7.3.1 Brief Introduction

Excited states may be obtained from density functional theory via linear response, ³²⁹ Dreuw A., Head-Gordon M.
Chem. Rev.
(2005), 105, pp. 4009. Link which for historical reasons is known as “time-dependent” (TD-)DFT. This should not be confused with the explicitly time-dependent methods that are discussed in Section 7.4, however linear-response DFT is nearly universally called TDDFT and we shall use that nomenclature as well. This approach 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

where the elements of the matrix $𝐀$ similar to those used at the CIS level, Eq. (7.11), but with an exchange-correlation correction. Elements of $𝐁$ are similar. Equation (7.15) is solved iteratively for the lowest few excitation energies, $\omega$. Alternatively, one can make a Tamm-Dancoff approximation (TDA) in which the “de-excitation” amplitudes $𝐘$ are neglected. ⁵⁴² Hirata S., Head-Gordon M.
Chem. Phys. Lett.
(1999), 314, pp. 291. Link In that case, the $𝐁$ matrix is not required and Eq. (7.15) reduces to a CIS-like equation $\mathrm{𝐀𝐱}=\omega 𝐱$.

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. ⁷²⁷ Laurent A. D., Jacquemin D.
Int. J. Quantum Chem.
(2013), 113, pp. 2019. Link The calculation of the nuclear gradients of full TDDFT and within the TDA is implemented. ⁷⁹² Liu F. et al.
Mol. Phys.
(2010), 108, pp. 2791. Link

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. ¹⁹⁶ Casida M. E. et al.
J. Chem. Phys.
(1998), 108, pp. 4439. Link ^{, 1275} Tozer D. J., Handy N. C.
J. Chem. Phys.
(1998), 109, pp. 10180. Link ^{, 710} Lange A., Herbert J. M.
J. Chem. Theory Comput.
(2007), 3, pp. 1680. Link The extent to which a particular excited state is characterized by charge transfer can be assessed using an a spatial overlap metric, $\mathrm{\Lambda }$, defined as ⁹⁸³ Peach M. J. G. et al.
J. Chem. Phys.
(2008), 128, pp. 044118. Link

where $O_{ia}$ is the spatial overlap of occupied MO ${\psi }_i$ with virtual MO ${\psi }_a$:

The absolute value signs are necessary since the occupied and virtual MOs are orthogonal, $⟨{\psi }_i|{\psi }_a⟩$, although Q-Chem includes the option to use squares of the MOs instead:

which is then used in place of $O_{ia}$ in Eq. (7.16). For the original version of the metric (using $O_{ia}$), Tozer and coworkers find that $0.45\le \mathrm{\Lambda }\le 0.89$ for localized valence excitations whereas Rydberg excitations lie in the range $0.08\le \mathrm{\Lambda }\le 0.27$, ⁹⁸³ Peach M. J. G. et al.
J. Chem. Phys.
(2008), 128, pp. 044118. Link and they suggest functional-specific cutoffs for when a particular excitation may have too much charge-transfer character for TDDFT results to be trusted. Note that while Q-Chem implements the definition in Eq. (7.16) that was proposed in Ref.  983 Peach M. J. G. et al.
J. Chem. Phys.
(2008), 128, pp. 044118. Link , the normalizing denominator in this expression is inconsistent for full TDDFT calculations (i.e., those not invoking the TDA), for which the normalization condition on $𝐱$ and $𝐲$ is ${\sum }_{ia}(x_{ia}^2-y_{ia}^2)=1$ rather than ${\sum }_{ia}(x_{ia}^2+y_{ia}^2)=1$. See Ref.  532 Herbert J. M.
Phys. Chem. Chem. Phys.
(2024), 26, pp. 3755. Link for a discussion.

Standard TDDFT also does not yield a good description of static correlation effects (see Section 6.13), because it is based on a single reference configuration of Kohn-Sham orbitals. A variant called spin-flip (SF) TDDFT has been developed to address this issue. ¹¹⁵⁶ Shao Y., Head-Gordon M., Krylov A. I.
J. Chem. Phys.
(2003), 118, pp. 4807. Link SF-TDDFT 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-TDDFT 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-TDDFT method with a non-collinear exchange-correlation potential, originally developed by Ziegler and co-workers, ¹³²⁸ Wang F., Ziegler T.
J. Chem. Phys.
(2004), 121, pp. 12191. Link ^{, 1150} Seth M., Mazur G., Ziegler T.
Theor. Chem. Acc.
(2011), 129, pp. 331. Link has also been implemented. ¹⁰⁷ Bernard Y. A., Shao Y., Krylov A. I.
J. Chem. Phys.
(2012), 136, pp. 204103. Link This non-collinear version sometimes improves upon collinear SF-TDDFT for excitation energies but contains a factor of spin density (${\rho }_{\alpha }-{\rho }_{\beta }$) in the denominator that sometimes causes stability problems. Best results are obtained using functionals with $\approx 50$% Hartree-Fock exchange, ¹¹⁵⁶ Shao Y., Head-Gordon M., Krylov A. I.
J. Chem. Phys.
(2003), 118, pp. 4807. Link ^{, 107} Bernard Y. A., Shao Y., Krylov A. I.
J. Chem. Phys.
(2012), 136, pp. 204103. Link behavior that was later explained on theoretical grounds. ⁵⁶⁶ Huix-Rotllant M. et al.
Phys. Chem. Chem. Phys.
(2010), 12, pp. 12811. Link Becke’s half-and-half functional BH&HLYP has become something of a standard approach when using SF-TDDFT. A spin-adapted version of SF-TDDFT has also been developed. ¹⁴³⁵ Zhang X., Herbert J. M.
J. Chem. Phys.
(2015), 143, pp. 234107. Link