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Physical Review A

Symmetry of the nodal hypersurface for 1s2p P3 helium

James B. Anderson

Published Apr 1, 1987• 25 citations
Publisher

Abstract

The wave function for the 1s2p ${\mathrm{}}^{3}$P 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.

DOI
10.1103/physreva.35.3550

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Analysis of the relationship between the structure of the symmetrical permutation group and the topological structure of the configuration space makes it possible to perform complete separation of spin and spatial variables for multielectron wave functions. This separation of variables allows one to explain the role of nodal surfaces in intepreting and understanding the physical sense of the Pauli principle for systems of fermions. The suggested hypothesis about the form of the nodal surfaces permits one to find these surfaces a priori for arbitrary multielectron systems. The use of nodal surfaces for constructing trial wave functions is demonstrated by the example of calculating the energy of the Li atom in the ground state.
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The Hooke's-law atom (hookium) provides an exactly soluble model for a two-electron atom in which the nuclear-electron Coulombic attraction has been replaced by a harmonic one. Starting from the known exact position-space wave function for the ground state of hookium, we present the momentum-space wave function. We also look at the intracules, two-electron probability distributions, for hookium in position, momentum, and phase space. These are compared with the Hartree-Fock results and the Coulomb holes (the difference between the exact and Hartree-Fock intracules) in position, momentum, and phase space are examined. We then compare these results with analogous results for the ground state of helium using a simple, explicitly correlated wave function.
For the SSS states of two-electron atoms, we introduce an exact and unique factorization of the internal eigenfunction in terms of a marginal amplitude, which depends functionally on the electron-nucleus distances r1{r}_{1}r1​ and r2{r}_{2}r2​, and a conditional amplitude, which depends functionally on the interelectronic distance r12{r}_{12}r12​ and parametrically on r1{r}_{1}r1​ and r2{r}_{2}r2​. Applying the variational principle, we derive pseudoeigenvalue equations for these two amplitudes, which cast the internal Schr\"odinger equation in a form akin to the Born-Oppenheimer separation of nuclear and electronic degrees of freedom in molecules. The marginal equation involves an effective radial Hamiltonian, which contains a nonadiabatic potential energy surface that takes into account all interparticle correlations in an averaged way, and whose unique eigenvalue is the internal energy. At each point (r1,r2)({r}_{1},{r}_{2})(r1​,r2​), such surface is, in turn, the unique eigenvalue in the conditional equation. Employing the ground state of He as prototype, we show that the nonadiabatic potential energy surface affords a molecularlike interpretation of the structure of the atom, and aids in the analysis of energetic and spatial aspects of the Coulomb correlation, in particular correlation-induced symmetry breaking and quantum phase transition.