 Let's see why Dirac was justified in saying that while we have a mathematical basis for describing the whole of chemistry, in practice the equations are too complex to be solved. Consider the hydrogen molecule, H2. With one electron this is just the hydrogen molecule ion that we considered previously. The Schrodinger equation has three terms for the first electron, representing the kinetic energy and its attraction to each of the protons. The second electron also needs three corresponding terms. Finally, there's a term for the repulsion between the protons and a term for the repulsion between the electrons. Each potential energy term in the Schrodinger equation corresponds to a line in this figure. Now consider the oxygen molecule, O2. With one electron this is similar to the hydrogen molecule ion, except the nuclei have eight protons each. The electron's nuclear attraction terms therefore contain a factor of eight. As we add electrons, each has three corresponding terms. For all sixteen electrons we need forty-eight terms in the Schrodinger equation. We have a single nuclear repulsion term containing a factor of eight times eight equals sixty-four due to the eight protons in each nucleus. Finally, we need a large number of terms that represent repulsion between electrons. There are terms for electron one's repulsion of electron two, three, four, through sixteen. Then terms for electron two's repulsion of electron three, four, five, through sixteen. And so on until repulsion between every possible pair of electrons is represented. For n electrons, there are one-half n times n minus one such terms. For the oxygen molecule with sixteen electrons, this means 120 electron repulsion terms. Even for simple molecules, the number of terms in the Schrodinger equation appears overwhelming. It certainly did to those working in the field more than eighty years ago. But today we have access to massive levels of computing power. An equation with hundreds or thousands, even millions of terms, should not be an insurmountable challenge. Indeed the classical problem of sixteen mutually repulsing electrons moving around two nuclei without radiation is easy to solve numerically. The solution consists of the description of the x, y and z coordinates of each electron through time. So although we can't solve the so-called classical n-body problem with pencil and paper, it's straightforward to generate and visualize numerical solutions, even with large numbers of particles. Unfortunately the same statement cannot be made for the quantum mechanical version of the problem. In quantum mechanics, instead of an electron trajectories, we have to find an an electron wave function. For n equals one, as with the hydrogen atom or hydrogen molecule ion, this is a probability amplitude spread throughout all three-dimensional space. For two electrons, the wave function will depend on the coordinates of both electrons. It's tempting to think that this just means that we need a second wave function for the second electron. Given the probability that electron one is in a first location and electron two is at a second location, it's simply described by the product of a probability factor for electron one times a probability factor for electron two. However, this is true only if the two electrons move independently without interacting. Since electrons do interact, the wave function cannot have this simple factored form. Instead, if we take electron one to be at a particular point in space, there will be some wave function describing the probability amplitude distribution for electron two. If we take electron one to be at some other point in space, the wave function for electron two might be different, in fact it almost certainly will be. In general, for every location of electron one, there is a different probability amplitude distribution, or wave function, for electron two. This is much more complicated than simply finding a wave function for each electron. If we need m samples or parameters to specify a one electron wave function, then in general, we need m to the n samples or parameters to specify an n electron wave function. The complexity of the wave function grows exponentially with the number of electrons. If we're looking for exact solutions, this is a showstopper, even for numerical solutions with massive computing power. An n electron wave function has to provide a probability amplitude for every possible location of electron one, and for each of those locations, for every possible location of electron two, and for each of those combinations, for every possible location of electron three, and so on. An n electron wave function is not a function of the three coordinates of normal space, but a function of the three n coordinates of the electrons. The challenge this presents was summed up by Douglas Hartree as follows. The full specification of a single wave function of neutral iron, which has 26 electrons, is a function of 78 variables. It would be rather crude to restrict to 10 the number of values of each variable at which to tabulate this function, but even so, full tabulation of it would require 10 to the 78th power entries. How big a number is 10 to the 78th power? About a billion times the number of atoms in the Milky Way Galaxy. Vladimir Falk put it this way. Since the wave function sought depends on the great number of variables, namely, there are as many of them as there are degrees of freedom in the n electron system, the exact solution of this problem encounters insuperable difficulties, and consequently, one needs to resort to approximate methods. One of the most important approximation methods in quantum chemistry was presented in 1928 in a two-part paper titled, The Wave Mechanics of an Atom with a Non-Coulomb Central Field, Part 1, Theory and Methods, and Part 2, Some Results and Discussion. Here Douglas Hartree outlined the central field approximation. Applied to the Schrodinger equation, this produced the Hartree equation, which was solved using the method of the self-consistent field. Let's see how, even in a time when numerical calculations had to be performed by hand, Hartree was able to obtain impressively accurate approximations to the wave function of many electron atoms. The central field approximation greatly simplifies the description of electron interactions. In a neutral atom with Z protons, each electron is attracted to the nucleus. This force depends only on the electron's location, but the electron is also repulsed by each of the other electrons. These electron repulsions are complicated because they depend on the locations of both electrons. Hartree's idea was to treat electron repulsion in an average sense. We treat nuclear attraction normally, but we treat the other electrons as if they were spread throughout space as a cloud of electric charge. The density of this charge cloud at a point is proportional to the probability of the electron being at that point, as given by the wave function. If this density is spherically symmetric, then the force it exerts on an electron points radially away from the nucleus, the center of the atom. This is a so-called central field. If the density is not spherically symmetric, then we average the force over all angles to obtain a central field. We do this for all the other electrons to get a total force on the electron of interest and the corresponding total potential energy. The average force exerted by the jth electron at a distance r from the nucleus is an equivalent charge q sub j over r squared. This equivalent charge is simply the probability that electron j is inside a sphere of radius r. In Hartree's theory, this is represented as the sum of the square of a radial density function p sub j over radii from 0 to r. The energy and angular momentum of an electron moving in a central field is conserved. In the central field approximation, therefore, we can treat each electron as being in an orbital with definite energy and angular momentum. The Schrodinger equation for n electrons has a kinetic energy term for each electron, a nuclear attraction term for each electron, and repulsion terms for every pair of electrons, one-half n times n minus one in all. Hartree converted this to an equivalent single electron equation by, roughly speaking, the following steps. Suppose we want an equation for electron one. We assume the distribution of the n minus one other electrons remains frozen. This allows us to subtract the constant kinetic energies of those electrons from both sides of the equation, leading the kinetic energy term for only electron one. Likewise, we assume all but one of the nuclear attraction energies are constant, and subtract these from both sides. Finally, all electron repulsion terms that don't involve electron one are taken to be constant, and subtracted from both sides. This leaves the nuclear attraction term for electron one, and the repulsion term for electron one in each of the n minus one other electrons. The central field approximation converts these latter terms into equivalent potentials, which are combined with a nuclear attraction term to get a total potential for electron one. The n result is a single electron Schrodinger equation, with the energy being that required to bring this electron from rest at infinity into the orbital it occupies in the atom. Consider the assumption that the orbitals of the other electrons remain frozen.