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metadynamics, and also to recover a truly equilibrium free energy profile $^{50,82,87}$. Among these, the approach proposed by Babin and coworkers 50 is in our opinion one of the most intuitive and versatile ones, since it allows in a simple way to simultaneously estimate the statistical error and to eliminate systematic errors introduced by the deposition protocol. The basic idea is the following: a metadynamics run is performed up to the build-up time t$_{b}$ , so that the whole collective variables space has been explored, and the total potential energy – sum of the physical and the bias potential – at the end of the run reads 

<img src='content_image/349826.jpg'>

At this point, a molecular dynamics simulation in the potential V is performed, much in the spirit of umbrella sampling, whereas the biasing potential has been determined in an adaptive way by the metadynamics run. The dynamics is characterized by an almost diffusive behavior in the collective variables space, since the meta-stabilities have been removed by the bias potential, provided that all the states of the new potential V ( x ) are separated by energy barriers comparable or lower than the thermal energy scale k b T . The deviation from a truly diffusive behavior is due by residual features of the free energy landscape which are originated from the statistical and systematic errors in the metadynamics run. During this equilibrium run, the phase space is sampled with probability density 

that can be estimated by computing the histogram ρ$_{bias}$ ( s ) = ( t − t$_{b}$ ) − 1 ∫ t t b δ [ s − s ( t $^{′}$)] dt ′ during the run. Eventually, the free energy profile is given, up to an additive constant, by 

<img src='content_image/349832.jpg'>

<img src='content_image/349827.jpg'>

In the exact expression Eq. 3, the term k b T ln ρ$_{bias}$ ( s ) can be regarded as a correction to what is the standard metadynamics estimate of the free energy landscape, F$_{meta}$ ( s ) = − U$_{bias}$ ( s ). This correction term compensates for the systematic errors introduced by the deposition protocol, which are not completely under control in the metadynamics run. In other words, from a simple metadynamics run there is no way to guarantee that the term ρ$_{bias}$ ( s ) has become a constant within statistical fluctuations. The corresponding statistical error can be estimated from the fluctuations of ρ$_{bias}$ ( s ) using standard error analysis. Differently from the approach involving only a metadynamics run, one should not worry about the speed of 

