Plane-wave Born Approximation

The factorisation is characteristic also of the plane-wave Born approximation, which is (10.30) with distorted waves replaced by plane waves. Here the two-electron T-matrix element is replaced by the two-electron potential matrix element (3.41). 

The generalized oscillator strength formulation of the plane wave Born approximation to the calculation of stopping power is modified by introducing radial Green s functions in place of the infinite sums over bound excited states and integrations over the continuum. Some properties of the resulting expressions are examined. 

Inner shell ionization of electrons to the continuum in ion-atom collisions can occur by two different processes. For low Z (projectile) particles on high Z2 (target) atoms the only available process is Coulomb excitation which is variously treated by plane wave Born approximation (PWBA), the binary encounter approximation (BEA), and the semiclassical approximation (SCA). When Z becomes comparable to 7/1 and the ion velocity v is lower than the velocity of the bound electron in question, v, the electrons adjust adiabati-cally to the approach of the two nuclei and enter molecular orbitals (MO) which in the limit of fused nuclei approach the atomic orbitals of the united atom Z = Z + Z2. This stacking of electrons can lead to a promotion of an innershell electron to the continuum or to a vacant outer orbital by direct curve crossing, rotational coupling, or radial coupling between molecular levels when such channels are available. 

The first (and still the foremost) quantum theory of stopping, attributed to Bethe [19,20], considers the observables energy and momentum transfers as fundamental in the interaction of fast charged particles with atomic electrons. Taking the simplest case of a heavy, fast, yet nonrelativistic incident projectile, the excitation cross-section is developed in the first Born approximation that is, the incident particle is represented as a plane wave and the scattered particle as a slightly perturbed wave. Representing the Coulombic interaction as a Fourier integral over momentum transfer, Bethe derives the differential Born cross-section for excitation to the nth quantum state of the atom as follows. 

In first Born approximation, the wave function is the product of a plane wave and a part which contains only the spin structure (ip = we1,

momentum conserving (5-functions, which are explicitly removed in the definiton of the T-matrix. The main point is now the connection between the 16 x 16 Tif = u 1u 2Tu U2 and its 8x8 form M. Defining in analogy to the one-particle case... 

Solution of the Kohn-Sham equations as outlined above are done within the static limit, i.e. use of the Born-Oppenheimer approximation, which implies that the motions of the nuclei and electrons are solved separately. It should however in many cases be of interest to include the dynamics of, for example, the reaction of molecules with clusters or surfaces. A combined ab initio method for solving both the geometric and electronic problem simultaneously is the Car-Parrinello method, which is a DFT dynamics method [52]. This method uses a plane wave expansion for the density, and the inner ions are replaced by pseudo-potentials [53]. Today this method has been extensively used for studies of dynamic problems in solids, clusters, fullerenes etc [54-61]. We have recently in a co-operation project with Andreoni at IBM used this technique for studying the existence of different isomers of transition metal clusters [62,63]. 

AIMD, CPMD still needs to treat a large number of integrals related to electronic coordinates. For these integrals, special tricks should be used. That is, the time-consuming integral treatments in conventional quantum chemistry should be avoided. For this reason, the current implementations of CPMD use two techniques plane waves as a basis set and the pseudopotential approximation. Of course, it should be borne in mind that CPMD was originally developed for solid state applications, an area where the use of plane waves is a traditional technique. 

In first Born approximation, the wave fnnction is the prodnct of a plane wave and a part which contains only the spin strnctnre (t/> = we, 4> = =... 

The evaluation of P requires knowledge about the photoelectron amplitude. It should, of course, be calculated as a continuum amplitude from the Dyson equation, but for a general molecule that is still a tough problem, and one proceeds by making more or less ad hoc choices. The perhaps simplest description of the photoelectron is v kf,r) — (27t) 5 ex.p ikf r). This choice of a plane wave is often referred to as the sudden approximation, or the zeroth-order Born approximation. If a primitive atomic orbital basis aj(r — Pa) is used. 

The density g(f) in Eqs. (7.1) and (7.2) came from a semi-classical treatment employing scattering centers. Quantum mecha,nically one frequently obtains an identical expression. In the quantum mechanical treatment one starts off with the matrix element between identical initial and final states of all the nucleons involved in the process. In those cases where the matrix element can be reduced to a sum of individual nucleon matrix elements and where plane waves can be employed for the scattered particle (Born approximation) one obtains expression (7.1) and (7.2). If the calculation are carried through considering all... 

Many approximation methods have been developed to solve Eq. (60) (Mott and Massey, 1965 Levine, 1969). A commonly used one is the Born approximation. Within the validity of the first Born approximation, we may replace T in Eq. (64) by the product of an incoming plane wave and the wavefunction of the initial unperturbed molecular state,... 

Within the framework of the Born-Oppenheimer, binary encounter, plane wave impulse and target Kohn-Sham [KS] (or target Hartree-Fock [HF]) approximations, and disregarding rotational wave functions, differential (e, 2e) ionization cross-sections are proportional to the square of a structure factor F (p), which is given by [1-3, 64] ... 

Other contributions which are related to the present analysis includes the work of Foldy [69], Waterman and Truell [258], Fikioris and Waterman [65], Twerski [230,231], Tsang and Kong [223,224] and Tsang et al. [227]. The case of a plane electromagnetic wave obliquely incident on a half-space with densely distributed particles and the computation of the incoherent field with the distorted Born approximation have been considered Tsang and Kong [225, 226]. 

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