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##### Solution of the minimization problem.

In quaternion algebra, the rotational minimization problem may now be phrased as follows:

| (4.60)

Since the matrix representing a normalized quaternion is orthogonal this may also be written as

| (4.61)

This follows from the simple fact that , if is normalized. Eq. (4.61) shows that the target function to be minimized can be written as a simple quadratic form in the quaternion parameters [55],

|  |  | (4.62)
|  |  | (4.63)

The matrices are positive semi-definite matrices depending on the positions and :

| (4.64)

The rotational fit is now reduced to the problem of finding the minimum of a quadratic form with the constraint that the quaternion to be determined must be normalized. Using the method of Lagrange multipliers to account for the normalization constraint we have

| (4.65)

This leads immediately to the eigenvalue problem

|  |  | (4.66)
|  |  | (4.67)

Now any normalized eigenvector fulfills the relation . Therefore the eigenvector belonging to the smallest eigenvalue, , is the desired solution. At the same time gives the average error per atom.

The result of RBT analysis is stored in a new trajectory file that contains only RBT motions.

Next: Parameters Up: Theory and implementation Previous: Quaternions and rotations. Contents