J. Am. Chem. SOC.1991, 113, 2919-2922
+ sin2 = 1 F5(a,/3,y) = cos2 p + sin2 /3 = 1 F6(a,P,y) = cos2 y + sin2 y = 1 F4(a,/3,y) = cosz CY
CY
we obtain a system of six simultaneous equations in six unknowns that can be treated as independent and are collected in a vector {cos a,sin a,cos /3, sin /3, cos y, sin y]: q 3 {q,] F,(q) = H,, n = 1-6 This equation set is solved iteratively. First, we choose an initial guess qcO)and evaluate h, = F,(q(O)). From the partial derivatives of the functions of F, with respect to the unknowns collected in q, we have
2919 AFn = CAqm(aFn/aqm) m
as the linear approximation to the changes in the values of F, that will be caused by a change Aq in the unknowns. To reach the solution, we would require AF,, = H, - h,, n = 1-6. These six linear equations are solved for the Aqmk Since the functions F, are not linear in q,, it is necessary to take q(l) = q(O) + Aq and iterate. In practice, damping is usually required so that we actually set AF, = (H, - h,)t for a suitably chosen 0 < t
