Completeness and orthonormality

We proceed by induction, assuming that the (A — l)-electron CSFs 1 (t) cs) constitute an orthonormal set of spin eigenfunctions. This is trivially true for one-electron CSFs. Next, we calculate the overlap between two arbitrarily chosen A-electron CSFs 6 (u) cs) and (v) cs). Expanding these CSFs according to (2.6.4), we obtain 

The second expression follows from the first since the creation operator to the right commutes or anticommutes with the second spin tensor of)erator, and likewise the annihilation operator to the left 

The first Kronecker delta arises since different un and vn lead to different spins in the N — 1)-electron tensor operators. The remaining Kronecker deltas arise since the N — l)-electron tensor operators are orthogonal by assumption if any of the pairs Uk and Vk are different, although the final spin symmetries are identical. We now write (2.6.22) in the more compact form 

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Assuming that the basis functions used in discretization ( < ) )) are complete and orthonormal, the wave function in Eq. [1] can be expressed in the following expansion ... 

The parameter is introduced to keep track of the order of the perturbation series, as will become clear. Indeed, one can perform a Taylor series expansion of the perturbed wave functions and perturbed energies using X to keep track of the order of the expansions. Since the set of eigenfunctions of the unperturbed SE form a complete and orthonormal set, the perturbed wave functions can be expanded in terms of them. Thus,... 

The above unconstrained optimization cannot be directly applied at intermediate values of A for which the Hamiltonian contains the unknown potential V. We found, however, that a simultaneous determination of and V can be achieved by performing the following constrained optimization. We assume that the trial many-body wavefunction results in the electron density rcA(r) and expand both nA(r) and the ground state density n(r) in a complete and orthonormal set of basis functions fs ... 

The set of plane waves is complete and orthonormal. Therefore, the Kohn-Sham orbitals can be expanded with plane waves ... 

For fhe model operator Ho, the Schrodinger equation has a complete (and orthonormal) set of eigenfunctions... 

This follows from the completeness and orthonormality of (See Problem 6-4.) Evaluation of Eq. (6-39) for the first few terms gives (Problem 6-22)... 

From a formal point of view, we can represent the spinors of any system of interest in terms of a complete orthonormal set of spinors located on a single center. These spinors include both bound state and continuum spinors and range over all angular momenta. In an application, the particular choice of spinors will be determined by the state of the atom whose core we wish to freeze. But for the purpose of analysis, it does not matter which spinors we choose provided the set is complete and orthonormal. 

Now, multiplying both sides of this last equation from the left by (pj and using the completeness and orthonormality properties of the unperturbed wavefunctions, we obtain... 

Let us now make the considerations more general. As was stated before, the set of eigenfunctions of the Hermitian operator H is complete and orthonormal - functions are orthogonal and normalized ... 

Each Lewis-type Tabc must therefore be complemented by two remaining three-center awttbond NBOs (labeled 3C in NBO output) to conserve basis completeness and orthonormality. In many cases, these valence antibond NBOs (as optimally chosen by the NBO program) correspond to two-center re-type Tabc " and three-center A-type Tabc linear combinations (cf. W B, p. 306ff),... 

We are now ready to separate the angular and the radial parts of rn- To this end the angular part of the Coulomb term is expanded in Legendre polynomials P (cos0), which form a complete and orthonormal set of eigenfunctions on the interval cos0 e] — 1,1[, and the radial part is expressed through coefficients a x) that will be determined shortly ... 

Completeness and orthonormality of in L fK ) then follow from completeness and... 

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