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Unexplored solutions in Hartree–Fock and Kohn–Sham equations

Mohamed Loutis, Matthias Ernzerhof

Published Jun 16, 2025• 0 citations

Abstract

Many problems in physical chemistry involve systems that are coupled to an environment, such as a molecule interacting with a nearby surface. In these cases, electron density may flow between the system and its surroundings, leading to metastable states and other open-system phenomena that cannot be captured by conventional Hermitian quantum mechanics. Non-Hermitian quantum mechanics (NHQM) [1] provides a natural framework for the description of open systems. Hartree-Fock (HF) and Kohn-Sham (KS) methods can be extended to the non-Hermitian domain [2–7]. Here, we show that even for nominally isolated systems, within HF and KS, each electron can be viewed as coupled to the remaining electrons, which serve as an environment. This viewpoint reveals that an electron can leave or rejoin the system, producing self-consistent complex potentials. These unconventional solutions are absent in the conventional closed-system HF and KS approach. We show that these additional solutions can be physically meaningful. Our results highlight the importance of incorporating non-Hermitian perspectives into electronic structure calculations, offering new insights and extending the range of problems HF and KS can be applied to. [1] Moiseyev, N. Non-Hermitian Quantum Mechanics, Cambridge University Press (2011). [2] Zhou, Y., Ernzerhof, M. J. Chem. Phys. 136, 094105 (2012). [3] Ernzerhof, M. J. Chem. Phys. 125, 124104 (2006). [4] Zhou, Y., Ernzerhof, M. J. Phys. Chem. Lett. 3, 1916–1920 (2012). [5] Burton, H. G. A., Thom, A. J. W. J. Chem. Theory Comput. 12, 167–173 (2016). [6] Burton, H. G. A., Gross, M., Thom, A. J. W. J. Chem. Theory Comput. 14, 607–618 (2018). [7] Burton, H. G. A., Thom, A. J. W., Loos, P.-F. J. Chem. Theory Comput. 15, 4374–4385 (2019).

DOI: 10.5281/zenodo.19874683

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Many problems in physical chemistry involve systems that are coupled to an environment, such as a molecule interacting with a nearby surface. In these cases, electron density may flow between the system and its surroundings, leading to metastable states and other open-system phenomena that cannot be captured by conventional Hermitian quantum mechanics. Non-Hermitian quantum mechanics (NHQM) [1] provides a natural framework for the description of open systems. Hartree-Fock (HF) and Kohn-Sham (KS) methods can be extended to the non-Hermitian domain [2–7]. Here, we show that even for nominally isolated systems, within HF and KS, each electron can be viewed as coupled to the remaining electrons, which serve as an environment. This viewpoint reveals that an electron can leave or rejoin the system, producing self-consistent complex potentials. These unconventional solutions are absent in the conventional closed-system HF and KS approach. We show that these additional solutions can be physically meaningful. Our results highlight the importance of incorporating non-Hermitian perspectives into electronic structure calculations, offering new insights and extending the range of problems HF and KS can be applied to. [1] Moiseyev, N. Non-Hermitian Quantum Mechanics, Cambridge University Press (2011). [2] Zhou, Y., Ernzerhof, M. J. Chem. Phys. 136, 094105 (2012). [3] Ernzerhof, M. J. Chem. Phys. 125, 124104 (2006). [4] Zhou, Y., Ernzerhof, M. J. Phys. Chem. Lett. 3, 1916–1920 (2012). [5] Burton, H. G. A., Thom, A. J. W. J. Chem. Theory Comput. 12, 167–173 (2016). [6] Burton, H. G. A., Gross, M., Thom, A. J. W. J. Chem. Theory Comput. 14, 607–618 (2018). [7] Burton, H. G. A., Thom, A. J. W., Loos, P.-F. J. Chem. Theory Comput. 15, 4374–4385 (2019).

Many problems in physical chemistry involve systems that are coupled to an environment, such as a molecule interacting with an adjacent surface, possibly resulting in meta-stable molecular states where electron density is transferred to the surface. Such systems can be described by non-Hermitian quantum mechanics (NHQM), where the Hamiltonian includes dissipative terms. Within NHQM, one can also formulate the Hartree-Fock (HF) and Kohn-Sham (KS) methods and, as in the conventional theory, an effective independent-particle picture is employed. The crucial observation of the present work is that even for systems that are not coupled to an environment, in the HF or KS equation a single electron is coupled to a bath of the remaining electrons which can act as an environment, opening up the possibility for the exchange of current density between the one-electron and the remaining N-1 electron system. The corresponding self-consistent states represent a new uncharted space of solutions to the HF and KS equations. We show that the additional solutions can have a physical interpretation and thus extend the range of problems HF and KS can be applied to. If open-system HF and KS calculations are performed, the new class of solutions is always encountered but this has also not been noted previously.

