Hartree-fock and Kohn-Sham Equations Reveal Non-Hermitian Solutions and Electron Density Transfer

The behaviour of electrons in complex systems often involves interactions with their surroundings, creating situations where energy can be lost or gained, and conventional methods of calculation struggle to provide accurate results. Matthias Ernzerhof, Mohamed Loutis, and Pierre-Olivier Roy, from Université de Montréal, alongside Didier Mayou at Institut Néel, now demonstrate that even seemingly isolated electrons within a molecule experience a similar interaction, effectively coupling to a ‘bath’ of other electrons.

This research reveals a previously uncharted space of solutions to the fundamental Hartree-Fock and Kohn-Sham equations, which underpin much of modern quantum chemistry, and shows these solutions possess a clear physical meaning. By recognising this inherent coupling, the team extends the applicability of these methods to a wider range of problems and explains previously unrecognised results from calculations designed to model open systems. Non-Hermitian Quantum Systems and Resonances This research extends traditional quantum mechanical methods, such as density functional theory, to investigate non-Hermitian systems, crucial for accurately describing unstable states and systems interacting with their surroundings. These systems include resonances, like excited states in molecules, and open quantum systems, such as molecules connected to electrodes in electronic devices. The core of this work lies in the development of a framework that directly describes resonances and open systems by allowing complex-valued densities and potentials, avoiding artificial adjustments typically needed to model these phenomena. Key achievements include the development of complex density functional theory (CDFT) as a rigorous framework for non-Hermitian systems, enabling the direct calculation of resonance energies and lifetimes without complex scaling or artificial potentials. This framework also provides a natural way to model open quantum systems by incorporating environmental effects directly into the calculations and connects CDFT to the physics of exceptional points, singularities that can dramatically alter system behavior. The research builds upon, and extends, the concept of holomorphic DFT, a related approach using complex-valued wavefunctions, and demonstrates the application of these methods to molecular electronics, highlighting the importance of accurately modeling electron transport through molecules. The research distinguishes between Hermitian and non-Hermitian Hamiltonians, explaining that while Hermitian Hamiltonians have real eigenvalues representing stable energy levels, non-Hermitian Hamiltonians can have complex eigenvalues representing unstable states with finite lifetimes. Complex density allows for the description of wavefunctions that decay or grow in time, and complex potential can model environmental effects or introduce artificial absorption. Extending the Hohenberg-Kohn theorems, which prove the ground state density uniquely determines the Hamiltonian, to complex densities is a central challenge in CDFT. Exceptional points represent points where multiple eigenvalues and eigenvectors coalesce, leading to enhanced sensitivity to perturbations. This work has significant implications for various applications, including calculating molecular resonances, describing scattering processes, modeling electron transport in molecular devices, and understanding open quantum systems.

This research represents a significant advance in quantum chemistry and physics, providing a powerful new tool for studying resonances, open systems, and other non-Hermitian phenomena with implications for molecular electronics, materials science, and chemical physics. In essence, the team argues for a more natural and rigorous way to treat open quantum systems and resonances by embracing complex-valued densities and potentials within the framework of density functional theory. Non-Hermitian Quantum Mechanics for Open Systems Scientists pioneered a novel approach to understanding open quantum systems by extending established Hartree-Fock and Kohn-Sham methods into the realm of non-Hermitian quantum mechanics. This work addresses systems where interaction with the environment is crucial, such as molecules interacting with surfaces, and opens new avenues for describing unstable states arising from electron transfer.

The team focused on the source-sink potential (SSP) model, an elementary system representing a molecule coupled to external contacts, to illustrate the principles of electron transport and current density. To model this system, researchers constructed a specific Hamiltonian matrix for a diatomic molecule, incorporating complex potentials to represent the coupling to external contacts. By setting specific parameters, the team derived a simplified Hamiltonian where the complex diagonal elements are zero and the off-diagonal elements are -1, effectively modeling electron density entering one atom and exiting the other after traversing the molecule. The study then demonstrated a surprising connection between this SSP model and the Fock matrix derived from Hartree-Fock theory for an isolated diatomic molecule. Researchers found that the mean field created by one electron in the Fock matrix closely mimics the environmental effects represented by the external contacts in the SSP model. This discovery suggests that the new solutions arising from non-Hermitian calculations are not limited to open systems but also emerge naturally when considering electron interactions within isolated molecules. Non-Hermitian Quantum Chemistry Reveals Hidden Solutions Scientists have achieved a significant breakthrough in quantum mechanics by extending established Hartree-Fock and Kohn-Sham methods to encompass non-Hermitian systems, revealing a previously uncharted space of solutions to fundamental equations. This work demonstrates that even isolated molecules can be treated as open quantum systems, exhibiting behavior typically associated with interactions with an external environment.

The team discovered that standard calculations, when performed within a non-Hermitian framework, consistently yield a new class of solutions that were previously overlooked. Researchers found that these additional solutions arise because a single electron within a molecule effectively interacts with the remaining electrons as if they constitute an environment, allowing for the exchange of current density. This conceptualization allows for the application of non-Hermitian quantum mechanics to systems not explicitly coupled to an external environment. The study establishes a connection between these newly discovered solutions and the behavior of open quantum systems, providing a physical interpretation for their complex energies, orbitals, and mean-field potentials. Furthermore, the team demonstrated that the non-Hermitian Hartree-Fock and Kohn-Sham theories for isolated systems are a special case of their previously developed open-system theories. By adapting the fundamental equations of quantum mechanics to accommodate complex densities, the researchers ensured a one-to-one correspondence between external potentials and electron densities, extending the validity of established density functional theory and Hartree-Fock methods. The work introduces a modified continuity equation, revealing that the imaginary part of the energy eigenvalues directly influences the electron density and current, demonstrating a fundamental link between complex energies and observable physical quantities.

Complex Solutions Reveal Accurate Electronic Structure This work presents a significant advancement in the established methods of Hartree-Fock and Kohn-Sham calculations, commonly used to determine the electronic structure of materials. Researchers discovered that even when considering isolated systems, a non-Hermitian approach to these calculations reveals a previously unrecognized set of solutions where density and orbital energies can be complex. These solutions, indicative of states with finite lifetimes, represent a broadening of the scope of problems to which these methods can be applied.

The team demonstrated that these newly discovered solutions provide a more accurate approximation to the true electronic structure, particularly in cases where conventional methods fail to capture essential features, such as specific conduction channels in molecular conductance calculations. This extension of Hartree-Fock and Kohn-Sham methods parallels the transition from restricted to unrestricted calculations, where relaxing constraints on the orbitals leads to improved approximations of the total energy. 👉 More information 🗞 The uncharted space of non-Hermitian solutions to the Hartree-Fock and Kohn-Sham equations 🧠 ArXiv: https://arxiv.org/abs/2512.07048 Tags: