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Physical Review Research

Geometric potential of the exact electron factorization: Meaning, significance, and application

Jakub Kocák, Eli Kraisler, Axel Schild

Published Jan 17, 2023• 10 citations
Publisher

Abstract

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 ${v}^{\mathrm{H}}$, which represents the energy of the environment, and in a potential ${v}^{\mathrm{G}}$, which has a geometrical meaning. In this paper, we focus on ${v}^{\mathrm{G}}$ and study its properties in detail. We investigate the geometric origin of ${v}^{\mathrm{G}}$ as a metric measuring the change of the environment, exemplify how translation and scaling of the state of the environment are reflected in ${v}^{\mathrm{G}}$, 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 ${v}^{\mathrm{G}}$ to provide an alternative interpretation to the Pauli potential in orbital-free density functional theory.

DOI
10.1103/physrevresearch.5.013016

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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.
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.
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.
A quantum system Q\mathcal{Q}Q is characterized by a single potential vvv and its eigenstates. While vvv is usually postulated for a given physical problem, it represents the interaction with an implicit environment E\mathcal{E}E. We use the exact factorization to show how vvv emerges if the quantum environment is explicitly taken into account. In general, each eigenstate of the supersystem $\mathcal{S}=\mathcal{Q}\ensuremath{\cup}\mathcal{E}$ corresponds to a different potential vj{v}_{j}vj​ and state ${\ensuremath{\chi}}_{j}$ of Q\mathcal{Q}Q. Such a state ${\ensuremath{\chi}}_{j}$ typically has no nodes and is the ground state of vj{v}_{j}vj​, even if the corresponding state of the supersystem is an excited state. There are however two exceptions. First, if the energy scale for exciting Q\mathcal{Q}Q is much smaller than for exciting E\mathcal{E}E, the potentials vj{v}_{j}vj​ are similar in shape and differ only by sharp spikes. An excitation of S\mathcal{S}S can then be viewed as an excitation of Q\mathcal{Q}Q with its environment being unaffected, and Q\mathcal{Q}Q is approximately described by a single spikeless potential vvv and its eigenstates. Second, ${\ensuremath{\chi}}_{j}$ can sometimes have exact nodes, e.g., due to the symmetry of the problem, and is the excited state of a spikeless potential. We explain and investigate the two cases with model systems to illustrate the intricacies of the separation of a quantum system from its environment. As an application, we use the equivalence of ${\ensuremath{\chi}}_{j}$ being either an excited state of a spikeless potential or the ground state of a spiky potential: For one-dimensional systems, we provide a method to calculate the location of the nodes of an excited state from the calculation of a ground-state wave function. This approach can also be conceptually useful for the computationally hard problem of calculating highly excited states or many-fermion systems.
, which are relevant to properly describe phenomena such as molecular dissociation with density-functional theory, to the case in which the electronic wavefunction corresponds to a complex current-carrying state. In such a case, an extra term in the form of a vector potential appears which cannot be gauged away. Both scalar and vector potentials are introduced via the exact factorization formalism which allows us to express the given Schrödinger equation as two coupled equations, one for the marginal and one for the conditional amplitude. The electronic vector potential is directly related to the paramagnetic current density carried by the total wavefunction and to the diamagnetic current density in the equation for the marginal amplitude. An explicit example of this vector potential in a triplet state of two non-interacting electrons is showcased together with its associated circulation, giving rise to a non-vanishing geometric phase. Some connections with the exact factorization for the full molecular wavefunction beyond the Born-Oppenheimer approximation are also discussed.
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.
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.