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Born Oppenheimer approximation (Read 3777 times)
Gerrit-Jan Linker
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Born Oppenheimer approximation
02.01.08 at 21:16:12
 
Born Oppenheimer approximation
 
In the Born Oppenheimer (BO) approximation only the movements of electrons are considered and the atoms are considered to be in a rigid, fixed position.  
 
It is assumed in the BO approximation that the movement of nucleii due to the movements of electrons can be neglected. Nucleii are much heavier than electrons. It is assumed that the molecular wavefunction can be approximated by the product of a electronic and a nuclear wavefunction.
 
The BO approximation is applicable if the potential energy surfaces of different states of the molecule are well separated. In regions where the potential energy curves of different states overlap the BO approximation is inapplicable.
 
The Born-Oppenheimer is sometimes also called the adiabatic approximation.
 
Related topics:
Condon Approximation
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1341844204/0#0
Conical intersection
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1289733283/0#0
PES: Potential energy surface
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1216671925/0#0
 
References:
European Summerschool in Quantum Chemistry 2013, Book I, p201
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« Last Edit: 25.03.15 at 11:03:26 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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Gerrit-Jan Linker
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Re: Born Oppenheimer approximation
Reply #1 - 01.03.08 at 11:36:07
 
The Born Oppenheimer approximation lead to the following electronic equations.
 
It is assumed that the total wave function can be written into a nuclear (N) and electronic (e) part:
Φ = ΦNΦe
 
It is further assumed that the Hamiltonian is also split into a nucear and electronic part:
H = HN + He
 
The nuclear coordinates are included into Φe as parameters. Therefore, for a given choice of the nuclear coordinates the electronic Schrodinger equation can be solved:
HeΦe = EeΦe
 
For the nuclear Schrodinger equation we cannot do this because the nuclear Hamiltonian also works on the electronic wave function. For the nuclear part of the Schodinger equation we can derive the following:
 
HΦ = EΦ <=>
(HN + HeNΦe = EΦNΦe <=>
HNΦNΦe + HeΦNΦe = EΦNΦe  
   Now we insert the solution of the electronic Schrodinger equation:
HNΦNΦe + EeΦNΦe = EΦNΦe  
   Nothing works on the electronic wavefunction anymore so we can write:
HNΦN + EeΦN = EΦN  <=>
(HN + EeN = EΦN
 
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« Last Edit: 06.02.15 at 09:53:35 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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