Excited States from Time-Dependent Density Functional Theory - Reviews in Computational Chemistry - page 111

Continuum

5 6S

24

4S

3 3P

23

22

2P

21

Singlet

2S

20

Triplet

Exact KS Exact TDDFT

Figure 4 Transitions for the helium atom using only ground-state DFT on the left and
TDDFT on the right. In both cases, the exact functionals have been used. The results for
employing the exact XC kernel in TDDFT linear response were deduced from
calculations using Ref. 193. In each pair of lines on the right, the triplet is below the
singlet.

Approximations
While all the equations above are formally exact, as in the ground-state
case, a TDDFT calculation requires an approximation for the unknown XC
potential. The most common approximation in TDDFT is the adiabatic
approximation 194,195 in which

v adia
XC

v g XC s ½

s ½

n ð rt Þ¼ n 0 ð r Þj n 0 s ð r Þ¼ n s ð rt Þ ½ 52

This means the XC potential at any time is simply the ground-state XC poten-
tial at that instant. This obviously becomes exact for sufficiently slow pertur-
bations in time, in which the system always stays in its instantaneous ground
state. Although most applications are not in this slowly varying regime, results
obtained within the adiabatic approximation are, nevertheless, remarkably
accurate in many cases.
Any ground-state approximation (LDA, GGA, hybrid) provides an adia-
batic approximation for use in TDDFT automatically. The most famous is the
adiabatic local density approximation (ALDA). 194,195 It employs the func-
tional form of the static LDA with a time-dependent density:

n s ¼ n s ð rt Þ

de unif
XC
dn s

v ALDA v unif

XC XC s ½

n ð rt Þ¼ ð n a ð rt Þ; n b ð rt ÞÞ ¼ ½ 53