J. Phys. Chem. 1081, 85,4015-4018
where m is the mass of an electron, h in the Planck’s constant divided by 27r. As an example, the a, and E,, (n = 1) are calculated for hexane ( 6 = 2.023); a. = 6.3 A and El = -0.1 eV. The obtained a, value seems to be too small to assume the WO picture, and thus the two-dimensional free-electron picture does not seem to hold in the pesent experimental conditions. However, the surface potential barrier must have the effect of confining electrons in the surface region of the solid. The confined electrons in the surface region may tunnel through the potential barrier
4015
to the favorable energy levels of trapping potential wells in the film and eventually reach the anode by traveling from one trapping site to another under the influence of the image potential. Acknowledgment. The author thanks Mr. M. Nara for performing some of the measurements. The financial support of the Grant-in-Aid from the Ministry of Education and of the 24th Nishina Memorial Fund are gratefully acknowledged.
Molecular Orbital Examination of Electron Density and Excitation Energy in Linear, Odd, Alternant Polyenes Paul E. Blatz Chemistty Department, University of Missouri-Kansas In Final Form: September 14, 1981)
City, Kansas City, Missouri 64 1 10 (Received: June 11, 1981;
The simple parameter set introduced by Roos for the PPP-MO calculation of even alternant polyenes has been employed successfully for odd alternant polyenylic cations. This is accomplished by modification of the W , term of the diagonal terms in the F matrix. The modification is a function of the previously calculated electron density and is a modified w technique. The final calculated excitation energies and electron densities for a series of three a,a,w,w-tetramethylpolyenyliccations are in excellent agreement with previously reported experimental values.
Introduction The long-range objective of this study is to develop a relatively simple parameter set that will allow the PPPl molecular orbital method to be used to more accurately examine the chromophore in visual pigments, the N retinylidene-n-alkyl (opsin) cation (NRAAH+ (opsin)). A simple parameter set for the evaluation of integrals in the PPP method was developed initially by Roos and Skancke; it has been extended and successfully applied to a large variety of even alternant conjugated compounds by a number of workers. However, the structural state of compounds related to visual photoreception, such as Nretinylidene-n-butylammoniumchloride (NRBAH+Cl)and NRAAH+ (opsin), is either that of an odd alternant system or one that lies between even and The parameter set developed in Stockholm for even alternant systems was found not to give satisfactory results for odd alternant systems. Consequently, the specific aim of this study was to develop a modified set of parameters, based on the Stockholm set, that would give satisfactory results in the evaluation of the integrals required in the PPP-MO method. This was accomplished in the theoretical study of a set of three symmetrical, a,a,w,w-tetramethyl carbon cations. The calculated absorption wavelength of the first singlet-singlet transition and the ground-state electron densities are compared with the experimental values obtained from absorption and NMR ~pectroscopy.~
In an earlier study, difficulties associated with the application of the PPP method to odd polyene cations were pointed out.6 Values of input parameters found acceptable for even polyenes give poor results for the calculated excitation energy and electron densities. This was corrected by a modification of the w technique first proposed by Wheland and Manna6 Navangul and Blatz5 adjusted the value of the Coulomb attraction integral in the following way: a p = a0 + (1 - Q , ) 4 0 (1) Correction of a is made a function of electron density, qc, previously calculated in the PPP method. The newly calculated electron densities for the undecapentaenylic cation are in good agreement with experimental values obtained from NMR chemical-shift data. However, this initial work did not include the contribution of methyl substituents found on the experimental compounds, nor did it attempt to apply the new parameter adjustment to additional members of the cation series. These shortcomings are overcome in this study.
(1)R. Pariser and R. G. Parr, J.Chem. Phys., 21, 466,767 (1953);J. A. Pople Trans. Faraday SOC.,49, 1375 (1953). (2)B. Roos and P. N. Skancke, Acta Chem. Scand., 21,233 (1967);B. Roos, ibid., 21, 2318 (1967). (3)P. E. Blatz, Photochem. Photobiol., 16,l (1972);H.V. Navangul and P. E. Blatz, J. Am. Chem. SOC.,100,4340 (1978). (4)T. S. Sorensen, J.Am. Chem. SOC.,87,5075 (1965).
