Sean Hanley
Quantum Research Report
December 21st 2006
Molecular Orbital Calculations, Hückel’s Molecular Orbital
Theory, Hückel’s Rule, Extended Hückel Theory, and Variations
of Hückel Calculations
The analysis of Hückel’s Molecular Orbital Theory and resulting rule will be
shown and interpreted with the his techniques used and presented. Hückel’s Rule of 4n +
2 π electrons will be examined with an explanation as to it becoming a rule. Aromaticity
as the main goal of Hückel’s rule will be given special attention. An overview of how the
quantitative Molecular Orbital Theory came to be created; and how it has changed over
the years to include the variations and the application to quantitative results using
Extended Hückel Theory. Further, analysis of future research and advances in Hückel’s
Rule will be presented, including all the newer equation to explain fullerene’s 2(n + 1)2 π
electrons. Thus giving an overall review and explanation of computational and qualitative
results of aromatic compounds.
Erich Hückel born in Germany on August 9th 1896, studying mathematics and
physics in Berlin. He is best renown in the field of chemistry and physics for two
contributions: 1. Debye- Hückel theory of electrolytic solutions. 2. Hückel method of
approximate molecular orbital calculations on  electron systems. In 1930 he published
his finding on  systems, in 1931 he generalized and expanded the system theory to
explain benzene. Due to his poor communication skills, his findings went unrecognized
for over 20 years. It was in 1951 that his 4n + 2 rule was finally clarified by Doering to
show if C=C bonds would show aromaticity. Hückel died on February 16th, 1980 with
very little worldwide recognition for his contributions to chemistry. Today chemistry
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students recognize his name and that his contributions to the field are quite substantial.
Had he survived, he would have propbably received the Nobel Prize in 1981.
Hückel’s rule fully stated is as follows; any conjugated monocyclic polyene that is
planar and has (4n+2)π and/or nonbonding electrons, with n = 0,1,2, etc., will exhibit the
special stability associated with aromaticity. This rule has substantial influence in the
area of organic chemistry in relation to cyclic conjugated  systems. After his retirement
in 1961, Hückel’s theory was being widely used for different  systems. It was in 1962
that Roald Hoffmann developed the Extended Hückel Theory which was a broader
generalization for nonplanar molecules. In 1981 Roald Hoffmann along with Kenichi
Fukui were awarded the Nobel Prize in chemistry developing a similar system in
conjugated  systems. Erich Hückel died the year before so he was unable to be awarded
the prize, or be nominated. Because of Hückel’s and his missing the Nobel Prize he is not
well known outside of the fields of chemistry and physics.
Hückel’s theory is formulated around the idea of Molecular Orbital Theory. This
delineated a way of combining atomic orbitals. Essentially, giving the details behind
structure and bonding of molecules. Molecular Orbital Theory (MO) was developed in
1929 by Sir John Lennard-Jones which combined linear atomic orbitals together. This
MO theory along with the Lewis structure theory were both being developed at the
simultaneously which compliment in each other in qualitative ways. Present day
molecular orbital theory is based around the Total Hamiltonian. This Total Hamiltonian
involves the following: 1. Kinetic energy of nuclei, 2. Kinetic energy of electrons, 3.
Potential energy of nuclear-nuclear repulsion, 4. Potential energy of nuclear-electron
attraction, 5. Potential energy of electron-electron repulsion.
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Some approximations to fit the Hamiltonian include the Born-Oppenheimer
Approximation, Orbital Approximation, and Hartree-Fock Approximation. The BornOppenheimer Approximation assumptions are the following: 1. Kinetic energy of nuclei
is equal to zero, 2. Kinetic energy of electrons is experimentally found to be a constant,
and sections 3., 4., and 5 dealing with the potential energies are in the Hamiltonian form
below.
H el 
 h2
8 2 m 2
n
n
N
n
N
 i2  e 2 Z I riI1   e 2 rij1
i 1
i 1 I 1
3.
i 1 j i 1
4.
5.
This approximation does not imply that the nuclei are moving, but that instead the
electrons respond to the nuclei. When a more exact solution is needed the proceeding
process repeated to give a better approximation to the actual geometry.
