Orthogonalization Schmidt

Imagine two vectors u and v, each of length 1 (i.e. normalized), with the dot product (m w = a. If a = 0, the two vectors are orthogonal. We are interested in the case a 0. Can we make such linear combinations of u and v, so that the new vectors, u and i/, will be orthogonal We can do this in many ways, two of them are called the Schmidt orthogonalization  

It is seen that Schmidt orthogonalization is based on a very simple idea. In Case I the first vector is left unchanged, while from the second vector, we cut out its component along the first (Fig. J.l). In this way the two vectors are treated differently (hence, the two cases above). 

In this book the vectors we orthogonalize will be Hilbert space vectors (see Appendix B), i.e. the normalized wave functions. In the case of two such vectors 4 and p2 having a dot product we construct the new orthogonal wave 

In case of many vectors the procedure is similar. First, we decide the order of the vectors to be orthogonalized. Then we begin the procedure by leaving the first vector unchanged. Then we continue, remembering that from a new vector we have to cut out all its components along the new vectors already found. Of course, the final set of vectors depends on the order chosen. 

The eigenvalues of S are always positive, therefore the diagonal elements of Sdiag can be replaced by their square roots, thus producing the matrix denoted by 

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The weakly occupied pre-NAOs on each centre are made orthogonal to the strongly occupied NAOs on the same centre by a standard Gram-Schmidt orthogonalization. 

The classical method is known as Schmidt orthogonalization. In the general step, the im column of A has added to it a linear combination of... 

Scattering processes, 586 Schiff, L. J., 437,444 Schiffer, M.% 363 SchUchting, H., 24 Schmidt orthogonalization, 65 Schonflies notation for magnetic point groups, 739... 

Based on the same two step proeedure as presented above for C2H4 (MCSCF ealeulations followed by Schmidt orthogonalization of Rydberg functions), a systematic search was conducted by progressively incorporating groups of orbitals in the active space. Two types of wave functions proved well adapted to the problem, one for in-plane excitations, the other for out-of-plane excitations from the carbene orbital. The case of the Ai states will serve as an illustration of the general approach done for all symmetries and wave functions. 

This construction is known as the Schmidt orthogonalization procedure. Since the initial selection for 0i can be any of the original functions ip, or any linear combination of them, an infinite number of orthogonal sets 0, can be obtained by the Schmidt procedure. 

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... 

Because of a different normalization , the coefficients of the parentheses are not identical for the Gram-Schmidt orthogonalization and for the recursion formula. 

The vectors generated by the Lanczos recursion differ from the Krylov vectors in that the former are mutually orthogonal and properly normalized, at least in exact arithmetic. In fact, the Lanczos vectors can be considered as the Gram-Schmidt orthogonalized Krylov vectors.27 Because the orthogonalization is performed implicitly along the recursion, the numerical costs are minimal. 

Investigation shows that N is far from unique. Indeed, if N satisfies Eq. (1.47), NU will also work, where U is any unitary matrix. A possible candidate for N is shown in Eq. (1.18). If we put restrictions on N, the result can be made unique. If N is forced to be upper triangular, one obtains the classical Schmidt orthogonalization of the basis. The transformation of Eq. (1.18), as it stands, is frequently called the canonical orthogonalization of the basis. Once the basis is orthogonalized the weights are easily determined in the normal sense as... 

If we choose our basis functions for a particular function space to be orthonormal (orthogonal and normalized) i.e. (/ /,) = J/, /, dr then, since the transformation operators are unitary ( 5-7), the representation created will consist of unitary matrices. This is proved in Appendix A.6-1. It should be stated that it is always possible to find an orthonormal basis and one way, the Schmidt orthogonalization process, is given in Appendix A.6-2. 

Usually on the first iteration of an SCF calculation W is computed by the Schmidt orthogonalization method but thereafter W is chosen to be the C matrix from the previous iteration. This produces an F matrix which is nearly diagonal so the Jacobi method becomes quite efficient after the first iteration. Further, in the Jacobi method, F is diagonalized by an iterative sequence of simple plane-rotation transformations... 

A fundamental requirement of the derivation of (III. 1) and (III.2) is that ma must be orthogonal to <(>0/8. If they are not initially orthogonal, they must be Schmidt orthogonalized in the normal way. However, such orthoganalization may lead to serious errors even in simple systems, as Bell and Kingston found for the helium l S- 3 5 transition.41... 

Since the two wave functions overlap, we can form from them two orthogonal wave functions using the Graham—Schmidt orthogonalization procedure,... 

The correction vector is Schmidt orthogonalized with the basis set and the resultant is normalized. This new vector Qm+ is used to augment the set Q,. The new basis, Q, i = 1,...,m + 1 is used to construct a new augmented small matrix a(" +1 Inverting a m+1 gives and this iterative procedure is continued until all components of the residue vector are found to be below a previously chosen threshold. If the set of vectors Qi becomes too large to handle, the procedure can... 

The matrix B transforms the STO basis to an AO basis. The 2j-functions are Schmidt orthogonalized to the ls-functions, and 2p-functions are aligned along the local atomic principal axis. S 1/2 (S is the overlap matrix) is the usual Lowdin ortho-gonalization. The following approximations are made ... 

Secondly, each of the singly occupied orbitals m n,.. ., N-m may be orthogonalized to the core, without changing the total wavefunction (70). This fact does not seem to be widely realized, but the proof of it is very simple let us replace each valence orbital 4 n (j m+, ..., N—m) by its counterpart < >, which is Schmidt orthogonalized to all the core orbitals,... 

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