Parity-Time Symmetry in Hartree-Fock Theory

PT-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 PT 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 spatially symmetry-broken unrestricted HF wave functions (i.e., diradical configurations) are PT-symmetric, while the spatially symmetry-broken restricted HF wave functions (i.e., ionic configurations) break PT-symmetry.