Some Examples of Molecule Orbitals

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Molecule orbitals are formed when atomic orbitals overlap. Mathematically, this is represented by a linear combination of atomic orbitals (as in the LCAO-MO method). Some possible classifications of orbitals are bonding and antibonding and - and -symmetry.

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This Demonstration shows the basic characteristics for a set of six molecules: the label, the description, the number of electrons in the chosen molecular orbital, and a 3D view of the probability density (with boundary surface, phase-coloring included) and also a ball and stick model for each example.

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Contributed by: Guenther Gsaller (October 2013)
Suggested by: Norbert Mueller
(Institute of Organic Chemistry, Johannes Kepler University, Linz, Austria, www.jku.at/orc)
Open content licensed under CC BY-NC-SA


Snapshots


Details

The atomic orbitals used in this Demonstration are the following Slater-type orbitals:

,

,

,

.

For ethene, the LCAO-MO method gives two normalized molecule orbitals:

,

.

The formula for the distance of two points in three-dimensional space can be used to rewrite the wavefunctions with the corresponding -Slater orbital ( … interatomic distance):

,

.

The diagrams for the molecular orbitals for dihydrogen, ethyne, and allene can also be written using the separation-dependent formula.

The simple Hückel method (SMO) is a rudimentary method for energies and orbitals of -electron systems. An essential step is solving the secular equations.

For cyclobutadiene, the secular equations are:

,

which can be reduced to

.

Taking symmetry into account, one can factorize this secular determinant into two separate determinants:

yielding the solutions , and .

The four corresponding LCAO molecular orbitals are found using cofactors:

,

,

,

.

An alternative set for cyclobutadiene is:

,

,

,

.

Similarly for benzene, from the secular determinant,

can be factorized to give the wavefunctions:

,

,

,

,

,

.

Cyclic -sytems (such as cyclobutadiene and benzene) can be rewritten with the corresponding -Slater orbitals in the complex exponential form:

,

with .

ChemSpider [1] is the source for the ball and stick figures. The ChemSpider IDs for each structure used are listed in the code for the Demonstration.

References

[1] Royal Society of Chemistry. ChemSpider. (Oct 8, 2013) www.chemspider.com.

[2] H. E. Zimmerman, Quantum Mechanics for Organic Chemists, London: Academic Press, 1975.

[3] A. Rauk, Orbital Interaction Theory of Organic Chemistry (2nd ed.), New York: John Wiley & Sons, 2000.



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