3 Theoretical Chemistry Applications of Molecular Mechanics Calculations ~~~~~ ~ ~ By M.B. HURSTHOUSE G. P. MOSS AND K. D. SALES Department of Chemistry Queen Mary College Mile End Road London El 4NS 1 Introduction The current state of theory suggests that if we knew the proper Hamiltonian operators and had the necessary computational apparatus and time all the results of chemistry should follow from the solution to the relevant Shroedinger equation. In particular we would be able to predict ground state geometries heats of formation and energies of different conformations of molecules. Of course ab initio cal-culations of this nature are not possible for most molecules of chemical interest and semi-empirical theories whilst being somewhat more successful are still a long way from this ideal.However force field calculations which adopt a much more pragmatic approach have already proved themselves. The idea of force field calculations originated in vibrational spectroscopy,’ when physicists attempted to predict vibrational frequencies with a simple model. A molecule is envisaged as a system of particles held together by forces with a potential energy V given by equation (1). The molecule contains N atoms the fii are force constants and the xi are the co-ordinates of the atoms with the convention that x1,x2,x3 are the Cartesian co-ordinates of atom 1; x4 xs x6 of atom 2 etc. The form of equation (1) pre-sumes that quadratic (so-called harmonic) terms are sufficient; for other than small displacements of the atoms cubic quartic and higher anharmonic terms may be required.In general there are i(3N -6)(3N-5) independent force constants and one of the simplest approximations to reduce this number is to assume central forces forces which act only along the lines joining pairs of atoms asif the molecule was composed of ions. Another idea the valence force approximation seems more reasonable to a chemist the forces are those which resist changes in valence bond lengths and bond angles. Finally Urey and Bradley2 proposed the use of a mixed potential function a function which is basically of the valence force type but which includes central force terms between 1,3non-bonded atoms. The name Urey-Bradley is attached to a force field which specifically includes such terms.See for example E. B. Wilson J. C. Decius and P. C. Cross ‘Molecular Vibrations’ McGraw-Hill New York 1955 Chap. 8; G. Herzberg ‘Molecular Spectra and Molecular Structure’ Van Nostrand New York 1945 Vol. 2 pp. 159 and 168. H. C. Urey and C. A. Bradley jun. Phys. Rev. 1931 38 1969. 23 M. B. Hursthouse G. P. Moss and K. D. Sales Chemists approached the same sort of idea from a desire to quantify the concept of strain energy or steric effect^.^ The original attempts were directed towards highly hindered biphenyls4 and somewhat later to the stable conformations of cy~loalkanes.~~ The various names under which this topic has been known reflect its history Westheimer Method Strain Energy Minimization Technique Force Field Calculations and Molecular Mechanics which is the accepted modern term.A molecular force field describes the potential energy of a molecule relative to the energy of a reference geometry which used to be called the strain-free or natural geometry terminology not now used because of the difficulty in defining these concepts. All the force fields used in practice employ parameters derived inductively by a systematic comparison of calculated and observed molecular properties. The ultimate aim is to cover all or most of organic structures with a reasonably limited set of transferable parameters. The properties one would like to predict include heats of formation conformational energies barriers to rotation geometries of various conformations in the ground state of molecules in crystals and of transition states between conformers.The Molecular Mechanics Method with carefully chosen parameters can predict these and other quantities accurately for a wide range of molecules. The subject has been reviewed several times.g In practice the method works as follows choose the type of force field required find the best parameters for this field and for the properties to be calculated and finally minimize the energy of the molecule by varying the positions of all the atoms. These stages will now be discussed separately. 