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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],
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
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
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pellegrini eric
2009-10-06