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Physical Review Letters

Unexpected Symmetry in the Nodal Structure of the He Atom

Dario Bressanini, PETER J. REYNOLDS

Published Sep 6, 2005• 47 citations
Publisher

Abstract

The nodes of even simple wave functions are largely unexplored. Motivated by their importance to quantum simulations of fermionic systems, we have found unexpected symmetries in the nodes of several atoms and molecules. Here, we report on helium. We find that in both ground and excited states the nodes have simple forms. In particular, they have higher symmetry than the wave functions they come from. It is of great interest to understand the source of these new symmetries. For the quantum simulations that motivated the study, these symmetries may help circumvent the fermion sign problem.

DOI
10.1103/physrevlett.95.110201

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The nodes of even simple wave functions are largely unexplored. Motivated by their importance to quantum simulations of fermionic systems, we have found unexpected symmetries in the nodes of several atoms and molecules. Here, we report on helium. We find that in both ground and excited states the nodes have simple forms. In particular, they have higher symmetry than the wave functions they come from. It is of great interest to understand the source of these new symmetries. For the quantum simulations that motivated the study, these symmetries may help circumvent the fermion sign problem.
Although quantum Monte Carlo is, in principal, an exact method for solving the Schroedinger equation, it is well-known that systems of Fermions still pose a challenge. Thus far all solutions to the "sign problem" remain inefficient (or wrong). The fixed-node approach, however, is efficient, and in many situations remains the best approach. If only we could find the exact nodes, or at least a systematic way to improve the nodes, we would, in effect, bypass the sign problem. Unfortunately, very little is known about wave function nodes, and a systematic study has never been attempted, despite the obvious consequences for improving quantum simulations that such knowledge might generate. In this paper we study the nodal surfaces of simple atomic systems.
The wave function for the 1s2p 3{\mathrm{}}^{3}3P state of the helium atom is found to have an unexpectedly simple nodal hypersurface, symmetric about the z axis when the position of one electron is fixed and independent of the interelectron distance. The six-dimensional configuration space of the electrons is divided by the surface into two geometrically similar regions, but the hypersurface is not a plane of symmetry and points on the hypersurface are not at equivalent sites in the two regions.
In a recent paper [Bressanini et al. Phys. Rev. Lett. 95, 110201 (2005)] it was pointed out that ``the nodes of even simple wave functions are largely unexplored.'' Here we show that for NNN-particle wave functions nodal surfaces arise from the spin and orbital entanglement of constituent two-particle wave functions and derive, for two-electron atoms, 11 exact nodal rules applicable in LSLSLS coupling. In addition, the ``higher symmetry'' identified numerically in the above paper is shown to be an approximate dynamical symmetry described by a molecular model or a classical unstable periodic orbit. We show that the analysis is readily extended to four-particle wave functions and consider the case of the hydrogen molecule, both fixed in space and rotating.
A fermion node is subset of fermionic configurations for which a real wave function vanishes due to the antisymmetry and the node divides the configurations space into compact nodal cells (domains). We analyze the properties of fermion nodes of fermionic ground state wave functions for a number of systems. For several models we demonstrate that noninteracting spin-polarized fermions in dimension two and higher have closed-shell ground state wave functions with the minimal two nodal cells for any system size and we formulate a theorem which sumarizes this result. The models include periodic fermion gas, fermions on the surface of a sphere, fermions in a box. We prove the same property for atomic states with up to 3d half-filled shells. Under rather general assumptions we then derive that the same is true for unpolarized systems with arbitrarily weak interactions using Bardeen-Cooper-Schrieffer (BCS) variational wave function. We further show that pair correlations included in the BCS wave function enable singlet pairs of particles to wind around the periodic box without crossing the node pointing towards the relationship of nodes to transport and many-body phases such as superconductivity. Finally, we point out that the arguments extend also to fermionic temperature dependent/imaginary-time density matrices. The results reveal fundamental properties of fermion nodal structures and provide new insights for accurate constructions of wave functions and density matrices in quantum and path integral Monte Carlo methods.
We study nodes of fermionic ground state wave functions. For two dimensions and higher we prove that spin-polarized, noninteracting fermions in a harmonic well have two nodal cells for arbitrary system size. The result extends to noninteracting or mean-field models in other geometries and to Hartree-Fock atomic states. Spin-unpolarized noninteracting states have multiple nodal cells; however, interactions and many-body correlations generally relax the multiple cells to the minimal number of two. With some conditions, this is proved for interacting two and higher dimensions harmonic fermion systems of arbitrary size using the Bardeen-Cooper-Schrieffer variational wave function.
Diffuse clusters of arbitrary numbers of helium atoms are treated in a mean-field theory of the many-body system. The nonlinear Hartree self-consistent-field equations are solved numerically to give their energy eigenvalues and eigenfunctions. Both isotopes 4He and 3He are treated according to their bosonic and fermionic statistics and symmetry requirements. Clusters of all sizes of 4He atoms are stable, but there is a minimum threshold of ∼70 atoms for the formation of 3He clusters. All the excited states in addition to the ground state were found. The wave functions are seen to have interesting differences from electronic wave functions in atoms.