PT-Symmetry in Hartree–Fock Theory

P T -symmetry — invariance with respect to combined space reflection P and time reversal T — provides a weaker condition than (Dirac) Hermiticity for ensuring a real energy spectrum of a general non-Hermitian Hamiltonian. PT -symmetric Hamiltonians therefore form an intermediate class between Hermitian and non-Hermitian Hamiltonians. In this work, we derive the conditions for PT-symmetry in the context of electronic structure theory, and specifically, within the Hartree–Fock (HF) approximation. We show that the HF orbitals are symmetric with respect to the P T operator if and only if the effective Fock Hamiltonian is PT -symmetric, and vice versa. By extension, if an optimal self-consistent solution is invariant under PT , then its eigenvalues and corresponding HF energy must be real. Moreover, we demonstrate how one can construct explicitly PT -symmetric Slater determinants by forming PT doublets (i.e. pairing each occupied orbital with its PT -transformed analogue), allowing PT -symmetry to be conserved throughout the self-consistent process. Finally, considering the H2 molecule as an illustrative example, we observe PT-symmetry in the HF energy landscape and find that the symmetry-broken unrestricted HF wave functions (i.e. diradical configurations) are P T -symmetric, while the symmetry-broken restricted HF wave functions (i.e. ionic configurations) break PT -symmetry.