Solvent reaction potential

The dynamics of reactions in solution must include an appropriate description of the solvent dynamics. To simplify this problem we start with some considerations supported by intuition and by some concepts described in the preceding sections. In the initial stages of the reaction the characteristic time is given by the nuclear motions of the solute, large enough to allow the use of the adiabatic perturbation approximation for the description of motions. In practice this means that the evolution of the system in time may be described with a time independent formalism, with the solvent reaction potential equilibrated at each time step for the appropriate geometry of the solute. 

With the superscript R we indicate that the corresponding operator is related to the solvent reaction potential, and with the subscripts r and rr the one- or two-electron nature of the operator. The convention of summation over repeated indices followed by integration has been adopted, p is the electron density operator and is the operator which describes the two components of the interaction energy we have previously called t/en and f/ne. In more advanced formulations of continuum models going beyond the electrostatic description, other components are collected in this term. yR is sometimes called the solvent permanent potential, to emphasize the fact that in performing an iterative calculation of P > in the BO approximation this potential remains unchanged. 

An ab initio version of the Mpller-Plesset perturbation theory within the DPCM solvation approach was introduced years ago by Olivares et al. [26] following the above intuitive considerations based on the fact that the electron correlation which modifies both the HF solute charge distribution and the solvent reaction potential depending on it can be back-modified by the solvent. To decouple these combined effects the authors introduced three alternative schemes ... 

In the previous contributions to this book, it has been shown that by adopting a polarizable continuum description of the solvent, the solute-solvent electrostatic interactions can be described in terms of a solvent reaction potential, Va expressed as the electrostatic interaction between an apparent surface charge (ASC) density a on the cavity surface which describes the solvent polarization in the presence of the solute nuclei and electrons. In the computational practice a boundary-element method (BEM) is applied by partitioning the cavity surface into Nts discrete elements and by replacing the apparent surface charge density cr by a collection of point charges qk, placed at the centre of each element sk. We thus obtain ... 

The solvent reaction potential can be partitioned into several contributions of different physical origin, related to electrostatic, repulsive, induction and dispersion interactions between solute and solvent. In the original polarizable continuum approach only the electrostatic and induction terms are explicitly considered as an interaction potential... 

The solution of these differential equations yields the total electrostatic potential at any point r. Assuming a linear response approximation the electrostatic component of solvation can be obtained as of the work necessary to generate the solvent reaction potential, which can be determined by simply computing the ratio of potential generated by the solute in vacuo to the total potential around the solute (Equation (4.34)). 

The mutual polarization process between the solute and the polarizable medium is obtained by solving a system of two coupled equations, i.e., the QM Schrodinger equation for the solute in presence of the polarized dielectric, and the electrostatic Poisson equation for the dielectric medium in presence of the charge distribution (electrons and nuclei) of the solute. The solute occupies a molecular shaped cavity within the dielectric continuum, whose polarization is represented by an apparent surface charge (ASC) density spread on the cavity surface. The solute-solvent interaction is then represented by a QM operator, the solvent reaction potential operator, Va, corresponding to the electrostatic interaction of the solute electrons and nuclei with the ASC density of the solvent. 

The formulation of the QM continuum models reduces to the definition of an Effective Hamiltonian, i.e. an Hamiltonian to which solute-solvent interactions are added in terms of a solvent reaction potential. This effective Hamiltonian may be obtained from the basic energetic quantity which has the thermodynamic status of free energy for the whole solute-solvent system and for this reason is called free energy functional, This energy... 

Basically, the continuum solvation models consider the solvent as a uniform polarizable medium with dielectric constant e, where the solute (M) is immersed inside a cavity. Once placed inside the dielectric, the solute charge distribution polarizes the medium that in turn acts back (reaction field) polarizing the solute molecule. The system is then stabilized by the electrostatic interaction between the polarized solute and the polarized medium. Calling the charge distribution of the solute and the solvent reaction potential, the electrostatic solute-solvent interaction can be written as... 

A many-body perturbation theory (MBPT) approach has been combined with the polarizable continuum model (PCM) of the electrostatic solvation. The first approximation called by authors the perturbation theory at energy level (PTE) consists of the solution of the PCM problem at the Hartree-Fock level to find the solvent reaction potential and the wavefunction for the calculation of the MBPT correction to the energy. In the second approximation, called the perturbation theory at the density matrix level only (PTD), the calculation of the reaction potential and electrostatic free energy is based on the MBPT corrected wavefunction for the isolated molecule. At the next approximation (perturbation theory at the energy and density matrix level, PTED), both the energy and the wave function are solvent reaction field and MBPT corrected. The self-consistent reaction field model has been also applied within the complete active space self-consistent field (CAS SCF) theory and the eomplete aetive space second-order perturbation theory. ... 

