Nonequilibrium solvation models

This model accounts only partially for the specific structure of liquid water, and to refine it, calculations within supermolecular and semicontinuum models were also performed. In these cases, the properties were computed for a cluster of five water molecules, simulating the inclusion of a first solvation shell. In the semicontinuum model, the cluster was immersed in the dielectric continuum. Because of the (prohibitive for the times) size of the cluster, it was possible to obtain only an uncorrelated result. On the other hand, a nonequilibrium solvation model was used in computing the orientational contribution of Equation (2.218). Finally, to determine mC(o>, T), an extensive property, a differential shell method was employed. 

Nonequilibrium solvation model for the electric dipole polarizability. b Result corrected for local field effects. c Ref.[27], mean value for T between 283.1 5 and 293.1 5 K. 

K.O. Sylvester-Hvid, K.V. Mikkelsen, D. Jonsson, P. Norman, H. Agren, Nonlinear optical response of molecules in a nonequilibrium solvation model, J. Chem. Phys. 109 (1998) 5576. 

Solvation Thermodynamics and the Treatment of Equilibrium and Nonequilibrium Solvation Effects by Models Based on Collective Solvent Coordinates... 

Nonequilibrium solvent effects can indeed by significant at the kcal level-maybe even at a greater level, but so far there is no evidence for that when the reaction coordinate involves protonic or heavier motions. Our goal in this section has been to emphasize just how powerful and general the equilibrium model is. In addition, in both the previous section and the present section, we have emphasized the use of models based on collective solvent coordinates for calculating both equilibrium and nonequilibrium solvation properties. 

Aguilar, M. A., Olivares del Valle, F. J. and Tomasi, J. Nonequilibrium solvation an ab initio quantum-mechanical method in the continuum cavity model approximation, J.Chem.Phys., 98 (1993), 7375-7384... 

In recent years many attempts have been made to extend the implicit solvent models to the description of time-dependent phenomena. One of these phenomena is nonequilibrium solvation [3] and it can be described effectively in a very simplified way, despite the fact that it actually depends on the details of the full frequency spectrum of the dielectric constant. Typical examples of nonequilibrium solvation are the absorption of light by the solute which produces an excited state which is no longer in equilibrium with the surrounding polarization of the medium [11-13], Another example is intermolecular charge transfer within the solute, also leading to a nonequilibrium polarization [14],... 

This analysis shows that in order to account properly for solvent polarity effects, a solvation model has to be characterized by a larger flexibility with respect to the same model for ground state phenomena. In particular, it should be possible to shift easily from an equilibrium to a nonequilibrium regime according to the specific phenomenon under scrutiny. In the following section, we will show that such a flexibility can be obtained in continuum models and generalized to QM descriptions of the electronic excitations. 

Still within continuum solvation models, Wang et al. [5] have used an ab initio SCRF Onsager model to compute vibrational frequencies at different levels of the ab initio QM molecular theory, the G-COSMO model has been used by Stefanovich and Truong to calculate vibrational frequencies at the DFT level [6], and the multipole SCRF model, developed by the group of Rivail, has been extended to the calculation of frequency shifts at the HF, MP2 and DFT levels, including nonequilibrium effects [7],... 

Also, because such derivatives are to be evaluated at the equilibrium geometry, a key point is the determination of that geometry on the solvated PES, which leads to the so-called indirect solvent effects , which still requires a viable method to calculate free energy gradients (and possibly hessians). The problem of the formulation of free energy derivatives within continuum solvation models is treated elsewhere in this book and for this reason it will not considered here. Instead, it is worth remarking in this context another implication of such a formulation, i.e. that a choice between a complete equilibrium scheme or the account for vibrational and/or electronic nonequilibrium solvent effects [42, 43] should be done (see below). 

We model the nonequilibrium solvation for a molecular state using the following interaction operator between the outer dielectric medium and the molecular system[2]... 

The SES, ESP, and NES methods are particularly well suited for use with continuum solvation models, but NES is not the only way to include nonequilibrium solvation. A method that has been very useful for enzyme kinetics with explicit solvent representations is ensemble-averaged variational transition state theory [26,27,87] (EA-VTST). In this method one divides the system into a primary subsystem and a secondary one. For an ensemble of configurations of the secondary subsystem, one calculates the MEP of the primary subsystem. Thus the reaction coordinate determined by the MEP depends on the coordinates of the secondary subsystem, and in this way the secondary subsystem participates in the reaction coordinate. 

A significant recent advance in continuum SD has been achieved by combining the solvation response expressions in terms of the solvent s(cu) with quantum mechanical (QM) electronic structure methodology for solvated species. Specifically, the polarizable continuum model (PCM) [51], which was originally developed to predict the electronic structure of solutes in polar media, has been extended to nonequilibrium solvation [52]. A review by Mennucci [8] describes this extension of PCM and its application to the evaluation of S(t). The readers are referred to that article for the outline of the overall approach and for the details of the methods used. 

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... 

The outline of the remainder of this contribution is as follows. In Section 3.7.2, we discuss radical anion dissociation in solution, in which a conical intersection has an important impact on the ground state reaction barrier, rate constant and reaction path, all of which are also influenced by nonequilibrium solvation. The excited electronic state conical intersection problem for the cis-trans isomerization of a model protonated Schiff base in solution is discussed in Section 3.7.3, focusing on the approach to, and passage through, the conical intersection, and the influence of nonequilibrium solvation thereupon. Some concluding remarks are offered in Section 3.7.4. We make no attempt to give a complete discussion for these topics, but rather focus solely on several highlights. Similarly, the references herein are certainly incomplete. We refer the interested reader to refs [1-9] for much more extensive discussions and references. 

Electron transfer (ET) reactions are analyzed by Newton in terms of continuum solvation models. Their role in the determination of the ET critical parameters (i.e. the solvent reorganization energy and the electronic coupling between the initial and final states) is analyzed using both an equilibrium and nonequilibrium solvation framework. 

PCM originated as a method to describe solvent effects on ground state molecules [2], but the extension to excited states was realized only after the original presentation, with a model [3], which introduced nonequilibrium effects in the solvent response for the optical processes of photon absorption and emission. The nonequilibrium solvation regime has later been applied to vibrational spectroscopies... 

R. Cammi, B. Mennucci, K. Ruud, L. Frediani, K.V. Mikkelsen, J. Tomasi, A second order, quadratically convergent multiconfigurational self-consistent field polarizable continuum model for equilibrium and nonequilibrium solvation. J. Chem. Phys. 117, 13 (2002)... 

M. A. Aguilar, F. J. Olivares del Valle, and J. Tomasi, /. Chem. Phys., 98, 7375 (1993). Nonequilibrium Solvation An Ab Initio Quantum-Mechanical Method in the Continuum Cavity Model Approximation. 

Fig. 15.3 The nonequilibrium solvation function S(t) (full lines) and the solvation correlation functions C(i) for a model solute ion of diameter 3.1 A in acetonitrile computed with the positive solute (dotted line) and neutral solute (dashed line). (From M. Maroncelli, J. Chem. Phys. 94, 2084 (1991).)...

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