Classical 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.

Miller W H 1971 Semiclassical nature of atomic and molecular collisions Accounts Chem. Res. 4 161-7 Miller W H 1974 Classical-limit quantum mechanics and the theory of molecular collisions Adv. Chem. Phys. 25 69-177... 

Miller W H 1974 Classical-limit quantum mechanics and the theory of molecular collisions Adv. Chem. 

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... 

In a classical limit of the Schiodinger equation, the evolution of the nuclear wave function can be rewritten as an ensemble of pseudoparticles evolving under Newton s equations of motion... 

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. 

Examples ofTypical QRA Objectives Types of Eacility Resotirces/Personnel Classical Limitations of QRA Example of Mortality Statistics Issues Affecting Perception of Risk Typical Pitfalls in Using QRA... 

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. 

In the high-temperature, classical limit this agrees with the finding in Ref. 343. In order to map the phase diagram in the Q -T plane, we have to set K = in (51) and solve for TJ as a function of 0J. This can be done numerically and the resulting phase diagram is shown in Fig. 14. 

In the high-temperature quasi-classical limit (fico [Pg.63]

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

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 ... 

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