The Orbital Approximation takes the molecule as a general series of one-electron
wave functions. For example:
 tot  a (1)  b (2)  c (3)  ...  n (i)
So the product of one-electron functions is called an orbital. One subset of this is that
factors are called the Determinantal Wavefunction. Since electrons are Fermions then
 must be antisymmetric. This then, will effect the interactions between two electrons
within the atom. The Slater determinant can be obtained from this assumption and can be
derived from any wavefunction. For example:
a (1) b (1) c (1) ... n (1)
a (2) b (2) c (2) ... n (2)
1
 norm 
 a (3) b (3) c (3) ... n (3)
n!
:
a (i )
:
b (i )
:
:
:
c (i ) ... n (i )
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The Slater determinant can be simplified using the ideas of normalization and
orthogonality. Due to the Pauli Principle some overlapping electrons will turn out to be
either zero or 1. This new principle will simplify the Slater determinant to a greater
extent.
The Hartree-Fock Approximation is yet another way to determine molecular
orbitals. Hartree-Fock states that the exact wavefunction can be expressed as a linear
combination of approximate determinantal wavefunctions. This approximation layers
upon the Orbital approximation using the  function above. From this, the Method of
Variations, Hartree-Fock Equation and Restricted Hartree-Fock Theory are derived. The
method of variations involves the following: 1. construct the determinantal wavefunction;
2. vary E with respect to each of the orbitals, thus setting the resulting equations equal to
zero; 3. solve the simultaneous set of equations to give the lowest energy. An example of
a simple equation from the Method of Variations is the following:
Eel   H el d
The Hartree-Fock Approximation is a very detailed and intensive mathematical
solution but a simple overview will be shown to generalize its results. This approximation
combines the Method of Variations stated in the paragraph above with the determinantal
wavefunction for a two electron system. Overall, the process of obtaining the energy
from a two electron system involves the following: 1. overlap integrals described earlier
to simplify the problem; 2. exchange integrals to include the two electron contribution.
The energy of the two electron system takes the following form:
n
n1
n
E   hi    ( J ij  K ij )
i 1
i 1 j i 1
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Quantum Research Report
The first summation involves the one electron contribution, the second two summations
involve the two electron term. Other considerations that need to be accounted for are the
two-electron contribution which involves the Coulomb and Exchange integrals. From
this, one can obtain the Fock operator which takes the following form:
n


h(1)   J j (1)  K j (1) i (1)   ii (1)
j 1




To obtain the lowest electron energy, one-electron wavefunctions (orbitals) need to be
eigenfunctions of the one-electron Fock operator. This will be at its lowest energy level
when the n orbitals satisfy    . The restricted Hartree-Fock Theory takes into account
the electron spin interaction and involves going into more detail about MO Theory.
From all of the above detailed steps one can obtain the wavefunction for a
molecule. To summarize this process there the four following steps: 1. choose a
molecular geometry and determine the determinantal wavefunction; 2. evaluate the
electronic energy of the system using one of the above approximation for example
Hartree-Fock; 3. vary and change these wavefunctions with respect to the orbitals; 4.
obtaining Eel can than add on the term of nuclear-nuclear repulsion, then solving through
until lowest energy level is determined. From these calculations the molecular orbitals are
obtained and a closer look into the one-electron orbitals, the Hückel theory, and other
theories are possible.
From the previous approximations the following have been developed to help
computational chemists attempting to obtain the lowest energy species. The perturbation
methods are Linear Combination of Atomic Orbitals- Molecular Orbital (LCAO-MO),
and the Many-Body Perturbation Theory, and the Semi-empirical methods, which include
Hückel Theory and the Extended Hückel Theory. The LCAO-MO method involves basis
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sets for its calculations and the orbitals are made up of both the Slater-type Orbitals and
the Gaussian-type Orbitals. One example of a perturbation method is called the MøllerPlesset Perturbation Theory (MPPT). This theory uses perturbations between the Fock
operator and the exact Hamiltonian to compute lowest energy levels.
