Page 72 - 71 the abc of dft_opt

17.3. GRADIENT ANALYSIS 143 144 CHAPTER 17. GRADIENTS
We define a density of reduced gradients, analogous to our density of r s ’s:
1 3
g 3 (s) = d r n(r) δ(s − s(r)), (17.12)
also normalized to N, and plotted for the Ar atom in Fig. 17.4. Note there is no contribution
40
Ar atom
35
30
25
g 3 (s) 20
15
10
5
0
0 0.5 1 1.5 2
s
Figure 17.4: g 3 (s) in Ar atom.
to atoms (or molecules) from regions of s = 0. Note also that almost all the density has
reduced gradients less than about 1.2, with a long tail stretching out of larger s.

Exercise 84 Gradient analysis for H atom
Calculate r s (r) and s(r) for the H atom, and make the corresponding g 1 (r s ) and g 3 (s) plots.
Why does the exponential blow up at large r not make the LDA or GEA energies diverge?
We now divide space around atoms into spheres. We will (somewhat arbitrarily) denote Figure 17.5: Distribution of densities and gradients in the N atom.
the valence region as all r values beyond the last minimum in s(r), containing 6.91 electrons
in Ar. We will denote the region of all r smaller than the position of the last peak in s as precisely the inner core peak, while in the g(s) figure, we see the contribution of the valence
being the core region (9.34 electrons). The last region in which s decreases with r is denoted electrons alone to the peak from the combined core-valence region.
the core-valence region, in which the transition occurs (1.75 electrons). We further denote We may use the s curve to make more precise (but arbitrary) definitions of the various
the tail region as being that part of the valence region where s is greater than its maximum shells. Up to the first maximum in s we call the core. The small region where s drops rapidly
in the core (1.54 electrons). is called the core-valence intershell region. The region in which s grows again, but remains
To see how big s can be in a typical system that undergoes chemical reactions, consider below its previous maximum, we call the valence region. The remainer is denoted the tail.
Fig. 17.5. On the left is plotted values of r s (top) and of s (bottom) as a function of r in an N We make these regions by vertical dashed lines, which have been extended to the upper r s
atom. Then on the right are plotted the distribution of values (turned sideways). We see that curve. Because that curve is monotonic, we can draw horizontal lines to meet the vertical
the core electrons have r s ≤ 0.5, while the valence electrons have 0.5 ≤ r s ≤ 2. The s-values lines along the curve, and thus divide the g(r s ) curve into corresponding regions of space.
are more complex, since they are similar in both the core and valence regions. We see that But we are more interested in atomization energies than atomic energies. In the case of a
almost all the density has 0.2 ≤ s ≤ 1.5. Because s is not monotonic, many regions in space molecule, there is no simple monotonic function, but one can easily extract the densities of
can contribute to g 3 (s) for a given s value. To better distinguish them, one can perform a variables. Having done this for the molecule, we can make a plot of the differences between
pseudopotential calculation, that replaces the core electrons by an effective potential designed (twice) the atomic curves and the molecular curve. Now the curves are both positive and
only to reproduce the correct valence density. This is marked in the two right-hand panels negative. We see immediately that the core does not change on atomization of the molecule,
by a long-dashed line. In the g 3 (r s ) figure, we see indeed that the pseudopotential is missing and so is irrelevant to the atomization energy. Interestingly, it is also clear that it is not