Mendeleev Communications Electronic Version, Issue 5, 1999 (pp. 171–212) Systematic estimation of isotropic hyperfine coupling constants for protons in free radicals Nikolai D. Chuvylkin*a and Andrei M. Tokmachevb a N. D. Zelinsky Institute of Organic Chemistry, Russian Academy of Sciences, 117913 Moscow, Russian Federation. Fax: +7 095 135 5328 b Department of Chemistry, M. V. Lomonosov Moscow State University, 119899 Moscow, Russian Federation.Fax: +7 095 939 4575 Modified formalism of consistent quantum-chemical estimations of contact interactions between magnetic moments of an unpaired electron and protons, when the spin populations of atoms are determined in the basis of symmetrically orthogonalised atomic orbitals, is presented and examined using the H2C=CH·a test free radical as an example.Radiospectroscopic investigations provide a wealth of information on the structures of molecular compounds. Thus, a simple EPR technique is widely used for studying radical systems. Experimental EPR spectra are often difficult to interpret.1 Current methods of quantum chemistry can give useful assistance in identifying spectra.2–5 Successful applications of these approaches require reliable techniques for computation of magnetic resonance parameters of radical systems.Bulky radical systems are of current practical interest. Thus, radical probes such as 2,2,6,6-tetramethylpyperidin-N-oxyl and 2-octyl-2,4,5,5-tetramethyl-3-imidazolin-N-oxyl at the surfaces of various catalysts were systematically examined.6 Analogous radical probes were added to protein molecules for investigating their conformational transformations.7 The commonly used ab initio methods even of relatively low levels are usually inapplicable in practice to these bulky systems because of technical difficulties. In the circumstances, alternative quantumchemical approaches to estimate the EPR spectra parameters of free radicals deserve special consideration. The reliable and rapid identification of highly informative EPR spectra is presently associated, as a rule, with the future progress and applications of the density functional theory8–11 or adequate semi-empirical quantum-chemical ZDO approaches,2,3,12,13 which are believed to be theoretically well-founded in the basis of Löwdin symmetrically orthogonalised atomic orbitals ( s).14 In the framework of semi-empirical ZDO approaches, when establishing structure–property correlations, the confusing problems of ascertaining the interrelations between the calculated wave functions and experimental radiospectroscopic parameters inevitably arise.Thus, to relate the isotropic hyperfine coupling (ihfc) constant aZn iso found by EPR to the quantum-chemically calculated spin population rs Zn of a valence s-AO of the particular n-th atom Zn, the following well-known expression is commonly used:2 Over a long period of time it was assumed that the proportionality coefficient K(Z) in this expression was dependent on only the type of the atom regardless of its chemical environment, i.e., of the compound in which it occurred.However, afterwards in calculations of proton ihfc constants by ZDO semi-empirical methods tremendously unlike values of the proportionality coefficient K(H) were suggested.15,16 Keeping this in mind, the opinion was expressed3 that for description of two fundamentally alternate mechanisms of spin density distribution (i.e., polarization and delocalization) the use of two corresponding highly different Ka(H) and Kb(H) values was reasonable.Regardless of the qualitative explanation3 of a need for using two proportionality coefficients Ka(H) and Kb(H), which diminishes the contradiction between recommendations of different authors,15,16 it was necessary to determine theoretically each of the coefficients using an adequate mathematical expression that takes into account the specificity of both mechanisms of spin distribution and conventional quantumchemical approximations.Recently, using model p- and s-electron hydrocarbon fragments as an example, the scheme of consistent quantum-chemical calculations of the proton ihfc constants aH iso for free radicals was considered,12 when the spin populations rs H were determined in the basis of Löwdin symmetrically orthogonalized s.Keeping in mind the fact that in a purely p-electron radical the ihfc with a proton results only from the exchange spin polarization, the expression was deduced in the framework of a valence bond (VB) approach with the inclusion of s–p- configuration