The Hückel method or Hückel molecular orbital theory, proposed by
Erich Hückel in 1930, is a simple
method for calculating molecular orbitals as linear combinations of atomic orbitals. The theory predicts the
molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of findin ...
s for
π-electrons in
π-delocalized molecules, such as
ethylene,
benzene
Benzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen atoms ...
,
butadiene
1,3-Butadiene () is the organic compound with the formula (CH2=CH)2. It is a colorless gas that is easily condensed to a liquid. It is important industrially as a precursor to synthetic rubber. The molecule can be viewed as the union of two vi ...
, and
pyridine
Pyridine is a basic heterocyclic organic compound with the chemical formula . It is structurally related to benzene, with one methine group replaced by a nitrogen atom. It is a highly flammable, weakly alkaline, water-miscible liquid with a ...
. It provides the theoretical basis for
Hückel's rule
In organic chemistry, Hückel's rule predicts that a planar ring molecule will have aromatic properties if it has 4''n'' + 2 π electrons, where ''n'' is a non-negative integer. The quantum mechanical basis for its formulation was ...
that cyclic, planar molecules or ions with
π-electrons are
aromatic
In chemistry, aromaticity is a chemical property of cyclic ( ring-shaped), ''typically'' planar (flat) molecular structures with pi bonds in resonance (those containing delocalized electrons) that gives increased stability compared to satur ...
. It was later extended to
conjugated molecules such as
pyridine
Pyridine is a basic heterocyclic organic compound with the chemical formula . It is structurally related to benzene, with one methine group replaced by a nitrogen atom. It is a highly flammable, weakly alkaline, water-miscible liquid with a ...
,
pyrrole and
furan that contain atoms other than carbon and hydrogen (
heteroatom
In chemistry, a heteroatom () is, strictly, any atom that is not carbon or hydrogen.
Organic chemistry
In practice, the term is usually used more specifically to indicate that non-carbon atoms have replaced carbon in the backbone of the molecula ...
s). A more dramatic extension of the method to include σ-electrons, known as the
extended Hückel method (EHM), was developed by
Roald Hoffmann
Roald Hoffmann (born Roald Safran; July 18, 1937) is a Polish-American theoretical chemist who won the 1981 Nobel Prize in Chemistry. He has also published plays and poetry. He is the Frank H. T. Rhodes Professor of Humane Letters, Emeritus, at ...
. The extended Hückel method gives some degree of quantitative accuracy for organic molecules in general (not just planar systems) and was used to provide computational justification for the
Woodward–Hoffmann rules The Woodward–Hoffmann rules (or the pericyclic selection rules), devised by Robert Burns Woodward and Roald Hoffmann, are a set of rules used to rationalize or predict certain aspects of the stereochemistry and activation energy of pericyclic rea ...
. To distinguish the original approach from Hoffmann's extension, the Hückel method is also known as the simple Hückel method (SHM).
In spite of its simplicity, the Hückel method in its original form makes qualitatively accurate and chemically useful predictions for many common molecules and is therefore a powerful and widely taught educational tool. It is described in many introductory
quantum chemistry and
physical organic chemistry
Physical organic chemistry, a term coined by Louis Hammett in 1940, refers to a discipline of organic chemistry that focuses on the relationship between chemical structures and reactivity, in particular, applying experimental tools of physical c ...
textbooks, and organic chemists in particular still routinely apply Hückel theory to obtain a very approximate,
back-of-the-envelope understanding of π-bonding.
Hückel characteristics
The method has several characteristics:
* It limits itself to
conjugated hydrocarbons.
* Only
π electron molecular orbitals are included because these determine much of the chemical and spectral properties of these molecules. The
σ electrons are assumed to form the framework of the molecule and σ connectivity is used to determine whether two π orbitals interact. However, the orbitals formed by σ electrons are ignored and assumed not to interact with π electrons. This is referred to as σ-π separability. It is justified by the
orthogonality of σ and π orbitals in planar molecules. For this reason, the Hückel method is limited to systems that are planar or nearly so.
