3 Theoretical Chemistry By H. S. RZEPA Department of Chemistry Imperial College of Science and Technology London SW7 2AY 1 Introduction The number of reports of theoretical studies by molecular orbital and related methods now appears to be doubling annually. A notable feature is the increased application of ab initio methods compared with semi-empirical and other more approximate procedures. Clementi' has reviewed the historical and recent development of reliable approximations to the Schroedinger equation. The practical limit for accurate calculations still unfortunately a long way off is put at molecules containing between 100 and 150 atoms. Various more approximate approaches for considering interactions in biologically important molecules are also discussed and it is significant that the number of such studies has recently been increasing.Jug has briefly summarized the various theoretical procedures which have been used for studying chemical processes' and has tabulated results that have been obtained for more than 120 reactions of various types. 2 Advances in Theoretical Techniques Geometry Optimization.-The optimization of molecular geometries using ab initio methods in which energy gradients are evaluated analytically and efficient mini- mization algorithms are employed is becoming increasingly common. The appli- cation of the Hellmann-Feynman theorem to the calculation of forces on atoms has been beset by problems of inadequacy of the basis set but it has 6een successfully a~plied,~ and may be one approach to the problem of reducing the computation time required for geometry optimization.Calculation of the force-constant (or Hessian) matrix has been greatly facilitated by the formulation of the second derivatives of the total energy with respect to the nuclear co-ordinates for ab initio wave function^.^ This allows very much more effective location and characterization of stationary points in a potential-energy surface and makes calculation of the vibrational frequencies and related thermodynamic properties straightforward. It may however not always be necessary to obtain all the elements of the Hessian matrix in order to identify whether or not a given stationary point is an energy minimum (when this matrix has E. Clementi J. Phys.Chem. 1980,84,2122. K. Jug Theor. Chim. Actu 1980,54,263. H. Nakatsuji,K. Kanda. andT. Yonezawa. Chem. Phys. Lett. 1980,75 340. J. A. Pople R. Krishnan H. B. Schlegel and J. S. Binkley Znt. J. Quantum Chem. Symp. 1979,13,225. H. S. Rtepa no negative eigenvalues) or a true transition state (when this matrix has one unique negative eigenvalue). Such information can also be obtained from the rank and signature of the Hessian matrix,' the evaluation of which may require only a few elements of this matrix. Although variable metric or quasi-Newtonian minimization methods (i.e. the Davidon-Fletcher-Powell) are best suited for the optimization of the geometries of molecules with less than about 100 degrees of freedom it has been suggested6 that the geometries of macromolecules are best optimized using selective constraints (e.g.bond lengths) in conjunction with steepest descent/conjugate gradient methods. Basis Sets.-The STO basis sets introduced by the Pople group more than ten years ago have become the most widely used in ab initio calculations. The most common combination has been the use of a STO-3G basis for optimizing the molecular structure followed by evaluation of relative energies using a 4-31G basis (for which the nomenclature 4-31G//STO-3G has been proposed). It is now known that programs which calculate derivatives analytically increase greatly in efficiency if the number of primitive gaussians in the basis set is reduced. Accordingly the Pople group have proposed' new basis sets for the first-row elements termed STO 3-21G which are of comparable quality to STO 4-3 1Gand clearly superior to STO-3G basis sets but which require only as much time as the latter for the evaluation of the energy derivatives.They suggest that for calculations which are dominated by derivative evaluation the 3-21G basis offers a superior alternative to the 3G basis. Huzinaga* has used the same philosophy in developing minimum contracted gaussian basis sets of double-5 quality for Li-Ne which promise to offer similar advantages and minimum gaussian basis sets for the first three rows of the Periodic Table which offer considerable advantages to the use of STO-3G basis sets for these elements. The Pople group have also introducedg 6-31 1G"" basis sets of triple-5 quality which have been developed for use in calculations which include corrections for electron correlation using Mgller-Plesset perturbation theory and which are claimed to overcome some of the problems encountered with earlier basis sets.The charge distribution in a molecule is commonly obtained using a 'Mulliken population analysis' of the ab initio wavefunction. The results however are very dependent on the basis set and several alternative partitioning schemes have been proposed which suffer to a lesser extent from this defect." Whereas a few years ago the prospective user of theoretical methods was faced by a bewildering choice of variously parameterized semi-empirical SCFmethods to this has now been added an equally large if not greater choice of variously optimized basis sets for use in ab inifio calculations! Electron Correlation.