DISCRETE AND CONTINUOUS doi:10.3934/dcds.2013.33.5319DYNAMICAL SYSTEMSVolume 33, Number 11&12, November & December 2013 pp. 5319–5325

UNDERSTANDING THOMAS-FERMI-LIKE APPROXIMATIONS:

AVERAGING OVER OSCILLATING OCCUPIED ORBITALS

John P. Perdew and Adrienn Ruzsinszky

Department of Physics and Quantum Theory GroupTulane University, New Orleans, LA 70123, USA

Abstract. The Thomas-Fermi equation arises from the earliest density func-

tional approximation for the ground-state energy of a many-electron system.Its solutions have been carefully studied by mathematicians, including J.A.

Goldstein. Here we will review the approximation and its validity conditions

from a physics perspective, explaining why the theory correctly describes thecore electrons of an atom but fails to bind atoms to form molecules and solids.

The valence electrons are poorly described in Thomas-Fermi theory, for two

reasons: (1) This theory neglects the exchange-correlation energy, “nature’sglue”. (2) It also makes a local density approximation for the kinetic energy,

which neglects important shell-structure effects in the exact kinetic energy that

are responsible for the structure of the periodic table of the elements. Finally,we present a tentative explanation for the fact that the shell-structure effects

are relatively unimportant for the exact exchange energy, which can thus bemore usefully described by a local density or semilocal approximation (as in

the popular Kohn-Sham theory): The exact exchange energy from the occu-

pied Kohn-Sham orbitals has an extra sum over orbital labels and an extraintegration over space, in comparison to the kinetic energy, and thus averages

out more of the atomic individuality of the orbital oscillations.

1. Introduction. Ordinary matter is made up of atoms, which bind or glue to-gether to form molecules and solids. The atom is so small that about 50,000,000atoms would fit on a line 1 cm long, but its radius is still 100,000 times bigger thanthat of the nucleus of charge +Z and mass M � 1 at its center. The nucleus of aneutral atom is surrounded by a kind of bee-swarm of Z electrons, each of charge-1 and mass m = 1 atomic units. The positive integer Z is the atomic number. Thenucleus is so massive that we can often think of it as a classical point-particle atrest, but the electrons are so light that they are fully quantum mechanical. Thereis a Coulomb or electrical attraction of each electron to the nucleus, and a Coulombrepulsion between every pair of electrons.

A theory of ordinary matter needs to predict the size of the atom (about 2 bohr= 1.058 Angstrom), and the magnitude of the total energy (about 1 hartree = 27.21eV or more) required to remove one or all the electrons from the nucleus, as wellas the much smaller atomization energies per atom needed to break a molecule orsolid up into free atoms. It should also predict the bond lengths and bond anglesdefining the nuclear framework of the molecule or solid.

2010 Mathematics Subject Classification. 41, 45, 46, 49, 81.Key words and phrases. Thomas-Fermi, kinetic energy, exchange energy, averaging, density

functional.

5319

5320 JOHN P. PERDEW AND ADRIENN RUZSINSZKY

The classical mechanics of Newton with the classical electrodynamics of Maxwellfails totally as a theory of matter, because in classical physics the electrons allradiate energy and fall inside the nucleus, leaving nothing of ordinary matter. Onlythe quantum mechanics discovered early in the 20th century can serve as a theoryof ordinary matter. There are two principal quantum effects that produce an atomof realistic size: (1) The uncertainty principle, which says that the more we tryto confine an electron the faster it moves to get away. This already produces anatom much bigger than the nucleus, but for all atoms with Z > 2 this atom is stilltoo small. (2) The Pauli exclusion principle, which says that a pair of electrons ofthe same spin (up-up or down-down) avoid one another even when we neglect theCoulomb repulsion between them.

Quantum theory [1] provides a Schroedinger equation whose eigenvalues are thepossible well-defined energies of the system. The eigenfunctions are the N-electronwavefunctions, which describe the bee-swarm of electrons. For N � 1, the wave-functions are functions of too many arguments, and thus difficult to calculate, store,or use. Typically to understand ordinary matter we only need the ground-state(most negative) total energy and the corresponding electron density n(~r), definedso that n(~r)d3r is the average number of electrons in an infinitesimal volume d3raround point ~r in three-dimensional space.

