LOCAL AD SEMI-LOCAL DESITY FUCTIOAL APPROXIMATIOS FOR EXCHAGE AD CORRELATIO: WHY DO THEY WORK, AD DO THEY WORK BEST AT ZERO TEMPERATURE? John P. Perdew and Stefan Kurth Department of Physics and Quantum Theory Group Tulane University ew Orleans, LA 708 USA Abstract Kohn-Sham spin density functional theory is a widely-used method of electronic structure calculation. The simple local spin density approximation for the ehange-correlation energy of a many-electron system, proposed over 30 years ago, has worked beyond all expectation, and is still the basis for much of solid state physics. The semi-local generalized gradient approximation has brought much of quantum chemistry under the rubric of density functional theory. Some old and new explanations for the success of these approximations will be reviewed. In particular, the sum rule on the ehangecorrelation hole will be displayed for systems of fixed or fluctuating electron number at zero or non-zero temperature. ITRODUCTIO In 965, Kohn and Sham derived exact selfconsistent field equations for the ground-state (T = 0) energy E and electron spin densities n (r), n (r) of electrons in an external potential v(r): n σ (r) = f ασ ψ ασ (r) (σ = or ), () α f ασ = θ(µ ε ασ ), () ( +v(r)+ d 3 r n(r ) r r + δe ) ψ ασ (r) = ε ασ ψ ασ (r), (3) δn σ (r) E = f ασ ψ ασ ασ ψ ασ + d 3 rn(r)v(r)+ d 3 r d 3 r n(r)n(r ) +E r r [n,n ]. (4) All equations are in atomic units ( h = m = e = ). So long as the external potential v(r) is spin-independent, only the total density n(r) = n (r)+n (r) (5)
is formally necessary, but approximations to the ehange-correlation energy functional E are more successfully constructed from the separate spin densities. The chemical potential µ of Eq. () must be adjusted to make d 3 rn(r) =. (6) Kohn and Sham also proposed the local spin density (LSD) approximation, E LSD [n,n ] = d 3 rn(r)ǫ (n (r),n (r)), (7) where ǫ (n,n ) is the accurately-known ehange-correlation energy per particle of the uniform electron gas. LSD is exact for densities that vary slowly over space, and is still widely and successfully used in solid state physics 3, 4. The second-order gradient expansion approximation (GEA), E GEA = E LSD + σ,σ d 3 n rc σσ σ n σ, (8) n /3 σ n /3 σ is almost never used, but the generalized gradient approximation (GGA) 5, 6 E GGA [n,n ] = d 3 rnǫ GGA (n,n, n, n ). (9) has been enthusiastically adopted in quantum chemistry 3, 4 (and to some extent solid state physics) since 990. GGA reduces the errors of LSD atomization energies by about a factor of five. Simple local(eq.(7)) and semi-local(eq.(9)) approximations have worked beyond all expectation. This article will summarize some old and new explanations for this success, and suggest that the high accuracy of these approximations at zero temperature may not carry over to non-zero temperature. EXPLAATIO FOR THE SUCCESS OF LOCAL AD SEMI-LOCAL APPROXIMATIOS AT ZERO TEMPERATURE The LSD approximation of Eq. (7) is exact for densities that vary slowly over space, but the densities of real atoms, molecules, and solids are not slowly varying. If they were, the GEA of Eq. (8) would work better than LSD, but in fact it works less well. Physical insight into the ehange-correlation energy E is provided by the coupling-constant integration 7, 8. Imagine a Hamiltonian Ĥλ depending upon a parameter λ: Ĥ λ = i + v λ (r i )+λ. (0) i i i r i r j The external potential v λ is adjusted to keep the ground-state spin densities n (r), n (r) independent of λ. At λ =, Ĥ λ describes the real interacting system, while at λ = 0 it describes the Kohn-Sham non-interacting system. From the ground-state wavefunction Ψ λ, one can find the pair density P λ (r,r ), defined so that P λ (r,r )d 3 rd 3 r is the joint probability to find an electron in d 3 r and another in d 3 r. If these two events were independent (as they typically are not), the pair density would factor as P λ (r,r ) = n(r)n(r ). Instead j i P λ (r,r ) = n(r)[n(r )+n λ (r,r )], ()