-The graphical unitary-group approaches to direct configuration interaction (CI) offer a systematic solution to the problem of electron P.Scharfenberg Theor. Chim. Acta 1980,58 73. W.F.van Gunsteren and M. Karplus J. Comput. Chem. 1980,1,266. 'J. S. Binkley J. A. Pople and W. J. Hehre J. Am. Chem. SOC.,1980,102.939. (a) H. Tatewaki and S. Huzinaga J. Cornput. Chem. 1980 1 205; (6)A. N.Tavouktsoglou and S. Huzinaga J. Chem. Phys. 1980,72 1385. R.Krishnan J. S. Binkley R. Seeger and J. S. Pople J. Chem. Phys. 1980,72 650. lo (a)J. B. Collins and A. Streitwieser,J. Comput. Chem. 1980,1,81;(6)M.Yanez R. F. Stewart and J. A. Pople Acta Crystallogr. Sect. A 1978,34,641.Theoretical Chemistry 17 correlation" and have the advantage of requiring a relatively simple data input. Schaefer and co-workers'2 have extended the utility of such methods by formulating the analytical gradients for an arbitrary closed-shell CI illustrating the procedure with a study of the conversion of HNC into HCN that included 12 497 configura-tions. Small but significant differences were found between the CI and the single- configuration SCF structures for the transition state. Multi-configuration (MCSCF) studies in which the orbital coefficients (in the Roothaan LCAO sense) and the CI coefficients are variationally optimal are also becoming comm~n.'~ These tech- niques are expected to have the greatest advantages in the study of those electronic states which are not dominated by a single configuration or of reactions in which the dominant electronic configuration may change.Pople and co-workers14 have concentrated on the application of MBller-Plesset perturbation theory at second and higher order for the evaluation of electron correlation energy and they have now reported the formulation of analytical gradients for this level of theory. There have been many advances in pseudopotential methods for the calculation of the wavefunctions of heavy atoms. For such atoms relativistic effects can be as important as electron-correlation effects and a glimpse of what we can expect in the future is illustrated by the relativistic effective-potential MCSCF calculations on thallium h~dride.'~ For elements of the first row such as carbon relative relativistic corrections for e.g.the 'Al and 3B1 states of methylene are now considered to be negligible,16 although such corrections to the total energy can be significant." Semi-empirical and Other More Approximate Methods.-Several authors have proposed improvements to existing zero differential overlap (ZDO) methods such as CNDO or MND0,I8 and ThielIg has dissected the differences between the INDO and the NDDO approximations. These are most pronounced for molecules with adjacent heteroatoms molecules with four-membered rings classical vis a vis non-classical carbonium ions and the relative energies of u-and w-orbitals. Jug and Nanda2' have presented a statistical analysis of the SINDO/l method which is based on the INDO approximation and which represents a considerable improvement over MIND0/3 whilst requiring fewer adjustable parameters.An alternative to the Hartree-Fock (HF) approach which has been borrowed from solid-state physics and much applied in calculations involving the transition metals is the Xa multiple-scattering method. This technique can be used to calculate the one-electron proper- ties of benzene and pyridine with an accuracy comparable with those obtained from l1 (a)B. 0.Roos and P. E. M. Siegbahn Inr. J. Quantum Chem. 1980,17,485;(b)P. E.M. Siegbahn J. Chem. Phys. 1980,72 1647; (c) B. R.Brooks W. D. Laidig P. Saxe and H. F. Schaefer 111 J. Chem. Phys. 1980,72 3837. B. R.Brooks W. D. Laidig P. Saxe J. D. Goddard Y. Yamaguchi and H.F. Schaefer 111 J. Chem. Phys. 1980,72,4652. 13 (a)H.J. Werner and W. Meyer J. Chem. Phys. 1980,73,2342; (b)B. 0.Roos,P. R.Taylor and P. E. M. Siegbahn Chem. Phys. 1980,48,157. l4 R.Krishnan H. B. Schlegel and J. A. Pople J. Chem. Phys. 1980,72,4654. l5 P.A. Christiansen and K. S. Pitzer J. Chem. Phys. 1980,73,5160. l6 (a)C. P. Wood and N. C. F'yper Chem. Phys. Lert. 1980,71,368;(b)E.R.Davidson D. Feller and P. Phillips ibid. 1980 76 416. *' 0.Matsuoka N. Suzuki T. Aoyama and G. Malli J. Chem. Phys. 1980 73 1320 1329. *' (a)P.G.Nelson J. Chem. Res. 1980 (S)106 (M)1701; (b)U.Dinur and B. Honig J. Chem. Phys. 1980 72 1817. l9 W.Thiel J. Chem. SOC.,Faraday Trans. 