Cross terms, force fields

In addition to these basic terms, force fields often have cross terms that combine the above interactions. For example there may be a term which causes an angle bend to interact with a bond stretch term (opening a bond angle may tend to lengthen the bonds involved). 

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). 

Extensive use of cross-terms is made by Haglcr s quantum mechanically derived force field CFF, because one of the intentions of this type of PEF is to obtain ac-... 

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. 

The consistent force field (CFF) was developed to yield consistent accuracy of results for conformations, vibrational spectra, strain energy, and vibrational enthalpy of proteins. There are several variations on this, such as the Ure-Bradley version (UBCFF), a valence version (CVFF), and Lynghy CFF. The quantum mechanically parameterized force field (QMFF) was parameterized from ah initio results. CFF93 is a rescaling of QMFF to reproduce experimental results. These force fields use five to six valence terms, one of which is an electrostatic term, and four to six cross terms. 

Empirical force field (EFF) is a force field designed just for modeling hydrocarbons. It uses three valence terms, no electrostatic term and five cross terms. 

MMl, MM2, MM3, and MM4 are general-purpose organic force fields. There have been many variants of the original methods, particularly MM2. MMl is seldom used since the newer versions show measurable improvements. The MM3 method is probably one of the most accurate ways of modeling hydrocarbons. At the time of this book s publication, the MM4 method was still too new to allow any broad generalization about the results. However, the initial published results are encouraging. These are some of the most widely used force fields due to the accuracy of representation of organic molecules. MMX and MM+ are variations on MM2. These force fields use five to six valence terms, one of which is an electrostatic term and one to nine cross terms. 

The Merck molecular force field (MMFF) is one of the more recently published force fields in the literature. It is a general-purpose method, particularly popular for organic molecules. MMFF94 was originally intended for molecular dynamics simulations, but has also seen much use for geometry optimization. It uses five valence terms, one of which is an electrostatic term, and one cross tenn. 

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... 

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. 

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... 

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. 

We have seen that for our calculations essentially two types of force fields have to be considered VFF- and UBFF-expressions. The main difference with repect to spectroscopic force fields consists in the superposition of nonbonded interactions. The force fields used so far for our purposes are almost exclusively simple valence force fields without cross terms, and a veriety of UB-force fields. Only recently could experiences be gathered with a valence force field that includes a number of important cross terms (79). Vibrational spectroscopic force fields of both types have been derived and tested with an overwhelming amount of experimental data. The comprehensive investigations of alkanes by Schachtschneider and Snyder (26) may be mentioned out of numerous examples. The insights gained from this voluminous spectroscopic work are important also when searching for suitable potentials for our force-field calculations. 

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... 

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. 

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