Scattering formalism

In a two-lowest-electronic-state Bom-Huang description for a chemical reaction, the nuclei can move on both of two corresponding PESs during the reaction, due to the electronically non-adiabatic couplings between those states. A reactive scattering formalism for such a reaction involving a triatomic system... 

At a certain stage in the refinement, the electron density map is interpreted using a model representation of the charge density distribution to extract the atomic coordinates. A commonly used scattering formalism is the independent-atom model (IAM), in which the total charge density in the crystal is approximated by the superposition... 

Mujica V, Kemp M, Ratner MA (1994) Electron conduction in molecular wires. I. A scattering formalism. J Chem Phys 101 6849... 

What guidance for improving the scattering formalism can be obtained from theory In the linear combination of atomic orbitals (LCAO) formalism, a molecular orbital (MO) is described as a combination of atomic basis function ... 

The density expression (3.15) contains several types of terms, which are to be considered in formulating an advanced X-ray scattering formalism ... 

In summary, the electron density for HF as described by Eq. (3.15) includes the effects of charge transfer between atoms, atomic orbital overlap, and preferential population of lone-pair orbitals, which are neglected in the independent-atom scattering formalism. 

In addition to the positional and thermal parameters of the atoms, least-squares procedures are used to determine the scale of the data, and parameters such as mosaic spread or particle size, which influence the intensities through multiple-beam effects (Becker and Coppens 1974a, b, 1975). It is not an exaggeration to say that modern crystallography is, to a large extent, made possible by the use of least-squares methods. Similarly, least-squares techniques play a central role in the charge density analysis with the scattering formalisms described in the previous chapter. 

The last term in Eq. (8.41) originates from the Slater-type monopole in the scattering formalism, with n0 as power of r, and Co as exponent. 

The quantum theory of molecular collisions in external fields described in this chapter is based on the solutions of the time-independent Schrodinger equation. The scattering formalism considered here can be used to calculate the collision properties of molecules in the presence of static electric or magnetic fields as well as in nonresonant AC fields. In the latter case, the time-dependent problem can be reduced to the time-independent one by means of the Floquet theory, discussed in the previous section. We will consider elastic or inelastic but chemically nonreac-tive collisions of molecules in an external field. The extension of the formalism to reactive scattering problems for molecules in external fields has been described in Ref. [12]. 

Starting from the knowledge of the transmission T(E), the calculation of the current can be pursued within the scattering formalism as presented in the previous section, and the results for the two models show an overall agreement with the experimental findings (see Fig. 23). 

Domany, E. Eutin-Wohlman, 0. and Mizrachi, L., "Multiple Scattering Formalism Application to Scattering by Two Spheres," J. Appl. Phys., 1984, pp. 132-136. 

What appears below is a brief discussion of neutron scattering formalism (for an in depth exposition see, for example, the classic paper by van Hove). We assume the scattering is from point particles, which is reasonable, because thermal neutron scattering involves wavelengths on the order of 1A and the length scale of the interaction between an atomic nucleus and the neutron is around five orders of magnitude smaller. If we consider an individual particle j at position rj at time t, the number density can be expressed as... 

Differential cross section for scattering Formally this is seen by differentiating (6.30) and using (6.34). 

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