2778
J. Phys. Chem. 1993, 97, 2778-2781
Relation between Predicted Soliton Charge-Density Wave Structure of Odd Polyene Anions and I3C NMR Chemical Shifts? J. R. Reimers,$ J. S. Craw,: and N. S. Hush’**.$ Department of Physical and Theoretical Chemistry and Department of Biochemistry, University of Sydney, Sydney, N S W 2006, Australia Received: November 30, I992
The charge-density wave and geometric soliton structures of a,w-diphenyl odd polyene ions ( C ~ H ~ ) ~ ( C H ) ~ , , + I with chain length 2n 1 = 1-21,29,3 1, and 41 are calculated using A M I . Qualitatively, the effects of a phenyl end group on a polyene chain are predicted to be similar to the effects of extending the chain by three C H units. Hence, phenyl substitution is not expected to affect the properties of an infinite chain; the predicted geometric soliton bond-length alternation is Ar, = 0.095 f 0.002 A and half-width is 1 = 7.0 f 0.2 while the entire net ionic charge is localized on the soliton whose charge-density-wave half-width is I = 9.1 f 0.3. For the shorter ions, experimental estimates of the charge-density-wave properties are made assuming various I3CNMR atomic chemical shift to atomic charge correlations. Use of the OBrien postulate, according to which individual charges and shifts correlate in the same way as do their averages, is shown to be suspect by the AM1 calculations and predicts 1 = 13 f 1.5; other correlations which relate the observed chemical shifts to AM1 charges predict that 1 could be as low as 7. It is demonstrated that it is important not to consider individual atomic charges and chemical shifts, which are affected by the problems associated with determining C C bond polarizations (or bond ionicities), but rather to combine atomic charges into average bond charges and atomic chemical shifts into average bond chemical shifts. Given this, the SuSchrieffer-Heeger soliton model is shown to provide an excellent description of the deduced charge-density wave.
+
1. Introduction
BOND LENGTHS
Ln
The electronic structure of polyene radicals and ions has been studied since the earliest days of molecular quantum chemistry (see ref 1 and refences therein). For much of this time, only the *-electron distribution, and the geometry consistent with it, was explicitly considered; each CH unit (containing the nuclei and the u electrons) was assumed to carry a charge of + l . We are concerned here with a,@-substitutedodd trans-polyene anions of the general formula R2(CH)2n+l-, as in Figure 1. In parts I2and 113of this series, we considered properties of the parent polyene ions with R = Hand of the Brooker (poly(methinecyanine)) ions with R = NH2;herein weconcentrateonpropertiesofthediphenyl derivatives R = CsHs. These ions areclosely related tonegativelydoped polyacetylene chains and are thus of much interest for the theory of molecular conductors (“molecular wires”) in molecular electronics, as well as the theory of conducting polymers. In the independent-electronHiickel approximation,with all bond lengths equal and assuming that there exist no end effects arising from the R groups, the highest filled molecular orbital in such ions is a nonbonding orbital, i.e., one with zero coefficients on alternate carbon centers. Moreover, the *-charge density q, at center i is, for a chain with 2n 1 carbon centers, -l/(n + 1) on the n + 1 alternate centers which include the chain center and zero in between. Thus, for R2C9H9-( n = 4), numbering the chain atoms linearly from -n to n as is indicated in Figure 1, centers i = -4, -2,0, 2, and 4 would have charge q, = -’Is, while centers i -3, -1, 1, and 3 would have charge 0. The picture is then of an electron charge-density wave (CDW) with amplitude oscillating between a characteristic constant value and zero for each chain length. The attractive simplicity of this description is lost when variations in carbonxarbon bond lengths are taken into account. Again, these have been the subject of much theoretical work,
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’ Part 111 of the series ‘Solitons in Finite and Infinite Length NegativeDefect trans-Polyacetylene and the Corresponding Brooker (Polymethinecyanine) Cations”. Department of Physical and Theoretical Chemistry + Department of Biochemistry. f
0022-3654/93/2091-2118$04.00/0
1
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POLYENE C H A I N
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POLYENE C H A I N
ATOMIC II CHARGES
,
,
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Figure 1. Sample soliton development in a polyene chain R z ( C H ) ~ ~ + , displaying the central charge-bearing soliton region, the end alternating single- and double-bond regions, the atom numbers i, the effect of the geometric soliton on the adjacent CC bond lengths, r, and the change density wave 4,. These are actually AMI results for C19Hll- (R = H, n = 9); see refs 2 and 3.
initially at the independent-electron level and subsequently using ab initio and related techniques. This has been summarized in part I* of this series. For our present purpose, the most important development was that introduced by Su,Schrieffer, and Heeger (SSH),4-6again using a one-electron Hamiltonian but now including a term allowing for coupling of electronic and nuclear motion. The description that emerges from this is a *-charge distribution resembling that of the simpler model in that it oscillates between zero and a finite value on alternate centers. However, there is a major qualitative change: the charge-density wave becomes concentrated in the center of the chain, the magnitude of the oscillation decreasing as sech2(i/l), where 1 is a measure of the half-width of the CDW. This is accompanied by a decrease in the difference between adjacent CC bonds as the center is approached; the resulting structure is termed a soliton. In the limit of large n, the soliton will be an isolated structure 0 1993 American Chemical Society
Structure of Odd Polyene Anions in the center of the molecule, flanked by chains in which the CC bonds alternate regularly in length between single and double bonds. All the charge of the ion is then localized on the central soliton. As an illustration, the AM 1-calculated *-charge-density wave3and the associated geometric soliton2are sketched in Figure 1 for the polyene anion C19H2- (n = 9, R = H). This account is generally accepted as being broadly correct, although calculations at the SCF and higher levels reveal that it has to be modified in a number of ways, as discussed in parts I2and 113and extensive references therein. One modification is that centers in the soliton region previously assigned zero charge densities will in general have small positive densities, as seen in Figure 1. Another is that the width of the geometric and chargedensity-wave solitons need not be identical. It should also be noted that examination of the Fock matrix for many-electron calculations suggests2 that the generation of the soliton structure is in fact driven by the Coulomb forces, these being mimicked in the independent electron calculations by inclusion of the electronvibrational coupling term. Detailed calculations of both charge-density wave and geometry have been made (see refs 2 and 3 and references therein). It is, however, not obvious how the former could be tested by comparison with experiment as they are, strictly speaking, not uniquely defined properties. The recent work of Tolbert and Ogle,7-9in which the I3C N M R chemical shifts of a,w-diphenyl polyene anions have been m e a ~ u r e dis, ~thus of much interest, as thesecould in principle provide a means of testing the CDW predictions. Indeed, this was the aim of Tolbert and Ogle's work,s and they have provided a useful start to this, obtaining an estimate of the soliton width without the need of a direct mapping of chemical shift onto atomic charge density. This work is very valuable, as it provides the first extensive set of experimental data from which such conclusions can be drawn. Our purpose here is to look in more detail into the character of the charge distributions and the ways in which this may be tested by comparison with N M R measurements. We develop a method for the analysis of the *-charge density which is insensitive to the differences between most of the charge-analysis schemes commonly available;ls20 hence, we do not enter into a detailed discussion of the relative theoretical standing of these different methods. The electrons in a molecule give rise to nuclear screening; as a result, the magnetic field experienced by a particular nucleus in an applied field is reduced by a factor 1 - u, where u is the shielding constant for that nucleus. 13C N M R chemical shifts depend upon the magnetic field at the I3Cnuclei and hence provide an indirect measure of the molecular electronic distribution. The shielding constant can be expressed as the sum of two terms:
