7.7 Restricted Open-Shell Kohn-Sham Method for Singlet Excited States

Singly excited states of closed-shell molecules cannot be described via a single non-aufbau filled Slater determinant as both the up and down spins are equally likely to be excited, leading to at least two configurations with equal weights. Triplet energies can nonetheless be found from a single determinant by switching from the $M_S=0$ subspace of the ground state to $M_S=\pm 1$ (i.e. by having both unpaired electrons have spins pointing in the same direction instead of having one up and one down spin). This tactic however does not work on singlet excited states, with non-aufbau filled configurations where only the up (or down) spin is excited being intermediate between singlet and triplet (and thus spin contaminated). This mixed state is not unlike spin-symmetry broken, unrestricted ground state solutions. An actual singlet energy can be obtained via approximate spin-purification post SCF, by removing the triplet contribution to the energy. The triplet energy thus has to be separately estimated with a second orbital optimization.

Restricted Open-Shell Kohn-Sham (ROKS) offers an alternative route to singlet excited states of this nature. The mixed non-aufbau configuration (with either the up or down spin being excited) is exactly halfway between a singlet and triplet when restricted open-shell orbitals are used, and has an energy $E_{mix}$. The triplet energy $E_T$ is also computable from a single determinant within the the $M_S=\pm 1$ subspaces. Consequently, ROKS optimizes a set of spin-restricted orbitals $\{{\varphi }_{ROKS}\}$ such that the spin-purified singlet energy $E_S=2E_{mix}[\{{\varphi }_{ROKS}\}]-E_T[\{{\varphi }_{ROKS}\}]$ is stationary. This therefore needs only one orbital optimization, in contrast to the two sets needed for the $\mathrm{\Delta }$SCF approach mentioned in the preceding paragraph. The structure of the ROKS Fock matrix however is more complex (see Ref. 501) on account of it’s natively two determinant nature. It is also important to note that this excited-state method is distinct from ROKS theory for open-shell ground states (which is single determinant and all the unpaired spins point in the same direction).

The implementation of ROKS excited states in Q-Chem largely follows the theoretical framework established by Filatov and Shaik²⁵⁶ and is described in detail in Ref. 501 for the case of the lowest excited singlet ($S_1$ state) with a DIIS based approach (see example  7.7 below for usage). ROKS for higher excited states was later accessed via the Square Gradient Minimization (SGM) approach, described in Ref. 334. Examples of using ROKS with SGM are provided in Section 7.8.

ROKS has been found to be significantly more accurate than TDDFT for describing charge-transfer states (see Ref. 336) and preliminary evidence shows the same to hold for Rydberg states. ROKS is also extremely accurate for core excitation energies, with the SCAN functional yielding errors below 0.5 eV for both the K and L edges of small molecules (see Ref. 333 for details). Examples of using ROKS/SGM to compute core excited states is given in Sec  7.8. Analytic excited state nuclear gradients are also available, enabling geometry optimization and molecular dynamics calculations as well; along with finite-difference frequency calculations. Users of the ROKS code are requested to cite Ref. 501. Ref. 334 should also be cited if the SGM implementation is employed, as well as Ref. 336 (for usage to charge-transfer states) and Ref. 333 (for application to core excitations).

The chief limitations of ROKS are that it can only describe states with one broken electron pair. It is consequently applicable only to certain excited states of closed-shell systems: all singlet single excitations well-described by a single natural transition orbital (NTO) pair, or higher singlets where only one electron pair is broken in total (like the ${}^1$B${}_{3g}$ doubly excited state of tetrazine). Fortunately, most charge-transfer and core-excitations do not require more than one broken electron pair, and so this limitation is not a major problem in practice.

To perform an ROKS excited state calculation, simply set the keyword ROKS = TRUE and ensure that UNRESTRICTED = FALSE. An additional keyword ROKS_LEVEL_SHIFT is included to assist in cases of DIIS convergence difficulties with a standard level-shift technique. It is recommended to perform a preliminary ground-state calculation on the system first, and then use the ground-state orbitals to construct the initial guess using SCF_GUESS = READ.