The Semi-empirical method is the most commonly used method to determine
structures of molecules. Since it takes a great deal of time for computers to do multiple
integrals an approximation should be used to speed the process up. The Semi-empirical
method involves a process that combines two realizations to simplify the Semi-empirical
process. First some of these integrals, can be entered earlier to simplify different research
problems even before one starts final calculation problems. Second to obtain the
solutions, the computational results were previously solved to the experimental data. One
common method used today is the Neglect of Differential Overlap (NDO) Method. Some
of its common variations include: 1. Complete NDO; 2. Intermediate NDO; 3. Modified
Intermediate NDO; 4. neglect of diatomic differential overlap. All of these methods are
employed at various in computational studies to obtain the ground state energies of
molecules.
Hückel Theory was one of the first Semi-empirical methods discovered in the
early 1930’s used to explain large organic molecules. Due to the sophistication of the
process detailed above and with all of the different methods, Hückel’s method had some
major approximations. His method applies to planar, conjugated systems which rely on
the assumption that  and
 electrons act independently. This theory is the simplest of
theories and includes the following approximation: 1. only considering the  electrons; 2.
ignoring the electron-electron repulsion; 3. all orbital overlap is ignored; 4. Coulomb
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terms are only considering the 8 closest neighbors. Thus making Hückel’s theory a very
good qualitative one to substantiate aromaticity and orbital symmetry arguments.
Ignoring the electron-electron repulsion does not generally give accurate quantitative
results. This theory does have one major advantage in that it can be accomplished by
hand with no need for computer involvement.
Continuing on, the Extended Hückel Theory (EHT) is similar to the Hückel
Theory but it goes beyond since it is no longer restricted to planar conjugated molecules.
Although, EHT is still limited because it does not take into account the electron-electron
repulsion issue. Unlike some new computer programs, this theory does take into account
the orbital overlap. EHT still fails in the sense that it lacks the quantitative accuracy as
computer methods but still gives qualitative results. This method is quickly calculated
and provides good qualitative results and geometries but the energies obtained are not
accurate. Overall the Hückel theory and EHT methods, are obsolete for determining
energies because they both lack the term for electron-electron repulsion. They still have
value in determining the geometry and the visualization of orbitals. These two methods
also lead to the development of the Semi-empirical methods first developed in the 1950’s.
A newer method for large organic molecules was developed by Pariser-Parr-Pople
Method (PPP) and used the ideas from Semi-empirical methods. This PPP method was
determined to give results for the basic structure and the reactivity of some organic
molecules. This method uses the zero-differential overlap (ZDO) attempted to obtain
characteristics of complex organic dyes, but they were unable to realize this outcome for
the PPP method. Pariser-Parr-Pople has widely used applications to theoretical and
quantum chemistry. The two papers proving the PPP method have been among the top
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five chemistry and physics citations reported, with 2,450 references in the years between
1961-1977.(3)
Recent discoveries of Fullerenes have looked at whether or not these Fullerenes
are actually showing aromaticity. These spherical Ih symmetrical Fullerenes have been
determined to have characteristics of 2( N  1) 2 rule. This Fullerene rule is very similar to
the Hückel rule 4( N  2) which has not distorted Dnh symmetry. The Dnh symmetry has
very strong diamagnetic ring currents, while the antiaromatic 4 N rule often distorted
shape to show paratropic character. These ideas were then implemented into the
Fullerene experiment involving diamagnetic currents to determine if they showed
aromaticity. Buckminsterfullerene (Buckyball) C60 shows both diamagnetic and
paratropic effects which lead to a minor change in the chemical shift showing no real
aromatic contribution. The neutral species of C60 shows no aromaticity, but the general
occupation of the valence shell with charged ions can show aromaticity.
Another section of the Fullerene rule 2( N  1) 2 depends on the occupation of the
electrons in the valence shell. These π electrons can be represented by wavefunctions
with a type of electron gas having angular momentum quantum values (l=0, 1, 2, 3, …).
