Quantum average

Cnm is the quantum average of proton coupling, and AG m is the activation barrier for n — m transition. With explicit computation of Cnm, and considering reaction symmetry within the Q mode, a new term, Ea = h2(y2/2inu, whose physical interpretation is the coupling term between Q mode and solvent polarity, is introduced, accompanied by removal of the summation term in (13) ... 

The quantum average A1/2 G characterises spin-spin interactions established in the absence of any chain fluctuations. The ratio defined by ... 

Thus, in the presence of damping, it appears that the classical limit of the quantum average (143) may be obtained in writing in a similar way ... 

Moreover, since we are interested in the classical limit situation, it is suitable to look at the high temperature limit of the average of the quantum number operator appearing in Eq. (145). The quantum average is... 

A pulse scheme recovering the zero-quantum Hamiltonian was proposed by Baldus and Meier.142 It is weakly dependent on spectral parameters and a faithful measure of internuclear distances. This sequence is based on the former rotor-synchronized R/L-driven polarization transfer experiments.143,144 It uses the LG or FS-LG, which is used to decouple the high-7 spins, and combined MAS and RF irradiation of low-7 spins to decouple the hetero-nuclear dipolar interactions. With phase-inversion and amplitude attenuation in the rotating frame and refocusing pulses in the laboratory frame part of the pulse sequence, a zero-quantum average Hamiltonian can be obtained with optimum chemical-shift/offset independence. 

Eq. (11-22) means that the ensemble-quantum averaged value of R is the trace of the product of the matrices of R and p, where the matrix of p is the linear average of the D matrices ... 

This property is quite remarkable In the large photon number regime the coherent quantum average on a number state gives the same result as the incoherent statistical average over coherent states. 

While the mapping from the quantum average to a classical average in (46) can be performed for any quantum system, it can happen in frustrated quantum magnets, that some of the weights W(C) in the quantum system are negative, as is shown in Fig. 11. 

The quantum averaging is performed over the initial state of the field (no matter pure or mixed). The initial values of the total energy and its first derivative (with respect to x) are given by... 

The validity of this description was discussed in the Introduction. The Bronsted coefficient can be quantitatively described in the adiabatic PT picture by any of three differences between the TS and the reactant of the separation in the solvent coordinate, of the electronic structure, and of the quantum-averaged nuclear structure [3, 4]... 

Here AiS - AEi is the solvent reaction coordinate distance between the TS and reactant, and AE - AE is the corresponding distance between the product and reactant, (cp) is the quantum average over the proton and H-bond vibrations of Cp, the limiting product contribution to the electronic structure. The electronic structure for each critical point (c = R, P, and ) is evaluated at the respective critical point position AE (AE = AEp, AEp, or A ) along the reaction coordinate. The structural element is the quantum-averaged proton distance (over both... 

We pause to remark that the Bronsted coefficient a has often been used to describe TS structure via the Hammond postulate [15] or the Evans-Polanyi relation [45], where a is viewed as a measure of the relative TS structure along the reaction coordinate, usually a bond order or bond length. The important point is that, although adiabatic PT has a quite different, environmental, coordinate as the reaction coordinate, Eq. (10.12) is consistent with that general picture, with a proper recognition that quantum averages are involved. 

As seen from Fig. 10.8, a key component of the TS structure variation is reflected in the variation of the ZPE along the reaction coordinate. This feature is incorporated in Eq. (10.14) since the force constant is the sum (via Eq. (10.6)) of the ZPE and variation [3, 4]. (We pause to note that Eq. (10.4) shows that the coefficients in the FERs Eqs. (10.5) and (10.6) are not the same [3, 4].) Further, since a is also directly related to the relative difference in structure between R and TS, i.e. the last expression in Eq. (10.12), the variation of ZPE versus AE directly correlates with structural variation along a reaction path . A comparison between a reaction path described with quantum averages via the adiabatic PT picture and those with a classical description is presented in Section 10.2.3.5. 

We have addressed this issue in Ref. [3b], where, for a model PT system in solution treated in the adiabatic PT fashion, we have generated a certain reaction path in the following fashion. At each value of the solvent coordinate AE, we have calculated the quantum averaged values and of the proton and H-bond coor-... 

Figure 10.11 Calculated quantum averages vs. for an 0---0 system (solid line) [3b] and the BEBO curve (dotted line).

The pulse sequence for a double-quantum filtration experiment utilizing the HORROR sequence is shown in Fig. 29. The weak on-resonance RF field leads to recoupling of the homonuclear dipolar interaction under MAS if the DQ-HORROR condition Wr = 2 is fulfilled (which may be seen as equivalent to a (homonuclear) n = 1/2 7 experiment). This resonance condition leads to a purely dipolar double-quantum average Hamiltonian which has to be calculated by averaging over two rotation periods. Similarly to... 

Electron-nucleus ESR, ENDOR, ESEEM 1-2 Electron-nucleus Quantum average... 

Finally, the obtained expression (4.424) is unfolded through replacing the coordinate observation with the atomic radius quantum average displacement respecting its instantaneous value (Putz, 2010c)... 

Knowing the first and second order quantum averages for the atomic radius of a Hydrogenic system written in terms of the principal and azimuth quantum numbers n and /, respectively (Morse Feshbach, 1953)... 

Finally, one would like to use the developed rules of radial wave-function integration to compute other integrals of interests. For instance, one may wish to compute the quantum average (the observed) of the applied potential ... 

Now, taken together, the quantum averaged path (4.576) and its dispersion (4.577) provide the average of the squared quantum paths, according to the general definition ... 

The cross terms are Weyl-symmetrized [33,90]. The EOM for these variables are derived by commutation with the Hamiltonian and quantum averaging over the time-independent wave function x(Q) as described in the detail in Refs. [56-58]. The higher order moments that may appear as the result of the commutation and extend beyond the specified set are decomposed into the products of the moments belonging to the set. The only difference from the ordinary QHD [56-59] is due to the presence of the q subsystem. It is coupled to the Q subsystem in the MF way, producing the following MF approximation for the time evolution of an arbitrary Q P moment... 

One particular strength of perturbation theory is its intuitive simplicity. That may not be apparent when you ve just emerged from two pages of calculus, but the first- and second-order corrections to the energy are very useful conceptually. As we ve seen, the first-order correction is essentially the quantum average value theorem applied to the perturbation Hamiltonian H. In many cases. 

However, the third working DFT-BEC connection can be established by employing the inter-bosonic quantum average (Putz, 201 Ib-c, 2012a)... 

Then, in the electronic states, characterized by wave function (2.14) the vibrational deviation (amplitude) is calculated through the average values formula of the observable (by quantum averaging) postulate of the quantum mechanics ... 

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