(5)H.V. Navangul and P. E. Blatz, J. Am. Chem. SOC.,95, 1508 (1973). (6) G. W. Wheland and D. E. Mann, J . Chem. Phys., 17,264 (1949). (7)Quantum Chemistry Program Exchange No. 167, U. MuellerWesterhoff, Chemistry Department, Indiana University, Bloomington,
Methods and Calculations The computer program, QCPE Program 167,7 follows the normal PPP approximations in the calculation of the ground state. The program contains an SCF routine in which the F matrix is solved iteratively until bond order converges to preselected values. The program also makes
IN.
(8)R. G.Parr, J. Chem. Phys., 20, 1499 (1952).
0022-3654/81/2085-4015$01.25/00 1981 American Chemical Society
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The Journal of Physical Chemistry, Vol. 85, No. 26, 1981
available a subroutine for calculation of the requested excited states. The elements of the CI matrix are calculated and diagonalized to obtain the oscillator strengths, transition moments, and direction angles; the excitation energies of the requested states are calculated in eV, nm, and IO-’ pm-’. As an additional option, the percent contributions of the individual configurations to the transitions are also calculated. The program was slightly modified to be consistent with inclusion of Stockholm parameters, which consisted of allowing the two-center repulsion integrals between bonded atoms to be accepted as direct input, whereas the repulsions for nonbonded centers were calculated by the hard-sphere approximation of Parr. The parameter set of Roos and Skancke was developed in accordance with the conclusions of Fischer-Hjalmar~,~ who critically examined the approximations of the PPP method. Integrals between bonded atoms, the resonance integral, P,, and the two-center electron repulsion integral, y,,, are transferrable from one molecule to another and are treated as functions of interatomic distance, R,,,.
P,, = Po + 6,(R,, - Ro)
(2)
+ 6,(R,, - Ro)
(3)
Y,v = Yo
Roos values of Po, 6,, yo, and 6, were obtained by fitting to experimental quantities. In the Fischer-Hjalmars analysis, the Coulomb attraction integral, a,, was found to be sensitive to its nearest neighbors; and in the RoosSkancke treatment this is handled through manipulation of a parameter, W,. The W, parameter originates in the Goeppert-Mayer and Sklar relationship.1°
w,= a, + (17 - lh,, + c
17vYpv
(4)
VfP
Blatz
in the Stockholm set. Thus, in the first calculation, according to eq 2,3, and 5, Ppv = Po, W , = Wo+ 2AW0, and y,, = yo (for bonded atoms). Values of y,,,for nonbonded atoms are determined by the hard-sphere method.s The computer program automatically converts the resulting set of bond orders into bond lengths using Coulson’s12relationship with the following numerical values: single bond, 1.543 A; double bond, 1.337 A; and constant = 0.823. Throughout all of the calculations, the bond length between substituent methyls and sp2 carbons is taken to be 1.52 8. The calculated set of bond lengths from the first calculation was then used to evaluate a second set of Ow,, y,,, and W,, according to the method of Roos, and a second PPP calculation wm made. The second set of bond lengths was then used to calculate a final set of input parameters which are, except for W,, used directly in the final PPP calculation. Parameter W , is modified by the electron density, qp, in a modified w technique. W,’ = w, + (17 - q,)wPo (6) Parameter 17 is assigned the average electron density of the odd system at C,, and the value of wP0 was set equal to 3.333.
