Stretch-bend-cross-term

A stretch-bend cross-term is now widely used in valence force Helds but is not needed in the Urey-Bradley force Helds. 

These potential energy terms and their attendant empirical parameters define the force field (FF). More complicated FFs which use different and/or more complex functional forms are also possible. For example, the simple harmonic oscillator expression for bond stretching can be replaced by a Morse function, Euorse (3), or additional FF terms may be added such as the stretch-bend cross terms, Estb, (4) used in the Merck molecular force field (MMFF) (7-10) which may be useful for better describing vibrations and conformational energies. 

MM2 [189] uses cubic anharmonic potential to represent the bond stretching, up to sixth power expansion for the valence angle bending, and harmonic field for the out-of-plane deformations. The stretch-bending cross term is included. 

In MM2 and MM3 [49,50 72 75] stretch-bend cross terms are used (Eq. 3.24), but in many other programs this is neglected. 

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. 

If the nonbonded potential does not include the 1,3 vicinal intramolecular atom pairs, a stretch/bend cross-term is added of the form Er e = -i-kr e(r-r0)2-(0-0o>2... 

The results given by a Urey-Bradley force field are very similar, although not identical, to those given by a force field utilizing stretch-bend cross terms. One is likely to need many cross terms in the force field anyway, apart from the stretch-bend interactions, so it is more convenient for most purposes just to add stretch-bend terms here as well. The Urey-Bradley force field is consequently seldom used anymore, but it is mentioned here because it is conceptually useful. 

Bond Stretch and Angle Bending Cross Term... 

Intensive use of cross-terms is important in force fields designed to predict vibrational spectra, whereas for the calculation of molecular structure only a limited set of cross-terms was found to be necessary. For the above-mentioned example, the coupling of bond-stretching (f and / and angle-bending (B) within a water molecule (see Figure 7-1.3, top left) can be calculated according to Eq. (30). 

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

In addition to these basic term s. force fieldsoften h ave cross term s that combine the above interactions. For example there may be a term which causes ati angle bend to interact with a bond stretch term (opening a bond angle may tend to lengthen the bonds in volved). 

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). 

Terms in the energy expression that describe how one motion of the molecule affects another are called cross terms. A cross term commonly used is a stretch-bend term, which describes how equilibrium bond lengths tend to shift as bond angles are changed. Some force fields have no cross terms and may compensate for this by having sophisticated electrostatic functions. The MM4 force field is at the opposite extreme with nine different types of cross terms. 

The components in i cross are usually written as products of Taylor-like expansions in the individual coordinates. The most important of these is the stretch/bend term which for an A-B-C sequence may be written as... 

Usually the constants involved in these cross terms are not taken to depend on all the atom types involved in the sequence. For example the stretch/bend constant in principle depends on all three atoms. A, B and C. However, it is usually taken to depend only on the central atom, i.e. = k , or chosen as a universal constant independent of atom type. It should be noted that cross tenns of the above type are inherently unstable if the geometry is far from equilibrium. Stretching a bond to infinity, for example, will make str/bend go towards — oo if 0 is less than If the bond stretch energy itself is harmonic (or quartic) this is not a problem as it approaches +oo faster, however, if a Morse type potential is used, special precautions will have to be made to avoid long bonds in geometry optimizations and simulations. 

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