Angle Bending

Similar to bond stretching, angle bending term is also an empirical function of angle deviating from the ideal angle value, i.e., \(\Delta\theta=\theta-\theta_0\). Terms from cubic to sextic are added to generalize the HARMONIC functional form.

\[U = k\Delta\theta^2(1+k_3\Delta\theta+k_4\Delta\theta^2+k_5\Delta\theta^3+k_6\Delta\theta^4).\]

MMFF force field has a special treatment for LINEAR angle, e.g., carbon dioxide. Since the ideal angle should always be \(\pi\) rad, the deviation can be approximated by

\[\Delta\theta =\theta-\pi =2(\frac{\theta}{2}-\frac{\pi}{2}) \sim 2\sin(\frac{\theta}{2}-\frac{\pi}{2}) =-2\cos\frac{\theta}{2}.\]

Only keeping the quadratic term, the angle bending term can be simplified to

\[U = 2k(1+\cos\theta).\]

The LINEAR angle type is a special case of the SHAPES-style Fourier potential function [1] with 1-fold periodicity, which is referred to as the FOURIER angle type in Tinker jargon and has the following form

\[U = 2k(1+\cos(n\theta-\theta_0)).\]

In addition, there is another IN-PLANE angle type for trigonal center atoms. One can project atom D to point X on plane ABC. Instead of angle A-D-B, the ideal and actual angle values are for angle A-X-B (see Fig. 1).

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Fig. 1 A trigonal center and an in-plane angle.

Bond Stretching Stretch-Bend Coupling