FORCE CONSTANTS By J. W. LINNETT M.B. D.PHIL. ( U m DEMONSYXUTOR IN CHEMISTRY AND FELLOW OF THE QUEEN’S COLLEGE OXFORD) Introduction THB vibration frequencies of a molecule having a particular shape depend (u) on the mame of the atoms in the molecule and (b) on the restoring foroee that come into play when the molecule is distorted from its equilibrium configuration. Therefore a study of the vibration frequencies supplies information about these restoring forces. Since it may be assumed t h a t the atomic masses are known the problem of aocounting for molecular vibration frequencies is one of finding out about the redoring forces. The most marked advances have been made by thinking of the molecule in terms of its valency bonds and supposing that when any valency bond (or angle) is distorted a restoring force comes into play to restore the bond (or angle) to its equilibrium value.It is found that the restoring force may be assumed to be proportional to the distortion and the proportionality constant relating the force and the distortion is called the force constant. However this simple picture of thinl$ng of the molecule only in terms of the tendency of its valency bonds to return to their origins1 lengths and positions when distorted is not a complete one. The result ie that much work has bean and is being done to try to discover how thh simple valency picture is best modified to account for the obser- vatione. When a suitable force field has been selected and tested for a given molecule we obtain from the measured vibration frequencies the numerical values of the force constants of its valency bonds and angles.The bond force constants have been used to assess the strength of the bonds and it has been found that the force constant of a bond between a given pbir of atome depends on the nature of the bond-or bond order. Force oonstanta have been used to assess bond order. It has also been found that other properties of the bond (such as the equilibrium length) may be related to its force constant and work has been done on these relationships. There has been much less development m our understanding of bending form constants and there is as yet little theory to account for their mrgnitudes. Vibratwn Frequency Calculations To obtain an expression for the frequencies in terms of (a;) and (b) it ia necemary to write expressions for the kinetic and potential energies of the molecule.To do this we have to choose a co-ordinate system to represent the positions of the atoms in the molecule. Various co-ordinate ryefsmr may be aelected but because under ordinary conditions a molecule dwbp mrmrincr cllose to its equilibrium form it is eaaieet to represent the 73 74 QUARTERLY REVIEWS internal configuration in terms of displacement co-ordinates which measure the displacements of the atoms from their equilibrium positions. It is simplest to use Cartesian displacement co-ordinates so that if there are N atoms in the molecule 3N co-ordinates are required xl yl xl x2 . . . xs. To make our formulz less lengthy it is convenient for us to represent these 3iV Cartesian co-ordinates by the symbols ql qz . . . qn (where n = 3N q1 = xl q2 = y1 .. . and qn = xAV). The kinetic energy of the system may then be written as T = Z$miqiz (Qi = dqi/dt) . - (1) where mi is the mass of the atom associated with the qi co-ordinate. We have in (1) an expression t o take account of the effect of (a;) on the internal movement,s of the molecule-the atomic masses are the coefficients in the kinetic energy (K.E.) expression. Now we must consider how ( b ) the restoring forces affect the internal motions. It is easiest to represent this by writing the potential energy rather than the restoring force as a function of the distortions. The potential energy and restoring force are intimately related since the force along any co-ordinate qi is given by - aV/aqi. The most general form of this potential energy (P.E.) function will be where the summation terms cover all combinations of the co-ordinates including squared terms.The zero of any potential-energy scale is arbitrary so we will choose our zero so that the potential energy is zero at the equili- brium configuration (i.e. V = 0 when all the qi’s are zero). Therefore Vo = 0. Also the equilibrium configuration must be a minimum of potential energy so aV/aqi = 0 for all qi’s when all the displacement co-ordinates are zero. Thirdly under ordinary conditions molecules are never distorted far from their equilibrium configurations (i.e. the qi’s are always small). This means that we may to a first approximation presume that the terms involving qiqjqk (product of three small numbers) are always small compared with terms involving qiqj (product of two small numbers).Equation (2) may therefore be simplified to V = Ziaijqiqj . (3) This means that the variation of the potential energy along any one co- ordinate is parabolic in form for if all the co-ordinates except qi are zero we have V = &z&-i.e. the potential energy is proportional t o the square of the displacement which is the equation of a parabola having its minimum a t the equilibrium configuration. Moreover since the force along the ith co-ordinate Fi = - aV/aqi = - uiiqi when all the other qj’s are zero the force is proportional to the displacement. The success that has attended the application of (1) and (3) to the calculation of molecular vibration frequencies implies that (3) is an adequate form for the P.E. function and that the proportionality between restoring force and displacement is in h c t a good approximation.Therefore all the Ai’s are zero. LINloETT FOECB CONSTANTS 76 The fact that reshoring force and displacement are proportional to o m another means that the mation will be simple harmonic. By equathg the product of and acceleration of each atom to the reeforing force act% on that atom as deduced from (3) and doing this for all the atoms in the molecule one can obtain an expression for the vibration fhquencies which ia of the form 1 4 . (4) where Y is the vibration frequency. The values of a, as etc. are dependent on and can be calculated from the atomic masses mi and the constants in the potential-energy function ail. Equation (4) is satisfied by n values of 9. Some of these are zero and correspond to the small translations and rotations of the molecule which have a “ vibration frequency ” of zero since translations and rotations do not involve any change in the potential energy.For a non-linear molecule (4) is satisfied by n - 6 (i.e. 3N - 6) positive non-zero