Use of Hiickel Molecular Orbital Theory in Interpreting the Visible Spectra of Polymethine Dyes An Undergraduate Physical Chemistry Experiment Donald A. Bahnick University of Wisconsin-Superior,
Superior, WI 54880
The particle in a box and Hiickel molecular orbital models are generally introduced in physical chemistry to illustrate quantum mechanical concepts and methodology. Laboratory application of the particle in a box model to rr electrons in molecules containing conjugated systems (free electron model) has been used to explain the relative wavelengths for the maxima in the visible region absorption bands of polymethine dyes ( I , 2) and the directional charl abacter of licrht absor~tion(3).The visible s ~ e c t r areeion sorption hand of a series of symmetrical polymethine dyes shifts toward loncrer - wavelenzths - as the lenzth - of the conjugated system increases. The physical chemistry laboratory experiment involves two parts:
-
-
measurement of the wavelengths for the absorption maxima in a series of structurallysimilar dyes relating the results to the calculated wavelengths for elmtranic transitions between the highest filled and lowest unfilled electronic energy levels predicted by the free electron model WE). The Hiickel molecular orbital model (HMO) can also be used to explain certain absorption bands in molecnles with conjugated systems as due to a + a* transitions between the highest-energy occupied molecular orbital (HOMO) a n d the lowest-enerev unoccu~iedmolecular orbital (LUMO) (4). We have introduced the use of HMO in the physical chemistry laboratory by having the students calculate the energies of the LUMO and HOMO for a series of conjugated dyes. Based on these results and the experimentally observed absorotion s~ectra.the relative enereies of the rr + rr* transitions predicted by HMO are related to the experimental results. -0
-
gated system in terms of a and p, solution of the following 4 x 4 determinant is required. X l O O
1x10 0 1 x 1 =O OOlx
(1)
where
The four nontrivial solutions to the secular determinant, in order of increasing energy, are E, = a +1.62
P
E2=a+0.618P E3=a-0.618P
E4 = a - 1.62 where a and p are negative numbers. El and EZare the energies of two bonding molecular orbitals containing the four rr electrons; E3 and E4 are the energies of two empty antibonding orbitals. The lowesbenergy electronic transition occurs between the HOMO (energy E2)and the LUMO (energy E3). The energy of the absorbed photon (AE) causing this rr + n* transition is given below. AE=E8-Ez=-1.236P (2) For linear polyenes, the energy difference AE is given by
Theorv The IIMO model as applied to conjugated n systems requires solution of secular determinants ta arrive at the aliowed quantized energies of the a electrons (4). These energies a r e expressed i n t e r m s of two parameters designated as
where n is the number of a centers in the molecule. A plot of AE (as experimentally obtained from electronic spectra) versus
a,the coulomb integral, which appmximates the energy
for a series of linear polyenes has been shown to fit a linear relationship (4). The s l o ~ eof the d o t allows calculation of 0. For the series ethyiene, butadiene, hexatriene, and octatetrene, a value for 0 of-60.5 kcaWmol was obtained. The same treatment applied to a series of polyene aldehydes gave a value of ,7O.4 kcaWmol for p. The HMO and FE models both predict decreasing AE values as the length of the conjugated rr system increases in' a linear polyene system. Recently the FE and HMO models were compared for long polyenes (5).Using the FE model and an average C-C bond length of 1.40 A in the polyenes, we estimate the Hiickel parameter P as -70.4 kcal/mol.
of an electron in a carbon 2p orbital p, the resonance or bond integral, which approximates the interaction energy between two atomic p, orbitals
Using HMO, we assign the value P to all resonance integrals corresponding to 2p orbitals on neighboring carbon atoms in the conjugated system, and zero is assigned to the integrals corresponding to the 2p orbitals of nonhonded carbon pairs. Streitweiser has extended the simple HMO model to allow for variations in the assigned integral values to reflect various bonding patterns and the presence of heteroatoms (4). For example, when using simple HMO to calculate the energies E of the four a electrons in the butadiene conju-
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Number 2 Februarv 1994
171
Experimental The visible spectra of three cyanine dyes were obtained by student teams using dilute solutions of the dyes in methanol a s described elsewhere (I, 2). A Perkin-Elmer model 552 spectrophotometer with rewrder was used to obtain the spectra. The three cyanine dyes (shown below) have conjugated chains between and including two nitrogen atoms.
1,1'diethyl4,2qanineiodide
1 , 1 ' d i e t h y l - 2 , P a ~ n i nchloride e
1,i'diethyl2,2dicarbocyanineiodide Commercially available software called HMO Calculator (Trinity Software, Campton, NH) available for MS-DOS or Apple Macintosh operating systems was used with a Macintosh wmputer. Students had been exposed to HMO theory and had carried out some calculations using the so&ware on some simple molecules before t h e laboratory period. While the spectra were being obtained, the structures of the dye cations were entered into the computer, and the calculations were carried out. The cations have 22,24, or 26 x electrons in 11,12, or 13 bonding molecular orbitals. The output gave hardcopy of the molecular structures with atom numbering system total n energies n energies of the 22,24, or 26 molecular orbitals linear comhination of atomic orbital expressions for the wavefunctions n charge densities n bond orders The software also allowed students to select simple HMO or modified Streitweiser methods to calculate the results.
