Valence bond theory

The valence bond theory (VB theory) , also valence structure theory , electron pair theory , Heitler-London theory or engl. Called Valence Bond Theory , it is a quantum mechanical approximation method developed by Walter Heitler and Fritz London in 1927 for the description of atomic bonds in systems with more than one atom with a dominant two-electron behavior (e.g. in molecules made up of two alkali atoms, the two "luminous electrons" are considered separately ). The standard case is the hydrogen molecule.

In this model, a valence bond is created between two atoms when each of these atoms provides an electron for a bond. These two electrons form an electron pair in a binding local state of the molecule . The two atomic nuclei in the molecule have a share in both electrons, i.e. (in the case of polar bonds) in their “own” electron and in the electron “of the partner”, while in the case of non-polar bonds one can only make statements about the symmetry of the system, since according to Pauli Principle a symmetrical two-electron spatial function must always be multiplied by an antisymmetrical two-electron spin function or, conversely, an anti-symmetrical spatial function with a symmetrical spin function. Because of the exchange symmetry of the system, the spatial function of the two-electron system is either symmetric (+ sign in the second term, so-called binding state) or antisymmetric (-sign, so-called antibonding state) in the absence of spin-dependent potentials. The energy curve that arises when the distance between the two atomic nuclei is varied is the driving force for the formation of molecules, whereby a singlet state is obtained in the case of a symmetrical spatial wave function (since the spin wave function of the two electrons must then be antisymmetric), in the case of a antisymmetric spatial wave function, on the other hand, one of three triplet states (see below). ${\ displaystyle E_ {| AB |}}$ ${\ displaystyle | AB |}$

The calculation of the molecular binding energy shows in detail that in the first case, i.e. for the singlet state of the two-electron system, a stable bond is obtained ("bonding state"), while for the triplet case, i.e. for an antisymmetric two-electron system Position function, not just a much higher energy result; but (in contrast to the singlet state with its pronounced minimum of the energy curve) now even a monotonically decreasing behavior of the energy curve results, which corresponds to an antibonding behavior.

One advantage of the valence bond theory is that the wave function can be understood as a linear combination of resonance structures, making it accessible to a direct chemical interpretation, which is much more difficult with molecular orbital-based approaches. Therefore, VB approaches are particularly suitable for gaining an understanding of molecules with an unusual electronic structure.

Singlet and triplet states of a two-electron system

The singlet spin function is when using the usual arrow symbols (whereby the first arrow refers to one electron, the second to the other):

{\ displaystyle \ left | S {=} 0, M {=} 0 \ right \ rangle = (1 / {\ sqrt {2}}) \ cdot (\ left | {{\ mathord {\ uparrow}} {\ mathord {\ downarrow}}} \ right \ rangle - \ left | {{\ mathord {\ downarrow}} {\ mathord {\ uparrow}}} \ right \ rangle)}

(i.e. the spin of the first electron points upwards, that of the second downwards or vice versa, with a minus in the linear combination; as I said: the whole thing multiplied by a symmetrical position function of the two electrons.)

The three triplet states are on the other hand (distinguished by the "magnetic quantum number" M ):

{\ displaystyle \ left | S {=} 1, M {=} {+ 1} \ right \ rangle = \ left | {{\ mathord {\ uparrow}} {\ mathord {\ uparrow}}} \ right \ rangle }

{\ displaystyle \ left | S {=} 1, M {=} 0 \ right \ rangle = (1 / {\ sqrt {2}}) \ cdot (\ left | {{\ mathord {\ uparrow}} {\ mathord {\ downarrow}}} \ right \ rangle + \ left | {{\ mathord {\ downarrow}} {\ mathord {\ uparrow}}} \ right \ rangle)}

{\ displaystyle \ left | S {=} 1, M {=} {- 1} \ right \ rangle = \ left | {{\ mathord {\ downarrow}} {\ mathord {\ downarrow}}} \ right \ rangle }

(e.g. both spins upwards or the first opposite to the second, now with a + in the linear combination; or; both spins downwards. As I said: the whole thing multiplied by an antisymmetric position function of the two electrons.)