Results Electron Density from NMR. Sorensen synthesized a series of three methyl-substituted linear polyenes which upon protonation give rise to a series of a,a,o,o-tetramethylpolymethines. The general structure of the series is given in structure I by 1,1,7,7-tetramethylheptatrienyl I
2
3
v
(5)
The second term is evaluated for each bonded neighbor of C, and then summed; all of the parameters are determined by experimental fitting. In subsequent work Roos also incorporated methyl substituents into the general scheme. This was done through the heteroatom model proposed by Matsen.l’ A detailed explaination and numerical values for the parameter sets may be found in ref 2. According to eq 2,3, and 5, P,, ypu,and W, are functions of bond length. Thus, in order to calculate a linear molecule such as hexatriene, a set of bond lengths must be available. Since bond lengths are known for hexatriene, this is no problem. Even if they are unknown, an initial set may be taken from butadiene with success. On the other hand, bond lengths are not available for linear odd alternant polyenes. This particular difficulty was handled in the following way. Procedure for Odd Alternant Polyenes. The general procedure consists of a series of three PPP calculations. The first calculation is used to generate a set of bond lengths for the odd system. A second calculation gives an improved set of bond lengths and a set of electron densities. These are used to obtain a third set of input parameters for the final calculation of the odd system. In the first PPP calculation, bond lengths between atoms in the T system are taken to be 1.397 A, the value of Ro (9)I. Fischer-Hjalmars, J. Chem. Phys., 42,1962 (1965). (10)M. Goeppert-Mayer and A. L. Sklar, J. Chem. Phys., 6, 645 (1938). (11)F. A. Matsen, J. Am. Chem. SOC.,72,5243 (1950).
t 5
6
7
8
9
I
I
LH;,
&i39,
Roos and Skancke evaluate the W , parameter in this way. W , = Wo + CqvAWp(RPv)
4
CH-Z-CH-CH-CH-CH-CH-C-CH,
I
cation (TMHTRC) (vide infra). The cations are perfectly symmetrical and relatively stable in acid solution so that the electronic absorption and the proton NMR spectra were obtained. I examined the relationship between the change in chemical shift between the polyene and cation and the change in electron density. However, for the work reported in this paper, it was necessary to determine the electron density on each carbon. This was accomplished by using the chemical-shift data from the first two compounds reported by Sorensen: 1,1,7,7-tetramethylheptatrienyl (I) and 1,1,9,9-tetramethylnonyltetraenyl (TMNTC) cations. Two kinds of carbon are recognized: those with an attached proton and those with no attached proton but an attached methyl group. The chemical shift for all atoms was summed and equated to one unit charge by employing two proportionality constants, kl and k2, for the two cations cited above. This was done for both compounds, giving two simultaneous equations: TMHTRC 9.76/12, + l.28/k2 = 1 (7) TNMTC
0.63/kl
+ 0.98/k2
=1
(8)
Equations 7 and 8 represent TMHTRC and TMNTC, and the constants were evaluated as 13.47 and 4.65, respecti~e1y.l~With these values and the relationship AC = A6/k, (9) the charge density of each carbon atom was evaluated for (12)C. A. Coulson, Proc. R. SOC.London, Ser. A, 169, 413 (1939). (13)G. Fraenkel, R. E. Carter, A. McLachlan, and R. H. Richards, J. Am. Chem. SOC.,82,5846 (1960);T.Schaefer and W. G. Schneider, Can. J.Chem. 41,966 (1963);C. MacLean and E. L. Makor, Mol. Phys., 4,241 (1961).
Examination of Linear, Odd,Alternant Polyenes
The Journal of Physical Chemistry, Vol. 85,No. 26, 1981 4017
TABLE I: Values of Electron Density for TMUPC from Three-Step MO Calculation and NMR electron density per atoma step 1 2 3 NMR
1 1.96 1.96 1.97 1.97
2
3
4
5
6
0.76 0.82 0.92 0.92
1.13 1.11 0.99 0.98
0.84 0.81 0.87 0.88
1.04 1.05 0.96 0.96
0.80 0.76 0.85 0.88
7 1.04 1.05 0.95 0.96
~
a Since electron density is symmetrical, only half of the atoms are shown.
TABLE 11: Values of Electron Density for Cation Series from MO Calculations and NMR electron density per atoma source
1
2
3
4
5
6
7
TMHTRC~ MO NMR
1.96 1.96
0.85 0.86
0.97 0.94
0.80 0.82
0.94 0.94
MO NMR
1.96 1.96
0.89 0.90
TMNTC‘ 0.98 0.85 0.96 0.85
0.95 0.94
MO NMR
1.97 1.97
0.92 0.92
0.82 0.85
TMUPC~ 0.99 0.87 0.96 0.85 0.95 0.98 0.88 0.96 0.88 0.96 Since electron density is symmetrical, only half of the atoms are shown. TMHTRC = 1,1,7,7-tetramethylhaptatrienyl cation. TMNTC = 1,1,9,9-tetramethylnonatetraenyl cation. TMUPC = l , l , l l , l l - t e t r a methylundecapentaznyl cation. TABLE 111: Calculated and Experimental Absorption Maxima and Oscillator Strengths for First Singlet Transition in Cation Series absorption maxima, nm
oscillator strength
cationa
calcd
exptl
calcd
exptl
TMHTRC TMNTC TMUPC
481 553 626
473 550 6 26
1.512 1.853 2.135
1.10
a
See footnotes b-d in Table 11.