2 Types of Force Field The potential energy of the molecule may be expressed as a function of all the internal co-ordinates and interatomic distances in the general manner of equation (2). v= v,+v,+v,+v,+v,b+v,+v,b+(v,,or vet) (2) The different symbols represent the molecular potential energy due to bond length changes (vb),to valence angle changes ( Vo) to dihedral angle changes (V,) to out-of-plane bending ( V‘) to non-bonded interactions ( Vnb), to electrostatic inter- actions ( Ve),to hydrogen bonding (Vhb),and to either 1,3-interactions (V13)or cross terms (Vet) These will each be discussed below.The inclusion of non-bonded interactions means that the force field is no longer of the simple harmonic valence or Urey-Bradley type is responsible for the non-transferability of spectroscopic force T. L. Hill J. Chem. Phys. 1946,14,465;‘Steric Effects in Organic Chemistry’ ed. M. S. Newman John Wiley New York 1956,Chap. 12,by F. H. Westheimer. F.H. Westheimer and J. E. Mayer J. Chem. Phys. 1946,14 733. N. L.Allinger I. Amer. Chem. SOC.,1959 81 5727. J. B. Hendrickson J. Amer. Chem. SOC.,1961,83,4537; 1962,843355; 1964,86,4854. J. B. Hendrickson J. Org. Chem. 1964,29,991. * K. B. Wiberg J. Amer. Chem. SOC.,1965,87 1070. (a) J. E.Williams P. J. Stang and P. von R. Schleyer Ann. Rev. Phys. Chem. 1968 19 531; (b)L.S. Bartell J. Chem. Educ. 1968 45 754; H.A.Scheraga Adv. Phys. Org. Chem. 1968,6,103; H. A. Scheraga Chem. Rev. 1971,71,195; M. I. Page, D.A.Brant Ann. Rev. Biophys. Bioeng. 1972,1,369; Chem. SOC.Rev. 1973,2,295; N. L.Allinger Ado. Phys. Org. Chem. 1974,13,1;C. Altona and D. H. Faber,*Topicsin Current Chem. 1974,45,1;(c) 0.Errner Structure and Bonding 1976.27,161;(d)J. D. Dunitz and H.B. Biirgi MTP Int. Rev. Sci. Physical Chern. Series Two Butterworths London 1975 Vol. 11 Chap. 4. Theoretical Chemistry Applications of Molecular Mechanics Calculations constants to calculations of geometry enthalpy efc.,but is essential if the field is to be transferable between systems of different strain. Bond Stretching.-The form of vb will be considered rather more fully than for subsequent terms because many of the ideas are common to them all. It is usually expressed by equation (3). bonds The symbols are b the bond length bo the reference bond length and Kb the harmonic force constant. Kband boare parameters to be adjusted for each different type of bond in the molecule; note therefore that the Kbare not force constants in the normal spectroscopic sense nor are the bo equilibrium .values of the b.Anharmonic terms of the type & 1 KI,(b -b0)3were introduced by Allinger." bonds Bond Angle Bending.-Equation (4) gives the common expression for Ve. Vo=$ C He(#-#o)2 (4) bond angles The symbols are analogous to those of equation (3); He and O0 are adjustable parameters. Anharmonic terms have been introduced. 11312~13 Bond Torsion.-Despite the fact that non-bonded interactions are considered in force field calculations it has always proved necessary to include a specific term to allow for changes in dihedral angles. The torsional potential energy for rotation about a carbon-carbon bond is commonly expressed by equation (5). v,= 1 H,(l+scosnf#I) dihedral angles The angle q5 is the dihedral angle H is an adjustable parameter and n and s are fixed by the nature of the bond as follows the periodicity of the torsional motion is given by n and s is -1 if the configuration of lowest energy has eclipsed atoms and +1 for the staggered case.Thus n = 3 s = +1 for ethane and n =2 s = -1 for ethene. An alternative way to express this interaction is by equation (6),which might be expected to hold if (4-do)is reasonably small. Terms additional to that in equation (5) have been considered. I4.l5 N. L. Allinger and J. T. Sprague J. Amer. Chem. SOC., 1972,94 5734. S. Lifson and A. Warshel J. Chem. Phys. 1968,49 5116. l2 N. L. Allinger M. T. Tribble M. A. Miller and D. H. Wertz J. Amer. Chem. SOC.,1971 93 1637. l3 E. M. Engler J.D. Andose and P. von R. Schleyer J. Amer. Chem. SOC., 1973 95,8005. l4 L. S. Bartell J. Amer. Chem. SOC.,1977,99,3279;N. L. Allinger D. Hindman and H. Honig J. Amer. Chem. Soc. 1977,99,3282. '' N. L. Allinger J. Amer. Chem. SOC.,1977,99,8127. 26 M. B. Hursthouse G.P.Moss and K. D. Sales Out-of-plane Bending.-This term is required when alkenes are being considered and allows for the motion shown in the figure. H \/ "H The usual expression for V is equation (7),the reference value of x the angle describing the out-of-plane bending being taken as zero. (7) out-of-plane bends Non-bondedInteractions.