* The method is based on applying the
variational method
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
to
linear combination of atomic orbitals
A linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry. In quantum mechanics, electron configurations of atoms are described as wavefun ...
and making simplifying assumptions regarding the overlap, resonance and Coulomb integrals of these atomic orbitals. It does not attempt to solve the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
, and neither the functional form of the basis atomic orbitals nor details of the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
are involved.
* For hydrocarbons, the method takes atomic connectivity as the only input; empirical parameters are only needed when heteroatoms are introduced.
* The method predicts how many energy levels exist for a given molecule, which levels are
degenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
and it expresses the molecular orbital energies in terms of two parameters, called α, the energy of an electron in a 2p orbital, and β, the interaction energy between two 2p orbitals (the extent to which an electron is stabilized by allowing it to delocalize between two orbitals). The usual sign convention is to let both α and β be negative numbers. To understand and compare systems in a qualitative or even semi-quantitative sense, explicit numerical values for these parameters are typically not required.
* In addition the method also enables calculation of
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in ...
for each atom in the π framework, the fractional
bond order between any two atoms, and the overall
molecular dipole moment
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
*An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system i ...
.
Hückel results
Results for simple molecules and general results for cyclic and linear systems
The results for a few simple molecules are tabulated below:
The theory predicts two energy levels for
ethylene with its two
π electrons filling the low-energy
HOMO
''Homo'' () is the genus that emerged in the (otherwise extinct) genus '' Australopithecus'' that encompasses the extant species ''Homo sapiens'' ( modern humans), plus several extinct species classified as either ancestral to or closely relat ...
and the high energy
LUMO
In chemistry, HOMO and LUMO are types of molecular orbitals. The acronyms stand for ''highest occupied molecular orbital'' and ''lowest unoccupied molecular orbital'', respectively. HOMO and LUMO are sometimes collectively called the ''frontie ...
remaining empty. In
butadiene
1,3-Butadiene () is the organic compound with the formula (CH2=CH)2. It is a colorless gas that is easily condensed to a liquid. It is important industrially as a precursor to synthetic rubber. The molecule can be viewed as the union of two vi ...
the 4 π-electrons occupy 2 low energy molecular orbitals, out of a total of 4, and for
benzene
Benzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen atoms ...
6 energy levels are predicted, two of them
degenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
.
For linear and cyclic systems (with ''N'' atoms), general solutions exist:
:* Linear system (
polyene/polyenyl):
.
:** Energy levels are all distinct.
:* Cyclic system,
Hückel topology (
annulene/annulenyl):
.
:** Energy levels
are each doubly degenerate.
:* Cyclic system,
Möbius topology (''hypothetical for N'' < 8):
.
:** Energy levels
are each doubly degenerate.
The energy levels for cyclic systems can be predicted using the mnemonic (named after the American chemist ). A circle centered at α with radius 2β is inscribed with a regular ''N-''gon with one vertex pointing down; the ''y''-coordinate of the vertices of the polygon then represent the orbital energies of the
'N''nnulene/annulenyl system. Related mnemonics exists for linear and Möbius systems.
The values of α and β
The value of α is the energy of an electron in a 2p orbital, relative to an unbound electron at infinity. This quantity is negative, since the electron is stabilized by being electrostatically bound to the positively charged nucleus. For carbon this value is known to be approximately –11.4 eV. Since Hückel theory is generally only interested in energies relative to a reference localized system, the value of α is often immaterial and can be set to zero without affecting any conclusions.
