We start with the variational principle. Given a hamiltonian \( H \) and a trial wave function \( \Psi_T(\boldsymbol{R};\boldsymbol{\alpha}) \), the variational principle states that the expectation value of \( \cal{E}[H] \), defined through
$$ \cal {E}[H] = \frac{\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R};\boldsymbol{\alpha})H(\boldsymbol{R})\Psi_T(\boldsymbol{R};\boldsymbol{\alpha})} {\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R};\boldsymbol{\alpha})\Psi_T(\boldsymbol{R};\boldsymbol{\alpha})}, $$is an upper bound to the ground state energy \( E_0 \) of the hamiltonian \( H \), that is
$$ E_0 \le {\cal E}[H]. $$