Page 17 - 71 the abc of dft_opt

34 CHAPTER 3. ONE ELECTRON
independent of the particular problem, i.e., we apply the same operation on a given trial
wavefunction, no matter what our physical problem is. But the potential energy functional
differs in each problem; it is not universal. This is analogous to trying to find the minimum
2
over x of f(x, y) = x /2 − xy. The first term is independent of y, so we could make a list
Chapter 3 of it value for every x, and then use that same list to find the minimum for any y, without
having to evaluate it again for each value of y.
Now our variational principle can be written by writing
One electron 1
2
E = min {T[φ] + V [φ]}, ∞ dx |φ(x)| = 1 (3.4)
φ −∞
We need simply evaluate the energy of all possible normalized wavefunctions, and choose the
1
In this chapter, we review the traditional wavefunction picture of Schr¨odinger for one lowest one.
particle, but introducing our own specific notation.
3.2 Trial wavefunctions
3.1 Variational principle
Next, we put trial wavefunctions into the variational principle to find upper bounds on the
We start off with the simplest possible case, one electron in one dimension. Recall, from ground-state energy of a system.
basic quantum mechanics, the Rayleigh-Ritz variational principle:
Exercise 9 Gaussian trial wavefunction
1 ∞
ˆ
2
E = min % φ | H | φ &, dx |φ(x)| = 1. (3.1) An interesting potential in one-dimension is just V (x) = −δ(x), where δ is the Dirac delta
φ −∞ −1/4 2
function. Using a Gaussian wavefunction, φ G (x) = π exp(−x /2), as a trial wavefunc-
We will use this basic, extremely powerful principle throughout this book. This principle says tion, make an estimate for the energy, and a rigorous statement about the exact energy.
that we can use any normalized wavefunction to calculate the expectation value of the energy
for our problem, and we are guaranteed to get an energy above the true ground-state energy. We can improve on our previous answers in two distinct ways. The first method is to
For example, we can use the same wavefunction for every 1-d one-electron problem, and get include an adjustable parameter, e.g., the length scale, into our trial wavefunction.
an upper bound on the ground-state energy.
Exercise 10 Adjustable trial wavefunction
Having learnt about functionals in Chapter ??, we may write this principle in the following Repeat Ex. (9) with a trial wavefunction φ G (αx) (don’t forget to renormalize). Find % T &(α)
useful functional form. For example, the kinetic energy of the particle can be considered as and % V &(α) and plot their sum as a function of α. Which value of α is the best, and what
a functional of the wavefunction:
is your estimate of the ground-state energy? Compare with previous calculation.
 
1 ∞ 1 d 2 1 1 ∞ 2
T[φ] = dx φ (x) −  φ(x) = dx |φ (x)| (3.2) Note that this has led to a non-linear optimization problem in α. Varying exponents in
∗
#

−∞ 2 dx 2 2 −∞
trial wavefunctions leads to difficult optimization problems.
where the second form is gotten by integration by parts, and the prime denotes a spatial To see how well we did for this problem, the next exercise yields the exact answer.
derivative. (This second form is much handier for many calculations, as you only need take
one derivative.) Thus, for any given normalized wavefunction, there is a single number T, Exercise 11 1-d Hydrogen atom
Use φ E (αx), where φ E (x) = exp(−|x|) as a trial wavefunction for the problem in Ex. (9).
which can be calculated from it, via Eq. (3.2). Similarly, the potential energy is a very simple
Find the lowest energy. This is the exact answer. Calculate the errors made in the previous
functional:
1 ∞ 2 exercises, and comment.
V [φ] = dx V (x) |φ(x)| . (3.3)
−∞
An alternative, simpler approach to improving our trial wavefunction is to make a linear
The kinetic energy of a given wavefunction is always the same, no matter what the prob-
combination with other trial wavefunctions, but not vary the length scale. Thus we write
lem we are applying that wavefunction to. The kinetic energy is a universal functional,
1 c !2000 by Kieron Burke. All rights reserved. φ trial (x) = c 0 φ 0 (x) + c 1 φ 1 (x), (3.5)
33