Guess the shape and phase of the MOs of butadiene using LCAO and Hückel's method?

Question

I used Hückel's method along with a Linear Combination of Atomic Orbitals (LCAO) to calculate an estimate for the orbital energies of cyclobutadiene ($\ce{C4H4}$) and butadiene ($\ce{C4H6}$). For butadiene, I ended up with the result

$E = \alpha \pm 1.62 \beta$ and $E = \alpha \pm 0.62 \beta$,

where $\alpha$ and $\beta$ are both negative (and are the Coulomb integral and the exchange integral, respectively). I'm fairly confident with the method and interpretation of these integrals so far, but I don't know how to use these results to "guess and draw" the shapes and phases of the molecular orbitals, and I'm also not sure how to tie in what $\alpha$ and $\beta$ represent.

As I understand things, in these molecules, hybridisation of the orbitals occur and $\mathrm{sp^2}$ orbitals form, taking up three electrons from each carbon atom and leaving one electron left over in the third p orbital. This p orbital is the one I assume is portrayed in the below images.

Could someone please give me a hint towards the following:

1. relating the molecular orbital shapes/phases with the Coulomb integral (overlap of one carbon's electron wavefunction with a neighbouring carbon's potential) and the exchange integral (the overlap of two electrons' wavefunctions in the field of one of the carbons
2. how to get any sort of information about the phases of the different orbitals. I understand that $E = \alpha + 2\beta$ will have the lowest energy, as both $\alpha$ and $\beta$ are negative, but how does that correspond to the phase configurations shown below? Why do the size of the circles in the first image vary?

Note: I've been sent here from the Physics Stack Exchange, please be nice. :) Thanks for any help! This is my first post so please also let me know if the posting etiquette passes muster.

Sources: First image, second image

Answer 1

The shape of the molecular orbitals is not guessed, it is determined by the LCAO-MO expansion coefficients which one needs to find out in the first place. The hint is that an eigenvalue equation is usually solved in two steps:

1. We start by writing down the eigenvalue equation in the matrix form and proceed then by writing down and solving the characteristic equation in which the secular determinant is equated with zero. By solving the characteristic equation we find the eigenvalues (energies).
2. Then we put eigenvalues (energies) back into the matrix eigenvalue equation and solve it for the coefficients (LCAO-MO expansion coefficients) that determine eigenvectors (molecular orbitals).

As far as I understand, the second step was not simply done yet. Note that any scalar multiple of an eigenvector is also an eigenvector with the same eigenvalue, so that eigenvectors are not unique. In solving the matrix eigenvalue equation one will find not the actual values of the LCAO-MO expansion coefficients, but rather some relationship between them which could be satisfied by a number of different coefficients. In principle one is free to choose any set of coefficients satisfying the relationship, and usually there is some obvious choice.

For the simplest case in which just two coefficient are related with each other through some ratio the obvious choice is to take some coefficient to be one, then the value of the second coefficient is nothing but the value of the ratio. One could also normalize the wave function then if desired.

Also note that one can use symmetry to simplify the math. OP did not mention were the symmetry adapted linear combinations (SALCs) of atomic orbitals used at the first step, so I mention just in case.

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P.S. Fully worked examples could be found in different places, for instance, here (PDF, 150 KB).