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arXiv:2111.09620

On the Geometric Potential and the Relationship between the Exact Electron Factorization and Density Functional Theory

Jakub Kocák, Eli Kraisler, Axel Schild

Published Nov 18, 2021• 1 citation
arXivHTMLPDF

Abstract

There are different ways to obtain an exact one-electron theory for a many-electron system, and the exact electron factorization (EEF) is one of them. In the EEF, the Schrödinger equation for one electron in the environment of other electrons is constructed. The environment provides the potentials that appear in this equation: A scalar potential $v^{\rm H}$ representing the energy of the environment and another scalar potential $v^{\rm G}$ as well as a vector potential that have geometric meaning. By replacing the interacting many-electron system with the non-interacting Kohn-Sham (KS) system, we show how the EEF is related to density functional theory (DFT) and we interpret the Hartree-exchange-correlation potential as well as the Pauli potential in terms of the EEF. In particular, we show that from the EEF viewpoint, the Pauli potential does not represent the difference between a fermionic and a bosonic non-interacting system, but that it corresponds to $v^{\rm G}$ and partly to $v^{\rm H}$ for the (fermionic) KS system. We then study the meaning of $v^{\rm G}$ in detail: Its geometric origin as a metric measuring the change of the environment is presented. Additionally, its behavior for a simple model of a homo- and heteronucler diatomic is investigated and interpreted with the help of a two-state model. In this way, we provide a physical interpretation for the one-electron potentials that appear in the EEF and in DFT.

