Unitary group adapted state specific multireference perturbation theory: Formulation and pilot applications

Abstract

We present here a comprehensive account of the formulation and pilot applications of the second-order perturbative analogue of the recently proposed unitary group adapted state-specific multireference coupled cluster theory (UGA-SSMRCC), which we call as the UGA-SSMRPT2. We also discuss the essential similarities and differences between the UGA-SSMRPT2 and the allied SA-SSMRPT2. Our theory, like its parent UGA-SSMRCC formalism, is size-extensive. However, because of the noninvariance of the theory with respect to the transformation among the active orbitals, it requires the use of localized orbitals to ensure size-consistency. We have demonstrated the performance of the formalism with a set of pilot applications, exploring (a) the accuracy of the potential energy surface (PES) of a set of small prototypical difficult molecules in their various low-lying states, using natural, pseudocanonical and localized orbitals and compared the respective nonparallelity errors (NPE) and the mean average deviations (MAD) vis-a-vis the full CI results with the same basis; (b) the efficacy of localized active orbitals to ensure and demonstrate manifest size-consistency with respect to fragmentation. We found that natural orbitals lead to the best overall PES, as evidenced by the NPE and MAD values. The MRMP2 results for individual states and of the MCQDPT2 for multiple states displaying avoided curve crossings are uniformly poorer as compared with the UGA-SSMRPT2 results. The striking aspect of the size-consistency check is the complete insensitivity of the sum of fragment energies with given fragment spin-multiplicities, which are obtained as the asymptotic limit of super-molecules with different coupled spins.

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

“…For a benchmark study of UGA-SSMRPT2 vis a vis FCI and other PTs we would like to refer the readers to our earlier publication. 68 Our findings indicate that PES computed using UGA-SSMRPT2 show more parallelity with those obtained from FCI and are also less erratic in behavior across the PES when compared with MRMP2 or the MCQDPT2 numbers. In addition, we may mention that since localized orbitals perform somewhat poorly at the equilibrium but very well at the asymptote, the quality of PES in terms of NPE and MAD is poorer in case of localized orbitals vis a vis natural orbitals.…”

“…68 It is well known that, if a theory is both size-extensive and invariant with respect to transformation within the class of core, active and virtual orbitals, it will be size-consistent. Multi-reference theories satisfying these two constraints are hard to come by, although the internally contracted multi-reference coupled cluster(icMRCC) theories 45,47,49 and NEVPT2 60,61 in their most rigorous form belong to this category.…”

“…55 The UGA-SSMRCC uses a normal-ordered multi-exponential Ansatz for the component of the wave operator, Ω µ , We now recapitulate briefly only the salient features of the development, referring to our very recent UGA-SSMRPT2 paper for the details. 68 Using any partitioning of the Hamiltonian, H = H 0 + V and considering the amplitudes of the T µ s to be of first order, the first order perturbative approximation to the UGA-SSMRCC is given by:…”

A Jeziorski-Monkhorst fully uncontracted multi-reference perturbative treatment. I. Principles, second-order versions, and tests on ground state potential energy curves

“…If one returns to methods using single-determinant perturbers, the origin of their size-inconsistency problem has been identified as due to the unbalance between the multi-determinant character of the zero-order wave function and the single determinant character of the perturbers. 34 It is in principle possible to find size consistent formulations but they require rather complex formulations, [35][36][37][38] and face some risk of numerical instabilities since they involve divisions by possibly small coefficients, the amplitudes of which may be small. Finally, one should mention a very recent approach based on the rewriting of the multi-reference linear coupled cluster equations in a stochastic framework of Full-CI Quantum Monte Carlo which also uses single Slater determinants as perturbers 39 .…”

A simple approach to the state-specific MR-CC using the intermediate Hamiltonian formalism

“…Contracted perturbative expansions, which perturb the multideterminant zero-order wave function under the effect of linear combinations of outer-space determinants have also been proposed. One may quote the CASPT2 method 19,20 , which uses a monoelectronic zero-order Hamiltonian, faces intruder state problems and is not size consistent, the NEVPT2 method [21][22][23] which uses a bi-electronic zeroorder Hamiltonian (the Dyall's one 24 ) and is size consistent and intruder-state free, and the method from Werner 25 , as well as the perturbation derived by Mukherjee et al [26][27][28] from their MRCC formalism.…”