Page 46 - 71 the abc of dft_opt

11.5. UNIFORM LIMIT 91 92 CHAPTER 11. SIMPLE EXACT CONDITIONS
If we apply LSD to the exact density, we make a large error, because E LSD [n] = E LDA [n A ]+ 4. What does the LO bound say for one electron?
E LDA [n B ], and we saw in a previous chapter the huge error LDA makes for the H-atom,
because it does not account for spin-polarization. We would much rather find E LSD =
2E LSD (H).
There is a way to do this, that has been well-known in quantum chemistry for years. If
one allows a Hartree-Fock calculation to spontaneously break spin-symmetry, i.e., do not
require the up-spin spatial orbital to be the same as that of the down-spin, then at a certain
separation, called the Coulson-Fischer point, the unrestricted solution will have lower energy
than the restricted one, and will yield the correct energy in the separated-atom limit, by
producing two H atoms that have opposite spins. Thus we speak of RHF and UHF. The
trouble with this is that the UHF solution is no longer a pure spin-eigenstate. Thus the
symmetry dilemma is that an RHF calculation produces the correct spin symmetry with the
wrong energy, while the UHF solution produces the right energy but with the wrong spin
symmetry.
Kohn-Sham DFT calculations with approximate density functionals have a similar problem.
The LSD calculation will spontaneously break symmetry at a Coulson-Fischer point (further
than that of HF) to yield the appropriate energy. Now the dilemma is no longer the spin
eigenvalue of the wavefunction, but rather the spin-densities themselves. In the unrestricted
LDA calculation, which correctly dissociates to two LSD H atoms, the spin-densities are
completely polarized, which is utterly incorrect.

11.5 Uniform limit

We have already used the argument that any local approximation must use the value for the
uniform limit as its argument. But more generally, we require
E XC [n] → V e unif (n) n(r) → n (constant) (11.7)
XC
where V is the volume of the system. This can be achieved, for example, by taking ever
larger boxes, but keeping the density of particles fixed. This limit is obviously gotten right by
LDA, but only if we use the correct energy density. Getting this right is relevant to accuracy
for valence electrons of simple bulk metals, such as Li or Al.

11.6 Questions

1. If I approximate the correlation energy as about 1 eV per electron, is that a size-consistent
approximation?
2. A spin-restricted LSD calculation for two H atoms 10 ˚ Aapart yields a different answer to
twice the answer for one H atom. Does this mean LSD is not size consistent?
3. Does LDA satisfy the LO bound pointwise? Does LSD?