## Initialization

Each inertia tensor is diagonalized by a rotational transformation to the body coordinate system:
 (11.14)

The transformation matrix is used to initialize rotational variables such that = . Values for the body-frame coordinates of group elements are obtained by
 (11.15)

The net force and torque acting on each body are determined by summing the force and torque acting on each of its constituent atoms. Center-of-mass variables are initialized with a two-step process. The initial center-of-mass velocities are determined from the atom properties VX,VY,VZ:
 (11.16)

These velocities are then used to advance the center-of-mass coordinates
 (11.17)

The more stable Euler-Cayley parameters (also referred to as quaternions) are used as rotational variables instead of the Euler angles , , (cf. Goldstein (1980)). They are defined in Eq. 2.1. The quaternions are initialized using a first-order approximation to their equation of motion:

 (11.18)

where is the four-vector (,, ,), is the four-vector (0,,,), and is the matrix that gives their time evolution:
 (11.19)

Thus one obtains
 (11.20)

The initial angular velocity follows directly from the initial angular momentum, which is determined by
 (11.21)

where is the momentum of the atom of the rigid body. The initial half-step advanced angular momentum can be expressed as
 (11.22)

and the first advancement of the center-of-mass coordinates can be written as
 (11.23)

Xplor-NIH 2023-11-10