With the neglect of relativistic terms assumed here, the energy of the system depends only on the position functions and is, as it turns out (see below), binding for the singlet case and antibonding (repulsive) for the triplet case. The Hamilton operator (energy operator) of the system takes into account the kinetic energy of the two electrons - the atomic nuclei are retained ( Born-Oppenheimer approximation ) - only all Coulomb interactions involving the two electrons, i.e. the location-dependent Coulomb attractions of both electrons through both atomic nuclei ( four contributions to the Hamilton operator) as well as the position-dependent mutual Coulomb repulsion of electrons 1 and 2 (only one contribution to ). ${\ displaystyle {\ mathcal {H}}}$

Basic model

The basic model was developed for the hydrogen molecule as it allows the simplest calculations:

Each of the two hydrogen atoms provides one electron for an electron pair bond .
By combining the s orbitals of the hydrogen atoms in which the electrons were originally located, molecular orbitals , an unoccupied antibonding molecular orbital and an occupied binding molecular orbital , in which the two electrons are then (spin- coupled , see above) as an electron pair, are created. (The spin coupling takes into account the Pauli principle .)
The theoretical energy of the bond and the probability of the electrons being in the binding molecular orbital can be determined using the wave function of this orbital.
However, the wave function of the binding molecular orbital is unknown and is approximated taking various factors into account until it is in satisfactory agreement with experimental results.
The two s orbitals of the originally individual hydrogen atoms serve as the starting point for the computational approximation .

Valence bond method for the hydrogen molecule

Hydrogen atom A has electron number 1 and the wave function . ${\ displaystyle \ Psi _ {A (1)}}$

Hydrogen atom B has electron number 2 and the wave function . ${\ displaystyle \ Psi _ {B (2)}}$

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

Elementary approximation

In the elementary approximation, it is completely disregarded whether and how the two atomic nuclei and electrons influence each other when they approach each other to form a bond. The wave function for a system of two atoms that do not influence each other is obtained from the wave functions of the individual hydrogen atoms: ${\ displaystyle \ Psi}$

{\ displaystyle \ Psi = \ Psi _ {\ mathrm {A (1)}} \ cdot \ Psi _ {\ mathrm {B (2)}}}

The binding energy and the nuclear distance that result from this hardly agree with the experimental results.

Exchange energy according to Heitler and London

In the molecular orbital, electron 1 does not always have to be in hydrogen atom A, just as electron 2 does not always have to be in hydrogen atom B. Both cases are rather equally probable because of the Pauli principle . Accordingly, a term for exchanged electrons is added which, for reasons of symmetry, is weighted either with plus 1 or with minus 1. At first it is by no means clear whether the first or second case is associated with binding or antibonding behavior; this only results from a specific invoice or a separate consideration, as carried out by Eugen Wigner .

So where a symmetric position function, (... + ...), is multiplied by a singlet spin function, an antisymmetric, (... - ...), on the other hand, is multiplied by a triplet spin function. As Eugen Wigner showed through the above consideration, the ground state of a two-electron system is always a singlet state (i.e. with a symmetrical position function, case +) and never corresponds to a triplet position function (case -).

{\ displaystyle \ Psi _ {\ pm} = \ Psi _ {\ mathrm {A (1)}} \ cdot \ Psi _ {\ mathrm {B (2)}} \ pm \ Psi _ {\ mathrm {B ( 1)}} \ cdot \ Psi _ {\ mathrm {A (2)}},}

The energy difference calculated with the Hamilton operator is called the exchange energy , whereby the sign factor ± occurs again due to the normalization condition of the Schrödinger functions, now in the denominator: ${\ displaystyle {\ mathcal {H}}}$ ${\ displaystyle E _ {+} - E _ {-} \ (= \ Delta E _ {\ mathrm {Exch.}})}$

{\ displaystyle \ Delta E _ {\ mathrm {Exch.}} = {\ frac {\ langle \ Psi _ {+} | {\ mathcal {H}} | \ Psi _ {+} \ rangle} {\ langle \ Psi _ {+} | \ Psi _ {+} \ rangle}} - {\ frac {\ langle \ Psi _ {-} | {\ mathcal {H}} | \ Psi _ {-} \ rangle} {\ langle \ Psi _ {-} | \ Psi _ {-} \ rangle}}.}

