Excited-state response theory within the context of the coupled-cluster formalism

Abstract

The propagation of general electronic quantum states provides information of the interaction of molecular systems with external driving fields. These can also offer understandings regarding non-adiabatic quantum phenomena. Well established methods focus mainly on propagating a quantum system that is initially described exclusively by the ground state wavefunction. In this work, we expand a previously developed formalism within coupled cluster theory, called second response theory, so it propagates quantum systems that are initially described by a general linear combination of different states, which can include the ground state, and show how with a special set of time-dependent cluster operators such propagations are performed. Our theory shows strong consistency with numerically exact results for the determination of quantum mechanical observables, probabilities, and coherences. We discuss unperturbed non-stationary states within second response theory and their ability to predict matrix elements that agree with those found in linear and quadratic response theories. This work also discusses an approximate regularized methodology to treat systems with potential instabilities in their ground-state cluster amplitudes, and compare such approximations with respect to reference results from standard unitary theory.

Second response theory: a theoretical formalism for the propagation of quantum superpositions

“…If known, Ψ(t) would determine all the measurable quantities of the system. Previous work [68] on SR CC theory showed how to propagate general arbitrary states by computing separately the CC representations of ⟨Ψ E | ÂH (t)|Ψ 0 ⟩, and ⟨Ψ E | ÂH (t)|Ψ E ⟩. In this current work we show how, through an integrated approach, an arbitrary initial state that includes a contribution from the ground state is propagated in SR CC theory.…”

“…However, the evolved 'right' excitation TD vector is not connected in a straightforward fashion to the EOM-CC eigen-vectors, nor the conjugate TD excitation vector. This motivates our goal, which is to describe the auxiliary observable shown in equation ( 8) through CC quantities, then use of the differential step followed by the limiting procedure presented in section 2.1 to compute ⟨Ψ(t)| Â|Ψ(t)⟩ (similarly as in [68]), without calculating the wavefunction Ψ(t) of course. It allows for a connection between EOM-CC eigen-vectors and general TD propagations, where one sets the values of the initial-state coefficients S, and {C N }.…”

“…where the subscript 'm' indicates the vectors are modified objects as they account for the new initial conditions (which depend on g L and g R ), xm = ∑ µ x m,µ τµ , and ⟨•⟩ 0 denotes average with respect to the reference (|0⟩): ⟨•⟩ 0 = ⟨0| • |0⟩. The terms x m,0 and λ m,0 are time-independent (this is a consequence of our TD variational method [68]). The phase follows the relation: ∂ t ϕ(t) = ⟨ λm (t)e −xm(t) Ĥ(t)e +xm (t)⟩ 0 .…”

“…The phase follows the relation: ∂ t ϕ(t) = ⟨ λm (t)e −xm(t) Ĥ(t)e +xm (t)⟩ 0 . This number, ϕ(t), does not influence the calculation of observable expectation values, so it will not be considered in detail in this work (its properties are presented in [68]). These modified operators determine the TD behavior of the following kept and its left conjugate:…”

“…The inner product of these two CC wavefunctions is ⟨Υ(t)|Φ(t)⟩ = ⟨ λm (t)⟩ 0 . As shown in [68], there are four (reduced to three in this work) TD excitation vectors that are needed to compute TD observables. These are: xr (t) = ∂ R xm (t), λl (t) = ∂ L λm (t), λr (t) = ∂ R λm (t), and λl,r (t) = ∂ 2 LR λm (t).…”