Primary eigenvectors

The first step of the analysis consists in the diagonalization of the matrix that represents the hole (eqn (10)) in appropriate basis (usually canonical orbitals). This primary diagonalization yields the set of eigenvalues and eigenfunctions that are, in the second step subjected to the isopycnic transformation. Its aim is to transform the primary eigenvectors, that are usually delocalized over the nearest neighborhood of the reference basin, into the set of more localized functions whose resemblance with localized orbitals often allows the association with classical concepts of bonds, lone pairs etc., in terms of which chemists are used to classify the molecular structures. 

Factor analysis of the covariance matrix indicates two pure components, as anticipated. The two primary eigenvectors for the two real eigenvalues were used to construct the abstract eigenspectra in Fig. 3.34. 

In Fig. 3.35, a plot of the four ordered pairs of the two primary eigenvector mixtures is shown on the eigenvector axis, clearly indicating a linear relationship between the eigenvectors. 

The simplest and most widely used chemometric technique is Principal Component Analysis (PCA). Its objective is to accomplish orthogonal projection and in that process identify the minimum number of sensors yielding the maximum amount of information. It removes redundancies from the data and therefore can be called a true data reduction tool. In the PCA terminology, the eigenvectors have the meaning of Principal Components (PC) and the most influential values of the principal component are called primary components. Another term is the loading of a variable i with respect to a PQ. 

Spectral data are highly redundant (many vibrational modes of the same molecules) and sparse (large spectral segments with no informative features). Hence, before a full-scale chemometric treatment of the data is undertaken, it is very instructive to understand the structure and variance in recorded spectra. Hence, eigenvector-based analyses of spectra are common and a primary technique is principal components analysis (PC A). PC A is a linear transformation of the data into a new coordinate system (axes) such that the largest variance lies on the first axis and decreases thereafter for each successive axis. PCA can also be considered to be a view of the data set with an aim to explain all deviations from an average spectral property. Data are typically mean centered prior to the transformation and the mean spectrum is used a base comparator. The transformation to a new coordinate set is performed via matrix multiplication as... 

The sums in Eq. (61) always run over the complete sets of eigenvectors of the corresponding Hilbert sp2ices. The sums over u, in particular, run over all exact states with N particles, p and A lable the iV + 1, and k and a the N — 1 particle states. As already mentioned in the discussion of the Hilbert space Y of Eq. (6), symmetry considerations may reduce the number of states that actually couple to the primary states. Along with a reduced Hilbert space Y, then also less terms are needed in the diagonal representation (61) of the extended operator H. 

The direct Cl method was designed with the primary aim of avoiding explicit construction of a huge Hamiltonian matrix in the Cl eigenvalue problem (10). Eigenvalues and eigenvectors are calculated directly from the molecular integrals and this is why this variant of the Cl approach is called direct. A detailed description of the method may be found in several review articles. Here we restrict ourselves to a brief presentation of the essence of the method and we also note briefly some computational aspects. 

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