Russian Chemical Reviews 70 (1) 1 ± 22 (2001) Correlation analysis in the chemistry of free radicals A R Cherkasov,MJonsson, V I Galkin, R A Cherkasov Contents I. Introduction II. Correlation analysis for reactions involving free radicals III. The use of r72 analysis in free-radical chemistry IV. Conclusion Abstract. correlation of use the on available now data Published Published data now available on the use of correlation analysis systematically. discussed are chemistry free-radical in analysis in free-radical chemistry are discussed systematically. The The scales previously proposed substituents of of scales of `radical' `radical' s-constants -constants of substituents proposed previously are as applicable is them of none that shown is It analysed. are analysed.It is shown that none of them is applicable as a general scale because almost in all cases, it is impossible to general scale because almost in all cases, it is impossible to separate contributions polar and radical proper the correctly separate correctly the proper radical and polar contributions to to the the to approach new A substituents. of effect overall the overall effect of substituents. A new approach to the quanti- quanti- tative structure the between relationship the of estimation tative estimation of the relationship between the structure and and reactivity processes free-radical in molecules of reactivity of molecules in free-radical processes called called r72-analysis is 238 includes bibliography The proposed. is proposed.The bibliography includes 238 references. references. I. Introduction Correlation analysis is a widely used mathematical formalisation of the chemical similarity principle. The foundations of the analysis were laid by Hammett, who discovered a correlation between the rate constants (k) or equili- brium constants (K) of reactions involving benzene derivatives and the ionisation constants of benzoic acids 1 kH log kX à rlog KX à rs, KH where the subscripts X and H refer to substituted and unsubsti- tuted derivatives and r is the reaction constant for the given series. The reaction series of dissociation of benzoic acids in aqueous solutions at 25 8C was chosen as the reference series, described by r=1, and the ratio logKX , KH was introduced as the s-constant, reflecting the influence of the substituent X in a set of similar reaction series.This empirical approach, which is also known as sr-analysis, became one of the first quantitative methods for describing the structure ± reactivity relationship for compounds and has served as the basis for the A R Cherkasov, V I Galkin, R A Cherkasov Department of Chemistry, Kazan State University, ul. Kremlevskaya 18, 420008 Kazan, Russian Federation. Fax (7-843) 231 54 16. Tel. (7-843) 231 54 16. E-mail: vladimir.galkin@ksu.ru (V I Galkin); rafael.cherkasov@ksu.ru (R A Cherkasov) MJonsson Department of Chemistry, Royal Institute of Technology, Teknikringen 56, S-100 44 Stockholm, Sweden. Fax (46-8) 790 87 72. Tel.(46-8) 790 91 23. E-mail: matsj@nuchem.kth.se Received 2 February 2000 Uspekhi Khimii 70 (1) 3 ± 27 (2001); translated by Z P Bobkova #2001 Russian Academy of Sciences and Turpion Ltd DOI 10.1070/RC2001v070n01ABEH000574 12 11 20 subsequent development of the general formal theory of the influence of substituents. The principles of linear free energy (LFE) and polylinearity (PL) provide the physical grounds for this approach, according to which the increment of the Gibbs energy change (DG) in a reaction or of the activation energy caused by the introduction of a substituent is represented as the additive sum dDG=dDGI+dDGS+dDGR , each of the terms reflecting a particular type of substituent effects, namely, inductive (I), steric (S) or resonance (R) effect. The multitude of inductive, steric and resonance constants determined from diverse experimental sources, together with some specific constants (for example, `phosphorus' or `silicon' constants) form dozens of scales which are widely used to describe the influence of substituents on the reactivity at the group level of additivity.Various aspects of the determination and versatile practical application of steric and inductive constants of groups and the concept of electronegativity tightly connected to them have been described in detail in our previous reviews.2± 5 It should only be noted that modern correlation analysis has long overstepped the limits of a purely empirical approach. The group constants of substituents have provided the experimental basis for the physical interpretation and for the development of various quantitative models describing the steric and electronic effects.Certainly, some key problems of the formal theory of inter- action and correlation analysis are still to be ultimately solved. For example, the nature of the inductive effect and the mechanism of its transfer remain one of the most debatable and topical problems of physical chemistry 2 (the state-of-the-art in this line of research has been discussed comprehensively in recent review publications 5 ±8). Nevertheless, it can be stated with confidence that the theoretical grounds of modern correlation analysis provide in general a clear picture and give good grounds for the extensive use of the substituent constants for description of the structure ± property relationships in the chemistry of organic and heteroorganic compounds.Perhaps, the only field of chemistry in which the use of correlation analysis has not yet provided good results is free-radical chemistry. The application of LFE and PL principles in this field of chemistry is still limited to a rather narrow range of compounds (mainly, aromatics). In this review, the current state of the use of correlation analysis for radical reactions is considered for the first time. In addition, we thought it necessary to present the generalising results obtained by the new approach to the estimation of the structure ± reactivity relationship in free-radical reactions which we have developed and called `r72 analysis'.2II.Correlation analysis for reactions involving free radicals 1. Methods for the description of the effects of substituents in homolytic processes based on kinetic parameters. s -Scales Despite the fact that the applicability of the Hammett equation to homolytic processes is sometimes debated,9 there exist quite a few examples of successful use of various polar constants for the description of free-radical reactions.10 ¡¾ 28 In addition to the proper Hammett s-constants,18 ¡¾ 29 which reflect the overall effect of substituents in aromatic systems, so- called dual s+ and s7 parameters of substituents, which imply direct polar (mesomeric) interaction of +M or 7M substituents with the reaction centre, are also actively used in the analysis of homolytic processes.The scale of s+-constants proposed by Brown and Okamoto,30 which is based on the rates of solvolysis of para- and meta-substituted cumyl chlorides, is believed to be more adequate in describing the properties of aromatic radicals than the scale of Hammett s-constants.31 In our opinion, the use of s+-constants is really more effective as far as the properties of radical cations are concerned but this is not that obvious for neutral radicals. Never- theless, radical bromination of substituted toluenes with bro- mine,24 N-bromosuccinimide 24 and bromotrichloromethane 32 as well as the quantitative interpretation of abstraction of a hydrogen atom by alkoxy radicals,11, 12, 33 chlorine atoms,34 and peroxy radicals 34 have become classical examples of successful use of the s+ constants in free-radical chemistry.Experimental results obtained by one of the authors of this review have confirmed the exceptionally high efficiency of the s+ constants in correlations with single-electron reduction potentials of various aromatic molecules and radicals.35 ¡¾ 40 Other researchers 18, 41 have also considered the Brown ¡¾ Okamoto scale to be the most suitable scale for the description of properties of free radicals; it is the scale of s+constants that has served as the base for the introduction of several specific radical constants of substituents s (they will be considered in detail in the subsequent sections of this review).The s7-constants found from the pK values for ionisation of substituted anilines, N,N-dimethylanilines and phenols 42, 43 have also been used successfully to describe a number of radical processes.29, 44 However, polar constants are finding fairly limited application in free-radical chemistry. The scope of application of particular scales to the description of properties of free radicals remains obscure, identification of statistically significant correla- tions in each successful case being rather a matter of luck than of a systematic approach. Additional difficulties arise when the corre- lation analysis technique is employed to describe the properties of non-aromatic free radicals. The main problem in the quantitative estimation of the effects of substituents in homolytic processes is, undoubtedly, the neces- sity of adequate description of their ability to delocalise the unpaired electron.Currently, this is one of the most topical and ambiguous problems in free-radical chemistry (see, for example, a monograph 45 and references therein). There is a common opinion that both donors and acceptors are capable of stabilising a radical centre;45 this is often used as an argument in favour of the introduction of special radical s constants of substituents, which are expected to describe quanti- tatively the ability of substituents to delocalise spin density. The methods of determination of s -constants can be classified into three main types. The first type includes methods based on the use of kinetic parameters of homolytic processes; the second type comprises physicochemical methods which provide direct infor- mation on the spin density distribution in radicals; and thermody- namic approaches operating with energy characteristics belong to the third type.Below we consider the existing approaches in accordance with this classification. The methods based on the kinetic parameters of homolytic processes are used most widely in determining the s -constants of A R Cherkasov,MJonsson, V I Galkin, R A Cherkasov substituents and are modifications of the classical correlation analysis, based on the use of the extended Hammett equation (1) log k a r1sX a r2s . k0 The essence of this approach is to separate the contribution of the proper radical stabilisation (s ) from the polar effect (sX); hence, it requires selecting a model reaction series in which the latter contribution would be small and could be determined reliably in terms of polar constants.The most widely used reactions involv- ing free radicals include electron transfer, free-radical transfer of a hydrogen atom and radical addition. The kinetic approach to the description of the effects of substituents in radical reactions was first employed by Alfrey and Price 46 in a study of the reactivity of vinylic monomers in copolymerisation. The researchers 46 did not use the classical logarithmic form of Eqn (1); however, the capacity of substituents for stabilisation of radicals was estimated quantitatively for the first time in terms of group constants Q, whereas the polar component was taken into account by the parameter e (the Q constants are given in Table 1) (2) k1 k2 a Q1 Q2 exp¢§e1Oe1 ¢§ e2U.A substantial disadvantage of this approach is the fact that the calculation of Q is based on the analysis of a set of data on copolymerisation, and introduction of new individual values requires a consistent procedure for recalculation of the whole array of Q. Walling et al.59 have critisised the theoretical substantiation of the Alfrey ¡¾ Price approach, which ignores the possibility of resonance stabilisation of a radical transition state. Later, the group parameters ER have been introduced using the relative reactivity of monomers (1/r) in the radical copoly- merisation of aryl-substituted methacrylates 47, 48, 60 (3) log 1r a rs a gER , R, where the g coefficient was taken to be 1, and the parameter ER was interpreted as the resonance component of the substituent effect in free-radical processes. Subsequently,49 this approach was subjected to severe criticism, the legitimacy of the ER and g values being prejudiced.The researchers 49 believed that the nature of ER is rather complex; therefore they proposed adjusted values, En free from the contribution of heterolytic resonance (see Table 1). A similar scale of resonance constants ED has been developed to describe the effects of para-substituents in the addition of trichloromethyl radical to substituted styrenes 53 (4) log kp a rsap OrspU a ED ; kH in this case, the polar effect was taken into account by both the Brown ¡¾ Okamoto constants and the Hammett sp-constants.Correspondingly, two sets of ED parameters for para-substituents (MeO, Cl, CN,NO2) in styrenes were obtained [ED(s) and ED(s+) in Table 1]. The ED constants of these substituents found using the s+ parameters are in good agreement with the Alfrey ¡¾ Price Q constants. A similar approach was used 54 to develop the t scale, which reflects the conjugation effects in radical arylation of substituted benzenes (5) log kp k a rsp a tp , where kp are the rate constants for the arylation into the para- position expressed in terms of the Hammett constants of para- substituents (sp) and the group parameters tp, which reflect the additional delocalisation effects of para-substituents.The r-con- stant appearing in Eqn (5) was determined on the basis of a similar reaction series of meta-arylation, the influence of substituentsTable 1. Constants characterising the radical-stabilisation properties of substituents (data of kinetic studies). R(p) logQ E Substituent see a 0.00 H 0.04 Me 0.13 OMe 0.21 NO2 0.27 CN ± ± ± ± ± ± ± ± 70.25 F 0.01 Cl 0.04 Br 0.07 I ± OH ± C(O)Me P(O)(OEt)2 ± ± ± ± ± ± ± ± ± ± ± 0.18 P(S)(OEt)2 ± ± ± ± ± ± ± ± ± ± ± 0.29 ± ± ± ± ± ± ± ± 0.12 PhSMe ± ± ± ± ± ± ± ± ± (0.49) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± SPhCF3 ± ± ± ± ± ± ± ± ± ± ± 0.08 ± OPh CO2Et ± ± ± ± ± ± ± ± ± ± ± 0.39 CO2Me ± ± ± ± ± ± ± ± ± ± ± 0.35 ± C(O)Ph But ±± ± ± ± ± ± ± ± ± ± ± ± ± 0.15 EtPri ±± SOMe ± SC(O)Me ± S(O)OMe SO2Ph ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± S(O)Ph SO3Me ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± OC(O)Me ± OC(O)Ph ± N=NPh NH2 ± ± ± ± ± ± ± ± ± ± ± 0.69 NMe2 ± SiMe3 ± ± ± ± ± ± ± ± ± ± ± 0.17 CH=CH2 ± ± ± ± ± ± ± ± ± ± ± 0.67 Note.The values which vary in original publications are given in parentheses. a From Ref. 46; b from Refs 47, 48; c from Refs 49 ± 52; d from Ref. 53; e from Ref. 54; f from Ref. 55; g from Refs 50 ± 52; h from Ref. 56; i from Ref. 57; j from Ref. 58. En ER(m) R see c see b see b 0.000 0.00 0.00 70.020 ± 0.03 70.008 ± 0.11 0.410 0.35 0.41 0.230 ± 0.24 0.062 0.08 0.10 0.072 ± 0.12 0.037 ± 0.12 ± ± 0.21 0.240 ± ± 70.147 ± 0.13 ± ± ± 0.014 ± 0.03 0.034 ± 0.03 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 70.110 ± 0.24 s s to tp ED(s+) ED(s) J FM see g see f see e see e see d see d 0.00 0.00 0.00 0.00 0.00 0.00 70.02 0.38(0.39) 0.50 0.09 0.11 0.16 70.12 0.31(0.42) ± 0.14 0.19 0.40 0.73(0.76) 0.27 0.88 0.90 0.27 0.27 (0.41) 0.34 ± ± 0.32 0.33 0.07(0.12) 0.06(0.18) 0.08 0.48 0.16 0.07 0.12 (0.20) ± ± ± ± 0.17 ± ± ± ± ± 0.16 ± ± 0.57 ± ± ± ± ± ± ± ± 0.53 0.39(0.42) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.13 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.18 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.18 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 70.11 ± ± ± ± ± ± ± ± ± ± ± ± 0.33 ± 0.90 ± ±± ± ± ± ± ± 0.535 js ¡ sj n s sC(p) C(m) see i see i see h 0.00 0.00 0.000 70.02(0.03) 0.11 0.170 70.02 0.24 0.250 70.11 0.57 0.440 70.12 0.46 0.260 70.05 70.08 0.065 70.04 0.12 0.065 ± 0.13 0.055 0.10 0.41 0.075 ± ± ± ± ± ± 70.11 70.06 ± 0.46 0.230 70.03 0.43 ± 70.07 ±±±0.01 70.07 ± 1.08 ± ±± 0.90 ±± s s JJ(m) JJ(p) see j see j 0.00 0.00 0.00 0.15 0.10 0.23 0.001 0.36 0.11 0.42 70.02 0.03 70.05 0.22 0.12 0.23 ± ± ± ± ± 0.54 ± ± ± ± ± 0.47 ± 0.62 70.07 70.01 ± ± 0.10 0.33 0.11 0.26 ±± 0.50 ± 0.38 70.10 ± ± ± ± ± 1.00 ± 0.31 ± ±4being entirely ascribed to the polar interaction.The parameters to for ortho-substitution were introduced by a similar procedure.The 4-XC6H4CH2 radicals, where X=H, NO2, Cl, Me, MeO, were investigated (regardless the nature of the attacking agent); in this case, the tp and to constants decreased in the series PhNO2 44 PhCl, PhOMe>PhMe (see Table 1). This model series has an important advantage: the measured kinetic param- eters vary over a broad range due to the direct conjugation of para-substituents with a cyclohexadienyl radical. R X +R H X However, it cannot be ruled out that, within the framework of the given reaction series, fairly reactive phenyl radicals would be involved with high probability in side homolytic properties. The methodology of the above approaches is evidently similar to that used in the classical equations of correlation analysis, which determine the inductive and resonance constituents of the polar effect of substituents in terms of Taft,42, 61 ¡À 63 Exner,64 ¡À 66 Yukawa ¡À Tsuno,67, 68 Swain ¡À Lupton,69 Dewar ¡À Grisdale,70 Charton 71 and Palm 72, 73 approaches.In all the studies described above, except for the study by Alfrey and Price,46 the researchers dealt only with aromatic systems. Therefore, the radical constants t, ER and ED are believed to contain a component due to the polar effect. Fisher et al.55, 74, 75 used radical bromination of 4-substituted 3-cyanotoluenes with N-bromosuccinimide as the model reaction to determine the `extra-resonance' (this name was proposed by the authors), i.e., the proper radical contribution (the polar contribu- tion is minimised).When considering the possible canonical forms of the transition state of hydrogen abstraction via the reaction YH+Z Y+H Z¡¦ 1b Y¡¦ HZ�¢ , 1c YHZ 1a the researchers suggested that the high electronegativity of the attacking bromine atom Z precludes a substantial contribution of structure 1c. In order to destabilise the polar form 1b to decrease its contribution, the electron-withdrawing CN group was intro- duced in the meta-position relative to the methyl group (substrate YH) of the substituted toluene. The radical intermediate 1a is stabilised additionally due to direct resonance. Assuming that the polar effects in the model system were thus minimised, Fisher et al.55, 74, 75 proposed a scale of s radical constants within the framework of a two-parameter equation including the Brown ¡À Okamoto constant s+ (6) log k �� rs�¢ �¢ s , k0 in which the parameter r was determined on the basis of a similar reaction series of 4-substituted toluenes.Fisher and Meierhoefer 55 did not use a series of meta- substituted toluenes as the standard for determining r on the ground that the corresponding parameter in Eqn (6) reflects, in their opinion, both the resonance and inductive polar effects. Thus, the differences between the effects of substituents in the free radical bromination of monosubstituted and 3-cyano-4-substi- tuted toluenes by N-bromosuccinimide were attributed to the stabilisation of the structure 1a; they were proposed as character- istics of the radical effects of para-substituents.In accordance with the results of this study, the ability of substituents to stabilise free radicals increases in the series F<OMe<Me<H<Cl<Ph< I <Br<NO2<N=NPh<CN<Ac, the s constants for F and MeO being negative [s (F)= 70.25, s (MeO)=70.12], which implies that these groups destabilise the radical reaction center (see Table 1).55 Meanwhile, it is obvious that the resonance stabilisatoion of substituted benzyl radicals by electron-withdrawing substituents (structures 2a and 2b) is similar to resonance delocalisation of the A R Cherkasov,MJonsson, V I Galkin, R A Cherkasov negative charge.This accounts for the most pronounced radical- stabilisation properties of substituents such as Ac, CN, N=NPh and NO2, which also exhibit a negative mesomeric effect. X Y X Y CH2 CH2 2b 2a In the case of stabilisation of the benzyl radical by the donor mesomeric effect of some substituents such as halogens (except for F), the researchers cited 55 discussed the formation of three contributing structures 2c, 2d and 2e. +X X X CH¡¦ CH2 CH2 2 2c 2e 2d Structure 2e is characterised by substantial charge separation, which makes it energetically less favourable. The electrically neutral structure 2d implies delocalisation of the unpaired electron because it has been added to the electron octet of the X atom. As Tsuno et al.68 rightly noted, elements of Period II, O and F, are incapable of this type of interaction, which accounts for the negative values of the corresponding radical constants.The presence of negative s values in the Fisher ¡À Meierhoefer scale hampers its quantitative comparison with other known scales, in most of which the radical constants can be only positive, implying that the radical reaction centres are stabilised by both donors and acceptors. It should also be noted that in the model system of disubstituted toluenes proposed, the possibility of resonance effect of para-substituents is retained, despite the minimisation of polar effects. The search for standard reaction series with a clear-cut radical character of the transition state, which rules out a noticeable influence of polar interactions, has led al.50, 51, 56, 76, 77 to consideration of the thermal dissociation of dibenzylmercury derivatives in solution. ArCH2HgCH2Ar ArCH2 +HgCH2Ar HgCH2Ar Hg+ArCH2 .It was found that homolysis occurs in two stages. The first, rate-determining stage is generation of benzyl and benzylmercurio radicals;52, 78, 79 it was suggested that in the transition state, one C¡ÀHg bond has been cleaved almost completely, which corre- sponds to the formation of a nearly free benzyl radical. The conventional sr-analysis, which makes use of the first- order rate constants for the reaction under consideration and the Hammett s-constants (subsequently, the s0 parameters were used), identified a linear correlation for meta-derivatives, whereas para-substituents of both electron-donating and electron-with- drawing nature accelerate homolysis, due to an additional stabi- lisation of benzyl radicals.In the equation (7) log k �� rs �¢r s k0 resulting from the correlation procedure, the parameter r was taken to be unity for reactions giving rise to free benzyl radicals in the rate-determining step. In terms of the regression analysis performed, the constants which were later designated by sJ (see Ref. 45) and which reflect the stabilising effect of substituents in the para-position of the benzyl radicals were determined from the deviations from the meta-correlation (see Table 1). The sJ values were found to be in good agreement with the Q, t, ED and ER scales (the correlation coefficients r varied from 0.83 to 0.92); as regards the Fisher ¡À Meierhoefer constant (sFM), only some qual- itative correspondence was found in this case (r=0.63).It has been proposed 56 to use the |s7s|/n scale, where s is the s+ or s7 constant, as a measure of radical stabilisation; for substituents containing a lone electron pair n=2, while for groups with a multiple bond or any other p-system, n=1. ThisCorrelation analysis in the chemistry of free radicals scale correlates well with the sJ values. As drawbacks of this approach, the authors note the difficulty of preparing a broad range of dibenzylmercury derivatives and the limited solubility of these compounds in octanol, used as the reaction medium.An attempt to select a perfect free-radical transition state is also the central point in Creary's approach,57, 80 ± 84 in which thermal rearrangement of 3-aryl-2,2-dimethylmethylenecyclopro- panes 4 has been used as a standard reaction series to introduce the scale of radical constants sC. Me Me CH2 H H X 4 Me Me H Me H H Ar X=NO2, NMe2, CH=CH2, C(Me)=CH2, Ph, cyclo-C3H5, 6 CH2SiMe3, SiMe3, SnMe3, HgCl, B(OCH2CH2O). The transition state of the rearrangement is of biradical nature; this has been confirmed in a series of studies.85 ± 94 Proceeding from the assumption that the biradical nature of the transition state rules out the possibility of substantial charge separation in it, Creary suggested using the logarithm of the relative rate constant for the reaction, krel, as a measure for the ability of substituents to stabilise the benzylic radical centre sC =log krel .The kinetic parameters of thermal rearrangement of 2-aryl- 3,3-dimethylmethylenecyclopropanes 4 into 2-arylisopropylide- necyclopropanes 5 can be easily determined by gas chromatog- raphy or NMR; this permits determination of radical constants for meta- and para-substituents of the aromatic ring within the framework of this approach. It is also possible to evaluate the stabilising influence on the radical centre of heteroorganic sub- stituents.45 It was found 57, 84 that electronegative substituents in the meta- position retard the rearrangement, which might point to an electron-deficient character of the biradical transition state.Most para-substituents, both electron-donating and electron- withdrawing ones have positive values of sC constants. An exception is F, for which sC= 70.08 (see Table 1); this was explained by the destabilising influence of F caused by the enhanced inductive effect. The influence of the para-methoxy group was regarded as stabilisation (sC=0.24), caused by the contribution of the canonical forms 7a,b. CHR H3C:O H3C:O 7a + CHR F ¡ 8 A similar structure for the para-fluorine-substituted system appears less favourable because of the high electronegativity of fluorine. From the molecular-orbital standpoint, the stabilising effect of the para-methoxy group can be due to the interaction of the radical centre with the filled non-bonding orbital of oxygen; Me CMe X 5Me H H CHX (8) á CHR ¡ 7b 5 however, the low energy of the filled 2p orbital of fluorine precludes manifestation of the effects of this type.As has been noted in a monograph,45 synthesis of a broad range of methylenecyclopropanes 4 is quite difficult; therefore, the reaction series proposed by Creary has no advantages in this respect over thermal dissociation of mercury dibenzyl derivatives. In addition, it is not quite clear to what extent the centre in the transition state 6 is a radical centre. Nevertheless, this approach has been successfully developed; it was shown in a recent pub- lication 94 that the introduction of nitrogen-containing substitu- ents into the para-position of the substrate 4 provides considerable radical stabilisation.Moreover, some substituents, for example, CH=NNMe2, N=NPh, N=N(O)But and CH=N(O)But (sC=0.92, 1.08, 1.08 and 1.13, respectively), displayed extraordi- nary radical-stabilisation capacity; they were called super-stabil- isers. Afree-radical transition state in aliphatic systems 45, 95 ± 100 has been modelled by pyrolysis of substituted azopropanes 9. Me Me Me Me C Me N N C C Me X X X 10 9 According to the researcher's assumption, this reaction series is characterised by slight steric and inductive interactions; therefore, the pronounced difference between the reaction rates (the range of variation of the rate constants is 10 9) is interpreted as being due exclusively to resonance stabilisation. Later, the kinetic character- istics of pyrolysis of the azopropanes 9 were found to be well correlated with the Creary radical constants sC, despite the substantial difference between the structures of the radical sub- strates 6 and 10 and the fact that the extents of influence of substituents on the radical centres are difficult to compare.83 The series of model aryl-substituted azoalkanes has been markedly extended 101 by the inclusion of polycyclic azoalkane derivatives,102 symmetrical azopropanes,103, 104 azoethanes,105, 106 azomethanes,98 azoneopentanes 107 and 3,5-diaryl-1-pyrazo- lines,98, 108 for which the kinetic parameters of pyrolysis have been found. It is worth mentioning that, only in terms of two- parameter correlations including polar constants s, was it possible to attain satisfactory relationships between the experimental relative rate constants for the thermolysis of azo derivatives and the corresponding Creary parameters sC.84 The equilibrium constant of the dissociation of substituted hexaarylethanes to triarylmethyl radicals has been used success- fully as a measure of radical-stabilising capacity of substituents.109 An association reaction, namely, thermal [2+2]-cycloaddi- tion