MOLECULAR ORBITALS AND ORGAfIC REACTIONS By R. D. BROWN M.Sc (THEORETICAL PHYSICS DEPT. KING'S COLLEGE STRAND LONDON W.C.2) 1. Introduction THE first attempt to apply the approximate quad um-mechanical treatment known as the method of orbitals to the prob1t:m of chemical reactivity in conjugated organic molecules was made over fifteen years ago.2 Since then and particularly during the last five years :onsiderable attention has been devoted to this hybrid of quantum mechanics and organic chemistry. Although other approximations have been invoked it has become clear that the molecular-orbital technique has partic1 lar advantages when one is dealing with the relatively complicated molecules which interest the organic chemist. Accordingly the present Review will be devoted pre- dominantly to the results obtained by this method.We have aimed a t something rather more critical than a mere enumeration of the relevant literature and an attempt has been made to incicate some of the aspects which call for further attention. The development of the molecular-orbital ti eory of organic reactions has proceeded along two rather distinct lines. On the one hand the approximation has been used to calculate quantiti1:s such as charge densities q,2 bond orders and free valences P,4 of the vsrious bonds and positions in a particular molecule. The assumption has tien been made that these quantities are an index of the reactivities of the F arious positions in certain chemical reactions. This procedure bears a close analogy to the widely used qualitative electronic theory in which the assumption is made that reactivity in ionic substitution reactions is determined by the electronic charge to be associated with a position a t the ;noment of reaction.We shall have to examine more closely the validity oE the assumptions implicit in this treatment of organic reactions. The second approach to the problem is pe:-haps best regarded as a simplified version of the theory of absolute reaction rates.5 The technique consists in calculating energies of activation foi the various positions in a molecule a necessarily oversimplified model being used for the activated complex. An indication of these simplifications s necessary in any review of this procedure which has been termed the locrdisation theory of organic reactions. Since in both of these treatments only the 3,-electrons are considered any effects due to o-electrons are assumed constant and hence ignored.'Lennard-Jones Trans. Faraday SOC. 1929 25 6ri8; Hund 2. Physik 1928 51 759; Mulliken Phys. Review 1928 32 186. aCoulson Proc. Roy. SOC. 1939 A 169 413. 41dem Trans. Paraday Xoc. 1946 42 265. Wheland and Pauling J . Amer. Chem. SOC. 1935 f17 2086. Glasstone Laidler and Eyring " The Theory of Rate Processes " McGraw- Hill 1941. 63 E 64 QUARTERLY REVIEWS Clearly a more comprehensive theory would include the latter effects but a t present the theoretical treatment of a-electrons has not been sufficiently developed to meet this requirement. However recent detailed investiga- tions of molecules such as benzene and ethylene have indicated that it is justifiable to neglect the interaction between n- and a-electrons when considering chemical reactivities.Likewise entropies of activation are assumed to be constant for a particular type of organic reaction; this has some experimental justification.' At present the calculation of entropies of large organic molecules and their activated complexes by means of partition functions is not feasible. Throughout the ensuing sections reference will frequently be made to relative reactivities of organic molecules. These will always be taken to be characterised strictly by the reaction rate constants under comparable conditions of solvent temperature etc. or alternatively by relative yields in competitive reactions. It is also convenient to partition the overall reactivity of a molecule into reactivities of the various positions (or pairs of positions in addition reactions) in the molecule on an experimental basis of relative yields of isomers.Before reviewing the achievements and defects of the theoretical methods it will be necessary to consider the precise significance of some of the molecular-orbital quantities which they employ. 2. The Molecular-orbital Method The molecular-orbital approximation has been adequately described elsewhere * g 9 lo and we shall here only consider the quantities involved in discussions of chemical reactivity. The n-electrons of the conjugated system are considered to occupy molecular orbitals y, which it is hoped will be reasonably well represented as a linear combination of atomic orbitals (L.C.A.O.) #j being a 2pz atomic orbital centred upon atom j . The coefficients crj have t o be determined in such a way that in (1) they make yr the best possible representation of the rth molecular orbital in this form.They are found by a standard technique 9 11 l2 which at the same time produces the energy of the rth molecular orbital. The coefficients are of course pure numbers but the energies * of the orbitals are found in terms of a quantity y termed the bond integral of a benzene bond [defined in (12)]. To carry out the calculations of energies and coefficients it is necessary to Altmann private communication. Stubbs and Hinshelwood J. 1949 571 ; Ingold and Nathan J. 1936 222 ; Wr = xjcrj+j * (1) Evans and Haman Trans. Paraday SOC. 1951 47 40. Coulson and Longuet-Higgins Proc. Roy. SOC. 1947 A 191 39. * Coulson Quart. Reviews 1947 I 144. lo Lennard-Jones ibid. 1949 A 198 1.l1 Eyring Walter and Kimball " Quantum Chemistry " Wiley 1944 p. 254. la Brown Australian J . Sci. Res. 1949 A 2 564. * The origin of the energy scale is frequently assumed to be such that the coulomb integral of carbon ac is zero. BROWN MOLECULAR ORBITALS AND OIbGANIC REACTIONS 65 ascribe to each atom a quantity ai called the crouZomb integral of the ith atom and to each bond quantities bij and X j termed respectively the resonance integrul and overlap integral of the boiid between atoms i and j. These three quantities are defined formally as ai = J +iX+i.dt * (2) Bij = J +iX+jj.dt i # 3 . . (3) Sij = J +i+j.dr . - (4) where X is the so-called Hamiltonian operator which corresponds to the total energy of the n-electron. The volume eement dz is a shorthand notation for dxdydz (or the equivalent in other co-ordinate systems) and the range of integration is from + co to - co for each variable.In the case of &j and Xij it can be shown that the valiie of the integrals is much greater when i and j are neighbours than otherw se so when values of these integrals are discussed it is nearly always tacitlJr assumed that i a n d j are neighbouring atoms of the conjugated system. In practice a( and /?ij are generally not evaluated from (2) and (3) but arc treated as empirical para- meters in the theory. In chemical applications t is particularly important to appreciate the physical significance of these 1)arameters. They are dis- cussed in detail below but first we must consicier some of the quantities which can be calculated once the Coefficients and energies have been determined.It has been found convenient to define quaitities termed charge den- sities ~ i ~ and mobile bond orders pig,' in terms of the coefficients. More specifically qi = Crvrcz . * (5) pij = Crv,CriCrj . * (6) where vr represents the number of n-electrons occupying the rth molecular orbital and in accordance with the Pauli prin,iple can have the values 0 1 or 2 only. For a conjugated system * of 2n atoms there will be 2n orbitals and for molecules in the ground statc which will be the state involved in most chemical reactions Y,. will be f for each of the n orbitals of lowest energy 0 for all other orbitals. Since t ne values of the coefficients cri depend upon the values assigned to R Pij and Xij for every atom i and bond i:j in the conjugated system qi and pij will also depend upon the values of these parameters.Two other quantities free valence F,4 and localisation energy,? have been employed for discussing Ehemical reactivitj . The free valence of the ith atom Fi is defined as pi = Nrnax. - cjpij - (7) where the summation is over all atoms j whicli are adjacent to i. N,,,. * Hydrogen atoms are not considered as part of the ctmjugated system ; e.g. phenol is regarded as containing a conjugated system of 7 atcms. t The symbol employed for this quantity depends upoii the type of localisation energy considered. Atom localisation energies are frequently d 3noted by A bond localisation energies by B and para-localisation energies by P. It ivould appear to be convenient to use the general symbol L when the type of localisation need not be specifled.In the literature 8 variety of symbols has been used in place of A. This point is mentioned again below. 66 QUARTERLY REVIEWS is a constant usually taken * to be d3. Localisation energies are calculated from the total n-electron energies of various conjugated systems. It is evident that if the energy of the rth molecular orbital is e then the total n-electron energy of the system.wil1 be E = Xrvrer the sum as usual being taken over all orbitals. The localisation energy of a position (or pair of positions) may be regarded as the energy required to localise a certain number of electrons (2 1 or 0) a t that position (or positions). When certain positions together with an appropriate number of electrons are thus localised they are effectively removed from the con- jugated system.The conjugated system (or sometimes two or more separate systems) which remains after this removal is termed the residuaZ moZecuZe.12 The localisation energy is thus the difference in energy of the residual molecule and the original molecule where L represents a localisation energy in general and the superscript r indicates a property of the residual molecule. When a n-electron is localised upon a carbon atom its energy ccc is zero (according to the conventional zero of energy ; see footnote p. 64) but if n-electrons are localised in a bond their energy is not then zero and this ethylene-type system is con- sidered as part of the residual molecule when calculating L. The bond localisa- tion energy l2 is the energy required to localise two of the n-electrons around two adjacent atoms of the conjugated system.The bond localisation energy of the 9 10-bond of phenanthrene is the energy required to isolate the 9- and the 10-position with two n-electrons from the rest of the molecule. The residual molecule is therefore comprised of two separate conjugated systems-ethylene and diphenyl-and the bond localisation energy is the n-electron energy of diphenyl and ethylene minus the n-electron energy of phenanthrene . It is important to assess the precise physical significance of these quan- tities since they have been very extensively used in the discussion of chemical properties of conjugated systems. Clearly the total n-electron energy E represents the best possible value that can be obtained for the energy of a particular system by means of the simple L.C.A.O.molecular-orbital technique. Parr 13 has indicated that energies so calculated are related to those derived by using Roothaan’s self-consistent-field L.C.A.O. molecular- orbital approximation 149 l5 if we re-interpret cci and /?ij as integrals involving L = E T - E The concept is perhaps made clearer by an example. Is J . Chem. Physics 1951 19 799. l4 Roothaan Rev. Mod. Physics 1951 23 69. l5Mulliken J. Chirn. physique 1949 46 497 675. * Unfortunately a wide variety of values of Nma. has been employed in the literature and particular care is needed to avoid errors due to using values from the literature which are not derived from the same value of Nmax.. There are also good reasons for employing other values of Nmx. when the ith atom does not have two neighbouring carbon atoms in the conjugated framework.Sometimes the a-bond contribution is included in p and then N-. has to be increased by 3. BROWN MOLECULAR ORBITALS AND ORGANIC REACTIONS 67 an operator M slightly different from X. There is scme indirect evidence 16 that for cyclic alternant hydrocarbon systems (i.e. not containing odd- membered rings) the molecular-orbital approximations to n-electron energies are remarkably self-consistent but a t present we cannot be sure that this applies also to radical systems ions or heteromolecules. The relation between qi and the charge density defined rigorously in terms of the square of the total n-electron eigenfunction I !P 12 is rather more subtle. If we suppose that the atomic orbitals are orthogonal i.e. Xij in (4) is zero for all i ;;f j then the total n-electron density is easily shown to be &qi suggesting that qi can be taken to give a useful measure of the fraction of the n-electron charge to be associated with atom i.How- ever the overlap integral is known to be appreciable for neighbouring atoms and strictly this requires a more elaborate definition l7 of qi than (5). For hydrocarbons the more elaborate definition of qi can be shown1* to give results identical with those of ( 5 ) but for heterocyclic systems this is unfortunately no longer true. The mobile bond order pij has no precise physical significance although it can be shown to be related to the length of the i:j-bond. It is probably more satisfying to associate qi and pij with the total n-electron energy through the identities 3 qi = aE/acci .