Costain rule

For example, rs coordinates are defined operationally as functions of ground-state planar second moments. The only uncertainty in an r% coordinate as an estimate of an rs coordinate is from experimental uncertainty in the planar second moments, and this is easy to compute from Eq. (10). As estimates of equilibrium values of the coordinates, however, rs coordinates contain an additional contribution to their uncertainty from the neglected pseudoinertial defects. This contribution may be estimated from the Costain rule [Eq. (14)]. 

The rc and p-Kr coordinates are determined by adjusting coordinates to fit moments of inertia and differences in moments of inertia, respectively. This is again an operational definition, so the only uncertainty is the result of experimental uncertainty in the rotational constants, and this may be treated by standard methods.18 However, the standard deviation between observed rotational constants and those computed from the final structure should not be used in such a calculation. In the comparison shown above the structural parameters of NF2CN are more reliable than those of PF2CN, but the deviation between observed and calculated moments is greater for NF2CN. The standard deviations of the rotational constants are best obtained from the fit of the rotational spectra. These values should be used to estimate the experimental uncertainties in rQ and p-Kr coordinates. The uncertainties in r0 or p-Kr coordinates as estimates of the equilibrium coordinates are very difficult to compute. The Costain rule is probably satisfactory for p-Kr coordinates, but less so for rQ coordinates. 

For atoms that lie close to a principal plane the Costain rule gives large uncertainties. If only one or two atoms lie close to a principal plane, a better estimate of the small coordinates may be obtained from Eq. (15) and/or Eq. (16) and correspondingly better estimates of the uncertainties in these coordinates may be obtained from derivatives of these equations and uncertainties in the other coordinates. 

We recommend that experimental uncertainties in the coordinates be assessed by Eq. (10) and the vibration/rotation contributions be assessed by the Costain rule [Eq. (14)1 or by first moment or product of inertia relations. Then, either the procedure introduced by Tobiason and Schwendeman20 should be used to propagate the uncertainties into distances or angles, or the two contributions should be added together and used with Eq. (18) to generate distance and angle uncertainties. 

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