= dd + dp
(1) The diamagnetic (Lamb) term (Td is a property of the ground state. The paramagnetic term up arises from mixing of ground and excited electronic states in the applied magnetic field, which reduces the "free circulation" of the electron. Calculations of proton and 13C chemical shifts can be carried out at the selfconsistent-field (SCF) level using (say) gauge-invariant atomic orbitals (e.g., refs 21-23); this can be extended to correlated calculations.24 In this procedure, each M O is expressed as a linear combination of atomicorbitals that have different individual gauges. The perturbed wave function in the presence of a uniform external field is solved iteratively by modified RoothaanZ5 equations, and the chemical shift is evaluated from the resultant perturbed density matrix. Results obtained from this procedure can26be very accurate. If, however, one wishes to obtain insight into the electronic structure dependence of nuclear shielding, an analytic perturbation approach is appropriate. Such interpretations are normally based on Ramsey's equations;*7 an interpretation of these equations in terms of molecular orbitals (MOs) was formulated by PopleZBand Karplus and Pople29 and has formed d
The Journal of Physical Chemistry, Vol. 97, No. 1I, 1993 2179 the basis of almost all subsequent analyses of shielding constants. Pople's independent electron model can be applied either at the abinitiolevelor at anapproximateSCFleve1. Typicalcalculations of the latter kind30 indicate either that nonlocal contributions to both Ud and up are approximately constant for a given nucleus in a range of fairly similar molecular environments or that their variations approximately cancel. Furthermore, the local contribution to bd is also essentially independent of the chemical environment of thenucleus. It is thus reasonable, for hydrocarbon molecules and ions, to assume that the I3C chemical shift (screening constant relative to that in a standard molecule) is dominated by the remaining local paramagnetic term. This, together with neglect of variations in ring currents, is in fact the usual practice. In the Pople-Karplus treatment of 13Cshifts in hydrocarbons, the magnitude of the paramagnetic shift was related to three factors: (i) the *-charge density at the nucleus, (ii) total mobile bond order for atoms linked to the one considered, and (iii) polarity of the C-H bond, if present. Attempts have been made to evaluate the relative importance of individual terms, but the most commonly followed approach has been an essentially empirical one, in which, following a suggestion by L a ~ t e r b u r , ~variations lJ~ in chemical shift are linearly correlated with variations only in charge density. Such a correlation can be attempted with theoretical densities, but a procedure which involves only experiment21 quantities has in fact been most widely employed. In this approach, the average '3C chemical shift in an ion 8 is correlated with the average ?r charge 4 per nucleus:
B = 8, + Cuq where the average charge is simply the total charge divided by the number of carbon centers. Although there is no formal justification for this approach, there seems to be no doubt that for certain classes of compounds, correlations of this type are in fact obtained (for early references see, e.g., refs 33 and 34). Spiesecke and S ~ h n e i d e and r ~ ~later Olah and Mateescd6 put averagecharge-chemical shift correlations on a more quantitative basis by considering a series of cyclic aromatic ions in which the charge densities at individual carbon atoms are equivalent by symmetry; OBrien et al.37subsequently extended this to include polyene anions, mostly substituted by one to three phenyl groups with theresult that (sometimesknownas the'sensitivity factor") was set as 156.3 ppm/e while 80 was set as 127.9 ppm. O'Brien et al. cautioned that this correlation applies specifically to planar, unbridged, all-carbon systems containing only hydrogens as substituents. In fact, this has to be further qualified. Edlund, Miillen, and c o - w o r k e r ~ for , ~ ~example, have found that in diand tetranegative ions of condensed aromatics, values of the sensitivity factor much less than the value of 160 ppm/e are obtained. A slightly different approach is to replace the constant 80 in eq 2 by a quantity for a given carbon framework, yielding
-
sN
(3) where SA and BN are respectively the average I3C shifts for ion A and the corresponding neutral molecule N. For the dianions and neutral species of anthracene and phenanthrene, for example, it is found for anthracene39 that BA = 1 15.1 ppm, BN = 127.9 ppm, and 4 = -0.143 e, yielding (Y' = 89.5 ppm/e, and for phenanthrene40 that 8 A = 120.8 ppm, 8 N = 128.1 ppm, and 4 = -0.143 e, yielding (Y' = 51 .Oppm/e. Both values of cu' are evidently much smaller than the value of 160 ppm/e usually employed. Even lower values of the sensitivity factor have been obtained for larger condensed aromatics, e.g., 28.0 and -1 ppm/e for the dianions of pyrene and acepleiadylene, respectively. Edlund et al.38note that there is a relationship between the IH ring current anisotropy and the chemical shift/charge ratio in these ions and suggest that these have a common origin in variations in the average excitation energy AE (especially for u r* transitions) which
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Reimers et al.
2780 The Journal of Physical Chemistry. Vol. 97, No. 1 I , 1993 enters the denominator in the Pople-Karplus equations for the paramagnetic contribution to the chemical shift, considered as the dominant term, and also enters the denominator of some terms in theoretical expressions for the paramagnetic ring currents. However, a basic understanding of the magnitude of the sensitivity factor in relation to structure has yet to be obtained. Based upon the accuracy of the SpieseckeSchneider correlation for a range of phenyl-substituted polyenes containing carbon atoms in quite different environments, OBrien et al. postulated that eq 2 for the average ?r charges and shifts arises because a parallel equation 6, = 8,
+ (Yql
(4)
actually applies to the individual shifts and *-charges a t center i. Clearly, if eq 4 is correct, then eq 2 follows immediately, but the converse of this need not necessarily apply. Their argument is weakened by the subsequent observation (e.g., refs 39 and 40) of ions which do not obey eq 2; Tolbert and Ogle,* however, have demonstrated that for the a,w-diphenyl polyene ions, an approximate linear correlation does exist between 6, and AMI -calculated qn although, as shown later, not necessarily one with the coefficients predicted by the O'Brien postulate. In summary, we may say (i) that there are reasonable grounds for assuming that the chemical shift is dominated by the local paramagnetic term crP,(ii) that there is good experimental evidence that for monocyclic or linear anions or systems formed by linkages of these, which include the a,w-diphenyl polyene anions, there is a strong correlation between average x charge per center and average IT chemical shift, and (iii) that this correlation must be regarded at present as strictly an empirical one, since for condensed ions very different values of the sensitivity factor are observed. Furthermore, it does not provide direct information about a correlation between charge and chemical shift on individual centers. How far such a correlation in fact is obtained for the a,@-diphenyl polyene anions is the subject of section 3; the charges thus obtained from the chemical shift data are also analyzed to obtain an estimate for the soliton width in these systems. The first task, however, is to clarify the concept of charge density in this context: this is discussed in section 2, and AM1 calculations are used to predict the soliton width. Conclusions concerning the structure of the charge-density wave in odd polyene anions and some general conclusions about the correlation of I3C chemical shifts with individual or integrated charge densities are presented in section 4.