Around 1926, Thomas [2] and Fermi [3] independently proposed the simplestreasonable theory for atoms. They posited a functional or rule that yields an energy

ETFv [n] (1)

=3(3π2)2/3

10

∫d3rn5/3(~r) +

1

2

∫d3r

∫d3r′n(~r)n(~r′)/|~r′ − ~r|+

∫d3rn(~r)v(~r).

The first term represents the kinetic energy of electron motion through the bee-swarm. The second is the electrical potential energy of electron-electron repulsion,when the bee-swarm is modeled by a rigid continuous distribution of charge density−n(~r). The last term is the potential energy of interaction between the electrons andan “external” potential v(~r), which for an atom is −Z/r. They further posited that,for fixed electron number N =

∫d3rn(~r) and fixed v(~r), the ground-state density

can be found by minimizing Eq. (1) over all non-negative n(~r). The resulting Eulerequation is the Thomas-Fermi equation

1

2(3π2)2/3n2/3(~r) +

∫d3r′n(~r′)/|~r′ − ~r|+ v(~r) = µ, (2)

where the chemical potential µ is the Lagrange multiplier for the constraint on theelectron number.

Eqs. (1) and (2) make a reasonable prediction for the electron density and totalenergy of an atom. Features of the solution of Eq. (2) and related equations havebeen investigated by J. A. Goldstein [4]–[7] and others.

2. Exact density functionals and local density approximations. Thomas-Fermi theory was “exactified” into modern density functional theory in the 1960’s.Hohenberg and Kohn [8] proved that there exists an exact density functional F [n]that can replace the first term on the right of Eq. (1) to yield the exact ground-statedensity and energy. This is “only” an existence theorem, since the exact F [n] isnot known in any calculable form, but it motivated the search for approximationsbetter than that of Eq. (1).

UNDERSTANDING THOMAS-FERMI-LIKE APPROXIMATIONS 5321

A major step was taken by Kohn and Sham [9]. They wrote

F [n] = Ts[n] + Exc[n], (3)

where

Ts[n] =1

2

occup∑α,σ

∫d3r|∇Ψα,σ(~r)|2 (4)

is the ground-state kinetic energy for non-interacting electrons of density

n(~r) =

occup∑α,σ

|Ψα,σ(~r)|2. (5)

The one-electron wavefunctions or Kohn-Sham spin orbitals Ψα,σ(~r) are defined forthe fictitious non-interacting system. They have spatial quantum numbers α andspin quantum numbers σ =up or down. They are eigenfunctions of an effectivedensity-dependent one-electron Hamiltonian, and thus are implicit functionals ofthe density n(~r). The one-electron Schroedinger equation must be solved selfcon-sistently. The Pauli exclusion principle demands that no more than one electroncan occupy a given spin orbital. Thus in the ground-state the orbitals are occupiedin order of increasing orbital energy eigenvalue, until all N electrons are assignedto orbitals. Since the orbitals are orthonormal, the higher-energy orbitals mustoscillate over space to be orthogonal to the nodeless lowest orbitals.

Thus Kohn and Sham [9] divided the exact Hohenberg-Kohn F [n] into a largeterm Ts[n] that can be treated exactly via the orbitals, and a smaller correctionterm Exc[n], the exchange-correlation energy, that must be approximated. Theyproposed a local density approximation for the latter:

ELDAxc [n] =

∫d3rn(~r)εunifxc (n(~r)), (6)

where εunifxc (n) is the known exchange-correlation energy per electron of an electrongas of uniform density n. Eq. (6) is clearly exact for an electron density that isconstant or slowly-varying over space. Modern density functional theory, a realistictheory of ordinary matter, typically starts from the approximation of Eq. (6) andadds various inhomogeneity corrections to it Although the exchange-correlationenergy is typically a relatively small part of the total energy, it is still “nature’sglue” [10]: Neglecting it leads to chemical bonds that are much longer and weakerthan real bonds.