of substituted a,b,b-trifluorostyrenes 11 58 has been used as a model series for determination of spin-delocalisation group con- stants sJJ (see Table 1).DD(+ or 7 ) dd+ dd7 CF Y CF2 F1 F3 D(+ or 7) d7 d+ C 2Y C F2 11 CF Y CF2 d F F Ar F Y CF D CF2 F F F F Ar F F + F CF D Y dCF2 Ar F Ar F 12 An advantage of this model reaction is that no side radical processes are involved.In addition, charge distribution in the transition state 12 is symmetric, which eliminates additional polar effects. Study of the kinetics of this reaction gave a two-parameter correlation equation6 (9) log kY a rmbsmb a r sJJ, kH which was employed for the introduction of spin-delocalisation constants of substituents sJJ. The smb parameter in Eqn (9) is identical to the group constant proposed previously, which is determined on the basis of fluorine chemical shifts in the 19F NMR spectra of compounds 11.58, 110 ¡¾ 113 The mb (multiple bond) subscrflects the hypothetical additional repulsive component of interaction between the double bond and the electron lone pair or p-electrons in the Y substituent in the compound 11 (Jiang et al.113 consider this to be a fundamental feature distinguishing the scale of smb constants from the Brown ¡¾ Okamoto s+ constants).Equation (9) provided the possibility of determining the sJJ constants for 32 substituents (see Table 1) at five different temperatures; this allowed the researchers to verify the adequacy of the correlation parameters proposed and to introduce an averaged scale. JJ the two-parameter equations r as a measure of the relative r contributions of radical effects in the corresponding model series.By the use of the sJJ scale, together with the s+ parameters, the kinetic data underlying other scales of s radical constants have been reproduced with rather high accuracy within the framework of a two-parameter equation. Assuming that the s parameters reflect the real ability of substituents to delocalise spin density, Dust and Arnold 114 used the ratio of the coefficients of Thus it was shown that the ratio r for the reaction series of r monosubstituted toluenes used by Fisher and Meierhoefer 55 to determine polar effects amounts to 4.8, whereas an analogous model series of substituted toluenes containing a cyano group in the meta-position is described by a ratio of polar to resonance effect equal to 1 : 1.51; in addition, the accuracy to which each contribution has been determined does not provide the possibility of their correct separation.It should be noted that many researchers 4, 52, 77, 114, 115 pro- posed their own scales of s constants which they considered to be the most suitable for estimating the radical character of transition states. For instance, Jackson et al.56 used the sJ scale and demonstrated that in the transition states of the model homolytic reactions that underlay the Fisher ¡¾ Meierhoefer radical constants sFM and the ER and ED parameters, the fraction of structures corresponding to the proper benzyl radicals is not more than 50%. The authors of most of the approaches discussed above have attempted to formulate the requirements to an ideal model system to be used for kinetic determination of radical constants.1. The substituents should interact directly with the radical centre. 2. The mechanism of the model reaction should be as clear as possible. 3. Side reactions should be minimised. 4. Model systems should be preferred in which substituents have a strong influence on the reaction rate and for which accessible and precise methods for the determination of relevant kinetic parameters exist. 5. The polar effects should be either absent or minimised, or be reliably separated from the effect of radical stabilisation. 6. The model compounds should be readily available and should contain diverse substituents of both electron-donating and electron-withdrawing natures.7. The role of other external and internal factors, for example, the solvent and the steric effect, should be minimised. 8. One of the most important features required for a free- radical reaction to be used as a model to describe the radical effects of substituents is that the nature of the transition state should be as radical as possible. According to the Hammond postulate,116 the last-mentioned condition (this is very important to emphasise) implies a late A R Cherkasov,MJonsson, V I Galkin, R A Cherkasov transition state of the process, which corresponds to a more endothermic generation of free radicals. This, in turn, stipulates a higher selectivity of the reaction.117 Therefore, the radical effects of substituents (in particular, the captodative effect { of radical stabilisation) are often considered in the context of selectivity of free-radical processes (see, for example, Refs 45, 118).It should be noted that in none of the above-listed approaches, the transition state can be considered completely radical�¢in any case, one should suggest the presence of charge separation and polar interactions. Therefore, the topicality of the question of to what extent a particular scale of radical constants is free from the polar constituent is beyond doubt. Apparently, the polar constit- uent can hardly be ruled out completely. 2. Spectroscopic determination of the capacity of substituents for delocalisation of spin density in free radicals Physicochemical methods such as EPR spectroscopy permit direct investigation of the properties of free radicals.According to the McConnel equation aH=QrpC, the hyperfine splitting (HFS) constants aH in the EPR spectra are directly proportional to r.118 For a number of compounds, the relative HFS constants have been found 119, 120 to correlate with the s and s7 polar constants; it was concluded 119 that delocalisation of an electron pair is a more substantial stabilising factor than spin density delocalisa- tion. A non-kinetic scale of the stability of alkyl radicals based on HFS constants of b-protons has been composed.121 The approach proposed by Arnold et al.122 ¡¾ 126 is best known of all; according to this approach, the HFS constants of a-protons are used as a measure of stabilisation energy, determined by delocalisation of spin density in substituted benzyl radicals 13 with respect to PhCH2.X1 CH2 13 X2 This method has obvious advantages, which include the absence of side radical reactions and the relative ease of generation of the required free radicals.45 In addition, the radical constants sa, defined by the equation (10) sa a 1 ¢§ aOHXU 0U , aOH where aOHXU and aOH0U are the HFS constants of the a-protons in the substituted and unsubstituted radicals, respectively, can be considered to be free from the polar effects characteristic of radical-like transition states and do not correlate with the sub- stituent effects in the corresponding diamagnetic initial com- pounds.The researchers 122 assumed that other factors which can potentially influence the HFS constants (hybridisation of the orbital bearing the unpaired electron, interaction with b-substitu- ents, steric effects reflected in the geometry of benzyl radicals 118) are insensitive to the substitution in the aromatic ring. The sa values thus obtained (Table 2) confirmed that the meta-substituents in the aromatic ring decrease delocalisation of the unpaired electron. The correlation established between the constants of meta-substituted benzyl radicals and the correspond- ing Hammett meta-constants (sm) made it possible to relate the destabilising effect of meta-substituents to their inductive elec- tron-withdrawing effect.45 However, it is not entirely clear why electronegative meta-substituents hamper delocalisation of spin density on the methylene carbon atom in the benzyl system.124 { Captodative effect is a joint influence of electron-donating and electron- withdrawing substituents on a radical reaction centre, resulting in a stabilisation greater than the sum of the two stabilising effects in the corresponding monosubstituted radicals.45Correlation analysis in the chemistry of free radicals Table 2.Constants characterising the radical-stabilisation properties of substituents (from EPR data and bond energies). Substituent HMe OMe NO2 CN FCl Br IOH C(O)Me P(O)(OEt)2 P(S)(OEt)2 Ph SMe SPh CF3 OPh CO2Et CO2Me C(O)Ph But Et Pri SOMe SC(O)Me S(O)OMe SO2Ph S(O)Ph SO3Me OC(O)Me OC(O)Ph N=NPh NH2 NMe2 SiMe3 CH=CH2 Note.Ed is the energy of homolytic dissociation of the C7X bond in substituted methanes. a From Refs 114, 125; b from Ref. 127; c from Ref. 45; d from Ref. 115. Presumably,124, 128 a decrease in the electron density by the inductive mechanism prevents the effective overlap of the inter- acting sites, i.e., of the orbital carrying the unpaired electron with the p-system of the benzene ring, or, in other words, it decreases the energy of the p-system and thus weakens the interaction between the CH2-group carbon and the aromatic ring. The effects of the OC(O)Me, CF3 and F groups located in the para-position were found to destabilise the radical centre, whereas other para-substituent according to the data obtained from EPR, favour spin density delocalisation.It has been suggested that a decrease in the spin density in the a-position of the benzyl radical would result in an increase in the p-resonance binding of the aromatic system with the methylene group and, correspondingly, in an increase in the barrier to rotation around the Carom±CH2 bond.45, 124 The p-component of the stabilisation energy (Es) of the p-radical was expressed via the rotation barriers of the CH2 group in the ZCH2 radicals with or without spin density delocalisation Es= V2(CH2)7V2 , where the corresponding rotation barriers (V2 and V 2 ) can be determined from EPR data. s DDp see b a(m) see a sa(p) see a 0.00 0.02 70.05 0.90 0.54 70.17 0.09 0.05 0.04 70.05 ± 0.000 70.001(0.002) 70.001 ± 70.039(70.026) 70.018(70.009) 70.001(70.007) ±±±± ±±± ±±± 0.11 ±± 70.014(70.017) 70.013(70.002) ± 70.004 ±±±±±±±±±± 0.53 ±±±±±±±±±±±±±0.30 0.000 0.015 0.034(0.018) ±0.043(0.040) 70.011 0.017(0.011) ±±±0.066 (0.060) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.063 0.058 0.001(70.009) 0.018 ±0.048(0.043) 0.064(0.055) 0.036(0.008) 0.012 0.009 0.006(0.018) 0.029 0.016(0.005) 0.018 0.026 0.003(0.013) 0.001(70.005) 0.000 ±±±±± 70.014 ±±±±±± ±±± 7 av RRSX see c Ed see c DDm see b ssee d 0.00 0.23 0.35 0.76 0.63 70.03 0.18 0.23 ±± 0.0 2.3 4.5 9.4 8.6 71.4 2.4 0.9 0.2 3.1 10.2 0.00 70.09 70.15 70.06 70.14 ± 70.08 ± 70.09 70.22 ± 0.58 104.4 100.3 93.3 ± 92.9 101.0 99.2 102.5 103.4 95.0 ± 0.59 0.55 ±0.09 ±± 11.2 10.7 10.7 70.3 4.9 ±7.9 9.3 3.0 1.5 ±4.1 6.7 3.6 4.5 5.9 2.9 0.9 0.0 ±8.4 8.9 3.5 12.8 0.54 0.65 0.13 ±±±±±±±±±±±±±±± 87.9 ±± 106.7 ±±±± 93.9 99.7 ±±±±±±±±±± 94.0 84.0 ± 86.7 ±±± 70.04 ±±±±±±±±±±±±±±±± 70.19 ±±± Jackson 129, 130 has noted that the range of HFS constants considered by Arnold et al.122 ± 126 is rather narrow and some individual values were determined with substantial errors.There- fore, he developed a new approach that provided more accurate determination of the HFS constants a(CH2) in the EPR spectra of benzyl radicals.116 The results 129, 130 are generally consistent with the sa scale. Yet another example of using quantitative parameters of EPR spectra is a study by Adam et al.,127, 131 ± 135 who made use of the electronic spin ± spin splitting parameters D. The parameter D is a sensitive function of the averaged distance between the unpaired electrons in a biradical (D!1/r3) which is proportional to the product of local spin densities in the triplet biradicals 14 and 15, chosen as the reference species to determine the capacity of the substituent X for spin density delocalisation.130 ± 132 3 3 X X X 14 158 Proceeding from the assumption that specific electronic effects (captodative stabilisation, spin polarisation, etc.) are relatively insignificant in 1,3-biradicals,136 Adam et al.127, 132 ± 134 proposed the scale of DD= DH7DX values (where H corresponds to the unsubstituted radical), which reflects delocalisation of the unpaired electron by the aryl group. Negative DD values, which should correspond to electron-donating groups, were found for all meta-substituents (including charged ones, viz., NHá3 , O7) and for para-substituents such as F, OMe, OC(O)Me and OH.The para-substituents NO2, CN, C(O)OMe, NH2 and CF3 exhibit the highest capacity for spin density delocalisation (see Table 1). The additivity of DD values found for monosubstituted and symmetrically disubstituted triplet biradicals 14 and 15 served as the basis for the assumption that each of them can be represented as composed of two fixed fragments of the cumyl radical. Hence, the researchers suggested 127, 132 that the DD scale could also be used for adequate description of benzyl monoradicals. This suggestion is supported by the fact that the parameters D for triplet biradicals are correlated with the a-spin densities and the energies of resonance stabilisation of the corresponding cumyl radicals.127, 132 ± 134 Comparison of the DD scale with known radical constants based on the properties of benzyl radicals revealed a relatively good correlation of this scale with the Fisher data (the correlation coefficient squared is r2=0.92); however, the discrepancies with the sJ , sC , and sJJ arrays are quite substantial (r2=0.33 ± 0.60).Subsequently, this approach has been extended to heteroaromatic p-systems such as pyridine and furan derivatives and some other.135 3. Thermodynamic approaches to the estimation of radical stabilisation. Bond energy as a measure of radical stabilisation Study of the effects of substituents in free radicals is tightly connected with the problem of quantitative evaluation of the energies of dissociation of chemical bonds. The energy of anR7R0 bond is known 136 to be determined by the enthalpy DH0 1 of homolytic dissociation of this bond, giving rise to free radicals R and R0 R+R0 .R7R0 In turn, the DH0 1 value is determined by the standard enthal- pies of formation of RR0, R and R0 DH10= DH0(R)+DH0(R0)7DH0(RR0) . Thus, the relative energies of the homolytic dissociation of bonds can serve as a measure of radicals stabilisation. On the basis of the foregoing, Benson and O'Neal 137, 138 have proposed estimating the stabilisation energy for p-resonance delocalised radicals R(p-CH2) as the difference between