Description of Rigid Body Units

A rigid body unit is a collection of point atoms whose local geometry is time invariant. One way to enforce this in a simulation is to impose a sufficient number of bond constraints between the atoms in the unit. However, in many cases this is may be either problematic or impossible. Examples in which it is impossible to specify sufficient bond constraints are

1. linear molecules with more than 2 atoms (e.g. CO$_2$)
2. planar molecules with more than three atoms (e.g. benzene).

Even when the structure can be defined by bond constraints the network of bonds produced may be problematic. Normally, they make the iterative SHAKE procedure slow, particularly if a ring of constraints is involved (as occurs when one defines water as a constrained triangle). It is also possible, inadvertently, to over constrain a molecule (e.g. by defining a methane tetrahedron to have 10 rather than 9 bond constraints) in which case the SHAKE procedure will become unstable. In addition, massless sites (e.g. charge sites) cannot be included in a simple constraint approach making modelling with potentials such as TIP4P water impossible. All these problems may be circumvented by defining rigid body units in terms of a center of mass (c.o.m) and an orientation and solving the resultant rigid body equations of motion. ^2.5

A rigid body has associated with it a rotational inertia matrix $\underline{\underline{\bf I}}$, whose components are given by $I_{\alpha\beta} = 1/2 \sum_{\rm sites} m r_{\alpha} r_{\beta}$ where the $r$ are the distance to the centre of mass and the $m$ are the site masses. In DL_POLY_2 we define the local body frame to be that in which the rotational inertia tensor $\hat{\mbox{$\underline{\underline{\bf I}}$}}$ is diagonal and the components satisfy $I_{xx} \ge I_{yy} \ge I_{zz}$. These three components are stored in the arrays rotinx, rotiny and rotinz for each unique type of rigid body in the system. The total mass of the rigid body unit, $M$, is stored in the array gmass and the location of sites with respect to the local body axes are stored in the arrays gxx, gyy and gzz.

The orientation of a local body frame with respect to the space fixed frame is described via a four dimensional unit vector, the quaternion $\mbox{$\underline{q}$} = [q_0,q_1,q_2,q_3]^T$. The Rotational matrix to transform from the local body frame to the space fixed frame is the unitary matrix

$$\mbox{$\underline{\underline{\bf R}}$} = \pmatrix{ q_0^2+q_1... ..._1q_3-q_0q_2) & 2(q_2q_3+q_0q_1) & q_0^2-q_1^2-q_2^2+q_3^2\cr}$$ (2.202)

so that if $\hat{\mbox{$\underline{d}$}_{\alpha}}$ is the position of a site in the local body frame with respect to its centre of mass, its position in the space fixed frame (w.r.t. its centre of mass) is given by

$$\mbox{$\underline{d}$}_{\alpha} = \mbox{$\underline{\underline{\bf R}}$} ~ \hat{\mbox{$\underline{d}$}_{\alpha}}$$ (2.203)