The classic limit

In the classical limit, the triplet of quantum numbers can be replaced by a continuous variable tiirough the transformation... 

If z = exp(pp) l, one can also consider the leading order quantum correction to the classical limit. For this consider tlie thennodynamic potential cOq given in equation (A2.2.144). Using equation (A2.2.149). one can convert the sum to an integral, integrate by parts the resulting integral and obtain the result ... 

Even the description of the solvent itself presents major theoretical problems the partition fiinction for a liquid can be written in the classical limit [1, 2] as... 

The fluctuation dissipation theorem relates the dissipative part of the response fiinction (x") to the correlation of fluctuations (A, for any system in themial equilibrium. The left-hand side describes the dissipative behaviour of a many-body system all or part of the work done by the external forces is irreversibly distributed mto the infinitely many degrees of freedom of the themial system. The correlation fiinction on the right-hand side describes the maimer m which a fluctuation arising spontaneously in a system in themial equilibrium, even in the absence of external forces, may dissipate in time. In the classical limit, the fluctuation dissipation theorem becomes / /., w) = w). 

The Onsager regression hypothesis, stated mathematically for the chemically reacting system just described, is given in the classical limit by... 

Figure A3.8.3 Quantum activation free energy curves calculated for the model A-H-A proton transfer reaction described 45. The frill line is for the classical limit of the proton transfer solute in isolation, while the other curves are for different fully quantized cases. The rigid curves were calculated by keeping the A-A distance fixed. An important feature here is the direct effect of the solvent activation process on both the solvated rigid and flexible solute curves. Another feature is the effect of a fluctuating A-A distance which both lowers the activation free energy and reduces the influence of the solvent. The latter feature enliances the rate by a factor of 20 over the rigid case.

The fonn of the classical (equation C3.2.11) or semiclassical (equation C3.2.11) rate equations are energy gap laws . That is, the equations reflect a free energy dependent rate. In contrast with many physical organic reactivity indices, these rates are predicted to increase as -AG grows, and then to drop when -AG exceeds a critical value. In the classical limit, log(/cg.j.) has a parabolic dependence on -AG. Wlren high-frequency chemical bond vibrations couple to the ET process, the dependence on -AG becomes asymmetrical, as mentioned above. 

As mentioned above, the correct description of the nuclei in a molecular system is a delocalized quantum wavepacket that evolves according to the Schrbdinger equation. In the classical limit of the single surface (adiabatic) case, when effectively 0, the evolution of the wavepacket density... 

This proves that the pseudoparticles in the quantum fluid obey classical mechanics in the classical limit. 

For the case of intramolecular energy transfer from excited vibrational states, a mixed quantum-classical treatment was given by Gerber et al. already in 1982 [101]. These authors used a time-dependent self-consistent field (TDSCF) approximation. In the classical limit of TDSCF averages over wave functions are replaced by averages over bundles of trajectories, each obtained by SCF methods. 

Therefore as n becomes large the classical limit is approached. 

The classical bath sees the quantum particle potential as averaged over the characteristic time, which - if we recall that in conventional units it equals hjk T- vanishes in the classical limit h- Q. The quasienergy partition function for the classical bath now simply turns into an ordinary integral in configuration space. 

The notation is meant to suggest that the frequency is variable and depends on the propagator matrix elements. The following criteria have proved valuable in choosing the variable coefficients of eq. IV.5 (1) at low temperature, the VQRS reference should weight the region around the potential minimum most heavily, and (2) at high temperature, our approximation should approach the classical limit ... 

As emphasized by Sadovskii and Zhilinskii [2], this latter point is important for quantum systems for which the lattice is too small to allow the constmction in Fig. 16a, because there is still a systematic reorganization of the spectra, involving transfer of individual levels or groups of levels from lower to upper bands, as y increases from 0 to 1. Figure 17 shows examples for n = A and i = j, 1, and, which illustrate the influence of quantum monodromy far from the classical limit. 

Often, we will be interested in how the velocities of molecules are distributed. Therefore we need to transform the Boltzmann distribution of energies into the Maxwell-Boltzmann distribution of velocities, thereby changing the variable from energy to velocity or, rather, momentum (not to be confused with pressure). If the energy levels are very close (as they are in the classic limit) we can replace the sum by an integral ... 

The final expression is the classical limit, valid above a certain critical temperature, which, however, in practical cases is low (i.e. 85 K for H2, 3 K for CO). For a homonuclear or a symmetric linear molecule, the factor a equals 2, while for a het-eronuclear molecule cr=l (Tab. 3.1). This symmetry factor stems from the indistinguishable permutations the molecule may undergo due to the rotation and actually also involves the nuclear partition function. The symmetry factor can be estimated directly from the symmetry of the molecule. 

Because the frequency of a weakly bonded vibrating system is relatively small, i.e. kBT hu we may approximate its partition function by the classical limit k T/hv, and arrive at the rate expression in transition state theory ... 

In the development of probability theory, as applied to a system of particles, it is necessary to specify the distribution of particles over die various energy levels of a system. The energy levels may be clearly separated in a quantized system or approach a continuum in the classical limit. The notion of probability is introduced with the aid of the general relation... 

The potential energy surface (PES) Up(Qk) for the excited electron state p has its minimum near the point Q (Fig. 1). In the classical limit, the electron transition may be treated as a continuous motion of the system on the lower PES, Ua, from the... 

The result for the transition probability in the classical limit had the form... 

If the dependence of nA and nB on q is taken into account in the calculation of the statistical operators for heavy particles, we obtain the improved Condon approximation (ICA) which differs from Eq. (17) only by the change of p and p°f to p, and pf, respectively. In the classical limit for p, and p/ the expression for the transition probability takes the form... 

The calculation of the integrals in Eq. (55) in the classical limit in the improved Condon approximation (for the nuclear subsystem) using the saddle point method leads to two coupled equations for the electron wave functions of the donor and the acceptor in the transitional configuration ... 

The height of the potential barrier separating the initial and final states of the nuclear subsystem decreases and, hence, the Franck-Condon factor increases (Fig. 6). In the classical limit, this results in a decrease of the activation free energy. 

In the classical limit, the transitional configuration q, Q in the Condon approximation is determined by the equations... 

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