Wave function mixed-spin state

An alternate model has been proposed (296-298) that attempts to interpret the data in terms of a single mixed state. For this model S is no longer a good quantum number. The unpaired electron spin density at the metal and the wave function that represents the single state are functions of temperature and pressure. This latter model was considered as a possibility by Leipoldt and Coppens. In their structure of the Fe(Et2Dtc)3 complex at 297 and 79°K they attempted an analysis of the temperature parameters at 297°K. The analysis could not distinguish between a mixed-spin state and a mixture of two different spin states (403). 

As seen in the radiationless process, intercombinational radiative transitions can also be affected by spin-orbit interaction. As stated previously, spin-orbit coupling serves to mix singlet and triplet states. Although this mixing is of a highly complex nature, some insight can be gained by first-order perturbation theory. From first-order perturbation theory one can write a total wave function for the triplet state as... 

It should be pointed out that a somewhat different expression has been given for the Knight shift [32] and used in the analysis of PbTe data that in addition to the g factor contains a factor A. The factor A corresponds to the I PF(0) I2 probability above except that it can be either positive or negative, depending upon which component of the Kramers-doublet wave function has s-character, as determined by the symmetry of the relevant states and the mixing of wavefunctions due to spin-orbit coupling. 

UHF, on the other hand, does optimize tire a and orbitals so tlrat they need not be spatially identical, and thus is able to account for both spin polarization and some small amount of configurational mixing. As a result, however, UHF wave functions are generally not eigenfunctions of the operator S-, but are contaminated by higher spin states. 

However, the problem of variational collapse typically prevents an equivalent SCF description for excited states. That is, any attempt to optimize the occupied MOs with respect to the energy will necessarily return the wave function to that of the ground state. Variational collapse can sometimes be avoided, however, when the nature of the ground and excited states prevents their mixing within the SCF formalism. This simation occurs most commonly in symmetric molecules, where electronic states belonging to different irreducible representations do not mix in the SCF, and also in any situation where the ground and excited stales have different spin. 

In the high-spin ferrous ion, spin-orbit interactions mix the ground state wave functions with the excited states. If the ground state is assumed to have dZ2 symmetry, then the following expressions apply for an ion in a crystal field with both rhombic and axial distortions (Edwards et al. (70). 

FIGURE 7.Ans.l (a) Resonance structures of pentadienyl radical and their symmetry properties, (b) The VB mixing of the resonance structures, (c) The quasiclassical (spin alternant) determinant that dominates the ground-state wave function, and the corresponding secondary determinants, and the resulting spin density distribution (p) in the ground state, (d) The spin distribution in the covalent excited state. 

If accurate electronic wave functions are available, Aa> and Aa> can be estimated from equations (7.109) and (7.120) respectively. All nearby electronic states which contribute by spin-orbit mixing to A(2) must be included if the result is to be reliable. 

The inversion operation i which leads to the g/u classification of the electronic states is not a true symmetry operation because it does not commute with the Fermi contact hyperfine Hamiltonian. The operator i acts within the molecule-fixed axis system on electron orbital and vibrational coordinates only. It does not affect electron or nuclear spin coordinates and therefore cannot be used to classify the total wave function of the molecule. Since g and u are not exact labels, it was realised by Bunker and Moss [265] that electric dipole pure rotational transitions were possible in ll], the g/u symmetry breaking (and simultaneous ortho-para mixing) being relatively large for levels very close to the dissociation asymptote. The electric dipole transition moment for the 19,1 19,0 rotational transition in the ground electronic state was calculated... 

The dominant electronic configuration for TiO in its X 3 A ground state may be written (core) (9a)1 (IS)1 where the 9a orbital is essentially the 4,v orbital of the Ti2+ ion and the IS orbital is essentially a 3d orbital of Ti2+. It is also necessary to provide wave functions for the low-lying excited electronic states of TiO because both the spin spin constant k and the A-doubling constant oA for the ground state depend upon mixing of excited states produced by a combination of the rotational and spin orbit interactions. Namiki, Saito, Robinson and Steimle [69] give an acceptable... 

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