Plane partitions and spin adapted quantum states

TL;DR

This paper introduces a combinatorial basis for the space of spin-adapted quantum states in quantum chemistry, using plane partitions and algebraic tools to facilitate analysis of electron configurations.

Contribution

It provides an explicit basis for the SU(2)-invariant space of electron pairs, connecting quantum state spaces with combinatorial objects like plane partitions and Dyck paths.

Findings

Constructed a basis via the excitation ring and plane partitions.

Computed a Gröbner basis and enumerated standard monomials.

Established a bijection with Dyck paths counted by Narayana numbers.

Abstract

We describe an explicit basis for the $\operatorname{SU}(2)$-invariant space of the exterior power $\wedge_{2k} \mathbb{C}^{2m}$ via the combinatorics of plane partitions. In quantum chemistry, this is the space of spin adapted quantum states of an electronic system with $m$ spin orbitals and $k$ electron pairs. We construct our basis by identifying the invariant space with an Artinian commutative ring called the excitation ring. We compute a Gr\"obner basis and enumerate its standard monomials via an explicit bijection to Dyck paths counted by the Narayana numbers.