Vector model of atoms

A brief review is given here of the spectroscopic vector model of an atom or ion. In crystal-field theory, the wave function of the isolated ion is taken as the unperturbed state, and the perturbing effect of the electric and magnetic fields is computed. Thus crystal field theory uses the language, nomenclature, and methods employed in the theory of atomic spectra. A complete discussion of these methods can be found in books by Condon and Shortley (5) and by Griffith (/). 

The Stokes parameters for the polarization of an electron beam can be represented in a Cartesian basis which also provides a convenient pictorial view for the polarization state of an electron beam. Since the polarization of an ensemble of electrons requires the determination of spin projections along preselected directions, the classical vector model of a precessing spin will first be discussed. Here the spin is represented by a vector s of length 3/2 (in atomic units) which processes around a preselected direction, yielding as expectation values the projections (in atomic units, see Fig. 9.1)... 

Figure 9.1 Vector model of the electron spin. Using atomic units, the spin is represented by a vector s of length y/3/2 which precesses around the z-axis. By looking at the respective projections of the precessing spin vector, the model provides two important properties the projection onto the z-axis leads to a sharp value, ms = 1/2 in the case shown (ms = —1/2 for a precession around the negative z-axis), but no sharp values exist for the projections in the xy-plane, i.e., for the projections onto the x- or y-axis one finds with equal probability...

In this volume, principal consideration is given to the lighter elements, so that the Russell-Saunders (549) vector model of the atom is used. In this model a multielectron atom is assumed to have the quantum numbers n, L = lif Ml, 8 = siy (or n, L, J = L + S, Mj). This implies stronger and Si-Sj coupling than U-Si coupling. It follows from Pauli s principle that for a closed shell =... 

For a detailed discussion of spectroscopic nomenclature and the vector model of the atom see Pauling and Goudsmit The Structure of Line Spectra. The triplet levels of helium were long called doublets, complete resolution being difficult. Their triplet character was first suggested by J. C. Slater, Proc. Nat. Acad. Set. 11, 732 (1925), and was soon verified experimentally by W. V. Houston, Phys. Rev. 29, 749 (1927). The names parhelium and orthohelium were ascribed to the singlet and triplet levels, respectively, before their nature was understood. 

Section 4.4 describes how the vector model of the atom can predict overall symmetries of many-electron atoms. A similar approach can be used to determine overall symmetries of molecular electronic states. 

Many-Electron Spatial Wavefunctions 157 TOOLS OF THE TRADE Photoeiectron Spectroscopy 163 Approximate Solution to the Schrodinger Equation 164 BIOSKETCH Syivia Ceyer 179 Spin Wavefunctions and Symmetrization 179 Vector Model of the Many-Electron Atom 186 Periodicity of the Elements 190 Atomic Structure The Key to Chemistry 191... 

The Hypothesis of Electron Spin, 124. Electronic States of Complex Atoms, 128. The Pauli Exclusion Principle, 129. The Calculation of Energy Levels, 132. Angular Momenta, 133. Multiplet Structure, 135. Calculation of the Energy Matrix, 143. Fine Structure, 151. The Vector Model of the Atom, 155. Selection Rules for Complex Atoms, 159. The Radial Portion of the Atomic Orbitals, 162. The Hartree Method, 163. The Periodic System of the Elements, 167. 

Fig.3.4. Vector model of a single-electron atom showing...

Magnetic moments and a vector model of a many-electron atom. The Lande factor... 

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

Molecules are usually represented as 2D formulas or 3D molecular models. WhOe the 3D coordinates of atoms in a molecule are sufficient to describe the spatial arrangement of atoms, they exhibit two major disadvantages as molecular descriptors they depend on the size of a molecule and they do not describe additional properties (e.g., atomic properties). The first feature is most important for computational analysis of data. Even a simple statistical function, e.g., a correlation, requires the information to be represented in equally sized vectors of a fixed dimension. The solution to this problem is a mathematical transformation of the Cartesian coordinates of a molecule into a vector of fixed length. The second point can... 

The range of systems that have been studied by force field methods is extremely varied. Some force fields liave been developed to study just one atomic or molecular sp>ecies under a wider range of conditions. For example, the chlorine model of Rodger, Stone and TUdesley [Rodger et al 1988] can be used to study the solid, liquid and gaseous phases. This is an anisotropic site model, in which the interaction between a pair of sites on two molecules dep>ends not only upon the separation between the sites (as in an isotropic model such as the Lennard-Jones model) but also upon the orientation of the site-site vector with resp>ect to the bond vectors of the two molecules. The model includes an electrostatic component which contciins dipwle-dipole, dipole-quadrupole and quadrupole-quadrupole terms, and the van der Waals contribution is modelled using a Buckingham-like function. 

Transient computations of methane, ethane, and propane gas-jet diffusion flames in Ig and Oy have been performed using the numerical code developed by Katta [30,46], with a detailed reaction mechanism [47,48] (33 species and 112 elementary steps) for these fuels and a simple radiation heat-loss model [49], for the high fuel-flow condition. The results for methane and ethane can be obtained from earlier studies [44,45]. For propane. Figure 8.1.5 shows the calculated flame structure in Ig and Og. The variables on the right half include, velocity vectors (v), isotherms (T), total heat-release rate ( j), and the local equivalence ratio (( locai) while on the left half the total molar flux vectors of atomic hydrogen (M ), oxygen mole fraction oxygen consumption rate... 

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