In chemistry, a molecular orbital is a region in which an electron may be found in a molecule.^[1] In general, atomic orbitals combine to form molecular orbitals. The probability of finding an electron in a given region can be obtained by solving the Schrödinger wave equation. Chemistry (from Greek χημεία khemeia[1] meaning alchemy) is the science of matter at the atomic to molecular scale, dealing primarily with collections of atoms, such as molecules, crystals, and metals. ... The Electron is a fundamental subatomic particle that carries an electric charge. ... In chemistry, a molecule is an aggregate of two or more atoms in a definite arrangement held together by chemical bonds [1] [2] [3] [4] [5]. Chemical substances are not infinitely divisible into smaller fractions of the same substance: a molecule is generally considered the smallest particle of a pure... In chemistry, an atomic orbital is the region in which an electron may be found around a single atom. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ...

Electron atomic and molecular orbitals

Image File history File links Electron_orbitals. ... Image File history File links Electron_orbitals. ...

Overview

In quantum chemistry, i.e. electronic structure theory, the molecular electronic states, which are the eigenstates of the electronic molecular Hamiltonian, are expanded (see configuration interaction expansion and basis) into linear combinations of anti-symmetrized products (Slater determinants) of one-electron functions. The spatial components of these one-electron functions are called molecular orbitals (MO). When one considers also their spin component, they are called molecular spin orbitals. Linus Pauling, as a pioneer of the valence bond theory, is one of the first quantum chemists. ... A quantum state is any possible state in which a quantum mechanical system can be. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... The electronic Hamiltonian for a multi-electron molecule in atomic units is: where is the vector position of electron with vector components in Bohr radii, is the charge of fixed nucleus a in units of the elementary charge, is the vector position of nucleus with vector components in Bohr radii. ... Configuration interaction (CI) is a post Hartree-Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born-Oppenheimer approximation for a quantum chemical multi-electron system. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... A Slater determinant (named after the physicist John C. Slater) is an expression in quantum mechanics for the wavefunction of a many-fermion system, which by construction satisfies the Pauli principle. ... The Electron is a fundamental subatomic particle that carries an electric charge. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ...

Most methods in computational chemistry today start by calculating the MOs of the system. A molecular orbital describes the behavior of one electron in the electric field generated by the nucleii and some average distribution of the other electrons. In the case of two electrons occupying the same orbital, the Pauli principle demands that they have opposite spin. Computational chemistry is a branch of chemistry that uses the results of theoretical chemistry incorporated into efficient computer programs to calculate the structures and properties of molecules and solids, applying these programs to real chemical problems. ... The Pauli exclusion principle, commonly referred to simply as the exclusion principle, is a quantum mechanical principle formulated by Wolfgang Pauli in 1925, which states that no two identical fermions may occupy the same quantum state. ...

Qualitative discussion

For an imprecise, but qualitatively useful, discussion of the molecular structure, the molecular orbitals can be obtained from the "Linear combination of atomic orbitals molecular orbital method" ansatz (using eventually the concept of hybridized orbitals). This article may be too technical for most readers to understand. ... Ansatz is a term (from German) often used by physicists. ... four sp³ orbitals three sp² orbitals In chemistry, hybridisation or hybridization (see also spelling differences) is the concept of mixing atomic orbitals to form new hybrid orbitals suitable for the qualitative description of atomic bonding properties. ...

In this approach, the molecular orbitals are expressed as linear combinations of atomic orbitals, as if each atom were on its own. In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In chemistry, an atomic orbital is the region in which an electron may be found around a single atom. ...

