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The Schrödinger equation can only be solved in very few cases. However, when one cannot solve the Schrödinger equation, not all is lost. One can still find approximate solutions or approximations. These approximations can be very good, and then there is no need for an exact solution. An important point when one looks for approximations is what is meant by a good and a bad approximation. To determine this one needs a criterion for the quality of an approximation. Such a criterion has some arbitrariness. If wave function Y1 gives a better approximation for the energy of a molecule than wave function Y2, but Y2 gives a better value for the dipole moment than Y1, which wave function is better depends on the property one is interested in. A criterion that uses the energy will say Y1 is better, but a criterion that uses the dipole moment will say Y2 is better. A criterion should not, however, say that F2 is better than F1 when F1 is an exact solution of the Schrödinger equation.
The variation principle is the most widely used criterion to get approximate solutions of the Schrödinger equation. The variation principle uses the expectation value of the Hamiltonian. This expectation value is defined in the bracket notation as
Here H is the Hamiltonian and Y is a wave function that we want to be as good an approximation of the solution of the Schrödinger equation as possible. If Yn are the solutions of the Schrödinger equation (the subscript n is a label to distinguish the different solutions) with energies En (HYn = EnYn), then
So the expectation value becomes equal to the energy of the solution of the Schrödinger equation. For a wave function that is not a solution of the Schrödinger equation the expectation value is often called the energy of that wave function.
The variation principle states that the expectation value of an arbitrary wave function is always larger or equal than the energy of the ground state; i.e., the lowest energy of a solution of the Schrödinger equation. Mathematically this is
with E0 the energy of the ground state. This inequality can be proven rigorously, but this is something that we will not do here.
The variation principle can be used as follows. Suppose that we have a number of wave functions F1,F2,... etc. For each of these wave functions we calculate the expectation value of the Hamiltonian. The wave function with the lowest energy (i.e., lowest value of the expectation value) is said to be the one that approximates the wave function of the ground state best. So the variation principle used in this way approximates only the ground state of a system. For the electronic wave function of atoms and molecules this is generally also the only wave function of interest.
It is possible to compare a much larger number of wave functions than the finite set one can handle with the procedure just described. Instead of using a set of wave functions, we write down a wave function that depends on a parameter. For each value of the parameter we get a different wave function. For example, suppose we don't know how to solve the Schrödinger equation for the hydrogen atom, but we think that the electron is more likely to be found near the nucleus. So we try a wave function of the form
| Y(x,y,z) = exp[-a(x2+y2+z2)]. |
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We have introduced a parameter a, because we do not know have rapidly the probability of finding the electron decreases away from the nucleus. The expectation value of the Hamiltonian for this wave function is
| áHñ = |
3h2a
8p2 m
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æ
ú
Ö
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We see that the energy of the wave function is now a function of the parameter a. The best wave function is the one that minimizes this energy. The result is
and the corresponding value for the energy
which is somewhat higher than the exact value of -me4/8e02h2.
The number of parameters in a wave function need not be restricted to one. It is possible to choose any number of parameters. The energy of the wave function then will be a function of all the parameters in the wave function. Minimizing this function again gives the best approximation for the ground state. Of course, the more parameters the more difficult it will be to find the minimum. If one tries to determine the minimum by equating the derivatives to the parameters to zero, then one finds not just the global minimum, but also local minima. These are often interpreted as approximations to excited states
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