Difference between revisions of "Orbital overlap"
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'''Orbital overlap''' is a concept in the [[Molecular orbital theory|molecular orbital]] description of [[chemical bond]]ing. In the simplest terms, a [[covalent bond]] can only form between two atoms if there is significant overlap between [[atomic orbital]]s on the atoms. | '''Orbital overlap''' is a concept in the [[Molecular orbital theory|molecular orbital]] description of [[chemical bond]]ing. In the simplest terms, a [[covalent bond]] can only form between two atoms if there is significant overlap between [[atomic orbital]]s on the atoms. | ||
| − | Orbital overlap can be expressed in qualitative terms through empirical rules, or quantitatively through [[overlap integral]]s ''S<sub>ab</sub>'' ( | + | Orbital overlap can be expressed in qualitative terms through empirical rules, or quantitatively through [[overlap integral]]s ''S''<sub>ab</sub> (where a and b represent atomic orbitals). |
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| + | It is often assumed that a greater orbital overlap leads to a stronger chemical bond, as in the [[angular overlap model]]. This is not always true, as strong chemical bonds can occur with very little orbital overlap ([[ionic bond]]s) and, conversely, there are situations where high orbital overlap does not lead to bonding, such as the hypothetical dihelium molecule He<sub>2</sub>. | ||
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| + | ==Empirical rules== | ||
| + | *Orbital overlap is poor if the two orbitals have very different energies. This is the basis behind the idea that chemical bonding involves the [[valence shell]], the outermost electrons. | ||
| + | *Orbital overlap is poor if the orbitals are very different in size. This can be seen in the reduction in stability of [[hydride]]s on descending a group in the [[p block]]. | ||
| + | *The orbital overlap depends on the type of orbital, with s > p > d > f if other factors are similar. Overlap is also more effective when it produces a σ bond than when it produces a π bond, leading to σ bonds being stronger than π bonds when other factors are equal. | ||
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| + | ==Overlap integral== | ||
| + | {{main|Overlap integral}} | ||
| + | The overlap integral ''S''<sub>ab</sub> is the name given to an integral of the form | ||
| + | :<math>S_{\rm ab} = \int \varphi_{\rm a}^\ast \varphi_{\rm b} {\rm d}\tau</math> | ||
| + | where φ<sub>a</sub> and φ<sub>b</sub> are the [[wavefunction]]s corresponding to the atomic orbitals a and b and dτ indicates that the integral is to be taken over all space.<ref>{{GoldBookRef|file=O04357|title=overlap integral, ''S''<sub>rs</sub>|accessdate=2010-09-12}}.</ref> In [[Dirac notation]], ''S''<sub>ab</sub> = <φ<sub>a</sub>|φ<sub>b</sub>>, usually written as simply <a|b>. | ||
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| + | ===Graphical representation=== | ||
| + | |||
| + | ===Basic results=== | ||
| + | #If a = b, i.e. the two orbitals are the same (and on the same atom), ''S''<sub>ab</sub> = 1.<ref group="Note" name="norm">Strictly speaking, this is only true if the orbitals are normalized. Indeed, ''S''<sub>ab</sub> = 1 is the condition for an orbital to be normalized. However, the use of normalized orbitals is universal in chemical calculations.</ref> This is a way of saying that each orbital overlaps perfectly with itself. | ||
| + | #For any set of [[hydrogenic orbital]]s on the same atom, ''S''<sub>ab</sub> = 0 if a ≠ b. This might seem counterintuitive at first sight, as surely a 2s orbital must "overlap" with a 1s orbital. | ||
| + | #Where a and b are orbitals on different atoms, ''S''<sub>ab</sub> lies between zero and one.<ref group="Note" name="norm"/><ref group="Note">This statement assumes that the wavefunctions corresponding to a and b are both real. Not all of the hydrogenic wavefunctions are real, but it is possible to take linear combinations to produce a basis set in which all the hydrogenic wavefunctions are both real and orthogonal: p<sub>''x''</sub> and p<sub>''y''</sub> orbitals are examples of such linear combinations.</ref> | ||
==Notes and references== | ==Notes and references== | ||
Latest revision as of 20:48, 11 September 2010
Orbital overlap is a concept in the molecular orbital description of chemical bonding. In the simplest terms, a covalent bond can only form between two atoms if there is significant overlap between atomic orbitals on the atoms.
Orbital overlap can be expressed in qualitative terms through empirical rules, or quantitatively through overlap integrals Sab (where a and b represent atomic orbitals).
It is often assumed that a greater orbital overlap leads to a stronger chemical bond, as in the angular overlap model. This is not always true, as strong chemical bonds can occur with very little orbital overlap (ionic bonds) and, conversely, there are situations where high orbital overlap does not lead to bonding, such as the hypothetical dihelium molecule He2.
Empirical rules
- Orbital overlap is poor if the two orbitals have very different energies. This is the basis behind the idea that chemical bonding involves the valence shell, the outermost electrons.
- Orbital overlap is poor if the orbitals are very different in size. This can be seen in the reduction in stability of hydrides on descending a group in the p block.
- The orbital overlap depends on the type of orbital, with s > p > d > f if other factors are similar. Overlap is also more effective when it produces a σ bond than when it produces a π bond, leading to σ bonds being stronger than π bonds when other factors are equal.
Overlap integral
The overlap integral Sab is the name given to an integral of the form
- <math>S_{\rm ab} = \int \varphi_{\rm a}^\ast \varphi_{\rm b} {\rm d}\tau</math>
where φa and φb are the wavefunctions corresponding to the atomic orbitals a and b and dτ indicates that the integral is to be taken over all space.[1] In Dirac notation, Sab = <φa|φb>, usually written as simply <a|b>.
Graphical representation
Basic results
- If a = b, i.e. the two orbitals are the same (and on the same atom), Sab = 1.[Note 1] This is a way of saying that each orbital overlaps perfectly with itself.
- For any set of hydrogenic orbitals on the same atom, Sab = 0 if a ≠ b. This might seem counterintuitive at first sight, as surely a 2s orbital must "overlap" with a 1s orbital.
- Where a and b are orbitals on different atoms, Sab lies between zero and one.[Note 1][Note 2]
Notes and references
Notes
- ↑ 1.0 1.1 Strictly speaking, this is only true if the orbitals are normalized. Indeed, Sab = 1 is the condition for an orbital to be normalized. However, the use of normalized orbitals is universal in chemical calculations.
- ↑ This statement assumes that the wavefunctions corresponding to a and b are both real. Not all of the hydrogenic wavefunctions are real, but it is possible to take linear combinations to produce a basis set in which all the hydrogenic wavefunctions are both real and orthogonal: px and py orbitals are examples of such linear combinations.
References
- ↑ overlap integral, Srs, <http://goldbook.iupac.org/O04357.html> (accessed 12 September 2010), Compendium of Chemical Terminology Internet edition; International Union of Pure and Applied Chemistry (IUPAC).
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