Spin polarization

The polarization of the photoelectron spin in atomic and molecular photoionization processes encodes supplementary information about the relative magnitudes of electron transition moments. In atoms, the spherical symmetry permits complete characterization of the transition moments that contribute to the total excitation cross section for each resonance (Heinzmann, 1980). However, in molecules, the Z-mixing caused by the absence of spherical symmetry makes it impossible to completely characterize a resonance excitation mechanism from partial cross sections, angular distribution, and spin-polarization measurements. 

The information obtainable from photoelectron polarization measurements is reviewed, for both atoms and molecules, by Heinzmann and Cherepkov (1996). Even at non-relativistic excitation energy, photoelectrons can be spin-polarized (Fano, 1969). For l / 0 atoms, due to the spin-orbit splitting of the initial atomic and/or the final ionic state, photoelectrons are in most cases highly spin-polarized (up to 100%) when photoexcited with circularly polarized light. Analogous effects occur in molecular photoionization, but systematic studies have only been made for hydrogen halide molecules, HX. The electronic ground state of HX+ is X2n. 

The first prerequisite for measurement of photoelectron spin-polarization is the ability to separately detect the photoelectrons ejected from the different fine-structure levels (e.g., 2n3/2 and 2n1/2 for HX+ X2n). When the molecule contains a heavy atom (e.g., large spin-orbit splitting), it becomes easier to use the electron kinetic energy to distinguish the photoelectrons ejected from the different fine structure channels. For spin-polarization analysis, the accelerated electron beam (20-120 keV) can be scattered by a thin gold foil in a Mott-detector. The spin-polarization is determined from the left-right (or up-down) asymmetry in the intensities of the scattered electrons (Heinzmann, 1978). Spin polarization experiments, however, are difficult because the differential spin-up/spin-down flux of photoelectrons is typically one thousandth that obtained when recording a total photoionization spectrum. 

Similarly to the angular asymmetry parameter, / , the spin-polarization parameters have signed values and their expressions, as function of contributing transition moments, in principle permit determination of the relative signs of these transition moments. However, in contrast to the situation for atoms, the number of relevant unknown parameters exceeds the number of experimentally measurable quantities. Nevertheless, measurement of both resonance widths and spin-polarization parameters can considerably narrow the assignment possibilities. This is particularly true in the region between the two ion-core fine-structure thresholds, for example, 2n3/2 and 2n1/2 of an AB+ 2n state. 

The spin-polarization parameters have the property of having opposite signs for different fine-structure components if the photoelectrons have the same kinetic energy, for example  

For cyclobutadiene, there is another interesting possibility which we have explored before with the case of methylene in Section 8.8. Is it possible to produce a stable structure by allowing the two highest energy electrons of the 4n species to separately occupy the orthogonal pair of degenerate orbitals with their spins parallel The result would be a triplet diradical species as in 12.11. For such an electronic 

A C-H sigma bonding orbital 0i, the p orbital 02. and the sigma antibonding orbital 03 (a) in the absence of spin polarization and (b) in the presence of spin polarization. The up- and down-spin arrows are used in (b) to indicate the up- and down-spin spatial orbitals, respectively. 

Chapter 8. Thus the Coulomb repulsion between 0 and j) has a stronger spin polarization effect than does the Coulomb repulsion between 02 and 0. We will employ the simplified approach described earlier in our later discussion of spin polarization.) 

In a u radical system, the amount of unpaired down-spin density (pn) ori a hydrogen atom, and hence the hyperfine splitting constant (oh) of the hydrogen atom in a ESR spectrum of the compound, is proportional to the amount of unpaired up-spin density (pc) the carbon atom to which the hydrogen atom is attached [30]. In other words. 

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By examining the expression for Q ( equation (B1.16.4)). it should now be clear that the nuclear spin state influences the difference in precessional frequencies and, ultimately, the likelihood of intersystem crossing, tlnough the hyperfme tenn. It is this influence of nuclear spin states on electronic intersystem crossing which will eventually lead to non-equilibrium distributions of nuclear spin states, i.e. spin polarization, in the products of radical reactions, as we shall see below. 

In the early 1990s, a new spin polarization mechanism was posPilated by Paul and co-workers to explain how polarization can be developed m transient radicals in the presence of excited triplet state molecules (Blattler et al [43], Blattler and Paul [44], Goudsmit et al [45]). While the earliest examples of the radical-triplet pair mechanism (RTPM) mvolved emissive polarizations similar in appearance to triplet mechanism polarizations, cases have since been discovered m which absorptive and multiplet polarizations are also generated by RTPM. 

While each of die previous examples illustrated just one of the electron spin polarization iiiechanisms, the spectra of many systems involve polarizations from multiple iiiechanisms or a change in meclianism with delay time. 

Closs G L 1969 A mechanism explaining nuclear spin polarizations In radical combination reactions J. Am. Chem. Soc. 91 4552-4... 

Atkins P W and Evans G T 1974 Electron spin polarization in a rotating triplet Mol. Phys. 27 1633—44... 

