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

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.

Figures (13)

• Figure 1: Illustration of Region \ref{['eq:spin condition']}, for the C atom with varying $N_\textrm{tot}$ and $M_\textrm{tot}$. The full gray circles correspond to pure states with integer $N_\uparrow$ and $N_\downarrow$. The cyan region corresponds to $N_\textrm{tot} \in [6,7]$ (i.e., C $\rightarrow$ C$^-$), the blue region -- to $N_\textrm{tot} \in [5,6]$ (i.e., C$^+$$\rightarrow$ C), and so on. The pure states that contribute to the ensemble ground state for $N_\textrm{tot} \in [5,6]$ are emphasized with black circles.
• Figure 2: Spin polarization $\zeta(r)$, for the C atom, with $\alpha=0.8$ and $M_\textrm{tot}=0$, for various values of the parameter $y$ defined in the text (see Legend).
• Figure 3: Left: Continuous lines represent the statistical weights $\lambda_{N_0,1}$ (dashed black), $\lambda_{N_0,0}$ (solid green) and $\lambda_{N_0,-1}$ (dotted blue), versus the spin $M_\textrm{tot}$, for the neutral C atom, analytically calculated from maximization of entropy for C (see Eq. \ref{['eq:C__maxS__x']}). Circles (of the same colors) correspond to the numerical solution of Eq. \ref{['eq:Smax_Mtot_condition']}; full overlap is observed. Right: Statistical weights $\lambda_{N_0,1}$, $\lambda_{N_0,0}$ and $\lambda_{N_0,-1}$ (same colors and dash types as on the left panel) obtained for minimal $\Delta S_z$
• Figure 4: Left: Entropy $\mathscr{S}$ versus the spin $M_\textrm{tot}$, for the neutral C atom, for the case of maximal entropy (thin blue line) and minimal $\Delta S_z$ (thick red line). Right: value of the deviation $\Delta S_z$ versus the spin $M_\textrm{tot}$, for the case of maximal entropy (thin blue line), lying in between the lines for minimal (thick red line) and maximal (dashed green line) values of $\Delta S_z$.
• Figure 5: Illustration of the proof of Statement 1: The ensembles $\hat{\Lambda}_\mathrm{out}$ (red circle), $\ket{\Psi_\mathrm{in}}$ (yellow circle, inside Region \ref{['eq:spin condition']}), $\hat{\Lambda}$ and $\hat{\Lambda}'$ (dark yellow and orange circles; coincide), and $\hat{\Lambda}_b$ and $\hat{\Gamma}_b$ (blue and cyan rhombuses; coincide at the edge of Region \ref{['eq:spin condition']}), depicted in the $N_\uparrow-N_\downarrow$ plane.