Homework 6: one-electron states of molecules and crystals.

One can write molecular wave-functions as linear combination of atomic wave-functions with different center positions. (In my experience, this is the most common way in which to construct them. See, for example, LCAO.)

For example, suppose your are given a generic attractive potential (in 1D) of $U(x)$ for which the electron ground state wave-function is $\psi_o (x)$, and the energy of an electron in this state is $E_o$. You are given all that:
i) an atom-like potential, $U(x)$ (attractive, negative, single well, symmetric, with its minimum at x=0 ),
ii) an atom-like ground state wave-function, $\psi_o (x)$ (a simple,  normalized, exponential function such as $e^{-|x|/a}/a^{1/2}$  or,  $e^{-x^2/(2a^2}/(a^{1/2} \pi^{1/4})$),
iii) the ground state energy of this atom-like state in this potential, $E_o$.
The idea is -- with these things as given, solved and known: how would you use them to construct one-electron wave-functions for molecules and crystals?

1. Suppose that and electron experiences the double potential,
$U_m (x) = U(x) + U(x-b)$, where U(x) has a simple minimum at x=0 and U(x-b) has an identical minimum at (x-b)=0.
a) Sketch $U_m (x)$
b) Write an expression for a wave-function that might be a reasonable guess for the ground state of this potential. Do this by combining atom-like wave-functions with different center positions.
c) Sketch your wave-function.
d) How would you normalize it, more or less? (You can assume that the atomic wave-functions are initially normalized.) Is it easy or difficult to normalize the molecular wave-function?
(please comment on this below. Is it hard or easy? why?)

2. For the same potential,
a) How might you make a molecular wave-function that would be a reasonable guess for the first excited state of this potential?
b) Sketch your wave-function.
c) how many nodes does this state have?

3. Now consider 4 identical potentials in a row.
For example,$U_m (x) = U(x) + U(x-b) + U(x-2b) + U(x-3b)$.
a) sketch $U_m (x)$.
b) Make a molecular wave-function that might be a reasonable guess for the ground state of this potential by combining atomic wave-functions with different center positions?
c) Sketch your wave-function.
d) Now many excited states can you make (using just the atomic ground state $\psi_o (x)$)?
e) Make them all (write an expression for each one) and sketch each one in a separate graph next to its wave-function. (hint: You are given the atomic wave-functions. What you need to do is write a plus or minus sign in from of them in an informed manner that, for example, makes the molecular states you are creating orthogonal to each other. You know you are done when you cannot make any more that are orthogonal to the ones you already have.)
f) Discuss their energy order. (Which is lowest energy? next lowest?, highest?, why?)
g) How many nodes does each one have?

4. Let's now consider many (N) such identical potentials, all in a row.
a) What does the ground state of this many-well system look like? How would you construct it?
b) Discuss here in this blog the excited states you can create. What are they like? How would you describe them?
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Problems 1-4 are very important. I would focus a lot of attention on them, especially 4. Problem 4 is too difficult and nuanced for any one person to do alone. It has many layers that can be gradually peeled back and explored. I think the best way to approach it is to post your thoughts and questions about it here, and then you can make progress on it by working together as a class. You will get beaucoup credit for your contributions here.
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5. I would like to add one more problem which is pretty interesting, but not as deep and nuanced as 4 (so i do not want to take too much attention away from 4 by adding this last problem).
a) Do a contour plot illustrating the nature of the state:
$\psi(x,y,z) = (e^{-(r/a)} + e^{-r_b/a})/\sqrt{2 \pi a^3})$,  (typo corrected Feb 15)
where
$r_b = \sqrt{(x-b)^2 + y^2 + z^2}$
(what plane did you choose to do your contour plot in? why?)