J. Phys. Chem. 1992, 96, 9232-9235
9232
Referenom .ad Notes (1) (a) Orville-Thomas, W. J., Ed. Internu1 Rotution in Molecules; John Wiley and Sons: New York, 1974. (b) Lambert, J. B. Top. Stereochem. 1971. ~.6. 19. (2) Weiss, S.;Leroi, G. E. J . Chem. Phys. 1968, 48, 962. (3) Lowe, J. P. Science 1973, 179, 527. (4) (a) Aler, T. D.; Gutowsky, H. S.;Vold, R. J. J. Chem. Phys. 1%7,47, 3130. (b) Abraham. R. J.: Loftus. P. Chem. Commun. 1974.180. (c) Ris&. 0.; Taylor, R. C. Spkrochim. Acta 1959,I5, 1036. (d) Dung, J. R.;Craven, S.M.; Hawley, C. W.; Bragin, J. J . Chem. Phys. 1972,57, 131. (e) Ward, C. R.; Ward, C. H. J. Mol. Specrrosc. 1964, 12, 289. (f) Durig, J. R.; Wurrey, C. J.; Bucy, W. E.; Sloan, A. E. Spectrochim. Acru 1976,32A, 175. (g) Dung, J. R.;Bucy, W. E.; Wurrey, C. J. J . Chem. Phys. 1974,60, 3293. (h) Dung, J. R.; Craven, S.M.; Lau, K. K.;Bragin, J. J. Chem. Phys. 1971, 54, 419. ( 5 ) Thompson, D. S.;Newmark, R.A,; Sederholm, C. H. J. Chem. Phys. 1%2,37,411. Newmark, R.A,; Stderholm, C. H. J. Chem. Phys. 1963,39, 3131; 1965,43,602. Weigart, F. J.; Winstead, M. B.; Garrels, J. I.; Roberts, J. D. J . Am. Chem. Soc. 1970,92,7359. Govil, G.; Bcmstein, H. J. J . Chem. ~
I~
Phys. 1968,49,911. Newmark, R. A,; Graves, R. E. J. Phys. Chem. 1968, 72, 4299. (6) Weigart, F. J.; Mahler, W. J . Am. Chem. Soc. 1972, 91, 5314. (7) The theoretical DNMR spectra were calculated by using a locally modified version of computer program DNMR3 written by Kleier, D. A. and Binsch, G.* A description of the modifications has been published? (8) Kleier, D. A,; Binsch, G. QCPE Program No. 165. (9) Bushweller, C. H.; Bhat, G.; Lctendre, L. J.; Brunelle, J. A.; Bilofsky, H. S.;Ruben, H.; Templeton, D. H.; Zalkin, A. J. Am. Chem. Soc. 1975,97, 65. (10) Jensen, F. R.;Smith, L. A.; Bushweller, C. H.; Beck, B. H. Rev. Sci. Insrrum. 1972, 43, 894. (1 1) Bovey, F. A.; Jelinski, L.; Mirau, P. A. Nuclear Mugnetic Resononce Spectroscopy, 2nd 4.; Academic Press, Inc.: New York, 1988. (12) Dupuis, M.; Spangler, D.; Wendoloski, J. Nurionul Resource jor Computurionul Chemistry Program QGO1.2, Revised by Schmidt, M.; Elbert,
S. (13) Davidson, E. R.; Feller, D. Chem. Rev. 1986, 86, 681. (14) McIver, J. W.; Komornicki, A. Chem. Phys. k r r . 1971, IO, 302.
Dependence of Spin Coupling In Non-Kekul6 Molecules on the ?r-Electron Network B. L.V. Prasad and T.P.Radhakrishnan* School of Chemistry, University of Hyderabad, Hyderabad 500 134, India (Received: May 6, 1992; In Final Form: July 9, 1992)
Several specifically chosen non-Kekulb r-electron biradicals are studied using semiempirical A M l / C I calculations to analyze. the dependence of the singlet-triplet gap on the topology and r-electron network. A simple algorithm is developed to predict the singlet-triplet gap wing spin densities at atomic sites and the r-electron pathways. The significance of this model in experimental studies of the ground-state spin of non-KekuW systems and the design of organic ferromagnets is outlined.
