Page 35 - 71 the abc of dft_opt

7.4. QUESTIONS 69 70 CHAPTER 7. KOHN-SHAM
• The exact kinetic energy is simply |E|, by the virial theorem for atoms. 3. The exchange energy of the orbitals is the same in Hartree-Fock and Kohn-Sham theory.
What is the relation between the two for the Hartree-Fock density?
• The interaction with the nucleus, V ext , is a little more than twice the kinetic energy, as
it must also overcome the (positive) V ee . 4. Considering the exact relations for the asymptotic decay of the density of Coulombic
systems, is there any significance to & HOMO for the KS system? (See section 14.2).
• The electron-electron repulsion is typically less than half the kinetic energy, but all three
are on the same scale. Recall from Thomas-Fermi theory that V ext = −7T/3 = −7V ee .
• The KS kinetic energy T S is almost as large as the true kinetic energy, and they differ by
less than 2%.
• The Hartree energy is typically a small overestimate of V ee .
• The exchange energy cancels a fraction of the Hartree energy, that fraction getting smaller
as Z increases.
• The kinetic contribution to the correlation energy, which is also the correlation contribu-
tion to the kinetic energy, is on the same scale as −E C .
• The potential contribution to the correlation energy, which is also the correlation contri-
bution to the potential is energy, is a little more than −2T C .
• The magnitude of all energy components grows with Z.

We end with an exercise designed to make you think about the trends.
Exercise 34 Energy components for atoms

Use the data in Table 7.3 to answer the following:
1. How close are the ratios of T to V ee to V ext to their Thomas-Fermi ideal values? What
is the trend with Z? Comment.
2. What is the ratio of |E X | to U? How does it change with Z? Comment.
3. Repeat above for T C versus T S . Comment.
4. What is the ratio T C versus |U C |? As a function of Z? Comment.

7.4 Questions

All questions are conceptual.
1. Define the ground-state Kohn-Sham wavefunction generating density n(r).
2. Consider two normalized orbitals in one dimension, φ 1 (x) and φ 2 (x), and the density
2
2
n = |φ 1 | + |φ 2 | . What is the Kohn-Sham kinetic energy of that density? How does
it change when we alter one of the orbitals? Repeat the question for the Kohn-Sham
exchange energy.