4.7.1 Introduction

The dual-basis approximation ⁷⁴⁶ Liang W. Z., Head-Gordon M.
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(2009), 5, pp. 1560. Link 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:

Here $𝐅$ (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 ²⁶⁵ Curtiss L. A. et al.
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(1998), 109, pp. 7764. Link ^{, 263} Curtiss L. A. et al.
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(2000), 112, pp. 7374. Link 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.