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

Abstract

We developed the quantal cumulant mechanics for treating multidimensional quantum many-body clusters including twobody 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 À6 . 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 ¼ 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.

Quantal Cumulant Mechanics as Extended Ehrenfest Theorem

“…Recently Shigeta and co-workers derived a general expression for the expectation value of an arbitrary operator by means of cumulants rather than moments [9][10][11][12][13][14][15][16][17][18][19]. For one-dimensional case, the expectation value of a differential arbitrary operator, A s ( q, p ) , that consists of the symmetric sum of power series of q and p is derived as…”

Semiquantal molecular dynamics simulations of hydrogen-bond dynamics in liquid water using multi-dimensional Gaussian wave packets

“…Note that the analogous approximations have previously been suggested within the framework of the non-Hamiltonian formulation. 52,54,59,60,84 In the single-particle approximation, we assume that the trial wave function of the system can be factorized into a direct product of correlated 3D Gaussian functions as…”

“…denotes the pair potential function depending only on the distance between the jth and kth atoms. Based on the facts that any central potential can be fitted by a sum of Gaussian functions centered at the origin and that a Gaussian in the distance q jk centered at the origin remains to be a Gaussian after transformation into the Cartesian coordinates, the pair potential can be approximated in terms of a sum of Gaussians as 52,54,59,60,84…”

“…jk }. The evaluation of the potential expectation is now reduced to that of the following Gaussian integral: 59,84 exp…”

“…72 Interestingly, the SQGWP theory is closely related to the second-order quantized Hamilton dynamics (QHD-2), [77][78][79][80] and is essentially equivalent to the second-order quantal cumulant dynamics (QCD-2). [81][82][83][84] It is also intriguing to note that the SQGWP framework has recently been extended to describe many electron systems 85,86 and electron-nuclear systems in the non-Born-Oppenheimer framework. 87 Although the SQMD simulation method with the TDH ansatz can treat large molecular systems, the rotational invariance of WPs should be retained within the Hartree approximation.…”