Chemistry Reference
In-Depth Information
1.5
Exact KS
LDA
1
0.5
0
0.6
0.8
1
1.2
1.4
1.6
1.8
1.5
Exact Calculations
Experiment
LDA/ALDA
1
0.5
0
0.6
0.8
1
1.2
1.4
1.6
1.8
ω
Figure 12
Optical intensities (in inverse hartrees) as a function of
o
(in hartrees) for the
He atom. (
Top panel
) The exact KS and LDA spectra. (
Lower panel
) The TDDFT
corrected spectra. LDA/ALDA results are from Ref. 247 but unshifted. The exact
calculations are from Ref. 248, multiplied by the density of states factor (see text), and
the experimental results are from Ref. 249.
theory. To see this, consider Figure 12, which shows both the bare KS response
and the TDDFT-corrected response as a function of
o
for the He atom. The
d
-function absorptions at the discrete transitions have been replaced by
straight lines, whose height represents the oscillator strength of the absorption
multiplied by the appropriate density of states.
250
In the top panel, just the KS
transitions are shown for both the exact KS potential and the LDA potential of
Figure 3. The exact curve has a Rydberg series converging to 0.904 h
(24.592 eV), which is the exact ionization threshold for He. The LDA curve,
on the other hand, has a threshold at just below 0.6 h. Nonetheless, its optical
absorption clearly mimics that of the exact system, even in the Rydberg series
region, and is accurate to about 20%. The TDDFT ALDA kernel corrections
are small and overcorrect the bare LDA results, but the LDA/ALDA spectra
still follows the exact one closely.
Why do the LDA spectra look so similar to the exact one? Is this just
a coincidence? Returning to Figure 3, we notice that the LDA (or a GGA)
potential runs almost exactly parallel to the true potential for
r
2, i.e., where
most of the density is. Thus, the scattering orbitals of the LDA potential, with
transition energies between 0.6 and 0.9 h, almost match exactly the Rydberg
orbitals of the exact KS potential with the same energy.
253
When defined
carefully, i.e., when we use phase space factors for the continuum relative to
bound states,
9
the oscillator strengths for both the LDA and exact KS
