My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 4.
ERRORS IN QMC SIMULATIONS
To understand the origin of the finite-size error, it is helpful to separate E EW
e−e into
Hartree and exchange-correlation terms. For this analysis, the two-electron density
is required:
∫
n(r, r ′ ) =
cell|Ψ(X)| ∑ 2 δ(r − r i )δ(r ′ − r j ) dX. (4.20)
i≠j
With this definition, the electron-electron energy becomes
Ee−e EW = 1 ∫∫
n(r, r ′ )v E (r − r ′ ) dr dr ′ + 1 Nξ. (4.21)
2
2
cell
If the electrons were completely uncorrelated, then the two-electron density n(r, r ′ )
would simply be the product of the one-electron densities n(r) and n(r ′ ). However,
the electrons are not uncorrelated; the relationship between one- and two-electron
densities defines the exchange-correlation hole:
n(r, r ′ ) = n(r)n(r ′ ) + n(r)n XC (r, r ′ ). (4.22)
The exchange-correlation hole describes the way that electrons avoid each other; the
reduction in the electron density at r when one electron is fixed at r ′ is described
by n XC (r, r ′ ). Integrating equation (4.22) with respect to r ′ reveals the important
property
∫
cell
n XC (r, r ′ ) dr ′ = −1. (4.23)
This is a manifestation of the fact that the hole consists of the absence of a single
electron from the overall density.
Applying this to equation (4.21) gives
Ee−e EW = 1 ∫∫
n(r)n(r ′ )v E (r − r ′ ) dr dr ′ + 1 ∫∫
2
2
cell
= U Ha + U EW
XC .
cell
n(r)n XC (r, r ′ )[v E (r − r ′ ) − ξ] dr dr ′
(4.24)
The first term, the Hartree energy, is the classical self-interaction energy per simulation
cell of a static periodic charge density n(r) (compare equation (2.11)). The
second term is the interaction energy of the electron with the exchange-correlation
hole; this dynamical correction appears because the electron motions are correlated.
64