THE ABSOLUTE INTENSITIES OF INFRARED ABSORPTION BANDS By D. STEELE (DEPARTMENT OF CHEMISTRY ROYAL HOLLOWAY COLLEGE ENGLEFIELD GREEN SURREY) THE intensity of absorption of infrared radiation by a given system is intimately related to the electronic charge movements during the associated vibrational quantum transition. Molecular deformations must involve bond deformations and are very unlikely to affect any but the valence-shell electrons. Consequently in principle absorption intensities could yield not only information on charge distributions in molecules but also infor- mation on the manner in which the valence electrons redistribute them- selves during molecular deformations. Since chemical reactions of neces- sity involve specific bond deformations such information could lead to a deeper understanding of reaction mechanisms.The equilibrium charge distributions can lead to a better understanding of the bonding. During recent years a large number of publications on the theory measurement and interpretation of the absolute intensities of infrared absorption bands has appeared in the literature. Many serious difficulties still beset the spectroscopist in the interpretation of the results but the information gleaned has reached the state where a survey of the gains the difficulties and the prospects can be usefully made. Experimental Techniques.-It has proved to be extremely difficult to make accurate absolute measurements of absorption intensities. The intensities are usually defined as (1) where c is the absorbent concentration I is the path length of the beam through the absorbing material (Io)y and (Iv) are the initial and final intensities of the beam of frequency v (expressed in cm.-l in all subsequent equations) and the integration is over the complete band.Such a definition follows readily from the usual exponential low of absorption. The ex- perimental difficulties were first made apparent by the early measurements of Bourginl and Bartholomk2 Independently they measured the intensity of the vibration-rotation band of hydrogen chloride in the gas phase and obtained results which differed by a factor of four. The major reason for this was that for finite slit-widths the beam is not monochromatic and consequently the measured fractional absorption of the sample [(TO-T)/TO], at a given frequency setting Y generally differs from the true transmission in such a way that the measured absorption value is too low.B~urgin,~ Bartholomt5,2 Penner and we be^-,^ Wilson and Wells,5 and others Gc = 1 /cl Sin [(I O ) l m v Id 1nv D. G. Bourgin Phys. Rev. 1927 29 794; ibid. 1928,32,237. E. BartholomC 2. phys. Chem. 1933 b23 131. D. G. Bourgin Phys. Rev. 1928,31 503. S . S. Penner and D. Weber J. Chem. Phys. 1951 19 801 817 974 ibid. 1953 21 E. B. Wilson and A. J. Wells J. Chem. Phys. 1946 14 578. 649. 21 22 QUARTERLY REVIEWS have investigated the conditions under which these two values approach one another and how the true absorption can be evaluated with an instru- ment of limited resolving power. In the procedure of Wilson and Wells the measured integrated optical density Jlog, (TOIT) d In V divided by the concentration is graphed against the concentration and the plot extra- polated to zero concentration.It was shown that the limiting value of the integral for zero concentration is equal to Jlog, (Io/I)>y d In v if (a) the incident intensity I, does not vary over the slit-width and (b) the resolving power is high compared with the variations in the absorption coefficient. In Bourgin’s method [ f(T,/T) dv]/c is graphed against c and the ratio extrapolated as in the Wilson-Wells technique to zero concentration. However the curvature of the plot is far greater and the extrapolation consequently less accurate. Since the aim of absorption intensity measurements is usually to study intramolecular properties it is necessary to carry out studies on the gaseous phase where intermolecular interactions are reduced to a minimum.Unfortunately the removal of intramolecular interactions results in sharp vibrational-rotational absorption lines. As a consequence condition (b) is difficult to attain. In order that this condition should be satisfied the rotational bands must be collision-broadened by adding a high pressure of a chemically inert non-absorbing gas or for the study of weak absorp- tion bands by self-broadening at high pressures. The pressure-broadening is considered to be sufficient when an increase of total pressure produces no further change in the molecular extinction coefficient E = l/cZ loglo(lo/l),ax. However even when the individual rotational lines are sufficiently broadened to yield an overall smooth absorption curve the band contour may still have sufficiently steep gradients to result in low measured values of the absorbance.This is particularly the case with bands having strong sharp Q branches (corresponding to no change in the rotational quantum number J ) such as the out-of-plane deformation bands in aromatic sys- tems. The pressure required to produce broad Q branches may be exces- sively high and it is to be noted that in such cases the extrapolation pro- cedure still may not be strictly valid if condition (b) has not been fully met. Consequently the use of instruments of high resolving power is really required in such cases. At very high pressures the molecule is being subjected to excessive col- lisional perturbations which are often undesirable. Thus it has been shown that absorption by the infrared-inactive a, mode of methane can be induced in this manner,6 and many examples are now known of simultane- ous transitions in mixed gases at high pressures.