Extended Huckel method
Encyclopedia
The extended Hückel method is a semiempirical quantum chemistry method, developed by Roald Hoffmann
since 1963. It is based on the Hückel method
but, while the original Hückel method only considers pi orbitals, the extended method also includes the sigma orbitals.
The extended Hückel method can be used for determining the molecular orbitals, but it is not very successful in determining the structural geometry
of an organic
molecule
. It can however determine the relative energy
of different geometrical configurations. It involves calculations of the electronic interaction
s in a rather simple way where the electron-electron repulsions are not explicitly included and the total energy is just a sum of terms for each electron in the molecule. The off-diagonal Hamiltonian matrix elements are given by an approximation due to Wolfsberg and Helmholz that relates them to the diagonal elements and the overlap matrix
element.
K is the Wolfsberg-Helmholtz constant, and is usually given a value of 1.75. In the extended Hückel method, only valence electrons are considered; the core electron energies and functions are supposed to be more or less constant between atoms of the same type. The method uses a series of parametrized energies calculated from atomic ionization potentials or theoretical methods to fill the diagonal of the Fock matrix. After filling the non-diagonal elements and diagonalizing the resulting Fock matrix, the energies (eigenvalues) and wavefunctions (eigenvectors) of the valence orbitals are found.
It is common in many theoretical studies to use the extended Hückel molecular orbitals as a preliminary step to determining the molecular orbitals by a more sophisticated method such as the CNDO/2
method and ab initio quantum chemistry methods
. Since the EHT basis set is fixed, the monoparticle calculated wavefunctions must be projected to the basis set where the accurate calculation is to be done. One usually does this by adjusting the orbitals in the new basis to the old ones by least squares method.
As only valence electron wavefunctions are found by this method, one must fill the core electron functions by orthonormalizing the rest of the basis set with the calculated orbitals and then selecting the ones with less energy. This leads to the determination of more accurate structures and electronic properties, or in the case of ab initio methods, to somewhat faster convergence.
The method was first used by Roald Hoffmann
who developed, with Robert Burns Woodward
, rules for elucidating reaction mechanism
s (the Woodward-Hoffmann rules
). He used pictures of the molecular orbitals from extended Hückel theory to work out the orbital interactions in these cycloaddition reactions.
A closely similar method was used earlier by Hoffmann and William Lipscomb
for studies of boron hydrides. The off-diagonal Hamiltonian matrix elements were given as proportional to the overlap integral.
This simplification of the Wolfsberg and Helmholz approximation is reasonable for boron hydrides as the diagonal elements are reasonably similar due to the small difference in electronegativity
between boron and hydrogen.
The method works poorly for molecules that contain atoms of very different electronegativity
. To overcome this weakness, several groups have suggested iterative schemes that depend on the atomic charge. One such method, that is still widely used in inorganic and organometallic chemistry is the Fenske-Hall method.
A recent program for the extended Hückel method is YAeHMOP which stands for "yet another extended Hückel molecular orbital package".
Roald Hoffmann
Roald Hoffmann is an American theoretical chemist who won the 1981 Nobel Prize in Chemistry. He currently teaches at Cornell University in Ithaca, New York.-Escape from the Holocaust:...
since 1963. It is based on the Hückel method
Hückel method
The Hückel method or Hückel molecular orbital method proposed by Erich Hückel in 1930, is a very simple linear combination of atomic orbitals molecular orbitals method for the determination of energies of molecular orbitals of pi electrons in conjugated hydrocarbon systems, such as ethene,...
but, while the original Hückel method only considers pi orbitals, the extended method also includes the sigma orbitals.
The extended Hückel method can be used for determining the molecular orbitals, but it is not very successful in determining the structural geometry
Molecular geometry
Molecular geometry or molecular structure is the three-dimensional arrangement of the atoms that constitute a molecule. It determines several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism, and biological activity.- Molecular geometry determination...
of an organic
Organic chemistry
Organic chemistry is a subdiscipline within chemistry involving the scientific study of the structure, properties, composition, reactions, and preparation of carbon-based compounds, hydrocarbons, and their derivatives...
molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
. It can however determine the relative energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
of different geometrical configurations. It involves calculations of the electronic interaction
Electronic correlation
Electronic correlation is the interaction between electrons in the electronic structure of a quantum system.- Atomic and molecular systems :...
s in a rather simple way where the electron-electron repulsions are not explicitly included and the total energy is just a sum of terms for each electron in the molecule. The off-diagonal Hamiltonian matrix elements are given by an approximation due to Wolfsberg and Helmholz that relates them to the diagonal elements and the overlap matrix
Overlap matrix
The overlap matrix is a square matrix, used in quantum chemistry to describe the inter-relationship of a set of basis vectors of a quantum system. In particular, if the vectors are orthogonal to one another, the overlap matrix will be diagonal. In addition, if the basis vectors form an...
element.
