1

The equation above is giving errors such as Undefined control sequence, Missing number, treated as zero, Illegal unit of measure (pt inserted)

\documentclass{article}
\usepackage {amsmath}
\begin {document}

\begin {equation}
\gamma_{\alpha , \beta} = \int \int' \Gamma [\vec{r},\vec{r}',n_0] \frac {F^{\alpha}_{00}(|\vec{r} - \vec{R_{\alpha} | ) F^{\beta}_{00}(|\vec{r}' - \vec{R_{\beta}} | ) } }{4\pi}
\end {equation}

\end {document}
2

you have mismatch with curly braces ... try

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{equation}
\gamma_{\alpha , \beta} =
    \int \int' \Gamma [\vec{r},\vec{r'},n_0]
\frac{F^{\alpha}_{00}(|\vec{r}-\vec{R}_{\alpha}|)F^{\beta}_{00}(|\vec{r'}-\vec{R_{\beta}}|)}
     {4\pi}
\end{equation}

\end{document}

enter image description here

or

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{equation}
\gamma_{\alpha , \beta} =
    \int \int' \Gamma [\vec{r},\vec{r}\,',n_0]
\frac{F^{\alpha}_{00}(|\vec{r}-\vec{R}_{\alpha}|)F^{\beta}_{00}(|\vec{r}\,'-\vec{R_{\beta}}|)}
        {4\pi}
\end{equation}

\end{document}

enter image description here

0

Probably you would have better to split your equation

\documentclass{article}%
\usepackage {amsmath}%
\begin {document}%
%
\newcommand{\leftmember}{\gamma_{\alpha , \beta}}%
\newcommand{\numerator}{\utilityF{\alpha} \utilityF{\beta} }%
\newcommand{\utilityF}[1]{F^{#1}_{00}(|\vec{r}-\vec{R}_#1|)}%
% easier to manage and improve when splitted
% for instance :
% \newcommand{\utilityF}[1]{F^{#1}_{00}\left(\left|\vec{r}-\vec{R}_#1\right|\right)}%
%
\begin{equation}
  \leftmember = \int \int' \Gamma [\vec{r},\vec{r}',n_0] \frac{\numerator} {4\pi}
\end{equation}
%
\end{document}

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