The Thomas-Fermi approximation of Eq. (1) can now be seen to involve twosevere approximations: use of a local density approximation for the kinetic en-ergy Ts[n] and neglect of the exchange-correlation energy Exc[n]. The exchange-correlation energy is negative, and corrects the second term on the right of Eq. (1):The electrons are really correlated charged particles in a bee swarm, not a rigiddistribution of charge density. They avoid one another like bees in a swarm, andthis avoidance lowers their electrostatic repulsion energy. There is more avoidancein a molecule or solid than in separate atoms, so the atoms bind to one anotherlargely through this effect. We can write Exc = Ex +Ec, where the exact exchange

5322 JOHN P. PERDEW AND ADRIENN RUZSINSZKY

energy

Ex[n] (7)

=− 1

2

∑σ

occup∑α

occup∑α′

∫d3r′

∫d3r′Ψ∗α,σ(~r)Ψα,σ(~r′)Ψα′,σ(~r)Ψ∗α′,σ(~r)/|~r′ − ~r|

arises from the Pauli exclusion principle. In the special case where N = 1 (a swarmof one), the exact exchange energy just cancels the second term on the right ofEq. (1). (This correct limit is built into the Fermi-Amaldi extension of Eq. (1).)The exact correlation energy cannot be expressed so simply, but it is also a doubleintegral over three-dimensional space, and it vanishes for N = 1. In the high-density limit, the kinetic energy dominates over exchange, and exchange dominatesover correlation. From Eq. (7), one can find

εunifx (n) = − 3

4π(3π2n)1/3. (8)

For large n, Eq. (8) is indeed dominated by tunifs (n) = (3/10)(3π2n)2/3. Theseexpressions clearly contain no shell-structure, because the orbital energies of theuniform electron gas are continuous and smooth (not discrete, like those of an atomor molecule)..

The kinetic and exchange energies have as a local length scale the local Fermiwavelength λF (~r) = 2π/kF (~r) where kF = (3π2n)1/3. Local density approximationsfor them are valid when the electron density varies slowly over the local Fermiwavelength:

|∇n|/[2kFn]� 1, |∇2n/(2kF )2n| � 1. etc. (9)

These conditions are satisfied in the high-density cores of atoms (but not too closeto the nucleus) [11]. Since the core dominates the total energy of an atom, Thomas-Fermi theory works reasonably for the total energy. In fact, the exact total energyof a neutral atom has the large-Z expansion [12, 13]

E = −aZ7/3 + bZ6/3 − cZ5/3 + . . . , (10)

where the first term on the right is given by Thomas-Fermi theory [14], the secondby the Scott correction from the region near the nucleus, the third from local-densityexchange (plus a smaller contribution from gradient corrections to the kinetic en-ergy), etc. Terms of higher order than those shown in Eq. (10) oscillate with Z, asa precursor of the periodic shell-structure variation of chemical properties over theperiodic table.

3. Periodic table of the elements: Shell structure. The n-th major periodof the table of the elements begins with the electron orbital configuration (ns)1

(one valence electron in an ns orbital of zero orbital angular momentum) and ends(for n > 1) with the configuration (np)6 (six valence electrons in np orbitals, eachwith orbital angular momentum one). The number of radial nodes of the valenceorbital, n − 1, increases by one from one major row to the next, leading to anabrupt change in valence-electron properties. The first ionization energy of a neutralatom is the energy that must be added to remove the least-bound electron. It isa valence-electron property that increases across each row of the periodic tablebut drops abruptly from one row to the next, and is one of the most importantatomic descriptors for chemical bonding. Numerical extrapolation of Kohn-Shamcalculations for hypothetical non-relativistic atoms with Z up to 3000 show that the

UNDERSTANDING THOMAS-FERMI-LIKE APPROXIMATIONS 5323

non-relativistic periodic table becomes perfectly periodic in the Z → ∞ [15]. Theatoms approach a large but finite size range, while the ionization energies and atomicsizes begin to repeat exactly from one row of the periodic table to the next. ThisPlatonic periodic table, like the real one we actually observe, is not found from theThomas-Fermi approximation, which neglects all shell-structure effects. However,the local density approximation for exchange gives nearly the exact-exchange (withno correlation) Kohn-Sham ionization energies as Z →∞ [15].