the dissociation energies Ed of the corresponding derivatives Es[R(p-CH2)]=Ed[R(CH27H)]7Ed[R(p-CH27H)]. Subsequently, the C7H bond energies have been widely used to estimate the stabilisation energies for carbon-centred radicals.For example, the resonance stabilisation energy of the benzyl radical can be represented as the difference between the energy of the benzylic bond in toluene and the energy of the C7H bond in ethane.45, 139, 140 The enthalpy of dissociation of the C7Hbond in the methane molecule Ed(CH3 ±H)=439 kJ mol71 is used most often as the reference value.140 ± 145 In one of the first studies dealing with this topic, the enthalpy of the isodesmic reaction (i.e., a reaction in which the number and character of bonds do not change on passing from the initial compounds to the products) 146 RH+ CH3 R +CH4 was identified as the stabilisation energy ER of the R radicals 146 ER(R )=DH0 2 =Ed(CH3±H)7Ed(R ± H).A R Cherkasov,MJonsson, V I Galkin, R A Cherkasov Subsequently, stabilisation energies of radicals expressed in terms of Ed , i.e., Es(R )=Ed(CH3±H)7Ed(R ± H), have been widely used for various correlations.138 ± 145 Bordwell et al. 147, 148 have proposed a scale of redox poten- tials IAO, which is also actually based on the relative energies of dissociation of bonds in substituted fluorenes. H H OMe OMe H7 +OMe The potentials were determined using the thermodynamic cycle (pKa) (Eox) H++A7 A +e7 H (Ered) HA A7 H+ + e7 H +A HA (Ed), in which the proton reduction potential Ered is a constant and the sum pKHA+Eox=IAO can be regarded as a measure of relative radical stability of the radical A.The oxidation potentials Eox of substituted fluorenide ions were determined experimentally in DMSO solutions, and the pK values of the conjugate acids were established on the basis of their correlation with Eox; this gave rise to a scale of IAO values for a number of substituents at the 2-, 4- and 7-positions expressed in kJ mol71 and reflecting the ability of these substituents to delocalise the unpaired electron in fluorenes. The application of this approach to 9-substituted fluorenes and to 3- and 4-substituted phenylacetonitriles has made it possible to analyse a wide spectrum of radical-stabilisation and -destabilisation effects.149, 150 Comparison of the IAO values for 4-substituted arylacetonitriles with the sJJ constants within the framework of a two-parameter equation containing the s+-parameter showed that the contributions of the polar and radical constituents to IAO are approximately equal.58 It has been proposed to use the Eox oxidation potentials of 3- and 9-substituted fluorenes 150, 151 and of a-substituted acetone and acetophenone derivatives 152 in DMSO as the basis for determination of the stabilisation energies ERS of the correspond- ing radicals DEd=1.37DpKHA +23.1DEox(A7)=ERS. The resulting values point to a stabilising effect of substituents at the 3-position in substituted fluorenyl radicals, except for F and PhSO2.Substitution into the 9-position both by donors and by acceptors results, as a rule, in system stabilisation. Most of 4-substituents (except for F and CF3) also exhibit pronounced stabilising properties.148 In general, it was concluded 150 that the majority of substituents play a dual role in the stabilisation of a free radical: they destabilise the radical due to the electron- acceptor influence and stabilise it due to delocalisation.In some studies, the radical-stabilisation energy of substituent X was set identical to the covalent component of the dissociation energy of the XR7Z bond, where R is a permanent molecular fragment (for example, the aromatic ring), while R7Z is the `indicator' bond.140, 153 ± 156 The notion `indicator' bond was introduced back in Pauling's studies;151 the stabilisation energy ERS was defined via the relative dissociation energies of bonds in symmetrically substituted molecules ERS=0.5 Ed(XR ± RX)70.5 Ed(HR ± RH).Correlation analysis in the chemistry of free radicals A similar approach has been proposed by Leroy,157, 158 who used the enthalpies of atomisation DHa together with the standard bond energy EAB to define the stabilisation energy Es Es=DHa7SNABEAB , where NAB is the number of AB bonds.This procedure was applied successfully both to free radicals and to diamagnetic molecules. Thermochemical and kinetic investigations of the rupture of the C7C bond have been employed to define the parameter ERS, introduced to reflect the difference between the stabilities of the hydrocarbon radical 16a and the corresponding substituted rad- ical 16b.117, 156, 159 ± 161 Z S Y C Y C X X 16a 16b X, Y, Z are hydrogen atoms or alkyl substituents and S is a functional group.Asubstantial drawback of the approaches in which the radical stabilisation is defined in terms of the relative bond energies and the reason why these approaches are severely criticised is the fact that the energy of homolytic dissociation of a bond is a relative value, i.e., the difference between the energies of the molecule and the corresponding radicals; therefore, the substituent effect can- not be attributed to the properties of the free radical alone.117, 143, 162, 163 It is well known that the dissociation energies of bonds depend on steric effects 116 and on polar effects;164 the presence of numerous correlations between Ed(CH) and the `ionic' constants of substituents s+ (see publications 23, 147, 165) is com- mon knowledge.In our opinion, this virtually rules out a legitimate separation of the polar and radical effects in this type of approach. Moreover, it was shown that in some cases, the effect of substituents can even prevail in the ground state; then the corresponding DEd values reflect destabilisation of the molecule rather than stabilisation of the free radical.163 It has been clearly demonstrated 143 using, among other reasons, vast literature data that relative bond energies cannot be related directly to the stability of radicals. 4. Calculation of the energies of stabilisation of radical centres Apart from experimental methods, there also exist theoretical methods for the determination of group parameters of substitu- ents reflecting their capacity for stabilisation of radical centres.Modern quantum-chemical approximations permit direct calculation of quantitative characteristics of radicals such as the total energy, the standard enthalpy of formation, the energy of the orbital with the unpaired electron and the spin and charge density, which make it possible to judge the stabilising influence of substituents. Many quantitative parameters employed to deter- mine the radical-stabilisation effects in terms of the experimental approaches described above such as the kinetic parameters of free- radical reactions and the overall thermodynamic characteristics, spin densities and rotation barriers in free radicals can be calculated quite reliably by quantum-chemical methods.Cer- tainly, any general description of theoretical approaches requires separate consideration. Within the framework of this review, we shall dwell only on those of them that have served as the base for the introduction of sets (scales) of energetic parameters of substituents reflecting their radical-stabilisation influence and implying consistent utilisation of the LFE principle. In studies by Pasto et al.,166, 167 the change in the total energy of species participating in the free-radical transfer of a hydrogen atom has been identified with the stabilisation energy ERS of the corresponding substituted methyl radical. The ERS values were calculated for a large number of substituents.The captodative 9 effect of radical stabilisation and the influence of the substituents attached to the reaction centre containing an unpaired electron were analysed from the additivity standpoint.166 The comparison of the calculated ERS values with the known scales demonstrated a relatively good quantitative agreement with the IAO parameters for 3- and 9-substituted fluorenes and 4-substituted phenylaceto- nitriles,148 with the relative constant of the thermal rearrangement of 3-aryl-2,2-dimethylmethylenecyclopropanes 57, 82, 83 and a-spin density in 4-substituted benzyl radicals.114 Within the framework of this approach, it was found that the radical-stabilisation effect markedly increases on passing from F- to O- and N-containing substituents.In the series Cl, S, P, the capability for spin density delocalisation increases due to the decrease in the energy gap between the vacant MO of the methyl radical and the MO occupied by the unpaired electron. Study of the effects of substitution with charged (+NR3, +SR2, +PR3) or electro- negative non-charged (CF3, SOR, SO2R) groups showed that they destabilised the radical. In addition, it was found that the energies of stabilisation of the radical centre calculated in terms of the approach proposed are exceptionally sensitive to the confor- mation of the free radical and the corresponding diamagnetic molecule. Isodesmic reactions (see Section II.3) proved convenient for the calculation of ERS(R ).44, 144, 168 ± 173 The enthalpy of hydrogen atom detachment in these reactions is a measure of the energy of stabilisation of the radical centre; the use of this value decreases the errors related to the basis set chosen and to electron correla- tion effects.174 The stabilisation energies of the radical centre ERS correlated with the barriers to rotation around the C7C bond in benzyl radicals were calculated by quantum-chemical methods, as well as the difference between the energies of the conformers correspond- ing to angles of rotation around the s-bond equal to 90 8 and 08.175 ± 177 ERS=DHf(90)7DHf(0). The ERS values thus obtained are well correlated with the parameters D of the biradicals 14 and 15 determined experimen- tally by EPR.127 5. Critical analysis of the existing approaches The attempts at systematisation of the existing radical scales of substituents made previously can be classified, in our opinion, into two main types, namely, the choice of the s -scale describing most accurately the standard homolytic reaction series and the develop- ment of a quantitative approach integrating the accumulated arrays of `radical' constants and combining them into some universal scale. We have already mentioned that the authors of many of the studies presented in the previous Sections proposed their own standard series; the corresponding arrays of the group parameters were regarded as most suitable for developing a scale that could be used as the reference in analogous approaches.However, the search for this optimal scale did not reveal obvious advantages of any of the known sets of radical parameters; therefore, in some cases, it was concluded 9 that development of universal radical constants s is impossible in principle. The attempts to find multiparameter correlations with the use of various combinations of polar and radical scales did not produce a satisfactory general result either. Moreover, the methodology of using the extended Hammett equation for the description of radical-stabilisation effects has been criticised.49 In view of the difficulty of using group radical constants to describe homolytic processes, Hansch and Leo 178 proposed that the polar parameters s should be used for this purpose whenever possible. Exner 179 also suggested that the sáp and sm,p constants suffice for interpretation of the free-radical reactivity.In another study,180 radical-stabilisation effects of substituents are described in terms of the quadratic function s2; this, in principle, contradicts the LFE formalism.10 Several prominent researchers attempted, in a joint review,45 to devise a universal scale of relative radical stabilisation (RRS), based on averaging of the above-mentioned scales of the Creary (sC ),57, 80 ± 84 Jackson (sJ ) 51, 52, 56, 76 ± 79 and Arnold (sa) con- stants,114, 122 ± 125 the kinetic parameters of radical fragmentation of bis-azo derivatives 96 ± 100 and a number of other analogous characteristics of free-radical reactions.45, 180 ± 184 It was suggested that statistical averaging would eliminate the errors inherent in each of basic experimental approaches.In addition, the average group RRS parameters are expected to reflect the degree of additivity of the group influences upon multiple substitution at the carbon radical centre and to describe the effect of captodative radical stabilisation. Positive averaged RRSX values (see Table 2), which imply radical-stabilisation properties, were found for a large number of diverse substituents. The only exceptions were F and CF3, for which the RRSX(F) values were71.4 and70.3, respectively. By means of the scale of averaged RRSX, the researchers 45 were able to identify a number of interesting correlations which point, in particular, to underestimation of the radical-stabilisation properties of electron-withdrawing substituents and overestima- tion of the role of electron donors in terms of some s -approaches.av scale was based 115 on two sets of The averaged radical s EPR characteristics of benzyl radicals taken from the litera- trure 115, 124 and on three `kinetic' arrays, namely, scales of the sC (see Refs 57, 80 ± 84) and sJ (see Refs 50 ± 52, 56, 76 ± 79) constants and kinetic data taken from the literature.101 The sav values (see Table 1) confirmed the radical-stabilisation properties of the vast majority of para-substituents including CF3. Only the averaged constant for p-F was found to be negative (sav =70.02). This deviation was explained by assuming that, under the action of the electronegative F atom, hybridisation of the carbon atom at the 4-position changes somewhat towards sp3; this disturbs the aromaticity of the benzene ring and, hence, destabilises the radical centre.Each of the definitions of the radical s constants character- ised in this Section suffers from its own specific disadvantages. They were noted above whenever appropriate. In addition, the methods of description of the radical effects of substituent can be subjected to a common criticism. It is expedient to arrange critical remarks in accordance with the above classification. 