Following a recent work [Gál, Phys. Rev. A64, 062503 (2001)], a simple derivation of the density-functional correction of the Hartree–Fock equations, the Hartree–Fock–Kohn–Sham equations, is presented, completing an integrated view of quantum mechanical theories, in which the Kohn–Sham equations, the Hartree–Fock–Kohn–Sham equations and the ground-state Schrödinger equation formally stem from a common ground: density-functional theory, through its Euler equation for the ground-state density. Along similar lines, the Kohn–Sham formulation of the Hartree–Fock approach is also considered. Further, it is pointed out that the exchange energy of density-functional theory built from the Kohn–Sham orbitals can be given by degree-two homogeneous N-particle density functionals (N = 1, 2, …), forming a sequence of degree-two homogeneous exchange-energy density functionals, the first element of which is minus the classical Coulomb-repulsion energy functional.

The exact static and time-dependent Kohn-Sham (KS) exchange-correlation potential is extremely challenging to approximate as it is a local multiplicative potential that depends on the electron density everywhere in the system. The KS approach can be generalized by allowing part of the potential to be spatially nonlocal. We take this nonlocal part to be that of unrestricted Hartree-Fock theory. The additional local correlation potential in principle ensures that the single-particle density exactly equals the many-body density. In our case, the local correlation potential is predominantly nearsighted in its dependence on the density and hence an (adiabatic) local-density approximation to this potential yields accurate ground-state properties and real-time densities for one-dimensional test systems.

Predictivity of the Kohn-Sham approach to dynamical problems, when regarded as an initial value problem in a time-dependent density functional framework, is analysed for a class of models for which the argument devised in the work of Maitra et al. (Phys. Rev. A 78, 056501 (2008), arXiv:cond-mat/0710.0018) for the standard electronic many-body problem does not apply. The original argument is here extended and revised. As a result, predictivity for this class of problems seems possible only at the price of introducing extra unknown functionals in the corresponding Kohn-Sham equation. Furthermore, the same argument, when applied to original electronic problem, suggests that the Hartree-exchange-correlation potential is not unambiguously identified by the contemporary and past densities and initial states, but also requires knowledge of the divergence of the contemporary Kohn-Sham current.

Density functional theory is currently the most widely applied method in electronic structure theory. The Kohn-Sham method, based on a fictitious system of non-interacting particles, is the work horse of the theory. The particular form of the Kohn-Sham wavefunction admits only idem-potent one electron density matrices whereas wavefunctions of correlated electrons in post-Hartree-Fock methods invariably have fractional occupation numbers. Here we show that by generalizing the orbital concept, and introducing a suitable dot-product as well as a probability density a non-interacting system can be chosen that can represent the one-electron density matrix of any system, even one with fractional occupation numbers. This fictitious system ensures that the exact electron density is accessible within density functional theory. It can also serve as the basis for reduced density matrix functional theory. Moreover, to aid the analysis of the results the orbitals may be assigned energies from a mean-field Hamiltonian. This produces energy levels that are akin to Hartree-Fock orbital energies such that conventional analyses based on Koopmans theorem are available. Finally, this system is convenient in formalisms that depend on creation and annihilation operators as they are trivially applied to single determinant wavefunctions.

This note is intended for expanding the details on the derivation and properties of density functional theory, in hope to make them more systematic, better motivated, and step-by-step for readers new to the domain. The note starts with basic concepts in quantum mechanics, then takes the step towards many-body systems using the tools of second quantization and Fock space, with some highlights on properties of the Coulomb system. Given these general technical preparations, the Hartree-Fock method is naturally unrolled, with expressions for various cases and quantities. Density functional theory is then presented, with a motivating reasoning of the Kohn-Sham formulation, and a connection and comparison of the expressions with the Hartree-Fock method. Lieb's celebrated 1983 paper on functional analysis on density functionals is also summarized.