Distribution spherically symmetric

The quadrupole is the next electric moment. A molecule has a non-zero electric quadrupole moment when there is a non-spherically symmetrical distribution of charge. A quadrupole can be considered to arise from four charges that sum to zero which are arranged so that they do not lead to a net dipole. Three such arrangements are shown in Figure 2.8. Whereas the dipole moment has components in the x, y and z directions, the quadrupole has nine components from all pairwise combinations of x and y and is represented by a 3 x 3 matrix as follows ... 

In the absence of an external force, the probability of moving to a new position is a spherically symmetrical Gaussian distribution (where we have assumed that the diffusion is spatially isotropic). 

Figure 3 Characteristic solid state NMR line shapes, dominated by the chemical shift anisotropy. The spatial distribution of the chemical shift is assumed to be spherically symmetric (a), axially symmetric (b), and completely asymmetric (c). The top trace shows theoretical line shapes, while the bottom trace shows rear spectra influenced by broadening effects due to dipole-dipole couplings.

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. 

Consider the lowest energy level of a hydrogen atom, n = 1. We have just learned that there are /7s levels with this energy, and since n = 1, there is but one level. It corresponds to an electron distribution that is spherically symmetrical around the nucleus, as shown in Figure 15-8. It is called the Is orbital. An electron moving in an s orbital is called an s electron. 

Show that the electron distribution is spherically symmetrical for an atom in which an electron occupies each of the three p-orbitals of a given shell. 

The quantity p2 as a function of the coordinates is interpreted as the probability of the corresponding microscopic state of the system in this case the probability that the electron occupies a certain position relative to the nucleus. It is seen from equation 6 that in the normal state the hydrogen atom is spherically symmetrical, for p1M is a function of r alone. The atom is furthermore not bounded, but extends to infinity the major portion is, however, within a radius of about 2a0 or lA. In figure 3 are represented the eigenfunction pm, the average electron density p = p]m and the radial electron distribution D = 4ir r p for the normal state of the hydrogen atom. 

This model of the hydrogen atom accordingly consists of a nucleus embedded in a ball of negative electricity—the electron distributed through space. The atom is spherically symmetrical. The electron density is greatest at the nucleus, and decreases exponentially as r, the distance from the nucleus, increases. It remains finite, however, for all finite values of r, so that the atom extends to infinity the greater part of the atom, however, is near the nucleus—within 1 or 2 A. 

The charge distribution of the nucleus was taken to be spherically symmetric and represented by a single Gaussian of the form ... 

The theory of Debye and Hiickel started from the assumption that strong electrolytes are completely dissociated into ions, which results, however, in electrical interactions between the ions in such a manner that a given ion is surrounded by a spherically symmetrical distribution of other ions mainly of opposite charges, the ionic atmosphere. The nearer to the central ions the higher will be the potential U and the charge density the limit of approach to the central ion is its radius r = a. 

For spherically symmetric nuclear charge distribution (Gaussian, Fermi, or point nucleus), the electric field at a point r outside the nucleus can be evaluated from Gauss law as... 

These nuclei (and they form by far the majority of the NMR-active nuclei ) are subject to relaxation mechanisms which involve interactions with the quadrupole moment. The relaxation times Tj and T2 (T2 is a second relaxation variable called the spin-spin relaxation time) of such nuclei are very short, so that very broad NMR lines are normally observed. The relaxation times, and the linewidths, depend on the symmetry of the electronic environment. If the charge distribution is spherically symmetrical the lines are sharp, but if it is ellipsoidal they are broad. 

As mentioned above and discussed in Chapter 2, atomic charges were often obtained in the past from dipole moments of diatomic molecules, assuming that the measured dipole moment equal to the bond length times the atomic charge. This method assumes that the molecular electron density is composed of spherically symmetric electron density distributions, each centered on its own nucleus. That is, the dipole moment is assumed to be due only to the charge transfer moment Mct. and the atomic dipoles Malom are ignored. 

Here r and v are respectively the electron position and velocity, r = —(e2 /em)(r/r3) is the acceleration in the coulombic field of the positive ion and q = /3kBT/m. The mobility of the quasi-free electron is related to / and the relaxation time T by p = e/m/3 = et/m, so that fi = T l. In the spherically symmetrical situation, a density function n(vr, vt, t) may be defined such that n dr dvr dvt = W dr dv here, vr and vt and are respectively the radical and normal velocities. Expectation values of all dynamical variables are obtained from integration over n. Since the electron experiences only radical force (other than random interactions), it is reasonable to expect that its motion in the v space is basically a free Brownian motion only weakly coupled to r and vr by the centrifugal force. The correlations1, K(r, v,2) and fc(vr, v(2) are then neglected. Another condition, cr(r)2 (r)2, implying that the electron distribution is not too much delocalized on r, is verified a posteriori. Following Chandrasekhar (1943), the density function may now be written as an uncoupled product, n = gh, where... 

