Page 39 - 71 the abc of dft_opt

78 CHAPTER 9. SPIN

1.6
Li atom
1.4 up
down
1.2
1
Chapter 9
0.8
0.6
Spin 0.4
0.2
0
0 1 2 3 4 5 6 7 8
In practice, people don’t use Kohn-Sham density functional theory, they use Kohn-Sham
Figure 9.1: Radial spin densities in the Li atom.
spin-density functional theory. For many problems, separate treatment of the up and
down spin densities yields much better results.
9.2 Spin scaling
We can easily deduce the spin scaling of orbital functionals, while little is known of the exact
9.1 Kohn-Sham equations
spin-scaling of correlation in general. By orbital functionals, we mean those that are a sum
We next introduce spin density functional theory, a simple generalization of density functional of contributions from the orbitals of each spin seperately.
theory. We consider the up and down densities as separate variables, defined as We do spin-scaling of the kinetic energy functional as an example. Suppose we know T S [n]
as a density functional for spin-unpolarized systems. Let’s denote it T unpol [n] to be very
1 1 2 S
n σ (r) = N dx 2 . . . dx N |Ψ(r, σ, x 2 , . . . , x N )| , (9.1) explicit. But since the kinetic energy is the sum of contributions from the two spin channels:
3
with the interpretation that n σ (r)d r is the probability for finding an electron of spin σ T S [n ↑ , n ↓ ] = T S [n ↑ , 0] + T S [0, n ↓ ] (9.2)
3
in d r around r. Then the Hohenberg-Kohn theorems can be proved, showing a one-to-
one correspondence between spin densities and spin-dependent external potentials, v ext,σ (r). i.e., the contributions come from each spin separately. Applying this to the spin-unpolarized
1 Similarly, the Kohn-Sham equations can be developed with spin-dependent Kohn-Sham case, we find
T unpol [n] = T S [n/2, n/2] = 2T S [n/2, 0] (9.3)
potentials. All modern density functional calculations are in fact spin-density functional S
calculations. This has several advantages: or T S [n, 0] = T unpol [2n]/2. Inserting this result back into Eq. (9.2), we find
S
1. Systems in collinear magnetic fields are included. 1 & unpol unpol '
T S [n ↑ , n ↓ ] = T [2n ↑ ] + T [2n ↓ ] (9.4)
2 S S
2. Even when the external potential is not spin-dependent, it allows access to magnetic
response properties of a system. This clearly yields a consistent answer for an unpolarized system, and gives the result for a
fully-polarized system:
3. Even if not interested in magnetism, we will see that the increased freedom in spin DFT T S pol [n] = T S unpol [2n]/2 (9.5)
leads to more accurate functional approximations for systems that are spin-polarized, Analogous formulas apply to exchange:
e.g., the Li atom, whose spin densities are plotted in Fig. 9.1.
1 & unpol unpol '
We will see in the next sections that it is straightforward to turn some density functionals E X [n ↑ , n ↓ ] = 2 E X [2n ↑ ] + E X [2n ↓ ] (9.6)
into spin-density functionals, while others are more complicated.
and
1 This is not quite true. There is a little more wriggle room than in the DFT case. For example, for a spin-up H atom, the spin down potential pol unpol
is undefined, so long as it does not produce an orbital whose energy is below -1/2. E X [n] = E X [2n]/2 (9.7)
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