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Monte Carlo sampling for atomic and molecular clusters with fixed energy and angular momentum
1.For an introduction to molecular dynamics (MD), and Monte Carlo methods see, for example, D. W. Heerman, Computer Simulation Methods (Springer-Verlag, New York, 1986);
1.or E. Blaisten-Barojas, in Elemental and Molecular Clusters, edited by G. Benedek, T. P. Martin, and G. Pacchioni (Springer-Verlag, Berlin, 1988).
1.Applications of MD to Ar clusters are considered by T. Beck, J. Jellinek, and R. S. Berry, J. Chem. Phys. 87, 545 (1987),
1.for example. A Monte Carlo treatment of Ar cluster properties is employed by H. Davis, J. Jellinek, and R. S. Berry, J. Chem. Phys. 86, 6456 (1987).
2.S. Nosé, Mol. Phys. 52, 255 (1984).
3.See, for example, J. W. Brady, J. D. Doll, and D. L. Thompson, J. Chem. Phys. 74, 1026 (1981).
4.E. S. Severin, B. C. Freasier, N. D. Hamer, D. L. Jolly, and S. Nordholm, Chem. Phys. Lett. 57, 117 (1978);
4.H. Schranz, S. Nordholm, and G. Nyman, J. Chem. Phys. 94, 1487 (1991).
5.G. Nyman, S. Nordholm, and H. W. Schranz, J. Chem. Phys. 93, 6767 (1990).
6.Some problems associated with Metropolis Monte Carlo are described in D. Chandler, Introduction to Modern Statistical Mechanics (Oxford University, New York, 1987).
7.R. S. Dumont, J. Chem. Phys. (submitted).
8.There are circumstances for which sampling of more than one set of momenta per configuration does give the most efficient method. Consider sampling with respect to a modified density such as where is a configuration-space function which is difficult to compute. This sort of sampling can arise in reaction dynamics studies. The method of this paper applies to this sampling if the configuration-space sampling is adapted to account for the additional density factor. Since the configuration-space density is computationally expensive, it is important to make the most of every sampled configuration. This is accomplished by generating multiple sets of momenta, and corresponding phase-space points, for every sampled configuration.
9.The Metropolis Monte Carlo method is reviewed in G. Bhanot, Rep. Prog. Phys. 51, 429 (1988);
9.and J. P. Valleau and S. G. Whittington, in Statistical Mechanics, Part A, edited by B. J. Berne (Plenum, New York, 1977).
10.The “inverse cumulative probability” sampling method is described in R. D. Richtmeyer, Principles of Advanced Mathematical Physics (Springer-Verlag, New York, 1978), Sec. 13.8.
11.A random vector u distributed uniformly on a sphere of radius r is generated as follows. Two random variates, and uniform on (0,1), are generated by usual means. They are ordered, and relabeled if necessary, so that They now partition the unit interval into three identically distributed random intervals, and correctly distributed u vector is constructed via and The signs are specified by independent random-number generation.
12.Isometric matrices are like projectors, except that they map vectors from one representation to another. For example, maps vectors represented in terms of the usual space-fixed Cartesian basis, to vectors represented in the basis of nonzero eigenvectors. is the projector onto the nonzero eigenspace of which operates solely within the space-fixed Cartesian representation.
13.The Metropolis process in configuration space can be modified to reject exact collinearity of the first three atoms. Since collinearity of three atoms is a measure zero circumstance, this modification has no effect on process statistics. Note that this proposed modification ensures that all inertia matrices, with are nonsingular, as required by Eq. (14);
13.see Ref. 17.
14.T. L. Hill, An Introduction to Statistical Thermodynamics (Addison-Wesley, Reading, MA, 1960).
15.For example, T. L. Beck and T. L. Marchioro, J. Chem. Phys. 93, 1347 (1990).
16.R. S. Dumont, J. Chem. Phys. 91, 6839 (1989).
16.See also D. Chandler, J. Chem. Phys. 68, 2959 (1978).
16.Another envisioned application of the E Jensemble sampling algorithm of this paper is the computation of statistical rate constants for unimolecular processes. See R. S. Dumont, J. Comput. Chem. 12, 391 (1991).
17.Suppose v is an eigenvector of associated with zero eigenvalue. In this case, Where is the angle between v and The above expression is positive unless for all k. That is, v must be parallel to every This is possible only if all the are parallel. Thus, is nonsingular unless all n atoms are collinear. Note that we have also shown that the inertia matrix is positive definite.
18.The β integral is given in F. Oberhettinger, Tables of Fourier Transforms and Fourier Transforms of Distributions (Springer-Verlag, New York, 1990). See p. 200, result 3.15, with
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