-These interactions are probably the most important of all the terms in the expression for V,and yet the model from which they are calculated is the most uncertain.The usual form assumed is either equation (8)or equation (9). vnb = {Nlr," -N2ri6} non-bonded pairs separated by three or more bonds v, = C {N~ ii e-''ii-N2r+) (9) non-bonded nails separated by ihree ormore bonds The distance between the atoms is rii,x is 9 or 12; the remaining symbols are adjustable constants. The second term in each equation is the familiar London dispersion force of attraction; the first expresses the repulsion at small internuclear distances by either the Lennard-Jones 12 or 9 potential (depending upon the value of x) or the modified Buckingham exponential potential. Even when the form of V, has been decided there is still a divergence of opinion as to its use. Some authors treat the parameters as adjustable for all interactions whereas others average those for H--Hand C--Cto give values for C--H.16Some foreshorten the distance when hydrogen is involved by moving the protons.12*17~18133 interactions are usually ignored (unless explicitly included in a Urey-Bradley potential? Vl,3),and even 1,4interactions are neglected on occasion on the grounds that the bond torsion term should account for changes in dihedral angle. Finally some calculations cut offthis interaction at separations greater than about 5 A even though the longer range London forces are still contributing. Electrostatic Interactions.-A simple Coulomb expression is used for the energy of interaction between partial charges 4i and qi separated by a distance rii [equation The 4i are regarded as variable parameters.When heteroatoms are present in the l6 T. L. Hill J. Chem. Phys. 1946,16,399; T. L. Hill 'Lectures on Matter and Equilibrium' Benjamin New York 1966 p. 46. l7 D. E. Williams J. Chem. Phys. 1965 43,4424. l8 S. Fitzwater and L. S. Bartell J. Amer. Chem. Soc. 1976,98,5107. Theoretical Chemistry Applications of Molecular Mechanics Calculations 27 molecule this term is usually incl~ded,'~ although it has been omitted.20 Partial charges in saturated hydrocarbons have also been considered. '1*21*22 Hydrogen Bonding.-These contributions may be very important in the con-formational analysis of certain types of molecular system for instance biopolymers.19*23 For most applications in organic chemistry they are ignored.1,34nteractions or Cross-terms.-These attempt to take into account explicitly geminal interactions with a Urey-Bradley potential as in equation (1 1). (11) all 1.3-all 1.3-interactions interactions The adjustable parameters are F F' and ro; r is the distance between the geminal atoms. Valence Force Fields can include a term which contains two types of change simultaneously see equation (12). v=,=CF(pi-poi)(pj-~oj) (12) 1.1 The pi are internal co-ordinates and F is an adjustable parameter. This term might for instance take into consideration the interaction between a CH stretch and an adjacent HCH angle bend. It has been suggested that 1,3-interactions are equivalent to cross-terms between adjacent co-ordinate~.~~ 3 Determination of Force Field Constants One should always remember that the functions used in force field calculations are empirical being picked because they are easy to handle numerically.Their ability to give a quantitative description of molecular systems is almost entirely dependent on the choice of parameters. Obviously from the outline above there are a large number of such constants and furthermore different parameter sets may give equally good descriptions for a change in one constant can often be taken up in another. Several good force fields have emerged over the last ten years from calculations on large sets of molecules together with thorough refinement on a trial-and-error basis. A more systematic approach has been developed by Lifson and co-workers:" as many experimental values for properties such as structural quantities thermo- chemical measurements etc.are taken the corresponding values calculated and the sum of the squares of the differences minimized in an iterative process by vary- ing the potential constants. Problems can arise because of correlations between the parameters but techniques have been developed to avoid these difficulties. l9 R. A. Scott and H. A. Scheraga,J. Chem. Phys. 1966,45,2091. 2o See for instance D. N. J. White and G. A. Sim Tetrahedron,1973,29,3933. 21 A.Warshel and S. Lifson J. Chem. Phys. 1970,53,582. 22 0.Ermer and S. Lifson J. Amer. Chem. SOC.,1973,95,4121. '' R.F.McGuire F. A. Momany and H. A. Scheraga,J. Phys. Chem. 1972,76 375. 