- (8) 2pij = aE/a& . (9) Again there is no clear-cut physical significance for free valence although it is now considered to characterise the " localisability " of an atom in the conjugated system (see Section 4). We intuitively associate free valence with the fact that an atom has not participated in bonding to the extent to which it is capable. Other molecular-orbital quantities which arise in connection with chemical properties are polarisabilities nitj and nij,mn,s termed atom and bond polaris- abilities respectively. * The former tells us the rate of increase in qi with increase in the coulomb integral of thejth atom while the latter represents the rate of change of mobile order of the i:j-bond with change in resonance integral of the m:n-bond. For our purposes they are most conveniently defined by ni,j = a2E/&iaaj .* (10) %j,mn = a2E/aBijaBmn * (11) Finally we must return to consider the coulomb resonance and overlap The coulomb integral ai is a negative quantity t with the dimensions integrals in more detail. 16Brown J. 1951 1612. Wheland J. Amer. Chem. SOC. 1942 64 900. 18 Chirgwin and Coulson Proc. Roy. SOC. 1950 A 201 196. * Self-atom polarisabilities are usually termed auto-polarisabilities in the French literature. t Strictly if the conventional zero is taken to be q (see footnote p. 64) coulomb integrals of some atoms e.g. boron will presumably be positive. However the elements commonly encountered in organic compounds-N 0 halogens-are all more electro- negative than C and have negative coulomb integrals. 68 QUARTERLY REVIEWS of energy and clearly can be considered as a property of the atom i.Values of a for various atoms of a conjugated system represent roughly their relative affinities for n-electrons ; i.e. we may regard Q as a measure of the electro- negativity of atom i. This will in general depend on the position of the atom in the conjugated system and also on the n-electron density in its neighbourhood qi as well as on its atomic number (i.e. nuclear charge). In any self-consistent calculation this dependence must be included. The dependence upon position has not been considered so far and probably would not have any significant eEect upon the prediction of chemical pro- perties. Several attempts 199 2o have been made to allow for the dependence of the coulomb integral upon qi but although it is clear that 01i is a function of q; it is not at present apparent what form this functional relationship must take.Consequently in the great majority of calculations of the various quantities described above it has been assumed that oli is a function only of the atomic number of i and perhaps of its nearest neighbours. To illustrate the magnitude of values used for the coulomb integrals of various atoms the values for C N and 0 which at present appear to be most reason- able are listed in Table 1. Most of the methods which have been used to estimate these quantities must unfortunately be relegated to the realm of inspired guesswork although Mulliken 1 5 9 21 has recently attempted to derive more reliable values from experimental ionisation potentials. TABLE I.* Representative values of the parameters a and 18 * Note that is customarily used in two distinct senses as the resonance integral of any arbitrary bond and also in the present Table as the resonance integral of a standard C:C bond (frequently taken to be a benzene bond).In this Review the two usages are distinguished by always using /3 with subscripts (e.g. Bij) for the former. The use of a single value for the coulomb integral of carbon can be shown 22 to lead to the self-consistent results of a uniform n-electron charge distribution in the case of a certain class of hydrocarbons called alternants (those whose conjugated system does not contain an odd-membered ring). This is no longer the case for heterocyclic systems or for non-alternant hydrocarbons such as azulene; i.e. atoms with the same assumed value of Q do not have the same qi.For this reason it is to be expected that the simple molecular-orbital results for alternant hydrocarbons will be rather inore reliable than those for heterocyclic and non-alternant systems. Originally Wheland The hetero-atom sulphur calls for special attention. 19Wheland and Mann J. Chem. Phys& 1949 17 264. 20Laforgue J. Chim. physique 1949 46 568. 21Mulliken Phys. Review 1948 74 736. 2 2 Coulson and Rushbrooke PTOC. Cumb. Phil. SOC. 1940 36 193. BROWN MOLECULAR ORBITALS AND ORGANIC REACTIONS 69 and Pauling attempted to treat the thiophen molecule by attributing an appropriate value to as and subsequent computations for thiazole and benzothiazole 23 24 followed along the same lines. The problem has been considered in detail by Long~et-Higgins,~~ who emphasised the importance of the 3d orbitals.According to him the sulphur atom is regarded approxi- mately as equivalent to a double bond with reduced resonance integrals.26 An alternative treatment by Moffitt 27 is not so easily applicable to large conjugated systems. Some calculations have been performed in which cci is assumed to depend upon the atomic numbers of all atoms adjacent to i (as well as of i itself of course). in an attempt to obtain charge densities which would predict the correct chemical reactivities.* For example the coulomb integral of carbon is sometimes considered to be greater in magnitude when it is adjacent to a hetero-atom than when it is surrounded only by other carbon atoms. Qualitatively this increase would be expected from the charge induced upon the carbon atom by the hetero-atom ; i.e.this treatment is equivalent to a very crude self-consistent treatment in which is allowed to depend somewhat upon qi. Since it can be shown 28 that the charges alternate as one proceeds outwards from a hetero-atom we might expect the coulomb integrals to be enhanced and reduced in magnitude alternately so the assumption 29 that cc = ac + anP for a carbon removed n from a hetero- atom 6 being a positive constant of the order of 1/3 is somewhat question- able. The basis of the latter assumption is experimental and depends upon the effect of substituents removed n atoms from the carboxyl group of a fatty acid upon the free energy of disso~iation.~~ However it is not easy to see how this phenomenon involving saturated carbon atoms is to be related to the coulomb integrals of unsaturated carbon atoms.It will appear in the following sections that as yet no discrepancy between values of theoretical quantities and observed chemical reactivities can be traced definitely to the neglect of this " inductive " effect. We therefore take the attitude that as few parameters as possible should be introduced into the theory and in later sections this additional parameter will be referred to as the auxiliary inductive parameter to distinguish it from the major hetero-atom parameters exemplified in Table 1. The resonance integral ,& also is a negative energy quantity which is to be regarded as a property of the bond between atoms i andj. Roughly This procedure was initiated by Wheland and Pauling 23Pullman and Metzger Bull.SOC. chim. 1948 15 1021. 24 Metzger and Pullman ibid. p. 1166. 26Evans and de Hem Acta Cryst. 1949 2 363. Proc. Roy. SOC. 1950 A 200 409. 28 Coulson and Longuet-Higgins ibid. 1947 A 192 16. 29Dewar J. 1949 463. 30 Branch and Calvin " The Theory of Organic Chemistry " Prentice-Hall 1941 p. 217. *They did not consider the possibility of crossing of energy curves discussed in section 6 when the charge densities would not be expected to predict the correct reactivities. 25 Trans. Paraday SOC. 1949 45 173. 70 QUARTERLY REVIEWS we may picture it as a measure of the strength of the n-electron bond between these atoms. It will mainly depend upon the length of the bond159 31 and the atomic numbers and n-electron densities of the atoms i andj. Starting with the assumption that all bonds have the same value of Pij we may calculate the mobile orders pij [equation (S)].These may then be used to estimate the bond lengths and so to obtain corrected values for the /Iij and this iterative process may be repeated until the values no longer alter. Plainly this is a laborious process for large conjugated systems and i t has been performed only in the case of a few relatively simple molecules.31t 32 The dependence of ,8ij upon qi and qj does not seem to have been considered but is presumably just as important as the dependence of ai or aj upon these quantities discussed above. The dependence upon atomic number of i and j has however been considered. For example the values of pij for C:N and C:O bonds relative to Pij for C:C has been taken into account by various Representative values of Pij for various bonds are listed in Table 1.They are based mainly upon rather crude thermo- chemical considerations 3 3 9 35 of bond energies and again must be regarded as rather tentative. Mulliken has discussed in detail the effects of various structural details upon resonance integrals,15 but his results concern reson- ance integrals in the more elaborate self-consistent molecular-orbital approximation and cannot be used directly for our present purposes. The overlap integral Xij is quite simply interpreted physically and has been the subject. of considerable attention the~retically.~~ 37 38 It repre- sents the amount to which the 2pn atomic orbitals on neighbouring atoms of the conjugated system overlap one another. In contrast t o the former two parameters it is possible to make quite reliable theoretical estimates of Xij.Most of the numerical calculations of Sij have been carried through by using the well-known Slater approximations to the 2pz atomic wave functions. In benzene Xij is found to be quite close to 0.25. The major uncertainty associated with the value of the overlap integral is that the effective nuclear charge of a carbon atom for a 2pn-electron is not very precisely known. Indeed this lack of knowledge of the effective nuclear charge is one of the central problems of all molecular quantum mechanics. In the case of hydrocarbons when all the coulomb integrals are equal and all resonance integrals are taken to be proportional to the corresponding overlap integrals which would appear to be a relatively good approximation and is the usual assumption made in treatments of both alternant and non-alternant hydrocarbons it can be shown18 that the values found for the qi and pij are independent of the constant of proportionality between pij and Xij.In particular this means that the results obtained by assuming 3 4 5 35 31Lennard-Jones Proc. Roy. SOC. 1937 A 158 280. 32Altmann Proc. Physical SOL 1950 A 63 1234. 33 Coulson Trans. Faraday SOC. 1946 42 106. 34 Longuet-Hjggins and Coulson ibid. 1947 43 87. 35 Orgel Cottrell Dick and Sutton ibid. 1851 47 113. 36Mulliken Rieke Orloff and Orloff J . Chem. Physics 1949 17 1248. 37Maccol1 Trans. Faraday SOC. 1950 46 369. 38Mulliken J . Arner. Chern. SOC. 1950 '72 4493. BROWN MOLECULAR ORBITALS AND ORGANIC REACTIONS 71 that Sij is zero will apply equally if this assumption is dropped.For hetero- cyclic molecules this is no longer the case however. In order to distinguish energy quantities computed with the assumption that X = 0 from the same quantities computed with 8 = 1/4 the former are usually represented by ordinary italic-type symbols and the latter by the same symbols with a prime. For example the total n-electron energy is represented as E or E’ according to the value assumed for S. In addition the energy unit in the former case is written as p the standard resonance integral while in the latter case it is written as y the standard bond integral. The two quantities are inter-related 39 y = p - s S a c . . (12)* On the scale of energy usually chosen ,8 and y are equivalent because ac is zero but in general they are only equal if X = 0.Since ,6 and y are always determined from experimental data the currently accepted values (p = - 17 kcal. ; y = - 34 kcal.) include contributions which help to correct for any absolute errors inherent in the molecular-orbital approxi- mations. Consequently the relationship between these “ experimental ” values is unknown and we are not justified for example in using these two values in (12) to compute ac. Finally it should be borne in mind that all of the molecular-orbital results discussed in subsequent sections of this review have been calculated under the assumption that cci and /& depend only upon the atomic numbers of i and j unless a contrary statement is made. In general discussions of localisation energies no distinction will be made between energies calculated with inclusion of overlap from those computed for X = 0 because the quali- tative results do not appear to be affected by this choice except in some unimportant cases connected with the Diels-Alder reaction.40 However the procedure used to obtain any numerical data quoted in subsequent sections will be indicated by the prime convention just described and by the appropriate use of /I or y .It should be noted that when localisation energies have been converted into relative rate constants the data atways have been derived by assuming X = l/4 since the inclusion of overlap seems to increase the quantitative significance of results.16 3. Kinetic Basis of the Theoretical Treatment Ideally rates of organic reactions should be discussed in terms of activa- tion energies and frequency factors or entropies of activation.The mole- cules encountered in this field are generally so complex that it is not at present feasible to attempt to evaluate the relevant partition functions owing to insufficient knowledge of vibrational frequencies and of geometry of activated complexes. Therefore the assumption has to be made that for a series of related reactions such as the addition of maleic anhydride to a series of similar polycyclic hydrocarbons the entropy of activation 39Mulliken and Rieke J. Amer. Chem. SOC. 1941 63 1770. 