Redundant parameters

In order to eliminate parameters that are correlated to each other, we calculate their Pearson correlation coefficients (25). Linearly uncorrelated parameters have Pearson correlation coefficients close to zero and likely describe different aspects of the phenotype under study (exception for non-linearly correlated parameters which cannot be scored using Pearson s coefficient). We have developed an R template in KNIME to calculate Pearson correlation coefficients between parameters. Redundant parameters that yield Pearson correlation coefficients above 0.4 are eliminated. It is important to visually inspect the structure of the data using scatter matrices. A Scatter Plot and a Scatter Matrix node from KNIME exist that allow color-coding the controls for ease of viewing. 

Thus the Hessian will become singular if we include rotations between the active orbitals. Redundant parameters must not be included in the Newton-Raphson procedure.They are trivial to exclude for the examples given above, but in more general cases a redundant variable may occur as a linear combination of S and T and it might be difficult to exclude them. One of the advantages of the CASSCF method is that all parameters except those given above are non-redundant. 

Let be the set of values k with this property, and define an algorithm A as A followed by the conversion of m and m into jU-tuples of numbers, and without the redundant parameter 1 . Both codej g gg and this conversion are injective. Hence for all fe e K ... 

Further redundancies are represented by rotations amongst the inactive (occupied) orbitals or amongst the secondary (virtual) orbitals as they do not change the quasienergy they are straightforwardly eliminated within the chosen parameterization. We will reserve indices i, j, k and I for inactive orbitals, indices a, b, c and d for secondary orbitals and indices p, q, r and, v for general orbitals. In terms of non-redundant parameters the k operator can then be written... 

Using a redundant internal coordinate set as in the case of Figures 3e - 3h, a significant improvement of all correlation patterns can be observed. This has to do with the fact that with increasing size of the redundant parameter set c-vectors adopt more the form of a-vectors [19]. For example, in the case of the noruedundant internal coordinate set the average overlap between adiabatic and c-vectors is 0.69, which means that the two types of internal mode vectors are... 

The principle of parsimony (de Noord, 1994 Flury and Riedwyl, 1988 Seasholtz and Kowalski, 1993) states that if a simple model (that is, one with relatively few parameters or variables) fits the data then it should be preferred to a model that involves redundant parameters. A parsimonious model is likely to be better at prediction of new data and to be more robust against the effects of noise (de Noord, 1994). Despite this, the use of variable selection is still rare in chromatography and spectroscopy (Brereton and Elbergali, 1994). Note that the terms variable selection and variable reduction are used by different researchers to mean essentially the same thing. 

Both from the explicit form of Eq. (3.14) or from the defining equations of the invariant manifold It we get the tangent space of the redundant parameter manifold (see Eq. (3.3))... 

A complete analysis of the redundancy problem for a nonlinearly parametrized wave function such as the MCSCF wave function is complicated and will not be attempted here. Instead, we shall be content with obtaining a set of simple conditions that are sufficient for identifying most (if not all) of the redundancies that occur in practice. As a starting point for our discussion, we classify a set of parameters A as redundant if there exists a smaller set B such that all states that can be described by A can also be described by B. For MCSCF wave functions in the form (12.2.1), this definition of redundancy means that, for any choice of c and of a redundant parameter set A, we may find a set of parameters k and c belonging to a smaller set B such that... 

To illustrate the identification of redundant parameters, we consider the three CSFs of Section 10.1.1. Since, according to Section 3.3, no diagonal elements Kpp appear in the special orthogonal orbital transformation, we need to examine only the nondiagonal elements. For the closed-shell CSF (10.1.2), we have for rotations among inactive orbitals... 

More generally, we conclude that the inactive-inactive and secondary-secondary rotations are always redundant, whereas the inactive-active, inactive-secondary and active-secondary rotations are always nonredundant. The active-active rotations, by contrast, may or may not be redundant depending on the structure of the CSF, and their redundancy must be established for each CSF separately. In Section 12.2.6, we shall see how redundant parameters can be identified for multiconfigurational wave functions and present a simple general prescription for identifying the nonredundant active-active rotations for single-configuration CSF wave functions. 

In any Newton-based optimization - which, as discussed in Section 10.8, implicitly or explicitly requires the inversion of the Hessian matrix - the inclusion of redundant parameters is not only unnecessary but also undesirable since, at stationary points, these parameters make the electronic Hessian singular. The singularity of the Hessian follows from (10.2.8), which shows that the rows and columns corresponding to redundant rotations vanish at stationary points. Away from the stationary points, however, the Hessian (10.1.30) is nonsingular since the gradient elements that couple the redundant and nonredundant operators in (10.2.5) do not vanish. Still, as the optimization approaches a stationary point, the smallest eigenvalues of the Hessian will tend to zero and may create convergence problems as the stationary point is approached. Therefore, for the optimization of a closed-shell state by a Newton-based method, we should consider only those rotations that mix occupied and virtual orbitals ... 

The unitary matrix exp(—ic ) transforms the inactive orbitals among themselves, the active orbitals among themselves, and the virtual orbitals among themselves. For the optimization of the wave function, it is convenient to set the redundant parameters equal to zero in each iteration since this choice of k simplifies the orbital-rotation operator ic. In Exercise 10.2, it is shown that, for a closed-shell state, the choice of k = 0 leads to transformed MOs that (in the least-squares sense) are as similar as possible to the original MOs - that is, to the transformed MOs that have the largest possible overlap with the original ones. 

Although we may keep the redundant parameters fixed (equal to zero) during the optimization of the Hartree-Fock state, we are also free to vary them so as to satisfy additional requirements on the solution - that is, requirements that do not follow from the variational conditions. In canonical Hartree-Fock theory (discussed in Section 10.3), the redundant rotations are used to generate a set of orbitals (the canonical orbitals) that diagonalize an effective one-electron Hamiltonian (the Fock operator). This use of the redundant parameters does not in any way affect the final electronic state but leads to a set of MOs with special properties. 

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