Orthogonal scheme

Given a set of independent vectors x1 x2,. .., xn, it is requested to form a new set yk,y2, y , of orthogonal vectors spanning the same space. The Gram-Schmidt orthogonalization scheme sets... 

To alleviate a number of these problems, Lowdin proposed that population analysis not be carried out until the AO basis functions tp were transformed into an orthonormal set of basis functions / using a symmetric orthogonalization scheme (Lowdin 1970 Cusachs and Politzer 1968)... 

Orthogonal scheme, 359 2-Oxacydopentylidene ortho esters, to protect 1,2-diols, 137... 

The performance of DORTHO, the double precision version of ORTHO, was a very pleasant surprise to the writers. Up through the fitting of fifteenth degree polynomials the first six digits of each coefficient were the same whether monomials or Chebyshevs were used for a coordinate system. This means that the internal orthogonalization scheme built into ORTHO and DORTHO functioned very effectively and that the double-precision arithmetic avoided meaningful roundoff errors. From this, we conclude that... 

To prevent contamination of the desired eigenvector by modes corresponding to overall translation or rotation, a simple orthogonalization scheme was applied both to the vector y and the derivative dX/dy. After the calculations described in this review were completed we switched all minimizations from conjugate-gradient to a new approach that is significantly faster [342]. 

U. As U is neither normalized nor orthogonal, we introduce an overlap matrix S defined in Equation 15 and use Lowdin s symmetric orthogonalization scheme to... 

A problem with the matrix elements we dealt with up to now is that in the limit p1 or p- x all of them reduce to matrix elements involving the dominant eigenstate, although symmetries might be used to yield other eigenstates besides the absolute dominant one. However, if symmetries fail, one has to employ the equivalent of an orthogonalization scheme, such as, discussed in the next section, or one is forced to resort to evolution operators that contain, in exact or in approximate form, the corresponding projections. An example of this are matrix elements computed in the context of the fixed-node approximation [18], discussed in Section VI.A.2. Within the framework of this approximation, one considers quantities of the form... 

Here it should be noted that the resulting orthogonal linear combinations depend greatly on the coordinates chosen since the orthogonalization scheme depends on the order of xl/im(hy) s. Usually the definition of the principal axis of a molecule as the z axis gives a chemically reasonable result. 

The use of metal reagents such as silver trifluoromethanesulphonate to remove 5-Acm and 5-Mob groups in acid media has been alluded to earlier (Section 3.1). Fujii and co-workers have developed a two-step procedure to prepare disulphides based on the idea that the mercaptide intermediate can then be oxidized directly, for example by treatment with DMSO (40,41,95). The oxidation is carried out in the presence of aqueous HCI, leading to concomitant precipitation of AgQ. These conditions are reported to avoid oxidative decomposition of Tip residues, and to not affect pre-existing disulphides formed in an orthogonal scheme (41). 

The orthogonalization scheme is summarized in Table 13. Here D represents initially selected descriptors, i =... 

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