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Secular equations (Read 4341 times)
Gerrit-Jan Linker
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Secular equations
19.02.08 at 21:15:00
 
Secular equations
 
Using the variationial theorem we can find eigenvalues and eigenvectors of the Hamiltonian. We can obtain them through the so called secular equations which are derived as follows:
 
E = <Φ|H|Φ> / <Φ|Φ>
 
where:
Φ is the trial wave function  
H is the Hamiltonian operator
E is the average energy of the system
 
A trial wave function Φ is chosen to be a linear combination of linear independent known functions satisfying the same boundary conditions as the exact wave function:  
Φ = ∑iaiψi  
 
Note that this is not the same as expanding Φ in the basis formed by eigenfunctions of H. In the expansion above <ψij> is probably not 0, the functions are linear independent but they do not form an orthonormal basis.  
 
E = <Φ|H|Φ> / <Φ|Φ>
= <∑iaiψi|H|∑jajψj> / <∑iaiψi|∑jajψj>
= ∑ija*iaji|H|ψj> / ∑ija*iajij>
= ∑ija*iajHij / ∑ija*iajSij
 
We arrive at E = = ∑ija*iajHij / ∑ija*iajSij
where:
Hij = <ψi|H|ψj> ; resonance integral
Hii = <ψi|H|ψi> ; corresponds to the energy of a single electron occupying this function i; essentially it is the ionisation potential of the AO (atomic orbital) in the environment of the surrounding molecule.
Sij = <ψij> ; overlap integral
ψi ; AO = atomic orbital i
 
The secular equations are derived from this expression. We are looking for a minimum of the energy (see the variation principle) so we seek:
δE/δak = 0 for all k
 
This leads to N equations with N unknowns ai
 

 
From linear algebra we know that there is a non-trivial solution (the trivial solutions are all coefficients ai=0) if the determinant formed from the coefficients of the unknowns is zero:
 
|Hki - ESki| = 0
 

 
This is the secular equation. It gives N roots E, N energies E where each value of Ej gives a different set of coefficients aij. These coefficients will define an optimal wave function Φj within the given basis set:  
Φj = ∑i=1..Naijψi
 
See also:
Variational theorem  
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1203365153
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« Last Edit: 21.02.08 at 08:13:28 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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