40Brown J. 1950 2730. *Unfortunately the symbols and y are generally used in the opposite sense in American literature. 72 QUARTERLY REVIEWS remains sensibly constant (or alternatively varies uniformly along the series). Experimental evidence in support of the assumption has already been mentioned,' and the point has also been discussed in textbooks ; 41 42 it would appear to be a reasonable working hypothesis in the absence of more detailed information.However the fact that analogous assumptions in the case of chemical equilibria lead one astray over the equilibrium between bromine and various substituted phenanthrenes 43 should serve as a warning. It is clear that in the future this aspect of organic reactions must be given further experimental and theoretical attention. If we agree to assume constancy of entropy factors then the relative reactivities of various positions in a given molecule or of two different molecules will be determined solely by the relative energies of activation Let us consider the variation in energy of a system as i t undergoes reaction choosing for example the addition of a molecule of maleic anhydride to the 9 10-positions of anthracene.In the course of reaction two of the n- electrons from the anthracene portion interact with the two n-electrons in the maleic anhydride to form two new 0-bonds ; a t the activated complex stage these will be only partly formed. In addition the 9- and the 10-atom of anthracene which were originally in a state of sp2 hybridisation and likewise the two carbon atoms of maleic anhydride pass over continuously into their final states of sp3 hybridisation. Presumably the hybridisation will be a t some intermediate stage in the activated complex. An absolute theoretical computation of the energy changes accompanying these processes would be intractable so the assumption must be made that they would be very much the same if we replace the anthracene portion by some other hydrocarbon.Clearly this assumption is much more likely to be valid if the other hydrocarbon is very similar to anthracene in general structure e.g. 1 2-benzanthracene but it would not be surprising if it is found that these energy changes are appreciably diflerent for a hydrocarbon such as butadiene. In view of these approximations which of course are included in any electronic theory of organic reactivity we must be guided j in the first instance by experiment ! in deciding whether these assump- 1 tions are reasonable. In the Diels- I Alder reaction of anthracene deriv- atives it does appear 16 that this approximation is justifiable. Course o f reaction - FIG.1 The foregoing assumptions reduce the problem to a consideration of the n-electrons only. The change in n-electron energy accompanying an organic reaction is indicated schematically in Pig. 1. The curve represents the total energy E of the system of n-electrons including those which 4 1 Hammett " Physical Organic Chemistry " McGraw-Hill 1940 Chap. IV. 4 2 Amis " Kinetics of Chemical Change in Solution " Macmillan 1949 Chap. VI. 43Fieser and Price J . Amer. Chem. SOC. 1936 58 1838. BROWN MOLECULAR ORBITALS AND ORGANIC REACTIONS 73 eventually become a-electrons. The stage a corresponds to the separated reactants e to the separated products and c to the activated complex or transition state. If we are comparing the reactivities of different positions in a given molecule or in different molecules then we shall have a separate energy curve for each possible reactive position.Since we are assuming constancy of entropy of activation the most reactive position will be that for which the increase in energy from a to c is least. It is not yet possible to make reliable computations of the z-electron energy for the activated complex except in a few special cases so we have to be content with two FIU. 2 FIG. 3 FIU. 4 FIU. 6 alternative approximations based upon idealised models of the transition state. These in effect are estimations of the increase in n-electron energy from a to b (the “ isolated molecule approximation ”) or from a to some part of the curve between c and e possibly a hypothetical extrapolation to d (the “localisation approximation ”). If therefore it is wished to discuss the relative reactivities of two different positions r and s towards a specific reagent the best that can be done by using the present theory is to determine the relative positions of the potential-energy curves for r and s at stage by or some stage after c.Various possibilities arise as illus- trated in Figs. 2-5 * which have been drawn so that in every case r reacts more rapidly than s. If r and 8 are in different molecules the curves will not in general start from the same point but to facilitate the comparison of the relative increases in energy it is convenient to subtract a constant energy from one curve so that the starting points again coincide. *The two curves are represented as starting from the common point a. 74 QUARTERLY REVIEWS Fig.2 represents a case where the theoretical treatment would lead to an incorrect prediction of relative reactivities unless the localisation approxi- mation applied to a stage of the reaction between stage c and the stage where the curves cross for the second time when the two approximations would give contradictory predictions. It is encouraging that so far there seems to be no definite example of this double crossing of energy curves but this may be due to the meagre quantitative experimental data available for the reactions so far studied in detail theoretically. Reactions characterised by energy curves of the type represented by Pig. 3 will be very well accounted for by the isolated molecule approxima- tion but the reliability of the localisation theory will depend upon the relative positions of the stage corresponding to the model assumed for the activated complex and the crossing point x'.Fig. 4 represents the opposite behaviour. Some examples of these types of theoretical behaviour have been found but all too frequently the relative reactivities have not been determined experimentally or the theoretical data are rendered uncertain by doubts as to the appropriate values for some of the The most satisfactory behaviour for theoretical treatments would be that illustrated in Fig. 5. Here quantities which tell us the relative positions of the energy curves a t any stage of the reaction path will be satisfactory criteria of both relative reaction rates and relative stabilities of products.* The Diels-Alder reaction and the osmium tetroxide oxidation appear to belong to this classification.It will be convenient as a shorthand notation to speak of a chemical " non-crossing rule ".f Thus if the isolated molecule approximation is in agreement with the observed reactivities we shall say that the non-crossing rule applies between stages a and c thus avoiding a more lengthy explanation in terms of the n-electron energy curves. It should be realised however that this " rule " is purely a description of the relationship between theoretical data and other theoretical data or experimental data although especially in the case of the Diels-Alder and osmium tetroxide reactions it appears to hold almost universally. We have several times referred to the isolated molecule approximation and the localisation approximation. We must now discuss these in more detail before considering their application to specific reactions.and ,&. 4. The Isolated Molecule Approximation The isolated molecule approximation represents an approximately quan- titative extension of the well-known and widely used qualitative theory of organic reactions 4 4 in which the reactivity of a position in an ionic reaction is considered to be determined by the n-electron charge a t that position at the moment of reaction. More rigorously it may be said that it represents 4 4 Ingold Chern. Reviews 1934 15 225. * The discussion of stabilities in terms of energy changes involves the same type of assumptions about entropy terms etc. as the discussion of rates of reaction in similar terms. This of course is in no way related to the well-known spectroscopic non-crossing rule.BROWN MOLECULAR ORBITALS AND ORGANIC REACTIONS 75 an approximate method of determining the relative positions at the point b of the n-electron energy curves for the attack of various positions in a molecule starting from the common point a of course (Fig. 1). Let us consider the effect of an approaching reagent. An incipient bond is formed between the reagent and the position undergoing substitution and this brings about a change in hybridisation of this position from the sp2 type representative of conjugated systems towards the sp3 type characteristic of saturated compounds. This may be regarded as approximately equiva- lent to a partial localisation of this position such as would be brought about if the resonance integrals of all bonds proceeding from this position were reduced in magnitude.Consider the change in n-electron energy due to approach of the reagent to the rth position of a molecule this position having as neighbours my n . . . We shall suppose that an effective change in the resonance integrals Prm Prn . . . has been brought about by the reagent. The energy change may be expanded as a Taylor series the first term being 6E = ~ m ( a E / a P r m ) ~ P r m = Z-n2prmSPmZ where the summation is over all neighbours m. If we suppose that the change in all resonance integrals is the same and equal to XP then this becomes 6E = constant - 2x. Fr. fl . * (13) If we recall that Pjj is always negative so that a reduction in its magnitude will require x to be negative also then the smallest increase in energy will occur for the greatest value of the free valence F,.On this basis we should expect the most reactive position initially to be that with the highest free valence F . Clearly if the reagent attacks several positions simultaneously (e.g. dienophils in diene reactions) then the most reactive set of positions will be that for which XF is greatest. This effect will come into play only when the reagent has approached quite close to the conjugated system because the formation of an incipient covalent bond is necessary to bring about an appreciable change in hybridisa- tion. However if the attacking reagent is a charged entity such as those commonly responsible for electrophilic substitution in aromatic compounds then another effect will arise a t an earlier stage in the reaction. If the reagent is a positive ion then as it approaches the conjugated system the n-electron system of the latter will be perturbed and will tend to move towards the approaching ion.A similar drift is obtained theoretically if the coulomb integral of the position being attacked is decreased," so it is a reasonable approximation to assume that the effect of the ion is to reduce the coulomb integral of the position under attack. We may expand the energy change as a Taylor series in dw. thus 6E = (dE/da,)dar + (d2E/da",(6a,)2/2 . . . = q r b + nr,r(dar)2/2 + - - - ' (14) * This nearly always corresponds to an increase in magnitude see footnote p. 67. 76 QUAflTEBLY REVIEWS where the identities (8) and (10) between differential coefficients and q and n have been employed. Since for a positive ion Sa is negative the smallest increase in E will occur for the largest value of q, i.e.the greater the charge density q, the greater the initial electrophilic reactivity of the rth position. In the case of nucleophilic substitution the coulomb integral of the position undergoing substitution will be decreased in magnitude i.e. 6a will be positive. This will mean that the smaller the charge density the greater the initial nucleophilic reactivity. If all charge densities happen to be the same then both electrophilic and nucleophilic reactivity will be determined initially by the self-polaris- ability nr,r. This will mean that in a given molecule the same position will initially be preferentially attacked in both classes of substitution. Since the self-polarisabilities are all negative * the most reactive position will be that whose polarisability is greatest in magnitude.In the case of ionic reagents there will also occur the partial localisation effect discussed above depending upon free valence. It is convenient to distinguish these two separate effects of reagents by terming the first one the localisability egect and the second the perturbability efect. For reactions such as homolytic substitution or the Diels-Alder addition where the attacking entity is a neutral molecule the perturbability effect will be negligible and the relative positions of the energy curves in the region b will be determined mainly by the relative localisabilities of the various positions (i.e. F or XFr). In heterolytic substitution the perturbabilities will be the first to assume importance and presumably play the dominant r61e in determining the relative positions of the energy curves in the region b for this type of reaction.However when the first-order perturbabilities of the positions (i.e. the qr) are the same so that the perturbabilities differ only a t the second order (the nr,r) it is not easy to assess the relative importance of localisabilities and this aspect of the theory of which the heterolytic substitution reactions of alternant hydrocarbons provide an important example calls for further investigation. It is to be expected that when the relative perturbabilities are in opposition to the relative localisabilities the crossing of energy curves as in Fig. 4 will be more likely. Some possible examples of this will be considered in a later section. 