The electron spin

It may seem surprising that in the preceding paragraphs we have never been concerned with the electron spin and the electron pair. 

Also with the older formulation of the Pauli principle we can understand that a penetration of the electron orbits, as in the attractive state, whereby the two electrons can, as it were, meet at one place, is only possible when there is also a difference in spin orientation. 

As already pointed out, terms such as wave function, electron orbit, resonance, etc., with which we describe the formulations and results of wave mechanics, are borrowed from classical mechanics of matter in which concepts occur which, in certain respects at least, show a correspondence to the wave mechanical concepts in question. The same is the case with the electron spin. In Bohr s quantum theory, Uhlenbeck and Goudsmit s hypothesis meant the introduction of a fourth quantum number j, which can only take on the values +1/2 and —1/2- In wave mechanics it means that the total wave function, besides the orbital function, contains another factor, the spin function. This spin function can be represented by a or (3, whereby, for example, a describes the state j = +1/2 and P that with s = —1/2. The correspondence with the mechanical analogy, the top, from which the name spin has been borrowed, is appropriate in so far that the laevo and dextro rotatory character, or the pointing of the top in the + or — direction, can be connected with it. A magnetic moment and a 

With two electrons 1 and 2 we can compose the following four spin functions oc(i) a (2) 

It is clear that the first three are symmetrical with respect to interchange of the electrons (1 )and (2). Only the latter is an tisymmetrical and therefore only this one can be combined with the symmetrical orbital function of the bonding state, which thus has a total spin moment S = o. The other three spin functions, multiplied by the antisymmetrical orbital function of the repulsive state, also give three different total wave functions. These three functions correspond to three states of equal energy (threefold degeneracy) with the total spin moment S = 1. 

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Initially, we neglect tenns depending on the electron spin and the nuclear spin / in the molecular Hamiltonian //. In this approximation, we can take the total angular momentum to be N(see (equation Al.4.1)) which results from the rotational motion of the nuclei and the orbital motion of the electrons. The components of. m the (X, Y, Z) axis system are given by ... 

If we allow for the tenns in the molecular Hamiltonian depending on the electron spin - (see chapter 7 of [1]), the resulting Hamiltonian no longer connnutes with the components of fVas given in (equation Al.4.125), but with the components of... 

Finally, we consider the complete molecular Hamiltonian which contains not only temis depending on the electron spin, but also temis depending on the nuclear spin / (see chapter 7 of [1]). This Hamiltonian conmiutes with the components of Pgiven in (equation Al.4,1). The diagonalization of the matrix representation of the complete molecular Hamiltonian proceeds as described in section Al.4,1.1. The theory of rotational synnnetry is an extensive subject and we have only scratched the surface here. A relatively new book, which is concemed with molecules, is by Zare [6] (see [7] for the solutions to all the problems in [6] and a list of the errors). This book describes, for example, the method for obtaining the fimctioiis ... 

The interaction of the electron spin s magnetic dipole moment with the magnetic dipole moments of nearby nuclear spins provides another contribution to the state energies and the number of energy levels, between which transitions may occur. This gives rise to the hyperfme structure in the EPR spectrum. The so-called hyperfme interaction (HFI) is described by the Hamiltonian... 

In equation (bl. 15.24), r is the vector coimecting the electron spin with the nuclear spin, r is the length of this vector and g and are the g-factor and the Boln- magneton of the nucleus, respectively. The dipolar coupling is purely anisotropic, arising from the spin density of the impaired electron in an orbital of non-... 

The simplest system exliibiting a nuclear hyperfme interaction is the hydrogen atom with a coupling constant of 1420 MHz. If different isotopes of the same element exhibit hyperfme couplings, their ratio is detemiined by the ratio of the nuclear g-values. Small deviations from this ratio may occur for the Femii contact interaction, since the electron spin probes the inner stmcture of the nucleus if it is in an s orbital. However, this so-called hyperfme anomaly is usually smaller than 1 %. 

While all contributions to the spin Hamiltonian so far involve the electron spin and cause first-order energy shifts or splittings in the FPR spectmm, there are also tenns that involve only nuclear spms. Aside from their importance for the calculation of FNDOR spectra, these tenns may influence the FPR spectnim significantly in situations where the high-field approximation breaks down and second-order effects become important. The first of these interactions is the coupling of the nuclear spin to the external magnetic field, called the... 

Phenomenologically, the FNDOR experiment can be described as the creation of alternative relaxation paths for the electron spins, which are excited with microwaves. In the four-level diagram of the system... 

