Degenerate wave function

If the solution of the zero-order Schiodinger equation [i.e., all teiins in (17) except V(r,Ro) are neglected] yields an/-fold degenerate electronic term, the degeneracy may be removed by the vibronic coupling tenns. If F) and T ) are the two degenerate wave functions, then the vibronic coupling constant... 

In the adiabatic approximation the form of the total wave function is restricted to one electronic surface, i.e. all coupling elements in eq. (3.12) are neglected (only the terms with i = i survive). Except for spatially degenerate wave functions, the diagonal first-order non-adiabatic coupling element is zero. 

We conclude that when the representation generated by the degenerate wave functions is reducible, the degenerate wave functions can always be chosen so that they divide up into subsets whose members transform... 

Strictly speaking the densities of the electronic cloud on the sites of the atomic nuclei, the so-called on-site density, need not be different for different components of a degenerate wave function. A simple counter-example is a orbital level in a cubic cluster. Let a,) denote a a-type atomic orbital on a given site i. The symmetry adapted linear combinations (SALC s) of these basis orbitals are given by (see Fig. 1) ... 

Thus Fi/ ei, is an eigenfunction of with eigenvalue ei- Let the orbital degeneracy of the term to which belongs be n. It follows from (15.4) that must be some linear combination of the n orbitally degenerate wave functions of the term that have the same value of as ... 

The first-order property with respect to an external field is the magnetic dipole moment m (eq. (10.10)). When field-independent basis functions are used, the HF magnetic dipole moment is given as the expectation value of the V2LG and S (total electron spin) operators over the unperturbed wave function, eqs (10.21) and (10.24). Since the Lg operator is imaginary it can only yield a non-zero result for spatially degenerate wave functions and the expectation value of S is only non-zero for non-singlet states. 

This perturbation acts at once on both degenerate wave-functions of the ground level in a unit cell, producing the correction energies ... 

FIGURE 18.20 Degenerate wave functions for one-quarter filling. 

Sometimes, there only is a finite number of supplementary solutions dictated by symmetry. Let us consider the case of three protons at infinite separation, occupying the vertices of an equilateral triangle. Let us put two electrons of the same spin into this system. One of the degenerate wave functions of the system is given by the Slater determinant ... 

Since the (25 -f 1)(2L -f 1) degenerate wave functions may be chosen to be real and since the expectation value of a Hermitian imaginary operator vanishes for real wave functions, we obtain for the spin-orbit operator... 

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