Hückel method

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The Hückel method is used to evaluate energy levels disposition in a molecule, following the linear combination of atomic orbitals theory (MO-LCAO). We'd have to solve <math> Hu=\epsilon u </math>. We do recall <math> \alpha \equiv \left\langle u^{p_{z}}_{j}\left| H \right| u^{p_{z}}_{j}\right\rangle </math> the diagonal elements of <math> H </math> and <math> \beta \equiv \left\langle u^{p_{z}}_{j^{\prime }}\left| H \right| u^{p_{z}}_{j}\right\rangle </math> its off-diagonal elements. Hence:  
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The Hückel method is used to evaluate energy levels disposition in a molecule, following the linear combination of atomic orbitals theory (MO-LCAO). We'd have to solve the stationary Schrödinger equation <math> Hu=\epsilon u </math>. We do recall <math> \alpha \equiv \left\langle u^{p_{z}}_{j}\left| H \right| u^{p_{z}}_{j}\right\rangle </math> the diagonal elements of the Hamilton operator <math> H </math> and <math> \beta \equiv \left\langle u^{p_{z}}_{j^{\prime }}\left| H \right| u^{p_{z}}_{j}\right\rangle </math> its off-diagonal elements, describing a first-neighbor coupling. Hence:  
  
 
<math display="block">  H = \begin{bmatrix}  \alpha  
 
<math display="block">  H = \begin{bmatrix}  \alpha  
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\\ \end{bmatrix}  </math>
 
\\ \end{bmatrix}  </math>
  
Hence: <math> H_{ii} =\alpha </math>, <math> H_{i,j+1}=H_{i+1,j}=\beta </math> and all the other elements are zeros, except for the case of circular molecules, where also the matrix element that correspond to the first/last atom bound is equal to <math> \beta </math>. Energy levels now can be found solving <math> H-\epsilon \mathbb {1}=0 </math>, that is:  
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Hence: <math> H_{ii} =\alpha </math>, <math> H_{i,j+1}=H_{i+1,j}=\beta </math>. Assuming that an atom couples only with its first neighbor, all the other elements are equal to zero, the only exeption being the case of circular molecules, where also the matrix element that correspond to the first/last atom interaction must be taken equal to <math> \beta </math>. Energy levels now can be found solving <math> H-\epsilon \mathbb {1}=0 </math>, that is:  
  
 
<math display="block">  \beta \begin{vmatrix}  \frac{\alpha -\epsilon }{\beta }  
 
<math display="block">  \beta \begin{vmatrix}  \frac{\alpha -\epsilon }{\beta }  

Revision as of 09:56, 9 April 2020

The Hückel method is used to evaluate energy levels disposition in a molecule, following the linear combination of atomic orbitals theory (MO-LCAO). We'd have to solve the stationary Schrödinger equation . We do recall the diagonal elements of the Hamilton operator and its off-diagonal elements, describing a first-neighbor coupling. Hence:

Hence: , . Assuming that an atom couples only with its first neighbor, all the other elements are equal to zero, the only exeption being the case of circular molecules, where also the matrix element that correspond to the first/last atom interaction must be taken equal to . Energy levels now can be found solving , that is:

Recall and solve in order to find energy levels. Remember that generally , and it is called hopping parameter.

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