Quantal cumulant mechanics and dynamics for multidimensional quantum many-body clusters

摘要

We developed the quantal cumulant mechanics for treating multidimensional quantum many-body clusters including two-body interaction such as the Morse potential. To evaluate an effective potential appearing in the actual calculation, a Gaussian fitting method was adopted to approximate the Morse potential. The number of the Gaussians that are required to reproduce the total energy of a classical three-dimensional (3D) Morse 3 (M_3) cluster is 31, where the error is 10~(-6). We compared structures of the classical M_3 cluster with those of quantum counterpart and found that the quantum structure have broad distribution due to zero point vibration effects. The symmetry of the cluster becomes lower from D_(3h) to D_(2h) by applying the diagonal approximation to the cumulant matrices. Conversely, the original and spherical approximation holds the symmetry constraint. We also perform the same analyses on 2D M_4 cluster with two different stable structures, where one has D_(3h) symmetry and the other D_(2h) symmetry. In the latter case, the diagonal approximation accidentally gives the same results as the original one, when two of three Cartesian axes coincide with the cluster symmetric axes. When the cluster rotates with respect to these axes, the results of the diagonal approximation deviate from those by the original one and the artificial symmetry breaking is also found. We also evaluate the optimized structures of the highly symmetric small Morse clusters M_n ranging from n = 4-7 and compare with those with the corresponding classical ones. We found that the errors of both the diagonal and spherical approximations in total energy decrease with the number of particles. This fact indicates that these approximations will be useful to investigate static and dynamical properties of many-particle quantum clusters with low computational cost.