values of 9. The positive values of 2/? give the vibration frequencies of the molecular system. For molecules which possess some symmetry (e.g. methyiene dichloride CH,Cl, which has two planes and a two-fold axis of symmetry) the algebraic equation (4) factorises. The reason for this is that if a molecule has for instance a plane of symmetry the fundamental vibrations must be either symmetric or antisymmetric to the plane and (4) factorises into two-one factor giving the frequencies of the symmetric vibrations and the other factor those of the antisymmetric vibrations.In methylene dichloride there are four symmetry types (i) symmetric to both planes ; (ii) anti- eymmetric to both; (iii) symmetric to the CH plane and antisymmetric to the CCl plane ; (iv) vice versa from (iii). The equation (4) in this case factorises into (i) a quartic (ii) a linear and (iii) and (iv) two quadratic equations in v 2 which give the nine vibration frequencies of the methylene dichloride molecule. This factorisation simplifies the calculation problem (two quadratics may be solved more quickly than a quartic). To calculate the fundamental vibration frequencies of a given molecule of a known configuration we have to know the constants in (1) and (3). The constants in (1) are the atomic masses which are always known. However to know the aij in (3) we have to make some assumptions regarding the way in which V varies with the distortion of the molecule.That is we have to make some assumptions regarding the force field existing within the molecule. That is knowing the atomic masses we use the vibration frequencies to obtain information about the potential-energy function of the molecule. However we find that the available data are inadequate for us to determine all the constants in (3) -i.e. all the aij’s. In methylene dichloride 1 E. T. Whittaker “ Analytical Dynamics,” Cambridge Univ. Press. * G. Henberg “ Infra-red and Raman Spectra of Polyatomic Molecules,” Van v a - a1.v2(n-1) + as .v%n-2) - . . . ~ - 1 .Y* + = O In practice the process is of course reversed. An example will show this. Nostrand Co. E. B. Wilson J . Chem. P h y a k 1939 7 1047.4 M. A. El’yashevich Compt. rend. A&. ScL U.R.S.S. 1940 28 604. rle QUARTERLY REVIEWS the numbr of possible independent constants for the most general tspe of the potentid-energy function (3) is seventeen. The number is redud compared with m uneymmetric pentatomic molecule by the symmetry. But methylene dichloride has only nine fundamental vibration hquenciea. One cannot determine more than nine comtants from nine observed quan- tities. In this case a way out of the difEculty is to examiqe the nine fundamental frequenciee of CHDClz and the nine of CD,Cl,-one could also use^ the chlorine isotopes but the percentage change of mass is smaller and the eflFect on the frequencies is therefore smaller. There are now a suEcient number of observed frequencies (27) to calculate the 17 constants.Also there is an internal check on the reliability of the treatment because there are more obeerved frequencies than adjustable constants. However it is not always possible to determine all the fundamental frequencies of a molecule and it is not always possible to make use of isotopes. Then the other way out of the difficulty is used. Instead of determining more vibration frequencies assumptions are made regarding the molecular force field (potential-energy function) which reduce the number of independent constants in (3). The two most important simplifications are the " simple valency force field " (S.V.F.F.) and the " simple central force field " (S.C.F.F.). SpeciaE Force Fields The assumption of the valency force field is that the only forces in the molecule are those associated with valency bonds.Though we know that in molecules non-bonded atoms do exert forces on one another it is reason- able to suppose that these may be neglected relative to the bond forces. The S.V.F.F. assumes that if a bond alters its length there is a force tending to restore it to its original length which is proportional to its change in length. It likewise supposes that if the angle between two bonds alters there is a force proportional to the change which tends to restore it to its original value. This means that on the basis of the S.V.F.F. the P.E. function (3) may be written as where AR is the change in the length of the bond " a " from the equilibrium value and Aa is the distortion in the angle " m ". The summations are over all bonds and angles.The constants k are called the force constants. Since - dV/dAR is the restoring force along the " a " bond it will be seen that when all the distortions other than dR are zero the restoring force according to (5) is - kaARa-i.e. the restoring force is proportional to the displacement. By geometry the dR and Aa may be represented in terms of the Cartesian displacement co-ordinates qi and so ( 5 ) may be converted into the form of (3) (i.e. V as a function of the $8) which may be used in conjunction with (1) to obtain (4). Let us see how this would help us with methylene dichloride which has fwo C-H bonds two C-C1 bonds one CleCl angle one HCH angle and four ClCH angles. Both C-H bonds must be the same and have the V = ZikaARi + E&kmA& . - ( 5 ) LINNETT FORCE CONSTANTS 77 same force conetant.Therefore the potential-energy function of methylene dichloride according to the S.V.F.F. is obtained in terms of $ve force constants. If all the nine fundamental frequencies have been found then the five constants may be determined. More important still because the nine frequencies are obtained in terms of less than nine constants a relation between the observed frequencies which is independent of the actual values of the force constants may be obtained. This may be used to check the reliability of tho initial assumption (a S.V.F.F.). Let us turn to a simpler example than methylene dichloride namely methane. On a S.V.F.F. this has one bond-stretching constant kcH and one angle constant kHCH. It has four fundamental frequencies which are observed in the Raman and the infra-red spectra to be 2914 1526 3020 and 1306 cm.