2. Calculated Molecular Geometries and Charges (a) Geometries. Geometry optimizations were performed using the AM1 4 1 method via the MOPS program package42for the odd a,w-diphenyl polyene ions (C6Hs)2(CH)2n+Iwith n = 0-10, 14, 15, and 20. For the largest ion, CZ,symmetry was assumed; for all other ions, no symmetry was assumed, and the resulting equilibrium geometries were found to be of Cz, symmetry. Considerable steric interaction is observed between phenyl ring hydrogens for the shortest ion, n = 0; this results in an AM1calculated central CCC bond angle of 134O, close to the value of 132' obtained from X-ray diffraction s t ~ d i e s . ~The 3 AM1 geometries are expected to be quite accurate as, for polyene chains, AM1 has been shown2 to reproduce to within 0.01 A a b initio SCF bond lengths obtained using a double-t basis; indeed, those calculations showed that the two sets, geometric solitons and charge-density waves, are very similar. Also, MNDO and a b initio STO-3G calculations g:ve comparable results44to those of AMI. The root-mean-square (RMS) difference in the calculated and observed4) bond lengths for the n = 0 ion is 0.02 A; the agreement is quite reasonable when one considers that the observed structure is measured in a condensed phase and is affected by the presence of nearby anions and counterions. The simplest description of the geometric soliton comes through use of the nearest-neighbor CC bond length alternations4 defined
10
1
I
I
I
I
1
2
3
Y
5
6
I
I
I
I
I
I
1
2
3
Y
5
6
1
Figure 2. For one-half of the polyene chains with i 2 0, bond-length alternations Ari and Lowdin net accumulated r charges Qi calculated using AM1 vs chain atom number i for some short ions with n 5 6: ( 0 ) a,w-diphenyl polyene anions; (-) for the corresponding atoms on the polyene anion containing three C H units in place of each phenyl ring. 1
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Figure 3. Bond-length alternations Ar, about chain atom i (only i Z 0 shown) calculated using AM I for the a,w-diphenyl polyene anions (C6H~)2(CH)2n+lwith n = 5 , 10, 15, and 20: ( 0 )AMI results: (-) fitsoftheAM1 data to thesoliton tanhfunction.eq(6). Theinsertsshow the soliton parameters AL and I plotted vs chain length n on a I/n scale: (0) a,w-diphenyl polyenes; ( 0 )polyenes H z ( C H ~ ( ~ +with ~ ) +threeextra ~C H units per half-chain; (-) AM1 model2 results for the extended polyenes.
about chain atom i (note that -n Ii In, as in Figure 1) as (5) hri = sign(i)lri.i+, - ri-l,j wherer,,jis the bond length fromcarbon i tocarbonj. Calculated valuesof the bond-length alternations for (C6H5)2(CH)2,,+,-with n = 2-6 are shown in Figure 2, while those for n = 5, 10, 15, and 20 are shown in Figure 3. Note that as all chains have C2, symmetry and Arj is an odd function of i, data for only one-half of the chains with i 1 Oare shown in these and subsequent figures.
Structure of Odd Polyene Anions Also shown in Figure 2 are the bond-length alternations calculated2 for the unsubstituted polyenes H Z ( C H ) ~ ( ~with + ~ three ~ - CH units extra in each chain half. Qualitatively, the results are very similar, indicating that theeffectofa phenyl groupon thestructure of the polyene chain is similar to the effect of three additional C H units. The SSH soliton theory expresses the bond-length alternation of an infinite chain (n = =) as
The Journal of Physical Chemistry, Vol. 97, No. 1I, 1993 2781 ?
\
!
-
l
l
l
l
l
l
-
Aii = Ai, tanh(i/l) where&, is thealtemation foran infinitechain (Le., thedifference between the single and double bond lengths in a regular polyene chain) and 1 is a characteristic length associated with the geometric kink (i.e., a measure of the soliton half-width). Shown also in Figure 3 are tanh functions obtained by fitting the calculated bond-length alternations. A least-squares procedure was used in the construction of this fit during which the last two data points at the end of the chain were ignored because of a chain-end effect. We see that the geometric properties of the finite-length ions are described quite accurately as a truncated soliton. In the inserts to this figure, the fitted values of Ar, and I are plotted as a function of chain length n on a 1/ n scale; there, they are compared to the values obtained2 for the unsubstituted polyene anions with three C H units extra per half-chain. For the shorter ions with n < 10, it is difficult to obtain unique values of the two parameters because only a small fraction of the soliton is revealed; for large n, the least-squares fitting procedure is stable and indicates again that the effect of a phenyl group on the polyene chain is similar to the effect of three additional C H units. Also shown in Figure 3 are the values of Ar, and I obtained using an empirical model parametrized2 so as to reproduce the AM1 Fock matrices for polyene ions. This model requires only a small fraction of the computational resources of an AM1 calculation, facilitating calculations up ton = 160, close to the infinite-chain limit. Using this model to extrapolate the AM 1 results indicated that both the unsubstituted polyenes and the awdiphenyl polyenes have, in their infinite-chain limits, identical central solitons with &, = 0.095 f 0.002 A and I = 7.0 f 0.2. (b) Charges. No uniquely-defined method exists for the calculation or observation of molecular atomic charges. At the simplest level, such charges constitute an approximation to the electrostatic potential energy surface of a m ~ l e c u l e . ~Before calculated atomic charges may be related to real physical quantities, it is important to select appropriate physical quantities whose calculated values are insensitive to changes in the method of charge calc~lation.~ Atomic charges are usually evaluated as sums over atomic orbital charges. The most common method used for calculating orbital charges is that of Mulliken;lo while this method is unstable when the basis set contains polarization or diffuse it is expected to give reasonable results for a minimal basis set, as used in AM 1. The most natural procedure arising from the AM1 approximations for the determination of atomic charges is that of Lowdin, in which the charge analysis is performed using Lowdin-orthogonalized atomic orbitals;' I while this method obscures the identity of the basis functions when large overlap is involved, it is expected to give reasonable results for the carbanion systems considered here. Three different combinations of orbital charges may be relevant to the interpretation of the N M R I3Cchemical shifts. These are the r-orbital charge, the total carbon charge (the sum of the chargesin all r a n d uorbitalson theoneatom),and thecombined carbon plus bonded-hydrogen charge. It is not clear a priori which of these three possible sets of charges relates most directly to the N M R experiment: while simple theories connect the N M R shift to only the r charge,g contributions from the polarity (ionicity) of the C H may also be important. Shown in Figure 4 are the AM1 charges calculated using Mulliken r charges, Lowdin carbon (C) charges, and Lowdin
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-.Y -.2 0 ,2 LOWOIN I CHARGE / I Figure 4. Plots of the Mulliken n charges, the Lbwdin CH charges, and the Lowdin C charges vs the Lbwdin n charges for the a,@-diphenyl polyenesanions (CbH5)2(CH)l,,+l-withn = 0-6: ( 0 )atomson thepolyene chain; (0) ipso bridge atoms; (+) remaining phenyl ring atoms. The solid lines are the lines of best fit for the polyene chain atoms; their coefficients are given in Table I.
