Stretch-bend interactions

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 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. 

Stretching constants in mdyn/A, deformation constants in mdyn A/ radian , and stretch-bend interaction constants in mdyn/radian. 

Ideally, the Stanger demonstration should be an expression of the stretch-bend interaction in the Bj normal mode in benzene. Thus, it is instructive to examine the nature of the stretch-bend interaction in typical monocyclic and bicyclic alkenes. The IR spectra of monocyclic and bicyclic alkenes shift hypsochromically for exomethylenes and bathochromically for endocyclic alkenes with decreasing ring size. " This effect has been shown to be dominated by mode-coupling for the monocyclic series, but consistent with a pure angular effect or the bicyclic series.Further investigation in this vein is likely to resolve the paradox initiated by Stanger s work. 

To obtain the anharmonic terms in the potential, on the other hand, the choice of coordinates is important 130,131). The reason is that the anharmonic terms can only be obtained from a perturbation expansion on the harmonic results, and the convergence of this expansion differs considerably from one set of coordinates to another. In addition it is usually necessary to assume that some of the anharmonic interaction terms are zero and this is true only for certain classes of internal coordinates. For example, one can define an angle bend in HjO either by a rectilinear displacement of the hydrogen atoms or by a curvilinear displacement. At the harmonic level there is no difference between the two, but one can see that a rectilinear displacement introduces some stretching of the OH bonds whereas the curvilinear displacement does not. The curvilinear coordinate follows more closely the bottom of the potential well (Fig. 12) than the linear displacement and this manifests itself in rather small cubic stretch-bend interaction constants whereas these constants are larger for rectilinear coordinates. A final and important point about the choice of curvilinear coordinates is that they are geometrically defined (i.e. independent of nuclear masses) so that the resulting force constants do not depend on isotopic species. At the anharmonic level this is not true for rectilinear coordinates as it has been shown that the imposition of the Eckart conditions, that the internal coordinates shall introduce no overall translation or rotation of the body, forces them to have a small isotopic dependence 132). 

Here, f, /i2, /i3 and /33 are the stretching, stretching-stretching interaction, stretching-bending interaction, and bending force constants, respectively, and r (equilibrium distance) is multiplied to make /i3 and /33 dimensionally similar to the others. Using matrix expression, Eq. (1-86) is written as... 

Force constants are often expressed in mdyn A-1 = aJ A-2 for stretching coordinates, mdyn A = aJ for bending coordinates, and mdyn = ajA-1 for stretch-bend interactions. See [17] for further details on definitions and notation for force constants. 

The recent inclusion of a specific stretch-bend interaction in calculations of molecular geometry and energy by Allinger et al. 22) did not necessitate major revisions as far as the stretching constants or zq are concerned. It is known from vibrational analyses of alkanes and cycloalkanes that when force fields specifically include Urey-Bradley forces, this must be accompanied by a vast reduction of the C—C stretching constant in order to reproduce the experimental C—C stretching frequencies. However, reduction of kcc also necessitates a substantial reduction of zo(CC) in calculation of geometry because the balance between the sum of the non-... 

Intimately connected to the choice of bending parameters is the choice between a general valence force (GVFF) expression (including a stretch-bend interaction) or a UBFF representation. The necessity for the inclusion of either a stretch-bend or a Urey-Bradley term to account for the abnormally long bonds in e.g. norbomane was pointed out by several authors 17,22,31), Several representations have been proposed normal non-bonded interactions 24) the classical UB expression 21) and whittled atoms 7,31,56) (smaller radii in the direction of the geminal atoms), but a further theoretical analysis of the UB potentials seems mandatory. 

Jones neglected the bend-bend and the stretch-bend interaction constants for the compounds M(CO)6 (M = Cr, Mo, or W), so that the F matrices for the a g, e, and vibrations included eleven valence force constants. Further restrictions were introduced by assuming a range of values for the MC-CO stretch-stretch interaction constants, so that the equations could be solved for the eight remaining force constants using the eight frequencies observed directly in the Raman and infrared spectra (195). Other force constant calculations have been made for the compounds M(CO)e (M = Cr, Mo, or W) (71, 72, 75, 109, 110, 266), Fe(CO)5 (141), and Ni(C0)4 (27, 28, 107) in which selected interaction constants were equated to zero. It is clear that, even in these complete calculations, considerable uncertainty arises in the values of the force constants because of these approximations. 

The effect of this term is very similar to the effect of the stretch-bend interaction. The distance between atoms 1 and 3 is forced toward tq. If either the angle or the bond length is distorted, the other one adjusts to make r nearer to ro, so as to keep the energy at a minimum, similar to what happens with the stretch-bend interaction. 

The most commonly used cross terms are stretch-bend, stretch-stretch for two bonds to the same atom, stretch-torsion, bend-torsion, and bend-bend for two angles with a common central atom. MM2 and MMFF94 include only stretch-bend interactions. The TRIPOS, AMBER, CHARMm, DREIDING, and UFF force fields have no cross terms. MM3 and MM4 include stretch-bend, bend-bend, and stretch-torsion terms. 

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