A Single Spinning Electron

The symmetry properties of the spin wave functions, a and are not covered by the Character Tables in Appendix A, but require the expansion of the usual symmetry point groups to double groups [4, p. 308ff.]. The imperfect analogy beween orbital- and spin-angular momentum can be drawn once more in order to provide an intuitive rationale for double groups, and to explain at the same time why we can do without them in the context of this book. 

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Since we are assuming no spiir-space interactions in the Hamiltonians the total wavefunction for a single spinning electron, the spin-orbital, is a simple product of the spatial part and a spin factor a or 0. Both of these functions have the same energy of course because of the form of the Hamiltonian. 

A similar examination of the wavefunction shows that this wavefunction describes an atom in which the total spin angular momentum is nearly twice as large as that of a single spinning electron. Such an atom is described as having two electrons of (approximately) parallel spins. As we have pointed out above no two-electron atom has been found to have parallel electron spins in the ground state. 

ENDOR transitions can be easily understood in temis of a simple system consisting of a single unpaired electron spin (S=2) coupled to a single nuclear spin (1=2). The interactions responsible for the various... 

When the states P1 and P2 are described as linear combinations of CSFs as introduced earlier ( Fi = Zk CiKK), these matrix elements can be expressed in terms of CSF-based matrix elements < K I eri IOl >. The fact that the electric dipole operator is a one-electron operator, in combination with the SC rules, guarantees that only states for which the dominant determinants differ by at most a single spin-orbital (i.e., those which are "singly excited") can be connected via electric dipole transitions through first order (i.e., in a one-photon transition to which the < Fi Ii eri F2 > matrix elements pertain). It is for this reason that light with energy adequate to ionize or excite deep core electrons in atoms or molecules usually causes such ionization or excitation rather than double ionization or excitation of valence-level electrons the latter are two-electron events. 

The bonding in a diatomic molecule with a single unpaired electron such as NO can be described in an analogous manner. In this case there are six electrons of a spin and five of /3 spin, with the following arrangement. 

Suppose we have an isolated system with a single unpaired electron and no hyper-fine interaction. Mononuclear low-spin Fe111 and many iron-sulfur clusters fall in this category (cf. Table 4.2). The only relevant interaction is the electronic Zeeman term, so the spin Hamiltonian is... 

Consider for example the simplest possible system consisting of the muon, an electron, and a single spin nucleus labelled i = n. Take the muon and nuclear hyperfine interactions to be istoropic. The level crossing of interest occurs near the field... 

As deduced from their magnetic moments [Con(salen)] /zeff=2.38 Mb [Con(saloph)] Meff 2.74 mb], 27 they are low spin complexes. Even if the spin-orbit contribution makes such value slightly greater than expected, it indicates an electronic configuration with a single unpaired electron (d7-dyz2dxz2dz22dxy ). 

Figure 3. Two electrons in three orbitals system. Configuration (a) is the reference configuration. Single electron excitations with spin-flip produce configurations (bf(g). Two-electron excitations with a single spin-flip produce configurations (h)-(j). Note that non-spin-flipping excitations or excitations that flip the spin of two electrons produce M = l configurations, which do not interact through the Hamiltonian with the final M =0 states, and thus are not...

Fig. 23. Energy level scheme of a single 3d electron showing the effect of crystalline fields (CF) of various symmetry. Electron occupation of levels is indicated by a circle in (d) and by arrows in (e) to denote spin polarization.

Figure 8.9 Schematic illustrations of spin states in a two dimensional periodic material. Circles indicate individual atoms and the dotted lines show a single supercell. In (a), all electrons are paired on each atom. In the remaining examples, a single unpaired electron exists on each atom. Examples of a ferromagnetic state, an antiferromagnetic state and a more complex mag netic state are shown in (b), (c), and (d), respectively.

For closed-shell molecules (in which all electrons are paired), the spin density is zero everywhere. For open-shell molecules (in which one or more electrons are unpaired), the spin density indicates the distribution of unpaired electrons. Spin density is an obvious indicator of reactivity of radicals (in which there is a single unpaired electron). Bonds will be made to centers for which the spin density is greatest. For example, the spin density isosurface for allyl radical suggests that reaction will occur on one of the terminal carbons and not on the central carbon. 

Calculated geometries for a small number of diatomic and small polyatomic free radicals are compared with experimental structures in Table 5-18. These have been drawn from a somewhat larger collection provided in Appendix A5 (Tables A5-50 to A5-57). Except for triplet oxygen, all radicals possess a single unpaired electron (they are doublets). The usual set of theoretical models has been examined. All calculations involve use of the unrestricted open-shell SCF approach, where electrons of different spin occupy different orbitals, as opposed to the restricted open-shell SCF approach, where paired electrons are confined to the same orbital (see Chapter 2 for more detailed discussion). 

An experimental determination of the spin density distribution in the allyl radical recently has become available (30) and is in fair agreement with the results of the valence bond calculation and in somewhat better agreement with an extended Hartree-Foek calculation to be described below. It is seen that although the allyl radical possesses only a single unpaired electron, the total calculated ir-electron spin density on the molecule is % to % units depending on the approximation employed. However, the relationship (75)... 

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