F(ac,t) = (constant)u2(ct,t), (15)

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1 VOL. 54, 1965 CHEMISTRY: GREEN AND MALRIEU 659 a solution of which is F(ac,t) = (constant)u2(ct,t), (15) and where u(a,t) must satisfy ii(a,t) = M(a,t)u(a,t). (16) Imposing the boundary condition that the space is asymptotically flat, lim F(at,t) = lim u2(ca,t) = 1, (17) to-axc t-b- )c let us write the integral equation which formally solves the problem as u(a,t) = uo(a) + 4 f (t - t')m(a,t')u(a,t')dt', (18) which can be solved by an iterative process. Thus the line element (11) can be determined for any specified density function since the functions u(a,t) can be found by iteration in equation (18) and then the components of the metric tensor determined by equation (15). The application of this solution to a cosmological model consisting only of clusters of galaxies is obvious. It is hoped that the computations will prove tractable. 1 Dingle, H., these PROCEEDINGS, 19, 559 (1933). QUANTUM CHEMICAL STUDIES OF CHARGE-TRANSFER COMPLEXES OF INDOLES* BY JACK PETER GREENt AND J. P. MALRIEU INSTITUT DE BIOLOGIE PHYSICO-CHIMIQUE, PARIS Communicated by Albert Szent-Gyorgyi, July 6, 1965 The high capacity of indoles to form charge-transfer complexes with various electron acceptors is well established'- but not fully understood. The capacity of a compound to donate electrons is generally associated with a low ionization potential, reflected in a low-energy coefficient of the highest occupied molecular orbital:'-' indoles are better donors than would be expected from this coefficient.8 It was suggested by Szent-Gy6rgyi and associates' that the ir electrons of indoles may be transferred in a "local" interaction probably at C-2 or C-3 (or both), which have high electron densities.8 Recently, measurements were made of the capacities of a group of methylated indoles to form charge-transfer complexes with a series of electron acceptors.5 Molecular orbital calculations were carried out in an attempt to find a relationship between the electronic structure of the indoles and their abilities to form charge-transfer complexes. Attention was focused on atom-localized indexes, some of which have been correlated with the formation of complexes between aromatic hydrocarbons and silver.10 The results of this work are presented here. Method8.-The energy of the highest occupied and lowest empty orbitals, electron density, bond order, and free valence were calculated by methods described;8 the significance of these

2 660 CHEMISTRY: GREEN AND MALRIEU PROC. N. A. S. values has been discussed.8 Frontier electron density, superdelocalizability, and approximate superdelocalizability, which are derived from hyperconjugation energy of the transition state," were calculated as described by Fukui and his associates.'2 The frontier electron theory, which is controversial,'3-'7 has yielded indexes that correlate with chemical activities.'8 According to this theory," the frontier electrons for an electrophilic reaction, which is the relevant reaction here, are the two 7r electrons occupying the highest molecular orbital in the ground state, and the density of the frontier electrons indicates the reactivity of different portions of the same molecule. Superdelocalizability is a second-order perturbation term indicating both the comparative chemical reactivities of different molecules and the stabilization energy in the formation of a complex with another molecule; for an electrophilic reaction, it is large when the electron density in the highest occupied orbital is large. The approximate superdelocalizability reflects the degree of contribution of the frontier orbitals to the superdelocalizability: if the frontier electrons play a more significant role in the course of reaction than do the other,r electrons, the approximate superdelocalizability would be a better index of reactivity than is superdelocalizability;" for the latter, by including the other electrons, can obscure the contribution of the frontier electrons. These three indexes are relatively simple to calculate and are formally similar to other indexes that have been described: frontier electron density to Brown's Z value, superdelocalizability to Wheland's localization energy, and approximate superdelocalizability to Dewar's approximate localization energy."' 18 Recent theoretical studies23 have demonstrated that superdelocalizability is involved in the calculation of localized charge-transfer energies. One may show that the second-order perturbation energy due to charge transfer is given by the summation over all the charge-transfer states, <0juj j>2, ije-eo where 0> represents the ground state of the molecules, u is the interaction Hamiltonian between the two molecules, and Ei - Eo is the difference between the energies of the i> and 0> states. The integrals <0 u i> may be developed in terms of molecular orbitals and then in terms of atomic orbitals. If the interaction is localized to the rth atom of the donor molecule, the sum reduces to (cr' <e> + <j>)2 iei - o where <e> and <j> are sums of coulombic and nuclear integrals, which depend only on the geometry of the complex and on the electronic properties of the acceptor molecule; the summation is over the occupied orbitals of the donor molecule. The first term of the summation gives (cr')2,ej - Eo which is closely related to superdelocalizability. These conclusions are independent of the Huckel method and of the hyperconjugation perturbation in the transition state. The Huckel method was employed with the parameters described,8 except that a value of was used for a H3 instead of for kcaliph, as appears by typographical error.8 A purely inductive model was used, as well as the hyperconjugative-inductive model of the methyl group8 (henceforth called the hyperconjugative model). For the inductive model, the value a, = -0.2 was assigned to the atom bearing the methyl group. The choice of the conventional tautomer"9 of indole, in which the proton is on the nitrogen, was determined both by the need for consistency (since compounds with a substituent on the nitrogen atom were analyzed) and by calculations showing that in indole, the selected tautomer has a higher resonance energy (3.39,) and a higher energy of the highest occupied molecular orbital (0.534 A) than the other tautomer, which has values of 2.62 j3 and 0.705,, respectively. Results.-Table 1 shows the maximum of the charge-transfer band obtained in the reaction between the indoles and chloranil, along with the calculated energies