‘ In a simultaneous transi- tion absorption occurs at a frequency v & vb where v and Vb are transi- tion frequencies for two different molecules and can be for two different R.Coulon B. Oksengorn J. Robin and B. Vodar J. Phys. Radium 1953 14 63. H. L. Welsh M. F. Crawford J. C. F. McDonald and D. A. Chisholm Phys. Rev. 1951 83 1264; J. Fahrenfort and J. A. A. Ketelaar J . Chem. Phys. 1954 22 1631. STEELE INFRARED INTENSITIES 23 chemical species. Such evidence indicates that pressure-broadening must be treated cautiously especially when dealing with weak bands and easily polarisable molecules. The sources of error in the Wilson-Wells and Bourgin procedures are so serious that efforts have been made to find alternative techniques.The dispersion of infrared rays has proved to be a very valuable tool in this connection since a vibrating electric moment in a molecule gives a con- tribution to the refractive index. Molecular vibrations are separable into normal modes each of which makes a contribution to the refractive index n of A i ( n - 1) =- where N is the number of molecules per ml. p is the molecular dipole moment,Qi is the ith normal co-ordinate vi is the frequency of the cor- responding vibration and v is that of the incident light (in cm.-l). (This formula known as the Kramers-Heisenberg formula assumes that the absorption line is infinitely narrow and has to be modified slightly for finite line widths). The infrared absorption intensity of a given fundamental band has been shown to be equal to (This formula is derived neglecting rotational quantisation.An exact summation of intensity over the rotational components of a parallel band of a symmetric rotor molecule* leads to a correction factor equivalent to multiplying the right-hand side of (3) by 2 B c [l + exp (- hv,/kT)] v [I - exp (- h v o / k ~ ) r + B the rotational constant is equal to h/8n2c IB where IB is the moment of inertia perpendicular to the axis of the top and c is the velocity of light. This factor is unlikely to lead to an error exceeding 5% in (ap/8Q)2 and is usually neglected on the excuse that the experimental uncertainty is generally of the same order of magnitude.) Thus the vibrational contribution to the refractive index is intimately related to the infrared absorption intensity and in fact these two pheno- mena are manifestations of the same property.Consequently absolute infrared absorption intensities can be deduced from dispersion studies. The particular advantage of the dispersion method as can be seen from equation (2) is that even for infinitely narrow absorption lines the change of the refraction with frequency is quite gradual. A typical refraction spectrum due to the R2-0 bands of H35Cl and H3'Cl is shown in Fig. 1,9 B. L. Crawford and H. L. Dinsmore J. Chern. Phys. 1950,18 1682. F. Legay Rev. Opt. (theor. instrum.) 1958 37 11. 24 QUARTERLY REVIEWS 2 I ‘ 0 0 9 x c P t - I - 2 -3 5 7 3 0 5725 5720 5715 FIG. 1. The vibration-rotation contribution to the refractive index by the R(2) bands of the 2-0 vibrationaL transitions of H35Cl and H3’Cl.The vertical lines indicated as R(2) represent the relative intensities of the corresponding absorption lines. and is compared with the corresponding absorption curve. It can be seen that the distance between opposite branches of the refraction curve is a function of the absorption intensity. The technique is only suitable at present for simple molecules with well-separated strong absorption bands. This is a very severe restriction but fortunately it is for such molecules that the Wilson-Wells procedure is most unsuitable. The results of the dispersion measurements generally compare quite favourably with those of the best absorption measurements and are usually though by no means always higher than their Wilson-Wells counterparts.An excellent sum- mary of measurements up to 1960 is given in ref. 10. Another recent technique capable of giving relatively accurate results is the curve-of-growth method.ll This involves making measurements at different path-lengths and allows the error due to finite slits to be eliminated if the band shape is known. It can be applied only if the individual rota- tional lines can be resolved which seriously restricts its applicability. Where it can be applied it is usual to assume that the lines can be des- lo J. H. Jaffe “Advances in Spectroscopy,” Interscience New York 1961 Vol. 2 p. 263. l1 S. S. Penner and €I. Aroeste J. Chem. Phys. 1955 23 2244. STEELE INFRARED INTENSITIES 25 cribed by the Lorentzian function. In such cases the technique would be expected to be of superior accuracy to the Wilson-Wells procedure.Consequently there was a great deal of consternation when the intensity of the 670 cm.-l band of carbon dioxide was measured in this way and a result obtained which was 50% higher than previous results.12 As pointed out by Kaplan and Eggers,12 this 670 cm.-l band is an extremely difficult band to study as far as the Wilson-Wells procedure is concerned since (a) it has a great deal of its intensity concentrated in a sharp Q branch (half-width ca. 0.35 cm.