The Geometric Potential of the Exact Electron Factorization: Meaning, significance and application

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