# Example 2: Transforming a bent three body system from Cartesian to internal coordinates

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The first task is determine the elements of the B matrix. We have:

where X12 = cosq and Y12 = sinq. Bx1 = cosq and Bx2 = -cosq, By1 = sinq and By2 = -sinq, for the 1-2 coordinate and since X23 = 1 and Y23 = 0 we have Bx2 = 1 and Bx3 = -1, for the 2-3 coordinate.

Given the above values we have for the angle coordinate Bx1 = -sinq/r12, Bx3 = 0 and Bx2 = sinq/r12. We also calculate By1 = cosq/r12, By3 = -1/r23, and therefore By2 = 1/r23 � cosq/r12.

The B-matrix is:

Dx1 | Dx2 | Dx3 | Dy1 | Dy2 | Dy3 | |

Dr12 | cosq | -cosq | 0 | sinq | -sinq | 0 |

Dr23 | 0 | -1 | 1 | 0 | 0 | 0 |

Dq | -sinq/r12 | sinq/r12 | 0 | cosq/r12 | 1/r23-cosq/r12 | -1/r23 |