Surfaces hamiltonian matrix elements

The two diabatic surfaces and wave functions are allowed to couple by way of a Hamiltonian matrix element denoted / ... 

The adsorption of H on Ni has been the subject of a recent EH-type calculation by Fassaert et al. (68). A finite-size representation consisting of up to 13 atoms was employed for the (111), (100), and (110) nickel surfaces. In this calculation, the d orbitals of nickel were taken as a linear combination of two Slater orbitals in order to improve the fit with more exact SCF atomic calculations, as described in Section 11.B.1. In addition, the diagonal Hamiltonian matrix elements were modified to depend on charge, similar to Eq. (11), in order to check the charge separation predicted by a noniterative calculation. 

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. 

A more elegant and convenient way than Bq.(2.174) to compute the surface Green s function from the bulk Green s functions, is to consider Hamiltonian matrix elements Hfj to belong to the Hamiltonian matrix of the closed chain and to consider... 

Several approaches have been used for determining functional forms for the pair sum Eq. [12]. Once the Hamiltonian matrix elements had been specified, Chadi, for example, used a near-neighbor harmonic interaction for covalent materials where the force constants and minimum energy distances were fit to bulk moduli and lattice constants, respectively. This expression was then used to predict energies and bond lengths for surfaces and related structures. More recently. Ho and coworkers have fit the pair sum to the universal binding energy relation. This reproduces not only lattice constant and bulk modulus, but also ensures reasonable nonlinear interatomic interactions that account for properties like thermal expansion. 

Conical intersections between electronically adiabatic potential energy surfaces are not only possible but actually quite frequent, if not prevalent, in polyatomic systems. Some examples are triatomic systems whose isolated atoms have 5 ground states [156, 157] such as H3 and its isotopomers (DH2, HD, HDT, etc.), LiH2 and its isotopomers, and tri-alkali systems such as Na3 and LiNaK. Many other kinds of polyatomic molecules also display such intersections. The reason is that they have three or more internal nuclear motion degrees of freedom, and only two independent relations between electronic Hamiltonian matrix elements are sufficient for the existence of doubly degenerate electronic energy eigenvalues as a result, these relations are easy to satisfy [157]. The systems C2H [159, 160], NH2 [161], NO2 [162, 163], and HO2 [164, 165] represent just a few examples. 

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... 

With the gas-phase potential surface we can obtain the solution Hamiltonians by eq. (3.23), adding the solvent-solute interaction to the classical part of the diagonal EVB matrix elements. That is, we use... 

Initially we consider a simple atom with one valence electron of energy and wave function which adsorbs on a solid in which the electrons occupy a set of continuous states Tj, with energies Ej. When the adsorbate approaches the surface we need to describe the complete system by a Hamiltonian H, including both systems and their interaction. The latter comes into play through matrix elements of the form Vai = / We assume that the solutions T j to this eigen value problem... 

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 simplest model of a solid is a linear chain of N atoms, with one end of the chain corresponding to the surface. If an atomic orbital r> (r = 1,..., N) is associated with the rth atom, then, in the tight-binding approximation, the matrix elements of the Hamiltonian for the solid, can... 

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. 

Using the properties of the Green s function (see Appendix B), the evaluation of the effect of distortion to transmission matrix elements can be greatly simplified. First, because of the continuity of the wavefunction and its derivative across the separation surface, only the multiplier of the wavefunctions at the separation surface is relevant. Second, in the first-order approximation, the effect of the distortion potential is additive [see Eq. (2.39)]. Thus, to evaluate the multiplier, a simpler undistorted Hamiltonian might be used instead of the accurate one. For example, the Green s function and the wavefunction of the vacuum can be used to evaluate the distortion multiplier. 

Consider a metal surface with one-electron states k> with energies ek, and an adsorbate with a single valence state a> of energy sa. When the adsorbate approaches the surface from far away, to a position just outside, the two sets of states are coupled by matrix elements Vak = , where H is the Hamiltonian of the combined system. If we expand the solutions /> of H in terms of the free adsorbate and surface solutions ... 

Suppose Va and V/, are diabatic surfaces which cross at some configuration and let cj/2 be the matrix element of the electronic Hamiltonian between the two diabatic states. In other words, the adiabatic curves are the eigenvalues of the potential energy matrix. 

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