Introduction There has been a renewed interest in conjugated non-Kekult systems in recent years, due to their relevance in the design of organic ferromagnetic materials. Special topological characteristics that lead to a high-spin ground state is the basis of the models proposed by Ovchinnikovl and Matagaa2A more recent proposal by Fukutome et al.' involving polaronic ferromagnetism is also a closely related one. Several experimental studies along these lines have been reporteda4 Coulson et al.? Ovchinnikov,' Klein? Borden and Davidson,' and Tyutyulkov* have addressed the problem of theoretically predicting the mode of spin coupling in non-KekuE systems. The former studies focusad on alternant systems, whereas Tyutyulkov has extended the Coulson-Rushbrooke theorem to nonaltemant systems and heterosystems as well. However, these rules are applicable to specific classes of non-Kekul€ systems, and several exceptional behaviors are also observed. Recently, one of us has proposed9 a general framework in which the problem of spin coupling in all kinds of non-Kekule systems can be viewed. The simple rule that we proposed to predict the ground-state spin of non-Kekule systems encompasses alternant, nonaltemant, and heteroatomic systems. We focus attention on the particular resonance structure (double-bond distribution) where the spins are localized at sites as close as possible and then analyze the coupling in terms of the rclectron network. The spin-polarization picture shows that it is ferromagnetic or antiferromagnetic, depending on whether the number of intervening r-electrons is odd or wen, respectively. We have also used our approach in the analysis of some novel systems.I0 Here we report the results of our investigation of some quantitative aspects of the spin rule. Semiempirical calculations were carried out on several classes of specially chosen biradical nonKekuE r-electron systems to study the dependence of the singlet-triplet gap on the topology of the systems and the r-network. Physically meaningful trends are discerned which can be qualitatively explained using the superexchange mechanism, which is the basis of our spin rule. A quantitative measure of the spin
coupling in biradicals is obtained through the application of the spin rule in conjunction with calculated atomic spin densities. It is found that this quantification of the spin coupling (in the resonance structure of the biradical in which the radical sites are at the smallest possible separation) is linearly related to the singlet-triplet gap. The relevance of our results to the experimental determination of ground-state spin from a knowledge of spin densities and to the design of organic ferromagnets is noted. Computational Details
All quantum chemical calculations were carried out using the AM1 semiempirical method." We considered both the UHF and open-shell RHF (with CI) procedures for the calculation of spin densities and singlet-triplet gaps. The latter was adopted for the calculations based on two observations: (i) the singlet-triplet gaps calculated using full CI bracketing the five frontier molecular orbitals were in very good agreement with the experimental and previous theoretical mults obtained through more rigorous computations (this was observed in our previous study as well9); (ii) for several systems where experimental fine structure splitting (D and E) values in the triplet states were available, the agreement with those calculated using the spin densities and geometries from the RHF/CI calculations were superior to those calculated using the UHF method (computational results submitted as supplementary material). We preferred the opm-shell RHF calculations for the singet state, since in most cases the open-shell singlets were lower in energy than the closed-shell ones, and also for the calculation of the singlet-triplet gaps based on the difference in heat of formation with that of the triplet state, this was considered more meaningful. Lahti and Ichimura12have also used this a p proach in their semiempirical study of non-Kekul€ systems. Throughout this paper, a positive & implies a triplet ground state. We have optimized the geometries separately for the singlet and triplet states within the symmetry constraints (see Figure 1) of the molecule. The geometries were compared in all cases and found to be very similar, except for one or two cases which are mentioned in the next section.
0022-3654/92/2096-9232$03.00/00 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No.23, 1992 9233
Dependence of Spin Coupling in Non-Kekulb Molecules
.
.
( e ) Sei
s
Figure 1. Molecules considered in this study, showing the relevant resonance structure with the resulting spin sites labeled. The point group symmetries are shown. a, b, c, d, and e refer to sets 1, 2, 3, 4, and 5, respectively (see text).