The sign convention comes from the theory of ferromagnetism and takes some getting used to here: Since in two-electron systems the triplet state (the state with the minus sign in the position function) always has the much higher energy, i.e. it is energetically unfavorable, the sign of the exchange energy for two-electron systems is always negative , which is valid for the antiferromagnetic states in the theory of ferromagnetism in the so-called Heisenberg model . Another common convention is obtained by adding a factor of 2 to the last equation on the left (or the penultimate equation on the right) . ${\ displaystyle \ Delta E _ {\ mathrm {Exch.}}}$

In any case, the positive coefficient is a good approximation of experimental results, while the negative sign leads to the aforementioned "unoccupied state".

shielding

The terms and only take into account that an electron shields the nuclear charge of a hydrogen nucleus. In the molecule, however, there are two nuclei and two electrons that shield the nuclear charges more strongly, which means that the effective nuclear charge is lower. ${\ displaystyle \ Psi _ {\ mathrm {A}}}$ ${\ displaystyle \ Psi _ {\ mathrm {B}}}$

The effective nuclear charge is taken into account in the above wave functions by dashed symbols. So we get the position functions again:

{\ displaystyle \ Psi _ {\ pm} = \ Psi _ {\ mathrm {A (1)}} ^ {\ prime} \ cdot \ Psi _ {\ mathrm {B (2)}} ^ {\ prime} \ pm \ Psi '_ {\ mathrm {B (1)}} \ cdot \ Psi _ {\ mathrm {A (2)}} ^ {\ prime}}

resonance

Theoretically, the hydrogen atoms in the compound can not only exchange their electrons, but there is also a small probability that sometimes both electrons are located on one of the hydrogen nuclei. Accordingly, ionic structures ( Hund-Mulliken , one-electron molecular functions) with weighting factors can be added to the non-ionic two-electron functions (two-electron Heitler-London atom functions) for the hydrogen molecule . ${\ displaystyle \ lambda _ {\ pm}}$

However, since the ionicity is low in the cases under consideration, this factor is generally small and, under certain circumstances, U. even completely negligible. So:

{\ displaystyle \ Psi _ {\ pm} = \ Psi '_ {\ mathrm {A (1)}} \ Psi' _ {\ mathrm {B (2)}} \ pm \ Psi '_ {\ mathrm {B (1)}} \ Psi '_ {\ mathrm {A (2)}} + \ lambda _ {\ pm} \ Psi' _ {\ mathrm {A (1)}} \ Psi '_ {\ mathrm {A (2)}} + \ lambda _ {\ pm} ^ {*} \ Psi '_ {\ mathrm {B (1)}} \ Psi' _ {\ mathrm {B (2)}}}

(The asterisk denotes conjugation (mathematics) .)

Using this simple wave function, the above-mentioned interpretability of the wave function in the form of resonance structures can already be recognized. The first two terms correspond to the purely covalent structure, the other two to the two ionic ones.

Here the deviation from experimental findings is already very small and after applying a wave equation with 100 correction terms, results are obtained that almost agree with the experimental findings. In addition, it must be taken into account that u. U. even with "rather bad" wave functions get quite good results for the energy.

It is also interesting that the simple Heitler-London model, in contrast to the Hartree-Fock approach of the molecular orbital theory , predicts the correct dissociation behavior for the hydrogen molecule. In the dissociation limit, the proportions of the ionic resonance structures must be zero ( homolytic cleavage ). The Heitler-London approach offers the degree of freedom that the coefficients of the ionic structures vanish variably within the framework of the optimization, while the Hartree-Fock wave function provides a 50:50 ratio between ionic and covalent components.

In any case, the basic model for hydrogen molecules has been continuously refined and the problem has been transferred to larger and much more complicated molecules, as well as to multiple bonds , not only in molecules but also in solids.

The approach of the valence bond method as well as the molecular orbital theory represent the basis of today's molecular modeling , which enables predictions and interpretations of many molecular structures and properties through computer-aided calculations.

