Exploration and Optimization of the One- and Many-particle Basis Sets in Modern Explicitly Correlated Electronic Structure

Abstract

Basis set incompleteness error (BSIE) remains one of the largest sources of error in modern electronic structure calculations. Despite the many development producing methods that circumvent the issue by using grids or real-space methods, the linear combination of atomic orbitals (LCAO) approach using Gaussian atomic basis sets is still the most popular method for the simplicity of the underlying technology and the low computational cost. Gaussian basis sets require either basis set extrapolation or the use of explicitly correlated wavefunctions that address the deficiencies of standard determinantal expansions at short interelectron distances in order to yield accurate electron correlation energies in molecules. Practical and robust explicitly correlated F12 methods require the use of standard or specialized atomic orbital (AO) basis sets that include diffuse AOs, even for neutral species. Although modern reduced-scaling formulations of explicitly correlated many-body methods have become routinely applicable to systems with hundreds of atoms, application of F12 methods to large molecular systems can be severely hampered due to the onset of poor numerical conditioning brought on by the presence of diffuse AOs in the F12-appropriate orbital basis sets. The development of practical explicitly correlated methods during the 1980s and 1990s was greatly accelerated in the following decade, consisting of a number of significant and often unrelated breakthroughs published by a number of researchers. The current state of these methods is an amalgam of the best approximations and adaptations of many of these contributing research directions. The determination of appropriate basis sets and development of basis sets specifically designed for F12 methods lead to empirical observations about the ideal composition of these basis sets, many of which remain relatively unexplored. The prodigious use of such methods in the modern context, especially for less well-behaved systems or for pushing the limits of attainable accuracy for thermodynamic data have revealed areas for growth in the most efficient use of basis sets for explicit correlation. First, we reexamine the optimal geminal exponent used in the correlation factor most widely used among explicitly correlated methods. These Slater-type F12 geminal lengthscales, originally tuned for the second-order Møller-Plesset F12 method are too large for higher-order F12 methods formulated using the SP (diagonal fixed-coefficient spin-adapted) F12 ansatz. The new geminal parameters reported herein reduce the basis set incompleteness errors (BSIE) of absolute coupled-cluster singles and doubles F12 correlation energies by a significant — and increasing with the cardinal number of the basis — margin. The effect of geminal reoptimization is especially pronounced for the cc-pVXZ-F12 basis sets (those specifically designed for use with F12 methods) relative to their conventional aug-cc-pV$X$Z counterparts. The BSIEs of relative energies are less affected but substantial reductions can be obtained, especially for atomization energies and ionization potentials with the cc-pVXZ-F12 basis sets. The new geminal parameters are therefore recommended for all applications of high-order F12 methods, such as the coupled-cluster F12 methods and the transcorrelated F12 methods. The geminal exponent relates to the many-particle basis, but the composition of the single-particle atomic basis is also important, specifically the use of diffuse AOs. Here we re-examine why diffuse AOs are necessary for application of F12 methods. To this end we develop a dual-basis formulation of traditional and F12 coupled-cluster singles and doubles (CCSD) methods in which the occupied and virtual (correlating) orbitals were expanded in separate AO basis sets. This indicates that the diffuse AOs are fundamentally important for the traditional (non-F12) description of dynamical correlation; the necessity of diffuse AOs in F12 calculations arises indirectly due to the dramatic reduction of the basis set error by the F12 terms such that the error due to the lack of diffuse AOs becomes comparable to the residual basis set incompleteness. The dual-basis CC methods are suggested as an important candidate formalism for accurate (in particular, F12) reduced-scaling many-body methods in molecules.