Page 38 - 71 the abc of dft_opt

8.3. UNIFORM ELECTRON GAS 75 76 CHAPTER 8. THE LOCAL DENSITY APPROXIMATION
in the correlation as r s → 0 is due to the logarithmic term (but is not as singular as exchange, 8.4 Questions
which is diverging as 1/r s ). At large r s , the two curves become comparable, but r s must be
larger than shown here. For typical valence electrons, r s is between 1 and 6, and correlation 1. Following the previous question, can you deduce the asymptotic form of the exchange-
is much smaller than exchange. correlation potential, i.e., v XC (r) as r → ∞? Would a local approximation capture this
Over the years, this function has become very well-known, by combining limiting informa- behavior? (See section 14.2).
tion like that above, with accurate quantum Monte Carlo data for the uniform gas. An early
popular formula in solid-state physics is PZ81, while the parametrization of Vosko-Wilkes-
Nusair (VWN) has been implemented in quantum chemical codes. Note that in the Gaussian
codes, VWN refers to an older formula from the VWN paper, while VWN-V is the actual
parametrization recommended by VWN. More recently, Perdew and Wang reparametrized the
data. These accurate parametrizations differ only very slightly among themselves.
An early but inaccurate extrapolation from the low-density gas, was given by Wigner:

e C (r s ) = −a/(b + r s ) (Wigner) (8.18)
where a = −.44 abd b = 7.8. This gives a ballpark number, but misses the logarithmic
singularity as r s → 0.
Unfortunately, we must postpone a general review of the performance of LDA until after
the next chapter. This is because we have only developed density functional theory so far,
and not accounted for spin effects. A slight generalization of LDA, called the local spin-
density approximation, is what is actually used in practice. In fact, many (if not most)
practical problems require this. For example, pure LDA performs badly whenever there is an
odd number of electrons, since it makes no distinction between polarized and unpolarized
densities. We emphasize this fact in the next few exercises.
Exercise 35 Hartree energy for an exponential

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1. For an exponential density, n(r) = Z /π exp(−2Zr), calculate (a) the Hartree poten-
tial and (b) the Hartree energy.
2. Find the exact exchange energy for the Hydrogen atom.
3. Find the exact exchange energy for the Helium atom, using the exponential approximation
of Chapter 4.
Exercise 36 LDA exchange energy for 1 or 2 electrons
1. Calculate the LDA exchange energy for the Hydrogen atom, and compare with exact
answer.
2. Find the LDA exchange energy for the Helium atom, using the exponential approximation
of Chapter X, and compare with exact answer.