High Resolution Image Download MS PowerPoint Slide Description of many-electron systems with a fractional electron number ( N tot ) and fractional spin ( M tot ) is of great importance in physical chemistry, solid-state physics, and materials science. In this Letter, we provide an exact description of the zero-temperature ensemble ground state of a general, finite, many-electron system and characterize the dependence of the energy and the spin-densities on both N tot and M tot, when the total spin is at its equilibrium value. We generalize the piecewise-linearity principle and the flat-plane condition and determine which pure states contribute to the ground-state ensemble. We find a new derivative discontinuity, which manifests for spin variation at a constant N tot, as a jump in the Kohn–Sham potential. We identify a previously unknown degeneracy of the ground state, such that the total energy and density are unique, but the spin-densities are not. Our findings serve as a basis for development of advanced approximations in density functional theory and other many-electron methods.
High Resolution Image Download MS PowerPoint Slide Description of many-electron systems with a fractional electron number ( N tot ) and fractional spin ( M tot ) is of great importance in physical chemistry, solid-state physics, and materials science. In this Letter, we provide an exact description of the zero-temperature ensemble ground state of a general, finite, many-electron system and characterize the dependence of the energy and the spin-densities on both N tot and M tot, when the total spin is at its equilibrium value. We generalize the piecewise-linearity principle and the flat-plane condition and determine which pure states contribute to the ground-state ensemble. We find a new derivative discontinuity, which manifests for spin variation at a constant N tot, as a jump in the Kohn–Sham potential. We identify a previously unknown degeneracy of the ground state, such that the total energy and density are unique, but the spin-densities are not. Our findings serve as a basis for development of advanced approximations in density functional theory and other many-electron methods.
Description of many-electron systems with a fractional electron number Ntot and fractional spin Mtot is of great importance in physical chemistry, solid state physics and materials science. In this Letter, we provide an exact description of the zero-temperature ensemble ground state of a general, finite, many-electron system, and characterize the dependence of the energy and the spin-densities on both Ntot and Mtot, when the total spin is at its equilibrium value. We generalize the piecewise-linearity principle and the flat-plane condition and determine which pure states contribute to the ground-state ensemble. We find a new derivative discontinuity, which manifests for spin variation at constant Ntot, as a jump in the Kohn-Sham potential. We identify a previously unknown degeneracy of the ground state, such that the total energy and density are unique, but the spin-densities are not. Our findings serve as a basis for development of advanced approximations in density functional theory and other many-electron methods.
The description of many-electron systems with a fractional electron number, Ntot, and fractional z-projection of the spin, Mtot, is of great importance in physical chemistry, solid-state physics, and materials science. In this study, we analyze the fundamental question of what the ensemble ground state of a general, finite, many-electron system at zero temperature is, with a given Ntot and Mtot, distinguishing between low- and high-spin cases (separated by the boundary spin MB). For the low-spin case, the general form of the ensemble ground state has been rigorously derived in Goshen and Kraisler [J. Phys. Chem. Lett. 15, 2337 (2024)], generalizing the piecewise linearity and the flat-plane conditions for many-electron systems. Here, we provide an alternative proof for this case, discuss the ambiguity in the description of the ground state, and show that this ambiguity can be removed via maximization of the system's entropy. For the high-spin case, we find that the form of the ensemble ground state strongly depends on the system in question. We prove three general properties that characterize the ground state at high spins and narrow down the list of pure states it may consist of. We illustrate the aforementioned properties of high-spin cases by examining the ensemble ground state when Mtot approaches MB from above during the addition of (a fraction of) an up- and down-electron to a given system. Furthermore, we relate the frontier orbital energies of Kohn-Sham (KS) density functional theory (DFT) to total energy differences at high spin values, particularly the ionization potential (IP), the fundamental gap, and the spin flip energies. Analyzing the frontier energies on both sides of each boundary in the total energy profile, where the energy slope changes abruptly, we derive expressions for new derivative discontinuities, which are predicted to appear as jumps in the corresponding KS potentials. In this way, we generalize the well-known IP theorem of DFT to cases with fractional electron number and to cases with high spin. Our analytical results are supported by an extensive numerical analysis of the Atomic Spectra Database of the National Institute of Standards. The new exact conditions for many-electron systems derived in this study are instrumental for the development of advanced approximations in DFT and other many-electron methods.
