Page 32 - 71 the abc of dft_opt

6.5. QUESTIONS 63 64 CHAPTER 6. DENSITY FUNCTIONAL THEORY
advantages and disadvantages of local approximations, since they apply in one dimension (no 1. How would you go about finding the potential from the wavefunction of a two-electron
shell structure) and without interaction. system? For two electrons, can more than one potential have the same ground-state
First we note that, for the particles in a box, we cheated slightly, by applying the local density?
approximation to the exact density. But if we’re making the local approximation for the
2. Define the ground-state wavefunction generating density n(r) without mentioning the
energy, where would we have gotten the exact density? A more honest calculation is to external potential.
find the self-consistent density within the given approximation. Repeating the arguments of
section 6.1, we find 3. When we say F[n] is a universal functional, exactly what do we mean? After all, since
1 % v ext (r) itself is a functional of n(r), is not the entire energy universal?
n(x) = 2(µ − v ext (x)) (6.23)
π
wherever the argument of the root is positive, zero otherwise, and µ is to be found by 4. Why do the HK theorems specify the ground-state and not, say, the first excited state?
normalization ( dx n(x) = N. 5. Using the local approximation for the one-dimensional kinetic energy developed in Chapter
!
For the particle in the box, v(x) vanishes inside the box, yielding 1, find the density for the one-dimensional harmonic oscillator, and evaluate the total
energy on that density. Compare with your results for the HO problem there.
N
n(x) = n = (6.24)
L
This is quite different from the exact density for one electron, but the true density approaches
it as N gets large. Using exactly this density, the energy contains only one term, the
3
2
2
asymptotic one, E = π N /(6L ).
Exercise 27 Energy from local approximation on exact density in box
A ‘fun’ math problem is to show that, using the local approximation evaluated on the exact
density for a particle in a box, the energy can be found analytically to be
E π 2 < 2 9 3 =
= N + N + (6.25)
N 6 8 8
You can check how it compares to your answers to Exercise X.
In particular, for one electron, the self-consistent solution is too small by a factor of 3!
Compare this with the 25% error using the exact density. This is because, although the local
approximation is good for the energy, its derivative is not so good, and the density is quite
sensitive to this.
Note that our reasoning applies to all bounded 1d problems, not just particles in boxes.

Exercise 28 Large N for harmonic oscilator
Calculate the self-consistent density of the local approximation for the harmonic oscillator,
and compare with your exact densities. Also, compare the energies.

6.5 Questions

These are all conceptual.