Let $i, j$ represent a wavefunction.
In Bra - Ket notation, I have an expression like
$\langle ij | g | i j \rangle $ which is also just $(\langle i | \otimes \langle j | ) | g | (|i \rangle \otimes |j\rangle)$
If the state $i$ were perturbed by $\delta i$:
$(\langle i + \delta i| \otimes \langle j |) | g | (|i + \delta i \rangle \otimes |j\rangle)$
I am struggling to expand this out. Any help with this is greatly appreciated.
First expand the ket-vector \begin{align*} (\langle \color{red}{i + \delta i}| \otimes \langle j |) | g | (|i + \delta i \rangle \otimes |j\rangle)&= \langle \color{red}{i} \otimes j | g | (|i + \delta i \rangle \otimes |j\rangle)+ \langle \color{red}{\delta i} \otimes j | g | (|i + \delta i \rangle \otimes |j\rangle) \end{align*} and then, analogously, the bra-vectors: \begin{align*} &\color{blue}{\langle i \otimes j | g | (|i + \delta i \rangle \otimes |j\rangle)}+ \color{red}{\langle \delta i \otimes j | g | (|i + \delta i \rangle \otimes |j\rangle)}\\ &\qquad\qquad\qquad= \color{blue}{\langle i \otimes j | g |i\otimes j\rangle+\langle i \otimes j | g |\delta i\otimes j\rangle}+\color{red}{ \langle \delta i \otimes j | g | i \otimes j\rangle+ \langle \delta i \otimes j | g | \delta i \otimes j\rangle}\,. \end{align*} Therefore the higher-order terms of $(\langle {i + \delta i}| \otimes \langle j |) | g | (|i + \delta i \rangle \otimes |j\rangle)$ are $$\langle i \otimes j | g |\delta i\otimes j\rangle+\langle \delta i \otimes j | g | i \otimes j\rangle+\mathcal O(\delta^2)$$