One-Dimensional Hamiltonians

The problem of interest in this section is defined by the simple one-dimensional Hamiltonian... 

The one-dimensional Hamiltonian H that assures us of getting the Gibbs measure as t —> 00 is therefore given by... 

The prediction of the giant resonance in microwave ionization starts from the one-dimensional Hamiltonian... 

To illustrate the appearance of resonances when a potential parameter is varied we consider the one-dimensional Hamiltonian [156],... 

As with trimethylene oxide, Wieser et al. have studied a number of deuterated derivatives of trimethylene sulfide67-69). The barriers derived vary over a range of 8 cm". This variation is probably due to the factors mentioned above for trimethylene oxide and gives an indication of the precision to which barriers may be determined using a simple effective one-dimensional Hamiltonian. 

The dynamical system thus reduces to a one-dimensional Hamiltonian one with a... 

In the absence of curvature coupling, this reduces to a simple one-dimensional Hamiltonian for motion along a vibrationally adiabatic potential,... 

In the one-dimensional algebraic model, the study of molecular vibrations requires proper treatment of bending modes as well. As alluded to earlier, the one-dimensional Hamiltonian operator is equally well suited for the description of both stretches and bends, by virtue of... 

Actually, the actions (/i, J2 are most generally line integrals and are constants of the motion for any system with a zeroth-order Hamiltonian that is decomposable into the sum of two one-dimensional Hamiltonians in some coordinate system, and this coordinate system need not be explicitly known. A more general way to write down the actions that refleas these facts is... 

Hn is a one-dimensional Hamiltonian describing the motion in the reaction coordinate and H is a Hamiltonian which describes the bound motion in the remaining orthogonal coordinates. The thermal flux operator can then be divided into two components ... 

The efficiency of the sampling scheme depends on the choice of the basis functions x/- If the Hamiltonian is separable and the xi. re eigenfunctions of the corresponding one-dimensional Hamiltonian, only a single sample is required since all samples yield equal results. Thus, the sampling scheme is ideal for the description of weakly correlated degrees of freedom. Since rotational and vibrational motion are often approximately separable, rotational motion can be described very efficiently employing the statistical approach. 

Let us consider a simple illustrative example, with orthogonal basis functions, and apply the method to both ordered and disordered systems. The Hamiltonian for these systems has zero matrix elements, j when ji-j 4. It is trivial to modify the computer programs to consider a greater or lesser number of diagonals. The matrix ha 1 f-bandwidth of 5 was chosen arbitrarily to demonstrate the technique for something other than the usual trivial case of the tridiagonal one-dimensional Hamiltonian. 

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