Condensed Matter Seminar: Ensemble approach to many-electron systems: from exact relations to practical density functional approximations

Density functional theory (DFT) is the leading theoretical framework used to describe the electronic structure of materials. 

The central approximation of the theory is that of the exchange-correlation (xc) energy functional, which is responsible for all the electron-electron interactions in the system, beyond classical electrostatic repulsion.  One approach to design accurate and globally applicable functionals is by satisfying exact properties of many-electron systems. An important set of properties, in which I will focus in my talk, stems from describing a many-electron system with a varying, possibly fractional, number of electrons, N, using the ensembe approach. 

In my talk, I first describe how the piecewise-linearity of the energy versus N results in a sharp spatial step of the effective Kohn-Sham potential in DFT, and will share with you our recent success to accurately describe this step, using the orbital-free DFT approach [1, 2, 3].  

Next, I will show how the piecewise-linearity condition is generalized to spin-dependent systems. Recently [4], we succeeded to exactly describe the ground state of a finite, many-electron system with a fractional electron number and fractional spin S, by an ensemble of pure states, and characterize the dependence of the energy and the spin-densities on both N and S. We show which pure states contribute to the ensemble, and which states do not; we find a new type of a derivative discontinuity, which manifests in the case of spin variation at constant N, as a jump in the Kohn-Sham potential. We demonstrate this property in approximate DFT calculations with hybrid xc approximations [5] and discuss deviations of common approximations from the expected exact behavior.  Interestingly, for fractional N and S, we find a previously unkown degeneracy of the ground state, where 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.

[1] E. Kraisler, M. J. P. Hodgson and E. K. U. Gross, From Kohn-Sham to many-electron energies via step structures in the exchange-correlation potential, J. Chem. Theory Comput. 17, 1390 (2021)

[2] E. Kraisler, A. Schild, Discontinuous behavior of the Pauli potential in density functional theory as a function of the electron number, Phys. Rev. Research 2, 013159 (2020)

[3] N. E. Rahat, E. Kraisler, Plateaus in the potentials of density-functional theory: analytical derivation and useful approximations, accepted to J. Chem. Theory Comput. (2025); doi: 10.1021/acs.jctc.4c01771

[4] Y. Goshen, E. Kraisler, Energy of a many-electron system in an ensemble ground-state, versus electron number and spin: piecewise-linearity and flat plane condition generalized, J. Phys. Chem. Lett. 15, 2337 (2024)

[5] A. Hayman, N. Levy, Y. Goshen, M. Fraenkel, E. Kraisler, T. Stein, Spin migration in density functional theory: energy, potential and density perspectives, J. Chem. Phys. 162, 114301 (2025)