Frozen Internal Rotation Approximation (FIRA)

Let us briefly turn to the UA butane model shown in Figure 2. As already noted, this model can be described [1] by only six ICs: three bond distances, two bond angles and one dihedral, where only the latter as a soft IC. Let us suppose that the JOYCEfitting includes two conformations, namely the trans equilibrium (C$_1$-C$_2$-C$_2$-C$_1$ = 180$^{\circ}$ ) and totally eclipsed (C$_1$-C$_2$-C$_2$-C$_1$ = 0$^{\circ}$ ) The central C$_2$-C$_3$ distance is different for the two conformations (2.90 Å  and 2.95 Å , respectively for the staggered and eclipsed [1]), whereas the two bending angles change by about 3$^{\circ}$ . Despite the dihedral angle is by far the most evident geometrical change on going from staggered to eclipsed conformation, relevant energy contributions occur even for the small changes of the other ICs: bond lengths and angles were found [1] to account for 3.4 and 2.9 kJ/mol, respectively. Consequently the torsional energy term of eq. (10) accounts for about 75% of the relative energy E(eclip)-E(stagg). Therefore the resulting pure torsional potential (eq. (10)) describes a lower barrier (18 rather than 24 kJ/mol), being the remaining gap accounted for the energy terms of the bond distances and angles.

This (rather obvious) finding has the unpleasant consequence that a good description of the large amplitude torsional geometrical movements cannot be achieved with high accuracy by simple FFs. Indeed, by using a class I FF (i.e. no coupling term), the fraction of the torsional energy connected with the changes of the other IC is completely loss, because there is no reason the bond lengths and angles change during the internal rotation (frozen rotation). In fact the information linking the dihedral to the other ICs in QM calculation is completely lost, since in central FFs the motion of one IC is independent from the position of the other ICs. The straightforward remedy for this problem would require the inclusion of a relevant number of coupling functions in equations (14) - (23), as done for example in the QMFF procedure [39], with the consequence of increasing the number of functions in the FF.

A more simple and direct solution is to ignore the changes of most of the ICs not directly involved in the internal rotation and, in case, retaining the changes of few pertinent ICs whose coupling term with the dihedral are included in the FF. This route has the effect of ascribing the torsional energy to the torsional term (10) only, whereas in the QM calculation it is distributed on several ICs since all the ICs are in principle coupled to each other. This method, which is implicitly adopted in partial parameterization of flexible molecules will be called FIRA: frozen internal rotation approximation.