Page 21 - Mechanism and Theory in Organic Chemistry

take various forms, ranging from complex ab initio calculations, which begin from
first principles and have no parameters adjusted to fit experimental data, to
highly approximate methods such as the Hiickel theory, which is discussed further
in Appendix 2. The more sophisticated of these methods now can give results
of quite good accuracy for small molecules, but they require extensive use
of computing eq~ipment.~ Such methods are hardly suited to day-to-day qualita-
tive chemical thinking. Furthermore, the most generally applicable and therefore
most powerful methods are frequently simple and qualitative.
Our ambitions in looking at bonding from the point of view of the quantum
theory are therefore modest. We want to make simple qualitative arguments that
will provide a practical bonding model.

Atomic Orbitals
The quantum theory specifies the mathematical machinery required to obtain a
complete description of the hydrogen atom. There are a large number of func-
tions that are solutions to the appropriate equation; they are functions of the x,
y, and z coordinates of a coordinate system centered at the nucleu~.~ Each of these
functions describes a possible condition, or state, of the electron in the atom, and
each has associated with it an energy, which is the total energy (kinetic plus
potential) of the electron when it is in the state described by the function in
question.
The functions we are talking about are the familiar Is, 2s, 2P, 3s,. . .
atomic orbitals, which are illustrated in textbooks by diagrams like those in
Figure 1.1. Each orbital function (or wave function) is a solution to the quantum
mechanical equation for the hydrogen atom called the Schrodinger equation.
The functions are ordinarily designated by a symbol such as g,, X, $, and so on.
We shall call atomic orbitals g, or X, and designate by a subscript the orbital
meant, as for example g,,,, g,,,, and so on. Later, we may abbreviate the notation
by simply using the symbols Is, 2s, . . . , to indicate the corresponding orbital
functions. Each function has a certain numerical value at every point in space;
the value at any point can be calculated once the orbital function is known. We
shall never need to know these values, and shall therefore not give the formulas;
they can be found in other source^.^ The important things for our purposes as
fiist, that t k e m e s are positive in certain regions ocspace and neg?
tive in other regions, and second, that the value of each function approaches zero

a A number of texts cover methods for obtaining complete orbital descriptions of molecules. Ex-
amples, in approximate order of increasing coverage, are (a) A. Liberles, Introduction to Molecular-
Orbital Theory, Holt, Rinehart, and Winston, New York, 1966; (b) J. D. Roberts, Notes on Mokcular
Orbital Theory, W. A. Benjamin, Menlo Park, Calif., 1962; (c) K. B. Wiberg, Physiral Organic
Chemistry, Wiley, New York, 1964; (d) A. Streitwieser, Jr., Molecular Orbital Theory for Organic
Chemists, Wiley, New York, 1961; (e) M. J. S. Dewar, The Molecular Orbital Theory of Organic
Chemistry, McGraw-Hill, New York, 1969; (f) P. O'D. Offenhartz, Atomic and Mokcular Orbital
Theory, McGraw-Hill, New York, 1970; (g) S. P. McGlynn, L. G. Vanquickenborne, M. Kinoshita,
and D. G. Carroll, Introdudion to Applied Quantum Chemistry, Holt, Rinehart, and Winston, New York,
1972.
Actually, the origin is at the center of mass, which, because the nucleus is much more massive than
the electron, is very close to the nucleus.
See, for example, Wiberg, Physical Organic Chemistry, pp. 17, 19, and 25.