Symmetric atom

The spherically symmetric atomic momentum densities, in contrast, exhibit monotonic as well as nonmonotonic behavior even in their ground states. Further, it was... 

Fig. 12.4 The (squared) frequency of the motion along the reaction coordinate q for a symmetric atom exchange reaction. L is the range of the chemical barrier region, and the frequency is shown in units of the mean frequency of the unperturbed solvent. The range of frequencies in the solvent is indicated as a solid bar. The negative values for the solvent correspond to unstable solvent modes.

The Fourier transform of the spherical atomic density is particularly simple. One can select S to lie along the z axis of the spherical polar coordinate system (Fig. 1.4), in which case S-r = Sr cos. If pj(r) is the radial density function of the spherically symmetric atom,... 

Fig. 4.41 (a) Cross-sectional view showing field variation across a plane. (b) In the absence of an applied field, the surface potential of an adatom, except at the plane edges, is symmetric. Atomic jumps are symmetric, (c) In an applied field, the surface potential of an adatom becomes inclined owing to the additional polarization binding. Atomic jumps are now asymmetric. 

By examining the size, shape, and orientation of the thermal ellipsoid associated with the bridging hydrogen atom, we concluded that the hydrogen atom in the bent Mo-H-Mo system is described preferably as an effectively symmetric atom oscillating around a single equilibrium point rather than being randomly distributed between two equilibrium positions in the crystal lattice. The thermal... 

For the Ain SALQ which must match the totally symmetric atomics orbital, it is obvious (as it was in the tetrahedral case) that all basis orbitals must enter with positive signs and equal weight. Thus, the normalized A,a. SALC has to be... 

The other mechanism is called the Fermi contact interaction and it produces the isotropic splittings observed in solution-phase EPR spectra. Electrons in spherically symmetric atomic orbitals (s orbitals) have finite probability in the nucleus. (Mossbauer spectroscopy is another technique that depends on this fact.) Of course, the strength of interaction will depend on the particular s orbital involved. Orbitals of lower-than-spherical symmetry, such as p or d orbitals, have zero probability at the nucleus. But an unpaired electron in such an orbital can acquire a fractional quantity of s character through hybridization or by polarization of adjacent orbitals (configuration interaction). Some simple cases are described later. 

It is conventional to label the internuclear axis in a diatomic molecule as z. Thus the three 2p(F) orbitals can be labelled 2p, 2py and 2p2 2px and 2py have their lobes directed perpendicular to the internuclear axis, and have nodal surfaces containing that axis, while 2pr clearly overlaps in o fashion with ls(H). (The reader may wonder whether this orientation of 2p, 2py and 2pz is obligatory, or whether it is chosen for convenience. For a spherically-symmetric atom, there are no constraints in choosing a set of three Cartesian axes. Any set of orthogonal p orbitals can be transformed into another equally acceptable set, by a simple rotation which does not change the electron density distribution of the atom. The overlap integral between a hydrogen Is orbital and the set of three 2p(F) orbitals is the... 

The electron density in an atom, molecule or crystal is described by a wave function which is subject to strict characteristic boundary conditions. As shown before (eqn. 5.3) an electron on a spherically symmetrical atom obeys the one-dimensional radial Helmholtz equation... 

The orbitals p and x are on C and, as before, k and / are on A and m, n on B. For the interaction of three spherically symmetric atoms the third-order dispersion nonadditivity contains the famous Axilrod-Teller-Muto triple-dipole interaction311,312. 

In the original work of Cortona,203 the subsystems correspond to spherically symmetric atoms in solids. This formalism can be also applied for polyatomic subsystems - interacting molecules in particular.204... 

For a spherically symmetric atom with radial wavefunction R(r), Eq. (10.6.16) simplifies to... 

For a symmetrical atomic or molecular system, these considerations place a severe restriction on the possible eigenfunctions of the system. All possible eigenfunctions must form bases for some irreducible representation of the group of symmetry operations. The form of the possible eigenfunctions is also determined to a large extent since they must transform in a quite definite way under the operations of the group. 

A spherically symmetrical atomic orbital one per energy level. 

A better known relation, also involving the density, is the theorem of Kato.11 For a spherically symmetric atom or ion, it relates the electron density at the nucleus to the derivative of p also taken at the nucleus through... 

The purpose of calculating Henry s Law constants is usually to determine the parameters of the adsorption potential. This was the approach in Ref. [17], where the Henry s Law constant was calculated for a spherically symmetric model of CH4 molecules in a model microporous (specific surface area ca. 800 m /g) silica gel. The porous structure of this silica was taken to be the interstitial space between spherical particles (diameter ca. 2.7 nm ) arranged in two different ways as an equilibrium system that had the structure of a hard sphere fluid, and as a cluster consisting of spheres in contact. The atomic structure of the silica spheres was also modeled in two ways as a continuous medium (CM) and as an amorphous oxide (AO). The CM model considered each microsphere of silica gel to be a continuous density of oxide ions. The interaction of an adsorbed atom with such a sphere was then calculated by integration over the volume of the sphere. The CM model was also employed in Refs. [36] where an analytic expression for the atom - microsphere potential was obtained. In Ref. [37], the Henry s Law constants for spherically symmetric atoms in the CM model of silica gel were calculated for different temperatures and compared with the experimental data for Ar and CH4. This made it possible to determine the well-depth parameter of the LJ-potential e for the adsorbed atom - oxygen ion. This proved to be 339 K for CH4 and 305 K for Ar [37]. On the other hand, the summation over ions in the more realistic AO model yielded efk = 184A" for the CH4 - oxide ion LJ-potential [17]. Thus, the value of e for the CH4 - oxide ion interaction for a continuous model of the adsorbent is 1.8 times larger than for the atomic model. 

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