Page 16 - 71 the abc of dft_opt

2.4. QUESTIONS 31 32 CHAPTER 2. FUNCTIONALS
of Lagrange multipliers. Following the example in the Appendix (A.1), construct a new Extra exercises
functional
B[r] = A[r] − µP[r], (2.18) 1. Consider a square of side 1. Check that the circle of the same perimeter has a larger
area.
where µ is at this point an unknown constant. Then extremize the new functional B[r], by
setting its functional derivative to zero: 2. For the unit square centered on the origin, check that Eqs. (2.2) and (2.4). are correct.
#2
3
δB δA δP r + r (2r − r ) Hint: You need only do the first octant, x > y > 0.
##
= − µ = r(θ) − µ = 0, (2.19)
#2 3/2
2
δr δr δr (r + r )
A simple solution to this equation is r(θ) = µ, a constant. This tells us that the optimum
shape is a circle, and the radius of that circle can be found by inserting the solution in the
constraint, P[r = µ] = 2πµ = l. The largest area enclosable by a piece of string of length l
2
is A[r = µ] = l /(4π).
Exercise 7 Change in a functional
Suppose you know that a functional E[n] has functional derivative v(r) = −1/r for the
density n(r) = Z exp(−2Zr), where Z = 1. Estimate the change in E when Z becomes
1.1.
Exercise 8 Second functional derivative
Find the second functional derivative of (a) a local functional and (b) T VW .
S
2.4 Questions

1. Compare the functional derivative of T VW [n] with T loc [n] for some sample one-electron
S S
problem. Comment.
2. If someone just tells you a number for any density you give them, e.g., the someone might
be Mother Nature, and the number might be the total energy measured by experiment,
devise a method for deducing if Mother Nature’s functional is local or not.
3. Is there a simple relationship between T S and ! dx n(x)δT S /δn(x)? First consider the
local approximation, then the Von Weisacker. Comment on your result.
4. For fixed particle number, is there any indeterminancy in the functional derivative of a
density functional?