When Hartree's method is extended to a Slater-determinant wave function the result is the Hartree-Fock
equations.
These were derived by Vladimir Fock and presented in 1929.
As a shorthand for a Slater determinant we'll write a Hartree product inside a Dirac ket
symbol.
Fock started with the assumption that the wave function of a many-electron system has
the form of a Slater determinant of one-electron orbitals.
He then sought those orbitals which minimize the expected value of the ground-state energy.
According to the variational method we discussed previously, this will be the optimal wave
function of this form.
Let's sketch out the resulting Hartree-Fock equation for one of the electrons in a Hydrogen
molecule which has two electrons and two protons.
We'll arbitrarily label this electron one.
For reference recall that the Schr�dinger equation for a single electron has the form
of a kinetic energy term plus a potential energy term equals a total energy term.
The kinetic energy of electron one is the normal kinetic energy operator applied to
the first orbital.
The attractive forces of the two nuclei each result in a negative potential energy term.
In an exact treatment the repulsion between the two electrons would be represented by
a positive potential energy term that depends on the positions of both electrons.
This is the primary source of overwhelming complexity in trying to exactly solve the
Schr�dinger equation for a multi-electron molecule.
Hartree's intuitive approximation was to treat electron repulsion in an average sense.
This same idea arises rigorously in the Hartree-Fock equations.
We treat electron 2 as a cloud of electric charge with a charge density given by the
magnitude squared of the second orbital wave function.
Summing, or integrating this over all space gives us the bracketed term shown here which
serves as an effective potential energy term for electron repulsion.
Now, the terms we've written so far are those we'd expect from Hartree's method.
But Fock found that the exclusion principle as enforced by using a Slater determinant
wave function requires an additional term.
This is called the exchange term because it's identical to the electron repulsion term with
an exchange of orbitals.
This is a strange expression and there is no classical analog in terms of a cloud of
electric charge.
This maybe isn't surprising given that it arises from the exclusion principle which
is a purely quantum mechanical effect with no classical analog.
With these equations in place we can now outline the Hartree-Fock method for obtaining an approximate
wave function for an N electron molecule.
We start with a guess for the N orbitals.
We then repeatedly solve for each of the N orbitals in terms of the others until we converge
to a self-consistent field.
For the kth orbital we solve the corresponding Hartree-Fock equation.
This has a kinetic energy term and a potential energy term for the attraction of electron
k to each of the M nuclei.
Here Z j is the number of protons in the M nucleus.
There are terms for the repulsion between electron k and each of the other N-1 electrons
and the corresponding exchange terms.
These correspond to the sum of kinetic and potential energy terms in the Schr�dinger
equation and this is set equal to a total energy term E k times psi k.
E k is the energy that would be required to remove an electron from the kth orbital assuming
the other orbitals do not change.
These equations, especially due to the exchange terms, are very challenging.
No practical procedures were available at the time nor are any available today to directly
solve these for molecules.
So, although there were some successful applications to atoms, the Hartree-Fock method remained
of mostly theoretical interest.
And, with the outbreak of world war 2 almost all research efforts were directed elsewhere.
After world war 2 not only were peacetime research programs revived, but digital computer
technology was becoming available.
This meant that it was now possible to solve types and sizes of problems that could not
be attacked with pencil and paper.
Around 1950 Clemens Roothaan reformulated the Hartree-Fock equations as the Roothaan
equations.
The idea was to represent electron orbitals in terms of so-called basis functions.
This converted the Hartree-Fock equations into algebraic equations that were ideally
suited for computer solution.
For example, we can represent an orbital as a linear combination of atomic orbitals on
different atomic centers.
In fact we already applied a simple version of this to the hydrogen molecule ion.
Suppose we are solving for the orbitals of the water molecule.
Basis function u 1 might be an s orbital centered on the oxygen nucleus.
Basis function u 2 could be an s orbital centered on one of the hydrogen nuclei and u 3 the
same but centered on the other hydrogen nucleus.
We could have more basis functions representing p-type orbitals centered on the various nuclei
and so on.
We then represent an unknown orbital as a linear combination of these known basis functions.
