Page 104 - 71 the abc of dft_opt

207 208 APPENDIX E. DISCUSSION OF QUESTIONS
2. The Kohn-Sham wavefunction of density n(r) is that normalized, antisymmetric wave-
function that has density n(r) and minimizes the kinetic operator.
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3. The Kohn-Sham kinetic energy is not 1 2 ! dx |φ | + |φ | , because these orbitals could
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have come from anywhere. There is no reason to think they are Kohn-Sham orbitals.
They might be, e.g., Hartree-Fock orbitals. The way to get T S is to construct the density
n(x) and then find v S (x), a local potential that, with two occupied orbitals, yields that
density. The kinetic energy of those two orbitals is then T S . Again, if we alter one
of the original orbitals, the change in T S is not the kinetic contribution directly due to
that orbital. Rather, one must calculate the new density, construct the new Kohn-Sham
potential, get the kinetic energy of its orbitals, and that tells you the change.
All the same reasoning applies for E X [n].
4. See above. One can do a HF calculation, yielding orbitals and density. Then E X HF is
simply the Fock integral of those orbitals. But this is not E X [n HF ], which could only be
found by finding the local potential v S (r) whose orbitals add up to n HF (r), and evaluating
the Fock integral on its orbitals.
5. The additional flexibility of spin-DFT over DFT means that its much easier to make good
approximations for spin-polarized systems.
6. Given v Sσ (r), solve the Kohn-Sham equations for up and down non-interacting electrons
in these potentials, and construct the total density. Then ask what single potential all
electrons must feel in the Kohn-Sham equations to reproduce that density. Obviously,
they coincide when the system is unpolarized, so that v Sσ (r) = v S (r), and when fully
polarized, v S↑ (r) = v S (r), v S↓ (r) = 0 or undetermined.
7. No. See further work whead.