Spin angular momentum operators

Electrons and most other fiindamental particles have two distinct spin wavefunctions that are degenerate in the absence of an external magnetic field. Associated with these are two abstract states which are eigenfiinctions of the intrinsic spin angular momentum operator S... 

Equation (9-392) together with (9-394) and (9-395) are the proofs of the assertions that x is the position operator in the Foldy-Wouthuysen representation.16 (Note also that x commutes with /J the sign of the energy.) We further note that in the FTP-representation the operators x x p and Z commute with SFW separately and, hence, are constants of the motion. In the F W-representation the orbital and spin angular momentum operators are thus separately constants of the motion. The fact that... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

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. 

Polyatomic molecules. The same term classifications hold for linear polyatomic molecules as for diatomic molecules. We now consider nonlinear polyatomics. With spin-orbit interaction neglected, the total electronic spin angular momentum operator 5 commutes with //el, and polyatomic-molecule terms are classified according to the multiplicity 25+1. For nonlinear molecules, the electronic orbital angular momentum operators do not commute with HeV The symmetry operators Or, Os,. .. (corresponding to the molecular symmetry operations R, 5,. ..) commute... 

Let us also recall that the one-electron submatrix element of spin angular momentum operator s1 and of scalar quantity s° (in h units) is given by... 

Notice that the latter nine operators are formed by all the possible products (hence the name of the formalism), two at time, of the three spin angular momentum operators of spin / and the three spin angular momentum operators of spin K. Vector representations of all these operators are shown in Fig. IX.2. 

Here are Luttinger parameters, and me is an electron effective mass, av, b, d and ac are Bir-Pikus deformation potentials. As0 is a spin-orbit splitting energy. L and a denote orbital and spin angular momentum operators, respectively. [Lf, LJ is defined as [Lf, LJ = (LjLj + LjLj)/2. The summation of i,j runs through x, y, z. 

We next use the Pauli matrix representations of the spin angular momentum operator components in the instantaneous molecule-fixed axis system from equation (2.92) to rewrite the above relationships ... 

We saw in section 3.3 that neither the orbital (TiL = R a P) nor the spin (1/2)7/a angular momenta commute with the Dirac Hamiltonian and are not therefore separate constants of motion, although their sum is. However, we can construct the mean orbital angular momentum and mean spin angular momentum operators in the same way as in equation (3.129). These operators are, respectively,... 

It is a fundamental fact of quantum mechanics, that a spin-independent Hamiltonian will have pure spin eigenstates. For approximate wave functions that do not fulfill this criterion, e.g. those obtained with various unrestricted methods, the expectation value of the square of the total spin angular momentum operator, (5 ), has been used as a measure of the degree of spin contamination. is obviously a two-electron operator and the evaluation of its expectation value thus requires knowledge of the two-electron density matrix. 

The difference is that B1 is applied in the transverse plane, for instance along the x-axis of the laboratory coordinate frame. Consequently, only the x-component of the spin angular momentum operator I defines the interaction energy together with the magnitude and time dependence of the x-component of B (cf. Section 2.2.1). 

Because the convenience of the one-electron formalism is retained, DFT methods can easily take into account the scalar relativistic effects and spin-orbit effects, via either perturbation or variational methods. The retention of the one-electron picture provides a convenient means of analyzing the effects of relativity on specific orbitals of a molecule. Spin-unrestricted Hartree-Fock (UHF) calculations usually suffer from spin contamination, particularly in systems that have low-lying excited states (such as metal-containing systems). By contrast, in spin-unrestricted Kohn-Sham (UKS) DFT calculations the spin-contamination problem is generally less significant for many open-shell systems (39). For example, for transition metal methyl complexes, the deviation of the calculated UKS expectation values S (S = spin angular momentum operator) from the contamination-free theoretical values are all less than 5% (32). 

The amounts of the three operators will vary with time during pulses and delays. This expression of the density operator as a combination of the spin angular momentum operators is exactly analogous to specifying the three components of a magnetization vector. 

L the total orbital angular momentum operator the total spin angular momentum operator... 

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