Excited States from Time-Dependent Density Functional Theory - Reviews in Computational Chemistry - page 112

Here e unif
XC

is the accurately known exchange-correlation energy density
of the uniform electron gas of spin densities n

ð n

a ;

n

b Þ

. For the time-dependent
exchange-correlation kernel of Eq. [40], Eq. [53] leads to

, n

a b

n s ¼ n 0 s ð r Þ

d 2 e unif
XC

f ALDA
XC

; rt 0 Þ¼ d ð3Þ ð r r 0 Þ d ð t 0 Þ

ss 0 ½

n 0 ð rt t ½ 54

dn dn

s s 0

The time Fourier transform of the kernel has no frequency dependence at all in
any adiabatic approximation. Via a Kramers-Kronig relation, this implies that
it is purely real. 190
Thus, any TDDFT linear response calculation is carried out in two steps:

1. An approximate ground-state DFT calculation is done, finding a self-
consistent KS potential. Transitions from occupied to unoccupied KS
orbitals provide zero-order approximations to the optical excitations.
2. An approximate TDDFT linear response calculation is done on the orbitals
of the ground-state calculation. This corrects the KS transitions into the
true optical transitions.

In practice both of these steps contain inherent errors. We shall dissect
the relative importance of both of these errors later in the chapter.

IMPLEMENTATION AND BASIS SETS

Time-dependent DFT has the ability to calculate various physical and
quantum quantities, and different techniques are sometimes favored for each
type. For some purposes as, for example, if strong fields are present, it can be
better to propagate forward in time the KS orbitals using either a real space
grid 196,197 or plane waves. 198 For finite-order response, Fourier transforming
to frequency space with localized basis functions may be preferable. 199 We dis-
cuss in detail below how the latter approach works, emphasizing the impor-
tance of basis set convergence.

Density Matrix Approach
We can write the dynamics of the TDKS systems in terms of the one-
particle density matrix g s ð rr 0 t Þ of the TDKS determinant, rather than the

orbitals. g s ð rr 0 t Þ has the spectral representation

X

N

g s ð rr 0 t Þ f j s ð r 0 t

Þ¼

1 f j s ð rt

Þ ½ 55

j ¼

which means the N . The eigenva-

lue of all TDKS orbitals, which is their occupation number , is always 1,

TDKS orbitals are the eigenfunctions of g s

s