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Suppose that we have two orbitals, c1 and c2, that interact. We assume that these orbitals are normalized, but they are not necessarily orthogonal. The (molecular) orbital j that is formed by the interaction of these two orbitals is a solution of the Fock equation.
As usual we write the solution as a linear combination of the interacting orbitals
Determining j then means determining the coefficients c1 and c2. We can do this by converting the Fock equation to a secular or eigenvalue equation. We do this in a number of steps. First we substitute the linear combination in the Fock equation. This gives us
| F[c1c1+c2c2] = c1Fc1+c2Fc2 = e[c1c1+c2c2]. |
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In the second step we multiply these equations from the left with the interacting orbitals. (Strictly speaking we should multiply with the complex conjugate of these orbitals. For convenience we're assume that the orbitals are real-valued. This is the normal situation.) This gives us two equations; one for multiplication with c1, and one with c2.
Then, the third step, we integrate over all coordinates of the electron. (This is the argument of the orbitals.) This gives us
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| c1 |
ó
õ |
dr c1Fc1 +c2 |
ó
õ |
dr c1Fc2 |
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| = e |
é
ë |
c1 |
ó
õ |
dr c1c1 +c2 |
ó
õ |
dr c1c2 |
ù
û |
, |
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| c1 |
ó
õ |
dr c2Fc1 +c2 |
ó
õ |
dr c2Fc2 |
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| = e |
é
ë |
c1 |
ó
õ |
dr c2c1 +c2 |
ó
õ |
dr c2c2 |
ù
û |
. |
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These equations are quite unwieldy. We can simplify them first by noting that
|
ó
õ |
dr c1c1 = |
ó
õ |
dr c2c2 = 1 |
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because we have assumed that these orbitals are normalized. A further simplification is obtained by introducing some new notation. We have atomic integrals
resonance integrals
| b = |
ó
õ |
dr c1Fc2 = |
ó
õ |
dr c2Fc1, |
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and overlap integrals
| S = |
ó
õ |
dr c1c2 = |
ó
õ |
dr c2c1. |
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(The two resonance integrals are equal because of a property of the Fock operator which is outside the scope of this course.) With this new notation we get
In the fourth step we rearrange this equations a bit and then rewrite them in matrix-vector notation. This then finally gives us
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ç
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ö
÷
ø |
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æ
ç
è |
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ö
÷
ø |
= e |
æ
ç
è |
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ö
÷
ø |
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æ
ç
è |
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ö
÷
ø |
. |
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This is the secular or eigenvalue equation.
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