Polarizable continuum model formulation

E. L. Coitino, J. Tomasi and R. Cammi, On the evaluation of the solvent polarization apparent charges in the polarizable continuum model A new formulation, J. Comput. Chem., 16... 

The approach which will be reviewed here has been formulated within the framework of the quantum mechanical polarizable continuum model (PCM) [7], Within this method, the effective properties are introduced to connect the outcome of the quantum mechanical calculations on the solvated molecules to the outcome of the corresponding NLO experiment [8], The correspondence between the QM-PCM approach and the semi-classical approach will also be discussed in order to show similarities and differences between the two approaches. 

J. Tomasi, R. Cammi and B. Mennucci, Medium effects on the properties of chemical systems An overview of recent formulations in the polarizable continuum model (PCM), Int. J. Quantum Chem., 75 (1999) 783-803. 

R. Cammi and J. Tomasi, Nonequilibrium solvation theory for the polarizable continuum model - a new formulation at the SCF level with application to the case of the frequency-dependent linear electric-response function, Int. J. Quantum Chem., (1995) 465-74 B. Mennucci, R. Cammi and J. Tomasi, Excited states and solvatochromic shifts within a nonequilibrium solvation approach A new formulation of the integral equation formalism method at the self-consistent field, configuration interaction, and multiconfiguration self-consistent field level, J. Chem. Phys., 109 (1998) 2798-807 R. Cammi, L. Frediani, B. Mennucci, J. Tomasi, K. Ruud and K. V. Mikkelsen, A second-order, quadratically... 

Abstract The computational study of excited states of molecular systems in the condensed phase implies additional complications with respect to analogous studies on isolated molecules. Some of them can be faced by a computational modeling based on a continuum (i.e., implicit) description of the solvent. Among this class of methods, the polarizable continuum model (PCM) has widely been used in its basic formulation to study ground state properties of molecular solutes. The consideration of molecular properties of excited states has led to the elaboration of numerous additional features not present in the PCM basic version. Nonequilibrium effects, state-specific versus linear response quantum mechanical description, analytical gradients, and electronic coupling between solvated chromophores are reviewed in the present contribution. The presentation of some selected computational results shows the potentialities of the approach. 

Here, we provide the theoretical basis for incorporating the PE potential in quantum mechanical response theory, including the derivation of the contributions to the linear, quadratic, and cubic response functions. The derivations follow closely the formulation of linear and quadratic response theory within DFT by Salek et al. [17] and cubic response within DFT by Jansik et al. [18] Furthermore, the derived equations show some similarities to other response-based environmental methods, for example, the polarizable continuum model [19, 20] (PCM) or the spherical cavity dielectric... 

F. J. Olivares del Valle and J. Tomasi, Chem. Phys., 150, 139 (1991). Electron Correlation and Solvation Effects. I. Basic Formulation and Preliminary Attempt to Include the Electron Correlation in the Quantum Mechanical Polarizable Continuum Model so as to Study Solvation Phenomena. 

Calculations of analytic excited state properties for correlated methods have been reported by several groups [107-118]. Excited state dynamic properties from cubic response theory were first obtained by Norman et al. at the SCF level [55] and by Jonsson et al. at the MCSCF [56] level, and in a subsequent study a polarizable continuum model was applied to account for solvation effects [119]. Hattlg et al. presented a general theory for excited state response functions at the CC level using a quasi-energy formulation [120] which was subsequently implemented and applied at the CCSD level [121, 122]. The first ID DFT calculation of dynamic excited state polarizabilities, which we will shortly review here, was presented in [58] for pyrimidine and -tetrazine utilizing the double residue of the cubic response function derived in Section 2.7.3. 

R. Cammi and J. Tomasi, Int. J. Quantum Chem., 29, 465 (1995). Nonequilibrium Solvation Theory for the Polarizable Continuum Model A New Formulation at the SCF Level with Application to the Case of the Frequency-Dependent Linear Electric Response Function. 

For solvation of small molecules, the polarizable continuum model (PCM) and its variants have been widely used for calculation of solvation energy. The conductor-like PCM (CPCM) model gives a concise formulation of solvent effect, in which the solvent s response to the solute polarization is represented by the presence of induced surface charges distributed on the solute-solvent interface. In this formulation, no volume polarization (extension of solute s electron distribution into the solvent region) is allowed. The induced surface charge counterbalances the electrostatic potential on the interface generated by the solute molecule. 

