The weak orthogonality functional in explicitly correlated pair theories

Abstract

Recent advances have seen the convergence of the R12 and Gaussian geminal explicitly correlated methods, such that the principal remaining distinction is the way in which the many-electron integrals are handled. Here we examine the weak orthogonality functional and the resolution of the identity and find that the first, although exact in the limit of infinite basis, introduces a conflict between the physical description of the electronic cusp and the satisfaction of the strong orthogonality constraint. This leads us to propose an improved weak orthogonality functional where the explicitly correlated pair functions are almost orthogonal to the occupied orbitals by construction. For applications where 95%-98% accuracy in the total correlation energy is sufficient, we recommend use of the strong orthogonality functional in combination with the resolution of the identity for three- and four-electron integral evaluations.

Perspective: Explicitly correlated electronic structure theory for complex systems

“…34 Recently, there has been a resurgence of interest in WOF, and Tew et al proposed an improved functional within MP2-GTG, the so-called intermediate orthogonal functional (IOF). 35 Another way to avoid four-electron integrals is to use a similarity-transformed Hamiltonian. This approach was first introduced by Hirschfelder to remove the electron-electron poles from the original Hamiltonian.…”

Second-order Møller–Plesset calculations on the water molecule using Gaussian-type orbital and Gaussian-type geminal theory

“…17 The GG1 and GG2 energies on the other hand converge very rapidly to the basisset limit, as shown in ref. 15.…”

“…We note, however, that the performance of the GG0 model is significantly improved by using the intermediate orthogonality (IO) functional in place of the WO functional used in our present implementation. 17 We have also calculated the MP2 correlation correction to the barrier to linearity of water. Based on our own estimates of this correction and the value given by Valeev, 34 we suggest the value À463 AE 5 cm À1 for this correction.…”

The geminal basis in explicitly correlated wave functions

“…Nonetheless, our results clearly demonstrate the immense redundancy between the conventional and geminal basis functions in explicitly correlated calculations. Furthermore, it also indicates that in GGn methods the geminal expansion should converge rapidly, provided that the strong orthogonality functional is used [27]. We postulate that for the development for future R12 methods it may be possible to capitalise on the redundancy between the geminal and conventional excitations and to construct smaller sets of virtual orbitals, perhaps in the fashion of Almlöfs' dual basis sets [52], as proposed in Refs.…”

“…The c kl ij are the R12 amplitudes that are to be determined.Q 12 ensures that the new excitations are strongly orthogonal on the Hartree-Fock reference state [27].…”