2 1980,76,302. *' K. Jug and D. W. Nanda Theor. Chim. Acra 1980,57,95,107,131.18 H. S. Rzepa ab initio HF wavefunctions but at significantly less computational cost.21 An LCAO variation of the Xa procedure fared much better than the multiple-scattering method in the calculation of the equilibrium internuclear distance and of the dipole moment of carbon monoxide.22 The purely parametric force-field methods continue to be developed for new elements and types of Parameters have been reported which give good results for the conformational properties of crown and for the strain energies in cyclic n~cleotides.~~~ 3 Electronic Structure and Geometries of Molecules” Ab initio STO 4-21G calculations have resolved serious discrepancies between two electron-diff raction studies of t~luene,~’ and a molecular-orbital-constrained elec- tron-diffraction (MOCED) method has been used to investigate the structure and conformational properties of 1-butene.26 The force field of uracil has been analysed with the aid of interaction co-ordinates calculated using the MNDO method and the results have been used to interpret the effects of hydrogen-bonding on the observed vibrational frequencies.The m.c.d. spectrum of cyclo-propane has been investigated using an ab initio CI method.28 The calculated spectrum was in good agreement with the observed one although the interpretation afforded by the calculations differed in many respects from that obtained through a moment analysis of the experimental spectrum. Molecular orbital methods have also frequently been applied to the interpretation of photoelectron ~pectra,~’ although the dangers of correlating an ordered set of MO energy levels with observed ionization energies have been empha~ized.~’ Numerous studies have been concerned with substituent effects including the relationship between the angle of tilt of a methyl group and hyperc~njugation,~~ the possibility of rr-electron donation by CN or CHO groups attached to unstable cationic and further discussion of the separation of u/rr interactions in saturated A model for retention of configuration in SN2reactions’in terms of a perturbational approach and STO-3G calculations reproduces the experimental trends in silicon without requiring either &orbitals on silicon or pseudorotation in the transition The ab initio approach to calculations on large molecules using 21 D.A. Case M. Cook and M. Karplus J. Chem. Phys. 1980,73,3294. 22 J. W. Mintmire and J. R. Sabin Chem. Phys. 1980,50,91. 23 (a)N.L. Allinger Adu. Phys. Org. Chem. 1979 $4,1; (b)M. J. Bovill D. J. Chadwick I. 0.Sutherland and D. Watkin J.Chem. SOC.,Perkin Trans. 2,1980,1529;(c)F.J. Marsh P. Weiner J. E. Douglas P. A. Kollman G. L. Kenyon and J. A. Gerlt J. Am. Chem. SOC.,3980,102 1660. 24 P. S. Bagus B. Liu A. D. McLean and M. Yoshidne Comp. Methods Chem. (Proc. Int. Symp.) 1980 203. 25 F.Pang J. E. Boggs P. Pulay and G. Fogarasi J. Mol. Struct. 1980,66,281. 26 D. Van Hemelrijk L. Van den Enden H. G. Geise H. L. Sellers and L. Schaefer J. Am. Chem. Soc. 1980,102,2189. ’’(a)B.I. Swanson T. H. Arnold M.J. S. Dewar J. J. Rafalko H. S. Rzepa and Y. Yamaguchi J. Am. Chem. SOC.,1978,100,771;(b)W. D. Bowman and T. G. Spiro J. Chem. Phys. 1980,73,5482. 28 E.Goldstein S. Vijaya and G. A. Segal J. Am. Chem. SOC. 1980,102,6198. 29 B.KovaE M. Mohraz E. Heilbronner V. Boekelheide and H. Hopf J. Am. Chem. SOC.,1980,102 4314. 30 E. Heilbronner and A. Schmelzer Nouv. J. Chim. 1980,423. A. Pross L.Radom and N. V. Riggs J. Am. Chem. SOC.,1980,102,2253. 32 M. Paddon-Row C. Santiago and K. N. Houk J. Am. Chem. SOC.,1980,102,6561. 33 K.B. Wiberg J. Am. Chem. SOC.,1980,102 1229. 34 Nguyen Trong Anh and C. Minot J. Am. Chem. SOC.,1980,102,103. Theoretical Chemistry molecular fragments continues to be de~eloped,~' and an interesting application of the Xa valence-bond method to the study of models of 2-Fe ferredoxin has satisfactorily accounted for the observed antiferromagnetic coupling between the two iron centres in the oxidized protein.36 Conformations and Intermolecular Interactions.-The theoretical prediction that glycine should have a low-energy conformer (1) with a small dipole moment (1.1D)37led to an intensive search for the expected weak transitions in the microwave spectrum.Such transitions have indeed now been located and the derived rotational constants show excellent agreement with the theoretical values. These results have important bearing on spectroscopic searches for interstellar glycine which is likely to exist largely as conformation (1). Force-field calculations on humulene (2) have revealed an unsuspected new low-energy conformer3' which may be important in several biosynthetic trans- formations.Gorenstein et al. have continued their investigation into stereoelectronic effects in phosphate esters phosphoramidates and pho~phonates.