Should Imaginary Energy Levels Be Counted in Huckel Approximation Calculations?

[physgirl]

Huckel approximation--pi-energy

Homework Statement

So in this question, I already found the solution to the matrix representing pi eletrons in naphthalene and solved for the energy levels, using a math program. I did get 10 values of x (where x is (alpha-E)/beta), which was what I was expecting, since naph has 10 carbons and 10x10 matrix was solved... but some of the values for x was a complex number (like x=1.6+0.17i, for instance)... so when I'm filling in the energy levels, would I count those imaginary values as energy levels?

I didn't initially, because it seemed weird to put electrons into imaginary energy levels but then when I was trying to calculate the energy value for delocalized pi electrons, I ended up with a positive value of energy, which didn't make sense to me since naphthalene is a conjugated molecule and should have a negative stabilizing energy... but then when I did fill in the imaginary energy levels, I -did- get a negative value for delocalization energy...

Can anyone explain to me why this is?
And let me know if that question just now didn't make sense...

Homework Equations

The Attempt at a Solution

 

[Gokul43201]

Homework Statement

So in this question, I already found the solution to the matrix representing pi eletrons in naphthalene and solved for the energy levels, using a math program. I did get 10 values of x (where x is (alpha-E)/beta), which was what I was expecting, since naph has 10 carbons and 10x10 matrix was solved... but some of the values for x was a complex number (like x=1.6+0.17i, for instance)... so when I'm filling in the energy levels, would I count those imaginary values as energy levels?

I think you're making a mistake in the calculation. Your matrix needs to be Hermitian, and hence will have only real eigenvalues. If you're getting complex eigenvalues, either there's an error in their calculation, or you've made a mistake in calculating the matrix elements and ended up with a non-Hermitian matrix.

That's only possible if :
1. You don't have $\beta_{ij}=\beta ^* _{ji}$, for all betas (which should all be identical within each triangle, since you'd be using the same p_z wavefunction everywhere)
OR
2. One of your alphas isn't real (again, there should be only one value of alpha).

If you don't have either of the above 2 mistakes, try solving the matrix equation using a different program (like Mathematica).

 

[physgirl]

I used a website that has a program that solves huckel determinant, got the function they give me and typed that into mathematica to solve (ive used mathematica for the first time and got very confused as to how to solve matrices there)... I think I set up the matrix in the website basically so that... there's a diagonal line of x's (x is what I defined before) from left upper corner to right lower corner, and there are 1's on position of any adjacent carbons... so if carbon 1 is adjacent to carbon 2 and carbon 9, I entered "1" in matrix position (1,2), (2,1), (1,9), and (9,1)... is there a mistake somewhere? (the website just gives the real values btw, which gives me just 6 energy levels, 2 of which are degenerate (x=0), and then realized that when I use mathematica, i get 4 complex values in addition)

EDIT addition: Actually, I tried different things on that website, and depending on how I number my carbons in the molecule, the resulting x values change... how do i know which to use?! the numbering system that gives 10 real x values? (which i haven't found yet...)

 

[Gokul43201]

You should not get complex eigenvalues if you entered the matrix elements correctly. Make sure that your matrix is indeed Hermitian, i.e., for every $A_{ij}=1$, you've got $A_{ji}=1$.

For cyclic compounds with a single n-carbon ring, the solution to the characteristic equation has a simple closed form:

$$E_m=\alpha + 2\beta cos(2\pi m/n)$$

I don't know of any such solutions for more than one ring.

 

[physgirl]

How do you know that the E_m equation you just gave only applies to 1-ring molecules?

 

[Gokul43201]

The secular determinant for a 1-ring molecule is unique since every C-atom has exactly 2 nearest neighbors. This isn't the case in any other n-ring molecule.

For example, in naphthalene, there are 2 C-atoms that have 3 nearest neighbors (while all the other C-atoms have only 2).