Valence bond theory

The valence theory, also VB theory, electron-pair theory, Heitler-London theory or English. Called Valence Bond Theory, one of Walter Heitler and Fritz London in 1927 developed quantum mechanical approximation method for the description of atomic bonds in polyatomic systems with dominant two-electron behavior (for example, are for molecules of two alkali atoms, the two " light-emitting electrons " considered separately ). The default case is the hydrogen molecule.

In this model, a valence bond between two atoms formed by the fact that each of the atoms providing an electron is a bond. These two electrons form an electron pair in a binding local state of the molecule. The two nuclei in the molecule have share two electrons, ie (with polar bonds ) at "own" electron and the electron " partner ", while one can make statements in non-polar bonds only on the symmetry of the system, since, according to the Pauli principle, a symmetrical two-electron spatial wave function always has to be multiplied by an antisymmetric two-electron spin function, or vice versa an antisymmetric spatial wave function with a symmetric spin function. Because of Vertauschungssymmetrie the system is the Ortsfunkion of the two- electron system either symmetric ( sign in the second term, so-called bonding state ) or antisymmetric ( characters, so-called anti- bonding state ) in any case in the absence of spin-dependent potentials. The energy curve produced by the variation of the distance of the two nuclei, is the driving force for the formation of molecules, one in the case of a symmetric spatial wave function takes a singlet state ( since then the spin wave function of the two electrons must be antisymmetric ) in the case of contrast, anti-symmetric spatial wave function of a three triplet states (see below).

The calculation of the molecular binding energy results in detail, that is, a stable bond receives ( "bonding state" ) in the first case, that is for the singlet state of the two- electron system, while the triplet case, that is, for an anti-symmetric two-electron local function, not only a much higher energy results; but ( in contrast to the singlet state with its pronounced minimum of the energy curve ) results now even a monotonically - decaying behavior of the energy curve, which corresponds to an antibonding behavior.

• 4.1 inner / outer orbital complexes
• 4.2 geometry
• 4.3 limits

Singlet and triplet states of a two-electron system

The singlet spin function is in use, the usual arrow symbols (with the first arrow refers to an electron, the second to the other ):

The three triplet states, however, are ( distinguished by the "magnetic quantum number " M ):

The energy of the system depends on the assumed here neglecting relativistic terms only of the local functions from, and is, as it turns out (see below), binding for the singlet case or for the triplet case antibonding ( repulsive ). The Hamiltonian ( energy operator ) of the system taking into account other than the kinetic energy of the two electrons - atomic nuclei are held ( Born- Oppenheimer approximation ) - Only the all Coulomb interactions with participation of two electrons, ie, the position-dependent Coulomb attractions of both electrons by both atomic nuclei ( four contributions to the Hamiltonian ) and also location-dependent mutual Coulomb repulsion of electrons 1 and 2 ( only one contribution to ).

Basic model

The basic model was developed for the hydrogen molecule, as it allows the most simple calculations:

• Each of the two hydrogen atoms is one electron for an electron pair bond available.
• By combining the s orbitals of the hydrogen atoms in which the electrons originally were incurred molecular orbitals (so- spin coupled, ) are an unoccupied antibonding molecular orbital and an occupied binding molecular orbital in which the two electrons then as an electron pair. ( The spin coupling takes into account the Pauli principle. )
• The theoretical energy of the bond and the probability of the electrons in the bonding molecular orbital can be determined on the wave function of this orbital.
• The wave function of the bonding molecular orbital, however, is unknown and is approximated by considering various factors until it agrees satisfactorily with experimental findings.
• As a starting point for the computational approach are the two s- orbitals of the original single hydrogen atoms.

Valence method for the hydrogen molecule

A hydrogen atom has electron 1 and the wave function ΨA (1).

Hydrogen atom has electron B No. 2 and the wave function ΨB (2).

The experimentally determined distance between the hydrogen nuclei in the molecule is 74 pm, the binding energy at -458 kJ · mol -1.

Elementary approximation

In the elementary approximation is entirely disregarded whether and how the two atomic nuclei and electrons affect each other when they approach each other to form a bond. The wave function Ψ for a system of two atoms that do not interact with each other, are obtained from the wave functions of the individual hydrogen atoms:

The binding energy and the internuclear distance, resulting from it, hardly consistent with the experimental findings.

Exchange energy by Heitler and London

In the molecular orbital electron is 1 nor 2 electron Both cases need not always be located at hydrogen atom A, always with hydrogen atom B. Rather, because of the Pauli principle equally likely. Accordingly, a term for reversed electron is added, which is weighted for reasons of symmetry with either plus 1 or minus 1. At first, none is reasonably clear whether the first or second case is associated with binding or antibonding behavior; the results only by specific invoice or by a separate unpublished consideration as it has done Eugene Wigner.