The infinite-Z limit of the ionization energy is 1.29 electron volts (eV) in theThomas-Fermi approximation, and 3.15 eV in the extended Thomas-Fermi (ETF)approximation which includes exchange in the local-density approximation. TheETF result is close to the infinite-Z sp row average (3.02 eV) of the Kohn-Shamexact-exchange ionization energy, but the shell-structure oscillations lead to a vari-ation from 1.4 eV at the start of a row to 4.3 eV at the end. Correlation appearsto increase the ionization energy by less than 1 eV. The results of Ref. [15] arenumerical, and rigorous derivations are a challenge to mathematical physics.

For an atom or atomic ion described at the level of selfconsistent exchange-only (no correlation), the virial theorem tells us that E = −Ts. The first ionizationenergy is then the amount by which Ts is reduced when the electron is removed. Theinfinite-Z valence electrons have moderately low densities for which the exchangecontribution to the ionization energy is roughly comparable to the kinetic energycontribution.

4. Why are the shell-structure oscillations important for the exact ki-netic energy but not for the exact exchange energy? Local density ap-proximations omit the contribution of shell-structure oscillation to the functional.While it is true that the kinetic energy dominates the exchange energy in the coreof an atom, the exchange energy can be almost as important as the kinetic en-ergy for valence-electron properties. Thus the success of Kohn-Sham methods [16](with exact kinetic energy and typically local or semilocal exchange-correlation) forvalence-electron properties, and the failure of local or semilocal approximations forthe kinetic energy [17] for the same properties (e.g., see section 3), seems to be aconsequence of the fact that the shell-structure oscillation of an atom or moleculeis much more important for the exact kinetic energy than for the exact exchange(or exchange-correlation) energy. Why? (Even when applied to the Kohn-Shamdensities, semilocal approximations for the kinetic energy fail for valence electrons[17].)

We suspect that the answer is something like this: The exact kinetic energy ofEq. (4) partially averages the oscillations of the occupied orbitals via one sum overorbital labels and one integral over three-dimensional space. But the exact exchangeenergy of Eq. (7) averages out still more of the individuality of the oscillations ofthe occupied orbitals, via a second sum over orbital labels and a second integral overspace. When enough individuality is washed out, what remains can be describedby the information transferred to an atom or molecule from a uniform electron gasin a local density or semilocal approximation. Moreover, the derivatives ∇ = ∂/∂~rthat appear only in the kinetic and not in the exchange energy may enhance theshell-structure oscillations. We are not sure how this explanation can be quantifiedor tested, but we welcome suggestions from mathematicians, physicists, or chemists.

The standard explanation [19]–[21] for the success of local and semilocal ap-proximations to the exchange and exchange-correlation energies invokes a sum rule

5324 JOHN P. PERDEW AND ADRIENN RUZSINSZKY

on the exchange-correlation hole density (corresponding to the “personal space”around an electron in the swarm), and thus invokes the double integration overthree-dimensional space that defines the exact functional. What has perhaps notbeen realized until now is the strong averaging-out of shell-structure effects for theseexact functionals that occurs as a result.

For recent progress toward accurate orbital-free electronic-structure methods, seeRef. [18].

Acknowledgments. Thanks first to Jerry Goldstein, who in the late 1970’s andearly 1980’s founded the Quantum Theory Group at Tulane that brought JohnPerdew’s and Alan Goodman’s people in Physics together with Jerry’s in Mathe-matics and with Mel Levy’s and Mike Herman’s in Chemistry for chalk talks anddiscussions. Jerry was interdisciplinary before interdisciplinarity became popular.JPP also thanks Kieron Burke for discussions of section 4, and Jianwei Sun for com-ments on the manuscript. This work has been supported in part by the NationalScience Foundation under Grant No. DMR-0854769 (JPP) and NSF CooperativeAgreement No. EPS-1003897, with support from the Louisiana Board of Regents(JPP and AR).

