An accidental derivation of metadynamics: Error estimates and exact results

We present a unique derivation of metadynamics. The starting point for the derivation is an on-the-fly reweighting scheme but through an approximation we recover the standard metadynamics and the well-tempered metadynamics in a general form while never appealing to the extended Lagrangian framework. This work leads to a more robust understanding of the error in the computed free energy than what has been obtained previously. Moreover, a formula for the exact free energy is introduced. The formula can be used to post-process any existing well-tempered metadynamics data allowing one, in principle, to obtain an exact free energy regardless the metadynamics parameters.