Boltzmann weighted averages

This methodology provides coupling constants with a mean error of 0.51 Hz, and when applied to a probe set of 61 conpUng constants from 37 different molecules (including a few that require a number of conformers and thus a Boltzmann-weighted averaging of the coupling constants), the mean error is only... 

Boltzmann-weighted average based on the G3 energies listed in Figure 2.2. 

Da Silva et al. " computed the optical rotation of three a and five p conformations of D-glucose in aqueous solution using a TDDFT/GLAO approach with either the aug-cc-pVDZ or 6-31-l-l-G(d,p) basis sets. The a conformations aU have large positive values of [a]o, while the P conformations have values that range from -10° to -1-80°. The Boltzmann-weighted average value is 62.6° or 58.8° with the aug-cc-pVDZ or 6-31-l-l-G(d,p) basis sets, respectively. These values are reasonably close to the experimental value of 52.T For those interested, Mennucci et al. have reviewed the application of computational models of solvent effects on chiroptical properties. 

If we have sufficiently sampled the possible arrangements of molecules in the simulation and have an accurate method to calculate their energy E, then the above formula will give a Boltzmann weighted average of the property X... 

The high pressure limiting rate coefficient is therefore a Boltzmann-weighted average of the energy dependent rate coefficient. 

The limit of nearly free relative orientations is not of much physical interest in the case of electrostatic forces. The intermolecular interaction is a Boltzmann-weighted average of the form... 

The luminescence lifetime of this complex at 77°K in glassy medium has been found to be 6.92 fisec, and the luminescence quantum yield is 0.496 (24), From these values, the intrinsic lifetime for the radiative decay of these levels at 77°K is estimated to be 14.0 jj.sec. Studies of the temperature dependence of the luminscence lifetime of similar complexes between 2° ond 100 °K indicate that the emission is caused by four thermally equilibrated levels which lie within roughly 100 cm" of one another (4), The lifetime and quantum yield data reported at 77°K are therefore representative of the Boltzmann weighted average values of these quantities for a manifold of four thermally equilibrated levels. 

Hence, the scattering problem involves finding expressions for< 5c >, which can be found by taking a Boltzmann weighted average over all fluctuations, where the fluctuations are treated as a continuum of concentration states, each with a free energy G( )... 

It is possible to run many N VE simulations with different energies and result could be used in a Boltzmann-weighted average as... 

The collisional cross section, which is normally given the symbol cr, is a measure of the effective size of the colliding particles, expressed as an area The link between the rate coefficient and a is k = , where v refers to fiie relative velocity of the colliding molecules and the brackets r resent a Boltzmann-weighted average of the cross section (which depends on the collision energy) and the relative collision velocity. 

The values shown in Table 7.2 cannot be directly compared with experimental liquid values, because the latter involve Boltzmann-weighted averaging over torsional motions that are rapid on the NMR time scale. However, for the coupling J[CH2,CH3] between methylene and methyl protons one can make an eyeball estimate by simply averaging the six vicinal couplings between methylene protons H(5), H(8) and methyl protons H(6), H(7), H(9), leading to the following... 

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