Feher G 1956 Observation of nuclear magnetic resonances via the electron spin resonance line Rhys. Rev. 103 834-7... 

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. 

As is well known, when the electronic spin-orbit interaction is small, the total electronic wave function v / (r, s R) can be written as the product of a spatial wave function R) and a spin function t / (s). For this, we can use either... 

Moreover, as also shown in Appendix C for the electronic spin, one has... 

The permutational symmetry of the rotational wave function is determined by the rotational angular momentum J, which is the resultant of the electronic spin S, elecbonic orbital L, and nuclear orbital N angular momenta. We will now examine the permutational symmetry of the rotational wave functions. Two important remarks should first be made. The first refers to the 7 = 0 rotational... 

Secondly, you must describe the electron spin state of the system to be calcn lated. Electron s with their individual spin s of Sj=l /2 can combine in various ways to lead to a state of given total spin. The second input quantity needed is a description of the total spin... 

Both of these integrals are zero due to the orthogonality of the electron spin states a and fd. 

Irons of benzene are distributed in pairs among its three bonding tt MOs giving a closed shell electron configuration All the bonding orbitals are filled and all the electron spins are paired... 

Secondly, you must describe the electron spin state of the system to be calculated. Electrons with their individual spins of sj=l/2 can combine in various ways to lead to a state of given total spin. The second input quantity needed is a description of the total spin S=Esj. Since spin is a vector, there are various ways of combining individual spins, but the net result is that a molecule can have spin S of 0, 1/2, 1,. These states have a multiplicity of 2S-tl = 1, 2, 3,. ..,that is, there is only one way of orienting a spin of 0, two ways of orienting a spin of 1/2, three ways of orienting a spin of 1, and so on. 

MOs around them - rather as we construct atomic orbitals (AOs) around a single bare nucleus. Electrons are then fed into the MOs in pairs (with the electron spin quantum number = 5) in order of increasing energy using the aufbau principle, just as for atoms (Section 7.1.1), to give the ground configuration of the molecule. 

The nature of the intermediates impHcated in the photooxidation of water with Ti02 has been identified in several reports using spin traps by the electron spin resonance (esr) technique under ambient conditions (53). No evidence for OH species, even at 4.2 K, was found (43), but the esr signal... 

Resonance theory can also account for the stability of the allyl radical. For example, to form an ethylene radical from ethylene requites a bond dissociation energy of 410 kj/mol (98 kcal/mol), whereas the bond dissociation energy to form an allyl radical from propylene requites 368 kj/mol (88 kcal/mol). This difference results entirely from resonance stabilization. The electron spin resonance spectmm of the allyl radical shows three, not four, types of hydrogen signals. The infrared spectmm shows one type, not two, of carbon—carbon bonds. These data imply the existence, at least on the time scale probed, of a symmetric molecule. The two equivalent resonance stmctures for the allyl radical are as follows ... 

The electron—photon coupling that forms the microscopic basis of MOKE makes it possible, in principle, to determine the electron spin-dependent band structure of elements and alloys. This is done by examining the dependence of the Kerr response on the wavelength of the incident light. 

The EPR spectrum of the ethyl radical presented in Fig. 12.2b is readily interpreted, and the results are relevant to the distribution of unpaired electron density in the molecule. The 12-line spectrum is a triplet of quartets resulting from unequal coupling of the electron spin to the a and P protons. The two coupling constants are = 22.38 G and Op — 26.87 G and imply extensive delocalization of spin density through the a bonds Note that EPR spectra, unlike NMR and IR spectra, are displayed as the derivative of absorption rather than as absorption. 

Although it is required to refine the above condition I in actuality, this rather simple but impressive prediction seems to have much stimulated the experiments on the electrical-conductivity measurement and the related solid-state properties in spite of technological difficulties in purification of the CNT sample and in direct measurement of its electrical conductivity (see Chap. 10). For instance, for MWCNT, a direct conductivity measurement has proved the existence of metallic sample [7]. The electron spin resonance (ESR) (see Chap. 8) [8] and the C nuclear magnetic resonance (NMR) [9] measurements have also proved that MWCNT can show metallic property based on the Pauli susceptibility and Korringa-like relation, respectively. On the other hand, existence of semiconductive MWCNT sample has also been shown by the ESR measurement [ 10], For SWCNT, a combination of direct electrical conductivity and the ESR measurements has confirmed the metallic property of the sample employed therein [11]. More recently, bandgap values of several SWCNT... 

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