Hi I need help deriving a result from the PMO theory. I'm currently reading the second edition of the book Orbital Interactions in Chemistry by Thomas A. Albright, Jeremy K. Burdett & Myung-Hwan Whangbo and, in order to understand the chapter about PMOs, I read and replicated the reasoning presented in the [Appendix I][1] to succesful results for all but one of the derivations. Bellow are the relevant equations and how I followed the operations presented in the book.
Starting from the pseudo-eigenvalue equation
\begin{equation}
(\textbf{h}-e_{i}\textbf{s})\textbf{T}_{i}=0 \tag{1}
\end{equation}
One can find, once the perturbation expansions are applied to the pseudo-eigenvalue, that the second-order coefficient has the form
\begin{equation}
-(e^{(1)}_{i}\tilde{\textbf{S}}+e^{(2)}_{i}\textbf{I})\textbf{T}^{(0)}_{i}+(\tilde{\textbf{H}}-e_{i}^{(0)}\tilde{\textbf{S}}-e_{i}^{(1)})\textbf{T}^{(1)}_{i}+(\textbf{e}^{\mathsf{o}}-e_{i}^{(0)}\textbf{I})\textbf{T}^{(2)}_{i}=0\tag{2}\end{equation}
Where the elements of the matrix of the zero-order solution $\textbf{T}^{(0)}$ are
\begin{equation}
T^{(0)}_{ij}= (\textbf{T}^{(0)}_j)^{\mathsf{T}}\textbf{I}\,\textbf{T}^{(0)}_i=\delta_{ij}
\tag{3}\end{equation}
Therefore, the column vectors $\textbf{T}^{(0)}_i$ are useful to extract the elements of the overlap, Hamiltonian operator and MOs energies matrices as follows
\begin{equation}
(\textbf{T}_j^{(0)})^{\mathsf{T}}\tilde{\textbf{S}}\,\textbf{T}_i^{(0)}=\tilde{S}_{ji}
\tag{4}\end{equation}
\begin{equation}
(\textbf{T}_j^{(0)})^{\mathsf{T}}\tilde{\textbf{H}}\,\textbf{T}_i^{(0)}=\tilde{H}_{ji}
\tag{5}\end{equation}
\begin{equation}
(\textbf{T}_j^{(0)})^{\mathsf{T}}\textbf{e}^{\mathsf{o}}\,\textbf{T}_i^{(0)}=e_{j}^{\mathsf{o}}\delta_{ji}
\tag{6}\end{equation}
Now, in order to solve the Eq.2, one takes advantage of the fact that any of the non-zero-order mixing coefficient vectors $\textbf{T}^{(n)}_{i}$ can be expanded as a linear combination of the zero-order vectors $\textbf{T}^{(0)}_k$
\begin{equation}
\textbf{T}^{(n)}_{i}=\sum_{k} a^{(n)}_{ki}\textbf{T}^{(0)}_k\tag{7}
\end{equation}
From previous derivations we know that $e^{\mathsf{o}}_{i}=e^{(0)}_{i}$, and that the value of the coefficient $a_{ji}^{(1)}$ is equal to
\begin{equation}
a^{(1)}_{ji}=\frac{\tilde{H}_{ij}-e^{\mathsf{o}}_{i}\tilde{S}_{ij}}{e^{\mathsf{o}}_i-e^{\mathsf{o}}_j}\hspace{20pt}i\neq j\tag{8}
\end{equation}
And thus Eq.2 can be written as