carbon-plus-hydrogen charges (CH), all plotted vs the Lowdin A charges, for the a,o-diphenyl polyene anions with n = 0-6. In this figure, the data points are marked as arising from atoms on the polyene chain, the ipso phenyl atoms directly linked to the chains, and the other atoms on the phenyl rings. The coefficients of the lines of best fit of the polyene chain atoms are shown in Table I: there, type x and type y refer to the charges used for the abscissa and the ordinate, respectively. Clearly, use of either the Mulliken or the Lowdin charges makes very little difference. The combined C H charges arevery similar to the r charges, indicating that the principal effect of the inclusion of the u charges is to increase C H bond polarity while slightly dissipating the T charges over a larger volume. The large constant term of -0.0946 e which appears in the Lowdin C charge to r-charge correlation is indicative of the magnitude of the u polarity of the C H bonds. Significant deviations of this correlation from a linear relation exist (seeFigure4); hence, it maybepossibletodeterminewhether the observed N M R shifts correspond more closely to the ?r or C charges. Note that the ipso and remaining phenyl atoms display significantly different correlations both from each other and from the chain atoms.
2782 The Journal of Physical Chemistry, Vol. 97,No. 1 I, 1993
Reimers et al.
TABLE I: Coefficients and E R O in~ the Linear R ession of the Plots (See Figure 4) of the Type y vs Type x x a i n Charges for the a,o-Diphenyl Polyenes (Cd-15)2(CH)2h,- with II = 0-6' type Y
x
const, e
slope
Mu1l.C Mull. A Mull. C H L0w.C Low.CH Mull. C Mull. C H
Low. C Low. A Low. C H Low.* Low.* Low. A Low. A
-0.0517 0.0011 0.0005 -0.0946 -0.0003 -0.1452 0.0002
0.9872 1.0105 0.9706 0.9512 0.8949 0.9379 0.8683
max error, e RMS error, e 0.0159 0.0025 0.0070 0.0346 0.0131 0.0498 0.0161
0.0041 0.0006 0.0023 0.0102 0.005 1 0.0141 0.0070
Mull. are Mulliken charges, Low. are charges from the Lowdinorthogonalized basis set, A are A-orbital charges, C are total carbon atom charges, and C H are combined carbon and hydrogen atom charges.
It is also important that charges obtained from the semiempirical AM1 method correlate with charges predicted a b initio and with analytical models for the charge distribution. As theorys suggests that it is the r charges which are of fundamental importance to the calculation of these N M R chemical shifts, henceforth we concentrate our attention upon the calculation of the r charges. Individual atomic r charges obtained using different methods such as A M I , ab initio, and the analytical SSH theory differ significantly primarily because these methods ascribedifferent polarityorbondionicity to theCCb0nds.j Indeed, most of the nonuniqueness (both experimental and theoretical) of atomiccharge stems from assumptions made (indirectly) about the CC bond ionicities. We have shown in part 113that it is possible to generate a function of the molecular charge distribution which removes most of the effects of bond ionicity by integrating the atomic charges along the polyene chain. This method needs a slight adaptation in order to be applied to the a,@-diphenyl polyenes to include the bridge atoms. A possible approach is to "open up" the phenyl rings, forming a "chain". An alternative approach, taken here because it leads to slightly better agreement between theory and experiment in section 3, is to construct bond charges from the atomic charges. The total charge is expressed in terms of point charges q'located in the center of each C C bond
(7) where q, is the atomic charge and m, is the number of C C bonds attached to that carbon center. (For the unsubstituted polyenes, it is necessary3 to include an extra bond charge past each end of the chain so that the terminal carbon atom also has msn = 2; for the phenyl rings, we define atoms n + 1 ton + 6 as the ipso, ortho, meta, para, meta', and ortho' atoms, respectively, so that the final charge is given by q'n+6= qip0/3 + qorlw/2.) Summing these bond charges from the central atom outward produces the net accumulated charges Q, Qi
= kq> j=O
that were exploited in part 11.3 They are insensitive to the type of electronic structure calculation, their being little difference between the AM1 and a b initio double-(basis net accumulated charges for the unsubstituted polyene anions.3 A vast range of charge analysis schemes alternative to the Mulliken and LBwdin methods exists; see, e.g., refs 12-20; while each of these schemes has its own particular advantages and disadvantages, a detailed comparison of their relative merits with respect to the problem of the determination of N M R chemical shifts is beyond thescopeof this paper. Instead, we havedeveloped a method above which is independent of the particular representation of the CC bond ionicities produced by any particular charge analysis method. As the N M R chemical shifts are believed* to be primarily related to r charges, our method
Figure 5. Net accumulated Lowdin A charge Q,about chain atom i (only i 1 0 shown) calculated using A M 1 for the a,@-diphenyl polyene anions ( C ~ H S ) ~ ( C H with ~ ~ +nI = - 2, 5, 10, 15, and 20: ( 0 )A M I results; (-) fits of the A M 1 data to the soliton tanh function, eq 9, assuming Qtol = -1. The insert shows theother soliton parameter Iplottedvs chain length non a 1lnscale: (0) apdiphenyl polyenes; ( 0 )polyenes H ~ ( C H ) Z ( ~ + ~ ) + I with three extra C H units per half-chain; (-) A M 1 model2 results for the extended polyenes.