5. The Localisation Theory Let us again consider our example of the addition of maleic anhydride to anthracene.In the activated complex the reactants are linked by two incipient a-bonds involving two of the original n-electrons of the anthracene system. These n-electrons will then be rather more localised than the rest in the region of these partial a-bonds and we may regard the formation of the activated complex as a process of partial localisation of two of the n-electrons at those points in the anthracene molecule a t which addition eventually takes place. It is not a t all easy to compute the energy of * Polarisabilities are generally calculated in units of I/& which is a negative reciprocal-energy unit. BROWN MOLECULAR ORBITALS AND ORGANIC REACTIONS 17 such a system of partially localised n-electrons but if we suppose this localisation process to proceed to completion it is a simple matter to com- pute the energy change or the locaZisation energy as it is frequently termed in terms of the energies of the original and residual molecules as explained in Section 2.It is this extreme localisation approximation which is the basis of the localisation theory of organic reactivity. Of all sets of positions which could possibly react with the attacking reagent the most reactive set will be that with the smallest corresponding localisation energy. For various possible sets of positions in the same molecule the most reacti.ve will be that for which the corresponding residual molecule is the most stable i.e. has the lowest n-electron energy or highest resonance energy.In the Diels-Alder reaction this complete localisation is achieved in the final adduct corresponding to position e in Fig. 1 so we are assuming that the heat of activation is related qualitatively to the heat of reaction i.e. it is assumed that the non-crossing rule mentioned in Section 3 holds between stages c and e so that the relative positions of the energy curves a t e indicate the relative positions a t c also. The above discussion of the localisation theory as applied to the Diels- Alder reaction applies equally well to the reaction of conjugated systems with osmium tetroxide the idealised model for the transition state again being equivalent to stage e in Fig. 1. The same is not true of the localisation treatment of other reactions in particular of aromatic substitution reactions.Let us consider the example of the nitration of benzene. The reaction proceeds by electrophilic attack on benzene by the NO$ i0n.~5 In this case the benzene system must supply both of the electrons to form the incipient a-bond so in the activated complex there will be a partial localisa- tion of two of the n-electrons a t the position undergoing substitution. Again in the localisation treatment the extreme view is taken that this localiaation is complete. The difference from the foregoing case is that this idealised model of the transition state does not correspond to the final product. It corresponds to a hypothetical extrapolation of the energy curve represented by the point d in Fig. 1. Here again the use of the localisation approximation to predict reactivities involves the assumption of a non-crossing rule this time between c and d.The thermal cis-trans-isomerisation of substituted ethylenes forms yet another class of reactions from the viewpoint of the localisation theory. If we consider say stilbene then in the localisation approximation it is assumed that in the activated complex the n-electrons as a whole undergo a partial localisation into two equal groups in two independent benzyl radicals. In this case the localisation model for the transition state cor- responds quite accurately to the actual activated complex c. We may thus classify the localisation treatments into three classes depending upon the model assumed for the activated complex ( a ) a hypo- thetical idealised model ( b ) the final product (c) the actual transition state.It will be convenient to consider reactions in each of these categories in turn. ai Braude Ann. Reports 1949 46 131. 78 QUARTERLY REVIEWS 6. Application to Specific Reactions Reactions of Class (a).-The most important reactions of this class are the aromatic substitution reactions. For our purposes they may be sub- divided into ionic (electrophilic and nucleophilic) and radical (homolytic) reactions. In the ionic reactions it will be remembered that the initial reactivities are determined primarily by the relative perturbabilities but that the localisabilities may become significant if the former happen to be very similar ; on the other hand for homolytic reactions the localisabilities will assume primary importance. In the localisation approximation the electrophilic and nucleophilic reactivities will be determined by the atom localisation energies A and A, while radical reactivities will be charac- terised by A,.If the isolated molecule approximation and the localisation theory differ in their predictions of relative reactivities we are faced with a crossing of energy curves as has already been discussed in Section 3. In such cases no definite theoretical predictions of reactivities can be made unless there is some evidence to indicate at what stage of the reaction the energy curves cross. Hence to summarise the difficulties which are to be associated with the theory of aromatic substitution it may be said that reliable predictions of the position of highest reactivity may be made ( a ) in electrophilic substitution when the position with greatest q is also that of smallest A, and the free valences do not vary greatly or the position with greatest q also has a high value of F ; ( b ) in nucleophilic substitution when one position has both the lowest q and lowest A, together with reservations upon F similar to those under ( a ) ; (c) in radical substitution when the position of highest free valence has the smallest value of A,.These conditions are fulfilled rather more frequently than might have been expected. We must now examine the theoretical results for specific conjugated systems in the light of the above discussion. It will be convenient to group the conjugated systems into separate types. Alternant hydrocarbons are those whose con- jugated systems contain only even-membered rings. Benzene naphthalene butadiene styrene and all common polycyclic hydrocarbons polyenes and arylpolyenes belong to this class.Such molecules have two important properties (i) For every position in the conjugated system qi is unity.22 This means that (cf. p. 76) the relative perturbabilities are characterised by the relative values of ni,i. (ii) For every position in the conjugated system Both (i) and (ii) require that electrophilic and nucleophilic substitution will occur preferentially at the same position. In (i) since the first-order perturbabilities are all equal the first-order localisabilities characterised by P may be of comparable importance to the second-order perturbabilities in determining the initial reactivity. We should expect however since the position of smallest A or A is also that of smallest A, that this position 46Brown Trans.Faraday SOC. 1950 48 146. Alternant hydrocarbons. A = A = Ar.46 BROWN MOLECULAR ORBITALS AND ORGANIC REACTIONS 79 would also have the largest value of P (if the isolated molecule and localisa- tion approximations are to be consistent for radical substitution) and this is found to be almost invariably the case. . It may be said then that the molecular-orbital theory of the reactivity of this class of hydrocarbons is completely self-consistent and so is likely to be very reliable. Some repre- sentative values of A( = /3 .ni,J A and P for the most reactive positions in some alternant hydrocarbons are listed in Table 2. TABLE 2. Alternant hydrocarbons Benzene. . . . . . Naphthalene . . . . Anthracene . . . . . Phenanthrene . . . . Diphenyl . . . . . Butadiene .. . . ' . Hexatriene . . . . . Position 1 0.398 0.443 0.526 0-442 0-423 47 0.626 0.685 2.536 2.299 2-013 2-31 3 2.400 4 7 1.644 1.524 I 0.399 0.452 0.520 0.451 0.437 0.838 0.861 * Data for A i = S . q r mostly taken from ref. (9). t Data from unpublished calculations by the Reviewer. 3 Recalculated for Nm=. = 4 3 . Energy of residual molecule taken from Whelax~d.~* The only alternant hydrocarbon at present known for which the theory is not self-consistent is diphenylene (I).46 In this case the position of greatest & and smallest A is position 2 while position 1 has the greabest free valence. The numerical values of all of these quantities are very similar for the 1- and the 2-position so the discrepancy is not very large. However diphenylene represents a case where the non-crossing rule breaks down for homolytic 'w( substitution and may possibly break down for ionic sub- stitutions ; it is unfortunate that the course of substitution in diphenylene has not yet been investigated experiment- ally.For all other alternant hydrocarbons so far investigated the non- crossing rule is valid. It will appear after discussion of other organic reactions that this is apparently a general property of alternant hydro- carbons. Conjugated hydrocarbons containing odd- membered rings are termed non-alternant hydrocarbons. Azulene acenaph- thylene and fluoranthene are examples which have been synthesised and various other systems have been considered theoretically. Neither of the properties (i) and (ii) of alternant hydrocarbons is applicable to non- alternants. Their particular theoretical interest arises from the fact that no parameters are required to allow for hetero-atoms while the lack of uniformity of electron densities provides a greater variety of reactive positions to be predicted and checked experimentally.Furthermore if it can be demonstrated that the molecular-orbital calculations are reliable in 6 \ (1) Non-alternant hydrocarbons. 47 Brown Experientia 1950 6 376. 48 J . Amer. Chem. Soc. 1941 63 2025. F 80 QUARTERLY REVIEWS this non-self-consistent field approximation where the theoretical method is least likely to be reliable (p. 68) they can then be applied to all other systems in particular hetero-systems in which further difficulties due to parameters arise with greater confidence. One of the most interesting molecules of this class is azulene (II).49 The free valences are greatest in positions 1 and 4 the former being also the position of greatest q and the latter of least q.Consequently the perturbabilities and localisabilities together point to preferential electrophilic attack of position 1 and nucleophilic substitution a t position 4. The atom localisation energies also indicate these respective reactivities so the non-crossing rule holds and these predictions of reactive positions can be made with confidence. The free valence of position 4 is just slightly greater than that of position 1 and the +position also has the smallest value of A, so again the non-crossing rule applies and we can confidently expect the 4-position to be most rapidly attacked by free radicals. Recently the chemistry of azulene has been studied experimentally and confirmation obtained for the high electrophilic reactivity of the 1-position.60 The hypothetical hydrocarbons pentalene (111) 46 51 and heptalene (IV) 46 61 represent cases where the perturbability and localisability effects work in opposition for electrophilic substitution.However the electron 6 *qL> - 8 7 (11) (111) (Iv) (V) densities are so widely different that they doubtless outweigh the free- valence differences. The atom localisation energies indicate the same relative positions of the energy curves as the charge densities i.e. the non-crossing rule again applies. Likewise for radical substitution the values of 3’ and A reveal that the non-crossing rule is valid. A rather different behaviour is shown by another non-alternant hydro- carbon fulvalene (V) which has also not been prepared although some dibenzo-derivatives 62 and the tetrabenzo-compound 53 are known.Here the non-crossing rule appears to break down in nucleophilic substitution. The free valences and charge densities of positions 1 and 2 61 are in opposition for nucleophilic attack and the latter differ in magnitude more than the former so we might expect the initial ease of attack to be characterised predominantly by the charge densities. These predict position 2 to be more rapidly attacked whereas the localisation energy is smaller for the l - p o ~ i t i o n . ~ ~ It might be thought that the higher terms in the perturb- 49 Brown Trans. Faraday XOC. 1948 44 984. 50Anderson and Nelson J . Amer. Chem. SOC. 1950 72 3824. 51 Brown Trans. Faraduy SOC.1949 45 296. 6 8 Vallette Compt. rend. 1951 232 1494. 