Hückel Theory in the 21st Century

 

Hückel Theory in the 21st Century

Principal Author

Martin Grayson MartinY 09:44, 29 September 2006 (UTC).

Preface

This topic is suitable for a wiki journal because it is a summary of the usefulness of a scientific backwater which modern computation might have rendered into a teaching only tool and the mathematical skills required to practice this basic theory are perceived as difficult by some students. However it is believed that there is a future for these kind of small expansion models both for teaching linear algebra and rationalizing phenomena.

Opinions and contributions are solicited.

Introduction

A one-pound theory can no more produce a ten-pound theorem than a one-hundred-pound pregnant woman can give birth to a two-hundred pound child, Gregory J. Chaitlin .

Hückel theory is a minimalist theory of molecular quantum mechanics which requires so little computation and has so few degrees of freedom in its parameter space that one view is that it is a totally obsolete model, belonging to the pre-computer days of pencil and paper. However this simplicity gives the method strengths as both a tool for modelling molecular electronic structure and for pedagogical purposes by providing an inexhaustible supply of degree level pen, paper and computer problems.

(There is another even more basic theory of conjugated systems, the Free-Electron Molecular-Orbital method (FEMO) whose applications are very elegantly described in Introduction to Applied Quantum Chemistry chapter 5 and the original references therein. However this method involves mathematics more familiar to physicists than chemists, and so is less suited to a chemistry course where the correspondence with the language of organic chemistry is stronger in Hückel theory.)

Though many Hückel calculations are way beyond pen and paper analysis there is little which cannot be computed with this model and it might be regarded as the zeroth order approximation of any kind of molecular orbital methodology. Hückel theory has been used to investigate the magnetic properties of large carbon sp² structures potentially as large as nanotubes, and the polarizabilities of similarly large structures can be calculated

In the 1960s Hückel theory was used by Hameka to investigate the polarization properties of polyacetylene. There can surprisingly be some advantages in using a theory with the minimal degrees of freedom. If a phenomenon should appear at this level it means you have a basic model of the physics behind the property and at least it cannot an be an artifact of the basis set or computational methodology. One might suggest that there are three levels of calculation which are really useful models:

• A low level which gives largely conceptual ideas such as Hückel theory or FEMO.
• A higher level method such as SCF or semiempirical calculations which give semi-quantitative answers and are subject to improvement to a limit by increasing the basis functions.
• The probably unobtainable, definitive ab initio model having most of the electron correlation such as a CCSD(T) calculation with a large basis.

Hückel Theory and Education

Hückel theory was invented in the 1930s , before the advent of computers but well into the era of linear algebra which was later to be brought to prominence in applied science by the invention of computers. Since the development of SCF programs Hückel theory has long been regarded as only a research curiosity. It is taught in all undergraduate courses because it is such a good exemplar of both molecular orbital theory and linear algebra and mathematical methods using characteristic polynomials, determinants and matrices can all be demonstrated as applied to familiar organic chemistry. Also the Hückel orbitals almost always have the same node structure as the SCF $\pi$-orbitals and so do not lead to misleading models. This means the method is very useful as pre-generator of molecular orbitals for appliying Woodward-Hoffmann rules

Some quite complex applications of molecular orbital theory can be rationalized essentially by Hückel theory, as when you form a parameterized phenomenological Hamiltonian in a small visualizable basis you get a set of equations identical to Hückel theory. In situations where there is fuzzy translational or spatial symmetry a small basis expansion with parameterization can give an effective model to rationalize experimental and complex calculation data

A model Hamiltonian for a system where there is perturbation which mixes an s-orbital with only the p_z orbital would look like this:

$\begin{vmatrix} \alpha_s -E & 0 & 0 & \beta \\ 0 & \alpha_{px} -E & 0 & 0 \\ 0 & 0 & \alpha_{py} -E &0 \\ \beta & 0 & 0 & \alpha_{pz} -E \\ \end{vmatrix} = 0$

The solution to this system immediately collapses to a quadratic equation.

