Revision as of 23:34, 23 May 2026

← Back to Foundations Book I

A linear combination of atomic orbitals (LCAO) is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry.^[1] In quantum mechanics, electron configurations 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, orbital wavefunctions are modified, i.e. the electron cloud shape is changed, according to the type of atoms participating in the chemical bond.

It was introduced in 1929 by Sir John Lennard-Jones 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 H₂⁺.^[2]^[3]

linear combination of atomic orbitals in the Quantum Collection.

Mathematical description

The reducible representation of the bonding of water with C_2v symmetry.

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:

ϕ_{i} = c_{1 i} χ_{1} + c_{2 i} χ_{2} + c_{3 i} χ_{3} + \dots + c_{n i} χ_{n}

or

ϕ_{i} = \sum_{r} c_{r i} χ_{r}

where $ϕ_{i}$ is a molecular orbital represented as the sum of n atomic orbitals $χ_{r}$ , each multiplied by a corresponding coefficient $c_{r i}$ , 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 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 functions which may or may not be centered on the nuclei of the component atoms of the molecule. 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 atoms since these are known analytically i.e. Slater-type orbitals but other choices are possible such as the Gaussian functions from standard basis sets or the pseudo-atomic orbitals from plane-wave pseudopotentials.

By minimizing the total energy 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, 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 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.

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 extended Hückel method and the Pariser–Parr–Pople method, provide some quantitative theories.

