Excited states may be obtained from density functional theory by time-dependent density functional theory,^{139, 233} 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.^{384} 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),^{385} 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,^{139} 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.^{544} The calculation of the nuclear gradients of full TDDFT and within the TDA is implemented.^{589}
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.^{137, 968, 528} 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).^{731} (However, see Ref. 819 for a cautionary note regarding this metric.)
Standard TDDFT also does not yield a good description of static correlation effects (see Section 6.10), 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.^{869} 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. Recently, we also implemented^{69} a SF-DFT method with a non-collinear exchange-correlation potential from Tom Ziegler et al.,^{1011, 866} which is in many case an improvement over collinear SF-DFT.^{869} Recommended functionals for SF-DFT calculations are 5050 and PBE50 (see Ref. 69 for extensive benchmarks). Spin-adapted version of SF-DFT was developed by John Herbert. See also Section 7.10.3 for details on wave function-based spin-flip models.