VOL. 54, 1965 CHEMISTRY: GREEN AND MALRIEU 661 TABLE 1 CHARGE-TRANSFER COMPLEXES AND ENERGIES OF THE HIGHEST OCCUPIED MOLECULAR ORBITALS OF INDOLES Maximum of charge- Energy of Highest Occupied

3 VOL. 54, 1965 CHEMISTRY: GREEN AND MALRIEU 661 TABLE 1 CHARGE-TRANSFER COMPLEXES AND ENERGIES OF THE HIGHEST OCCUPIED MOLECULAR ORBITALS OF INDOLES Maximum of charge- Energy of Highest Occupied Molecular Orbital transfer band with chloranil (6 units) Derivative of indole (cm-, X 10-3)* Hyp.t Ind.t 2,3-Dimethyl 15.15K(" 0, ,2-Dimethyl ,5-Dimethyl Methyl Methyl Methyl Unsubstituted indole t * These observations, which appear in graphical form,5 were kindly sent to us by Foster. On reexamination of his results, he has concluded in a personal communication that indole is the weakest donor in this series. t The usual hyperconj ugative-inductive model was used. The inductive model. TABLE 2 CHARGE-TRANSFER COMPLEXES AND LOCALIZED ELECTRONIC INDEXES OF INDOLES Reactivity Index at C-3 of Indole Ring Total Charge in Frontier Electron Approximate Derivative Ground State Density* Superdelocalizability* Superdelocalizability* of indole Hyp. Ind. Hyp. Ind. Hyp. Ind. Hyp. Ind. 2,3-Dimethyl ,2-Dimethyl ,5-Dimethyl Methyl Methyl Methyl Unsubstituted indole * For an electrophilic reaction. of the highest occupied orbitals; Table 2 shows the relevant reactivity indexes at C-3. Other indexes at C-3 (charges of the positive ion, bond order, free valence) showed no correlation with activity, nor did any index on the other atoms. The energy of the highest occupied molecular orbital did not correlate exactly with donor ability (Table 1): 3-methylindole, in both the hyperconjugative and inductive model, and 1-methylindole in the hyperconjugative models are poorer donors than the calculated energies would indicate. The total charge on C-3 showed no correlation with donor capacity (Table 2). In all indoles, the frontier electron density for an electrophilic reaction was highest on C-3, as has been shown previously for unsubstituted indole.20 The density at this position did not parallel the donor capacities of the indoles. Donor capacity was, however, correlated with superdelocalizability for an electrophilic reaction at C-3, in the inductive model; and with approximate superdelocalizability for an electrophilic reaction at this same position in both the inductive and hyperconjugative models. In all indoles both indexes were highest at C-3 (Table 2). Superdelocalizability of the bonds between adjoining atoms, including that of C-2 and C-3, and the sums of superdelocalizability of atoms in each ring system or of adjoining atoms, including those of C-2 and C-3, showed no correlation with donor capacities. Discussion.-Although the energies of the highest occupied molecular orbitals almost paralleled the donor capacity of the indoles (a correlation as good as those usually obtained in such studies), the reactivity indexes at a single atom, C-3, better

