Quadratic stretch

Each molecular mechanics method has its own functional form MM+, AMBER, OPES, and BIO+. The functional form describes the analytic form of each of the terms in the potential. Eor example, MM+ has both a quadratic and a cubic stretch term in the potential whereas AMBER, OPES, and BIO+ have only quadratic stretch terms. The functional form is referred to here as the force field. Eor example, the functional form of a quadratic stretch with force constant Kj. and equilibrium distance rQ is ... 

Each force field includes a set of atom types. Consider the quadratic stretch term shown above. In principle, every different bond in every molecule would have its own parameters rg and Kj.. This... 

The cubic stretch term is a factor CS times the quadratic stretch term. This constant CS can be set to an arbitrary value by an entry in the Registry or the chem. ini file. The default value for MM2 and MMh- is CS=-2.0. 

Figure 6. Quadratic stretching force constants, X(MX), calculated for the MX bonds comprising coordination polyhedra for nitride, oxide and sulfide molecules plotted against the geometry optimized bond lengths, R(MX) cdculated with molecular orbital methods (Hill 1995). The triangles denote MX bonded interactions involving row-one M- and X-atoms, the squares denote MX interactions involving row-one and row-two M- and X-atoms and the circles denote bonded interactions involving row-two M- and X atoms.

Figure 7. Quadratic stretching force constants, /c(MX), used to construct Figure 6 (Hill 1995) plotted against the compressibilities estimated for the coordination polyhedra with an expression defined by Hazen and Prewitt (1977). The symbols are defined in the legend of Figure 6.

Rahman potential is an extension of the SPC model. The extension is the inclusion of quadratic stretch and bend terms, so that internal degrees of freedom are present. The model was used to examine the effect of the liquid environment on internal modes, and the pair functions were found to be close to those for the SPC model. In contrast. Teleman et al., ° who used the same SPC model with a harmonic intramolecular potential, reported some large effects on properties. However, Barrat and McDonald find that flexibility has a small effect on calculated properties, stating... 

Th c fun ction al form for bon d stretch in g in HlOa, as in CHARMM, is quadratic only and is identical to that shown in equation (1 1) on page I 75. Th e bond stretch in g force con stan ts are in units of... 

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. 

A cubic bond-stretching potential passes through a maximum but gives a better approximation to the Morse e close to the equilibrium structure than the quadratic form. 

In the case of ethylene, because of 2-fold symmetry, odd terms drop out of the series, V3, V5,... = 0. In the case of ethane, because of 3-fold symmeti-y, even temis drop out, V2, V4,... = 0. Terms higher than three, even though permitted by symmetry, are usually quite small and force fields can often be limited to three torsional terms. Like cubic and quaitic terms modifying the basic quadratic approximation for stretching and bending, terms in the Fourier expansion of Ftors (to) beyond n = 3 have limited use in special cases, for example, in problems involving octahedrally bound complexes. In most cases we are left with the simple expression... 

A larger value for the bending force constant K0 leads to a greater tendency for the angle to remain at its equilibrium value 0g. There may be cubic, quartic, etc. terms as with the corresponding bond stretch term in addition to the quadratic term shown here. 

The functional form for bond stretching in AMBER is quadratic only and is identical to that shown in equation (11) on page 175. The bond stretching force constants are in units of kcal/mol per A and are in the file pointed to by the QuadraticStretch entry for the parameter set in the Registry or the chem.ini file, usually called =>istr.txt(dbf). 

Figure 1. Bottom panel OH stretch frequencies, to , for water clusters and the surrounding point charges, versus electric field ) (in atomic units). The solid line is the best quadratic fit. Top panel Dipole derivative, / ], (relative to the gas phase value) for water clusters and the surrounding point charges, versus electric field ). The solid line is the best linear fit.

Figure 2. The histogram is the distribution of OH stretch frequencies for the water clusters and surrounding point charges, and the solid line is the distribution of frequencies from the quadratic electric field map.

The stretching energy is given by a sum of quadratic (harmonic) and cubic teims ... 

---