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 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.12, the value $e_i^{(2)}\delta_{ji}$ vanishes since we stablished that $j\neq i$. And from the second sum, 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 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