Page 73 - 71 the abc of dft_opt

17.3. GRADIENT ANALYSIS 145 146 CHAPTER 17. GRADIENTS
0

& X (r) -0.4

-0.8
He atom
X
LDA
-1.2
0 1 2 3
r
Figure 17.7: Exchange energy per electron in the He atom, both exactly and in LDA. The nucleus is at r = 0.

cancellation of errors throughout the system, i.e., LDA is not describing this quantity well at
each point in the system, but it does a good job for its integral. As we will show below, a
pointwise analysis of energy densities turns out to be, well, somewhat pointless...

17.4 Questions
1. If GEA overcorrects LDA, what can you say about how close the system is to uniform?
2. Do you expect LDA to yield the correct linear response of a uniform electron gas?

3. Sketch the gradient and density analysis for the ionization energy for Li.
Figure 17.6: Differences in distribution of densities and gradients upon atomization of the N 2 molecule.

only the valence electrons (r s > 0.8) that contribute, but also the core-valence region, which
include r s values down to 0.5. Finally, note that the r s values contributing to this energy also
extend to higher r s regions.
To summarize what we have learned from the gradient analysis:
• Note again that nowhere in either the N or Ar atoms (or any other atom) has s << 1,
the naive requirement for the validity of LSD. Real systems are not slowly-varying. Even
for molecules and solids, only a very small contribution to the energy comes from regions
with s < 0.1.
• A generalized gradient approximation, that depends also on values of s, need only do
well for s ≤ 1.5 to reproduce the energy of the N atom. To get most chemical reactions
right, it need do well only for s ≤ 3.
We end by noting that while this analysis can tell us which values of r s and s are relevant
to real systems, it does not tell us how to improve on LSD. To demonstrate this, in Fig.
17.7, we plot the exchange energy per electron in the He atom. One clearly sees an apparent