where n λ (r,r ) is the density at r of the ehange-correlation hole about an electron at r. Then E = d 3 r n(r) d 3 r n (r,r ), () r r where n (r,r ) = 0 dλ n λ (r,r ) = n x (r,r )+n c (r,r ). (3) The coupling-constant average in Eq. (3) is a trick to include the kinetic energy of correlation in a form that looks like a potential energy. The ehange hole n x in Eq. (3) is the integrand in the non-interacting or λ = 0 limit. ote that electron conservation d 3 r P λ (r,r ) = n(r)[ ] (4) implies the sum rule d 3 r n λ (r,r ) =. (5) The local spin density approximation to the hole is clearly n LSD (r,r ) = n unif (n (r),n (r); r r ), (6) where n unif (n,n ; r r ) is the hole density in an electron gas with uniform spin densities n, n. Because this is a possible physical system, its hole respects many of the same conditions as the exact hole of the real system: (a) LSD obeys the sum rules d 3 r n x (r,r ) =, (7) d 3 r n c (r,r ) = 0, (8) which constrain 8 the integral of Eq. () to reasonable values. (b) LSD respects 9 the negativity of the exact ehange hole, n x (r,r ) 0. (9) (c) The LSD on-top ehange hole density n x (r,r) is exact 0. This is why accurate approximations for E require the spin densities n and n. (d) The LSD on-top correlation hole density n c (r,r) is not exact, but is still very accurate and provides the missing link between real systems and the uniform electron gas. Because n (r,r) is accurate in LSD, so is the Coulomb cusp of n (r,r ). The second-order density-gradient expansion (GEA) for the hole density improves upon LSD for small r r, but gives unphysical results for large r r 5, 9. Because the GEA hole is a truncated expansion, and not the hole of any physical system, it violates conditions (a) and (b) above. A non-empirical derivation of the generalized gradient approximation of Eq. (9) starts from the gradient expansion of the hole density, then cuts off its spurious large- r r contribution to restore conditions (a) and (b). Essentially the same ǫ GGA (n,n, n, n ) has been derived more simply 6 from general constraints on E [n,n ], without appeal to the hole. 3
A hierarchy of equations which starts with LSD and proceeds through GGA will probably be completed by some fully-nonlocal approximation of high accuracy. The hole constraints (a) - (d) may find a last hurrah in the construction of such an approximation. In summary, LSD and GGA work well outside their formal domain of validity because they are conserving approximations which retain important features of the exact ehange-correlation energy. O-ZERO TEMPERATURES AD OPE SYSTEMS Formal Kohn-Sham theory was extended to open systems at non-zero temperatures by Mermin 3. The basic structure of Eqs. () - (0) is preserved, with a few changes: (i) Within the grand-canonical ensemble, n (r) = n (r) = n(r)/ and ε α = ε α = ε α. Thus, for spontaneously-magnetized systems, the LSD or GGA ontop hole density will be less accurate for the ensemble than for the pure-state density functional theory. (ii) The occupation numbers become Fermi functions f α = exp[(ε α µ)/kt]+. (0) (iii) Inputs such as ǫ (n) now depend 4, 5 upon the temperature T. Interesting applications of this formalism to metallic