Page 71 - 71 the abc of dft_opt

17.2. GRADIENT EXPANSION 141 142 CHAPTER 17. GRADIENTS

n & local % error GEA % error exact For correlation, the procedure remains as simple in principle, but more difficult in practice.
2 0.4 6.28 -13 7.33 2 7.22 Now, there are non-trivial density- and spin- dependences, and the terms are much harder to
2 0.6 6.28 -24 8.80 7 8.26 calculate in perturbation theory. In the high-density limit, the result may be written as
2 0.8 6.28 -34 11.31 18 9.55 ∆E GEA = 2 1 d r n(r)φ(ζ(r))t (r). (17.11)
3
2
4 0.4 6.28 -33 10.48 12 9.34 C 3π 2
%
4 0.6 6.28 -48 16.34 36 12.04 Here t = |∇n|/(2k s n), where k s = 4k F /π is the Thomas-Fermi screening length, the
4 0.8 6.28 -58 26.39 76 15.00 natural wavevector scale for correlations in a uniform system. The spin-polarization factor is
6 0.4 6.28 -47 15.73 32 11.94 φ(ζ) = ((1 + ζ) 2/3 + (1 − ζ) 2/3 )/2. This correction behaves poorly for atoms, being positive
6 0.6 6.28 -61 28.90 77 16.32 and sometimes larger in magnitude than the LSD correlation energy, leading to positive
6 0.8 6.28 -70 51.52 146 20.93 correlation energies!
8 0.4 6.28 -57 23.07 56 14.76
Exercise 82 GEA correlation energy in H atom
8 0.6 6.28 -70 46.50 124 20.80 Calculate the GEA correction to the H atom energy, and show how ∆E GEA scales in the
8 0.8 6.28 -77 86.71 221 27.04 C
high-density limit.
Table 17.4: Perimeter’s of shapes for various * and n, both exactly and in local approximation and in GEA.
17.3 Gradient analysis
Then, if LSD was moderately accurate for inhomogeneous systems, GEA, the gradient ex-
pansion approximation, should be more accurate. The form of these gradient corrections can Having seen in intimate detail which r s values are important to which electrons in Fig. 15.2,
be determined by scaling, while coefficients can be determined by several techniques. we next consider reduced gradients. Fig. 17.3 is a picture of s(r) for the Ar atom, showing
For both T S and E X , defined on the Kohn-Sham wavefunction, the appropriate measure of
2
the density gradient is given by Ar atom
1.5
s(r) = |∇n(r)|/(2k F (r)n(r)) (17.8)
This measures the gradient of the density on the length scale of the density itself, and has s(r) 1
the important property that s[n γ ](r) = s[n](γr), i.e., it is scale invariant. It often appears in
the chemistry literature as x = |∇n|/(n 4/3 ), so that x ≈ 6s. The gradient expansion of any 0.5
functional with power-law scaling may be written as:
1 ( ) 0
3
2
A GEA [n] = d r a(n(r)) 1 + Cs (r) (17.9) 0 0.5 1 1.5 2 2.5 3
The coefficient may be determined by a semiclassical expansion of the Kohn-Sham density r
matrix, whose terms are equivalent to a gradient expansion. One finds Figure 17.3: s(r) in Ar atom.
C S = 5/27, C X = 10/81 (17.10) how s changes from shell to shell. Note first that, at the nucleus, s has a finite moderate
value. Even though the density is large in this region, the reduced gradient is reasonable.
(In fact, a naive zero-temperature expansion gives C X = 7/81, but a more sophisticated
Exercise 83 Reduced gradient at the origin
calculation gives the accepted answer above). The gradient correction generally improves Show that at the origin of a hydrogen atom, s(0) = 0.38, and argue that this value wont
both the non-interacting kinetic energy and the exchange energy of atoms. The improvement
change much for any atom.
in the exchange energy is to reduce the error by about a factor of 2.
For any given shell, e.g. the core electrons, s grows exponentially. When another shell begins
Exercise 81 For the H-atom, calculate the gradient corrections to the kinetic and exchange to appear, there exists a turnover region, in which the gradient drops rapidly, before being
energies. (Don’t forget to spin-scale). dominated once again by the decay of the new shell. We call these intershell regions.