Molecules are of course atoms that are held together by shared valence electrons. That is, most of each atom is pretty much as it would be if the atom were isolated, but one or a few electrons are located in regions (for example, between two atomic cores) where they lead to an overall attractive effect. To see roughly how this works, you might consider the case of two protons and two electrons at the corners of a square. If you calculate (classically) the total electrostatic energy of this arrangement, you will find that it is negative - the system is bound.

Molecules have several kinds of motion, with quite different energy
scales. Assume that a couple of nuclei and a couple of electrons are
confined to a volume with characteristic size (a couple of Å,
typically). Then the uncertainty principle requires that at a minimum,
, so the typical electron energies will be of order

while the purely nuclear motion of rotation for nuclei of mass will have sizes like

The nuclear rotation energy may also be estimated using our previous solution for the rigid rotator, in which the energy of the lowest level was found to be of order . Thus nuclear rotation will involve energies that are of order times smaller than electronic energies.

The nuclei may also participate in vibrations, in which they interact
directly with the electron cloud. The energies of such motions are
larger than , and may be estimated by treating the
interaction between electrons and nuclei as simple harmonic motion
with a spring constant , the same (of course) for both parties. In
this case the ratio of electronic () to nuclear vibrational
energies () will be of order

Thus we have three rather different energy scales, rotational, vibrational, and electronic, a fact which will help us to study simple molecular structure.

Consider the Schrödinger equation for a system of two identical
nuclei and two electrons, in the centre of mass system of the nuclei,
neglecting spin:

where

is the separation of the nuclei, and and are the positions of the two electrons.

We will solve this system by a separation and some approximations.
Because the electrons are *much* lighter than the nuclei, the
nuclei will hardly move at all in the time that an electron takes to
``orbit'' once, so let's suppose that we can solve the simpler problem
of electron motion in the presence of motionless nuclei separated by a
fixed , for which the electronic wave equation is

The energies of this system depend parametrically on . Because these wave functions form a complete set, we may expand the exact wave function in them:

The coefficients are the wave functions describing the nuclear motion when the electrons are in the state .

To find equations satisfied by the 's, put into
the exact Schrödinger equation and projects using each of the
's:

Using the equation satisfied by the 's, we get

The complicated part of this is the action of on the product , since both depend on :

We now make the *adiabatic* or *Born-Oppenheimer*
approximation by assuming that we can neglect
compared to
. In this case, we get the *nuclear
wave equation*

The equation for is in the form of a wave equation with the function playing the role of potential energy. Let us consider the specific case of the electrons in a state of zero orbital angular momentum. In this case, is a function only of the radial variable , and the wave equation for becomes one for a spherically symmetric potential. In this case, the Hamiltonian commutes with and . The simultaneous eigenfunctions of these operators are the spherical harmonics, and have eigenvalues and .

Since there is no privileged direction in space, the total energy of
the system cannot depend on , but it does depend on . (You may
wonder why the direction of is not a privileged direction
now. It is for the electron cloud, but in a spherically symmetric
electron potential, it is just a coordinate as far as the nuclei are
concerned. Thus can rotate in space, and the nuclei may have
non-zero angular momentum.) It will be found that there is also
another quantum number in the system, , which acts as a principal
quantum number and will be found to number vibrational states. Thus we
may write (dividing the radial function by to get a simple form
for the resulting equation)

We substitute this into the wave equation for and find that the functions satisfy

Now in a bound molecular state, the energy varies from the
sum of the energies of the two atoms separately for large , through
a local minimum, to a large positive value as the two nuclei get close
to one another. We may approximate the minimum region of by a
second order expansion:

where

In the same spirit, evaluate the term with at , and call this quantity :

where is the moment of inertia for the reduced mass and is known as the

the familiar simple harmonic oscillator equation, so we see immediately that the energies are given by

Because of the relative sizes of and , each vibrational level is split into a number of closely spaced rotational levels.

For the electron cloud, the (slowly rotating) internuclear axis picks out a direction in space which we take as the -axis. The
electronic structure of the molecule is invariant under rotation
around this axis, so commutes with . However, ,
and do not commute with . The electronic eigenstates
of may also be made to be eigenstates of , so that

where is the absolute value of the projection of the total electronic angular momentum on the internuclear axis. We label states of the electron cloud by their values with a system like that used in atomic spectroscopy: states are called , , , etc.

Diatomic molecules are symmetric to reflections through planes
including , such as the plane. Thus if operator
does this,

but since ,

which means that - if - converts an eigenstate with eigenvalue into one of . But because of commutation of with , both states have the same energy, and such states are degenerate: two states of different eigenvalues have the same energy (spin effects may break this, an effect called -doubling).

If *is* 0, the state is not degenerate, and can
only multiply it by a constant. Since , such states are
either symmetric or anti-symmetric to reflection through a plane, and
one distinguishes and states.

