Power Series Greens Function

The computational effort required by an exact quantum calculation grows exponentially with the size of the system. Accordingly, the amount of information obtained from a quantum calculation grows exponentially as well. The most extreme case is encountered when one studies the dynamics of a pure state in which all the relative phase information is required. It therefore seems reasonable that the treatment of mixed states, which provides less detailed dynamical information, should be less computationally demeinding and thus more appHcable for the study of larger chemical systems. For example, the canonical rate constant for a bimolecular chemical reaction can be expressed as 

The calculation of N(E) by an absorbing boundary condition Green s function relies on the construction and inversion of a non-Hermitian Hamiltonian matrix. 

Forming the matrix representation of the Hamiltonian operator and manipulating the Hamiltonian matrix to obtain the observable of interest can be computationally intensive. A discrete variable representation [53-55] (DVR) can ameliorate both of these difficulties. That is, the construction of the Hamiltonian matrix is particularly simple in a DVR because no multidimensional integrals involving the potential function are required. Also, the resulting matrix is sparse because the potential is diagonal, which expedites an iterative solution [37, 38]. In the present research we use a sinc-function based DVR (vide infra) first developed by Colbert and Miller [56] for use in the 5—matrix version of the Kohn variational principle [57, 58], and used subsequently for 5—matrix calculations [37, 38] in addition to N(E) calculations [23]. This is a uniform grid DVR which is constructed from an infinite set of points. It is 

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W. H. Thompson and W. H. Miller, State-specific reaction probabilities with a power series Green function, Chem. Phys. Lett. 206 123 (1993). 

We develop in Chapter 3 a time dependent calculation of the ABC Green s function on a grid, called the power series Green s function (PSG). We compute the cumulative reaction probability for the collinear H+H2 test problem. The similarity of our approach in Chapter 3 to modern path integral methods is also discussed. 

We now give the sinc-function based DVR of the power series Green s function. For simplicity, we restrict our attention to a one-dimensional system. The multidimensional generalization is straightforward, and will be given afterwards. Letting... 

To conclude the description of the power series Green s function (PSG), we wish to underscore how the recursive calculation proceeds. To compute the column of the ABC Green s function, denoted by Gj with elements given by (Gj)j/ = G(qj/,qj), one forms the dot product of the matrix in Eq. (3.20) with the column of the identity matrix. As such, the column of G(jE) is... 

Figure 3.2 Cumulative reaction probability for the coUinear H+H2 reaction. The circles are the J —matrix propagation results (summed over final states) by Bondi et aL, and the line is the power series Green s function results. Excellent agreement is obtained over the entire energy range. The staircase structure is evident, with N E) = 1 when the is one open channel, and = 1.6 with two open channels. The non-monotonic increase indicates recrossing dynamics, and the peak at = 0.87 eV indicates a collision complex for H3.

A brief summary of the power series Green s function appeared previously in the following article ... 

The main problem now is to calculate the action of the Green s function onto the initial state xo- The standard strategy is to expand G in a power series of H. For vanishing A, a highly efficient expansion exists [215,235] in terms of Chebyshev polynomials, Tn H), which is similar to the one used in short-time wave packet propagations [166,171]. For finite A, this expansion has to be modified to account for the absorbing potential. As was shown by Mandelshtam and Taylor [221], the analytically continued Chebyshev polynomials, can be used for this purpose. If the initial... 

The two important convergence parameters introduced by the power series expansion of the ABC Green s function are the total propagation time T, and the time step At. The total time T represents the time required for reaction and absorption by e(q). The time step At is the duration for which the STP is a faithful representation of the propagator. These are both a function of the dynamics and the choice of absorbing potential. We measure At in units of a fundamental small time given by... 

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