Possible erratum in solutions to Orbital Interactions in Chemistry 2ed
I ran the calculations through the MATLAB command window and obtained the same answers as you:

&gt;&gt; H = [-13.60 -14.18 0; -14.18 -13.60 -14.18; 0 -14.18 -13.60]
&gt;&gt; S = [1 0.596 0; 0.596 1 0.596; 0 0.596 1]
&gt;&gt; energies = eig(H,S)

energies = 

  -18.2615
  -13.6000
   41.0717


On the other hand, if you assume that $S_{12} = 0$ i.e. $S = I_3$ (3x3 identity matrix), then you get the same result as the authors:

&gt;&gt; energies = eig(H)       % or eig(H, eye(3))

energies = 

  -33.6535
  -13.6000
    6.4535


I find it slightly telling that the authors explicitly comment that the bonding orbital is stabilised by exactly as much as the antibonding orbital is destabilised. If you solve the secular equations without explicitly substituting in numbers, you should find that this can only be true if the overlap matrix is set to the identity matrix.

The same is true of part (e); you only get the authors' answer by setting $S = I_3$.

I would consider this an erratum in the provided answers.



To find the eigenvalues, use instead [energies, MOs] = eig(H,S). You can try this block of code online. Note that this website has Octave instead of MATLAB, so energies and MOs need to be reversed:

H = [-13.60 -14.18 0; -14.18 -13.60 -14.18; 0 -14.18 -13.60];
S = [1 0.596 0; 0.596 1 0.596; 0 0.596 1];
[MOs, energies] = eig(H,S);
disp(energies);
disp(MOs);


energies is a 3x3 matrix with the eigenvalues as the diagonal elements. MOs is a 3x3 matrix, where each column vector is the MO corresponding to the relevant eigenvalue (i.e. the left-most column is the MO corresponding to the first eigenvalue in energies).