4.5 Converging SCF Calculations 4.5.17 Internal Stability Analysis and Automated Correction for Energy Minima 4.6 Large Molecules and Linear Scaling Methods

4.5.18 Orbital Energy Scaling in Canonical ROSCF Calculations

(September 23, 2025)

For high-spin half-filled open-shell systems, canonical ROSCF calculations according to Plakhutin and Davidson ¹⁰⁴⁸ Plakhutin B. N., Davidson E. R.
J. Chem. Phys.
(2014), 140, pp. 014102. Link can be performed by setting ROSCF_DODS = TRUE. The canonical ROSCF method provides the same total wavefunction and, thus, total energy and correct $\langle\hat{S}^{2}\rangle$ behavior as “conventional” ROSCF. Other than “conventional” ROSCF, it features two different sets of orbitals for $\alpha$ and $\beta$ electrons, as unrestrcited SCF does, which fulfill Koopmans’ theorem. Because of this, canonical ROSCF is also called ROSCF-DODS, where DODS stands for “different orbitals for different spins”.

In general, ROSCF calculations can be hard to converge, often because of violations of the aufbau principle. In ROSCF calculations, these situations can be avoided when applying an intermediary scaling to the open-shell diagonal blocks of the $\alpha$ and $\beta$ RO Fock matrices, chosen such that the “correct” orbital ordering is restored during the SCF iterations. The procedure is similar to level-shifting (Section 4.5.5) but much simpler. It exploits the fact that in ROSCF calculations, the diagonal blocks of the RO Fock matrices can be arbitrarily scaled without affecting the total wave function.

As an example, consider a system with a single unpaired $\alpha$ electron, where the highest $\alpha$ MO belongs to the doubly-occupied orbital space and the lowest $\alpha$ MO belonging to the (doubly-) unoccupied orbital space. Suppose that these orbitals have energies of $-0.5$ and $0.5~{}E_{h}$ , respectively. If the unpaired electron has an orbital energy of $-0.6~{}E_{h}$ , it might be arbitrarily shifted, e.g., to $0.3~{}E_{h}$ by scaling its energy with a factor of $0.5~{}E_{h}$ , corresponding to a setting of ROSCF_DIAG_SCALE_A = 50. This makes life easier for the SCF algorithm.

The obvious caveat of this procedure is that the converged orbital energies need to be known in order to choose the correct scaling factors. In practice, one can try different combinations of ROHF_DIAG_SCALE_A and ROHF_DIAG_SCALE_B, if poor convergence behavior is encountered.

4.5.18.1 Job Control

ROSCF_DIAG_SCALE_A

ROSCF_DIAG_SCALE_A
       When performing canonical restricted open-shell SCF calculations, scale the open-shell diagonal $\alpha$ Fock matrix block (and, thus, singly occupied $\alpha$ orbital energies) during the SCF iterations in order to mitigate aufbau principle violations in case of poorly converging systems. After convergence, proper canonical orbital energies are restored.
TYPE:
       INTEGER
DEFAULT:
       50 Scale $\alpha$ orbital energies by a factor of 0.5 which often leads to improved convergence behavior.
OPTIONS:
       $n$ Scale $\alpha$ orbital energies by a factor of $n\cdot 10^{-}2$ .
RECOMMENDATION:
       In case of convergence issues, try series of different values, e.g., $\pm 800$ , $\pm 400$ , $\pm 200$ , $\pm 100$ , $\pm 50$ , $\pm 20$ , $\pm 10$ .

ROSCF_DIAG_SCALE_B

ROSCF_DIAG_SCALE_B
       The same as ROSCF_DIAG_SCALE_A, except that singly unpccupied $\beta$ orbital energies are manipulated during the SCF iterations.
TYPE:
       INTEGER
DEFAULT:
       50 Scale $\beta$ orbital energies by a factor of 0.5.
OPTIONS:
       $n$ Scale $\beta$ orbital energies by a factor of $n\cdot 10^{-}2$ .
RECOMMENDATION:
       See ROSCF_DIAG_SCALE_A for recommendations.

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4.5.18 Orbital Energy Scaling in Canonical ROSCF Calculations

4.5.18.1 Job Control