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 Schrödinger equation for one electron, in the environment of all other electrons, is formulated. The influence of the environment is reflected in the potential vH, which represents the energy of the environment, and in a potential vG, which has a geometrical meaning. In this paper, we focus on vG and study its properties in detail. We investigate the geometric origin of 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, 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 to provide an alternative interpretation to the Pauli potential in orbital-free density functional theory.