Hückel Theory and Graph Theory

Hückel theory has only one basis function per atom site (node) and so there is very close correspondence between Hückel theory and graph theory . (However a minimal number of degrees of freedom is not confined to Hückel theory or Graph Theory. A small basis SCF calculation on a highly symmetric molecule such as benzene can have remarkably few free coefficients unconstrained by symmetry .)

Hückel Theory and Electrical Properties

The first literature example found of Hückel theory with a parameterization of dipole integrals was due to Hameka who used it to investigate the polarization properties of polyacetylene.

Champagne and André have revisited this polyacetylene problem using the modern computational MP2 method thereby gaining a little more insight and methodological confidence than was possible using Hückel theory. However Hameka was able to use Hückel theory for the special case of infinite systems to give analytical expressions which have uses in rationalizing the optical behaviour of crystals.

As the polarizability $\alpha$ scales with, and sometimes is given in units of, molecular volume the spatial representaion of a Hückel calculation might be expected to give some sort of valid representation of the polarizablity tensor. The elements for the $\pi$ part of the system should have the right shape and approximate magnitude as the accurate tensor shows this to be the case.

The expression for the dipole moment integrals in Hückel Theory is:

$< \chi _{i} |x| \chi _{j}> = \delta _{ij} \lambda x _{i}$

where $\chi _{i}, \chi_{j}$ are the AOs on the $\pi$ atoms i and j respectively. $x_i$ is the atomic coordinates which will usually but need not be be two dimensional.

As shown there Hückel calculations produce results which are certainly not realistic for the higher polarizabilities $\beta$ and $\gamma$. However these calculations represent the nonlinear responses of an abstract geometric model which still retains qualitative analogies with reality. There are examples of basic two and three state model systems being applied to practical optical problems in references .

Hückel Theory and Dipole Moments

HT predicts that non-alternant hydrocarbons have a permanent dipole moment which is caused only by conjugation effects, not by electronegativity as all the atoms are sp² carbons. Azulene is a very good example as it has a five membered ring fused to a seven membered ring. Seven membered rings like to get a positive charge , this way getting 6 $\pi$ electrons. Five membered rings similarly take a negative charge getting 6 $\pi$ electrons that way. Azulene does indeed behave like that giving a value for $\mu$ of 21.48 Cm x $10^{30}$ by HT but the value for a minimal basis SCF is a more realistic 5.90.

The other way a HT model of a molecule can have a dipole is via the heterocyclic atom parameters which have been added to HT.

Hückel Theory and Linear and Nonlinear Polarizabilities

The excited states which are formed in HT are not good states in that they are just a one-electron function which progressively has one more node than the state below. There is no optimization. Hoewever it is possible to choose $\lambda$ to be -1 so that the perturbations from analytical theory correspond to the perturbations from a finite field so that the excited states have some meaning. HT tends to exaggerate the polarization caused by perturbations. The same reservation applies to using the single electron swap to generate a model of the excited state. (This is not however different to what is done in text book molecular orbital diagrams and occupations.)

Hückel Theory, Excitations and Intensities

Using the paradigm explained in reference and molecular mechanics geometries from MACROMODEL with the MM2^* force field several transition matrix elements were calculated. $< \Psi^0|r| \Psi^1>$ is the matrix elements between the ground state and the first excited state, (in the case of benzene this is doubly degenerate). In benzene $< \Psi^0|r| \Psi^1>$ = 1.3194 au. (intensity 1.7408.) In planar biphenyl $< \Psi^0|x| \Psi^1>$ = 2.3918 au. (intensity 3.28625 x benzene.) $< \Psi^0|y| \Psi^1>$ and $< \Psi^0|x| \Psi^2>$ are zero. (Remember that the intensity comes out as a function of the square of the matrix element in the derivation of Fermi's Golden Rule.) In 5-member jointed biphenyl, (a biphenyl forced to be planar by adding two -CH₂- groups at the ring connection, $< \Psi^0|x| \Psi^1>$ = 2.2528 au. (intensity 2.9154.) $< \Psi^0|y| \Psi^1>$ and $< \Psi^0|x| \Psi^2>$ are zero. These zero matrix elements are provable by simple group theory i.e. the triple direct product contains no $A_1$ representations. The reduction in intensity on linking with -CH₂- groups is entirely due to the slight contracting of the molecular geometry as only the $< \phi_n|x| \phi_m>$ integrals change!

