Ramsey equations

Measurement of Kd for any ion pair using the Fuoss formulation enables the degree of dissociation of that pair to be assessed, and this knowledge is essential in evaluating reactivity. The theory automatically provides a value for A0 for the salt and, if Kd is determined at a series of temperatures, then the corresponding enthalpy and entropy of dissociation can also be calculated. Use of the Denison and Ramsey equation then conveniently provides for an estimate of Kd in other... 

Though the data here are scant, they show also that increasing the polarity of the solvent (CH2C12 — liquid S02) increases the dissociation constant of a given salt, as predicted by the Denison and Ramsey equation (—LnKAcc 1/D). This result is more apparent from studies of quaternary ammonium salts in more polar solvents and solvent mixtures [e.g. Ref. 53)]. The previous data on sul-phonium salts (Table 1) in methylene chloride and nitrobenzene also confirm this conclusion. 

Data for equivalent conductances at infinite dilution A0, and in particular single ion contributions AJ and Aq, have been deliberately excluded from this report being of more interest to the electrochemist than to the polymer kinetidst. However, it is worthwhile pointing out that conductance data on tri-isoamyl-n-butyl ammonium tetraphenylborate and hexachloroantimonate in CH2C1 has been obtained (26), allowing single ion conductances for all the ions in Tables 4 and 5 to be estimated. Thus the effective Stoke s radii (r and r ) can be calculated (as outlined in Section III.B) and compared with the effective interionic distance (r+ and 6 ) in the ion pairs, as calculated from the simplified Denison and Ramsey equation. This correlation is displayed in Table 6. 

Mathias and Gil have attempted to explain the observed high-field shifts of the signals of protonated azines as arising from one of two effects. Either a decrease of the paramagnetic term in Ramseys equation, or a change in bond order of the bond,... 

Nowadays, many electronic structure codes include efficient implementations [37—41] of the Ramsey equations [42] for the calciflations of nonrelativistic spin—spin coupling constants. A vast number of publications devoted to the calculation of/-couplings can be found in the Hterature, covering different aspects such as the basis set effects [43-55], the comparison of wave function versus density functional theory (DFT) methods [56-60], or the choice of exchange-correlation functional in DFT approaches [61-68]. Excellent recent reviews of Contreras [69] andHelgaker [70] cover these particular aspects. 

It was the Al spectra for the A1X4" (X Cl, Br, I) compounds that in 1968 provided the first theoretical basis for the NHD shielding pattern (4). Since deshielding is proportional to through the paramagnetic term of the Ramsey equation expressed in the form of eq 3, the ligand whose binding causes... 

A generalized equation was suggested by Karplus [51] that takes into account the paramagnetic term (cpara) of the basic Ramsey equation [52] that links the three components of the shielding of a nucleus ... 

A compromise between the opposing interpretations of Pople and Mason is offered by Webb and co-workers (46). These authors conclude that it is improper to equate the diamagnetic terms used by the two authors, since in contrast to Pople s o A, Flygare s contains significant and variable nonlocal contributions and is comparable rather to the gauge-dependent term in the original Ramsey expression (29). 

Equations relating and the spin-rotation constants have been given by Ramsey 9, 77, 78) for linear molecules and by Flygare 24) for symmetric top, spherical top, and asymmetric molecules. A simple expression for the general case, and neglecting vibrational effects, is... 

To account for the term A S /R, and for any other entropy or energy contributions which may arise, e.g. Van der Waal s interactions between ions, Ramsey (45) has re-written the original equation as follows,... 

A more quantitative attempt at interpreting platinum chemical shifts requires that we consider some form of the Ramsey (48) equation, which describes the resonance frequency, v9 in terms of the paramagnetic screening term, o>- QAB is a charge-density, bond-order matrix, AE is the average excitation energy, and r represents a distance from the nucleus for, in this case, a given d electron. 

In the past fifty years, the NMR Chemical Shielding have evolved from corrections to the measurement of nuclear magnetic moments as quoted from Ramsey s 1950 original papers (Phys. Rev. 77, 567 and 78, 699), to one of the most important tools for structural elucidation in many branches of chemistry. There are no simple relationships between molecular structure and chemical shifts. Their dependence on the molecular electronic and geometrical structure can be derived via complex quantum mechanical equations. 

The G-BASE project has used several statistical packages to perform this nested ANOVA analysis (e.g., Minitab and SAS). It currently uses an MS Excel procedure with a macro based on the equations described by Sinclair (1983) in which the ANOVA is performed on results converted to logio (Johnson, 2002). Ramsey et al. (1992) suggest that the combined analytical and sampling variance should not exceed 20% of the total variance with the analytical variance ideally being <4%. 

The subscripts a and n are used by Ramsey [52] and others, but we use the more explicit subscripts parallel and perpendicular. The paramagnetic part of the susceptibility tensor, given in equation (8.116), has a zero component in the z direction because it depends upon the mixing brought about by the x and y components of the orbital angular momentum L Hence ... 

For the 1 X ground state of CsF, A = () in this equation. This result should now be compared with the term used by English and Zorn [51] and Ramsey [52] to represent the quadrupole interaction, which is... 

There is considerable controversy over the value of the radius of the hydrated proton (51-57), and it has been reported from 0.945-4.5 A. However, overwhelming experimental as well as theoretical work seems to suggest that the hydrated proton species is H904+ with a radius of 3.0 A. Hydrated anionic radii (r.) were calculated using this value of the hydrated proton, and these are compiled in Table VIII. What do these radii mean The term (l/r+ + l/r ) in Equation 32 really implies the sum of reciprocals of the minimum distances of approach of the two ions or some function of it. In the Dennison and Ramsey (58) treatment of the Bom model this term is 1/r, where r is the minimum distance of approach of the two ions. Implicit in the expression (l/r+ + 1/r.) is the assumption that both ions are monatomic and spherical therefore, the minimum distance of approach of the two ions may be apportioned as the hydronium ion radius and the anion radius. Although it might be possible to assign a value to the hydronium ion radius, the anionic radius... 

Octahedral cobalt complexes, CoL, have three filled t2 and two empty e molecular orbitals. The Co chemical shifts of several such complexes are linear with the wavelength of the first absorption (longest wavelength) in the UV/visible spectrum. Explain the linearity in terms of Ramsey s equation for shielding. 

The various diamagnetic and paramagnetic terms in equation (1) are not directly comparable to those terms bearing these names in Ramsey s theory. 

Most Pt complexes have low-lying excited electronic states, and Pt NMR shifts are dominated by the paramagnetic term in Ramsey s equation ( ). The Pt(IV) halides alone span some 12,500 ppm (8) ... 

A general theory of chemical shifts of magnetic nuclei has been given by Ramsey, and the screening constant for a particular nucleus has been conveniently expressed in terms of three main contributions by equation (9)—... 

---