Reaction field models Hamiltonians

The salient features of quantum formulation of Onsager reaction field model (dipole model) is described here. In this method, the reaction field is treated as perturbation to the Hamiltonian of the isolated molecule. If H0 is the Hamiltonian of the isolated molecule and HR[ is the reaction field [21], the Hamiltonian of the whole system (Hlol) is represented as... 

Hannachi, Y., and Angyan, J. G., The role of induction forces in infra-red matrix shifts Quantum chemical calculations with reaction field model Hamiltonian, J. Mol. Struct. (Theochem) 232,97-110(1991). 

If the species is charged then an appropriate Bom term must also be added. The reaction field model can be incorporated into quantum mechanics, where it is commonly referred to as the self-consistent reaction field (SCRF) method, by considering the reaction field to be a perturbation of the Hamiltonian for an isolated molecule. The modified Hamiltonian of the system is then given by ... 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

In the quantum mechanical continuum model, the solute is embedded in a cavity while the solvent, treated as a continuous medium having the same dielectric constant as the bulk liquid, is incorporated in the solute Hamiltonian as a perturbation. In this reaction field approach, which has its origin in Onsager s work, the bulk medium is polarized by the solute molecules and subsequently back-polarizes the solute, etc. The continuum approach has been criticized for its neglect of the molecular structure of the solvent. Also, the higher-order moments of the charge distribution, which in general are not included in the calculations, may have important effects on the results. Another important limitation of the early implementations of this method was the lack of a realistic representation of the cavity form and size in relation to the shape of the solute. 

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],... 

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 basic features of ET energetics are summarized here for the case of an ET system (solute) linearly coupled to a bath (nuclear modes of the solute and medium) [11,30]. We further assume that the individual modes of the bath (whether localized or extended collective modes) are separable, harmonic, and classical (i.e., hv < kBT for each mode, where v is the harmonic frequency and kB is the Boltzmann constant). Consistent with the overall linear model, the frequencies are taken as the same for initial and final ET states. According to the FC control discussed above, the nuclear modes are frozen on the timescale of the actual ET event, while the medium electrons respond instantaneously (further aspects of this response are dealt with in Section 3.5.4, Reaction Field Hamiltonian). The energetics introduced below correspond to free energies. Solvation free energies may have entropic contributions, as discussed elsewhere [19], Before turning to the DC representation of solvent energetics, we first display the somewhat more transparent expressions for a discrete set of modes. 

The free energies of the initial (i) and final (f) states, the so-called diabatic states in the ET process (discussed in more detail in Section 3.54, Reaction Field Hamiltonian, Electronic Structure models), are given by [28]... 

Up to this point, solute charge densities (p) have been accepted as known . Here we consider the Hamiltonian underlying the reaction field (RF) models, and the role of timescales in defining the relevant densities [50-52], For the conventional RF model, in which solute and solvent electrons are assumed to have comparable timescales, Equations (3.86)-(3.89) may be expressed in terms of mean field RF potentials, RF and RF, due, respectively, to optical and inertial modes [12,14] ... 

The solvent effects are often described within a semiempirical selfconsistent reaction field theory (SCRF)248. In this theory the free energy of solvation is obtained from a set of selfconsistent equations describing the interaction of the solute (denoted by S) with the solvent modeled by a polarizable continuum characterized by a dielectric constant e. In the SCRF formalism, as developed by Rivail and collaborators249- 250 the solute-solvent system is modeled by a polarizable continuum (characterized by a dielectric constant e) in which the solvent molecule is immersed within an ellipsoidal cavity251,252. The Hamiltonian describing the solute in the cavity is given by,... 

In the absence of spectral information in the gas phase, it is common to compare calculated features of the vibrational spectrum to data measured in rare gas matrix, the premise being that the latter medium perturbs the H-bonded system as little as possible. The influence of the medium was considered via a self-consistent reaction-field formalism wherein inductive interactions between the polar system and the polarizable medium are incorporated into a model Hamiltonian ". The calculations made use of the 6-31G basis set at the SCF level. 

Generalized Reaction Fields from Surface Charge Densities Ab initio formulations of the PCM model discussed earlier, undertaken primarily by Tomasi and co-workers (see, e.g.. Refs. 72, 73, 266, 267), have very recently been implemented into four different semiempirical packagcs.- - Available codes include MOPAC,30o,325 a locally modified s version of MOPAC, oo and VAMP.302 While the model used by Negre et al. o NDDO Hamiltonians follows exactly the derivation of Equations [23] and [27], those of Wang and... 

The most serious limitation remaining after modifying the reaction field method as mentioned above is the neglect of solute polarizability. The reaction field that acts back on the solute will affect its charge distribution as well as the cavity shape as the equipotential surface changes. To solve this problem while still using the polarizable continuum model (PCM) for the solvent, one has to calculate the surface charges on the solute by quantum chemical methods and represent their interaction with the solvent continuum as in classical electrostatics. The Hamiltonian of the system thus is written as the sum of the Hamilton operator for the isolated solute molecule and its interaction with the macroscopic... 

Despite the demands presented by such a calculation, a number of researchers have used ab initio models to treat the electronic and nuclear degrees of freedom for the quantum motif in molecular mechanics, energy minimization studies. Examples of this include the self-consistant reaction field methods developed by Tapia and coworkers [42-44], which represent only the quantum motif explicitly and use continuum models for the environmental effects (classical and boundary regions), and the methods implemented by Kollman and coworkers [45] in their studies of condensed phase (chemical and biochemical) reaction mechanisms. In both of these implementations the expectation value of the quantum motif Hamiltonian, defined in Eqs. (11) and (14) above, is treated at the Hartree Fock level with relatively small basis sets. 

The evidence presented in Section VI shows the importance of incorporating this strain energy term in the perturbation term of the Hamiltonian. In this work only the results of Eq. (46) were discussed for cases where the reaction field can represent long-range interactions. Although this equation was derived for the equilibrium case, the time dependence of the tensile (or shear) converts this into a time-dependent case. The agreement between the DiMarzio-Bishop model is also a verification of the basic assumption. Consequences of a fortuitous cancellation of terms in the denominator, that is, (eq -t- 2), due to the acceptance of the reaction field, was studied. 

---