Coalescence point

Hence the activation energy barrier to dimethylamino group rotation in dimethylacetamide (41) is calculated from equation 9 with k,. = 17.8 at the coalescence point 353 K (Fig. 2.26) ... 

As a result of the small, but apparent single bond character of the triafulvene C3/C4 bond and the good stabilization of the transition state of the rotation established earlier, rotation around this bond should be lower in energy in comparison to simple ethylene derivatives183. In fact, 1H-NMR spectra of several types of asymmetrically substituted triafulvenes 219-224 proved to be temperature-dependent and showed reversible coalescence phenomena at definite temperatures diagnostic for internal rotation processes. These were characterized by the free enthalpy of activation AG at the coalescence point of appropriate substituent signals61. ... 

To improve on the wave function one has to accept that the standard multideterminantal expansion [Eq. (13.3)] is unsuitable for near-exact but practical approximations to the electronic wavefunction. The problem is dear from a simple analysis of the electronic Hamiltonian in Eq. (13.2) singularities in the Coulomb potential at the electron coalescence points necessarily lead to irregularities in first and higher derivatives of the exact wave function with respect to the interpartide coordinate, rj 2. The mathematical consequences of Coulomb singularities are known as electron-electron (correlation) and electron-nuclear cusp conditions and were derived by... 

Although this equation only applies when the coalescence point is reached, rate constants for the exchange between two or more exchanging sites are accessible by analysis of line widths at half height, Avip, and shift differences, Ar, in Hz. The comparison between the experimental spectrum and the spectrum calculated by use of a simulation package for line shapes provides the mechanism for determining the rate constant of exchange [161, 162]. 

To expose the essence of the R12 method of Kutzelnigg [19], consider the simplest two-electron system, the helium atom in its ground state. The exact wave function in the vicinity of an electron-electron coalescence point r can be expressed [13] as... 

Another critical advance responsible for the success of MP2-R12 is the introduction of nonlinear correlation functions, in particular, the Slater-type correlation function, 1— exp(—yr 2), of Ten-no [29]. It is asymptotically linear in ru near the coalescence point and, hence, satisfies the cusp condition in the leading order. Unlike the linear correlation function, the Slater-type correlation function... 

This gives the rate of a symmetric A B exchange at the coalescence point, i.e. when the minimum between the two peaks involved disappears (64)... 

The dimethylamino group rotation follows a first-order rate law the exchanging methyl protons show no coupling and their singlet signals are of the same intensity. Under these conditions, equation 7 2 35-37 affords the rate constant kr at the coalescence point Tc ... 

A procedure often used to evaluate t is to estimate the temperature at which the lines coalesce . The disadvantage of this method is that the coalescence point is usually not very sharply defined. 

Fig. 13.12 ESED of noble metals is a tri-potential sequence (inset), (a) Eirst the HOPG step edges are oxidized to form carboxyl groups, disrupting edge sp bond, (b) deposition is initiated with nucleation sites at the edge plane from a low potential pulse, (c) followed by a low overpotential step for subsequent growth of the nucleation sites to the coalescing point...

The ethoxy-initiated, methoxy-terminated oligomer of chloral has a small energy difference between the right-handed and left-handed helical states (AH = 0.498 kcal/mol, AS = 0.238 cal/mol K for the hexamer). It was possible to determine the fraction of both states in solution by NMR spectroscopy at temperatures below the coalescence point.282... 

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