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

Abstract

Processes related to electronically excited states are central in many areas of science, however accurately determining excited-state energies remains a major challenge in theoretical chemistry. Recently, higher energy stationary states of non-linear methods have themselves been proposed as approximations to excited states, although the general understanding of the nature of these solutions remains surprisingly limited. In this Le er, we present an entirely novel approach for exploring and obtaining excited stationary states by exploiting the properties of non-Hermitian Hamiltonians. Our key idea centres on performing analytic continuations of conventional quantum chemistry methods. Considering Hartree-Fock theory as an example, we analytically continue the electron-electron interaction to expose a hidden connectivity of multiple solutions across the complex plane, revealing a close resemblance between Coulson-Fischer points and non-Hermitian degeneracies. Finally, we demonstrate how a ground-state wave function can be morphed naturally into an excited-state wave function by constructing a well-de ned complex adiabatic connection.Introduction.-Electronic excited states are central in chemistry, physics and biology, playing a role in key processes such as photochemistry, catalysis, and solar cell technology. However, de ning e ective methods that reliably provide accurate excited-state energies remains a major challenge in theoretical chemistry. Two of the most widely-used approaches to obtain excited-state energies are i) the time-dependent (TD) version of density-functional theory (DFT) which relies on the linear response formalism, and ii) the equation-of-motion (EOM) ansatz of coupled cluster (CC) theory.In particular, TD-DFT has practically revolutionised computational chemistry due to its user-friendly black-box nature compared with the more computationally expensive multi-con gurational methods (such as CASPT2 and NEVPT2) where one must choose an active space based on chemical intuition. Despite their success, fundamental de ciencies associated with TD-DFT and EOM-CC remain. For example, excited states presenting double excitation character 1-8 -which have a key role in the faithful description of many physical and chemical processes -are notoriously di cult to model using conventional single-reference methods such as adiabatic TD-DFT or EOM-CC. Although some viable and promising alternative approaches have been developed -for example spin-ip, 5 dressed TD-DFT 2 or ensemble DFT 6 -each faces major limitations.At present, most excited-state techniques, including TD-DFT and EOM-CC, are built upon a single reference Slater determinant, o en corresponding to a Hartree-Fock (HF) solution. As an inherently non-linear method, similar to CC 9 and GW, 10-17 HF can produce a multitude of distinct stationary states. In recent years, multiple HF states have themselves been proposed as approximations to excited states. [18][19][20][21] However, these solutions do not necessarily share the symmetries of the exact Hamilto...

A General Approach for Multireference Ground and Excited States using Non-Orthogonal Configuration Interaction

“…52 Each individual h-HF state is given by a di erent branch of a Riemann surface, with Coulson-Fischer points forming isolated exceptional points corresponding to the branch points of the Riemann surface. 52 Signi cantly, we can use this continuous topology of h-HF states to follow solutions around a Coulson-Fischer point by tracing a suitable λ-trajectory in the complex plane. We illustrate this idea by considering the multiple HF solutions of H 2 in a minimal basis set (STO-3G) using two orbitals φ α and φ β parameterised by the complex angle θ,…”

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 .|.…”

“…(28)]. Stationary points that do not correspond to P T -symmetric states (including the sb-RHF and h-UHF solutions) occur in pairs which are interconverted by the action of P T , and the onset of P T -symmetry breaking coincides with the disappearance of the h-RHF or the sb-UHF solutions at Coulson-Fischer (quasi-exceptional) points 17 in a similar manner to other types of symmetry-breaking in HF theory. [10][11][12]…”

“…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. 17.…”

A General Approach for Multireference Ground and Excited States using Non-Orthogonal Configuration Interaction

“…we have shown that multiple h-HF states form a continuous interconnected manifold in the complex λ-plane. 52 Each individual h-HF state is given by a di erent branch of a Riemann surface, with Coulson-Fischer points forming isolated exceptional points corresponding to the branch points of the Riemann surface. 52 Signi cantly, we can use this continuous topology of h-HF states to follow solutions around a Coulson-Fischer point by tracing a suitable λ-trajectory in the complex plane.…”