The spin coupling was analyzed using a series of algorithms. In the simplest case, the coupling of spins localized as close as poaPible in the relevant resonance structure alone was considered. In more detailed treatments, all pairwise spin couplings were considered. Weighting by the appropriate topological distance or the number of intervening *-electrons were studied. Several distance dependences were analyzed. As we discuss below, one of the simplest procedures worked best. The correlation of the singlet-triplet gaps with the quantity representing spin coupling was analyzed using standard statistical procedures. R d b ind Discussion We have considered five sets of non-Kekulb systems in this study. They share the common feature that the connectivity of the *-network enforces a unique shortest topological distance between the unpaired electron sites (smallest number of bonds
separating them) for each molecule. Further, within every set, there is a regular gradation in this topological distance as well as the number of intervening r-electrons so that the variation of the singlet-triplet gaps with respect to either can be analyzed. The systems were chosen with odd and even numbers of intervening *-electrons between the radical sites so that the mode of coupling can also be analyzed. The 26 systems considered are given in Figure 1, and the calculated singlet-triplet gaps are provided in Table I. The first set (Figure la) consists of open-chain branched molecules with the radical sites f o d to be at the extreme ends. The topological distance increases from 1 to 5 in the 5 molecules, 80 also the number of intervening r-electrons, n, thus oscillating between odd and even. Molecules 4 and 5 were chosen instead of 2,3,4,5-tetramethylenehexatriene and 2,3,4,5,6-pentamethyleneheptatriene, since the latter two are f o r d to be non-
9234 The Journal of Physical Chemistry, Vol. 96, No. 23, 1992
-
TABLE I: Am (Es ET in kd/mol) Calculated Using the AMl/CI Method for the Mokculea in Figure 1 molecule &T, kcal/mol molecule AST, kcal/mol 1 16.5 14 0.0 2 -6.3 15 11.8 3 0.8 16 -4.8 4 -0.7 17 -0.7 5 0.0 18 3.2 6 -9.6 19 5.3 7 1.3 20 -18.9 8 -0.3 21 -6.3 9 0.2 22 -4.6
10 11 12 13 2O 15
-1 0
0.0
23 24 25 26
25.6 -2.8 4.3
r
-5.1 -2.3 4.8 3.2
I
d n
L (0)Set 1
-10
30
-lo
I
b
t
"n
(c) Sot 3
FIpure 2. Variation of Asr with the number of intervening r-electrons, n,, between the radical sites in (a) set 1, (b) set 2, and (c) set 3. The line is only a guide to the eye.
planar due to steric reasons and the optimization of planar structures lead to unphysical geometries, making a serial trend study meaningless. The singlet-triplet gaps are plotted in Figure 2a. The expected damped oscillation is clearly observed. The second set (Figure lb) is very similar to the first one except for the fact that inclusion of 0 atoms allows switching of the spincoupling mode without affecting the topological distance between the spin sites. This exercise brought to light an interesting feature. For the same topological distance, the singlet-triplet gap even depends quantitatively on the number of r-electrons between the radical sites. We discuss below the correlation of the singlet-triplet gaps to the measure of spin coupling between the localized spin sites, which is controlled by the intervening number of *-electrons. Figure 2b shows the singlet-triplet gap variation in set 2. Set 3 (Figure IC) consists of poly(cyc1obutadienes) with methylene group at the ends, and the topology ensures increasing separation between the radical sites along the series. Once again, the singlet-triplet gaps vary as in the previous sets (Figure 2c), though a minor discrepancy was observed here; i.e., Asr was higher in magnitude for 13 than for 12. Sets 4 and 5 (Figure Id and le) consist of cyclic systems with methylene groups attached to the ring. Set 4 is all hydrocarbons, and set 5 consists of heterocyclic rings involving oxygen atoms. In some of these rings, odd as well as even *-electron pathways exist between the radical sites in the same molecule. Molecule 11 is included in set 3 as well as set 4, since it is logically the first
Prasad and Radhakrishnan TABLE Ik Calculated Spin Densities at Sites 1 and 2 of the Molecules in Figure 1 spin density spin density molecule site 1 site 2 molecule site 1 site 2 1 0.33 0.33 14 0.13 0.13 2 0.24 0.24 15 0.26 0.26 3 0.26 0.26 16 0.21 0.21 4 0.25 0.25 17 0.21 0.21 5 0.25 0.25 18 0.22 0.22 6 0.44 0.44 19 0.35 0.35 7 0.39 0.26 20 0.42 0.42 8 0.42 0.42 21 0.23 0.23 9 0.39 0.39 22 0.23 0.23 10 0.38 0.38 23 0.37 0.25 11 0.34 0.34 24 0.24 0.21 12 0.24 0.24 25 0.26 0.26 13 0.23 0.23 26 0.22 0.23 ~~
