Cubic stretching constant

Hach molecular mechanics method has its own functional form MM+. AMBER, OPL.S, and BIO+. The functional form describes the an alytic form of each of th e term s in th e poteri tial. For exam pie, MM+h as both a quadratic and a cubic stretch term in th e poten tial whereas AMBER, OPES, and BIO+ have only c nadratic stretch term s, I h e functional form is referred to here as the force field. For exam pie, th e fun ction al form of a qu adratic stretch with force constant K, and equilibrium distance i q is ... 

The default parameters for bond stretching are an ec iiilibriiim bond length an d a stretch in g force eon starit. fb e fun etion al form isjiist that of the. M.M+ force field including a correction for cubic stretches. The default force constant depends only on the bond... 

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. 

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). 

Kuchitsu and co-workers5 7 were the first to introduce what is perhaps the simplest and most generally useful model, in which they assume all anharmonic force constants in curvilinear co-ordinates to be zero with the exception of cubic and quartic bond-stretching constants. These may be estimated from the corresponding diatomics, or from a Morse function, or they may be adjusted to give the best fit to selected spectroscopic constants to which they make a major contribution. This is often called the valence-force model. It is clear from the results on general anharmonic force fields quoted above that this model is close to the truth, and in fact summarizes 80 % of all that we have learnt so far about anharmonic force fields. 

From calculations made for a number of simple molecules, it has become clear that in the cubic and quartic part of the potential written in curvilinear valence-force coordinates, the diagonal bond-stretching force constants (fm and fmr) are much larger than the bending and interaction constants. On this observation is based the simplest model potential, the anharmonic simple valence-force (SVF) model that consists of a complete harmonic potentialf with only the diagonal cubic and quartic stretching constants) and fmr added,... 

The Cl + HC1 quantized transition states have also been studied by Cohen et al. (159), using semiclassical transition state theory based on second-order perturbation theory for cubic force constants and first-order perturbation theory for quartic ones. Their treatment yielded 0), = 339 cm-1 and to2 = 508 cm"1. The former is considerably lower than the values extracted from finite-resolution quantal densities of reactive states and from vibrationally adiabatic analysis, 2010 and 1920 cm 1 respectively (11), but the bend frequency to2 is in good agreement with the previous (11) values, 497 and 691 cm-1 from quantum scattering and vibrationally adiabatic analyses respectively. The discrepancy in the stretching frequency is a consequence of Cohen et al. using second-order perturbation theory in the vicinity of the saddle point rather than the variational transition state. As discussed elsewhere (88), second-order perturbation theory is inadequate to capture large deviations in position of the variational transition state from the saddle point. 

C Domingo, S Montero. Experimental determination of CH-stretching-bending-bending cubic force constants of ethane from Raman intensity analysis. J Chem Phys 86 6046-6058, 1987. 

The values given in parentheses are the deviations from the analytic cubic force constants. Displacement sizes for analytic first derivative (energy point) calculations were 0.01 ( 0.02) A for stretches and 0.02 (0.04) rad for bends. 

A larger value for the stretch force constant Kj. leads to a greater tendency for the bond to remain at its equilibrium distance rg Higher powers of r - rg, giving cubic, quartic, or higher terms are also common. A Morse function might also be employed. 

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. 

Some additional comments regarding Equation 4 are in order. The factor 143.88 converts the units to kcal/mol. There are two additional constants. The first is ks, which is the stretching force constant parameter in units of md A 1. The second constant is cs, which is the cubic term with a unitless value of 2.55. When a Morse potential is expanded in a power series, the factor 7/12 is obtained. 

Computer simulations of a range of properties of block copolymer micelles have been performed by Mattice and co-workers.These simulations have been based on bead models for copolymer chains on a cubic lattice. Types of allowed moves for bead chains are illustrated in Fig. 3.27. The formation of micelles by diblock copolymers under weak segregation conditions was simulated with pairwise interactions between A and B beads and between the A bead and vacant sites occupied by solvent, S (Wang et al. 19936). This leads to the formation of micelles with a B core. The cmc was found to depend strongly on fVB and % = x.w = %AS. In the range 3 < (xlz)N < 6, where z is the lattice constant, the cmc was found to be exponentially dependent onIt was found than in the micelles the insoluble block is slightly collapsed, and that the soluble block becomes stretched as Na increases, with [Pg.178]

The discussion so far may be summarized as follows. There are two reasons for using curvilinear co-ordinates to represent the anharmonic force field of a polyatomic molecule, despite their apparent complexity. The first is that it is only in this way that we obtain cubic and quartic force constants which are independent of isotopic substitution. The second is that in terms of curvilinear bond-stretching and angle-bending co-ordinates we obtain the simplest expression for the force field, in the sense that cubic and quartic interaction terms are minimized. The first reason is compulsive the second reason is not compulsive, but it does make the curvilinear co-ordinates very desirable. 

One conclusion is clear the dominant cubic and quartic interaction force constants are those associated with bond stretching, and these are not dissimilar to those of the corresponding diatomics. The same conclusion follows from a study of all other published data, and comparisons between bondstretching anharmonicity in related molecules are discussed further below (see Table 15). 

Table 12 shows the normal co-ordinate force constants for H20, calculated from the force field of Table 10, and it shows the major contributions of the internal co-ordinate force constants / to each . This table illustrates one important and general point that large contributions to the cubic and quartic values come from the quadratic / values. For example, the bending coordinate Q2, in which the atoms move in straight-line displacements, involves a positive displacement in the bond stretch Sr which increases as the square of... 

A body will obey Young s modulus only if it is stretched or compressed within its elastic limit if this limit is exceeded, structural failure ensues. For a one-dimensional system, or for a cubic crystal, Young s modulus reduces to the Hooke s law constant kH ... 

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