$662 CHEMISTRY: GREEN AN\D MALRIEU PROC. N. A. S. reflected donor capacity.$

4 662 CHEMISTRY: GREEN AN\D MALRIEU PROC. N. A. S. reflected donor capacity. This relationship obtained with the approximate superdelocalizability in both the hyperconjugative and inductive models of the methyl group, and with superdelocalizability only in the inductive model. There is no a priori reason for preferring either model.9' 21, 22 The correlation of donor ability with these indexes, which have been demonstrated to be relevant for localized charge-transfer processes in the theoretical study23 quoted iil Methods, supports the ideal that charge transfer takes place in a local interaction. In the compounds studied here, the contribution to the second-order perturbation energy would derive mainly from C-3. The presence of a methyl group at C-3 does not prevent an electrophilic substitution at C-3,24 and hence would not prevent a charge-transfer reaction at C-3. Crystallographic studies show that the plane-to-plane superimposition of two chemical species in a charge-transfer complex may be distorted so that the planes are no longer parallel.25 Or, more likely, as has been shown especially in complexes with chloranil, parallelism obtains but the two species are oriented so that only parts of the molecules overlap.26 In either event, the geometry could favor the greater contribution of a single atom to the stabilization energy. With indoles, this atom would be C-3. The fact that the charge-transfer complex formed between indole and tetracyanoethylene decomposes into 3-tricyanovinylindole27 supports this suggestion. This hypothesis can be tested both by a more rigorous theoretical analysis in which the acceptor is also considered and by laboratory experiments testing the relative donor capacities of indoles as predicted by the hypothesis. Toward the latter end, calculations were made on a series of indoles which, to our knowledge, have not been studied as electron donors. In Table 3 the compounds are listed in order of predicted donor ability based on the calculation of approximate superdelocalizability of C-3. By this index, in both the hyperconjugative and inductive models, the predicted order is: 2,6-dimethyl- > 2,7-dimethyl- > = 3,7- dimethyl- > 1,6-dimethyl- > 1,7-dimethyl- > 4, 7-dimethylindole. Superdelocalizability (inductive model) yields an identical order, except that it suggests that the donor capacities of 2,7-dimethyl- and 3,7-dimethylindole might be experimentally discernible, whereas those of 1,6-dimethyl- and 1,7-dimethylindole should probably not: 2,6-dimethyl- > 2,7-dimethyl- > 3,7-dimethyl- > 1,6-dimethyl- > = 1,7- dimethyl- > 4,7-dimethylindole. Significantly different are the predictions based on calculation of the energy of the highest occupied molecular orbital: 3,7-dimethyl- > 4,7-dimethyl- > = 1,7-dimethyl- > = 1,6-dimethyl- > = 2,6-dimethyl- > 2,7- dimethylindole. It has been pointed out28 that for maisy biologically active compounds, only tenu- TABLE 3 SOME QUANTUM CHEMICAL CALCULATIONS OF INDOLES FOR THE PREDICTION OF DONOR ABILITY Energy of Highest.- Reactivity Index of C-3 of Indole Ring- Occupied Molecular Orbital Approximate (#-units) Superdelocalizability Superdelocalizability Derivative Hyp. Ind. Hyp. Ind. Hyp. Ind. 2,6-Dimethyl ,7-Dimethyl ,7-Dimethyl ,6-Dimethyl ,7-Dimethyl ,7-Dimethyl (

VOL. 54, 1965 CHEMISTRY: GREEN AND MALRIEU 663 ous evidence has been offered to support the idea that they form charge-transfer complexes. With in doles, this evidence is unequivocal.

