Role of the Exchange Correlation Energy: Nature s Glue STEFAN KURTH, JOHN P. PERDEW Department of Physics and Quantum Theory Group, Tulane University, New Orleans, Louisiana 70118 Received 11 March 1999; accepted 1 September 1999 ABSTRACT: In the Kohn Sham density functional theory of ground-state electronic structure, only the exchange correlation energy E xc must be approximated. Although E xc is not typically a large component of the total energy, it is the principal ingredient of the glue that binds atoms together to form molecules and solids. To illustrate this fact, we present self-consistent results for atomization energies of molecules and for surface energies and work functions of jellium, calculated within the Hartree approximation, which neglects E xc. The Hartree world displays weak bonding between atoms, low or negative surface energies, and work functions that are close to zero. Other aspects of the Hartree world can be deduced from known size effect relationships. The mechanism behind the glue role of exchange and correlation is the suppression of Hartree charge fluctuations. c 000 John Wiley & Sons, Inc. Int J Quant Chem 77: 814 818, 000 Key words: Hartree approximation; density functional theory; exchange correlation energy; jellium surface energy; bonding Introduction Kohn Sham density functional theory [1 3] is the most widely used method to find the ground-state electronic structure of atoms, molecules, and solids. In this theory, the density of electrons of spin σ, n σ (r) = α θ(µ ε ασ ) ψασ (r), (1) Correspondence to: J. P. Perdew. Contract grant sponsor: National Science Foundation. Contract grant number: DMR98-1060. Contract grant sponsor: Petroleum Research Fund. Contract grant number: ACS-PRF #33001-AC6. is constructed from orbitals that solve the selfconsistent one-electron Schrödinger equation ( 1 + v(r) + d 3 r n(r ) ) r r + vσ xc (r) ψ ασ (r) = ε ασ ψ ασ (r). () In Eq. (), v(r) is the external potential, and n(r) = n (r) + n (r). The total energy is E = θ(µ ε ασ ) d 3 r 1 ψασ (r) ασ + d 3 rn(r)v(r) + 1 d 3 r d 3 r n(r)n(r ) r r + E xc [n, n ] + E nn, (3) where E nn is the Coulomb repulsion of the nuclei. International Journal of Quantum Chemistry, Vol. 77, 814 818 (000) c 000 John Wiley & Sons, Inc.
ROLE OF THE EXCHANGE CORRELATION ENERGY The exchange correlation energy E xc in Eq. (3) accounts for three distinct physical effects: The exchange energy E x 0 corrects the spurious selfinteraction of one electron with itself and also contains the effects of the Pauli exclusion principle. The correlation energy E c = E xc E x 0 accounts for the effects of Coulomb correlation upon the manyelectron wave function. The exchange correlation potential in Eq. () is the functional derivative v σ xc (r) = δe xc δn σ (r). (4) Apart from the practical need to approximate the functional dependence of E xc upon the spin densities n (r) and n (r), Kohn Sham theory is exact for n, n,ande. While it tends to be a relatively small part of the total energy of Eq. (3), especially for the heavier elements, E xc plays an extremely important role in physics and chemistry: It is the major ingredient of the glue that binds atoms together to form molecules and solids. The purpose of this short article is to stress the importance of E xc by showing what the world would be like in the Hartree approximation, which neglects both E xc and v σ xc (r). This is not the approximation originally proposed by Hartree [4], which includes an orbital-dependent self-interaction correction for both E xc and v σ xc (r), except as discussed in the Appendix. The Hartree approximation can be simplified even further by making the Thomas Fermi approximation [5, 6] in which the orbital kinetic energy of Eq. (3) is replaced by its local spin density approximation: T TF s [n, n ] = 1 TTF s [n ] + 1 TTF s [n ], (5) Ts TF [n] = 3 ( 3π ) /3 d 3 rn 5/3 (r). (6) 10 In the opposite direction, the Hartree approximation can be refined by a sequence of approximations for E xc. The simplest is the local spin density (LSD) approximation [1]: E LSD xc [n, n ] = d 3 ( rn(r)ɛ xc n (r), n (r) ), (7) where ɛ xc (n, n ) is the exchange correlation energy per particle for an electron gas with uniform spin densities n and n. Next, in order of sophistication, is the generalized gradient approximation (GGA) [7], E GGA xc [n, n ] = d 3 rf(n, n, n, n ). (8) The meta-gga [8] makes use of additional local information, such as the Laplacian n σ or the kinetic energy density that appears in the first term of Eq. (3). These approximations are potentially exact for an electron gas of uniform or slowly varying density, and those functionals that are constructed nonempirically are typically exact or nearly exact in this limit. Near the upper level of sophistication is an approach [9] that treats exchange exactly, long-range correlation in the random-phase approximation (RPA), and the short-range correction to RPA in GGA. Atomization