Quadratic interaction terms

Can the relationship be approximated by an equation involving linear terms for the quantitative independent variables and two-factor interaction terms only or is a more complex model, involving quadratic and perhaps even multifactor interaction terms, necessary As indicated, a more sophisticated statistical model may be required to describe relationships adequately over a relatively large experimental range than over a limited range. A linear relationship may thus be appropriate over a narrow range, but not over a wide one. The more complex the assumed model, the more mns are usually required to estimate model terms. 

Among the most widely used ab initio methods are those referred to as Gl" and 02." These methods incorporate large basis sets including d and / orbitals, called 6-311. The calculations also have extensive configuration interaction terms at the Moller-Plesset fourth order (MP4) and fiirther terms referred to as quadratic configuration interaction (QCISD). ° Finally, there are systematically applied correction terms calibrated by exact energies from small molecules. 

Suppose, now, that the response from the experiment can be adequately represented by a model as in equation (22) but with the addition of pure quadratic and interaction terms for the design variables, x. For n design variables, X/, Xj,.. ., and m environmental variables, z, z, . . ., z , it is supposed that the model for the experiment is... 

In the remainder of this section it is desired to obtain the relative, constant-pressure heat capacity of the liquid at x=j and the concentration fluctuation factor for all compositions. Since the latter equation is complicated, it is not written out in full here. This has been done in Eqs. (37)-(45) of the paper by Liao et al. (1982) for the special case that 14 = 34 = 0 and / l3 is the only nonzero cubic interaction term, i.e., the version of the model applied here to the Ga-Sb and In-Sb binaries. Bhatia and Hargrove (1974) have given equations for the composition fluctuation factor at zero wave number for the special cases of complete association or dissociation and only quadratic interaction coefficients. 

The model Mr+1 contains the r-th degree term in the mixture components only along with the product of this term with the first degree terms in the Zj s. For example, a planar or first-degree model in the mixture components, and a main effects only model in the process variables, is y=M1+i+e. A planar model in the Xj s, combined with a main effect plus first-order interaction effects model in the Zj s, would be y=Mi+i+Mi+2+ . The model containing up to quadratic blending terms by main effects in the Zfs is defined as y=Mi+i+M2+l+H. This continues, up to the complete 2q+n-2n term model that is defined as ... 

Note that in the vicinity of the point considered, the potential V2 up to quadratic terms included does not depend on qy i.e., this potential is on the verge of the dynamical instability with respect to a shift in the q2 direction. Note also that the Jahn-Teller distortion parameter p0 increases if the quadratic interaction parameter w2 decreases. The reason for such a decrease may be, as it was already noticed above, the pseudo-Jahn-Teller effect. 

C(r)pH are derived from the independant variables (temperature (T), rhamnose concentration C(r) and pH. Thus the model is composed of a constant, 3 linear, 3 quadratic and 3 variable interaction terms. The models were refined by eliminating those terms which were not statistically significant. The resulting mathematical equations may be graphically represented as a response surface as shown in Figure 1. 

Thus reaction flavor generation may well be suitably investigated by systematic changes in all of the reaction condition variables in addition to the one-variable-at-a-time approach which is commonly employed. Such systematic change requires a suitable design such as the central composite factorial design which was used for the estimation of both primary, quadratic and variable interaction terms. 

In all cases, graphical or other methods should be used to access the adequacy of the fit obtained. These examinations often uncover residual patterns that may indicate the suitability of using a transformation, or some kind of weighting, or adding extra variables such as quadratic or interaction terms. Unfortunately, inference becomes almost impossible if the total subset of available predictors is augmented subjectively in this way. 

Intercept Linear terms Quadratic terms Interaction terms ... 

This model consists of 10 terms, impossible if only seven experiments are performed. How can the number of terms be reduced Arbitrarily removing three terms such as die quadratic or interaction terms has little theoretical justification. A major problem with the equation above is that the value of A2 depends on x and. t 2, since it equals 1 — A — a 2 so diere are, in fact, only two independent factors. If a design madix consisting of die first four terms of the equation above was set up, it would not have an inverse, and the calculation is impossible. The solution is to set up a reduced model. Consider, instead, a model consisting only of the first three terms ... 

A 10 parameter model is to be fitted to the data, consisting of the intercept, all single factor linear and quadratic terms and all two factor interaction terms. Set up the design matrix, and by using the pseudo-inverse, calculate the coefficients of the model using coded values. 

Using coded values, determine tire optimum conditions as follows. Discard the two interaction terms that are least significant, resulting in eight remaining terms in the equation. Obtain tire partial derivatives with respect to each of tire three variables, and set up three equations equal to zero. Show that tire optimum value of the third factor is given by —b jilb ), where the coefficients correspond to tire linear and quadratic terms in tire equations. Hence calculate tire optimum coded values for each of tire three factors. 

It is possible to convert die model of question 1 to a seven term model in two independent factors, consisting of two linear terms, two quadratic terms, two linear interaction terms and a quadratic term of the form v1v2(jr1 + jc2). Show how the models relate algebraically. 

Table 4 Summary of important electronic energies, for the interacting states of the fluorobenzene radical cations including the quadratic coupling terms (QVC). The diagonal values represent the minima of the diabatic potential energies, off-diagonal entries are minima of the corresponding intersection seams. Three dots (...) indicate missing results...

Third, in most cases in this paper, only the linear vibronic terms are considered. Meanwhile, it is known (Bersuker and Polinger, 1982, 1983) that the quadratic terms of vibronic interactions may be of primary importance in vibronic problems. Therefore the evaluation of the electric properties of molecules considering also the quadratic vibronic terms (in the cases where it has not been done yet) may result in new effects and regularities in this area. 

Again, this latter effect is of some importance. Referring to Table 4.10 and Eq. (4.7), we see that the vibrational dependence of the quartic term in the effective potential function is quite small, indeed within the quoted uncertainty. For cyclobutane, the reduced quartic potential constant is 26.15 0.07 cm-1 for the ground state and 26.12 0.07 cm-1 for the first excited state of the i>14 mode. On the other hand, the effect on the quadratic term is more noticeable, as expected from Eq. (4.7). For the ground state of p14, it is - 8.87 0.03 cm-1 compared to - 8.76 0.04 cm-1 for the excited state. From these data, we may conclude that the sign of the coefficient of the interaction term Q24Z2 is positive. 

Start with a model with only linear terms. If this gives sufficient infonnation for the choice of suitable experimental conditions, there is no need to try a more complicated model. If the linear model gives a poor description of the response, it is alway possible to run a set of complementary experiments to augment the model by adding also interaction terms. If this still gives a poor fit, then yet another set of complementary experiments can be run to include quadratic terms. This means that it will be possible to adjust the number of individual experiments to cope with the slope, the twist, and the curvature of the response surface. 

This last interaction term is harmonic becanse it is quadratic in the vibrational coordinates and harmonic terms are by definition those that are of degree less or eqnal to 2. go is defined in eq. (7.A5) of the appendix. The effect of this harmonic interaction term is to shift the centre of the band by a quantity eqnal in wavennmber to -( > (gg WqI c), with c the velocity of light. In the case of a mixed dimer the effect of this last term is, as already mentioned, negligible, because the two vibrations are not resonant. The magnitude of this term can conseqnently be taken equal to 110 cm in the case of (CD3COOH)2 and 80 cm in the case of (CD3COOD)2. These are values that are consistent with eq. (7.A7). The comparison of the bands of these mixed dimers with those of homodimers conseqnently conveys usefnl information on these stretching vibrations of H-bonds, more particnlarly on their harmonic interactions. 

---