This l=0 s shell is comparable to the s atomic orbital in its structure. The major difference
between these is the center of the l=0 shell should be approaching zero density of
electrons. This Fullerene approximation also applies to the other shells with their
counterpart atomic orbital. It can be observed in the attached Figure 1.(4) C60 depends
strongly on the valence shells and the electron filling of these valence shells to show
10
aromaticity. The charged C60 structure shows that the most aromaticity is the C 60
charged species, showing a substantial stabilizing effect of approximately   81.4 . This
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stabilizing amount is due to the closed shell of the h shell. This h shell has extremely
stabilizing factors that show the aromaticity. In table 1. attached the effects of different
ions show aromaticity or do not show aromaticity.(4) Hirsch’s was able to apply his
results to the experimental section and able to determine this new Fullerene rule.(4) This
2( N  1) 2 rule can even be applied universally; applicable for all conjugated  systems,
even inorganic compounds which may distort the shape but still show these Fullerene
aromaticity results.
Even at this date there are no widely accepted results for value for the aromatic
stabilization energy (ASE), even for the common C60 structure. Cyrański’s paper is
looking at the Fullerenes in general not just for the icosahedral shape. The Fullerene rule
as stated above does not apply to C72 which has been experimentally shown to be the
most aromatically stabilized. Estimation for the C60 structure stabilizing energy has been
proposed. These results show E  379 kJ mol-1 which would have a substantially
stabilizing effect.(5) The results contradicted the Hirsch paper so this stabilization energy
is quite controversial. Cyrański looks at the strain in the bonds of the C=C interactions
and finds they are computational determined. Cyrański calculated that these strains factor
into lowering the stabilization of aromaticity.
Overall the future research into this area of chemistry is continuing. Researchers
are still trying to obtain a general value for the aromatic stabilization energy (ASE) for
the common C60 Fullerene. The research is heading towards the general approach to
finding results of aromaticity in Fullerenes and of finding definitive results dealing with
ASE numbers to give quantitative details. Qualitative results in 2000 show the
2( N  1) 2 rule to have positive outcome in showing aromaticity. Yet a general rule needs
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to be determined as a guide for more general systems. These rules are leading to chemists
ability to determine aromaticity in general for all conjugated systems. This then is the
universal goal for researchers into the field of chemistry. Quantum chemistry is
expanding today and being used to determine structure and reactivity of molecules.
Erich Hückel has been an influential figure in the areas of organic and quantum
chemistry. His findings in quantum chemistry are substantial. Even though recognition of
his findings are not well acknowledged outside of the fields of chemistry and physics. His
passing in 1980 was a tragedy in that he was never recognized for his contributions to
chemistry. Roald Hoffmann continued with Hückel’s results and developed the Extended
Hückel Theory winning him the 1981 Nobel Prize. The advancement of the Molecular
Orbital Theory caused the quantitative ideas to become obsolete. The use of Hückel’s
rule still plays a major role in the qualitative area of findings.
Over the past years the use of technology and computers has made Hückel’s rule
obsolete but his influence in the field of chemistry is still felt. The theory behind his
approach was the first time anyone looked into larger organic molecules, and thus leading
to newer and better variations. The influence of Erich Hückel will be felt for many years,
especially in the field of organic chemistry which uses his rule for qualitatively finding if
a compound is aromatic. The findings of Erich Hückel will be continued to be expanded
upon and the cornerstone of his rule will be a substantial element for the future of
quantum chemistry and aromaticity.
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Figure 1.
Table 1.
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Work Citied
(1) "Erich Hückel- Scientist." Chemistry. 27 May 2004. BBC. 14 Dec 2006
<http://www.bbc.co.uk/dna/h2g2/A2390005>.
(2) Singleton, Scott. "Physical Organic Chemistry." Chem 445 . 29 Nov 1999.
Rice university. 14 Dec 2006 <http://www.owlnet.rice.edu/~chem445/>.
(3) Anders, U. "John A. Pople ." The Origin of PPP Theory. 10 Mar 2002.
www.Quantum-Chemistry-History.com. 14 Dec 2006 <http://www.quantumchemistry-history.com/Popl_Dat/Pople2.htm>.
(4) Hirsch, A.; Chen, Z.; Jiao, H. Angew. Chem. Int. Ed. 2000, 39, No. 21, 3917
(5) Cyrański, M.; Howard, S.; Chodkiewicz, M. Royal Chem. Soc. 2004, 2458
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