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Linear combination of atomic orbitals molecular orbital method

 
Linear Combination of Atomic Orbitals Molecular Orbital Method

 
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Linear combination of atomic orbitals molecular orbital method
 
 
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A linear combination of atomic orbitals or LCAO is a quantum superposition
Quantum superposition

Quantum superposition is the fundamental law of quantum mechanics. It defines the allowed state space of a quantum mechanical system.In Probability theory, every possible event has a non-negative real number between zero and one associated to it, the probability, which gives the chance that it happens....
 of atomic orbitals and a technique for calculating molecular orbital
Molecular orbital

In chemistry, a molecular orbital is a mathematical function that describes the 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 finding an electron in any specific region....
s in quantum chemistry
Quantum chemistry

Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry....
. In quantum mechanics, electron configurations of atoms are described as wavefunctions. In mathematical sense, these wave functions are the basis set
Basis set (chemistry)

A basis set in chemistry is a set of functions used to create the molecular orbitals, which are expanded as a linear combination of atomic orbitals of such functions with the weights or coefficients to be determined....
 of functions, the basis functions, which describe the electrons of a given atom. In chemical reactions, orbital wavefunctions are modified, i.e.






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A linear combination of atomic orbitals or LCAO is a quantum superposition
Quantum superposition

Quantum superposition is the fundamental law of quantum mechanics. It defines the allowed state space of a quantum mechanical system.In Probability theory, every possible event has a non-negative real number between zero and one associated to it, the probability, which gives the chance that it happens....
 of atomic orbitals and a technique for calculating molecular orbital
Molecular orbital

In chemistry, a molecular orbital is a mathematical function that describes the 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 finding an electron in any specific region....
s in quantum chemistry
Quantum chemistry

Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry....
. In quantum mechanics, electron configurations of atoms are described as wavefunctions. In mathematical sense, these wave functions are the basis set
Basis set (chemistry)

A basis set in chemistry is a set of functions used to create the molecular orbitals, which are expanded as a linear combination of atomic orbitals of such functions with the weights or coefficients to be determined....
 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
Electron cloud

Electron cloud is not a term used by the Nobel Prize laureate and acclaimed educator Richard Feynman in The Feynman Lectures on Physics for discussing "exactly what is an electron?"....
 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
Linus Pauling

Linus Carl Pauling was an United States scientist, peace activist, author and list of educators. He was one of the most influential chemists in history and ranks among the most important scientists in any field of the 20th century....
 for H2+.

A mathematical description is



or



where (phi
Phi (letter)

Phi , pronounced [] in Modern Greek language and as [] in English, is the 21st letter of the Greek alphabet. In modern Greek, it represents [], a voiceless labiodental fricative....
) is a molecular orbital
Molecular orbital

In chemistry, a molecular orbital is a mathematical function that describes the 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 finding an electron in any specific region....
 represented as the sum
SUM

SUM can refer to:* The State University of Management* Soccer United Marketing* StartUp-Manager...
 of n atomic orbital
Atomic orbital

An atomic orbital is a mathematical function that describes the 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 specific region around the atom's nucleus....
s (chi
Chi (letter)

Chi is the 22nd letter of the Greek alphabet, pronounced as [kai] in English. Its value in Ancient Greek was an aspirated voiceless velar plosive ....
), each multiplied by a corresponding coefficient . The coefficients are the weights of the contributions of the n atomic orbitals to the molecular orbital. The Hartree-Fock
Hartree-Fock

In computational physics and computational chemistry, the Hartree-Fock method is an approximate method for the determination of the Stationary state wavefunction and Stationary state energy of a Many-body problem....
 procedure is used to obtain the coefficients of the expansion from the Hartree-Fock procedure.

The orbitals are thus expressed as linear combination
Linear combination

In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.Most of this article deals with linear combinations in the context of a vector space over a field , with some generalisations given at the end of the article....
s of basis function
Basis function

In mathematics, particularly numerical analysis, a basis function is an element of the Basis for a function space. The term is a degeneration of the term basis vector for a more general vector space; that is, each function in the function space can be represented as a linear combination of the basis functions....
s, and the basis functions
Basis set (chemistry)

A basis set in chemistry is a set of functions used to create the molecular orbitals, which are expanded as a linear combination of atomic orbitals of such functions with the weights or coefficients to be determined....
 are one-electron
Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
 functions centered on nuclei
Atomic nucleus

The nucleus of an atom is the very dense region, consisting of nucleons , at the center of an atom. Although the size of the nucleus varies considerably according to the mass of the atom, the size of the entire atom is comparatively constant....
 of the component atom
Atom

|-! bgcolor=gray | Properties|-||}The atom is a basic unit of matter consisting of a dense, central atomic nucleus surrounded by a electron cloud of electric charge electrons....
s of the molecule
Molecule