5 VOL. 54, 1965 CHEMISTRY: GREEN AND MALRIEU 663 ous evidence has been offered to support the idea that they form charge-transfer complexes. With in doles, this evidence is unequivocal. Charge-transfer reactions may be important in biological systems7' 29, 31 and some of the biological activities of indoles32-3 may depend on their capacities to form such molecular complexes. In most biologically active indoles, the C-3 atom is linked to an ethylamino group, which should impede contact36 between the electron acceptor and the C-3 atom unless the ethylamino group is fixed in vivo. A study of biologically active indoles by molecular orbital methods should reveal the contribution to biological activity of the various atoms, including C-3. In view of the diverse actions of indoles and their differing orders of activities at different biological sites, it would be puzzling if one index, such as energy of the highest occupied molecular orbital, or any index at any single atom could correlate with all the activities. M\ore likely, the configuration of biological receptors at each site differs, and hence the type of index and/or the site(s) of the reactive atom(s) in the indole probably differs at different sites. The specificity of the action is further indicated by the fact that many substances with demonstrated capacity to form charge-transfer complexes do not have the biological actions of indoles. It is conceivable that the energy of the highest occupied molecular orbital influences all the biological effects of indoles by determining the closeness of approach of the indole to the receptor and thus facilitates specific atomic interactions. A quantum mechanical approach has given added insight into the molecular forces by which cells bind 5-hydroxytryptamine and other biogenic amines.37 Such an approach can help to delineate the molecular characteristics that determine specific activities and, while rationalizing the presently chaotic relationship between structure and biological activity, also permit inferences about the chemical nature of the biological receptor. Summary.-The capacity of a series of indoles to form charge-transfer complexes was not exactly correlated with the energies of the highest occupied molecular orbitals, but was instead correlated with electrophilic superdelocalizability of the C-3 atom. These results support the idea that this atom is involved in a localized charge-transfer complex, and this possibility is discussed. Some biological implications of this work are also suggested. Note added in proof: C. R. Merril, S. H. Snyder, and D. Bradley have obtained results similar to those presented here (Nature, in press). The authors are grateful to B. Pullman for encouragement and advice, and to R. Astic for help in the calculations. * Supported by research grants from the National Institute of General Medical Sciences (GM ), the American Heart Association (64-G-217), and the Eleanor Roosevelt Cancer Foundation (International Union Against Cancer). t Holder of a Research Career Program Award (2-K3-GM ) from the National Institute of General Medical Sciences. Present address: Department of Pharmacology, Cornell University Medical College, New York 21, N. Y. 1 Szent-Gyorgyi, A., I. Isenberg, and J. McLaughlin, these PROCEEDINGS, 47, 1089 (1961). 2 Fujimori, E., these PROCEEDINGS, 45, 133 (1959). 3 Harbury, H. A., K. F. Lanoue, P. A. Loach, and R. M. Amick, these PROCEEDINGS, 45, 1708 (1959). 4 Cilento, G., and P. Tedeschi, J. Biol. Chem., 236, 907 (1961}> 5 Foster, R., and P. Hanson, Trans. Faraday Soc., 60, 2189 (1964). 6 Shifrin, S., Biochemistry, 3, 829 (1964).

664 CHEMISTRY: BECK AND ORGEL PROC. N. A. S. 7Mulliken, R. S., J. Am. Chem. Soc., 74, 811 (1952). 8Pullman, B., and A. Pullman, Quantum Biochemistry (New York: Interscience Publishers, 1963).

