 When Hartree's method is extended to a slater-determinate wave function, the result is the Hartree-Fauch equations. These were derived by Vladimir Fauch and presented in 1929. As a shorthand for a slater-determinate, we'll write a Hartree product inside a Dirac ket symbol. Fauch started with the assumption that the wave function of a many electron system has the form of a slater-determinate of one electron orbitals. He then sought those orbitals which minimized the expected value of the ground state energy. According to the variational method we discussed previously, this will then be the optimal wave function of this form. Let's sketch out the resulting Hartree-Fauch 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 Schrodinger 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 resulted 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 Schrodinger equation for a multi-electron molecule. Hartree's intuitive approximation was to treat electron repulsion in an average sense. This same idea rises rigorously in the Hartree-Fauch equations. We treat electron two 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 overall 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 Fauch 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's 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-Fauch method for obtaining an approximate wave function for an an 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-Fauch 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 sub j is the number of protons in the jth nucleus. There are other 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 Schrodinger equation. This is said to be equal to a total energy term ek times psi k. Ek is the energy that would be required to remove an electron from the kth orbital assuming the other orbitals do not change, that is, they remain frozen. 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-Fauch method remained of mostly theoretical interest. And with the outbreak of World War II, almost all research efforts were directed elsewhere. After World War II, 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-Ruten reformulated the Hartree-Fauch equations as the Ruten equations. The idea was to represent electron orbitals in terms of so-called basis functions. This converted the Hartree-Fauch 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've already applied a simple version of this to the hydrogen molecule ion. Suppose we're solving for the orbital of the water molecule. Basis function U1 might be an S orbital centered on the oxygen nucleus. Basis function U2 could be an S orbital centered on one of the hydrogen nuclei and U3 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 then is to solve for the coefficients C1, C2, and so on by solving the Ruten equations. A great advantage is that the sums or integrals over all space that appear in the Hartree-Fauch equations are now in terms of the known basis functions instead of the unknown orbitals. So they 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 Ruten 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 Boyce 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 of 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. Boyce 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 Ruten equations can then readily be 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 the 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 peak density near the nucleus, while the Gaussian-type orbital is more diffuse. However, a linear combination of two Gaussian-type orbitals gives a better approximation, and a combination of three 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 two-lobed P-type orbital. A combination of four Gaussians can be used to create a four-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's built from a linear combination of simple primitive Gaussian-type orbitals. With the Hartree-Fock equations, as reformulated by Routen, and using Boyd's 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 computing 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 solved for so far, the hydrogen molecule ion. This chemical bond is due to two protons being surrounded by 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 two, four, six, eight, ten, and twelve, 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 GAB added 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. We can make two of these with our twelve 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. Here 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 three 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 two 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 three or more atoms, the geometry gets more complicated. Consider the water molecule H2O. 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 ten electrons. Assuming two electrons can fit into an orbital, one spin up and the other spin down, water should have five 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 two-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 two-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 five occupied orbitals, we can now calculate the total electron density due to all ten 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 its 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're 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 pentascene, a flat molecule with 36 nuclei, 22 carbon nuclei arranged in five linked rings, surrounded by 14 hydrogen nuclei. An animation of total electron density, as predicted by Hartree-Fock theory, shows how the electrons effectively glue the components of this molecule into place. Using atomic force microscopy, researchers at IBM Zurich have imaged a single pentascene molecule. In this image, the carbon rings and even the hydrogen atom bonds are clearly visible. Pentascene'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. Here's a recent paper in which a supercomputer was used to perform Hartree-Fock calculations on components of protein molecules having almost 3,000 atoms. So with this theory, it's even possible to apply quantum mechanics to aspects of the chemistry of living organisms. Returning to Dirac's 1929 quote, In the decade since then, there has been impressive progress towards overcoming the difficulties he noted in implying 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.