concentrates on the meaningful physical properties which charge analysis methods have in common rather than concentrating upon the poorly-defined aspects with respect to which they differ. Our calculations for carbon charges, even after transformation to average bond charges, do involve the CH bond ionicities directly, and hence, these results are subject to any errors present in the AM 1 Hamiltonian and the Mulliken or LiSwdin analysis methods. Often, AM1 underestimates the number of electrons notionally attached to hydrogen in a C H indeed the AM1 hydrogen positive charges for the unsubstituted polyene anionsZare in excess of those calculated from double-( SCF a b initio wave functions and show increased sensitivity to chemical environment. The AM1 charges appear qualitatively reasonable, however, for this model system. Another model system relevant to the ions studied herein is the benzene molecule. AM1 predicts for benzene qH = 0.120 e; this is slightly less than the value of 0.16 e required for an atomic charge distribution to reproduce the observed4* benzene quadrupole moment and is consistent with the values of 0 . 1 2 4 16 e 0btained~~J0 from a b initio calculations using extensive basis sets and sophisticated charge analysis schemes. AM1 net accumulated LBwdin ?r charges for the chain atoms of some short a,w-diphenyl polyenes are given in Figure 2, along with the results3 for the corresponding positions on the unsubstituted polyene anions with three extra C H groups per halfchain. Note that as Qi is an odd function of i, only atoms with i L 0 are shown in this figure. As was found previously for the geometric properties, the principal effect of a phenyl ring on the properties of the polyene chain are similar to the effects of adding three extra C H atoms. We interpret the SSH4 soliton theory result that the charge q, is zero on every alternate atom as a specific consequence of the predicted bond ionicity. Neither the bond charges q', nor the net accumulated charges Q, deduced from their theory suffer from such a constraint; indeed, it can be deduced that the net accumulated charges of an infinite polyene are predicted to be Qi =
%
tanh(
f)
(9)
where Qtol is the total r charge on the ion and I is the soliton half-width. This function is readily applied to finite-length ions and to the combined C H charges. For the ions considered herein, the total charge is , . Q = -1 less any charge deemed to be permanently associated with the phenyl rings. The AM1 net accumulated Lawdin r charges for various a,wdiphenyl polyene ions are plotted in Figure 5 along with their best
The Journal of Physical Chemistry, Vol. 97, No. 11, 1993 2783
Structure of Odd Polyene Anions fit to eq 9. This figure demonstrates that all of the charge is associated with the soliton, and hence, we put Q,o, = -1. Thus, only one free parameter remains to be determined by the numerical fitting procedure, and hence, this procedure is considerably more stable than the two-parameter fit described earlier for the bondlength displacements. This has the consequence that reliable results can be obtained for ions of shorter length. Deduced halfwidths 1 are shown in the insert of Figure 5 as a function of chain length, n, plotted on a l / n scale, along with the values obtained3 for the unsubstituted polyenes containing three extra C H units. Again, very good agreement is seen, indicating that the effect of a phenyl ring on the polyene chain is qualitatively similar to the effect of three additional CH units. Also, the results for the unsubstitutedpolyenesobtainedusing the AM1 model2Jareshown in the insert to Figure 5 . Simpleextrapolationof theAM1 results ton = predicts that 1 = 9.6. The AM1 model predicts, however, that 1 should become constant as 1/ n 0; indeed, such a result is expected on theoretical grounds as the end effect decreases dramatically in the range once the soliton is completely formed.* A better estimate of the soliton width for an infinite ion would be the AM1 model result 1 = 9.4 itself, but, as this model overestimates the half-width of the smaller ions by 0.3, weestimate that the AM1 half-width of the charge distribution in the unsubstituted and a,w-diphenyl polyenes is 1 = 9.1 f 0.3. Note that while the SSH soliton theory predicts the correct qualitative shape of the geometric soliton and of the charge-density wave, unlike AM1 it predicts that they have identical soliton half-widths.
I
z i-
'
-
'
'
I
'
"
g
'
I
'
9Y.8ql
132.8 t l 8 8 . ' t i
-: DAi nnn --- 201 -
-
3. Interpretation of Observed NMR Chemical Shifts We now address the problem of inverting the observed7 '3C chemical shift data to obtain atomic charges for the chain carbon atoms and hence to estimate the half-width of the soliton chargedensity wave. It has been shown by Tolbert and Ogle7 that the average chemical shift 8 correlates with the average a charge 4 in accord with eq 2, with 80 = 132.8 ppm and = 188.4 ppm/e. They also have showns that the postulate of O'Brien et al.37that the individual chemical shifts are correlated in the same fashion with the individual atomic charges (eq 4) is worthy of consideration. We plot in Figure 6 the individual shifts 6, as a function of the individual AM1 a or C charge and, in the inserts to this figure, the average shifts as a function of the average charges. While a qualitative relationship
'
- _ - 6,. 127.9 t
Xn-3
-__-
-
6,- 136.8 t
90.9q,
2 I- T 186.5 + Y31.Y: -: bAi nnn --- 021 Om nn -- 56 - Xn-3
-0.Y
-0.2 -0.1 0 LOWDIN C CHARGE q / e
0.1
-0.3
Figure 6. For the a p d i p h e n y l polyene anions ( C ~ H S ) ~ ( C H ) ~ ,with ,+,n = 0 - 6, the main plots showing the correlation between the individual observed 'T chemical shifts and the individual AM 1-calculated Lowdin Tor C charges. Theinsertsprovidea X2.5 scaleexpansionoftheindicated regions of the main plots and show the correlation between the average I3C chemical shifts and the average 7r or C charge. On both the main plots and the inserts are shown the lines of best fit of the data (- -) for the individual shift/change correlation for all ions; (-) for the average shift/charge correlation for ions n = 1-6.
-
1
4,= 6, + w,
(10) does exist, the deduced values of the parameters 60 = 127.9 ppm and a = 94.8 ppm/e for the a-charge correlation are clearly very different from those predicted by the OBrien postulate. AM1 indicates that eq 2 is not a fundamental relationship but rather results from a balance between many competing effects. We conclude that use of this equation to generate individual atomic charges (and hence soliton properties) is likely to give erroneous results; a discussion of the estimated reliability of this AM1based conclusion is given in section 4. Here, we proceed with the O'Brien approach in order to obtain a crude estimate of the halfwidth. We use eq 8 to produce the net accumulated charges from the OBrien-postulate-derivedatomic charges. As the total charge on the ion produced by this method is not precisely -1, we renormalize the resultant charges; this renormalization has only a minor effect due to the accuracy of eq 2. The results are shown in Figure 7 for the apdiphenyl polyene ions with n = 3-6. Also shown in this figure are the curves resulting from the least-squares fit of these data to the soliton equation, eq 9. We assume that Q,,,= , -1 and exclude the last two data points from the fit, as they are subject to a chain-end effect.*J The resulting soliton halfwidths 1 are shown in the insert of Figure 7, and the data for n = 4-6 are extrapolated to n = w . Given the noise present in the data, the small size of the ions studied, and the complicated n dependence predicted by AM1 (see Figure 5 ) , an error bar for this extrapolation is difficult to obtain. We estimate the charge-
\
Y
5
2
-6
Figure 7. Renormalized net accumulated charges Q,about chain atom i (only i L 0 shown) obtained from the observed 13CN M R shifts' using the O'Brien postulate, eq 4, for the a,w-diphenyl polyene anions ( C 6 H ~ ) 2 ( C H ) 2 nwith + l - n = 3-6: ( 0 )these results; (-) fits of this data, excluding the last two data points per ion, to the soliton tanh function, . , = -1; (- -) a line simply connecting the last two eq 9, assuming Q observed data points to the soliton curve. The insert shows the soliton parameter I plotted vs chain length n on a l / n scale and a linear extrapolation of the data for n = 4-6 to n = a.
