MO THEORY
I.
Shapes and Signs on Atomic Orbitals:
Atomic orbitals are visual pictures of mathematical formulas, which, (when squared =
Ψ
2
), tell us
the probability of finding an electron at each possible position in space. These functions, like any
mathematical function, have both positive and negative signs. For example, an stype orbital is
represented by a mathematical formula that is everywhere positive or everywhere negative. It
makes no difference which, since the actual probability of finding the electron at a particular set of
coordinates is given by the square of the orbital – i.e., the probability is always a positive value.
The spherical shape of an stype orbital tells us that the probability of finding the electron at a
point in space depends only on how far the electron is from the nucleus, and not on the direction
from the nucleus.
However, ptype and dtype orbitals have nonspherical shapes. They have lobes which point at
various different directions in space. Thus there will be a different probability for finding the
electron in one direction, than there will be in finding it in another direction.
The stype and ptype orbitals, with their accompanying signs, are shown below. Note that if we
reverse ALL the signs in one of these orbitals, that does not change the probability distribution,
since the probability will be given by squaring each picture. [And thus all negative lobes will
become positive, once we square the wavefunction – i.e., the orbital.]
+
+
•
–
•
+
+
•
–
–
s
p
x
type
p
y
type
p
z
type
Note: in the pictures above, the three ptype orbitals are arbitrarily assigned as
p
x
,
p
y
, and
p
z
. Each
will always be aligned along whichever coordinate axis that is given in its name – i.e., the
p
x
type
orbital is always aligned along the
x
axis direction; the
p
y
type orbital is always aligned along the
y
axis direction; and the
p
z
type orbital is always aligned along the
z
axis direction. Also, all three
porbitals have identical forms – only their directions are different. [The
p
z
orbital has its lobes
coming in and out of the paper.]
II.
How to Combine Atomic Orbitals to Create Molecular Orbitals:
Now, consider what happens when we bring two atoms together within a bond length of each
other
. Each atom brings with it all its atomic orbitals. The best overlap between them will occur
when the lobes are aligned in the same directions, so we will bring together
s
type with
s
type;
p
x

type with
p
x
type, etc. Also, we will be sure to apply the
Conservation of Orbitals Rule:
that is,
if we mix together 2 atomic orbitals, we must get 2 new molecular orbitals. [In general, if we mix
together
n
atomic orbitals, we must get
n
new molecular orbitals].
For the
s
type orbitals, there
are 4 different ways those orbitals can approach each other with their various signs: both can have
+ signs; both can have – signs; the leftone can be + and the right –; or the left one can be – and
the right +:
– 1 –