Solvent reaction field

Sanchez Marcos, E., Pappalardo, R. R. and Rinaldi, D. Effects of the solvent reaction field on the geometrical stmetures of hexahydrate metallic cations, J.Phys. Chem., 95 (1991), 8928-8932... 

In all cases the dielectric constant used is that of the pure solvent. Neglect of the solute is usually justified by its low concentration and the assumption that any necessary correction would be additive. In at least a few cases where the first two expressions have been employed the linearity of the results is to some extent dependent on how closely the refractive index of the solute meets the conditions n2 = 2.0 or 2.5 a situation not always recognized by the investigators. In one instance attempts have been made to clarify the role of solvent reaction field by examining solutes with different dipole moment orientations relative to the bonds involving coupled atoms. 

The reorganization energy term derives from the solvent being unable to reorient on the same timescale as the electron transfer takes place. Thus, at the instant of transfer, the bulk dielectric portion of the solvent reaction field is oriented to solvate charge on species A, and not B, and over the course of the electron transfer only the optical part of the solvent reaction field can relax to the change in tire position of the charge (see Section 14.6). If the Bom formula (Eq. (11.12)) is used to compute the solvation free energies of the various equilibrium and non-equilibrium species involved, one finds that... 

The second step solves the integral equations using the boundary elements previously introduced. The result of this second step is the evaluation of the various contributions of different physical origin (electrostatic, repulsion, dispersion, cavitation) which determine the solvent reaction field. This second step depends (at least for the electrostatic part) on the level of description of the molecular structure. 

The quantum mechanical (QM) (time-independent) problem for the continuum solvation methods refers to the solution of the Schrodinger equation for the effective Hamiltonian of a molecular solute embedded in the solvent reaction field [1-5]. In this section we review the most relevant aspects of such a QM effective problem, comment on the differences with respect to the parallel problem for isolated molecules, and describe the extensions of the QM solvation models to the methods of modern quantum chemistry. Such extensions constitute a field of activity of increasing relevance in many of the quantum chemistry programs [6],... 

The nonlinear nature of the Hamiltonian implies a nonlinear character of the Cl equations which must be solved through an iteration procedure, usually based on the two-step procedure described above. At each step of the iteration, the solvent-induced component of the effective Hamiltonian is computed by exploiting the first-order density matrix (i.e. the expansion Cl coefficients) of the preceding step. In addition, the dependence of the solvent reaction field on the solute wavefunction requires, for a correct application of this scheme, a separate calculation involving an iteration optimized on the specific state (ground or excited) of interest. This procedure has been adopted by several authors [17] (see also the contribution by Mennucci). 

A further issue arises in the Cl solvation models, because Cl wavefunction is not completely variational (the orbital variational parameter have a fixed value during the Cl coefficient optimization). In contrast with completely variational methods (HF/MFSCF), the Cl approach presents two nonequivalent ways of evaluating the value of a first-order observable, such as the electronic density of the nonlinear term of the effective Hamiltonian (Equation 1.107). The first approach (the so called unrelaxed density method) evaluates the electronic density as an expectation value using the Cl wavefunction coefficients. In contrast, the second approach, the so-called relaxed density method, evaluates the electronic density as a derivative of the free-energy functional [18], As a consequence, there should be two nonequivalent approaches to the calculation of the solvent reaction field induced by the molecular solute. The unrelaxed density approach is by far the simplest to implement and all the Cl solvation models described above have been based on this method. 

The Cl relaxed density approach [18] should give a more accurate evaluation of the reaction field, but because of its more involved computational character it has been rarely applied in Cl solvation models. The only notably exception is the Cl methods proposed by Wiberg at al. in 1991 [19] within the framework of the Onsager reaction field model. In their approach, the electric dipole moment of the solute determining the solvent reaction field is not given by an expectation value but instead it is computed as a derivative of the solute energy with respect to a uniform electric field. 

In Equation (1.161) (or equivalently in Equation (1.165) for the nonequilibrium case) we have shown that excited state free energies can be obtained by calculating the frozen-PCM energy E s and the relaxation term of the density matrix, PA (or P 1) where the calculation of the relaxed density matrices requires the solution of a nonlinear problem in which the solvent reaction field is dependent on such densities. 

Methods based on the solvent reaction field philosophy differ mainly in (i) the cavity shape, and (ii) the way the charge interaction with the medium is calculated. The cavity is differently defined in the various versions of models it may be a sphere, an ellipsoid or a more complicated shape following the surface of the molecule. The cavity should not contain the solvent molecules, but it contains within its boundaries the solute charge distribution. 

The applications of continuum models to the study of solvent induced changes of the shielding constant are numerous. Solvent reaction field calculations differ mainly in the level of theory of the quantum mechanical treatment, the method used for the gauge invariance problem in the calculations of the shielding constants and the approaches used for the calculations of the charge interaction with the medium. 

