Orthonormalization 4-spinors

The orthonormality restriction for the spinors, Eq. (8.108), may be split into two parts in the case of atoms. The product ansatz for the spinor automatically yields orthonormal angular parts (coupled spherical harmonics cf. chapter 9). But these do not contain information about the principal quantum numbers in the composite indices i and j. For this reason, the restriction to orthonormal spinors results in the orthonormality restriction for radial functions... 

As seen from equation (50), the ESC Hamiltonian is energy dependent and Hermitian. For a fixed value of E, the ESC Hamiltonian can be diagonalized and the resulting solutions, in principle, form a complete orthonormal set. The eigenfunctions of are identical to the large component of the Dirac spinor. When Z — 0, equations (38) and (44) give us the similarity transformed Hamiltonian... 

To set up the total time-independent wave function of the many-electron system the independent-particle model is used in general, resulting in an antisymmetrized Hartree product of four-component orthonormal one-electron functions. Independent of the system (atom, molecule or solid) these four-component single electron functions, the spinors, may be written as... 

In analysing the variation of the energy functional with respect to orbital variations, we need to maintain the orthonormality of the spinor basis. It is convenient to do this by constructing a unitary transformation U = I -t- T such that the set p = 1,..., A/1 WqUqp is an improved basis. 

Since the positive- and negative-energy solutions for all possible p form a complete orthonormal basis set, we may write the most general free-particle wave function as a superposition of the basis spinors. 

The orthonormality of the spherical harmonics and of the spin eigenvectors psnts (with s = 1/2 and OTs = 1/2) ensures orthonormality of the Pauli spinors,... 

In order to calculate expectation values for a wave function of the structure given in Eq. (8.100), it is convenient to introduce orthonormality constraints on the one-electron spinors... 

Then, we can benefit largely from factors that become zero upon resolution of an expectation value over two Slater determinants because of the orthogonality of two different spinors. Also, the Slater determinants turn out to be orthonormal then. 

To determine a set of spinors ipi for a given expression of the total electronic energy, we again apply the variational principle (see chapter 4). Thus, the energy as a functional of the spinors , [ i/>i ] is minimized. This variation must be carried out under the constraint that the orbitals remain orthonormal. Therefore we define a Lagrange functional L as... 

Given a trial orthonormal set of one-particle radial spinors r) (6... 

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. 

This is not as bad as it sounds. If we ensure that the spinors are orthonormal on the valence projection operator,... 

Because we have a complete orthonormal set of spinors, we can expand the function fv in this set ... 

If we expand ( )r in the complete orthonormal set of spinors, the contributions to frr that come from core spinors are made positive by the terms from Brr, and the remaining contributions sum to a value that is less negative than It must also be true that fan > Cy. The terms from the residual Fock operator are small, so the second derivative is positive and the minimum, with a small gradient, must be nearby. In this case also, the deviation of the valence pseudospinor from the valence spinor causes mixing of other spinors to compensate for the change. 

Electron density from 2c-spinors. Most of the time, the V-electron ground state wave functions are approximated by an antisymmetrized product of N orthonormal single-electron functions (spin-orbitals) and are expressed in terms of a Slater determinant y/>. The electron density is then the expectation value of the one-electron density operator ... 

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