Page 40 - 71 the abc of dft_opt

9.3. LSD 79 80 CHAPTER 9. SPIN
9.3 LSD 0

The local spin density (LSD) approximation is the spin-scaled generalization of LDA. As input, -0.05
we need the energies per particle of a spin-polarized uniform gas. The formula for the exchange
-0.1 C
energy is straightforward, but the correlation energy, as a function of spin-polarization, for X
the uniform gas, must be extracted from QMC and known limits. -0.15
Begin with exchange. In terms of the energy density of a uniform gas, Eq. (9.6) becomes:
-0.2
1 & unpol unpol '
e X (n ↑ , n ↓ ) = e (2n ↑ ) + e (2n ↓ ) (9.8) uniform gas
2 X X -0.25 pol
Now we know e unpol (n) = A X n 4/3 , so -0.3 unpol
X
# 4/3 4/3 $ 0 1 2 3 4 5 6
e X (n ↑ , n ↓ ) = 2 1/3 A X n + n (9.9)
↑ ↓
Figure 9.2: Effect of (full) spin polarization on exchange and correlation energies of the uniform electron gas.
Now we introduce a new, useful concept, that of relative spin polarization. Define
9.4 Questions
ζ = (9.10)
n ↑ − n ↓
n
as the ratio of the spin difference density to the total density. Thus ζ = 0 for an unpolarized All are conceptual.
system, and ±1 for a fully polarized system. In terms of ζ,
1. Why does LSD yield a more accurate energy for the Hydrogen atom than LDA?
n ↑ = (1 + ζ) n/2, n ↓ = (1 − ζ) n/2, (9.11)
2. State in words the relation between v S [n](r) and v Sσ [n ↑ , n ↓ ](r). When do they coincide?
Inserting these definition into Eq. (9.9), we find, for the energy density:
3. Deduce the formula of T S loc for unpolarized electrons in a one-dimensional box.
4/3
4/3
& X (n, ζ) = & unpol (n) (1 + ζ) + (1 − ζ) (9.12)
X
2
Thus our LSD exchange energy formula is 4/3 4/3
3
E LSD [n ↑ , n ↓ ] = A X 1 d r n 4/3 (r) (1 + ζ(r)) + (1 − ζ(r)) (9.13)
2
X
where ζ(r) is the local relative spin polarization:
n ↑ (r) − n ↓ (r)
ζ(r) = (9.14)
n(r)
Unfortunately, the case for correlation is much more complicated, and we merely say that
there exists well-known parametrizations of the uniform gas correlation energy as a function of
spin polarization, & unif (n, ζ). In Fig. 9.2, we show what happens. When the gas is polarized,
C
exchange gets stronger, because electrons exchange only with like spins. On the other hand,
exchange keeps the electrons further apart, reducing the correlation energy. This effect is
typical of most systems, but this has not been proven generally.
Exercise 37 Relative spin polarization
Sketch ζ(r) for the Li atom, as shown in Fig. 9.1.
Exercise 38 Spin polarized Thomas-Fermi
Convert the Thomas-Fermi local approximation for the kinetic energy in 3D to a spin-density
functional. Test it on the hydrogen atom, and compare to the regular TF.