Orthogonality condition

Since the Jacobi polynomials belong to a class of orthogonal polynomials, they satisfy the orthogonality condition 

The integration is defined in the domain [0,1]. Outside this domain, orthogonality cannot be guaranteed. Any physical systems having a finite domain can be easily scaled to reduce to the domain [0,1]. 

The weighting function for this particular orthogonality condition defined with reference to the Sturm-Liouville equation is 

The exponents a and /3 are seen to dictate the nature of the orthogonal Jacobi polynomials. 

There are N equations of orthogonality (Eq. 8.83) because y = 0,1,2. N - 1, and there are exactly N unknown coefficients of the Jacobi polynomial of degree N to be determined 

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The orthogonality condition assures one that the lowest energy state will not converge to core-like states, but valence states. The wavefimction for the solid can be written as... 

This is the process that makes the method a Galerldn method. The basis for the orthogonality condition is that a function that is made orthogonal to each member of a complete set is then zero. The residual is being made orthogonal, and if the basis functions are complete and you use infinitely many of them, then the residual is zero. Once the residual is zero, the problem is solved. 

Comparing this with (A. 10), it is seen that is the velocity gradient when U = I, i.e., in a pure rotation. It is easily seen by differentiating the orthogonality condition (A.14i) that is antisymmetric. Analogously, the tensor / will be defined by... 

To build up vin the cluster function (1) we use the functions (PvA vA2---fvBi 9vs2 -- all of which satisfy the strong orthogonality condition in the sense of to (2), but do not satisfy the strong orthogonality needed for (1) We therefore consider the linear combination... 

The orthogonalization condition on the off-diagonal elements correctly yields 2 N(N - l)/2 real conditions. The assumption of the hermiticity property of the scalar product in the subspace of N dimensions, would lead finally to, Kc,R = 2NM - N2 in the case of a complex , and not 2Kc,c, as had been claimed. 

The orbitals containing the bonding electrons are hybrids formed by the addition of the wave functions of the s-, p-, d-, and f- types (the additions are subject to the normalization and orthogonalization conditions). Formation of the hybrid orbitals occurs in selected symmetric directions and causes the hybrids to extend like arms on the otherwise spherical atoms. These arms overlap with similar arms on other atoms. The greater the overlap, the stronger the bonds (Pauling, 1963). 

The orthogonality conditions (cuab c coabc ) = toA Bc tOABC > = 0 allow one to establish that ca = cc and cb = 2/X — X/2, and therefore (taking account of the normalization condition) that... 

Solution Real solutions of the orthogonality condition (4.27) are obtained only if... 

The second term of (13) can be evaluated with the help of (8) and the knowledge that the RS can only involve, at most, two-electron operators. If any VRS contains only two-electron operators and the R are constrained by the strong orthogonality condition (11) then it is obvious that only the first two terms in the expansion (8) of Ax give rise to non-zero contributions to... 

As noted earlier, we limit ourselves arbitrarily, but judiciously, to orthonormal orbital sets in this function space, which implies the orthogonality conditions of Eq. (6). This equation represents 1/2 N(N + l)/2 conditions for the N2 matrix elements of T. Thus an orthogonal transformation of degree N contains N(N -1)/2 arbitrary parameters. Hence there exist N(N -1)/2 de-... 

The eigenvectors are the rows or the colunrn of an orthogonal matrix A that diagonalizes M. They fulfill the orthogonality condition... 

In this equation we first insert the identity matrix I before and after M , and then apply m times the operation (7.2.7) to produce We next apply the orthogonality condition (7.2.8), and the fact that all our vectors and matrices are real, hence (lla,) = (a,II). Finally, we use the expression for the GPF obtained in Eq. (7.1.25). 

Differentiation of bi-orthogonality conditions (2.186)-(2.189) with respect to an arbitrary soft coordinates cf yields the relations... 

The exact many-electron wave function for an excited state, kj, f / 0, satisfies orthogonality conditions with respect to other many-electron state including the ground state, ko. For example, for the first excited state with many-electron wave function we have... 

The annihilation of either one of the two determinants in (20) leads to fulfillment of the orthogonality condition (19). From energy considerations and previous computational experience, we impose the orthogonality restrictions only via the first determinant which is associated with the a set and involves the occupied orbital highest in energy. 

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