Bonding cross-terms

Additional terms, including out-of-plane interactions, hydrogen bonding, cross terms etc., can also be included (see Section 3.2). The general approach of mole-... 

In the framework of molecular mechanics, the atoms in molecules are treated as rubber balls of different sizes (atom types) joined by springs of varying length (bonds). The potential energy of a system can be expressed as a sum of valence (or bond), cross-term, and nonbond interactions, which together are commonly referred to as force fields ... 

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). 

Bond Stretch and Angle Bending Cross Term... 

A stretch-torsion cross term can be used to model the stretching of a bond that occurs in ai eclipsed conformation. Two possible functional forms are ... 

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. 

Assisted model building with energy refinement (AMBER) is the name of both a force field and a molecular mechanics program. It was parameterized specifically for proteins and nucleic acids. AMBER uses only five bonding and nonbonding terms along with a sophisticated electrostatic treatment. No cross terms are included. Results are very good for proteins and nucleic acids, but can be somewhat erratic for other systems. 

Note also that different analytical forms are used for some of the terms (angles harmonic in cos 0 rather than 0, cosine expansions for dihedrals, r or exp(—yfr,y) instead of repulsions, etc.). Some force fields also have cross-terms between different degrees of freedom. Additional terms may be present for out-of-plane bendings, hydrogen bonds, etc. 

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. 

Another type of correction, which is related to cross terms, is the modification of parameters based on atoms not directly involved in the interaction described by the parameter. Carbon-carbon bond lengths, for example, become shorter if there are electronegative atoms present at either end. Such electronegativity effects may be modelled by adding a correction to the natural bond length based on the atoms which are attached to the A-B bond. 

Although cross terms between the bonded potentials are part of all force fields designed to aclfieve high accuracy, the coupling between the geometry and the atomic charges is rarely addressed. From electronic structure calculations it is known that the optimum set... 

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