5.4 Beyond DFT: one- and two-particle excitations 105
Such a many-body theory was formulated by Hedin in 1965 [ 28 ], for reviews see
Refs. [ 29 , 30 ].Weshallnotgointothedetailsofthemany-bodytheorybehind
this approach, but consider a particular, relatively simple form, the COHSEX
approximation in some detail (we shall explain the name COHSEX below). This
approximation can be understood quite well without going into the formal theory.
Suppose we put an extra unit charge into the system. This charge will occupy some
state with orbital wave functionψ(r). We could describe the interaction between
this electron and the resident electrons by Hartree–Fock terms, i.e. a Coulomb inter-
action and an exchange interaction. However, although exchange is treated correctly
(apart from neglecting screening effects, see below), the Coulomb interaction will
push the resident electrons away from the visitor and thereby lower the interaction
between visitor and residents.
Let us first neglect screening. The electrostatic energy is then given by
EES(r)=∫
n(r′)v(r,r′)|ψ(r)|^2 d^3 rd^3 r′, (5.54)where the interactionv(r,r′)is the ‘bare’ Coulomb interaction potentialv(r,r′)=
1 /|r−r′|. Screening can be viewed from two different standpoints. The first is
to consider the change nin charge distribution due to the presence of the new
electron. The second view is to take for the potential felt by this electron a screened
potentialw(r)which falls off more rapidly than the bare potential. Obviously, the
two are connected.
For the correction to the energy due to the change n(r)in the charge distribution
we write:
E=∫
n(r′)v(r,r′)|ψ(r)|^2 d^3 rd^3 r′. (5.55)However, this result is wrong, because the response nto the test charge is propor-
tional to that charge! Therefore, if we integrate the energy up over the extra charge
put into the system, we get a prefactor of 1/2:
E=1
2
∫
n(r′)v(r,r′)|ψ(r)|^2 d^3 rd^3 r′. (5.56)In order to get a handle on the screened potentialw, we note that it is defined as
the potential measured atr′given the fact that there is a test point charge atr.We
therefore may write:
w(r′,r)=∫
δ(r−r′′)v(r′′,r′)d^3 r′′+∫
n(r′′|r)v(r′′,r′)d^3 r′′=v(r,r′)+∫
n(r′′|r)v(r′′,r′)d^3 r′′. (5.57)Here, n(r′|r)is the change in the charge density atr′due to a unit test charge
placed atr. The induced charge density n(r)due to a charge distribution|ψ(r)|^2