24 S.Califano PureAppl.Chem. 1969,18,353. M. B. Hursthouse G. P.Moss and K. D. Sales 4 Energy Minimization The major problem of Molecular Mechanics is how to minimize the potential energy of a molecule by varying the atomic Fositions. All the methods available are iterative and rely upon guessing a starting geometry computing the gradients of the energy with respect to changes in the atomic co-ordinates and thereby estimating the best direction in which to move the atoms to reduce the energy. This new position is used to start the procedure all over again. The Newton-Raphson pr~cedure~~(NR) gives the correction to the Cartesian co-ordinates of all the atoms Sx from a starting set xo as equations (13)-(15). sx = -F-'D(xo) a* v F..=-" ax,ax The procedure has to be iterative because of the approximations involved in deriving these equations; it is iterated until all the gradients are zero i.e.until D(xo)has zero elements. The elements of F may be evaluated analytically or more usually numerically. The technique is very good if a reasonable geometry is guessed to begin with (i.e.one which is fairly close to the minimum). Otherwise different techniques must be used and usually have been in the literature. The method of steepest descents is equivalent to setting F equal to a diagonal matrix with elements 1/L(a scaling factor) when 8x is given by equation (16). The methods of parallel tangents and of pattern search have been discussed by Schleyer ~?tal.'~ AllingerI2 has adopted a different modification of the NR method in which each atom is considered in turn.Other changes include taking F to be a diagonal matrix" with elements Ei=d2 V/dx or a block diagonal matrixZ0*26 with elements Ei= d2V/dxi axj for i j =s3 (i.e. each atom has a block to itself). As mentioned above the NR method is sensitive to poor starting positions and is therefore often used together with one of the other techniques which converge more slowly but are not so sensitive. For example the steepest descents method may be used until the energy change at a given iteration is about 0.1 kcal mol-' A-' when the NR method will quickly converge to changes of kcal mol-' Having found a minimum one has to determine if it is unique and if not to find other local minima to determine the absolute minimum.One technique is to start a new search from each of a number of points chosen randomly or in some predeter- mined manner and check that each search reaches the same minimum. The big drawback is the amount of work required. False minima can occur because of stationary points such as saddle points or can be an artefact of an inadequate minimization technique. The NR procedure is the only one which guarantees a 25 See for example reference 9(c). 26 N. L. Allinger and G. A. Lane J. Amer. Chem. SOC.,1974,96,2937. Theoretical Chemistry Applications of Molecular Mechanics Calculations Table 1 Some force fields for saturated hydrocarbons" Field 1 ? 3 4 5 6 7 8 9 10 Reference 28 29 30b 12' 22' 13' 31' 32 18' 15' -590.4 655.2 662.4 654.0 662.4 600 662.4 554.4 662.4 -1.056 1.090 1.094 1.105 1.100 1.056 1.100 1.0203 1.113 -316.8 633.6 633.6 645.3 633.6 300 633.6 337.0 633.6 -1.240 1.530 1.512 1.501 1.520 1.250 1.520 1.166 1.523 0.0230 0.0228 0.0223 0.0088 0.0241 0.0145 0.0158 0.0144 0.0154 0.0140 ---0.0007 -0.0004 -112.0 109.47 -111.2 106.4 109.2 108.2 109.47 109.0 112.0 109.47 107.9 108.5 109.6 109.1 109.41 109.1 109.47 109.4 0.0230 0.0140 0.0267 0.0105 0.0268 0.0175 0.0140 0.0176 0.0141 0.0158 ----0.0005 -0.0005 -112.0 109.47 -107.8 112.39 109.5 -109.0 109.47 110.0 112.0 109.47 109.5 112.8 109.18 109.0 109.47 109.2 109.47 109.4 108.2 109.47 109.5 108.4 -109.2 -109.2 109.47 109.4 0.0230 0.0301 0.0351 0.0175 0.0284 0.0250 0.0274 0.0240 0.0276 0.0197 lccc HBn ---0.0004 -0.0007 - 112.0 109.47 -110.2 110.5 110.4 112 110.4 109.47 109.5 Itlo 110.7 109.47 -110.6 -110.1 -110.1 109.47 109.5 -109.47 111.0 109.47 109.47 109.5 -109.5 109.47 109.5 2.65 2.73 2.10 0.50 2.85 0.68 2.60 0.22 0.34 0.237 2300 6591 2650 49680 903.8 6515 5524 2604 2120 13630 HH 3.60 4.08 3.74 9.06 9.00 3.75 4.00 3.90 3.4 4.17 49.2 49.2 27.4 1.54 28.3 86.0 49.2 28.4 48.0 77.1 4012 44710 4320 69077 4979 4875 67925 4866 19498 13340 VL CH 3.40 2.04 3.42 9.06 9.00 3.58 4.40 3.60 3.75 3.59 125 125 138 2.14 163 84.7 126 84.5 540 144 7000 2199300 14976 96048 27372 15466 99787 19531 72317 12760 I cc 3.2 12 3.1 9.06 9.00 3.12 4.00 3.10 3.75 3.16 325 325 641 2.97 943 619 322 782 400 298 The units throughout this table are kcal mol-' for energies 8 for length and degrees for angles.The fields have been derived for different purposes which explains why certain entries are blank and usually have extra detail which is given in the following notes. The symbols are defined in equations (3)-(6) (a) and (9). 