Dual-Basis Pairings for 6-31G* Calculations:
While the above results for analytic gradients showed promise for large target basis sets, the timing performance for small basis sets (of the 6-31G* ilk) was,frankly, not stellar. The cost of the underlying small basis set (for both SCF and response calculations) was simply too high. Given that the ratio of basis set size was close to unity—as opposed to more successful pairings, where the ratio was 2–3—some work remained to be done in this area. With only a single set of polarization (d) functions to eliminate, 6-31G* was not an ideal candidate for dual-basis implementation, yet its widespread use compelled us to try.

The solution for this regime was, in essence, to make 6-31G smaller. Its "31" split-valence structure was re-contracted into a 6-4G minimal basis. Importantly, the exponents and the necessary relative weightings were retained so that it remained a proper subset of 6-31G by primitives. As such, the large-basis steps remain identitical in cost to a 6-31G/6-31G* pairing, but the small-basis steps are reduced to a minimal-basis calculation.

The basis set was, therefore, re-optimized to fit the energies of a small set of atomic and small molecular systems, similar to the original optimization of 6-31G*. The resulting basis was tested for reaction energies, molecular structures, and even harmonic frequencies. While DB results were not exact reproductions of the target basis, they effectively captured the lion's share of the difference between single-basis calculations in either basis set alone.

Timings were also much-improved. For the glycine hexadecapeptide (115 atoms), a 6-4G/6-31G** derivative calculation is reduced to less than half the cost of the SCF alone in the target basis set. Reference:
"Dual-basis self-consistent field methods: 6-31G* calculations with a minimal 6-4G primary basis"
R. P. Steele and M. Head-Gordon. Mol. Phys. 105 2455 (2007) [Peter Pulay Special Issue].