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19.1 Averaging Data Structures

There are three types of averaging data structures – one for all the internal coordinates, one for the positional time distributions for all the atoms, and one for velocity time distributions for all the atoms.

For internal coordinates, there is one averaging data structure called ICAV (Internal Coordinate AVerages). It contains the minimum and maximum values as well as measures of the first four moments of the time distribution for the geometries and energies for the bonds, bond angles, torsion angles, and improper torsion angles in the structure.

The measures of the first four moments is as follows: For the first moment, the average is used.

                                     2    1/2
     The second moment is ( < (X-<X>)  > )
                                                   3    1/3
     The measure of the third moment is ( < (X-<X>)  > )
                                                    4    1/4
     The measure of the fourth moment is ( < (X-<X>)  > )

For averaging atomic positions and velocities, a complication arises because each position or velocity is specified by three coordinates which do not change independently of each other. In computing second and higher order moments, we must maintain all the cross correlations. These cross correlations form a tensor whose order is the same as the moment. Fortunately, the second moment tensor can be diagonalized so that the cross correlations are zero. In other words, the fluctuations can be resolved into independent components. The transformed second moment defines a set of principal axes. The principal axis can then be used to define a rotation which can be applied to the third and fourth moments as well.

The most important aspect of this transformation from the data management point of view is that it is not easily inverted; one cannot conveniently go from the principal axis set of moments to the cartesian axis set. Therefore, the averaging data structures for positions and velocities of atoms consist of two data structure each. The ones for the coordinates are AVX (AVerage X) and PAX (Principal Axis X). The ones for the velocities are called AVV (AVerage Velocity) and PAV (Principal Axis Velocity).

The AV data structures are required only for accumulating averages for an dynamics runs. The BUILD command only requires the PA data structures. The analysis facility automatically constructs a new PA data structure from an AV data structure whenever necessary.

If it is necessary to rotate all the coordinates in a trajectory with respect to a particular coordinate set, the COMPARE command may be used. See Reorienting a Coordinate Trajectory, for details. If it is necessary to merge several trajectories together, the MERGE DYN command in the main part of CONGEN can be used, see Manipulating Trajectories.