We calculated the atomization energy of aluminum clusters (Al 2 -Al 7 ) with several multilevel methods, including MCG3/3 and G3X, that have been previously shown to provide high accuracy for atomization energies. We used the results to test a number of hybrid density functional theory (HDFT) methods and found that the PBE0 method is in best agreement with the accurate methods. We then used the PBE0/MG3 method to develop a more extensive data set for the energies of small aluminum clusters (Al 2 -Al 13 ), and this was used to test a number of semiempirical methods, in particular Austin model 1 (AM1), modified neglect of differential overlap (MNDO), modified symmetric-orthogonalized intermediate neglect of differential overlap (MSINDO) with and without d-functions, parametrized model 3 (PM3), and the tight-binding total energy (TBTE) method, for geometries, energies, and multiplicities of Al clusters. The AM1 model and MSINDO model are the most accurate of the semiempirical methods for energetics, and PM3 is the most accurate method for geometries.

A stochastic model of a continuous nondemolition observation of a free quantum Brownian motion is presented. The nonlinear stochastic wave equation describing the posterior dynamics of the observed quantum system is solved in a Gaussian case for a free particle of mass m > 0. It is shown that the dispersion of the wave packet does not increase to infinity like for the free unobserved particle but tends to the finite limit τ 2 ∞ = ( /2λm) 1/2 where λ is the accuracy coefficient of an indirect nondemolition measurement of the particle position, and is Planck constant.

Nineteen analytic potential energy functions (PEFs) for aluminum (three pairwise additive ones, six nonpairwise additive ones with three-body terms, and ten embedded atom-type PEFs) were obtained from the literature. The PEFs were tested and reparametrized using a diverse training set that includes 20 potential energy curves and a total of 224 geometries for five aluminum clusters Al N (N ) 2, 3, 4, 7, and 13) computed using hybrid density functional theory, as well as the experimental face-centered cubic cohesive energy and lattice constant. The best PEFs from the literature have mean unsigned errors (MUEs) over the clusters in the data set of ∼0.12 eV/atom. The best reparametrized PEFs from the literature have MUEs of 0.06 eV/atom. The data set is also used to develop, parametrize, and systematically study the effectiveness of several functional forms designed specifically to model many-body effects in clusters, including bond angle, screening, and coordination number effects; a total of eighteen new PEFs are proposed and tested. The best potential overall has an MUE of 0.05 eV/atom, explicitly includes screening and coordination number effects, features linear scaling, and incorporates the accurate two-body and bulk limits.

A stochastic model for the continuous nondemolition ohservation of the position of a quantum particle in a potential field and a boson reservoir is given. lt is shown that any Gaussian wave function evolving according to the posterior wave equation with a quadratic potential collapses to a Gaussian wave packet given by the stationary solution of this equation.. * M.I.E.M., D. Vusovsky 3/12,