-1.5-' If the force field in methane were really of the S.V.F.F.type it can be shown that the following relation would hold For methane the left-hand side (L.H.S.) is 0.89 and the R.H.S. 0.94 which is fairly good agreement showing that the S.V.F.F. is quite a good approxi- mation for methane. However for CCl the L.H.S. is 2-44 while the R.H.S. is 2.92 which is not such good agreement the difference having risen to about 20%. Other tetrahalides AX, are rather like carbon tetrachloride. Before discussing the merits of the S.V.F.F. we will discuss the S.C.F.F. This assumes that the only forces in molecules are those between atoms along the line joining them whether the atoms are bonded or not. It supposes that there are no angle forces like those of the S.V.F.F.The forces are supposed proportional to the displacement and the potential energy is therefore proportional to the squares of the displacements where ARAB is the change in the distances between atoms A and B from the equilibrium value and kAB is the force constant. The summation is made over all pairs of a,toms. Thus in methylene dichloride there are five constants For the two C-H bonds for the two C-C1 bonds for the H-H separation for the CI-Cl separation and for the four C1-H separations. If we return to the tetrahedral molecules the S.C.F.F. gives an expression similar to (6) which may be used to :eat the validity of the assumptions v - Z ~ C ~ ~ A R ~ ~ . - (7) For methane the L.H.S. is imaginary because 4v is greater than v ao the S.C.F.F.gives a poor account of methane. For carbon tetrachloride the L.H.S. is 7.96 and the R.H.S. 2.92. For SnBr, the L.H.S. is 2-14 and the R.H.S. 1.57 and this last gives the best agreement of all the tetra- hedral molecules. The general conclusion which is supported by much R. G. Dickinson R. T. Dillon and F. Rasetti Physical Rev. 1929 34 682. J. P. Cooley Astrophys. J. 1926 62 73. 6 G. E. MacWood and H. C. Urey J . Chem. Pihysics 1936 4 402. 78 QUARTERLY REVIEWS other evidence is that the S.V.F.F. is closer to the truth than the S.C.F.F. (see W. G. Penney and G. B. B. M. Sutherland for triatomic molecules and J. B. Howard and E. B. Wilson It is noteworthy that on the whole in the tetrahedral molecules the S.V.F.F. becomes less successful as we pass from the hydrides to the halides (e.g.CH to CCl,) while the S.C.F.F. becomes more successful. This suggests that the forces between the non-bonded halogen atoms are more important than those between the non-bonded hydrogen atoms as would be expected. A further objection to the S.C.F.F. is that it does not account for the bending vibrations of linear molecules since no interatomic distances change in such vibrations. Also it fails to account for the out-of-plane vibrations of planar molecules (e.g. CH,O). However in this last case the S.V.F.F. only accounts for such vibrations if some rather more complicated bending terms are included. The general conclusion of this phase of the study of molecular force fields .has been that we should base any improvements or refinements on fields of the S.V.F.F.type rather than on those of the S.C.F.F. type Before leaving our consideration of the S.V.F.F. it is worth mentioning that some workers have made use of the unmodified S.V.F.F. in the following way. The symmetrical non-linear triatomic molecule AB (H,O type) has on the S.V.F.F. two force constants. It has three fundamental vibration frequencies. The expression for these frequencies depends on the atomic masses which may be taken as known the two force constants and the BAB angle. Because there are three frequencies the two force constants may be eliminated and an expression obtained for the BAB angle in terms of the three frequencies. So the angle may be calculated from the observed vibration frequencies.10 However the reliability of the result depends on the reliability with which the S.V.F.F.may be applied to the particular molecule. Because the S.V.F.F. is known t o be only an approximate representation of the force field for most molecules the author considers that this is a dangerous method to use to determine bond angles in molecules. The S.V.F.F. has been extended by including in the P.E. function “ cross terms ” so that (5) becomes for molecules of the AX type) V = ZikJR; + aZ&kmA& + C&k,bARaARb + Z+krn,tAamAan + -WbmARaAarn . * (9) The P.E. function is still quadratic and will lead to an equation of the form of (3) when the dR’s and A x ’ s are replaced by q displacement co- ordinates. One may suppose that the cross-term ARa.Aarn is included in the P.E. function to account for the fact that when the bond a changes its length the equilibrium value of the bond angle a is affected.For example it may be that when the length of the C-C1 bond in CH,Cl changes the conBguration of the methyl group which has a minimum potential energy also changes. Such eEects can be allowed for by including these cross-terms and much use has been made of them. However by reason * Proc. Roy. Soc. 1936 A 156 654. lo D. M. Simpson Trans. Faraday SOC. 1945 41 209. J . Chern. Physics 1934 2 620. LMNETT FORCE CONSTANTS 79 of the scarcity of our data (number of frequencies available) we cannot usually introduce all the possible cross-terms. Thus in the molecule AB (H,O type) we could have besides the two S.V.F.F. constants two constants associated with the two types of cross-terms one between the two bonds and the other between the bond and the angle.So the most general form of the valency force field requires the use of four constants. If we cannot make use of isotopes we have only three fundamental frequencies and so can determine only three constants. Therefore in most cases it is necessary to limit the number of cross-terms that we introduce into the V.F.F. potential energy function. The difficulty is then to know which cross- terms to include and which to omit. It must be admitted that there is no sure way of doing this. The only course to adopt is to include those cross-terms which seem to account best most reasonably and most easily for the departures from the S.V.F.F. but it does seem sometimes that there is little physical explanation for some cross-terms that are introduced and that the introduction of these cross-terms has resulted in the V.F.F.system becoming rather artificial It must be stressed that it is never satisfactory to use as many unknown constants as there are observed fre- quencies because there is no check then whether the right set of cross-terms has been selected. That is there is no check on the reliability of the P.E. function that has been used and a careful check is very important as we do not yet know with any exactness the true nature or relative importance of the various interatomic forces that may occur in molecules. Ethylene formaldehyde and similar molecules have been treated by P.E. functions of this kind by H. W. Thompson and J. W. Linnett.ll3 l2 Z. I. Slawsky and D. M. Dennison,13 and J. W. Linnett