Cross-terms

In the harmonic approximation, V does not involve the cross-term pj-Pe because pj- and are the symmetry coordinates. It is thus of the form... 

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-... 

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... 

If vve now expand the expression for the energy as for the ground state, terms analogous to the electron-nucleus and electron-electron interactions can again be obtained. However, the cross-terms are no longer equal to zero as was the case for the ground state, because the... 

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 ... 

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). 

We shall concenPate on the potential energy term of the nuclear Hamiltonian and adopt a sPategy similar to the one used in simplifying the equation of an ellipse in Chapter 2. There we found that an arbiPary elliptical orbit can be described with an arbiParily oriented pair of coordinates (for two degrees of freedom) but that we must expect cross terms like 8xy in Eq. (2-40)... 

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. 

Optimized potentials for liquid simulation (OPES) was designed for modeling bulk liquids. It has also seen significant use in modeling the molecular dynamics of biomolecules. OPLS uses five valence terms, one of which is an electrostatic term, but no cross terms. 

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