DOI
10.48550/arxiv.2111.09620

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There are different ways to obtain an exact one-electron theory for a many-electron system, and the exact electron factorization (EEF) is one of them. In the EEF, the Schrödinger equation for one electron in the environment of other electrons is constructed. The environment provides the potentials that appear in this equation: A scalar potential vHv^{\rm H}vH representing the energy of the environment and another scalar potential vGv^{\rm G}vG as well as a vector potential that have geometric meaning. By replacing the interacting many-electron system with the non-interacting Kohn-Sham (KS) system, we show how the EEF is related to density functional theory (DFT) and we interpret the Hartree-exchange-correlation potential as well as the Pauli potential in terms of the EEF. In particular, we show that from the EEF viewpoint, the Pauli potential does not represent the difference between a fermionic and a bosonic non-interacting system, but that it corresponds to vGv^{\rm G}vG and partly to vHv^{\rm H}vH for the (fermionic) KS system. We then study the meaning of vGv^{\rm G}vG in detail: Its geometric origin as a metric measuring the change of the environment is presented. Additionally, its behavior for a simple model of a homo- and heteronucler diatomic is investigated and interpreted with the help of a two-state model. In this way, we provide a physical interpretation for the one-electron potentials that appear in the EEF and in DFT.
The theoretical and computational description of materials properties is a task of utmost scientific and technological importance. A first-principles description of electron-electron interactions poses an immense challenge that is usually approached by converting the many-electron problem to an effective one-electron problem. There are different ways to obtain an exact one-electron theory for a many-electron system. An emergent method is the exact electron factorization (EEF) -- one of the branches of the Exact Factorization approach to many-body systems. In the EEF, the Schrödinger equation for one electron, in the environment of all other electrons, is formulated. The influence of the environment is reflected in the potential vHv^{\rm H}vH, which represents the energy of the environment, and in a potential vGv^{\rm G}vG, which has a geometrical meaning. In this paper, we focus on vGv^{\rm G}vG and study its properties in detail. We investigate the geometric origin of vGv^{\rm G}vG as a metric measuring the change of the environment, exemplify how translation and scaling of the state of the environment are reflected in vGv^{\rm G}vG, and explain its shape for homo- and heteronuclear diatomic model systems. Based on the close connection between the EEF and density functional theory, we also use vGv^{\rm G}vG to provide an alternative interpretation to the Pauli potential in orbital-free density functional theory.
The theoretical and computational description of materials properties is a task of utmost scientific and technological importance. A first-principles description of electron-electron interactions poses an immense challenge that is usually approached by converting the many-electron problem to an effective one-electron problem. There are different ways to obtain an exact one-electron theory for a many-electron system. An emergent method is the exact electron factorization (EEF) -- one of the branches of the exact factorization approach to many-body systems. In the EEF, the Schr\"odinger equation for one electron, in the environment of all other electrons, is formulated. The influence of the environment is reflected in the potential vH{v}^{\mathrm{H}}vH, which represents the energy of the environment, and in a potential vG{v}^{\mathrm{G}}vG, which has a geometrical meaning. In this paper, we focus on vG{v}^{\mathrm{G}}vG and study its properties in detail. We investigate the geometric origin of vG{v}^{\mathrm{G}}vG as a metric measuring the change of the environment, exemplify how translation and scaling of the state of the environment are reflected in vG{v}^{\mathrm{G}}vG, and explain its shape for homo- and heteronuclear diatomic model systems. Based on the close connection between the EEF and density functional theory, we also use vG{v}^{\mathrm{G}}vG to provide an alternative interpretation to the Pauli potential in orbital-free density functional theory.
Providing sufficient accuracy at low-computational cost, not surprisingly, density-functional theory ( DFT ) is a standard tool in theoretical material science. Similarly, quantum-electrodynamical density-functional theory ( QEDFT ) provides a promising route to describe experiments at the interface of quantum optics and quantum chemistry. Formally, in the Kohn-Sham ( KS ) formulation of DFT and QEDFT the exact many-particle Schrödinger equation is replaced by a set of coupled non-linear one-particle differential equations. The electron-electron, the electron-ion and the electron-photon interactions of the interacting system are mimicked by their corresponding effective KS potential with an, in principle unknown exchange-correlation ( xc ) term. In practice, DFT and QEDFT require approximations for this xc potential. The quality of which fully determines how much the DFT and QEDFT observables deviate from the exact ones. Understanding the shortcomings and improving the quality of approximate functionals makes the study of the exact functional of model systems indispensable. Here, we employ exact diagonalization for the Hamiltonian of different types of real-space lattice models in one spatial dimension to access the ground-state wave function, and thus to explicitly construct the exact functionals of DFT and QEDFT . To study the functional behavior of purely electronic systems, we consider two singlet-state electrons moving in an external potential with two connected atomic wells of variable depth and distance. To study coupled electron-photon systems, we consider the Rabi Hamiltonian with variable electron-photon coupling strength. By varying the model parameters, we systematically construct the functionals for systems from weak to strong coupling between the building blocks of the global system. Using the concept of the effective density-to- potential map, we discuss the intrasystem steepening which is an exact condition on the xc functional that captures the static correlation present in the system and carve out how it relates to the well-known intersystem derivative discontinuity. We further construct arbitrary observables as functionals of the internal variables which are directly affected by such features of the effective density-to-potential map. By constructing the reduced dipole map in the four-particle case of coupled electron-photon system, we show that the density-to-potential map helps to identify and understand recently discovered physical effects in the field of strongly coupled electron-photon systems. This work provides a contribution to the development of approximate function- als required in practical density-functional calculations of arbitrary experimentally accessible observables. Both the intrasystem steepening and the intersystem derivative discontinuity play a crucial role for systems with highly-localized electrons. Including both features in approximate functionals will improve the accu- racy of DFT evaluation of observables for such systems. Those results provide novel insight into the properties of the exact xc potentials in QEDFT . This work opens a new route to construct new functionals using coarse-grained or embed- ding approaches within the general and exact frameworks of DFT and QEDFT.
Aiming to combine density functional theory (DFT) and wave-function theory, we study a mapping from the many-body interacting system to an effectively interacting Kohn-Sham system instead of a noninteracting Kohn-Sham system. Because a ground state of effectively interacting systems requires having a solution for the correlated many-body wave functions, this provides a natural framework to many-body wave-function theories such as the configuration interaction and the coupled-cluster method in the formal theoretical framework of DFT. Employing simple one-dimensional two-electron systems---namely, the one-dimensional helium atom, the hydrogen molecule, and the heteronuclear diatomic molecule---we investigate properties of many-body wave functions and exact exchange-correlation potentials of effectively interacting Kohn-Sham systems. As a result, we find that the asymptotic behavior of the exact exchange-correlation potential can be controlled by optimizing that of the effective interaction. Furthermore, the typical features of the exact noninteracting Kohn-Sham system, namely, a spiky feature and a step feature in the exchange-correlation potential for the molecular dissociation limit, can be suppressed by a proper choice of the effective interaction. These findings open a possibility to construct numerically robust and efficient exchange-correlation potentials and functionals based on the effectively interacting Kohn-Sham scheme.
Abstract Density functional theory (DFT) is usually formulated in terms of the electron density as a function of position n(r). Here we discuss an alternative formulation of DFT in terms of the orbital occupation numbers {n α } associated with a local-orbital orthonormal basis set {ϕ α }. First, we discuss how the building blocks of DFT, namely the Hohenberg–Kohn theorems, the Levy–Lieb approach and the Kohn–Sham method, can be adapted for a description in terms of {n α }. In particular, the total energy is now a function of {n α }, E[{n α }], and a Kohn–Sham-like Hamiltonian is derived introducing the effects of the electron–electron interactions via effective potentials, { V α eff = ∂ E e e [ { n β } ] / ∂ n α } . In a second step we consider the Hartree and exchange energies and discuss how to describe them, in the spirit of a DFT approach, in terms of the orbital occupation numbers. In this contribution special attention is paid to the description of the (intra-atomic) correlation energy and corresponding correlation potentials {V corr,α }. For this purpose, a model system is analyzed in detail, whereby an atomic Hamiltonian interacts with the environment via a simplified model; the use of this model allows us to obtain the correlation energy and potentials (in terms of {n α }) for different cases corresponding to low, intermediate and high electron correlations.
For a three-electron system with finite-strength interactions confined to a one-dimensional harmonic trap, we solve the Schrödinger equation analytically to obtain the exact solutions, from which we construct explicitly the simultaneous eigenstates of the energy and total spin for the first time. The solutions for the three-electron system allow us to derive analytic expressions for the exact one-particle Green's function (GF) for the corresponding two-electron system. We calculate the GF in frequency domain to examine systematically its behavior depending on the electronic interactions. We also compare the pole structure of non-interacting GF using the exact Kohn-Sham (KS) potential with that of the exact GF to find that the discrepancy of the energy gap between the KS system and the original system is larger for a stronger interaction. We perform numerical examination on the behavior of GFs in real space to demonstrate that the exact and KS GFs can have shapes quite different from each other. Our simple model will help to understand generic characteristics of interacting GFs.