Roughly speaking, β physically represents the energy of stabilization experienced by an electron allowed to delocalize in a π molecular orbital formed from the 2p orbitals of adjacent atoms, compared to being localized in an isolated 2p atomic orbital. As such, it is also a negative number, although it is often spoken of in terms of its absolute value. The value for , β, in Hückel theory is roughly constant for structurally similar compounds, but not surprisingly, structurally dissimilar compounds will give very different values for , β, . For example, using the π bond energy of ethylene (65 kcal/mole) and comparing the energy of a doubly-occupied π orbital (2α + 2β) with the energy of electrons in two isolated p orbitals (2α), a value of , β, = 32.5 kcal/mole can be inferred. On the other hand, using the resonance energy of benzene (36 kcal/mole, derived from heats of hydrogenation) and comparing benzene (6α + 8β) with a hypothetical "non-aromatic 1,3,5-cyclohexatriene" (6α + 6β), a much smaller value of , β, = 18 kcal/mole emerges. These differences are not surprising, given the substantially shorter bond length of ethylene (1.33 Å) compared to benzene (1.40 Å). The shorter distance between the interacting p orbitals accounts for the greater energy of interaction, which is reflected by a higher value of , β, . Nevertheless, heat of hydrogenation measurements of various
polycyclic aromatic hydrocarbon
A polycyclic aromatic hydrocarbon (PAH) is a class of organic compounds that is composed of multiple aromatic rings. The simplest representative is naphthalene, having two aromatic rings and the three-ring compounds anthracene and phenanthrene. ...
s like naphthalene and anthracene all imply values of , β, between 17 and 20 kcal/mol.
However, even for the same compound, the correct assignment of , β, can be controversial. For instance, it is argued that the resonance energy measured experimentally via heats of hydrogenation is diminished by the distortions in bond lengths that must take place going from the single and double bonds of "non-aromatic 1,3,5-cyclohexatriene" to the delocalized bonds of benzene. Taking this distortion energy into account, the value of , β, for delocalization without geometric change (called the "vertical resonance energy") for benzene is found to be around 37 kcal/mole. On the other hand, experimental measurements of electronic spectra have given a value of , β, (called the "spectroscopic resonance energy") as high as 3 eV (~70 kcal/mole) for benzene. Given these subtleties, qualifications, and ambiguities, Hückel theory should not be called upon to provide accurate quantitative predictions – only semi-quantitative or qualitative trends and comparisons are reliable and robust.
Other successful predictions
With this caveat in mind, many predictions of the theory have been experimentally verified:
* The HOMO–LUMO gap, in terms of the β constant, correlates directly with the respective
molecular electronic transitions observed with
UV/VIS spectroscopy. For linear
polyene
In organic chemistry, polyenes are poly- unsaturated, organic compounds that contain at least three alternating double () and single () carbon–carbon bonds. These carbon–carbon double bonds interact in a process known as conjugation, result ...
s, the energy gap is given as:
:
:from which a value for β can be obtained between −60 and −70
kcal
The calorie is a unit of energy. For historical reasons, two main definitions of "calorie" are in wide use. The large calorie, food calorie, or kilogram calorie was originally defined as the amount of heat needed to raise the temperature of o ...
/
mol (−250 to −290
kJ/mol).
* The predicted molecular orbital energies as stipulated by
Koopmans' theorem
Koopmans' theorem states that in closed-shell Hartree–Fock theory (HF), the first ionization energy of a molecular system is equal to the negative of the orbital energy of the highest occupied molecular orbital (HOMO). This theorem is named aft ...
correlate with
photoelectron spectroscopy
Photoemission spectroscopy (PES), also known as photoelectron spectroscopy, refers to energy measurement of electrons emitted from solids, gases or liquids by the photoelectric effect, in order to determine the binding energies of electrons in th ...
.
* The Hückel
delocalization energy correlates with the experimental
heat of combustion. This energy is defined as the difference between the total predicted π energy (in
benzene
Benzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen atoms ...
8β) and a hypothetical π energy in which all
ethylene units are assumed isolated, each contributing 2β (making benzene 3 × 2β = 6β).
