Spin orbital, spatial extent

Configurations of different spin or spatial symmetry may not mix. For configurations which are related by a single electron transfer the extent of mixing is related to the overlap of the orbitals which differ in one-electron occupancy. For example, DA and D+A for the face-on dimerization of two ethylenes do not mix significantly since the 0 and < orbitals cannot... 

The preceding development of the HF theory assumed a closed-shell wavefunction. The wavefunction for an individual electron describes its spatial extent along with its spin. The electron can be either spin up (a) or spin down (P). For the closed-shell wavefunction, each pair of electrons shares the same spatial orbital but each has a different spin—one is up and the other is down. This type of wavefunction is also called a (spin)-restricted wavefunction since the paired electrons are restricted to the same spatial orbital, leading to the restricted Hartree-Fock (RHF) method. 

Thus this model maps density over atoms rather than spatial coordinates. If overlap is included some other definition of charge density such as Mulliken s17 may be employed. Eq. (30) and (31) are then used with this wave function to calculate the hyperfine constants as a function of the pn s. If symmetry is high enough, there will be enough hyperfine constants to determine all the p s, otherwise additional approximations may be necessary. For transition metal complexes, where spin-orbit effects are appreciable, it is necessary to include admixtures of excited-state configurations that are mixed with the ground state by the spin-orbit operator. To determine the extent of admixture, we must know the value of the spin-orbit constant X and the energy of the excited states. 

In the non-relativistic domain one-electron operators can be classified as triplet and singlet operators, depending on whether they contain spin operators or not. In the relativistic domain the spin-orbit interaction leads to an intimate coupling of the spin and spatial degrees of freedom, and spin symmetry is therefore lost. It can to some extent be replaced by time-reversal symmetry. We may choose the orbital basis generating the matrix of Hx to be a Kramers paired basis, that is each orbital j/p comes with the Kramers partner = generated by the action of the time-reversal operator We can then replace the summation over individual orbitals in (178) by a summation over Kramers pairs which leads to the form... 

This correlation exists despite any a priori reason why Np should always behave as a rare-earth element where the large orbital contribution unambiguously determines the hyperfine field Hhf. Such a type of relation is less reliable in transition metals due to variations of the localization of the d states, where the spatial extent of the spin density can differ markedly according to bonding conditions. 

In addition, the associated spin orbitals (with a maximum occupation number of one electron) acquire different spatial extents, and they are drawn in Figure 2.25. At first sight, it is difficult to recognize that the low-lying a spin orbitals (a) are more spatially contracted than the j6 spin orbitals (b) but an orbital difference plot (c) immediately shows that the oc spins dominate in the region close to the nucleus, where they experience a higher nuclear potential. The analysis may also be carried out numerically, but we will not do so here. 

The first-row transition metals have the special 3d orbital set which is not shielded by any symmetry-related orbital closer to the atomic nucleus such that the 3d orbitals are fairly contracted. This is nicely reflected in the 3d/id I, orbital exponents given in Table 2.1. Small changes to the shieldings of the 3d set caused by spin-polarization give rise to comparably large changes in their energies and spatial extents. The heavier transition metals, on the other hand, have inner d functions which help to screen their valence d orbitals. Therefore, perturbations from spin-polarization do not have such a dramatic effect upon these well-shielded orbitals. 

Up until now we have discussed the general methods for computing the cluster wavefunctions we now consider how the wavefunctions can be analyzed to obtain insights into the nature of chemical interactions at surfaces. In the introduction, we pointed out that the most commonly used method of analysis is the Mulliken population analysis and that this method of analysis may give misleading results. One alternative to a population analysis to get information about the charge associated with a given atom is the orbital projection approach. Here, one takes an atomic or molecular orbital, projection operator, P(( ) = spin orbital. The expectation value of P(v>) taken with respect to the cluster wavefunction provides a measure of the extent to which

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