1. Kinetic methods. It is known that the classical s-constants of electron-withdrawing substituents, which stabilise an electron- enriched reaction centre, are positive, whereas electron-donating groups, which stabilise electron-deficient systems, have negative polar constants.In turn, the sign of the parameter r of the correlation equation reflects the character (either electrophilic or nucleophilic) of the reaction, while the magnitude of r shows the degree of charge separation in the transition state. It is commonly known that the polar constants of substituents correlate equally adequately with the properties of the transition state and the ground state (numerous polar scales are based on the physico- chemical, for example spectroscopic, properties of neutral mole- cules). Thus, successful description of energy characteristics of heterolytic processes in terms of group constants is normally explained by assuming either that the substituent effects in the transition and ground states are correlated or that the latter can be neglected.The kinetic methods of determination of substituent effects in homolytic processes also deal with the difference between the energy of the ground and radical-like transition states. However, a significant specific feature of free-radical reactions is that the natures of the initial species (or products) and the transition state in any homolytic process are fundamen- tally different because of the presence of an unpaired electron; the effects of substituents on the energies of the ground and transition states differ not merely quantitatively, as in the case of heterolytic processes, but also qualitatively.Furthermore, the numerous examples cited in the preceding Sections point to the importance of substituent effects in the ground state of homolytic processes, which precludes the possibility of neglecting them. In addition, A R Cherkasov,MJonsson, V I Galkin, R A Cherkasov even if the polar interactions in the ground state are assumed to be proportional to those in the radical-like state, their role in delocalisation of the proper unpaired electron still remains obscure. Strictly speaking, the magnitudes of radical s parameters established by kinetic methods often reflect the differences between the influence of substituents on the initial and transition states of a particular reaction but not the effectiveness of delocal- isation of the unpaired electron by one or another substituent.In this connection, the numerous examples of successful use of polar constants alone for the description of free-radical processes seem quite obvious and the lack of correlation between radical scales is quite logical. Thus, reasonable separation of the effects related to charge stabilisation and to delocalisation of the unpaired electron within the framework of kinetic approaches appears unlikely, if achievable at all. It is also worthy of note that the role of other factors (steric effects and, especially, solvent effects) has been interpreted fairly arbitrarily in most of the approaches considered. As a rule, the authors assumed that the homolytic mechanism of the standard reaction allows the solvent effects to be completely neglected.However, as has already been noted, in reality, none of the kinetic approaches allows complete elimination of the influence of polar interactions (which can depend substantially on the properties of the medium); this casts doubt on the proper methodology of determination of many s -scales. 2. Spectroscopic methods. Methods based on the use of EPR for determination of the spin density distribution are free from the drawbacks associated with the substituent effects in the corre- sponding molecules. Meanwhile, the question of whether spin density delocalisation follows a linear correlation with the stabi- lisation energy of a free radical remains relevant.In addition, it is unclear to what degree the spin density distribution is governed by the delocalisation contribution and to what degree, by the spin- polarisation contribution.45 And, finally, what is the role of polar effects (for example, resonance effects) in the spin density distri- bution? Is it proper to consider them insignificant and related exclusively to charge stabilisation? In our opinion, the following circumstance is also rather important. Let us assume that the spectral s -constants do actually reflect the capacity of substituents for delocalisation of the unpaired electron. Then it is clear that valid considerations about the mechanisms of delocalisation of an unpaired electron by substituents can be constructed on their basis and the s values can be used to verify the adequacy of similar approaches to the solution of problems for which the correlation analysis methods are meant. However, the scope of applicability of these spectral radical scales still remains vague, while their predictive capacity and practical application hardly possess considerable advantages.In any case, when describing real homolytic reaction series, the presence of polar interactions (which are always quite probable, as we have repeatedly emphasised above) implies the use of a two- parameter correlation equation containing both radical and polar constants. One or another polar scale can be preferred and the relationship between the polar and radical components of the stabilisation effect can be identified only on the basis of a statistical selection procedure.From a practical point of view, this brackets the s constants obtained by spectroscopic methods with those determined by kinetic methods. 3. Thermodynamic methods. The approaches based on the use of thermodynamic methods are criticised especially vigo- rously. As has been mentioned above, determination of radical- stabilisation effects in terms of thermodynamic parameters (bond energies) corresponding to diamagnetic substrates does not permit one to rule out the substituent effects related to the molecules or to distinguish the proper radical stabilisation from polar or steric factors. Strictly speaking, these approaches are rather suitable for discussing the relative effects of substituents on the energies of covalent bonds; this, in itself, is a complex problem not ultimately solved.It is also noteworthy that, within the framework ofCorrelation analysis in the chemistry of free radicals thermodynamic approaches, the radical stabilisation effect is largely replaced by the notion of radical stability. Strictly speak- ing, this is not the same; below we shall consider this point in more detail. To conclude this section, it can be stated that none of the existing methods of sr-description of substituent effects in free- radical processes is free from serious drawbacks and, as a rule, each method operates within a narrow reaction series. The lack of correspondence between radical scales and the different in kind interpretations of the radical-stabilisation properties of the same substituents in terms of different approaches make difficult the understanding of the physical meaning of radical constants and preclude the possibility of their joint application. It cannot be ruled out that the correlation analysis formalism cannot, in principle, be applied in full measure to the description of homolytic processes because the additivity of group interac- tions is there violated. An especially clear example is the captoda- tive stabilisation effect, or push ± pull effect, repeatedly mentioned above.4. Captodative effect. The existence of an extra-stabilisation effect peculiar to bifunctional radicals was first assumed by Dewar.185 A quantitative proof of the unusual stability of these radicals was later presented in studies by Balaban et al.,186, 187 who used the term `push ± pull resonance.' Later, Katritzky et al.188 ± 190 arrived at the conclusion that radical centres bearing simultaneously an electron-donating and an electron-withdraw- ing group are subject to an extra-stabilisation effect, which was called merostabilisation (later, captodative effect).45,191 ± 194 Asimple substantiation of the captodative stabilisation can be derived from the numbers of contributing structures [in the example presented below, there are two structures for the amino- (17) or the cyano-derivative (18)] corresponding to monosubsti- tuted radicals and to a disubstituted radical containing both electron-donating and electron-withdrawing groups [five contri- buting structures are presented for the amino-cyano-substituted radicals 19].+ N C N C¡, 17 C C N , C C N 18 + 7 N C N C N C N C N C N C 19 C C N +C N+ C N¡. N¡ For disubstituted radicals containing either only amino- (20) or only cyano groups (21), three contributing structures can be proposed + 7 7 C C N N N C N N N+ , 20 N C C C N N C C C N 21N C C C N . Thus, stabilisation of these species due to delocalisation would be somewhat weaker than is expected based on the sum of the substituent effects in the monosubstituted analogues. The over-additive stabilisation effect in radical species con- taining both electron-donating and electron-withdrawing groups has been repeatedly confirmed experimentally.The enhanced reactivity of captodative alkenes has been noted in radical 193 ± 199 and [2+2]-cycloaddition 45, 199 ± 202 reactions. The captodative acceleration was considered to be the reason explaining the results 11 obtained in studies of cyclisation of 6-substituted hex-5-en-1-yl radical,203 homosolvolysis of some alkyl bromides on treatment with di-tert-butylnitroxide,204 thermal rearrangement of substi- tuted 2-arylmethylenecyclopropanes,205 isomerisation of diaste- reomeric substituted cyclopropanes 206, 207 and thermal homolysis of the C7C bond in hexa-1,5-dienes 207 and dibenzyls.208, 209 EPR data for substituted benzyl radicals point to enhanced stabilisation of those radicals that contain both electron-donating and electron-withdrawing substituents, whereas two identical substituents decrease radical stabilisation.114 Analysis of the barriers to internal rotation and the energy barriers to E,Z- isomerisation of substituted allylic radicals determined from EPR data also attests in favour of captodative interactions.210 Study of the barriers to rotation around the C7N bond in substituted aminoalkyl radicals has led to similar conclusions.211 In several publications,161, 212, 213 the captodative effect was sub- stantiated from the standpoint of orbital views.Calculations of the stabilisation effects in free radicals also predict an enhanced stabilisation of captodatively substituted derivatives.45, 163, 214, 215 However, a number of experimental and theoretical studies have not elucidated any significant distinctive features of capto- datively substituted radicals and demonstrated strictly additive influence of the corresponding substituents at a radical reaction centre.216 ± 219 Moreover, some researchers cast doubt on the existence of an additional captodative stabilisation.Within the context of the present consideration, we would only like to mention that, under the assumption of the existence of captoda- tive stabilisation of free radicals, the use of additive correlation equations based on LFE and polylinearity principles to describe homolytic processes should be subject to certain, perhaps quite substantial, constraints. Thus, analysis of the published data available to date provides grounds for the following conclusions.1. The classical polar constants s are not fully applicable to the description of free-radical processes and do not reflect the radical-stabilisation effects of substituents. In all probability, one can speak about a disproportion of substituent effects in hetero- lytic and homolytic processes. Although quite a few examples of successful use of polar constants in free-radical chemistry have been reported, the existing scales of specific radical constants s are not general. The lack of correspondence between the scales and vagueness of their physical meaning might be due to the constrains inherent in the use of the LFE principles and correla- tion analysis in free-radical chemistry and violation of the additvity of the group influence in paramagnetic systems rather than to the drawbacks of the methods of their determination.2. It is obvious that an approach, new in principle, to the description of substituent effects in free-radical systems needs to be developed. It is important that this approach should possess predictive capacity and provide a quantitative estimate of the character of interaction of substituents with the radical reaction centre. However, it is quite probable that the physical nature of this interaction would not be clarified completely but it would be possible to obtain a satisfactory quantitative description of the structure ± reactivity relationship as applied to the chemistry of free radicals.III. The use of r72 analysis in free-radical chemistry 1. Modelling of inductive and steric effects Previously, we have developed quantitative procedures for deter- mination of the steric and inductive effects of substituents in terms of discrete atomic contributions. Detailed description and theo- retical substantiation of the models proposed, aspects of their practical use and the inductive electronegativity concept based on them have been presented in our previous reviews.2±6 In the context of this review, we considered it pertinent to mention the main theses of these approaches.12 Steric interactions were determined in terms of the model of frontal steric effect in which the steric effect of a substituent is represented as a result of mechanical shielding of the reaction centre by the surrounding atoms ia1 Xn R2iR0S a , (11) 4r2i where R0S is the steric constant, n is the number of atoms in the substituent, Ri is the radius of an ith atom and ri is the distance from the ith atom to the reaction centre.It should be emphasised that in terms of this approach, the steric effect of any substituent at any reaction centre can be predicted with high accuracy on the basis of only fundamental parameters (the size and the distance to the reaction centre) of the atoms forming the substituent. The R0S parameters calculated for a broad range of the most frequently encountered groups follow a good correlation with the Taft steric constants (N=35, R=0.9854, S=0.141).In the model of inductive effect, the inductive constant s* of an n-atomic substituent was represented at the atomic additivity level (12) r2iia1 s aXn sAOiU , where sA is an empirical constant introduced in the model of inductive effect and reflecting the ability of an atom to exhibit the inductive effect, which depends on the nature and valence state of the atom. The sA values have been established for a wide range of elements; the inductive constants calculated theoretically using these values (altogether 426 substituents) have formed a high- quality correlation with the corresponding experimental data (N=426, R=0.9910, S=0.190). The practical use of this additive approach in the description of reactivity and investigation of the mechanisms of reactions involving organic and heteroorganic compounds proved fairly successful.This has also contributed to the solution of a number of important theoretical problems related to inductive interactions such as the inductive effect of alkyl substituents, the presence or absence of linearity in the inductive effects of substituents on carbon and heteroatom reaction centres and some others. The models developed make it possible to describe quantitatively only the inductive and steric effects; no approach based on the modelling principle for estimation of resonance interactions has been devised yet. Nevertheless, as shown in our study,4 the additive model of inductive effect describes rather adequately the electronic interactions in various conjugated systems and only some of them, subjected to the influence of direct polar conjuga- tion, can provide exceptions.2. Combinational approach. r72-Analysis. Ionisation energies of amines The new topological approach developed on the basis of the above models, which allows the substituent effect to be expressed in terms of discrete atomic contributions, was first reported in our publication and called r72-analysis.220 Since both the steric and inductive constants were expressed in Eqns (11) and (12) as functions of 1/r72, the two-parameter Taft equation (13) Y a r i i Xs a dXEs can be represented in terms of discrete atomic contributions (14) rc¢§i i6arc Y ¢§ Y0 aXN¢§1 ei , r2 where N is the number of atoms in the molecule, rc is the atom chosen as the reaction centre, rrc7i is the distance between atom i and the reaction centre.The dependent parameter Y can be A R Cherkasov,MJonsson, V I Galkin, R A Cherkasov represented by any physical value which is used in terms of a general correlation equation (13) (the logarithm of the rate or equilibrium constant, the reaction or activation energy, etc.), Y0 is the same parameter for a compound chosen as the standard in this reaction series, and e is a parameter reflecting the capacity of an atom of a definite type for exhibiting intramolecular effects determining the Y values. The approximation we used permits one, first of all, to verify whether it is possible, in principle, to introduce the operational atomic constants ei corresponding to atoms of different types and depending on the nature and the valence state.The next step is to determine their physical meaning. The formalism of this approach implies that each atom encountered in all the molecules of the reaction series can, in principle, be regarded as a hypothetical reaction centre. In this case, for each molecule contained in the reaction series, the other N71 atoms are treated as a single sub- substituent. After the reaction centre for a given series has been chosen, the matrix of sums 1 , r2 k k rc¢§m X rc¢§mk corresponding to the types of atoms present should be composed. A table should be made up, containing N vertical matrix elements (rows), where N is the number of molecules in the reaction series, and M horizontal elements (columns),where M is determined by the types of atoms represented in the molecules of the series. The k value in the matrix of sums corresponds to the number of atoms of type m in molecule n, and r2 are the distances between type m atoms and the reaction centre rc in molecule n.If no type m atoms are present in molecule n, the corresponding matrix element is taken to be zero. As an example, Table 3 gives the r72-matrix for the reaction series including the MeH2C , ClH2C, and Me(NH2)HC radicals in which the carbon bearing the unpaired electron is chosen as the reaction centre. In real calculations, the number of horizontal elements should exceed the number of vertical elements; then the columns of this matrix can be regarded as sets of independent arguments and the Y7Y0 array can be regarded as an array of dependent values of multiparameter linear regression (14).Table 3. r72-Matrix for the reaction series containing the MeH2C , ClH2C and Me(NH2)HC radicals. Type of atom Radical H C(sp3) Cl N(sp3) 0 0 MeH2C 1 1:542 2 3 1:092 a 2:162 2 0 0 ClH2C 1 1:762 1:092 0 Me(NH2)HC 1 1:462 1 1:542 1 3 2 1:092 a 2:162 a 2:042 Thus, if the corresponding multiparameter correlation can be established with a satisfactory accuracy, its linear coefficients are the operational atomic constants ei corresponding to particular types of atoms. It is clear that this scheme should be effective in the description of those reaction series that can be analysed quantita- tively using inductive and steric group constants.The method developed for the description of substituent effects has several obvious advantages over classical correlation analysis. First, it is free from the restrictions associated with the choice of an appropriate scale of substituents because it deals directly with interatomic distances and is conformationally sensi- tive. Second, the use of a simple mathematical apparatus allows one to analyse quickly and effectively indefinitely large sets ofCorrelation analysis in the chemistry of free radicals parameters. Third, within the framework of this model, it becomes possible to consider all the potential reaction centres of the series, which can provide additional information on the mechanism of the process under study.This is important in those cases where interpretation of the mechanism does not appear unambiguous and variation of the reaction centre could permit decision in favour of one of the most probable mechanisms. We implemented the algorithm outlined above as a product called RMATRIX, which was developed on the basis of the MATLAB program.221 The RMATRIX program allows import- ing of intramolecular distances from HyperChem or from similar files containing data on the spatial structure of the molecules of a reaction series; then it composes the r72-matrix corresponding to the reaction centre chosen for each molecular series, and, finally, it establishes statistically the operational atomic increments e for the series.3. The physical meaning of the operational atomic parameters A drawback of r72-analysis is that it does not allow direct consideration of resonance interactions because they cannot be described certainly by any distance function. However, a more substantial problem is that the physical meaning of the opera- tional atomic parameters ei, defined by Eqn (14), remains vague and does not provide the possibility of resolving the substituent effects into the inductive and steric components. The analysis of the ei values presented below is pertinent in this connection. The correlations established previously in terms of the steric and inductive models can serve as the basis for this consideration.Thus it has been found 4 that the sA constants for a broad range of elements are correlated with the difference between Pauling's electronegativities Dwi ¡À rc of a given element A and the atomic reaction centre (rc) (this difference reflects the ability of atomAto displace electron density) and with the squared covalent radius of the A atom (which reflects its capacity for charge delocalisation) 4, 5, 222 sA �� 7:84Dwi¡¦rcR2i . The back calculation of the w values using this relation makes it possible to introduce a new scale of `inductive' electronegativ- ities, which is purely empirical because it is based on the inductive constants describing a huge array of reaction series.4, 5 This scale is, by definition, in good agreement with Pauling's electronegativ- ities, except that for carbon.According to the `inductive' scale, the electronegativity of carbon is 2.1. This value appears to be more plausible than wP=2.55 proposed in the Pauling scale.2, 223, 224 The latter value implies that carbon should be more electro- negative than, say, phosphorus (wP=2.2) or iodine (wP=2.4), which is not the case in reality. The low polarity of the C7Hbond also indicates that the Pauling wP value for carbon is overesti- mated. Subsequently, a scale of group inductive electronegativ- ities and the concept of inductive chemical hardness, which interpret a number of widely used reactivity indices using simple geometrical characteristics of bonded atoms, have been developed in terms of this approach.2 These concepts were considered in detail in our review 2 and in a number of publications.223 ¡À 231 On the basis of established correlations, the Taft inductive constant can be expressed in terms of fundamental characteristics of the atoms of substituents and the reaction centre, namely, electronegativities, radii and interatomic distances (15) r2ii s �� 7:84XDwR2i.By superposition of Eqns (11) ¡À (14), the atomic parameters ei can be represented as the sum of inductive and steric constituents(16) ei �� aDwi¡¦rcR2i �¢ bR2i . Evidently, if a satisfactory correlation of type (16) has been established, it becomes possible not only to calculate the unknown ei values on its basis but also to express the dependent parameters 13 Y in terms of inductive and steric Taft constants as an ordinary two-parameter equation (13).If the `inductive' electronegativity of the reaction centre atom is unknown, the e value can be found using the two-parameter correlation (17) ei �� a0wiR2i �¢ b0R2i , where the coefficient b0 includes the wrc value. It should be emphasised that correct separation of the inductive and steric effects requires that the nature (electronega- tivity) of the reaction centre be known exactly and, in any case, it implies certain assumptions. Nevertheless, the corresponding atomic operational constants ei still can be employed to predict the unknown Y values for related molecular systems consisting of atoms with known ei even in those cases where correlations (15) and (16) cannot be established. Except for the inapplicability for the description of the resonance effect, the method under discussion does not have any serious limitations and can be used successfully even in those cases where the standard empirical scales are ineffective.The quantita- tive description of the vertical and adiabatic ionisation potentials of amines performed in our study exemplifies the practical application of the r72-scheme.221 Previously, the dependence of the ionisation potentials of a limited range of amines on their structure has been interpreted with utilisation of a large number of independent parameters both those reflecting particular inductive, resonance and polarisation effects and those chosen rather arbitrarily.232 7e +N NWe attempted to consider the ionisation potentials of amines in terms of the unified approach developed and composed the corresponding r72-matrix in which the nitrogen atom was chosen as the reaction centre.The adiabatic and vertical ionisation potentials (I ad, I vert) of amines in the gas phase, taken from a publication,233 were analysed in terms of the parameters of the basic equation (14). As a result, reliable multiparameter correla- tions were established for the sets of adiabatic and vertical potentials IRvert 3N �� �¢ const1 (R=0.954, S=0.331, N=287), (18) XN¡¦1 i vert i i r2i IRad XN¡¦1 i ad ir2ii��1 i i 3N �� �¢ const2 (R=0.960, S=0.341, N=231), (19) where N is the number of atoms in the amine molecule and i vert and i ad are the operational atomic parameters reflecting the capacity of the atom of a definite type for participation in intra- molecular interactions, which determine the I vert and I ad values, respectively.The following fact attracts attention: although we analysed arrays of the absolute values of I ad and I vert not referred to any standard, the free terms of multiparameter correlations (18) and (19) determined statistically actually reproduced the correspond- ing ionisation potentials of the ammonia molecule. The values of the i ad/vert parameters were then analysed using Eqn (16); the following correlations were identified: (20) i ad=4.55 (w72.79)R2, (21) i vert=4.38 (w72.98)R2. Thus, the correlations found allow the calculation of the unknown parameters i ad/vert and, hence, the previously unknown ionisation potentials of amines from atom electronegativities and covalent radii.Moreover, if the role of steric effect during electron transfer is assumed to be insignificant (which appears quite reasonable), values of 2.790.7 and 2.980.7 found using these14 relations are in a fairly good agreement with the inductive electro- negativity of nitrogen (2.56, see Ref. 223). 