Skip to main content
Chemistry LibreTexts

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

  • Page ID
    2246
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    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


    Example 2: Transforming a bent three body system from Cartesian to internal coordinates is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Stefan Franzen.

    • Was this article helpful?