Equation - brackets

I do not know where \expval comes from but with braket you could do

\documentclass{article}
\usepackage{amsmath}
\usepackage{braket}
\begin{document}
\begin{equation}\label{E}
E_0^{(1)} = \Braket{\Phi | \frac{e'^2}{\left|\vec r_1 - \vec r_2\right|} | \Phi}
\end{equation}
\end{document}

If you're using the physics package, then it looks like you have to pass two * arguments to \expval to allow it to auto-resize the brackets.

As for your second equation, it is missing an argument. You had \expval**{stuff \Phi}, but you need to provide another argument: \expval**{stuff}{\Phi}.

\documentclass{article}
\usepackage{physics}
\begin{document}
\begin{equation}\label{E}
E_0^{(1)} = \expval**{\frac{e'^2}{|\vec{r}_1-\vec{r}_2|}}{\Phi}
\end{equation}
\begin{equation}\label{E2}
E_0^{(1)} = 4 \pi e'^2 \expval**{\frac{1}{|\vec{r}_2|} \sum_{l=0}^\infty \sum_{m=-l}^l \left(\frac{|\vec{x}^{(1)}|}{|\vec{x}^{(2)}|} \right)^l}{\Phi}
\end{equation}
\end{document}