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A Correction for the Hartree-Fock Density of States for Jellium Without

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A Correction for the Hartree-Fock Density of States for Jellium Without A correction for the Hartree-Fock Density of States for Jellium without Screening Alexander I. Blair, Aristeidis Kroukis and Nikitas I. Gidopoulos Department of Physics, Durham University, South Road, Durham, DH1 3LE, United Kingdom 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. INTRODUCTION Hartree Fock 4 Free Electron 0 F The uniform electron gas, or jellium model, is an /ε 2 )] F archetypal example in solid-state physics and many-body k ( ε − theory. Its treatment, in the Hartree Fock (HF) approxi- ) 0 k ( ε mation, can be found in classic textbooks [1–6], where, we [ learn that the HF equations applied to the ground state −2 of the jellium, admit plane wave solutions with energy- 0.0 0.5 1.0 1.5 2.0 wavevector dispersion relation given by, k/k 3 F 2 2 2 Hartree Fock k kF k k kF + k ε(k)= 1+ F − ln . (1) Free Electron 2 − π 2kkF kF k − 2 /N 3 2 0 F kF is the Fermi wavevector, kF =3π ( N/V ). The single- ε )) k particle energy ε(k) is the sum of the free-electron energy, ( ε 2 ( 1 k /2, and the single-particle exchange energy. The Fermi g wavevector kF is often expressed in terms of the mean radius per particle r = 3 9π/4k3 [5]; for typical values s F 0 of r in metals, the two terms in (1) are comparable in −3 −2 −1 0 1 2 3 s [ε(k)−ε(k )]ε 0 p F F size. It is well known in the literature that the dispersion relation (1) has a logarithmically divergent derivative at Figure 1: Solid lines show ground-state HF results for the Fermi energy, shown in Fig. 1. Another marked dif- jellium, compared to free-electron results in dotted 0 2 ference between the free electron result and the HF so- lines. (rs/a0 = 4, εF = kF /2.) Top: Energy vs wave lution for jellium, evident in Fig. 1, is the considerably vector dispersion relation ε(k). The logarithmic arXiv:1502.01479v1 [cond-mat.str-el] 5 Feb 2015 increased bandwidth of the HF dispersion. Finally, it divergence in the derivative, dε/dk, is marked with a is well known that in the HF approximation the DOS triangle (N). Bottom: DOS, showing the unphysical for jellium vanishes at the Fermi level (Fig. 1), since the zero at the Fermi level for jellium. DOS is inversely proportional to the derivative of the dis- persion. The zero in the DOS at the Fermi level suggests that jellium is a semimetal, in obvious disagreement with duction of electronic many body correlation effects [1–7], experimental evidence for simple metals, such at sodium which screen the bare Coulomb potential and thus elimi- or aluminium, which are described accurately by the jel- nate the unphysical divergent derivative of the dispersion lium model. relation, the zero in the DOS at the Fermi level, and also In the literature, the qualitatively wrong description of reduce the bandwidth of the HF dispersion relation of jellium in the HF approximation is attributed to the long jellium. range of the Coulomb repulsion [1–6]. It is well known In an effort to understand whether HF’s lack of screen- that the flawed description can be corrected by the intro- ing actually plays a role, we revisit the HF study of jel- 2 lium, attempting to correct the qualitative errors of the [12], and independently by Gross, Oliveira, Kohn [13]. HF description, but without including any form of elec- These variational principles can be derived as special tronic correlation. For this purpose, we employ Slater’s cases from the Helmholtz variational principle in statis- hyper-HF (HHF) theory for the ground and the excited tical mechanics [14, 15]. In particular, the inequality in states of an N-electron system [8]. Specifically, we use (2) arises as the high temperature limit of the Helmholtz the single-particle HHF equations by Gidopoulos and variational principle. Theophilou [9, 10], who considered an N-electron system Optimisation of the R spin-orbitals