Page 12 - 71 the abc of dft_opt

1.6. QUESTIONS 23 24 CHAPTER 1. INTRODUCTION
that the exact energy is 8. The simplest density functional approximation is a local one. What form might a cor-
π 2 - 2 π 2 rection to T S loc take? (see section 17.2 for the answer).
N
E = j = N(N + 1)(2N + 1)/6 (1.14)
2
2L j=1 2L 2 9. In the chapter and exercises, T loc does very well. But there are cases where it fails quite
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3
2
whose leading contribution is simply π N /6. Since the density has become uniform (to this badly. Can you find one, and say why?
order), the local approximation becomes exact, yielding the constant. 10. Consider a single electron in an excited state, with density n (x). Will T loc [n ] provide
∗
∗
S
Later in the book, we will see how, if we had an extremely accurate non-interacting kinetic an accurate estimate of its kinetic energy?
energy density functional in three dimensions, we could revolutionize electronic structure
calculations, by making them very fast. This is because we would then have a method for
calculating the density and ground-state energy of systems which involved solving a single
self-consistent integro-differential equation for any electronic system. This would be a true
density functional calculation. Most present calculations do not do this. By solving an
effective single-particle equation for the orbitals (the Kohn-Sham equation), they find T S
exactly, but at large computational cost (for large systems). So now you even know an
unsolved problem in density functional theory.
Now answer the following simple questions as well as you can, justifying your answers in
each case.
1.6 Questions
We begin with conceptual questions, that are designed to test your understanding of the
material presented in the chapter.
1. Why is a Kohn-Sham calculation much faster than a traditional wavefunction calculation?
2. If you evaluate the kinetic energy of a Kohn-Sham system, is it equal to the physical
kinetic energy?
3. Repeat above question for %1/r&, which can be measured in scattering experiments.
4. Why is the density cubed in the local approximation for T S ? (see section ?? for the
answer).
We continue with more creative questions, designed to make you think about what is in
the chapter.
3
5. Suppose you’d been told that T loc is proportional to ! dxn (x), but were not told the
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constant of proportionality. If you can do any calculation you like, what procedure might
you use to determine the constant in T loc ? (see section 11.5 for the answer).
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6. In what way will T loc change if spin is included, e.g., for two electrons of opposite spin
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in a box? (see section 9.2 for the answer).
7. If we add an infinitesimal to n(x) at a point, i.e., &δ(x) as & → 0, how does T loc change?
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(see section 2.2 for the answer).