not lower than that of inverted chairs. On the other hand, some results appear to be not compatible with this set of estimates. Ab ab-initio calculation 78 assigned a much higher free energy to glucose inverted chairs, while other data form AFM spectroscopy of CMA 73,74 go into the opposite direction. By and large, we feel confident that the two series 62,63 represent reasonable approximations of pyranoses free energy difference between chair conformers, and in this work we will use these values as a reference when testing new force field parameter sets. 

## III. METHODS 

## A. Refining Metadynamics with Umbrella Sampling 

The term metadynamics identifies a number of techniques that have been devised dur- ing the last decade to accelerate dynamics for systems displaying meta-stabilities. Various, slightly different algorithms have been proposed in literature, that exploit the same under- lying idea of the first metadynamics implementation 80 (sometimes referred to as discrete metadynamics): the Lagrangian version $^{81}$, which relied on the introduction of fictitious de- grees of freedom associated to the reaction coordinates; the direct metadynamics $^{82}$, which is probably the most employed variant; the well-tempered metadynamics $^{83}$, which assures a convergent estimate of the free energy profile. All these variants share the usage of a time-dependent biasing potential, U$_{bias}$ [ s ( x ) , t ], to ease the exploration of the phase space along suitably chosen collective variables, s ( x ), in turn themselves functions of the atomic coordinates x . The collective variables must represent all slow degrees of freedom charac- terizing the system, in order for metadynamics – as well as any other free energy profile reconstruction method – to be meaningful $^{84}$. 

The direct metadynamics approach shares many traits with other enhanced sampling methods, such as the Local Elevation method 85 or the adaptive umbrella sampling 86 (detailed lists of enhanced sampling techniques based on molecular dynamics can be found in Refs. 43 and 84), but the fact that the free energy landscape in metadynamics is usually estimated as the negative of the biasing potential in a single, out-of-equilibrium sweep, makes the technique susceptible, at least in principle, of errors introduced by the deposition protocol. Distinct approaches have been devised to estimate the statistical and systematic errors in 

