Bending angle

The deviation of angles from their reference values is also frequently described using a Hooke s law or harmonic potential  

The contribution of each angle is characterised by a force constant and a reference value. Rather less energy is required to distort an angle away from equilibrium than to stretch or compress a bond, and the force constants are proportionately smaller, as can be observed in Table 4.2. 

The quadratic angle bending term in MM+ is identical to that of equation (12) on page 175, apart from a factor 1/2. Three Gq values 

MMh- also includes a sextic angle bending term. The final form for the angle bending energy is  

The sextic bending term is a scale factor SF times the quadratic bending term. This constant SF can be set to an arbitrary value by an entry in the Registry or the chem.ini file. The default value for 

MMh- is SF = 7.0 x 10 . The constant 0.043828 converts the MMh-bending constants expressed in millidyne-Aper radian to 

The MM+ force field uses special values for the bending force constants when the atoms are in a three- or four-membered ring. 

A similar equation holds for angle bending (Eq. 2.43), where 0 is the value of the angle being evaluated, is analogous to a force constant, and 0o is the natural bond angle. 

there is a pair of parameters for each kind of angle. As with bond stretching, this parabolic-type function is often not optimal, and so a cubic term is added (Eq. 2.44). 

The simplest form for a torsional potential function is Eq. 2.45, where n is the foldedness of the barrier, and B = 1. If B = -1-1, then the staggered form of the bond is preferred, whereas if B = -1, the eclipsed form of the bond is preferred. 

When do we ever want 6 = -1 We want it for C-C double bonds, as in ethylene or benzene Remember, molecular mechanics knows nothing about tt bonds or molecular orbitals. We have to explicitly tell it that a double bond wants to be planar (i.e., eclipsed with a twofold barrier). Again, every particular torsion type has its own set of parameters. 

More modern force fields have found that an expanded torsional equation is beneficial (Eq. 2.46). 

As for bond stretching, the simplest description of the energy necessary for a bond angle to deviate firom the reference value is a harmonic potential following Hooke s law, as shown in Eq. (22). 

For every type of angle including three atoms, two parameters (force constant fe and reference value 0q) are needed. Also, as in the bond deformation case, higher-order contributions such as that given by Eq. (23) are necessary to increase accuracy or to account for larger deformations, which no longer follow a simple harmonic potential. 

Force fields like MM3, MM4, CFF, or MMFF therefore use cubic and/or quartic or even higher contributions, up to the sixth power. Special attention has to be paid in the case of reference angles approaching 180°, e.g., for molecules with linear firag-ments such as acetylene compounds. In this circumstance, replacing Eq. (23) by a 

However, as more and better structural data have become available since the original formulation of MM2, it became desirable to introduce a cubic term, which has a substantial effect when the bending is more than about 10-15°. Still higher terms have also been used in MM3, but they are not well defined. 

Angle bending has been divided into in-plane and out-of-plane bending modes for planar trigonal centers such as a carbonyl carbon. This division was 

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In this article we describe an extension of SISM to a system of molecules for which it can be assumed that both bond stretching and angle bending describe satisfactorily all vibrational motions of the molecule. The SISM presented here allows the use of an integration time step up to an order of magnitude larger than possible with other methods of the same order and complexity. 

SISM Treatment of Bond Stretching and Angle Bending Terms... 

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). 

Figure 7-13. Cross-terms combining internal vibrational modes such as bond stretch, angle bend, and bond torsion within a molecule.

Directly bonded (a 1-2 bond stretch relationship) Geniinal to each other (a 1-3 angle bending relationship)... 

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... 

Like MM2, MM+ in eludes eonplin g bcLween bond slrcLcb ing an d angle bending. If the angle is defined lo include aloms i, j, and k, where k is lb e cen iral atom, ih en MM + couples slrelch in g of the ik and jk bonds wilb ibc angle ... 

Thii functional form for angle bending in GIO+ is quadratic only and IS identical wiLh that shown in equation (12) on page 175. The... 

A typical force field model for propane contains ten bond-stretching terms, eighteen angle-bending terms, eighteen torsional terms and 27 non-bonded interactions. 

In a Urey-Bradley force field, angle bending is achieved using 1,3 non-bonded interaction rather than an explicit angle-bending potential. The stretch-bond term in such a forci field would be modelled by a harmonic function of the distance between the 1,3 atoms ... 

SHAPES angle-bending term is very similar ... 

Three-body and higher terms are sometimes incorporated into solid-state potentials. The Axilrod-Teller term is the most obvious way to achieve this. For systems such as the alkali halides this makes a small contribution to the total energy. Other approaches involve the use of terms equivalent to the harmonic angle-bending terms in valence force fields these have the advantage of simplicity but, as we have already discussed, are only really appropriate for small deviations from the equilibrium bond angle. Nevertheless, it can make a significant difference to the quality of the results in some cases. 

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