Born type approximations

The basic ideas of constructing Born type approximations for the vector wavefield are very similar to those considered in Chapter 9 for an electromagnetic field. I will provide below a brief description of the corresponding approximations. 

We can derive the extended Born approximation of Habashy et al. (1993) and Torres-Verdin and Habashy (1994) by replacing the total field in the integral (14.80), not by the incident field, as in the Born approximation, but by its projection onto a scattering tensor T (r, u)  

We obtain the expression for the scattering tensor by approximating u(r,o ) in the integral (14.80) by its value at the peak point r = of the Green s tensor  

We can introduce the quasi-linear (QL) approximation of the vector wavefield assuming that the scattered field u inside the inhomogeneous domain is linearly proportional to the incident field u (see Chapter 9 for comparison)  

Substituting formula (14.93) into (14.80), we obtain the QL approximation (r) for the scattered field  

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One branch of chemistry where the use of quantum mechanics is an absolute necessity is molecular spectroscopy. The topic is interaction between electromagnetic waves and molecular matter. The major assumption is that nuclear and electronic motion can effectively be separated according to the Born-Oppenheimer approximation, to be studied in more detail later on. The type of interaction depends on the wavelength, or frequency of the radiation which is commonly used to identify characteristic regions in the total spectrum, ranging from radio waves to 7-rays. 

Before we can discuss the recent developments further, we must discuss terminology. This will also provide a guide to earlier literature as well as to the classification we use in our review of recent papers (Section lOd). As mentioned, the total wave function in the Born-Oppenheimer method. is expressed as a product of electronic and vibrational wave functions. What has resulted is that different types of electronic wave functions have been used (which is not necessarily confusing), and that in many cases a particular selection has been called the Born-Oppenheimer approximation (which has led to confusion). We discuss here only the two predominant choices of... 

In this case the zero-order electronic wave functions are, in principle, referred to a Hamiltonian that contains the potential from the ions at their actual positions, i.e., the electrons follow the ionic motion adiabatically. Since both these approximations are sometimes referred to as the Born Oppenheimer approximation, this has led to confusion in terminology for example, Mott (1977) refers to the Born-Oppenheimer approximation, but gives wave functions of the adiabatic type, whereas Englman (1972) differentiates between the two forms, but specifically calls the static form the Bom Oppenheimer method. [We note that, historically, the adiabatic form was first suggested by Seitz (1940)—see, for example, Markham (1956) or Haug and Sauermann (1958)]. In this chapter, we shall preferentially use the terminology static and adiabatic. [Note that the term crude adiabatic is also sometimes used for the static approximation, mainly in the chemical literature—see, for example, Englman (1972, 1979).]... 

Bohr relationship, 95 Boltzmann distribution law, 51, 52 Bond clevage, Norrish Type I, 238 Norrish Type II and III, 240 Bond dissociation energy, 6 Bond length, 29, 93 Born-Oppenheimer approximation, 29, 97... 

In molecular orbital (MO) theory, which is the most common implementation of QM used by chemists, electrons are distributed around the atomic nuclei until they reach a so-called self-consistent field (SCF), that is, until the attractive and repulsive forces between all the particles (electrons and nuclei) are in a steady state, and the energy is at a minimum. An SCF calculation yields the electronic wave function 4C (the electronic motion being separable from nuclear motion thanks to the Born-Oppenheimer approximation). This is the type of wave function usually referred to in the literature and in the rest of this chapter. 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

Type II quantum chaos is discussed in Section 4.2. It arises naturally in molecular physics in the form of the dynamic Born-Oppenheimer approximation (Bliimel and Esser (1994)). In the dynamic Born-Oppenheimer approximation chaos may occur in both the classical and the quantum subsystem, although neither the classical nor the quantum systems by themselves are chaotic. Type II quantum chaos was also identifled in a nuclear physics context (Bulgac (1991)). 

In Section 11.1 we discuss recent advances in quantum chaology, i.e. the semiclassical basis for the analysis of atomic and molecular spectra in the classically chaotic regime. In Section 11.2 we discuss some recent results in type II quantum chaos within the framework of the dynamic Born-Oppenheimer approximation. Recent experimental and theoretical results of the hydrogen atom in strong microwave and magnetic fields are presented in Sections 11.3 and 11.4, respectively. We conclude this chapter with a brief review of the current status of research on chaos in the helium atom. 

The time-dependent perturbation theory of the rates of radiative ET is based on the Born-Oppenheimer approximation [59] and the Franck Condon principle (i.e. on the separation of electronic and nuclear motions). The theory predicts that the ET rate constant, k i, is given by a golden rule -type equation, i.e., it is proportional to the product of the square of the donor-acceptor electronic coupling (V) and a Franck Condon weighted density of states FC) ... 

All aspects of molecular shape and size are fully reflected by the molecular electron density distribution. A molecule is an arrangement of atomic nuclei surrounded by a fuzzy electron density cloud. Within the Born-Oppenheimer approximation, the location of the maxima of the density function, the actual local maximum values, and the shape of the electronic density distribution near these maxima are fully sufficient to deduce the type and relative arrangement of the nuclei within the molecule. Consequently, the electronic density itself contains all information about the molecule. As follows from the fundamental relationships of quantum mechanics, the electronic density and, in a less spectacular way, the nuclear distribution are both subject to the Heisenberg uncertainty relationship. The profound influence of quantum-mechanical uncertainty at the molecular level raises important questions concerning the legitimacy of using macroscopic analogies and concepts for the description of molecular properties. ... 

The Born-Oppenheimer approximation is based on this assumption and enables reduction of the mathematically intractable spectral eigenvalue problem to a set of separable spectral problems for each type of motion. According to this approximation, energy levels associated with each type of motion are proportional to the ratio of electronic mass (mg) to the nuclear mass (Mn). This ratio, f quite smaller than unity, is given by Eq. (2) ... 

The Born-Oppenheimer approximation is thus valid provided the term C R) can be neglected. This term is indeed small, and is usually neglected in calculations of this type one of the reasons is that these integrals depend upon 1/M or jfi which is much smaller than 1/mo, e.g. one of the terms in the double summation of (2.22) is... 

In this review, almost all of the simulations we have described use only classical mechanics to describe the nuclear motion of the reaction system. However, a more accurate analysis of many reactions, including some of the ones that have already been simulated via purely classical mechanics, will ultimately require some infusion of quantum mechanical methods. This infusion has already taken place in several different types of reaction dynamics electron transfer in solution, > i> 2 HI photodissociation in rare gas clusters and solids,i i 22 >2 ° I2 photodissociation in Ar fluid,and the dynamics of electron solvation.22-24 Since calculation of the quantum dynamics of a full solvent is at present too time-consuming, all of these calculations involve a quantum solute in a classical solvent. (For a system where the solvent is treated quantum mechanically, see the quantum Monte Carlo treatment of an electron transfer reaction in water by Bader et al. O) As more complex reaaions are investigated, the techniques used in these studies will need to be extended to take into account effects involving electron dynamics such as curve crossing, the interaction of multiple electronic surfaces and other breakdowns of the Born-Oppenheimer approximation, the effect of solvent and solute polarization, and ultimately the actual detailed dynamics of the time evolution of the electronic degrees of freedom. 

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