Molecular orbital theory

The molecular orbital theory (abbreviated MO theory ) is the second Valenzstrukturtheorie ( VB theory ) to describe one of two complementary ways, the electronic structure of molecules. In the MO method, the atomic orbitals of all atoms are mixed and split it into bonding and antibonding delocalized over the molecule molecular orbitals on. The procedure was developed ( a little later than the VB method ) by Friedrich Hund and Robert S. Mulliken and is now used for most quantum chemical calculations.

• 5.1 σ - π model
• 5.2 τ - model
• 5.3 delocalization

• 6.1 hydrogen
• 6.2 oxygen
• 6.3 1,3-Butadiene

Physical explanation

A n -electron wave function, when the spinning is not observed, the general shape. The product of the complex conjugate function is real and outputs the probability density again, the first electron of the location to find the 2- te at the location, etc..

The exact wave function can not be found analytically. A goal-oriented simplification is to be independent, the electrons statistically. Mathematically, this means to use a product approach. This approach is known as the Hartree product. The enter the residence areas on the individual electrons. They are referred to as molecular orbitals. To comply with the Pauli principle, the wave function is taken as the Slater determinant ( a sum of n products). Then, the electrons are indistinguishable and switch between all orbitals. This ensures that the wave function is antisymmetric under the exchange of two electrons as it needs to apply to fermions.

Besides the fact that MO schemes generally can not play the real situation, it should be noted that they are not uniquely determined in the MO theory. The decisive factor is the sum of all squared orbitals, the electron density (which is also the basis of the density functional theory ). Mathematically speaking, the wave function is invariant to a unitary linear transformation. An example of this are the two specified models for the description of the double bond.

Mathematical Foundations

We are looking for solutions of the Schrödinger equation of a molecule. The bills, however, are much more difficult to perform than in an isolated atom. Normally, if there is more than one electron is considered, there is no assignable analytically exact solutions in the sense of a three -body problem. Therefore, approximate methods must be used. In return, the VB and MO methods, which lead to similar results are.

Important is the Born- Oppenheimer approximation, according to which the electron and nuclear motion can be considered in isolation, because the electrons move much faster compared to the core. Thus electron distribution and vibration can be treated separately.

For the approximate determination of the molecular orbitals of the Rayleigh - Ritz principle serves. This means that when making any function the expected value of the Hamiltonian, the expectation value is greater than or equal to the expected value of the eigenfunction of the Hamiltonian with the lowest eigenvalue. So you have to select in an extreme value problem of the function with the lowest energy expectation value. This is then probably the best approximation.

Just a complete set of basis functions selected to form the expected value for a general linear combination of these, and then to minimize the expected value is too complicated a task. It reduces to simplify the problem, the many-electron problem obtained after the Born- Oppenheimer approximation to a one-electron problem. One way to do this is the Hartree -Fock self- consistent field method, which, since it is a nonlinear problem must be solved iteratively. The solutions of this equation are Einelektronenwellenfunktionen called orbitals. The principle is that affects the averaged potential of all the other electrons on each electron. The other electrons in turn are located in the orbitals describing the Hartree- Fock equation, which is why it is called the method including self - Consisting -field method.

MO method

The MO method (of English. Molecular orbital ) by Friedrich Hund and Robert Sanderson Mulliken assigns all the electrons of the molecule to a set of molecular orbitals. The illustration is done by electron clouds, which are mostly delocalized over the entire molecule.

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

Basically, any function could be used as a base. Good solutions with little computational effort can be obtained when physically meaningful functions are used. For this, suitable as Lennard -Jones noted as the first, the atomic orbitals that describe correctly the electrons in isolated atoms. This is called LCAO. In order to improve the atomic orbitals can be varied or additional functions included in the basis set.

MO, the process can be intuitively understood with small symmetrical molecules. For reasons of symmetry, the molecular orbitals of the addition or subtraction of the atomic orbitals arise. For more complex molecules, the molecular orbitals are composed as a linear combination of different atomic orbitals. In fact, interact already in the second period, the 2s and 2p orbitals, so that already there would get more complicated linear combinations. When conjugated π systems, the Hückel approximation, a method for roughly determining MOs dar.

A fundamental error of this method is that the electrons are statistically independent of each other (up to compliance with the Pauli principle ) as. Much more sophisticated correlated calculations, v. a CI (English for configuration interaction ), note the electron correlation.

Drawing of LCAO-MO diagrams

Qualitative LCAO-MO diagrams can also be drawn without invoice. It should be noted that in the linear combination of two AOs a bonding MO with lower energy than the underlying AO and an antibonding MO are formed with higher -lying energy than the higher lying AO. The splitting is determined in the first approximation of the overlap. So you can, for example, predict that a σ - bond splits more than one π bond.

σ - bond

As a σ - bond is a bond is known, which is rotationally symmetrical to the bond axis. In other words, orbitals with the magnetic quantum number ml = 0 combined, that is, s-, p-, dz2 orbitals and mixtures (hybrids) of these. Hybrid orbitals are commonly referred to as q independent of the hybridization orbitals.

Examples:

• The hydrogen molecular orbital formed by overlap of the 1s orbital of the hydrogen atoms. The small circles correspond to the bond distance, the large circles the atomic radius.
• In the water molecule the 1s orbitals of two hydrogen atoms combine with one sp3 hybrid orbital of the oxygen atom to two σ - bonds. The four orbitals of the bonding and nonbonding electron pairs are oriented to the corners of a tetrahedron. The red marked pairs of electrons are in the atomic orbitals and the gray pairs of electrons in the molecular orbitals.
• In hydrogen fluoride, the spherical 1s orbital of the hydrogen atom combines with the dumbbell-shaped px orbital of the fluorine atom to a molecular orbital with orbital unequal halves. ( The nonbonding py and pz orbitals are not shown. )
• In ethyne two sp hybrid orbitals of the carbon atoms combine to form a molecule orbital, the other hybrid orbitals form with the 1s orbitals of the hydrogen atoms also connect a molecular orbital, the py and pz orbitals of carbon atoms that have not been used for hybridization of the atomic orbitals, perpendicular to the bond axis and form two π bonds (see below). ( In the figure, the p- orbitals are in addition to the molecular orbitals as black lines)

The molecular orbital of the double bond

It is symmetrical with respect to the bond axis.

