# Molecular orbital theory

The molecular orbital theory ( MO theory for short ) is, in addition to the valence bond theory (VB theory), one of two complementary ways of describing the electronic structure of molecules . In the MO process, molecular orbitals delocalized over the molecule are formed by a linear combination of the atomic orbitals of all atoms in a molecule. A distinction is made between binding and antibonding molecular orbitals. The method was developed (a little later than the VB method) by Friedrich Hund and Robert S. Mulliken and is used today for most quantum chemical calculations.

## Physical explanation

A time-independent -electron- wave function has the general form if the spin is ignored . The product with the complex conjugate function is real and represents the probability density of finding the first electron at that point , the second at that point , and so on. ${\ displaystyle n}$${\ displaystyle \ Psi ({\ vec {r}} _ {1}, \ ldots {\ vec {r}} _ {n})}$${\ displaystyle \ Psi ^ {*} ({\ vec {r}} {1}, \ ldots {\ vec {r}} _ {n}) \ cdot \ Psi ({\ vec {r}} _ { 1}, \ ldots {\ vec {r}} _ {n})}$${\ displaystyle {\ vec {r}} _ {1}}$${\ displaystyle {\ vec {r}} _ {2}}$

The exact wave function can not be found analytically for multi-electron systems . A useful simplification is to view the electrons as stochastically independent . Mathematically, that means using a product approach . This approach is also known as the Hartree product . They indicate the location areas for the individual electrons. They are called molecular orbitals. In order to adhere to the Pauli principle , the wave function is used as a Slater determinant (an antisymmetrized product approach). This ensures that the wave function is antisymmetric with the interchange of two indistinguishable electrons, as it must be for fermions . ${\ displaystyle \ Psi ({\ vec {r}} _ {1}, \ ldots {\ vec {r}} _ {n}) = \ prod _ {i = 1} ^ {n} \ Psi _ {i } ({\ vec {r}} _ {i})}$${\ displaystyle \ Psi _ {i} ({\ vec {r}} _ {i})}$

Apart from the fact that MO schemes represent a simplified, model-like description, it should be noted that they are not clearly defined in MO theory. The only decisive factor is the sum over all squared orbitals, the electron density (this is also the basis for density functional theory ). Mathematically speaking, the wave function is invariant to a unitary linear transformation. In addition to the orbitals, which are often referred to as canonical MOs, various other representations can be found, e.g. B. Localized Molecular Orbitals (LMOs).

## Mathematical basics

We are looking for solutions to the time-independent Schrödinger equation of a molecule. However, the calculations are much more difficult to carry out than with an isolated atom. In the normal case, when more than one electron is considered, there are no analytically specifiable exact solutions in the sense of a three-body problem . Therefore, approximation methods must be used. The VB and MO processes are suitable for this and lead to similar results.

The Born-Oppenheimer approximation is important , according to which the electron and nuclear movement can be viewed approximately decoupled. Put simply, this is due to the fact that the electrons move much faster compared to the nucleus. Thus, electron distribution and vibration can be treated separately.

The Rayleigh-Ritz principle is used for the approximate determination of the molecular orbitals . This means that if one forms the expected value of the Hamilton operator with an arbitrary function, the expected value is greater than or equal to the expected value of the eigenfunction of the Hamilton operator with the lowest eigenvalue. So you have to select the function with the lowest expected energy value in an extreme value problem. This is then probably the best approximation.

Simply selecting a complete set of basis functions, forming the expectation for a general linear combination of them, and then minimizing the expectation is too complicated a task. To simplify the problem, the multiple-electron problem obtained according to the Born-Oppenheimer approximation is reduced to a one-electron problem . One possibility for this is the Hartree-Fock-Self-Consistent-Field-Method , which, since it is a non-linear problem, has to be solved iteratively . The solutions to this equation are one-electron wave functions, so-called orbitals. The principle is that the average potential of all other electrons acts on each electron. The other electrons, in turn, are located in the orbitals described by the Hartree-Fock equation, which is why the method is also called the self-consistent field method.

## MO procedure

MO method (from English molecular orbital ) according to Friedrich Hund and Robert Sanderson Mulliken assign all electrons of the molecule to a set of molecular orbitals. The orbitals can be visualized by showing their isosurfaces . The visualization is then sometimes reminiscent of electron clouds .

Molecular orbitals can be used as linear combinations on a finite basis. Then the molecular orbitals are determined in an extended eigenvalue problem . As suggested by Lennard-Jones, the atomic orbitals of the isolated atoms in the sense of the LCAO approximation (from English Linear Combination of Atomic Orbitals ) can be used as a basis.

In principle, any functions could be used as a basis. Good solutions with little computational effort are obtained if physically meaningful functions are used. As Lennard-Jones was the first to establish, the atomic orbitals, which correctly describe the electrons in isolated atoms, are suitable for this. One then speaks of LCAO. For improvement, the atomic orbitals can also be varied or further functions can be included in the basic set.

MO processes can be understood intuitively for small symmetrical molecules. For reasons of symmetry, the molecular orbitals result from the addition or subtraction of the atomic orbitals. In conjugated π systems, the Hückel approximation is a method for the rough determination of π MOs.

