Holomorphic Hartree–Fock Theory: The Nature of Two-Electron Problems

Abstract

We explore the existence and behaviour of holomorphic restricted Hartree-Fock (h-RHF) solutions for two-electron problems. Through algebraic geometry, the exact number of solutions with n basis functions is rigorously identified as 1 2 (3 n − 1), proving that states must exist for all molecular geometries. A detailed study on the h-RHF states of HZ (STO-3G) then demonstrates both the conservation of holomorphic solutions as geometry or atomic charges are varied and the emergence of complex h-RHF solutions at coalescence points. Using catastrophe theory, the nature of these coalescence points is described, highlighting the influence of molecular symmetry. The h-RHF states of HHeH 2+and HHeH (STO-3G) are then compared, illustrating the isomorphism between systems with two electrons and two electron holes. Finally, we explore the h-RHF states of ethene (STO-3G) by considering the π electrons as a twoelectron problem, and employ NOCI to identify a crossing of the lowest energy singlet and triplet states at the perpendicular geometry.

PT-Symmetry in Hartree–Fock Theory

“…In contrast, the complex-symmetric inner product x|y C = x y requires complex-symmetric density D (k) = C (k) (C (k) ) and Fock matrices F (k) = (F (k) ) , with energies that are complex in general. e complex-symmetric formulation of HF is used in holomorphic HF theory to ensure solutions exist over all geometries, [13][14][15]17 and for describing resonance phenomena in non-Hermitian approaches. 16 In what follows, we employ the complex-symmetric inner product .|.…”

“…Holomorphic HF (h-HF) theory, for example, is formulated by analytically continuing real HF theory into the complex plane without introducing the complex conjugation of orbital coe cients. [13][14][15] e result is a non-Hermitian Hamiltonian and an energy function that is complex analytic with respect to the orbital coe cients. In addition, non-Hermitian HF approaches are extensively used to study unbound resonance phenomena where they occur in nature.…”

“…For example, through the polynomial nature of the h-HF equations, a mathematically rigorous upper bound for the number of real RHF solutions can be derived for two-electron systems. 15 Moreover, by scaling the electron-electron interactions using a complex parameter λ, h-HF theory reveals a deeper interconnected topology of multiple HF solutions across the complex plane. 17 By slowly varying λ in a similar (yet di erent) manner to an adiabatic connection in KS-DFT (without enforcing a density-xed path), 18 one can then "morph" a ground-state wave function into an excited-state wave function of a di erent symmetry via a stationary path of h-HF solutions, as we have recently demonstrated for a very simple model [19][20][21][22][23] in Ref.…”

Complex adiabatic connection: A hidden non-Hermitian path from ground to excited states

“…21,[27][28][29][30][31] Despite signi cant progress, however, our understanding of the general nature of multiple solutions remains surprisingly limited. 18,20,23,[31][32][33][34][35][36][37][38][39][40][41] In this Le er, we propose a totally novel approach for exploring multiple solutions in electronic structure methods.…”

“…To our knowledge, the multiple solutions to the non-linear HF equations remain unexplored in the framework of analytic continuation. Since the conventional complex Hermitian extension of HF theory violates the Cauchy-Riemann conditions (resulting in functions that are not complex analytic), we rely here on the holomorphic HF (h-HF) approach 31,38,39 originally developed as a method for analytically continuing real HF solutions beyond the Coulson-Fischer points at which they coalesce and vanish. 45,46 In h-HF theory, the complex conjugation of orbital coe cients is simply removed from the conventional HF equations, resulting in a non-Hermitian Hamiltonian and an energy function that is complex analytic with respect to the orbital coe cients.…”