Frontier Orbital Analysis of the Bathochromic Shift Anthony J. Duben Computer Science Department. Southeast Missouri State University, Cape Girardeau, MO 63701
-
When benzene is substituted by a single group, the 'Bz, 'AI, transition (also known as the B band) undergoes a bathochromic shift from 254 nm to longer wavelengths. Some (1-4). texts in which this nhenomenon is discussed are refs. ~-~ ,~ ~. Examples oft his s h i t are given in the table; this data has been taken from Jaffee and Orchin ( 5 ) .u,ho. in turn. ouoted Doub and Vandenbelt ( 6 ) .Water, witha trace of mkt'hanol to enhance solubilitv. -. was used as the solvent for all of the substances listed. The shift to loneer waveleneths occurs reeardless of the character of the sibstituent "electron w%hdrawing" or "electron donating." This fact causes some surmise amone students who have had a good introduction to the reaction chemistry of aromatic systems where this distinction is meaningful. In order to interpret this phenomenon, Cooper (4) presents a figure (originally given by Matsen ( 7 ) )in which the Hiickel r energy levels of the aromatic compound are plotted against the electronegativity of the substituent. This figure is redrawn as Figure 1. There are seven orbitals of T symmetry in the substituted molecule. Five of the orbitals have energies given by the solid lines in the fieure. Thev differ from the orbital energies ~~~-~ " ~of- ~ benzene (dotted lines) since four of the original benzene orbitals can interact with thr substituent. Twoof t h ~orbitals . in the substituted molecule will have energies of either a + fl or a - since each enerev has an orbital that has a nodal olane containing the substit&; no interaction will occur. The T svstem will have eieht electrons occunviue the lowest four orbitals. These will b i the three orbi&is Golid lines) starting a t a 28, a 0, and a units of enerev and the unperturbed a &er& orbital. The highest&cupied molecular orbital (HOMO) will be the orbital originating with a , the additional orbital introduced by the substituent. The lowest unoccupied molecular orbital (LUMO) will be the unperturbed a - fl orbital originally present in the unsubstituted benzene. The effect of substitution will be t o destabilize the a - 9, orbital that can interact with the substituents. The description of the bathochromic shift using Figure 1 involves the comparison of the energy gap in benzene a 8 - ( a - fl) = 28 against the gap in the figure between the HOMO and LUMO. Based on the identification of the HOMO and LUMO in the previous paragraph, the gap between these two molecular orbitals will always be less than 28. The tran~~
~~~~~~~~
~~~~
sition energy of the substituted molecule will be less than that for benzene itself, and the wavelength corresponding to the transition should be larger than that for benzene. Figure 1is the result of a 7 X 7 matrix diagonalization. I t lacks impact for many students because of the dimensionality of the problem and the automatic, iterative nature of the diagonalization procedure used to obtain the energies plotted. A more direct and immediate physical interpretation of the B-band Transltlons In MonosubstHuledBenzenes * (5. 6 ) Substlt~ent H CI
& OH 0% NH2 CN S02NH2
cb COOH
Max. Wavelength (nm)
Extinction Coefficient
254 263.5 261 270 269 280 27 1 264.5 261 273
204 190 192 1450 1480 1430 1000 740 225 970
~~
~
~
W a W was he wrlvent fwall wiVl some memanal added toenhlnce wrbbllity.
A
+ +
+
+
Figure 1. The energy levels of lhe CeHsXsystem as a hrnctionof h (taken from Matsen (7)).
