Dynamics on statistical samples of potential energy surfaces

Prior work [K. D. Ball and R. S. Berry, J. Chem. Phys. 109, 8541 (1998); 109, 8557 (1998)] has demonstrated that master equations constructed from a complete set of minima and transition states can capture the essential features of the relaxation dynamics of small systems. The current study extends this work by examining robustness of master equations based only on statistical samples of the surface topography, to make it possible to work with larger systems for which a full topographical description is either impossible or infeasible. We ask whether such "statistical" master equations can predict relaxation on the entire potential energy surface. Our test cases are Ar-11 and Ar-13, for which we have extensive databases: 168 geometrically distinct minima and 1890 transition states for Ar-11, and 1478 minima and 17,357 saddles for Ar-13 which we assume represent complete set of stationary points. From these databases we construct statistical sample sets of transition sequences, and compare relaxation predictions based on these with those obtained from the master equations representing the full potential surfaces, and with results of molecular dynamics simulations. The slowest, rate-controlling relaxation timescale converges at moderate temperatures as the number of sequences in a sample reaches similar to 1000, approaching convergence for as few as 100 sequences. The asymptotic value of the slowest nonzero relaxation rate is essentially identical to that from the full potential energy surface. Equilibrium properties from the statistical samples match those of the full surface. To achieve convergence within a factor of 2 of full-surface rates, the number of sequences required is approximately the same for Ar-13 as for Ar-11. Precise convergence, however, appears to scale as the number of stationary points. These results reveal how the reliability and precision of kinetic predictions from statistical master equations depends on the size of the statistical database. (C) 1999 American Institute of Physics. [S0021-9606(99)00629-7].