A priori estimation of scale and overall anisotropic temperature factors from the Patterson origin peak

An idea due to D. Rogers [Computing Methods in Crystallography (1965), edited by J. S. Rollett, pp. 117-148. Oxford: pergamon Press] has been developed and implemented. The method is an advantageous alternative to Wilson plot or K-curve scaling of intensity data. On the relative experimental scale the structure factor can be written in matrix notation as F(h) = kappa -1 sigma j fj(h) exp (2 pi ih tau xj) exp (-h tau bjh); and the squared structure-factor magnitude can be written as magnitude of F(h)2 = kappa -2 exp (-2h tau bh) [sigma j fj2+ 2 sigma j sigma k greater than jfjfk exp [2 pi ih tau (xj-xk)]], if a a common, or average, anisotropic temperature factor is factored out of the atomic summations. The fj2 summation corresponds to the Patterson origin peak, and the fjfk double summation to the off-origin Patterson peaks. A tovariate Gaussian density function, P(u)-Pmin = Po exp (-u tau pu), is fitted by least squares to the origin peak from a Patterson synthesis with coefficients magnitude of F2 meas/sigma jf2j. Fourier inversion of the fitted Gaussian gives the scale and thermal parameters, k2 = (detp)1/2/(pi 3/2 Vcell Po) and b = (pi 2/2)p-1. The fit of the parameter Pmin is constrained by the condition that Pmin = -F(000)2/(k2Vcell sigma j Zj2), and thus only po and the six coefficients pij (i less than j = 1,2,3) are independent parameters.