2:
It can be seen that
for all . Also,
has a minimum at a
particular value of , which corresponds to the equilibrium bond length
while
exhibits no such minimum. The location of the minimum and depth of
the well will be the prediction of the equilibrium b
Parameter and error
Suppose we guessed, instead, a trial wave function of the form:
The potential and this trial wavefunction are illustrated in the figure below:
figure=lec10_fig3.eps,height=4.0in,width=4.5in
We now regard
as a variational parameter. Thu
LCAO approximation
The LCAO approximation employed above leads to only a qualitatively correct
description of chemical bonding in the H molecule ion. The bond length and
binding energy are quantitatively off from the exact values. In order to see how to
i
The harmonic oscillator
We will use the harmonic oscillator Hamiltonian in order to illustrate the procedure of
using the variational theory. The Hamiltonian we wish to consider, therefore, is
Suppose that we do not know the exact ground state solution of
Hamiltonian matrix elements
The overlap, , and Hamiltonian matrix elements will now be computed explicitly.
For the overlap, , the integral that needs to be performed is
This is integral is most easily performed in the confocal elliptic coordinate system
The H molecule ion
The H molecule ion is the simplest example of a chemical bond. It is a particularly
important example to study because it is an analytically solvable problem, therefore,
any approximation method applied to it can be assessed against the
Denominator Conditions
these conditions yield two equations for
or, since
and
:
,
These may be written as a matrix equation:
which is called a generalized eigenvalue equation. In matrix notation is becomes
where and are the Hamiltonian and overlap matrice
Coulomb integral:
Explicitly, the Coulomb integral is
which, again, is most easily evaluated using the confocal elliptic coordinates.
Transforming into this coordinate system gives
where the fact that
performed straightforwardly, to yield:
has been used.
Contours and Origins
The state
, which corresponds to the energy
called a bonding state. The state
admits a chemical bond and is, therefore,
, which corresponds to the energy
does not admit a
chemical bond and is, therefore, called an anti-bonding state.
Bonding and anti-bonding orbitals
We look, first, at the form of the orbitals that correspond to the energies
These can be obtained by solving for the variational coefficients,
given by the matrix equation:
For example, using
are obtained:
The overall con
The trial wavefunction: a linear combination of atomic
orbitals
Physically, when the two protons are far apart, and the electron is close to one or the
other proton, the ground state wavefunction of the system should resemble that of
a
orbital of hydrogen