Gram-Schmidt procedure

We never explicitly utilize the LMOs of the surroundings. They could be determined in a manner similar to the local space LMOs except that they must be orthogonal to the latter. In principle, the usual Gram-Schmidt procedure could be employed for that purpose. Finally, it is convenient to symmetrically orthogonalize the local space LMOs which involves only local space operations since, from Eq. (19), S M0 = BL Rl B ... 

In particular, the specification or modification of the d mole fraction variables, Xi, is done in the (d - l)-dimensional hyperplane orthogonal to the d-dimensional vector (1,1,..., l). lhis procedure ensures that the constraint Xi = 1 is maintained. This subspace is identified by a Gram-Schmidt procedure, which identifies a new set of basis vectors, u,, that span this hyperplane. Figure 3 illustrates the geometry for the case of three composi-... 

The remaining u,, 1 < i < d, are chosen to be orthogonal to u,/, so that they lie in the allowed hyperplane. Indeed, the u, form an orthonormal basis for the composition space. This orthonormal basis is identified by the Gram-Schmidt procedure. First, the original composition basis vectors are defined... 

As we have not found a proof, this is now demonstrated. Property (2) implies that the eigenvectors o) of X form an orthonormal set if the nondegenerate ones are unity normed and if the Gram-Schmidt procedure is applied to the degenerate ones. The formal representation of X in this set is then Y = a>A a. This relation and... 

First one can build up other effective Hamiltonians based on hierarchized orthogonalization procedures. The Gram-Schmidt procedure is recommended if one starts from the best projected wavefunctions of the bottom of the spectrum. Thus one can obtain a quite reliable effective Hamiltonian with well behaved wavefunctions and good transferability properties (see Section III.D.2). The main drawback of this approach is that the Gram-Schmidt method, which involves triangular matrices, does not lead to simple analytical expressions for perturbation expansions. A partial solution to these limitations is brought about by the new concept of intermediate Hamiltonian,... 

In terms of this notation, the Gram-Schmidt procedure is as follows ... 

Gram-Schmidt procedure An algorithm for obtaining a set of orthonormal functions from a set of signals. 

The first three states of the method of moments [8] characterize the space of the three relevant quasi-bound states (the model space). Since these states are non-orthogonal, the Gram-Schmidt procedure applied to 1), H l) and provide the orthonormalized states i), i — 1,2,3). In this basis the matrix representation of the exact energy-dependent Hamiltonian (13) may be written as... 

The Gram-Schmidt procedure provides a simpler ordu onalization scheme that leads to a hermitian effective Hamiltonian. Since and we already orthogonal to the other projections, we only have to worry about and This means that the coefficients of 4 are defined by that is, if I l = a ( afe + 6a ) -I- p ( aa -P bb )... 

The process of orthgonalization is somewhat analogous to the orthgonalization of vectors, known as the Gram-Schmidt procedure. Construction of orthogonal descriptors 2, 2, 2,... from given descriptors D, D, D,... is accomplished by calculating the residuals between the descriptors. " By definition, the residual of a correlation between two descriptors is the part of one descriptor that does not correlate with the other. 

There are six quadratic polynomials, which are given in Table 1 as basis vectors 64 - eg. A Gram-Schmidt procedure ensures that all the basis vectors are mutually... 

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