In this work, we analyze the fundamental question of what is the ensemble ground state of a general, finite, many-electron system at zero temperature, with a given, possibly fractional, electron number Ntot and a given z-projection of the spin, Mtot, distinguishing between low- and high-spin cases. For the low-spin case, the general form of the ensemble ground state has been rigorously derived in J. Phys. Chem. Lett. 15, 2337 (2024), finding the presence of an ambiguity in the ground state. Here we further discuss this ambiguity, and show that it can be removed via maximization of the entropy. For the high-spin case, we find that the form of the ensemble ground state strongly depends on the system in question. Furthermore, we prove three general properties which characterize the ensemble, and narrow the list of pure states it may consist of. We relate the frontier Kohn-Sham orbital energies to total energy differences, providing a generalization of the ionization potential theorem to systems with arbitrary fractional Mtot. Furthermore, we derive expressions for new derivative discontinuities, which appear as jumps in the KS potentials when crossing a boundary in the N↑-N↓ plane. Our analytical results are supported by an extensive numerical analysis of the Atomic Spectra Database of the National Institute of Standards. The new exact conditions for many-electron systems derived in this work are instrumental for development of advanced approximations in density functional theory and other many-electron methods.
The exact conditions for density functionals and density matrix functionals in terms of fractional charges and fractional spins are known, and their violation in commonly used functionals has been shown to be the root of many major failures in practical applications. However, approximate functionals are designed for physical systems with integer charges and spins, not in terms of the fractional variables. Here we develop a general framework for extending approximate density functionals and many-electron theory to fractional-charge and fractional-spin systems. Our development allows for the fractional extension of any approximate theory that is a functional of G(0), the one-electron Green's function of the non-interacting reference system. The extension to fractional charge and fractional spin systems is based on the ensemble average of the basic variable, G(0). We demonstrate the fractional extension for the following theories: (1) any explicit functional of the one-electron density, such as the local density approximation and generalized gradient approximations; (2) any explicit functional of the one-electron density matrix of the non-interacting reference system, such as the exact exchange functional (or Hartree-Fock theory) and hybrid functionals; (3) many-body perturbation theory; and (4) random-phase approximations. A general rule for such an extension has also been derived through scaling the orbitals and should be useful for functionals where the link to the Green's function is not obvious. The development thus enables the examination of approximate theories against known exact conditions on the fractional variables and the analysis of their failures in chemical and physical applications in terms of violations of exact conditions of the energy functionals. The present work should facilitate the calculation of chemical potentials and fundamental bandgaps with approximate functionals and many-electron theories through the energy derivatives with respect to the fractional charge. It should play an important role in developing accurate approximate density functionals and many-body theory.
A new approach for the calculation of the linear response of both the ground and thermal-equilibrium states of many-electron systems is proposed via an integral formulation of the density-functional theory. Based on the path-integral representation of the one-particle Green's function, the frequency-dependent linear-response function for the Kohn-Sham noninteracting system is expressed as an explicit functional of the Kohn-Sham local potential, instead of the summation over the single-electron orbitals. Thus the dynamic linear density response can be determined as the solution of self-consistent equations which only need as input the total electron density of the ground or the equilibrium state. Exchange and correlation effects are incorporated. Because orbitals are not employed, the formulation provides the possibility for calculations of linear-response properties, such as dynamic polarizability and photoabsorption cross sections of systems with very many electrons. The present formulation can also be applied to general fermions.
For a linear combination of electron densities of degenerate ground states, it is shown that the value of any energy functional is the ground state energy, if the energy functional is exact for ground state densities, size consistent, and translational invariant. The corresponding functional of kinetic and interaction energy is the linear combination of the functionals of the degenerate densities. Without invoking ensembles, it is shown that the energy functional of fractional number electrons is a series of straight lines interpolating its values at integers. These results underscore the importance of grand canonical ensemble formulation in density functional theory.