The simplest general MO procedure is the Hartree-Fock (HF) method . A fundamental flaw of this method is that the electrons (apart from adherence to the Pauli principle ) are seen as statistically independent of each other. Correlated calculations based on the HF method , v. a. CI (for configuration interaction ), also consider the electron correlation.

### Draw LCAO-MO diagrams

Qualitative LCAO-MO diagrams can also be drawn without an invoice. It should be noted that with the linear combination of two AOs, a binding MO with lower energy than the lower AO and an antibonding MO with higher energy than the higher AO are formed. As a first approximation, the split is determined by the overlap integral of the atomic orbitals to be combined. So you can z. B. predict that a σ-bond splits more strongly than a π-bond.

## σ orbital

 σ orbital Molecular orbital of H 2 sp-MO (e.g. hydrogen fluoride)

An orbital that is rotationally symmetrical to the bond axis is called a σ orbital. It therefore only occurs in systems with 2 atoms. The σ-orbital is formed as a combination of orbitals with the magnetic quantum number m l = 0, i.e. H. s, p and d z 2 orbitals.

Examples:

• The hydrogen molecular orbital is created by the overlap of the 1s orbitals of the hydrogen atoms.
• In hydrogen fluoride , the spherical 1s orbital of the hydrogen atom connects with the dumbbell-shaped 2p x orbital of the fluorine atom to form a molecular orbital with unequal orbital halves. In reality, the not shown 2s orbital of the fluorine atom also has a significant share in the bond orbital. (The nonbinding 2p y and 2p z orbitals are also not shown.)

## Examples in chemical compounds

### hydrogen

hydrogen
binding antibonding

Additive superposition of the wave function,
where a and b are the positions of the protons

Subtractive superposition of the wave function

Binding molecular orbital

Antibonding molecular orbital
Occupation of the molecular orbitals of hydrogen and helium
hydrogen helium

Occupation with hydrogen

Occupation at helium

The individual electrons necessary for binding are located in the 1s orbital of the two atoms H a and H b , which is described by the eigenfunctions ψ a (1s) and Ψ b (1s).

The addition of the wave functions ψ a (1s) + ψ b (1s) results in a rotationally symmetrical binding molecular orbital (σ (1s)) with increased charge density between the nuclei of the binding partners. The attraction of the nuclei by the charge holds the molecule together.

The subtraction of the wave functions ψ a (1s) - ψ b (1s) results in an antibonding molecular orbital (σ * (1s)) with a nodal plane between the nuclei of the binding partners. The resulting low electron density between the nuclei causes the atoms to be repelled.

The molecular orbitals (like the atomic orbitals) can be occupied by a maximum of two electrons with opposite spins. Since each hydrogen atom provides one electron, the binding molecular orbital in the lowest-energy ground state is occupied by an electron pair, while the antibonding remains empty. (In the excited state, the binding and the antibonding molecular orbital are each occupied by one electron.)

Another example is helium . Here every 1s orbital is already occupied with an electron pair. When combining these atomic orbitals, both the binding and the antibonding molecular orbital would have to be occupied with one pair of electrons each. Their effects would cancel each other out, there is no bond.

### oxygen

LCAO-MO scheme of triplet oxygen: occupation of energy levels

The LCAO-MO scheme can be derived qualitatively as described above. Every oxygen atom has six valence electrons on the second main energy level in the ground state. The twelve valence electrons of an O 2 oxygen molecule are distributed to the four bonding (σ s , σ x , π y and π z ) and three of the four antibonding molecular orbitals (σ s * , π y * , π z * ). Since two antibonding orbitals are occupied by only one electron (a “half bond”), a double bond results.

In the ground state, a triplet state, di-oxygen has two unpaired electrons with parallel spins according to Hund's rule. This electron distribution explains the paramagnetism and diradical character of oxygen. Interestingly, the diradical character lowers the reactivity, since a concerted reaction would contradict the preservation of spin. The excited singlet oxygen is particularly reactive.

Another consequence of the MO occupation is that it is difficult to give a correct Lewis formula for O 2 . Either the diradical character is neglected or the double bond.

The π system of butadiene is made up of 4 p z orbitals, which are each occupied by one electron at the beginning. These 4 atomic orbitals are now linearly combined to form four molecular orbitals. The coefficients are obtained by variational minimization of the energy with the Rayleigh-Ritz principle z. B. with the Hückel or Hartree-Fock method , or by symmetry-adapted linear combination (SALK). This creates the orbitals drawn on the right. The red / blue color indicates whether the orbital had a negative or positive sign before it was squared . It has no physical relevance.

Each of these orbitals can be occupied by 2 electrons. So the two lower orbitals are fully filled and the two upper ones remain empty. The orbital is energetically particularly favorable, in which the coefficients of the p z orbitals have the same sign and therefore the electrons can move almost freely over the whole molecule.

One recognizes the property demanded by SALK that all symmetry elements of the molecule are preserved in each molecular orbital. You can also see how the number of node levels increases with increasing energy.