ϕi to minimise described by a Hamiltonian H and then variationally op- the sum of the energies on the l.h.s. of{ (2)} leads to the timised the average energy Φ H Φ of all configu- following single-particle equations for the spatial part of nh n| | ni rations (N-electron Slater determinants Φn) constructed the spin-orbitals [9] (in atomic units): from a basis set of R spin-orbitals,P R N. ≥ 1 2 + V (r) ϕ (r) −2∇ ext i THE HYPER-HARTREE-FOCK EQUATIONS 1 R FOR JELLIUM + J (r) δ K (r) ϕ (r)= λ ϕ (r) , (3) Λ j − sj ,si j i i i j=1 X The aim in HHF theory is to obtain approximations, at the HF level of description, for the ground and the where, excited states of an N-electron system. These states R 1 Λ= − , (4) are represented by N-electron Slater determinants, con- N 1 structed from a common set of spin-orbitals. Obvi- − ously, to have the flexibility to describe excited states, and the number of spin-orbitals (R) must exceed the num- d3r′ J (r)ϕ (r) ϕ (r′) 2 ϕ (r) , (5) ber of electrons. For example, say we are interested to j i ≡ r r′ | j | i approximate the ground and excited states of the he- Z | − | d3r′ lium atom. In the HF ground state of He, the 1s orbital K (r)ϕ (r) ϕ∗(r′)ϕ (r′)ϕ (r) , (6) j i ≡ r r′ j i j (ϕ1s) is doubly occupied. To study a couple of excited Z | − | states, we need at least one more spin-orbital and the are the Coulomb and exchange operators respectively. next one is: ϕ↑ . With the three available spin-orbitals, 2s Vext signifies the attractive potential of the nuclear 1s↑, 1s↓, 2s↑, we can form three configurations for the He ↓ ↑ charge. For R = N, Eqs. 3 reduce to the familiar ground- atom (two-electron Slater determinants): Φ1 = [1s , 1s ], ↓ ↑ ↑ ↑ state HF equations. In Eqs. 3, the orbitals ϕi, with Φ2 = [1s , 2s ], Φ3 = [1s , 2s ]. In HHF theory, we vari- i = 1,...,R, are correctly repelled electrostatically by ationally optimise the three common spin-orbitals simul- a charge of N 1 electrons. In contrast to the ground- taneously, by minimising the sum of the expectation val- state HF case− [16], this holds true even for the orbitals ues 3 Φ H Φ . The minimisation leads to the HHF i=1h i| | ii that are not occupied in the HHF ground-state Slater de- single-particle equations for the three spin-orbitals. It terminant as long as these orbitals are variationally op- turnsP out that these equations resemble the ground-state timised, i.e., for ϕi, with N <i R. Furthermore, the HF equations for the lithium atom (three electrons) but ≤ orbitals that are left variationally unoptimised, i.e., ϕi, with a weakened Coulomb repulsion between the three with i > R, are repelled by a charge of N 1+(1/Λ) elec- electrons, to keep balance with the nuclear charge which trons. In the HHF equations, the well-known− asymmetry has remained that of the He nucleus. in the treatment of the variationally optimised and unop- In general, in HHF theory [8, 9] for an N-electron sys- timised orbitals by the nonlocal exchange operator [16] tem, one considers a set of R orthonormal spin-orbitals, is still present, but softened (for large Λ), compared with with R N. On this spin-orbital basis set one may ground-state HF. We note that for R>N, Koopmans’ define, D≥= R!/(N!(R N)!), N-electron Slater deter- − theorem [19, 20] ceases to hold for the HHF equations. minants. The HHF equations (3) have the form of ground-state The derivation of the single-particle HHF equations in HF equations for a virtual system of R-electrons, where Ref. 9 is based on Theophilou’s variational principle [11], the electronic Coulomb repulsion is multiplied by 1/Λ: 1/ r r′ Λ−1/ r r′ . Therefore, the calculation D D | − | → | − | Φ H Φ E(0) , (2) of the optimal spin-orbitals to represent the ground and h n| | ni≥ n n=1 n=1 excited-states of an N-electron system in the HHF ap- X X proximation, is reduced to the calculation of the ground- 0 where En are the D lowest eigenvalues of the N- state HF orbitals of a fictitious system with a greater electron{ Hamiltonian} H.
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