l4 have treated the methyl halides CH3X in this way and a difference between the two methods of approach brings out an important point.15 Both treatments used the P.E.function (10) which it is to be noted had been arrived at quite independently I‘ = zikCHd&H + *kCXd&X + z@HCHdakCH + z*kHcxdaLc + Z#’dRcxdaHCX (10) There are four S.V.F.F. squared terms and one cross-term involving the constant E’. Slawsky and Dennison supposed that kcH and EHcH were the same in the four methyl halides as in methane. So they calculated kcH and En, from the frequencies of methane. Then for each methyl halide they adjusted the remaining three constants to give the six observed fre- quencies as well as possible. The check on the P.E. function was in 1311 cases quite good. Linnett on the other hand concluded from a study of ethane that a P.E. function of the above type was a satisfactory one to use.He then deduced separately for each methyl halide the five con- stants in (10). Because four of the six frequencies are determined almost entirely by four of the constants there is very little check on the P.E. l1 J . 1937 1376. l2 J. 1937 1384. 13 J . Chem. Physics 1939 7 509. l5 D. M. Dennison Rev. Mod. Physics 1940 12 175. l4 Ibid. 1940 8 91. 80 QUARTERLY REVIEWS function for each CH,X molecule separately. The reliability of the method depends on whether the successful test with ethane is enough. However in this case the two approaches do agree quite well. Linnett h d s that kHcH is virtually constant throughout the series as assumed by Slawsky and Dennison. However he finds that the consiant kCH increases from 4.71 to 5-00 x lo5 dynes/cm.on passing from methyl fluoride to the iodide. The study of the methyl halides therefore raises the question To what extent is it justifiable to use a force constant determined in one molecule for an apparently identical bond in another molecule ? One would have said that the C-H bonds in the methyl halides were all very nearly the mme but the above treatment suggests that their force constants do in fact change not inconsiderably from one halide to another (6%). The conclusion would therefore seem to be that we must be very careful in transferring a constant determined in one molecule to another even if the two bonds appear similar though the results for the kHCH constant show that in certain cases such a transfer is satisfactory. F. Stitt l6 treated ethane with a V.F.F.of the ty-pe represented by (9). By using the determined frequencies of both C,H and C,D Stitt was able to introduce a large number of cross-terms (6) into his P.E. function and dealt with this molecule most completely and very satisfactorily. B. L. Crawford and S. R. Brinkley l7 studied together acetylene ethane methyl- acetylene dimethylacetylene hydrogen cyanide methyl cyanide and the methyl halides. As far as possible they transferred both squared and cross-term constants from molecule to molecule. For example they used the force constants they found necessary for the methyl group in ethane for that group in all the other molecules studied-and similarly for the acetylenic and cyanide groupings. That is they went a stage further than Slawsky a1 4 Dennison in transferring cross-term as well as squared-term constants f:*om molecule to molecule.The agreement they obtained between calculated and observed frequencies was most satisfactory. They calculated for all the molecules 84 frequencies with 31 constants which implies that the procedure adopted namely transfer of constants was reliable. Lin- nett l8 treated methyl- and dimethyl-acetylene and methyl cyanide with a much simpler force field based on the one he had successfully used for ethane. He used an entirely new set of constants for methyl- and dimethyl- acetylene and calculated the 25 frequencies of these two molecules satis- factorily using 11 adjustable constants. Linnett obtained a value for the C-C bond force constant in these acetylene derivatives which was dieerent from that obtained by Crawford and Brinkley and showed that his value was more in agreement with the observed bond length the correctness of Douglas Clark's empirical relationship (see later) being assumed.He therefore questioned whether Crawford and Brinkley were justified in transferring constants so liberally from one molecule to another. The transfer of constants in the series of chlorinated derivatives of methane was studied by H. H. Voge and J. E. Rosenthal,19 but the number of frequencies l6 J . Chem. Physics 1939 7 1115. l7 Ibid. 1941 9 69. l8 J . Chem. Physics 1936 4 137. Trans. Faraday SOC. 1941 37 469. LINNET"! FORCE CONSTANTS 81 only exceeded the number of adjustable constants by one so that the test cannot be regarded as very convincing. A quite different but very realistic potential-energy function has been used by H.C. Urey and C. A. Bradley 2o for a number of molecules of the CCl type (tetrahedral). They superimposed on the S.V.F.F. (one bond and one bending constant in this case) a repulsion potential of the form V' = a/- between the non-bonded atoms. Using the four frequencies of the tetrahedral molecules they are able to calculate the two S.V.F.F. constants and a and n. In fact they assumed n to be 7 throughout and found that they could account for the four frequencies each of CCl, SiCl, TiCl, SnCl, CBr, and SnBr with the three adjustable constants (a and the two S.V.F.F. constants for each molecule). They found that the repulsions which had to be assumed between the non-bonded atoms were of the same order as those required for similar repulsions in other circum- stances.Urey and Bradley also considored the ions SO; and C10; and found that by adding an additional Coulombic repulsion force between the oxygen atoms the observed frequencies could be accounted for. Rosenthal has treated the tetrahalides with a more general potential-energy function and has considered her results in the light of Urey and Bradley's assumptions. This section has shown repeatedly that the problem in'obtaining force constants is How is one to choose a sufficiently general force field and yet not have too many unknown constants for a particular molecule ? The ideal method of approach is to obtain as many frequencies of a given structure as possible by using isotopic molecules. For instance I?. Miller and B. L. Crawford 21 have been able to deduce and check the most general form of the P.E.function for the non-planar distortions of the benzene molecule by using the vibration frequencies of benzene and a number of its deuterated derivatives. However