Dual-Basis HF/DFT First Derivatives:
Optimizing molecular structures and performing molecular dynamics simulations both require first derivatives of the potential energy surface with respect to the position of nuclei. Qualitatively, analytic derivatives require two things: Derivatives of the two-electron integrals and derivatives of the MOs. For standard SCF calculations, the optimized orbitals dictate that the latter is not required (within linear response). Thus, only a single set of integral derivatives is required. Dual-basis SCF methods, however, deal with unconverged orbitals. Thus, a response term is required in the gradient. Though seemingly a fatal complication, this response term can be solved entirely in the small basis. The savings in the integral derivative term (stemming from the structure of the density matrix), combined with the savings in the underlying energy calculation, still lead to cost speedups in the gradient by a factor of 3-5.

As might have been guessed from the promising energy results, errors in molecular structures due to the DB approximation are quite tolerable. Relative to target-basis structures, errors are on the order of a few thousandths of an Angstrom. Furthermore, structures relative to experimental values are essentially indistinguishable. (The 6-31G/6-31G** pairing systematically—though fortuitously—outperforms its single-basis counterpart.) Therefore, for negligible tradeoff in accuracy, molecular geometries may be obtained roughly 35-75% faster.

Reference:
"Dual-basis Analytic Gradients: 1. Self-Consistent Field Theory"
R. P. Steele, Y. Shao, R. A. DiStasio, Jr., and M. Head-Gordon. J. Phys. Chem. A 110 13915 (2006).