Our problem is solve for the coefficients c 1 , c 2 and so on by solving the Roothaan
equations.
A great advantage is that the sums or integrals over all space that appear in the Hartree-Fock
equations are now in terms of the known basis functions instead of the unknown orbitals
and can, in principle, be done once and for all rather than having to be repeated at each
step of the self-consistent field method.
Unfortunately, the logical choice of basis functions, hydrogen atom type orbitals generally
lead to intractable calculations that keep us from even setting up the Roothaan equations.
Fortunately around the same time it was pointed out that so-called Gaussian basis functions
produce tractable equations.
This insight was due to Francis Boys and was the last key piece of the puzzle.
So-called Slater type orbitals such as we find in the hydrogen atom are characterized
by an exponential decrease with distance from the nucleus.
That distance is given by the square root in this expression.
These have the form e to the minus alpha r where alpha is some constant and r is the
distance of the electron from the nucleus.
It's the square root that creates the mathematical difficulties.
Boys suggested the use of Gaussian type orbitals, which lack the square root and have the form
of e to the minus alpha r squared.
He showed that the mathematical calculations needed to set up the Roothaan equations can
then be readily done, leading to a practical method for implementation on a digital computer.
Although Gaussian type orbitals have convenient mathematical properties they aren't a good
approximation to the types of orbitals that actually form in atoms.
Here we plot as blue dots the Slater type orbital e to the minus r which is the exact
solution of the hydrogen atom ground state.
The best Gaussian type orbital approximation to this is solid red curve.
The agreement is not good.
These 3D scans illustrate the Slater type orbital on top and the Gaussian type orbital
on the bottom.
The Slater type orbital has a much more peaked density near the nucleus while the Gaussian
type orbital is more diffuse.
However a linear combination of 2 Gaussian type orbitals gives a better approximation
and a combination of 3 Gaussians does better still.
So given enough Gaussian type orbitals it should be possible to accurately represent
the types of orbitals that actually arise in atoms and molecules.
To go beyond the spherical symmetry of s orbitals we can combine Gaussians, offset from one
another and with coefficients of different signs.
The result is a 2-lobed p-type orbital.
A combination of 4 Gaussians can be used to create a 4-lobed d-type orbital.
More complicated spatial variations become possible as we add more primitive Gaussians.
For example, here's an orbital with a fairly complicated three-dimensional structure that
is built from a linear combination of simple, primitive Gaussian type orbitals.
With the Hartree-Fock equations, as reformulated by Roothaan, and using Boys' idea of Gaussian
basis functions, it became possible to perform so-called ab initio quantum chemistry calculations.
Or, calculations from the beginning or from first principles.
Given enough basis functions (and enough computer power) a computer can solve any chemistry
problem we throw at it.
Let's do an ab initio calculation for the only molecule we've solve for so far, the
hydrogen molecule ion.
This chemical bond is due to 2 protons being surrounded a single electron whose exact orbital
is shown here.
A calculation using a single Gaussian gives a poor representation of this.
But as we increase the number of Gaussians to 2
4 6
8 10
and 12 our result more and more closely approaches the exact wave function.
So, with a finite number of simple Gaussians we can construct a highly accurate approximation
to the true molecular wave function.
Now we'll use the open-source program ergo SCF to do Hartree-Fock calculations on a few
different molecules.
We'll also use the open source package gabedit to visualize the results.
Quantum chemistry not only has to explain why certain molecules form but also why others
do not.
Imagine we have four protons, four electrons and four neutrons and we want to build a molecule.
Two protons and two neutrons form a helium nucleus.
Adding two electrons we have a helium atom.
W e can make 2 of these with our 12 particles.
We could also take a proton and an electron to form a hydrogen atom and from the remaining
particles form a lithium atom, in this case lithium 7 which is the most common isotope.
He we plot the interactions between lithium and hydrogen atoms as predicted by the Hartree-Fock
method.
The contours correspond to different constant values of electron density.
When far apart we have isolated atoms.
But as they come closer molecular orbitals are able to form.
Plotting the total energy versus nuclear separation we find a minimum near 3 Bohr radii.