General description of molecules

In general, the VB wave function can be written as a linear combination of the individual VB structures:

{\ displaystyle \ Psi _ {VB} = \ sum _ {i} c_ {i} \ Phi _ {i}}

The weight of the individual resonance structures can be calculated using the so-called Chirgwin-Coulson formula:

${\ displaystyle \ omega _ {i} = c_ {i} ^ {2} + \ sum _ {i \ neq j} c_ {i} c_ {j} S_ {ij}}$

Where represents the overlap integral between the structures. In order to obtain the VB energy and the coefficients, the wave function is inserted into the Schrödinger equation and the energy expectation value is calculated variably (analogous to the derivation of the configuration interaction ). As usual in quantum chemistry , the individual VB structures are written as determinate wave functions, but using atomic orbitals (as opposed to molecular orbitals in MO theory). As usual, the atomic orbitals are represented as linear combinations of basis functions. As part of the VBSCF method (valence bond self consistent field calculation), both the expansion coefficients of the VB wave function and the basic functions of the individual atomic orbitals are variably optimized. ${\ displaystyle S_ {ij}}$

Using all relevant ( linearly independent ) structures, the approach is equivalent to full CI.

There are various program packages with which VB calculations can be carried out, such as B. XMVB and a variety of different ab intio methods in addition to the already mentioned VBSCF, such as. B. BOVB or VBCI.

Pauling's theory of complexes

The coordinative bond is present in complexes made up of a central atom and a certain number of ligands . This type of bond does not come about because both reaction partners, i.e. the central atom and ligand, each provide one electron, but rather because the ligand alone brings two electrons and thus forms a bond to the central atom.

If a ligand supplies two electrons, and n is the number of binding ligands, then the central atom gets n × 2 electrons that it has to accommodate somewhere. The empty outer orbitals of the central atom are available for accommodation:

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

Inner / outer orbital complexes

The explanation of why some ligands produce outer or inner sphere complexes was only possible with the crystal field or ligand field theory . Here the concept of high spin and low spin complexes was introduced according to the magnetic properties of such complexes.

Example:

The Fe ²⁺ - cation has 6 electrons in the 3 d orbital, so a 3 d ⁶ - configuration .

With six water ligands there are 12 electrons. In this case, the configuration of the cation is retained: 3 d ⁶ 4 s ² 4 p ⁶ 4 d ⁴ .

With six cyanide ligands there are also 12 electrons. Here the “original occupation” of the central atom changes to: 3 d ¹⁰ 4 s ² 4 p ⁶ .

geometry

In order to form bonds between the central atom and ligands, hybrid orbitals are formed by the central atom according to the VB theory , the number of which corresponds to the number of electron pairs that the ligands make available for binding.

Depending on the type and, above all, the number of ligand electron pairs, certain d (inner), s , p and d (outer) orbitals of the central atom are used for hybridization , resulting in characteristic coordination geometries:

6 ligands : 2 d (inner) + 1 s + 3 p = 6 d ²sp ³ → octahedron ( inner orbital complex )

6 ligands : 1 s + 3 p + 2 d (outer) = 6 sp ³ d ² → octahedron ( outer orbital complex )

4 ligands : 1 s + 3 p = 4 sp ³ → tetrahedron

4 ligands : 1 d (inside) + 1 s + 2 p = 4 dsp ² → planar square

etc.

Limits

The VB theory is well suited for determining complex geometries and for explaining magnetic phenomena. However, other phenomena, such as the color of complexes, can only be explained with great effort using the VB theory. To this end models such as the crystal field theory or the ligand field theory or molecular orbital theory more appropriate.

literature

Charles A. Coulson: The chemical bond . Hirzel, 1969.
James E. Huheey, Ellen Keiter, Richard L. Keiter: Inorganic Chemistry. Principles of structure and reactivity . 3. Edition. Gruyter, 2003, ISBN 3-11-017903-2 .
AF Holleman , E. Wiberg , N. Wiberg : Textbook of Inorganic Chemistry . 101st edition. Walter de Gruyter, Berlin 1995, ISBN 3-11-012641-9 .
Philippe C. Hiberty, Sason S. Shaik: A Chemist's Guide to Valence Bond Theory , John Wiley & Sons, 2008 ( limited preview in Google Book search).

