arXiv:1502.01479v1 [cond-mat.str-el] 5 Feb 2015 ftejlim di ln aesltoswt energy- with by, solutions given wave state relation plane ground dispersion wavevector admit the to jellium, applied the equations of HF we where, the [1–6], that textbooks learn classic approxi- in (HF) found Fock be Hartree can the mation, in treatment, Its many-body and theory. physics solid-state in example archetypal aisprparticle per radius wavevector k ag fteCuobrplin[–] ti elknown intro- well the by is corrected It be can [1–6]. description flawed repulsion the long Coulomb that the the to of attributed is range approximation HF the in jellium jel- the by accurately model. described lium sodium are at which such aluminium, metals, or simple with disagreement for suggests obvious evidence level in Fermi experimental semimetal, the a at is DOS jellium the that dis- in the the zero of The DOS since derivative 1), the persion. the to (Fig. approximation proportional level inversely Fermi HF is it the DOS the at Finally, vanishes in jellium that considerably for dispersion. known the HF well is the is 1, so- of Fig. HF bandwidth in the evident increased and jellium, result dif- marked for electron Another lution free the 1. between Fig. in ference at shown derivative energy, divergent Fermi logarithmically the a has (1) relation size. of atceenergy particle k F 2 h nfr lcrngs rjlimmdl san is model, jellium or gas, electron uniform The / nteltrtr,teqaiaieywogdsrpinof description wrong qualitatively the literature, the In dispersion the that literature the in known well is It r orcinfrteHrreFc est fSae o Jell for States of Density Hartree-Fock the for correction A steFriwavevector, Fermi the is ,adtesnl-atceecag nry h Fermi The energy. exchange single-particle the and 2, s ε ( nmtl,tetotrsi 1 r oprbein comparable are (1) in terms two the metals, in k = ) k k 2 iia riat fteH olclecag prtr not operator, literature. exchange the in nonlocal known HF are the artifactcorrelation, an of is artifacts approximation HF similar ground-state the of failure iprinrlto n h eoi h O r tl present th still that are find DOS we the in jellium, zero of the states and excited relation the dispersion and state ground ihvle fwvvco n nry elcerfo h Fer i the but from fixed, clear not well varia is energy, of and boundary wavevector set this of the of values between location high boundary The the orbitals. to HHF jellium of energy and wi disagreement attributed qualitative is in failure are qualitative – this Currently, level dispersio Fermi metals. energy the the at of (DOS) derivative divergent – predictions 2 F eateto hsc,Dra nvriy ot od Durh Road, South University, Durham Physics, of Department ecnld ht ahrta h ako cenn nteH e HF the in screening of lack the than rather that, conclude We unifor the for calculation (HF) Hartree-Fock the revisit We − mlyn ltrshprHrreFc HF qain,d equations, (HHF) hyper-Hartree-Fock Slater’s Employing ε sotnepesdi em ftemean the of terms in expressed often is ( k k π INTRODUCTION stesmo h reeeto energy, free-electron the of sum the is ) F r s  = + 1 p lxne .Bar rsedsKoksadNktsI Gidopoulos I. Nikitas and Kroukis Aristeidis Blair, I. Alexander 3 k 9 F 2 k 2 π/ F kk 3 − 4 3 = F k k F 3 2 π ln 5;frtpclvalues typical for [5]; 2 (

N/V k k F F − + .Tesingle- The ). k k

 . (1) n culypasarl,w eii h Fsuyo jel- of study HF the revisit we role, a plays actually ing of relation dispersion HF also the and of level, jellium. Fermi bandwidth the at the DOS reduce the in zero dispersion the the relation, of elimi- derivative thus divergent and unphysical the potential nate Coulomb [1–7], bare effects the correlation screen which body many electronic of duction iue1 oi ie hwgon-tt Frslsfor results HF ground-state show lines Solid 1: Figure iegnei h derivative, the in divergence ie.( lines. rage( triangle na ffr oudrtn hte Fslc fscreen- of lack HF’s whether understand to effort an In elu,cmae ofe-lcrnrslsi dotted in results free-electron to compared jellium, otelc fsreigi h Fequations. HF the in screening of lack the to etrdseso relation dispersion vector eainadvnsigdniyo states of density vanishing and relation n fisnnoa xhneoeao.Other operator. exchange nonlocal its of u hfe rmteFriwavevector Fermi the from shifted but , lcrngs rjlimmdl whose model, jellium or gas, electron m ilvlo jellium. of level mi

