Page 27 - 71 the abc of dft_opt

5.6. QUESTIONS 53 54 CHAPTER 5. MANY ELECTRONS
spherical systems is 4. Which is bigger, the kinetic energy of the true wavefunction or that of the HF wavefunc-
d log(n(r)) tion, for an atom?
κ(r) = (1/2) , (5.20)
dr
5. What do you expect happens to the correlation energy of the two-electron ions as Z →
since κ(0) = −Z, while κ(r) → −α, as r → ∞. The ground-state density can always be ∞?
reconstructed from
1
∞ 6. What do you expect happens to the correlation energy of the four-electron ions as Z →
n(r) = C exp( dr 2κ(r )) (5.21)
#
#
r ∞?
where the constant is determined by normalization.
In a one-dimensional world with delta function interactions, these conditions remain true,
except for the details of the power law in front of the exponential in Eq. (5.18). In particular,
we have already seen the cusp in the orbital for 1-d hydrogenic atoms. We may see dramatic
evidence for these conditions in our HF solution of the 1-d He atom, by plotting κ(x) on the
left of Fig. 5.2. Near the nucleus, the exact cusp condition is satisfied, so that κ equals -2,
but at large distances, it tends to a smaller constant, determined by the ionization potential.
We can also clearly see the difference between the approximate and exact HF solutions here,
which was not so visible in the last figure. On the right, we have also plotted κ(r) for real
He, and see that the effect of correlation on the density is extremely small: The HF and exact
densities are very close.
-1.3 -1.3
exact
-1.4 approx -1.4
-1.5 -1.5 exact
HF
-1.6 -1.6
κ
-1.7 -1.7
-1.8 -1.8
-1.9 -1.9
1d He atom in HF He atom
-2 -2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10
x r
Figure 5.2: κ in (a) 1d He atom in HF approximation, both exact and approximate; (b) in real He, in HF and exactly

5.6 Questions

All the questions below are conceptual.
1. Does each orbital in a HF calculation satisfy the same Schr¨odinger equation?

i=1 i ?
2. What is the relationship between E HF and " N &
3. Speculate on why the correlation energy of Li is about the same as that of He.