Ferrighi, L., Frediani, L., Fossgaard, E., and Ruud, K. [2006]. Parallelization of the integral equation formulation of the polarizable continuum model for higher-order response functions, J. Chem. Phys. 125, p. 154112. 

The aim of this book is to present the basic aspects of the molecular response function theory for molecular systems in solution described with the Polarizable Continuum Model, giving special emphasis both to the physical basis of the theory and to its quantum chemical formalism. The QM formalism will be presented in the form of the coupled-cluster theory, as it is the most recent and less known formulation for the QM calculation of molecular properties within the PCM... 

The induced dipole formulation is an example of the so-called polarizable embedding but it is not the only possible choice. There are in fact various alternatives schemes to simulate the polarization of the MM subsystem, such as the fluctuating charges [24, 25], the classical Drude oscillators [26], or the Electronic Response of the Surroundings (ESR) [27] which mixes a non-polarizable MM scheme with a polarizable continuum model characterized by a dielectric constant extrapolated at infinite frequency. 

Valle." The two approaches are inserted in the framework of the MPE model of the Nancy group, and of the ASC model (in the formulation known by the acronym PCM, polarizable continuum model), respectively. Here we shall not compare the features of the two models (for a more detailed analysis see Section 4) but we only stress that, even if the differences in the practical implementation and in calculations are noticeable, the formal elaboration is nevertheless similar in the two cases namely it allows one to include a dispersive term in the solute Hamiltonian, together with the term associated with the solute-solvent electrostatic perturbation. 

Rizzo reviews in a unitary framework computational methods for the study of linear birefringence in condensed phase. In particular, he focuses on the PCM formulation of the Kerr birefringence, due to an external electric field yields, on the Cotton-Mouton effect, due to a magnetic field, and on the Buckingham effect due to an electric-field-gradient. A parallel analysis is presented for natural optical activity by Pecul Ruud. They present a brief summary of the theory of optical activity and a review of theoretical studies of solvent effects on these properties, which to a large extent has been done using various polarizable dielectric continuum models. 

Over the last years, the basic concepts embedded within the SCRF formalism have undergone some significant improvements, and there are several commonly used variants on this idea. To exemplify the different methods and how their results differ, one recent work from this group [52] considered the sensitivity of results to the particular variant chosen. Due to its dependence upon only the dipole moment of the solute, the older approach is referred to herein as the dipole variant. The dipole method is also crude in the sense that the solute is placed in a spherical cavity within the solute medium, not a very realistic shape in most cases. The polarizable continuum method (PCM) [53,54,55] embeds the solute in a cavity that more accurately mimics the shape of the molecule, created by a series of overlapping spheres. The reaction field is represented by an apparent surface charge approach. The standard PCM approach utilizes an integral equation formulation (IEF) [56,57], A variant of this method is the conductor-polarized continuum model (CPCM) [58] wherein the apparent charges distributed on the cavity surface are such that the total electrostatic potential cancels on the surface. The self-consistent isodensity PCM procedure [59] determines the cavity self-consistently from an isodensity surface. The UAHF (United Atom model for Hartree-Fock/6-31 G ) definition [60] was used for the construction of the solute cavity. 

The analysis presented so far on the difrerent specificities of LR and SS descriptions of excitation processes within QM/continuum approaches also ap-phes to the polarizable QM/MM approaches. In those cases, however, the picture is simpler because there is no need to partition the polarization into dynamic and inertial terms as in continuum models, since the inertial (nuclear) degrees of freedom are considered expUcidy through the fixed multipolar expansion while the dynamic response is represented by the polarizable term, such as the induced dipoles in the ID formulation described earlier. 

The simple virtual charge model discussed by Constanciel and Tapia [6] has been developed into an extended generalized Born (EGB) approach. Different approximations have been proposed. Constanciel [40] has analyzed the theoretical basis used as foundations for empirical reaction field approximations through the continuum model to the surrounding medium. Artifacts in the EGB scheme have been clearly identified. The new approximate formulation proposed derives from an exact integral equation of classical electrostatics following a well defined procedure. It is shown there how the wavefunction of solvated species imbedded in cavities formed by interlocking sphere in a polarizable continuum can be computed. 

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