~' The results of STO-3G calculations suggest that elimination of methoxide anion is generally favoured by about 40 kJ mol-' in cases where an adjacent lone pair on N or 0 is antiperiplanar with respect to the leaving group [i.e. (3a) us (3b)l. A number of studies of conformations and hydrogen-bonding in peptides have appeared. Ab initio calculations in particular are often in close agreement with results obtained from crystal structures of protein^.^' Haddon41 has used simple Huckel theory to study the intrinsic ability of a functional group to develop symmetrical hydrogen-bonding.The eigenvalues of the bond-bond polarizability matrix have been shown to provide a measure of the degree of distortion or asymmetry of the hydrogen bond. Maleate anion (4) for example is known to have a symmetric hydrogen bond and was found to have the lowest such eigenvalue of all the compounds studied. Ab initio methods are generally more reliable than semi- empirical procedures in their treatment of hydrogen In one interesting study ab initio calculations suggest that (5a) is 7kJ mol-' more stable than (5b) as a result of hydrogen-bonding between the hydroxyl group and the d~uble-bond.~~ It has been demon~trated~~ that the considerable variations in the calculated STO-3G 35 D.Spangler and R. E. Christoffersen Int. J. Qudntum Chem. 1980 17 1075. 36 J. G.Norman P. B. Ryan and L. Noodleman J. Am. Chem. SOC.,1980,102,4279. 37 (a)R.D. Suenram andF. J. Lovas J. Am. Chem. SOC.,1980,102.7180;(b)L.Schaefer H. L. Sellers F. J. Lovas and R. D. Suenram ibid. p. 6566. 38 H. Shirahama E. Osawa and T. Matsumoto J. Am. Chem. Soc. 1980,102,3208. 39 D.G. Gorenstein B. A. Luxon and E. M. Goldfield. J. Am. Chem. SOC.,1980 102 1757. 40 (a)D.Peters and J. Peters J. Mol. Struct. 1980,62,229; (c) E.L.Mehler J. (b)ibid. 1980,68,243,255; Am. Chem. SOC.,1980,102,4051. 41 R. C. Haddon J. Am. Chem. SOC.,1980,102,1807. 42 S.Scheiner Theor. Chim. Actu 1980 57 71. 43 K.Morokuma and G. Wipff Chem. Phys. Lett. 1980,74,400. 44 Y.-C.Tse M. D. Newton and L. C. Allen Chem. Phys. Lett. 1980,75 350. 20 H. S. Rzepa geometries for hydrogen-bonded dimers of H20 and MeOH are largely attenuated with larger basis sets such as 4-31Gor 6-31G". Numerous studies of intermolecular interactions have been de~cribed.~' These include the interactions between biomolecules and polar amino-acid~,~'~ between morphine45b or and water and model systems of choline The ultimate aim of such studies is the development of reliable structure-activity relationships. Ab initio/CI techniques have been used to investigate the So S1 and T1 hates of ethyl bacteriochlorophyllide-a and the related radical and the necessity of CI in the theoretical study of several fundamental problems of chemical carcinogenesis has been empha~ized.~~' Neutral Species.-An interesting example of the problems associated with the interpretation of wavefunctions of molecules is illustrated by three studies of methyl-lithium Collins and Streitwieser,"" on the basis of double-5 a6 initio calculations and integrated electron populations claim the C-Li bond to be wholly ionic; a study using basis sets of triple-5 quality concluded that this bond is 60% and an INDO study of the 7Li-'3C n.m.r.coupling constants suggests that the value obtained is typical of a covalent bond.466 The Schleyer group continues to predict unusual structures for lithiated specie^.^' For example of the several stable calculated structures for C2Li6 (6) contains essentially a single CGC bond and (7)a single C-C bondq4'" The species (8) appears to be an example of an aromatic compound which can be regarded as both a Huckel and a Mobius and the most stable isomer of SiHzLiF has been predicted to have carbenoid character (9) and to be a good candidate for experimental ~bservation.~~" F H \ H/c=si (9) 4s (a)E.Clementi G. Corongiu and G. Ranghino J. Chem. Phys. 1981,74,578 and references therein; (6) A. Agresti F. Buffoni J. J. Kaufman and C. Petrongolo Mol. Pharmacol. 1980,18,461; (c)Y. Orita A. Ando H. Abe S. Yamabe H. Berthod and A. Pullman Theor. Chim. Acta 1979 54 73; (d)H. Johansen S. Rettrup and B. Jensen Theor. Chim. Acta 1980,55,267; (e)J. D. Petke G. M. Maggiora L. S. Shipman and R. E. Christoffersen Phoiochem. Photobiol.1980,32 399; (f)J. J. Kaufman Int. J. Quantum Chem. Quantum Biol. Symp. 1979,6 503. 46 (a)G. D. Graham D. S. Marynick and W. N. Lipscomb J. Am. Chem. SOC.,1980,102,4572; (b)T. Clark J. Chandrasekhar and P. von R. Schleyer J. Chem. Soc. Chem. Commun. 1980,672. 47 (a)A. J. Kos D. Poppinger P. von R. Schleyer and W. Thiel TetrahedronLett. 