0 0 electronic of lack the with associated r (ε()) ε [()−( )] s heprmna vdnefrsimple for evidence experimental th − 0 1 2 3 iegn eiaieo h energy the of derivative divergent e − utos h elkonqualitative well-known the quations, inlyotmsdadunoptimised and optimised tionally a ecoe olea arbitrarily at lie to chosen be can t eoa h em ee o jellium. for level Fermi the at zero /a N 2 2 4 0 rvdvrainly osuythe study to variationally, erived 0 3 m H L,Uie Kingdom United 3LE, DH1 am, . 0 .Bto:DS hwn h unphysical the showing DOS, Bottom: ). 0 4, = reElectron Free Fock Hartree − reElectron Free Fock Hartree 2 ε 0 . F 0 5 − u ihu Screening without ium = 1 [ ε ( k ) F 2 − ε / 1 ( 0 . . o:Eeg swave vs Energy Top: 2.) 0 ε dε/dk )] ( ε k .Telogarithmic The ). 0 1 smre iha with marked is , 1 . 5 2 2 . 0 3 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 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. number of electrons R N, and scaled down electronic An extension of the variational principle, with unequal Coulomb repulsion. A≥ related approach is the “super- weights in the sums in (2) was proposed by Theophilou hamiltonian method” by Katriel [22, 23]. 3

Finally, before applying the HHF equations to jellium, Λ (See Fig. 2). For increasing Λ, the bandwidth we remark that correlated, approximate eigenstates of of→ the ∞ HHF dispersion, λ(k), decreases compared to the the Hamiltonian H can be obtained by diagonalising ground-state HF dispersion, ε(k). For Λ the ex- → ∞ the matrix Φn H Φm [9], where Φn are the N-electron change term in HHF dispersion vanishes and λ(k) reduces HHF Slaterh determinants.| | i This configuration-interaction to the free-electron result. method employs the HHF spin-orbitals, which are opti- mised to represent on equal footing the ground and the Importantly, the wave vector at which the logarithmi- excited states of H. cally divergent derivative occurs is shifted from kF to kR, such that the divergence no longer occurs at the Fermi energy of the physical N-electron system, when the num- Solution of the HHF equations for jellium. ber of optimised orbitals is R>N.

Similarly to the HF ground state, the HHF equations for jellium admit plane wave solutions. It follows that the ground-state N-electron Slater determinant and its total energy are the same in the HF and HHF approximations. Although the HF and HHF equations for jellium admit the same solution for the orbitals, the dispersion relations 0 for the single-particle energies ε(k) and λ(k) differ. In

particular, the HHF dispersion, λ(k), results from an op- timisation that involves a broader range of wavevectors − than the HF dispersion ε(k). 0 Following the standard treatment in textbooks [1–6], it is straightforward to work out directly the solution of the HHF Eqs. 3. Here, we exploit the similarity of Eqs. 3 − with ground-state HF equations of an R-electron system, to obtain that the HHF dispersion relation, λ(k), will 0.0 0. .0 . .0 . be given by (1) with the single-particle exchange energy scaled down by the factor 1/Λ: 3 2 2 2 k k k k k + k λ(k)= R 1+ R − ln R . (7) 2 − Λπ 2kk k k  R R −  kR is the Fermi wave vector of the virtual R-electron system, λ 3 2 R k =3π . (8) R V

Dividing kR/kF , and taking the thermodynamic limit, N, V , with the ratio Λ fixed, we obtain: → ∞ −3 − − 3 1/3 kR =Λ kF . (9) λ−λ Substitution of the above into Eq. (7) yields the desired expression for the single particle energy levels of jellium, Figure 2: Excited-state HF results for jellium, for in terms of Λ and the Fermi wavevector kF of the actual various Λ = R/N, compared to free electron results 0 2 N-electron system: (dotted lines). (rs/a0 = 4, εF = kF /2.) Top: Energy vs wave vector dispersion relations λ(k). When Λ = 1, k2 k N λ(k)= Λ−2/3 F (10) λ(k)= ε(k). Triangles ( ) mark logarithmic divergence 2 − π in dλ/dk at Fermi level of fictitious R-electron system. Λ2/3k2 k2 Λ1/3k + k Bottom: DOS g(λ(k)), showing the zero at the Fermi 1+ F − ln F . × 2Λ1/3kk Λ1/3k k level of the fictitious R-electron system. F F ! −

