Some quantum mechanical considerations

For a quantum mechanical state function the RS of eq. (3.5.7) requires multi- 

We already know from the invariance of the scalar product under symmetry operations that spatial symmetry operators are unitary operators, that is they obey the relation R R = R R1 E, where E is the identity operator. It follows from eq. (3.5.7) that the set of function operators / are also unitary operators. 

Exercise 3.6-1 Prove that the function operators / are unitary. 

In quantum mechanics the stationary states of a system are described by the state function (or wave function) ip( x ), which satisfies the time-independent Schrodinger equation 

Here x) stands for the positional coordinates of all the particles in the system, E is the energy of the system, and 77 is the Hamiltonian operator. Since a symmetry operator merely rearranges indistinguishable particles so as to leave the system in an indistinguishable configuration, the Hamiltonian is invariant under any spatial symmetry operator R. Let tpi denote a set of eigenfunctions of H so that 

---

J.O. Hirschfelder, Some quantum mechanical considerations in the theory of reactions involving an activation energy, J. Chem. Phys. 7 (1939) 616. 

Here, we first discuss preliminary matters regarding some of the distribution functions used later. Because our exposition in Section V-B is founded on the modification of a classical equation, we include the classical distribution functions as well. For both classical and quantum mechanical considerations, we take a system to be a macroscopic object (preferably in gaseous form). A hypothetical collection of such similar systems we call an ensemble. 

The more general question of whether chemistry is reducible to quantum mechanics is more subtle and requires a consideration of the nature of ab initio calculations. According to some authors, the question even depends on the problems in the foundations of quantum mechanics 44). 

There are several different ways in which quantum mechanics has been applied to the problem of relating the barrier to the frequency separation of the spectroscopic doublets. These are all approximation procedures and each is especially suitable under an appropriate set of circumstances. For example one may use perturbation theory, treating either the coupling of internal and external angular momenta, the molecular asymmetry, or the potential barrier as perturbations. Some of the different treatments have regions of overlap in which they give equivalent results choice is then usually made on the basis of convenience or familiarity. Extensive numerical tabless have been prepared which simplify considerably the calculations. 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

Transition state theory, as embodied in Eq. 10.3, or implicitly in Arrhenius theory, is inherently semiclassical. Quantum mechanics plays a role only in consideration of the quantized nature of molecular vibrations, etc., in a statistical fashion. But, a critical assumption is that only those molecules with energies exceeding that of the transition state barrier may undergo reaction. In reality, however, the quantum nature of the nuclei themselves permits reaction by some fraction of molecules possessing less than the energy required to surmount the barrier. This phenomenon forms the basis for QMT. ... 

To summarize, in this article we have discussed some aspects of a semiclassical electron-transfer model (13) in which quantum-mechanical effects associated with the inner-sphere are allowed for through a nuclear tunneling factor, and electronic factors are incorporated through an electronic transmission coefficient or adiabaticity factor. We focussed on the various time scales that characterize the electron transfer process and we presented one example to indicate how considerations of the time scales can be used in understanding nonequilibrium phenomena. 

---