'Kb for CH is for >CH2 and % CH groups; that for -CH3 is 676.8 A Urey-Bradley potential [equation (1l)]was included. Cross terms [equation (12)] were included. The bond lengths were assumed invariant in field 1 at CC = 1.533A CH = 1.109 A. Fields 9 and 10 included anharmonic terms in V,. 'The numbers to the right of B0 refer to the substitution pattern of the carbon atom 1 is primary 2 secondary etc. A different anharmonic term was used for field 10. Field 2 used equation (6) for V, and field 10 the additional terms fV,(l+ cos 4) +fV2(-cos 24).V, is given by equation (9)for all fields except for number 5 which used equation (a) and number 2 which used equation (8) for the C-€ interaction. Fields 4 and 5 used a potential derived from four independent parameters only [see discussion after equation (9)}. M. B. Hursthouse G. P.Moss and K.D. Sales proper minimum for although the gradients (8V,8xi)are zero at a maximum as well as a minimum the F matrix has negative eigenvalues at improper minima. For example a one dimensional partial maximum is a saddle point which can be recognized because F will have one negative eigenvalue a fact which can be used to find transition states. Furthermore although the energy may not change appreci- ably for a given change of the atomic positions the partial derivatives may change considerably and lead to sizeable adjustments to for instance torsion angles.27 Thus in conclusion it is worth noting that if the NR method has not been used to refine the minimum the final geometries obtained with a given force field may not be correct in that they may not correspond to the partial derivatives being zero.5 Stereochemistry Molecular mechanics calculations were originally devised to analyse saturated hydrocarbons. These compounds still excite interest especially when strained or sterically crowded. For example cis-1,4- and trans- 1,2-di-t-butylcyclohexanewere shown to prefer a twist conformation of the ring;33 or 1,1,2,2-tetra-t-butylethane does not adopt an alternating arrangement between the groups at each end when viewed in a Newman pr~jection.~~ Steric hindrance from the alkyl groups of a series of methyl ketones showed a good correlation with Taft’s E-scale of steric parameter^.^^ In a study of a peri-substituted naphthalene with two t-butyl groups it is predicted that the ring system will be n~n-planar.~~ Large rings may exist in many different conformations.Molecular mechanics calculations offer one of the few methods of analysing the various alternatives and allow an estimate of the energy barrier for their interconversion. For example calculations by Anet3’ on 11- 12- 13- and 15-membered ring cycloalkanes cyclo-octa-1,3- and -1,4-diene and cyclododeca-1,5,9-trienehave been used to analyse their low temperature ‘H and 13Cn.m.r.spectra. After a similar examination of humulene (1)d~awa~~ showed that the more stable conformers were also those required for cyclization in the biosynthesis of the illudane group of sesquiterpenoids. 27 D. N. J. White and 0.Ermer Chem. Phys. Letters 1975,31,111;J. M. A.Baas B. van de Graaf A. van Veen and B. M. Wepster Tetrahedron Letters 1978 819. 28 J. B. Hendrickson J. Amer. Chem. SOC.,1967,89,7036,7043,7047. 29 E.J. Jacob H. B. Thompson and L. S. Bartell J. Chem. Phys. 1967,47,3736. 30 S. Chang D. McNally S. Shary-Tehrany M. J. Hickey and R. H. Boyd 1.Amer. Chem. Soc.,1970,92 3109. 31 R.L.Hilderbrandt J. D. Wieser and L. K. Montgomery J. Amer. Chem. SOC.,1973.95,8598. 32 D.N.J. White and M.J. Bovill J.C.S. Perkin 11 1977 1610. 33 B. van de Graaf J. M. A. Baas,and B. M. Wepster Rev. Trav. Chim. 1978,97,268. 34 W. D.Hounshell D. A. Dougherty and K. Mislow J. Amer. Chem. Soc. 1978,100,3149. 35 J.-E.Dubois J. A. MacPhee and A. Panaye Tetrahedron Letters 1978,4099. 36 J. Handal J. G. White R. W. Franck Y. H. Yah and N. L. Allinger J.Amer. Chem. Soc. 1977,99,3345. 37 F.A. L.Anet and I. Yavari J.Amer. Chem. SOC.,1977,99,6986; 1978,100,7814; F.A. L.Anet and T. N. Rawdah 1978,_100,5003,7166,7810. 38 H. Shirahama E. Osawa and T. Matsumoto Tetrahedron Letters 1978 1987. Theoretical Chemistry Applications of Molecular Mechanics Calculations 31 In a study3' of some substituted trans-2-decalones molecular mechanics cal- culations of the conformation were combined with ab initio methods (STO-3G) to calculate the energy.It was found that a significant proportion of flexible conformers should be present with 3a-methyl and l,l,lO-trimethyl cases but not with the la-methyl compound. This result should be contrasted with the study of 4,4-dimethylandrostran-3-one,where a chair conformer was predicted although a twist ring A was found for the A5 ana10gue.