The QM continuum solvation methods (QM/CSM) have a more simple physical and computational structure, as no explicit molecular degrees of freedom of the solvent enter into the calculation. The procedure is based on the definition of an effective Hamiltonian for the molecular solute, which is composed by the Hamiltonian of the isolated solute accompanied by a solute-solvent integral operators, with a nonlinear kernel, and describing the solute in the presence of the solvent reaction potential. The solution of the corresponding nonlinear Schrodinger equation, obtained at ab-initio QM level and with an iterative procedure, determines the properties of the molecular solute in the presence of the solvent, with a complete description of the solvent effects. 

In the QM/GB method, the molecular charge distribution of the solute is reduced to a sum of atomic point charges, each placed within a sphere of appropriate radius, and the solvent reaction potential produced by these charges is evaluated with an approximated formula generalizing the Bom expressions for a single charge within a sphere, and introduced into the Hamiltonian of the solute (for a review see [26]). 

As other QM continuum models, the PCM model requires the solution of two coupled problems an electrostatic classical problem for the determination of the solvent reaction potential Va induced by the total charge distribution and a quantum mechanical problem for the determination of the wavefunction I of the solute described by the effective QM Hamiltonian (1.1). The two problems are nested and they must be solved simultaneously. 

With respect to other QM continuum models, the PCM method represents of the interaction operator Vi l ) (i.e. of the solvent reaction potential Va) in terms of an apparent surface charge (ASC) charge distribution a spread on the boundary F of the cavity (C) hosting the solute M. 

The physics of the solute-solvent interaction potential V F) of Eq.(l.l) is simple. The total charge distribution pu of the molecular solute polarizes the dielectric medium, which in turn becomes source of an additional electrostatic potential Va (the solvent reaction potential) for the electrons and nuclei of the molecular solute. V If) can be written as... 

The solvent reaction potential Vo in Eq,(1.2) is determined by solving the classical electrostatic Poisson equation which governs total electrostatic potential V = Vm + Vff (Vm is the electrostatic potential produced by the electronic and nuclear charge distribution of the solute). The Poisson problem has the form of a partial differential equations with domain in the whole three-dimensional space [2] ... 

The solve the Poisson problem (1.3), the PCM model exploits an integral representation of the solvent reaction potential Vo ... 

The PCM coupled-cluster (PCM-CC) theory [9-13] introduces an explicit description of the coupling between the electron-correlation (dynamic) of the molecular solute and the solute reaction potential. The electron-correlation modifies the charge distribution Pm of the solute. The changes in charge distribution pm modify the solvent reaction potential Va, which in turn influence the electron-correlation. If the coupling between the dynamical electronic correlation of the solute and the polarization of the solvent is neglected the dynamic electron-correlation of the molecular solutes is evaluated in the presence of the fixed Hartree-Fock solvent reaction potential. This approximated form of the PCM-CC theory is denoted with the acronym PTE (i.e. Perturbation Theory on the Energy) which derives from a many-body perturbation analysis of the solute-solvent interaction [14]. 

The explicit equations for the PCM-CC-PTE approximation can be obttiined from the PCM-CCSD-PTED equations by neglecting all the explicit terms involving the correlation components of the solvent reaction potential.The resulting PTE-PCM-CCSD equations have the same form of the CCSD equations for an isolated molecule, with the only difference that now the Fock matrix elements and the MO refer to those of the solvated molecules. 

The first term of Eq.(4.16) corresponds to the excitation energies of the solute in the presence of the PCM-CC fixed solvent reaction potential the second term represents a solute-solvent contribution determined by the interaction of the right transition density of the molecular solute represented by T... 

Solvent reaction potential, 1-3, 6, 8, 10 State specific excitation energies, 37, 56 Static first hyper-polarizability, 14 SVEP method, viii... 

The partition of O Eq. 28.14 is based on the so-called normal ordered operators and, Qn (Cammi 2009). Specifically, Qn(A, T) is the component of the solvent reaction potential due to the correlation CC electronic density, and H(0)n is the normal ordered form of Hamiltonian of the solute in presence of the frozen Hartree-Fock reaction field Qhf-... 

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