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Excitation Energies and Transition Moments from the PCM Linear Response Functions

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Book cover Molecular Response Functions for the Polarizable Continuum Model

Part of the book series: SpringerBriefs in Molecular Science ((BRIEFSMAGNET))

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Abstract

This chapter considers the properties of the molecular solute in electronic excited states determined from the linear response functions described in the previous Chap. 3. Transition energies and transition moments are determined from a generalized eigenvalue equations, and the first-order properties in electronic excited states are expressed as analytical gradients of the corresponding transition energies with respect to suitable perturbations.

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Notes

  1. 1.

    Here we consider the linear response function for the electronic ground state of the molecular solute.

  2. 2.

    The amplitudes of de-excitation operator \({\fancyscript{M}}=\sum _K\mu _K\tau ^{\dagger } _K\) are determined by the perturbation independent equation

    $$\begin{aligned}&<HF|(1+\varLambda )[[e^{-T}H_N e^{T},\tau _K], R_f]|HF >\\&+ < HF|(1+\varLambda )|e^{-T}[Q_N^{|\omega |},\tau _K]e^T|HF > \cdot < HF|(1+\varLambda )|e^{-T}[V_N, R_f]e^T|HF >\\&+<HF|M[e^{-T}H_N e^{T},\tau _K]|HF> \\&+ <HF|M e^{-T}Q_N^{|\omega |}e^T|HF>\cdot <HF|(1+\varLambda )|e^{-T}[V_N,\tau _K]e^T|HF>+\omega \, \mu _K = 0 \end{aligned}$$
  3. 3.

    The coefficients of the \({\fancyscript{Z}}_f\) and \({\fancyscript{T}}_f\) operators are determined from the following set of equations:

    $$\begin{aligned} 0 =&< HF|L_f\left[ [e^{-T}H_N e^T,\tau _p], R_f\right] |HF>\\&+< HF|(1+\varLambda ) [e^{-T}\mathbf{Q}_Ne^T,\tau _p]|HF >\cdot < HF|L_f[e^{-T}\mathbf{V}_N e^T, R_f ]|HF >\\&+ < HF|L_f [e^{-T}\mathbf{Q}_N^{|\omega |} e^T,\tau _p]|HF >\cdot < HF|(1+\varLambda )e^{-T}[[\mathbf{V}_N,\tau _p], R_f]e^T|HF >\\&+ < HF|{\fancyscript{Z}}_f [e^{-T}{H}_{N}e^T,\tau _p]|HF >\\&+ < HF|(1+\varLambda )e^{-T}[[{H}_{N},\tau _p],{\fancyscript{T}}_f]e^T|HF > \end{aligned}$$

    and

    $$\begin{aligned} 0 =&< HF|L_f\left[ e^{-T} < HF|\tau _q^{\dagger } e^{-T}\mathbf{Q}_N e^T|HF >\cdot \mathbf{V}_N e^T, R_f\right] |HF >\\&+ < HF|L_f e^{-T}\mathbf{Q}_N^{|\omega |} e^T|HF >\cdot < HF|\tau _q^{\dagger } e^{-T}[\mathbf{V}_N, R_f]e^T|HF >\\&+ < HF|{\fancyscript{Z}}_f e^{-T} > HF|\tau _q^{\dagger } e^{-T}\mathbf{Q}_N e^T|HF >\cdot \mathbf{V }_Ne^T|HF >\\&+ < HF|\tau _q^{\dagger } e^{-T}[{H}_{N},{\fancyscript{T}}_f]e^T|HF >\\&+ < HF|(1+\varLambda )e^{-T}[>HF|\tau _q^{\dagger } e^{-T}\mathbf{Q}_N e^T|HF > \cdot \mathbf{V }_N,{\fancyscript{T}}_f]e^T|HF > \end{aligned}$$
  4. 4.

    Note that, as the PCM-CC Eqs. (1.24, 1.28), and the PCM-LR-CC Eqs. (4.8) and (4.9) are assumed to be satisfied, the Lagrangian (4.18) is equal to the excitation energy, \(F=\omega _f\).

  5. 5.

    The one particle density matrix elements \(\gamma ^f_{rs},\gamma ^{0f}_{rs}\) and \(\gamma ^{f0}_{rs}\) are defined as:

    $$\begin{aligned} \gamma ^f_{rs} =&< HF|L_f\left[ e^{-T}\{\tau _p^{\dagger } \tau _q\}e^{T}, R_f\right] |HF >\\&+ < HF|(1+\varLambda )e^{-T}[\{\tau _p^{\dagger } \tau _q\},{\fancyscript{T}}_f]e^T|HF >\\&+ < HF|{\fancyscript{Z}}_f e^{-T}\{\tau _p^{\dagger } \tau _q\}e^{T}|HF >\\ \gamma ^{0f}_{rs}=&< HF|(1+)e^{-T}[\{\tau _p^{\dagger } \tau _q\}, R_f]e^T|HF >\\&\gamma ^{f0}_{rs}= < HF|L_f e^{-T}\{\tau _p^{\dagger } \tau _q\}e^{T}|HF > \end{aligned}$$

    where \(\{\tau _p^{\dagger } \tau _q\}e^{T}|HF\) denote a normal ordered sequences of the creation/annihilation operators.

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  1. Department of Chemistry, University of Parma, Parco Area delle Scienze 17/A, 43100, Parma, Italy

    Roberto Cammi

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Cammi, R. (2013). Excitation Energies and Transition Moments from the PCM Linear Response Functions. In: Molecular Response Functions for the Polarizable Continuum Model. SpringerBriefs in Molecular Science(). Springer, Cham. https://doi.org/10.1007/978-3-319-00987-2_4

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