Quantum-Mechanical Consideration

In the above semiclassical approximation, nuclear motion has been treated classically. We now consider the general case in which the motions of both electrons and nuclei obey the laws of quantum mechanics. Denoting by x s the nuclear coordinate normal to the intersection line L, and by z the set of electron coordinates, we write according to (3 1) the wave function for a given qiaantum state n of the total system 

Introducing (I6I.II) in (2.1) yields two coupled differential equations for the nuclear motion /112/ 

Sufficiently far from the classical turning points and X2 (at which quasiclassical (BWK) wave functions 

A simplified solution to the problem may be found, as shown by NIKITIN et al./113/, using the linear potential functions (154.11) which represent an approximation of the diabatic potential curves in the region of the crossing point x = x. Choosing this point as origin of the coordinate system, we set x = 0 and V-j (x ) = 2(Xq) = 0 to obtain the equations 

We assume further that the exchange integral (162.11) is a constant V12 = V.j2(0) is then convenient to introduce the momentum repre- 

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An expression for the short-range repulsive force (which arises from the interpenetration of the electron clouds of the two atoms) can also be derived from quantum-mechanical considerations" as... 

In the preceding sections it has been shown that in a large number of crystals containing covalent bonds, cited as examples, the number and distribution of the bonds are in good agreement with a classification deduced from quantum mechanical considerations, and, moreover, there exist regularities in the observed interatomic distances which may be expressed by assigning covalent radii to the atoms, dependent in a... 

Quantum mechanical considerations predict that a jr-electron system containing six electrons should be particularly stable.205 Examples of conspicuously stable six electron systems are benzene and the cyclopentadienyl anion. The cycloheptatrienylium cation is also stable, presumably for the same reason.206-207... 

A representation of all of the elementary reactions that lead to the overall chemical change being investigated. This representation would include a detailed analysis of the kinetics, thermodynamics, stereochemistry, solvent and electrostatic effects, and, when possible, the quantum mechanical considerations of the system under study. Among many items, this representation should be consistent with the reaction rate s dependence on concentration, the overall stoichiometry, the stereochemical course, presence and structure of intermediate, the structure of the transition state, effect of temperature and other variables, etc. See Chemical Kinetics... 

Quantum mechanical considerations show that, like many other atomic quantities, this angular momentum is quantized and depends on I, which is the angular momentum quantum number, commonly referred to as nuclear spin. The nuclear spins of / = 0, 1/2, 1, 3/2, 2. .. up to 6 have been observed (see also Table 1). Neither the values of I nor those of L (see below) can yet be predicted from theory. 

A well-known expression (Marcus, 1956) arises when the energy changes in the solvent-ion distances in the rearrangements concerned with electron transfer are taken to be harmonic, i.e., that Ux=kx2. This expression has been discussed (Levich and Dogonadze, 1983) in a quantum mechanical context with the implication that its derivation depends on quantum mechanical considerations. However, the association... 

Not only must the difference E2 - E1 be correct for absorption but also there must always be a change in the dipole moment of the molecule in going from one energy level to another. Only when this is true can the electric field of the light wave interact with the molecule. A further limitation comes from the symmetry properties of the wave functions associated with each energy level. Quantum mechanical considerations... 

RESONANCE. 1. In chemistry, resonance (or mesomerism) is a mathematical concept based on quantum mechanical considerations (i.e.. die wave functions of electrons). It is also used to describe or express the true chemical structure of certain compounds that cannot be accurately represented by any one valence-bond structure. It was originally applied to aromatic compounds such as benzene, for winch there are many possible approximate structures, none of which is completely satisfactory. See also Benzene. 

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

Quantum mechanical considerations indicate that it is necessary to keep the volume of the system constant in order for the energy levels of the system to he unchanged in the process. 

Molecular-dynamic simulations are characterized by a solution of Newton s laws of motion for the molecules travelling through the zeolite pore system under control of the force field given by the properties of the host lattice, by interactions between the host and the molecules, and by interactions between the molecules. To date this has been possible only for the diffusion of simple molecules (e.g. methane or benzene) inside a zeolite lattice of limited dimensions [29, 37, 54], To take into account the effects of a chemical reaction as well would require quantum-mechanical considerations however, such simulations are in their infancy. 

Of course, repulsive interactions play a role as well. When molecules are squeezed together, the nuclear and electronic repulsions and the rising electronic kinetic energy begin to dominate over the attractive forces. An expression for the repulsive force between two molecules can be derived from quantum-mechanical considerations as ... 

In addition, there is an exchange energy, which arises from purely quantum mechanical considerations. This energy depends on the number of possible exchanges between two electrons with the same energy and the same spin. 

Since a nucleus is a system having atomie dimensions, quantum mechanical considerations limit its orien-... 

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