Solvent solute wavefunction

In our discussion the usual Born-Oppenheimer (BO) approximation will be employed. This means that we assume a standard partition of the effective Hamiltonian into an electronic and a nuclear part, as well as the factorization of the solute wavefunction into an electronic and a nuclear component. As will be clear soon, the corresponding electronic problem is the main source of specificities of QM continuum models, due to the nonlinearity of the effective electronic Hamiltonian of the solute. The QM nuclear problem, whose solution gives information on solvent effects on the nuclear structure (geometry) and properties, has less specific aspects, with respect the case of the isolated molecules. In fact, once the proper potential energy surfaces are obtained from the solution of the electronic problem, such a problem can be solved using the standard methods and approximations (mechanical harmonicity, and anharmonicity of various order) used for isolated molecules. The QM nuclear problem is mainly connected with the vibrational properties of the nuclei and the corresponding spectroscopic observables, and it will be considered in more detail in the contributions in the book dedicated to the vibrational spectroscopies (IR/Raman). This contribution will be focused on the QM electronic problem. 

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). 

The key differences between the PCM and the Onsager s model are that the PCM makes use of molecular-shaped cavities (instead of spherical cavities) and that in the PCM the solvent-solute interaction is not simply reduced to the dipole term. In addition, the PCM is a quantum mechanical approach, i.e. the solute is described by means of its electronic wavefunction. Similarly to classical approaches, the basis of the PCM approach to the local field relies on the assumption that the effective field experienced by the molecule in the cavity can be seen as the sum of a reaction field term and a cavity field term. The reaction field is connected to the response (polarization) of the dielectric to the solute charge distribution, whereas the cavity field depends on the polarization of the dielectric induced by the applied field once the cavity has been created. In the PCM, cavity field effects are accounted for by introducing the concept of effective molecular response properties, which directly describe the response of the molecular solutes to the Maxwell field in the liquid, both static E and dynamic E, [8,47,48] (see also the contribution by Cammi and Mennucci). 

In general, any satisfactory theoretical calculation of a nuclear coupling constant requires reliable calculation of the molecular wavefunction. As a consequence, a realistic approximation to the actual charge distribution in the carbohydrate molecule must presumably enter any theoretical model that attempts to provide a quantitative interpretation of solvent effects. The simplest treatments, and those that have been proposed most frequently to account for the solvent effect in the absence of specific effects, are those in which the solvent is treated as a continuum surrounding the solute molecule. Several different models where the solvent dependence of coupling interactions is related to the polarity of the medium have been proposed.78-79 The solvation theory80,81 has been successfully used within the FPT formalism to interpret the effect of solvent on Jc H and 3/CH. On the basis of this model, the Hamiltonian of a particular molecule includes the solvent-solute interaction term //so,v ... 

The essence of the basic model is then to describe the process of mutual solute-solvent interaction, and to extract from the final solute wavefunction, all the... 

Fig. 7 shows the behaviour of these charges with variation of the CC1 distance. As was to be expected in going towards the products, namely CH3NH3 and Cl-, Qn tends to a minimal value while qci tends to unity. The effect of water, the solvent, results in a further enhancement of qci and in a further reduction of qjy with the increase of CC1 distance. This effect is completely due to the solvent polarization when the ionic products begin to form. It is important to note that such small changes in the Mulliken charges (27) and (28) correspond instead to a strong change in the value of the energy along the reaction path (see Fig. 6). Repulsion and dispersion contributions to the solvation free energy do not have any visible effects on the solute wavefunction in this reaction.

As an example of application of the method we have considered the case of the acrolein molecule in aqueous solution. We have shown how ASEP/MD permits a unified treatment of the absorption, fluorescence, phosphorescence, internal conversion and intersystem crossing processes. Although, in principle, electrostatic, polarization, dispersion and exchange components of the solute-solvent interaction energy are taken into account, only the firsts two terms are included into the molecular Hamiltonian and, hence, affect the solute wavefunction. Dispersion and exchange components are represented through a Lennard-Jones potential that depends only on the nuclear coordinates. The inclusion of the effect of these components on the solute wavefunction is important in order to understand the solvent effect on the red shift of the bands of absorption spectra of non-polar molecules or the disappearance of... 

The use of a Cl description of the solute wavefunction, instead of MO treatment, has allowed a further development of the continuum medium model. In fact, in its traditional versions, solvent electrons are considered in terms of the fast polarization field defined by the solute charge density. The limitations of this classical description become clear if we consider... 