6 664 CHEMISTRY: BECK AND ORGEL PROC. N. A. S. 7Mulliken, R. S., J. Am. Chem. Soc., 74, 811 (1952). 8Pullman, B., and A. Pullman, Quantum Biochemistry (New York: Interscience Publishers, 1963). 9 Streitweiser, Jr., A., Molecular Orbital Theory for Organic Chemists (New York: John Wiley and Sons, 1962). lo Fukui, K., A. Imamura, T. Yonezawa, and C. Nagata, Bull. Chem. Soc. Japan, 34, 1076 (1961). 11Fukui, K., T. Yonezawa, and C. Nagata, J. Chem. Phys., 26, 831 (1957). 12Ibid., 27, 1247 (1957). 13Pullman, B., J. Chem. Phys., 31, 551 (1959). 14 Greenwood, H. H., J. Chem. Phys., 31, 552 (1959). 15Sung, S. S., 0. Chalvet, and R. Daubel, J. Chem. Phys., 31, 553 (1959). 16 Dewar, M. J. S., Advan. Chem. Phys., 8, 65 (1964). 17 Fukui, K., T. Yonezawa, and C. Nagata, J. Chem. Phys., 31, 550 (1959). 18 Fukui, K., in Molecular Orbitals in Chemistry, Physics, and Biology, ed. P. O. Lowdin and B. Pullman (New York: Academic Press, 1964), p Sumpter, W. C., and J. M. Miller, Heterocyclic Compounds with Indole and Carboxole Systems (New York: Interscience Publishers, 1954). 20 Fukui, K., T. Yonezawa, C. Nagata, and H. Shingu, J. Chem. Phys., 22, 1433 (1954). 21Schubert, W. M., R. B. Murphy, and J. Robins, Tetrahedron, 17, 199 (1962). 22 Dewar, M. J. S., Hyperconjugation (New York: Ronald Press, 1962). 23 Claverie, P., J. P. Malrieu, and S. Diner, in preparation. 24Hinman, R. L., and E. R. Shull, J. Org. Chem., 26, 2339 (1961). 25 Harding, T. T., and S. C. Wallwork, Acta Cryst., 8, 787 (1955). 26 Wallwork, S. C., J. Chem. Soc., 494 (1961). 27 Foster, R., and P. Hanson, Tetrahedron, 21, 255 (1965). 28 Pullman, B., Abstracts, 6th International Congress of Biochemistry, 10, 1 (1964). 29 Pullman, B., and A. Pullman, these PROCEEDINGS, 44, 1197 (1958). 3 Karreman, G., I. Isenberg, and A. Szent-Gyorgyi, Science, 130, 1191 (1959). 31 Szent-Gyorgyi, A., An Introduction to a Submolecular Biology (New York: Academic Press, 1960). 32 Kaminer, B., Biochim. Biophys. Acta, 56, 14 (1962). 33 Kanner, L. C., and L. M. Kozloff, Biochemistry, 3, 215 (1964). 34 Merril, C. R., and S. H. Snyder, in preparation. 35 Snyder, S. H., and C. R. Merril, these PROCEEDINGS, 54, 258 (1965). 36 Merrifield, R. E., and W. D. Phillips, J. Am. Chem. Soc., 80, 2778 (1958). 37 Green, J. P., in Mechanisms of Release of Biogenic Amines, ed. B. Uvnas (London: Pergamon Press, 1965). THE FORMATION OF CONDENSED PHOSPHATE IN AQUEOUS SOLUTION BY A. BECK AND L. E. ORGEL THE SALK INSTITUTE FOR BIOLOGICAL STUDIES, LA JOLLA, CALIFORNIA Communicated by Renato Dulbecco, June 30, 1966 In view of the central role of adenosine triphosphate and related compounds in metabolism, the abiological synthesis of condensed phosphates is relevant to the problem of the origins of life.'-3 Here we describe experiments on the condensation of phosphate to pyrophosphate and of linear triphosphate to trimetaphosphate in aqueous solution. We have used three condensing agents-potassium cyanate,

Proceedings of the NATIONAL ACADEMY OF SCIENCES. (A Commentary)... E. C. Slater and W. C. Hulsnann Commentary... J. S.

Proceedings of the NATIONAL ACADEMY OF SCIENCES Volume 47 * Number 8 * August 15, 1961 SYMPOSIUM ON ENERGY-LINKED BIOLOGICAL REACTIONS BY INVITATION OF THE COMMITTEE ON ARRANGEMENTS FOR THE AUTUMN MEETING