Monte Carlo algorithm for calculating moments of atomic transition arrays

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Abstract
A new method for calculating moments of atomic transition arrays is described. It is based on the collective vector method described in earlier publications. In this new approach a single collective state vector is generated from a single parent state vector encompassing the entire parent basis. The amplitudes of the basis vectors comprising the parent state vector are randomized. Thus it is representative of the entire parent manifold. We show that a statistical estimate of the transition moments of the array is given by a suitable combination of moments of the matrix element of the E1 (electric dipole) operator between the representative parent state vector and the collective state vector derived from it. Although the detailed characteristics of the dispersion in these statistical results are determined by the detailed characteristics of the Hamiltonian and the model space, we find that overall the dispersion decreases inversely as the square root of the dimension of the model space. This is in keeping with results obtained earlier for the Hamiltonian moments of nuclear systems and means, speaking broadly, the larger the problem the more accurate the method.