The fundamental limitation of the conjugate gradient method is that it requires, in general, n cycles to reach the minimum. We need a procedure which will perform most of the function minimization in the first few cycles.
The eigenvalues of the normal matrix (Leunberger, 1973) provide information about how a method will refine parameters in the early cycles. The normal matrix describes the shape of the minimum of the function, and its eigenvalues define how oblong the neighborhood of the minimum is. Because the normal matrices for the functions usually minimized in macromolecular refinement are nearly diagonal there is a close correspondence between the eigenvectors and the parameters of the model. For a perfectly diagonal normal matrix, the eigenvectors are the axes of parameter space and the diagonal elements, or curvatures, are the eigenvalues.
The method of steepest descent works best when all the eigenvalues or diagonal elements are equal. If they are not equal the parameters with the largest curvatures dominate. The conjugate gradient method must infer the differences in curvature from the history of the search but this takes more cycles than we give the method in practise.
This problem is especially serious when positional parameters are compared to thermal parameters. The curvatures for positional parameters are much larger than those for thermal parameters; therefore, refinement of thermal parameters is blocked by the influence of the positional parameters. This effect is usually avoided by refining thermal parameters with the positional parameters held constant and vise versa.
A more intractable problem arises because the curvatures associated with numerically large thermal parameters are much smaller than those of smaller thermal parameters. In all models produced by refinement using the conjugate gradient method and methods which simplistically incorporate curvatures, the large thermal factors are poorly refined and probably should have even larger values than those obtained during the refinement process. In addition atom types with many electrons, such as sulfur and iron, have large curvatures. The thermal factor shifts of these atoms will be overestimated, resulting in an oscillation about the correct value.