Chemistry Reference
In-Depth Information
marginally here, suggesting that more accurate XC kernels are required to
account properly for valence-Rydberg mixing.
Analyzing the Influence of the XC Kernel
As mentioned earlier, the TDDFT kernels used in practice are local or
semilocal in both space and time. Even hybrids are largely semi-local, as
they only mix in 20-25% exact exchange.
In these practical calculations, both the ground-state XC potential and
TDDFT XC kernel are approximated. A simple way to separate the error in
the XC kernel is to examine a test case where the exact KS potential is known.
Figure 5 shows the spectrum of He using the exact KS potential, but with the
ALDA XC kernel. The ALDA XC kernel does rather well
232
(very well, as we
shall see later when we examine atoms in more detail), and very similar results
are obtained with standard GGAs.
The errors in such approximate kernels originate from the locality in
space and time. We can test the error of locality separately from the local
time error for the He atom, by studying the exchange limit for the XC kernel.
For two spin-unpolarized electrons,
f
x
ΒΌ
j
r
r
0
j
, i.e., it exactly cancels
half the Hartree term. Most importantly, it is frequency-independent so that
there is no memory, meaning that the adiabatic approximation is exact. In Fig-
ure 6, we compare ALDAx, which is the ALDA for just exchange, to the exact
exchange result for He. ALDA makes noticeable errors relative to exact
exchange, showing that nonlocality in space is important in this example.
Thus, the hybrid functionals, by virtue of mixing some fraction of exact
exchange with GGA, will have slightly different potentials (mostly in the
asymptotic region), but noticeably different kernels.
1
=
2
25
Continuum
5
6S
24
4S
3
3P
23
22
2P
21
2S
20
Exact KS
ALDAxc
Exact TDDFT
19
Figure 5
Spectrum of helium calculated using the ALDA XC kernel
232
with the exact KS
orbitals.
