Beyer-Swinehart direct count

The same expression holds for the transition state, except that the sum is over s - 1 oscillators and the frequencies are the vf. The Beyer-Swinehart algoritlnn [38] makes a very efficient direct count of the number... 

The most accurate procedure for determining the density of harmonic vibrational states is by the direct count method. A particularly clever scheme for doing this was proposed by Beyer and Swinehart (1973). As demonstrated by Gilbert and Smith (1990), this approach is based on the convolution of state densities. Suppose that the system consists of s harmonic oscillators with vibrational frequencies, co, = v,/c (cm" )- Each will have a series of equally spaced states located at , = nco, (n = 0, 1,. . . ). We choose the zero of energy at the molecule s zero point energy, and divide the energy into bins. The vibrational frequencies must be expressed as integral numbers of bin sizes, for example, as multiples of 10 cm for a 10 bin size. A convenient bin size is 1 cm so that the s frequencies can be simply rounded off to the nearest wave-number. 

The quantum mechanical trace in equation (17) is most readily carried out for separable Hamiltonians. This became feasible for collections of harmonic oscillators with the advent in 1973 of the discrete convolution algorithm of Beyer and Swinehart, which was generalized to the case of arbitrary separable Hamiltonians by Stein and Rabinovitch. These direct-count methods are exceedingly fast and exact for separable Hamiltonians. Direct counts can also be implemented for energy levels derived from perturbative expansions of the Hamiltonian. 

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