The computed with the Hamiltonian energy difference is called exchange energy, because of the normalization condition of the Schrödinger functions once more the sign factor ± occurs now in the denominator:

The sign convention is derived from the theory of ferromagnetism and is here getting used to: As in two-electron systems always the triplet state (the state with the minus sign in the space function ) has the much higher energy, that is energetically unfavorable, the sign of the exchange energy for two-electron systems is always negative what is considered in the theory of ferromagnetism in the so-called Heisenberg model for the antiferromagnetic states. Another common convention is obtained by placing in the last equation on the left side (or in the penultimate equation on the right side ) in front of a factor of 2.

Here can be found in any case to the positive coefficient have a good approximation of the experimental results, whereas the negative sign leads to the above-mentioned " vacant " state.

Shielding

The terms and only consider that an electron shields the nuclear charge of the hydrogen nucleus. In the molecule, however, there are two nuclei and two electrons which screen the nuclear charges altogether stronger, reducing the effective nuclear charge is less.

The effective nuclear charge is taken into account in the above wave functions crossed symbols. So we obtain again the local functions:

Resonance

Theoretically, the hydrogen atoms can not only share their electrons in the composite, but also there is the small chance that sometimes both electrons are on one of the hydrogen nuclei. Accordingly, can be used for the hydrogen molecule ionic structures (dog - Mulliken, one-electron molecular functions) should add with weight factors to the nonionic two- electron functions ( two- electron - Heitler-London atomic functions ).

However, since the ionicity is small in the cases considered, this factor is generally small and may even completely negligible. So:

( The asterisk denotes the complex conjugate. )

The deviation of experimental results is very low, and after application of a wave equation with 100 correction terms to get results that are almost consistent with the experimental findings. In addition, it should be noted that one may even with " pretty bad" wave functions get quite good results for the energy.

The basic model for hydrogen molecules was at any rate has continued to refine and transfer the problem to larger and much more complicated molecules, as well as multiple bonds, not only in molecules, but also in the solid state.

The approach of Valenzstrukturmethode, as well as the molecular orbital theory are the basis of today's molecular modeling, which enables computer-based calculations and predictions interpretations of many molecular structures and properties.

Pauling's theory of complexes

In complexes of the central atom and a certain number of ligands, the coordinate bond is present. This type of binding is not due to the fact that both reaction partners, ie the central atom and ligand each set, one electron is available, but by the fact that the ligand alone brings two electrons and thus forms a bond to the central atom.

When a ligand provides two electrons, and n is the number of the binding ligands, is then given to the central atom × 2 n electrons must place it somewhere. To accommodate the empty outer orbitals of the central atom are available:

• First period of the transition metals: ( outside) 4 d, 4 p, 4 s, 3 d (lbs)
• Second Period of transition metals: ( outside) 5 d, 5 p, 5 s, 4 d ( inside)
• Third period of the transition metals: ( outside) 6 d, 6 p, 6 s and 5 d ( inside)

Inner / outer orbital complexes

The elucidation why some ligands produce outer- or inner- sphere complexes, only succeeded with the crystal field or ligand field theory. Here the concept of high spin and low spin complexes was introduced corresponding to the magnetic properties of such complexes.

Example:

The Fe2 cation has 6 electrons in the 3d orbital, ie a 3d6 configuration.

With six water ligands 12 electrons are added. In this case, the configuration of the cation is obtained by: 4s2 4p6 3d6 4d4.

With six cyanide ligands also 12 electrons are added. Here, the " original cast " of the central atom changed to: 3d10 4s2 4p6.

Geometry

To bond formation between the central atom and ligand hybrid orbitals on the part of the central atom are formed according to the VB theory, the number of which corresponds to the amount of electron pairs that provide the ligand for binding available.

Used, s-, p- and d ( outside) orbitals of the central atom, resulting in characteristic coordination geometries - Depending on the nature and especially the number of ligand electron pairs for hybridization certain d (in):

• 6 ligands: 2 · d ( s) 1 · s 3 · p = 6 · d2sp3 → octahedron (inner orbital complex)

• 6 ligands: 1 3 · s · p 2 · d ( outside) = 6 · sp3 d2 → octahedron (outer orbital complex)

• 4 ligands: 1 3 · s · p = 4 · → sp3 tetrahedron

• 4 ligands: 1 · d ( s) 1 2 · s · p = 4 · dsp2 → planar square

Etc.

Confines

The VB theory is well suited for the determination of complex geometries and to explain the magnetic phenomena. Other phenomena, such as the color of complexes, however, can be explained by the VB theory only at great expense. For this purpose, models such as the crystal field theory and ligand field theory or the molecular orbital theory are better suited.