The Quantum Mechanics of Some Simple Systems

V. The Quantum Mechanics of Some Simple Systems The Free Particle, 68. The Particle in a Box, 70. The Rigid Rotator, 72. The Rigid Rotator in a Plane, 75. The Harmonic Oscillator, 75. 

Given the generalizing power of so simple a relation as the n + rule, it is natural to wonder whether a theoretical derivation or explanation for it can be given. If such a derivation from quantum mechanics were available it would strengthen the view that we possess a good theoretical understanding of the periodic system. What is not generally appreciated in this context is that the frequently encountered textbook explanation of the periodic system has some important limitations. 

This chapter builds an understanding of atomic structure in four steps. First, we review the experiments that led to our current nuclear model of the atom and see how spectroscopy reveals information about the arrangement of electrons around the nucleus. Then we describe the experiments that led to the replacement of classical mechanics by quantum mechanics, introduce some of its central features, and illustrate them by considering a very simple system. Next, we apply those ideas to the simplest atom of all, the hydrogen atom. Finally, we extend these concepts to the atoms of all the elements of the periodic table and see the origin of the periodicity of the elements. 

This restriction rules out all discrete models exclusively based on semiempirical force fields, leaving among the discrete models the MC/QM and the MD/QM procedures, in which the second part of the acronyms indicates that the solute is described at the quantum mechanical (QM) level, as well as the full ab initio MD description, and some mixed procedures that derive the position of some solvent molecules from semiclassical simulations, replace the semiclassical description with the QM one, and repeat the calculation on these small supermolecular clusters. The final stage is to perform an average on the results obtained with these clusters. These methods can be used also to describe electronic excitation processes, but at present, their use is limited to simple cases, such as vertical excitations of organic molecules of small or moderate size. This limitation is due to the cost of computations, and there is a progressive trend toward calculations for larger systems. 

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