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Physical Review A

Time-reversal symmetry in general coupled cluster theory

Timo Fleig

Published Jun 4, 2008• 10 citations
Publisher

Abstract

In this paper, the symmetry property of time reversal, which is a consequence of the fundamental quantum-mechanical microreversibility principle, is analyzed in the context of relativistic open-shell and multireference coupled cluster theory. The fact is discussed that in projective theories, in contrast to variational theories such as configuration interaction models, the implementation of time-reversal symmetry is far less straightforward and even depends on the projective model theory in question. An operator basis of time reversal even $n$-body operators is introduced which allows for the formulation of a fully time-reversal-adapted and practical general coupled cluster theory, i.e., covering open-shell models and various types of multireference approaches. The resulting model theory is noncommuting but truncating, albeit at a higher order of commutator nesting.

DOI
10.1103/physreva.77.062503

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In this paper, the symmetry property of time reversal, which is a consequence of the fundamental quantum-mechanical microreversibility principle, is analyzed in the context of relativistic open-shell and multireference coupled cluster theory. The fact is discussed that in projective theories, in contrast to variational theories such as configuration interaction models, the implementation of time-reversal symmetry is far less straightforward and even depends on the projective model theory in question. An operator basis of time reversal even nnn-body operators is introduced which allows for the formulation of a fully time-reversal-adapted and practical general coupled cluster theory, i.e., covering open-shell models and various types of multireference approaches. The resulting model theory is noncommuting but truncating, albeit at a higher order of commutator nesting.
In this work we present a coupled-cluster theory for the propagation of multireference electronic systems initiating at general quantum mechanical states. Our formalism is based on the infinitesimal analysis of modified cluster operators, from which we extract a set of additional operators and their equations of motion. These cluster operators then can be used to compute electronic properties of interest in a size extensive manner. The equations derived herein are also studied for the free propagation of linear superpositions. In this regime, we derive asymmetric matrix elements for observables, and a resymmetrization factor so they become consistent with their quantum mechanical counterparts. Our formalism is applied to a four-electron/four-level model with a Hubbard-like Hamiltonian, where we show that it reproduces the numerically exact results for observables, populations, and coherences.
We introduce a simple coupled cluster approach for multireference and open-shell systems. In the restricted active space coupled cluster framework, the cluster operator only includes excitations from doubly occupied to virtual orbitals, and this ensures the individual operators commute among each other. The ground and excited states are obtained by diagonalisation of the coupled cluster similarity transformed Hamiltonian in a restricted active space. We demonstrate the framework for the RAS-CCSD model, where single and double excitations are included in the cluster operator, and likewise for a complete active space. We consider systems with multireference character and open-shell systems in a spin-adapted manner. The approach captures significantly more dynamical correlation than without the coupled cluster similarity transformation.
In nonrelativistic quantum mechanics and in relativistic quantum field theory, the time coordinate ttt is a parameter and thus the time-reversal operator T\mathcal{T}T does not actually reverse the sign of ttt. In contrast, in the five-dimensional approach to relativistic quantum mechanics introduced by Feynman, time ttt is a quantum-mechanical operator. In this paper it is shown how one can use this five-dimensional approach to extend T\mathcal{T}T and PT\mathcal{P}\mathcal{T}PT symmetry from nonrelativistic to relativistic quantum mechanics and implement time-reversal as an operation that effects $\mathcal{T}\text{ }t\mathcal{T}=\ensuremath{-}t$ just as P\mathcal{P}P effects $\mathcal{P}x\mathcal{P}=\ensuremath{-}x$, with PT\mathcal{P}\mathcal{T}PT thus effecting $\mathcal{P}\mathcal{T}\text{ }{x}_{\ensuremath{\mu}}\mathcal{P}\mathcal{T}=\ensuremath{-}{x}_{\ensuremath{\mu}}$. Some illustrative relativistic quantum-mechanical models are constructed whose associated Hamiltonians are non-Hermitian but PT\mathcal{P}\mathcal{T}PT symmetric, and it is shown that for each such Hamiltonian the energy eigenvalues are all real.
In the nonrelativistic Schrödinger equation, the total spin S and spin projection M are good quantum numbers. In contrast, spin symmetry is lost in the presence of spin-dependent interactions, such as spin-orbit couplings in relativistic Hamiltonians. Therefore, the relativistic density matrix renormalization group algorithm (R-DMRG) only employing particle number symmetry is much more expensive than nonrelativistic DMRG. In addition, artificial breaking of Kramers degeneracy can happen in the treatment of systems with an odd number of electrons. To overcome these issues, we propose time-reversal symmetry adaptation for R-DMRG. Since the time-reversal operator is antiunitary, this cannot be simply achieved in the usual way. We introduce a time-reversal symmetry-adapted renormalized basis and present strategies to maintain the structure of basis functions during the sweep optimization. With time-reversal symmetry adaptation, only half of the renormalized operators are needed, and the computational costs of Hamiltonian-wavefunction multiplication and renormalization are reduced by half. The present construction of the time-reversal symmetry-adapted basis also directly applies to other tensor network states without loops.
Time is used in non-relativistic quantum theory as a classical concept. On the one hand, time is the parameter entering Schrödinger's equation and measured by an external laboratory clock. But it could be considered that time also plays the role of an observable when a given event occurs (e.g., when a particle crosses through a potential barrier), and one would expect that like every observable, it should be represented in the theory by an operator whose properties are predictors of the outcome of time measurements made on physical systems. Several time operators have been proposed in different publications, but still, there is not a widely accepted consensus. That subject, among others concerned with time in quantum mechanics, is analyzed in the present chapter, where we focus on the analysis of time-reversal symmetry in non-relativistic quantum systems. The search for symmetries in Physics helps to study the structure of theories. Briefly speaking, a symmetry is a transformation that does not change the structure of the theory. We start with the analysis of Schrödinger’s Equation verifying that for obtaining a time-symmetric description of the dynamics of a particle, the operation of time-reversal must include conjugation represented by the action of an antiunitary operator acting over a wave function. In particular, we analyze quantum systems whose evolution is determined by a set of external parameters varying cyclically, discussing how even though these parameters return to their initial values, the state of the system acquires a geometric phase, which depends on the path followed by the external parameters in the parameter space and which in general can be associated with the lack of the time-reversal symmetry of the system.
Reduced density matrix functional theory (RDMFT) and coupled cluster theory restricted to paired double excitations (pCCD) are emerging as efficient methodologies for accounting for the so-called non-dynamic electronic correlation effects. Up to now, molecular calculations have been performed with real-valued orbitals. However, before extending the applicability of these methodologies to extended systems, where Bloch states are employed, the subtleties of working with complex-valued orbitals and the consequences of imposing time-reversal symmetry must be carefully addressed. In this work, we describe the theoretical and practical implications of adopting time-reversal symmetry in RDMFT and pCCD when allowing for complex-valued orbital coefficients. The theoretical considerations primarily affect the optimization algorithms, while the practical implications raise fundamental questions about the stability of solutions. In particular, we find that complex solutions lower the energy when non-dynamic electronic correlation effects are pronounced. We present numerical examples to illustrate and discuss these instabilities and possible problems introduced by N-representability violations.