Spin-orbit interactions

The electron has a spin angular momentum s and an associated magnetic moment /ig, i.e., 

As the electron moves in the electric field of the nucleus it is subject to a magnetic field B , which is proportional to the angular momentum 1 of the electron (Fig.2.4). The electron then has an orientational energy Eg in the field 

For the corresponding quantum numbers describing the length of the vector (e.g., j = [j(j+l)] /, dimensionless angular momentum The normally occurring h in the length expression of an angular momentum vector is included in ) we have 

Here we have inserted the quantum-mechanical squares or, to use a more modern term, used first-order perturbation theory. The so-called fine-structure splitting between the two possible j levels is given by 

For the corresponding quantmn numbers describing the length of the vector (e-g j = j j + dimensionless angular momentum The normally oc- 

Level ordering in alkali atom doublet series. (The asterisk indicates that the lowest member of the Rubidium series is inverted) 

The main effect of taking spin-orbit interaction into account will be an admixture of singlet character to triplet states and triplet character to singlet states. The spin-orbit coupling Hamiltonian can to a good approximation be described by an effective one-electron operator Hso  

The molecule is approximated by a set of shielded atoms, each center giving rise to a spherical electric field. Z, eif can be determined by the Slater rules, and li and Si are the orbital and spin angular momentum operators. 

First-order perturbation theory is then apphed to derive the nominal singlet ground state and first excited triplet functions. Pure spin states are no longer possible. 

The transition moment from So to the component Fs of Ti is obtained from  

The oscillator strength of a specific ATi- - So transition and the radiative lifetime of a sublevel f sTi can now be calculated from  

The term c in Equation 8-59 is the speed of light. The angular momentum of a particle moving about a circular loop is the particle s mass times the square of the radius of the loop times the frequency, mr o. The angular momentum of an electron in an orbit is determined by the D and operators hence. Equation 8-59 can be rewritten as follows for an electron in a hydrogen orbital. 

As can be seen by Equation 8-60, the magnetic dipole moment is proportional to the angular momentum of the electron. Since the angular momentum of an electron will be in units of h, it is convenient to collect the constant terms in Equation 8-60 and define a new constant called the Bohr magneton, b- 

Electronic magnetic dipole moments in molecules and atoms are measured in terms of Bohr magnetons in the same way that angular momentum is measured in terms of h. 

The same analysis can now be done for the intrinsic spin of an electron. The magnetic moment as a result of the intrinsic spin will be directly proportional to the angular momentum of the intrinsic spin, S.  

The expression for the magnetic dipole moment for the intrinsic spin of an electron is similar to that of an electron in its orbit except that an additional term g, is needed. The additional term is needed because the simple model of a circulating electron used to obtain Equation 8-60 does not apply to the intrinsic spin of an electron. 

There is, however, one more interaction which should be (at least phenomenologically) included into the magnetic Hamiltonian the spin-orbit interaction. 

If we apply a central field (which is exactly true for a free atom and a good approximation for an atom in a ligand field), the 4V(r) is a scalar function. The electric field strength becomes 

The interaction energy of the electron spin-internal magnetic field BN is given by the operator 

This equation defines the spin-orbit coupling constant (r) as a function of the electron-nuclear distance, for which the approximate expression is 

We have abstracted so far from the so-called Thomas precession. This originates in the relativistic transformations which account for the fact that the electron is moving in a curved path around a fixed nucleus. If an axis of the gyroscope obeys an own dynamical precession with the Larmor angular velocity coL = (e/m)B, then the corrected precession in the inertial system associated with the fixed nucleus is (o = (oL + o r, the Thomas precession being 

The atomic Hamiltonian (11.1) does not involve electron spin. In reality, the existence of spin adds an additional term, usually small, to the Hamiltonian. This term, called the spin-orbit interaction, splits an atomic term into levels. Spin-orbit interaction is a relativistic effect and is properly derived using Dirac s relativistic treatment of the electron. This section gives a qualitative discussion of the origin of spin-orbit interaction. 

When a proper relativistic derivation of the spin-orbit-interaction term Hs.o. in the atomic Hamiltonian is carried out, one finds that for a one-electron atom (see Bethe and Jackiw, Chapters 8 and 23) 

Calculating the spin-orbit interaction energy Es q by finding the eigenfunctions and eigenvalues of the operator H(ii.i) + where (n.i) is the Hamiltonian of Eq. (11.1), is difficult. One therefore usually estimates Es o. liy using perturbation theory. Except for heavy atoms, the effect of Hs.o. is small compared with the effect of and first-order 

For a many-electron atom, it can be shown (Bethe and Jackiw, p. 164) that the spin-orbit interaction energy is 

The ground electron configuration is 15 25 2 . Table 11.2 gives 5, D, and as the terms of this configuration. By Hund s rule, is the lowest term. [Alternatively, a diagram like (11.48) could be used to conclude that P is the lowest term.] The P term has L = 1 and S = 1, so the possible J values are 2,1, and 0. The levels of P are P2, P, and Pq). The 2p subshell is more than half filled, so the rule just given predicts the multiplet is inverted, and the 2 level lies lowest. This is the ground level of O. 

EXAMPLE Find the ground level of the oxygen atom. 

TrOr = Trcr, = Trcr = 0 and their determinants equal — 1 

The spin magnetic moment Ms of an electron interacts with its orbital magnetic moment to produce an additional term in the Hamiltonian operator and, therefore, in the energy. In this section, we derive the mathematical expression for this spin-orbit interaction and apply it to the hydrogen atom. 

According to the Biot and Savart law of electromagnetic theory, the magnetic field B at the fixed electron due to the revolving positively charged nucleus is given in SI units to first order in y/c by 

The spin-orbit energy —Ms B may be related to the spin and orbital angular momenta through equations (7.1) and (7.29) 

This expression is not quite correct, however, because of a relativistic effect in changing from the perspective of the electron to the perspective of the nucleus. The correction, known as the Thomas precession, introduces the factor on the right-hand side to give 

M = Cl I + C20x + ClOy + C4Oz where c, C2, C3, C4 are complex constants. 

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Because the spin-orbit interaction is anisotropic (there is a directional dependence of the view each electron has of the relevant orbitals), the intersystem crossing rates from. S to each triplet level are different. 

In this section, the spin-orbit interaction is treated in the Breit-Pauli [13,24—26] approximation and incoi porated into the Hamiltonian using quasidegenerate perturbation theory [27]. This approach, which is described in [8], is commonly used in nuclear dynamics and is adequate for molecules containing only atoms with atomic numbers no larger than that of Kr. 

Figure 1- Representatiori of degenerate states from nonrelativistic components, (a) Degenerate zeroth-order states at (b) Spin-orbit interaction splits 11 state, (c) With full spin-orbit...

In the nonrelativistic case much has been, and continues to be, learned about the outcome of nonadiabatic processes from the locus and topography of seams of conical intersection. It will now be possible to describe nonadiabatic processes driven by conical intersections, for which the spin-orbit interaction cannot be neglected, on the same footing that has been so useful in the nonrelativistic case. This fully adiabatic approach offers both conceptual and potential computational... 

The expressions for the rotational energy levels (i.e., also involving the end-over-end rotations, not considered in the previous works) of linear triatomic molecules in doublet and triplet II electronic states that take into account a spin orbit interaction and a vibronic coupling were derived in two milestone studies by Hougen [72,32]. In them, the isomorfic Hamiltonian was inboduced, which has later been widely used in treating linear molecules (see, e.g., [55]). 

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... 

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