Boltzmann average

LW) interactions refer to the purely physical London s (dispersion), the Keesom s (polar) and Debye s (induced polar) interactions and correspond to magnitudes ranging from approximately 0.1 to 10 kJ/mol (but in rare cases may be higher). The polar forces in the bulk of condensed phases are believed to be small due to the self-cancellation occurring in the Boltzmann-averaging of the multi-body... 

In Sect. 2.1, the electron transfer rate was defined as the Boltzmann average of transition probabilities, which were calculated through time-dependent perturbation theory by using the Born-Oppenheimer and Frank-Condon approx-... 

Proton NMR coupling constants are calculated as Boltzmann averages over a discrete number of states, in the frame of the RIS theory. Different models are used for these calculations. 

E nt - E ot is small compared to the other terms, allowing them to replace h ) - E nt + E ot with a Boltzmann averaged E. Combining the two equations, they then obtained... 

The bond lengths were calculated as the Boltzmann average of the geometric properties of the individual vibronic energy levels. 

The rate constant can be expressed in terms of the potential of mean force at the activated complex. This potential may, e.g., be defined such that the gradient of the potential gives the average force on an atom in the activated complex due to the solvent molecules, Boltzmann averaged over all configurations. 

The bracket indicates a Boltzmann average over the solvent degrees of freedom with the reaction coordinate at the top of the barrier (x = 0). We have only included diagonal terms in the sum, since the ensemble average will be zero for all off-diagonal terms. Then... 

The PCM/DFT model failed to predict the intrinsic rotation (i.e. the specific rotation extrapolated to infinite dilution) of (R)-3-methylcyclopentanone dissolved in carbon tetrachloride, methanol and acetonitrile [68], This molecule has been investigated because it exists in both an equatorial and an axial form, allowing researchers to investigate the interplay of solvent and conformational effects. The conformer populations used in the Boltzmann averaging were derived from IR absorption and VCD spectra. The deviation of the calculated optical rotation from experiment was found actually to be larger when IEF-PCM was used to account for direct solvent effects (and geometry relaxation) on the optical rotation than when the gas-phase values were used. 

The expression van der Waals attraction is widely used and is here defined as the sum of dispersion forces [9], Debye forces [17] and the Keesom forces [18]. Debye forces are Boltzmann-averaged dipole-induced dipole forces, while Keesom forces are Boltzmann-averaged dipole-dipole forces. The interaction for all three terms decays as 1 /r6, where r is the separation between the interacting particles, and they are combined into one term with the proportionality constant denoted the Hamaker constant. In order to determine the van der Waals force there are at least two approaches, either to calculate the force between two particles assuming that the interaction is additive, (this is usually called the Hamaker approach) or to use a variant of Lifshitz theory. 

After a proper Boltzmann averaging of the integral cross section over the kinetic energy in the center of mass system, ECM, one get the rate constants,... 

Fig. 7 Boltzmann averaged calculated and experimental ORD of 2,3-hexadiene. Data to prepare the figure were taken from [135]...

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