12
W
E HAVE SEEN HOW QUANTUM MECHANICS provides tools for
understanding some simple systems, up to and including the hydro-
gen atom itself. An understanding of the H atom is a crucial point because it
is real,not a model system. Quantum mechanics showed that it can describe
the hydrogen atom like Bohr’s theory did. It also describes other model sys-
tems that have applications in the real world. (Recall that all of the model
systems—particle-in-a-box, 2-D and 3-D rigid rotors, harmonic oscillators—
could be appliedto real systems even if the real systems themselves weren’t
exactly ideal.) As such, quantum mechanics is more applicable than Bohr’s
theory and can be considered “better.” We will conclude our development of
quantum mechanics by seeing how it applies to more complicated systems:
other atoms and even molecules. What we will find is that explicit, analytic
solutions to these systems are not possible, but quantum mechanics does
supply the tools for understanding these systems nonetheless.12.1 Synopsis
In this chapter, we will consider one more property of the electron, which is
called spin. Spin has dramatic consequences for the structure of matter, con-
sequences that could not have been considered by the standards of classical
mechanics. We will see that an exact, analytic solution for an atom as simple
as helium is not possible, and so the Schrödinger equation cannot be solved
analytically for larger atoms or molecules. But there are two tools for studying
larger systems to any degree of accuracy: perturbation theory and variational
theory. Each tool has its advantages, and both of them are used today to study
atoms and molecules and their reactions.
Finally, we will consider in a simple way how quantum mechanics con-
siders a molecular system. Molecules can get very complicated. However, we
canapply quantum mechanics to molecules. We will finish this chapter with
an introduction to molecular orbitals and how they are defined for a very
simple molecule, H 2 . Simple as this system is, it paves the way for other
molecules.370
12.1 Synopsis
12.2 Spin
12.3 The Helium Atom
12.4 Spin Orbitals and the
Pauli Principle
12.5 Other Atoms and the
Aufbau Principle
12.6 Perturbation Theory
12.7 Variation Theory
12.8 Linear Variation Theory
12.9 Comparison of Variation and
Perturbation Theories
12.10 Simple Molecules and the
Born-Oppenheimer
Approximation
12.11 Introduction to LCAO-MO
Theory
12.12 Properties of Molecular
Orbitals
12.13 Molecular Orbitals of Other
Diatomic Molecules
12.14 Summary