Knowledge of the nodes of the many-fermion wavefunction would make possible exact calculation of the properties of fermion systems by Monte Carlo methods. Little is known about nodes of many-body fermion systems even though the one-dimensional case is ubiquitous in textbooks on quantum mechanics. The nodes referred to here are the nodes of the exact many-body wavefunction and are very different from the nodes of orbitals.
In the absence of a rigorous simulation method for fermion systems, the fixed-node approximation has been found to be a useful and powerful approach. One assumes knowledge of where the exact wavefunction is positive and negative based on the nodes of a trial wavefunction. The Schrödinger equation in imaginary time is solved by simulating the diffusion process with branching within the regions bounded by the assumed nodes.
For the ground state, Ceperley (1991) has proved that ground state nodal cells have the tiling property (i.e., there is only one type of nodal cell, all other cells being related by permutational symmetry). The tiling property is the generalization to fermions of the theorem that a bosonic ground state is nodeless.
The nodal hypervolumes of a series of atomic N-body Hartree-Fock level electronic wavefunctions have been mapped by using a Monte Carlo simulation in 3N-dimensional configuration space (Glauser et al., 1992). The basic structural elements of the domain of atomic and molecular wavefunctions have been identified as nodal regions and permutational cells (identical building blocks). The results of this study on lithium-carbon indicate that Hartree-Fock wavefunctions generally consist of four equivalent nodal regions (two positive and two negative), each constructed from one or more permutational cells.
A generalization of the fixed-node method has been proposed that could solve the fermion problem at finite temperature if only the nodes of the fermion density matrix were known (Ceperley, 1991).
References
Ceperley, D.M., 1991, J. Stat. Phys. 63:1237.
Glauser, W.A., W.R. Brown, W.A. Lester, Jr., D. Bressanini, B.L. Hammond, and M.L. Koszykowski, 1992, J. Chem. Phys. 97:9200.
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