If the molecule has two nuclei with identical charge (a homonuclear molecule), the centre is also a symmetry point, and states have parity. States of even (g) and uneven (u) parity are denoted by , , etc. A homonuclear diatomic molecule has four non-degenerate sigma states, , , , and .

Finally, each electronic eigenstate has a total spin , with
eigenvalues of of
. The value of
is given as a left superscript (the *multiplicity*) on the
designation. Thus, since most molecular ground states (often labelled
X) are states of zero total spin, the complete label for a
ground state could be X or X
.

We may start our study of electronic wave functions with the simplest
molecule of all, H, with two protons and one electron. To
simplify writing, we will switch to Hartree's dimensionless *atomic units*, in which the units of mass, charge, angular momentum,
and length are chosen to be , , , and
, with
the result (you may check this) that the units for time, velocity, and
energy become
,
, and
,
where is the ground state binding energy of H. In these units,
the electronic wave equation becomes

where and are the distances to the electron from nuclei A and B, is the distance to the electron from the centre of mass, and is the internuclear separation. Note that the energy operator could be with respect to , , or .

We will solve this equation using the simple approximation of a linear
combination of atomic orbitals (LCAO). If the nuclei are far apart,
the electron will be attached to one (say A), with wave function

To give the wave function the correct symmetry about the midpoint of , construct the functions

and

These wave functions will only be accurate for large , but may be used as trial wave functions in a variational solution, in which the true ground state energy should be less than

The denominator of this expression is

This integral may be taken over , , or , since all the integrals extend over all space. Because of the normalization of , we find

where

using the result for two-centre integrals proved by B & J (appendix 9).

The numerator is

Again we use the two-centre integrals, as well as the fact that the 's are solutions of

where the kinetic energy operator may be with respect to , , or , since all measure the position of the electron. The final result is

The variation of , using the symmetric wave function, shows a minimum below of 1.77 eV at Å, and thus is a bonding molecular orbital. The minimum energy and equilibrium separation are reasonable (though not very accurate) estimates of the exact values of 2.79 eV at 1.06 Å.The curve of has no minimum; it is an anti-bonding state. The symmetric bonding state is held together by an excess of charge between the two protons, while in the anti-bonding state the electron spends most of its time away from the centre of the molecule.

This problem may be solved exactly (although numerically, not analytically) through the use of confocal elliptic coordinates, which are also used in the evaluation of the two-centre integrals.

With two electrons, we must consider the effects of the exclusion principle, which states that eigenstates must be antisymmetric to exchange of two identical fermions. The eigenstates of the two-electron cloud are products of spatial wave functions and spin functions. How does the spin affect the situation?

The total spin operator is

where operates only on electron 1, and only on electron 2. Individual electrons have spin eigenfunctions , , etc, as previously discussed. We need to find spin eigenstates of both and of . Try the simple combinations

It is easily shown by operating on these functions that each is an eigenstate of with eigenvalues respectively 1, 0, 0, and -1. However, when we operate with , using the relations (proved in B & J, appendix 4, with the aid of raising and lowering operators)

we find that neither nor is an eigenstate of . But it is easy to make linear combinations of these two functions that

and

It is easily shown that has eigenvalues of and of and , while has and . We now have four normalized and orthogonal spin eigenstates that we will denote by for clarity. is antisymmetric to particle exchange, while the other three states are symmetric. We call the one state a spin singlet, while the other three states form a spin triplet.

In solving (approximately) the problem of the electron cloud structure for the two-electron system H, we have the obvious choice of using as a basis set the molecular orbitals found for H, or the atomic orbitals of the individual H atoms. Either provides a useful basis for solution. We will start with the molecular orbitals.

Recall that we have two molecular orbitals and
(which are even and odd under reflection through the
midpoint of ). The product
is symmetric under exchange of electrons 1 and 2, as are the
combinations
and
. On
the other hand, the function
is antisymmetric under
exchange of the electrons. Thus we may form (only) four orthogonal
eigenfunctions that are antisymmetric under exchange:

where in can take the values 0 and . It is easy to see that and are states, while is a state and the three states are states.

To get a bonding state, we need the two electrons to prefer to be between the two protons. If they both occupy the bonding orbitals of the H system, and have opposite spins (so that the exclusion principle will not require them to occupy different spatial states), we guess that we should get maximum bonding. Thus we guess that may a reasonable approximation to the ground state of H.

We now need to solve the equation

where is given by

Using the fact that the individual molecular orbitals are solutions of

one finds that the Rayleigh-Ritz variational functional is given by

If one uses the approximate LCAO molecular orbitals found above, the binding energy is calculated to be 2.68 eV (compared to an accurate value of 4.75 eV), with a computed equilibrium of 0.8 Å, compared to an accurate value of 0.74 Å.

The trial wavefunction that we have used may be written as the sum of
two terms,

where

and

represents the situation in which one electron is associated with

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