4. Ionisation energies of free radicals The possibility of quantitative determination of the energies of oxidation of amines to the corresponding radical cations using the approach we developed points to its adequacy and applicability to free-radical processes.As noted above, the examples of quantita- tive interpretation of the reactivity of free radicals currently available are limited, as a rule, to aromatic systems. We attempted to apply the r72-analysis to electron transfer, which is one of the least studied homolytic reactions from the standpoint of `radical' effects of substituents. We did not restrict ourselves to narrow series of aromatic compounds but considered single-electron processO- centred radicals both belonging to aromatic and aliphatic ser- ies.233, 234 The single-electron ionisation potentials of free radicals in the gas phase were taken from a database 235 and analysed in terms of our approach.a. Carbon-centred radicals In accordance with the approach developed, the ionisation potentials (I ) and electron affinities (Ae) of carbon-centred radicals were analysed using the equations i and IR ¢§ IMe a r2ii(R Xea AeOR U ¢§ AeOMe U a i(R Xe¢§ in which the I and Ae values for the methyl radical are included as reference values provided that the carbon atom bearing the unpaired electron is taken to be the reaction centre. Table 4. Operational atomic parameters eaemp found from experimental ionisation potentials and the corresponding eacalc values calculated using correlation (17). Atom Radical centre CNO C(arom.) S CHC(arom.) C(sp2) C(sp) Cl Br FSONN(sp) CHNC(arom.) CHOCl FBr ICHSFC(arom.) (22) i (23) , r2i R2 wi 0.593 0.090 0.449 0.449 0.390 0.980 1.299 0.409 1.082 0.436 0.490 0.3025 0.593 0.090 0.490 0.449 0.593 0.090 0.436 0.980 0.410 1.300 1.769 0.449 0.593 0.090 1.082 0.410 0.449 2.10 2.10 2.45 2.25 2.65 3.09 2.97 3.93 2.67 3.05 2.56 6.76 2.10 2.10 2.56 2.45 2.10 2.10 3.05 3.09 3.93 2.97 2.80 2.45 2.10 2.10 2.67 3.93 2.45 A R Cherkasov,MJonsson, V I Galkin, R A Cherkasov The multiparameter correlations (22) and (23) were estab- lished with high accuracy.The operational parameters e+ and e7 found from relations (22) and (23) are presented in Tables 4 and 5 together with the corresponding values for the inductive electro- negativities and the covalent radii of atoms.By comparing them in terms of a type (17) relation, we expressed the e+parameters in the formeai a O1:09 0:33UwiR2i ¢§ O5:49 0:97UR2i (the O, N and S atoms were not included in the correlation) or eai a 1:09Owi ¢§ 5:04UR2i . Thus, the ionisation potentials of C-centred radicals in the gas phase can be expressed in terms of fundamental parameters of atoms Owi ¢§ 5:04UR2i. I a IMe a 1:09 r2iia1 XN Relations (24) and (25) demonstrate that the vast majority of atoms [except highly electronegative ones such as sp-hybridised N atom (w=6.76)] introduced in the environment of the C reaction centre decrease the corresponding ionisation energy relative to that of the methyl radical.The deviation of the atomic parameters e+ for O,Nand S atoms from the general pattern of dependence is apparently due to the resonance effect, which cannot be described from the standpoint of the given approach, as has been repeatedly mentioned above. To elucidate the reasons for this deviation, a special detailed analysis is required; we plan to perform it subsequently. In addition, analysis of the predicted ionisation potentials demonstrates a substantial deviation of the Ipred value for the methyl radical, which is a typical problem for this type of approach. Since the methyl radical was chosen as the reference species (DI=0), its e+ parameter should, by definition, be equal eaemp eacalc 71.860.11 70.250.04 71.450.14 71.780.14 70.800.39 71.910.19 72.730.23 70.780.10 76.610.58 73.770.27 76.150.26 5.888 76.2021.361 70.2170.485 75.9962.652 71.9341.088 75.4070.534 0.0220.327 73.6630.799 76.3071.052 70.4980.748 78.9421.244 714.251.51 74.0710.348 73.3701.831 0.0031.089 78.6802.598 70.8141.848 71.90 70.29 71.27 71.37 70.55 72.09 72.94 70.50 72.78 70.95 71.33 0.562 75.70 70.87 74.46 74.14 76.22 70.94 72.93 76.45 71.34 79.17 713.66 74.09 74.57 70.69 77.25 71.88 73.19 72.4290.972 (24) (25) eaemp ¢§ eacalc 0.04 0.04 70.18 70.41 70.25 0.18 0.21 70.28 73.83 72.82 74.82 5.326 70.502 0.653 71.536 2.206 0.813 0.962 70.733 0.143 0.842 0.228 70.59 0.019 1.2 0.693 71.43 1.066 0.761Correlation analysis in the chemistry of free radicals Table 5.Operational atomic parameters e¢§emp found from experimental electron affinities and the corresponding e¢§calc values calculated using correlation (17). Atom Radical centre CNO C(arom.) S CHC(arom.) C(sp2) Cl Br FOCHC(arom.) CHOCl FBr ICHSOC(arom.) or nearly equal to zero, which is not always fulfilled. The e+(H) value found statistically is markedly smaller than zero, as well as the calculated DI(CH3) value. Taft 236 was faced with similar difficulties when using the methyl group as the reference for the scale of inductive constants s*.There are also other factors determining the so pronounced difference between the experimental and calculated ionisation potentials of methyl radicals in the construction of a scale of radical constants. Detailed analysis of these factors is beyond the scope of this review. When considering the e7 parameters found from the electron affinities of carbon-centred radicals, the following correlation was established: e¢§i a O1:64 0:43UwiR2i ¢§ O3:17 1:26UR2i or e¢§i a 1:64Owi ¢§ 1:93UR2i . As for ionisation potentials, this correlation permits the electron affinity parameters of carbon-centred radicals to be expressed at the atomic level on the basis of fundamental atomic characteristics, namely, electronegativities, sizes and interatomic distances Ae a AeOMe U a 1:64 ia1 XN The ionisation potentials and electron affinities of carbon- centred radicals calculated from the atomic inductive electro- negativities and covalent radii are listed in Tables 6 and 7; the statistical parameters of correlations are presented in Table 8, and the operational parameters e+ and e7 are given in Tables 4, 5.b. Deviations from the general dependence As noted above, the presence of substantial resonance interactions in radicals can result in deviation of their characteristics predicted in terms of the additive scheme in question from the correspond- R2 wi 0.593 0.090 0.449 0.449 0.980 1.299 0.409 0.436 0.593 0.090 0.449 0.593 0.090 0.436 0.980 0.410 1.300 1.769 0.449 0.593 0.090 1.082 0.436 0.449 2.10 2.10 2.45 2.25 3.09 2.97 3.93 3.05 2.10 2.10 2.45 2.10 2.10 3.05 3.09 3.93 2.97 2.80 2.45 2.10 2.10 2.67 3.05 2.45 (26) (27) Owi ¢§ 1:93UR2i.r2i e¢§emp e¢§calc 70.260.14 70.010.04 0.600.06 0.510.37 2.220.17 2.080.23 1.040.08 70.160.44 70.3960.255 70.0080.093 0.6870.091 70.1040.155 0.0250.098 71.6540.257 1.3280.340 0.9360.242 1.8390.402 2.3380.488 0.3790.111 70.410.386 0.17 0.03 0.38 0.24 1.87 2.22 1.35 0.80 70.39 70.059 0.69 0.19 0.029 0.66 1.527 1.07 1.83 2.11 0.34 70.008 70.001 72.09 71.36 70.52 70.120.261 71.840.457 71.950.482 70.020.236 ing experimental values.An alternative and, perhaps preferable, explanation of most of the deviations observed in the description of redox properties of carbon-centred radicals is that in some radicals, the carbon atom bearing the unpaired electron is not the ionisation centre. For example, in the case of ionisation potentials, the opera- tional atomic parameters e+ for S, N and O atoms deviate from the linear dependence (24), and the effects of the corresponding substituents observed experimentally exceed those predicted based on the e+ values. The experimental values of the ionisation potential Iexp might correspond to the I value of a heteroatom (S, N, O) present in the carbon-centred radicals and having a lone electron pair rather than to the I value of the radical centre.A similar deviation from the linear dependence has been observed in a study of the single-electron reduction potentials of arylmethyl- chalcogenide radical cations in aqueous solutions.40 In addition, electron-withdrawing substituents (NO2, CN, Hal) are able to enter into reduction processes and, hence, only for some of the carbon-centred radicals considered, do the Ae values correspond completely to the ionisation of the radical centre. For some strong electron acceptors, a saturation effect also cannot be ruled out. Thus, most of the observed deviations of the predicted ionisation potentials and electron affinities of substituted car- bon-centred radicals are most likely due to non-homogeneity of the data arrays considered.It should also be borne in mind that the ionisation potentials and the electron affinities of radicals match the energy differences between the radical and the corre- sponding cation in the former case and between the anion and the radical in the latter case. Hence, the energy effects of substituents described by Eqns (22) and (23) can refer both to the radical and to the corresponding ion. The positive values of e+ and e7 imply that the radical is stabilised by the substituent in question more efficiently than the cation as compared with the CH3/CHa3 pair (for ionisation potentials), and that the anion is stabilised by the given substituent more efficiently than the radical, compared with the CH¢§3 /CH3 system (in the case of electron affinity).Note that, the use of the notion of relative stabilisation can also imply lesser 15 e¢§emp ¢§ e¢§calc 70.43 70.04 0.22 0.27 0.35 70.14 70.31 70.96 70.006 0.051 70.003 70.294 70.004 72.314 70.199 70.134 0.009 0.228 0.039 70.403 70.121 0.263 70.596 0.50016 Table 6. Ionisation potentials (in eV) of C-, S-, N- and O-centred radicals determined experimentally (Iexp), predicted (Ipred) using relation (14) and calculated (Icalc) from Eqns (25), (28), (30), (32). Radical 3H5) MeNHCH2 Me(NH2)CH Me2NCH2 Me2(NH2)C (H2C=CH)2CH Me2(HC:C)C (cyclo-C5H9) HSCH2 MeSCH2 H3C Me3C MeEtCH PrnCH2 PriCH2 MeCH2 Me2CH EtCH2 HOCH2 Me(HO)CH MeOCH2 Me(MeO)CH NCCH2 H2NCH2 Cl3C Cl2HC ClH2C F3C F2HC FH2C Br3C Br2HC BrH2C MeCF2 MeCHF H2C=CHCH2 (cyclo-C CH2=C(Me)CH2 (H2C=CH)MeCH (cyclo-C4H7) H2C=CHCH2CH2 (cyclo-C5H9) H2C=CHCMe2 H2C=CHCHEt MeCH=CHCHMe H2C=CHCH(Me)CH2 H2C=C(Et)CH2 (C6H5) PhCH2 HO (ref) a HOO ClO FO BrO IO MeO EtO PrnO PhO HS (ref) a MeSS FS MeS EtS PrnS Iexp Ipred 6.1 5.72 5.74 4.98 7.3 7.44 7.22 7.32 7.07 9.21 77.41 87.81 8.47 7.74 8.14 7.51 6.77 7.13 6.4 9.9 6.45 7.99 8.4 8.8 8.57 8.79 97.59 8.13 8.67 8.05 8.26 8.25 7.87 7.95 7.52 7.41 8.05 7.11 6.78 7.19 7.39 7.71 7.66 8.31 7.24 13.04 11.35 10.89 12.78 10.46 9.66 10.39 9.47 9.1 8.56 10.43 8 10.16 9.38 8.92 8.73 5.9 5.7 5.7 5.4 7.25 7.44 77.54 6.85 9.84 6.7 7.25 8.02 7.93 8.12 7.37 8.09 7.56 6.85 6.9 6.5 9.9 6.29 8.06 8.32 8.75 8.76 8.78 9.04 7.5 8.3 8.61 7.92 7.93 8.18 8.18 7.9 7.49 7.54 8.04 7.21 7.13 7.3 7.07 87.9 8.32 7.24 13.017 11.35 10.885 12.78 10.46 9.66 10.72 9.11 9.2 8.56 10.429 8 10.16 9.262 9.6 8.2 A R Cherkasov,MJonsson, V I Galkin, R A Cherkasov Iexp7Ipred 70.20 70.02 70.04 0.42 70.05 0 70.22 0.22 70.22 0.63 70.3 70.16 0.02 0.12 70.36 70.37 70.05 0.05 0.08 70.23 0.1 0 70.16 0.07 70.08 70.05 0.19 70.01 0.04 70.09 0.17 70.06 70.13 70.33 70.07 0.31 70.05 70.03 0.13 70.01 0.1 0.35 0.11 70.32 0.29 0.24 0.01 70.01 70.02 000000.33 70.36 0.11 0000 70.12 0.68 70.53 Icalc Iexp7Icalc 8.23 7.86 7.86 7.11 7.73 7.51 7.38 8.44 8.19 9.11 6.88 7.28 7.88 7.68 8.37 7.62 8.03 8.82 8.07 8.43 7.68 9.18 8.6 7.82 8.25 8.68 9.03 9.06 9.08 7.42 7.99 8.55 8.31 8.34 8.42 7.77 8.11 7.67 7.29 8.04 6.98 6.93 7.33 7.54 7.69 7.81 8.46 7.42 11.99 11.45 10.84 12.38 10.4 9.8 9.31 8.13 7.62 8.14 10.04 8.05 9.81 8.64 7.93 7.61 72.33 72.16 72.16 71.71 70.48 70.07 70.38 70.91 71.34 0.73 70.18 70.03 0.14 0.25 70.25 70.25 0.06 71.25 71.22 71.53 71.18 0.72 72.31 0.24 0.07 0.07 70.27 70.28 70.04 0.08 0.31 0.06 70.39 70.41 70.24 0.41 70.21 70.18 0.25 00.23 0.2 70.03 70.47 0.31 0.09 70.14 70.19 1.024 70.095 0.048 0.399 0.064 70.137 1.406 0.978 1.581 0.425 0.386 70.052 0.35 0.625 1.673 0.594Correlation analysis in the chemistry of free radicals Table 6 (continued).Radical PhS H2N (ref) a MeNH Me2N PhNH H2N7NH a The reference species in the corresponding reaction series. Table 7. Electron affinities (in eV) of C-, S-, N- and O-centred radicals determined experimentally [Ae(exp)], predicted [Ae(pred)] using relation (14) and calculated [Ae(calc)] from Eqns (27), (29), (31) and (33).Radical H3C MeCH2 MeCH2CH2 Me2CH MeEtCH Me3C H2C=CHCH2 PhCH2 Ph2CH Ph3C F3C F2HC FH2C Cl3C Cl2HC ClH2C Br3C BrH2C CF3CF2 MeOCH2 HO (ref) HOO ClO FO BrO IO MeO EtO PrnO PhO PriO ButO HS (ref) MeSS MeS EtS PrnS PhS HSS HOS PriS ButS H2N (ref) MeNH Me2N PhNH PhMeN Ph2N Iexp Ipred 8.63 10.36 7.55 4.75 8.26 7.61 8.63 10.78 6.7 5.17 8.26 7.61 Ae(pred) Ae(exp) 0.05 70.05 70.09 70.16 70.2 70.27 0.36 0.62 1.2 1.77 1.76 1.19 0.62 2.23 1.5 0.78 1.79 0.63 1.76 70.02 1.83 1.08 2.28 2.27 2.35 2.38 1.62 1.72 1.79 2.25 1.84 1.91 2.25 1.80 2.12 2.05 2.02 2.26 1.86 1.65 1.98 1.90 0.77 0.60 0.43 1.57 1.40 2.37 0.08 70.26 70.07 70.32 70.12 70.16 0.36 0.91 1.36 1.56 1.84 1.21 0.20 2.17 1.58 0.80 1.73 0.82 1.81 70.02 1.83 1.08 2.28 2.27 2.35 2.38 1.62 1.72 1.79 2.25 1.84 1.91 2.31 1.75 1.87 1.95 2.00 2.26 1.90 1.65 2.02 2.07 0.78 0.45 0.38 1.70 1.65 2.18 Iexp7Ipred 00.43 70.85 0.43 00Ae(exp)7Ae(pred) 0.03 70.21 0.02 70.16 0.08 0.11 0.00 0.29 0.16 70.21 0.08 0.02 70.42 70.06 0.08 0.02 70.06 0.19 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.07 70.05 70.25 70.10 70.02 0.00 0.05 0.00 0.04 0.16 0.01 70.15 70.05 0.13 0.25 70.19 17 Icalc Iexp7Icalc 0.81 1.698 70.003 0.846 3.442 0.222 7.82 9.08 6.7 4.32 4.82 7.39 Ae(calc) Ae(exp)7Ae(calc) 0.14 0.21 0.24 0.27 0.30 0.34 0.27 0.49 0.84 1.19 2.27 1.56 0.85 1.89 1.31 0.73 1.91 0.73 2.26 0.52 1.86 2.14 2.34 2.34 2.35 2.33 1.94 1.98 1.99 2.21 2.01 1.96 2.32 1.81 2.31 2.31 2.31 1.93 1.81 1.87 2.31 2.31 0.66 0.50 0.34 1.48 1.32 2.30 70.06 70.47 70.31 70.59 70.42 70.50 0.09 0.42 0.52 0.37 70.43 70.35 70.65 0.28 0.27 0.07 70.18 0.09 70.45 70.54 70.03 71.06 70.07 70.06 0.00 0.05 70.32 70.26 70.20 0.04 70.17 70.05 0.00 70.06 70.45 70.36 70.32 0.32 0.09 70.22 70.30 70.24 0.12 70.05 0.04 0.22 0.33 70.1218 Table 8.Statistical parameters for correlations of type (14) determined for various classes of free radicals. Series Type of radicals a and sC¢§, which N R S/ eV make it possible to calculate the I and Ae values for carbon- C-centred 0.2345 0.1995 0.9768 0.9816 48 20 IAe S-centred 0.6105 0.1491 0.9278 0.8390 7 10 IAe O-centred 0.3557 0.1150 0.9938 0.9836 10 12 IAe N-centred 1.041 0.2179 0.968 0.9744 56 IAe destabilisation. In a published discussion 9¡¾11 dealing with the influence of substituents on bond energies, similar considerations are presented concerning the situation where the effects observed can be attributed both to the RX molecule and the R radical.Equation (25) indicates that a substituent atom should have an abnormally high electronegativity (w >5.04) to stabilise a radical more effectively than the corresponding cation. In turn, relation (27) illustrates the fact that for most of atoms (w >1.93), stabilisation of anions predominates. Thus, it can be concluded that the described substituent effects which determine the electron potentials of free radicals refer to stabilisation of the correspond- ing ions. The role of steric interactions can be elucidated by comparing the adiabatic ionisation potentials, studied within the framework of the approach under discussion, with the corre- sponding vertical ionisation potentials.The adiabatic potentials reflect the difference between the energies of the radicals and the cation having the most favourable (equilibrium) spatial structure, whereas vertical potentials imply identical geometries of these species. Thus, the