Page 58 - 71 the abc of dft_opt

14.3. DERIVATIVE DISCONTINUITIES 115 116 CHAPTER 14. DISCONTINUITIES
from each other. For example, take the H-He + case. At large distances, the interacting atom, in contact with a bath of electrons, eg a metal. One finds that the step in the potential
wavefunction for the two electrons should become two (unbalanced) Slater determinants, induced by adding an infinitesimal of charge is I-EA.
making up a Heitler-London type wavefunction, yielding a density that is the sum of the But LDA is a smooth functional, with a smooth derivative. Thus its potential cannot
isolated atom densities, centered on each nucleus. In our world of 1-d illustrations, this will discontinuously change when an infinitesimal of charge is added to the atom. So it will be in
be error about (I-EA)/2 for the neutral, and the reverse for the slightly negatively charged ion.
n(x) = exp(−2|x|) + 2 exp(−4|x − L|) (14.4) Comparing the potentials for the Ne atom, we see in Fig. 14.3 how close the LDA potential
is in the region around r = 1, relevant to the 2s and 2p electrons. However, it has been
where the H-atom is at 0 and the He ion is at L. For this two electron system, we find
% shifted down by -0.3, the LDA error in the HOMO orbital energy. Since Ne has zero electron
the molecular orbital as φ(x) = n(x)/2 and the Kohn-Sham potential by inversion. This
affinity, this is close to (I-EA)/2 for this system, since I=0.8. This rationalizes one feature of
the poor-looking potentials within LDA.
3 0

n(x) 2 v XC (r) -0.5

-1
1d H-He+ density
1
-1.5
Ne atom
exact
LDA-0.3
0 -2
-1 0 1 2 3 4 5 0 0.5 1 1.5 2
v S (x) 4 r
0 Figure 14.3: XC potential in the Ne atom, both exact and in LDA, in the valence region, but with LDA shifted down by
-0.3.
If the functional is generalized to include fractional particle numbers via ensemble DFT,
-4
1d H-He+ potential these steps are due to discontinuities in the slope of the energy with respect to particle number
at integer numbers of electons. Hence the name.
-8

-12 14.4 Questions
-1 0 1 2 x 3 4 5
+
Figure 14.2: Density and Kohn-Sham potential for 1d H-He . 1. What kind of chemical systems will suffer from strong self-interaction error?
is shown in Fig. 14.2, for L = 4. The apparent step in the potential between the atoms 2. Which is better, to get the right energy and wrong symmetry, or vice versa?
occurs where the dominant exponential decay changes. This is needed to ensure KS system
3. Is Koopmans’ theorem satisfied by LDA?
produces two separate densities with separate decay constants. Elementary math then tells
you that the step will be the size of the difference in ionization potentials between the two 4. If one continues beyond x = 5 in Fig. 14.2, what happens?
systems.
More generally, for e.g., NaCl being separated, the step must be the difference in I-E.A.,
where E.A. is the electron affinity. This is the energy cost of transferring an infinitesimal of
charge from one system to another, and the KS potential must be constructed so that the
molecule dissociates into neutral atoms. But the same reasoning can be applied to a single