Blattler C and Paul H 1991 CIDEP after laser flash irradiation of benzil in 2-propanol. Electron spin polarization by the radical-triplet pair mechanism Res. Chem. Intermed. 16 201-11... 

Goudsmit G-H, Paul H and Shushin A I 1993 Electron spin polarization in radical-triplet pairs. Size and dependence on diffusion J. Phys. Chem. 97 13 243-9... 

Closs G L, Forbes M D E and Norris J R 1987 Spin-polarized electron paramagnetic resonance spectra of radical pairs in micelles. Observation of electron spin-spin interactions J. Phys. Chem. 91 3592-9... 

Closs G L and Forbes M D E 1991 EPR spectroscopy of electron spin polarized biradicals in liquid solutions. Technique, spectral simulation, scope and limitations J. Phys. Chem. 95 1924-33... 

Norris J R, Morris A L, Thurnauer M C and Tang J 1990 A general model of electron spin polarization arising from the interactions within radical pairs J. Chem. Phys. 92 4239—49... 

Koga T, Ohara K, Kuwata K and Mural H 1997 Anomalous triplet mechanism spin polarization... 

Salikhov K M, Molin Yu N, Sagdeev R Z and Buchachenko A L 1984 Spin Polarization and Magnetic Effects in Radical Reactions (Amsterdam Elsevier)... 

Hess H F, Kochanski G P, Doyle J M, Greytak T J, and Kleppner D 1986 Spin-polarized hydrogen maser Phys.Rev. A 34 1602-4... 

One consequence of the spin-polarized nature of the effective potential in F is that the optimal Isa and IsP spin-orbitals, which are themselves solutions of F ( )i = 8i d >i, do not have identical orbital energies (i.e., 8isa lsP) and are not spatially identical to one another (i.e., (l)isa and (l)isp do not have identical LCAO-MO expansion coefficients). This resultant spin polarization of the orbitals in P gives rise to spin impurities in P. That is, the determinant Isa 1 s P 2sa is not a pure doublet spin eigenfunction although it is an eigenfunction with Ms = 1/2 it contains both S = 1/2 and S = 3/2 components. If the Isa and Is P spin-orbitals were spatially identical, then Isa Is P 2sa would be a pure spin eigenfunction with S = 1/2. 

There are a number of other technical details associated with HF and other ah initio methods that are discussed in other chapters. Basis sets and basis set superposition error are discussed in more detail in Chapters 10 and 28. For open-shell systems, additional issues exist spin polarization, symmetry breaking, and spin contamination. These are discussed in Chapter 27. Size-consistency and size-extensivity are discussed in Chapter 26. 

The advantage of unrestricted calculations is that they can be performed very efficiently. The alpha and beta orbitals should be slightly different, an effect called spin polarization. The disadvantage is that the wave function is no longer an eigenfunction of the total spin <(5 >. Thus, some error may be introduced into the calculation. This error is called spin contamination and it can be considered as having too much spin polarization. 

ROHF does not include spin polarization. Thus, it is not useful for some purposes, such as predicting EPR spectra. Also because of this, it cannot reliably predict spin densities. 

A UHF wave function may also be a necessary description when the effects of spin polarization are required. As discussed in Differences Between INDO and UNDO, a Restricted Hartree-Fock description will not properly describe a situation such as the methyl radical. The unpaired electron in this molecule occupies a p-orbital with a node in the plane of the molecule. When an RHF description is used (all the s orbitals have paired electrons), then no spin density exists anywhere in the s system. With a UHF description, however, the spin-up electron in the p-orbital interacts differently with spin-up and spin-down electrons in the s system and the s-orbitals become spatially separate for spin-up and spin-down electrons with resultant spin density in the s system. 

Concluding this section, two interesting variants of the STM should be addressed. The spin-polarized STM (SPSTM), which works with a ferromagnetic tip, can be used to probe surface magnetism with high resolution [5.47, 5.48]. Other modifications of the STM involve electromagnetic radiation, whereby two basic concepts can... 

In this exercise, we will be studying the spin polarization in a series of molecules of the form CH2=CH-XH . Our study will have two parts ... 

Optimize the structure of acetyl radical using the 6-31G(d) basis set at the HF, MP2, B3LYP and QCISD levels of theory. We chose to perform an Opt Freq calculation at the Flartree-Fock level in order to produce initial force constants for the later optimizations (retrieved from the checkpoint file via OptsReadFC). Compare the predicted spin polarizations (listed as part of the population analysis output) for the carbon and oxygen atoms for the various methods to one another and to the experimental values of 0.7 for the C2 carbon atom and 0.2 for the oxygen atom. Note that for the MP2 and QCISD calculations you will need to include the keyword Density=Current in the job s route section, which specifies that the population analysis be performed using the electron density computed by the current theoretical method (the default is to use the Hartree-Fock density). 

Once you have determined the appropriate level of theory, predict the spin polarizations for these other substituents CH2, Mg, Be and S. 

We also plotted the electron spin polarization, by itself (top row) and projected onto the electron density isosurface for the molecules containing the CH2, O and Be substituents (the orientation of the atoms in the plots is indicated at the left) ... 

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