Page 103 - 71 the abc of dft_opt

205 206 APPENDIX E. DISCUSSION OF QUESTIONS
Chapter 2 Chapter 3

1. Compare the functional derivative of T VW [n] with T loc [n] for some sample one- 1. Suggest a good trial wavefunction for a potential that consists of a negative delta
S S
electron problem. Comment. function in the middle of a box of width L.
%
You might take either the particle in the box or the harmonic oscillator or the one- The simplest guess is just that of the particle in the box, 2/L cos(πx/L). Its energy is
2
2
dimensional hydrogen atom. You should find that a plot of the two functional derivatives obviously π /2L − 2Z/L, where Z is the strength of the delta function. Much better
gives you two very different curves in each case. To understand this, think of functions. (in fact, exact) is to take linear combinations of the delta-function solutions, exp(−α|x|)
Imagine one function being a horizontal line, and a second function having weak but and exp(α|x|) vanishing at the edges.
rapid oscillations around that line. The second function is a good approximation to first
2. What is the effect of having nuclear charge Z *= 1 for the 1-d H-atom?
everywhere, but its derivative is very different.
It alters the length scale of the wavefunction without changing its normalization, i.e.,
2. Devise a method for deducing if a functional is local or not. √
Z exp(−Z|x|). We see shortly how this can be a handy trick.
Take the second functional derivative. If its proportional to a delta function, the function
is local. Note that just saying if the integrand depends only on the argument at r is not 3. What is the exact kinetic energy density functional for one electron in one-dimension?
enough, since integrands can change within functionals, e.g., by integrating by parts. It is the von Weisacker functional,
1 1 ∞ 1 1 ∞
2
2
!
3. Is there a simple relationship between T S and dx n(x)δT S /δn(x)? T VW = dx |φ (x)| = dx |n (x)| /n(x). (E.2)
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#
S
2 −∞ 8 −∞
You should have found that the integral is three times the functional for the local ap-
proximation, but equal to the functional for von Weisacker. 4. Is the HF estimate of the ionization potential for 1-d He an overestimate or under-
estimate?
4. For fixed particle number, is there any indeterminancy in the functional derivative
The ionization potential is I = E 1 − E 2 for a two-electron system, where E N is the
of a density functional? energy with N electrons. Since E HF ≥ E, with equality for one electron, I HF < I.
3
!
If the particle number does not change, then d r δn(r) = 0. Thus addition of any con-
stant to a functional derivative does not alter the result, so that the functional derivative 5. Consider the approximate HF calculation given in section 4.2. Comment on what it
is not determined up to a constant. does right, and what it does wrong. Suggest a simple improvement.
The HF calculation correctly writes the wavefunction as a spatially symmetric singlet, but
incorrectly approximates that as a product of separate functions of x 1 and x 2 (orbitals).
It correctly has satisfies the cusp condition at the nucleus, and changes decay at large
distance, unlike the scaled orbital solution. But its energy is not low enough.
6. Which is bigger, the kinetic energy of the true wavefunction or that of the HF wave-
function for 1d He? Hooke’s atom?
The virial theorem applies to this problem (see section X), and, since all potentials are
homogeneous of order−1, E = −T = −V/2, where V includes both the external and
the electron-electron interaction. Since E HF > E, then T HF < T. For Hooke’s atom,
the situation is more complicated, since there are two different powers in the potentials.
Chapters 4-6
1. The ground-state wavefunction is that normalized, antisymmetric wavefunction that has
density n(r) and minimizes the kinetic plus Coulomb repulsion operators.