The linear combination of atomic orbitals approximation for molecular orbitals was introduced in 1929 by Sir John Lennard-Jones. His ground-breaking paper showed how to derive the electronic structure of the fluorine and oxygen molecules from quantum principles. This qualitative approach to molecular orbital theory represents the dawn of modern quantum chemistry. This article may be too technical for most readers to understand. ... 1929 (MCMXXIX) was a common year starting on Tuesday (link will take you to calendar). ... John Edward Lennard-Jones (October 27, 1894 - November 1, 1954) was a mathematician who held a chair of theoretical physics at Bristol University, and then a chair of theoretical science at Cambridge University. ... General Name, Symbol, Number fluorine, F, 9 Chemical series halogens Group, Period, Block 17, 2, p Appearance Yellowish brown gas Atomic mass 18. ... General Name, Symbol, Number oxygen, O, 8 Chemical series Nonmetals, chalcogens Group, Period, Block 16, 2, p Appearance colorless (gas) very pale blue (liquid) Atomic mass 15. ...

Some properties:

• The number of molecular orbitals is equal to the number the atomic orbitals included in the linear expansion,
• If the molecule has some symmetry, the degenerate atomic orbitals (with the same atomic energy) are grouped in linear combinations (called symmetry adapted atomic orbitals (SO)) which belong to the representation of the symmetry group, so the wave functions that describe the group is known as symmetry-adapted linear combinations (SALC).
• The number of molecular orbitals belonging to one group representation is equal to the number of symmetry adapted atomic orbitals belonging to this representation,
• Within a particular representation, the symmetry adapted atomic orbitals mix more if their atomic energy level are closer.

In mathematics, the general features of the representation theory of a finite group G, over the complex numbers, were discovered by Ferdinand Georg Frobenius in the years before 1900. ... The symmetry group of an object (e. ... In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function ψ defined over a portion of space normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared... In mathematics, the general features of the representation theory of a finite group G, over the complex numbers, were discovered by Ferdinand Georg Frobenius in the years before 1900. ... A quantum mechanical system can only be in certain states, so that only certain energy levels are possible. ...

Examples

H₂

As a simple example consider the hydrogen molecule, H₂, with the two atoms labelled H' and H". The lowest-energy atomic orbitals, 1s' and 1s", do not transform according to the symmetries of the molecule. However, the following symmetry adapted atomic orbitals do: General Name, Symbol, Number hydrogen, H, 1 Chemical series nonmetals Group, Period, Block 1, 1, s Appearance colorless Atomic mass 1. ...

The symmetric combination (called a bonding orbital) is lower in energy than the basis orbitals, and the antisymmetric combination (called an antibonding orbital) is higher. Because the H₂ molecule has two electrons, they can both go in the bonding orbital, making the system lower in energy (and hence more stable) than two free hydrogen atoms. This is called a covalent bond. Antibonding (or anti-bonding) is a type of chemical bonding. ... This article does not cite its references or sources. ...

H₃

On the other hand, consider the hypothetical linear molecule of H₃ with the atoms labelled H, H' (the central atom), H" and equal H-H' and H'-H" distances. Then we would expect three linear combinations:

Two electrons occupy the symmetric bonding orbital and the third one occupy the non bonding orbital.

Rare gases

Now let's move to larger atoms. Considering a hypothetical molecule of He₂, since the basis set of atomic orbitals is the same as in the case of H₂, we find that both the bonding and antibonding orbitals are filled, so there is no energy advantage to the pair. HeH would have a slight energy advantage, but not as much as H₂ + 2 He, so the molecule exists only a short while. In general, we find that atoms such as He that have completely full energy shells rarely bond with other atoms. (In fact there is not a single stable molecule containing He, Ne or Ar except short-lived Van der Waals complexes.) Van der Waals bonding, also known as London force, instantaneous dipole effect, and induced dipole interaction, is an intermolecular force or interatomic force that causes an attraction between temporarily induced dipoles in nonpolar molecules and atoms. ...

Inner shells

Inner shell orbitals should not be included in the LCAO expansion. Molecular structure relies on the outermost (valence) electrons of the atoms, which are usually of comparable energy. The linear combination of atomic orbitals molecular orbital method (usually called the LCAO MO method) is a technique for calculating molecular orbitals in quantum chemistry. ...

