$\mathcal{P}\mathcal{T}$-Symmetry in Hartree-Fock Theory

Abstract

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

12 pages, 5 figures

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• Holomorphic Density Functional Theory
• The uncharted space of non-Hermitian solutions to the Hartree-Fock and Kohn-Sham equations
• Extremely broken generalized $\mathcal{P}\mathcal{T}$ symmetry