\begin{equation}
-(e^{(1)}_{i}\tilde{\textbf{S}}+e^{(2)}_{i}\textbf{I})\textbf{T}^{(0)}_{i}+\sum_{k} a^{(1)}_{ki}(\tilde{\textbf{H}}-e_{i}^{\mathsf{o}}\tilde{\textbf{S}}-e_{i}^{(1)})\textbf{T}^{(0)}_k+\sum_{k} a^{(2)}_{ki}(\textbf{e}^{\mathsf{o}}-e_{i}^{\mathsf{o}}\textbf{I})\textbf{T}^{(0)}_k=0\tag{9}\end{equation}
The objective now is to find $a^{(2)}_{ki}$. Left multiplying Eq.9 by $(\textbf{T}^{0}_{j})^{\mathsf{T}}$ where $j\neq i$, we obtain
\begin{equation}
\begin{split}
-(\textbf{T}^{(0)}_j)^{\mathsf{T}}(e^{(1)}_{i}\tilde{\textbf{S}}+e^{(2)}_{i}\textbf{I})\textbf{T}^{(0)}_{i}+\sum_{k} a^{(1)}_{ki}(\textbf{T}^{(0)}_j)^{\mathsf{T}}(\tilde{\textbf{H}}-e_{i}^{\mathsf{o}}\tilde{\textbf{S}}-e_{i}^{(1)})\textbf{T}^{(0)}_k
\\+\sum_{k} a^{(2)}_{ji}(\textbf{T}^{(0)}_j)^{\mathsf{T}}(\textbf{e}^{\mathsf{o}}-e_{i}^{\mathsf{o}}\textbf{I})\textbf{T}^{(0)}_k=0\end{split}\tag{10}\end{equation}
\begin{equation}
-(e^{(1)}_{i}\tilde{S}_{ji}+e^{(2)}_{i}\delta _{ji})+\sum_{k} a^{(1)}_{ki}(\tilde{H}_{jk}-e_{i}^{(0)}\tilde{S}_{jk}-e_{i}^{(1)}\delta_{jk})+\sum_{k} a^{(2)}_{ki}(e^{\mathsf{o}}_{j}-e_{i}^{\mathsf{o}})\delta_{jk}=0\tag{11}\end{equation}
Here I separate the second sum of Eq.11 to show later the difference between what I found and what the book presents
\begin{equation}\begin{split}
-(e^{(1)}_{i}\tilde{S}_{ji}+e^{(2)}_{i}\delta _{ji})-\sum_{k} a^{(1)}_{ki}e_{i}^{(1)}\delta_{jk}+\sum_{k}a^{(1)}_{ki}(\tilde{H}_{jk}-e_{i}^{(0)}\tilde{S}_{jk})
\\+\sum_{k} a^{(2)}_{ki}(e^{\mathsf{o}}_{j}-e_{i}^{\mathsf{o}})\delta_{jk}=0\end{split}\tag{12}\end{equation}
In the first term of Eq.16, the value $e_i^{(2)}\delta_{ji}$ vanishes since we stablished that $j\neq i$. And from the second sum, the the only surviving term is $a_{ji}^{(2)}(e^{\mathsf{o}}_{j}-e_{i}^{\mathsf{o}})$ and using another previous result from the book $e_{i}^{(1)}=\tilde{H}_{ii}-e_i^{\mathsf{o}}\tilde{S}_{ii}$, we obtain
\begin{equation}\begin{split}
a^{(2)}_{ji}(e^{\mathsf{o}}_{i}-e_{j}^{\mathsf{o}})=
-(\tilde{H}_{ii}-e_i^{\mathsf{o}}\tilde{S}_{ii})\tilde{S}_{ji}
-\sum_{k}a_{ki}^{(1)}(\tilde{H}_{ii}-e_i^{\mathsf{o}}\tilde{S}_{ii})\delta_{jk}\\+\sum_{k}a^{(1)}_{ki}(\tilde{H}_{jk}-e_{i}^{\mathsf{o}}\tilde{S}_{jk})\end{split}\tag{13}
\end{equation}
\begin{equation}\begin{split}
a_{ji}^{(2)}=-\left(\frac{\tilde{H}_{ii}-e^{\mathsf{o}}_{i}\tilde{S}_{ii}}{e^{\mathsf{o}}_{i}-e_{j}^{\mathsf{o}}}\right)\tilde{S}_{ji}
+\sum_{k\neq i}\frac{(\tilde{H}_{ik}-e^{\mathsf{o}}_{i}\tilde{S}_{ik})(\tilde{H}_{jk}-e_{i}^{\mathsf{o}}\tilde{S}_{jk})}{(e^{\mathsf{o}}_i-e^{\mathsf{o}}_k)(e^{\mathsf{o}}_{i}-e_{j}^{\mathsf{o}})}
\\+\sum_{k\neq i}\frac{(\tilde{H}_{ik}-e^{\mathsf{o}}_{i}\tilde{S}_{ik})(\tilde{H}_{ii}-e_{i}^{\mathsf{o}}\tilde{S}_{ii})}{(e^{\mathsf{o}}_i-e^{\mathsf{o}}_k)(e^{\mathsf{o}}_{i}-e_{j}^{\mathsf{o}})}\delta_{jk},\hspace{20pt}j\neq i
\end{split}\tag{14}\end{equation}
It is here, in Eq.14, that the book's result and mine differ. In the book, the correct expression is (they use $b^{(2)}_{ki}$ instead of $a^{(2)}_{ki}$)
\begin{equation}
b_{ji}^{(2)}=-\left(\frac{\tilde{H}_{ii}-e^{\mathsf{o}}_{i}\tilde{S}_{ii}}{e^{\mathsf{o}}_{i}-e_{j}^{\mathsf{o}}}\right)\tilde{S}_{ji}
+\sum_{k\neq i}\frac{(\tilde{H}_{ik}-e^{\mathsf{o}}_{i}\tilde{S}_{ik})(\tilde{H}_{jk}-e_{i}^{\mathsf{o}}\tilde{S}_{jk})}{(e^{\mathsf{o}}_i-e^{\mathsf{o}}_k)(e^{\mathsf{o}}_{i}-e_{j}^{\mathsf{o}})},\hspace{20pt}j\neq i
\tag{15}
\end{equation}
The difference between the results being the sum
\begin{equation}
\sum_{k\neq i}\frac{(\tilde{H}_{ik}-e^{\mathsf{o}}_{i}\tilde{S}_{ik})(\tilde{H}_{ii}-e_{i}^{\mathsf{o}}\tilde{S}_{ii})}{(e^{\mathsf{o}}_i-e^{\mathsf{o}}_k)(e^{\mathsf{o}}_{i}-e_{j}^{\mathsf{o}})}\delta_{jk},\hspace{20pt}j\neq i
\tag{16}\end{equation}
And for it to vanish, for every term of the sum $k\neq j$ or, in a equivalent manner, the only surviving term of the sum, that in which $k=j$, should be zero (Eq.17). Which I do not really understand why it should be either way.
\begin{equation}
\sum_{k\neq i}\frac{(\tilde{H}_{ik}-e^{\mathsf{o}}_{i}\tilde{S}_{ik})(\tilde{H}_{ii}-e_{i}^{\mathsf{o}}\tilde{S}_{ii})}{(e^{\mathsf{o}}_i-e^{\mathsf{o}}_k)(e^{\mathsf{o}}_{i}-e_{j}^{\mathsf{o}})}\delta_{jk}=\frac{(\tilde{H}_{ij}-e^{\mathsf{o}}_{i}\tilde{S}_{ij})(\tilde{H}_{ii}-e_{i}^{\mathsf{o}}\tilde{S}_{ii})}{(e^{\mathsf{o}}_{i}-e_{j}^{\mathsf{o}})^2}=0,\hspace{20pt}j\neq i
\tag{17}\end{equation}
On the other hand, from the sources that the book employs, there is a [Seo-Papoian-Hoffmann paper][3] that carries out a similar derivation. Their second-order coefficient of the perturbation expansion (mantaining the notation consistent with the book, the one in the paper is different) is
\begin{equation}
\sum_{k}\left[\left(e_j^{\mathsf{o}}-e_i^{\mathsf{o}}\right)\delta_{jk}t^{(2)}_{ki}+\left(\tilde{H}_{jk}-e_i^{\mathsf{o}}\tilde{S}_{jk}-e_i^{(1)}\right)t_{ki}^{(1)}-\left(e_i^{(2)}\delta_{jk}+e_{i}^{(1)}\tilde{S}_{jk}\right)t_{ki}^{(0)}\right]=0
