Page 31 - 71 the abc of dft_opt

6.3. THOMAS-FERMI 61 62 CHAPTER 6. DENSITY FUNCTIONAL THEORY
the density decays with a power law, not exponentially. But overall, the trends will be 2. Show that the external potential goes as
approximately right.
B 7/3 7/3
The Thomas-Fermi equation for neutral densities, Eq. (6.15), is usually given in term of a V ext = − Z = 1.79374 Z (6.21)
a
dimensionless function Φ(x) satisfying:
3. By using the virial theorem, deduce the behavior of the Hartree energy as Z → ∞.
4
5 Φ 3
5
##
Φ (x) = 6 (6.16) 4. Using table 5.3, compare with energies along first row and for the noble gas atoms.
x
Comment.
where Φ(0) = 1 and Φ(∞) = 0. The distance x = Z 1/3 r/a, where
5. An accurate radial density for the Xe atom is given in Fig. 6.1. An accurate parametriza-
= 2/3
1 3π
<
a = = 0.885341, (6.17) tion of Φ is
2
2
2 4 Φ(y) = (1 + α y + β y exp(−γ y)) exp(−2α y) (6.22)
and the density is given by where y = √ x and α = 0.7280642371, β = −0.5430794693, and γ = 0.3612163121.
Z 2 < Φ = 3/2
n(r) = (6.18) Plot the approximate TF density on the same figure, and comment.
4πa 3 x
Its been found that at the solution for neutral atoms, Φ (0) = −B, where B = 1.58807102261, 6. Calculate the expectation value %1/r& for Thomas-Fermi. Do atomic radii grow or shrink
#
and the asymptotic behavior is the trivial solution: with Z? Explain why.
144 To see the quality of our TF solution, we calculate the κ(r) function of the last chapter,
Φ(x) = , x → ∞. (6.19)
x 3 and compare with that of the exact solution. We see that, although it messes up all the
These results, and the density of Fig. 6.1, are needed for the next exercise. 0
90
80
70 -5
Xe atom in LDA
60
4πr 2 n(r) 50 -10 Kappa in Ar atom
40
30
-15
20
10 exact
approx TF
0 -20
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3
r Figure 6.2: κ(r) for the Ar atom, both exactly and in Thomas-Fermi calculation of Exercise ??.
Figure 6.1: Radial density in the Xe atom, calculated using LDA.
details (nuclear cusp, shell structure, decay at large distances), it has a remarkably good
overall shape for such a simple theory.
Exercise 26 Thomas-Fermi for atoms
Lastly, Teller showed that molecules do not bind in TF theory, because the errors in TF
are far larger than the relatively small binding energies of molecules.
1. Using the differential equation and integrating by parts, find the TF kinetic energy. Using
the virial theorem for atoms, show that
6.4 Particles in boxes
3 B 7/3 7/3
E = − Z = −.7845 Z (6.20)
7 a Before closing this chapter, let us return to the examples of the introduction and understand
This is the exact limiting behavior for non-relativistic atoms as Z → ∞. how they are connected to the Thomas-Fermi model. They highlight more clearly the relative