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Number 5 May 1985
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bathochromic shift can be made by using frontier orbital calculations. Previous analyses of the ultraviolet spectra of substituted benzenes have been oublished bv Matsen (7) and I'etruska (8, 9). Although these references discuss the problem a t length, many students could find the symbolism and mathematics intimidating. The frontier orbital analysis has the virtue that it requires only simple mathematics and no computer programming. Numerical values can be obtained using only a hand calculator.
ular orbital may be calculated using the following pair of equations
Frontier Orbital Analysts In order to apply the frontier orbital method to this problem, we first conceptually divide the molecule into two portions:
and
(1) a two-atom segment consisting of the ring carbon to which the
substituent is attached and the substituent itself as a wudo-atom that will have a D-norbital capable of interactina . with the ring carbon p orbit& and (2) a five-atom segment consisting of the remaining ring atoms. This group of atoms will be an alternant hydrocarbon. The frontier orbitals under consideration are the nonbonding orbital of the five-carbon fragment and the antibondine orbital of the two-atom fraement. Thev will be allowed & interact and give rise to orbctals extending over both soecies as tbev come to form the monosubstituted benzenes. central assumption of the frontier orbital method is that the effects on the bonding will involve only the previously identified fragment orbitals. Lower-lying filled orbitals and empty orbitals of hieher energy are of much less importance. Since the problem i t hand spkcifica~~y relates to the longest wavelength transition between the HOMO andLUMO in the singly substituted benzene, the use of the fragments' frontier orbitals seems entirely appropriate. The analysis includes conjugation of the? electrons of the substituent with those of the ring. The substituent is assumed t o contribute two electrons to the ?r system. The noubonding molecular orbital of the five-carbon fragment has an energy of a,the energy of an isolated carbon atom p orbital. I t will contain a single electron. l'he coefficients of the two end atoms in this orbital will be (31-' l . The frontier orbital of the two atom C - X fragment will he its antihondinr orbital. It will also contain a sinale electron since it will be&sumed that the carbon in this fGgment will have contributed one electron and the substituent pseudoatom two elrctrons to the C-X fragment a system. l'he inductive and coniueative effects caused by the substituent will be accounted for in the calculation of the a orbitals of the C-X frament.'l'he Huckel coulomb integral for the carbon will be a . ~ h coulomb e integral for the subsiituent will be a h o where p is the bond integral and h is the parameter indicating either electron-withdrawing (h > 0) or electron-donating (h < 0) properties (10). The secular determinant t o be evaluated for the C-X fragment will be
A
+
in which atom 1 is X and atom 2 is the carbon. Changing variable t o y = ( a - E)l@generates the modified secular equation
c, + CJy) = 0
(4)
c; + c: = 1
(5)
C, = (1 + y2)-'n
(6)
and Solution yields
C, = -yo
The lower sien (-) refen to the bondine orbital. and the uooer .. sign (t)to iheantibonding orbital. The coefficients associated with each atom in earh molec374
Journal of Chemical Education
(7)
The two fragmenw will be allowed to interact no that the two end atoms of the five-carbon chain ioin with the carbon of the C-X fragment to form the singly substituted benzene molecule (Fig. 2). Since each fragment contributes one electron to the composite system, the lower energy (bonding) or-
Figure 2. m e twmation of bonds between the fragments.
bital of the composite system will contain both electrons. The higher energy (antibonding) orbital will be empty. The bonding orbital formed in the interaction of the frontier orbitals will become the highest occupied molecular orbital (HOMO) of the monosubstituted benzene. The emotv antibonding brbital formed from the frontier orbitalswifi have an enerev ereater (more oositive) than that of the unnerturbed anti6;nhing orbital or energy a - 3 of the pentad'Lenyl fragment. This unoerturbed orbital will br the lowest unoccuoied molecular orbital (LUMO) of the composite system, and i i will serve as the destination of the electron excited from the HOMO. This result in consistent with the results obtained from the variational calculations of Matsen (7) discussed in the introduction. The interaction of the two fragments' frontier orbitals generates a second 2 X 2 secular determinant:
in which the subscript c indicates the pentadienyl fragment and subscripts indicates the substituent C-X fragment. The over la^ iuteerals are taken to be unitv in the diaeonal elements and zero i n k e off-diagonal eleme&. The interaction energy between the two frontier orbitals is
-
H,=H,=Zp@
(9)
in which p is the bond order of each bond which developy hetween the frontier orhitalsof the twofraements.The factor of 2 originates in the fact that two bonds &e formed as seen in Figure 2. The bond order p is given by P =
with solutions
+ yZ)-'"
cnbcmti
(10)
C.b = (3)-1/2, the coefficient of the terminal carbons in the pentadienyl fragment. Cmti is the carbon coefficient in the C-X fragment antibonding orbital. Substitution of previously obtained results yields
p = (30
+ yf))-'IZ
(11)
in which y, is the molecular orbital energy for the C-X fragment antibonding orbital.