~' Hexaphenylethane notwithstanding claims to the contrary has not yet been synthesized. However it is predicted41 that it should be stable enough to isolate although it should have an elongated central C-C bond. Examination of related molecules with other aryl systems suggested that they too should have a similar long bond; however with 9,9'-bitriptycyl this bond was found4* to be only 1.558& compared with a calculated value of 1.589 A.With a conjugated polyene it is necessary to distinguish between the various bonds along the chain. This was acc~mplished~~ by assuming the bond stretching force constant was proportional to the bond order calculated by quantum mechanics (VESCF). In a study of some substituted di- and tri-enes S-cis conformers were shown to be present to a significant extent. The predicted conformation of a diene can then be used to calculate the U.V. absorption maxima.44 A similar study of enones4' was used to modify the Woodward-Fieser rules. It distinguished between S-cis- and S-trans-enones and suggested the parent molecules absorb at 215 and 209 nm respectively.Values are also calculated for all intermediate conformations of an a-,@- a@-,@p-,and a@@-substituted enone. Molecular mechanics calculations in general refer to isolated molecules at 0 K. Hence caution must be exercised in assuming these conclusions also apply at room temperature in the liquid phase. In a study of solvent interactions with molecules containing two polar groups it was necessary to consider not only dipole terms but also quadrupole contribution^.^^ 6 Elements other than Carbon and Hydrogen Most force fields were initially developed for hydrocarbons and have since been extended to include other elements. Replacement of a carbon atom by Si Ge Sn or Pb was examined by Mislo~,~' who showed that (M(Bu') or M(SiMe3)4 adopt a ground state with T symmetry previously unknown with organic molecules.The related (Bu')~S~H is interesting as it exhibits4' correlated rotation of the t-butyl groups; a prediction confirmed by low temperature I3C n.m.r. Tetra-alkyldisilane 39 M. Askari N. S. Ostlund and L. Schafer J. Amer. Chem. Soc. 1977,99 5246. 40 U.Burkert and N. L. Allinger Tetrahedrp 1978 34 807. 41 W. D. Hounshell D. A. Dougherty J. P. Hummel and K. Mislow J. Amer. Chem. Soc. 1977,99.1916. 42 M. H. P. Ardebili D. A. Dougherty K. Mislow L. H. Schwartz and J. G. White J. Amer. Chem. SOC. 1978,100,7994. 43 J. C.Tai and N. L. Allinger J. Amer. Chem. Soc. 1976,98,7928. 44 N.L.Allinger and J. C. Tai J. Amer. Chem. SOC.,1977,99,4256. 45 T. Liljefors and N. L. Allinger J. Amer. Chem. Soc.1978,100,1068. 46 L.DoJen-MikoviE and N. L. Allinger Tetrahedron 1978,34,3385; N. L.Allinger L. DoSen-Mibvik J. F.Viskocil jun. and M. T. Tribble ibid. p. 3395. 47 L. D. Iroff and K. Mislow J. Amer. Chem. Soc. 1978,100,2121. 48 W. D.Hounshell L. D. Iroff,R.J. Wroczynski and K. Mislow I.Amer. Chem. SOC.,1978,100,5212;R. J. Wroczynski L. D. Iroff and K. Mislow J. Org. Chem.. 1978,43,4236. M. B. Hursthouse G. P.Moss and K. D. Sales was found to prefer a gauche-conformation although anti-forms may be present. In the case of (BU')~S~HS~H(BU')~ the gauche-conformer was calculated49 to be 49.7 KJ mol-' more stable than the dnti-form. An additional complication occurs when alcohols or ethers are considered due to the oxygen lone pair electrons.However inclusion of them as a pair of additional pseudoatoms results in a better prediction of stereochemistry and conformational energy differences5' An alternative treatment" of lone pairs with 1,3-dioxans and alcohols is to consider them as point charges taken from quantum mechanical calculations (CND0/2) preferably with additional weak van der Waals interactions between the lone pairs. Other elements such as halogen,53 or phosphoruss4 are treated by adjustment of the normal parameters. With quinquevalent phosphorus the unusual rectangular pyramidal structure of (C61-&02)2PCH3 was predi~ted.