7.3.1 Brief Introduction

Excited states may be obtained from density functional theory via linear response, ³⁴⁰ 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. ⁵⁵⁹ 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. ⁷⁴⁵ 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. ⁸¹⁰ Liu F. et al.
Mol. Phys.
(2010), 108, pp. 2791. Link

Standard TDDFT also does not yield a good description of static correlation effects (see Section 6.14), 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. ¹¹⁸⁸ 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, ¹³⁶⁵ Wang F., Ziegler T.
J. Chem. Phys.
(2004), 121, pp. 12191. Link ^{, 1182} Seth M., Mazur G., Ziegler T.
Theor. Chem. Acc.
(2011), 129, pp. 331. Link has also been implemented. ¹¹² 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, ¹¹⁸⁸ Shao Y., Head-Gordon M., Krylov A. I.
J. Chem. Phys.
(2003), 118, pp. 4807. Link ^{, 112} Bernard Y. A., Shao Y., Krylov A. I.
J. Chem. Phys.
(2012), 136, pp. 204103. Link behavior that was later explained on theoretical grounds. ⁵⁸³ 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. ¹⁴⁷⁷ Zhang X., Herbert J. M.
J. Chem. Phys.
(2015), 143, pp. 234107. Link