Computational Physics

5.4 Beyond DFT: one- and two-particle excitations 103

because they impose this property in their form of the energy-functional (based
on a density operator form), which need not describe the pure-state functional of
DFT [27].
The property we have just derived – the Kohn–Sham energy of the highest occu-
pied level gives us the ionisation energy – is satisfied very well for extended systems,
but poorly for molecules, where the highest occupied Kohn–Sham energy (called
the highest occupied molecular orbital, or HOMO) is generally found a few eV
abovethe experimental value. Hartree–Fock usually gives a much better value. The
reason why DFT fails so badly in practice lies in the poor asymptotic behaviour of
the available exchange-correlation potentials, which, among other imperfections,
do not cancel the self-interaction and hence give an incorrect asymptotic form of
the Kohn–Sham potential. In our derivation, this asymptotic behaviour played a
crucial role. The fact that we do not have the exact exchange correlation functional
at our disposal therefore is a serious handicap in describing the spectra of atoms
and molecules.
There is a way around this: given the fact that DFT is very good at calculating
ground state energies, we can perform two calculations: one forNelectrons, and
one forN−1 (for the electron addition energies, the second calculation would
be performed forN+1 electrons). The difference in the total energies then gives
the ionisation (or electron addition) energy. Instead of these two energies, it is also
possible to do one calculation at half filling of the highest occupied (or lowest
unoccupied) level. The Kohn–Sham energy of that level is the derivative of the total
energy with respect to the charge, so that we can predict the ionisation energy from

EGS(N)−EGS(N− 1 )=

[

∂Etot

∂N

]

N+ 1 / 2

=εKSN(N+ 1 / 2 ). (5.53)

A similar procedure gives the electron addition energy. This method is known as
delta-SCF.

5.4.2 General theories for excitation energies

Looking at what causes a system which is in the ground state to go to an excited
state, we conclude that there should always be some time-dependent perturbation
to the Hamiltonian which is responsible for this. Therefore, we should consider the
response of the system to an external, time-dependent perturbation. The standard
approach is to consider the response to a monochromatic perturbation with fre-
quenceω. However, if the response of the system to a perturbation is faster than the
typical period of the perturbation, we may consider a time-independent approach.
An electron which has been excited to a higher energy level will return to its
ground state orbital after some time. This finite lifetime gives rise to a finite width of