Reaction-surface Hamiltonian method

Carrington and Miller (235) developed a method called the reaction-surface Hamiltonian for reactions with large amplitudes perpendicular to the reaction path and for some types of reactions with bifurcation of the reaction path. In contrast to the reaction-path Hamiltonian method, in the reaction-surface Hamiltonian method two coordinates are extracted from the complete coordinate set. One coordinate describes motion along the reaction path and the second one describes the large-amplitude motion. Potential energy in space of the remaining 3JV — 8 coordinates perpendicular to the two-dimensional reaction surface is approximated by quadratic functions. It... 

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... 

The BCRLM is by its very nature constrained to treating collinearly dominated reaction processes. One could extend the method to non-colllnear systems by Including effective potential terms and more complicated kinetic energy operators to represent the motion of the reacting system along its (bent) minimum energy path from reactants to products. This is indeed an example of the Carrington and Miller reaction surface Hamiltonian theory, which at present is probably the most fruitful approach for noncollinear systems. 

Thus, the apparently most accurate theoretical estimate of the barrier to proton transfer in a malonaldehyde molecule, determined as a difference between the energies of the structures XIa and XIc, is so far 4.3-5.0 kcal/mol. This value explains well fast (k > 10 s" ) tautomerization XIa F XIb observed in solution by the NMR method. Note, however, that calculations by means of a reaction surface Hamiltonian constructed for malonaldehyde [63] gave the barrier of 6.6 0.5 kcal/mol. 

The approach presented above is referred to as the empirical valence bond (EVB) method (Ref. 6). This approach exploits the simple physical picture of the VB model which allows for a convenient representation of the diagonal matrix elements by classical force fields and convenient incorporation of realistic solvent models in the solute Hamiltonian. A key point about the EVB method is its unique calibration using well-defined experimental information. That is, after evaluating the free-energy surface with the initial parameter a , we can use conveniently the fact that the free energy of the proton transfer reaction is given by... 

Figure 4-2. Computed potential energy surface from (A) ab initio valence-bond self-consistent field (VB-SCF) and (B) the effective Hamiltonian molecular-orbital and valence-bond (EH-MOVB) methods for the S 2 reaction between HS- and CH3CI...

In order to obtain the potential energy surfaces associated with chemical reactions we, typically, need the lowest eigenvalue of the electronic Hamiltonian. Unlike systems such as a harmonic oscillator and the hydrogen atom, most problems in quantum mechanics cannot be solved exactly. There are, however, approximate methods that can be used to obtain solutions to almost any degree of accuracy. One such method is the variational method. This method is based on the variational principle, which says... 

We have described a mixed MOVB model for describing the potential energy surface of reactive systems, and presented results from applications to SN2 reactions in aqueous solution. The MOVB model is based on a BLW method to define diabatic electronic state functions. Then, a configuration interaction Hamiltonian is constructed using these diabatic VB states as basis functions. The computed geometrical and energetic results for these systems are in accord with previous experimental and theoretical studies. These studies show that the MOVB model can be adequately used as a mapping potential to derive solvent reaction coordinates for... 

We would like to complete this section by briefly describing some of the recent developments on electronically non-adiabatic reactions. From the standpoint of the coupled-channels method, there is in principle no added difficulty in treating more than one electronic state of the reactive system. This may be done, for example, by keeping electronic degrees of freedom in the Hamiltonian and expanding the total scattering wavefunction in the electronic states of reactants and products. In practice, however, some new difficulties may arise, such as non-orthogonality of vibrational states on different electronic potential surfaces. There is at present a lack of quantum mechanical results on this problem. 

---