clusters 6, liquid metals 7, and plasmas 7 have been made. The LSD (Eq. (7)) and GGA (Eq. (9)) approximations should not be expected to work so well at non-zero T as they do at T = 0, because of the on-top hole problem mentioned above and because exact constraints like condition (a) of section are no longer satisfied. (Partial compensation for this loss may arise from the fact that the ehange and correlation holes become more short-ranged as the temperature increases - a favorable development for LSD and GGA.) At non-zero temperature, expectation values are taken not over a single pure state but over an ensemble of states Ψ νλ with probabilities w νλ = exp[ (E νλ µ ν )/kt]/z. The ensemble pair density is P λ (r,r ) = ν = ν w νλ P νλ (r,r ) w νλ n νλ (r)[n νλ (r )+n νλ (r,r )] = n(r)[n(r )+n λ (r,r )], () where n(r) = ν w νλ n νλ (r). If the ehange-correlation hole density n λ (r,r ) for the ensemble were the same as the ensemble average of the hole density n λ (r,r ) = ν w νλ n νλ (r,r ), () then conditions (a) and (b) of the previous section would be respected. But in fact where n λ (r,r ) = n λ (r,r ) + ν [n νλ (r) n(r)] w νλ [n νλ (r )+n νλ n(r) (r,r )], (3) d 3 r n λ (r,r ) = + ν w νλ n νλ (r) n(r) [ ν ]. (4) 4
The sum rule (4) was derived in Ref. 8. The corresponding sum rule for the energetically-important system-averaged hole is ( ) d 3 r d 3 r n(r)n λ (r,r ) = + ν w νλ ( ν ) = +kt ( / µ) λ,t (5) The last term of Eq. (5) is positive for a system with fluctuating electron number, such as an ion in a plasma. The softness ( / µ) λ,t / depends upon the energylevel structure of Ĥ λ, is therefore not predicted exactly by LSD or GGA, and equals (3π ) /3 n /3 /π = 3/(kT F ) for a non-interacting uniform electron gas of density n in the limit T 0. Eq. (5) also holds for a classical liquid (see Eqs. (.8) and (.9) of Ref. 9), and for a Bose system (see Eqs. (5.76) of Ref. 0).. Acknowledgments This work was supported in part by the ational Science Foundation under Grant o. DMR 95-353 and in part by the Deutsche Forschungsgemeinschaft. REFERECES. W. Kohn and L.J. Sham, Phys. Rev. 40, A 33 (965).. D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45, 566 (980). 3. R.M. Dreizler and E.K.U. Gross, Density Functional Theory (Springer, Berlin, 990). 4. R.G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, ew York, 989). 5. D.C. Langreth and M.J. Mehl, Phys. Rev. B 8, 809 (983). 6. J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (996), and references therein. Erratum 78, 396 (997). 7. D.C. Langreth and J.P. Perdew, Solid State Commun. 7, 45 (975). 8. O. Gunnarsson and B.I. Lundqvist, Phys. Rev. B 3, 474 (976). 9. J.P. Perdew, Phys. Rev. Lett. 55, 665 (985). 0. T. Ziegler, A. Rauk, and E.J. Baerends, Theoret. Chim. Acta 43, 6 (977).. K. Burke, J.P. Perdew, and D.C. Langreth, Phys. Rev. Lett. 73, 83 (994).. J.P. Perdew, K. Burke, and Y. Wang, Phys. Rev. B 54, 6533 (996). 3..D. Mermin, Phys. Rev. 37, A 44 (965). 4. U. Gupta and A.K. Rajagopal, Phys. Rev. A, 79 (988). 5. U. Gupta and A.K. Rajagopal, Phys. Rep. 87, 59 (98). 6. M. Brack, O. Genzken, and K. Hansen, Z. Phys. D, 65 (99). 7. C. Dharma-wardana and F. Perrot, in Density Functional Theory, Vol. B337 of ATO ASI Series, edited by E.K.U. Gross and R. Dreizler (Plenum, ew York, 995), p. 65. 8. J.P. Perdew, in Density Functional Methods in Physics, Vol. B3 of ATO ASI Series, edited by R.M. Dreizler and J. da Providencia (Plenum, ew York, 985), p. 65. 9. P.A. Egelstaff, An Introduction to the Liquid State (Academic Press, London, 967). 0. A.K. Rajagopal, Adv. in Chem. Phys. 4, 59 (980). 5