I have another SALC question. @Andrew and @user5713492 did a phenomenal job clarifying my previous confusion with Methane T2 SALC production, so I attempted to apply this to a new point group, and have hit another wall.
Utilizing the full projection operator, I can perform the following analysis to get one of the e1 SALC orbitals in C5v (assuming a purely sigma framework through 1s orbitals as basis on the ligands):
The e1 matrix representation can be generated from {x,y}, resulting in:
$$\begin{array}{cccccc}E&\sigma_{v}&C_{5}&C_{5}'\\ \begin{bmatrix}1&0\\0&1\\\end{bmatrix}& \begin{bmatrix}-1&0\\0&1\\\end{bmatrix}& \begin{bmatrix}Cos(72)&-Sin(72)\\Sin(72)&Cos(72)\\\end{bmatrix}& \begin{bmatrix}Cos(-72)&-Sin(-72)\\Sin(-72)&Cos(-72)\\\end{bmatrix} \end{array}$$
$$ \begin{array}{cccccc}C_{5}^2&C_{5}'^2\\ \begin{bmatrix}Cos(144)&-Sin(144)\\Sin(144)&Cos(144)\\\end{bmatrix}& \begin{bmatrix}Cos(-144)&-Sin(-144)\\Sin(-144)&Cos(-144)\\\end{bmatrix} \end{array} $$
$$ \begin{array}{cccccc}\sigma_{v}'&\sigma_{v}''\\ \begin{bmatrix}-Cos(-144)&Sin(-144)\\Sin(-144)&Cos(-144)\\\end{bmatrix}& \begin{bmatrix}-Cos(72)&Sin(72)\\Sin(72)&Cos(72)\\\end{bmatrix} \end{array} $$
$$ \begin{array}{cccccc}\sigma_{v}'''&\sigma_{v}''''\\ \begin{bmatrix}-Cos(-72)&Sin(-72)\\Sin(-72)&Cos(-72)\\\end{bmatrix}&\begin{bmatrix}-Cos(144)&Sin(144)\\Sin(144)&Cos(144)\\\end{bmatrix} \end{array} $$
Using the labeling scheme below (ignoring the $\psi$1 1s orbital at the top of the pentagonal pyramid) I can also generate the transformation series of each of these basis functions. Let's do $\psi$2 first:
$$ \begin{array}{cccccc}E&\sigma_{v}&C_{5}&C_{5}'&C_{5}^2&C_{5}'^2&\sigma_{v}'&\sigma_{v}''&\sigma_{v}'''&\sigma_{v}''''\\ \begin{bmatrix}\psi2\\\end{bmatrix}&\begin{bmatrix}\psi2\\\end{bmatrix}&\begin{bmatrix}\psi3\\\end{bmatrix}&\begin{bmatrix}\psi6\\\end{bmatrix}&\begin{bmatrix}\psi4\\\end{bmatrix}&\begin{bmatrix}\psi5\\\end{bmatrix}&\begin{bmatrix}\psi3\\\end{bmatrix}&\begin{bmatrix}\psi4\\\end{bmatrix}&\begin{bmatrix}\psi5\\\end{bmatrix}&\begin{bmatrix}\psi6\\\end{bmatrix} \end{array} $$
Now, when I choose to project using the matrix elements (1,1) and the basis set $\psi$2, I get the following:
$$ SALC_{e1}'\propto C(72)(\psi3-\psi4)+C(-72)(\psi6-\psi5)+C(144)(\psi4-\psi6)C(-144)(\psi5-\psi3) $$
$$
\propto \psi3+\psi6-\psi4-\psi5
$$
Where C($\theta$) and S($\theta$) are cosine and sine of theta respectively.
This SALC is obviously transforming as $y$ in our chosen coordinate frame (+y up the page, -y down the page, +x to the right and -x to left).
My confusion starts when I take any other combination of the $\psi$ projections and multiply by any other (row,column) combination of the matrix elements. Intuition tells me that I should either get the complementary SALC which transforms as x, or get the y-like SALC again. This is not true.
For instance, if I choose $\psi3$ as my starting point, and do (row,column) = (1,2), I get:
$$ \begin{array}{cccccc}E&\sigma_{v}&C_{5}&C_{5}'&C_{5}^2&C_{5}'^2&\sigma_{v}'&\sigma_{v}''&\sigma_{v}'''&\sigma_{v}''''\\ \begin{bmatrix}\psi3\\\end{bmatrix}&\begin{bmatrix}\psi6\\\end{bmatrix}&\begin{bmatrix}\psi4\\\end{bmatrix}&\begin{bmatrix}\psi2\\\end{bmatrix}&\begin{bmatrix}\psi5\\\end{bmatrix}&\begin{bmatrix}\psi6\\\end{bmatrix}&\begin{bmatrix}\psi2\\\end{bmatrix}&\begin{bmatrix}\psi3\\\end{bmatrix}&\begin{bmatrix}\psi4\\\end{bmatrix}&\begin{bmatrix}\psi5\\\end{bmatrix} \end{array} $$
and
$$ SALC_{e1}'\propto S(72)(\psi2+\psi3-2\psi4)+S(144)(\psi6-\psi2) $$
$$ \propto a\psi2+b\psi3-2\psi4+c\psi6 $$
Or
Which does not, at least at first glance, appear to satisfy our requirements of orthogonality, at least at position $\psi$5.
Can anyone indicate what I've done wrong here? Or show where I strayed off the correct path? Because I am quite confused. I would like to stick to the full projection method if at all possible, as I understand that that method should work, meaning I've just messed up somewhere.
Thanks!