9772

Some Examples of Molecule Orbitals

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.
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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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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