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IPM: Independent Particle Model (Read 3917 times)
Gerrit-Jan Linker
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IPM: Independent Particle Model
25.12.07 at 22:09:47
 
IPM: Independent Particle Model
 
The molecular Schrodinger equation describes the motion of all electrons and nuclei in the molecule. The molecular hamiltonian is inseperable because of the nucleus-electron interactions. With the Born Oppenheimer approximation the nuclear coordinates become parameters to the molecular Hamiltonian. We are left with an electronic Hamiltonian. This Hamiltonian is still inseparable due to the electron-electron repulsion.
 
In the IPM the Schrodinger equation Hf=Ef is solved in which the interactions between electrons are removed. If we have a 2 electron system H(1,2) = H(1) + H(2) + g(1,2) where H(i) is the hamiltonian operator for electron i and g(1,2) describes the interaction of electrons 1 and 2. In the IPM the term g(1,2) is neglected. The hamiltonian then becomes  
 
[1]       H(1,2) = h(1) + h(2)  
 
We look for solutions of the schrodinger equation using the hamiltonian as [1] in the form of
[2]       f(r1,r2) = f(r1)f(f2)  
where each factor only contains the variables for one electron.
Such a product of orbital functions is called a Hartree product. See the topic on the Hartree product for more details.
 
Substitution of [1] in the Schrodinger equation:
 
[3]       (h(1) + h(2)) f(r1,r2) = E f(r1,r2)
 
Substitution [2] this becomes,
[4]       (h(1) + h(2))f(r1)f(r2) = Ef(r1)f(r2)
 
or       h(1)f(r1)f(r2) + h(2)f(r1)f(r2) = Ef(r1)f(r2)
 
or       h(1)f(r1)f(r2) / f(r1)f(r2) + h(2)f(r1)f(r2) / f(r1)f(r2) = E
 
Since H(1) only works on electron 1 we can rewrite to:
 
[5]       h(1)f(r1) / f(r1) + h(2)f(r2) = E
 
If equation [5] is to result in a constant the first and second factor must also be a constant:
     e1 + e2 = E
 
When then get the following equations
                 e1 = h(1)f(r1) / f(r1)  
                 e2 = h(2)f(r2) / f(r2)  
 
or             h(1)f(r1) = e(1)f(r1)
            h(2)f(r2) = e(2)f(r2)
 
If the electrons were not interacting we should never have to solve anything more than a 1 electron eigenvalue equation.  
 
Source:
Methods of molecular quantum mechanics, R. McWeeny, second edition.
 
See also:
Molecular Hamiltonian
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1251548847
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« Last Edit: 28.12.09 at 18:07:13 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
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Gerrit-Jan Linker
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Posts: 75
IPM: Examples
Reply #1 - 27.08.09 at 12:43:18
 
IPM: Examples
 
Examples of the Independent Particle Model or Independent Particle Approximation is the Hartree Fock (HF) and Density Functional Theory (DFT)
 
See also:
Hartree Fock theory
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1198498430/0#2
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Gerrit-Jan Linker
Linker IT Software
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