I'm trying to understand the conformal transformation law of Ricci curvature. It states that if the metric $g$ now changes to $e^{2\phi}g$, then $$Ric(e^{2\phi}g)=Ric^g-(n-2)Hess^g(\phi)-\Delta_g(\phi)g-(n-2)|grad^g(\phi)|^2g+(n-2)d\phi\otimes d\phi$$
I am trying to understand why that is. It seems to me that $Ric(X,Y)=g\big(R(X\wedge e_i),(Y\wedge e_i)\big)$. Hence, if the metric has now changed to $e^{2\phi}g$, then $$Ric^{e^{2\phi}g}(X,Y)=e^{2\phi}g(R(X\wedge e_i),(Y\wedge e_i))=e^{2\phi}Ric(X,Y)$$ Why is this not true?