One-parameter linear regression model applied to eq 4 using the observed wavelengths for the maxima of the cyanine dye absorption bands and HMO-determined Darameteffi. See the table for data used in obtaining this plot. With application of the FE model, the h values for the dyes are related to the chain lengths for the conjugated systems through ( I )
where p is the number of carbon atoms in the conjugated system (between the nitrogen atoms in the cyanine dye cations); and a is a wnstant for the dye series used to adjust the box width to give the best agreement betweencom~ u t e dand ex~erimentalh values. ' Students cdlculated average vvalurs for P (eq 4, and a (eq 5, from the observed h values for the three dyes. Substitution of these average values i n the same equations gave calculated h values. An alternate method of obtaining P is to plot the experimental h values versus -1lN. The slope of this plot equals 119,600lP. The graphical approach has the advantage of visual inspection of the linear relationship given by eq 4. Results The average value for P was computed a s -77.55 kcaVmol (s= 2.49) with ~impleHMO methods -75.39 keallmol (s = 2.64) with Streitweiser modified parameters The average value for n was
Comparison of Observed and Calculated Wavelength Absorption Maxima for Cyanine Dyes
Data Treatment With the HMO model, the excitation energies for the lowest energy electronic transitions (AE)are equal to -NP, where N is a number determined by the difference in energy between the HOMO and the LUMO. Because the enof a n absorbed photon of wavelength h a t the ergy maximum of the absorption band is equal to hclh, with the known values of the constants h (Planck's wnstant) and c (speed of light), the value of h can be computed using
(a)
119,600 (Wx nm) %om)= NP (kJ)
172
Journal of Chemical Education
h(nm) 1 2 3 1'
-AE
A
523.0 604.5 707.5
0.7318P 503.6 0.6010!3 613.3 0.5098 8 723.0 =1 .t'diethyl-2,Z'cyanine iodide
-AE
A
0.7555 P 501.9 0.61448 617.2 0.5256 P 721.4
2 = 1.1'-diethyl-2.2'-carbocyaninechloride
3 = 1.1'-diethyl-2.2'-dicarbocynine iodide b~alculated with eq 4 and p =-75.39 kcal/mol. %atahad with eq 4 and p --77.55 kcalimol. '~alculatedwith eo 5 and cr = 1.292.
1 483.9 611.1 738.4
With the use of a regression through the origin model with eq 4, the corresponding p values were determined as -78.02 and-75.86 kcal/mol. The plot ofthe results using eq 4 and the simple HMO-determined N values is shown in the figure. Observed and calculated h values are compared in the table. The calculated values were obtained using the average a and p values and eqs 4 and 5. Results are presented using both the simple HMO- and Streitweiser-modified calculations. Comparison of the observed and calculated h values indicates better agreement using the HMO model than with the FE model.
thermochemical data (4, 6).The differences largely arise from the approximations used in the estimation methods. Conclusion Use of the HMO calculations in the experiment provides an active laboratory learning format for student exposure to molecular orbitals, relative electronic energies, and n + n* electronic transitions. They expanded student learning of the inherent concepts over the brief exposure in lecture. Student interest in quantum mechanical methodology was stimulated by conducting the computer calculations and fitting the results to experimental observations.
Discussion
The FE and HMO models both explain the relative wavelength maxima for the visihle-region absorption spectra of the cyanine dyes. The calculations pertain to gas-phase molecules, and the presence of a solvent will cause solvatochromatic shifts that will affect the deviation between theoretical and experimental results. Values of p obtained empirically from absorption spectra (spectroscopic p) are higher than values estimated from
Literature Cited 1. Shoemaker, D. P;Garland, C. w ; Nibler, J. W. E*p.timnts in Physical chemistry; Mdjraw-Hill: New Yo*, 1989: p 44C. 2. Sime, R. J. Physlml Chemistry. MdhaLs. nxhnigurs, and Experimnb: Saunders: Philadebhia, 1 9 9 t 0 681. 3. Natsrsjan,L.V:Robhson.M.;Blankenship, E.J Chem. Edve 1981,60,241-243. 4 . Streifwieaer, A. M o h l o r O ~ b i l dTheory for Ogonir Chemiefs: Wiley: New York, ,%I. ~
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5Taubmann. T. J. ChamEduc. 1982,bS, 96-97. 6. Cotton,F A. Chemiml Applimtions ofCmvp Theory, 3rd ed.: mley: NewYork, 1990.
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