The Spin Part of

Equation (3.1.2) is the nonrelativistic Hamiltonian. This means that the spin-dependent part of the Hamiltonian (Hso spin-orbit and Hss spin-spin) has been neglected. The electronic angular momentum quantum numbers, which are well-defined for eigenfunctions of nonrelativistic adiabatic and diabatic potential curves, are A, E, and 5 (and redundantly, Q = A + E). 

The relativistic Hamiltonian may be defined by adding Hso to Hel. The eigenfunctions of this new Hamiltonian are the relativistic wavefunctions, i,n, which define the relativistic potential curves 

Function type Definition of Definition of Et R) Curve type 

Adiabatic functions ( f Hel f) = 0 for all j i Ef=Ef° + ( rd T d)r Born-Oppenheimer potential curve Adiabatic potential curve 

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Before going on to calculate the energy levels it is necessary to digress and briefly describe the wavefunction. The spin Hamiltonian only operates on the spin part of the wavefunction. Every unpaired electron has a spin vector /S = with spin quantum numbers ms = + and mB = — f. The wavefunctions for these two spin states are denoted by ae) and d ), respectively. The proton likewise has I = with spin wavefunctions an) and dn)- In the present example these will be used as the basis functions in our calculation of energy levels, although it is sometimes convenient to use a linear combination of these spin states. 

The basic idea of the slow-motion theory is to treat the electron spin as a part of the lattice and limit the spin part of the problem to the nuclear spin rather than the IS system. The difficult part of the problem is to treat, in an appropriate way, the combined lattice, now containing the classical degrees of freedom (such as rotation in condensed matter) as well as quantized degrees of freedom (such as the electron Zeeman interaction). The Liouville superoperator formalism is very well suited for treating this type of problems. 

In those graphs, only the spin part of the four spin-orbitals appearing in eq. (10) is made explicit. 

The electric dipole moment operator is independent of spin and, in the absence of spin-orbit coupling, the spin part of the magnetic moment operator will have no influence on MCD. We shall therefore neglect the spin part of Eq. (8) until we come to spin-orbit-induced MCD in Section II.A.4. 

If energetics is dominated by a spatially uniform magnetization M, the spin part of the free energy density in the magnetic field H can be written in the form... 

In the above equations, integration over the spin parts of the MOs has been carried out and the refer only to the space part of the MOs. 

The superscripts stand for the spin part of the wave functions. The coefficients Aj, Bj, and C are taken to be real, and their values for azidoferrihemoglobin are given in Table 3. Fig. 24 shows an energy diagram for the three states. 

Symmetry dictates that the representations of the direct product of the factors in the integral (3 /T Hso 1 l/s2) transform under the group operations according to the totally symmetric representation, Aj. The spin part of the Hso spin-orbit operator converts triplet spin to singlet spin wavefunctions and singlet functions to triplet wavefunctions. As such, the spin function does not have a bearing on the symmetry properties of Hso- Rather, the control is embedded in the orbital part. The components of the orbital angular momentum, (Lx, Ly, and Lz) of Hso have symmetry properties of rotations about the x, y, and z symmetry axes, Rx, Ry, and Rz. Thus, from Table 2.1, the possible symmetry... 

Equation [172] or related expressions (Table 10) are applied extensively when evaluating the spin part of spin-orbit matrix elements, for configuration interaction (Cl) wave functions. The latter are usually provided for a single Ms component only. 

Note, that it is not possible to compute the nonvanishing MEs of a triplet-triplet coupling by using Ms = 0 wave functions, because the spin part of... 

When the orbitals are ordered so that the first two are the inner orbitals and, if a valence orbital is even-numbered (odd-numbered), its symmetry-equivalent counterparts also are even-numbered (odd-numbered), then the spin part of the SC wavefunction is dominated by the perfect-pairing Yamanouchi-Kotani (YK) spin function, with a coefficient exceeding 0.99. The coefficients of the other 13 YK functions are all smaller than 0.01. 

The spin part of this expression is readily evaluated ... 

The spin part of the Hamiltonian in the solid is identical to that of the AX system, and the solution is the same—a pair of lines for A and a pair for X, as in Fig. 6.2. However, the separation is dependent on orientation of the vector r joining the spins relative to B0, because the effective coupling is D(3 cos2 0—1), rather than D alone. As indicated in Fig. 7.2, the lines cross as 0 is varied, becoming coincident when 0 = 54.7°, the angle for which the term (3 cos2 0 — 1) = 0. This angle that will appear frequently in our later discussion. 

Multiple pulse (MP) sequences work on the spin part of the interaction Hamiltonian, and are based on the original ideas of Ostroff and Waugh75 and... 

Spin-allowed electronic transition An electronic transition which does not involve a change in the spin part of the wavefunction. 

In this way, it is hence possible to separate the orbital parts and the spin parts of the Fock-Dirac operator in a simple way. The... 

Electron-spin-based phenomena are attracting attention for the possibility of logic devices based on spin rather than on the flow of charge for possible quantum information applications. Retention of a spin lifetime in the material is important for these considerations. The conduction-band states at the zone center are approximately spin eigenstates, and therefore, they have relatively long spin lifetimes because they are made up of s-type orbitals. This is in contrast to valence-band states that are made up of p-type orbitals for which the spin-orbit interaction mixes the orbital part of the p state with the spin part of the electron wave... 

Pj,(f) is the spin part of the density matrix where the spin phase appears as... 

For an evaluation of the matrix element < (n,.n ) soP(- )> of Ihe operator given in Equation (1.42), it is convenient to first determine the effect of the spin part of the three components t Ij, and /x of the scalar product t Son the triplet function and to perform the spin integration, which reduces the number of terms considerably. In this way one obtains the following ... 

Let us analyze the principal components of the magnetic susceptibility tensor (/n and (j.). The spin part of the magnetic susceptibility is isotropic, so the anisotropy... 

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