Cross terms in 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). 

A number of more general force fields for the study of small molecules are available that can be extended to biological molecules. These force fields have been designed with the goal of being able to treat a wide variety of molecules, based on the ability to transfer parameters between chemical systems and the use of additional terms (e.g., cross terms) in their potential energy functions. Typically, these force fields have been optimized to... 

The influence of bilinear cross terms of this type in force field caculations has been studied systematically only once so far (79). They are standard for vibrational-spectroscopic force field expressions (20), and accordingly vibrational frequencies depend considerably more sensitively on cross terms than e.g. conformational parameters. An example for the significant influence of cross terms also with respect to the latter is described in Section 6.1.3. 

In Eq. (65) it is to be understood that the terms in the first line include cross-terms between similar coordinates, i.e., Ar, Ar, A0i Aflj, etc., although these are assumed to be physically significant only for i close toj. A similar assumption is made for the cross-terms in the second line of Eq. (65). In both cases, experience with such force fields provides a guide as to which off-diagonal terms to include terms in Ar Aft, have been found to have a minor effect on the frequencies, and terms for which the Jacobian elements (see Section II,D,3) for all modes are small are usually negligible. The units for the force constants areFr, mdyn/A Fra, mdyn all others, mdyn A. 

Fig. 4.13. Schematic illustration of the cross terms believed to be most important in force fields. (Adapted from Dinar U and A T Hagkr 1991. New Approaches to Empirical Force Fields In Reviews m Computational Chemistry, Lipkowitz K B and D B Boyd (Editors) New York, VCH Publishers, pp 99-164 )...

It should be mentioned that the two contributions can be completely separated because they have different symmetries, i.e., there are no density/orientation cross terms in the perturbation expansion involved in the calculations. The density component comprises three mechanisms of ET activation (i) translations of permanent dipoles, (ii) translations of dipoles induced by the electric field of the donor-acceptor complex (or the chromophore), and (iii) dispersion solute-solvent forces. On the other hand, it appears that in the orientational part only the permanent dipoles (without inductions) are involved. 

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. 

A force field which includes anharmonic terms in the potential functions, and explicit cross terms in the force constant matrix is sometimes referred to as Class 2. These kinds of force fields are popular today for accurate work, although Class 1 force fields, because of their speed, are still widely used for studies on macromolecules. In general the physics (mechanics) required for Class 1 and Class 2 force fields is now considered to be pretty straightforward and well understood. 

We have suggested " that a Class 3 force field be defined as one which contains chemical effects, in addition to the physical effects so far discussed. Chemical effects would include such things as hyperconjugation, the electronegativity effect, the anomeric and Bohlmann effects, and so on. These effects depend upon exactly which atom occupies a position, that is, they depend not only on ordinary mechanical quantities, but also on specific properties of oxygen, for example, compared with nitrogen, or with carbon. These effects can, of course, be properly represented by suitable cross terms in the force constant matrix. The origins of these terms, however, have a definite chemical basis. 

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. 

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