Matrix elements Zeeman perturbation

There is no need to pass from the basis set of the ATs L,Mi,S,Ms) to the basis set of the atomic multiplets (LS),J,Mj) since such a unitary transformation does not lead to a gain in the computational effort. In the basis set of the L,Mi, S, Ms) functions, the operator Vax is diagonal but the operator Hso has off-diagonal matrix elements. In contrast, in the basis set of the (LS),J,Mj) kets, the operator Hso is diagonal, but the operator Vax has off-diagonal matrix elements. Therefore, none of these basis sets is appropriate for considering the Zeeman term as a small perturbation. 

Here, co represents the Euler angles (orbital Zeeman interaction, we see that it has off-diagonal matrix elements which link electronic states with A A = 0, 1, as well as purely diagonal elements. It is clearly desirable to remove the effect of these matrix elements by a suitable perturbative transformation to achieve an effective Zeeman Hamiltonian which acts only within the spin-rotational levels of a given electronic state rj, A, v), in the same way as the zero-field effective Hamiltonian in equation (7.183). 

In this appendix we collect together tables of matrix elements of r, r2, r3, z, z2, rz, and r(r2 — z2) in the scaled hydrogenic basis and perturbation corrections to the ground-state energy for the LoSurdo-Stark, Zeeman, screened Coulomb, and charmonium Hamiltonians and the one-electron diatomic ion. 

Matrix Elements of the Zeeman Perturbation W = r(r2 — z2) in the Scaled Hydrogenic Basis"... 

The asymmetric top matrix elements of the complete effective rotational Hamiltonian are then calculated and the eigenvalues are determined by a perturbation treatment. In view of the smallness of the Zeeman contributions, a first order perturbation treatment in the asymmetric top basis is sufficient in most cases so that only the diagonal elements of. f zeeman= g + x + TS need to be calculated explicitly. Symmetry considerations and order of magnitude estimates will prove helpful in this context. 

More subtle are the spin-orbit-induced heavy-atom chemical shifts at the atom nearest to the heavy one, or at more remote nuclei. If one uses the same Hamiltonians as Ramsey, one must go to third-order perturbation theory, with one Zeeman, one hyperfine, and one spin-orbit matrix element [22]. For a recent discussion on the nature of this shift, see Ref. [23]. It was also noted that an analogous effect, a heavy-atom shift on the heavy atom can occur, for instance on the Pb(II) nucleus in PbR compounds. The early semiempirical calculations suggested that the Zeeman-SO-Fermi contact cross term, zero in Ramsey s theory, could then become the dominant contribution to the Pb chemical shift [24]. 

To evaluate the effeet of spin-orbit coupling we will start writing down the first-order corrected wave functions of the Ms = 5 sublevels, then calculate the matrix elements of the Zeeman Hamiltonian (Eq.2.23) and compare these to the matrix elements of the (effective) spin-only Zeeman Hamiltonian given in Eq. 2.48 to find analytical expressions for the diagonal elements of g. With fZ, 5 as perturbation operator, the wave functions that describe the lowest two levels become... 

In most experiments the Zeeman splitting is small compared with the fine-structure separations and the effect of the magnetic field can be obtained by calculating the diagonal matrix elements of the perturbation operator. It can be shown (Problem 3.16) that the energy of the sub-level y LSJMj> is changed by an amount... 

The z-component of the Zeeman matrix appeared only at the diagonal of the local matrix HJ. However, after the transformation into the molecular-state basis the off-diagonal elements become non-zero. For small magnetic fields the Zeeman term can be treated as a perturbation and then second-order perturbation theory can be applied in order to determine the magnetic energy levels... 

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