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The Journal of Chemical Physics

Coupled cluster response functions revisited

Thomas Bondo Pedersen, Henrik Koch A

Published May 15, 1997• 148 citations
Publisher

Abstract

We introduce an inherently real coupled cluster time-dependent expectation value of a Hermitian operator. Based on the expansion of this expectation value in orders of the generally time-dependent perturbation, we subsequently identify the coupled cluster time-independent expectation value, the linear response function, and the quadratic response function. The response functions and their residues behave physically correctly. Spectroscopic observables are identified as residues, whereas the identification of individual transition matrix elements is prohibited. Thus the unphysical behavior of previously published coupled cluster response functions may be viewed not as a consequence of the projection, but rather that identifications are made on the basis of an unphysical expectation value or quasienergy.

DOI
10.1063/1.473814

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We introduce an inherently real coupled cluster time-dependent expectation value of a Hermitian operator. Based on the expansion of this expectation value in orders of the generally time-dependent perturbation, we subsequently identify the coupled cluster time-independent expectation value, the linear response function, and the quadratic response function. The response functions and their residues behave physically correctly. Spectroscopic observables are identified as residues, whereas the identification of individual transition matrix elements is prohibited. Thus the unphysical behavior of previously published coupled cluster response functions may be viewed not as a consequence of the projection, but rather that identifications are made on the basis of an unphysical expectation value or quasienergy.
We introduce a time-dependent coupled cluster based Lagrangian that includes orbital rotation. This Lagrangian is shown to give gauge invariant response properties for one-electron operators in the limit of a complete one-electron basis. The pole structure of the linear response function is compatible with that of the exact response function and the notorious problem of unphysical second-order poles in the Brueckner coupled cluster response theory is not present in this model. The total energy of the model is identical to the coupled cluster model using optimized orbitals recently revived by Sherrill et al. [J. Chem. Phys. 109, 4171 (1998)]. The model provides a straightforward approach for calculating magnetic response properties in a gauge invariant manner using coupled cluster type wave functions.
Equations are derived for the time evolution of time-dependent vibrational coupled cluster (TDVCC) wave functions covering both the TDVCC ket state and the associated so-called Λ bra state. The equations are implemented in the special case of both the Hamiltonian and the cluster operator containing at most two-mode coupling terms. The nontrivial behavior of the evolution of norm, energy, and expectation values due to the nonunitary time-evolution of the nonvariational TDVCC theory is analyzed theoretically and confirmed in numerical experiments that also include time-dependent Hamiltonians. In the spirit of time-independent size-consistency analysis, the separability of both the coupled cluster and Λ states for noninteracting systems is studied. While the coupled cluster state clearly has the correct behavior, the behavior of the Λ state is more intricate, and the consequence for different properties is shown theoretically and numerically. Overall, the numerical experiments show that TDVCC in incomplete expansions gives higher accuracy than a standard linear variational wave function parameterization with the same number of independent parameters, while equivalent results are obtained for complete expansions. The efficiency of the methodology is illustrated in computations on polycyclic aromatic hydrocarbons with up to 156 modes.
We compute nuclear response functions by solving the time-dependent A-body Schrödinger equation, recording the time-dependent transition moment and extracting spectral information via Fourier transforms. The solution of the time-dependent many-body problem accounts for correlations on top of the mean field by taking advantage of a time-dependent formulation of coupled-cluster theory. As a validation, we focus on electric dipole transitions in 4^44He and 16^{16}16O and compare moments of the response function distribution to the results of an equivalent static framework, finding negligible discrepancies. We investigate how proton and neutron densities evolve in time, and we see the traditional picture of soft and giant dipole resonances as collective oscillations of protons and neutrons emerging from our calculations in 16^{16}16O and 24^{24}24O. This method also allows us to investigate the behavior of the nucleus in the presence of a strong electric field. In that regime, the behavior of the system becomes chaotic. Qualitatively, the spectral information obtained in this limit is in line with previous time-dependent mean-field results.
We present the diagrammatic theory of the irreducible self-energy and Bethe-Salpeter kernel that naturally arises within the Green’s function formalism for a general N -body non-Hermitian interaction. In this work, we focus specifically on the coupled-cluster (CC) self-energy generated by the similarity transformation of the electronic structure Hamiltonian. The main result of this work is a Green’s function formalism for non-Hermitian many-body quantum systems. We develop the biorthogonal quantum theory to construct dynamical correlation functions where the time dependence of operators is governed by a non-Hermitian Hamiltonian. We extend the Gell-Mann and Low theorem to include non-Hermitian interactions and to generate perturbative expansions of many-body Green’s functions. We introduce the single-particle coupled-cluster Green’s function and derive the perturbative diagrammatic expansion for the non-Hermitian coupled-cluster self-energy in terms of the “noninteracting” reference Green’s function, Σ ̃ [ G 0 ] . From the exact equation of motion of the single-particle coupled-cluster Green’s function, we derive the self-consistent renormalized coupled-cluster self-energy, Σ ̃ [ G ̃ ] , and demonstrate its relationship to the perturbative expansion of the self-energy, Σ ̃ [ G 0 ] . Subsequently, we show that the usual electronic self-energy can be recovered from the coupled-cluster self-energy by neglecting the effects of the similarity transformation. We show how the coupled-cluster ground-state energy can be obtained from the coupled-cluster self-energy and provide an overview of the relationship between approximations for the coupled-cluster self-energy, ionization potential/electron affinity equation-of-motion coupled-cluster theory, and the G 0 W 0 approximation. As a result, we introduce the CC- G 0 W 0 self-energy by leveraging the connections between Green’s function and coupled-cluster theory. Finally, we derive the diagrammatic expansion of the coupled-cluster Bethe-Salpeter kernel.
An approximation scheme for dynamic response functions within the coupled cluster or exp(T)\mathrm{exp}(T)exp(T) method of Coester and K\"ummel is described in this paper. Particular attention is given to the relationship between the "vacuum amplitude" $〈0|U(t,\ensuremath{-}\ensuremath{\infty})|0〉$ for the time development of an exact eigenstate of a many-electron Hamiltonian in the presence of an adiabatically switched-on perturbaction, and the amplitude $〈{\ensuremath{\Phi}}_{0}|U(t,\ensuremath{-}\ensuremath{\infty})|0〉$ for the time evolution into an independent-particle reference state $|{\ensuremath{\Phi}}_{0}〉$, which is not orthogonal to ∣0〉|0〉∣0〉. The latter amplitude plays a prominent role in the time-dependent coupled-cluster calculation of response functions.