Cubic term

It might be asked what happens when one adds further couplings beyond the quadratic one In the next higher order one finds a scalar cubic term of the form ... 

An undesirable side-effect of an expansion that includes just a quadratic and a cubic term (as is employed in MM2) is that, far from the reference value, the cubic fimction passes through a maximum. This can lead to a catastrophic lengthening of bonds (Figure 4.6). One way to nci iimmodate this problem is to use the cubic contribution only when the structure is ,utficiently close to its equilibrium geometry and is well inside the true potential well. MM3 also includes a quartic term this eliminates the inversion problem and leads to an t". . 11 better description of the Morse curve. 

It was shown above that the cubic term in the potential function for the anharmonic oscillator cannot, for reasons of symmetry, contribute to a first-order perturbation. However, if the matrix elements of = ax3 are evaluated, it is found that this term results in a second-order correction to the... 

As an illustration consider then a zero-field Hamiltonian for S = 4 in which we have retained only the familiar axial D- and rhombic E-term plus the cubic terms that split the non-Kramer s doublets in first order ... 

If the intradoublet splitting is distributed around a nonzero value A i.e., if a rhombic E-term and/or higher order cubic terms (cf. Section 8.1) are nonzero, then we have... 

In the study of Watkins et al. (1990), the anharmonic coupling was identified with the cubic term of an expansion of the potential energy,... 

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. 

The algebraic vibrational analysis should be compared with the vibrational analysis carried out using the Dunham expansion. The quality of the fit of Table 4.8 is equivalent to that of a Dunham expansion with cubic terms... 

Calculations of vibrational spectra of bent triatomic molecules with second order Hamiltonians produce results with accuracies of the order of 1-5 cm-1. An example is shown in Table 4.9. These results should again be compared with those of a Dunham expansion with cubic terms [Eq. (0.1)]. An example of such an expansion for the bent S02 molecule is given in Table 0.1. Note that because the Hamiltonian (4.96) has fewer parameters, it establishes definite numerical relations between the many Dunham coefficients similar to the so-called x — K relations (Mills and Robiette, 1985). For example, to the lowest order in l/N one has for the symmetric XY2 case the energies E(vu v2, V3) given by... 

The correction to the coefficient of in the dynamic dilution (cubic) term in the potential (compare Eq. 22 for the pure star case) arises from the way the difference in arm molecular weights affects the fraction of unrelaxed arm at the same timescale. 

It must be emphasized that t enters into the expression for 0(j, where it results in increased shielding with increasing effective nuclear charge. However, it only enters as a first-order term, so its effect is much less significant than in the already dominant Op term, where it enters as a cubic term. 

A linear term has been observed in the specific heats of the larger metal cluster compounds measured down to 20 mK [49, 56], The value of the linear term of these cluster compounds, which have metal cores of Pt309 and Pd 551, is only a fraction ( 1/3) of the bulk value. We might extrapolate to AU55, and use a fraction smaller than 1/3. But even using 1/3, the linear term would only become equal to the cubic term at about 15 mK. 

Even in these instances, however, there is some utility to considering die Morse function. If we approximate the exponential in Eq. (2.5) as its infinite series expansion tiuncated at the cubic term, we have... 

Here "he quantity q is a measure of the relative rates of the quadratic and cubic terms... 

The values of t/) for HC135 are listed in Table 4.3, which also gives the first and second differences. We see that A2o0 is nearly constant, and that vexe is about 51 j cm-1. The small variation in the A2o0 values can be attributed to the cubic term (4.79) in the vibrational energy, and to experimental error, perhaps. 

Before we go on to consider functional forms for all of the components of a molecule s total steric energy, let us consider the limitations of Eq. (2.2) for bond stretching. Like any truncated Taylor expansion, it works best in regions near its reference point, in this case req. Thus, if we are interested primarily in molecular structures where no bond is terribly distorted from its optimal value, we may expect Eq. (2.2) to have reasonable utility. However, as the bond is stretched to longer and longer r, Eq. (2.2) predicts the energy to become infinitely positive, which is certainly not chemically realistic. The practical solution to such inaccuracy is to include additional terms in the Taylor expansion. Inclusion of the cubic term provides a potential energy function of the form... 

In their turn, the equations (4) and (5) correspond to taking account of two more terms of this expansion respectively. (It was erroneously stated in [41] that the coefficient of the cubic term of equation (5) equals -0.01, which was then copied in all the experimental studies that used this approximation.) However, it is not always acknowledged that these two equations also assume the absence of aggregation of defects the authors [41] propose using of equation (5) up to the saturation concentration. Therefore it is not justified to extrapolate the rate of accumulation of defects to a zero value for obtaining n0 from equations (3)-(5) (cf. [43,44,46]). Actually the saturation concentration can exceed several times the value predicted by these equations. 

---