Particle–hole symmetry in many-body theories of electron correlation

ABSTRACT

Second-quantised creation and annihilation operators for fermionic particles anticommute, but the same is true for the creation and annihilation operators for holes. This introduces a symmetry into the quantum theory of fermions that is absent for bosons. In ab initio electronic structure theory, it is common to classify methods by the number of electrons for which the method returns exact results: for example Hartree–Fock theory is exact for one-electron systems, whereas coupled-cluster theory with single and double excitations is exact for two-electron systems. Here, we discuss the generalisation: methods based on approximate wavefunctions that are exact for n-particle systems are also exact for n-hole systems. Novel electron correlation methods that attempt to improve on the coupled-cluster framework sometimes retain this property, and sometimes lose it. Here, we argue for retaining particle–hole symmetry as a desirable design criterion of approximate electron correlation methods. Dispensing with it might lead to loss of n-representability of density matrices, and this in turn can lead to spurious long-range behaviour in the correlation energy.

KEYWORDS:

• Coupled cluster
• distinguishable cluster
• particle–hole symmetry
• electron correlation
• n-representability

Acknowledgments

We are indebted to our friend and colleague Martin Schütz, who prematurely passed away in February 2018 after a battle against a severe illness, for illuminating discussions on this and other topics of quantum chemistry over the years. Two of us also owe a debt of gratitude to Martin for his support and supervision throughout our scientific careers.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. In quantum chemistry, it is common to use a third alternative, the particle–hole formalism, in which determinants are produced by creating pairs of particles and holes in a Fermi vacuum, and we adopt this more conventional formalism in what follows. We use the conventional notation for spin-orbital indices such that i, j, … denote occupied; a, b, … denote virtual; and p, q, … denote arbitrary spin-orbitals.

2. Conventionally, only the first two terms in Equation (16(16) ) are referred to as ladder diagrams. However, since the four diagrams of Equation (16(16) ) are interrelated by the LPH symmetry, for convenience we apply the term ‘ladder diagram’ to all of them.

Additional information

Funding

Deutsche Forschungsgemeinschaft [grant number US 103/1-2]; H2020 European Research Council [grant number ASES 320723].