Electron orbital motions

Hfi includes a nuclear Zeeman term, a nuclear dipole-dipole term, an electron-nuclear dipole term and a term describing the interaction between the nuclear dipole and the electron orbital motion. 

In those calculations, the contributions from electronic orbital motion (induced by spin-orbit mixing) were estimated from crystal field theory (for the copper atom) or were neglected (for the nitrogen and hydrogen atoms). Here I discuss for the first time direct calculations of these contributions to the copper and nitrogen hyperfine tensors, as well as to the molecular -tensor. 

It can be shown from the commutation relations that a function cannot be an eigenfunction of more than one component of M, but can be an eigenfunction of M2 and one component of M simultaneously. To find expectation values from these wave functions, we need to define an operation of integration. We shall let the symbol (9A m > Mx9Am) represent such an operation. For electron-orbital motion where s a function of xyz, this operation becomes... 

The first term on the right-hand side arises from external electric fields. The second (B) term arises from external magnetic inductions interacting with electronic orbital motion. The SL term arises from electron spin-orbital motion interactions. The Z term arises from the Zeeman interaction between electron spin and the external electric field. H s arises from electron spin-electron spin interactions and includes all hyperfine terms arising from nuclear spins. 

The complete wave function of the hydrogen molecule must describe the electronic orbital motion, the electronic spin orientation, the vibration of the nuclei, the rotation of the nuclei and the nuclear spin orientation. As a first approximation the various forms of motion must be considered as being independent of each other and the complete wave function may thus be represented as the product of five separate functions ... 

For the normal electronic state of the molecule, since Pauli s principle must be obeyed, the entire function must be antisymmetric in the two electrons, i,e. when the electrons are transposed, the sign of the wave function must change. Let us consider the symmetry of each of the above five wave functions with reference to the two electrons. The electron orbital motion function is (see Chapter 3),... 

In this Hamiltonian (5) corresponds to the orbital angular momentum interacting with the external magnetic field, (6) represents the diamagnetic (second-order) response of the electrons to the magnetic field, (7) represents the interaction of the nuclear dipole with the electronic orbital motion, (8) is the electronic-nuclear Zeeman correction, the two terms in (9) represent direct nuclear dipole-dipole and electron coupled nuclear spin-spin interactions. The terms in (10) are responsible for spin-orbit and spin-other-orbit interactions and the terms in (11) are spin-orbit Zeeman gauge corrections. Finally, the terms in (12) correspond to Fermi contact and dipole-dipole interactions between the spin magnetic moments of nucleus N and an electron. Since... 

Indeed, Dunham s energy-level formula [Eq. (2.1.1)] is based both on the concept of a potential energy curve, which rests on the separability of electronic and nuclear motions, and on the neglect of certain couplings between the angular momenta associated with nuclear rotation, electron spin, and electron orbital motion. The utility of the potential curve concept is related to the validity of the Born-Oppenheimer approximation, which is discussed in Section 3.1. 

In the application of the quantum theory to the simplest example of a diatomic molecule, the important new factor is the existence in the molecule of an axis defining a specific direction. An atom possesses no such axis. There exists therefore for the molecule a quantum number A which measures the number of units of angular momentum in the component of the electronic orbital motion projected along the axis joining the nuclei. According as A = 0, 1, 2,..., the state is called S, II, A,..., by analogy with the atomic states 8, P, D,..., which are determined by the values of I (p. 199). 

Corresponding to a given molecular electronic term, there are in general several electronic states. Ihe interactions between electronic spin and electronic orbital motion and between electronic and nuclear motions split the energies of these states. These splittings are usually small. 

FIGURE 33-2 G rieration of atomic magnetic moments by (a) electron orbital motion around the nucleus (b) electron spin around its axis of rotation. 

Most of our attention thus far has been with wavefunction symmetry and energy. However, understanding atomic spectroscopy or interatomic interactions (in reactions or scattering) requires close attention to angular momentum due to electronic orbital motion and spin. In this section we will see what possibilities exist for the total electronic angular momenta of atoms and how these various states are distinguished symbolically. 

The magnetic moments arising from electron orbital motion and from electron spin can interact. This feature of atomic structure, and of molecular electronic structure, too, is termed spin-orbit interaction. Since the magnetic dipoles due to spin and orbital motion are proportional to their respective angular momentum vectors, the interaction... 

Hmag from external magnetic fields interacting with electronic orbital motion,... 

Diamagnetism results from changes in electron orbital motion that are indnced by an external field. The effect is extremely small (with susceptibilities on the order of -10 ) and in opposition to the applied field. All materials are diamagnetic. 

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