Ionic bonds

see main article ionic bond

When the energy difference between the atomic orbitals of two atoms is quite large, one atom's orbitals contribute almost entirely to the bonding orbitals, and the other's almost entirely to the antibonding orbitals. Thus, the situation is effectively that some electrons have been transferred from one atom to the other. This is called a (mostly) ionic bond. Electron configurations of lithium and fluorine. ... Electron configurations of lithium and fluorine. ...

More quantitative approach

To obtain quantitative values for the molecular energy levels, one needs to have molecular orbitals which are such that the configuration interaction (CI) expansion converges fast towards the full CI limit. The most common method to obtain such functions is the Hartree-Fock method which expresses the molecular orbitals as eigenfunctions of the Fock operator. One usually solves this problem by expanding the molecular orbitals as linear combinations of gaussian functions centered on the atomic nuclei (see linear combination of atomic orbitals and basis set (chemistry)). The equation for the coefficients of these linear combinations is a generalized eigenvalue equation known as the Roothaan equations which are in fact a particular representation of the Hartree-Fock equation. A quantum mechanical system can only be in certain states, so that only certain energy levels are possible. ... Configuration interaction (CI) is a post Hartree-Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born-Oppenheimer approximation for a quantum chemical multi-electron system. ... Full configuration interaction (or full CI) is a linear variational approach which provides numerically exact solutions (within the given one-particle basis set) to the electronic Schrödinger equation. ... In computational physics and computational chemistry, the Hartree-Fock (HF) or self-consistent field (SCF) calculation scheme is a self-consistent iterative variational procedure to calculate the Slater determinant (or the molecular orbitals which it is made of) for which the expectation value of the electronic molecular Hamiltonian is minimum. ... In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ... In quantum mechanics, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of basis vectors. ... Gaussian curves parametrised by expected value and variance (see normal distribution) A Gaussian function (named after Carl Friedrich Gauss) is a function of the form: for some real constants a > 0, b, and c. ... A semi-accurate depiction of the helium atom. ... The linear combination of atomic orbitals molecular orbital method (usually called the LCAO MO method) is a technique for calculating molecular orbitals in quantum chemistry. ... In modern computational chemistry, quantum chemical calculations are typically performed within a finite set of basis functions. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... The Roothaan equations are a representation of the Hartree-Fock equation in a non orthonormalized basis set which can be of Gaussian type. ... In computational physics and computational chemistry, the Hartree-Fock (HF) or self-consistent field (SCF) calculation scheme is a self-consistent iterative variational procedure to calculate the Slater determinant (or the molecular orbitals which it is made of) for which the expectation value of the electronic molecular Hamiltonian is minimum. ...

See also

• Atomic orbital
• Computational chemistry
• Electron configuration
• HOMO/LUMO
• MO diagram
• Molecular radiation
• Quantum chemistry

In chemistry, an atomic orbital is the region in which an electron may be found around a single atom. ... Computational chemistry is a branch of chemistry that uses the results of theoretical chemistry incorporated into efficient computer programs to calculate the structures and properties of molecules and solids, applying these programs to real chemical problems. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... HOMO and LUMO are acronyms for highest occupied molecular orbital and lowest unoccupied molecular orbital, respectively. ... A molecular orbital diagram or MO diagram for short is a simple 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 (LCAO method) in particular [1] [2] [3]. This tool is very well... This page meets Wikipedias criteria for speedy deletion. ... Linus Pauling, as a pioneer of the valence bond theory, is one of the first quantum chemists. ...

References

1. ^ Daintith, J. (2004). Oxford Dictionary of Chemistry. New York: Oxford University Press. ISBN 0198609183.

External links

• Visualizations of some atomic and molecular orbitals (Note: These visualisations run only on Apple Mac.)
• Java molecular orbital viewer shows orbitals of hydrogen molecular ion.