TMHTRC, TMNTC, and TMUPC (l,l,ll,ll-tetramethylundecapentaenylic cation); these are reported in Tables I and 11. MO Calculation Results. The MO calculation results are discussed most satisfactorally in terms of the starred and unstarred convention. The numbering system employed is shown from TMHTRC in structure I. Even carbon atoms are starred and correspond with low electron density; odd atoms, except for methyl groups, are unstarred and correspond to high electron density (see Table I). Tables I and IV show the resulting changes in electron density and bond length in the stepwise calculation of TMUPC. The first set of calculated densities, obtained by assuming equal bond lengths, shows a large difference between starred and unstarred atoms and is in poor agreement with values obtained from NMR. Even when the new set of Yimprovednbond lengths is used (see Table
IV, TMUPC-l), there is little or no improvement in electron density and no convergence toward the NMR values. However, when the calculated densities are used in eq 6 to give corrected values of W,, the charge densities change dramatically and conform closely with the NMR values; this is shown in step 3, Table I. Thus, in organic cations, the calculated charge densities are dependent on a charge correction of the W, term. Inspection of the bond-length data in Table IV (TMUPC-1-3) shows that bond lengths undergo their greatest change in the first calculation step. Adjusting the W, term in the third step has almost no additional impact on bond lengths. Examining the fluctuation of the calculated ,A, of the first singlet-singlet transition of TMUPC is also informative. The following procedure was employed. In the third calculation step, Po is adjusted to -2.07 eV in order to of 626 nm. This value of Po obtain the experimental ,A, gives a A, of 599 nm in the first step, and the value drops shows 30 nm to 569 nm in the second step. Thus, the A, considerable sensitivity to changes in bond length. When the W, parameter is modified by charge in the third step, increases 57 nm to 626 nm. Thus, the wavelength the A, is sensitive to alteration in both bond length and charge density. The techniques applied in the calculation of TMUPC were next applied to TMHTRC and TMNTC. Final calculated electron densities are compared with those from NMR in Table 11. The technique brings the calculated values into very close correspondence with those from NMR for all three cations. As expected, all members of the series show considerable charge alternation, the positive charge being greatest on the starred atoms. Furthermore, when the starred-atom convention is applied to odd alternant hydrocarbons, two subclasses are possible: either a starred or an unstarred atom may reside at the center of the T system. These additional subclassifications are important since they correlate with certain molecular properties: electron density and bond length. TMHTRC and TMUPC are of type u, an unstarred atom at the center, and, as expected, the electron density is high on the central atoms of each cation, 0.94 and 0.95, respectively. On the other hand, TMNTC is an example of a type-s cation, the central atom starred, and, as predicted, has a lower electron density, 0.85. The first singlet-singlet transition is V N. Since the species is a positively charged, odd, alternant polyene hydrocarbon, the electronic transition is from the highest bonding MO, $Bo, to the nonbonding MO, $mo, $NBO $BO. All three cations were calculated in the all-trans, all-trans geometry which belongs to point group C2”. The ground state is of symmetry species Al and the excited state derived from the described one-electron transition is of symmetry species B2, giving rise to a strongly allowed transition (see calculated oscillator strengths in Table 111). Specifically, for TMHTRC, the transition is described as 1Bz($12$2$3$4NB)1A1($12$22$32). The calculated excitation energies, in terms of wavelength, are reported and compared with experimental values in Table 111. The agreement is very good for this linear series. Absorption
-
-
TABLE IV: Calculated Bond Lengths (A I of Cations
a
cations bondsa 1-2 2-3 TMHTRC 1.490 1.370 TMNTC 1.494 1.364 TMUPC-1 1.490 1.368 TMUPC-2 1.493 1.361 TMUPC-3 1.497 1.360 Since cations are symmetrical, only half of the bonds are shown.
