Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2601.03012

Many-electron systems with fractional electron number and spin: exact properties above and below the equilibrium total spin value

Yuli Goshen, Eli Kraisler

Published Jan 6, 2026• 0 citations
arXivHTMLPDF

Abstract

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 $N_{tot}$ and a given $z$-projection of the spin, $M_{tot}$, 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 $M_{tot}$. Furthermore, we derive expressions for new derivative discontinuities, which appear as jumps in the KS potentials when crossing a boundary in the $N_{\uparrow}$-$N_{\downarrow}$ 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.

DOI
10.48550/arxiv.2601.03012

Related papers

Many-electron systems with fractional electron number and spin: Exact properties above and below the equilibrium total spin value

Yuli Goshen, Eli Kraisler

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.

The Journal of Chemical Physics•Mar 11, 2026•1 citation

Ensemble Ground State of a Many-Electron System with Fractional Electron Number and Spin: Piecewise-Linearity and Flat-Plane Condition Generalized

Yuli Goshen, Eli Kraisler

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.

The Journal of Physical Chemistry Letters•Feb 22, 2024•8 citations

Energy of a many-electron system in an ensemble ground-state, versus electron number and spin: piecewise-linearity and flat plane condition generalized

Yuli Goshen, Eli Kraisler

Description of many-electron systems with a fractional electron number $N_\textrm{tot}$ and fractional spin $M_\textrm{tot}$ 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 $N_\textrm{tot}$ and $M_\textrm{tot}$, 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 constant $N_\textrm{tot}$, 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.

arXiv:2308.03465•Aug 7, 2023•0 citations

Thermodynamic extension of density-functional theory. II. Finite-temperature ensemble spin-density functional theory

Robert Balawender, A. Holas

The formalism developed in the first paper of the series [arXiv:0901.1060] is applied to two thermodynamic systems: (i) of three global observables (the energy, the total electron number and the spin number), (ii) of one global observable (the internal electron energy) and two local (position-dependent) observables (the total electron density and the spin density). The two-component potential of the many-electron system of interest is constructed of a scalar external potential and a collinear magnetic field (coupled only with the spin operator). Various equilibrium characteristics of two systems are defined and investigated. Conditions for the equivalence between two systems (the same equilibrium density matrix demanded) are derived and thoroughly discussed. The applicability of the Hohenberg-Kohn theorem is extended to the thermodynamic spin-density functional theory. Obtained results provide a rigorous mathematical foundation for future derivation of the zero-temperature limit of this theory and determination of its chemical reactivity indices.

arXiv:0904.3990•Apr 25, 2009•5 citations

Extension of many-body theory and approximate density functionals to fractional charges and fractional spins

Weitao Yang, Paula Mori-Sánchez, Aron J. Cohen

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.

arXiv:1305.5194•The Journal of Chemical Physics•Sep 13, 2013•36 citations

Spin gaps and spin-flip energies in density-functional theory

Klaus Capelle, Giovanni Vignale, Carsten A. Ullrich

Energy gaps are crucial aspects of the electronic structure of finite and extended systems. Whereas much is known about how to define and calculate charge gaps in density-functional theory (DFT), and about the relation between these gaps and derivative discontinuities of the exchange-correlation functional, much less is known about spin gaps. In this paper we give density-functional definitions of spin-conserving gaps, spin-flip gaps and the spin stiffness in terms of many-body energies and in terms of single-particle (Kohn-Sham) energies. Our definitions are as analogous as possible to those commonly made in the charge case, but important differences between spin and charge gaps emerge already on the single-particle level because unlike the fundamental charge gap spin gaps involve excited-state energies. Kohn-Sham and many-body spin gaps are predicted to differ, and the difference is related to derivative discontinuities that are similar to, but distinct from, those usually considered in the case of charge gaps. Both ensemble DFT and time-dependent DFT (TDDFT) can be used to calculate these spin discontinuities from a suitable functional. We illustrate our findings by evaluating our definitions for the Lithium atom, for which we calculate spin gaps and spin discontinuities by making use of near-exact Kohn-Sham eigenvalues and, independently, from the single-pole approximation to TDDFT. The many-body corrections to the Kohn-Sham spin gaps are found to be negative, i.e., single-particle calculations tend to overestimate spin gaps while they underestimate charge gaps.

arXiv:0911.1712•Physical Review B•Mar 16, 2010•47 citations
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 NtotN_{tot}Ntot​ and a given zzz-projection of the spin, MtotM_{tot}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 MtotM_{tot}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_{\uparrow}N↑​-N↓N_{\downarrow}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 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.
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 NtotN_\textrm{tot}Ntot​ and fractional spin MtotM_\textrm{tot}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 NtotN_\textrm{tot}Ntot​ and MtotM_\textrm{tot}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 NtotN_\textrm{tot}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 formalism developed in the first paper of the series [arXiv:0901.1060] is applied to two thermodynamic systems: (i) of three global observables (the energy, the total electron number and the spin number), (ii) of one global observable (the internal electron energy) and two local (position-dependent) observables (the total electron density and the spin density). The two-component potential of the many-electron system of interest is constructed of a scalar external potential and a collinear magnetic field (coupled only with the spin operator). Various equilibrium characteristics of two systems are defined and investigated. Conditions for the equivalence between two systems (the same equilibrium density matrix demanded) are derived and thoroughly discussed. The applicability of the Hohenberg-Kohn theorem is extended to the thermodynamic spin-density functional theory. Obtained results provide a rigorous mathematical foundation for future derivation of the zero-temperature limit of this theory and determination of its chemical reactivity indices.
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.
Energy gaps are crucial aspects of the electronic structure of finite and extended systems. Whereas much is known about how to define and calculate charge gaps in density-functional theory (DFT), and about the relation between these gaps and derivative discontinuities of the exchange-correlation functional, much less is known about spin gaps. In this paper we give density-functional definitions of spin-conserving gaps, spin-flip gaps and the spin stiffness in terms of many-body energies and in terms of single-particle (Kohn-Sham) energies. Our definitions are as analogous as possible to those commonly made in the charge case, but important differences between spin and charge gaps emerge already on the single-particle level because unlike the fundamental charge gap spin gaps involve excited-state energies. Kohn-Sham and many-body spin gaps are predicted to differ, and the difference is related to derivative discontinuities that are similar to, but distinct from, those usually considered in the case of charge gaps. Both ensemble DFT and time-dependent DFT (TDDFT) can be used to calculate these spin discontinuities from a suitable functional. We illustrate our findings by evaluating our definitions for the Lithium atom, for which we calculate spin gaps and spin discontinuities by making use of near-exact Kohn-Sham eigenvalues and, independently, from the single-pole approximation to TDDFT. The many-body corrections to the Kohn-Sham spin gaps are found to be negative, i.e., single-particle calculations tend to overestimate spin gaps while they underestimate charge gaps.