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Suppose we have a normalized wave function Y for two electrons that is a product of two normalized functions y1 and y2, each depending on just one electron.
A product is the simplest way to combine orbitals for different orbitals, because of the following reason. The absolute square of the wave function |Y|2 is a probability distribution that tells us what the probabilities are of finding the electrons at certain positions. Because Y is a product we have |Y|2 = |y1|2|y2|2, which is a product of two probability distributions; one for each electron. This means that the electrons are statistically independent; one does not affect the other.
Although a product is simple, it does not fulfill the Pauli-principle, which states that
must hold. Swapping the electrons in the product would lead to
| y1(1)y2(2) = -y2(1)y1(2), |
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but this is only possible if y1(1)y2(2) = 0. However, Y = 0 is not a physically acceptable wave function.
We can, however, combine the original product and the result of the swap of the electrons as follows.
| Y(1,2) º y1(1)y2(2)-y2(1)y1(2). |
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For this wave function the Pauli-principle does hold. It is an acceptable wave function unless y1 and y2 are linearly dependent, because in that case we have Y = 0. The wave function Y will not be normalized in general. However, when y1 and y2 form an orthonormal set, then a good normalized two-electron wave function is given by
| Y(1,2) = |
1
Ö2
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[y1(1)y2(2)-y2(1)y1(2)]. |
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With more than two electrons the ad hoc approach we followed above becomes very cumbersome. We can, however, anticipate the result by rewriting the result for two electrons. We can write for two electrons
where we have used a determinant. If we have N electrons and N orbitals y1,y2,...,yN, forming an orthonormal set (i.e.,
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ó
õ |
dr ds yi(r,s)yj(r,s) = dij, |
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then we write analogously
| Y(1,2,...,N) = |
1
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ê
ê
ê
ê
ê
ê |
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ê
ê
ê
ê
ê
ê |
. |
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The resulting wave function is called a Slater-determinant . The factor 1/Ö[N!] takes care of the normalization of Y. Because of the properties of a determinant, the wave function fulfills the Pauli-principle. Swapping two electrons is the same as swapping two columns. It is known from linear algebra that a determinant then changes sign. Another consequence of the form of the wave function is that when two orbitals are equal, say yi = yj, that two rows, i and j, become equal, so that we get Y = 0. More generally we have that if y1,y2,...,yN form a linear dependent set, the resulting wave function vanishes. Not all wave functions can be written as a Slater-determinant, but we can write any wave function as a linear combination of Slater-determinants. The properties of a determinant make sure that the Pauli-principle holds for such a linear combination as well. |
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