Spin of a Single Electron

The solution to the spin problem followed when Dirac generalized the SE to a rela-tivistically invariant equation (1928). In the Dirac equation, the spin-orbit coupling appears as a natural part of the formalism. The Dirac wave function r has four components, but only two of them refer to electrons. The other two coordinates are positron coordinates. It is reasonable to assume that our particle is 100% an electron, with no probability density for being a positron. 

The electronic components correspond to electronic spin. We may assume that the spin is either up or down  

It is important to check that the new object in Equation 1.57 has the same mathematical properties as an ordinary wave function. In the scalar product (v /i /2). we need to replace the complex conjugate of the simple function rgi by the transposed complex conjugate of the vector representing pi  

From this equation, we may conclude that spin-up and spin-down wave function components never mix. 

We now need operators that operate on row vectors. An operator for a row vector is a 2 X 2 square matrix. In addition to the unit matrix e, Dirac used the Pauli spin matrices  

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Each hSOi is essentially the same operator acting on the coordinates and spins of a single electron. Thus, the argument i can be dropped for clarity. Each hso operator in turn is given (approximately) as sum of contributions of the atoms A in the molecule ... 

Let us consider more specifically wave functions depending on the coordinates (and possibly on the spin) of a single electron. Such functions are called orbitals (or, if spin is explicitly included, spin orbitals ). According to their behaviour under a reflection with respect to the nuclear plane of a planar molecule, they are classified as o or n orbitals one has... 

The spin of a single electron is always s = 1/2. No orbital reflection symmetry index is needed for a Rydberg orbital because all A = 0 states have E+ symmetry and the reflection symmetry of u> = 0 states is (—1)J. ... 

We have described the spin of a single electron by the two spin functions a(this section we will discuss spin in more detail and consider the spin states of many-electron systems. We will describe restricted Slater determinants that are formed from spin orbitals whose spatial parts are restricted to be the same for a and p spins (i.e., xi = il iP ), Restricted determinants, except in special cases, are not eigenfunctions of the total electron spin operator. However, by taking appropriate linear combinations of such determinants we can form spin-adapted configurations which are proper eigenfunctions. Finally, we will describe unrestricted determinants, which are formed from spin orbitals that have different spatial parts for different spins (i.e., fjS ). 

The explicit form of the spin eigenfunction psnis depends on the system under consideration. Since we will mostly consider ensembles of electrons in this account, the spin of a single electron is most important to us. Experimentally, it is found that an electron is to be described by a spin quantum number s = 1/2. The eigenvalue equations may be written in vector form with the electron spin operator... 

The separable, analytic functions 7/with / an integer are the spherical harmonics described in Chapter 4 and long known to classical physics. Indeed, the symbol 7/, has come to stand for those functions. For systems involving half-integer / and m values, no analytical functions of the usual sort can be written. Instead, matrices and vectors are used that manifest the correct relationships. Thus, for the spin of a single electron, Pauli used the following representation ... 

Electronic wavefunctions symbolized in this text as I e(ri, S], ra, S2,..., r , s ) depend on the spatial (r) and spin (s) variables of all the m electrons. The electron density on the other hand depends only on the coordinates of a single electron. I discussed the electron density in Chapter 5, and showed how it was related to the wavefunction. The argument proceeds as follows. The chance of finding electron 1 in the differential space element dti and spin element ds] with the other electrons anywhere is given by... 

Electron spin resonance (or electron paramagnetic resonance) is now a well-established analytical technique, which also offers a unique probe into the details of molecular structure. The energy levels involved are very close together and reflect essentially the properties of a single electronic state split by a small perturbation. 

As a particularly simple example of this formalism, let us consider the spin states of a single electron with respect to the 2-direction. In occupation number representation these states may be written as... 

Fig. 2.1 (A) Representation of the electronic ground state for a closed-shell system in which all of the lowest energy MOs contain two electrons of opposite spin. (B) Two example configurations for singlet excited states of the QM system that involve promotion of a single electron to a previously unoccupied, or virtual, MO. Note that the spins of the two unpaired electrons are antiparallel.

The assumption of a single electron spin and a single T2 holds usually for S = 1/2 and for S > 1 in certain limits. Let us assume that the instantaneous distortions of the solvation sphere of the ion result in a transient ZFS and that the time-dependence of the transient ZFS can be described by the pseudorotation model, with the magnitude of the transient ZFS equal to At and the correlation time t . The simple picture of electron relaxation for S = 1 is valid if the Redfield condition (Att <5c 1) applies. Under the extreme narrowing conditions ((Os v 1), the longitudinal and transverse electron spin relaxation rates are equal to each other and to the low-field limit rate Tgo, occurring in Eqs. (14) and (15). The low field-limit rate is then given by (27,86) ... 

The extension to other cases is straightforward but tedious, and the principal results for low-spin octahedral species are summarised in Table 2, which shows some interesting features. At this level of discussion, R loss is never assisted. The question of demotion only arises where the t2g subshell is less than full. Two-electron demotion is is only possible for R loss from cf , cf, and systems, and in all of these it is actually term-term assisted. R" loss is assisted by the demotion of one electron fotd, d, or d curves, but among these it is only term-term assisted for d. R loss is clearly assisted by the demotion of a single electron for all d" (n < 6), but is only term-term assisted for n = 1 and n = 2. (These predictions are quite different from those of Ref. which refers exclusively to second order terms in R loss). The only configurations with n < 6 for which no process shows first order term-term assistance are d and d. This is a gratifying result and tends to promote confidence in the usefulness of the theory. The relative ease of preparation of Cr(III) alkyl complexes has often been noted and t/ is exemplified by the Co(IV) alkyls now known to be accessible by electrochemical oxidation of Co(III) Presumably a parallel chemistry of Fe(III) awaits discovery. 

A soluhon of the equahons arising in the Hartree-Fock approximahon takes the form of a set of spin-orbitals - each consishng of a funchon of coordinates X, y, and 2 describing the spahal distribution of a single electron and an in-... 

Demirplak and Rice developed the counter-diabatic control protocol while studying control methods that efficiently transfer population between a selected initial state and a selected target state of an isolated molecule [11-13]. The protocol has been studied for manipulation of atomic and molecular states [11, 12, 19] and spin chain systems [20, 21]. Experiments with the counter-diabatic protocol have been demonstrated for the control of BECs [22] and the electron spin of a single nitrogen-vacancy center in diamond [23]. The counter-diabatic field (CDF) protocol is identical with the transitionless driving protocol, independently proposed by Berry a few years later [24]. A discussion of the relationship between these approaches and several of the other proposed shortcuts to adiabaticity can be found in the review by Torrontegui and coworkers [10]. 

The simplest process, among those noted above, is ionization of a single electron. The true vertical ionization potential (/H) for the ionization of an electron from the spin-orbital IH) is given by... 

The exclusion principle is one example of an emergent property. It cannot be predicted, or even formulated, from all the known properties of a single electron, but emerges as an inevitable property of a world collective of electrons, or other fermions - particles with half a unit of spin angular momentum. 

Fig. 53. Tetrahedral-site distortions due to ordering of a single electron. (Ordering of a single hole gives distortions of opposite sign.) (a) Jahn-Teller ordering into a drv orbital, (b) Spin-orbit ordering into the (dyz db idtx) orbitals.

At this point, it is helpful to collect all the various terms in the zero-field effective Hamiltonian together. We remind ourselves that the effective Hamiltonian is an operator which is confined to act only on the rotational, electron spin and nuclear spin states spanning a given vibrational level of a single electronic state to, A) ... 

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