Energies, Surface Energies, and Work Functions The atomization energy of a molecule is the minimum total energy that must be added to break the molecule into separate atoms. Table I shows the atomization energies of seven typical molecules, calculated in various approximations and compared to experiment. All calculations have been performed at the observed equilibrium geometries, and the contribution from zero-point vibration has been removed from the experimental atomization energies [7]. These calculations have been done selfconsistently with the CADPAC [10] program, using a Gaussian basis set of triple-zeta quality that includes p and d polarization functions for hydrogen and d and f polarization functions for the first-row elements. By Teller s theorem [11], atoms do not bind to form molecules within the Thomas Fermi approximation. The Hartree approximation, which neglects E xc, and the unrestricted Hartree Fock approximation, which neglects E c, produce binding, but it TABLE I Atomization energies (in kcal/mol) from various self-consistent calculations, using experimental molecular geometries and static nuclei. Molecule Hartree UHF LSD GGA expt H 57 84 113 105 109 N 11 115 67 43 9 O 35 33 175 144 11 F 37 78 53 39 NH 3 94 01 337 30 97 H O 76 155 67 34 3 HF 50 97 16 14 141 a UHF is unrestricted Hartree Fock. Experimental values were taken from [7] (1 kcal/mol = 1.594 10 3 hartrees.) INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 815
KURTH AND PERDEW is far weaker than in reality. LSD overbinds, and GGA reduces but does not entirely eliminate this overbinding tendency. The surface energy of a solid is the energy per unit area needed to cleave a macroscopic surface. The simplest model of a solid is jellium, in which a positive background of uniform density n = 3 (9) 4πr 3 s is neutralized by valence electrons. Table II shows the surface energy of jellium, calculated selfconsistently as in the work of Lang and Kohn [1, 13]. The negative surface energies at small r s reflect a deficiency of the jellium model (instability to spontaneous cleavage), which can be rectified by the more sophisticated stabilized jellium model [14, 15]. But the Hartree surface energies, like those [13] of the Thomas Fermi approximation, are all negative and seriously too low. The work function [1, 16] is the minimum energy needed to remove an electron through the surface. Table II shows that the work function, which is identically zero [13] in the Thomas Fermi approximation, is close to zero in the Hartree approximation but realistic in LDA and GGA. These self-consistent Hartree results are prefigured in a variational calculation [17] with a simple analytic model for the electron density profile at the jellium surface. Hartree electron densities are too diffuse, making the electrostatic dipole moment at the jellium sur- TABLE II Total surface energies σ (in ergs/cm )andwork functions W (in ev) of the jellium surface from various self-consistent calculations. r s σ Hartree σ LSD σ GGA W Hartree W LSD W GGA.00 4790 864 957 0.06 3.79 3.65.07 4088 607 690 0.07 3.77 3.64.30 506 104 164 0.00 3.67 3.57.66 175 171 133 0.06 3.51 3.43 3.00 76 4 119 0.08 3.33 3.8 3.8 477 19 00 0.10 3.1 3.16 4.00 186 163 154 0.13.90.87 5.00 63 98 93 0.15.5.51 6.00 5 59 57 0.15.8. a Work functions calculated from Koopmans and displacedprofile change in self-consistent field [16] expressions are in agreement (1 erg/cm = 6.43 10 7 hartrees/bohr ;1eV= 0.03675 hartree). face, D = 4π dx x ( n(x) nθ(x) ), (10) too large. And the Hartree approximation contains no driving force for magnetism or spin polarization, other than an external magnetic field or an odd electron number. The too-low Hartree atomization energies of Table I, and the too-low Hartree surface energies of Table II, are related by the formulas of the following section. Size Effect Relationships The size effect formulas presented here are approximate relationships between the properties of a macroscopic solid metal and the properties of a finite spherical cluster of radius R. The smallest cluster is the atom, whose volume 4πR 3 /3 = z/n is assumed to be the volume per atom of the bulk solid. (Here z is the valence and n the average valence electron density.) These formulas will help to describe the Hartree world in the next section. The energy required to create the positively curved surface of a spherical cluster is [18] σ 4πR + γ πr, (11) and the energy to create the negatively curved surface of a vacancy is [18] σ 4πR γ πr, (1) where σ is the surface energy and γ the curvature energy [15, 18]. The cohesive energy (atomization energy per atom of the solid) is estimated by applying Eq. (11) to a single atom. The first ionization energy I and electron affinity A of a cluster are [19, 0] I = W + ( 1 + c) /(R + δ), (13) A = W + ( 1 + c) /(R + δ), (14) where W is the bulk work function; c and δ are small, material-dependent parameters [0]. These equations may also be applied to a single atom. Conclusions: The Hartree World It is sometimes suggested that Kohn Sham theory works because the exchange correlation energy, which must be approximated, is small. This statement is only partially true. In fact, the Hartree 816 VOL. 77, NO. 5
