## Physical Chemistry: The Simple Hückel Method (Part I)

The Hückel molecular orbital method is a quantum mechanics approach for calculating the energies of molecular orbitals of π electrons in conjugated hydrocarbon systems, such as ethylene, benzene, and butadiene.

It was proposed by Erich Hückel in 1930, and, subsequently extended and improved but other scientists. It provides the theoretical foundation for Hückel’s rule for the aromaticity of (4n + 2) π electron cyclic, planar systems. These are some of my slides on this topics. In the future, I will add some more explanations. Hückel assumed that for unsaturated organic compounds the π electrons can be treated separately by those involved in the $\sigma$ bonds. In fact, a π orbital is antysymmetric for reflection through the plane of the molecule whilst a $\sigma$ one is symmetric.

If the assumption is true, in term of energy, we have that $E_{tot}=E_{\pi}+E_{\sigma}$. This means that the wavefunction of the molecule is given by the product of the wavefunction describing the $k/sigma$ and $(k-1) \pi$ electrons:

$\psi(1, 2, \dots,n) = \psi_{\pi}(1, 2,\dots,k)\psi_{\sigma}(k+1, \dots, n)$

We can also assume that the molecular wave function of the whole system can be approximated as a product of 1-electron wave function orbitals

$\psi(r_1,r_2, \dots, r_n) = \psi_1(r_1)\psi_2(r_2)\psi(r_3) \dots \psi(r_N)$

The energy of a sysmt can be evaulated using tha Hamiltonian quantum operator that is defined as

$H=K_n + K_e + V_{ne} + V_{ee} + V_n$

with

• $K_n$ : the kinetic energy of the nucleus.
• $K_e$ : the kinetic energy of the electrons.
• ${V}_{ne}$: the proton-electron attraction potential energy.
• $V_n$ : the proton-proton repulsion potential energy.
• ${V}_{ee}$ : the electron-electron repulsion energy.

In the case of a molecule composed by M atoms, the total hamiltonian can be written as

For a molecular sytem with π electrons, we can further distinguish these electron from the $\sigma$ ones as

For not interaction electrons, we can also assume that the total energy of the system can be calculate by a total Hamiltonian (see my blog on the classical mechanics) operator can be expressed as

$\hat H = \hat h_1 + \hat h_2 + \dots + \hat h_n$

$\hat h_i = -\frac{1}{2} \nabla_i^2 - \sum_{j=1}^{N}\frac{Z_j}{r_{i,j}}$

and the eigenstates of each electron can be calculate by the Schroedingen equations

${\hat h_i} \psi_i(r_i)=\epsilon_i\psi_i(r_i)$

As further assumption the wavefunction of the $\pi -$electrons can be factorized as follow

Using this approximations the total electronic energy from the $\pi -$ electrons can be expressed as

We are now going to consider a simple molecular system

We are now going to consider a simple molecular system to show how to identify the basic atomic orbitals and to construct the Hückel determinant. An allyl molecules has the structural formula $H_2C=CH-CH_2 H$ and consists of a methylene bridge ($-CH_2-$) attached to a vinyl group ($-CH=CH_2-$). Its name, derived from the Latin word for garlic, Allium sativum, was given in 1844 by Theodor Wertheim when he was able to isolate an allyl derivative from garlic oil.

We consider only the three orbitals $p_Z$ of each carbon since they are antisymmetric for reflection along the molecular plane. The total molecular orbital can be considered as a linear combination of the atomic orbitals.

The total molecular orbital can be considered as a linear combination of the atomic orbitals.

$\psi(1,2,3)=c_1\chi_1+c_2\chi_2+c_3\chi_3$

In order to find the energy of the molecular system, we are going to use the result of the so called variational theorem.

The variation theorem states that given a system with a Hamiltonian operator $H,$ then if $\Psi$ is any well-behaved function that satisfies the boundary conditions of the Hamiltonian, then

where $E_0$ is the true value of the lowest energy eigenvalue of $H.$ This expression is also called Rayleigh ratio.

The variational principle allows us to calculate an upper bound for the ground state energy by finding the trial wavefunction $\Psi_{trial}$ for which the integral is minimised. The principle assure us that only if the function is exact $E=E_0$ in all the other cases $E>=E_0$.

Linear variation method

We are going to use the so-called linear variation method (or Rayleigh-Ritz method) in which the linear variation function is a linear combination of $k$ linearly independent atomic orbital functions that satisfy the boundary conditions of the problem ($\Psi_{trial}$). The coefficients ($c_i$) of the trial wavefunction are parameters to be determined by minimising the variational integral.

By inserting the function $\Psi$ of the allyl molecule in the Rayleigh ratio, the numerator become

$< \Psi | \hat H | \Psi >= <\sum_i c_i^*\chi_i|\hat H| \sum c_j\chi_j>=\sum_i \sum_j c_i^* c_j <\chi_i|\hat H | \chi_j> = \sum_i \sum_jc_i^* c_j H_{ij}$

where $H_{ij}$ is the element $(i,j)$ of the Hamiltonian matrix.

and the denominator

$<\Psi | \Psi >= <\sum_i c_i^*\chi_i | \sum c_j\chi_j>= \sum_i \sum_j c_i^* c_j <\chi_i|\chi_j>=\sum_i \sum_jc_i^* c_j S_{ij}$

where $S_{ij}$ is the element $(i,j)$ of the overlap matrix.

Therefore we can now write

$E=\frac{\sum_i \sum_jc_i^* c_j H_{ij}}{\sum_i \sum_jc_i^* c_j S_{ij}}$

or in the rearranged form as

$E(\sum_i \sum_jc_i^* c_j S_{ij})=\sum_i \sum_jc_i^* c_j H_{ij}$ (1)

Now, we need to find the value of the coefficients $c_i$ that minimize the function $E(c_1, \dots, c_k)$ using the variational method. This can be obtained using the definition of a mimimum of a function, namely by calculating the equations

$\frac{\partial E}{\partial c_i}=0$

for all the functions i.

By differentiating both the sides of (1) and using the properties of $S_{ij}$ and $H_{ij}$ and of the coefficients $c_i$, we finally obtain the so-called secular equation

$\sum_i c_i(H_{ik}-ES_{ik})=0$

for all the k.

We therefore have a set of k simultaneous homogeneous equation (secular equations) in k unknowns. In the case of the allyl system the system is the follwing

For a non-trivial solution (i.e. $c_i \neq 0$ for all i), the determinant of the secular matrix must be equal to zero

Therefore we can set up the values of $H_{ij}$ integral in Hückel determinant as constants and on the bases of the molecular connectivity as

and for the overlap integral

With these semplification the Hückel determinant for the allyl molecule become

Here some more example of Hückel determinat for simple unsaturated molecules

In the second part of this article, we are going to learn how to solve this determinantal equation to find the energy of the molecular orbitals.

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## REFERENCES

J.P. Lowe Quantum Chemistry. 1993, Academic Press.

## About Danilo Roccatano

I have a Doctorate in chemistry at the University of Roma “La Sapienza”. I led educational and research activities at different universities in Italy, The Netherlands, Germany and now in the UK. I am fascinated by the study of nature with theoretical models and computational. For years, my scientific research is focused on the study of molecular systems of biological interest using the technique of Molecular Dynamics simulation. I have developed a server (the link is in one of my post) for statistical analysis at the amino acid level of the effect of random mutations induced by random mutagenesis methods. I am also very active in the didactic activity in physical chemistry, computational chemistry, and molecular modeling. I have several other interests and hobbies as video/photography, robotics, computer vision, electronics, programming, microscopy, entomology, recreational mathematics and computational linguistics.
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