The strength of the quadrupolar interaction is proportional to the quadrupole moment Q of a nucleus and the electric field gradient (EFG) [21-23]. The size of Q depends on the effective shape of the ellipsoid of nuclear charge distribution, and a non-zero value indicates that it is not spherically symmetric (Fig. 1). 

The physical origin of the dispersion interaction is often described in terms of a quasi-classical induced-dipole-induced-dipole picture. The quantum-mechanical fluctuations of the electronic distribution about its spherically symmetric average can be pictured as leading to an instantaneous (snapshot) dipole /za(mst) on monomer a, which in turn induces an instantaneous dipole tb(mst) on b. Thus, if the dipole fluctuations of the two monomers are properly correlated, a net attraction of the form (5.25) results. As remarked by Hirschfelder et al,28... 

For a spherically symmetric charge distribution, an exact relationship between the electrostatic potential and the electron density is the Poisson equation ... 

The ionic model, developed by Bom, Lande, and Lennard-Jones, enables lattice energies (U) to be summed from inverse square law interactions between spherically symmetrical charge distributions and interactions following higher inverse power laws. Formation enthalpies are related to calculated lattice energies in the familiar Bom-Haber cycle. For an alkali fluoride... 

The reason is that these alleged kp values are mostly composite, comprising the rate constants of propagation of uncomplexed Pn+, paired Pn+ (Pn+A ), and Pn+ complexed with monomer or polymer or both, without or with an associated A" [17]. Even when we will eventually have genuine kp values for solvents other than PhN02, it will not be possible to draw many (or any ) very firm conclusions because the only theoretical treatments of the variation of rate constants with solvent polarity for (ion + molecule) reactions are concerned with spherically symmetrical ions, and the charge distribution in the cations of concern to us is anything but spherically symmetrical. 

The nuclei of such type of isotopes possess essentially a spherically symmetrical charge distribution, and... 

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In our non-BO calculations performed so far, we have considered atomic systems with only -electrons and molecular systems with only a-electrons. The atomic non-BO calculations are much less complicated than the molecular calculations. After separation of the center-of-mass motion from the Hamiltonian and placing the atom nucleus in the center of the coordinate system, the internal Hamiltonian describes the motion of light pseudoelectrons in the central field on a positive charge (the charge of the nucleus) located in the origin of the internal coordinate system. Thus the basis functions in this case have to be able to accurately describe only the electronic correlation effect and the spherically symmetric distribution of the electrons around the central positive charge. 

As for the possible correlation between geometry and electronic structure, consider the variation of ionic radii with atomic number in the first row transition metal series If the points for Ca, Mn, and Zn are connected, i. e., for atoms with a spherically symmetrical distribution of d electrons, the ionic radii of the other atoms are smaller than interpolation would yield from the Ca-Mn-Zn line. The nonuniform distribution of d electrons around the nuclei is assumed to be the reason for this contraction of the ionic radii. The data available so far on the bond lengths for the vapor-phase dichlorides are seen in Fig. 8. 

Figure 3 presents the NMRD curves of Pu " " (5/ , Hs/2) and Np (5/, %/2)-These two ions have non-spherically symmetric distribution of their unpaired electronic spins. Also included is the NMRD curve of Pr which is the lanthanide analog of Pu " " (12). All the ions are very poor relaxation agents... 

Equation (2.18) represents a linearized Boltzmann distribution. It contains two unknown variables, p(r) and >) (r). It is possible to reduce the problem of two unknown variables to a problem with one unknown variable by introducing a second equation expressing the relationship between the variables p(r) and t) (r). This second equation is known as the Poisson equation. The Poisson equation for spherically symmetrical charge distribution is given as... 

A term used in electrostatic descriptions of ions to denote the continuous electric charge density [p(r)] surrounding an ionic species. On average, an ion will be surrounded by a spherically symmetrical distribution of counterions that form its ion atmosphere. See Hydration Atmosphere... 

The Laplace-Young equation refers to a spherical phase boundary known as the surface of tension which is located a distance from the center of the drop. Here the surface tension is a minimum and additional, curvature dependent, terms vanish (j ). The molecular origin of the difficulties, discussed in the introduction, associated with R can be seen in the definition of the local pressure. The pressure tensor of a spherically symmetric inhomogeneous fluid may be computed through an integration of the one and two particle density distributions. 

Hansen, N. K., Study of the Electron Density Distribution in Molecular Crystals by Analysis of X-ray Diffraction Data Using Non-Spherically Symmetric Scattering Functions, Thesis, University of Arhus, Denmark (1978). 

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