Magnetic Hamiltonians

Magnetic Hamiltonians are defined for a desired group of N electronic states obtained in the ab initio calculation, to which a pseudospin S (it reduces to a true spin S in the absence of spin-orbit coupling) is subscribed according to the relation N = 2S + 1. For instance, the two wave functions of a I[Pg.161]

Parameters of pseudospin magnetic Hamiltonians (see the description in Section 6.3.1, points 1-4). It is important to note that the multiplicity of the pseudospin can become quite large in the case of polynuclear compounds with weakly interacting magnetic sites. 

Table 1 Explanation of the magnetic Hamiltonian and the spin Hamiltonian...

Most of the variational treatments of spin-orbit interaction utilize one-component MOs as the one-particle basis. The SOC is then introduced at the Cl level. A so-called SOCI can be realized either as a one- or two-step procedure. Evidently, one-step methods determine spin-orbit coupling and electron correlation simultaneously. In two-step procedures, typically different matrix representations of the electrostatic and magnetic Hamiltonians are chosen. 

J.-P. Malrieu, The Hierarchy of VB Determinants and How to Exploit it Through Magnetic Hamiltonians, in Valence Bond Theory and Chemical Structure, D. J. Klein and N. Trinajstic (eds.), Elsevier, 1990, pp. 135-175. 

Therefore the hyperfine coupling A tensor is often given in units of Tesla ("A /gNA N IT =1.36 MHz for Fe). The magnetic Hamiltonian is given as... 

For the discussion of hard pulses and chemical shift evolution, the Hamiltonian is transformed into the rotating frame. The only change is in the form of the magnetic Hamiltonian HM, which becomes35... 

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

We have obtained so far that the magnetic Hamiltonian Hmg B, jle) contains the following terms... 

INCLUSION OF THE NUCLEAR SPIN 3.5.1 Magnetic Hamiltonian with nuclear spin... 

When the explicit expansion of the magnetic Hamiltonian is considered... 

INCLUSION OF THE ELECTRON AND NUCLEAR SPINS 3.6.1 Magnetic Hamiltonian with electron and nuclear spins... 

In the third, the most general case the energy of the system depends upon the external magnetic field, the magnetic moments of nuclei and the electronic magnetic moment. The following expansion of the magnetic Hamiltonian is assumed... 

The number in the magnetic Hamiltonian indicates an ordering number of the Hamiltonian term according to Table 4.6.) When the derivatives are completed one gets... 

The (spin) Zeeman term entering the total magnetic Hamiltonian is expressed... 

In order to follow the procedure in more detail the case of S = 1 will serve as an example. The magnetic Hamiltonian in the parallel direction (, z) has the following matrix elements... 

Eigenvalues of the magnetic Hamiltonian matrices for E = 0 shifted to an appropriate zero level... 

The magnetic Hamiltonian in the perpendicular direction ( L, x) has the following matrix elements... 

The magnetic Hamiltonian in the second perpendicular direction (J, y) is complex Hermitian... 

The magnetic energy levels are then obtained as the eigenvalues of the magnetic Hamiltonian... 

The g-tensor is obtained by evaluating the matrix elements of the magnetic hamiltonian (18.105) using the wave functions of the lowest levels that result from solution of the eigenvalue problem for %cef (18.109). Illustrative calculations of the g-factors for lanthanide ions in various cases are given, for example, by Al tshuler and Kozyrev (1974) and by Abragam and Bleaney (1970), among many others. 

The magnetic Hamiltonians describe the interaction of electrons with the intramolecular perturbation, that is, the intrinsic magnetic dipoles m/, via the vector potential XXi A , and with an external, spatially uniform and time-independent magnetic field B = V x A ,... 

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