References and footnotes

↑ ^a ^b Philippe C. Hiberty and Sason Shaik: Valence Bond Theory, Its History, Fundamentals and Applications: A Primer . In: Kenny B. Lipkowitz, Raima Larter and Thomas R. Cundari (eds.): Reviews in Computational Chemistry . John Wiley & Sons, Ltd, 2004, ISBN 978-0-471-67885-4 , pp. 1–100 , doi : 10.1002 / 0471678856.ch1 (English).
↑ In quantum mechanics one would speak of “perturbation theory of the 0th order”, i.e. H. worse than the standard, the "first order perturbation theory".
↑ See for example DC Mattis : The theory of magnetism , Berlin etc., Springer 1988, ISBN 0-387-15025-0 .
↑ See e.g. B. the chapter on perturbation theory of the first order in all standard textbooks of quantum mechanics.
^ ^A ^b ^c Sason S. Shaik, Philippe C. Hiberty: A Chemist's Guide to Valence Bond Theory . John Wiley & Sons, 2007, ISBN 978-0-470-19258-0 .
↑ JH van Lenthe, GG Balint-Kurti: The valence-bond self-consistent field method (VB-SCF): Theory and test calculations . In: The Journal of Chemical Physics . tape 78 , no. 9 , May 1, 1983, ISSN 0021-9606 , p. 5699-5713 , doi : 10.1063 / 1.445451 .
↑ Jiabao Li, Roy McWeeny: VB2000: Pushing valence bond theory to new limits . In: International Journal of Quantum Chemistry . tape 89 , no. 4 , 2002, p. 208-216 , doi : 10.1002 / qua.10293 .
↑ Lingchun Song, Yirong Mo, Qianer Zhang, Wei Wu: XMVB: A program forab initio nonorthogonal valence bond computations . In: Journal of Computational Chemistry . tape 26 , no. 5 , 2005, ISSN 0192-8651 , p. 514-521 , doi : 10.1002 / jcc.20187 .
↑ Philippe C. Hiberty, Sason Shaik: Breathing-orbital valence bond method - a modern valence bond method did includes dynamic correlation . In: Theoretical Chemistry Accounts . tape 108 , no. 5 , November 1, 2002, ISSN 1432-2234 , p. 255-272 , doi : 10.1007 / s00214-002-0364-8 .

[:1-1] Philippe C. Hiberty and Sason Shaik: Valence Bond Theory, Its History, Fundamentals and Applications: A Primer . In: Kenny B. Lipkowitz, Raima Larter and Thomas R. Cundari (eds.): Reviews in Computational Chemistry . John Wiley & Sons, Ltd, 2004, ISBN 978-0-471-67885-4 , pp. 1–100 , doi : 10.1002 / 0471678856.ch1 (English).

[2] In quantum mechanics one would speak of “perturbation theory of the 0th order”, i.e. H. worse than the standard, the "first order perturbation theory".

[3] See for example DC Mattis : The theory of magnetism , Berlin etc., Springer 1988, ISBN 0-387-15025-0 .

[4] See e.g. B. the chapter on perturbation theory of the first order in all standard textbooks of quantum mechanics.

[:0-5] A ^b ^c Sason S. Shaik, Philippe C. Hiberty: A Chemist's Guide to Valence Bond Theory . John Wiley & Sons, 2007, ISBN 978-0-470-19258-0 .

[6] JH van Lenthe, GG Balint-Kurti: The valence-bond self-consistent field method (VB-SCF): Theory and test calculations . In: The Journal of Chemical Physics . tape 78 , no. 9 , May 1, 1983, ISSN 0021-9606 , p. 5699-5713 , doi : 10.1063 / 1.445451 .

[7] Jiabao Li, Roy McWeeny: VB2000: Pushing valence bond theory to new limits . In: International Journal of Quantum Chemistry . tape 89 , no. 4 , 2002, p. 208-216 , doi : 10.1002 / qua.10293 .

[8] Lingchun Song, Yirong Mo, Qianer Zhang, Wei Wu: XMVB: A program forab initio nonorthogonal valence bond computations . In: Journal of Computational Chemistry . tape 26 , no. 5 , 2005, ISSN 0192-8651 , p. 514-521 , doi : 10.1002 / jcc.20187 .

[9] Philippe C. Hiberty, Sason Shaik: Breathing-orbital valence bond method - a modern valence bond method did includes dynamic correlation . In: Theoretical Chemistry Accounts . tape 108 , no. 5 , November 1, 2002, ISSN 1432-2234 , p. 255-272 , doi : 10.1007 / s00214-002-0364-8 .