Difference between revisions of "Orbital overlap"
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===Basic results=== | ===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">Strictly speaking, this is only true if the | + | #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. | + | #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== | ||
Revision as of 20:05, 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).
Contents
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. 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
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