The Dual-Basis Idea
Self-consistent field (SCF) calculations form the basis of electronic structure theory as well as our common qualitative picture of chemistry—namely, molecular orbitals (MOs). The SCF wavefunction is a direct product of these MOs:

These orbitals are, in general, complicated objects with no compact analytic form. They are typically expanded in a basis of atom-centered Gaussian basis functions (AOs), such as the commonly used Pople-style (6-31G*) and Dunning-style (cc-pVDZ) basis sets:

The computational bottleneck for most systems becomes the formation of the two-electron repulsion integrals, formally an process, where is the size of the AO basis. While natural sparsity and linear scaling algorithms reduce the scaling with respect to system size, the scaling with respect to basis set size remains fourth-order.

In fact, many improvements in the subsequent correlation calculation—such as RI-MP2—have left the underlying SCF the computational bottleneck. As a practical example, consider an MP2 calculation of an alanine tetrapeptide using the cc-pVQZ basis. The combination of so-called "local" methods with the RI approximation (RI-TRIM-MP2) reduces the cost of the correlation calculation to a mere 2 hours. However, the underlying SCF to obtain the MOs takes 6 days!

The dual-basis SCF (DB-SCF) method is a simple, cost-effective alternative designed to tackle this basis set size scaling. Essentially perturbation theory for the basis set problem, DB-SCF combines a full, iterative calculation in a small basis set with a subsequent non-iterative correction in the target basis.

Therefore, instead of the 10-15 SCF cycles typically required for convergence, only a single step is required in the target basis set. Integral screening further reduces the cost of this lone Fock build. Average savings in the neighborhood of 90% have been demonstrated. To be useful, such an approximation must be an accurate representation of the target-basis result. As shown in the following table, dual-basis density functional theory (DFT) bond-breaking energies are extremely accurate and exhibit errors that are orders of magnitude smaller than use of the smaller basis set alone. Results are shown for atomization energies per bond on the G3 test set (223 molecules) and involve quantities ranging from roughly 10-300 kcal/mol.

Target Basis	RMS Error (kcal/mol)
6-311++G(3df,3pd)	0.071
cc-pVTZ	0.025
cc-pVQZ	0.029