^{1,2,a)}, Jeppe Olsen

^{1}and Poul Jørgensen

^{1}

### Abstract

The time-dependent Schrödinger equation for a time-periodic perturbation is recasted into a Hermitian eigenvalue equation, where the quasi-energy is an eigenvalue and the time-periodic regular wave function an eigenstate. From this Hermitian eigenvalue equation, a rigorous and transparent formulation of response function theory is developed where (i) molecular properties are defined as derivatives of the quasi-energy with respect to perturbation strengths, (ii) the quasi-energy can be determined from the time-periodic regular wave function using a variational principle or via projection, and (iii) the parametrization of the unperturbed state can differ from the parametrization of the time evolution of this state. This development brings the definition of molecular properties and their determination on par for static and time-periodic perturbations and removes inaccuracies and inconsistencies of previous response function theory formulations. The development where the parametrization of the unperturbed state and its time evolution may differ also extends the range of the wave function models for which response functions can be determined. The simplicity and universality of the presented formulation is illustrated by applying it to the configuration interaction (CI) and the coupled cluster (CC) wave function models and by introducing a new model—the coupled cluster configuration interaction (CC-CI) model—where a coupled cluster exponential parametrization is used for the unperturbed state and a linear parametrization for its time evolution. For static perturbations, the CC-CI response functions are shown to be the analytical analogues of the static molecular properties obtained from finite field equation-of-motion coupled cluster (EOMCC) energy calculations. The structural similarities and differences between the CI, CC, and CC-CI response functions are also discussed with emphasis on linear versus non-linear parametrizations and the size-extensivity of the obtained molecular properties.

F.P. and P.J. acknowledge support from The European Research Council under the European Union’s (EU) Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 291371 and P.J. and J.O. acknowledge support from The Danish Council for Independent Research—Natural Sciences. F.P. also acknowledges support form the Polish National Science Centre (Project No. 3714/B/H03/2011/40).

I. INTRODUCTION II. THE TIME-DEPENDENT SCHRÖDINGER EQUATION FOR A PERIODIC PERTURBATION AS A HERMITIAN EIGENVALUE EQUATION A. The steady-state wave function and its eigenvalue equation B. The regular wave function and its eigenvalue equation C. The Hermiticity of the time-dependent Hamiltonian D. The Hermitian eigenvalue equation in the limit of a time-independent perturbation E. The variational principle for the quasi-energy F. Molecular response properties as derivatives of the eigenvalue of the Hermitian eigenvalue equation G. Size-extensivity of the quasi-energy and molecular properties III. RESPONSE FUNCTIONS FROM THE HERMITIAN EIGENVALUE EQUATION SOLVED USING THE VARIATIONAL PRINCIPLE A. Transformation of the quasi-energy from the time domain to the frequency domain 1. Parametrization of the quasi-energy in the time domain 2. Perturbation expansion of time-dependent wave function coefficients in the frequency domain 3. The quasi-energy in frequency domain B. Molecular response properties from an order expansion of the quasi-energy in the frequency domain 1. Order expansion of the quasi-energy in the perturbation strengths 2. Stationary conditions for the quasi-energy and its perturbation components C. Response functions through fourth order 1. Energy and first-order molecular properties 2. Second-order molecular properties 3. Third-order molecular properties 4. Fourth-order molecular properties D. Molecular properties from the quasi-energy compared to the standard definition IV. RESPONSE FUNCTIONS FROM THE HERMITIAN EIGENVALUE EQUATION SOLVED VIA PROJECTION A. Parametrization of the regular wave function 1. Parametrization of the unperturbed state 2. Parametrization of the time evolution B. Projected Hermitian eigenvalue equation C. Stationary conditions for the quasi-energy Lagrangian D. Complex quasi-energy Lagrangian and its stationary conditions 1. Complex quasi-energy Lagrangian 2. Stationary conditions for the complex quasi-energy Lagrangian E. Molecular properties from the quasi-energy Lagrangian 1. Molecular properties from an order expansion of the quasi-energy Lagrangian 2. Stationary conditions for the perturbation components of the quasi-energy Lagrangian V. QUASI-ENERGY LAGRANGIANS FOR THE CI, CC-CI AND CC WAVE FUNCTION-MODELS A. Wave function models in terms of different parametrizations of the unperturbed state and the time evolution B. Simplifications in the complex quasi-energy Lagrangian in the frequency domain VI. MOLECULAR PROPERTIES FOR THE CI MODEL A. Parametrization of the CI model B. Energy and first-order molecular properties C. Second-order molecular properties D. Third-order molecular properties E. Fourth-order molecular properties F. The CI Jacobian as a part of the CI eigenvalue equation 1. The CI eigenvalue equation 2. The CI Jacobian eigenvalue equation G. Comparison of response functions obtained from the quasi-energy and from the quasi-energy Lagrangian VII. MOLECULAR PROPERTIES FOR THE CC-CI MODEL A. Parametrization of the CC-CI model B. Energy and first-order molecular properties C. Second-order molecular properties D. Third-order molecular properties E. Fourth-order molecular properties F. The CC Jacobian as a part of the CC-CI eigenvalue equation 1. The CC-CI eigenvalue equation 2. The CC Jacobian eigenvalue equation VIII. MOLECULAR PROPERTIES FOR THE CC MODEL A. Parametrization of the CC model B. Energy and first-order molecular properties C. Second-order molecular properties D. Third-order molecular properties E. Fourth-order molecular properties F. The CC Jacobian IX. MOLECULAR PROPERTIES FOR THE EOMCC MODEL X. COMPARISON OF THE CI, CC-CI, CC, AND EOMCC MOLECULAR RESPONSE PROPERTIES A. Static molecular properties from finite field energy calculations in the CI, CC-CI and CC models 1. Standard finite field energy calculations of static molecular properties 2. Finite field energy calculations for the CC-CI model 3. Static CC-CI molecular properties from finite field EOMCC energy calculations B. Size-extensivity of molecular properties in the CI, CC-CI, and CC models 1. Separability of the reference and time-dependent states 2. Separability of the Jacobian 3. Size-extensivity of molecular properties C. Linear versus exponential parametrization in the CI, CC-CI and CC models D. Comparison of CC-CI and EOMCC molecular response properties XI. SUMMARY, CONCLUSION, AND OUTLOOK

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