53de la Harpe and van Dorp Ber. 1875 8 1048. BROWN MOLECULAR ORBITALS AND ORGANIC REACTIONS 81 ability expansion (14) are sufficiently different for the two positions to explain a crossing of the energy curves at some stage between b and d but the self-polarisabilities are very similar indeed.54 Consequently we appear to have here a case where the discrepancy between q and A is due to the localisability effect. We cannot be sure of the relative positions of the energy curves at c and no prediction of the position of highest nucleophilic activity is possible from these data. This illustrates how careful one must be in the correlation of theoretical data with chemical reactivities. For non-alternant hydrocarbons it may be said then that the non- crossing rule frequently applies but that exceptions do occur sufficiently frequently to cast doubt upon predictions of reactivity based upon q F or A alone.Incomplete theoretical data are available for many other non-alternant hydro~arbons.5~9 66 57 Heterocyclic S~tems.-Practically all of the conjugated heterocyclic systems contain either entirely six-membered rings or else one or more five-membered rings. The former may be regarded as being derived from alternant hydrocarbons by replacement of one or more carbon atoms by hetero-atoms and the latter are derived similarly from non-alternant radicals or molecules. Since the greatest amount of theoretical attention has been given to molecules such as pyridine and quinoline which belong to the first category we shall for the most part consider compounds of this class.The application of the molecular-orbital approximation to these hetero- sys tems involves further difficulties beyond those encountered for hydro- carbons. The major problem is to decide upon an appropriate set of parameters for the hetero-atoms. Superimposed upon this is the additional difficulty already present for non-alternant hydrocarbons that the simple molecular-orbital procedure is not self-consistent (p. 68). The use of values of parameters (including values of p or y ) determined by comparison with experiment doubtless introduces a partial correction for absolute errors inherent in the L.C.A.O. approximation. Consequently values which have been deduced from studies of say dipole moments will not necessarily be the most suitable for theoretical treatments of chemical reactivities.In keeping with our previously expressed opinion (p. 69) we shall give most attention to those results in which a minimum of parameters have been used i.e. one parameter for the coulomb integral of each type of hetero- atom and perhaps one for each type of bond (C:N C:O etc.) as exemplified in Table 1. Sandorfy 68 has shown that if nitrogen is taken to be more electronegative than carbon (Le. if aN = ac + hp and h is positive) then for a wide range of positive values of h and of pCN the charges at the 2- 3- and 4-positions are a11 less The pyridine molecule has been studied in detail. 64Brown Nature 1950 165 566. 66Bandorfy Compt. rend. 1948 227 198; 1950 230 961. s6 Pullman Pullman and Rumpf Bull. SOC. chim.1948 15 757. 67Brown J. 1961 2391. 6 8 B ~ U . SOC. chim. 1949 18 616. 82 QUARTERLY REVIEWS than unity * ( L e a the isolated molecule is deactivated relatively to benzene) the 3-position having the highest charge density. The effect upon the free valences is not so clear-cut but in general the 2- and 4-positions have higher values than the 3-position. Here then the localisabilities are in opposition to the perturbabilities for nucleophilic attack. The atom localisation energies,? A, A, and A, have been obtained 59 only for one set of para- meters (those of Table 1). They predict a similar reactivity to the q and P namely electrophilic substitution preferentially at position 3 homolytic and nucleophilic attack most rapidly at the %position and so the non- crossing rule again holds.A and q both indicate that the electronic activity of the 3-position of pyrj,dine is similar to that of a benzene position. How- ever this is not apparent experimentally because the common electrophilic reactions take place in acid media in which the pyridine is converted into a much less reactive pyridinium ion. The predictions are in accord with the known properties of pyridine bromination nitration and sulphonation occurring at the 3-position pre- dominantly,60 and amination methylamination and dimethylamination in the 2-’p0sition.~O The formation of 2-vinylpyridine from ethylene and pyridine in a hot tube 61 possibly indicates the position of highest homolytic activity. Quinoline has also been studied fully the parameters of Table 1 being used.62 The most interesting point is perhaps the prediction of the relative reactivities of the various positions towards electrophilic substitution.The theoretical results make it likely that the energy curves will cross soon after stage b because although position 5 has a lower electron density than the 6- or the 8-position it has the greatest & and highest free valence. The atom localisation energies show that at stage d its energy curve is below those of all positions except position 8. The experimentally deter- mined course of nitration indicates that these are also the relative positions at stage c. Thus from a combination of experimental and theoretical data we have established a crossing of energy curves between stages b and c. It is evident however that in circumstances such as these it is not possible to make definite theoretical predictions of relative reactivhies by using the present theoretical techniques.The theoretical results are more satis- factory for radical and nucleophilic attack pointing to highest reactivity in both cases at the 4-position. An added difficulty to the theoretical treatment of electrophilic reactivity of nitrogen heterocyclic compounds is that in many cases salt formation doubtless occurs so that for example the nitration of quinoline perhaps 59Yvan Compt. rend. 1949 229 622. 60 Beilstein “ Orgmische Chemie ” Springer Vol. XX. 6 1 Ladenburg Ber. 1887 20 1643. e2Sandorfy and Yvan Cornpt. rend. 1949 229 715. * Sandorfy used an auxiliary inductive parameter Q = + @?/lo and without t In French literature these are usually termed “ potential barriers ” a somewhat this the 3-position would actually suffer a very slight activation.misleading name. BROWN MOLECULAR ORBITALS AND ORGANIC REACTIONS 83 occurs by attack of nitronium ion upon the quinolinium ion. This will be much less reactive than quinoline owing to its charge and its electronic properties will be different; e.g. the appropriate value for the coulomb integral of the nitrogen will differ from that in quinoline itself. The relative importance of such complications can only be ascertained by further detailed studies both theoretical and experimental of relative reactivities of the various positions in other nitrogen heterocycles under varying experimental conditions. Incomplete theoretical data for some other nitrogen heterocycles are available in the 2 3 9 2 4 9 3 4 9 63 The only oxygen heterocyclic system which seems to have been studied in detail is ~ - p y r o n e .~ ~ The non-crossing rule holds and the data agree well with the known chemistry of the molecule. For theoretical purposes cyclic conjugated ketones such as cyclopenta- dienone and cycloheptatrienone are analogous to hetero-systems. A com- plete theoretical treatment of these systems e4 and of tropolone 65 has been carried through the latter including the auxiliary inductive parameter. Mobile bond orders which could if necessary be converted into free valences are available for some related compounds.33 the electron densities of PhX where X represents halogen OH or NH and supplies two n-electrons to the conjugated system were computed for a series of values of ax PCx being assumed equal to /?.A similar but more detailed study of the eaect of varying the parameters ax and PCx has been carried out more recently by Sandorfy.58 He used an auxiliary inductive parameter (cf. footnote p. 81) but this does not affect the qualitative results appreciably. For our purposes these results may be summarised quite simply the values of q for the o- and p-positions are varied from considerably more than unity to somewhat below unity by appropriate alteration of the parameters. The charge a t the m-position never varies appreciably from unity for any set of parameters which could apply to a substituent more electronegative than carbon. Similar data for free valences show that for the m-position P is almost independent of the parameters for electronegative substituents while a more appreciable variation occurs for the o- and p-positions.A difficulty arises in that we cannot choose a set of parameters which will cause an overall deactivation of the benzene ring in the isolated molecule owing to the lack of transmission to the m-position. This is similar to the state of affairs found for pyridine (p. 81). Although the q indicates slight activation (if no auxiliary parameter was introduced) the A indicates considerable deactivation relatively to benzene. In such a case the deactivation is to be attributed to the localisability effect. Calculations of A for various values of ax have been performed by Dewar.66 He assumed that pCx =/? and used an auxiliary inductive Benzene Derivatives.-In the original work of Wheland and Pauling 6 3 Longuet-Higgins J .Chim. physique 1949 46 246. 64Brown J. 1951 2670. 66 J. 1949 463. 6 5 Dewar Nature 1950 166 790. 84 QUARTERLY REVIEWS parameter but he claims that the results are not affected qualitatively by the value assigned to this parameter and so we may assume them to be valid even when it is equated to zero. His results are shown in Fig. 6 in which h represents the value assigned to ax i.e. ax = UC + hS From comparison with the value of A for benzene itself i t is clear that for sufficiently large values of h (say greater than 2) the atom localisation energies will predict opsubstitution with deactivation while for smaller values of the parameter the same type of substitution will occur with activation. This means that if we draw the n-electron energy curves for the o - m- and p-positions on the same diagram and also the curve for 0 + I h +2 +3 FIG.6 benzene itself then for certain values of h the curves will cross the benzene curve between b and d while for other values the non-crossing rule will be valid. It cannot be said however that the results just discussed provide a complete picture of benzene substitution. The localisation energies lead one to suppose that very electronegative substituents will cause deactivation while less electronegative substituents will bring about activation. This requires chlorine bromine and iodine to be more electronegative than oxygen and nitrogen which is not in accordance with the order of electro- negativities ordinarily 68p 69 It is evident that this branch of aromatic substitution deserves further theoretical attention ; i t is not 67 Mulliken J.Chem. Physics 1934 2 782. 68 Pauling “The Nature of the Chemical Bond” Cornell 1940. 69 Bellugue and Daudel Rev. sci. 1946 84 541. BROWN MOLECULAR ORBITALS AND ORGANIC REACTIONS 85 unlikely that the effect on A of varying pcx will provide a solution of the electronegativity difficulty,* but this will require more detailed knowledge of the effect of effective nuclear charge and bond length upon the resonance integral (cf. p. 69). Further if ax is to be related to the electronegativity of X we should use electronegativity data for the appropriate valence states rather than for the free atom in its ground state which is the quantity generally listed in electronegativity tables. It would seem better to attempt to obtain a satisfactory set of parameters for each type of substituent by using experimental relative reactivities as a basis.At present we are a long way from this goal owing mainly to lack of adequate theoretical but also of quantitative experimental,? data. It can be said however that there is as yet no indication of the need for an auxiliary inductive parameter and more rapid progress is likely to be made if this additional complication is avoided until convincing proof of its presence is obtained. The theoretical results also can be applied to nucleophilic and radical substitution but in view of the uncertainties just discussed there is little point in considering these reactions separately. Calculations have been made for other types of substituents ; e.g. Wheland and Pauling con- sidered the nitro- and carboxy-groups as 4-electron substituents and the nitroso- cyano- and formyl groups as 2-electron substituents.Wheland l 7 has computed atom localisation energies A’ for some of these systems but he used a large number of parameters and it is not certain that the sets chosen were particularly appropriate. Clearly all these types of substituents deserve further theoretical attention and will doubtless offer further opportunities for selection of appropriate sets of parameters for the various atoms forming the substituents. Dewar 66 has considered the case where the substituent X has a vacant 2pz orbital i e . a 7-atom system of 6 n-electrons. At present there do not appear to be any known groups of this type whose phenyl derivatives have been investigated experimentally. Useful contributions to the theory of aromatic substitution have been made by Coulson and Longuet-Higgins 70 and by L~nguet-Higgins,~~ who have developed simple methods of obtaining charge densities and atom localisation energies of various positions in hetero-systems relatively to their values in the isoconjugate hydrocarbon (Le.the hydrocarbon with the same number of n-electrons and the same number and arrangement of atoms). These are described in the next section on comparative calculations. 