- Hij = K Sij (Hii + Hjj)/2
K is the Wolfsberg-Helmholtz constant, and is usually given a value of 1.75. In the extended Hückel method, only valence electrons are considered; the core electron energies and functions are supposed to be more or less constant between atoms of the same type. The method uses a series of parametrized energies calculated from atomic ionization potentials or theoretical methods to fill the diagonal of the Fock matrix. After filling the non-diagonal elements and diagonalizing the resulting Fock matrix, the energies (eigenvalues) and wavefunctions (eigenvectors) of the valence orbitals are found.
It is common in many theoretical studies to use the extended Hückel molecular orbitals as a preliminary step to determining the molecular orbitals by a more sophisticated method such as the CNDO/2
CNDO/2
CNDO is the abbreviation for Complete Neglect of Differential Overlap. Although CNDO is based on quantum chemistry, it is more specifically one of the first semi-empirical quantum chemistry methods. It uses two approximations:...
method and ab initio quantum chemistry methods
Ab initio quantum chemistry methods
Ab initio quantum chemistry methods are computational chemistry methods based on quantum chemistry. The term ab initiowas first used in quantum chemistry by Robert Parr and coworkers, including David Craig in a semiempirical study on the excited states of benzene.The background is described by Parr...
. Since the EHT basis set is fixed, the monoparticle calculated wavefunctions must be projected to the basis set where the accurate calculation is to be done. One usually does this by adjusting the orbitals in the new basis to the old ones by least squares method.
As only valence electron wavefunctions are found by this method, one must fill the core electron functions by orthonormalizing the rest of the basis set with the calculated orbitals and then selecting the ones with less energy. This leads to the determination of more accurate structures and electronic properties, or in the case of ab initio methods, to somewhat faster convergence.
The method was first used by Roald Hoffmann
Roald Hoffmann
Roald Hoffmann is an American theoretical chemist who won the 1981 Nobel Prize in Chemistry. He currently teaches at Cornell University in Ithaca, New York.-Escape from the Holocaust:...
who developed, with Robert Burns Woodward
Robert Burns Woodward
Robert Burns Woodward was an American organic chemist, considered by many to be the preeminent organic chemist of the twentieth century...
, rules for elucidating reaction mechanism
Reaction mechanism
In chemistry, a reaction mechanism is the step by step sequence of elementary reactions by which overall chemical change occurs.Although only the net chemical change is directly observable for most chemical reactions, experiments can often be designed that suggest the possible sequence of steps in...
s (the Woodward-Hoffmann rules
Woodward-Hoffmann rules
The Woodward–Hoffmann rules devised by Robert Burns Woodward and Roald Hoffmann are a set of rules in organic chemistry predicting the stereochemistry of pericyclic reactions based on orbital symmetry. These include electrocyclic reactions, cycloadditions , sigmatropic reactions, and group transfer...
). He used pictures of the molecular orbitals from extended Hückel theory to work out the orbital interactions in these cycloaddition reactions.
A closely similar method was used earlier by Hoffmann and William Lipscomb
William Lipscomb
William Nunn Lipscomb, Jr. was a Nobel Prize-winning American inorganic and organic chemist working in nuclear magnetic resonance, theoretical chemistry, boron chemistry, and biochemistry.-Overview:...
for studies of boron hydrides. The off-diagonal Hamiltonian matrix elements were given as proportional to the overlap integral.
-
- Hij = K Sij.
This simplification of the Wolfsberg and Helmholz approximation is reasonable for boron hydrides as the diagonal elements are reasonably similar due to the small difference in electronegativity
Electronegativity
Electronegativity, symbol χ , is a chemical property that describes the tendency of an atom or a functional group to attract electrons towards itself. An atom's electronegativity is affected by both its atomic number and the distance that its valence electrons reside from the charged nucleus...
between boron and hydrogen.
The method works poorly for molecules that contain atoms of very different electronegativity
Electronegativity
Electronegativity, symbol χ , is a chemical property that describes the tendency of an atom or a functional group to attract electrons towards itself. An atom's electronegativity is affected by both its atomic number and the distance that its valence electrons reside from the charged nucleus...
. To overcome this weakness, several groups have suggested iterative schemes that depend on the atomic charge. One such method, that is still widely used in inorganic and organometallic chemistry is the Fenske-Hall method.
A recent program for the extended Hückel method is YAeHMOP which stands for "yet another extended Hückel molecular orbital package".