~~" Carbonyl compounds such as carboxylic acids or amide~~~ have also been examined. In a study of germacranolide (2) it was correctly predicted5' that hydrolysis and re-lactonization would favour the formation of a lactone to C-8 rather than to C-6.0 R=OH R=H 7 Reaction Mechanisms Molecular mechanics calculations have been used in two different ways for the analysis of reaction mechanisms. One approach is to predict the most probable pathway for a rearrangement when there are many possible alternative reactions. In the other type of study steric factors or transition states are analysed to predict the kinetics and stereospecificity of the reaction. Examir~ation~~ of the 69 isomers of tricycloundecane which do not have three- or four-membered rings or alkyl groups suggests a preferred pathway for the known conversion of tricyclo[6.2.1 .02*']undecane into 1-methyladamantane (Scheme 1)by 49 S.G. Baxter D. A. Dougherty J. P. Hummel J. F. Blount and K. Mislow J. Amer. Chem. SOC.,1978 100,7795. N.L. Allinger and D. Y. Chang J. Amer. Chem. SOC.,1976,98,6798. 51 U. Burkert Tetrahedron 1977,33,2237; 1979 35 209. 52 N. L. Allinger J. Kao H.-M. Chang and D. B. Boyd Tetrahedron 1976 32 2867. 53 A. J. Meyer J. Mol. Struct. 1977,40 127. 54 (a)J. A. Dieters J. C. Galluicci,T. E. Clark and R. R. Holmes J.Amer. Chem. SOC.1977,99,5461; (6)N. L. Allinger and H. von Voithenberg Tetruhedron 1978,34,807. 55 N.L. Allinger and S.H. M. Chang Tetrahedron 1977 33 1561. 56 D. N. J. White and M. H. P. Guy J.C.S. Perkin II 1975,43. 57 M. H. P.Guy G. A. Sim and D. N. J. White J.C.S. Perkin 11 1976 1917. E. Osawa K. Aigami N.Takaishi Y. Inamoto Y. Fujikura Z.Majerski P. von R. Schleyer E. M. Engler and M. FgrcaSiu J. Amer. Chem. SOC. 1977,99,5361. Theoretical Chemistry Applications of Molecular Mechanics Calculations Scheme 1 a series of 1,2-shifts. With pentacycloundecane the most stable isomer was predic- ted’’ to be D3 trishomocubane (4); a result confirmed by the AlBr,-catalysed isomerization of a suitable precursor. Steric effects in the SN2displacement of bromide from bromoalkanes by bromide were examined by De Tar et a1.,60 who found a close correlation between the transition state energy and the observed kinetics. Rates of solvolysis of polycyclic alcohol derivatives were similarly correlated with the strain of the system.61 The rate of oxidation of secondary alcohols to ketones was related to the structure of both alcohol and ketone.62 Reduction of cyclic ketones by hydride in general gives a mixture of epimeric alcohols.The yields of each isomer and the kinetics were predicted63 with reason- able precision by molecular mechanics calculations on the transition state. With nucleophiles larger than hydride the limiting step is controlled by steric factors in the approach of the reagent. Calculations based on the cone of access to the carbonyl group correlate closely with the observed stereo~electivity.~~ 8 X-Ray Structures One of the basic problems in the application of molecular mechanics calculations to the study of molecular structure and conformation is the choice of a suitable starting model especially if the molecule has a considerable degree of fiexibility.One simple solution has been to use the results of crystallographic investigations. Not only is this very convenient from a practical point of view (a set of atomic co-ordinates are easily deduced from the unit cell fractional co-ordinates) but it immediately leads to a means of studying the problem of how similar are the crystal and gas phase molecular structures. 59 G. J. Kent S. A. Godleski E. Osawa and P. von R. Schleyer J. Org. Chern. 1977,42,3852. 6o D. F. de Tar,D. F. McMullen and N. W. Luthra J. Amer. Chem. Soc.,1978,100,2484. 61 M. R. Smith and J. M. Harris,J. Org. Chem. 1978,43 3588; D. Fkcqiu ibid. p. 3878. 62 P. Miiller and J.-C. Perlberger 1Amer. Chem. Soc. 1976,98 8407. 63 J.-C. Perlberger and P.Muller J. Amer. Chem. SOC.,1977,99 6316. W. T. Wipke and P. Gund J. Amer. Chem. Soc. 1976,98,8107. M. B. Hursthouse G. P.Moss and K.D. Sales Molecular mechanics calculations of varying degrees of sophistication have been applied in this area. Huler and War~hel~~ have described a method for the simul- taneous calculation of the effect of inter- and intra-molecular forces on crystal packing which allows for the study of effects of intermolecular forces on molecular conformation using the potential surfaces of the quantum mechanical extension of the consistent force field to conjugated molecules. The applicability of the method to both rigid and flexible molecules is demonstrated by calculations on benzene biphenyl and p-ionylidenecrotonic acid.A variety of conformations are found for acetylcholine (AcOCH2CH2NMe3X) as its halide salt and lattice energy calculations have been made in order to examine the preferred conformation found for each halide.66 Similar studies have been made on crystals containing adrenaline (5),67and one significant result to come from this work is that the requirement