Quantum-mechanical treatment

The quantum mechanical treatment of a hamionic oscillator is well known. Real vibrations are not hamionic, but the lowest few vibrational levels are often very well approximated as being hamionic, so that is a good place to start. The following description is similar to that found in many textbooks, such as McQuarrie (1983) [2]. The one-dimensional Schrodinger equation is... 

A quantum mechanical treatment of molecular systems usually starts with the Bom-Oppenlieimer approximation, i.e., the separation of the electronic and nuclear degrees of freedom. This is a very good approximation for well separated electronic states. The expectation value of the total energy in this case is a fiinction of the nuclear coordinates and the parameters in the electronic wavefunction, e.g., orbital coefficients. The wavefiinction parameters are most often detennined by tire variation theorem the electronic energy is made stationary (in the most important ground-state case it is minimized) with respect to them. The... 

It is true that the structure, energy, and many properties ofa molecule can be described by the Schrodingcr equation. However, this equation quite often cannot be solved in a straightforward manner, or its solution would require large amounts of computation time that are at present beyond reach, This is even more true for chemical reactions. Only the simplest reactions can be calculated in a rigorous manner, others require a scries of approximations, and most arc still beyond an exact quantum mechanical treatment, particularly as concerns the influence of reaction conditions such as solvent, temperature, or catalyst. 

In the classical picture of an electron orbiting round the nucleus it would not surprise us to discover that the electron and the nucleus could each spin on its own axis, just like the earth and the moon, and that each has an angular momentum associated with spinning. Unfortunately, although quantum mechanical treatment gives rise to two new angular momenta, one associated with the electron and one with the nucleus, this simple physical... 

From a quantum mechanical treatment the magnitude of the angular momentum due to the spin of one electron, whether it is in the hydrogen atom or any other atom, is given by... 

In an approximation which is analogous to that which we have used for a diatomic molecule, each of the vibrations of a polyatomic molecule can be regarded as harmonic. Quantum mechanical treatment in the harmonic oscillator approximation shows that the vibrational term values G(v ) associated with each normal vibration i, all taken to be nondegenerate, are given by... 

These calculations began in 1927 with Heitler and London s approximate quantum mechanical treatment of the H2 molecule, which led to Eq. (5-13) for the energy. 

We now need to investigate the quantum-mechanical treatment of vibrational motion. Consider then a diatomic molecule with reduced mass /c- His time-independent Schrodinger equation is... 

The concept of a potential energy surface has appeared in several chapters. Just to remind you, we make use of the Born-Oppenheimer approximation to separate the total (electron plus nuclear) wavefunction into a nuclear wavefunction and an electronic wavefunction. To calculate the electronic wavefunction, we regard the nuclei as being clamped in position. To calculate the nuclear wavefunction, we have to solve the relevant nuclear Schrddinger equation. The nuclei vibrate in the potential generated by the electrons. Don t confuse the nuclear Schrddinger equation (a quantum-mechanical treatment) with molecular mechanics (a classical treatment). 

The requirement of an accurate global energy surface is even more important for a quantum mechanical treatment than for the classical case, since the wave function depends on a finite part of the surface, not just a single point. The updating of the positions and velocities is computationally inexpensive in the classical case, once the... 

Since the elementary quantum-mechanical treatment does not seem to give a high enough barrier, various treatments of the problem have been proposed which use empirical data such as bond dipole moments and steric repulsive forces. These treatments do not introduce any new forces which would not be included in a proper quantum-mechanical analysis, but they attempt to short-circuit these difficult and uncertain calculations. 

At longer distances weak attractions are expected due to induced dipoles and so-called dispersion forces. These are not included in the quantum-mechanical treatment outlined above but could be covered in principle by the inclusion of large numbers of excited... 

Both quantum mechanical and classical theories of Raman scattering have been developed. The quantum mechanical treatment of Kramers and Heisenberg 5) preceded the classical theory of Cabannes and Rochard 6). 

The third common level is often invoked in simplified interpretations of the quantum mechanical theory. In this simplified interpretation, the Raman spectrum is seen as a photon absorption-photon emission process. A molecule in a lower level k absorbs a photon of incident radiation and undergoes a transition to the third common level r. The molecules in r return instantaneously to a lower level n emitting light of frequency differing from the laser frequency by —>< . This is the frequency for the Stokes process. The frequency for the anti-Stokes process would be + < . As the population of an upper level n is less than level k the intensity of the Stokes lines would be expected to be greater than the intensity of the anti-Stokes lines. This approach is inconsistent with the quantum mechanical treatment in which the third common level is introduced as a mathematical expedient and is not involved directly in the scattering process (9). 

The first and second spectra of plutonium are probably the most thoroughly studied of any in the periodic table insofar as experimental description of the observed spectra and the term analysis is concerned, but a detailed quantum mechanical treatment has been handicapped by their great complexity. Fortunately, the lowest odd and lowest even configurations for both Pu I and Pu II are relatively simple, and parametric studies of the lowest levels of the 5f67s2, 5f56d7s2 and... 

Kahn, S. U. M. Quantum Mechanical Treatments in Electrode Kinetics 31... 

But why linearly and why with a slope of-1, or something thereabout, the reader may righteously ask. In anticipation of the quantum mechanical treatment in Chapter 5 we can briefly discuss here a simple electrostatic model which fully accounts for the observed behaviour. After all, as the detailed quantum mechanical treatment has shown, direct electrostatic... 

The whole question is clarified when considered in relation to the foregoing quantum mechanical treatment of the electron-pair bond. For the iron-group elements the following rules follow directly from that treatment and from the rules of line spectroscopy. 

From such crude data as are to be found in the literature we can calculate approximate values of the equilibrium constants, and hence of the free energies of dissociation for the various hexaarylethanes. From our quantum-mechanical treatment, on the other hand, we obtain only the heats of dissociation, for which, except in the single case of hexaphenylethane, we have no experimental data. Thus, in order that we may compare our results with those of experiment, we must make the plausible assumption that the entropies of dissociation vary only slightly from ethane to ethane. Then at a given temperature the heats of dissociation run parallel to the free energies and can be used instead of the latter in predicting the relative degrees of dissociation of the different molecules. 

The quantum-mechanical treatment previously applied to benzene, naphthalene, and the hydrocarbon free radicals is used in the calculation of extra resonance energy of conjugation in systems of double bonds, the dihydro-naphthalenes and dihydroanthracenes, phenylethylene, stilbene, isostilbene, triphenylethylene, tetraphenylethyl-... 

There are two principal methods available for the quantum mechanical treatment of molecular structure, the valence bond method and the molecular orbital method. In this paper we shall make use of the latter, since it is simpler in form and is more easily adapted to quantitative calculations.3 We accordingly consider each electron... 

This qualitative description of the interactions in the metal is compatible with quantum mechanical treatments which have been given the problem,6 and it leads to an understanding of such properties as the ratio of about 1.5 of crystal energy of alkali metals to bond energy of their diatomic molecules (the increase being the contribution of the resonance energy), and the increase in interatomic distance by about 15 percent from the diatomic molecule to the crystal. 

The quantum mechanics treatment of diamagnetism has not been published. It seems probable, however, that Larmor s theorem will be retained essentially, in view of the marked similarity between the results of the quantum mechanics and those of the classical theory in related problems, such as the polarisation due to permanent electric dipoles and the paramagnetic susceptibility. f Thus we are led to use equation (30), introducing for rK2 the quantum mechanics value... 

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