@@ Line 10: / Line 10: @@
 <div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
-A '''linear combination of atomic orbitals''' ('''LCAO''') is a [[Physics:Quantum superposition|quantum superposition]] of atomic orbitals and a technique for calculating [[Chemistry:Molecular orbital|molecular orbital]]s in [[Chemistry:Quantum chemistry|quantum chemistry]].<ref>Huheey, James. ''Inorganic Chemistry:Principles of Structure and Reactivity''</ref> In quantum mechanics, electron configurations of atoms are described as wavefunctions. In a mathematical sense, these wave functions are the [[Chemistry:Basis set|basis set]] of functions, the basis functions, which describe the electrons of a given atom. In chemical reactions, orbital wavefunctions are modified, i.e. the electron cloud shape is changed, according to the type of atoms participating in the chemical bond.
+A '''linear combination of atomic orbitals''' ('''LCAO''') is a [[Physics:Quantum superposition|quantum superposition]] of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry.<ref>Huheey, James. ''Inorganic Chemistry:Principles of Structure and Reactivity''</ref> In quantum mechanics, electron configurations 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, orbital wavefunctions are modified, i.e. the electron cloud shape is changed, according to the type of atoms participating in the chemical bond.
-It was introduced in 1929 by Sir [[Biography:John Lennard-Jones|John Lennard-Jones]] with the description of bonding in the diatomic molecules of the first main row of the [[Chemistry:Periodic table|periodic table]], but had been used earlier by [[Biography:Linus Pauling|Linus Pauling]] for H<sub>2</sub><sup>+</sup>.<ref>[[Biography:Friedrich Hund|Friedrich Hund]] and Chemistry, [[Biography:Werner Kutzelnigg|Werner Kutzelnigg]], on the occasion of Hund's 100th birthday, [[Chemistry:Angewandte Chemie|Angewandte Chemie]], 35, 572–586, (1996), {{doi| 10.1002/anie.199605721}}</ref><ref>{{cite journal | last=Mulliken | first=Robert S. | title=Spectroscopy, Molecular Orbitals, and Chemical Bonding | journal=Science | publisher=American Association for the Advancement of Science (AAAS) | volume=157 | issue=3784 | date=1967-07-07 | issn=0036-8075 | doi=10.1126/science.157.3784.13 | pages=13–24| pmid=5338306 | bibcode=1967Sci...157...13M|url=https://www.nobelprize.org/uploads/2018/06/mulliken-lecture.pdf}}</ref>
+It was introduced in 1929 by Sir [[Biography:John Lennard-Jones|John Lennard-Jones]] with the description of bonding in the diatomic molecules of the first main row of the periodic table, but had been used earlier by [[Biography:Linus Pauling|Linus Pauling]] for H<sub>2</sub><sup>+</sup>.<ref>[[Biography:Friedrich Hund|Friedrich Hund]] and Chemistry, [[Biography:Werner Kutzelnigg|Werner Kutzelnigg]], on the occasion of Hund's 100th birthday, Angewandte Chemie, 35, 572–586, (1996), {{doi| 10.1002/anie.199605721}}</ref><ref>{{cite journal | last=Mulliken | first=Robert S. | title=Spectroscopy, Molecular Orbitals, and Chemical Bonding | journal=Science | publisher=American Association for the Advancement of Science (AAAS) | volume=157 | issue=3784 | date=1967-07-07 | issn=0036-8075 | doi=10.1126/science.157.3784.13 | pages=13–24| pmid=5338306 | bibcode=1967Sci...157...13M|url=https://www.nobelprize.org/uploads/2018/06/mulliken-lecture.pdf}}</ref>
 </div>
 <div style="width:300px;">
-[[File:File not found.png|thumb|280px|No image available.]]
+[[File:CharakterH2Oa.svg|thumb|280px|linear combination of atomic orbitals in the Quantum Collection.]]
 </div>
@@ Line 22: / Line 22: @@
 == Mathematical description ==
-[[File:CharakterH2Oa.svg|thumb|500x500px|The [[reducible representation]] of the bonding of water with [[Cyclic group|C<sub>2v</sub> symmetry]]. ]]
+[[File:CharakterH2Oa.svg|thumb|500x500px|The reducible representation of the bonding of water with C<sub>2v</sub> symmetry. ]]
 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:
@@ Line 31: / Line 31: @@
 : <math>\ \phi_i = \sum_{r} c_{ri} \chi_r </math>
-where <math>\ \phi_i </math> is a molecular orbital represented as the sum of ''n'' [[Physics:Atomic orbital|atomic orbital]]s <math>\ \chi_r </math>, each multiplied by a corresponding coefficient <math>\ c_{ri} </math>, 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 [[Physics:Hartree–Fock method|Hartree–Fock method]] is used to obtain the coefficients of the expansion.
+where <math>\ \phi_i </math> is a molecular orbital represented as the sum of ''n'' atomic orbitals <math>\ \chi_r </math>, each multiplied by a corresponding coefficient <math>\ c_{ri} </math>, 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 is used to obtain the coefficients of the expansion.
-The orbitals are thus expressed as [[Linear combination|linear combination]]s of [[Basis function|basis function]]s, and the basis functions are single-[[Physics:Electron|electron]] functions which may or may not be centered on the [[Physics:Atomic nucleus|nuclei]] of the component [[Atom|atom]]s of the [[Physics:Molecule|molecule]]. 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|hydrogen-like atom]]s since these are known analytically i.e. [[Chemistry:Slater-type orbital|Slater-type orbital]]s but other choices are possible such as the [[Physics:Gaussian orbital|Gaussian functions]] from [[Standard basis|standard basis]] sets or the pseudo-atomic orbitals from plane-wave pseudopotentials.
+The orbitals are thus expressed as linear combinations of basis functions, and the basis functions are single-electron functions which may or may not be centered on the nuclei of the component atoms of the molecule. 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 atoms since these are known analytically i.e. Slater-type orbitals but other choices are possible such as the Gaussian functions from standard basis sets or the pseudo-atomic orbitals from plane-wave pseudopotentials.
 [[File:MO_Diagram.svg|thumb|250x250px|Example of a molecular orbital diagram.]]
-By minimizing the total [[Physics:Energy|energy]] of the system, an appropriate set of [[Coefficient|coefficient]]s of the linear combinations is determined. This quantitative approach is now known as the Hartree–Fock method. However, since the development of [[Chemistry:Computational chemistry|computational chemistry]], 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 [[Physics:Level repulsion|level repulsion]] 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 [[Chemistry:Koopmans' theorem|Koopmans' theorem]].
+By minimizing the total energy 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, 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 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.
-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|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.
+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.
-[[Chemistry:Molecular orbital diagram|Molecular orbital diagrams]] provide simple qualitative LCAO treatment. The [[Physics:Hückel method|Hückel method]], the extended Hückel method and the [[Physics:Pariser–Parr–Pople method|Pariser–Parr–Pople method]], provide some quantitative theories.
+Molecular orbital diagrams provide simple qualitative LCAO treatment. The Hückel method, the extended Hückel method and the Pariser–Parr–Pople method, provide some quantitative theories.
 == See also ==

Physics:Quantum linear combination of atomic orbitals: Difference between revisions

Revision as of 23:34, 23 May 2026

Mathematical description

See also

Table of contents (217 articles)

Index

Full contents

External links

References

Navigation menu

Search