Understanding Thomas-Fermi-Like Approximations: Averaging Over Oscillating Occupied Orbitals

Home , Electronic structure

DISCRETE AND CONTINUOUS doi:10.3934/dcds.2013.33.5319 DYNAMICAL SYSTEMS Volume 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 Group Tulane University, New Orleans, LA 70123, USA

Abstract. The Thomas-Fermi equation arises from the earliest density functional 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 the core 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’s glue”. (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 be more usefully described by a local density or semilocal approximation (as in the popular Kohn-Sham theory): The exact exchange energy from the occupied Kohn-Sham orbitals has an extra sum over orbital labels and an extra integration 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 together to form molecules and solids. The atom is so small that about 50,000,000 atoms would ﬁt on a line 1 cm long, but its radius is still 100,000 times bigger than that of the nucleus of charge +Z and mass M 1 at its center. The nucleus of a neutral 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. The nucleus is so massive that we can often think of it as a classical point-particle at rest, but the electrons are so light that they are fully quantum mechanical. There is a Coulomb or electrical attraction of each electron to the nucleus, and a Coulomb repulsion 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 = 27.21 eV or more) required to remove one or all the electrons from the nucleus, as well as the much smaller atomization energies per atom needed to break a molecule or solid up into free atoms. It should also predict the bond lengths and bond angles deﬁning 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 Maxwell fails totally as a theory of matter, because in classical physics the electrons all radiate energy and fall inside the nucleus, leaving nothing of ordinary matter. Only the quantum mechanics discovered early in the 20th century can serve as a theory of ordinary matter. There are two principal quantum effects that produce an atom of realistic size: (1) The uncertainty principle, which says that the more we try to confine an electron the faster it moves to get away. This already produces an atom much bigger than the nucleus, but for all atoms with Z > 2 this atom is still too small. (2) The Pauli exclusion principle, which says that a pair of electrons of the same spin (up-up or down-down) avoid one another even when we neglect the Coulomb repulsion between them. Quantum theory [1] provides a Schroedinger equation whose eigenvalues are the possible well-defined energies of the system. The eigenfunctions are the N-electron wavefunctions, which describe the bee-swarm of electrons. For N 1, the wavefunctions 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), defined so that n(~r)d3r is the average number of electrons in an infinitesimal volume d3r around point ~r in three-dimensional space. Around 1926, Thomas [2] and Fermi [3] independently proposed the simplest reasonable theory for atoms. They posited a functional or rule that yields an energy TF Ev [n] (1) 3(3π2)2/3 Z 1 Z Z Z = d3rn5/3(~r) + d3r d3r0n(~r)n(~r0)/|~r0 − ~r| + d3rn(~r)v(~r). 10 2 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 and an “external” potential v(~r), which for an atom is −Z/r. They further posited that, for fixed electron number N = R 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 Euler equation is the Thomas-Fermi equation 1 Z (3π2)2/3n2/3(~r) + d3r0n(~r0)/|~r0 − ~r| + v(~r) = µ, (2) 2 where the chemical potential µ is the Lagrange multiplier for the constraint on the electron number. Eqs. (1) and (2) make a reasonable prediction for the electron density and total energy of an atom. Features of the solution of Eq. (2) and related equations have been 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-state density and energy. This is “only” an existence theorem, since the exact F [n] is not known in any calculable form, but it motivated the search for approximations better 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

occup 1 X Z T [n] = d3r|∇Ψ (~r)|2 (4) s 2 α,σ α,σ is the ground-state kinetic energy for non-interacting electrons of density

occup X 2 n(~r) = |Ψα,σ(~r)| . (5) α,σ

The one-electron wavefunctions or Kohn-Sham spin orbitals Ψα,σ(~r) are defined for the fictitious non-interacting system. They have spatial quantum numbers α and spin quantum numbers σ =up or down. They are eigenfunctions of an effective density-dependent one-electron Hamiltonian, and thus are implicit functionals of the density n(~r). The one-electron Schroedinger equation must be solved selfcon- sistently. The Pauli exclusion principle demands that no more than one electron can occupy a given spin orbital. Thus in the ground-state the orbitals are occupied in order of increasing orbital energy eigenvalue, until all N electrons are assigned to orbitals. Since the orbitals are orthonormal, the higher-energy orbitals must oscillate over space to be orthogonal to the nodeless lowest orbitals. Thus Kohn and Sham [9] divided the exact Hohenberg-Kohn F [n] into a large term Ts[n] that can be treated exactly via the orbitals, and a smaller correction term Exc[n], the exchange-correlation energy, that must be approximated. They proposed a local density approximation for the latter: Z LDA 3 unif Exc [n] = d rn(~r)εxc (n(~r)), (6)

unif where εxc (n) is the known exchange-correlation energy per electron of an electron gas of uniform density n. Eq. (6) is clearly exact for an electron density that is constant or slowly-varying over space. Modern density functional theory, a realistic theory of ordinary matter, typically starts from the approximation of Eq. (6) and adds various inhomogeneity corrections to it Although the exchange-correlation energy is typically a relatively small part of the total energy, it is still “nature’s glue” [10]: Neglecting it leads to chemical bonds that are much longer and weaker than real bonds. The Thomas-Fermi approximation of Eq. (1) can now be seen to involve two severe approximations: use of a local density approximation for the kinetic energy 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 rigid distribution of charge density. They avoid one another like bees in a swarm, and this avoidance lowers their electrostatic repulsion energy. There is more avoidance in a molecule or solid than in separate atoms, so the atoms bind to one another largely through this eﬀect. We can write Exc = Ex + Ec, where the exact exchange 5322 JOHN P. PERDEW AND ADRIENN RUZSINSZKY energy

