Full canonical form

In conclusion, the Box-Wilson composite design is a convenient method for modeling of product yield as a function of reaction parameters (independent variables) especially when their number exceeds 2. In the dehydrogenation of n-decane, effect of reaction parameters on monoene selectivity / yield and n-decane conversion are represented satisfactorily by full II degree polynomial equations. The canonical form of the equations in the present study is indicative of an approximately stationary ridged system, with the reaction parameters close to center of design being optimum for monoene yield at conversion levels of 12 - 13 %. The polynomial equations were found to be consistent, with mechanistic considerations. 

Figure 6.8c displays the calculations in the form resulting from the application of the canonical orthonormalization procedure of Section 3.6 and Table 6.1 shows the full iterative process for the dihydrogen calculation within the HFS-SCF approximation based on the spreadsheet design in fig6-8.xls. 

We shall consider first the one-electron aspects of the model. The spin space and the orbital space are both two-dimensional. We find an amusing parallel between the three canonical basis sets that can be used to span the former (adapted to the x, y, and z molecular axes) and the three that can be used to span the latter (delocalized, complex, and localized). More general basis sets in either space can be produced by applying the rotation operator for a particle of spin 1/2. Since we shall eventually deal with the exact solutions both in the spin space and in the real space (equivalent to full Cl), the choice of the one-electron basis is immaterial. In practice, as we construct the two-electron wave functions from one-electron wave functions, we have to choose the latter somehow, and this choice is frequently dictated by convenience. For instance, the two-dimensional active space of orbitals may have been defined for us by an open-shell SCF calculation on the triplet state of a series of related biradicaloids, which yielded two singly occupied orbitals in each case, but not necessarily in either the most localized or the most delocalized form, or even in similar forms for the different mole-... 

In calculating the canonical Hartree-Fock orbitals, we carry out a full diagonalization of the Fock matrix. Complete diagonalization is not necessary, however, to ensure that the elements of the electronic gradient vanish. Indeed, any set of orbitals that brings the Fock matrix into block-diagonal form with vanishing occupied-virtual elements... 

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