Torsion-bend cross term

The number and type of cross terms vary among different force fields. Thus, AMBER2 contains no cross terms, MM23 uses stretch-bend interactions only and MM34 uses stretch-bend, bend-bend and stretch-torsion interactions. Cross terms are essential for an accurate reproduction of vibrational spectra and for a good treatment of strained molecular systems, but have only a small effect on conformational energies. 

In MM3 a cross term involving torsional angles is also used[75] but in other programs it is neglected (exceptions are anomeric effects, e.g., with sugars). In other programs[52,5gJ, 1,3-interactions can be selectively included, e.g., to model the geometry of the chromophore of metal complexes. In these cases, an additional correction via a stretch-bend cross term is probably redundant. 

Figure 7-13. Cross-terms combining internal vibrational modes such as bond stretch, angle bend, and bond torsion within a molecule.

Torsion-bend and torsion-bend-bend terms may also be included the latter, for example would couple two angles A-B-C and B-C-D to a torsion angle A-B-C-D. Maple, Dinu and Hagler used quantum mechanics calculations to investigate which of the cross term are most important and suggested that the stretch-stretch, stretch-bend, bend-bend stretch-torsion and bend-bend-torsion were most important [Dinur and Hagler 1991 (schematically illustrated in Figure 4.13). 

More complicated cross-terms between the different intramolecular degrees of freedom are also employed in some force fields, but we will not consider them in the following. The dihedral term may also include four-center improper torsion or out-of-plane bending interactions that occur at sp2 hybridized centers.29... 

EFF (Empirical force field) [186] has been designed just for modeling hydrocarbons. It uses the quartic anharmonic potential for the bond stretching, and the cubic anharmonic for the valence angle bending. No out of plane or electrostatic terms are involved, although the cross terms, except torsion-torsion and bend-torsion ones, are included. 

The MM4 method [ 191 ] is still under development in terms of extending the number of chemical elements it is parametrized for. The initial published results are encouraging. It uses the sixth power expansion, both for the stretching and for the bending and the improper torsion representation for the out-of-plane deformations. The idea of using the bond-dipole based electrostatic term is abandoned in the MM4 whereas the whole collection of the cross terms is included. 

A separation of the through-bond interactions (Ed) into bond stretching (Eb), angle bending (Eg), out-of-plane deformation (E ), torsional angle rotation (Ef) and other terms is only possible if these terms are not coupled, and this is most likely if the force constants are very different. This is generally true and leads to largely decoupled local oscillators. In cases where this requirement is violated correction terms have to be added. This can be done by the inclusion of cross-terms which will be discussed later. 

The strategy we have taken in the determination of the stretching force constants is as follows First, we determined a complete set of force constants in the S0 state, including the stretching, in-plane bending, out-of-plane wagging, and torsional force constants of Urey-Bradley-Shimanouchi (UBS) type, and also, non-UBS cross-terms, many of which were transferred from those determined by Saito... 

A more refined force field will also consider interactions or cross terms, such as stretch-bend, torsion-stretch, etc. These are usually small, and they can be neglected in the first approximation. Finally, other ad hoc terms such as hydrogen bonding have been used to take into account phenomena that are not properly accounted for otherwise. 

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