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arXiv:cond-mat/0001207 · Reviews of Modern Physics

Molecular dynamics for fermions

Hans Feldmeier, Jürgen Schnack

Published Jul 1, 2000• 230 citations
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Abstract

The time-dependent variational principle for many-body trial states is used to discuss the relation between the approaches of different molecular-dynamics models that describe indistinguishable fermions. Early attempts to include effects of the Pauli principle by means of nonlocal potentials, as well as more recent models that work with antisymmetrized many-body states, are reviewed under these premises.

DOI
10.1103/revmodphys.72.655

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The time-dependent variational principle for many-body trial states is used to discuss the relation between the approaches of different molecular-dynamics models that describe indistinguishable fermions. Early attempts to include effects of the Pauli principle by means of nonlocal potentials, as well as more recent models that work with antisymmetrized many-body states, are reviewed under these premises.
classical many body models supplemented by repulsive momentum-dependent potentials to simulate the Pauli and Heisenberg principles have been use with some success for nuclear and atomic bound state and collision problems. They are capable of describing mean ground state properties, hydrodynamics, shocks (if warranted by the physics), viscosity, correlations, clustering, fragmentation, etc. We have become interested in the Feldmeier Gaussian packet formulation since it is based on a variational principle using trial wave functions. We discuss some limitations of the model and discuss further directions of investigation.
Enforcing the Pauli principle in many-body systems of fermions to ensure an antisymmetric wave function is typically a numerically expensive task. Numerical methods, such as coupled-cluster, correlated basis function, Monte Carlo, and traditional configuration interaction, scale with powers of NNN that make the determination of the energy spectrum of these systems a challenge. In a recent paper, we successfully obtained energies for large systems of fermions in the unitary regime in which we applied the Pauli principle ``on paper'' without obtaining an explicit wave function. This approach, an extension of the symmetry invariant perturbation method to fermions, applies the Pauli principle by imposing restrictions on the quantum numbers used to obtain the energy. As a followup to this study, we perform an explicit test of the validity of this application of the Pauli principle by comparing our energy results against the energies corresponding to explicitly antisymmetrized wave functions for an exactly solvable system of harmonically confined, harmonically interacting fermions. Our results show that our simple method of applying the Pauli principle selects from a spectrum of possible many-body energies only those energies that correspond to explicitly antisymmetrized wave functions. Our results were tested for values of NNN through $\ensuremath{\approx}10 000$ and for weak and strong interactions both repulsive and attractive.
Fermionic Molecular Dynamics (FMD) models a system of fermions by means of many-body states which are composed of antisymmetrized products of single-particle states. These consist of one or several Gaussians localized in coordinate and momentum space. The parameters specifying them are the dynamical variables of the model. As the repulsive core of the nucleon-nucleon interaction induces short range correlations which cannot be accommodated by a Slater determinant, a novel approach, the unitary correlation operator method (UCOM), is applied. The unitary correlator moves two particles away from each other whenever their relative distance is within the repulsive core. The time-dependent variational principle yields the equations of motion for the variables. Energies of the stationary ground states are calculated and compared to exact many-body results for nuclei up to Ca 48. Time-dependent solutions are shown for collisions between nuclei.
For a given statistical state at time t0{t}_{0}t0​, the expectation value of some observable at a later time t1{t}_{1}t1​ is expressed in a variational form. Different trial choices for the quantities to be varied (state and observable) generate different approximations, in which the evolution of the state is optimally fitted to the measured quantity. Examples are given in the context of mean-field theories.