* Molecules with molecular orbitals paired up such that only the sign differs (for example α ± β) are called ''
alternant hydrocarbons'' and have in common small
molecular dipole moment
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
*An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system i ...
s. This is in contrast to non-alternant hydrocarbons, such as
azulene
Azulene is an organic compound and an isomer of naphthalene. Naphthalene is colourless, whereas azulene is dark blue. Two terpenoids, vetivazulene (4,8-dimethyl-2-isopropylazulene) and guaiazulene (1,4-dimethyl-7-isopropylazulene), that featur ...
and
fulvene
Fulvene (pentafulvene) is a hydrocarbon with the formula (CH=CH)2C=CH2. It is a prototype of a cross-conjugated hydrocarbon. Fulvene is rarely encountered, but substituted derivatives (fulvenes) are numerous. They are mainly of interest as ligand ...
that have large
dipole moments. The Hückel theory is more accurate for alternant hydrocarbons.
* For
cyclobutadiene
Cyclobutadiene is an organic compound with the formula . It is very reactive owing to its tendency to dimerize. Although the parent compound has not been isolated, some substituted derivatives are robust and a single molecule of cyclobutadiene is ...
the theory predicts that the two high-energy electrons occupy a degenerate pair of molecular orbitals (following from
Hund's rules
In atomic physics, Hund's rules refers to a set of rules that German physicist Friedrich Hund formulated around 1927, which are used to determine the term symbol that corresponds to the ground state of a multi-electron atom. The first rule is e ...
) that are neither stabilized nor destabilized. Hence the square molecule would be a very reactive triplet
diradical
In chemistry, a diradical is a molecular species with two electrons occupying molecular orbitals (MOs) which are degenerate. The term "diradical" is mainly used to describe organic compounds, where most diradicals are extremely reactive and i ...
(the ground state is actually rectangular without degenerate orbitals). In fact, all cyclic conjugated hydrocarbons with a total of 4''n'' π-electrons share this molecular orbital pattern, and this forms the basis of
Hückel's rule
In organic chemistry, Hückel's rule predicts that a planar ring molecule will have aromatic properties if it has 4''n'' + 2 π electrons, where ''n'' is a non-negative integer. The quantum mechanical basis for its formulation was ...
.
*
Dewar reactivity numbers deriving from the Hückel approach correctly predict the reactivity of aromatic systems with
nucleophiles
In chemistry, a nucleophile is a chemical species that forms bonds by donating an electron pair. All molecules and ions with a free pair of electrons or at least one pi bond can act as nucleophiles. Because nucleophiles donate electrons, they are ...
and
electrophiles
In chemistry, an electrophile is a chemical species that forms bonds with nucleophiles by accepting an electron pair. Because electrophiles accept electrons, they are Lewis acids. Most electrophiles are positively charged, have an atom that carri ...
.
*The benzyl cation and anion serve as simple models for arenes with electron-withdrawing and electron-donating groups, respectively. The π-electron population correctly implies the ''meta''- and ''ortho-''/''para-''selectivity for electrophilic aromatic substitution of π electron-poor and π electron-rich arenes, respectively.
Application in Optical Activity Analysis
The analysis of the optical activity of a molecule depends to a certain extent on the study of its chiral characteristics. However, for achiral molecules applying pesudoscalars to simplify the calculations of optical avtivity cannot be achieved due to the lack of spatial average.
Instead of traditional chiroptical solution measurements, Hückel theory helps focus on oriented π systems by separating from σ electrons especially in the planar,
-symmetric cases. Transition dipole moments derived by multiplying each wavefunction of individual planar molecule one by one, contribute to the directions of the most optical activity, where sit at the bisectors of two orthogonal ones. Despite of the zero value for the trace of the tensor, cis-butadiene shows considerable offdiagonal component which was computated as the first optical avtivity evaluation of achiral molecule.
By using 3,5-dimethylene-1-cyclopentene as an example. Transition electric dipole, magnetic dipole and electric quadrupole moments interactions result in optical rotation(OR), which can be described by both tensor components and chemical geometries. The in phase overlap of two molecular orbitals yield negative charge while depleting charge out of phase. The movement can be interpreted quantitatively by corresponding π and π* orbitals coefficients.