REFERENCES

[1] A. Messiah, “Quantum Mechanics,” Dover, 1999.

[2] L. H. Thomas, The calculation of atomic fields, Proc. Cambridge Philos. Soc., 23 (1926),542–548.

[3] E. Fermi, Un metodo statistico per la determinazione di alcune proprieta dell atomo, Rend.

Accad. Naz. Licei, 6 (1927), 602–607.[4] J. A. Goldstein and G. R. Rieder, Some extensions of Thomas-Fermi theory, Lecture Notes

in Mathematics, 1223 (1986), 110–121.

[5] J. A. Goldstein and G. R. Rieder, Recent rigorous results in Thomas-Fermi theory, LectureNotes in Mathematics, 1394 (1989), 68–82.

[6] P. Benilan, J. A. Goldstein and G. R. Rieder, Nonlinear elliptic system arising in electron-

density theory, Communications in Partial Differential Equations, 17 (1992), 2079–2092.[7] G. R. Rieder, J. A. Goldstein and N. Naima, A convexified energy functional for the Fermi-

Amaldi correction, Discrete and Continuous Systems, 28 (2010), 41–65.[8] P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev., 136 (1964), B864–

B871.

[9] W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation, Phys.Rev., 140 (1965), A11333–A1138.

[10] S. Kurth and J. P. Perdew, Role of the exchange-correlation energy: Nature’s glue, Int. J.

Quantum Chem., 77 (2000), 819–830.[11] J. P. Perdew, L. A. Constantin, E. Sagvolden and K. Burke, Relevance of the slowly-varying

electron gas to atoms, molecules, and solids, Phys. Rev. Lett., 97 (2006), 223002, 4 pages.

[12] J. Schwinger, Thomas-Fermi model: The leading correction, Phys. Rev. A, 22 (1980), 1827–1832; Thomas-Fermi model: The second correction, ibid., 24 (1981), 2353–2361.

[13] B. G. Englert and J. Schwinger, Statistical atom: Some quantum improvements, Phys. Rev.

A, 29 (1984), 2339–2352; Semiclassical atom, ibid., 32 (1985), 26–35.[14] E. H. Lieb, The stability of matter , Rev. Mod. Phys., 48 (1976), 553–569.

[15] L. A. Constantin, J. C. Snyder, J. P. Perdew and K. Burke, Ionization potentials in the limitof large atomic number , J. Chem. Phys., 133 (2010), 241103, 4 pages.

[16] J. P. Perdew and S. Kurth, Density functionals for non-relativistic Coulomb systems in thenew century, in “A Primer in Density Functional Theory” ( eds. C. Fiolhais, F. Nogueira andM. Marques), Lecture Notes in Physics, 620 (2003), 1–55.

[17] J. P. Perdew and L. A. Constantin, Laplacian-level density functionals for the kinetic energy

density and exchange-correlation energy, Phys. Rev. B, 75 (2007), 155109, 9 pages.[18] A. Cangi, D. Lee, P. Elliott, K. Burke and E. K. U. Gross, Electronic structure via potential

functional approximations, Phys. Rev. Lett., 106 (2011), 236404, 4 pages.

UNDERSTANDING THOMAS-FERMI-LIKE APPROXIMATIONS 5325

[19] D. C. Langreth and J. P. Perdew, The exchange-correlation energy of a metallic surface,Solid State Commun., 17 (1975), 1425–1429.

[20] O. Gunnarsson and B. I. Lundqvist, Exchange and correlation in atoms, molecules, and

solids, Phys. Rev. B, 13 (1976), 4274–4298.[21] D. C. Langreth and J. P. Perdew, Exchange-correlation energy of a metallic surface:

Wavevector analysis, Phys. Rev. B, 15 (1977), 2884–2901.

Received December 2011 for publication.

E-mail address: perdew@tulane.edu

E-mail address: aruzsinszky@gmail.com