\tag{18}\end{equation}
In accord with the book, $t_{ji}^{(0)}=\delta_{ji}$ and solving Eq.22 for $t^{(2)}_{ji}$
\begin{equation}\begin{split}
\sum_{k}\left(e_j^{\mathsf{o}}-e_i^{\mathsf{o}}\right)\delta_{jk}t^{(2)}_{ki}+
\sum_{k}\left(\tilde{H}_{jk}-e_i^{\mathsf{o}}\tilde{S}_{jk}-e_i^{(1)}\delta_{jk}\right)t_{ki}^{(1)}\\-
\sum_{k}\left(e_i^{(2)}\delta_{jk}+e_{i}^{(1)}\tilde{S}_{jk}\right)\delta_{ki}=0
\end{split}\tag{19}\end{equation}
\begin{equation}\begin{split}
\left(e_i^{\mathsf{o}}-e_j^{\mathsf{o}}\right)t^{(2)}_{ji}=
-\sum_{k}e_i^{(1)}\delta_{jk}t_{ki}^{(1)}-
\sum_{k}e_i^{(2)}\delta_{jk}\delta_{ki}-\sum_{k}e_{i}^{(1)}\tilde{S}_{jk}\delta_{ki}\\+
\sum_{k}\left(\tilde{H}_{jk}-e_i^{\mathsf{o}}\tilde{S}_{jk}\right)t_{ki}^{(1)}
\end{split}\tag{20}\end{equation}
\begin{equation}
t^{(2)}_{ji}=
-e_i^{(1)}\left(\frac{t_{ji}^{(1)}+\tilde{S}_{ji}}{e_i^{\mathsf{o}}-e_j^{\mathsf{o}}}\right)+
\sum_{k}\left(\frac{\tilde{H}_{jk}-e_i^{\mathsf{o}}\tilde{S}_{jk}}{e_i^{\mathsf{o}}-e_j^{\mathsf{o}}}\right)t_{ki}^{(1)},\hspace{20pt}j\neq i
\tag{21}\end{equation}
But now, I lack a term because the expression in the paper is actually
\begin{equation}
t^{(2)}_{ji}=-\frac{1}{2}\tilde{S}_{ii}t_{ji}^{(1)}
-e_i^{(1)}\left(\frac{t_{ji}^{(1)}+\tilde{S}_{ji}}{e_i^{\mathsf{o}}-e_j^{\mathsf{o}}}\right)+
\sum_{k}\left(\frac{\tilde{H}_{jk}-e_i^{\mathsf{o}}\tilde{S}_{jk}}{e_i^{\mathsf{o}}-e_j^{\mathsf{o}}}\right)t_{ki}^{(1)},\hspace{20pt}j\neq i
\tag{22}\end{equation}
I do not know what I am doing wrong, if there is a fundamental part of the theory I don't understand or if my math is just plain wrong. I would love the assistance of any one who can help me, I could not thank you enough!
1.Albright, T.A., Burdett, J.K. and Whangbo, M.-H. (2013). Appendix I: Perturbational Molecular Orbital Theory. In Orbital Interactions in Chemistry (eds T.A. Albright, J.K. Burdett and M.-H. Whangbo). https://doi.org/10.1002/9781118558409.app1
2.Seo, D.-K., Papoian, G. and Hoffmann, R. (2000), Generalized perturbational molecular orbital (PMO) theory†. Int. J. Quantum Chem., 77: 408-420. https://doi.org/10.1002/(SICI)1097-461X(2000)77:1<408::AID-QUA41>3.0.CO;2-1
[1]: https://onlinelibrary.wiley.com/doi/epdf/10.1002/9781118558409.app1
[2]: https://drive.google.com/file/d/1zAhAj7EZo5WWLfBqSX-s-U9UXkj5Ntry/view?usp=sharing
[3]: https://roaldhoffmann.com/sites/default/files/fromd6/453s_0.pdf