The experimental angle between the rings in biphenyl as quoted in reference is 44.4$^o$. (This reference also contains analytical torsonal potentials for substituted biphenyls. They do not change much on meta and para substitution.) Ab initio calculations of this energy surface have also been done . (A proper ab initio treatment of this molecular excitation has been done using the highly correlated method CASPT2 . The calculations here seem rather ridiculous by comparision.)

In HT only if you change the $\beta$ representing the inter-ring bond by multiplying by $\cos \theta$ will the intensity change! Here is the excitation energy and matrix element as a function of angle:

90   5.6000    1.3194
80   5.2798    1.5048
70   4.9794    1.6843
60   4.7094    1.8496
50   4.4766    1.9947
40   4.2850    2.1156
30   4.1361    2.2103
20   4.0302    2.2779
10   3.9669    2.3184
 0   3.9459    2.3319

When functionalizing polyphenylenes at the end group, HT predicts that push / pull groups affect the dipole and first hyperpolarizability at 1st order but the polarizability and $\gamma$ at 2nd order .

It would be nice to examine the experimental grounds for this hypothesis.

The derivative of the dipole moment as a function of the Coulomb Integral at the ortho, meta and para positions is respectively, 0.812518, 0.855764 and 1.244874. Not surprisingly this suggests the para position causes the most functionalization. Rather more surprisingly the ortho and meta positions are almost equally effective.

Hückel Theory, Semi-empirical Methods and QM/MM

The current development of super-computers has brought calculations at the SCF level of ab initio theory into possibility for all systems being considered as possible NLO molecules and complexes. However there is still the issue of electron correlation and the convergence of the basis set. Semi-empirical methods can be applied to very large molecules but these molecules are also accessible to methods employing a hierarchical mix of molecular mechanics and ab initio which the author believes is the method of choice as it retains the limits and convergence properties of ab initio theory with the huge extensibility of semiempirical and molecular mechanics theories. Because of this it is currently unclear what the future use of AM1 style methods will be in modelling NLO molecules.

Hückel Theory and the SCF Method

HT's first relation to the SCF method is that the phenomenological Hamiltonian can be taken to be the Fock operator, not the kinetic energy and the attraction term, therefore HT includes the SCF in its parameterization.

The modification to HT, the $\omega$-technique of Wheland and Mann includes iteration and so can be regarded as a kind of proto-SCF where one is iterating Hückel theory to optimise a parameter, $\omega$, which describes electron-electron repulsion. Wheland / Mann calculations are almost never performed now and it is an open question of whether the methodology is of any value as a progamming exercise in iteration for graduate students as a small SCF program for helium might be a better use of time. The method does however damp down some of the excessive charge separation generated by the basic HT method.

The Wheland-Mann method modifies the Hückel theory $\alpha$ parameter ths:

$\alpha_\mu = \alpha_0 + ( 1 - q_\mu) \omega \beta$

Where q is the value of $2c^2$ at each atom, i.e. for an sp² carbon atom in an alternant hydrocarbon it is 1. $\omega$ is conventionally 1.4 to give agreement with experiment.

The Hückel problem is then solved and the system is iterated until the valus of $q_\mu$ stop changing, so here is the analogy with the SCF procedure. $\omega$ is an optimisable electron repulsion parameter which has the effect of reducing the electron grabbing power of atoms with a $\delta -ve$ charge and increasing the electronegativity of the $\delta +ve$ atoms.