Orthonormality, conditions for

It should be noted that the properties must be chosen differently in order to satisfy the orthonormality condition for each definition of the scalar product. 

Example 4.4-3 Using the partial character table for C3v in Table 4.3, show that the character systems ixi and xf satisfy the orthonormality condition for the rows. 

General linear relations between the elements of the HOs residing on a heavy atom as taken in the quaternion form represent some interest. The orthonormality condition for the HOs written in the quaternion form allows us to establish the shape of the hybridization tetrahedra through eq. (3.61). On the other hand, the 4 x 4 matrix formed by HOs expansion coefficients is orthogonal not only with respect to rows, each representing one HO, but also with respect to columns, so that ... 

Iris type of constrained minimisation problem can be tackled using the method of Lagrange nultipliers. In this approach (see Section 1.10.5 for a brief introduction to Lagrange nultipliers) the derivative of the function to be minimised is added to the derivatives of he constraint(s) multiplied by a constant called a Lagrange multiplier. The sum is then et equal to zero. If the Lagrange multiplier for each of the orthonormality conditions is... 

The basis functions constructed in this manner automatically satisfy the necessary boundary conditions for a magnetic cell. They are orthonormal in virtue of being eigenfunctions of the Hermitian operator Ho, therefore the overlapping integrals(6) take on the form... 

The molecular Hartree-Fock equations are obtained by finding the condition for the energy to be a minimum, 6E — 0, and at the same time demanding that the molecular orbitals be orthonormal, (i j) = StJ. 

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-... 

Minimization of the functional (41) has to be performed under the orthonormality requirement in Eq. (4) for the NSOs, whereas the ONs conform to the N-representability conditions for D. Bounds on the ONs are enforced by setting rii = cos y, and varying y,- without constraints. The other two conditions may easily be taken into account by the method of Lagrange multipliers. 

The elements of D represent the sum over all unit cells of the interaction between a pair of atoms. D has 3n x 3n elements for a specific q and j, though the numerical value of the elements will rapidly decrease as pairs of atoms at greater distances are considered. Its eigenvectors, labeled e ( fcq), where k is the branch index, represent the directions and relative size of the displacements of the atoms for each of the normal modes of the crystal. Eigenvector ejj Icq) is a column matrix with three rows for each of the n atoms in the unit cell. Because the dynamical matrix is Hermitian, the eigenvectors obey the orthonormality condition... 

The orthonormality condition (5 11) results in the following condition for the tranformation matrices ... 

In theoretical atomic spectroscopy usually a phase system for the wave functions is chosen which ensures real values of the CFP. In this case the transformation matrices will acquire only real values, too. Let us notice that the transformation matrix in (12.12), according to (12.4), is reciprocal to that in (12.11). Due to the orthonormality of the sets of wave functions, these matrices obey the orthonormality conditions ... 

As Dewar points out in ref. [30a], this derivation is not really satisfactory. A rigorous approach is a simplified version of the derivation of the Hartree-Fock equations (Chapter 5, Section 5.2.3). It starts with the total molecular wavefunction expressed as a determinant, writes the energy in terms of this wavefunction and the Hamiltonian and finds the condition for minimum energy subject to the molecular orbitals being orthonormal (cf. orthogonal matrices, Section 4.3.3). The procedure is explained in some detail in Chapter 5, Section 5.2.3)... 

For the Hiickel calculations, the remainder of the Setup worksheet is devoted to the imposition of the orthonormality condition on the 5Hg[a] functions of Table ALL This condition in Hiickel theory requires only matrix multiplications between the matrix of coefficients and its transpose, with stepwise imposition of, for example, Gram-Schmidt orthogonalization until... 

Inserting (2.7) into (2.1) and using the orthonormality conditions leads to a set of equations for the b coefficients... 

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