Delocalization energy, π-bond orders, and π-electron populations
The delocalization energy, π-bond orders, and π-electron population are chemically significant parameters that can be gleaned from the orbital energies and coefficients that are the direct outputs of Hückel theory. These are quantities strictly derived from theory, as opposed to measurable physical properties, though they correlate with measurable qualitative and quantitative properties of the chemical species. Delocalization energy is defined as the difference in energy between that of the most stable localized Lewis structure and the energy of the molecule computed from Hückel theory orbital energies and occupancies. Since all energies are relative, we set
without loss of generality to simplify discussion. The energy of the localized structure is then set to be 2β for every two-electron localized π-bond. The Hückel energy of the molecule is
, where the sum is over all Hückel orbitals,
is the occupancy of orbital ''i'', set to be 2 for doubly-occupied orbitals, 1 for singly-occupied orbitals, and 0 for unoccupied orbitals, and
is the energy of orbital ''i''. Thus, the delocalization energy, conventionally a positive number, is defined as
:
.
In the case of benzene, the occupied orbitals have energies (again setting
) 2β, β, and β. This gives the Hückel energy of benzene as
. Each Kekulé structure of benzene has three double bonds, so the localized structure is assigned an energy of
. The delocalization energy, measured in units of
, is then
.
The π-bond orders derived from Hückel theory are defined using the orbital coefficients of the Hückel MOs. The π-bond order between atoms ''j'' and ''k'' is defined as
:
,
where
is again the orbital occupancy of orbital ''i'' and
and
are the coefficients on atoms ''j'' and ''k'', respectively, for orbital ''i''. For benzene, the three occupied MOs, expressed as linear combinations of AOs
, are:
:
,
E=2\beta">math>E=2\beta
:
,
E=\beta">math>E=\beta
:
,
E=\beta">math>E=\beta
Perhaps surprisingly, the π-bond order formula gives a bond order of
:
for the bond between carbons 1 and 2. The resulting total (σ + π) bond order of
is the same between any other pair of adjacent carbon atoms. This is more than the naive π-bond order of
(for a total bond order of
) that one might guess when simply considering the Kekulé structures and the usual definition of bond order in valence bond theory. The Hückel definition of bond order attempts to quantify any additional stabilization that the system enjoys resulting from delocalization. In a sense, the Hückel bond order suggests that there are four π-bonds in benzene instead of the three that are implied by the Kekulé-type Lewis structures. The "extra" bond is attributed to the additional stabilization that results from the aromaticity of the benzene molecule. (This is only one of several definitions for non-integral bond orders, and other definitions will lead to different values that fall between 1 and 2.)
The π-electron population is calculated in a very similar way to the bond order using the orbital coefficients of the Hückel MOs. The π-electron population on atom ''j'' is defined as
:
.
The associated Hückel Coulomb charge is defined as
, where
is the number of π-electrons contributed by a neutral, sp
2-hybridized atom ''j'' (we always have
for carbon).
For carbon 1 on benzene, this yields a π-electron population of
:
.
Since each carbon atom contributes one π-electron to the molecule, this gives a Coulomb charge of 0 for carbon 1 (and all other carbon atoms), as expected.
In the cases of benzyl cation and benzyl anion shown above,
:
and
,
:
and
.
Mathematics behind the Hückel method
The mathematics of the Hückel method is based on the
Ritz method
The Ritz method is a direct method to find an approximate solution for boundary value problems. The method is named after Walther Ritz, and is also commonly called the Rayleigh–Ritz method and the Ritz-Galerkin method.
In quantum mechanics, ...
. In short, given a basis set of ''n'' normalized atomic orbitals
, an ''ansatz'' molecular orbital
is written down, with normalization constant ''N'' and coefficients
which are to be determined. In other words, we are assuming that the molecular orbital (MO) can be written as a linear combination of atomic orbitals, a conceptually intuitive and convenient approximation (the
linear combination of atomic orbitals
A linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry. In quantum mechanics, electron configurations of atoms are described as wavefun ...
or LCAO approximation). The
variational theorem states that given an eigenvalue problem
with smallest eigenvalue
and corresponding wavefunction
, any normalized trial wavefunction
(i.e.,
holds) will satisfy
:
,
with equality holding if and only if
. Thus, by minimizing