Chemistry Reference
In-Depth Information
density functional theory predicts only ground-state properties, not electronic
excitations.
Time-dependent density functional theory (TDDFT),
11-15
in contrast,
applies the same philosophy as ground-state DFT to time-dependent problems.
Here, the complicated many-body time-dependent Schr ¨ dinger equation is
replaced by a set of time-dependent single-particle equations whose orbitals
yield the same time-dependent density. We can do this because the Runge-
Gross theorem
16
proves that, for a given initial wave function, particle statis-
tics and interaction, a given time-dependent density can arise from, at most,
one time-dependent external potential. This means that the time-dependent
potential (and all other properties) is a functional of the time-dependent
density.
Armed with a formal theorem, we can then define time-dependent
Kohn-Sham (TDKS) equations that describe noninteracting electrons that
evolve in a time-dependent Kohn-Sham potential but produce the same den-
sity as that of the interacting system of interest. Thus, just as in the ground-
state case, the demanding interacting time-dependent Schr
¨
dinger equation
(TDSE) is replaced by a much simpler set of equations. The price of this enor-
mous simplification is that the exchange-correlation piece of the Kohn-Sham
potential has to be approximated.
The most common time-dependent perturbation is a long-wavelength
electric field, oscillating with frequency
. In the usual situation, this field
is a weak perturbation to the molecule, and one can therefore perform a
linear response analysis. From the linear response, we can extract the optical
absorption spectrum of the molecule due to electronic excitations. Thus,
linear response TDDFT can be used to predict the transition frequencies to
electronic excited states (along with many other properties), and this has
been the primary use of TDDFT so far, with many applications to large
molecules.
Figure 1 compares TDDFT and experiment for the electronic Circular
Dichroism (CD) spectrum of the chiral fullerene C
76
. A total of 240 optically
allowed transitions were required to simulate the spectrum. The accuracy is
clearly good enough to assign the absolute configuration of C
76
. TDDFT cal-
culations of this size typically take less than a day on low-end personal com-
puters.
A random walk through some recent studies using TDDFT gives some
feeling for the breadth of applications. Most are in the linear response
regime. In inorganic chemistry, the optical response of many transition metal
complexes
20-35
has been calculated, as have some X-ray absorption
spectra.
36,37
In organic chemistry, heterocycles
38-43
have been examined
among others,
44-46
including the response of thiouracil,
47
s
-tetrazine,
48
and annulated porphyrins.
49
We also see TDDFT's use in studying various
fullerenes.
50-55
TDDFT is also finding many uses in biochemistry
56-66
where,
o
