the Ewald interaction to try and describe the exchange-correlation interaction between electrons in different simulation cells therefore appears unnecessary. The essential requirements of the electron-electron interaction are simply that (i) it correctly describes the Hartree energy and (ii) each electron interacts with its exchange-correlation hole via the full 1/r interaction. In fact it appears that it is the extra terms in the expansion of the Ewald interaction, Eq.() that are introducing finite size effects into the calculations.