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If there would be no interactions between the electrons in an atom or molecule, the Schrödinger equation for all electrons could be reduced into Schrödinger equations for each electron. The proof of this statement uses separation of variables. The Schrödinger equation for all electrons can be written as
| H(1,2,...,N)Y(1,2,...,N) = EelY(1,2,...,N). |
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In this equation we have used an index, 1, 2, ..., N, to indicate the coordinates and the spin of an electron. The energy Eel is the electronic energy. To get the total energy of a molecule the repulsion between the nuclei has to be added. The functions that depend on just one electron are the orbitals . Using orbitals to construct more-electron wave functions is a normal procedure in quantum chemistry.
When there is no interaction between the electrons the Hamiltonian can be written a
| H(1,2,...,N) = h(1)+h(2)+...+h(N), |
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where h consists of operators for the kinetic energy and the interactions with the nuclei of just one electron. Because there are no terms in the Hamiltonian depending on two or more electrons, we can use separation of variables; i.e., we write
| Y(1,2,...,N) = y1(1)y2(2)...yN(N). |
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where the lower case y's are the orbitals we mentioned above; i.e., electronic wave functions depending on the coordinates and spin of just one electron. Substitution of the expressions for H and Y in the Schrödinger equation yields
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| y2(2)y3(3)...yN(N)[h(1)y1(1)] |
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| +y1(1)y3(3)...yN(N)[h(2)y2(2)] +... |
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| +y1(1)y2(2)...yN-1(N-1)[h(N)yN(N)] |
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Dividing this by y1(1)y2(2)...yN(N) then gives
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h(1)y1(1)
y1(1)
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+ |
h(2)y2(2)
y2(2)
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+... + |
h(N)yN(N)
yN(N)
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= Eel. |
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On the left-hand-side of this equation we have terms that each depend at most on just one electron. On the right-hand-side we have an energy that doesn't depend on any electron. This equation can therefore only hold when also each term on the left-hand-side does not depend on any electron. This means that we must have
or
where en is a constant. We see that we have obtained one-electron Schrödinger equations. The orbital energies en are related to the electronic energy;
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