Excited States from Time-Dependent Density Functional Theory - Reviews in Computational Chemistry - page 109

transforming in time, the central equation of TDDFT linear response 189 is a

Dyson-like equation for the true w of the system:

ð d 3 r 1 ð d 3 r 2 w S ss 1 ð rr 1 o Þ

X

w ss 0 ð rr 0 o Þ¼ w S ss 0 ð rr 0 o Þþ

s 1 s 2

1

w s 2 s 0 ð r 2 r 0 o Þ

j þ

f XC s 1 s 2 ð r 1 r 2 o Þ ½ 42

j r 1 r 2

It is to be noted here that:

1. The XC kernel is a much simpler quantity than v XC s ½ n ð rt Þ because the

kernel is a functional of only the ground-state density.
2. The kernel is nonlocal in both space and time. The nonlocality in time manifests
itself as a frequency dependence in the Fourier transform, f XC ss 0 ð rr 0 o Þ .

3. When f XC is set to zero in Eq. [42], physicists call it the random-phase
approximation (RPA). The inclusion of f XC is an exactification of RPA, in
the same way that the inclusion of v XC ð r Þ

in ground-state DFT was an

exactification of Hartree theory.
4. The Hartree kernel is instantaneous; it is local in time, i.e., it has no
memory and will have no frequency dependence when Fourier transformed
to the frequency domain. Thus,

it is given exactly by an adiabatic

approximation.
5. The frequency-dependent kernel is a very sophisticated object since its
frequency dependence makes the solution of an RPA-type equation yield
the exact

(including all vertex corrections at every higher order term). The
kernel defies physical intuition; thus arguments based on the structure of
the TDDFT equations are at best misleading. If any argument cannot be
given in terms of many-body quantum mechanics, Eq. [42] cannot help.
6. The kernel is, in general, complex, with real and imaginary parts related via
a Kramers-Kronig relation. 190

w

rr 0 o Þ , are the excitation frequencies

of the true system. In order to extract these frequencies Casida 191 used ancient
RPA technology to produce equations in which these poles of

The poles of the linear susceptibility, w ð

are found as the
solution to an eigenvalue problem. In order to see this, first do an expansion of
the density change in the basis of KS transitions. We write

w

d

n s ð rt

Þ as

X
q ½

P q s ð o Þ q s ð r Þþ d n s ð r o Þ¼ P q s ð o Þ

s ð r Þ

½ 43

q

where . This representation is used to solve Eq. [36]

self-consistently using Eq. [39], and yields two coupled matrix equations: 192

q ¼ð a ; i Þ if q ¼ð i ; a Þ

X
Y

¼ d v
d v

AB
B A

1 0

o ½ 44

0 1