The electrostatic term, C/eie, in Eq. (58) is given formally in terms of the solute wavefunction i//s, the sum over solvent atomic charges qh and the sum over solute nuclear charges Zj by... 

These methods combine a QM description of the solute with a classical treatment of the solvent, which can be represented as a polarizable continuum (SCRF methods) or as discrete classical particles (QM/MM methods). In both cases the solute wavefunction is allowed to relax by the effect of the solvent reaction field, which makes possible to account for polarization effects. Furthermore, changes in molecular properties induced by solvent can be easily determined from the wavefunction of the solute in solution, which is a clear advantage with respect to pure classical methods. 

The main characteristics of ASEP/MD are (1) A reduced number of quantum calculations, that permits to increase the description level of the solute electronic stmcture which, in fact, can be described at the same level as in gas phase studies. (2) Since the solvent is described through MM force fields, there exists a great flexibility to include both bulk and specific interactions into the calculations. (3) At the end of the procediu e the solute wavefunction and the solvent stmcture become mutually equilibrated, i.e., the solute is polarized by the solvent and the solvent stmcture is in equilibrium with the polarized solute charge distribution. (4) Finally, the method permits to perform in an efftcient way optimizations on flee energy surfaces. 

MP methods have been developed for both spin-restricted HF (RHF), spin-unrestricted HF (UHF), and restricted open-shell HF (ROHF) wavefunctions to investigate both closed- and open-shell systems (see Table 2). For the calculation of electron systems with multireference character such as biradicals various multireference state (MRS) MP methods have been developed (Table 2). All these methods describe atoms, molecules, and reaction systems in the gas phase. However, many chemical reactions take place in solution phases. For this purpose, MP methods are available that start from a solvent corrected wavefunction where mostly polarizable continuum models are used (see Self-consistent Reaction Field Methods). 

In common with similar approaches that relate solvent accessible surface to cavity free energy90-93, the simple SMI model required careful parameterization, and assumed that atoms interacted with solvent in a manner independent of their immediate molecular environment and their hybridization76. In more recent implementations of the SMx approach, ak parameters are selected for particular atoms based on properties determined from the SCF wavefunction that is evaluated during calculation of the solute and solvent polarization energies27. On the other hand, the inclusion of more parameters in the solvation model requires access to substantial amounts of experimental data for the solvation free energies of molecules in the training set94 95. 

As a simple example of a QM/MM Car-Parinello study, we present here results from a mixed simulation of the zwitterionic form of Gly-Ala dipeptide in aqueous solution [12]. In this case, the dipeptide itself was described at the DFT (BLYP [88, 89 a]) level in a classical solvent of SPC water molecules [89b]. The quantum solute was placed in a periodically repeated simple cubic box of edge 21 au and the one-particle wavefunctions were expanded in plane waves up to a kinetic energy cutoff of 70 Ry. After initial equilibration, a simulation at 300 K was performed for 10 ps. 

The study of this problem is an example of the usefulness of CS ab initio methods. It is computationally easy to repeat calculations of wavefunction, energy and all the above mentioned properties for MS solutes with an increasing number n of solvent molecules and to determine at what n value the saturation for this effect is reached. Calculations on MS systems show other interesting aspects of the problem. The n S molecules must be inserted in the solvent as a supermolecule. In fact MM descriptions or Hartree QM descriptions (without exchange) have no effect on this correction. The quality of the wavefunction seems not to be important for the correction (it is important, however, for the main calculation of the property) calculations with an ONIOM scheme [26] with the solvent molecules kept at a low HF description gives the same accurate description as the full high level QM calculations [24],... 

Summing up, the structure of the effective Hamiltonian of Equation (1.107) makes explicit the nonlinear nature of the QM problem, due to the solute-solvent interaction operator depending on the wavefunction, via the expectation value of the electronic density operator. The consequences of the nonlinearity of the QM problem may be essentially reduced to two aspects (i) the necessity of an iterative solution of the Schrodinger Equation (1.107) and (ii) the necessity to introduce a new fundamental energetic quantity, not described by the effective molecular Hamiltonian. The contrast with the corresponding QM problem for an isolated molecule is evident. 

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. 

Bianco et al. [23] proposed a direct VB wavefunction method combined with a PCM approach to study chemical reactions in solution. Their approach is based on a Cl expansion of the wavefunction in terms of VB resonance structures, treated as diabatic electronic states. Each diabatic component is assumed to be unchanged by the interaction with the solvent the solvent effects are exclusively reflected by the variation of the coefficients of the VB expansion. The advantage of this choice is related to its easy interpretability. The method has been applied to the study of the several SN1/2 reactions. 

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