Time-dependent electric and magnetic fields

FORMAL FOUNDATIONS OF DENSITY FUNCTIONAL THEORY FOR TIME-DEPENDENT ELECTRIC AND MAGNETIC FIELDS... 

Unpolarized light is thought to consist of an infinite number of time-dependent electric and magnetic fields... 

Classical electrodynamics, i.e.. Maxwell s unquantized theory for time-dependent electric and magnetic fields is inherently a covariant relativistic theory— in the sense of Einstein and Lorentz not Newton and Galilei — fitting perfectly well to the theory of special relativity as we shall understand in chapter 3. In this section, only those basic aspects of elementary electrodynamics will be... 

Furthermore, the electrons are imder the influence of a generic, time-dependent potential, V r, t). The Hamiltonian (4.2) is completely general and describes a wealth of physical and chemical situations, including atoms, molecules, and solids in arbitrary time-dependent electric or magnetic fields, scattering experiments, etc. In most of the situations dealt with in this chapter we will be concerned with the interaction between a laser and matter. In that case, we can write the time-dependent potential as the sum of the nuclear potential and a laser field, + leaser- The term U n accounts... 

In this short review, a brief overview of the underlying principles of TDDFT has been presented. The formal aspects for TDDFT in the presence of scalar potentials with periodic time dependence as well as TD electric and magnetic fields with arbitrary time dependence are discussed. This formalism is suitable for treatment of interaction with radiation in atomic and molecular systems. The Kohn-Sham-like TD equations are derived, and it is shown that the basic picture of the original Kohn-Sham theory in terms of a fictitious system of noninteracting particles is retained and a suitable expression for the effective potential is derived. 

The Time Dependent Processes Section uses time-dependent perturbation theory, combined with the classical electric and magnetic fields that arise due to the interaction of photons with the nuclei and electrons of a molecule, to derive expressions for the rates of transitions among atomic or molecular electronic, vibrational, and rotational states induced by photon absorption or emission. Sources of line broadening and time correlation function treatments of absorption lineshapes are briefly introduced. Finally, transitions induced by collisions rather than by electromagnetic fields are briefly treated to provide an introduction to the subject of theoretical chemical dynamics. 

The ease of time-varying charge displacement, measured as the time-dependent dielectric or magnetic permittivity (or permeability), is expressed by the dielectric function e and magnetic function /x. Both e and // depend on frequency both measure the susceptibility of a material to react to electric and magnetic fields at each frequency. For succinctness, only the dielectric function and the electrical fluctuations are described in the rest of this introductory section. The full expressions are given in the application and derivation sections of Levels 2 and 3. 

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. 

The right-hand side term is the sum of the energies of the electric and magnetic fields. It expresses the time dependence of the energy of the electromagnetic wave, hence, the parameter S represents the energy flow per unit time and area. 

It should be apparent that when a solution has been obtained for harmonic fields a solution can also be derived for any arbitrary time dependence through the use of the Fourier transform. Most frequently the electric and magnetic field vectors cannot be completely described by using only a single spatial component. For this reason a solution can turn out to be very cumbersome. Some simplification can be obtained by making use of various auxiliary functions. There are two ways in which such auxiliary functions can be introduced. One approach follows from use of the third equation in the set 1.274 ... 

Of interest here are the effects of sfafic electric and magnetic fields and of cycle-averaged interaction with ac-fields. I do not discuss the theme of interaction with time-dependent pulses, for which the theoretical requirement is the proper time-dependent solution of the polyelectronic TDSE (see Chapter 6). 

The ab initio computation of molecular properties - including those associated with time-dependent external electric and magnetic fields - has advanced significantly in the last several decades, yielding accurate models for linear, quadratic, and higher-order response functions. When electron correlation effects play a pivotal... 

Ions can be con6ned in traps created by potential wells of electric and magnetic fields. Depending on the quality of the vacuum, the trapping time can be very long—hours, days, even months. Such an arrangement has several advantages compared to experiments with ion beams or ions in resonance cells ... 

Eq. (7.8) is the most general covariant form of the inhomogeneous Maxwell equations, which immediately imply the continuity equation dy.j = + div = 0 of section 5.2.3, and Eq. (7.9) is the covariant time-dependent Dirac equation in the presence of external electric and magnetic fields. The homogeneous Maxwell equations are automatically satisfied by the sole existence of... 

In its broadest sense, spectroscopy is concerned with interactions between light and matter. Since light consists of electromagnetic waves, this chapter begins with classical and quantum mechanical treatments of molecules subjected to static (time-independent) electric fields. Our discussion identifies the molecular properties that control interactions with electric fields the electric multipole moments and the electric polarizability. Time-dependent electromagnetic waves are then described classically using vector and scalar potentials for the associated electric and magnetic fields E and B, and the classical Hamiltonian is obtained for a molecule in the presence of these potentials. Quantum mechanical time-dependent perturbation theory is finally used to extract probabilities of transitions between molecular states. This powerful formalism not only covers the full array of multipole interactions that can cause spectroscopic transitions, but also reveals the hierarchies of multiphoton transitions that can occur. This chapter thus establishes a framework for multiphoton spectroscopies (e.g., Raman spectroscopy and coherent anti-Stokes Raman spectroscopy, which are discussed in Chapters 10 and 11) as well as for the one-photon spectroscopies that are described in most of this book. 

---