-
density-wave sotiton half-width obtained from use of the O'Brien hypothesis as 1 = 13 f 1.5. Such a value is consistent with the earlier result of 14 f 2 estimated by Tolbert and Ogle* without explicit conversion of the observed chemical shifts into atomic charges.
2184 The Journal of Physical Chemistry, Vol. 97, No. 11, 1993 --_
61- 7 7 . 9 t 12Y.7qj 61- 83.6 t 133.lq;
.,..
A n- 1 X
n- 3
,,
Figure 8. For the a,w-diphenyl polyene anions (CbH5)2(CH)2n+l-with n = I , 3, and 6 thecorrelation between the individual observed 'ICchemical shifts and the individual A M I-calculated C charges for the polyenechain atoms only. The lines indicate the correlation for each individual ion obtained by fitting data for all ions to eq 1 I ; see Table I1 for the values of the parameters.
In an alternative treatment, we construct an accurate function which expresses the observed chemical shifts in terms of the AM 1 calculated charges. The simplest approach of this type is to apply eq 10 to fit the AM1 calculated A or C H charges. As shown in the previous section, the AM 1 A and CH charges are very similar; hence, werestrict discussion to the A charges only. The correlation shown in Figure 6 with bo = 127.9 ppm and CY = 94.8 ppm/e leads to an extrapolated infinite polyene soliton half-width of 1 = 8 f 1, much narrower than that obtained assuming the O'Brien postulate. Because of the renormalization used to ensure that the total ion charge is -1, 1 obtained using eq 10 is actually independent of CY. Hence, we see that varying 60from the O'Brienpostulate value of 132.8 ppm to the AM1 value of 127.9 ppm reduces 1 from 13 to 8! Increasing 60to 136.9 ppm only increases 1 by 1 unit, however. Clearly, the determination of the soliton width is quite sensitive to the choice of model used to determine charges from chemical shifts. The deviations of the observed chemical shifts from the AM1 charges seen in Figure 6 show regular systematic trends. One possible reason is that the correlation is a function of the types of atoms involved, Le., that there is a different correlation for the ipso, ortho, meta, and para phenyl positions, as well as for atoms displaced a given number of bonds along the chain from the ipso atoms. Alternatively or concurrently, another possible reason is that the correlation may depend on the chain length n. From the Karplus and P ~ p l model, e ~ ~ one expects that CY depends upon the u-x* and A-u* energy gaps, and these gaps decrease as n increases. Here, we modify eq 10, obtaining
Reimers et al. Using eq 11, we fitted the observed chemical shifts to AM1 charges manipulated in a total of 54different ways. Thevariations considered were (i) Lowdin or Mulliken charges; (ii) C, CH, or A charges; (iii) three different linear combinations of the charges-the atomic charges, q, themselves, the bond charges q',, and the net accumulated charges Q,-and (iv) three different constraints to the parameter set-unconstrained (eight-parameter set), no chain-length correlations (four-parameter set with both u and bsth b parameters set to zero), and no atom-type or chainlength correlations (two parameters, reverting back to eq 10). The results for the Lowdin A and C charges are given in Table 11; the results for the Mulliken and CH variants are only trivially different from these results and are not given explicitly. The resulting parameters, the R M S error in the calculated and observed chemical shifts, the correlation of the average calculated chemical shift with the average A charge, and the extrapolated infinite-chain soliton half-width 1 are given in the table. The net carbon charges do not sum to the net ion charge, and hence, the renormalization procedure used above for the A charges cannot be directly applied in order to determine the soliton parameters from the C charges. First, we assume that the average CH bond ionicity
pertains to each of the individual 2n + 11 CH bonds of a particular ion. The charge attributed to the soliton is then given as the total C charge less this average ionicity per attached CH bond,
where nH = 0 for an ipso atom and 1 otherwise. These solitonattributed C charges do sum to the net ion charge, so eqs 8 and 9 can be directly applied. This approach is justified by the reasonablecorrelation seen between the A and C charges in Figure 4 and Table I; the value of 4.0946 e quoted for the constant term in Table I is the average CH ionicity for all ions considered. Table I1 shows that it is always considerably more difficult to fit atomic charges qi to atomic observed chemical shifts than it is to fit average bond charges q: to average bond chemical shifts. This arises because the CC bond ionicities, obtained unreliably by both theory and experiment, contribute to the atomic charges and shifts but not to the average bond charges and shifts. It is usually more difficult to fit the net accumulated charges Qi to the net accumulated observed chemical shifts; the summation procedure provides a weighting of the data biased toward the atoms a t the center of the chain and hence to the accurate determination of the soliton half-width I; the close similarity of these two sets of results for the more-accurate fits is encouraging. Finally, it is also always more difficult to fit the C charges than the r charges. Note especially the calculated slope 2 for the CY 6, = 60 average shift-average charge correlation: fitting the C charges 1 + a / n 1 b/nqi results in poor to very poor agreement of this calculated quantity where u and b are new empirical parameters controlling the chainwith the experimental value of 188.4 ppm/e, while fitting the A length dependence; in addition, we allow different coefficients charges gives good to excellent results. for atoms as belonging to the polyene chain or to a phenyl ring, The least error and the most accurate average shift-average resulting in a total of eight parameters. As an example of the *-charge correlation are produced when the net accumulated operation of this function, the observed chemical shifts and AM 1 chemical shifts are correlated with the net accumulated Lawdin calculated C charges for the chain atoms of ions with n = 1, 3, A charges. From Table 11, the RMS error in the fit is just 0.8 and 6 are shown in Figure 8. Also shown in this figure are the ppm; 60differs from the observed value by only 0.1 ppm, 2 differs predicted linear correlations for these ions obtained from eq 11 by only 0.3 ppm/e, and the deviation of the shortest ion n = 0 with u = 0.376 and b = 0.406; they reveal the systematic trends from the average shift-average *-charge correlation is accurately in the correlation. Note that in this approach, the apparently reproduced. Using this correlation, the renormalized net accuanomalous properties of the shortest ion, including its observed mulated charges Qiobtained from the I3Cchemical shift data are deviation from the average correlation eq 2, are seen to arise plotted as a function of chain atom number i in Figure 9. Also naturally as an extrapolation of the individual charge/shift shown in this figure are the tanh functions obtained by assuming correlations for the larger ions. It is not necessary to p o ~ t u l a t e ~ - ~ Q,,,= -1 and fitting I using eq 9. Similar to the results (Figure that steric interactions in the crowded n = 0 ion are solely 7) obtained using the O'Brien postulate, the last two data points responsible for these anomalies. per ion show an end effect and are excluded from this fit. The