this method of approach is not always possible and we are often forced to use simplified forms. For such purposes the modified valency force systems are the most satisfactory. Moreover the chemist always finds such systems more useful than any other even the more general ones because it gives him results in terma of valency bonds with which he is accustomed and equipped to deal. It is the determination of the force constants of individual bonds in the molecule which the chemist particularly requires for with these he may assess similarities and differences between bonds in different molecules.However these bond-force constants must be obtained by using a P.E. function (a force field) whose reliability for the molecule or molecules in question has been satisfactorily tested. G. Glockler and J. Y. Tung 22 have suggested that it is convenient in the case of for example triatomic molecules of the water type which on the general V.F.F. have four force constants (see p. 79) to plot three of the force constants against the fourth. One can then see quickly and easily what are the possible sets of values of the four constants though which four is the correct set cannot be decided with any certainty. Glockler *O Physical Rev. 1931 38 1969. 21 Ibid. 1945 13 388. 21 J . Chem. PhyeiCS 1946 14 282. F 82 QUARTERLY REVIEWS and Tung in addition suggest an arbitrary method of deciding which set of force constants is the correct one.However there does not seem to be any sound basis for the method suggested and the author agrees with D. M. Simpson 23 that “ the information obtained by using it should not be considered entirely reliable.” Nature of Experimental Data The vibration frequencies that are used in force-constant calculations are obtained mostly from infra-red 24 and Raman spectra,25-28 though ultra-violet fluorescence and resonance spectra may be used to a limited extent. In what has been said in previous sections it has been presumed that all the vibration frequencies of the molecule under consideration had or could be determined. Except for the simpler molecules this by no means represents the position. It is often difficult to decide for instance which infra-red bands are fundamentals and which combinations or over- tones.In addition when a molecule has some symmetry it is important to assign each observed fundamental frequency to its proper symmetry class. This is necessary so that we may know which frequencies are to be accounted for by a given factor of the algebraic equation (4). The assignment of frequencies to their proper symmetry classes is made possible by the application of selection rules which tell us which vibrations are forbidden to appear in the infra-red spectrum and which in the Raman spectrum.2 The polarisation of the scattered radiation in the Raman effect may also be used to help us make the correct assignment. The contours or fine structure of the infra-red bands may be used similarly. But even with all these aids it is often impossible to decide what the correct assign- ment is.For water,29 acetylene,30 ethylene,31 carbon dioxide,32 the methyl hallides,2 and many other molecules of a similar complexity all the frequencies are known and correctly assigned. For ethylene oxide cy~Zopropane,3~ the vinyl halide~,~4 p r ~ p y l e n e ~ ~ and even ozone 109 37 no complete and certain assignment has been made. The present position is conveyed by a few examples. 23 J . Chem. Physics 1946 14 294. 24 R. B. Barnes R. C. Gore U. Liddel and V. Z. Williams “ Infra-red Spectro- 25 J. H. Hibben “ The Raman Effect and its Chemical AppliCations,” Reinhold 86 G. B. B. M. Sutherland “ Infra-red and Raman Spectra,” Methuen. 27 G. Glockler Rev. Mod. Physics 1943 15 111. 28 D. M. Dennison ibicl.1931 3 280. ae B. T. Darling and D. M. Dennison Physical Rev. 1940 57 128. 30 G. Herzberg and J. W. T. Spinks 2. Physik 1934 92 87. 81 G. K. T. Corm and G. B. B. M. Sutherland Proc. Roy. Soc. 1939 A 173 172. a2 A. Adel and D. M. Dennison Physical Rev. 1933 43 716. 83 J. W. Linnett J. Chmn. Physics 1938 6 692. 34H. W. Thompson and P. Torkington Trans. Far&y SOC. 1945 41 236. 35 E. B. Wilson and A. J. Wells J. Chem. Physics 1941 9 319. 96 K. S. Pitzer ibid. 1944 12 310. 37 A. Adel and D. M. Dennison ibid. 1946 14 379. scopy,” Reinhold Publishing Gorp. Publishing Corp. LINNETT FORCE CONSTANTS 83 The accuracy of the determination of the observed frequencies is variable. Those obtained from the Raman spectra are accurate to 1 cm.-l if a reason- ably good spectrometer has been used.In the infra-red if the fine structure can be accounted for the centre of the .band may be placed even more accurately. However some bands have very complex contours and it is often difficult t o place the centre of such bands accurately. The same is even more true if bands overlap. Other difficulties such as resonance between vibrational energy levels may also make it difficult to determine the classical fundamental frequencies. The theory of molecular vibrations considers the isolated molecule. However the Raman spectrum is very often obtained by using the liquid and it is uncertain how much the internal vibration frequencies are aEected in this state by the intermolecular forces. In some cases the effect seems to be slight. Thus the totally symmetric vibration of the benzene ring gives a Raman shift of 993 cm.-l if the gas and 994 cm.-l if the liquid is used.On the other hand the observed value of the C-C1 valency vibration in CH3C1 changes from 732 to 709 cm.-l on passing from the gas to the liquid.2 So far it has not been found possible to assess what the change is likely to be for a given vibration but in most cases it is probable that the percentage error from this source is small. The assumption in (3) that the potential energy is a quadratic function of the displacement co-ordinates is an approximation. This means that the motion is not strictly simple harmonic and the observed frequencies are different from the values they would have if the displacements were infinitesimal. The assessment of this anharmonicity correction necessitates the observation of a large number of overtone and combination frequencies and is hardly ever possible.So we are forced t o use the uncorrected frequencies which are observed as if they were the fundamental frequencies for S.H.M In doing this we introduce an error into the derived force constants which may be considerable. For instance for the water molecule the anharmonicity correction averages 4%. z9 This means that the force constants derived by using the positions of the infra-red bands will be 8% different from the true force constants derived for vibrations involving infinitesimal displacements. For nitrous oxide the error in the force constants would be less-about 374.2 It must always be realised that the force constants derived from observed frequencies are subject to this error.When comparing force constants of similar bonds in different molecules it is presumed that the anharmonicity correction is likely to be very much the same in all cases. Thus for the C-H valency vibration in HCN the anharmonicity correction to the frequency is 4% (3312 to 3452) whereas for the 0-H vibration in water it is 43% (3652 to 3825). For all bonds between the same pair of atoms (e.g. all C-H bonds) it is even more likely to be the same. Uses of Force Constants Before considering the uses of force oonstants we will 8e0 what is the The bond-force constants are order of magnitude of these quantities. 