This predicts the existence of the Lithium hydride molecule and gives us a good approximation
to its bond length and dissociation energy.
Repeating the process for 2 helium atoms we obtain an energy curve without a minimum.
Therefore Hartree-Fock theory predicts that a chemical bond between two Helium atoms will
not form which is true under normal conditions.
With 3 or more atoms the geometry gets more complicated.
Consider the water molecule H 2 O . At the upper left we have a top down view
of the molecule.
In the lower middle a view at 45 degrees and in the upper right an end-on view at 90 degrees.
Water has 10 electrons.
Assuming two electrons can fit into an orbital, one spin up and the other spin down, water
should have 5 occupied orbitals.
The lowest energy orbital is essentially a spherical cloud around the oxygen nucleus.
These so-called core electrons don't contribute significantly to the chemical bonds.
The next highest energy orbital is a slightly triangular shaped cloud that engulfs all three
nuclei.
The next orbital is a 2 lobed structure.
The wave function is positive in the red region and negative in the blue with one hydrogen
nucleus in each lobe.
The next orbital is also 2 lobed but with a different orientation.
Both hydrogen nuclei are in the red lobe while the oxygen nucleus is in the blue lobe.
The highest energy occupied orbital is a p-type orbital centered on the oxygen nucleus that
does not envelop the hydrogen nuclei and contributes little to the chemical bonds.
Having solved for the 5 occupied orbitals we can now calculate the total electron density
due to all 10 electrons.
Here we show a red and white stick model of the water molecule on which we impose a surface
of constant electron density.
The highest densities are near the oxygen nucleus.
For decreasing densities the surface grows and eventually envelops the hydrogen nuclei.
By repeating this process for various positions of the hydrogen nuclei until we find an energy
minimum, we can determine the molecular geometry.
Here again is the highest-energy occupied molecular orbital.
Just as for the hydrogen atom we can calculate orbitals corresponding to excited states.
The next highest energy state, the lowest-energy unoccupied molecular orbital, is shown here.
Now we might adopt a seeing is believing attitude and remain skeptical about the extent to which
the results of these mathematical calculations represent something physically real.
The scanning tunneling microscope, which won it's designers Nobel prizes in 1986, has enabled
imaging at atomic-scale resolution.
Recently a team of researchers has used this technique to image a single water molecule.
By adjusting the tunneling voltage they were able to couple the probe to both the highest-energy
occupied and lowest-energy unoccupied molecular orbitals and obtain images of these.
The agreement between experiment, shown at center, and theory, shown at right, is stunning.
We are able to actually see single molecular orbitals.
The size of molecules that can be analyzed using the Hartree-Fock method is limited only
by our computing power.
Here's a diagram of pentacene, a flat molecule with 36 nuclei. 22 carbon nuclei arranged
in 5 linked rings surrounded by 14 hydrogen nuclei.
An animation of total electron density shows how the electrons effectively glue the components
of this molecule into place.
Using atomic-force microscropy researchers at IBM Zurich have imaged a single pentacene
molecule.
In this image the carbon rings and even the hydrogen atom bonds are clearly visible.
Pentacene's molecular orbitals have also been imaged.
Here is its highest occupied molecular orbital as predicted by Hartree-Fock theory.
And here is the Hartree-Fock lowest energy unoccupied molecular orbital.
Scanning tunneling microscope images verify the multi-lobed structure of these orbitals.
We can literally see that although the Hartree-Fock method provides only an approximate solution
to a molecular wave function, it can be a very good, and therefore very useful, approximation
indeed.
And, other methods continue being developed to obtain even better approximations.
The size of molecules that be treated by this theory are primarily limited by the amount
of available computing power.
Here is a recent paper in which a supercomputer was used to perform Hartree-Fock calculations
on components of protein molecules with almost 3,000 atoms.
So it even possible to apply quantum mechanics to aspects of the chemistry of living organisms.
Returning to Dirac's 1929 quote, in the decades since then there has been impressive progress
toward overcoming the difficulties he noted in applying the laws of quantum chemistry.
Today quantum chemistry stands as one of the tremendous achievements of quantum theory,
of immense practical as well as theoretical importance.