Cross term function

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. 

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... 

Even for such a simple molecule, which 1 deliberately constrained to be lineiir and where I assumed that the harmonic approximation was applicable, the potential energy function will have cross-terms. 

Sinee the probability distribution functions (1) and (2) are even functions of Xia and Xg, respectively, the cross-term integral containing Xm and Xe to the first power will vanish.)... 

In the familiar standard Huckel treatment2174 of planar 7r-systerns, the basis functions are atomic orbitals (AO) 2pz /x = (plL, z, being the coordinate perpendicular to the molecular plane. In analogy to the previous model, the basis energies of the atomic orbitals and the cross terms between neighbouring pairs of AOs, [Pg.203]

Although diagrams like Fig. 6.1 are especially convenient to illustrate the qualitative features of TST and VTST, the solution of the equations of motion in (rAB,rBc) coordinates is complicated due to cross terms coupling the motions of the different species. It is for that reason we introduced mass scaled Jacobi coordinates in order to simplify the equations of motion. So, one now asks what does the potential function for reaction between A and BC look like in these new mass scaled Jacobi coordinates. To illustrate we construct a graph with axes designated rAB and rBc within the (x,y) coordinate system. In the x,y space lines of constant y are parallel to the x axis while lines of constant x are parallel to the y axis. The rAB and rBc axes are constructed in similar fashion. Lines of constant rBc are parallel to the rAB axis while lines of constant rAB are parallel are parallel to the rBc axis. From the above transformation, Equations 6.10 to 6.13... 

For the point group D4h, the atomic d-orbital functions belong to four different group-theoretical representations. When the same representation occurs more than once, as it does, for example, in trigonal point groups, M-1 will contain cross terms between orbitals of the same symmetry, as shown in Table 10.3(a). In this case, we are interested in the population of the symmetry-adapted orbitals ysk, such as defined for the trigonal case by expression (10.2). The symmetry-adapted orbitals are linear combinations of the original functions, that is,... 

Though the term functionally graded materials is relatively young, the concept is not. Nature has been producing FGMs for a long time. Figure 1.82 shows a cross... 

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. 

Note that the correlation function involves cross terms... 

With this consideration die relaxation equation will give rise to a set of coupled equations involving the time autocorrelation function of the density and the longitudinal current fluctuation, and also there will be cross terms that involve the correlation between the density fluctuation and the longitudinal current fluctuation. This set of coupled equations can be written in matrix notation, which becomes identical to that derived by Gotze from the Liouvillian resolvent matrix [3]. 

---