difference between the adiabatic and vertical potentials can be attributed to structural reorganisation of a free radical caused by its oxidation. The adiabatic and vertical potentials of the methyl radical are known to be equal. This is apparently due to the fact that both the radical and the cation are planar. The other primary, secondary and tertiary carbon-centred radicals are non-planar, unlike the corresponding cations.The difference between their adiabatic and vertical potentials remains approximately constant (0.3 ¡¾ 0.4 eV) and does not change significantly on passing from primary to secondary or tertiary radicals, i.e., the steric contribution to the adiabatic ionisation potentials is insignificant but they do contain a constant component due to the energy of spatial reorganisation. The absence of this component in the case of the methyl radical can be the main reason for the observed difference between the predicted and experimental I(CH3) values. The foregoing attests that steric interactions have no influence on the ionisation potentials and electron affinities of free radicals and the corresponding effect of substituents has a purely inductive nature.It is common knowledge that cations are stabilised by elec- tron-donating substituents and anions are stabilised by electron- withdrawing substituents. According to Eqn (15), the atomic inductive effect is proportional to the difference between the electronegativities of the given atom and the reaction centre. Hence, it is reasonable to assume that the electronegativity of the central atom (carbon) increases in the series anion < radical < cation, while the inductive effect of the substituent atoms switches from electron-withdrawing to electron-donating influence, and only in extreme cases [for example, for N(sp), which has wind=6.76], is the effect of the atom adjacent to the cationic centre maintained as the electron-withdrawing influence.Thus, parameters of 5.04 and 1.93 in Eqns (24) and (26) can actually be taken as the inductive electronegativity of the reaction centre, i.e., of positively and negatively charged carbon, respectively. A R Cherkasov,MJonsson, V I Galkin, R A Cherkasov c. The group sCa and sC¢§ scales The empirical operational parameters e+ and e7 established have also been used to introduce group scales called sC centred radicals in terms of an additive scheme.233 a A 0 aPsC I a I e a A0e aPsC¢§ These parameters were calculated for the most frequently encountered organic substituents with the assumption of the standard geometries and using the e+ and e7 values for atoms contained in these substituents.These radical scales, based on electron transfer reactions and considering the substitution effect at the group level of additivity, reflect, as do the Hammett s-constants, the overall influence of substituents on a radical reaction centre without separation of the inductive, resonance, and steric constituents. The use of group sCa and sC¢§ scales simplifies the procedure of calculation of the ionisation energies of radicals and eliminates the necessity of determining interatomic distances. Similarly, we calculated the ionisation energies of O-, S- and N-centred radicals in terms of the basic equation (14). The atomic operational parameters e+ and e7 found in this way were then analysed using relation (16).The corresponding equations for radicals of various types are presented below in the same order which was accepted for carbon-centred radicals.234 d. Oxygen-centred radicals For oxygen-centred radicals, the operational parameters e+, which determine the relative contributions of atoms to the ionisa- tion energy of the system, are described with high accuracy by means of atomic electronegativities and the covalent radii in accordance with Eqn (16) eai a O3:94 1:01UwiR2i ¢§ O18:77 2:93UR2i or eai a 3:94Owi ¢§ 4:76UR2i . The corresponding atomic parameters e7 are determined in a similar way e¢§i a O1:25 0:27UwiR2i ¢§ O2:31 0:79UR2i or e¢§i a 1:25Owi ¢§ 1:84UR2i .Thus, the ionisation energy of the alkoxy radicals I(RO ) can be presented by the relation (28) , Owi ¢§ 4:76UR2iIORO U a IOHO U a 3:94 r2iia1 XN and the electron affinity of oxygen-centred radicals is determined in terms of atomic parameters A (29) Owi ¢§ 1:84UR2i. eORO U a AeOHO U a 1:25 r2iia1 XN e. Sulfur-centred radicals The atomic operational constants e+ and e7 of sulfur-centred radicals determined from the ionisation potentials and the elec- tron affinity correlate with the atomic electronegativities and atomic radii with a lower accuracy eai a O1:71 2:19UwiR2i ¢§ O11:29 5:93UR2i or eai a 1:71Owi ¢§ 6:61UR2i ,Correlation analysis in the chemistry of free radicals e¢§i a O¢§3:27 1:41UwiR2i a O6:84 3:68UR2i or e¢§i a ¢§3:27Owi ¢§ 2:09UR2i .Nevertheless, the expressions obtained which relate the ion- isation energies of sulfur-centred radicals to parameters of the atoms forming them appear well substantiated (30) Owi ¢§ 6:61UR2i. IORS U a IOHS U a 1:71 r2iia1 XN The electron affinity of sulfur-centred radicals is found from the relation A (31) Owi ¢§ 2:09UR2i. eORS U a AeOHS U ¢§ 3:27 r2iia1 XN ea f. Nitrogen-centred radicals A similar correlation obtained for nitrogen-centred radicals is unsatisfactory because of the deficiency and great inaccuracy of the corresponding experimental data; therefore, the available data on the ionisation of N-centred radicals cannot serve as the base for discussion.Nevertheless, we thought it fit to present the relations determining the e+ and e7 parameters for atoms making up N-centred radicals. As in the above cases, the electronegativities and the sizes of atoms forming the corresponding N-centred radical were used. i a O1:14 10:70UwiR2i ¢§ O12:01 24:9UR2i or ea e¢§i a 1:14Owi ¢§ 10:55UR2i , i a O6:24 0:41UwiR2i a O13:76 0:92UR2i , or e¢§i a 6:24Owi ¢§ 2:20UR2i . The ionisation energies and electron affinities of nitrogen- centred radicals can be written as follows: (32) Owi ¢§ 10:55UR2i, IOR2N U a IOH2N U a 1:14 r2iia1 XN A (33) Owi ¢§ 2:20UR2i. eOR2N U a AeOH2N U a 6:24 r2iia1 XN The ionisation potentials and the electron affinities of S-, N- and O-centred radicals calculated on the basis of the atomic inductive electronegativities and covalent radii are given in Tables 6 and 7.The electronegativities of ionic forms of the C, S, N and O atoms considered as reaction centres, obtained in accordance with the proposed procedure, are given in Table 9. Table 9. Electronegativities and statistical parameters for correlations of type (16) determined for various types of free radicals. N R S/ eV Type of radicals Parameter wrc e+ C-centred e7 0.9509 0.266 0.9502 0.328 8 a 7 b 5.04 1.93 e+ S-centred e7 0.9368 1.3768 0.8739 0.5329 55 6.61 2.09 e+ O-centred e7 0.9891 0.7342 0.9808 0.1975 87 b 4.76 1.84 e+ N-centred 10.55 2.20 e7 0.8393 1.9877 0.9977 1.041 43 a Except for parameters for the O, N, S and N(sp) atoms; b except for parameters for the O atom.19 The same Table contains the statistical parameters of the corre- sponding correlations of type (16) found for the the operational atomic parameters e+ and e7 of the series studied. 5. The stability of free radicals in redox processes We have already noted that the problem of radical stability, especially within the framework of the thermodynamic approaches proposed here, largely overlaps the problem of radical reactivity and is considered from the standpoint of the influence of substituents. The stability of free radicals itself in terms of the relative energies of covalent bonds in the corresponding systems with filled electron shells is normally considered separately from the known effect of steric shielding of the unpaired electron by the molecular environment, decreasing the reactivity of the radical.45, 114, 237 In our opinion, the concept of radical stability (reactivity) should be supplemented by one more type, namely, stability of free radicals in redox processes (subsequently referred to as redox stability).As has already been noted, single-electron transfer is one of the most frequently encountered types of reaction in homolytic chemistry. Meanwhile, neither kinetic nor thermodynamic methods for the description of the radical effects of substituents take into account the redox stability of free radicals or provide the possibility of quantitative interpretation of the corresponding single-electron potentials.However, it is absolutely clear that the lower the ionisation potential and the higher the electron affinity of a free radical, the more easily it would enter into single-electron transfer reactions. Thus, the redox stability of a free radical can be matched to its absolute chemical hardness (Z) Z a I ¢§ Ae, 2 where I and Ae are the single-electron ionisation potential and the electron affinity of the radical in the gas phase determined experimentally. Let us consider carbon-centred radicals. Using relations (25) and (26), we express the relative chemical hardness of a carbon- centered radical (Z) in terms of fundamental atomic characteristics used to describe its single-electron potentials. Owi ¢§ 4:22UR2iZORU a ZOMeU ¢§ 0:28 r2iia1 XN This relation reflects the trend of the redox stability of C-centred radicals to decrease under the influence of the vast majority of atoms present in the environment of a carbon reaction centre; only the most electronegative elements increase the ZORU value with respect to the chemical hardness of the methyl radical ZOMeU.As noted above, the stability of carbon-centred radicals is often expressed as a derivative of the dissociation energy of the C7H bonds. A lower energy implies higher radical stabillity. However, a low bond energy often corresponds to a low chemical hardness of the corresponding radical. In conformity with our approach, this, conversely, implies a lower redox stability of the radical.Hence, the criteria used to describe radical stability should be chosen with caution. 6. Energies of dissociation of C¡¾H bonds An attractive aspect of our approach is, in our opinion, the possibility of using it successfully to describe dissociation energies of bonds Ed. We demonstrated this taking carbon ¡¾ hydrogen bonds as an example.238 We obtained a large array of Ed(CH) values using the following relation i E (34) . dOR3CHU ¢§EdOCH4U a i C¢§i X d r220 Here Ed(CH4) is the energy of the C7H bond in the methane molecule, Ed(R3CH) is the energy of the C7H bond in the corresponding derivative; the correlation parameters are: N=72, R=0.9773, S=8.484 kJ mol71. Equation (34) allowed high-accuracy determination of atomic constants which were designated by d and corresponded to the operational parameters ei in the base equation (14).We were unable to determine the parameters d using the inductive and steric constituents and a correlation of type (15). Thus, the relationship between the redox and thermodynamic [expressed in terms of relative Ed(CH)] stabilities of free radicals remains uncertain. Nevertheless, the operational values d can be used successfully to calculate unknown energies of the C7H bonds. IV. Conclusion To summarise, it can be stated that the use of polar constants for the description of the reactivity and physicochemical properties of free radicals is rather limited, although in some cases, it is quite successful.The scales of specific radical constants s are not general either. The absence of coordination between them and the ambiguity of the physical meaning of the s radical constants might be due not only to the drawbacks of methods of their determination but rather to the limitations of principle inherent in the application of the linear methodology of correlation analysis to the free-radical chemistry and to the violation of additivity of the group influence in paramagnetic systems. Thus, the problem of using correlation analysis in the chem- istry of free radicals is far from being solved; in this respect `homolytic' chemistry is markedly inferior to the `heterolytic' chemistry of organic compounds. The new approaches to the quantitative description of the structure ± reactivity (property) relationships for the participants of homolytic processes which we propose give hope for elimination of this objectively formed unbalance and make the quantitative organic chemistry more harmonious in some respect. The model we developed, based on the consideration of discrete contributions of atoms, depending on the remoteness, to the substituent effects makes it possible to predict with high accuracy the quantitative characteristics of free radicals obtained experimentally.The method implies taking account of the influ- ence of the molecular environment on the reaction centre without preliminary resolution of the effect into electronic and steric constituents. This approach provides the possibility of analysing efficiently diverse experimental results, irrespective of the nature of reaction centres.Within the framework of our approach, we have considered only one of the three main types of free-radical processes. The quantitative description of the ionisation potentials and the electron affinities of diverse C-, N-, S- and O-centred radicals and the ionisation energies of aliphatic and aromatic amines permitted us to identify the physical nature of the corresponding substituent effects and to demonstrate the versatility of this approach in interpretation of the results of quantitative inves- tigations in the chemistry of organic and heteroorganic free radicals. In our opinion, of fundamental importance is the established fact that the substituent effects determining the electronic poten- tials of free C-, N-, O- and S-centred radicals are correlated with the stability of the corresponding ions and can be regarded as purely inductive.Nevertheless, correct quantitative determination of these values requires that the electronegativities of the ionic forms of C, N, O and S acting as the reaction centres be established. á and sC¡ scales introduced on the basis of atomic The sC operational parameters e+ and e7, which determine the overall effects of substituents in single-electron transfer reactions involv- ing C-centred radicals, exhibit high predictive capacity, although they do not clarify the physical nature of the corresponding interactions. 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