σ - π model

A double bond consists of a σ - bond, and of a π - bond, wherein the binding partners in the sp2 hybridised state are present, three hybrid orbitals point in the corners of an equilateral triangle, perpendicularly to this is the pz orbitals that is not used for the hybridization. The σ - bond is formed by overlap of two hybrid orbitals, the π bond is formed by overlap of the two pz orbitals. Since both pz orbitals must be parallel to each other, creates a new molecular orbital with a nodal plane. The π - molecular orbital is a combination of orbitals with | formed = 1 | ml. It contains a nodal plane in the core axis.

Ethene example: The two halves of the π - molecular orbital lie above and below the plane of the σ - bonds (blue, CC and HC -sigma bonds are only shown as black lines).

For example ethyne: The bonding in ethyne ( acetylene trivial name ), which contains a triple bond, is composed of a bond which is located between the internuclear axis, and two bonds together.

τ - model

A method for the description of double bonds rarely used is the τ - model. s and p orbitals are first mixed (both C- atoms are sp ³- hybridized ) and composed of the two hybrid orbitals of the double bond. The τ - bonds are formed by overlap of two hybrid orbitals are formed two mirror-image molecular orbitals ( " banana bonds "). It is shown that the τ - model bond angles and lengths are available to fit again.

The distinction makes sense only in the VB theory. In a LCAO method, the two models merge into each other, as is obtained in both cases in total, the same electron density. This is the only relevant.

Delocalization

Delocalisation occurs when a molecule contains several double bonds, which are conjugated. This means that between them always exactly is a single bond. For this, all pz orbitals must be parallel to each other and in the neighborhood. Then all the pz orbitals can be combined into a single molecular orbital, which can be proved by quantum mechanics.

Examples in chemical compounds

Hydrogen

The information necessary for binding lone are located in the 1s orbital of the two atoms Ha and Hb, which by the eigenfunctions ψa (1s) and Ψb (1s) is described.

The addition of the wave functions ψa (1s ) ψb (1s) resulting in a rotationally symmetrical binding molecule orbital ( σ (1s) ) with high charge density between the nuclei of the binding partner. By the attraction of the cores by the charge, the molecule maintains together.

The subtraction of the wave functions ψa (1S ) - ψb (1s) obtained an anti -binding molecule orbital ( σ * (1s )) with a nodal plane between the cores of the binding partner. Due to the resulting low electron density between the nuclei leads to a rejection of the atoms.

The molecular orbitals (such as the atomic orbitals ) with a maximum of two electrons of opposite spin are occupied. Since each hydrogen atom in each case represents an electron available, the bonding molecular orbital is occupied in the lowest energy ground state with an electron pair, while the antibonding remains empty. ( In the excited state, the bonding and the antibonding molecular orbital is occupied by one electron each. )

Another example is helium. Here, each 1s orbital is already occupied by an electron pair. With the combination of these atomic orbitals would have both the bonding and the antibonding molecular orbital are occupied with one electron pair. Their effects would cancel each other, it is concluded no bond.

Oxygen

The LCAO - MO scheme can be derived as described above qualitatively. Each oxygen atom has six valence electrons in the second principal energy level in the ground state. The twelve valence electrons of an oxygen molecule, O2 are the four binding ( σs, σx, πy and πz ) and three of the four antibonding molecular orbitals ( σs *, * πy, πz *) distributed. Since two antibonding orbitals are occupied by only one electron (a " half bond " ), results in a double bond.

Dioxygen has in the ground state, a triplet state, according to the Hund's rule two unpaired electrons parallel spins. Through this electron distribution, the paramagnetism and the diradical character of the oxygen can be explained. Interestingly, lowers the diradical responsiveness, as a concerted reaction of the spin conservation would disagree. Especially reactive is the excited singlet oxygen.

Another consequence of the MO- cast is that it is difficult for O2 to provide a correct Lewis formula. Either the diradical character is neglected or the double bond.

1,3-butadiene

The π - system of butadiene is composed of 4 pz orbitals that are occupied at the beginning of each with one electron. This 4 atomic orbitals are now linearly combined to four molecular orbitals. The coefficients are obtained by variational minimization of the energy with the Rayleigh - Ritz principle, for example, with the Hückel or the Hartree- Fock method, or by symmetry-adapted linear combination ( SALK ). The drawn right orbitals arise. The red / blue coloring indicates whether the Orbital had a negative or positive sign before squaring. Physically, it has no relevance.

Each of these orbitals can be occupied by two electrons. The two lower orbitals therefore are fully filled and the top two remain empty. Is energetically favored the orbital in which the coefficients of the pz orbitals have the same sign and therefore the electrons can move almost freely over the entire molecule.

One recognizes demanded by SALK property that preserves all the symmetry elements of the molecule in each molecular orbital. Furthermore, you can see how with increasing energy increases the number of nodal planes.

Bond order

The bond order refers to the number of effective bonds in one molecule. She is half the difference of the binding and the anti-bonding valence electrons (see bond order ). It is easy to read because it is equal to the number of chemical bonds in the Lewis structure of the compound.