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

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.