Page 24 - Mechanism and Theory in Organic Chemistry

13
Molecular Orbitals
applications, one may simply use orbitals of the shapes found for hydrogen to
describe behavior of electrons in all the atoms.
Ground and Excited States
We know that an electron ina hydrogen atom in a stationary state will be described
by one of the atomic orbital functions y,,, p,,, p21)x, and so fortl~.~ We can make
this statement in a more abbreviated form by saying that the electron is in one of
the orbitals y,,, y,,, 932px,. . ., and we shall use this more economical kind of
statement henceforth.
The orbital that has associated with it the lowest energy is y,,; if the electron
is in this orbital, it has the lowest total energy possible, and we say the atom is in its
electronic ground state. If we were to give the electron more energy, say enough to
put it in the 932px orbital, the atom would be in an electronic excited slate. In general,
for any atom or molecule, the state in which all electrons are in the lowest pos-
sible energy orbitals (remembering always that the Pauli exclusion principle
prevents more than two electrons from occupying the same orbital) is the elec-
tronic ground state. Any higher-energy state is an electronic excited state.
An Orbital Model for the Covalent Bond
Suppose that we bring together two ground-state hydrogen atoms. Initially, the
two electrons are in p,, orbitals centered on their respective nuclei. We shall call
one atom A and the other B, so that the orbitals arc p,,, and y,,,. \iVhen the
atoms are very close, say within 1 A (= lo-* cm) of each other, each electron
will feel strongly the attractive force of the other nucleus as well as of its own.
Clearly, then, the spherical p,, orbitals will no longer be appropriate to the
description of the electron motions. We need to find new orbital functions appro-
priate to the new situation, but we would prefer to do so in the simplest way
possible, since going back to first principles and calculating the correct new orbi-
tal functions is likely to prove an arduous task.
We therefore make a guess that a possible description for a new orbital
and
function will be obtained bv finding at each point in space the value of
of p,,, and adding the two numbers tog-ether. This process will give us a new
orbital function, which, since y,,, and y,,, are both positive everywhere, will
also be positive everywhere. Figure 1.3 illustrates the procedure. Mathematically,
the statement of what we have done is Equation 1.4:
(1.4)
+MO = VISA + qls~
The symbol MO means that the new function is a molecular orbital; a molecular
orbital is any orbital function that extends over more than one atom.Since_the-
technical term for a sum-of f~inctions of the type 1.4 is .alinear co_mbin.akio~,..thepco=
cedure of adding up atomic orbital functi~ns~is-calledlzaear combination ... c$atomic
.orbitals, or LCA0.-
This simple procedure turns out to fit quite naturally into the framework of
the quantum theory, which with little effort provides a method for finding the
We assume from here on that the reader is familiar with the number and shape of each type of
atomic orbital function. This information may be found in standard introductory college chemistry
texts.