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118 CHAPTER 15. ANALYSIS TOOLS
3
Ar atom
2.5
2
Chapter 15 r s (r) 1.5
1
Analysis tools 0.5
0
0 0.5 1 1.5 2 2.5 3
r
In this chapter, we introduce a variety of tools that help analyze how approximate density
Figure 15.2: r s (r) in Ar atom.
functionals work.
This has the simple interpretation: g(r s )dr s is the number of electrons in the system with
15.1 Enhancement factor Seitz radius between r s and r s + dr s , and so satisfies:
1
∞
dr s g(r s ) = N. (15.2)
1.5 0
This is plotted in Fig. 15.3. The different peaks represent the s electrons in each shell. If
1.4
45 Ar atom
40
F unif (r s ) XC 1.3 35
30
1.2
g 1 (r s ) 25
1.1 uniform gas 20
unpolarized
polarized 15
1 10
0 1 2 3 4 5 6
5
r s 0
0 0.5 1 1.5 2 2.5 3
Figure 15.1: Enhancement factor for correlation in a uniform electron gas as a function of Wigner-Seitz radius for unpolarized
(solid line) and fully polarized (dashed line) cases.
r s
Figure 15.3: g 1 (r s ) in Ar atom.
we integrate this function forward from r s = 0, we find that it reaches 2 at r s = 0.15, 4
15.2 Density analysis
at 0.41, 10 at 0.7, and 12 at 0.88; these numbers represent (roughly) the maximum r s in a
given shell. The 1s core electrons live with r s ≤ 1.5, and produce the first peak in g 1 ; the
What we can do is ask how accurately we need to know the uniform gas inputs. Recall the
2s produce the peak at r s = 0.2, and the 2p produce no peak, but stretch the 2s peak upto
radial density plot of the Ar atom, Fig. 5.1. Now we plot the local Wigner-Seitz radius,
r s (r), in Fig. 15.2, and find that the shells are less obvious. The core electrons have r s ≤ 1, 0.7. The peak at about 0.82 is the 3s electrons, and the long tail includes the 3p electrons.
Why is this analysis important? We usually write
while the valence electrons have r s ≥ 1, with a tail stretching toward r s → ∞. To see the
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distribution of densities better, we define the density of r s ’s: E LDA = 1 d r n(r) & unif (r s (r)), (15.3)
XC
XC
1
3
g 1 (r s ) = d r n(r) δ(r s − r s (r)). (15.1) where & unif is the exchange-correlation energy per particle in the uniform gas. But armed with
XC
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