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Energy-Level Diagrams and Molecular Orbitals for Conjugated Polyenes

Hückel molecular orbital (HMO) energy diagrams can be constructed using linear combinations of
π
-orbitals for conjugated polyenes. Quantum chemistry packages have been used to model the molecules and molecule-orbitals via RHF/6-31G(d) or U-B3LYP/6-31G(d).
This Demonstration shows characteristics for a set of seven conjugated polyenes: the HMO energy diagram, a 3D view of the probability density (with boundary surface, phase-coloring included), and a ball-and-stick model for each example.

Details

The simple Hückel method (SMO) is a rudimentary method for energies and orbitals of
π
-electron systems.
Possible sets of corresponding LCAO molecular orbitals for cyclobutadiene and benzene are
ψ
1
=
1
2
(
χ
1
+
χ
2
+
χ
3
+
χ
4
)
,
ψ
2
=
1
2
(
χ
1
-
χ
2
-
χ
3
+
χ
4
)
,
ψ
3
=
1
2
(
χ
1
+
χ
2
-
χ
3
-
χ
4
)
,
ψ
4
=
1
2
(
χ
1
-
χ
2
+
χ
3
-
χ
4
)
,
and
ψ
1
=
1
6
(
χ
1
+
χ
2
+
χ
3
+
χ
4
+
χ
5
+
χ
6
)
,
ψ
2
=
1
2
(
χ
2
+
χ
3
-
χ
5
-
χ
6
)
,
ψ
3
=
1
12
(2
χ
1
+
χ
2
-
χ
3
+2
χ
4
-
χ
5
+
χ
6
)
,
ψ
4
=
1
12
(2
χ
1
-
χ
2
-
χ
3
+2
χ
4
-
χ
5
-
χ
6
)
,
ψ
5
=
1
2
(
χ
2
-
χ
3
+
χ
5
-
χ
6
)
,
ψ
6
=
1
6
(
χ
1
-
χ
2
+
χ
3
-
χ
4
+
χ
5
-
χ
6
)
.
To visualize the different linear combinations of wavefunctions for
π
-electron systems of conjugated polyenes, a shaded version of the
p
z
orbital is used
,
. For example, the coefficients in the linear combinations for cyclobutadiene are all equal by size and differ only by sign. In benzene they can be ordered by size into three groups. In an HMO energy diagram for benzene, a combination of three different relative sizes of
,
is used. The last column in the energy diagram shows no symbol or the symbols
↑
and
⥮
to indicate that the orbital
ψ
i
is not occupied, singly occupied, or doubly occupied.
ChemSpider[1] is the basis for the coordinates in the input files of the nonradical structures. The ChemSpider IDs for each structure are listed in the following table.
molecule
ch-id
ethene
6085
butadiene
7557
1,3,5-hexatriene
15830
cyclobutadiene
20626
benzene
236
An input file for the allyl- and 1,4-pentadienyl-radical is created in Avogadro beginning with a sketch[6].
The non-radicals' optimizations are run in GAMESS-US with an RHF/6-31G(d) model via GamesQ[7].
For both radicals, an optimization with unrestricted Hartree–Fock is necessary. The model of choice is U-B3LYP/6-31G(d).
Here is a table for the final energy from the log-files of the optimizations for each molecule.
molecule
model
fin.energy[Hartree]
r.m.s.[Hartree]
ethene
RHF/6-31G(d)
-78.03
0.0000219
allyl
UHF-B3LYP/6-31G(d)
-117.18
0.0000472
butadiene
RHF/6-31G(d)
-154.92
0.0001377
pentadienyl
UHF-B3LYP/6-31G(d)
-194.54
0.0001311
hexatriene
RHF/6-31G(d)
-231.81
0.0000598
cyclobutadiene
RHF/6-31G(d)
-153.64
0.0000199
benzene
RHF/6-31G(d)
-232.70
0.0000339
From each log-file the last frame is exported as a mol-file via MacMolPlt[8]. The Demonstration "Displaying Molecules with Multiple Bonds" by Bianca Eifert makes it possible to show the double-bonds of molecules in Mathematica[9]. Through slightly modifying her code, the modeled molecules can be imported as a mol-file with double-bonds on.
The corresponding molecule-orbitals for the molecules are created in Gabedit[10].
The results are copied to this Demonstration into the initalization code and named MOOrbitals and MOMolecules.

References

[1] Royal Society of Chemistry. ChemSpider. (Apr 29, 2014) www.chemspider.com.
[2] A. Rauk, Orbital Interaction Theory of Organic Chemistry, 2nd ed., New York: John Wiley & Sons, 2000.
[3] E. Heilbronner and H. Bock, Das HMO-Modell und seine Anwendungen—Grundlagen und Handhabung, Weinheim: Verlag Chemie GmbH, 1970.
[4] J. Jensen. Molecular Modeling Basics. (Mar 19, 2014) molecularmodelingbasics.blogspot.co.at.
[5] "Density Functional Theory." (Mar 19, 2014) www.cup.uni-muenchen.de/oc/zipse.
[6] M. Hanwell, D. Curtis, D. Lonie, T. Vandermeersch, E. Zurek, and G. Hutchison, "Avogadro: An Advanced Semantic Chemical Editor, Visualization, and Analysis Platform," Journal of Chemoinformatics, 4(17), 2004 pp. 1758–2946. doi:10.1186/1758-2946-4-17.
[7] M. Schmidt, K. Baldridge, J. Boatz, S. Elbert, M. Gordon, J. Jensen, S. Koseki, N. Matsunaga, K. Nguyen, S. Su, T. Windus, M. Dupuis, and J. Montgomery, "General Atomic and Molecular Electronic Structure System," Journal of Computational Chemistry, 14, 1993 pp. 1347–1363. doi:10.1002/jcc.540141112.
[8] B. Bode and M. Gordon, "MacMolPlt," Journal of Molecular Graphics and Modelling, 16, 1998 pp. 133–138.
[9] B. Eifert. "Displaying Molecules with Multiple Bonds" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/DisplayingMoleculesWithMultipleBonds.
[10] A. Allouche, "Gabedit—A Graphical User Interface for Computational Chemistry Softwares," Journal of Computational Chemistry, 32, 2011 pp. 174–182. doi:10.1002/jcc.21600.

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