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In the Schrödinger equation
for a molecule the Hamiltonian has a large number of terms. (We will discuss here the more general case of a molecule. For an atom just restrict in all of the following the number of nuclei to one.) To make things as easy as possible we distinguish between terms corresponding to a kinetic energy and terms corresponding to a potential energy. We also distinguish between electrons and nuclei. The Hamiltonian can then be written as
In this expression Tn and Te contain kinetic energy terms, and Vnn, Vne, and Vee contain potential energy terms. The subscripts denote if we are dealing with an energy contribution of nuclei (Tn and Vnn), electrons (Te and Vee), or both (Vne). (Note the relation between the terms in the Hamiltonian and the contributions to the total energy of a molecule.)
The potential energy terms are the easiest, because they are all familiar Coulomb interactions. Remember that the interaction energy for two charges q1 and q2 a distance r apart is given by the Coulomb expression
We will see that all the potential energy terms are of this form. The easiest is the repulsion between the electrons. The charge of each electron is -e, so the potential energy for the interaction between electrons i and j equals
| gij(ee) = |
1
4pe0
|
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e2
rij
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, |
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where rij is the distance between the electrons. The total energy for the interactions between the electrons is then simply the sum of the interaction energies of all electron pairs; i.e.,
| Vee = |
N-1
å
i = 1
|
|
N
å
j = i+1
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gij(ee), |
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with N the total number of electrons. For the nuclei we find something similar. If we write the charge of nucleus a as Za e with Za the atom number, then
| gab(nn) = |
1
4pe0
|
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Za Zb e2
Rab
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, |
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is the interaction energy between nuclei a and b with Rab the distance between these nuclei. The total interaction between the nuclei is then
| Vnn = |
K-1
å
a = 1
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|
K
å
b = a+1
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gab(nn), |
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with K the number of nuclei. For the interaction between the nuclei and the electrons we get
| Vne = |
K
å
a = 1
|
|
N
å
i = 1
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gai(ne), |
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with
| gai(ne) = - |
1
4pe0
|
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Za e2
Rai
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, |
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with Rai the distance between nucleus a and electron i. Note the minus sign because of the opposite charges.
The kinetic energy has second derivatives. For the kinetic energy of electron i we get a term ti(e) in the Hamiltonian of the form
| ti(e) = - |
h2
8p2 m
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|
é
ê
ë |
¶2
¶xi2
|
+ |
¶2
¶yi2
|
+ |
¶2
¶zi2
|
ù
ú
û |
. |
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In this expression m is the electron mass, and the derivatives are with respect to the coordinates of electron i. Adding the terms of all electrons gives
For the kinetic energy of nucleus a we get a term ta(n) in the Hamiltonian of the form
| ta(n) = - |
h2
8p2 Ma
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|
é
ê
ë |
¶2
¶Xa2
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+ |
¶2
¶Ya2
|
+ |
¶2
¶Za2
|
ù
ú
û |
. |
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In this expression Ma is the mass of nucleus a, and the derivatives are with respect to the coordinates of that nucleus. Adding the terms of all nuclei gives
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