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Bond Length, Bond Angle, and Dihedral Coordinates

Given the Cartesian positions \(\mathbf{R}_i\) of the atoms involved, the three basic internal coordinates that can be scanned are defined as follows.

Bond Length

The distance between two atoms \(i\) and \(j\):

\[ r_{ij} = \left| \mathbf{R}_i - \mathbf{R}_j \right| \]

Scanning \(r_{ij}\) — as in this tutorial's O–H bond scan — probes stretching, compression, dissociation, and proton-transfer coordinates.

Bond Angle

The angle at atom \(j\) formed by atoms \(i\), \(j\), and \(k\):

\[ \theta_{ijk} = \arccos\left( \frac{(\mathbf{R}_i - \mathbf{R}_j)\cdot(\mathbf{R}_k - \mathbf{R}_j)}{\left|\mathbf{R}_i - \mathbf{R}_j\right|\left|\mathbf{R}_k - \mathbf{R}_j\right|} \right) \]

Scanning \(\theta_{ijk}\) probes bending and inversion barriers — for example, the planar-to-pyramidal inversion of an amine nitrogen.

Dihedral (Torsion) Angle

The angle between the planes defined by atoms \((i,j,k)\) and \((j,k,l)\), built from the normal vectors of each plane:

\[ \mathbf{n}_1 = (\mathbf{R}_j - \mathbf{R}_i) \times (\mathbf{R}_k - \mathbf{R}_j), \qquad \mathbf{n}_2 = (\mathbf{R}_k - \mathbf{R}_j) \times (\mathbf{R}_l - \mathbf{R}_k) \]
\[ \phi_{ijkl} = \arccos\left( \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{|\mathbf{n}_1||\mathbf{n}_2|} \right) \]

with the sign of \(\phi\) fixed by the sign of \(\mathbf{n}_1 \cdot (\mathbf{R}_l - \mathbf{R}_k)\). Scanning \(\phi_{ijkl}\) probes torsional and rotational barriers and conformational interconversion.

All three coordinate types are scanned with exactly the same input syntax in Gaussian — only which formula is being stepped through changes.