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184 CHAPTER 24. PERFORMANCE
At this point, we have surveyed all that one needs to perform a TDDFT calculation
of excitation energies using a modern code, either a quatum chemistry one or a real-time
evolution. The remainder of this chapter, and all of the next, are devoted to delving deeper
into understanding how TDDFT works and what are its current limitations.
Chapter 24 In this section, we report calculations for the He and Be atoms using the exact ground-
state Kohn-Sham potentials, the three approximations to the kernel mentioned in the previous
section, and including many bound-state poles in Eq. (??), but neglecting the continuum.
Performance The technical details are given in Ref. [?]. Table I lists the results, which are compared with
a highly accurate nonrelativistic variational calculations[?, ?] In each symmetry class (s, p,
and d), up to 38 virtual states were calculated. The errors reported are absolute deviations
from the exact values. The second error under the Be atom excludes the 2s → 2p transition,
24.1 Sources of error
for reasons discussed in the next section.
24.2 Poor potentials The effect of neglecting continuum states in these calculations has been investigated by van
Gisbergen et al.[?], who performed ALDA calculations from the exact Kohn-Sham potential in
A problem was noticed early on. Our favorite approximations to the exchange-correlation a localized basis set. These calculations were done including first only bound states, yielding
energy yield fairly lousy exchange-correlation potentials. These potentials are especially un- results identical to those presented here, and then including all positive energy orbitals allowed
pleasant at large distances from Coulombic systems. They are too shallow, making Koopman’s by their basis set. They found significant improvement in He singlet-singlet excitations,
theorem fail drastically. This means that all the higher states are too shallow. In fact the especially for 1s → 2s and 1s → 3s. Other excitations barely changed. Assuming inclusion
Rydberg series, i.e., an infinite sequence of excitations as E → 0 from below, characteristic of the continuum affects results with other approximate kernels similarly, these results do not
of the −1/r decay, does not exist at all. How can one correct KS transition frequencies if change the basic reasoning and conclusions presented below, but suggest that calculations
they do not exist? including the continuum may prove to be more accurate than those presented here.
The standard answer has been to asymptotically correct the potentials, to make them mimic
the true KS potential more accurately. A variety of schemes exist that do this to varying 24.4 Atoms
degrees of accuracy. However, we note that all such schemes then produce a potential that
is not a functional derivative of an exchange-correlation energy. 24.5 Molecules

24.6 Strong fields
24.3 Transition frequencies

In any event, for many systems of real interest, it is the low-lying valence excitations that are
of interest, and in these regions, the potentials are quite accurate. Table X shows...
Hybrid functionals in TDDFT: An important point to note here concerns imple-
mentation of hybrid functionals in TDDFT. There are two ways a hybrid might be coded,
which in the case of excitations, can yield very different results. In the first, ’correct’ DFT
approach, the optimized effective potential method should be used, to guarantee that the
effective potential is always a local, multiplicative one, as required by Kohn-Sham theory.
However, in practice, Hartree-Fock calculations are much easier, and have been around since
time immemorial. Thus the potential produced in many quantum chemical codes consists
of a fraction of the HF non-local potential. This has negligible effect for ground-state and
occupied orbitals, but a large effect on the unoccupied levels.
183