A linear combination of atomic orbitals or LCAO is a
quantum superposition of
atomic orbitals
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any sp ...
and a technique for calculating
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 in
quantum chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
. In quantum mechanics,
electron configurations
In atomic physics and quantum chemistry, the electron configuration is the distribution of electrons of an atom or molecule (or other physical structure) in atomic or molecular orbitals. For example, the electron configuration of the neon at ...
of atoms are described as
wavefunctions. In a mathematical sense, these wave functions are the
basis set of functions, the basis functions, which describe the electrons of a given atom. In
chemical reactions
A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaking ...
, orbital wavefunctions are modified, i.e. the
electron cloud
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any spe ...
shape is changed, according to the type of atoms participating in the
chemical bond
A chemical bond is a lasting attraction between atoms or ions that enables the formation of molecules and crystals. The bond may result from the electrostatic force between oppositely charged ions as in ionic bonds, or through the sharing of ...
.
It was introduced in 1929 by Sir
John Lennard-Jones
Sir John Edward Lennard-Jones (27 October 1894 – 1 November 1954) was a British mathematician and professor of theoretical physics at the University of Bristol, and then of theoretical science at the University of Cambridge. He was an imp ...
with the description of bonding in the diatomic molecules of the first main row of the periodic table, but had been used earlier by
Linus Pauling for H
2+.
Mathematical description
![CharakterH2Oa](https://upload.wikimedia.org/wikipedia/commons/9/9f/CharakterH2Oa.svg)
An initial assumption is that the number of molecular orbitals is equal to the number of atomic orbitals included in the linear expansion. In a sense, ''n'' atomic orbitals combine to form ''n'' molecular orbitals, which can be numbered ''i'' = 1 to ''n'' and which may not all be the same. The expression (linear expansion) for the ''i'' th molecular orbital would be:
:
or
:
where
is a molecular orbital represented as the sum of ''n''
atomic orbital
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any spe ...
s
, each multiplied by a corresponding coefficient
, and ''r'' (numbered 1 to ''n'') represents which atomic orbital is combined in the term. The coefficients are the weights of the contributions of the n atomic orbitals to the molecular orbital. The
Hartree–Fock method
In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.
The Hartree–Fock method often ...
is used to obtain the coefficients of the expansion.
The orbitals are thus expressed as
linear combinations of
basis functions, and the basis functions are single-
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
functions which may or may not be centered on the
nuclei of the component
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
s of the
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
. In either case the basis functions are usually also referred to as atomic orbitals (even though only in the former case this name seems to be adequate). The atomic orbitals used are typically those of
hydrogen-like atom
A hydrogen-like atom (or hydrogenic atom) is any atom or ion with a single valence electron. These atoms are isoelectronic with hydrogen. Examples of hydrogen-like atoms include, but are not limited to, hydrogen itself, all alkali metals such a ...
s since these are known analytically i.e.
Slater-type orbital Slater-type orbitals (STOs) are functions used as atomic orbitals in the linear combination of atomic orbitals molecular orbital method. They are named after the physicist John C. Slater, who introduced them in 1930.
They possess exponential deca ...
s but other choices are possible such as the
Gaussian functions from standard basis sets or the pseudo-atomic orbitals from plane-wave pseudopotentials.
![MO_Diagram](https://upload.wikimedia.org/wikipedia/commons/e/e7/MO_Diagram.svg)
By minimizing the total
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
of the system, an appropriate set of
coefficients of the linear combinations is determined. This quantitative approach is now known as the Hartree–Fock method. However, since the development of
computational chemistry
Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of m ...
, the LCAO method often refers not to an actual optimization of the wave function but to a qualitative discussion which is very useful for predicting and rationalizing results obtained via more modern methods. In this case, the shape of the molecular orbitals and their respective energies are deduced approximately from comparing the energies of the atomic orbitals of the individual atoms (or molecular fragments) and applying some recipes known as
level repulsion Level repulsion is the quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantu ...
and the like. The graphs that are plotted to make this discussion clearer are called correlation diagrams. The required atomic orbital energies can come from calculations or directly from experiment via
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 ...
.
This is done by using the symmetry of the molecules and orbitals involved in bonding, and thus is sometimes called ''symmetry adapted linear combination'' (SALC). The first step in this process is assigning a
point group to the molecule. Each operation in the point group is performed upon the molecule. The number of bonds that are unmoved is the character of that operation. This reducible representation is decomposed into the sum of irreducible representations. These irreducible representations correspond to the symmetry of the orbitals involved.
Molecular orbital diagrams provide simple qualitative LCAO treatment. The
Hückel method
The Hückel method or Hückel molecular orbital theory, proposed by Erich Hückel in 1930, is a simple LCAO MO Method, method for calculating molecular orbitals as linear combinations of atomic orbitals. The theory predicts the molecular orbitals f ...
, the
extended Hückel method
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 ...
and the
Pariser–Parr–Pople method
In molecular physics, the Pariser–Parr–Pople method applies Computational Chemistry#Semiempirical methods, semi-empirical quantum mechanical methods to the quantitative prediction of electronic structures and spectra, in molecules of interest i ...
, provide some quantitative theories.
See also
*
Quantum chemistry computer programs
Quantum chemistry computer programs are used in computational chemistry to implement the methods of quantum chemistry. Most include the Hartree–Fock (HF) and some post-Hartree–Fock methods. They may also include density functional theory (DF ...
*
Hartree–Fock method
In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.
The Hartree–Fock method often ...
*
Basis set (chemistry)
In theoretical and computational chemistry, a basis set is a set of functions (called basis functions) that is used to represent the electronic wave function in the Hartree–Fock method or density-functional theory in order to turn the parti ...
*
Tight binding
In solid-state physics, the tight-binding model (or TB model) is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at eac ...
*
Holstein–Herring method The Holstein– Herring method, also called the surface Integral method, or Smirnov's method is an effective means of getting the exchange energy splittings of asymptotically degenerate energy states in molecular systems. Although the exchange energ ...
External links
* LCAO @ chemistry.umeche.maine.ed
Link
References
{{DEFAULTSORT:Linear Combination Of Atomic Orbitals Molecular Orbital Method
Chemical bonding
Physical chemistry
Electronic structure methods