External field interaction

This corresponds to the principle of minimal coupling, according to which the interaction with a magnetic field is described by replacing in the Hamiltonian operator the canonical momentum p by the kinetic momentum 11 = p — f A(x). Other types of external-field interactions include scalar or pseudoscalar fields and anomalous magnetic moment interactions. The classification of external fields rests on the behavior of the Dirac equation rmder Lorentz transformations. A brief description of these potential matrices will be given below. 

An explanation of the bifurcation can be found from the angular density (lower panel of Fig. 36). Because we start from the rotational ground state, the first excitation step prepares a wavepacket with the rotational quantum number 7=1. Then, the density, initially, is proportional to Tio(0,0) cos (0) 2. It is seen that the density changes with time and that a depletion at angles smaller than 7i/2 occurs, which goes in hand with a concentration of density at a value of n. It is now straightforward to find a classical interpretation of the (radial) density bifurcation in regarding the classical force which stems from the external field interaction and acts in the radial direction ... 

The definition of vibrational intensities for molecules in solution requires some modifications with respect to the isolated system. The presence of the solvent in fact modifies not only the solute charge distribution but also the probing electric field acting on the molecule. As we shall see in the following sections, this is a problem of general occurrence when an external field interacts with a molecule in a condensed phase (historically it is known as local field effect ). 

Then, employing the dipole form of the external field interaction [Eq. (3.43)], the Schrodinger equation for the nuclear wavefunctions in the diabatic representation is given in the matrix form. 

Some of the terms included in the Breit-Pauli Hamiltonian also describe small interactions that can be probed experimentally by inducing suitable excitations in the electron or nuclear spin space, giving rise to important contributions to observable NMR and ESR parameters. In particular, for molecular properties for which there are interaction mechanisms involving the electron spin, also the spin-orbit interaction (O Eqs. 11.13 and O 11.14) becomes important The Breit-Pauli Hamiltonian in O Eqs. 11.5-11.22, however, only includes molecule-external field interactions through the presence of a scalar electrostatic potential 0 (and the associated electric field F) and the appearance of the magnetic vector potential in the mechanical momentum operator (O Eq. 11.23). In order to extract in more detail the interaction between the electronic structure of a molecule and an external electromagnetic field, we need to consider in more detail the form of the scalar and vector potentials. 

Even for a single radical tire spectral resolution can be enlianced for disordered solid samples if the inliomogeneous linewidth is dominated by iimesolved hyperfme interactions. Whereas the hyperfme line broadening is not field dependent, tire anisotropic g-matrix contribution scales linearly with the external field. Thus, if the magnetic field is large enough, i.e. when the condition... 

Our discussion of elecfronic effects has concentrated so far on permanent features of the cliarge distribution. Electrostatic interactions also arise from changes in the charge distribution of a molecule or atom caused by an external field, a process called polarisation. The primary effect of the external electric field (which in our case will be caused by neighbouring molecules) is to induce a dipole in the molecule. The magnitude of the induced dipole moment ginj is proportional to the electric field E, with the constant of proportionahty being the polarisability a ... 

Polarizability Attraction. AU. matter is composed of electrical charges which move in response to (become electrically polarized in) an external field. This field can be created by the distribution and motion of charges in nearby matter. The Hamaket constant for interaction energy, A, is a measure of this polarizability. As a first approximation it may be computed from the dielectric permittivity, S, and the refractive index, n, of the material (15), where is the frequency of the principal electronic absorption... 

These concepts play an important role in the Hard and Soft Acid and Base (HSAB) principle, which states that hard acids prefer to react with hard bases, and vice versa. By means of Koopmann s theorem (Section 3.4) the hardness is related to the HOMO-LUMO energy difference, i.e. a small gap indicates a soft molecule. From second-order perturbation theory it also follows that a small gap between occupied and unoccupied orbitals will give a large contribution to the polarizability (Section 10.6), i.e. softness is a measure of how easily the electron density can be distorted by external fields, for example those generated by another molecule. In terms of the perturbation equation (15.1), a hard-hard interaction is primarily charge controlled, while a soft-soft interaction is orbital controlled. Both FMO and HSAB theories may be considered as being limiting cases of chemical reactivity described by the Fukui ftinction. 

We consider first the polarizability of a molecule consisting of two or more polarizable parts which may be atoms, bonds, or other units. When the molecule is placed in an electric field the effective field which induces dipole moments in various parts is not just the external field but rather the local field which is influenced by the induced dipoles of the other parts. The classical theory of this interaction of polarizable units was presented by Silberstein36 and others and is summarized by Stuart in his monograph.40 The writer has examined the problem in quantum theory and finds that the same results are obtained to the order of approximation being considered. 

Before embarking on the problem of the interaction of the negaton-positon field with the quantized electromagnetic field, we shall first consider the case of the negaton-positon field interacting with an external, classical (prescribed) electromagnetic field. We shall also outline in the present chapter those aspects of the theory of the S-matrix that will be required for the treatment of quantum electrodynamics. Section 10.4 presents a treatment of the Dirac equation in an external field. 

The formulation of the theory outlined above is particularly well-suited for the description of scattering processes, i.e., experiments consisting of the preparation of a number of physical, free noninteracting particles at t — oo, allowing these particles to interact (with one another and/or any external field present), and finally measuring the state of these particles and whatever other particles are present at time t = + co when they once again move freely. The infinite time involved... 

For a discussion of the quantized electromagnetic field interacting with a given (prescribed) external current, see ... 

Although the previous development was based on the assumption that H(t) = 27(0), it can readily be verified that the formulae (10-148)-(10-152) are valid more generally. In particular for the negaton-positon field interacting with an external electromagnetic field, we have... 

The above rules are readily applied to the case of the electron-positron field interacting with an external field for which... 

Navier-Stokes equations, 24 Negative criterion of Bendixon, 333 Negaton-positon field in an external field, 580 interacting with electromagnetic field, Hamiltonian for, 645 interaction with radiation field, 642 Negaton-positon system, 540 Negaton scattering by an external field, 613... 

---