The thermal equilibrium aspects of the time dependent Hartree and the locally enhanced sampling approximations: Formal properties, a correction, and computational examples for rare gas clusters

The thermal properties of the ‘‘trajectory bundles’’ and the LES version of the classical Hartree approximation are investigated. A systematic protocol to find an effective Lagrangian from which the trajectory bundles equations of motion can be derived is introduced. It is shown that the trajectory bundles give incorrect virials. The errors are proportional to the number of copies. A correction is proposed that is based on a binary collision model. The corrected LES approximation is tested by calculating the following properties: (i) The expectation values of the kinetic energy and the virials; (ii) the spatial correlation function; (iii) the relaxation time of excess local heat to the ‘‘bath;’’ and (iv) the diffusion time for a He atom through a Xe cluster. The collision correction of LES (cLES) significantly improves the representation of the above four properties compared to exact calculations.