Technology Report pubs.acs.org/jchemeduc
A Simple Hü ckel Molecular Orbital Plotter Raghunathan Ramakrishnan* Department Chemie, Theoretische Chemie, Technische Universität München, D-85748 Garching, Germany S Supporting Information *
ABSTRACT: A program is described and presented to readily plot the molecular orbitals from a Hückel calculation. The main features of the program and the scope of its applicability are discussed through some example organic molecules.
KEYWORDS: Upper-Division Undergraduate, Organic Chemistry, Computer-Based Learning, MO Theory
T
he Hückel molecular orbital (MO) theory1 serves as an excellent starting point to learn some of the essential aspects of computational quantum chemistry. Several extensions, analyses, and novel applications of Hückel theory have been previously discussed in this Journal.2−8 Many implementations of Hückel theory are freely available on the Internet9 with which one can readily perform various calculations. A large fraction of a typical output of a Hückel calculation deals with numerical tabulation of the eigenvectors of the Hückel Hamiltonian matrix that represent the MOs. Except for simple molecules, such as ethylene, interpretation of the eigenvectors is nearly impossible without using graphical aids such as handdrawn sketches of MOs. Here a program is described that can serve as a utility tool to analyze results of a Hückel calculation. More specifically, given a set of two-dimensional Cartesian coordinates (i.e., x and y), the program enables one to readily plot the Hückel MOs of an arbitrary planar molecule. The necessity of the atomic coordinates for plotting the MOs is not surprising because in a simple Hückel approach, all that is required are the connectivities between atoms. The style of the MO plot employed here is the top-view approach where the planar conjugated pi system is viewed through an axis perpendicular to the molecular plane. Figure 1 shows such a plot for the MOs of ethylene along with a more familiar side-view plot of the same orbitals. The size of the red and blue circles in the top-view provides a measure of the contribution of the corresponding 2p orbital of the atomic center to the particular MO. The applicability of the top-view MO plots may be appreciated by considering larger systems where the side-view plots are less effective and numerical tabulations of eigenvectors are hard to interpret. For example, consider the somewhat larger molecule styrene whose highest occupied molecular © XXXX American Chemical Society and Division of Chemical Education, Inc.
Figure 1. Side-view (left) and top-view (right) plots of the bonding and the antibonding pi MOs of ethylene.
orbital (HOMO) is shown in Figure 2. This plot contains the following two essential features: 1. The relative signs of the MO coefficients mapped by two different colors. 2. The relative magnitude of the MO coefficients implied by the size of the circles. When focusing on these two aspects of a top-view plot, the dominant character of the MO can easily be deduced.
Figure 2. A top-view plot of the HOMO of styrene according to a Hückel calculation. The numbers given are the elements of the corresponding eigenvector of the Hamiltonian matrix.
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dx.doi.org/10.1021/ed300085g | J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
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The MO plotter program presented here is written in the language GNU-Octave10 that generates a set of scripts, one for each MO, which are executed by the program gnuplot11 to create the plots. The GNU-Octave program, which is named as “gen_script.m”, requires two input matrices, namely, (i) the x and y coordinate matrix of all the atoms and (ii) the Hückel Hamiltonian matrix. Considering ethylene as an example, these two matrices are entered into “gen_script.m” in the GNUOctave format as follows
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) McQuarrie, D. A.; Simon, J. D. Physical Chemistry: A Molecular Approach; Viva Books: New Delhi, 2008; Chapter 10. (2) Dalton, C. J.; Friedrich, L. E. J. Chem. Educ. 1975, 52, 721−724. (3) Taubmann, G. J. Chem. Educ. 1992, 67, 96−97. (4) Bahnick, D. A. J. Chem. Educ. 1994, 71, 171−173. (5) Healy, E. F. J. Chem. Educ. 1995, 72, A120−A121. (6) Nordholm, S.; Bäck, A.; Bacskay, G. B. J. Chem. Educ. 2007, 84, 1201−1203. (7) Teslja, A. J. Chem. Educ. 2011, 88, 1340−1340. (8) Moore, B. G. J. Chem. Educ. 2000, 77, 785−789. (9) For example HMO v1.0: Huckel molecular orbital program, http://www.unb.ca/fredericton/science/chem/ajit/f_hmo.htm (accessed Nov 2012). (10) GNU Octave: An open source high-level interpreted language, http://www.gnu.org/software/octave/ (accessed Nov 2012). (11) Gnuplot: An open source interactive plotting program, http:// gnuplot.sourceforge.net/, http://www.gnuplot.info (accessed Nov 2012).
XY = [ 1.00000 0.00000 ‐1.00000 0.00000 ];
and
H=[ 0 ‐1 ‐1 0 ]; The overall procedure for creating the MO plots is given by the following three simple steps. 1. Fill in the matrices XY and H in “gen_script.m”. 2. Execute “gen_script.m” using GNU-Octave to prepare a set of gnuplot scripts “plot_1.gnu”, and so on. 3. Execute the script “plot_1.gnu” using gnuplot to get the plots “plot_1.eps” and so on. The quality of a plot can be controlled by the following optional inputs in “gen_script.m” LWIDTH = 20.0; SCALE = 1.0; CMINUS = ’red’; CPLUS = ’blue’; X_MAX = max(max(XY)); X_MIN =-max(max(XY)); Y_MAX = max(max(XY)); Y_MIN =-max(max(XY)); The first option LWIDTH controls the thickness of the bonds for which one may use a somewhat larger value such as 20.0 especially when preparing smaller plots. The value of the data SCALE scales the radius of the circles for which the default value of 1.0 seem to suffice for most purposes. The keywords CMINUS and CPLUS control the colors of positive and negative phases of the MOs for which one may use contrasting colors such as the default options of red and blue. Finally, the keywords X_MAX, X_MIN, Y_MAX, and Y_MIN represent the maximum and minimum x and y coordinates, which define the size of the overall plot. When these values are hard to predict, one may use standard values deduced from the atomic coordinate with the maximum magnitude, that is, max(max(XY)).
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Technology Report
ASSOCIATED CONTENT
* Supporting Information S
Three “gen_script.m” files that can be used to prepare Hückel MOs of the molecules ethylene, styrene, and [18]annulene. This material is available via the Internet at http://pubs.acs.org. B
dx.doi.org/10.1021/ed300085g | J. Chem. Educ. XXXX, XXX, XXX−XXX