3-4
4-5
5-6
6-7
1 . 4 29 1.439 1.434 1.446 1.446
1.398 1.387 1.384 1.378 1.379
1.411 1.418 1.423 1.442
1.400 1.399 1.399
-
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Blatz
The Journal of Physical Chemistry, Vol. 85,No. 26, 1981
intensity was also calculated and reported as oscillator strength. Final calculated bond lengths are reported in Table IV. First, it is noted that, for the set TMUPC-1-3, the bond lengths quickly converge to self-consistency. Furthermore, all molecules, both types s and u, show alternation; the degree of alternation is greater toward the ends of the chains and it is greater in longer chains. The length of the bond projecting from the central atom is related to the compound type. In TMHTRC and TMUPC, type u, the central bonds are, of course, equal and very close to the aromatic values 1.398 and 1.399, 8, respectively. The central bonds in TMNTC, type s, have values of 1.411 A, considerably higher than those in type-u cations. Furthermore, the calculated bond alternation is greater throughout type s than in type-u cations.
Discussion In order to be consistent with the Roos method for obtaining parameters (see eq 2,3, and 6), a relationship was sought between bond length and charge adjustment. Attempts were unsuccessful. However, the modified w technique used here, in which the calculated electron density is used to obtain a corrected electron density, is simple enough to complement the Stockholm parameter scheme. Consequently, the scheme is now expanded to include cations. The utility of this technique is demonstrated by its ability to predict accurately electron densities and excitation energies for a series of three cations. It may be of some concern that the PPP method, which so accurately predicts the properties of even a systems, does not predict those of odd systems equally well. Examination of a more complete expression for the diagonal terms in the F matrix can be revealing.
F,, =
w,- (9, - l)Y,,
+
w , , y 4 ,
+
P V ”
- B ” h C V - c, (10)
The term C,, the penetration term, and (q - l)y, are usually neglected in PPP calculations. Indeed, there is no provision for them in QCPE Program 167, the program used in the present work. In even systems, since the electron occupancy, q, is taken as 1, 7 - 1is zero, and (7 - l)y, vanishes. In order to employ the heteroatom me-
thod for the inclusion of methyl groups, one must adjust F ., In this case, q is taken as 2, and W , is appropriately aftered to reflect (q - l)y,,. This is true whether or not the heteroatom is included as a part of an even or odd a system. The penetration term appears to present additional concerns in odd a systems. Since, in an even system, all of the atoms are approximately equivalent, the penetration terms are small and consistent from atom to atom. Thus, the penetration term either is taken to be zero or is compensated for empirically in other parameters. QCPE Program 167 has no provision for penetration terms. The penetration term is expressed in eq 11. Here the electron
c,
=
[C(WPL)+ C(h.41 A
(11)
IrV
on atom p is interacting on adjacent centers. Because in an odd system the electron density is both variable and alternating, the penetration term for each atom would not be small and consistent. As mentioned above, when Roos parameters are used in the first PPP calculation, the C, terms are missing, and it might be postulated that the resulting calculated electron densities will be distributed as a function of the imperfection caused by these missing terms. On the other hand, if a corrected F,, term is made a function of the previously calculated electron density, it might be anticipated that the corrected F,, term would appropriately adjust the electron density in the ensuing calculation. In the modified w technique each term in the diagonal of the F matrix is adjusted by adjusting W, according to eq 6. In the introduction the long-range objective was given as the development of a relatively simple parameter set that would allow further theoretical studies of NRAAH+ (opsin), the chromophore in visual pigments. As a first step in that direction, the newly developed parameter set, i.e., the Stockholm set adjusted by the modified w technique, has been applied to the successful calculation of the retinylic and anhydroretinylic cations. This will be reported in a subsequent article. Acknowledgment. I am especially grateful to the Arts and Sciences College of UMKC for making available computer funds necessary for the completion of this research.
