Boltzmann factor field energy

In molecular doped polymers the variance of the disorder potential that follows from a plot of In p versus T 2 is typically 0.1 eV, comprising contributions from the interaction of a charge carrier with induced as well as with permanent dipoles [64-66]. In molecules that suffer a major structural relaxation after removal or addition of an electron, the polaron contribution to the activation energy has to be taken into account in addition to the (temperature-dependent) disorder effect. In the weak-field limit it gives rise to an extra Boltzmann factor in the expression for p(T). More generally, Marcus-type rates may have to be invoked for the elementary jump process [67]. 

Fig. 4. Free-energy storage per monomer unit against monomer-flow field interaction energy a = exp (—AEg/kuT) with AEg = relative energy of the gauche with respect to the trans conformation P = exp (- AUfri( 1/kBT) = Boltzmann factor for the monomer-flow field interaction energy (AUfrii.t)...

Clearly, if a situation were achieved such that exceeded Np, the excess energy could be absorbed by the rf field and this would appear as an emission signal in the n.m.r. spectrum. On the other hand, if Np could be made to exceed by more than the Boltzmann factor, then enhanced absorption would be observed. N.m.r. spectra showing such effects are referred to as polarized spectra because they arise from polarization of nuclear spins. The effects are transient because, once the perturbing influence which gives rise to the non-Boltzmann distribution (and which can be either physical or chemical) ceases, the thermal equilibrium distribution of nuclear spin states is re-established within a few seconds. 

H-bonded systems may require additional diffuse or polarization functions. For example, the 6-311++G(d,p) basis set had been found to be suitable for H-bonded systems [78-81], It may be necessary to include Basis Set Superposition Errors (BSSE) [82] and Zero-Point-Energy (ZPE) corrections in evaluating the relative stabilities. Such corrections are often of the same magnitude as the energy differences among the dominant conformers. Moreover, the relative conformer energies may also differ noticeably with the basis sets used. All these factors will affect the Boltzmann factors predicted for different conformers and therefore the appearance of the population weighted VA and VCD spectra. Thus, an appropriate selection of DFT functionals and basis sets is very important for VCD simulations. A scale factor of 0.97-0.98 is usually applied to the calculated harmonic frequencies to account for the fact that the observed frequencies arise from an anharmonic force field instead of a harmonic one. A Lorentzian line shape is typically used in simulations of VA and VCD spectra. The full-width at half maximum (FWHM) used in the spectral simulation is usually based on the experimental VA line widths. 

This set of equations connects Planck s photon energy Ep with Einstein s mass/en-ergy equivalence, with Boltzmann s kinetic energy, with the kinetic energy of a particle and with the kinetic energy of an electron in an electric field of a voltage U of 1 V. The most important conversion factors used in photochemistry and photophysics are collected in Tab. 3-2. 

Dealing with the energy in the field 0 as a perturbation to the zero enargy Z7(t, 0) we have, on expanding the Boltzmann factors in (280) in a series in powers of 09 and on insertion into (278) ... 

If one plotted piT) in an Arrhenius diagram and determined the apparent activation energy A from the tangent at a given temperature T, one would obtain A(F) = (8/9)o cr. For (t=0.1 eV and 7 =29.5 K, A=0.35eV. A is, thus, always a multiple of cr. On the other hand, since any polaron-binding energy Ep would enter the Boltzmann factor for the jump rate as p/2 in the low-field limit, the disorder contribution to A would dominate even if cr and Ep were comparable. 

It is assumed at this point that the energy of interaction of the molecule and the field can be neglected in the exponent of the Boltzmann factor. An investigation shows that this assumption is valid... 

The equilibrium constants determined in Eqs. (8) and (9) are actually Boltzmann factors (statistics not included), but they indicate that in this case the heavier isotope prefers to occupy the hydride site. In contrast, both Field and Crabtree had previously observed the opposite effect, e.g., = 1.3 for ReD2(HD) - ReHD(D2) in [ReH2(H2)(CO)(PMe2Ph)3]+.57,58 However, detailed analysis of chemical shift and coupling constant data as a function of temperature in the Ir complex has not been carried out in these systems. The isotope effect was simplistically interpreted to be a consequence of a greater vibrational zero-point energy difference between Re(i/2-HD) and Re(f/2-D2) relative to Re-H and Re-D.58 These systems are quite complex... 

At the atomic and molecular levels, some of the charge displacements will reach saturation at high E-field strength levels (Section 8.4.1). The alignment of dipoles in a polar dielectric will reach a maximum when the field energy is of the same order of magnitude as the Boltzmann factor kT. 

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