-+- +
The Journal of Physical Chemistry, Vol. 97, No. 11, 1993 2785
Structure of Odd Polyene Anions
TABLE II: Parameters Obtained by Fitting the Polyene Chain and the Phenyl Observed Chemical Shifts to Various Charges’ Using Equation 11 and the Associated RMS Errors in the Fitsb charge type
60,ppm
c 41
136.6 134.6 140.3 141.2 143.2 144.9 160.2 168.6 155.1 128.0 126.2 131.0 129.2 129.2 132.6 131.6 133.4 133.7 132.8
c 4: C Qt r
4,
r
4:
7
Q,
chain atom parameters a,ppm/e a
OBrien
92.2 88.8 88.4 122.6 136.4 110.0 246.3 281.8 168.0 95.1 85.5 85.0 118.3 118.5 87.8 155.3 160.3 99.7 188.4
0 0 0.376 0 0 0.514 0 0 0.625 0 0 0.311 0 0 0.403 0 0 0.416
0
b
0 0 0.406 0 0 1.111 0 0 1.478 0 0 0.380 0 0 0.433 0 0 0.689 0
phenyl ring atom parameters 80,ppm a,ppm/e a 136.6 137.6 138.4 141.2 138.9 138.0 160.2 147.9 140.6 128.0 129.5 129.6 129.2 130.1 128.8 131.6 129.3 128.5 132.8
92.2 93.8 94.1 122.6 102.5 96.2 246.3 163.8 111.5 95.1 131.6 144.4 118.3 150.8 134.1 155.3 122.1 130.0 188.4
0 0 0.023 0 0 0 0 0 -0.014 0 0 0.002 0 0 -0.039 0 0 -0.040 0
b
RMS error,ppm
0 0 0.021 0 0 0 0 0 -0.172 0 0 0.215 0 0 -0.319 0 0 -0.332 0
3.3 3.1 1.8 2.4 2.2 1.2 7.9 3.2 1.4 3.5 2.6 1.6 1.8 1.7 0.8 4.6 2.8 0.8 2.9
av correlation 80,ppm &ppm/e 125.1 123.6 129.7 125.9 125.6 129.8 129.5 131.6 131.8 128.0 127.8 132.0 129.2 130.3 132.4 131.6 132.1 132.7 132.8
40.2 11.3 125.4 53.5 49.6 130.8 107.5 160.6 167.4 95.0 89.8 172.8 118.3 138.3 181.5 155.2 174.1 187.1 188.3
I 11.3 15.5 9.2 12.6 12.0 7.8 14.9 11.5 8.6 8.2 14.4 8.9 10.2 11.2 7.7 12.2 9.5 7.6 12.8
OBrien = from use of the OBrien postulate, eq 4; r = r charge; C = carbon charge; q, are atomic charges; are average bond charges; QI are net accumulated charges. Also given are the coefficients from the calculated-average-shift to average-n-charge correlation, and the charge density wave soliton half-width extrapolated to an infinite chain, 1. I
Oh
i
+:
w
1
2
3
.
Y
5
6
7
Figure 9. Renormalized net accumulated charges Qi about chain atom i (only i 2 0 shown) obtained from the observed N M R shifts7 using eq 11 fitted using no constraints to the A M I Lowdin r net integrated charges: ( 0 )these results; (-) fits of this data, excluding the last two data points per ion, to the soliton tanh function, eq 9, assuming Qlol = -1; (- -) a line simply connecting the last two observed data points to the soliton curve. The insert shows the soliton parameter I plotted vs chain length n on a l / n scale and a linear extrapolation of the data for n=4-6ton=-.
-
deduced values for the ions n = 4-6 are extrapolated to n = in the insert to this figure and yield a value of I = 7 for the half-width of the charge-density wave of an infinite chain. This is nearly half that obtained using the OBrien postulate and is much closer t6 the value obtained by using eq 10 with a set to reflect the chemical shift to AM 1 ?r-atomic-charge correlation shown in Figure 6.
4. Conclusions Atomic charges are not uniquely defined, and it is possible to arrive at quite different results depending on the particular theoretical or experimental technique used for their determination. We haveshown, however, that a major cause of this nonuniqueness arises from the way in which the polarization of the chemical bonds, the bond ionicity, is treated; consequently, comparison not of the atomic charges themselves, but rather of the average charges distributed about chemical bonds leads to very similar conclusions about the charge distribution independently of the technique used to determine it. Indeed, bond charges obtained
from a b initio calculation^,^ semiempirical AM 1 calculations, and analytical models such as the SuSchrieffer-Heeger soliton t h e ~ r yobtained ,~ using different charge-determination schemes, agree very well not only with each other but also with bond charges derived from I3C chemical shift experiments. Further, we have shown that the running sum of the bond charges, the net accumulated charges along the chain, provides a weighting of the atomic charges most suited to the determination of the solitonic charge-density-wave properties. For the a,w-diphenyl polyene anions of finite size n, AM1 calculations show that the properties of the polyene chain are very similar to thoseof that section of thechainof theunsubstituted polyene containing three CH units in place of each phenyl ring. Consequently, the properties of the infinite-chain a,w-diphenyl polyene are predicted to be identical with those of the infintechain unsubstituted polyene, as would be formed by, e.g., ndoping of polyacetylene. The result is that the infinite ion is predicted by AM1 to display a geometric soliton with bond-length alternation Ar, = 0.095 f 0.002 A and half-width I = 7.0f 0.2 and to display a slightly broader charge-density wave of halfwidth 1 = 9.1 f 0.3 within which all of the net -1 ionic charge is localized. Note that the phenyl groups are predicted to interact with the atoms at the ends of the polyene chains but that such interactions should2.3decay exponentially into the chain, resulting in a localization of the phenyl properties. Interaction between the two phenyl groups a t each end of the ions would be expectedS1 to be anomalously large for ions smaller than about n = 15 but to decay rapidly once the soliton is completely formed and the regular polyene chain starts to grow. Attempts to determine atomic charges and hence soliton properties from experimental I3CN M R chemical shift data give results which quantitatively are sensitive to the method used. All methods, however, verify that the polyene chains display soliton structure in their charge-density waves of the type predicted by the AM1 calculations and the SSH model. The O’Brien po~tulate,)~ that the individual atomic charges and chemical shifts correlate in precisely the same fashion as the average charges and shifts, is not supported by AM1 calculations combined with Tolbert and Ogle’s’ observed chemical shift data. Verification of the O’Brien postulate is, in fact, a two-step process: first, it must be established that a linear relation exists between the atomic charges and the chemical shifts and, second, that the coefficients in this linear relation are the same as those found in the SpieseckeSchneider correlation of the average ?r
Reimers et al.
