Integration orthogonal functions

Any absolutely integrable, bounded function, j x), can be expanded in terms of a set of orthogonal functions 15). If one writes... 

Nb)i and Hi) ptflal being the difference of two functions integrable in the sense of Eq. 7, is also integrable in this sense. Therefore, ptpal is TV-representable and the Levy constrained search can be performed. Let s consider all possible sets of orthogonal functions such that ... 

Finally calculate the normalization integral for the orthogonal function, 2s-A. I s), in the usual manner. 

Calculate normalization and overlap integrals for the orthogonal functions as in cells G 15 to G 17. 

The frequency distribution A(v) can be obtained by performing a Fourier transformation [3] on equation 6A.7. This theorem states that, the periodic, orthogonal functions [3] I(t ) and B (v) are related by the symmetrical integral equations... 

The only way for the differential overlap to be zero in Ju is for Xa or xb, or both, to be identically zero in dv. Zero differential overlap (ZDO) between Xa and xb in all volume elements requires that Xa and Xb can never be finite in the same region, that is, the functions do not touch. It is easy to see that, if there is ZDO between Xa and xb (understood to apply in all dv), then the familiar overlap integral S must vanish too. The converse is not tme, however. S is zero for any two orthogonal functions even if they touch. An example is provided by an s and a p function on the same center. 

The eigenfunctions of the Hamiltonian operator can be shown to form a complete set of orthogonal functions. By orthogonal we mean that the product of any two different members of such a set ( , and Tf), when integrated over all space, is zero ... 

For the lcTg+ orbital the overlap integral quantifies the build up of charge in the internuclear region. When the nuclei are far apart, Su will tend to zero and Mg becomes 1A/2, the value found for two equally weighted orthogonal functions in Chapter 6. As the nuclei approach, some of the density is described by the overlap integral, and so Mg is reduced to ensure the wavefunction remains correctly normalized. 

The expansion in Legendre polynomials or more generally spherical harmonics is chosen because they are orthogonal functions. The coefficients Ul in the expansion can be obtained by multiplying both sides of Eq. (24) by Pl(cos0) and integrating over 0, with the result ... 

To normalize the wave function, we integrate the function of Eq. (18.2-7) over the space coordinates and spin coordinates of both particles and set the result equal to unity. If the orbitals are normalized the first term and the fourth term both give unity after integration. Because the hydrogen-like spin orbitals are orthogonal to each other, the second and third terms give zero after integration. The result is... 

Two functions and are orthogonal if the product integrated over all configuration space, vanishes. 

A function T is normalized if the product T integrated over all configuration space is unity. An orthonormal set contains functions that are normalized and orthogonal to each other. 

In comparison with the ordinary OZ equation, there is an additional integration with respect to the particle size. To proceed further, this integration can be removed by applying the method proposed by Lado [81,82]. Thus, we expand the (j-dependent functions in orthogonal polynomials 7 = 0,1,2,..., defined such that... 

These new basis functions can easily be shown to be orthonormal. It also turns out that two-electron integrals calculated using these orthogonalized basis functions do indeed satisfy the ZDO approximation much more closely than the ordinart basis functions. 

In the case that the x s are individually normahzed but not necessarily orthogonal then the overlap integrals between the basis functions have to be taken into account. If we write the matrix of overlap integrals S and its determinant det S then... 

This implies that the absolute values of the overlap integrals clk are vanishing as A[Jk—Ek]l for Jk->Ekt ensuring automatic orthogonality between the function 0k and all the lower exact eigenfunctions F0, Wv. . Wk x when Jk = Ek. Substituting this... 

As before, we can directly show that this function is automatically orthogonal with respect to 0X and 0 2 and that the normalization integral is given by the expression... 

This procedure of multiplication by one of the members of the orthogonal set of functions and integration can be continued. The equations developed above will be sufficient to obtain the Navier-Stokes... 

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