Dual-Basis MP2:
Second-order perturbation theory (MP2) is the simplest means to add electron correlation to a HF calculation:

Many recent developments in peturbative correlation calculations—in particular, the "resolution-of-the-identity" (RI) approximation—have made the correlation part of these methods faster than the underlying SCF calculation for many systems of interest. Thus, the dual-basis (DB) approximation is well-suited for these correlation calculations. In a DB-RI-MP2 calculation, the reference HF calculation is replaced by its DB counterpart:


The DB approximation leads to speedups of a factor of 8-10 in the SCF, while the RI approximation drastically reduces the cost of the correlation calculation. Overall savings for accurate, large-basis calculations were demonstrated to be on the order of 95%.

Most importantly, this speed comes with a marginal tradeoff in accuracy. On the G3 set of 225 molecules, for example, bond-breaking errors were <0.2 kcal/mol, for quantities that range from tens to thousands of kcal/mol across the set. Results shown below are relative to the target, single-basis results. The last two entries demonstrate that dual-basis approximation recovers >97% of the error due to using the small basis set alone.

Target Basis Set	Small Basis Set	RMS Error per bond (kcal/mol)
6-311++G(3df,3pd)	6-311+G*	0.170
cc-pVTZ	rcc-pVTZ	0.083
cc-pVQZ	rcc-pVQZ	0.076
—	6-311+G* alone	11.099
—	cc-pVTZ alone	2.227

Furthermore, energies in relative conformations of alanine tetrapeptides were shown to be only 0.014 kcal/mol and qualitatively consistent with full, target-basis MP2 calculations.

Reference:
"Dual-basis second-order Moller-Plesset perturbation theory: A reduced-cost reference for correlation calculations"
R. P. Steele, R. A. DiStasio, Jr., Y. Shao, J. Kong, and M. Head-Gordon. J. Chem. Phys. 125 074108 (2006).