## Binding order

The bond order describes the number of effective bonds between two atoms. It is half the difference between the number of bonding and antibonding valence electrons. If mesomerism does not occur, i.e. H. only in simple cases is it equal to the number of dashes in the Lewis notation of the compound.

## history

The MO theory goes back to the work of Friedrich Hund, Robert S. Mulliken, John C. Slater and John Lennard-Jones . The MO theory was originally called the Hund-Mulliken theory . According to Erich Hückel , the first quantitative application of MO theory was developed in 1929 by John Lennard-Jones. The publication predicted a triplet state in the ground state of the oxygen molecule, which explains its paramagnetism . Two years later the phenomenon was also explained by the VB theory. In 1933, the MO theory was already recognized as valid.

In 1931 Erich Hückel applied the MO theory to determine the MO energies of π-electron systems on unsaturated hydrocarbons and described the Hückel molecular orbitals , which could explain aromaticity . The first exact calculation of the MO wave function of hydrogen was carried out in 1938 by Charles Coulson. From around 1950, molecular orbitals were treated as an eigenfunction ( Hartree-Fock method ). The Hartree-Fock approximation was originally developed for individual atoms and expanded with basic sets , which was described as Roothaan-Hall equations . As a result, various ab initio calculations as well as semi-empirical methods were developed. In the 1930s, the ligand field theory was developed as an alternative to the crystal field theory.

## literature

Commons : Molecular Orbitals  - collection of images, videos and audio files

## Individual evidence

1. ^ Ian Fleming, Ian Fleming, Podlech, Joachim: Molecular orbitals and reactions of organic compounds . 1st edition. Wiley-VCH-Verl, Weinheim 2012, ISBN 978-3-527-33069-0 , pp. 1-11 .
2. ^ Frank Jensen: Introduction to Computational Chemistry . Wiley, 2007, ISBN 978-0-470-05804-6 , pp. 304-308 .
3. ^ Charles, A. Coulson: Valence . Oxford at the Clarendon Press, 1952.
4. ^ Mulliken, Robert S .: Spectroscopy, Molecular Orbitals, and Chemical Bonding . (pdf) In: Elsevier Publishing Company (ed.): Nobel Lectures, Chemistry 1963-1970 . , Amsterdam 1972.
5. Erich Hückel: Theory of free radicals of organic chemistry . In: Trans. Faraday Soc . 30, 1934, pp. 40-52. doi : 10.1039 / TF9343000040 .
6. JE Lennard-Jones: The electronic structure of some diatomic molecules . In: Trans. Faraday Soc . 25, 1929, pp. 668-686. doi : 10.1039 / TF9292500668 .
7. ^ Coulson, CA Valence (2nd ed., Oxford University Press 1961), p. 103.
8. ^ Linus Pauling: The Nature of the Chemical Bond. II. The One-Electron Bond and the Three-Electron Bond . In: J. Am. Chem. Soc. . 53, 1931, pp. 3225-3237. doi : 10.1021 / ja01360a004 .
9. George G Hall: Lennard-Jones Paper of 1929 Foundations of Molecular Orbital Theory . In: Advances in Quantum Chemistry . August 22, . bibcode : 1991AdQC ... 22 .... 1H . doi : 10.1016 / S0065-3276 (08) 60361-5 .
10. Erich Hückel: Quantum theoretical contributions to the benzene problem. I. The electronic configuration of benzene and related compounds . In: Journal of Physics . tape 70 , no. 3 , March 1, 1931, p. 204-286 , doi : 10.1007 / BF01339530 . Erich Hückel: Quantum theoretical contributions to the benzene problem. II quantum theory of induced polarities . In: Journal of Physics . tape
72 , no. 5 , May 1, 1931, p. 310-337 , doi : 10.1007 / BF01341953 . Erich Hückel: Quantum theoretical contributions to the problem of aromatic and unsaturated compounds. III . In: Journal of Physics . tape
76 , no. 9 , September 1, 1932, p. 628-648 , doi : 10.1007 / BF01341936 . Erich Hückel: Quantum theoretical contributions to the problem of aromatic and unsaturated compounds. IV The free radicals of organic chemistry . In: Journal of Physics . tape
83 , no. 9 , September 1, 1933, pp. 632-668 , doi : 10.1007 / BF01330865 .
11. ^ Hückel Theory for Organic Chemists , CA Coulson , B. O'Leary and RB Mallion, Academic Press, 1978.
12. ^ CA Coulson: Self-consistent field for molecular hydrogen . In: Mathematical Proceedings of the Cambridge Philosophical Society . 34, No. 2, 1938, pp. 204-212. bibcode : 1938PCPS ... 34..204C . doi : 10.1017 / S0305004100020089 .
13. ^ GG Hall: The Molecular Orbital Theory of Chemical Valency. VI. Properties of Equivalent Orbitals . In: Proc. Roy. Soc. A . 202, No. 1070, Aug. 7, 1950, pp. 336-344. bibcode : 1950RSPSA.202..336H . doi : 10.1098 / rspa.1950.0104 .
14. ^ A b Frank Jensen: Introduction to Computational Chemistry . John Wiley and Sons, 1999, ISBN 978-0-471-98425-2 .