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\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}

\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!

\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}

\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\neq i}\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!

UPDATE: I found the one from the paper! My math was incomplete. The sum in Eq. 21 is absorbing the term I lack. The sum runs over all $k$, including $k=i$, but the form of $t^{(1)}_{ki}$ is not the same for the case $k=i$ and $k\neq i$, so I should have expanded the sum as the addition of the diagonal term with rest of the sum

\begin{equation}\begin{split} t^{(2)}_{ji}= -e_i^{(1)}\left(\frac{t_{ji}^{(1)}+\tilde{S}_{ji}}{e_i^{\mathsf{o}}-e_j^{\mathsf{o}}}\right)+ \left(\frac{\tilde{H}_{ji}-e_i^{\mathsf{o}}\tilde{S}_{ji}}{e_i^{\mathsf{o}}-e_j^{\mathsf{o}}}\right)t_{ii}^{(1)}\\+ \sum_{k\neq i}\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 \end{split}\tag{23}\end{equation}

And while diagonal term is $t_{ii}^{(1)}=-\frac{1}{2}S_{ii}$, the fraction that accompanies it is equal to $t_{ji}^{(1)}$, arriving at Eq.22!

I am still trying to arrive at Eq.15, though, I will update if and when I do.

One can find, once the perturbation expansions are applied to the pseudo-eigenvalue, that the second-order coefficient has the form

One can find, once the perturbation expansions are applied to the pseudo-eigenvalue equation, that the second-order coefficient has the form

In accord with the book, $t_{ji}^{(0)}=\delta_{ji}$ and solving Eq.22 for $t^{(2)}_{ji}$

In accord with the book, $t_{ji}^{(0)}=\delta_{ji}$ and solving Eq.18 for $t^{(2)}_{ji}$