84 QUARTERLY REVIEWS iisually measured in dynes/cm. and measure the restoring force that would come into play if the bond were stretched 1 cm. if the law of force assumed persisted t o such large separations.The force constant of the C-C bond in ethane is 4.5 x 105 dynes/cm. and constants as low as 2 x lo5 and as high as about 20 x 105 are known. If the C-C bond in ethane whose equilibrium length is 1-55 A. is stretched by 0.1 A. the restoring force exerted (- kAR) is 4.5 x 10-4 dyne and the potential energy (@AR2) is 2-25 x This energy per molecule corresponds to 3260 cals. per g.-mol. and if the force of N molecules in the same configuration could be added together the force would correspond to 2-73 x loll tons weight per g.-mo1.-nearly a million million tons weight. Angle-bending constants are measured in dpe-cm./radian (Le. ergs/radian). The bending constant of the HcH angle is about 0-5 x 10-11 dyne-cm./radian. So if the angle is distorted by 0.1 radian (5.73") the potential energy increases by 0.25 x erg.This is equivalent to 362 cals. per g.-mol. The force on the hydrogen atom which is a t the end of the C-H bond whoselength is 1.09 A. is k.Aa/R and this is 0.46 x loF4 dyne. We see that the forces necessary to distort a molecule by bending the boiids is of the order of a tenth of the force necessary to alter the length of the bonds by a similar amount. Because of this the vibrations which involve changes in a bond length have a higher frequency than those which involve mainly a bending of that bond. Much more interest has naturally been focused on the bond-stretching constants than on the bending constants and we will consider the former now. The following table records a number of these force constants. To show the variation that may obtain in the results for the force constants of a particular bond one may quote the figures for the C-Cl bond in methyl chloride these are 3-44,38 3 ~ 6 1 ~ ~ 4.42,13 3.35,14 3.64,l7 and 3.37 x lo5 dynes per It has been pointed out that force constants are related to the positions of the atoms in the Periodic Table.Thus the force constants of the bonds B-H C-H N-H O-H and F-H are 3.6 5.0 6.5 7.6 and ca. 9.0 x lo5 (all uncorrected for anharmonicity). Reference to the table will show other series. This relation to the Groups the atoms occupy in the Periodic Table appears in the empirical formulze of Douglas Clark Linnett and Gordy. Reference to the table shows that force constants vary with bond order. For those bonds which involve carbon as one member the force constants of single bonds are about 5 x lo5 or rather less those of double bonds about 10 x lo5 and those of triple bonds about 15 x 105 dynes per cm.or rather more. The variation of force constant with bond order for bonds between a given pair of atoms means that the force constant may be used to assess bond character in the way that bond lengths have been so widely employed. An example of this is provided by carbon dioxide. In carbon monoxide the CO force constant is 18.5 x lo5 dynes per cm. and in form- aldehyde it is about 12.3 x lo5. In carbon dioxide it is 15.5 x lo5. This 38 G. B. B. M. Sutherland and D. M. Dennison Proc. Roy. SOC. 1935 A 148 250. 89 H. D. Noether J . Chem. Physics 1942 10 664. erg per molecule greater than the equilibrium energy. The last is probably the best.LINNETT FORCE CONSTANTS 85 TABLE This table lists the bond-stretching force constants in dynes per cm. x 10-6 ; i.e. a force constant of 5.0 x lo5 is recorded as 5-0. The constants for diatomio molecules are corrected for anharmonicity but those for polyatomio moleaules are not. The probable correction is that 8% should be added to the uncorrected constants of those bonds of which one member is hydrogen and 2 or 3% should be added in other o m s (see text). Bond. H-H Li-Li Na-Ne K-K Rb-Rb cs-cs Li-H Na-H Rb-H K-H CS-H F-H CI-H Br-H I-H N-N N-N 0-0 0-0 c-0 c-0 N- 0 c-s P-N P-P s-s c1-c1 Br-Br 1-1 I-c1 I-Br B-H G H G H C-H G H C-H C-H C-H C-H C-H G H N-H Molecule. H2 Na2 KZ Rb cs LiH NaH K H RbH CsH HF HC1 HBr HI N2 N; 0 0,' co co + NO cs P N P Z Sa c1 Bra I ICl IBr CH CH,F CH ,C1 CH,Br CH,I C2H6 CZH HCN CH,O NH Li B2H6 c2H4 k x dynes/cm.5.76 0.25 0.17 0.10 0.08 0.07 1-03 0.78 0.56 0.51 0.47 9.7 5.2 4.1 3.1 22.8 20.0 11.8 16.5 18.9 19-7 15.9 8.4 10.1 5.5 5.0 3.3 2-5 1-7 2.4 2.1 3.6 5.0 4.7 4.9 4.95 5.0 5.1 5.1 5.9 5.9 4.4 6.5 Bond. N-H N-H N-H O-H O-H B-H 8e-H P-H Si-H c-c c-c c-c c-c c-c c-c c-c c-c C-C c-c C-C c-c c-c C-N C-N N-C C-N C-N C-N C-N c-0 c-0 c-0 c-0 c-s C-F c-Cl C-Br c-I c-CI C-Br GI G N Molecule. CH,*NH NH? NH,*CO*NH H2O CH ,*OH H2S H,Se PH3 SiH C2Hz C2H6 C2H4 C6H6 CH ,C$H CH ,-CC*CH CH,.C:CH CH ,*CE*CH CH,*CN CH,:CO CH,:C:CH C2N2 HCN CH,*CN CH,*NF CH ,*NC C2N2 ClCN BrCN ICN CH,O CH,:CO COZ cs c302 c30Z CH 3F CH,C1 CH ,Br CH,I ClCN BrCN ICN k x lo-' dynes/cm. 6.3 5.4 6.4 7.6 7.6 3.1 3.1 2.7 4.5 9.8 15.6 7.6 5.5 5.5 15-3 15.3 5.3 9.8 9.7 14-9 6.7 18-1 17.5 16-3 5.45 1'7.5 16.7 16.8 16.8 12-3 12.3 15.5 14.2 7.5 5.6 3.4 2.9 2.3 5.3 4-2 2.9 4.0 shows a t once that the C-0 bond in carbon dioxide is intermediate between that in formaldehyde and that in carbon monoxide.12 It may therefore be said fo be more than a double but less than a triple bond.A similar 86 QUARTERLY REVIEWS example is provided by cyanogen chloride,40 ClCN in which the C-Cl bond has a force constant of 5.3 x lo5 as against 3.4 x lo5 for the C-C1 bond in methyl chloride. This shows that the C-Cl bond in cyanogen chloride is stronger than a single bond an observation which can be accounted for by supposing that the real electronic structure is intermediate between and Several other similar examples of the use of force constants in this way may be quoted (c30,,41 C2N2,'* C6H6,42 methyl- and dimethyl-acetylenes 18) but in fact the method has not been used as much as it might have been because of the difficulty of finding a satisfactory tested P.E.function for all but the simplest molecules. It is to be hoped that as our knowledge of molecular-force fields increases this method of assessing bonds will be used more and more. The force constant of the C-H bond can be determined with fair accuracy in a large number of molecules. This is possible because the C-H bond vibration frequency is so much greater than all the other vibration frequencies of many organic molecules that it can be treated ~eparately.