Combining explicitly correlated R12 and Gaussian geminal electronic structure theories

Explicitly correlated R12 methods using a single short-range correlation factor (also known as F12 methods) have dramatically smaller basis set errors compared to the standard wave function counterparts, even when used with small basis sets. Correlations on several length scales, however, may not be described efficiently with one correlation factor. Here the authors explore a more general MP2-R12 method in which each electron pair uses a set of (contracted) Gaussian-type geminals (GTGs) with fixed exponents, whose coefficients are optimized linearly. The following features distinguish the current method from related explicitly correlated approaches published in the literature: (1) only two-electron integrals are needed, (2) the only approximations are the resolution of the identity and the generalized Brillouin condition, (3) only linear parameters are optimized, and (4) an arbitrary number of (non-)contracted GTGs can appear. The present method using only three GTGs and a double-zeta quality basis computed valence correlation energies for a set of 20 small molecules only 2.2% removed from the basis set limit. The average basis set error reduces to 1.2% using a near-complete set of seven GTGs with the double-zeta basis set. The conventional MP2 energies computed with much larger quadruple, quintuple, and sextuple basis sets all had larger average errors: 4.6%, 2.4%, and 1.5%, respectively. The new method compares well to the published MP2-R12 method using a single Slater-type geminal (STG) correlation factor. For example, the average basis set error in the absolute MP2-R12 energy obtained with the exp(-r12) correlation factor is 1.7%. Correlation contribution to atomization energies evaluated with the present method and with the STG-based method only required a double-zeta basis set to exceed the precision of the conventional sextuple-zeta result. The new method is shown to always be numerically stable if linear dependencies are removed from the two-particle basis and the zeroth-order Hamiltonian matrix is made positive definite.