Spin-orbit coupling Hamiltonian equations

Our choices of relativistic isolated cluster Hamiltonian allow us to write the embedded-cluster Hamiltonian of equation 9.1 as a sum of spin-free and spin-orbit coupling Hamiltonians,... 

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],... 

II electronic states, 638-640 vibronic coupling, 628-631 triatomic molecules, 594-598 Hamiltonian equations, 612-615 pragmatic models, 620-621 Kramers doublets, geometric phase theory linear Jahn-Teller effect, 20-22 spin-orbit coupling, 20-22 Kramers-Kronig reciprocity, wave function analycity, 201 -205 Kramers theorem ... 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

Semiclassical studies of the propagation of coherent states have proven useful in many circumstances see, e.g., (Klauder and Skagerstam, 1982 Perelomov, 1986). Here we consider spin-orbit coupling problems that result from the Dirac equation either in a semiclassical or in a non-relativistic approximation (see, e.g., the Hamiltonians (30) and (31)). The Hamiltonians H that arise in such a context can be viewed as Weyl quantisations of symbols... 

The ZFS representation in the spin Hamiltonian of equation (81) is that for axial symmetry. The parameter D arises (Section 6.4.4.6) from the action of spin—orbit coupling in second order. 

In Section 1.4, we discussed the history and foundations of MO theory by comparison with VB theory. One of the important principles mentioned was the orthogonality of molecular wave functions. For a given system, we can write down the Hamiltonian H as the sum of several terms, one for each of the interactions which will determine the energy E of the system the kinetic energies of the electrons, the electron-nucleus attraction, the electron-electron and nucleus-nucleus repulsion, plus sundry terms like spin-orbit coupling and, where appropriate, other perturbations such as an applied external magnetic or electric field. We now seek a set of wave functions P, W2,... which satisfy the Schrodinger equation ... 

The last term in Equation (2.36), Agoz/soc, is a second order contribution arising from the coupling of the orbital Zeeman (OZ) and the spin-orbit coupling (SOC) operators. The OZ contribution in the system Hamiltonian is ... 

Hess et al.119 utilized a Hamiltonian matrix approach to determine the spin-orbit coupling between a spin-free correlated wave function and the configuration state functions (CSFs) of the perturbing symmetries. Havriliak and Yarkony120 proposed to solve the matrix equation... 

Moriya (451) has extended the Anderson formalism for superexchange to include spin-orbit coupling in the perturbation Hamiltonian that appears in the transfer integral bij of equation 100. To first order, this leads to a second-order perturbation in the energy of the form... 

The energy of the helium atom calculated above is the first-order energy, which differs from the true energy by an amount called the correlation energy this is a measure of the tendency of the electrons to avoid each other. The simplest improvement to the trial wave function is to allow Z in (6.29) to be a variable parameter, which we call (not to be confused with the spin-orbit coupling parameter in equation (6.20)) Z in the Hamiltonian (6.23) remains the same. The expression for the calculated energy,... 

The spin-orbit coupling constant A is the coefficient of the operator term LZSZ in the effective Hamiltonian. It arises from the operator 3t in equation (7.71) we have seen in section 7.4.4 how the matrix elements of this operator lead to first-, second- and higher-order contributions in the effective Hamiltonian,... 

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] ... 

With only minor modifications, the spin-independent relativistic corrections can be put into this Schrodinger equation and so be absorbed in the definition of the basis orbitals ipi the spin orbit coupling potential, however, is treated explicitly in the effective Hamiltonian. 

In the case of the 5 = 5/2 ground states of high-spin Fe complexes, there is a ZFS of the Ms = 1/2, 3/2, 5/2 microstates due to spin-orbit coupling based on the axial D) and rhombic E) ZFS parameters within the spin Hamiltonian, H, with the central metal (Figure 13) and equation (22) ... 

The basic CDFT Hamiltonian in Equation (5.6) does not rule out the existence of a finite orbital magnetic moment in the nonrelativistic limit. With the help of model calculations, it could be demonstrated that this is not the case for bulk Fe, Co and Ni (Ebert et al. 1997a). Starting an SCF calculation with a finite spin-orbit-induced orbital current density and switching off the spin-orbit coupling during the SCF-cycle the orbital magnetic moment vanished (see also next section). 

A different approach is chosen when the screening of nuclear potential due to the electrons is incorporated in /z . Transformation to the eigenspinor basis is then only possible after the DHF equation is solved which makes it more difficult to isolate the spin-orbit coupling parts of the Hamiltonian. Still, it is also in this case possible to define a scalar relativistic formalism if the so-called restricted kinetic balance scheme is used to relate the upper and lower component expansion sets. The modified Dirac formalism of Dyall [24] formalizes this procedure and makes it possible to identify and eliminate the spin-orbit coupling terms in the selfconsistent field calculations. The resulting 4-spinors remain complex functions, but the matrix elements of the DCB Hamiltonian exhibit the non-relativistic symmetry and algebra. 

The quasi-relativistic Hamiltonians obtained using the general ansatz have usually a metric that include spin-orbit contributions. This can be a undesirable situation since in a perturbation study of spin-orbit effects, the addition of the spin-orbit coupling requires reorthogonalization of the orbitals [69]. However, as seen in equation (18), the relativistic correction term (V) consists of two contributions f and B. B is several orders of magnitude less significant than f. Furthermore, the B operator can also be separated into scalar relativistic and spin-orbit contributions. 

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