In this chapter we first examine how the concepts of quantum mechanics
can be extended to molecules in order to develop a quantitative description
for chemical bonds. Schrödinger’s equation is written for the hydrogen
molecule, leading to expressions for valence bonds and the Hückel model.
A central feature of proteins and other biological systems is building from
repetitive units, such as amino acids or nucleic acids. The covalent geo-
metry and other properties of the units are derived from the principles
of quantum mechanics applied to molecules. However, the overall mole-
cule is too large to be treated from a quantum-mechanical standpoint as
many different interactions influence protein structures, including steric
interactions, hydrogen bonds, electrostatic interactions, and hydrophobic
effects. These interactions have pronounced influences on bond angles
as the barriers to rotations are relatively low. For large molecules with
many possible bond angles, there are many possible protein conformations.
Although detailed quantum-mechanical descriptions of proteins are not
feasible, the effects of the many interactions can be modeled as potentials
that can then be used to describe the possible conformations of proteins.
Whereas the thermodynamically favored conformations are closely related,
the ability of theoretical calculations to predict only one stable conforma-
tion that matches the biologically observed structure is discussed.
Schrödinger’s equation for a hydrogen molecule
The simplest molecule, the hydrogen molecule, is used to introduce the con-
cept of applying Schrödinger’s equation to a molecule. For the hydrogen
molecule, we must insert into Schrödinger’s equation all of the interactions
involving the nuclei and electrons. For H 2 there are two electrons, which
are labeled 1 and 2, and two nuclei, labeled A and B (Figure 13.1). Only
the electrons are considered to be moving, and so only the two electrons
contribute to the kinetic energy: