documentclass[12pt]{article}
usepackage{amsmath}
usepackage{amssymb}
title{A test file}

begin{document}
maketitle

*Show_first_line = True*
*Show_second_line = False*

section{Density functional theory}

*if Show_first_line == True: #Now the following formula should be compiled*
     $$
     i hbar frac{partial Phi(mathbf{r}, mathbf{R} ; t)}{partial t}=left(-frac{hbar^{2}}{2 M} frac{partial^{2}}{partial mathbf{R}^{2}}-frac{hbar^{2}}{2 m} frac{partial^{2}}{partial mathbf{r}^{2}}+V(mathbf{r}, mathbf{R})right) Phi(mathbf{r}, mathbf{R} ; t)
     $$
*else:*
     $$
     i hbar frac{partial Phi(mathbf{r}, mathbf{R} ; t)}{partial t}=left( -frac{hbar^{2}}{2 m} frac{partial^{2}}{partial mathbf{r}^{2}}+V(mathbf{r}, mathbf{R})right) Phi(mathbf{r}, mathbf{R} ; t)
     $$

*if Show_second_line == True # In case it is false, nothing should be compiled*
     $$
     mathrm{M} gg mathrm{m} text { the Born-Oppenheimer approximation }
     $$

$$
i hbar frac{partial Phi(mathbf{r}, mathbf{R} ; t)}{partial t}=
left( -frac{hbar^{2}}{2 m} frac{partial^{2}}{partial mathbf{r}^{2}}+V(mathbf{r}, mathbf{R})right) Phi(mathbf{r}, mathbf{R} ; t)
$$

$$
M ddot{mathbf{R}}=-frac{partial E(mathbf{R})}{partial mathbf{R}} \
left(-frac{hbar^{2}}{2 m} frac{partial^{2}}{partial mathbf{r}^{2}}+V(mathbf{r}, mathbf{R})right) Psi(mathbf{r} | mathbf{R})=E(mathbf{R}) Psi(mathbf{r} | mathbf{R})
$$

end{document}