~ Linnett 43 calculated the force constants of a number of C-H bonds and found that they varied from 4.4 x 105 in aldehydes to 5-9 x lo5 dynes per cm.in acetylene. It is interesting and surprising that the force constant of this bond which is in all cases written as C-H varies over a range of ca. 30%. It appears that three factors affect the force constant ( a ) The nature of the bond orbital -whether sp sp2 or sp3 hybridised ; ( b ) the electrostatic state of the bond this being affected by neighbouring groups 44 (cf. Gordy) ; ( c ) resonance with various ionic structures. Other M-H bonds were studied with similar results but the treatment of the N-H and O-H bonds has been improved by R. E. Richards.45 Another important feature of force constants is the relation they bear to bond lengths. R. M. Badger 46 first pointed out from a study of diatomic molecules that as the bond between a given pair of atoms becomes stronger (i.e.the force constant bigger) its equilibrium length becomes shorter and he proposed the empirical relationship k = A / ( R - B)3 * ( 1 1 ) where R is the equilibrium bond length and A and B are constants. He found that their values depended on the positions in the Periodic Table of the atoms forming the diatomic molecule and gave a table of suitable values for A and A second relation was proposed by C. H. Douglas CIark,4s who suggested that \ / /cl=c=N \ I 1 - C l - C ~ - - One special case may be mentioned. kR = C - (12) ~~ 40 H. W. Thompson and J. W. Linnett J . 1937 1399. 4 2 R. C. Lord and n. H. Andrews J . Physiccll Chenz. 1938 41 149. 4 3 J. W. Linnett Traw. Furaday SOC. 1045 41 223. 4 4 H. C. Longuet-Higgins ibid. p. 233. 4fi J . Chem. Physics 1934 2 128.48 Phil. Mug. 1934 18 459; 1936 22 1137. I d e m ibid. p. 1291. 4 5 Trans. Famdny SOC. in the press. 47 Ibid. 1935 3 510. LINNETT FORCE CONSTANTS 87 The value of the constant C depends also on the position in the Periodic Table of the atoms forming the molecule and Douglas Clark and K. R. Webb 49 have given formulae from which C may be determined. Othef relations have been suggested by H. S. Allen and A. K. Longair,60 and by M. L. Huggim61 Linnett 52 suggested a potential-energy function for diatomic molecules which with various empirical assumptions led to a relation between k and Re.53 The author considers that the most useful of these relationships is that of Douglas Clark because it combines sufficient accuracy with great simplicity. For instance it gives better results than Badger’s equation and it has the great advantage that there is only one constant (C) instead of two ( A and B).G. B. B. M. Sutherland 54 has given an explanation of the Douglas Clark relationship and this has been discussed by Linnett.62 These relationships were developed for diatomic molecules but have been tested for individual links in polyatomic molecules. Badger found that his relation was quite successful with C-H C-0 C-S and S-0 links in triatomic molecules and H. W. Thompson and J. W. Linnett 55 found that both it and Douglas Clark’s relation gave good results for C-H G O and C-C links in a variety of molecules. J. J. Fox and A. E. Martin 66 pointed out that for C-C links in polyatomic molecules kRi was more nearly constant than kR:. It seems to the author that a modified Douglaa Clark relation IcR; = C may be useful n and C being fixed by reference to perhaps three molecules for which k and Re are both known.Then in other molecules k may be used to calculate Re for bonds between the same pair of atoms. There is no doubt that this possibility of deducing Re from k is very valuable as it can be used to check results obtained by more direct methods (spectroscopic and electron diffraction) and there are cases (e.g. some G-H bonds) where the direct methods cannot be employed. Badger considered also the application of his equation to molecules of the AX and the BX type in which repulsion between the non-bonded atom is ~onsiderable.~’ He found that he could account for his results if he supposed that the repulsions between the X atoms caused a considerable stretching of the AX (or BX) bond.Thus for CCl he concluded from the force constants obtained by J. E. Rosenthal 57 that the C-cl bond was stretched by 0.19 A. from the value it would have had if there had been no repulsion between the X atoms. This is a surprising result because it implies that the GC1 bond in methyl chloride in which such stretching must be very small and that in CCl have about the same length only because of a balancing of various quite large factors. 49 Trans. Faraday SOC. 1941 37 293. 61 J . Chem. Physics 1935 3 473; 1936 4 308. 5 a Trans. Faraduy Soc. 1940 36 1123. 63 J. W-. Linnett ibid. 1942 38 1. 64 Proc. I d . Acad. Sci. 1938 8 341. 6 6 J.,. 1937 1396. 66 J. 1939 884. ~57 Physical Rev. 1934 46 1934. Phil. Mag. 1935 19 1032. 