member of both series. In set 4, the intervening number of relectrons between the radical sites (the two paths are considered) increase as (l,l), (1,2), (2,2), (2,3), and (3,3). The net couplings are ferromagnetic for 11, 15, and 18 and antiferromagnetic for 16 and 17 (Table I). Here the topological distances scale as the number of intervening r-electrons, and one could conclude that the couplings over shorter distanm are stronger. In set 5 , on the other hand, the number of intervening r-electrons increases as (1,2), (2,2), ( 2 3 , (2,3), W),(2,3), (3,3), and (3,3), whereas the topological distances vary as (2,2), (2,2), (2,3), (2,4), (2,3), (3,3), (3,3), and (3,4). The sign of the coupling (Table I) is invariably controlled by the smaller of the numbers of intervening r-electrons. The magnitude of the spin coupling between the localized spin sites could be controlled by the topological distance or the number of intervening r-electrons. An analysis of set 5 clarifies which of the two the more dominating influence is, as discussed below. After having satisfied ourselves that the spin couplings in these non-Kekul6 systems are as expected on the basis of the spin rule? we proceeded to analyze the magnitudes of the singlet-triplet gaps in terms of the topology and *-electron network. Using the AMl/CI method, we carried out open-shell RHF calculations on the non-Kekul€ systems. The spin densities at localized radical sites 1 and 2 of the molecules shown in Figure 1 obtained from the triplet state calculations are collected in Table 11. The largest spin densities are invariably found at the sites which are expected to be the radical sites on the basis of the resonance structure with minimum separation between them. In the simplest approach, the spin coupling of the radical sites can be envisaged as a product of these spin densities weighted by the inverse of the topological distance between them or the number of intervening r-electrons. The sign of the coupling was assigned as positive or negative (fem/antiferro) on the basis of whether the number of intervening r-electrons is odd or even. When multiple paths as in a ring exist, a net spin coupling is evaluated as the algebraic sum. We investigated several different ways of weighting the spin couplng. When the spin coupling was assumed to be inversely proportional to the topological distance, the correlation with the singlet-triplet gap (multiple regression coefficient, about 0.9) was not very satisfactory. Several variations on this theme were tried, like powers of the topological distance, negative exponential variation in the distance, etc. We also attempted a definition of the spin coupling as an algebraic sum of all the pairwise couplings in the molecule (involving spin densities at all sites), each one weighted with the various dependences on the topological distances tried above. None of these definitiohp of the spin coupling gave a better correlation with the singlet-triplet gaps listed in Table I. Next we tried weighting the spin coupling (SC)in the simplest approach of considering only the spin densities at the sites marked in Figure 1 with the inverse of the number of intervening ?relectrons. When multiple paths of similar topological distance are present, an algebraic sum over the paths was used:
Dependence of Spin Coupling in Non-KekuM Molecules TABLE IIk Regreusion C d c i e n t for Linerr Fit of Am with SC for Seis 1-5 md the Full Set aa Well 18 with Certain Molecules Excluded (See Text for Explmrtioa) no. of multiple set molecules regression coeff 1 5 0.980 2 5 0.846 2 (excluding 7) 4 0.829 3 4 0.996 4 5 0.956 4 (excluding 15) 4 1 5 8 0.982 0.946 full 26 0.964 full (excluding 7 and 15) 24
.ow
where n, = number of intewening *-electrons between the radical sites, p1 and p2 are the spin densities at sites 1 and 2 (Figure l), and P are the paths between the radical sites. The multiple regression coefficient for the correlation of the spin coupling with the singlet-triplet gaps for each of the sets is given in Table 111. Table I11 also provides the regression coefficient for all 26 molecules treated as a single set. Except for set 2, all the sets show very good correlation, and the regression coefficient for the complete set is also remarkable, considering the varieties of systems considered. The singlet-triplet gap AsT for the complete set can be fit by AST (kcal/mol) = 0.193 112.6SC
+
where SC represents the spin coupling calculated by using eq 1. It must be noted here that set 5 as well as molecules like 7 and 8 in set 2 gave a clue to the fact that the spin couplings are controlled to a larger extent by the number of intervening uelectrons rather than the topological distance between the spin sites. More detailed algorithms where all pairwise spin couplings were considered weighted with the inverse of the number of intervening r-electrons produced less satisfactory correlation with ASP
