Page 37 - 71 the abc of dft_opt

8.3. UNIFORM ELECTRON GAS 73 74 CHAPTER 8. THE LOCAL DENSITY APPROXIMATION
8.3 Uniform electron gas We thus write 1
3
E C LDA = d r n(r) & unif (r s (r)) (8.14)
C
The local approximation is exact for the special case of a uniform electronic system, i.e.,
where & unif (r s ) is the correlation energy per electron of the uniform gas. A simple way to
one in which the electrons sit in an infinite region of space, with a uniform positive external C
potential, chosen to preserve overall charge neutrality. The kinetic and exchange energies think about correlation is simply as an enhancement over exchange:
of such a system are easily evaluated, since the Kohn-Sham wavefunctions are simply Slater & unif (r s ) = F XC (r s ) & unif (r s ) (8.15)
XC X
determinants of plane waves. The correlation energy is extracted from accurate Monte Carlo
calculations, combined with known exact limiting values. and this enhancement factor is plotted in Fig. 15.1. There are several important features to
the curve:
If we repeat the exercise above in three dimensions, we find that it is simpler to use
plane-waves with periodic boundary conditions. The states are then ordered energetically by • At r s = 0 (infinite density), exchange dominates over correlation, and F X = 1.
momentum, and the lowest occupied levels form a sphere in momentum-space. Its radius is
• As r s → 0 (high density), there is a sharp dive toward 1. This is due to the long-ranged
the Fermi wavevector, and is given by
nature of the Coulomb repulsion in an infinite system, leading to
2
k F = (3π n) 1/3 , (8.9) unif
& (r s ) → 0.0311 ln r s − 0.047 + 0.009r s ln r s − 0.017r s (r s → 0) (8.16)
C
where the different power and coefficient come from the different dimensionality. Note that we The logarithmic divergence is not enough to make correlation as large as exchange in this
have now doubly-occupied each orbital, to account for spin. Dimensional analysis produces
limit, but we will see later that finite systems have finite correlation energy in this limit.
the same form for the kinetic energy as in 1D, but matching to the uniform gas yields a
different constant: • In the large r s limit (low-density)
2
3
3
T TF [n] = 3 1 d r k (r)n(r) = A S 1 d r n 5/3 (r) (3D unpolarized) (8.10) & unif (r s ) → − d 0 + d 1 − .... (r s → ∞) (8.17)
S F C 3/2
10 r s rs
2 2/3
where A s = (3/10)(3π ) = 2.871. where the constants are d 0 = 0.896 and d 1 = 1.325. The constant d 0 was first deduced
For the exchange energy of the uniform gas, we simply note that the Coulomb interaction by Wigner from the Wigner crystal for this system. Note that this means correlation
unif
has dimensions of inverse length. Thus we know that its energy density must be proportional becomes (almost) as large as exchange here, and so F XC (r s → ∞) = 1.896. Note also
to k F , leading to 1 that the approach to the low-density limit is extremely slow.
3
E LDA [n] = A X d r n 4/3 (r) (8.11)
X
0
Evaluation of the Fock integral Eq. (5.13) for a Slater determinant of plane-wave orbitals
yields the exchange energy per electron of a uniform gas as -0.05
& unif (n) = 3k F , (8.12) -0.1
X
4π
-0.15
so that A X = −(3/4)(3/π) 1/3 = −0.738.
Correlation is far more sophisticated, as it depends explictly on the physical ground-state -0.2
wavefunction of the uniform gas. Another useful measure of the density is the Wigner-Seitz -0.25 uniform gas
C
radius X
< 3 = 1/3 1.919 -0.3
r s = = (8.13) 0 1 2 3 4 5 6
4πn k F
Figure 8.2: Exchange and correlation energies per particle for a uniform electron gas.
which is the radius of a sphere around electron such that the volume of all the spheres matches
the total density of electrons. Thus r S → 0 is the high-density limit, and r s → ∞ is the low The uniform gas exchange and correlation energies/particle are plotted in Fig 8.2, as a
density limit. function of r s . The exchange is very simple, being proportional to 1/r s . The sharp downturn