7.7.6 Freeze-and-Release Calculations

The $\mathrm{\Delta }$SCF approach recasts the excited-state problem as variational optimization of the electronic energy belonging to one or more electronic configurations generated using a single set of MOs. Although this procedure can yield accurate results for excited-state energies and properties, ¹⁴³ Bogo N., Stein C. J.
Phys. Chem. Chem. Phys.
(2024), 26, pp. 21575. Link variational convergence on the excited-state electronic hypersurface is more difficult than for ground-state calculations. Furthermore, the performance of variational optimization algorithms is intrinsically sensitive to their initial guess, meaning that the ground-state MOs are not always suitable for starting the calculation of an excited state in which the polarization of the electron density changes significantly, e.g., a charge-transfer excitation. The use of a poor guess may result in convergence to another electronic state, like in variational collapse to the ground state, slow convergence, or no convergence at all. Because of that, it is desirable to develop guess-refinement algorithms to improve the quality of the initial guess and, consequently, the convergence of the excited-state electron density.

Recently, Schmerwitz et al. proposed to perform a constrained optimization on the ground-state MOs (frozen step), whose result is then used to set up a full MO relaxation (release step). ¹²¹² Schmerwitz Y. L. A., Selenius E., Levi G.
J. Chem. Theory Comput.
(2026), 22, pp. 3571. Link A constrained optimization algorithm inspired by this contribution was implemented in Q-Chem, ¹⁴⁴ Bogo N. et al.
Phys. Chem. Chem. Phys.
(2025), 27, pp. 17533. Link named the FRZ MO guess. This method was applied to the calculation of challenging CT excitations in supramolecular complexes in combination with the SGM algorithm of Section 7.7.5. ⁵¹¹ Hait D., Head-Gordon M.
J. Chem. Theory Comput.
(2020), 16, pp. 1699. Link We call this a freeze-and-release electronic structure method, FRZ-SGM.

To set up an electronic structure calculation with constrained orbitals, it is necessary to group the MO rotations in active and frozen when updating the $𝐂$ matrix with a unitary transformation. In restricted Kohn–Sham (RKS) theory, this matrix is partitioned into occupied and unoccupied (virtual) sub-spaces. We start by describing the simplest case of the RKS matrix and then extend our formulation to unrestricted Kohn–Sham (UKS) theory. It is important to manually reorder the MO column vectors in the $𝐂$ matrix at the beginning of a calculation to have the right occupied/virtual MOs frozen. In frozen RKS, the $𝐂$ matrix has the structure outlined in Fig. 7.1, where the size of the frozen blocks is given by the a_frozen and va_frozen $rem variables, for the doubly-occupied and virtual blocks, respectively. It is up to the user to reorder the MOs to have the excited electron(s) in the rightmost block of the D block and the hole(s) in the rightmost block of the V block; this is done using the $reorder_alpha input section. This structure will be used to construct the MO rotations, as well as to calculate the corresponding gradients and steps. In the simple example shown in panel A of Fig. 7.1, a double excitation is computed by exciting one $\alpha$ and one $\beta$ electron from MO no. 2 to MO no. 4, and the MOs are sorted in the order 1-4-3-5-6-2. The D${}_{\mathrm{f}}$ and V${}_{\mathrm{f}}$ blocks contain only MOs no. 4 and 2, respectively. The block structure emerging for the $𝐂$ matrix by partitioning into active and frozen parts is depicted in panel B of Fig. 7.1, using the general notation introduced above.

This structure is extended to the UKS electronic structure. In the UKS case, two separate RKS calculations are performed on the $\alpha$ and $\beta$ MO sets. Since one can have excitations of $\alpha$ and $\beta$ electrons, the b_frozen and vb_frozen are introduced to freeze the $\beta$ MOs. The $\beta$ MOs can be reordered using the $reorder_beta input section.