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82 CHAPTER 10. PROPERTIES
10.2 Densities and potentials

The self-consistent LSD density is always close to the exact density, being hard to distinguish
by eye, as seen in Fig. 10.1. However, when we calculate κ(r), as defined earlier, we can
Chapter 10 12 Ne atom
exact
10
LDA
Properties 8
4πr 2 n(r) 6
4
In this chapter, we survey some of the more basic properties that are presently being 2
calculated with electronic structure methods, and include how well or badly LDA does for
0
these. The local density approximation to exchange-correlation was introduced by Kohn 0 0.5 1 1.5 2
and Sham in 1965, and (the spin-dependent generalization) has been one of the most
r
succesful approximations ever. Until the early 90’s, it was the standard approach for all
Figure 10.1: Radial density of the Ne atom, both exactly and from an LDA calculation
density functional calculations, which were often called ab initio in solid state physics. It
remains perhaps the most reliable approximation we have. see the difference in the asymptotic region: Note that the density integrates to 9.9 at about
-1
-1.2
10.1 Total energies
-1.4
κ(r)
For atoms and molecules, the total exchange energy is typically underestimated by about
-1.6
10%. On the other hand, the correlation energy is overestimated by about a factor of 2 or 3.
Since for many systems of physical and chemical interest, exchange is about 4 times bigger -1.8 Ne atom
than correlation, the overestimate of correlation compliments the underestimate of exchange, exact
LDA
and the net exchange-correlation energy is typically underestimated by about 7%. -2
0 1 2 3 4 5
r
Exercise 39 Wigner approximation for He atom correlation energy:
By approximating the density of the He atom as a simple exponential with Z eff = 7/4 as we Figure 10.2: Logarithmic derivative of density for Ne atom, both exactly and in LDA. The curves are indistinguishable close
to the nucleus, where they both reach -10.
found in Chapter 4, calculate the LDA correlation energy using the Wigner approximation,
Eq. (8.18). r = 2.68. So for all the occupied region, even κ(r) is extremely accurate in LDA, and all
integrals over the density are very accurate. For example, %1/r& is 31.1 exactly, but 31.0 in
LDA.
E X E C E XC
The Kohn-Sham potentials reflect this similarity. In Fig. 10.3, we show both the LDA
atom LDA exact error % LDA exact error % LDA exact error %
and exact potentials, and how close they appear. However, note that they are dominated
He -0.883 -1.025 -14 % -0.112 -0.042 167 % -0.996 -1.067 -7 %
by terms that are (essentially) identical in the two cases. The largest term is the external
Be -2.321 -2.674 -13 % -0.225 -0.096 134 % -2.546 -2.770 -8 % potential, which is −10/r. This determines the scale of the figure. The next term is the
Ne -11.021 -12.085 -9 % -0.742 -0.393 89 % -11.763 -12.478 -6 %
Hartree potential, plotted in Fig. 10.4. This is essentially the same in both cases, since the
Table 10.1: Accuracy of LDA for exchange, correlation, and XC for noble gas atoms. densities are almost identical. Again note the scale.
81