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General Response Theory for the Polarizable Continuum Model

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Molecular Response Functions for the Polarizable Continuum Model

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

Abstract

This chapter presents the general aspects of the response theory for molecular solutes in the presence of time-dependent perturbing fields: (i) the non-equilibrium solvation, (ii) the variational formulation of the time-dependent non-linear QM problem, and (iii) the connection of the molecular response functions with their macroscopic counterparts. The linear and quadratic molecular response functions are described at the coupled-cluster level.

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Notes

  1. 1.

    The concept of non-equilibrium solvation has been introduced to describe the solvent polarization in processes involving dynamic, or sudden, variations of solute charge distribution of the solute, and it takes into account that during the time-scale of a fast event not all the degrees of freedoms of the solvent molecules (nuclear, translational, rotational,vibrational; electronic) are able to respond to the variations of the charge distribution of the solute (see Appendix).

  2. 2.

    We assume that \(V^{\prime }(t)\) is applied adiabatically so that it vanishes at \(t=-\infty .\)

  3. 3.

    The time-dependent wavefunction (3.8) are also known as Floquet states [24–28], a particular set of solutions of the time-dependent Schrödinger equation for systems under the influence of an external time-dependent periodic perturbation.

  4. 4.

    The time-average over a period T for a general time-dependent function \(g(t)\) is defined as

    $$\begin{aligned} \{g(t)\}_T=\frac{1}{T}\int ^{T/2}_{-T/2}g(t)dt \nonumber \end{aligned}$$
  5. 5.

    Being the perturbing operator \(V(t)\) Hermitian, we have that: \(X^\dag =X\), \(\omega _{-j}=-\omega _j\), \(\varepsilon (\omega _j)^*=\varepsilon (\omega _{-j}). \)

  6. 6.

    We here consider the Molecular Orbital (MO) “unrelaxed” approach in which the reference state does not depend on the perturbation \(V(t).\)

  7. 7.

    The expansions (3.18a and 3.18b) imply an expansion in time-dependent polarization charges of (3.15):

    $$\begin{aligned} {\bar{ \mathbf Q}}_N(t)={\bar{ \mathbf Q}}_N(t)^{(0)}+{\bar{ \mathbf Q}}_N(t)^{(1)}+\cdots \nonumber \end{aligned}$$

    where \({\bar{ \mathbf Q}}_N(t)^{(n)}\) are terms of n-th order in the perturbation.

  8. 8.

    The variational criteria at the zero and first order are satisfied when the Coupled-cluster wavefunctions are solution of the PCM coupled-cluster Eqs. (1.24) and (1.28).

  9. 9.

    Third- and second-order derivatives of the \(T,\varLambda \) amplitudes are eliminated, respectively, by the zero- and first order stationarity of the quasi-free-energy functional \(\varDelta G_{CC}.\)

  10. 10.

    The polarization charges operator \({\mathbf Q}_N^{|\omega _X\,+\,\omega _Y|}\) is evaluated from Eq. (3.27) using the dielectric permittivity \(\varepsilon (|\omega _X+\omega _Y|)\) of the solvent at the frequency \(|\omega _X+\omega _Y|\). Therefore, Eq. (3.34a) is able to describe the non-equilibrium solvation effects in the quadratic response function describing general second-order molecular processes [4] (see Appendix B in Ref. [19]).

  11. 11.

    The contribution is due to the effect of the boundary condition of the cavity on the Maxwell field.

  12. 12.

    The charges \(\tilde{\mathbf{Q}}^{\mathbf{E}}_i\) are obtained as solution of the electrostatic problem (i.e. the Laplace problem) describing the Maxwell field \({\mathbf{E}}\) in the presence of the void cavity. The corresponding integral equation with domain on the PCM cavity boundary \(\Gamma \) is:

    $$\begin{aligned} \left( 2\pi \frac{\epsilon +1}{\epsilon -1} +D^*\right) \sigma _i^E(\mathbf{s})=-E_in_i(\mathbf{s})\ \mathbf{s}\subset \Gamma \end{aligned}$$

    where \(E_i\) and \(n_i\) are, respectively the i-th Cartesian component of the Maxwell field \(\mathbf{E}\) and of a unit vector \(\mathbf{n}(r)\) normal to the cavity surface at the point \(\mathbf{s}\), and \(\sigma _i^E(\mathbf{s})\) is the apparent surface charge density determined by the Maxwell field.

    The discretization of the the apparent surface charge density \(\sigma _i^E(\mathbf{s})\) leads then to the discrete set of charges \(\tilde{\mathbf{Q}}^{\mathbf{E}}_i\).

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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). General Response Theory for the Polarizable Continuum Model. 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_3

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