Spin-Orbit Coupling Term

Here and H describe radicals A and B of the radical pair and He the interaction of their electrons. The other terms in equation (15) are H g, the spin orbit coupling term, H g and Hgj, representing the interaction of the externally applied magnetic field with the electron spin and nuclear spin, respectively Hgg is the electron spin-spin interaction and Hgi the electron-nuclear hyperfine interaction. 

While the Hamiltonian operator Hq for the hydrogen atom in the absence of the spin-orbit coupling term commutes with L and with S, the total Hamiltonian operator H in equation (7.33) does not commute with either L or S because of the presence of the scalar product L S. To illustrate this feature, we consider the commutators [L, L S] and [S, L S],... 

It comprises the non-relativistic Hamiltonian of the form pf/2me + V and the relativistic correction terms, such as the mass-velocity operator —pf/8m c2, the Darwin term proportional to Pi E and the spin-orbit coupling term proportional... 

The examination takes place in two stages, one corresponding to the formal interelectronic repulsion component of the Hamiltonian HER and the second to the spin-orbit coupling term Hes. As will be pointed out, in principle, and in certain cases in practice, it is not proper to separate the two components. However, the conventional procedure is to develop HLS as a perturbation following the application of Her. That suffices for most purposes, and simplifies the procedures. Any interaction between the d- or/-electron set and any other set is ignored. It is assumed that it is negligible or can be taken up within the concept of an effective d-orbital set. 

The treatment of intersystem crossing, that is, of the k c and kbisc paths in equation (22), is in principle similar to that for knr.There is an added spin-orbit coupling term in the evaluation of the electronic factor in the treatment of the crossing probability. Although theory has provided a useful framework for discussing nonradiative processes, it has not so far been of much diagnostic or predictive value in the case of coordination compounds. 

Although the predominant part of the nuclear potential is the part described above that is derived from the central force produced by the average effects of all other nucleons in the system on the individual nucleon under observation, evidence exists that nonsymmetnc tensor and spin-orbit coupling terms must be included in the description of the nuclear potential. 

The spin-orbit coupling term in the Hamiltonian induces the coupling of the orbital and spin angular momenta to give a total angular momentum J = L + S. This results in a splitting of the Russell-Saunders multiplets into their components, each of which is labeled by the appropriate value of the total angular momentum quantum number J. The character of the matrix representative (MR) of the operator R(0 n) in the coupled representation is... 

TABLE 1 Experimental or theoretical values of the crystal field and spin-orbit coupling terms appearing in EQN (1) for wurtzite GaN estimated to be strain free. 

In the present treatment, we retain essentially all the diagonal matrix elements of X these are the first-order contributions to the effective electronic Hamiltonian. There are many possible off-diagonal matrix elements but we shall consider only those due to the terms in Xrot and X o here since these are the largest and provide readily observable effects. The appropriate part of the rotational Hamiltonian is —2hcB(R)(NxLx + NyLy). The matrix elements of this operator are comparatively sparse because they are subject to the selection rules AA = 1, A,Y=0 and AF=0. The spin-orbit coupling term, on the other hand, has a much more extensive set of matrix elements allowed... 

The only new term to be considered, therefore, is the third-order spin orbit coupling term, whose matrix elements in a case (b) basis are evaluated as follows ... 

The effective Hamiltonian used to analyse the spectrum was that previously described for the 4E state of CH, equation (9.148), but with the addition of two extra terms. The first is a fourth-order spin orbit coupling term, described by Brown and Milton [71] ... 

The spin-orbit coupling term is straightforward, with case (b) matrix elements ... 

Bernhardt and Comba have combined molecular mechanics and the angular overlap model (AOM) to predict the d-d electronic spectra of several hexaamine compounds.The force field used, which had been described earlier, is appropriate for the modeling of Cu(II), Ni(II) (S = 1), Co(III), Fe(III), Cr(III), Zn(II), and Rh(III) complexes of amines, carboxylates, pyridines, and thia ethers. The AOM requires several parameters including metal-ligand cr-and TT-bonding terms [e and e ), interelectronic repulsion terms P- and F" ), and spin-orbit coupling terms Q. The current application uses the empirical relationship e r) = C/r to determine the value of e, where r is the bond distance and C is a constant whose value increases with alkyl substitution. This choice is consistent with the view that N-alkylation increases donor strength. The d-d electronic spectra of a variety of Cr(III), Co(III), and Ni(Il) hexaamine complexes are predicted to within 1000 cm. ... 

In the case of intersystem crossing transitions between states of different multiplicity, an additional spin-orbit coupling term Ago has to be considered. From the perturbational expansion it follows that the contribution... 

However, the 5 state is viewed to play a minor role here in spite of the fact that it crosses the l1 A, S+,3 A) potential energy curves (Figure 6). The off-diagonal spin-orbit coupling terms between this quintet and these electronic states are computed to be equal to zero (for 1A, E+) or close to zero (for 3A) [17]. 

The standard term symbol gives a pre-superscript of 2S 11, where S is the total spin. The major symbol is the total angular momentum. The post-superscript and subscript are a symmetry term and a spin orbital coupling term. The electronic configuration determines the term symbol. 

Hochlaf, M., Taylor, S., Eland, J. H. D., Quartet States of the Acetylene Cation Electronic Structure Calculations and Spin Orbit Coupling Terms, J. Chem. Phys. 2006, 125, 214301/ 214301 214301/214308. 

Recently it has been shown [21] that the spectrum of this g-deformed, 3-dimensional harmonic oscillator (Q30) reproduces very well that of the modified harmonic oscillator introduced by Nilsson [12, 15] without the spin-orbit coupling term. Since the Nilsson model without spin-orbit coupling is essentially the Nilsson-Clemenger model used for the description of metal clusters [11], it is worth examining whether the Q30 model can be used to reproduce the magic numbers and some other properties of simple metal clusters. 

Equation (74) bears a great similarify with that for the modified harmonic oscillator suggested by Nilsson and Clemenger [11, 12] where the spin-orbit coupling term is omitted ... 

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