ROLE OF THE EXCHANGE CORRELATION ENERGY world, which neglects exchange and correlation, would be quite different from the real world. In the Hartree world, atoms bind, but only weakly. Bond lengths and lattice constants can also be significantly too long. In fact, for the jellium model, there is no equilibrium r s in the Hartree approximation; the positive background expands without limit. Exchange and correlation are needed to stabilize jellium at r s = 4.1 bohrs, near the density of metallic sodium. In the Hartree world, as in the Thomas Fermi world, the surface energies of metals are seriously too low and even negative, and work functions are close to zero. It then follows from the size effect relationships discussed above that cohesive energies, vacancy formation energies, first ionization energies, and electron affinities are all seriously too low in the Hartree world. At the observed bond lengths or lattice constants, Hartree electrical resistivities are too small [1], and Hartree phonon frequencies for metals are somewhat too high []. Hartree solids are nonmagnetic in the absence of external magnetic fields. The exchange correlation energy is the glue that holds the real world together; the actual glue mechanism is discussed in the Appendix. Appendix: Exchange and Correlation Suppress Hartree Charge Fluctuations In an extended periodic system (e.g., a macroscopic solid), solutions of the Hartree equations given in the first section are also solutions of the original equations of Hartree [4] since the selfinteraction correction vanishes for an extended orbital. (Of course, there may be other solutions of the original equations in which the orbitals break symmetry and localize.) Atoms and molecules may be repeated periodically on a lattice with an enormous lattice constant. There are two different ways to think about the Hartree approximation for the electron electron repulsion energy 1 d 3 r d 3 r n(r)n(r )/ r r : (A1) (1) The electrons are infinitesimal particles that form a continuous charged fluid of charge density n(r). () The electrons are particles of finite charge ( 1) that form a classical ideal gas of nonuniform number density n(r), in which the probability of simultaneously finding electrons in volume elements d 3 r and d 3 r is n(r)n(r ) d 3 rd 3 r. These two pictures are very different. In particular, a small finite volume fragment of the system has no charge fluctuation in the first picture, and a strong charge fluctuation [3] in the second picture. Random clusters of electrons have high electrostatic energy. To reconcile the two pictures, start with the second one (classical ideal gas of finite electrons) and imagine that each electron is divided into m equal parts, with no change in the charge density. The electrostatic energy remains as in (A1), the root-mean-square number fluctuation [3] in a fixed volume element increases by m,butthecharge fluctuation decreases by m/m, and tends to zero as m. In the latter limit, the two pictures become the same. In the second picture, exchange and correlation reduce the Hartree charge fluctuations [3, 4] at fixed electron charge, and so lower the energy. This lowering is greater when the electrons are brought closer together, explaining the glue role of exchange and correlation. ACKNOWLEDGMENTS This work was supported in part by the National Science Foundation under Grant No. DMR98-1060 and in part by the Petroleum Research Fund under Grant ACS-PRF #33001-AC6. One of us (J.P.) acknowledges discussions with Marvin Lee. References 1. Kohn, W.; Sham, L. J. Phys Rev A 1965, 140, 1133.. Parr,R.G.;Yang,W.Density-FunctionalTheoryofAtoms and Molecules; Oxford University Press: New York, 1989. 3. Dreizler, R. M.; Gross, E. K. U. Density Functional Theory; Springer: Berlin, 1990. 4. Hartree, D. R. Proc Cambridge Phil Soc 197, 4, 111. 5. Thomas, L. H. Proc Cambridge Phil Soc 197, 3, 54. 6. Fermi, E. Z Physik 198, 48, 73. 7. Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys Rev Lett 1996, 77, 3865; Ibid 1997 (E), 78, 1396, and references therein. 8. Perdew, J. P.; Kurth, S.; Zupan, A.; Blaha, P. Phys Rev Lett 199, 8, 544; Ibid 1999 (E), 8, 5179. 9. Kurth, S.; Perdew, J. P. Phys Rev B 1999, 59, 10461. 10. Amos,R.D.;Alberts,I.L.;Andrews,J.S.;Colwell,S.M.; Handy, N. C.; Jayatilaka, D.; Knowles, P. J.; Kobayashi, R.; Laming,G.J.;Lee,A.M.;Maslen,P.E.;Murray,C.W.; Palmieri,P.;Rice,J.E.;Sanz,J.;Simandiras,E.D.;Stone,A.J.; Su,M.-D.;Tozer,D.J.CADPAC6:TheCambridgeAnalytical Derivatives Package Issue 6.0; Cambridge, 1995. 11. Teller, E. Rev Mod Phys 196, 34, 67. INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 817
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