7.3.1 Brief Introduction to TDDFT

Excited states may be obtained from density functional theory by time-dependent density functional theory,^{174, 290} 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

where the elements of the matrix $𝐀$ similar to those used at the CIS level, Eq. (7.11), but with an exchange-correlation correction.⁴⁶⁹ Elements of $𝐁$ 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),⁴⁷⁰ in which the “de-excitation” amplitudes $𝐘$ are neglected, the $𝐁$ matrix is not required, and Eq. (7.15) reduces to $\mathrm{𝐀𝐱}=\omega 𝐱$.

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,¹⁷⁴ 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.⁶³⁸ The calculation of the nuclear gradients of full TDDFT and within the TDA is implemented.⁶⁹⁸

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.^{172, 1130, 621} 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).⁸⁶⁵ (However, see Ref. 966 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.¹⁰²² 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,^{1180, 1019} has also been implemented.⁹³ 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,^{1022, 93} behavior that was later explained on theoretical grounds by Casida and co-workers.⁴⁹³ 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.¹²⁷³ See also Section 7.10.5 for details on wave function-based spin-flip models.