# A correction for the Hartree-Fock density of states for jellium without screening

TL;DR: The Hartree-Fock (HF) calculation for the uniform electron gas, or jellium model, is revisited and it is concluded that the well-known qualitative failure of the ground-state HF approximation is an artifact of its nonlocal exchange operator.

Abstract: We revisit the Hartree-Fock (HF) calculation for the uniform electron gas, or jellium model, whose predictions—divergent derivative of the energy dispersion relation and vanishing density of states (DOS) at the Fermi level—are in qualitative disagreement with experimental evidence for simple metals. Currently, this qualitative failure is attributed to the lack of screening in the HF equations. Employing Slater’s hyper-Hartree-Fock (HHF) equations, derived variationally, to study the ground state and the excited states of jellium, we find that the divergent derivative of the energy dispersion relation and the zero in the DOS are still present, but shifted from the Fermi wavevector and energy of jellium to the boundary between the set of variationally optimised and unoptimised HHF orbitals. The location of this boundary is not fixed, but it can be chosen to lie at arbitrarily high values of wavevector and energy, well clear from the Fermi level of jellium. We conclude that, rather than the lack of screening in the HF equations, the well-known qualitative failure of the ground-state HF approximation is an artifact of its nonlocal exchange operator. Other similar artifacts of the HF nonlocal exchange operator, not associated with the lack of electronic correlation, are known in the literature.

## Summary (1 min read)

### INTRODUCTION

- The uniform electron gas, or jellium model, is an archetypal example in solid-state physics and many-body theory.
- The singleparticle energy ε(k) is the sum of the free-electron energy, k2/2, and the single-particle exchange energy.
- It is well known in the literature that the dispersion relation (1) has a logarithmically divergent derivative at the Fermi energy, shown in Fig.
- Finally, it is well known that in the HF approximation, the density of states (DOS) for jellium vanishes at the Fermi level (Fig. 1), since the DOS is inversely proportional to the derivative of the dispersion.

### THE HYPER-HARTREE-FOCK EQUATIONS FOR JELLIUM

- These states are represented by Nelectron Slater determinants, constructed from a common set of spin-orbitals.
- Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.
- The minimisation leads to the HHF single-particle equations for the three spin-orbitals.
- These variational principles can be derived as special cases from the Helmholtz variational principle in statistical mechanics.

### DISCUSSION

- In metals, screening is an important effect that reduces the range of the effective repulsion between electrons, shielding any charge at distances greater than a characteristic screening length.
- This understanding of HF’s failure is further supported by the softening of the divergence in the slope of ε(k), after replacing the bare Coulomb potential in the HF nonlocal exchange term by a screened Coulomb potential.
- Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.
- The authors find that the wellknown anomalies of the HF description of jellium are still present in the solution of the HHF equations.

Did you find this useful? Give us your feedback

...read more

##### Citations

15 citations

9 citations

9 citations

8 citations

##### References

5,080 citations

580 citations

### "A correction for the Hartree-Fock d..." refers background in this paper

...al [17] disproved the widely held view [1], that the HF nonlocal exchange potential decays as (−1/r) at large distances....

[...]

257 citations

245 citations