1980,21,2151; (b)A. J. Kos and P. von R. Schleyer J. Am. Chem. SOC.,1980,102,7928; (c)T. Clark and P. von R. Schleyer J. Organomet. Chem. 1980,191,347. Theoretical Chemistry Among the many studies of silicon compounds is the prediction48 that silaethyne will rearrange with no activation barrier to the silylene (lo) and STO 3-21G calculations49 do not encourage the hope that silatetrahedrane may be more stable relative to silacyclobutadiene than the carbon analogues predicting an energy difference between the two species of more than 125 kJ mol-’.The surprising result that tetra-t-butylcyclobutadieneis essentially square and not rectangular has received theoretical upp port.'^ The energy required to distort rectangular to square cyclobutadiene has been estimated at about 42 kJ mol-’ and this would be more than offset by the relief of steric compression of the substituents. The radical cation of tetra-t-butyltetrahedrane (11) has been calculated by the MNDO method not to be stable and to rearrange (with no energy barrier) to the cyclobutadiene radical cation,’l which is fully in accord with both the photoelectron spectrum and other known properties of (11).The singlet state of cyclopropyne has been investigated using a very large basis set and extensive CLS2It has been predicted to be unstable rearranging to vinylidene (H2C=C=C:) whereas the triplet state is marginally stable and was found to have a calculated C=C vibrational wavenumber of 1840cm-l. A study of the geometry of norbornene and norbor- nadiene at the STO-3G level has revealed that the m-system is expected to be significantly n~n-planar.’~ The olefinic CH bonds are bent endo by 3-5 O and the greater exo development of the ~~~-0rbital has been used to rationalize the observed preference for electrophilic exo-substitution. The apicophilicity of electronegative substituents in five-co-ordinate phosphorus has been the subject of several papers.Two independent ab initio calculations using double-5 basis sets with polarization functions were found to predict that (12) and (13) are essentially i~oenergetic.’~ Although an earlier study using a 4-31G basis had predicted (12) to be 32 kJ mol-’ more stable this is probably due to inadequate optimization of the geometry.’’ The two forms have been predicted to interconvert via a turnstile mechanism with a barrier of about 20 kJ mol-’. A comprehensive study of carbonyl ylides using the 4-3 lG//STO-3G method with 3x3 CI has suggested that the parent compound (14) is planar with a substantial barrier to rotation about the C-0 bond and a small barrier to inversion 48 A. C. Hopkinson and M. H. Lien J. Chem. Soc. Chem. Commun. 1980 107. 49 M. S.Gordon J.Chem. SOC.,Chem. Commun. 1980,1131. ” W.T.Borden and E. R. Davidson J. Am. Chem. SOC.,1980,102,7958. 51 (a)H. Bock B. Roth and G. Maier Angew. Chem. Int. Ed. Engl. 1980,19,209;(6)E. Heilbronner,T. B. Jones A. Krebs G. Maier K. D. Malsch J. Pocklington and A. S. Schmelzer J. Am. Chem. SOC. 1980,102,564. s2 P.Saxe and H. F. Schaefer 111 J. Am. Chem. Soc. 1980,102,3239. 53 G. Wipff and K. Morokuma Tetrahedron Lett. 1980,21,4445. ” (a)R.Hoeller and H. Lischka J. Am. Chem. SOC.,1980,102,4632;(6)H.J. Bestmann J. Chandrasek- har W. G. Downey and P. von R. Schleyer J. Chem. Soc. Chem. Commun. 1980,978. ” H. J. Bestmann Pure Appl. Chem. 1980,52,771. H. S. Rzepa about the central oxygen atom. Substituent effects upon the barrier to the formation of (14) from oxiran the barrier to rotational isomerization of the C-0 bond the reactivity and regioselectivity of cycloaddition reactions and the fragmentation of (14) were A variety of methods ranging from simple Huckel to MIND0/3 and STO-3G have been used to predict the relative stabilities and the spectral properties of an almost completely unknown class of compounds the az~loquinones.~~ Of the classical isomers for which a KekulC structure is possible 1,5-azuloquinone (15) was suggested as the most promising candidate for synthesis whereas of the non-classical forms 1,3-azuloquinone (16) proved by far the most stable The first such compound recently synthesized was in fact 1,2-a~uloquinone.~~ 0 0 The evergreen problem of bond alternation in [18]annulene has been studied by an ab initio method.58 The bond-alternating form was predicted to be 150 kJ mol-' more stable than the delocalized form; a result similar to earlier MIND0/3 cal- culations but in apparent disagreement with the experimental evidence.There seems little doubt that electron correlation is unusually important in cyclically conjugated hydrocarbons and that correct geometries are only obtained when CI or other approaches are used.59 Charged Species.