The optimized geometries of the singlet and triplet states of all molecules were compared (in terms of the minimum average deviation of the atomic positions) using the display options of the PCMODEL molecular mechanics program.” The geometries of all molecules except 7 and 15 showed an average deviation of less than 0.03 A, whereas the average deviation was 0.30 A for 7 and 0.06 A for 15. Therefore, we considered correlation of the calculated spin-coupling parameters to the singlet-triplet gaps excluding these molecules. In Table 111, we have shown in parentheses the resulting regression coefficients obtained for set 2 and set 4 as well as for the complete set. The melation decreases slightly for set 2 but increases drastically for set 4 as well as the complete set. A qualitative explanation for the SC dependence on the number of intervening ?r-elcctrons seen above m y be formulated as follows. The spin coupling over a pathway of several u-electrons can be viewed as arising out of a series of kinetic-exchange interactions14 between near-neighbor sites. The kinetic exchange is directly proportional to the square of the transfer integrals and inversely proportional to the on-site Coulomb repulsion energy. A larger number of *-electrons on the path could decrease the transfer integral since the nuclear charges on the path are screened more effectively. This should lead to a net decrease in the exchange interaction. It cannot be discounted that in the examples we have considered, the change in the number of u-electrons at a site is d a t e d with oxygen atom substitution of the carbon atom and, hence, the moditication of the transfer integral as well as the on-site Coulomb repulsion. Therefore, delineating the reasons for the observed effect of intervening r-electrons on the spin coupling is complicated.
Conclusion We have considered in this study several specifically chosen biradial non-Kekulb *-systems to understand the way in which the unpaired electrons couple and the effect of topology and the *-electron network on the magnitude of the spin coupling. The
The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 9235 results reconfirm the sign of coupling that other authors as well as we have proposed and further reveal the dependence of the magnitude of the spin coupling on the number of *-electrons in the coupling path. This leads to a convenient way of predicting semiquantitativelythe singlet-triplet gap in these systems. This should be of considerable interest in experimental studies to determine the ground-state spin. By using the simple approach developed here, experimental spin densities, where available, may be used to calculate the singlet-triplet gap to determine the ground-state spin or to design further experiments to determine the same. As an illustration of this point, we consider the case of tetramethyleneethane. Most theoretical treatment~~.~.~J~ predict a singlet ground state for this molecule, whereas the experimental claims16have been to the contrary. Using the experimental hyperfine spectrum reported in ref 16b, we can calculate a spin density of 0.23 on each of the methylenic carbon atoms of tetramethyleneethane and, using eqs 1 and 2, predict the singlettriplet gap to be -3.2 kcal/mol. This is in general agreement with previous theoretical calculations but is again inconsistent with the experimental observation of triplet signal at low temperatures. A knowledge of the strength of spin interaction in a system reflected in the singlet-triplet gap will be of great use in the design of appropriate candidates as model systems to be explored in the organic ferromagnet research. Of course, for the ferromagnet, one has to design S i t e spin systems, but understanding the spin coupling at a microscopic level can be quite crucial, and studies such as these should prove quite useful.
Acknowledgment. We thank the Council for Scientific and Industrial Research, New Delhi, for financial support through Grant 1(1185)/90-EMR-II and a junior research fellowship for B.L.V.P. Fruitful discussions with Dr. M. Durgaprasad are also gratefully acknowledged. Supplementary Material Available: D and E values for several biradicals calculated using spin densities from AM1 UHF and open-shell RHF/CI procedures compared to the experimental and previous theoretical results (4 pages). Ordering information is given on any current masthead page.
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