Intemuclear distances distributions

Sketch potential energy curves for the following states of CdH, Br/ and CH, given their intemuclear distances r, and suggest qualitative intensity distributions in the v" = 0 progressions for transitions between the states observed in absorption ... 

Rotational-echo double-resonance (REDOR)(75,79) is a new solid-state NMR technique which is sensitive to through-space carbon-nitrogen interactions between selectively 13C and 15N-enriched sites separated by up to 5A (20-22). The parameter directly measured in a REDOR experiment is the heteronuclear dipolar coupling constant DCN, which is in itself proportional to the inverse third power of the intemuclear distance, rCN. It is this dependence on (icn)3 which accounts both for REDOR s ability to accurately measure short distances and its insensitivity to longer-range interactions. As a technique which can probe, in detail, intermolecular interactions over a distance range of 5A, REDOR is well suited to studying the distribution of small selectively-labeled molecules in polymer delivery systems. 

In general, all observed intemuclear distances are vibrationally averaged parameters. Due to anharmonicity, the average values will change from one vibrational state to the next and, in a molecular ensemble distributed over several states, they are temperature dependent. All these aspects dictate the need to make statistical definitions of various conceivable, different averages, or structure types. In addition, since the two main tools for quantitative structure determination in the vapor phase—gas electron diffraction and microwave spectroscopy—interact with molecular ensembles in different ways, certain operational definitions are also needed for a precise understanding of experimental structures. 

Equations (5.2)—(5.4) and Figs. 5.1-5.3 illustrate the nature of the structural observables obtained from gas-electron diffraction the intensity data provide intemuclear distances which are weighted averages of the expectation values of the individual vibrational molecular states. This presentation clearly illustrates that the temperature-dependent observable distribution averages are conceptually quite different from the singular, nonobservable and temperature independent equilibrium distances, usually denoted r -type distances, obtained from ab initio geometry optimizations. 

Fig. 2.2. Electron ionization can be represented by a vertical line in this diagram. Thus, ions are formed in a vibrationaUy excited state if the intemuclear distance of the excited state is longer than in the ground state. Ions having internal energies below the dissociation energy D remain stable, whereas fragmentation will occur above. In few cases, ions are unstable, i.e., there is no minimum on their potential energy curve. The lower part schematically shows the distribution of Franck-Condon factors, fyc, for various transitions.

Figures 4 and 5 show not only the experimental distributions but also the distributions calculated for the best model of tetramethylsilane, which is characterized by the following bond lengths and bond angle and Td symmetry and staggered methyl conformation, Si-C 1.877(4)A, Si-H 1.110(3)A, and Si-C-H 111.0(2)°. These are so-called average parameters lyide infra). The radial distribution is convenient to visually inspect the validity of a model and to read off some principal intemuclear distances, but the quantitative determination of all the parameters is done on the basis of the molecular intensities. The refinement of parameters usually starts from an initial set of parameters. The expression of the molecular intensities is a non-linear relationship, a good choice of the initial parameters will ensure that the calculation reaches the global rather than a local minimum.

The vapor sample under investigation may not eontain only one kind of speeies. It is desirable to learn as mueh as possible about the vapor composition from independent sources, but here the different experimental conditions need to be taken into account. For this reason, the vapor composition is yet another unknown to be determined in the electron diffraction analysis. Impurities may hinder the analysis in varying degrees depending on their own ability to scatter electrons and on the distribution of their own intemuclear distances. In case of a conformational equilibrium of, say, two conformers of the same molecule may make the analysis more difficult but the results more rewarding at the same time. The analysis of ethane-1,2-dithiol data collected at the temperature of 343 kelvin revealed the presence of 62% of the anti form and 38% of the gauche form as far as the S-C-C-S framework was concerned. The radial distributions calculated for a set of models and the experimental distribution in Figure 6 serve as illustration. 

In the case of polyatomic molecules the radial distribution curve as deduced from electron-diffraction gas experiments may also be considered as a kind of a weighted sum of p ip curves for the internal motion in the molecule, but here all intemuclear distances are inseparably mixed together in a one-dimensional representation. For a rigid molecule, such as carbon tetrachloride or benzene, electron diffraction may produce quite accurate information as to the geometry of the molecule. As to the internal motion of the molecule, vibrational amplitudes may be deduced and compared with the corresponding data, differently but usually considerably more accurately, obtained by spectroscopic methods. How this is actually done in practice is perhaps most elegantly described by S. Cyvin4). 

