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Using a product form
| Y(t1,t2,...,tN,) = y1(t1)y2(t2) ... yN(tN) |
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as an approximation for the solution of the Schrödinger equation is not acceptable. This is because for such a form the Pauli principle does not hold, which means that for such a wave function different electrons can have different properties. This is something we do not want, even for approximate wave functions. Electrons are identical particles.
We can change the product form so that we get a wave function for which the Pauli principle does hold. This leads to the following approximation.
| Y(t1,t2,...,tN,) = |
ê
ê
ê
ê
ê |
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ê
ê
ê
ê
ê |
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We see that we now approximate the wave function by a Slater determinant . We now proceed as with the product form. We use the variation principle to find the orbitals yn that give a lowest value for the expectation value of the Hamiltonian. This is an even lengthier and more difficult procedure than for the product form, but again the result is not so difficult to understand. The best orbitals are again solutions of the Fock equation.
Here too the Fock operator contains three terms, which we can identify as a term corresponding to the kinetic energy of one electron, a term for the interaction of one electron with the nuclei, and and a term for the interaction of one electron with all other electrons. The first two terms are the same as for the product form. The last term is different.
If we operate with the term corresponding to the interaction with the other electrons on the orbital yn we get
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1
4pe0
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å
m
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ó
õ |
dt¢ |
e2|ym(t¢)|2
|r-r¢|
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yn(t) |
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| - |
1
4pe0
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å
m
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ó
õ |
dt¢ |
e2ym(t¢)*yn(t¢)
|r-r¢|
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ym(t). |
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The first term in this expression is the same as the term we have already seen when we used a product form as approximation. (There may seem to be an extra contribution from m = n, but, if we put m = n in both terms in the expression above, we see that we get two contributions that cancel each other.) It is called the Coulomb interaction . The second term is new. It is called the exchange interaction . The Coulomb interaction can be interpreted is a classical interaction between charges; one being an electron and the other the charge cloud of the (other) electrons. A similar interpretation cannot be done for the exchange interaction. Nevertheless this interaction is extremely important for electron systems. The exchange interaction is about the same order of magnitude as the Coulomb interaction, but is has an opposite sign. The Coulomb interaction is always a positive contribution to the energy, whereas the exchange interaction is always negative; i.e., it lowers the energy. Note that the integrals contain also an integration over the spin variable. For the Coulomb term this integration can be ignored, because we can assume that the spin part of ym is properly normalized. For the exchange term the integration is important. If the spin parts of ym and yn are orthogonal (say, one is spin up or a and the other spin down or b), then the integration over the spin makes the integral equal to zero.
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