7.8.3 Restricted Open-Shell Kohn-Sham Method (ROKS)

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.

The restricted open-shell Kohn-Sham (ROKS) method 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 $E_{\text{mix}}$. The triplet energy $E_{\text{T}}$ is also computable from a single determinant within the the $M_S=\pm 1$ subspaces. Consequently, ROKS optimizes a set of spin-restricted orbitals $\{{\varphi }_{\text{ROKS}}\}$ such that the spin-purified singlet energy $E_{\text{S}}=2E_{\text{mix}}[\{{\varphi }_{\text{ROKS}}\}]-E_{\text{T}}[\{{\varphi }_{\text{ROKS}}\}]$ 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, by virtue of the two-determinant nature of the equations. ⁵⁴¹ Hirao K., Nakatsuji H.
J. Chem. Phys.
(1973), 59, pp. 1457. Link ^{, 676} Kowalczyk T. et al.
J. Chem. Phys.
(2013), 138, pp. 164101. Link It is also important to note that this excited-state method is distinct from ROKS theory for open-shell ground states, which is a single-determinant method corresponding to the high-spin state with multiple unpaired spins.

The implementation of ROKS excited states in Q-Chem largely follows the theoretical framework established by Filatov and Shaik; ³⁶¹ Filatov M., Shaik S.
Chem. Phys. Lett.
(1999), 304, pp. 429. Link see Ref.  676 Kowalczyk T. et al.
J. Chem. Phys.
(2013), 138, pp. 164101. Link for the case of the lowest excited singlet ($S_1$ state) with a DIIS-based approach. An example is provided below. ROKS for higher excited states is possible using either the squared-gradient approach (Section 7.8.4), the maximum overlap method (Section 7.6), or else state-targeted energy projection (Section 7.8.5).

ROKS has been found to be significantly more accurate than TDDFT for describing charge-transfer states, ⁴⁷⁵ Hait D. et al.
J. Chem. Theory Comput.
(2016), 12, pp. 3353. Link 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 K- and L-edge excitations of small molecules. ⁴⁷² Hait D., Head-Gordon M.
J. Phys. Chem. Lett.
(2020), 11, pp. 775. Link Examples of using ROKS/SGM to compute core-excited states are provided in Section 7.8.4. Analytic nuclear gradients (in the excited state) 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.  676 Kowalczyk T. et al.
J. Chem. Phys.
(2013), 138, pp. 164101. Link , and in addition Ref.  473 Hait D., Head-Gordon M.
J. Chem. Theory Comput.
(2020), 16, pp. 1699. Link if the SGM implementation is employed, as well as Ref.  475 Hait D. et al.
J. Chem. Theory Comput.
(2016), 12, pp. 3353. Link for charge-transfer states and Ref.  472 Hait D., Head-Gordon M.
J. Phys. Chem. Lett.
(2020), 11, pp. 775. Link for application to core excitations.

The chief limitation of ROKS is 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.

Furthermore, ROKS often suffers from an un-physical mixing between the open-shell orbitals when they belong to the same spatial symmetry, which often manifests in $\pi ⟶{\pi }^{*}$ and $1s⟶3s$ excitations. ³⁷⁸ Friedrichs J., Damianos K., Frank I.
Chem. Phys.
(2008), 347, pp. 17. Link ^{, 676} Kowalczyk T. et al.
J. Chem. Phys.
(2013), 138, pp. 164101. Link This is associated with the inherent ambiguity in defining an effective Fock opereator for a two-determinant wave function. ⁵⁴¹ Hirao K., Nakatsuji H.
J. Chem. Phys.
(1973), 59, pp. 1457. Link ^{, 378} Friedrichs J., Damianos K., Frank I.
Chem. Phys.
(2008), 347, pp. 17. Link Convergence is often difficult for the states where this mixing manifests, and the overlap with the ground state may be significant. Two protocols can be used to alleviate this issue. The first is a level-shift technique that splits the energy between the two singly-occupied orbitals thereby supressing their mixing. ⁶⁷⁶ Kowalczyk T. et al.
J. Chem. Phys.
(2013), 138, pp. 164101. Link It is only available with DIIS and for the ROKS implementation on the old SCF engine (GEN_SCFMAN = FALSE). The second is a total supression of the mixing between the sinlgy-occupied orbitals. While this may disregard the correct variational condition, ⁵⁴¹ Hirao K., Nakatsuji H.
J. Chem. Phys.
(1973), 59, pp. 1457. Link it produces excited states with much smaller overlaps with the ground state and provides smoother convergence. See 7.8.3 for a demonstration.

To perform an ROKS excited state calculation, simply set the keyword ROKS = TRUE and ensure that UNRESTRICTED = FALSE. 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.