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28 CHAPTER 2. FUNCTIONALS
integrate over the entire range of angles. The contribution to the area is that of a triangle
of base r cos(dθ) ≈ r and height r(θ) sin(dθ) ≈ rdθ, yielding
1 1 1 1 2π
2
A[r] = rdθr = dθ r (θ) (2.2)
2 2 0
Chapter 2
For an abitrary r(θ) the integral above can be evaluated to give the area that is enclosed by
the curve. Thus this functional maps a real function of one argument to a number, in this
Functionals case the area enclosed by the curve. Similarly, the infinitesimal change in the perimeter is
just the line segment in polar coordinates:
2 2 2
δP = dr + r (θ)dθ. (2.3)
1 In this chapter, we introduce in a more systematic fashion what exactly a functional
Now, since r = r(θ), we can write dr = dθ (dr/dθ), yielding
is, and how to perform elementary operations on functionals. This mathematics will
be needed when we describe the quantum mechanics of interacting electrons in terms of P[r] = 1 2π dθ % r (θ) + (dr/dθ) . (2.4)
2
2
density functional theory. 0
In both cases, once we know r(θ), we can calculate the functional.
2.1 What is a functional? The area is a local functional of r(θ), since it can be written in the form:
1 2π
A[r] = dθ f(r(θ)), (2.5)
A function maps one number to another. A functional assigns a number to a function. For 0
example, consider all functions r(θ), 0 ≤ θ ≤ 2π, which are periodic, i.e., r(θ + 2π) = r(θ). 2
where f(r) is a function of r. For the area, f = r /2. These are called local because,
Such functions describe shapes in two-dimensions of curves which do not “double-back” on
inside the integral, one needs only to know the function right at a single point to evaluate
themselves, such as in Fig. 2.1. For every such curve, we can define the perimeter P as the
the contribution to the functional from that point. On the other hand, P is a semi-local
functional of the radius, as it depends not only on r, but dr/dθ.
We have already come across a few examples of density functionals. In the opening
chapter, the local approximation to the kinetic energy is a local functional of the density,
3
with f(n) = 1.645n . On the other hand, for any one-electron system, the exact kinetic
energy is a semi-local functional, called the von Weisacker functional:
Figure 2.1: A 2D curve which is generated by a function r = r(θ) T VW [n] = 1 1 ∞ dx n #2 , (2.6)
S
8 −∞ n
length of the curve, and the area A as the area enclosed by it. These are functionals of r(θ), #
where n (x) = dn/dx. Later we will see some examples of fully non-local functionals.
in the sense that, for a given curve, such as the ellipse
7
2
2
r(θ) = 1/ sin (θ) + 4 cos (θ) (2.1) 2.2 Functional derivatives
there is a single well-defined value of P and of A. We write P[r] and A[r] to indicate this When we show that the ground-state energy of a quantum mechanical system is a functional
functional dependence. Note that, even if we don’t know the relation explicitly, we do know of the density, we will then want to minimize that energy to find the true ground-state density.
it exists: Every bounded curve has a perimeter and an area. To do this, we must learn how to differentiate functionals, in much the same way as we learn
For this simple example, we can use elementary trigonometry to deduce explicit formulas how to differentiate regular functions in elementary calculus.
for these functionals. Consider the contribution from an infinitesimal change in angle dθ, and To begin with, we must define a functional derivative. Imagine making a tiny increase in
1 c !2000 by Kieron Burke. All rights reserved. a function, localized to one point, and asking how the value of a functional has changed due
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