Orthogonalized quadratic term

We note that the coefficients for the quadratic terms are the same in both cases. However, the best fitting functions have a constant intercept and varying slopes, while the functions based on the orthogonalized quadratic term has a constant slope and varying intercept. 

We are interested only in the temperature dependence of the position and the width of the ZPL. We suppose that coordinates q in the linear term (vq) in equation (5) are orthogonal to the coordinates contributing to the quadratic term (qwq)/2. In this case the linear term does not contribute to these characteristics of the ZPL. This allows us to exclude this term from the further consideration. 

The next step in the development and implementation of the MR ccsd method is to include the quadratic terms and, in general, non-linear terms. Here, we should mention the orthogonally spin-adapted MR ccsD-1, mr ccsd-2 and MR ccsd-3 approximations developed by Paldus et al. [105] and tested for the H4 model system. The first two approximations were designed just for testing purposes in order to better assess the importance of various non-linear terms. All three approximations are extensive. They differ by the presence of quadratic and bi-linear terms in the direct component, as well as in the coupling terms in the equation for cluster amplitudes (4.87). To be more precise, in addition to absolute and linear terms, the MR ccsd- 1 method contains the quadratic term involved in the direct term the MR ccsd-2 method contains the quadratic term involved in both the direct component, as well as in the coupling terms, and, finally, the MR ccsd-3 method represents a fully quadratic MR ccsd approximation which considers all bi-linear terms. The main conclusions to be drawn from these studies are that the inclusion of quadratic terms eliminates the singular behaviour of the linear mr ccsd approximation, mr l-ccsd, (even at the mr ccsd- 1 level) and that the inclusion of bi-linear components usually further improves the results. 

There is another alternative set of contrasts which involves the four factors and some two-way interactions between them. In Table 18.10 the quadratic terms have been omitted. These contrasts, however, are not orthogonal. Half normal plots can be constructed for these sets of contrasts but their interpretation should be treated with caution because of the lack of orthogonality. 

Orthogonal polynomials are particularly useful when the order of the equation is not known beforehand. The problem of finding the lowest-order polynomial to represent the data adequately can be achieved by first fitting a straight line, then a quadratic curve, then a cubic, and so on. At each stage it is only necessary to determine one additional parameter and apply the f-test to estimate the significance of each additional term. 

The estimation was of the parameters of a second-degree polynomial fitted to 20 equally spaced points. Thus three coefficients were calculated corresponding to the constant term and the linear and quadratic orthogonal polynomials. The coefficients, obviously uncorrelated, were rescaled to have variance 1 under least squares and the Gaussian distribution. The discussion will be in terms of the sum of these three variances. 

By a near-quadratic surface, we mean a PES like that of Eq. (4.9) or (4.14), where the PES is taken to be a second-order Taylor expansion in directions orthogonal to the reaction path, although we will include cases where some higher-order terms are included. 

Follow through the derivation of the orthogonality-constrained variations of the matrices R, R2 in (6.5.14), and generalize the result to include terms quadratic in x, X2, x. 

---