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##### Optimal superposition.

We consider a given time frame in which the atomic positions of a (sub)molecule are given by . The corresponding positions in the reference structure are denoted as . For both the given structure and the reference structure we introduce the yet undetermined centroids and , respectively, and define the deviation

| (4.45)

Here is a rotation matrix which depends on also yet undetermined angular coordinates which we chose to be quaternion parameters, abbreviated as vector . The quaternion parameters fulfill the normalization condition [56]. The target function to be minimized is now defined as

| (4.46)

where are atomic weights (see Section 4.2.1). The minimization with respect to the centroids is decoupled from the minimization with respect to the quaternion parameters and yields

|  |  | (4.47)
|  |  | (4.48)

We are now left with a minimization problem for the rotational part which can be written as

| (4.49)

The relative position vectors

|  |  | (4.50)
|  |  | (4.51)

are fixed and the rotation matrix reads [56]

| (4.52)

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