Page 19 - 71 the abc of dft_opt

3.6. QUESTIONS 37 38 CHAPTER 3. ONE ELECTRON
p
In the special case in which v(x) = Ax , % x dV & = p% V &, so that
dx
2T = pV (3.16)
This can save a lot of time when doing calculations, because one only needs either the kinetic
or the potential energy to deduce the total energy. Note that, for these purposes, the δ-fn
acts as if it had power p = −1.
The theorem is also true for three-dimension problems, and also true for interacting prob-
lems, once the potential is interpreted as all the potential energy in the problem. Thus, for
any atom, p = −1, and E = −T = −V/2, where V is the sum of the external potential and
the electron-electron repulsion.
Exercise 14 Virial theorem for one electron
Check Eq. (3.16) for the ground state of (a) the 1d H atom, (b) the 1d harmonic oscillator,
(c) the three-dimensional Hydrogen atom. Also check it for Ex. (9) and (10).
Some approximations automatically satisfy the virial theorem, others do not. In cases that
do not, we can use the virial theorem to judge how accurate the solution is. In cases that do,
we can use the virial theorem to check we dont have an error, or to avoid some work.

3.6 Questions

All the questions below are conceptual.
1. Suggest a good trial wavefunction for a potential that consists of a negative delta function
in the middle of a box of width L.

2. Sketch, on the same figure, the 1d H-atom for Z = 1/2, 1, and 2. What happens as
Z → ∞ and as Z → 0?
3. What is the exact kinetic energy density functional for one electron in one-dimension?
4. How would you check to see if your trial wavefunction is the exact ground-state wave-
function for your problem?
p
5. Consider what happens to the potential v(x) = Ax as p gets large. Can the virial
theorem be applied to the particle in the box (even if it yields trivial answers)?