2786 The Journal of Physical Chemistry, Vol. 97, No. 11, 1993
charges with the average chemical shifts. As shown by Tolbert and Ogle,* a qualitative linear relation does in fact exist between individual chemical shifts and charges, but significant systematic deviations from the relationship occur, and it is unlikely that this relationship could be useful as a tool for the assignment of peaks in N M R spectra. Addressing the second issue, AM1 predicts that both the slope and the intercept of the individual chargeindividual shift correlation are significantly different from those of the average charge-average shift correlation, eq 2. Indeed the average correlation appears to originate as the combination of the average of many different aspects of the individual charge to chemical shift correlation. Hence, it is anticipated that families of compounds similar to the ones studied herein should also show SpieseckeSchneider correlations with different coefficients to the (nonfundamental) coefficients used herein, as is observed.3740 It is, in principle, possible that the OBrien postulate is in some sense valid and that the problems evident in Figure 6 originate in shortcomings in the AM 1-calculated charges. Interpreted literally, theO’Brien postulate refers to thecorrelation of xcharges with chemical shifts. The AMl-based argument that this postulate is incorrect is indeed very strong: the alternate AM1derived correlation successfully reproduces all of the experimental data (i.e,, the Spiesecke and Schneider correlation, eq 2, and the deviations observed from it for the shortest ion), AM1 x charges are closely related to a b initio x charges for these systems, and the analysis used is insensitive to assumptions made as to the CC bond ionicities. Our conclusion would be incorrect if, e.g., say 0.2 ecould be transferred from the x-carbon orbitals tononvalence hydrogen orbitals of A symmetry. Although simple theories* indicate that perturbations to the u electrons such as C H bond ionicities should not effect the chemical shift, more complete treatments29 do allow for such effects. To show that such high-order effects do in fact pertain to this problem is a difficult and complicated task, beyond the scope of this paper. Here, we seek experimental evidence to suggest that such effects could occur and consequently raise the possibility that the O’Brien relationship might be essentially correct with the modification that net carbon charges are used in place of x charges. The AM 1 calculations, however, discount this possibility, predicting that the linear correlation between individual shifts and individual charges is reduced in quality in going from x charges to carbon charges and predicting that the resulting linear coefficients are even further removed from the coefficients of the average shift-average charge relationship. These AM1 calculations explicitly include the effects of the C H bond ionicities and hence are likely to be ~ n r e l i a b l e . ~It~is, ~difficult ~ to envisage that any charge calculation scheme could produce CH bond ionicities which cancel the systematic deviations observed between the individual x-charge densities and the observed chemical shifts. Simple chemical arguments suggest a very different picture: all of the hydrogen atoms are in similar environments, and hence, in a zeroth-order approximation, all C H bonds would be expected to show the same ionicity, independent of the x charge. Different calculation methods could give quite different values for the CH bond ionicity, but approximately the same value would be expected for all bonds, and the individual charge to individual chemical shift correlation would not be improved. Indeed, while the AM1 calculations reported here and elsewhere2 for the unsubstituted polyenes show some variations in the CH ionicity with environment, double-{ SCF ab initio calculations for the unsubstituted polyenes2 show very littlevariation, suggesting that the magnitude of the possible effects are in fact overestimated by our AM1 calculations. The question as to the value of the slope found in this correlation is a very difficult one to answer. As the Spiesecke-Schneider correlation is very sensitive to average changes of the order of 0.01 e per center, use of methods other than AM1 will produce dramatically different slopes from the one predicted by AMI
and from each other. It is conceivable, though rather unlikely, that some other method could give similar slopes for the average charge-average shift correlation as for the individual chargeindividual shift correlation. Another problem relating to the difficulty of obtaining quantitative estimates of soliton properties from the I3C NMR chemical shift data is that, using existing N M R techniques,7,* the largest ion which can be studied is n = 6. In such a short ion, the soliton is just starting to grow and is still shorter than its AM 1 estimated half-width of approximately 7-9. Hence, considerable extrapolation is required in order to estimate properties of an infinite ion, and AM1 calculations indicate that the calculated properties do not vary simply, say, as a linear function of 1/ n but rather undergo considerable changes as an isolated soliton forms (cf. Figure 3 and Figure 5 ) . Direct application of the O’Brien hypothesis, using linear extrapolation of thesoliton half-width as a function of 1/ n , results in an estimated half-width of thecharge-density waveofan infinite soliton as 1 = 13. Use of the eight-parameter AM1 chargeN M R shift correlations allows this value to fall to as low as 1 = 7, however, and arguing that the corrections predicted by AM1 should be at least qualitatively correct, we conclude that the actual value should lie somewhere within this range. It is not possible to decide whether the experimental data confirm the AM1 prediction that the charge-density wave is broader than the geometric soliton or the SSH soliton model prediction that these phenomena display identical widths. Tolbert and Ogles avoided all of the above-mentioned problems and devised a method for estimating the width of an infinite soliton directly from the N M R data. Their approach was to linearly extrapolate the I3C N M R chemical shifts for the phenylring atoms, estimating the chain length at which these shifts equal the shifts observed in even a,w-diphenyl polyenes. While this provides an excellent qualitative measureof the soliton width, it is not clear to which point on the soliton curves that this length corresponds (e.g., a width at half-maximum, 1 standard deviation, etc.). Further, this measure of the width is really a probe of the tail of the soliton, detecting when this tail ceases to affect the phenyl rings; it is not a property of the bulk of the soliton. Calculations for the unsubstituted polyenes (e.g., ref 3, Figure 5) indicate that the soliton tail is slightly extended compared to the bulk of the soliton described by the tanh function. Also, the Tolbert and Ogle measure is really that of the sum of the width of the soliton plus the width of the end effect (about 2 units) through which the phenyl groups perturb the properties of the polyene chain. Hence, we expect that the Tolbert and Ogle estimate of a soliton half-width of 14 f 2 is in fact an overestimate. The AM1 soliton charge density wave half-width of 9 is not necessarily inconsistent with this Tolbert and Ogle measure. It would be very useful if experimental methods could be devised for determining the I3C N M R chemical shifts for polyene ions up to at least n = 10. This would reduce the burden placed on extrapolations of the data. Similarly, it would be fruitful to apply ab initio methods directly to predict the chemical ‘shift for ions of this size. The OBrien postulate should be treated with caution, and individual atomic charges obtained from observed chemical shift data using this postulate may be quantitatively considerably in error.
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