Exact method for the calculation of pseudorotation parameters P, τ_m and their errors. A comparison of the Altona–Sundaralingam and Cremer–Pople treatment of puckering of five-membered rings

A comparison of the two methods available for describing the puckering of five-membered rings - the Altona-Sundaralingam (A-S) [Altona & Sundaralingam (1972). J. Am. Chem. Soc. 94, 8205-8212] pseudorotation parameters τ_m, P and the Cremer-Pople (C-P) [Cremer & Pople (1975). J. Am. Chem. Soc. 97, 1354-1358] puckering parameters q, φ - shows that they are the same for all practical purposes but both have minor shortcomings. In the A-S method, the value of τ_m is somewhat dependent on the choice of the origin atom, while in the C-P method the normal to the mean plane, rather than being fixed, precesses as the ring puckering changes. We present here an exact procedure for obtaining the A-S parameters τ_m, P that correspond to the amplitude and phase of the Fourier wave of period 4π/5, thus making τ_m independent of the origin atom. Formulae are also given for the estimation of the standard deviations in the A-S and C-P parameters.