In chemistry, a molecule is defined as a sufficiently stable, electric charge neutral group of at least two atoms in a definite arrangement held together by very strong chemical bonds....
. The atomic orbitals used are typically those of hydrogen-like atom
Hydrogen-like atom

A hydrogen-like atom is an atom with one electron and thus is isoelectronic with hydrogen. Except for the hydrogen atom itself these atoms carry the positive charge e, where Z is the atomic number of the atom....
s since these are known analytically i.e. Slater-type orbital
Slater-type orbital

Slater-type orbitals are functions used as atomic orbitals in the linear combination of atomic orbitals molecular orbital method. They are named after the physics John C....
s but other choices are possible like Gaussian functions
Gaussian orbital

In computational chemistry and molecular physics, Gaussian orbitals are function s used as atomic orbitals in the linear combination of atomic orbitals molecular orbital method for the computation of electron orbitals in molecules....
 from standard basis sets
Basis set (chemistry)

A basis set in chemistry is a set of functions used to create the molecular orbitals, which are expanded as a linear combination of atomic orbitals of such functions with the weights or coefficients to be determined....
.

By minimizing the total energy
Energy

In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....
 of the system, an appropriate set of coefficient
Coefficient

In mathematics, a coefficient is a constant multiplication factor of a certain object. For example, in the expression 9x2, the coefficient of x2 is 9....
s of the linear combinations is determined. This quantitative approach is now known as the Hartree-Fock
Hartree-Fock

In computational physics and computational chemistry, the Hartree-Fock method is an approximate method for the determination of the Stationary state wavefunction and Stationary state energy of a Many-body problem....
 method. However, since the development of computational chemistry
Computational chemistry

Computational chemistry is a branch of chemistry that uses computers to assist in solving chemical problems. It uses the results of theoretical chemistry, incorporated into efficient computer programs, to calculate the structures and properties of molecules and solids....
, 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 orbital
Molecular orbital

In chemistry, a molecular orbital is a mathematical function that describes the 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 finding an electron in any specific region....
s and their respective energies are deduced approximately from comparing the energies of the atomic orbital
Atomic orbital

An atomic orbital is a mathematical function that describes the 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 specific region around the atom's nucleus....
s 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
Koopmans' theorem

Koopmans' theorem is an approximation in molecular orbital theory, such as density functional theory, or Hartree-Fock theory, in which the first ionization energy of a molecule is equal to the energy multiplied by -1, of the highest occupied molecular orbital , and the electron affinity is the negative of the energy of the lowest unoccupied,...
.

This is done by using the symmetry of the molecules and orbitals involved in bonding. The first step in this process is assigning a point group
Point group

In mathematics, a point group is a group of geometric symmetry leaving a point fixed....
 to the molecule. A common example is water, which is of C2v symmetry. Then a reducible representation of the bonding is determined demonstrated below for water
Water

Water is a common chemical substance that is essential for the survival of all known forms of life. In typical usage, water refers only to its liquid form or States of matter, but the substance also has a solid state, ice, and a gaseous state, water vapor or steam....
:

Lcaowater1
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.

MO diagram
MO diagram

A molecular orbital diagram or MO diagram for short is a qualitative descriptive tool explaining chemical bonding in molecules in terms of molecular orbital theory in general and the Linear combination of atomic orbitals molecular orbital method in particular ....
s provide simple qualitative LCAO treatment.

Quantitative theories are the Huckel method, the extended Huckel method
Extended Huckel method

The extended H?ckel method is a semiempirical quantum chemistry method, developed by Roald Hoffmann since 1963. It is based on the H?ckel method but, while the original H?ckel method only considers pi orbitals, the extended method also includes the sigma orbitals....
 and the Pariser–Parr–Pople method
Pariser–Parr–Pople method

In molecular physics, the Pariser?Parr?Pople method applies Computational Chemistry#Semiempirical methods quantum mechanical methods to the quantitative prediction of electronic structures and spectra, in molecules of interest in the field of organic chemistry....
.

See also

  • Quantum chemistry computer programs
    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 and some post-Hartree-Fock methods....
  • Hartree-Fock
    Hartree-Fock

    In computational physics and computational chemistry, the Hartree-Fock method is an approximate method for the determination of the Stationary state wavefunction and Stationary state energy of a Many-body problem....
  • Basis set (chemistry)
    Basis set (chemistry)

    A basis set in chemistry is a set of functions used to create the molecular orbitals, which are expanded as a linear combination of atomic orbitals of such functions with the weights or coefficients to be determined....
  • Tight binding


External links

  • LCAO @ chemistry.umeche.maine.edu