interaction: where d(H) is the Hartree–Fock value (508 G) of the proportionality coefficient, and Shs is the overlap integral of a hybrid h-AO of a heavy atom (C) and 1s-AO of an H atom.From equation (2) it follows that for Shs ª 0.75 (typical case) the values of the Ka(H) and d(H) are practically indistinguishable. At the same time, it was shown12 that in the framework of the MO LCAO theory when passing from overlapping AOs to symmetrically orthogonalized s the expression for the ihfc constant aH iso did not undergo any changes in the case of p-electron radicals.At the same time, it was established12 that the ihfc constants for protons in typical s-electron radicals, resulting predominantly from spin delocalization, were to be computed with the proportionality coefficient Kb(H) higher than the Hartree–Fock coefficient because its expression with even reduced account of only the one-electron contribution r0 H into the spin population of hydrogen 1s-AO assumed the substantially modified form in the basis of symmetrically orthogonalised s: For typical values of the overlap integral Shs the proportionality coefficient Kb(H) is within the range 700–800 G.This AO aZn iso = K(Z)rs Zn (1) AO Ka(H) = d(H) (1+S2 hs), 2 3 (2) AO 60 50 40 30 20 10 0 90 100 110 120 130 140 150 160 170 180 1 2 DE/kcal mol–1 qa/° Figure 1 Relative total energies of the vinyl s-electron radical as functions of the angle CCHa: (1) MNDO/UHF, (2) UHF/6-31G**.AO Kb(H) = d(H) 2 . 1 2 1 1 – Shs 1 1 + Shs + (3)Mendeleev Communications Electronic Version, Issue 5, 1999 (pp. 171–212) is consistent with the data of ref. 17 wherein it has been found that the best coincidence of semi-empirically calculated ihfc constants and experimental ones for protons of a number of s-radicals is observed when Kb(H) = 743 G.With the use of the restricted Hartree–Fock (RHF) method, the calculated one-electron spin densities RHFr0( ) are always not negative whereas real physical experiments also detect the negative total spin densities r( ).The appearance of negative r( ) values is owing to the additional polarization contribution rp( ) caused mainly by exchange electronic correlation.2 This correlation suggests that the electrons with different spins become differently distributed in the space. In practical quantum-chemical calculations of magnetic resonance parameters of free radicals with allowance for electron correlation effects, the UHF method is most widely used.In this method, the one-electron (delocalization) and polarization contributions to the spin density are not partitioned.2 Note that because of proper symmetry properties of wave functions the polarization component rp( ) is usually overestimated in UHF calculations. The separation of the UHF spin densities into the delocalization and polarization contributions is possible by the natural orbital analysis within the scope of UHF method.2,18,19 At the same time, a simple relation between the UHF and projected unrestricted Hartree–Fock (PUHF) wave functions is well known.18,19 From these considerations a simple way was found to partition the spin density calculated by the UHF method into contributions from the spin polarization and spin delocalization mechanisms.For doublet radicals, the polarization correction takes the form:18 The delocalization contribution is then approximated as the difference: The validity of this approach was confirmed18 by comparing the delocalization components UHFr0 calculated by equation (5) with those (RHFr0) obtained by the open-shell RHF method.2 The result is that equation (5) becomes: Because of the physical simplicity, equation (6) can be quite useful in systematic analysis of the ihfc constants for protons in bulky radical systems. In the case of the RHF approach (e.g., CNDO/SP variant2) the polarization correction is estimated, as a rule, with the use of the perturbation theory.For convenience, we used expression (6).However, it was taken into account that the polarization component rp( ) of the total spin density can be partitioned into contributions of so-called spin [rsp( )] and exchange [rep( )] polarizations,2 and the component rep( ) is the only one for p-radicals and much more significant than rsp( ) for s-radicals. Consequently, supposing the correction sprs H to be negligible for s-radicals and not entering into a contradiction with the results,12 we have, when estimating the polarization contribution paH iso with equation (1), to multiply the spin population prs H obtained using equation (6) by the Hartree–Fock proportionality coefficient Ka(H) = 508 G.With the aim to test the presented approach to evaluating the ihfc constants for protons, in the framework of the MNDO approximation,20 we consider the vinyl radical, which is commonly regarded as a ‘criterion’.2,3,17,18 This approach based on equations (1)–(3) and (6), the standard MNDO parametrization, and the Dewar half-electron method21 is below referred to as MNDORU.When the geometry of free radicals is inadequately known, the magnetic-resonance parameters are evaluated by quantumchemical methods with great errors.3 Unfortunately, experimental data on the geometry of free radicals are either absent at all or often ambiguous.2 At the same time, because of the high sensitivity of calculated magnetic-resonance characteristics to changes in the structural parameters and the ability of NDO approaches to interpret radiospectroscopic data with acceptable accuracy, if the supposed structures of free radicals are close to the real ones, these methods make it possible to predict the electronic structure and geometry of free radicals more exactly than an energy optimization procedure.3 Like the majority of other radicals, the geometry parameters of a vinyl radical were not determined experimentally.Table 1 contains these parameters computed by us using both semiempirical MNDO and ab initio methods in the 6-31G** basis. The two sets of structural characteristics are similar; however, the data for the CCHa angle are dramatically different.The procedure of energy variation can reliably determine the gasphase geometries of free radicals only at a level of nonempirical calculations in extended basis sets in quality comparable to the 6-31G** basis.Therefore, we can believe that real structural parameters of the vinyl radical are closer to the ab initio calculated data (Table 1). Table 2 includes, together with the experimental aH iso values, the delocalization (0aH iso) and spin-polarization (paH iso) contributions and total (aH iso = 0aH iso + paH iso) ihfc constants estimated according to the above scheme for the specified (ab initio optimised) radical geometry.As can be seen, the MNDO/RHF method did not reproduce the recorded ihfc constants for the protons in the vinyl radical even qualitatively. After taking into account the polarization contribution, the great discrepancies between theoretical and experimental values for cis- and transprotons became much lower, but for the proton Ha calculated ihfc constant was found to have a wrong sign.It is problematic that the reversed sign [aH iso(a) = –1.3 G] was also obtained in ab initio calculations, whereas the INDO and CNDO/SP methods led to a satisfactory quantitative description of this constant [aH iso(a) = 10.7 G and aH iso(a) = 13.4 G, respectively].22 The contradiction with the experiment can be removed by a change in geometry parameters.Taking into account the data in Table 1, it is reasonable to remove the strong discrepancy between the calculated and experimental values by variation of the valence angle CCHa º qa keeping in mind the pronounced angular dependence of the calculated ihfc constant for the a-proton in the vinyl radical.2,3,17 The remaining (more ‘rigid’) structural parameters may be fixed in the course of this variation.Relative total energies of the vinyl radical calculated by MNDO/UHF and UHF/6-31G** methods are plotted in Figure 1 as functions of angle qa. As can be seen in Figure 1 (curve 1), DE remains practically constant over a wide range of qa. Hence, it is reasonable to accept the degree of agreement of theoretical r r r r r Table 1 Bond lengths (Å) and angles (°) in the vinyl radical, optimised by semi-empirical and ab initio methods. Parameter Method MNDO 6-31G** CHa 1.049 1.070 CHcis 1.091 1.075 CHtrans 1.091 1.078 CC 1.307 1.341 CCHa 180.0 135.6 CCHcis 122.9 121.4 CCHtrans 122.9 122.0 UHFrp(r) = [UHFr(r) – PUHFr(r)] 3 2 (4) UHFr0(r) = UHFr(r) – UHFrp(r) (5) UHFr(r) – RHFr0(r) = UHFrp(r) (6) r r r r r aEquations (1)–(3).Table 2 Experimental data23 and theoretical quantitiesa estimated for ab initio optimised radical geometry. Nucleus Shs K(H) MNDORU Exp. 0aH iso paH iso aH iso aH iso Ha 0.661 791 28.3 –31.4 –3.1 13.4 Hcis 0.648 773 18.3 12.2 30.6 37.0 Htrans 0.649 774 52.6 9.2 61.8 65.0 C C Htrans Hcis HaMendeleev Communications Electronic Version, Issue 5, 1999 (pp. 171–212) and experimental magnetic-resonance parameters for one or another radical geometry as an adequate test of authenticity