Excited states may be obtained from density functional theory by time-dependent
density functional theory,
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which calculates poles
in the response of the ground state density to a time-varying applied electric
field. These poles are Bohr frequencies, or in other words the excitation
energies. Operationally, this involves solution of an eigenvalue equation
$$\left(\begin{array}{cc}\hfill \mathbf{A}\hfill & \hfill \mathbf{B}\hfill \\ \hfill {\mathbf{B}}^{\u2020}\hfill & \hfill {\mathbf{A}}^{\u2020}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \mathbf{x}\hfill \\ \hfill \mathbf{y}\hfill \end{array}\right)=\omega \left(\begin{array}{cc}\hfill -\mathrm{\U0001d7cf}\hfill & \hfill \mathrm{\U0001d7ce}\hfill \\ \hfill \mathrm{\U0001d7ce}\hfill & \hfill \mathrm{\U0001d7cf}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \mathbf{x}\hfill \\ \hfill \mathbf{y}\hfill \end{array}\right)$$ | (7.15) |
where the elements of the matrix $\mathbf{A}$ similar to those used at the CIS
level, Eq. (7.11), but with an exchange-correlation
correction.
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Elements of $\mathbf{B}$ are similar.
Equation (7.15) is solved iteratively for the lowest few excitation
energies, $\omega $. Alternatively, one can make a CIS-like Tamm-Dancoff
approximation (TDA),
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in which the “de-excitation”
amplitudes $\mathbf{Y}$ are neglected, the $\mathbf{B}$ matrix is not required, and
Eq. (7.15) reduces to $\mathrm{\mathbf{A}\mathbf{x}}=\omega \mathbf{x}$.
TDDFT is popular because its computational cost is roughly similar to that of
the simple CIS method, but a description of differential electron correlation
effects is implicit in the method. It is advisable to only employ TDDFT for
low-lying valence excited states that are below the first ionization potential
of the molecule,
or more conservatively, below the first
Rydberg state, and in such cases the valence excitation energies are often
remarkably improved relative to CIS, with an accuracy of $\sim $0.3 eV for many
functionals.
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The calculation of the nuclear gradients of
full TDDFT and within the TDA is implemented.
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On the other hand, standard density functionals do not yield a potential with
the correct long-range Coulomb tail, owing to the so-called self-interaction
problem, and therefore excitation energies corresponding to states that sample
this tail (e.g., diffuse Rydberg states and some charge transfer excited
states) are not given accurately.
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The
extent to which a particular excited state is characterized by charge transfer
can be assessed using an a spatial overlap metric proposed by Peach, Benfield,
Helgaker, and Tozer (PBHT).
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(However, see
Ref. 935 for a cautionary note regarding this metric.)
Standard TDDFT also does not yield a good description of static correlation
effects (see Section 6.12), because it is based on a single
reference configuration of Kohn-Sham orbitals. A new variation of
TDDFT called spin-flip (SF) DFT was developed by Yihan Shao, Martin
Head-Gordon and Anna Krylov to address this issue.
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SF-DFT is
different from standard TDDFT in two ways:
The reference is a high-spin triplet (quartet) for a system with an even (odd) number of electrons;
One electron is spin-flipped from an alpha Kohn-Sham orbital to a beta orbital during the excitation.
SF-DFT can describe the ground state as well as a few low-lying excited states,
and has been applied to bond-breaking processes, and di- and tri-radicals with
degenerate or near-degenerate frontier orbitals.
A SF-DFT method with a non-collinear
exchange-correlation potential, originally developed by Ziegler and
co-workers,
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has also been implemented.
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This non-collinear version sometimes improves upon
collinear SF-DFT for excitation energies but contains a factor of spin density
(${\rho}_{\alpha}-{\rho}_{\beta}$) in the denominator that sometimes causes stability
problems. Early experience with SF-DFT suggested that best results are obtained
using functionals with $\approx 50$% Hartree-Fock exchange,
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behavior that was later explained on theoretical grounds by Casida and
co-workers.
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Becke’s half-and-half functional BH&HLYP has become something of a standard
approach when using SF-DFT. A spin-adapted version of SF-DFT has been developed
by Zhang and Herbert.
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See also Section 7.10.4 for details on wave function-based spin-flip models.