Ex[n] (7) occup occup Z Z 1 X X X 3 0 3 0 ∗ 0 ∗ 0 = − d r d r Ψ (~r)Ψ (~r )Ψ 0 (~r)Ψ 0 (~r)/|~r − ~r| 2 α,σ α,σ α ,σ α ,σ σ α α0 arises from the Pauli exclusion principle. In the special case where N = 1 (a swarm of one), the exact exchange energy just cancels the second term on the right of Eq. (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 double integral over three-dimensional space, and it vanishes for N = 1. In the high- density limit, the kinetic energy dominates over exchange, and exchange dominates over correlation. From Eq. (7), one can find 3 εunif (n) = − (3π2n)1/3. (8) x 4π unif 2 2/3 For large n, Eq. (8) is indeed dominated by ts (n) = (3/10)(3π n) . These expressions clearly contain no shell-structure, because the orbital energies of the uniform electron gas are continuous and smooth (not discrete, like those of an atom or molecule).. The kinetic and exchange energies have as a local length scale the local Fermi 2 1/3 wavelength λF (~r) = 2π/kF (~r) where kF = (3π n) . Local density approximations for them are valid when the electron density varies slowly over the local Fermi wavelength: 2 2 |∇n|/[2kF n] 1, |∇ n/(2kF ) n| 1. etc. (9) These conditions are satisfied in the high-density cores of atoms (but not too close to 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 energy of 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 second by the Scott correction from the region near the nucleus, the third from local-density exchange (plus a smaller contribution from gradient corrections to the kinetic energy), etc. Terms of higher order than those shown in Eq. (10) oscillate with Z, as a precursor of the periodic shell-structure variation of chemical properties over the periodic table.

3. Periodic table of the elements: Shell structure. The n-th major period of 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, each with orbital angular momentum one). The number of radial nodes of the valence orbital, n − 1, increases by one from one major row to the next, leading to an abrupt change in valence-electron properties. The first ionization energy of a neutral atom is the energy that must be added to remove the least-bound electron. It is a valence-electron property that increases across each row of the periodic table but drops abruptly from one row to the next, and is one of the most important atomic descriptors for chemical bonding. Numerical extrapolation of Kohn-Sham calculations 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]. The atoms approach a large but finite size range, while the ionization energies and atomic sizes begin to repeat exactly from one row of the periodic table to the next. This Platonic periodic table, like the real one we actually observe, is not found from the Thomas-Fermi approximation, which neglects all shell-structure effects. However, the local density approximation for exchange gives nearly the exact-exchange (with no correlation) Kohn-Sham ionization energies as Z → ∞ [15]. The infinite-Z limit of the ionization energy is 1.29 electron volts (eV) in the Thomas-Fermi approximation, and 3.15 eV in the extended Thomas-Fermi (ETF) approximation which includes exchange in the local-density approximation. The ETF result is close to the infinite-Z sp row average (3.02 eV) of the Kohn-Sham exact-exchange ionization energy, but the shell-structure oscillations lead to a variation from 1.4 eV at the start of a row to 4.3 eV at the end. Correlation appears to increase the ionization energy by less than 1 eV. The results of Ref. [15] are numerical, 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 ionization energy is then the amount by which Ts is reduced when the electron is removed. The infinite-Z valence electrons have moderately low densities for which the exchange contribution to the ionization energy is roughly comparable to the kinetic energy contribution.

4. Why are the shell-structure oscillations important for the exact kinetic energy but not for the exact exchange energy? Local density approximations omit the contribution of shell-structure oscillation to the functional. While it is true that the kinetic energy dominates the exchange energy in the core of an atom, the exchange energy can be almost as important as the kinetic energy for valence-electron properties. Thus the success of Kohn-Sham methods [16] (with exact kinetic energy and typically local or semilocal exchange-correlation) for valence-electron properties, and the failure of local or semilocal approximations for the kinetic energy [17] for the same properties (e.g., see section 3), seems to be a consequence of the fact that the shell-structure oscillation of an atom or molecule is much more important for the exact kinetic energy than for the exact exchange (or exchange-correlation) energy. Why? (Even when applied to the Kohn-Sham densities, semilocal approximations for the kinetic energy fail for valence electrons [17].) We suspect that the answer is something like this: The exact kinetic energy of Eq. (4) partially averages the oscillations of the occupied orbitals via one sum over orbital labels and one integral over three-dimensional space. But the exact exchange energy of Eq. (7) averages out still more of the individuality of the oscillations of the occupied orbitals, via a second sum over orbital labels and a second integral over space. When enough individuality is washed out, what remains can be described by the information transferred to an atom or molecule from a uniform electron gas in a local density or semilocal approximation. Moreover, the derivatives ∇ = ∂/∂~r that appear only in the kinetic and not in the exchange energy may enhance the shell-structure oscillations. We are not sure how this explanation can be quantified or tested, but we welcome suggestions from mathematicians, physicists, or chemists. The standard explanation [19]–[21] for the success of local and semilocal approximations 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 over three-dimensional space that defines the exact functional. What has perhaps not been realized until now is the strong averaging-out of shell-structure effects for these exact functionals that occurs as a result. For recent progress toward accurate orbital-free electronic-structure methods, see Ref. [18].

Acknowledgments. Thanks ﬁrst to Jerry Goldstein, who in the late 1970’s and early 1980’s founded the Quantum Theory Group at Tulane that brought John Perdew’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 and discussions. 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 National Science Foundation under Grant No. DMR-0854769 (JPP) and NSF Cooperative Agreement 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, Lecture Notes 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 limit of 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 the new century, in “A Primer in Density Functional Theory” ( eds. C. Fiolhais, F. Nogueira and M. 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: [email protected] E-mail address: [email protected]