-The stereoelectronic properties of CH(OH),NH,' and CH(OH)(NH2)0- as models for tetrahedral intermediates in acid- and base- catalysed hydrolysis of amides have been investigated with partial optimization of the geometries at the STO-3G and 4-31G The C-N bond is calculated to be very long and weak when it is antiperiplanar to two lone pairs of electrons.Protonation on the nitrogen atom leads to a marked amplification of this effect. Experimental evidence that the allenic anion is bent i.e. (17) and not linear i.e. (18) is supported by double-l/SCEP ab initio calculations,6' which also predict a barrier to inversion of 29 kJ mol.-' H / (18) " K. N. Houk N. G. Rondan C. Santiago,C. J. Gallo R. W. Gandour and G. W. Griffin J. Am. Chem. Soc. 1980,102,1504. " L. T. Scott M. D. Rozeboom K. N. Houk T. Fukunaga H. J. Lindner and K. Hafner,J. Am. Chem. Soc. 1980,102,5169. '* R. C. Haddon Chem.Phys. Lett. 1980,70,210. "M.J. S. Dewar and M. L. McKee Pure Appl. Chem. 1980,52 1431. 6o J. M. Lehn and G. Wipff J. Am. Chem.SOC.,1980,102,1347. J. K. Wilmshurst and C. E. Dykstra. J. Am. Chem. Soc.. 1980,102,4668. Theoretical Chemistry The novel non-classical ion (19) has been suggested on the basis of MIND0/3 calculations to account for scrambling of 13C in cyclopentyl cation.62 MINDO/3 4-31G//STO-3G and 6-31G*//STO-3G calculations have indicated that the two vinyl cations (20) and (21) may be unusually stabilized in the former by hypercon- jugation of the cyclopropyl ring with the vacant vinylic p-orbital and in the latter by the non-classical structure that is formed.63 The authors of the adjacent paper in the same journal independently present experimental evidence of the unusual stability of (21).64 Radom and co-w~rkers~~ have studied 17 isomers of C,&Of.and found five to be relatively low-energy species and possible candidates for detection by ICR techniques. In a later paper such evidence is indeed presented for the detection of one of these i.e. (22).65b Substituent effects in the various intermediates in the Birch reaction have been studied with the 4-3 lG//STO-3G method including benzene radical anions,66a cyclohexadienyl radicals,66b and cyclohexadienyl anions.66c The homoaromatic species (23) was calculated by various methods to be between 143 and 180 kJ mol-’ higher in energy than cyclohexadienyl anion itself. Carbenes and Open-Shell Species.-Double-l ab initio calculations with extensive CI predict a barrier of about 21 kJ mol-’ to the interconversion of the triplet states of (24) and (25) in agreement with experiments which indicated that the barrier to geometrical isomerization in triplet carbalkoxycarbenes is larger than their barrier to intermolecular reaction.67 The singlet states were calculated to have only one non-planar low-energy form.The barrier to the [l 21 shift of hydrogen in the triplet nitrene (26) was calculated68 to be as high as 200 kJ mol-’. The species should be stable in a matrix in contrast to the singlet ‘A’ state where the [1,2] shift of hydrogen was predicted to occur with no barrier. Carbenes of the type CX2 where X is less electronegative than carbon have been predicted to be linear,69 although Pauling has put forward an explanation in terms of resonance theory and the number of available orbitals on X.70 The ‘ll’ground states and ‘2’excited states of conjugated N and 0 radicals have been investigated by ab initio methods.71A marked dependence of the calculated geometry on the basis set used was found particularly with N radicals.Ab initio 62 W. Franke H. Schwarz H. Thies J. Chandrasekhar P. von R. Schleyer W. J. Hehre M. Saunders and G. Walker Angew. Chem. Int. Ed. Engl. 1980,19,485. 63 Y.Apeloig J. B. Collins D. Cremer T. Balley E. Haselbach J. A. Pople J. Chandrasekhar and P. von R. Schleyer J. Org. Chem. 1980,45 3496. W. Franke H. Schwarz and D. Stahl J. Org. Chem. 1980,45,3493. 65 (a)W. J. Bouma J. K. MacLeod and L. Radom J.Am. Chem. Sac. 1980,102,2246; (6) B. C. Baumann J. K. MacLeod and L. Radom ibid. p. 7927. 66 A. J. Birch A. L. Hinde and L. Radom J.Am. Chem. Soc. 1980 102 (a) p. 3370; (b) p. 4074; (c) p. 6430. 67 K. S. Kim and H. F. Schaefer 111 J. Am. Chem. SOC.,1980,102,5389. J. Demuynck D. J. Fox Y.Yamaguchi and H. F. Schaefer 111 J. Am. Chem. SOC.,1980,102,6204. 69 W. W. Schoeller I. Chem. Soc. Chem. Commun. 1980,124. 70 L. Pauling J. Chem. SOC.,Chem. Commun. 1980,688. ’’ N. C. Baird and K. F. Taylor Can. J. Chem. 1980,58 733. 24 H. S. Rzepa calculations72a predict that the energy separation between the ll and X states of (27) is 115 kJ mol-’ compared with 58 kJ mol-’ obtained using the MNDO method.72b The two states in such systems eventually become degenerate if the angle at the nitrogen atom is increased to about 140 ’. The results of STO 4-31G calculations suggest that the simultaneous stabilization of a radical by a T-donor and a T-acceptor substituent is significantly greater than the additive effects of the individual groups.73 Earlier failures to find this effect theoretically were ascribed to incomplete optimization of the geometry.The magni- tude of the stabilization is about 71 kJ mol-’ for (28) and 14 kJ mol-’ for (29). (27) The occupation numbers of natural orbitals (NOONS) have been proposed as a quantitative criterion for biradical chara~ter.’