Nodal surfaces and interdimensional degeneracies

TL;DR

This paper investigates the topology of wave function nodes in multi-electron systems, demonstrating cases where non-interacting and exact wave function nodes coincide and proposing a conjecture for more complex configurations, linked to interdimensional degeneracies.

Contribution

It provides rigorous proofs for node equivalence in certain electron configurations and introduces the concept of interdimensional degeneracies explaining these features.

Findings

Non-interacting wave function nodes match exact nodes in specific configurations

Interdimensional degeneracies explain the limited cases with known exact nodes

Conjecture that nodes for sp^3 configuration are also exact, supported by numerical evidence

Abstract

The aim of this paper is to shed light on the topology and properties of the nodes (i.e. the zeros of the wave function) in electronic systems. Using the "electrons on a sphere" model, we study the nodes of two-, three- and four-electron systems in various ferromagnetic configurations ($sp$, $p^2$, $sd$, $pd$, $p^3$, $sp^2$ and $sp^3$). In some particular cases ($sp$, $p^2$, $sd$, $pd$ and $p^3$), we rigorously prove that the non-interacting wave function has the same nodes as the exact (yet unknown) wave function. The number of atomic and molecular systems for which the exact nodes are known analytically is very limited and we show here that this peculiar feature can be attributed to interdimensional degeneracies. Although we have not been able to prove it rigorously, we conjecture that the nodes of the non-interacting wave function for the $sp^3$ configuration are exact.