Perturbation theory Zeeman effect

This is the spin Zeeman effect and in perturbation theory [69] gives the nonzero ground-state energy ... 

We calculate the effects of the Hamiltonian (8.105) on these zeroth-order states using perturbation theory. This is exactly the same procedure as that which we used to construct the effective Hamiltonian in chapter 7. Our objective here is to formulate the terms in the effective Hamiltonian which describe the nuclear spin-rotation interaction and the susceptibility and chemical shift terms in the Zeeman Hamiltonian. We deal with them in much more detail at this point so that we can interpret the measurements on closed shell molecules by molecular beam magnetic resonance. The first-order corrections of the perturbation Hamiltonian are readily calculated to be... 

The rotational mixing of the J = 2 and 3 levels by the electronic Zeeman effect, (9.6), may be calculated by second-order perturbation theory, noting that the separation between the two levels is 6/+, where Bq is the rotational constant in the v = 0 level the result is... 

They considered an increasing spin perturbation H2 that may reduce the original symmetry to only the second operation, or in other words, the irreducible structure of subspaces for II are decomposed into smaller non-decomposing components of H +H2. This theory, then, also explained the splitting of spectral terms by a perturbation that produces spin differences naturally. Empirically, such a phenomenon has been observed in the anomalous Zeeman effect, as spectral... 

Radford (1961, 1962) and Radford and Broida (1962) presented a complete theory of the Zeeman effect for diatomic molecules that included perturbation effects. This led to a series of detailed investigations of the CN B2E+ (v — 0) A2II (v = 10) perturbation in which many of the techniques of modern high-resolution molecular spectroscopy and analysis were first demonstrated anticrossing spectroscopy (Radford and Broida, 1962, 1963), microwave optical double resonance (Evenson, et at, 1964), excited-state hyperfine structure with perturbations (Radford, 1964), effect of perturbations on radiative lifetimes and on inter-electronic-state collisional energy transfer (Radford and Broida, 1963). A similarly complete treatment of the effect of a magnetic field on the CO a,3E+ A1 perturbation complex is reported by Sykora and Vidal (1998). The AS = 0 selection rule for the Zeeman Hamiltonian leads to important differences between the CN B2E+ A2II and CO a/3E+ A1 perturbation plus Zeeman examples, primarily in the absence in the latter case of interference effects between the Zeeman and intramolecular perturbation terms. 

The second-order effects comprise three contributions. The second-order Zee-man and hyperfine interactions involve obvious extensions of second-order Rayleigh-Schrodinger perturbation theory using the appropriate operators. If we introduce an arbitrary gauge origin Ra, and the variable tq — r — Rq, the second-order term involving both Zeeman and hyperfine operators is... 

The result follows from spin orthogonality. It is perfectly clear from experiments on atoms and molecules (Zeeman effect) that singlet-triplet and other apparently spin-forbidden transitions do occur, so we are led to assume that the spin and orbital motions of an electron are not uncoupled. In the above example, the transition moment will never vanish identically if the state with spin-orbit coupling operator ffjo- Assume that the state with index n and spin function a-j can interact with another state, say tn, with spin perturbation theory, the corrected state xi l is given by... 

Stark effect The splitting of lines in the spectra of atoms due to the presence of a strong electric field. It is named after the German physicist Johannes Stark (1874-1957), who discovered it in 1913. Like the normal Zeeman effect, the Stark effect can be understood in terms of the classical electron theory of Lorentz. The Stark effect for hydrogen atoms was also described by the Bohr theory of the atom. In terms of quantum mechanics, the Stark effect is described by regarding the electric field as a perturbation on the quantum states and energy levels of an atom in the absence of an electric field. This application of perturbation theory was its first use in quantum mechanics. 

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]. 

In this chapter, we review electronic structure in hydrogenlike atoms and develop the pertinent selection rules for spectroscopic transitions. The theory of spin-orbit coupling is introduced, and the electronic structure and spectroscopy of many-electron atoms is greated. These discussions enable us to explain details of the spectra in Fig. 2.2. Finally, we deal with atomic perturbations in static external magnetic fields, which lead to the normal and anomalous Zeeman effects. The latter furnishes a useful tool for the assignment of atomic spectral lines. 

Within degenerate perturbation theory at the nonrelativistic level, there are in principle two contributing terms arising from expectation values of the spin-Zeeman (O Eq. 11.50) and the magnetic dipole operator (O Eq. 11.44), respectively. The latter can be shown to be zero, and the effect of the spin-Zeeman operator is to recover the free-electron g factor, ge. Thus, within a purely nonrelativistic picture, there would be no effect of the electronic structure of the molecule on the interaction between an external magnetic field and the magnetic moment of the unpaired electron. 

If we examine Fig. 3 we see that only two of the four possible transitions are allowed, namely, those in which the nuclear spin does not change its orientation. The energy difference between these two transitions is defined as the hyperflne constant, usually symbolized by A in units of megahertz (MHz) or gauss (G). Since Ami = 0, the effect of the nuclear Zeeman term in the spin Hamiltonian will always cancel out for first-order spectral transitions. Thus, this term can be neglected in the Hamiltonian when one is considering only first-order spectra. However, it should be cautioned that if one considers spectra in which the perturbation theory approach must be carried out to second order or if one considers spin relaxation, which will be discussed later, the full spin Hamiltonian must be used. 

G. D. Bent, G. F. Adams, R. J. Bartlett, and G. D. Purvis, Many-body perturbation theory electronic structure calculations for the methoxy radical. I. Determination of Jahn-Teller energy surfaces, spin-orbit splitting, and Zeeman effect, manuscript submitted. 

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