First and second derivative matrix elements for linear and out‐of‐plane bending motion

Simplified formulas for first and second derivatives of the internal coordinates with respect to Cartesian coordinates are reported for linear and out‐of‐plane bending motion. They are expressed in a local coordinate system and then rotated to a space‐fixed Cartesian coordinate system. For linear motion the important points are: (1) the invariance of the energy with respect to translational and rotational coordinate transformations and (2) the presentation of derivatives of the energy in terms of (α − π)2 rather than (α − π), where α is the bond angle, to avoid indeterminant forms. By factoring the second derivatives of the energy with respect to (α − π) and sinα, analysis in this local coordinate system avoids division by sinα as α → π. The formulas describe linear systems without the need to project the motion onto two perpendicular planes. When the angle is exactly π two degenerate directions arise, and the coordinate axes perpendicular to the axis of the linear sequence of atoms may be chosen arbitrarily. Further refinements and clarifications of methods reported previously to obtain matrix elements for linear and out‐of‐plane motion are presented.