7OJ. 1949 971. 7 1 J . Chern. Physics 1950 18 283.. * Coulson and de Heer (private communication) have shown that the degree of conjugation of X with Ph is governed by pCx as much as by ax. t For example quantitative data for the relative reactivities of the compounds listed in Table 2 and related alternant hydrocarbons would show whether the atom localisation energies have any quantitative significance (if the plot of activation energy versus A were linear the slope would provide a value of /3 appropriate for substitution reactions).Further the present theory is more satisfactory for homoly-tic substitution for which the non-crossing rule generally holds and even qualitative data for unsub- stituted aIternant hydrocarbons and heterocyclic systems would supply a valuable test of the theory. 86 QUARTERLY REVIEWS Reactions of Class (b).-The Diels-Alder reaction the osmium tetroxide oxidation and the ozonolysis reaction are reactions falling in this classi- fication. In contrast to those of class (a) these reactions are characterised by the fact that in the localisation treatment the residual molecules are always comprised of conjugated systems with an even number of atoms (except that in the case of some non-alternant systems the residual molecule may consist of two separate systems with an odd number of atoms each).This has certain advantages for a theoretical treatment of the effect upon reactivity of introduction of hetero-atoms substituents etc. which is discussed in the next section. It will be convenient to consider each of these reactions in turn. The Diels-Alder Reaction.-The reaction is characterised by addition of a dienophil (e.g. maleic anhydride) to a pair of positions of relative para-orientation in a six-membered ring or 1 4- in a polyene chain. Much discussion has taken place as to whether the addition occurs in one stage or by a two-stage mechanism the best evidence 72 73 favouring the former.It has also been debated whether the reaction is to be regarded as proceed- ing through a biradical mechanism or an ionic mechanism.74 In the theoretical treatment now to be discussed it is assumed that the addition is a one-stage process. If this be the case it is not possible to distinguish between an ionic and a biradical mechanism for the following reason. In the activated complex the diene and dienophil are linked by two incipient 0-bonds to which each component has contributed two electrons. Since electrons are indistinguishable it is meaningless to enquire which of these quasi-o-electrons comes from the diene and which from the dienophil and the distinction between an ionic and a biradical transition state vanishes. If we consider first the isolated molecule approximation then we expect the perturbability eEect to be insignificant owing to the absence of a coulomb field from the dienophil.The predominant effect will be the localisability and since two positions are attacked simultaneously this will be char- acterised by the sum of the two corresponding free valences 23'. A more detailed investigation leads to the same conclusion.75 The basis of the localisation treatment of the diene synthesis has been outlined in Section 3 for the specific example of addition of maleic anhydride to anthracene. For addition to the 9 10-positions of anthracene the residual molecule is comprised of two separate benzene systems and in general it can be said that the residual molecules corresponding to addition t o the reactive positions in all polycyclic hydrocarbons are comprised of two separate smaller polycyclic systems.The calculation of the relevant localisation energies (p. 66) termed para-localisation energies P is therefore rather simpler than for other reactions. Calculation of P for various pairs of positions in a considerable number of polycyclic hydrocarbons 76 reveals that in every case the pair of positions for which P is least in a particular 7 2 Bergmann and Eschinazi J . Amer. Chem. SOC. 1943 65 1405. 7 3 Wassermann J . 1942 612. 7 4 Kloetzel " Organic Reactions " Wiley 1948 Vol. IVY p. 8. 75Brown J. 1951 3129. 76 Idem J. 1950 691. BROWN MOLECULAR ORBITALS AND ORGANIC REACTI[ONS 87 molecule is the one at which maleic anhydride adds (if reaction occurs at all). Further the para-localisation energies for the most reactive pair of positions in a series of hydrocarbons correctly predict the order of reactivities of these compounds ; e.g.the rapid increase down the polyacene series is adequately explained in this way. As an example the para-localisation energies of the most reactive positions in 1 2-benzonaphthacene 4 5-benzochrysene and 1 2-3 4- dibenzanthracene are listed in Table 3. The data correctly predict the TABLE 3. Para-localisation energies Hydrocarbon 1 Position p(- y ) 1 1 2-Benzonaphthacene . . . . 6 11 2.03 4 5-Benzochrysene . . . . . I 3 6 1 Kit 1 1 2-3 4-Dibenzanthracene . . . 9 10 I I relative reactivities of these three compounds as found experimentally,77 and in addition the position of reaction in benzonaphthacene is correctly predicted to be 6 11 rather than 5 12 (I" = - 2 .1 1 ~ ) . Localisabilities (XF) also show a good correlation with the observed relative reactivities when the relevant free valence data are available. In all cases so far examined the non-crossing rule is valid between b and e for various pairs of positions in diEerent molecules as well as in the same molecule. The strict non-crossing between c and e suggests that the struc- tures of the activated complex and the final product differ only in degree and that the n-electron energy of the former is very closely proportional to that of the latter. More definite evidence of this type of relationship is found for the theoretically related osmium tetroxide oxidation reaction. The application of the localisation theory to the diene reactions of polyenes and arylpolyenes has also been considered.4O For these molecules there is the added complication that the conjugated systems are not as rigid as those of polycyclic compounds. If addition is to occur at a butadiene- like portion of the molecule then this portion must be in the boat-configura- tion ; addition to the chair-form is stereochemically impossible. The importance of this has been demonstrated e.g. in the case of some conjugated unsaturated fatty Consequently the comparison of theoretical data with experimental work is not very satisfactory for this class of compound particularly as in much of the experimental work the relevant stereochemistry of the compounds employed is unknown. Little attention has been paid experimentally to Diels-Alder reactivities of heterocyclic systems.It can be shown on the basis of the general theory of effects of hetero-atoms outlined in the next section that particularly in the case of nitrogen heterocyclic compounds little difference in reactivity is to be expected from that of the isoconjugate hydrocarbon unless a hetero- atom happens to occupy one of the positions normally reactive in the 77 Clar and Lombardi Ber. 1932 65 1411. 78~013 Mikusch Angew. Chem. 1950 62 475. 88 QUARTERLY REVIEWS hydrocarbon. Some isolated cases of diene reactions of heterocyclic com- pounds have been reported in the l i t e r a t ~ r e ~ ~ ~ 8o but so far no systematic study has been accomplished. One factor which presumably is present in almost all organic reactions but which reveals itself most plainly in the diene synthesis has not been considered in the present theoretical treatment.In the activated complex of the reaction of say maleic anhydride and anthracene the maleic an- hydride portion will be the seat of a considerable electrical dipole (or multipole) while the anthracene residual molecule contains a number of easily polarisable n-electrons. Since these portions are much closer together than in the approach of non-reacting molecules of a gas there will arise an appreciable stabilisation of the transition state Le. a lowering of the activation energy due to induction forces between the two parts. The general Diels-Alder adduct with maleic anhydride can be represented as in (VIa) or (VIb). The relative rates of formation will be determined by the relative magnitudes of the induction forces of the maleic anhydride dipole with X and with Y.If X is unsaturated or conjugated and Y is not then owing to the higher polarisability of the former (VIa) will be preferentially formed. This is not taken into account in the localisation theory of the reaction. However in the case of polycyclic hydrocarbons both X and Y will be aromatic systems. Since the induction forces fall o f f rapidly with distance we might expect a very similar interaction energy whether X (or Y) is a benzene or naphthalene or larger conjugated system ; i.e. it is reasonable to assume that the lowering of the activation energy is approximately constant in all polycyclic hydrocarbons and the same for both stereoisomers. On the other hand when X or Y or both are not so extensive the inter- action energies will depend much more upon the nature of these groups.We should expect then that in this latter event one or other of the stereo- isomers will be preferentially formed and that the localisation treatment of the reactivity will be less reliable because the induction energy will no longer remain roughly constant. It is thus to be expected that the localisa- tion treatment of polyenes and arylpolyenes will be less satisfactory than for polycyclics even after the stereochemical difficulties connected with the former (p. 87) are clarified. Experimentally these considerations are Huisgen Annalen 1949 564 16. Etienne and Legrand Compt. rend. 1951 232 1223. BROWN MOLECULAR ORBITALS AND ORGANIC REACTIONS 89 the basis of a very extensive literature on the steric course of the diene synthesis.Some attempts to estimate the induction energies for cyclo- pentadiene have been made,s1* 82 a semi-classical treatment being used ; values of the correct order of magnitude (ca. 5 kcal.) were obtained. A quantum-mechanical treatment of this aspect of the Diels-Alder or other organic reactions would not be feasible at present. Localisation energies have been found to be quantitatively related to reaction-rate constants for the osmium tetroxide reaction (see below) and this might be expected to apply also to the diene synthesis. It is to be hoped that relevant experimental measurements of rates will soon become available in order to test this point. This would also be an indirect test of the assumption of constancy of induction energies but the latter could be tested directly by a determination of the relative proportions of stereo- isomers formed in the addition of maleic anhydride to say benzanthracene.Oxidation by Osmium Tetroxide.-The details of the theoretical treatment of this reaction of polycyclic hydrocarbons follow rather closely those out- lined above for the diene reaction. In this case we imagine the approach of the osmium tetroxide molecule to bring about a partial localisation of two of the hydrocarbon’s n-electrons in the region about one of the bonds. These electrons take part in the formation of two incipient a-bonds. As has already been discussed in full for the Diels-Alder reaction (p. 86) it is meaningless to enquire which of these quasi-a-electrons comes from the osmium tetroxide and which from the hydrocarbon; we are not justified in classifying the reaction as ionic or biradical.Again the attacking molecule is electrically neutral so we might expect the perturbability effect to be insignificant. The reagent however always attacks a bond and we might approximate to the commencement of reaction by assuming that the resonance integral & of the bond is increased in magnitude by approach of the osmium tetroxide thus tending to draw n-electrons into i t and facilitating formation of the linking a-bonds.* At the same time the change in hybridisation (the localisability effect) which now involves two adjacent positions will be present. It is easy to show 83 that in this case the perturbability is characterised by the mobile bond order pij and the localisability by Fg + Fj(ZF). The localisation process is equivalent to the localisation of one of the bonds of the hydrocarbon.In the localisation theory the reactivities of the various bonds are thus characterised by the bond localisation energy B,12 84 the computation of which has already been discussed (p. 66). The relative reactivities of all bonds in all polycyclic hydrocarbons containing five or fewer rings have been discussed in terms of B.84 Where the relevant data are available it has been found that perturbabilities and also localis- abilities if used separately as criteria of reactivity lead to exactly the same 81 Wassermann J. 1935 828. 83Brown J . 1951 1950. 84 Idem J. 1950 3249. * A mechanism involving a change in pi of the opposite sign is also conceivable. The main grounds for rejecting it are that it would require an inverse correlation of pdj with reactivity to that actually observed.82 Idem ibid. p. 1511. 90 QUARTERLY REVIEWS predicted order of reactivity as does B data for a considerable number of hydrocarbons having been examined.83 Here again we see that at least for alternant hydrocarbons the non-crossing rule applies strictly to the reaction and hence any one of By p and XB’ may be used as criteria of reactivity. On the assumption that B’ represents the variable portion of the acti- vation energy of the reaction it is possible to derive relative rate constants for comparison with those determined e~perimentally.