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

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Understanding Thomas-Fermi-Like Approximations: Averaging Over Oscillating Occupied Orbitals 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 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 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 occu- pied 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 to- gether 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 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), 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)=j~r0 − ~rj + 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)=j~r0 − ~rj + 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] = d3rjrΨ (~r)j2 (4) s 2 α,σ α,σ is the ground-state kinetic energy for non-interacting electrons of density occup X 2 n(~r) = jΨα,σ(~r)j : (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)"xc (n(~r)); (6) unif where "xc (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 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 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)=j~r − ~rj 2 α,σ α,σ α ,σ α ,σ σ α α0 arises from the Pauli exclusion principle.
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