Skip to main content
Chemistry LibreTexts

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:

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