Chemistry Reference
In-Depth Information
0.5
0.4
0.3
0.2
B
e
Atom
LB94
KS
OEP
0.1
-0.20
-0.15
-0.10
-0.05
0.00
Energy (a.u.)
Figure 10
The Be
p
quantum defect of LB94, exact exchange (OEP), and KS, and their
best fits. While both functionals give the correct asymptotic behavior of the KS potential,
we can learn more about their performance by calculating the quantum defect.
compared with the exact KS quantum defect. Any approximate XC kernel
should produce accurate corrections to the ground-state KS quantum defect,
which are typically on the scale of Figure 9.
To demonstrate the power of QD analysis, we test two common approx-
imations to the ground-state potential, both of which produce asymptotically
correct potentials (exact exchange
240
(see the discussion on approximate func-
tionals above and LB94
230
). Exact exchange calculations are more CPU
demanding than are traditional DFT calculations, but they are becoming pop-
ular because of the high quality of the potential.
241,242
In comparison, LB94
provides an asymptotically correct potential at little extra cost beyond tradi-
tional DFT.
218,229,243
In Figure 10 we show the
p
Be quantum defect obtained
with LB94, OEP, and exact KS potentials. Note the high quality of the exact
exchange potential; the quantum defect curve is almost identical to the exact
one, offset by about 0.1. Contrarily, the quantum defect of LB94 is poor for all
cases studied.
244,245
Figure 10 shows that just having an asymptotically correct
potential alone is not sufficient to get a good quantum defect.
Testing TDDFT
To see how well TDDFT really does, we can plot quantum defects for
atoms, using the He atom as our prototype in this section. In Figure 11, we
plot first the KS quantum defect and the exact singlet and triplet lines, as
was done in Figure 9. We consider the Hartree approximation, which is
equivalent to setting the XC kernel to zero. This approximation changes the
position of the singlet curve but leaves the triplet curve unchanged from its KS
