The hydrogen molecule ion H is the *only* molecule
for which we can solve the electronic Schrödinger equation
exactly. Note that it has just one electron! In fact, there
are no multi-electron molecules we can solve exactly.
Thus from H on up to more complicated molecules, we only
have approximate solutions for the allowed electronic energies
and wave functions. Before we discuss these, however, let us
examine the exact solutions for H starting with
a brief outline of how the exact solution is carried out.

The figure below shows the geometry of the H molecule ion and
the coordinate system we will use.

The nuclear-nuclear term is a constant, and we can define the potential energy relative to this quantity.

The energy is not a simple function of energies for the , , and directions,
so we try another coordinate system to see if we can simplify the problem.
In fact, this problem has a natural cylindrical symmetry (analogous to the
spherical symmetry of the hydrogen atom) about the -axis. Thus, we
try cylindrical coordinates. In cylindrical coordinates the distance
of the electron from the -axis is denoted , the angle is
the azimuthal angle, as in spherical coordinates, and the last coordinate
is just the Cartesian coordinate. Thus,

Using right triangles, the distance and can be shown to be

The classical energy becomes

First, we note that the potential energy does not depend on , and the classical energy can be written as a sum of a and dependent term and an angular term. Moreover, angular momentum is conserved as it is in the hydrogen atom. However, in this case, only

and is given by

which satisfies the required boundary condition . Note that the angular part of this problem is exactly like the particle on a ring problem from problem set # 4.

Unfortunately, what is left in and is still not that
simple. But if we make one more change of coordinates, the
problem simplifies. We introduce two new coordinates and
defined by

Note that when , the electron is in the plane. Thus, is analgous to in that it varies most as the electron moves along the axis. The presence or absence of a node in the plane will be an important indicator of wave functions that lead to a chemical bond in the molecule or not. The coordinate , on the other hand,is minimum when the electron is on the axis and grows as the distance of the electron from the axis increases. Thus, is analogous to . The advantage of these coordinates is that the wave function turns out to be a simple product

which greatly simplfies the problem and allows the exact solution.

The mathematical structure of the exact solutions is complex and
nontransparent, so we will only look at these graphically, where
we can gain considerably insight. First, we note that the
quantum number largely determines how the solutions appear.
First, let us introduce the nomenclature for designating
the orbitals (solutions of the Schrödinger equation, wave functions)
of the system

- 1.
- If , the orbitals are called orbitals, analogous to the orbitals in hydrogen.
- 2.
- If , the orbitals are called orbitals, analogous to the orbitals in hydrogen.
- 3.
- If , the orbitals are called orbitals, analogous to the orbitals in hydrogen.
- 4.
- If , the orbitals are called orbitals, analogous to the orbitals in hydrogen.

- I.
- A greek letter, , , , , ....
depending on the quantum number .
- II.
- A subscript qualifier g or u depending on how an orbital
behaves with respect to a spatial reflection or
*parity*operation . If

then is an even function of , so we use the ``g'' designator, where g stands for the German word*gerade*, meaning ``even''. If

then is an odd function of , and we use the ``u'' designator, where u stands for the German word*ungerade*, meaning ``odd''. - III.
- An integer in front of the Greek letter to designate the
energy level. This is analogous to the integer we use in
atomic orbitals (1s, 2s, 2p,...).
- IV.
- An asterisk or no asterisk depending on the presence or absence
of nodes between the nuclei. If there is significant amplitude between the
nuclei, then the orbital favors a chemical bond, and the orbital
is called a
*bonding orbital*. If there is a node between the nuclei, the orbital does not favor bonding, and the orbital is called an*antibonding orbital*.

These orbitals are depicted in the figure below: The and orbitals are the lowest in energy, however, note that the contains one more node than the orbital, hence it has a higher energy. Similarly for the and orbitals. The former has two nodes while the latter has three and, therefore, it is of higher energy. The next set of orbitals are displayed in the right panel. In this set of orbitals, the is the lowest energy with a single node. The number of nodes increases as we go up in energy in this subset of orbitals. In addition, in all of the orbitals, the nodal structure reveals the bonding/antibonding character of each orbital. When there is significant amplitude between the nuclei, the orbital is a bonding orbital, otherwise, when there is a node there, it is an antibonding orbital.

What do bonding and antibonding orbitals mean in terms of the
corresponding energy levels. Consider just the first two
energy levels
and
corresponding
to the and orbitals. The ground-state
orbital is bonding and the first excited state
is antibonding. In the figure below, we plot the
energy levels
and
as
functions of the internuclear separation :