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:
<UL>
 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.
</ul>
===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 pz 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.
===References===
Category:Interpretations in the Physical and Computational Sciences submitted article