Figure 3. Energy levels of me C.H,X system as a function of h. Solid linesInvolved VariatioMl 601utions of the seven a m system. Solid bisngl-itab in the frontierorbital analysis. Vertical solid lines show the transition between orbitais obtained in lhe full variational solution. Vertical dashed lines show the transition between orbitals obtained in the frontier orbital solution. The energies used in the secular equation above are and
The results of the frontier orbital calculations are represented as solid triangles. Orbital enerries had been determined for values of h which were multiples i f 0.5. The seven triangles a t each value of h include the HOMO and the hieh-enerw antihonding orbital resulting from the direct int&actionyf the f r a m e n t frontier orbitals, the four other oentadienvl and t i e fragment orbitals including the LI:MO at z = +i.~, low enerm tmndinn orbital from the suhstiturnt calculation. In two cases, there is an accidental degeneracy of bonding orbitals at z = -1.0 for h = 0 and z = -2.0 for h = 1.5: only one triangle is used in those two cases. Close agreement is found between the sets of triangles depicting frontier orhital results and the solid lines representing the variational results. The use of simple mathematics can reproduce, at least, the qualitative features of the variational results. Bathochromic shifts are found for hoth methods. The transitions are shown as vertical solid lines from the variational results and as dashed lines from frontier orhitals. Regardless of the sign or magnitude of h, the energy gap will he less than two units of z. the transition enerw -" for unsuhstituted benzene. In conclusion, it appears that the bathochromic shift of the longest wavelength band observed when benzene is suhstituted can be demonstrated theoreticallv using a simole frontier orhital analysis. The shift occurs regardlessof whether the substituent is an "electron-withdrawina" or an "electron-donating" group. Acknowledgment
Substitution of these quantities into the secular equation and changing the variable to z = ( a - E)IP yields solutions of the form
The negative sinn is used to calculate the enerv of the HOMO of the monosubstituted henzene. The positive sign is uiedto calculate the energy of the antibonding orhital. Discussion
The results of this calculation are presented graphically in Figure 3. The orhital energies have been scaled to units of z (a - E)I& The solid lines are a molecular orbital energies for a full 7 X 7 secular determinant diagonalization. The solid lines for positive h values had been seen in Figure 1.These curves had been extended to include negative h values.
The author thanks Robert W. Looyenga of the South Dakota School of Mines and Technology who posed this prohlem. The problem originally was a student's question in a spectral interpretation course. (11 Sllver8tein.R. M.,and BwIer, G.C.."SpedromBtric Identification ofOrganicCompounda," Wiley. New York. 1964,~. 101. (2) Dyer, J. R., '"Appkatiana of Absorption Spetmsmpy of Organic Compouods;. Prentice-Hd, E n g l e w d Cliffs, NJ, 1965,~. 16. (8) Parikh, V. M., "Absorption SpeNoseopy of Organic Molecule,"Addison-Wesley. Reedin., MA, 1974. pp. 3635. (4) Cooper, J. W., "Spedroscopic Technique8 for Organic Chemist*." Wiley, New York,
19s" nn 760.147 ... .,-=. .... ... (51 Jaffee, H.H..and Orchin, M.,"Thmy and A p p l i ~ t i i i ofUltraviolet s Spetrasmpy," Wiley. New York. 1962,~. 257. (6) Doub, L.,and Vandenhi%J. M., J. A m r . Chem. Sm.. 69, n14 (194'7). (7) Matsen.F. A,. J. Amer C k m . Sac.. 72.521 119MI.
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Number 5
May 1985
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