~ The method has been illustrated using the pPP/.rr-CI approach on a number of conjugated hydrocarbons in their ground and excited states and on several hypothetical Mobius-type systems and species with orthogonal double-bonds. An important study of the photoisomeriza- tion of polyenes has been carried out at the STO 4-31G The transition states for this process in butadiene hexatriene and (30) and the geometries of the lowest triplet state of these species were located and full normal-co-ordinate analyses were performed.MIND0/3 calculations for several larger polyenes gave qualitatively similar results. Integrals of the Franck-Condon type were evaluated from the calculated vibrational force field and the results indicated that the energy of the triplet state is funnelled mainly into the C(l)-C(2) torsional motion of the ground state in the deactivation process. 4 Dynamic Processes aiid Reaction Hypersurfaces New approaches to the problem of locating saddle points in an energy surface continue to be One such method involves the calculation of the energy of ‘hypercubes’ of space in conjunction with ‘global’ interpolation technique^.'^" A welcome development is the increasingly common characterization of stationary points via calculation of the force-constant matrix.The information that is available from this approach is illustrated by an ab inifiostudy of the rearrangement of MeNC to MeCN.77 The calculated activation energy includes corrections for zero-point energy and is within 10 kJ mol-’ of the experimental value. The vibrational analysis but not the structure is reasonably consistent with previous purely empirical estimates used in RRKM calculations of the kinetics of this reaction. At 500 K the experimental and the calculated value (ab initio) of the pre-exponential factor for ” (a) Y.Apeloigand R.Schreiber J.Am. Chem. SOC.,1980,102,6144;(b)T.Clark ibid. 1979,101,7746. 73 D. Crans T. Clark and P. von R. Schleyer Tetrahedron Lett. 1980,21 3681. 74 D.Doehnert and J. Koutecky J. Am. Chem. SOC.,1980,102,1789. ’* I. Ohmine and K. Morokuma J. Chem. Phys. 1980,73 1907. 76 (a) D.Feller W. T. Borden and E. R. Davidson J. Comput. Chem. 1980 1 158;(b) H.Cardy D. Liotard A. Dargelos and E. Poquet Now.J. Chim. 1980,4,751. ” P.Saxe Y.Yamaguchi P. Pulay and H. F. Schaefer 111 J. Am. Chem. SOC.,1980 102,3718. Theoretical Chemistry 25 this reaction agree remarkably well. Ab initio theory has also been used to evaluate tunnelling effects and the corresponding corrections to E for proton-transfer reaction^'^' and the equilibrium constant in isotopic exchange reactions.78 A number of reactions of fundamental importance to organic chemistry have been the subject of systematic and high-quality ab initio calculations.The cycloaddition reaction of ethyne to HCNO was calculated at the 4-31G/ single determinantal level to proceed concertedly via a relatively symmetrical transition The lengths of the new C-C and C-0 bonds in the transition state were calculated to be 2.185 8 and 2.219 8,respectively although the stretching force-constant for the C-C bond was ten times larger than for the C-0 bond. Contrary to a previous suggestion by McIverg0 that symmetric structures were unlikely to be genuine transition states the force-constant matrix corresponding to the structure that was located had only one negative eigenvalue.Iriclusion of extensive CI without further optimization of geometry decreased the barrier to the reaction two-fold (from 128 to 67 kJ mol-') and the authors merely speculate that CI would not have any substantial effect on the geometry or on the properties of the force-constant matrix! The reaction of borane with ethene has been re-investigated at the ab initio 4-31G leveL8' A .rr-complex is formed in the early stages of the reaction and the transition state corresponds to a four-centre structure involving concerted formation of the CH and C-B bonds and cleavage of the B-H and C=C bonds. This scheme differs significantly in several details from earlier theoretical studies. The Wittig reaction between H3P=CH2 and formaldehyde was studied with a double-t plus polarization-function basis An oxaphosphetan ring (12) is formed initially with a very small energy barrier followed by concerted dissociation into the products accompanied by a reorganization of the co-ordination of the phosphorus atom.An orbital-correlation diagram for the reaction