The Quantum-Mechanical Basis

This chapter introduces the quantum mechanics required for the analyses in this text. The state of an electron is represented by a wave funetion ji. Kach observable is represented by an operator O. Quantum theory asserts that the average of many measurements of an observable on electrons in a certain state is given in terms of these by ji 0 d r. The quantization of energy follows, as does the determination of states from a Hamiltonian matrix and the perturbative solution. The Pauli principle and the time-dependence of the state are given as separate assertions. 

In the one-eleetron approximation, electron orbitals in atoms may be classified according to angular momentum. Orbitals with zero, one, two, and three units of angular momentum are called s, p, d, and/orbitals, respectively. Electrons in the last unfilled shell of s and p electron orbitals arc called valence electrons. The principal periods of the periodic table contain atoms with diflering numbers of valence electrons in the same shell, and the properties of the atom depend mainly upon its valence, equal to the number of valence electrons. Transition elements, having different numbers of d orbitals or/orbitals filled, are found between the principal periods. 

When atoms are brought together to form molecules, the atomic states become combined (that is, mathematically, they are represented by linear combinations of atomic orbitals, or LCAO s) and their energies are shifted. The combinations of valence atomic orbitals with lowered energy are called bond orbitals, and their occupation by electrons bonds the molecules together. Bond orbitals are symmetric or nonpolar when identical atoms bond but become asymmetric or polar if the atoms are different. Simple calculations of the energy levels are made for a series of nonpolar diatomic molecules. 

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The theory of crystal growth accordingly starts usually with the assumption that the atoms in the gaseous, diluted, or hquid mother phase will have a tendency to arrange themselves in a regular lattice structure. We ignore here for the moment the formation of poly crystalhne solids. In principle we should start with the quantum-mechanical basis of the formation of such lattice structures. Unfortunately, however, even with the computational effort of present computers with a performance of about 100 megaflops... 

We see that it is a consequence of the Pauli principle and bond formation that the electrons in most molecules are found as pairs of opposite spin—both bonding pairs and nonbonding pairs. The Pauli principle therefore provides the quantum mechanical basis for Lewis s rule of two. It also provides an explanation for why the four pairs of electrons of an octet have a tetrahedral arrangement, as was first proposed by Lewis, and why therefore the water molecule has an angular geometry and the ammonia molecule a triangular pyramidal geometry. The Pauli principle therefore provides the physical basis for the VSEPR model. 

Thus far our examination of the quantum mechanical basis for control of many-body dynamics has proceeded under the assumption that a control field that will generate the goal we wish to achieve (e.g., maximizing the yield of a particular product of a reaction) exists. The task of the analysis is, then, to find that control field. We have not asked if there is a fundamental limit to the extent of control of quantum dynamics that is attainable that is, whether there is an analogue of the limit imposed by the second law of thermodynamics on the extent of transformation of heat into work. Nor have we examined the limitation to achievable control arising from the sensitivity of the structure of the control field to uncertainties in our knowledge of molecular properties or to fluctuations in the control field arising from the source lasers. It is these subjects that we briefly discuss in this section. 

Figure 4.6 schematically represents the quantum mechanical basis of the Franck-Condon principle for radiative transitions. As a result, the process of... 

Fig. 4.7 Visualization of the quantum mechanical basis for a slow rate of radiationless transitions...

The primary difference between covalent and ionic bonding is that with covalent bonding, we must invoke quantum mechanics. In molecular orbital (MO) theory, molecules are most stable when the bonding MOs or, at most, bonding plus nonbonding MOs, are each filled with two electrons (of opposite spin) and all the antibonding MOs are empty. This forms the quantum mechanical basis of the octet rule for compounds of the p-block elements and the 18-electron rule for d-block elements. Similarly, in the Heider-London (valence bond) treatment... 

This, plus the quantization of the normal modes of vibration of the electromagnetic radiation field (just demonstrated), form, together, the quantum-mechanical basis for the wave-particle duality A wave can become a particle, and vice versa, but you can never make a simultaneous experiment to test both the wave and the particle nature of the same system. 

A simple example—the quantum mechanical basis for macroscopic rate equations... 

R. F. W. Bader. The quantum mechanical basis of conceptual chemistry. Monatsh Chem. 136, 819-854(2005). 

Polarizability plays the central role in the most universal intermolecular force. Up to this point, we ve discussed forces that depend on an existing charge, of either an ion or a polar molecule. But what forces cause nonpolar substances like octane, chlorine, and argon to condense and solidify Some force must be acting between the particles, or these substances would be gases under any conditions. The intermolecular force primarily responsible for the condensed states of nonpolar substances is the dispersion force (or London force, named for Fritz London, the physicist who explained the quantum-mechanical basis of the attraction). 

Effective medium theory was originally introduced in the early 1980s to describe chemisorption of gas atoms on metal surfaces. It has since been developed as a relatively efficient method for describing bonding in solids, particularly metals, and therefore has found considerable use in materials model-ing. It also forms the quantum mechanical basis for the more empirical and widely used embedded-atom method discussed below. Specific implementations of effective medium theory for materials simulation have been developed by Norskov, Jacobsen, and co-workers and by DePristo and co-workers. ... 

The above considerations provide the quantum-mechanical basis of a collision theory of chemical reactions. In order to calculate the reaction velocity, however, a treatment on the basis of statistical physics is also necessary. 

This chapter, which describes the application of the theory of atoms in molecules to the chemistry of the alkanes, gives only a curtailed account of the quantum mechanical basis of the theory, as full accounts have been presented elsewhere The development of the theory of molecular structure is presented in somewhat more detail and the application begins with a review of the topological features of molecular charge distributions and the associated definitions of atoms, bonds and molecular structure. 

For atomic and molecular systems, we actually have such expressions They come from the application of quantum mechanics to the translations, rotations, vibrations, and electronic states of atoms and molecules. Admittedly, Boltzmann didn t have quantum mechanics, because he developed the rudiments of statistical mechanics about 50 years before quantum mechanics was formulated. In fact, some ofhis expressions are incorrect by not including Planclfs constant (Boltzmann was unaware of its existence for most ofhis life). But in the calculation of thermodynamic values, the Planck s constants cancel. Their omission was, ultimately, unnoticed. However, in the material to come, we will use the quantum-mechanical basis of energy levels. 

The quantum mechanical basis for coqugation is discussed in Chapter 10. 

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