Quadrupole moment derivative

A dipole moment of 0.5491 D was calculated for NH2 at the ab initio SCF Cl level [17] similar results are given in [12, 18]. Quadrupole moments, derivatives of dipole moments, and the polarizability were also calculated [17, 18]. 

Quantitative similarities of molecules can easily be recognized if it is possible to define quantities for molecular parts which are additive as well as transferable. Such quantities can be derived from transferable molecular orbitals because any one-electron property, such as dipole moment, quadrupole moment, kinetic energy, is a sum of the corresponding contributions from all molecular orbitals in a system, if such orbitals are chosen mutually orthogonal. Thus, for each transferable orthogonal molecular orbital there exists, e.g., a transferable orbital dipole moment. Since chemists appreciate additive decompositions of... 

Another parameter that one can extract from a Mossbauer spectrum is the quadrupole splitting. The 3/2 state in either iron or tin is degenerate with respect to an asymmetric electrostatic field, and in such a field these levels will be split into dz 3/2 and 1/2 levels. One can observe transitions either to or from these two levels to the ground state, and this is the quadrupole splitting. It is actually e qQ, where eq is the electrostatic field gradient—i,e., the second derivative of the potential with respect to the coordinate—and eQ is the nuclear quadrupole moment. The typical quadrupole split spectrum for iron is shown in Figure 6, in which the cubic (octahedral) symmetry around the iron atom is de-... 

All the methods used in this study are response methods. They deserihe the response of an ohservahle sueh as an eleetrie dipole moment /I or quadrupole moment to an external or internal perturhation, e.g., an eleetrie field or field gradient. Response funetions originated in various diseiplines in physies. In statistieal physies, they were used as time-eorrelation functions in the form of Green s functions [44,45]. Linderherg and Ohrn first showed the usefulness of this idea for quantum chemistry [46]. Since then response functions have been derived for many types of electronic wavefunctions. Four of these methods are employed here. 

Though the traceless moments can be derived from the unabridged moments, the converse is not the case because the information on the spherically averaged moments is no longer contained in the traceless moments. The general relations between the traceless moments and the unabridged moments follow from Eq. (7.3). For the quadrupole moments, we obtain with Eq. (7.2) ... 

In the derivation of the traceless quadrupole moments from the electrostatic moments, the spherical components are subtracted. Thus, the quadrupole moments can be derived from the second moments, but the opposite is not the case. Spackman (1992) notes that the subtraction introduces an ambiguity in the comparison of quadrupole moments from theory and experiment. The spherical component subtracted is not that of the promolecule, but is based on the distribution itself. It is therefore generally not the same in the two densities being compared. On the other hand, the moments as defined by Eq. (7.1) are based on the total density without the intrusion of a reference state. 

In this expression, the dipole dipole interactions are included in the electrostatic term rather than in the van der Waals interactions as in Eq. (9.43). Of the four contributions, the electrostatic energy can be derived directly from the charge distribution. As discussed in section 9.2, information on the nonelectrostatic terms can be deduced indirectly from the charge density. The polarizability a, which occurs in the expressions for the Debye and dispersion terms of Eqs. (9.41) and (9.42), can be expressed as a functional of the density (Matsuzawa and Dixon 1994), and also obtained from the quadrupole moments of the experimental charge density distribution (see section 12.3.2). However, most frequently, empirical atom-atom pair potential functions like Eqs. (9.45) and (9.46) are used in the calculation of the nonelectrostatic contributions to the intermolecular interactions. 

Another important nuclear characteristic is the nuclear quadrupole moment which, possessed by nuclei for which 7 1, has given rise to the important field of nuclear quadrupole resonance spectroscopy. A major importance of the quadrupole moment with respect to NMR absorption resides in the effects of quadrupole coupling constants on nuclear relaxation times and, therefore, on the line widths and saturation characteristics of NMR absorption (9). In addition, in favorable situations, quadrupole coupling constants can be derived from the characteristics of nuclear resonance of quadrupolar nuclei 127). Some examples of these effects will be described in Sections III, IV and VI of this chapter. 

Boron compounds contain two isotopes B10 and B11 of natural abundances 19% and 81%, respectively. Although both these isotopes possess magnetic moments, the Bn nucleus is better suited to the high resolution experiment because of its (1) greater natural abundance, (2) smaller quadrupole moment, and (3) larger nuclear moment. Because of the broad range of structures possible in boron compounds, particularly the hydrides, there has been considerable NMR work done in this field to confirm previously proposed structures and in a few cases to first establish geometry of a compound. The B11 spectra of tetraborane and a tetraborane derivative arc considered below. 

Due to the electric quadrupole interaction, the Mi = 1/2 and Mi = 3/2 components of the 7 = 3/2 state of 57Fe split up, giving rise to the quadrupole splitting. Derived from the interaction of the nuclear quadrupole moment with the electric field gradient at the iron nuclei, AEq provides information about the asymmetry of the electron density around the iron nucleus. The electric field gradient at the iron nucleus can be calculated to obtain AEq (97). Since both 6 and AEq are related to the electron density at the nucleus, basis sets with an enlarged flexibility at the core region... 

One of the simplest orientational-dependent potentials that has been used for polar molecules is the Stockmayer potential.48 It consists of a spherically symmetric Lennard-Jones potential plus a term representing the interaction between two point dipoles. This latter term contains the orientational dependence. Carbon monoxide and nitrogen both have permanent quadrupole moments. Therefore, an obvious generalization of Stockmayer potential is a Lennard-Jones potential plus terms involving quadrupole-quadrupole, dipole-dipole interactions. That is, the orientational part of the potential is derived from a multipole expansion of the electrostatic interaction between the charge distributions on two different molecules and only permanent (not induced) multipoles are considered. Further, the expansion is truncated at the quadrupole-quadrupole term. In all of the simulations discussed here, we have used potentials of this type. The components of the intermolecular potentials we considered are given by ... 

The microwave spectrum of the complex between ethylene and ketene (and of deuter-ated derivatives) reveals56 a crossed structure (22), while the ketene/acetylene complex shows a planar geometry57. This difference in geometry is explained by the different quadrupole moments of two unsaturated hydrocarbons. 

Nickel, palladium and platinum - The low natural abundance of the 61Ni isotope means that there is a paucity of data on NMR spectroscopy of this element, although much useful information can be derived from spectra of ligand nuclei (e.g. Zschunke et al., 1992 Oik et al., 1992). In the case of 105Pd, the large value of the quadrupole moment presents additional difficulties for direct spectroscopic measurements and, as with nickel, NMR studies have concentrated on the characterisation of ligand nuclei. An example here is the application of two-dimensional NMR to investigate the structure of a nucleic acid palladium complex (Bichenkova et al., 1992). 

Vanadium-51 is a spin 7/2 nucleus, and consequently it has a quadrupole moment and is frequently referred to as a quadrupolar nucleus. The nuclear quadrupole moment is moderate in size, having a value of -0.052 x 10 2S m2. Vanadium-51 is about 40% as sensitive as protons toward NMR observation, and therefore spectra are generally easily obtained. The NMR spectroscopy of vanadium is influenced strongly by the quadrupolar properties, which derive from charge separation within the nucleus. The quadrupole moment interacts with its environment by means of electric field gradients within the electron cloud surrounding the nucleus. The electric field gradients arise from a nonspherical distribution of electron density about the... 

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