can be con- structed in which all occupied orbitals of the reactant can be correlated with all such orbitals in the product. This is due to the polar nature of the molecules involved and it contrasts to the situation of a symmetry-forbidden [2 + 2,] addition. Harding et al. have presented a tour de force with their study of the potential-energy surface of H4C0.82Eight minima and six transition states were located using basis sets of up to 6-3 1G** quality and fourth-order Mgller-Plesset correction for the correlation energy.Vibrational frequencies and zero-point energies were evaluated from a full normal-co-ordinate analysis using the analytically derived force-constant matrix. At this level energies of reactions were predicted to be within 21 kJmol-' of the experimental values whilst the calculated vibrational frequencies were about 10 to 15% too large. Among the results obtained was that the formation of hydroxycarbene from photolysis of formaldehyde could be competitive with the dissociation into HZ and CO. Morokuma et al. have investigated the photochemistry of HFCO in a similar manner.83 In this case elimination of CO is predicted to occur rather than rearrangement to the carbene and a normal-co-ordinate analysis suggests that the HF will be formed with considerable vibrational energy.A 78 R. F. Hout M. Wolfsberg and W. J. Hehre J. Am. Chem. SOC.,1980,102,3296. 79 A.Komornicki J. D. Goddard and H. F. Schaefer 111 J. Am. Chem. SOC.1980,102 1763. J. W. McIver Acc. Chem. Res. 1974,7,72. S. Nagase N.K.Ray and K. Morokuma J. Am. Chem. SOC.,1980,102,4536. 82 L.B.Harding H. B. Schlegel R. Krishnan. and J. A. Pople J. Phys. Chem. 1980,84 3394. 83 K.Morokuma S. Kato and K.Hirao J. Chem. Phys. 1980,72,6800. 26 H.S. Rzepa normal-co-ordinate analysis of the transition state for the elimination of HF from fluoroethaneg4 suggests that here too the HF that is formed will be vibrationally excited. A study at the double-[/CI level of the Wolff rearrangement" reaffirms that oxiren is barely stable requiring only 8 kJ mol-' to rearrange to formylmethylene which itself is not stable to rearrangement to keten.None of the postulated transition states however was characterized by inspection of the corresponding force-constant matrices. After a detailed investigation of the ene reaction of '02with olefins it was concluded that the major pathway involves biradical intermediates and the results show how many aspects of the stereospecificity and regiospecificity can be under- stood in terms of biradical-like intermediates or transition states. One important point that emerges is that normal Markownikoff directing effects are not expected in the addition of '02 to unsymmetrical olefins.86 The hydration of formaldehyde to give methanediol was calculated (using a 4-31G basis) to proceed uia a concerted proton transfer and formation of a C-0 bond [see (31)] with a calculated activation energy of 184 kJ mol-'.The addition of a single extra water molecule changed the character of the transition state dramatically [see (32)] and lowered the activation energy c~nsiderably.~~ Prior protonation of the carbonyl group or deprotonation of the water was found to remove the barrier for the process whilst the potential-energy surface for the formation of the zwitterion H&-CH,o was found to be completely repulsive. 4-H (31) An entirely different approach to nucleophilic addition reactions involved the development of intermolecular SCF perturbation theory for weak interactions." The interaction energy was expressed as the sum of electrostatic exchange and exchange-repulsion charge-transfer and polarization terms and the approach was illustrated with a discussion of the orientation of a nucleophile in carbonyl addition reactions.Whereas the frontier-orbital (ie.charge-transfer) term alone was inadequate in describing the interactions it proved a useful means of expressing Baldwin's rules for such reactions in qualitative MO terms. Another approach to the problem of solvation has been to use the valence-bond concept of ionic- covalent resonance to develop a simple model for comparing activation energies of reactions in solution with that in an enzyme. The method was used to investigate the potential-energy surface of simple general acid catalysis in solution and in ~ysozyme.~~ 84 S.Kato and K. Morokuma J. Chem. Phys. 1980,73 3900. *' K.Tanaka and M. Yoshimine J. Am. Chem. SOC.,1980,102,7655. 86 L.B.Harding and W. A. Goddard J. Am. Chem. SOC.,1980,102,439. " (a)I. H. Williams D. Spangler D. A. Femec G. M. Maggiora and R. L. Schowen J. Am. Chem. SOC. 1980,102,6619;(b)I. H.Williams G. M. Maggiora and R. L. Schowen ibid. p. 7831. 88 A. J. Stone and R. W. Erskine J. Am. Chem. SOC.,1980,102,7185. 89 A.Warshel and R. M. Weiss J. Am. Chem. SOC.,1980,102,6218.