Stretch-Bends

We have seen that resonance couplings destroy quantum numbers as constants of the spectroscopic Hamiltonian. Widi both the Darling-Deimison stretch coupling and the Femii stretch-bend coupling in H2O, the individual quantum numbers and were destroyed, leaving the total polyad number n + +... 

By combining the Lagrange multiplier method with the highly efficient delocalized internal coordinates, a very powerfiil algoritlun for constrained optimization has been developed [ ]. Given that delocalized internal coordinates are potentially linear combinations of all possible primitive stretches, bends and torsions in the system, cf Z-matrix coordinates which are individual primitives, it would seem very difficult to impose any constraints at all however, as... 

A restrain t (not to be confused with a Model Builder constraint) is a nser-specified one-atom tether, two-atom stretch, three-atom bend, or four-atom torsional interaction to add to the list ol molec-11 lar mechanics m teraction s computed for a molecule. These added iiueraciious are treated no differently IVoin any other stretch, bend, or torsion, except that they employ a quadratic functional form. They replace no in teraction, on ly add to the computed in teraction s. 

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). 

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. 

If atom i or atom j is a hydrogen, the deformation (r-r ) is considered to be zero. Thus, no stretch-bend interaction is defined for XH2 groups. The stretch-bend force constants are incorporated... 

The constant 2.51118 converts between MM-t stretch-bend force constants expressed in millidynes per radian and HyperChem s default, kcal per degree. 

D. Frequencies Molecules vibrate (stretch, bend, twist) even if they are cooled to 0 K. This is the basis of infrared/Raman spectroscopy, where absorption of energy occurs when the frequency of molecular... 

The components in i cross are usually written as products of Taylor-like expansions in the individual coordinates. The most important of these is the stretch/bend term which for an A-B-C sequence may be written as... 

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. 

While the vibrations (stretching, bending, torsion) in high symmetrical rings (Ss, Ss, S12) are almost uncoupled [80], the vibrations in the low symmetrical Sy ring are heavily mixed, especially the bending and torsional modes [81]. 

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