of the vinyl radical geometry. Figure 2 depicts the calculated angular dependence of oneelectron (curve 1) and spin-polarization (curve 2) components and the total ihfc constant (curve 3) for the vinyl s-radical. Curve 3 in Figure 2 decays monotonically as the varied angle increases, whereupon a fair agreement with the experiment is achieved for qa ª 110–125° rather than qa ª 135°. The former value is energy-acceptable (Figure 1) and permissible from the point of view of computing errors of the ab initio method24 and a possible matrix effect on the constant under discussion.Anyhow, the qa angle estimated roughly in this manner proved to be much closer to the ab initio calculated one than the qa angle optimised by the MNDO/UHF method.The evaluated ihfc constants for the cis- and trans-protons distant from the radical centre in the vinyl radical range from 23.8 to 27.1 G and from 62.1 to 61.8 G at qa ª 110–125°, respectively. Therefore, with the account of spin polarization in the MNDORU approximation, all above experimental values of ihfc constants can be, in principle, reproduced with the use of geometry parameters of the vinyl radical, which are different only slightly from those fully optimised by us with the UHF/6-31G** method and accepted as authentic (Table 1).The same has been recently demonstrated13 by the results of similar MNDORU calculations carried out for a test representative sample of 17 p- and s-electron free radicals.Thus, the modified scheme12 of logically consistent quantumchemical calculation of ihfc constants for protons in free radicals when the spin populations of atoms are determined in the basis of Löwdin symmetrically orthogonalised s quite naturally removes the seeming contradiction in the applications15,16 of two tremendously different proportionality coefficients in equation (1) for estimating these constants by most commonly used semi-empirical methods of quantum chemistry.We can also conclude that in calculations of ihfc constants for protons even in s-electron radicals the spin-polarization effects must be borne in mind. Reliable determinations of geometry parameters of free radical systems by modern quantum-chemical methods require not only findings of a minimum on the potential energy surface but also concurrent comparisons of theoretical and experimental radiospectroscopic characteristics.References 1 Yu. V. Rakitin, G. M. Larin and V. V. Minin, Interpretatsiya spektrov EPR koordinatsionnykh soedinenii (Interpretation of EPR spectra of coordination compounds), Nauka, Moscow, 1993, p. 399 (in Russian). 2 G. M. Zhidomirov, P. V. Schastnev and N. D. Chuvylkin, Kvantovokhimicheskie raschety magnitno-rezonansnykh parametrov (Quantumchemical calculations of magnetic resonance parameters), Nauka, Novosibirsk, 1978, p. 368 (in Russian). 3 N. D. Chuvylkin, I. Yu. Shchapin, V. L. Klochikhin, V. A. Tikhomirov and V. I. Feldman, Vestn.Mosk. Univ., Ser. 2, Khim., 1992, 33, 307 (in Russian). 4 N. D. Chuvylkin, A. M. Tokmachev, A. V. Fionov and E. V. Lunina, Izv. Akad. Nauk, Ser. Khim., 1997, 1743 (Russ. Chem. Bull., 1997, 46, 1649). 5 N. D. Chuvylkin, A. M. Tokmachev, A. V. Fionov and E. V. Lunina, Mendeleev Commun., 1997, 191. 6 E. V. Lunina, in Kataliz (Catalysis), eds. O. A. Petrii and V. V. Lunin, Izdatel’stvo MGU, Moscow, 1987, p. 287 (in Russian). 7 E. G. Rosantsev and V. D. Sholle, Organicheskaya khimiya svobodnykh radikalov (Organic chemistry of free radicals), Khimiya, Moscow, 1977, p. 344 (in Russian). 8 C. Adamo, V. Barone and A. Fortunelli, J. Chem. Phys., 1995, 102, 384. 9 J. M. Martell, R. J. Boyd and L. A. Ericsson, J. Phys. Chem., 1995, 99, 623. 10 P. J. O’Malley and D. A. Ellson, Chem.Phys. Lett., 1996, 260, 492. 11 P. J. O’Malley, Chem. Phys. Lett., 1996, 262, 797. 12 N. D. Chuvylkin and A. M. Tokmachev, Izv. Akad. Nauk, Ser. Khim., 1999, 245 (Russ. Chem. Bull., 1999, 48, 245). 13 N. D. Chuvylkin and A. M. Tokmachev, Izv. Akad. Nauk, Ser. Khim., 1999, 1459 (in Russian). 14 R. D. Brown and K. R. Roby, Theor. Chim. Acta, 1970, 16, 175. 15 P. Bischof and G. Friedrich, J. 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Pople, Ab initio Molecular Orbital Theory, WIP, New York, 1986. 60 50 40 30 20 10 0 –10 –20 –30 –40 100 110 120 130 140 150 160 170 180 qa/° aiso(a)/G 1 2 3 Figure 2 Delocalization (1) and spin-polarization (2) contributions and the total (3) ihfc constant for the a-proton in the vinyl s-electron radical as functions of the angle CCHa. AO Received: 5th January 1999; Com. 99/1421