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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.