~~ 86 87 The results of such a comparison are presented in Table 4. It is unfortunate that the conjugated systems involved are already so complex that recourse is always made to approximate methods for computing B‘.Two different approxi- mate methods have been used both sets of results being given in the Table. The agreement with experiment is extraordinary when the relative simplicity of the localisation theory is borne in mind. Hydrocarbon Benzanthracene . . . . . . . Phenanthrene . . . . . . . . 3 4-Benzopyrene . . . . . . . 1 :2-5 6-Dibenzanthracene . . . . 5 6-Benzochrysene . . . . . . TABLE 4. Relative rate constants ~ -~ (a) (1) 0.13 1.8 1.3 0.04 I Predicted * Observed 1 0.1 2.0 1.3 slow * The conjugated systems involved are so large that approximate methods have been employed ( u ) using the general approximate theory of the effect of annelation on reactivity ; 88 ( b ) 8 F g statistical relations to compute conjugation energies 12 and annelation energies.The theoretical treatment involving By p and XF is of course equally applicable to polyenes and arylpolyenes subject to the same stereochemical reservations as in the case of the Diels-Alder reaction. Apart from the isolated example of the dinaphthylethylenes this field has received little experimental attention however. The effect of introduction of a hetero-atom upon the reactivity of a hydrocarbon is discussed in the next section. In this reaction also the problem of stabilisation of the transition state by dispersion forces arises. However the environments of reactive bonds in polycyclic hydrocarbons are so similar that the assumption of constancy of this effect is doubtless a good one. 0xonoZysis.-The theoretical treatment of the osmium tetroxide reaction can be carried over completely to the ozonolysis reaction.The experimental data are less clear-cut than for the osmium tetroxide oxidation owing in 85Badger and Reed Nature 1948 161 238. 86 Badger J. 1949 456. 8*Brown J. 1951 1955. QOBadger Nature 1950 165 647. 87 Badger and Lynn J. 1950 1726. Idem Trans. Furaduy Soc. 1950 46 1013. BROWN MOLECULAR ORBITALS AND ORGANIC REACTIONS 91 large measure to the low yields of products isolated but important informa- tion has nevertheless been accumulated and for a number of molecules the most reactive bond has been determined.g1* 92 93 From the theoretical viewpoint the osmium tetroxide reaction is more interesting owing to the relative ease with which rate constants can be obtained and freedom from side reactions. However ozonolysis experiments may assume importance for those molecules which react with osmium tetroxide only with very great difficulty (naphthalene benzene derivatives triphenylene).Addition of Diazo-compounds.-Diazoacetic ester and related compounds also seem to be bond reagents analogous to osmium tetroxide. This type of addition reaction has not however been investigated extensively for polycyclic hydrocarbons .94 Oxidation with Per-acids.-This reaction may also be analogous to the osmium tetroxide reaction. The available experimental data 95 96 do not cover polycyclic hydrocarbons and it is for these that the theoretical data are most reliable owing to the relative rigidity of the molecules. Reactions of Class (c).-The only reaction of this class which appears to have been investigated theoretically is the thermal cis-trans-isomerisation of ethylene derivatives.cis-trans-Isomerisation.-The general reaction may be represented as W X \ /y \ /y c:c --+ c:c / \ X 2 W z / \ In the activated complex the planes wCx and yCz will be perpendicular so that if w x y and z are conjugated systems the transition state will be comprised of two separate conjugated systems wCx and yCx. The difference in energy of the original molecule and the residual molecule comprised of two separate conjugated systems will be a measure of the activation energy. It will be noticed that the conjugated systems of the residual molecule are of the diarylmethyl type. Molecular-orbital calcula- tions of this difference in energy W have been perf~rmed,~' only the n-electrons being considered. W might be expected to give a qualitative representation of relative activation energies but can have little quanti- tative significance since the change in compressional energy of the central C-C a-bond has not been taken into account.The latter quantity will depend upon the length of the ethylenic bond in the planar form of the molecule which in turn could be estimated from its mobile bond order p . A rough computation using the C-C stretching force constant for ethane 98 indicates that this compressional energy is of the order of 10 kcal. Q2 Idem Bull. Xoc. chim. 1950 17 996. Q1 Wibaut Experientia 1949 5 337. Q3 Boer Sixma and Wibaut Rec. Trav. chim. 1951 70 509. 94Badger Quart. Reviews 1951 V 147. Q5Swern J. Amer. Chem. Soc. 1947 69 1692. Q6 Idem Chem. Reviews 1949 45 1. Q7 Gold Trans.Paraday Soc. 1949 45 191. O* Herzberg " Infra-red and Raman Spectra of Polyatomic Molecules " Van Nostrand 1946 p. 193. 92 QUARTERLY REVIEWS Various qualitative conclusions may however be drawn from the results e.g. that the cis-trans-activation energy is decreased by successive substitution of phenyl groups into ethylene but there are very few experi- mental data for comparison. Furthermore the energetics are complicated by steric interactions between the various substituents. A related problem is the hindered rotation about bonds of low mobile order diphenyl derivatives providing a well-known example. In this case the quantity W is more commonly termed the conjugation energy between the two conjugated systems. It is often represented by the symbol C and it is possible to calculate approximate conjugation energies from the appropriate self-polarisabilities.lzp 99 100p101 7.Changes in Reactivity due to Structural Changes So far we have been concerned with the absolute calculation of quantities whose relative magnitudes for various positions (or pairs of positions) are then regarded as criteria of the relative reactivities of these positions. I n many cases however we wish to know only the change in the reactivity of a certain position in a molecule when say a hetero-atom is introduced a t some other part of the conjugated system or when a benzene ring is fused to one of its bonds. It is therefore useful to have a calculus for direct computation of the change in a given theoretical quantity arising from such structural changes. Much of this section of the theory of organic reactions depends upon the expansion of changes in a theoretical quantity in terms of a Taylor series in the parameters expressing the structural change.We shall consider each of the fundamental theoretical quantities in turn but consider only the case where the structural changes are effected upon an alternant hydrocarbon. I n some cases the extension to non-alternant hydrocarbons hetero-molecules etc. is quite straightforward. Charge Densities.-Coulson and Longuet -Higgins 70 have dealt very fully with the case where the change in the qi is brought about by introduction of one or more hetero-atoms into the conjugated system. The change in structure arising from the replacement of carbon atom i by a hetero-atom can be represented approximately as a change in the coulomb integral ai and in the resonance integrals & for all j adjacent to i.To the first approximation i t is found that the latter changes have no effect in alternant hydrocarbons and the charge density change is 6qr = n r i d q . (15) The relevant mutual polarisabilities have been obtained for benzene naphthalene anthracene phenanthrene and some polyenes 28 70 and an extensive compilation of data for dq for various aza-derivatives of naph- thalene anthracene and phenanthrene has been p~blished.'~ Although this type of calculation is essentially a differentia1 method i t assumes the character of an absolute calculation in the special case of Heer Phil. Mag. 1950 41 370. loo Brown unpublished work. 101Coulson and de Hem Tram. Paraday SOC. 1961 47 681. BROWN MOLECULAR ORBITALS AND ORGANIC REACTIONS 93 alternant hydrocarbons considered here owing to the fact that the original charge densities are all unity.If another alternant hydrocarbon system is fused to any bond of an alternant hydrocarbon the resultant molecule will also be an alternant hydrocarbon so the effect of this structural alteration is to leave the charge densities unaltered at unity. If a non-alternant hydrocarbon system is involved in this annelation however the charge densities will differ from unity and no general method has yet been developed to enable the charge density changes in the alternant hydrocarbon portion to be evaluated directly. A study of absolute calculations of q for various molecules leads to the conclusion that the fusion of an alternant hydrocarbon to a non- alternant hydrocarbon results in a smoothing of the charge density fluctua- tions in the latter a t the expense of a small variation from unity in the charge densities of the former.Similar conclusions may be drawn for the annelation of a hetero-molecule to an alternant hydrocarbon system. The charge densities may also be afTected by introduction of sub- stituents into the alternant hydrocarbon. Vinyl phenyl and other alternant hydrocarbon substituents will leave the q unaltered but substituents con- taining hetero-atoms will cause some change in the charge densities. These latter structural changes are probably best treated as equivalent to the introduction of one or more hetero-atoms into a slightly different alternant hydrocarbon system ; e.g. benzaldehyde may be regarded as styrene with the b-carbon replaced by oxygen chlorobenzene as the negative benzyl anion (whose charge densities are very simply computed lo2 lo3) with the " methyl " carbon replaced by chlorine.In all these cases where the change may be regarded as produced by introduction of a hetero-atom it may be possible to deduce useful qualitative results from the molecular-orbital rule of alternating polarity. 28 Bond Orders.-The change in mobile bond orders arising from introduc- tion of a hetero-atom into an alternant hydrocarbon may be expanded in a Taylor series in 8% and S&j in the usual way. If only tho fist-order terms are retained the result is where as usual the summation is over all j which are neighbours of i. The coefficient of 6% is zero for alternant hydrocarbons and so does not appear in (16).For illustration the changes in mobile order of some of the bonds of naphthalene due to introduction of a hetero-atom at an or-position are listed in Table 5 where the change in the resonance integrals of the 1 2- and 9 l-bonds has been taken to be Jcp. The results are of some interest in connection with the recent study of the ozonolysis of quin~line.~~* lo* In naphthalene these four bonds are equivalent by sym- metry and have therefore the same mobile order (0-725) and free-valence sum (0.856) and for CN bonds k is generally assumed to be positive.35 @pm = XjzrqijaBij - * (16) lo2 Longuet-Higgins Nature 1950 168 139. 1031dem J. Chm. Physics 1950 10 265. lo4 Wibaut and Boer Proc. K . Ned. Aka&. Wetensch. 1950 53 19. 94 QUARTERLY REVIEWS The isolated molecule approximation provides an explanation of the pre- ferential ozonolysis of the benzene ring only if the increased reactivity in the 7 8-bond due to the increased localisability is sufficient to outweigh the increased perturbability of the 3 4-bond.The numerical values make this quite plausible. The effects upon p of annelation or of introducing substituents may be estimated only indirectly and need not be considered here in view of their relatively slight importance. 1 2 . . . . 5 6 . . . . 7 8 . . . . 3 4 . . . . TABLE 5. Bond order and ZP changes quinoline + 0-028k - 0.065k 1 + 0-005k + 0.004k + 0.046k ' + 0.027k + 0.050k - 0.024k i I I I 1 I- I I I Free Valence.