Surface Hamiltonian

In this section we will consider the same Hamiltonian as above, in which we add a surface Hamiltonian to account for a specific interaction with the wall. Our Hamiltonian is then given by = coui ideai loc nonioc surf j stead of (24) we will Consider a... 

It is seen that the symmetry of the non-coulombic non-local interaction in the bulk phase forces the symmetry of the localized interaction with the wall. If we omitted the surface Hamiltonian and set / = 0 we would still obtain the boundary condition setting the gradient of the overall ionic density to zero. The boundary condition due to electrostatics is given by... 

Fig. 5.4. Initial and final wavepackets of LiH excited by the same quadratically chirped pulse as that in Fig. 5.3. Initial wavepacket refers to the wavepacket propagated up to the pulse center according to the ground surface Hamiltonian Hg excited wavepacket (approximate) refers to the result obtained using the level approximation as Pi2 x) g(x, 0) 2 and excited wavepacket (exact) refers to the numerical solution of (5.1). The latter two are backward-propagated to the pulse center at time t = tp according to the excited state Hamiltonian He...

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... 

We have taken the nc arest integer value of T and ignored the resulting small error in the centrifugal potential in applying boundary conditions[45, 19]. Also, we have noticed that the eigenstates of the surface Hamiltonian defined in Eq. (20) approach the asymptotic vibrational/rotational states for large p such that... 

There is a range of iterative diagonalization routines to choose between, including classical orthogonal polynomial expansion methods [48], Davidson iteration[58] and Krylov subspace iteration methods. Here the popular Lanezos method[59] will be discussed in the context of finding the eigenstates of the surface Hamiltonian appearing in the hyperspherical coordinate method. 

As an alternative that solves the kinetic coupling problem. Miller and co-work-ers suggested an all-Cartesian reaction surface Hamiltonian [27, 28]. Originally this approach partitioned the DOF into atomic coordinates of the reactive particle, such as the H-atom, and orthogonal anharmonic modes of what was called the substrate. If there are N atoms and we have selected reactive coordinates there will he Nyi = 3N - G - N-g harmonic oscillator coordinates and the reaction surface Hamiltonian reads... 

Another obvious defect of both the RLM and BCRLM models is that they assume a collinearly dominated reaction intermediate. While the potential energy surfaces for many collision systems do favor collinear geometries, there are of course many reactions which do not. Extensions of the BCRLM model are therefore needed to treat noncollinear systems, perhaps along the lines defined by the Carrington and Miller reaction surface Hamiltonian theory. 

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. 

LAM = large amplitude motion RP = reaction path RPH = reaction path Hamiltonian RSH = reaction surface Hamiltonian SAM = small amplitude motion SRP = specific reaction parameter SRPH = solution reaction path Hamiltonian. 

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