Polyspherical Coordinates

The VCHAM programs are able to work with polyspherical parameterisations based on Jacobi, Radau and valence vectors.

The input syntax used to define these vectors is described in the following.


Jacobi Vectors

In both VCPNT and VCTRANS the desired set of Jacobi vectors for an N-atom system is specified using the syntax

jacobi_define
A1
A2
A3
.
.
.
AN
end-jacobi_define

Here, the first vector connects atoms A1 and A2. The second vector connects the centre of mass of the A1, A2 subunit and atom A3, and so on.


Radau Vectors

In both VCPNT and VCTRANS the desired set of Radau vectors for an N-atom system is specified using the syntax

radau_define
A1
A2
A3
.
.
.
AN
end-radau_define

Here, atom A1 is the heliocentre. The first vector connects atom A2 and the canonical point. The second vector connects A3 and the canonical point, and so on


Valence Vectors

In both VCPNT and VCTRANS the desired set of valence vectors for an N-atom system is specified using the syntax

valence_define
A1 A2
A3 A4
.
.
.
Am An
end-valence_define

Here, the first vector points from atom A2 to atom A1, the second vector points from atom A4 to atom A3, and so on.


Definition of the Polyspherical Coordinates

For an N-atom system there exists N-1 vectors Ri. The body-fixed (BF) frame used is defined such that the vector RN-1 lies along the z-axis of the BF frame, and the vector RN-2 lies in the x,z-plane of the BF frame. The 3N-6 internal coordinates used are then the N-1 vector lengths Ri, the N-2 planar angles θi, between the vectors RN-1 and Ri, and the N-3 dihedral angles φi, between the two vectors Ri and RN-2 and RN-1.

The coordinates are arranged such that:

The coordinates 1 to N-1 correspond to the vector lengths Ri;

The coordinates N to 2N-3 correspond to the planar angles θi

The coordinates 2N-2 to 3N-6 correspond to the dihedral angles φi.