4.5.18 Orbital Energy Scaling in Canonical ROSCF Calculations

For high-spin half-filled open-shell systems, canonical ROSCF calculations according to Plakhutin and Davidson ¹⁰⁴⁵ 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 $⟨{\widehat{S}}^2⟩$ 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 wavefunction.

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

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.