The dispersion relation in Eq. (10) reduces to the ground-state Hartree-Fock result for Λ = 1, and to the free electron dispersion relation, λ(k)= k2/2, in the limit The DOS, g(λ)δλ, can be obtained directly from Eq. 10 4

[2] and is given by the parametric equation: ing the unoccupied orbitals too diffuse, and raising their energy eigenvalue to unphysically high values [16]. V k2 It follows that the energy to excite an electron from an g(λ(k)) = 2 π (dλ/dk) occupied HF orbital ϕi to a virtual HF orbital ϕa, keep- −1 2 2 2 ing the other occupied orbitals frozen, must be smaller V k 1 kR kR + k kR + k than the eigenvalue difference εa εi. This is because = 2 k 2 ln (11). π − Λπ k − 2k kR k − "  # the energy εa incorporates the Coulomb interaction of − the orbital ϕa, hosting the electron after excitation, with 1/3 g(λ(k)) is expressed in terms of kR (rather than Λ kF ) all the occupied orbitals, including the orbital ϕi accom- to keep the notation simple. For any finite Λ, the DOS modating the same electron before excitation. In this still vanishes. However, as shown in Fig. 2, the zero in sense, the Coulomb interaction of the orbitals ϕi and ϕa the DOS occurs at the Fermi energy of the fictitious R- can be interpreted as a form of self-interaction raising electron system, λ(kR), rather than the Fermi energy of the energy of the virtual level εa. It is similar to the the physical system λ(kF ). “ghost” self-interaction discussed in Ref. 21. This inter- pretation is consistent with viewing the virtual HF en- ergies as single-particle levels of the N-electron system. DISCUSSION Alternatively, by extending the proof of Koopmans’ the- orem [19], it can be shown that εa is equal to the negative In metals, screening is an important effect that reduces of the electron affinity of the system to bind an electron the range of the effective repulsion between electrons, at the virtual orbital ϕa, see e.g. Ref. 20. The interpreta- shielding any charge at distances greater than a charac- tion of the virtual energies as negative affinities amounts teristic screening length. In the literature of many-body to regarding the unoccupied levels as virtual levels of an theory [1, 3–6] and solid-state physics [2–4], where jellium (N + 1)-electron system. is a paradigm, the qualitatively flawed description of met- We note that for jellium, where the occupied and vir- als by the HF approximation is attributed to the long- tual orbitals are plane waves, the self-interaction error range nature of the Coulomb interaction, which, com- of the virtual energy levels (discussed above) vanishes bined with the neglect of correlation, deprives from the in the thermodynamic limit. Therefore, it makes sense HF equations the flexibility to model the phenomenon of to consider that the HF unoccupied levels represent vir- screening. This understanding of HF’s failure is further tual single-particle levels of the N-electron system, and supported by the softening of the divergence in the slope to study their dispersion relation and density of states. of ε(k), after replacing the bare Coulomb potential in A consequence of the asymmetry in the treatment of the HF nonlocal exchange term by a screened Coulomb occupied versus unoccupied orbitals in the HF equations, potential [2]. is presented by Bach et al. [18], who prove that, for On the other hand, in the theoretical chemistry liter- a finite system, the highest occupied spin-orbital in the ature, it is well known that the HF nonlocal exchange ground-state unrestricted-HF Slater determinant, is non- term, in finite systems, gives rise to several counterintu- degenerate: a nonzero gap separates it from the lowest itive results, reminiscent of the HF anomalies in jellium, unoccupied spin-orbital, even in systems with an odd which are not associated with HF’s lack of correlation. number of electrons, in contrast to physical expectation. For example, Handy et. al [17] disproved the widely Recently, Hollins et al., using two different meth- held view [1], that the HF nonlocal exchange potential ods, the optimised effective potential, or exact exchange decays as ( 1/r) at large distances. In particular, in method [24], and the local Fock exchange potential − Ref. 17 it was demonstrated that the asymptotic decay (LFX) method [25], showed that it is possible to omit cor- of an occupied HF orbital ϕi with eigenvalue εi is not relation and still obtain an accurate dispersion relation exp( √ 2ε r), as would be expected from the ( 1/r) ε(k) for simple metals, provided the exchange potential ∼ − − i − tail of the exchange potential. On the contrary, in HF term is localv ˆx = vx(r) [25]. The LFX potential, defined all the occupied orbitals ϕi decay uniformly at large dis- as the local exchange potential with the HF ground-state tances away from the system, regardless of their energy density [25, 26], is particularly interesting in our context: eigenvalue. even though the determination of the LFX potential is As already mentioned, it is also widely known that based on the same ground-state HF calculation that gives the HF exchange operator deals with the occupied and a very poor prediction for the dispersion of simple metals unoccupied orbitals in the ground-state HF Slater de- (e.g. Na, Al), the band-structure of the LFX potential terminant in an asymmetric way [16]: for an N-electron [25] almost coincides with the band-structure of the local system, the (variationally optimised) occupied orbitals density approximation in density functional theory [27], are repelled electrostatically by a charge of N 1 elec- which, by construction, is very accurate for these sys- trons, while the (variationally unoptimised) unoccupied− tems. With regard to the vanishing of the HF DOS of orbitals feel the stronger repulsion of N electrons, mak- metals at the Fermi energy, Hollins et al. [25] argue that 5 it appears to be an artifact of the HF nonlocal exchange for any deep and profound explanation”. operator. Our work supports this point of view, by studying the nonlocal HF exchange term directly. We find that the well-known anomalies of the HF description of jellium are The Quantum Mechanics of Many-Body still present in the solution of the HHF equations. How- [1] D.J. Thouless Systems Second Edition, Pure and Applied Physics Se- ever, the location of the divergent derivative of λ(k) and ries, Academic Press Inc., New York (1972) the zero in the DOS are no longer at the Fermi level of the [2] N.W. Ashcroft, and N.D. Mermin, Solid State Physics, actual N-electron system. Instead, they are positioned Saunders College Publishing, Harcourt, Inc., Orlando at the border separating the variationally optimised and (1976) unoptimised orbitals. By choosing to variationally opti- [3] C. Kittel, Quantum Theory of Solids, John Wiley & Sons, mise a very large number of orbitals, the unphysical zero Inc., New York (1976) Many-Body Theory of Solids An Introduction in the DOS can be pushed to very high energies, avoid- [4] J.C. Inkson Plenum Press, New York (1984) ing completely the window of single-partice energies that [5] P. Fulde, Electron Correlations in Molecules and Solids, can be of any relevance to the ground state of jellium. Springer, Berlin, Heidelberg (1995) Therefore, it is no longer justified to relate the mobile di- [6] T. Lancaster, S.J. Blundell Quantum field theory for the vergence in the derivative of the HHF dispersion and the gifted amateur, Clarendon Press, Oxford (2014) travelling zero in the HHF DOS of jellium, with the lack [7] M.A. Ort˘iz, and R.M. M´endez-Moreno, Phys. Rev. A, of electronic correlation in the HHF approximation, even 36, 888, (1987); doi:10.1103/PhysRevA.36.888 though these anomalies can still be removed by screening [8] J.C. Slater The Self-consistent Field for Molecules and Solids, Vol. 4 of the series Quantum Theory of Molecules the Coulomb repulsion in the exchange term. and Solids, McGraw-Hill Book Inc. (1974) This strengthens the view that the failure of the [9] N. Gidopoulos and