-Free valence is defined directly in terms of mobile bond order [equation (7)] so if it is possible to estimate changes in the latter arising from some structural alteration then the changes in the F are easily found.We need not discuss such calculations separately. Examples of the type of numerical results obtained are given in Tables 5 and 6. Localisation Energies. -The change in localisation energy accompanying structural alterations has been studied in some detail for the case where the original system is an alternant hydrocarbon.ss The effect of replacing carbon atom i by a hetero-atom is given by where the superscript T indicates that the quantity refers to the residual molecule and the change in the coulomb integral 6 q has been taken to be h/3. It can be shown that particularly when the hetero-atom is nitrogen the effect of change in pij will usually be negligible.S8 In the Diels-Alder and osmium tetroxide reactions the residual molecule will always be one or more alternant systems with as many n-electrons as carbon atoms and in this case both qr and q will be unity so the first term vanishes.This also applies to radical sub- stitution. The conclusion is then that A, B and P are little affected by introduction of hetero-atoms the small changes involved being determined mainly by the polarisability magnitudes A. We have already seen that hetero-atoms have little effect upon the free valences (see Tables 5 and 6 bearing in mind that ii is of the order of O - l ) so it may be concluded that introduction of hetero-atoms has only a small effect upon these reactions. The investigation of the rate of oxidation of a dibenzacridine by osmium tetroxide 87 tends to support this theoretical prediction.Of course the above treatment breaks down if the hetero-atom is introduced at the position which is most reactive in the parent hydrocarbon. When L represents A or A the residual molecule is a carbonium ion Several possibilities arise. BROWN MOLECULaR ORBITALS AND OWANICY REACTIONS 95 or a carbanion respectively. In these cases the qr are no longer unity but Longuet-aggins has shown that they may be calculated very readily even for large conjugated systems.lo2~ lo3 If only the fist term in the series (17) is taken into account i.e. the perturbation caused by the hetero-atom is assumed to be small then this provides a very simple and convenient method for computing changes in the atom localisation energy of a given position in an alternant hydrocarbon due to introduction of one or more hetero-atoms at other positions.However it should be borne in mind that in (17) h will generally be of the order of unity and the second term may thus sometimes assume some importance. Since this is a differential calculation and the localisation energies in the original hydrocarbon will differ from position to position we cannot be sure that the most reactive position in the parent hydrocarbon will still be the most reactive one in the heterocyclic compound * unless we know both the localisation energies for the original hydrocarbon and the appro- priate value of h for the particular hetero-atom or atoms involved. For comparison of the theoretical results with experimental reactivities it is therefore useful to ensure that the position of attack is predetermined.For example we might study the hydrolysis of chlorine-substituted hetero- systems; for simplicity we assume that the interaction of the substituent with the z-electron system is constant in the series of compounds se1ected.t We give as an example in Table 6 a series of compounds originally chosen by Longuet-Higgins.lo2 The data differ somewhat from his values because he has employed the auxiliary inductive parameter for the hetero-atom. The table also includes the charge differences dq and free valence differences M' calculated by the methods described above. The effects of two or more hetero-atoms are assumed to be additive. The results in Table 6 are compatible with the order of reactivities 4-chloroquinazoline > 4-chlorocin.noline > 1-chloronaphthalene found ex- perimentally.lO63 log However they indicate that 4-chloroquinoline will hydrolyse with a similar ease to 4-chlorocinnoline whereas in fact the hydrolysis of the latter is more rapid than that of the former.If an auxiliary inductive parameter is introduced this anomaly disappears but it is dangerous to infer from this single piece of evidence that the theory is materially improved by inclusion of the additional parameter. Since h is of the order of unity (Table 1) the second term in (17) cannot in general be negligible and it is equally possible that the discrepancy is due to the simplification of neglecting terms in (17) other than the first. Here again further experimental work on the subject would be valuable. The change in localisation energies of an alternant hydrocarbon Y when another alternant hydrocarbon system X is fused to one of its 106Morley and Simpson Nature 1949 164 105.loo Keneford Morley Simpson and Wright J. 1950 1104. * Apart from the w e where the most reactive position in the hydrocarbon is activated at least as much as any other possible position of attack. TThis is justifiable since the hetero-atoms have only a small effect upon the self- polarisabilities which determine the conjugation energy of the substituent with the conjugated system. G 96 QUARTERLY REVIEWS bonds has also been discussed fully.loo If the a b-bond of Y is involved then apart from a small correction term it can be shown that or if the overlap integral is included SL = - 1*077(pab - P L ) ~ SL' = - 1*034(pah - p&)y . * (18) The unusual feature of these equations is that the change in localisation energy is in the first approximation independent of the nature of the system X which is fused to the u b-bond of the system.A convenient example TABLE- - 6. Reactivities of chlorinuted heterocycles in nucleophilic subat itution Compound 1 -Chloronaphthalene 4-Chloroquinoline . 4-Chloroisoquinoline 1 -Chloroisoquinoline 4-Chlorocinnoline . 4-Chloroquinazoline 4-Chlorophthalazine . . . . . . . . . . . . . . . . . . . . . . . . . . . . (0 ) - 4h/ll 0 - 4h/ll - 4h/ll - 8h/ll - 4h/ll 694 (0) * - 0.139h + 0.018h - 0.213h * - 0.121h - 0.352h - 0.196h (0) * - 0.072k + 0.048k + 0-070k * - 0.024k - 0.002k + 0.118k * In these cases the changes in q and P refer to the 1- and not the 4- position is provided by the bond localisation energy Bg:lO of the 9 10-bond of phenanthrene the residual molecule for this bond being diphenyl.Values of p - pr are listed in Table 7 for annelation to the various bonds of phenanthrene. From (18) it is clear that when p - pr is positive annelation makes the 9 10-bond less reactive while when p - pr is negative the TABLE 7. Annelation in phenanthrene Bond i 1 2 . . . . + 0.046 2 3 . . . . 1 -0.037 1 3 4 . . . . . +0*026 reactivity is increased. Thus annelation to the 1 2- and the 3 4-bond of phenanthrene will decrease the reactivity of the 9 10-bond while annelation to the 2 3-bond will increase the reactivity. Thus without further computation we may predict that benzanthracene and benzonaph- thacene (obtained by annelation of benzene and naphthalene respectively to the 2 3-phenanthene bond) will react more rapidly than phenanthrene with osmium tetroxide while chrysene and 3 4-benzophenanthrene (derived by annelation of benzene to the 1 2- and the 3 4-bond respec- tively) will be less reactive.In the last two cases the decreased reactivity is offset by the production of another reactive bond which by symmetry is equivalent to the one under consideration but it can be shown that BROWN MOLECULAR ORBITALS AND ORGANIC REACTIONS 97 this will not be sufficient to produce an overall increase in reactivity unless p - p' is less than about O-Ol.* It has been shown that on the basis of (18) it is a simple matter to estimate relative rate constants for reaction of various polycyclic hydrocarbons with osmium tetroxide or equally to estimate relative equilibrium constants for Diels-Alder addition of maleic anhydride.The effects of substituents upon the localisation energies of conjugated systems can be treated formally in terms of conjugation energies. It emerges that where Ci is the conjugation energy of the substituent when attached to the ith position in the conjugated system and Ci is that for the correspond- ing position in the residual molecule. The utility of (19) arises from the fact that conjugation energies may be calculated approximately from the self-polarisabilities of the positions involved. Since conjugation energies do not vary appreciably for a given substituent the change in localisation energy predicted by (19) will generally be very small. However am im- portant exception arises if the ith position happens to be the reactive one in the molecule .In such a case the residual molecule will not contain i and Ci will then be zero. Conjugation energies are generally positive (see below however) so that for this particular case (19) predicts a large decrease in reactivity ; e.g. anthracene undergoes Diels-Alder addition in the 9 10-positions and in agreement with the above discussion 9-phenyl- anthracene and 9 10-diphenylanthracene show progressively decreasing reactivities relatively to anthracene.lo7 6 13-Diphenylpentacene also affords an example of the deactivation of the 6- and the 13-position by the attached phenyl groups.108 In contrast to phenyl substituents methyl groups in the 9- and 10- positions in anthracene109 and benzanthracene,llO and in the 1- and the 4-position in naphthalene 111 enhance the Diels-Alder reactivity of these positions.We deduce then that the conjugation energy of a methyl group with an aromatic system is negutiue.t It is possible to give an interpretation of the varying reactivities of a number of methyl- and bromine-substituted benzanthracenes with osmium tetroxide in terms of conjugating powers of the various positions in the conjugated systems involved but further theoretical work on alkyl and other substituents is required before an adequate treatment of the effect of such groups on reactivity can be developed. S L = C i - G . (19) lo' Bachmann and Kloetzcl J. Amer. Chem. Soc. 1938 60 481. lo* Allen and Bell ibid. 1942 64 1253. loBBachmann and Kloetzel J. Org. Chem. 19.38 3 55. llOBachmann and Chemerda J. Amer. Ghem.~S'oc. 1938 60 1023. ll1 Kloetzel and Herzog ibid. 1950 72 1991. * This of course is the value of p - pr' which corresponds to a change in B' equivalent to a factor of 2 in the reaction rate constant. t This anomalous state of affairs doubtless arises from the assumption that variations in x a electron interactions need not be taken into account for alkyl derivatives. 98 QUARTERLY REVIEWS 8. Further Developments of the Theory Further developments in the foregoing theory of organic reactions must inevitably lie in several different directions. There are clearly some funda- mental aspects which need attention. Apart from the general problem of possible improvement to the simple molecular-orbital approximation without accompanying increase in the complexity of the computational technique there is the possibility of introducing more accurate models for the activated complex.In addition to the energetics of reactions there is the assumption of constancy of entropy of activation. The experimental investigation of this point involves only the determination of relative reaction rates a t more than one temperature so it is to be hoped that such information will soon be forthcoming. Throughout the preceding sections the need for further t heoret'ical work has been indicated in various connections and a decision as to the value of the present theory particularly for aromatic substitution reactions cannot be given until the relevant theoretical data are forthcoming. The possibility of extending the theory to cover other types of organic reaction is self-evident and it seems possible to interpret carbonyl addition reactions and decarboxylation in terms of the present theory.For the latter reaction and to a less extent the homolytic substitution reaction the theory cannot be tested adequately until more experimental data on relative reactivities are available. All too frequently the experiment'al data are available only for systems containing hetero-atoms or substituents and are of little value to the theoretician bent upon developing a theory of the reaction. Let us consider the ideal requirements of the theoretician. To avoid stereochemical uncertainties cyclic systems must be studied. Initially the appropriate parameters for hetero-atoms or substituents will not be known so it is important first to test the theory with experimental data on the smaller cyclic hydrocarbons (e.g.benzene naphthalene an- thracene). The most crucial qualitative test is provided by investigation of pairs of hydrocarbons of very similar reactivities (e.g. benzanthracene and 1 2-5 6-dibenzanthracene in the osmium tetroxide reaction). The next test is for possible quantitative significance of the theoretical quantities which involves the determination of the appropriate value of y (or b). The most severe test of this is provided by hydrocarbons of widely differing reactivities. The next logical step is to determine sets of parameters appropriate to the various hetero-atoms and the most satisfactory experi- mental information for this purpose is the change in reactivity of a hydro- carbon occasioned by replacement of a carbon atom by a hetero-atom.Here again the parameters can be determined most accurately when the hetero-atom produces the greatest change in reactivity. Finally in t'hp study of the effect of alkyl and other substituents it is important to use small parent hydrocarbons because various theoretical methods may have to be tried to approximate to the effects of the substituents and if the parent hydrocarbons are too large the theoretical labour involved will be prohibitive. BROWN MOLECULAR ORBITALS AND ORUANIC REACTIONS 99 There are many reasons why such an ideal is not likely to be achieved. Many of the theoretically most interesting compounds may be difficult to prepare in the required quantities or their synthesis may not have been developed. Alternatively they may be too unreactive to be studied con- veniently or may show a tendency to undergo side reactions.Again it may not be possible to study all compounds under comparable conditions owing to their insolubility in certain solvents. Clearly then there must be some compromise between the experimentalists and the theoreticians. But it should be realised that it is unfair to criticise a theory if its shortcomings are to be traced to the lack of crucial quantitative experiment'al data to determine the various parameters which of necessity appear in the theory. The R'eviewer records his thanks to Professor D. H. Hey who has read the manuscript and made helpful suggestions and to Professor C. A. Coulson whose careful perusal of the text has helped to eliminate some obscurities.