Averaging thermal

In the harmonic approximation the functions Xi and Xf are products of harmonic oscillator functions. We therefore specify the initial state by a set of quantum numbers n — (ni, ri2. n/v), and those for the final state by m = (mi,m2. tun)- So the nuclear wavefunctions are henceforth denoted by Xi,n and Xf,m- Equation (19.21) tells us how to calculate the rate of transition from one particular initial quantum mode n to a final quantum state m. This is more than we want to know. All we are interested in is the total rate from any initial state to any final state. The ensemble of reactants is in thermal equilibrium therefore 

Now it becomes apparent why it was useful to replace the delta function by its Fourier transform. The wavefUnctions Xin are products of harmonic oscillator functions, the Hamiltonians Hi and H/ are sums of harmonic oscillator terms. Therefore the terms in the brackets factorize in the form  

The terms Xk(t) can be evaluated analytically [3,4], but the result is not physically transparent. Instead we will first consider a classical mode j, for which hu C kT. The Hamiltonians Hij and Hfj can then 

We identify Aj = mjUtjAj/2 as the contribution of the mode j to the energy of reorganization [see Eq. (6.5)]. The thermal averaging is simplified by the fact that the expression does not depend on the nuclear momenta, which dropped out when the two Hamilton functions were subtracted. Explicitly we have  

We have dropped the index j momentarily to avoid cluttering the equations. Z is the partition function in coordinate space  

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For the case of nonzero temperatures the vacuum averages in Eq.(7) should be replaced by thermal averages over phonon populations. Using (7) and (5) we obtain that the scattering of an exciton in the effective medium by the perturbation fi — v z)) is described by the following self-consistent condition... 

Substitution of this for the golden-rule expression (1.14) together with the renormalized tunneling matrix element from (5.60) gives (5.64), after thermally averaging over the initial energies E-,. In the biased case the expression for the forward rate constant is... 

Thermal averages in the ensemble with constant pressure p are given via the corresponding partition function dVexp[—/3pF]Z(A, F, T). 

From this expression we obtain the thermal average... 

There are therefore four adjustable parameters per atom in the refinement (xy, yy, Zj, By). In the computer experiments we have carried out to test the assumptions of the nucleic acid refinement model we have generated sets of observed structure factors F (Q), from the Z-DNA molecular dynamics trajectories. The thermal averaging implicit in Equation III.3 is accomplished by averaging the atomic structure factors obtained from coordinate sets sampled along the molecular dynamics trajectories at each temperature ... 

Assuming that the spin conversion is a nonadiabatic process, the macroscopic rate constant may be expressed, following Levich [125], in terms of the thermally averaged transition probability, the averaging being extended over the initial vibronic levels, as ... 

In first-order perturbation theory, the rate can be calculated in a straightforward manner. The rate for a transition from a metal state k to the reactant orbital a is, after thermal averaging [Schmickler, 1996],... 

The generalization to the case of a thermally averaged parent state describes an interesting modulation curve that reflects in position and width the rotational eigenvalue spectrum of the resonant intermediate [31]. This structure has been observed in studies of HI ionization in Ref. 33. A schematic cartoon depicting the excitation scheme and the form of the channel phase for the case of a thermally averaged initial state is shown in Fig. 5g. 

X-ray crystallographic experiments measure the intensity of the diffraction peaks resulting from the X-rays scattered by electron clouds, which is related to the thermal average of electron density distributions in the crystal by a Fourier transform ... 

Note that in this approximation the incoherent scattering measures the time-dependent thermally averaged, mean square displacement <(rd(t) — (O))2). 

Due to the tensor character of D jK the thermal average is Eq. (96) leads to bulky expressions. However, it was shown [100] that (96) can be approximated at high accuracy by double sums of pre-averaged terms... 

During the last decades, a large body of structural information has been derived from gas-electron diffraction studies. The corresponding results are nearly exclusively reported in the literature in terms of r distances, or the equivalent thermal average intemuclear distances, which are denoted r. The r distances are defined by the relation, r = r — If. Alternative methods for interpreting gas-electron diffraction data are possible, for example, in terms of -geometries5, but they are currently too complex to apply in routine stmctural analyses, because they require detailed information on the molecular potential energy surface which is not usually available. 

Equations (5.7)—(5.9) define distances between average nuclear positions. A different type of average is obtained when the intemuclear distances, not the positions, are averaged. The meaning of the subtle shift in language is clear when the mathematical relation is considered. Thermal average intemuclear distances, or r -parameters, are related to re in the following way ... 

The brackets on the left of (9.15) indicate the fully coupled thermal averages, involving the actual interactions between the solution and the distinguished molecule of the joint system, specifically with the distinguished solute present. Equation (9.12) made a preliminary presentation of this result. 

The normalization integrals for the averages in the numerator and denominator cancel each other, leaving the traditional expression for the thermal average of F with the distinguished molecule present in the solution. This expression for the average will prove helpful several times below. The PDT is discussed extensively in Chap. 9, and in [29],... 

The function U fXj is called the PMF it was first introduced by Kirkwood to describe the structure of liquids [61]. It plays the role of a free energy surface for the solute. Notice that the dynamics of the solute on the free energy surface W(X) do not correspond to the true dynamics. Rather, an MD simulation on 1T(X) should be viewed as a method to sample conformational space and to obtain equilibrium, thermally averaged properties. 

The actual evaluations of Edd, II, and EHH are complex. Note that for Edd, CKJ uses Buckingham s (1957) prescription for the number of dipoles in the first solvation shell and considers both the thermally averaged dipole moment and the induced moment. The polarization energy is obtained from Land and O Reilly (1967). 

In conventional theories of rate processes, the temperature T is usually involved. The involvement of T implicitly assumes that vibrational relaxation is much faster than the process under consideration so that vibrational equilibrium is established before the system undergoes the rate process. For example, let us consider the photoinduced ET (see Fig. 5). From Fig. 5 we can see that for the case in which vibrational relaxation is much faster than the ET, vibrational equilibrium is established before the rate process takes place in this case the ET rate is independent of the excitation wavelength and a thermal average ET constant can be used. On the other hand, for the case in which the ET is much faster than vibrational relaxation, the ET takes place from the pumped vibronic level (or levels) and thus the ET rate depends on the excitation wavelength and often quantum beat will be observed. 

In this case, the thermal average rate constant given by Eq. (3.20) becomes... 

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