Wavefunction, broken symmetry

As observed in many theoretical studies of symmetric radicals (92), a HF wavefunction describing it ->- tt excited states preferentially localizes the hole on one moiety leading to a broken symmetry structure of the states with unequal bond lengths for the minimum energy structure. Similarly, for the n tt ... 

The broken symmetry wavefunction is not itself a pure spin state. However, spin projection techniques allow the approximate energies and properties of the correct spin states to be calculated. 

Despite the publication of these papers, as was indicated in the introduction, most of the later publications have focused on the calculation of dinuclear complexes employing the broken-symmetry approach proposed by Noodleman et al. In this approach the J value involves the calculation of the energy difference between the high-spin state and a low-spin solution that corresponds to a broken symmetry wavefunction in the case of symmetric homodinuclear complexes. From now on, it will be employed the expressions for the Hamiltonians indicated in Eqs. (1) and (2) Eq. (3) was kept for historical reasons. A general expression, (see Eq. 4) can be proposed for any dinuclear complex using the original broken-symmetry approach proposed by Noodleman [26] ... 

Perhaps the greatest need for Brueckner-orbital-based methods arises in systems suffering from artifactual symmetry-breaking orbital instabili-ties, " ° where the approximate wavefunction fails to maintain the selected spin and/or spatial symmetry characteristics of the exact wavefunction. Such instabilities arise in SCF-like wavefunctions as a result of a competition between valence-bond-like solutions to the Hartree-Fock equations these solutions typically allow for localization of an unpaired electron onto one of two or more symmetry-equivalent atoms in the molecule. In the ground Ilg state of O2, for example, a pair of symmetry-broken Hartree-Fock wavefunctions may be constructed with the unpaired electron localized onto one oxygen atom or the other. Though symmetry-broken wavefunctions have sometimes been exploited to produce providentially correct results in a few systems, they are often not beneficial or even acceptable, and the question of whether to relax constraints in the presence of an instability was originally described by Lowdin as the symmetry dilemma. ... 

In the tight-binding method, the wavefunctions are constructed of localized atomic levels, but an electron in such a level will be found with equal probability in any cell of a monatomic crystal [22, p. 185]. Nevertheless, these freely moving electrons do not necessarily contribute to electric conduction, since in the semiclassical picture of a filled band half the electrons move to the right and half to the left and this remains true even if an electric field is applied. For partially filled bands, the exact half/half-symmetry is broken by an electric field, and a current will flow. Note that half-filled bands can give an electric current but cannot... 

Figure 7.8. Simulated fijll- and half-field powder pattern ESR spectra of triplet states for different values of E/D (see text) [59] (a) = 0 the triplet wavefunction is axially symmetric. The pattern contains singularities at // (hv D/2 - /8hv)/gli and steps at D)/gP. (b) 0 E <. D/i the axial symmetry of the wavefunction is broken. The singularities occur at...

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