Magnetic perturbation operators

The definitions of the first and second order magnetic perturbation operators are given helow. In the nonrelativistic formalism these operators are two-component operators, in the Kutzelnigg formalism all operators are to he multiplied hy the four-component matrix. All operators are given in the atomic unit system and we do not apply QED corrections so that the free electron g-factor is precisely equal to 2. 

It is also possible to use the transformed magnetic perturbation operator of section 13.7. The use of this operator is discussed in section 19.5. 

As it has been shown previously [21 ], at the ZORA level the magnetic perturbation operators are the same no matter if the minimal substitution is made in the Dirac Hamiltonian before... 

The theory of including magnetic perturbations has been discussed earlier.(11,14-16) In Dirac theory, external fields appear through the operator... 

Zeeman Interaction. We now wish to find the effect of applying a magnetic field upon the doubly degenerate ground state of c/1 in a tetragonal field. The perturbation operator for this can be written... 

Consider a dn configuration present in a crystal field that leaves the ground state nondegenerate except for spin. The ground state then consists of (25+ l)-spin states and the effect of the spin-orbit interaction plus the magnetic field can be computed using first- and second-order perturbation theory. If we take as the perturbation operator... 

Table 1 Nonrelativistic one-electron magnetic terms in the Hamiltonian. Their derivatives with respect to /m and/or or [ip enter the expressions for the nuclear shielding and spin-spin coupling tensors via the perturbation operators ) ancj g is the spin-operator for an electron, va a distance vector with respect to nucleus A etc. ...

For a perturbing electric field in the v-direction we have V = W = Dv and W — Y = 0, while for a magnetic field in the v-direction we have for the imaginary magnetic moment operator W = —V = +MV and V + W = 0. A nonzero frequency couples the symmetric and the antisymmetric part of the perturbed density matrix, whereas in the static case the two equations in (16) are not coupled. For comments on the apparent lack of symmetry for the perturbation equations for static electric and magnetic fields see [46]. 

The perturbation operator 3 is proportional to the absolute value of the magnetic field strength H75 ... 

While the chemical interpretation of the e parameters is a matter of real concern to us, there are also several other difficulties which are, however, more apparent than real. Consider the question of the calculation of magnetic properties in transition metal complexes - paramagnetic susceptibilities and e.s.r. g values. In contrast to the study of eigenvalues for optical transition energies, these require descriptions of the wavefunc-tions after the perturbation by the ligand field, interelectron repulsion and spin-orbit coupling effects. In susceptibility calculations it is customary to use Stevens orbital reduction factor k in the magnetic moment operator... 

The presence of a magnetic field introduces two new terms, being linear and quadratic in the field. The B-L term is the orbital analogue of the Zeeman effect for the electron spin, discussed in section 8.1 (eq. (8.19)). The second-order property is the magnetizability which according to eqs. (10.17) and (10.18)/(10.27) contains contributions from both linear and quadratic perturbation operators. The operator is (half) the angular moment L, while the P operator may be written as... 

Perturbation theory can be formulated in terms of a complete set of functions Wo,..., Wt, which may be eigenfunctions of Ho (although this is not necessary and may in fact yield a poor convergence). If an external perturbation is apphed, for example an electric or magnetic field, a second perturbation operator must be added to Ho and the Hamiltonian becomes... 

We shall limit attention to molecular properties arising from the introduction of external electric and magnetic fields through minimal coupling (118). In the next section we will consider specific fields for the moment we will focus on the general forms. The introduction of external fields leads to perturbation operators on the form... 

The generic form of the perturbation part of the total Hamiltonian is expressed by (20). In this section we consider specific forms of the perturbation operators Hx appearing in this expression. Note that these operators are time-independent any time-dependence of the perturbation is expressed by the exponentials appearing in the Fourier transform of V(t). We will consider the perturbation operators arising from nuclear spins as well as external electric and magnetic fields. 

First order relativistic one-electron perturbation operators from the introduction of a uniform electric field E, a uniform magnetic field B and a nuclear magnetic dipole moment Mjf. , gives the time reversal symmetry of the operator and jt = -eca-, is the current density operator. 

Second order non-relativistic one-electron perturbation operators from the introduction of a uniform magnetic field B and a nuclear magnetic dipole moment Mjf. 

It does, probably not create any confusion that the same symbol Q is used for an operator acting on 4-component spinors and one acting on 2-component spinors. The case of a magnetic perturbation with 12 nondiagonal will be discussed in the next subsection. We start from... 

A J = 0 this means that, in the absence of nuclear spin and external perturbations (electric or magnetic fields, electromagnetic radiation fields, or collisions with other molecules), the total angular momentum of the molecule remains well defined. Even if the perturbation operator includes J+ or J-, this operator cannot change the value of J. Even in case (b), J (as well as N) remains well defined. Perturbations (denoted by ) correspond to an interaction between two levels, as opposed to an electric dipole transition (denoted by —) between two levels. 

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