Why doesn't the Λ n E n ϕ n {\displaystyle {{\Lambda }_{n}}{{E}_{n}}{{\phi }_{n}}} part wanishes? It is on both sides.
Where does m M I ∫ V N ( ϕ k ∗ ∇ R I ϕ l d 3 N r ) ∇ R I {\displaystyle {\frac {m}{{M}_{I}}}\int _{{V}^{N}}{\left(\phi _{k}^{*}{{\nabla }_{{\mathbf {R} }_{I}}}{{\phi }_{l}}{{d}^{3N}}r\right)}{{\nabla }_{{\mathbf {R} }_{I}}}} come from?