Hamiltonian matrices, potential energy surfaces

The previous treatment relied on the assumption that the transition occurs on a single potential energy surface V(x) characterized by a barrier separating two wells. This potential is actually created from the terms of the initial and final electronic states. The separation of electron and nuclear coordinates in each of these states gives rise to the diabatic basis with nondiagonal Hamiltonian matrix... 

In such a case the last choice is to take the direction of the eigenvector of the only one nonzero eigenvalue of the rank one Hessian matrix of the difference between the two adiabatic potential energies [51]. In the vicinity of conical intersection, the topology of the potential energy surface can be described by the diadiabatic Hamiltonian in the form... 

The obstacle to simultaneous quantum chemistry and quantum nuclear dynamics is apparent in Eqs. (2.16a)-(2.16c). At each time step, the propagation of the complex coefficients, Eq. (2.11), requires the calculation of diagonal and off-diagonal matrix elements of the Hamiltonian. These matrix elements are to be calculated for each pair of nuclear basis functions. In the case of ab initio quantum dynamics, the potential energy surfaces are known only locally, and therefore the calculation of these matrix elements (even for a single pair of basis functions) poses a numerical difficulty, and severe approximations have to be made. These approximations are discussed in detail in Section II.D. In the case of analytic PESs it is sometimes possible to evaluate these multidimensional integrals analytically. In either case (analytic or ab initio) the matrix elements of the nuclear kinetic energy... 

The empirical valence bond (EVB) approach introduced by Warshel and co-workers is an effective way to incorporate environmental effects on breaking and making of chemical bonds in solution. It is based on parame-terizations of empirical interactions between reactant states, product states, and, where appropriate, a number of intermediate states. The interaction parameters, corresponding to off-diagonal matrix elements of the classical Hamiltonian, are calibrated by ab initio potential energy surfaces in solu-fion and relevant experimental data. This procedure significantly reduces the computational expenses of molecular level calculations in comparison to direct ab initio calculations. The EVB approach thus provides a powerful avenue for studying chemical reactions and proton transfer events in complex media, with a multitude of applications in catalysis, biochemistry, and PEMs. 

A normal-mode representation of the Hamiltonian for the reduced system involves the diagonalization of the projected force constant matrix, which in turn generates a reduced-dimension potential-energy surface in terms of the mass-weighted coordinates of the reaction path [64] ... 

If, 2 3 are electronic orbital functions for a molecule with internal coordinates Q and Hamiltonian H then the potential energy surfaces of the molecule are eigenvalues of the matrix given in equation 10 ... 

The valence-bond approach plays a very important role in the qualitative discussion of chemical bonding. It provides the basis for the two most important semi-empirical methods of calculating potential energy surfaces (LEPS and DIM methods, see below), and is also the starting point for the semi-theoretical atoms-in-molecules method. This latter method attempts to use experimental atomic energies to correct for the known atomic errors in a molecular calculation. Despite its success as a qualitative theory the valence-bond method has been used only rarely in quantitative applications. The reason for this lies in the so-called non-orthogonality problem, which refers to the difficulty of calculating the Hamiltonian matrix elements between valence-bond structures. 

With these matrix elements in hand, we can constmct solutions to the coupled differential equations 2.33. However, to understand the character of the potential surface, it is useful to construct adiabatic potential energy surfaces. This means that, for a fixed relative position of the molecules (R, 6), we find the energy spectrum of Equations 2.33 by diagonalizing the Hamiltonian + Hs, where is a... 

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