88 QUARTERLY REVIEWS W.Gordy 68 has recently introduced a relation between the force con- stant bond order ( N ) bond length (Re) and the electronegativities (zA and zB) of the bonded atoms. The relation is of the form k = aN(zA.zB/Ri)f + b . * (13) where a and b are constants (for most pairs of atoms 1.67 and 0.30 respec- tively). Gordy shows that (13) may be applied widely to bonds in diatomic and polyrttomic molecules when the bond is not distorted by forces be- tween non-bonded atoms. He gives a table of electronegativities and uses measured values of Re to predict k when N is known. In other cases he predicts Re from k when N is known and in a few cases he determines the bond order when k and Re are both known (e.g. BrCN). He also con- siders the effect of a charge being located on the atoms and shows that a positive charge by affecting the electronegativity terms will cause an increase in the force constant if N remains the same.There has been no detailed work on the relation between the heat of dissociation of a bond and its force constant. Fox and Martin 56 for C-C bonds and G. Glockler and G. Matlack 59 for 0-0 bonds have shown that the graphs of both k and D against Re are smooth curves. This implies a smooth relation between k and D. For C-C bonds Fox and Martin pointed out that kE/D was a constant quantity. Linnett 53 used the P.E. function he had suggested to calculate the heats of dissociation of some diatomic molecules from the observed force constants but the data available were not accurate enough to provide a satisfactory test. It may also be noted that the C-H bond in methane has both a higher force constant and a higher diesociation energy than the C-H bond in ethane.Force constants have been used to calculate unobserved frequencies. It may happen that for a given molecule one or two vibration frequencies cannot be determined. In such circumstances it may be possible to test a P.E. function with the frequencies that have been observed use them to calculate the force constants and with these to calculate the unobserved frequency or frequencies. Before Stitt’s l6 complete treatment of ethane this approach was employed for that molecule but the frequency predicted by this method was later found to be quite wrong. This use of force constants will become increasingly valuable when we know more about molecular force fields for very often it is impossible to determine all the vibration frequencies experimentally.Yet for certain purposes such as the calculation of thermodynamic quantities it is necessary to know all the molecular frequencies. The possibility of calculating the frequencies of one molecule by using force constants obtained from another similar one is very liable to error because of the uncertainty of transferring force constants (see p. 80). Eventually it may be hoped that we shall be able to predict the values of force constants in molecules thaf have not been examined experimentally and so calculate their vibration frequencies. It was pointed out on p. 79 that cross-terms probably account for J . Chem. Physics 1946 14 305. 59 Ibid. p. 503. LINNETT FORCE CONSTANTS 89 the effect of a change of one part of the molecule on the configuration of another part.It has been suggested that the large cross-term constant for the interaction between the two bonds in CO is to be explained by the resonance in this molecule. When one C-0 bond lengthens it favours one of the single-triple bond structures (0-0) with the result that a shorten- ing of the other G O bond more readily accompanies a lengthening of the first C-0 bond. Bending Force Constants Up to the present less consideration has been given to bending-force constants than to valency-stretching force constants. The reason for this is that it has been impossible to account for the changes that occur in these constants. For instance although it is found that the HcH angle has a bending constant which is constant throughout the methyl halides (0.5 x dyne-cm./radian),l' yet the HeH angle in ethylene has a much smaller bending constant (0.36 x 10-11).l1 Admittedly the electron distribution in the two cases is Werent because the bond hybridisation is different but the change is nevertheless surprisingly big. Again the C - d H bending constant is 0-66 x 10-11,14 the C=C/ 0.6 x 10-11,11 but the CEC-H is 0.24 x 10-l1.l7 It is surprising that the change from the double to the triple bond is so much greater than the change from the single to the double. Also it might have been expected that the change would have been in the other direction and that the angle involving the triple bond would have been the most rigid. This change in the rigidity of the above angles may possibly be explained by the fact that in ethane there are four atoms round the carbon in ethylene three and in acetylene only two.Regularities can nevertheless be observed in bending constants. Thus the X - d H constants in methyl fluoride chloride bromide and iodide are 0.9,0-7,0.62 and 0.55 x 10-11. Where both bonds are multiple as in carbon dioxide the bending constant is 0.75 x It is interesting that this is smaller than that of F-dH which involves only single bonds. On comparing similar bond arrangements round the same atom it is found that the bending constant of C-C-H is 0.25 and of C-=C-C 0-35 x dyne-cm./radian. l7 Some of the above irregularities in bending constants certainly arise because some are obtained by using unsatisfactory force fields. The results obtained for bending constants are because they are small relative to the stretching constants much more dependent on the type of cross-terms that are introduced and because of this it is probable that mmy of the values obtained are unreliable.When a given type of force field is used throughout a series of molecules (e.g. the methyl halides) it is found that similar angles (e.g. HeH angles) have a single value for their bending constants throughout the series and also that a graded series of angles (e.g. X d H ) have a steadily varying value of the bending constant. This suggests that we H 90 QUARTERLY REVIEWS may hope for advances in the interpretation of the bending constants. This may be easier when more is known of the forces between non-bonded atoms. Summary In this report as much space has been spent on an examination of the force fields that have been employed as has been spent on the uses of the force constants obtained.Unfortunately this represents the position. More time has been spent since 1930 in examining the merits of various force fields than in using the constants obtained for the elucidation of chemical problems. Moreover the position today is still that for many polyatomic molecules we are by no means sure what sort of force field is best. It is in this direction that development must come first and only after this can we hope to extend the chemical applications of force con- stants. This development will bc aided by the investigation of the spectra of isotopic molecules (13C 15N etc. being used as well as deuterium) since these will provide for any given molecular system more measured vibration frequencies. The extension of our Itnowledge of force fields in molecules may then increase our understanding not only of the valency bond but also of the other forces that are exerted between the component atoms of a molecule. In conclusion I wish to thank Professor C. N. Hinshelwood and Mr. R. P. Bell for the helpful advice they have given me.