Separation theorems for singular values of matrices and their applications in multivariate analysis

Separation theorems for singular values of a matrix, similar to the Poincaré separation theorem for the eigenvalues of a Hermitian matrix, are proved. The results are applied to problems in approximating a given r.v. by an r.v. in a specified class. In particular, problems of canonical correlations, reduced rank regression, fitting an orthogonal random variable (r.v.) to a given r.v., and estimation of residuals in the Gauss-Markoff model are discussed. In each case, a solution is obtained by minimizing a suitable norm. In some cases a common solution is shown to minimize a wide class of norms known as unitarily invariant norms introduced by von Neumann.     

Near-infrared absorption in symmetric squarylium and croconate dyes: a comparative study using symmetry-adapted cluster-configuration interaction methods

Symmetric croconate (CR) and squarylium dyes (SQ) are well-known near-infrared (NIR) dyes and, in general, are considered to be donor-acceptor-donor type molecules. It is established in the literature that CR dyes absorb in a longer wavelength region than the corresponding SQ dyes. This has been attributed to the CR ring being a better acceptor than the SQ ring. Thus increasing the donor capacity should lead to a bathochromic shift in both SQ and CR. On the other hand, some experiments reported in the literature have revealed that increasing the conjugation in the donor part of the SQ molecule leads first to red shift, which upon a further increase of the conjugation changes to a blue shift. Hence, to understand the role of the central ring and the substitutions in the absorption of these dyes, we carried out high-level symmetry-adapted cluster-configuration interaction (SAC-CI) calculations of some substituted SQ and CR dyes and compare the absorption energy with the existing experimental data. We found that there is very good agreement. We also carried out SAC-CI calculations of some smaller model molecules, which contain the main oxyallyl substructure. We varied the geometry (angle) of the oxyallyl subgroup and the substitution in these model molecules to establish a correlation with the bathochromic shift. We found that the charge transfer is very small and does not play the key role in the red shift, but on the other hand, the perturbation of the HOMO-LUMO gap (HLG) from both the geometry and substitution seems to be responsible for this shift. We suggest as a design principle that increasing the donor capacity of the groups may not help in the red shift, but introducing groups which perturb the HLG and decrease it without changing the MO character should lead to a larger bathochromic shift.     

Role of the oxyallyl substructure in the near infrared (NIR) absorption in symmetrical dye derivatives: A computational study

It is well-known from experimental studies that the oxyallyl-substructure-based squarylium and croconium dyes absorb in the NIR region of the spectrum. Recently, another dye has been reported (J. Am. Chem. Soc. 2003, 125, 348) which contains the same basic chromophore, but the absorption is red-shifted by at least 300 nm compared to the former dyes and is observed near 1100 nm. To analyze the reasons behind the large red shift, in this work we have carried out symmetry-adapted cluster-configuration interaction (SAC-CI) studies on some of these NIR dyes which contain the oxyallyl substructure. From this study, contrary to the earlier reports, it is seen that the donor groups do not seem to play a major role in the red-shift of the absorption. On the other hand, on the basis of the results of the high-level calculations carried out here and using qualitative molecular orbital theory, it is observed that the orbital interactions play a key role in the red shift. Finally, design principles for the oxyallyl-substructure-based NIR dyes are suggested.     

Potassium dichromate as an oxidimetric reagent

A critical and historical review is made of the use of potassium dichromate in oxidimetry.