Aspects of Size-Consistency of Orbitally Noninvariant Size-Extensive Multireference Perturbation Theories: A Case Study Using UGA-SSMRPT2 as a Prototype

Abstract

Profiling a potential energy surface (PES), all the way to dissociate a molecular state into particular fragments and to display real or avoided crossings, requires a multireference description and the maintenance of size-consistency. The many body methods, which suit this purpose, should thus be size-extensive. Size-extensive theories, which are invariant with respect to transformation among active orbitals are, in principle, size-consistent. Relatively cheaper size-extensive theories, which do not possess this invariance, can still be size-consistent if the active orbitals are localized on the asymptotic fragments. Such methods, if perturbative in nature, require the use of an unperturbed Hamiltonian, which has orbital invariance with respect to the transformation within active, core, and virtual orbitals. The principal focus of this paper is to numerically realize size-consistency with localized active orbitals using our recently developed orbitally noninvariant Unitary Group Adapted State Specific Multireference second order Perturbation Theory (UGA-SSMRPT2) as a prototype method. Our findings expose certain generic potential pitfalls of size-extensive but orbitally noninvariant MRPT theories, which are mostly related to the inability of reaching proper localized active orbitals in the fragments due to the artifacts of the orbital generation procedure. Despite the invariance of the zeroth order CAS function, lack of invariance of the MRPT itself then leads to size-inconsistency. In particular, reaching symmetry broken fragment active orbitals is an issue of concern where suitable state-averaging might ameliorate the problem, but then one has to abandon full orbital optimization. Additionally, there can be situations where the orbitals of the fragment reached as an asymptote of the supermolecule are not the same as those obtained from the optimization of the fragments individually and will require additional transformation. Moreover, for a certain PES, one may either abandon the use of optimized orbitals for that state to preserve proper symmetry and degeneracy in the fragment orbitals or be satisfied with the use of optimized orbitals, which generate broken symmetric orbitals in the fragmentation limit. All these pathologies are illustrated using the PES of various electronic states of multiply bonded systems like N2, C2H2, HCN, C2, and O2. Subject to such proviso, the UGA-SSMRPT2 turns out to be an excellent theory for studying the PES leading to fragmentation of strongly correlated systems satisfying the requirements of size-consistency with localized active orbitals. An unexpected spin-off of our studies is the realization that the size-inextensive MRMP2, which bears a close structural similarity with our theory, might under certain situations display size-consistency. We analyze this feature concretely in our paper. Our studies may serve as a benchmark for monitoring numerically the size-consistency of any state specific multireference theory which is size-extensive but not invariant with respect to trans...

Multireference perturbation theory with improved virtual orbitals for radicals: More degeneracies, more problems

“…Due to the explicit connective nature of the working equation(s), the parent SSMRPT method [12] furnishes size-consistent description of any system composed of noninteracting fragments, provided that the reference space adopts a direct product form. [15,31] Note that SSMRPT does not conserve orbital invariance of the underlying CAS wavefunction. If the CAS reference function fails to satisfy the separability property, the reference energy itself is not sizeconsistent which is not corrected by the addition of electron correlation (except full CI).…”

“…Designing methods which can accurately treat MR‐systems at an affordable computational price present a challenge to the practitioners of electronic structure theory . Various workers in the realm of electronic structure theory have suggested different approaches to address these problems . Reviews of various MRPT methods can be found in Ref.…”

“…We will examine this issue in a future work. Due to the explicit connective nature of the working equation(s), the parent SSMRPT method furnishes size‐consistent description of any system composed of noninteracting fragments, provided that the reference space adopts a direct product form . Note that SSMRPT does not conserve orbital invariance of the underlying CAS wavefunction.…”

“…There is yet another prominent problem associated with the size‐consistency of a noninvariant MRPT, viz., the effect of the difference in the nature of symmetry breaking of the active orbitals in the fragments and the super‐molecule. Mukherjee and coworkers argued that the asymptotes of some energy surfaces generated via MRMP2 method displays exact size‐consistency, although MRMP2 method can be shown to be size‐inconsistent in general. It is worth mentioning that the invariance of reference function as well as definition of the unperturbed Hamiltonian are two important inter‐related issues of ensuring size‐consistency property.…”

Ring coupled-cluster doubles correction to geminal wavefunctions

“…Extensivity is always an issue to consider, especially for a CC theory. While the expressions reported above appear connected at first glance, spin cumulants represent a non trivial problem [44,45] and for this reason the analysis is put off to further study.…”