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

The first task is determine the elements of the B matrix. We have:
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where X12 = cos
q 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 = -sin
q/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:
|
D x1 |
D x2 |
D x3 |
D y1 |
D y2 |
D y3 |
|
|
D r12 |
cosq |
- cosq |
0 |
sinq |
- sinq |
0 |
|
D r23 |
0 |
-1 |
1 |
0 |
0 |
0 |
|
Dq |
-sinq/r12 |
sinq/r12 |
0 |
cosq/r12 |
1/r23-cosq/r12 |
- 1/r23 |