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Highlights in theoretical chemistry

Aspects of size extensivity in unitary group adapted multireference coupled cluster theories: the role of cumulant decomposition of spin-free reduced density matrices

Rahul Maitra, Debalina Sinha, Sangita Sen, Debashis Mukherjee

Published Jul 11, 2014• 0 citations
Publisher
DOI
10.1007/978-3-662-48148-6_3

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Exploration of Various Aspects of UGA-SUMRCC: Size Extensivity, Possible Use of Sufficiency Conditions, and an Extension for Direct Determination of Energy Differences

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The Unitary Group Adapted State Universal Multireference Coupled Cluster (UGA-SUMRCC) theory, recently developed by us (J. Chem. Phys.2012, 137, 074104), contains exactly the right number of linearly independent cluster operators. This avoids any redundancy of the excitation manifold in a way exactly paralleling the traditional spin-orbital based SUMRCC. The choice of the linearly independent cluster operators inducing the same change of orbital occupancy becomes increasingly cumbersome if we go over to the cases of active CSFs with more than two active quasiparticles. In the present development, we explore several aspects of the UGA-SUMRCC theory: (a) The first is a variant where we have deliberately incorporated redundancy of the cluster amplitudes to simplify the working equations and have shown that it can serve as a very good approximation to the parent UGA-SUMRCC theory for states with more than two valence occupancies. This in turn suggests that it could be a useful avenue to pursue for arbitrary mh-np situation since the working equations assume simpler algebraic structure in such cases. (b) The analyses of the aspects of size extensivity are known to involve greater complexity if they involve various reduced density matrices (RDMs), since the RDMs are not size-extensive quantities. We have presented the proof for UGA-SUMRCC starting from equations containing h-p RDMs via a decomposition involving products of size-extensive cumulants and argue that it has relevance for general cases beyond the h-p model spaces. (c) A useful extension of UGA-SUMRCC lies in formulating the theory for direct calculations of energy differences of spectroscopic interest such as excitation energies, ionization potentials, and electron affinities relative to a closed shell ground state, thus providing attractive alternatives to other allied methods such as SAC-CI, CC-LRT, EOM-CC, STEOM-CC, or ADC. This extension, called UGA-based Quasi-Fock MRCC by us, also leads to exact cancellation of common correlation terms between the initial and final states. Taking a cue from the hierarchical development in Fock-space theories but keeping in mind the advantages of a state-universal (equivalently called a valence specific) theory, our formulation proposes a spin-adapted, accurate, and compact scheme for studying such energy differences. Our results demonstrate superior performance of the method as compared to EOM-CC.

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We propose a spin-free approach to the cumulant decomposition of reduced density matrices of singlet and spin-rotation or SU(2) invariant ensemble of nonsinglet states as in [W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 110, 2800 (1999); 116, 4787 (2002)]. We provide a simple recursive procedure to obtain expressions which relate different spin components of spin-orbital reduced density matrices and cumulants of such states to the spin-free counterparts. These results are used to set up a spin-summation procedure to arrive at a definition of spin-free cumulants of any order. Alternatively, an analytic formula for the spin-free form resulting from a spin summation involving product of two spin-orbital cumulants is derived and its utility in spin-free cumulant decomposition of reduced density matrices is demonstrated. This leads to suitable definitions of spin-free analog of multireference normal ordering and the associated Wicks theorem. The results of this formulation are expected to be useful in investigations of spin-free multireference internally contracted coupled-cluster methods where cumulant approximations to the active reduced density matrices are employed.
The Unitary Group Adapted State Universal Multireference Coupled Cluster (UGA-SUMRCC) theory, recently developed by us (J. Chem. Phys.2012, 137, 074104), contains exactly the right number of linearly independent cluster operators. This avoids any redundancy of the excitation manifold in a way exactly paralleling the traditional spin-orbital based SUMRCC. The choice of the linearly independent cluster operators inducing the same change of orbital occupancy becomes increasingly cumbersome if we go over to the cases of active CSFs with more than two active quasiparticles. In the present development, we explore several aspects of the UGA-SUMRCC theory: (a) The first is a variant where we have deliberately incorporated redundancy of the cluster amplitudes to simplify the working equations and have shown that it can serve as a very good approximation to the parent UGA-SUMRCC theory for states with more than two valence occupancies. This in turn suggests that it could be a useful avenue to pursue for arbitrary mh-np situation since the working equations assume simpler algebraic structure in such cases. (b) The analyses of the aspects of size extensivity are known to involve greater complexity if they involve various reduced density matrices (RDMs), since the RDMs are not size-extensive quantities. We have presented the proof for UGA-SUMRCC starting from equations containing h-p RDMs via a decomposition involving products of size-extensive cumulants and argue that it has relevance for general cases beyond the h-p model spaces. (c) A useful extension of UGA-SUMRCC lies in formulating the theory for direct calculations of energy differences of spectroscopic interest such as excitation energies, ionization potentials, and electron affinities relative to a closed shell ground state, thus providing attractive alternatives to other allied methods such as SAC-CI, CC-LRT, EOM-CC, STEOM-CC, or ADC. This extension, called UGA-based Quasi-Fock MRCC by us, also leads to exact cancellation of common correlation terms between the initial and final states. Taking a cue from the hierarchical development in Fock-space theories but keeping in mind the advantages of a state-universal (equivalently called a valence specific) theory, our formulation proposes a spin-adapted, accurate, and compact scheme for studying such energy differences. Our results demonstrate superior performance of the method as compared to EOM-CC.