Chemistry Reference
In-Depth Information
transitions are about the same. This similarity is no coincidence but, due to the
lack of a derivative discontinuity in LDA, the potential LDA differs from the
exact one by roughly a constant.
The ''fruitfly'' of TDDFT benchmarks is the
p
!
p
transition in
benzene. This transition occurs at about 5 eV in a ground-state LDA calcula-
tion, and ALDA shifts it correctly to about 7 eV.
233
Unfortunately, this value
lies in the LDA continuum, which starts at about 6.5 eV! This shift is an
example of the same general phenomenon we saw above for He, where
LDA has pushed some oscillator strength into the continuum, but its overall
spectra remains about right.
We can take the observation that the LDA potential is roughly a constant
shift from the exact KS potential in a core region and go one step further to
deduce the energies of individual transitions. While the existence of a quantum
defect requires a long-ranged potential, its value is determined by the phase
shift caused by the deviation from
r
behavior in the interior of the
atom. The
quantum defect extractor
(QDE)
251
is a formula for extracting
the effective quantum defect from a scattering orbital of a short-ranged KS
potential, such as that of LDA. The QDE is
1
=
n
r
n
;
n
Þ
d
ln
f
1
n
1
r
U
ð
2
;
2
r
=
¼
½
62
dr
U
ð
1
n
;
2
;
2
r
=
n
Þ
p
2
n
Þ
1
, with
n
¼ð
Here
k
¼
j
E
j
is written as
k
¼ð
n
m
n
Þ
, where
n
numbers
the bound state, and
m
n
is the quantum defect;
U
is the confluent hypergeo-
metric function.
252
If the extractor is applied to an orbital of a long-ranged
potential, it rapidly approaches its quantum defect.
The results of the QDE for the He atom, applied to both the exact KS
potential and the LDA potential are plotted in Figure 13. The LDA potential
0.4
LDA
0.3
0.2
Exact
0.1
0
0
1
2
3
4
5
6
r
Figure 13
Solution of Eq. [62] for
m
as a function of
r
for the He atom. The
n
¼ 20
orbital was used for the exact case, and the scattering orbital or energy
E
¼
I
þ
E
LDA
1
s
was
used for the LDA.
