The dual-basis
approximation^{Liang:2004a, Steele:2006a, Steele:2006b, DiStasio:2007a, Steele:2007, Steele:2009}
to self-consistent field (HF or DFT) energies provides an efficient means for
obtaining large basis set effects at vastly less cost than a full SCF
calculation in a large basis set. First, a full SCF calculation is performed
in a chosen small basis (specified by BASIS2). Second, a single
SCF-like step in the larger, target basis (specified, as usual, by
BASIS) is used to perturbatively approximate the large basis energy.
This correction amounts to a first-order approximation in the change in density
matrix, after the single large-basis step:

$${E}_{\text{total}}={E}_{\text{small basis}}+\text{tr}{[(\mathrm{\Delta}\mathbf{P})\mathbf{F}]}_{\text{large basis}}.$$ | (4.55) |

Here $\mathbf{F}$ (in the large basis) is built from the converged (small basis) density matrix. Thus, only a single Fock build is required in the large basis set. Currently, HF and DFT energies (SP) as well as analytic first derivatives (FORCE or OPT) are available.

Note: As of version 4.0, first derivatives of unrestricted dual-basis DFT energies—though correct—require a code-efficiency fix. We do not recommend use of these derivatives until this improvement has been made.

Across the G3 set^{Curtiss:1991, Curtiss:1998, Curtiss:2000} of 223
molecules, using cc-pVQZ, dual-basis errors for B3LYP are 0.04 kcal/mol
(energy) and 0.03 kcal/mol (atomization energy per bond) and are at least an
order of magnitude less than using a smaller basis set alone. These errors are
obtained at roughly an order of magnitude savings in cost, relative to the
full, target-basis calculation.