The main differences are between X-ray diffraction (which probes nuclear positions via electron location) on the one hand and electron diffraction, microwave spectroscopy and neutron diffraction (which probe nuclear positions more directly), on the other hand. The differences result from (1) the fact that X-ray diffraction measures distances between mean nuclear positions, while the other methods measure essentially average distances, and (2) from errors in intemuclear distances caused by the nonisotropic (uneven) electron distribution around atoms. The mean versus average distinction is illustrated here ... 

The radial distribution function is the principal entity in the use of X-ray and neutron diffraction data to determine a structure. Write an expression for the number of particles, B, in a spherical shell of radius r with respect to a reference particle. Calculate the number of particles in that shell, assuming that the material concerned has a density of 1.6 that the first shell outside the reference atom S is at least a distance of 0.20 nm from the latter (intemuclear distance) and that the g. - r relation is idealized to a square box, height 2.0 and width 0.10 nm. 

Lithium-doped BPO4, another candidate ceramic electrolyte material for lithium batteries has been studied by Li NMR relaxation and linewidth measurements of samples with Li doping levels up to 20 mol % (Dodd et al. 2000). Comparison of the NMR data with values of the second moment calculated for both random and homogeneous models of Li distribution indicate the existence of Li clusters with an intemuclear distance of 3A, possibly consisting of 1 Li ion fixed at a boron vacancy with additional 2 Li ions in the conduction channels surrounding the vacancy. The atomic jump time, determined from measurements of the Li motional narrowing behaviour, indicate a maximum in the Li ionic mobility at the 10 mol % doping level (Dodd et al. 2000). 

This can be rationalized in one of two ways. First, the effect of electron correlation is to keep electrons apart, which therefore tends to increase intemuclear distances to accommodate the expansion of the electron distribution. The other equally satisfactory argument is that antibonding orbitals are mixed into the wave function by the correlation treatment. 

In order to give an explicit form to the potentials Up, Goeppert-Mayer and Sklar assumed that the a electron distribution around each atom is the same as in a molecule with infinitely large intemuclear distances the potential Up is then given by the Hartree-Fock potential for the atom P in the appropriate valence state 4> for instance, in the case of the carbon atom in the valence state (V4, s pxpypz)... 

The basis of the use of GED as a structural tool lies in the relationship between the intensity of scattered electrons as a function of scattering angle, 0, and P(r), the probability flmction which expresses the distribution in intemuclear separation. The molecular parameters determined by this relationship are ry, the effective intemuclear distance between atoms i and j, and /jj, the amplitude of vibration (sometimes also denoted as Mjj). 

The radial distribution curve has the advantage of being more understandable than the intensity curve since it has a peak for each intemuclear distance. The peak is rather narrow and is centered around every... 

Intemuclear distances t Mean square amplitudes Electron Electronic distribution excitations... 

Finally, there is one further source of information on the harmonic force field that has been used occasionally, namely mean square amplitudes of vibration in the various intemuclear distances, as observed by gas-phase, electron-diffraction techniques. These can be measured experimentally from the widths of the peaks observed in the radial distribution function obtained from the Fourier transform of the observed diffraction pattern. They are related to the harmonic force field as follows.23 If < n > denotes the mean square displacement in the distance between atoms m and /t, then the mean amplitudes <2 > are given as the diagonal elements of a matrix 2, where... 

When an ionic salt such as NaCl melts, the ionic lattice (see Figure 5.15) collapses, but some order is stiU retained. Evidence for this comes from X-ray diffraction patterns, from which radial distribution functions reveal that the average coordination number (with respect to cation-anion interactions) of each ion in liquid NaCl is 4, compared with 6 in the crystalline lattice. For cation-cation or anion-anion interactions, the coordination number is higher, although, as in the solid state, the intemuclear distances are larger than for cation-anion separations. The solid-to-liquid transition is accompanied by an increase in volume of il0-15%. The number of ions in the melt can be determined in a similar way to that described in Section 8.8 for H2SO4 systems in molten NaCl, v = 2. 

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