The solvent reaction field calculations involve several different aspects. We would like concentrate on the points required to make these models successful as well as on the facts that limit their accuracy. One of them is the shape of the molecular cavity, which can be modelled spherically or according to the real shape of the solute molecule. First, we discuss the papers in which spherical cavity models were applied. The studies utilizing the solute-shaped cavity models are collected the second group. Finally, the approaches employing explicit treatment of the first-solvation shell molecules combined with the continuum models are discussed. 

A similar system to that discussed in ref. [44] (tetrazine, tetrazole and pyrrole) has been studied by Manalo et al. [47] by means of the CSGT/ASC method at the B3LYP/6-311++G(2d,2p) level. The cavity was defined by using the Pauling radius for each solute atom. In this paper the effects of geometric relaxation (indirect effects) are found to be small, and the direct influence of the intensity of the solvent reaction field on the shielding constants dominates. However, the indirect effect has been found to be important for N, A-dimethylacetamidine in IEF-PCM calculations [48],... 

By taking as a reference the calculation in vacuo, the presence of the solvent introduces several complications. In fact, besides the direct effect of the solvent on the solute electronic distribution (which implies changes in the solute properties, i.e. dipole moment, polarizability and higher order responses), it should be taken into account that indirect solvent effects exist, i.e. the solvent reaction field perturbs the molecular potential energy surface (PES). This implies that the molecular geometry of the solute (the PES minima) and vibrational frequencies (the PES curvature around minima in the harmonic approximation) are affected by the presence of a solvating environment. Also, the dynamics of the solvent molecules around the solute (the so-called nonequilibrium effect ) has to be... 

Other methods of including nonequilibrium solvation are reviewed elsewhere [86], and the reader is also referred to selected relevant and more recent original papers [66,88-100], Particularly relevant to the present volume are methods that introduce extra degrees of freedom by using the solvent reaction field not only at the current value of R but also at nearby values [65,66], Many of the approaches introduce finite-time effects and additional degrees of solvent freedom by introducing different time scales for electronic and atomic polarization [88-97,99,100],... 

In this framework, the requirement needed in order to incorporate the solvent effects into the reactant (and product) wavefunctions is automatically fulfilled by using the effective Hamiltonian defined in Equation (3.155) and by adopting an iterative procedure until the wave-function and the solvent reaction field induced by the Cl density matrix of the state of interest reach self-consistency. One must note that this procedure is valid for ground and excited states fully equilibrated with the solvent, while the inclusion of nonequilibrium effects needs some further refinements, as indicated, for example, in ref. [35],... 

The methodology that uses the dielectric model is instead the simpler and in principle the more suitable for the study of chemical reactions involving large molecular systems. In 1998, Amovilli et al [13] developed a computer code in which the solvent reaction field, including all the basic solute-solvent interactions, has been considered for Complete Active Space Self Consistent Field (CASSCF) calculations. 

Gallicchio E, Linda Yu Zhang, Levy RM (2002) The SGB/NP Hydration Free Energy Model Based on the Surface Generalized Bom Solvent Reaction Field and Novel Nonpolar Hydration Free Energy Estimators. J. Comput. Chem. 23 517-529... 

Jensen, L., Duijnen P.Th. van and Snijders J.G., A discrete solvent reaction field model for calculating molecular linear response properties in solution. J.Chem.Phys. (2003) 119 12998-13006. 

Jensen, L. and Duijnen P. Th. van, The Discrete Solvent Reaction Field model A Quantum me-chanics/Molecular mechanics model for calculating nonlinear optical properties of molecules in the condensed phase., in Atoms, molecules and clusters in electric fields. Theoretical approaches to the calculation of electric polarizability, G. Maroulis, Editor. 2006, Imperial College Press London, p. 1-43. 

Wong MW, Wiberg KB, Frisch M (1991b) Hartree-Fock second derivatives and electric field properties in a solvent reaction field Theory and application. J Chem Phys 95 8991-8998... 

This demonstrates that especially for systems with very flat potential energy surfaces of polar bonds, the interpretation of precise X-ray data has to be carried out very carefully and all possible perturbations of the environment must be taken into account. The comparison with the structures of free molecules is not appropriate. On the other hand, the theoretical model for the correct interpretation of such structures in polar mediums must be expanded for example, with the calculation of an Onsager solvent reaction field. 

The gradient of the excitation energy includes two explicit PCM contributions, but the solvent reaction field also implicitly affects Eq. (7-22) through P4 and W ... 

M. W. Wong, K. B. Wiberg, and M. j. Frisch,/. Chem. Phys., 95, 8991 (1991). Hartree-Fock Second Derivatives and Electric Field Properties in a Solvent Reaction Field Theory and Application. 

The cavity surface is then subdivided into small domains, called surface tesserae, used to express as finite sums all the surface integrals needed to compute the solvent reaction field, as explained below. The final result is depicted in figure 3, where we show the GePol cavity for p-alanine subdivided into tesserae with average area of 0.4 and of 0.2 respectively. 

---