Orthonormality relation

In view of the orthonormality relations, the summation on the left-hand side may be simplified as follows... 

The electronic Hamiltonian and the corresponding eigenfunctions and eigenvalues are independent of the orientation of the nuclear body-fixed frame with respect to the space-fixed one, and hence depend only on qx. The index i in Eq. (9) can span both discrete and continuous values. The v /f, ad(r q J form a complete orthonormal basis set and satisfy the orthonormality relations... 

A complete orthonormal set of n-component vectors is a set vi,v2,---, vn, satisfying the orthonormality relations... 

It is interesting to examine the momentum-space orthonormality relations of the Fourier transforms of hydrogenlike Sturmian basis sets. If we let... 

The momentum-space orthonormality relation for hydrogenlike Sturmian basis sets, equation) 17), can be shown to be closely related to the orthonormality relation for hyperspherical harmonics in a 4-dimensional space. This relationship follows from the results of Fock [5], who was able to solve the Schrodinger equation for the hydrogen atom in reciprocal space by projecting 3-dimensional p-space onto the surface of a 4-dimensional hypersphere with the mapping ... 

Equation (24) can be derived from the theory of hyperspherical harmonics and Gegenbauer polynomials but for readers unfamiliar with this theory, the expansion can be made plausible by substitution into the right-hand side of equation (23). With the help of the momentum-space orthonormality relations, (17), it can then be seen that right-hand side of (23) reduces to the left-hand side, which must be the case if the integral equation is to be satisfied. Let us now consider an electron moving in the attractive Coulomb potential of a collection of nuclei ... 

Goscinski s treatment of the orthonormality relations of Sturmian basis sets [3] is easy to generalize and we can see by an argument analogous to equations (9) and (10) that when P ,... 

Sturmian basis set obeys a potential-weighted orthogonality relationship analogous to equation (10). This still does not tell us how to normalize the functions, and in fact the choice is arbitrary. However, it will be convenient to choose the normalization in such a way that in momentum space the orthonormality relations become ... 

Making use of the potential-weighted orthonormality relation (58) and the subsidiary conditions (62) and (63), we obtain ... 

This special normalization is convenient because it leads to momentum-space orthonormality relations of the form ... 

Exercise. Solve (6.2). [Hint Determine the normal modes obeying the boundary conditions, find an orthonormality relation between them and their adjoints, and assume that they constitute a complete set in the interval (L, R) so that the initial condition can be satisfied.]... 

Finally, some remarks will be made concerning the dimension of wavefunctions. The bound-state orbitals are subject to the orthonormality relation... 

Before we leave the C2V character table, it is noted that the characters of all groups satisfy the orthonormal relation ... 

The form of the bonding HOs may be, nevertheless, specified on the basis of the orthonormality relations for the HOs, which are consequences of the group 50(4) structure of the hybridization manifold. According to eq. (3.61) the interhybrid angle for the bonding HOs is given by ... 

Putting Di = [Di Ci] 1Di, one gets particularly [Di CJ = 1. It is evident that, in the distinct case, the operators T and Td have the same eigenvalues and that one has the bi-orthonormalization relation ... 

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