Page 47 - 71 the abc of dft_opt

94 CHAPTER 12. SCALING

2.0

Chapter 12

Scaling

0.0
!5.0 5.0
1
In this chapter, we introduce the concept of scaling the density, as the natural way to
understand the most important limits about density functionals. Figure 12.1: Cartoon of the one-dimensional H 2 molecule density, and scaled by a factor of 2.
However, if the potential energy is homogeneous of degree p, i.e.,
12.1 Wavefunctions
p
V (γx) = γ V (x), (12.6)
It will prove very useful to figure out what happens to various quantities when the coordinates
are scaled, i.e., when x is replaced by γx everywhere in a problem, with γ a positive number. then
For one electron in one dimension, we define the scaled wavefunction as −p
V [φ γ ] = γ V [φ]. (12.7)
1
φ γ (x) = γ 2 φ(γx), 0 < γ < ∞, (12.1)
Consider any set of trial wavefunctions. If scaling any member of the set leads to another
where the scaling factor out front has been chosen to preserve normalization: member of the set, then we say that the set admits scaling.
1 1 1
∞ 2 ∞ 2 ∞ # 2
1 = dx |φ(x)| = dx γ|φ(γx)| = dx |φ(x )| . (12.2)
#
Exercise 43 Scaling of trial wavefunctions
−∞ −∞ −∞
A scale factor of γ > 1 will squeeze the wavefunction, and of γ < 1 will stretch it out. Do the sets of trial wavefunctions in Ex. 10 admit scaling?
√
For example, consider φ(x) = exp(−|x|). Then φ γ (x) = γ exp(−γ|x|) is a different
√
wavefunction for every γ. For example, φ 2 (x) = 2 exp(−2|x|). In Fig. 12.1, we sketch The only important change when going to three dimensions for our purposes in the pre-
a density for the one-dimensional H 2 molecule of a previous exercise, and the same density ceding discussion is that the scaling normalization factors change:
scaled by γ = 2, 3 3
n γ (x) = γn(γx). (12.3) φ γ (r) = γ 2 φ(γr) n γ (r) = γ n(γr) (12.8)
Note how γ > 1 squeezes the density toward the origin, keeping the number of electrons When including many electrons, one gets a similar factor for each coordinate:
fixed. We always use subscripts to denote a function which has been scaled.
What is the kinetic energy of such a scaled wave-function? Ψ γ (r 1 , . . . , r N ) = γ 3N/2 Ψ(γr 1 , . . . , γr N ), (12.9)
1 1 ∞ γ 1 ∞ γ 2 1 ∞
2
# 2
2
2
T[φ γ ] = dx |φ (x)| = dx |φ (γx)| = dx |φ (x )| = γ T[φ], (12.4) suppressing spin indices. We can they easily evaluate our favorite operators. It is simple to
#
#
#
#
γ
2 −∞ 2 −∞ 2 −∞ show:
i.e., the kinetic energy grows quadratically with γ. The potential energy is not so straightfor- 2
ward in general, because it is not a universal functional, i.e., it is different for every different T[Ψ γ ] = γ T[Ψ] (12.10)
problem. and
1 1 x
∞ 2 ∞ # 2
#
V [φ γ ] = dx |φ γ (x)| V (x) = dx |φ(x )| V ( ) = % V (x/γ). & (12.5) V ee [Ψ γ ] = γ V ee [Ψ] (12.11)
−∞ −∞ γ
1 c !2000 by Kieron Burke. All rights reserved. while V ext has no simple rule in general.
93