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

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 $v^{\mathrm{H}}$ representing the energy of the environment and another scalar potential $v^{\mathrm{G}}$ 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 $v^{\mathrm{G}}$ and partly to $v^{\mathrm{H}}$ for the (fermionic) KS system. We then study the meaning of $v^{\mathrm{G}}$ 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.

arXiv commentarXiv admin note: substantial text overlap with arXiv:2010.14885

References in corpus (19)

• Exact factorization of the time-dependent electron-nuclear wavefunction
• Laplacian-level density functionals for the kinetic energy density and exchange-correlation energy
• Issues and Challenges in Orbital-free Density Functional Calculations
• Non-empirical Generalized Gradient Approximation Free Energy Functional for Orbital-free Simulations
• Semilocal Pauli-Gaussian Kinetic Functionals for Orbital-Free Density Functional Theory Calculations of Solids
• Numerical Methods for the Inverse Problem of Density Functional Theory
• Molecular geometric phase from the exact electron-nuclear factorization
• Constraint-based, Single-point Approximate Kinetic Energy Functionals
• How is the derivative discontinuity related to steps in the exact Kohn-Sham potential?
• Exact single-electron approach to the dynamics of molecules in strong laser fields
• Asymptotic analysis of the Berry curvature in the $E\otimes e$ Jahn-Teller model
• Universal nature of different methods of obtaining the exact Kohn-Sham exchange-correlation potential for a given density
• Response potential in the strong-interaction limit of DFT: Analysis and comparison with the coupling-constant average
• Density-inversion method for the Kohn-Sham potential: role of the screening density
• Kohn-Sham potential for a strongly correlated finite system with fractional occupancy
• Asymptotic nodal planes in the electron density and the potential in the effective equation for the square root of the density
• Methods to generate the reference total and Pauli kinetic potentials
• Accurate effective potential for density amplitude and the corresponding Kohn-Sham exchange-correlation potential calculated from approximate wavefunctions
• Bypassing the energy functional in density functional theory: Direct calculation of electronic energies from conditional probability densities