value of 10.43 kJ/mol for the free energy of $^{1}$C$_{4}$ , while in Ref. 43 this value ranged from 16.0 to 16.5 kJ/mol (see Tab.4 in Ref. $^{43}$) The difference between these two approaches is quite important, as it translates to inverted chair populations of ≃ 1% and ≃ 0 . 1% for the results of this work and of Ref.43, respectively. 

We give here a tentative explanation of this difference. It has to be noticed that the difference is most probably not related to which set of collective variables (Pickett dihedrals versus Cremer–Pople coordinates) has been used to enhance the sampling, because both represent every slow degree of freedom and allow to span the puckering conformational space in an ergodic way. Systematic errors related to the accelerated dynamics should have been eliminated in both approaches, since both sampling methods provide an unbiased estimate, thanks to the equilibrium sampling. 

Among the possible reasons are the use of different algorithms for the simulation of isothermal–isobaric conditions, the different system size, cut-off and long range corrections. Another explanation could be related to the way different states are defined: partitioning the conformational space using the three Pickett angles might lead to a miscounting of states, when using convex shapes like cubes or ellipsoids to perform cuts, as free energy basins in the three-dimensional space of Pickett angles can have have more complex, non-convex shape (compare for example Figs 2c and 3c in Ref. 43). By employing Cremer–Pople coordinates only two coordinates, θ and φ , need to be taken into account. Performing the appropriate cuts to identify the different states is far a simpler task in two dimensions than in three, also taking into account that only straight cuts along the θ and φ directions are needed to identify in a proper way the different basins. Still, more investigation would be needed to address properly this subject, which is ot of the scope of this work. 

## V. PUCKERING FREE ENERGY OF ALDOPYRANOSES 

## A. The 45a4 Parameter Set 

Apart from the problems related to a correct definition of different puckered states, it is clear that the calculation of the free energy profile of β -d- Glc alone cannot be considered exhaustive nor satisfactorily. The theoretical estimates predict a great variety in the confor- mational free energies of the 16 stereoisomers of β -d- Glc, and puckering properties have to be 

