Computational Physics

5.4 Beyond DFT: one- and two-particle excitations 107

The total change in the density is therefore given by

n(r′)= 2

∑

jocc.

ψj∗(r′)

ψj(r′)

= 2

∑

j,k

′

∫ ψ∗

j(r

′)ψk(r′)ψ∗

k(r

′′)ψj(r′′)

Ej−Ek

w(r′,r)d^3 r′′, (5.63)

where the prime with the sum indicates that the indexjruns over occupied, andk
over unoccupied levels. Putting this back into the rightmost term ofEq. (5.61)and
using the equality between the second and third expression in this equation yields

ε(r′,r)−δ(r′,r)= 2

∑

j,k

′∫∫ψj(r)ψk∗(r)ψk∗(r′′)ψj(r′′)

(Ej−Ek)|r−r′|

d^3 r′′. (5.64)

In this derivation we have assumed that the effects of the new electron on the resident
one can be completely described in terms of the Hartree term in the Hamiltonian.
This is known as therandom phase approximation[32].
Hybertsen and Louie have implemented the full GW approximation into an LDA
framework [33, 34 ], and obtained energy spectra with excellent agreement with
experiment. The static COHSEX approximation is only a first step in this procedure.
It is possible to replace the relation (5.60) by a local one:

v(r,r′)=w(r,r′)ε(r,r′), (5.65)

which is sometimes done for convenience. The detailed structure overlooked in this
approximation is denoted aslocal field effects. From the work of Hybertsen and
Louie it is clear that local field effects have a major impact on the energy spectra.
We see that a particle which is added to the system will influence the behaviour of
the other particles. If we could switch off the interaction between the particles, the
newly added particles would occupy sharp energy levels, and the new particle on its
own would completely determine the new level. Landau [35] analysed the many-
body behaviour of liquid helium-3 and argued that if we had a knob with which we
could tune the interactions, the spectrum would change in acontinuousway. That
is, for no interaction, the spectrum consists of a series of delta-functions, which
start to broaden and shift when the interactions are switched on. The corresponding
excitations involve, as we have seen, the presence of a new particle (or, in the
case of two-particle problems, a particle occupying a new state), accompanied by a
slight change of the orbitals of the other particles. This excitation is called aquasi-
particle. Quasi-particle excitations can be analysed in terms of many-body Green’s
functions [36].