of minimizing the lattice energy overrides the require- ment of minimizing the conformational energy. The presence of different con- formers in different crystal structures especially in systems containing ions is therefore not surprising. Studies on molecular crystals also show that the balance between crystal and intermolecular forces is very fine. Vos and co-workers have made an extensive study of molecules with folded conformations using only non-bonded interactions.The most recent work has shown that p-dimethylaminobenzyl-p-nitrophenylsulphone68 and p-dimethylaminophenyl-N-methyl-N-( p-nitrophenylsulphonylmethy1)car- bamate6’ have a folded conformation whereas p-chlorophenyl-p-methoxybenzyl-sulphone68 has a stretched conformation. Potential energy calculations however predict a folded conformation for all three although the stretched conformation is only -4 kJ mol-’ higher energy. Presumably crystal packing forces can provide the necessary difference. The importance of the attractive component of van der Waal’s forces and the interplay between the preferred free-molecule and crystal con- formations is well demonstrated by 1,3,5-trineopentylbenzenederivatives.Previous n.m.r. studies and molecular mechanics calculations on TNB (the parent compound) Me,TNB and Me,TNB showed a preference for a rotamer with all three neopentyl groups on the same side of the ring and it was suggested that attractive steric forces between the bulky hydrocarbon residues were significant in this choice.70 The crystal structure of 2,4,6-tribromo-TNB71 interestingly contains two independent rotamers-the preferred one and also one with a neopentyl group on the opposite 65 E. Huler and A. Warshel Acru Cryst. 1974 B30 1822. J. Caillet P. Claveni and B. Pullmann Actu Cryst. 1978 B34 3266. 67 J. Caillet P. Claveni and B. Pullmann Actu Cryst.,1976 B32 2740. 68 I. Tickle J. Hess A. Vos,and J. B.F. N. Engberts J.C.S. Perkin II.1978 460. 69 R.J. J. Visser A. Vos and J. B. F. N. Engberts J.C.S. Perkin II 1978,634. ’O R.E. Carter and P. Stilbs J. Amer. Chem. Soc. 1976,98,7515. 71 B. Aurivillius and R. E. Carter J.C.S. Perkin [I 1978 1033. Theore tica 1 Chemistry Applications of Molecular Mechanics Calculations side of the ring to the other two. Again crystal packing forces are assumed to help stabilize this latter conformation which lies about 4 kJ mol-’ above the preferred one. The effect on the molecular geometry of peri-substitution in naphthalenes has been studied crystallographically and by molecular mechanics calculations with an ad hoc force field.72 The results indicate that the C(8)-C(9)-C(l) angle is the most constant indicator of the steric crowding due to the peri-substitution and the difference in steric energies of 19 kJ mol-1 for the 1-methyl and 1,g-methyl deriva- tives of a 2-naphthyl acetate give a good idea of the energies involved.Conformational studies of cyclic molecules provide a very useful application of molecular mechanics calculations. For such systems differences between experi- mentally determined crystal and calculated gas phase conformations are generally small but for some ring systems several conformers may lie close in energy. The structure analysis of and molecular mechanics calculation on bicyclo- [4.4.l]undecane-l,6-di01’~have provided interesting information not only on the conformation of the cycloheptane rings (which are locked into C2 symmetric twist/chair forms) but also on the ten-membered ring the conformation of which was used as a starting point for strain energy minimization studies and which was found to resemble a low-energy cyclodeca- 1,6-diene conformer.A of a cyclodeca-1,5-diene ring system in the germacranolide costunolide (3)shows that the calculated free molecule conformation is only slightly perturbed by crystal packing forces-the main distortion being a small deviation from the prefer- red C2symmetry. Quite large differences in strain energies of various final con- formers were found for the ten-membered ring systems in the sesquiterpenes agerol diepoxide (6) and ageratriol (7)but again the crystal conformations correspond quite closely to the calculated minimum energy conforrner~.~~ 0’ 72 D.N. J. White J. Carnduff M. H. P. Guy and M. J. Bovill Acta Crysf.,1977 B33 2986. 73 D. N. J. White and M. J. Bovill Actu Crysr. 1977 B33. 3029. 74 M. J. Bovill P. J. Cox P. D. Cradwick M. H. P. Guy G. A. Sim and D. N. J. White Actu Cryst. 1976 B32,3203. ’’ W. Messerotti V. M. Pagnoni R. Trave R. Zanasi G. D. Andreetti G. Bocelli and P. Sgarabotto J.C.S. Perkin 11 1978 217.