A. Theophilou, Phil. Mag. Part B, ground-state HF solution for jellium is also an artifact 69, 1067-1074, (1994); doi:10.1080/01418639408240176. of the nonlocal exchange operator. At least part of the [10] At the time of publication of Ref. 9, its authors were not explanation of the divergent derivative of the disper- aware of Slater’s HHF scheme. sion seems to be the asymmetry in the treatment of the [11] A.K. Theophilou, J. Phys. C: Solid State Physics, 12, variationally optimised and unoptimised orbitals: as the 5419, (1979) [12] A.K. Theophilou, in The Single-Particle Density in wavevector k crosses kR from below, the plane wave solu- Physics and Chemistry, Edited by N.H. March and B. tions of the excited-state HHF equations are subjected to Deb, Academic Publishing, London (1987) a discontinuous drop of the nonlocal exchange term and [13] Gross, E. K. U. and Oliveira, L. N. and Kohn, W., Phys. a correspondingly discontinuous increase in the Coulomb Rev. A, 37, 2805, (1988) doi: 10.1103/PhysRevA.37.2805 repulsion from the Hartree term. This raises the single- [14] A. K. Rajagopal and F. A. Buot, Phys. Rev. A, 51, 1770 particle eigenvalue λ(k) to higher energies, diminishing (1995) the DOS in the neighborhood of λ(k ). The same mech- [15] E. Pastorczak, N.I. Gidopoulos and K. Pernal, Phys. Rev. R A, 87, 062501, (2013); doi:10.1103/PhysRevA.87.062501. anism operates in the divergent slope of the ground-state [16] Davidson, Ernest R., Rev. Mod. Phys., 44, 451, (1972); HF dispersion ε(k). doi: 10.1103/RevModPhys.44.451 In conclusion, we suggest that the qualitative failure [17] Handy, Nicholas C. and Marron, Michael T. and Sil- of the HF approximation for jellium is unrelated to the verstone, Harris J., Phys. Rev., 180, 45, (1969); doi: lack of correlation in the HF approximation. Instead, 10.1103/PhysRev.180.45 this failure is another example in the list of counterintu- [18] V. Bach, E.H. Lieb, M. Loss, J.P. Solovej, Phys. Rev. Lett., 72, 2981, (1994) doi: 10.1103/Phys- itive results caused by the nonlocality of the HF exchange RevLett.72.2981 operator. Our work complements the work of Hollins et [19] T. Koopmans, Physica 1, 104113 (1934); al. in Ref. 25 and our conclusion is contrary to what is doi:10.1016/S0031-8914(34)90011-2 currently written in almost any textbook in the fields of [20] A. Szabo and N.S. Ostlund, “Modern Quantum Chem- many body theory and solid state physics. To the best istry”, Macmillan Publishing Co. Inc., New York (1996). of our knowledge, Slater gave the only hint so far in the [21] N. I. Gidopoulos, P. G. Papaconstantinou, and E. K. U. literature that the HF failure may be unrelated to elec- Gross, Phys. Rev. Lett. 88, 033003 (2002) [22] J. Katriel, J. Phys. C: Solid St. Phys., 13 L375 (1980) tronic correlation. In his textbook on “Insulators, Semi- [23] J. Katriel, Int. J. Quantum Chem. 23, 1767 (1983) conductors and Metals” [28] he writes that “it cannot be [24] T.W. Hollins, S.J. Clark, K. Refson and N.I. Gidopoulos, in accordance with experiment to write the total energy Phys. Rev. B 85, 235126 (2012). of the electronically excited state of the crystal as a sum [25] T.W. Hollins, S.J. Clark, K. Refson and N.I. Gidopoulos, of one-electron energies of the type of ...” (cf. the HF so- to appear. lutions). “A great deal of effort has gone into explaining [26] I.G. Ryabinkin, A.A. Kananenka and V.N. Staroverov, this apparent paradox connected with the free-electron Phys Rev Lett 111, 013001 (2013). [27] J. Kohanoff, N.I. Gidopoulos, “Density functional theory: theory of electrons in metals. ... It is the impression of basics, new trends and applications”, in the Handbook of the present author, however, that we do not need to look 6

Molecular Physics and Quantum Chemistry, Edited by of the series Quantum Theory of Molecules and Solids, Stephen Wilson, Vol. 2, Part 5, Chapter 26, pp 532568, McGraw-Hill Book Inc. (1967) [See page 263.] John Wiley & Sons, Ltd, Chichester (2003) [28] J.C. Slater Insulators Semiconductors and Metals Vol. 3