# Hückel method

Marcoballa (talk | contribs) m (Tag: Visual edit) |
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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: | + | The Hückel method is used to evaluate energy levels disposition in a molecule, following the linear combination of atomic orbitals theory (MO-LCAO). Let us assume a linear chain of identical atoms. 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>, representing the energy of the isolated atom, 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. Therefore, we can fill a matrix with the following elements: <math> H_{ii} =\alpha </math>, <math> H_{i,j+1}=H_{i+1,j}=\beta </math>. |

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+ | Assuming that an atom couples significantly only with its first neighbor, all the other matrix elements are equal to zero. The only exeption is represented by 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>. 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> | ||

− | + | 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 } |

## Latest revision as of 10:04, 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). Let us assume a linear chain of identical atoms. We'd have to solve the stationary Schrödinger equation . We do recall the diagonal elements of the Hamilton operator , representing the energy of the isolated atom, and its off-diagonal elements, describing a first-neighbor coupling. Therefore, we can fill a matrix with the following elements: , .

Assuming that an atom couples significantly only with its first neighbor, all the other matrix elements are equal to zero. The only exeption is represented by the case of circular molecules, where also the matrix element that correspond to the first/last atom interaction must be taken equal to . Hence:

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*.