-l); (b) to measure it it is necessary to remove absorption due to atmospheric carbon dioxide; and ( c ) it lies in a range of the spectrum where sodium chloride prisms begin to absorb appreciably and where the dispersion of potassium bromide prisms is low. This means that all the serious problems characteristic of this technique are in force for this case.Crawford and his co-workers13 have remeasured the intens- ity by the Wilson-Wells procedure exercising great care to overcome these problems and have obtained a result very close to that of the curve-of-growth method (see Table 1). Also they pointed out that cakcula- TABLE 1. Measured intensities of 15p band of carbon dioxidc. Ref. 15 16 17 18 12 13 Intensity (103cm.2/mole) 7.40 6.28 5.41 6.02 8.07 8-09 Method C-0-G W-W W-W D C-0-G W-W C-o-G curve-of-growth. W-W Wilson-Wells. D dispersion. tions made by Kostkowski and B a d 4 on the functional dependence of the errors in measuring intensities of individual rotational lines should be applicable to measurements on sharp Q branches. Using Kostkowski and Bass’s results and estimating the pressure-broadened line-widths from collision theory they showed that the pressure-broadening in previous determinations of the intensity (at total pressures of up to 5 atm.) had been inadequate.At the pressures of about 68 atm. that they had employed the error resulting from slit-widths should be negligible. Also they had failed to observe any induced absorption. These careful measurements indicate that the Wilson-Wells procedure is capable of reasonable accuracy (within 2-3 %) if sufficient care is exercised. Interpretation of Results.-The interpretation of the measured intensities in terms of bond properties is best appreciated by considering what can be deduced with and without assumptions from the intensity measurements. Equation (3) is derived on the assumption that the dipole moment p can l2 L.D. Kaplan and D. F. Eggers J. Chem. Phys. 1956,25 876. l3 J. Overend M. J. Youngquist E. C . Curtis and B. Crawford J . Chem. Phys. 1959 l4 H. J. Kostkowski and A. M. Bass J. Opt. Soc. Amer. 1956 46 1060. l5 L. D. Kaplan J. Chem. Phys. 1947 15 809. l6 A. M. Thorndike J. Chem. Phys. 1947 15 868. l7 D. F. Eggers and B. L. Crawford J. Chem. Phys. 1951 19 1554. 30 532. Values reviewed by 0. Fuchs 2. Physik 1927,46 519. 26 QUARTERLY REVIEWS be expanded as a Taylor series in terms of displacements from the equilibrium positions and all but the first derivatives can be neglected. That is p = po -I- 2 ( a ~ / a Q ~ ) ~ Q k -/- higher terms (negligible). k Q k represents the molecular distortion in the vibration k (i.e. normal co- ordinate for k).This is the assumption of electrical harmonicity which is true only to a first approximation. The intensity of infrared combination bands ought to be zero in this approximation. This is certainly not so but the intensities are usually far less than those of fundamentals unless a combination band gains intensity from a fundamental of the same sym- metry by resonance. This confirms the validity of the assumption. The error involved is certainly much less than the present experimental errors and those which are to be discussed later arising from the uncertainty in the normal co-ordinate Q. If the molecule has any symmetry the vibrations can be separated into independent groups each group being characterised by the behaviour of its constituents towards the symmetry elements.This means that the normal co-ordinates Qk separate according to this behaviour. For example in carbon dioxide there are two non-degenerate vibrations and one doubly degenerate vibration. The linear molecule has three planes of symmetry mutually perpendicular and passing through the carbon nucleus. This also implies a centre of symmetry and rotation axes but a consideration of the symmetry of the vibrations with respect to the planes suffices for our present purpose. Each vibration must be symmetric or antisymmetric with respect to a particular plane. If a vibration is sym- metric with respect to the XZ and YZ planes (axes defined in Fig. 2) then i 0-2 Y FIG. 2. The vibrational modes of carbon dioxide and the definition of the Cartesian axes. clearly this cannot involve movements of the nuclei in the 2 direction.Consequently only CO stretching motions belong to the symmetry classes involving these symmetry characteristics. In addition the CO stretching STEELE INFRARED INTENSITIES 27 vibrations must be symmetric or antisymmetric with respect to the X Y plane. This means that the CO bonds must vibrate either in phase (sym- metric stretch) or out of phase (antisymmetric stretch). Similar reasoning shows that the only deformation which can be antisymmetric with respect to either the XZ or the YZ plane is that of the 0-C-0 angle. Thus in this simple case the normal co-ordinates are apart from normalisation factors dr + 4 r 2 dr - dr2 and da. In general the symmetry of Qk and hence of ap/aQk is known but in solving for the dipole gradients from the measured intensities using equation (3) there are sign ambiguities arising from taking the square root of the intensities.Furthermore the exact form of the vibrations Qk must be determined before any further inter- pretation of the gradients is possible. That is it is necessary to obtain a relationship between the normal co-ordinates Q and a set of co-ordinates such as Cartesians which are defined with respect to the molecular axes. Provided that the potential energy can be expressed as a function of all possible distortions of the molecule molecular-vibration theory will yield the required relationship as well as the vibration frequencies. In order to know the potential energy it is sufficient to know the force field i.e.,the force constants connecting all atoms in the molecule. Unfortunately these are known with precision for very few molecules.A simple example is that for the harmonic oscillator for which the vibrational frequency v is given by 1 2 n v = - J ( k / p ) and V = &q2 (4) where k is the force constant for the distortion q and p is the reduced mass of the system. Such equations describe to a first approximation the vibrational motions of a diatomic molecule. Purely theoretical approaches to evaluating the force constants of the system are impossible at present except for the simplest of molecules. Force constants have been calculated for a number of simple systems such as LiHI9 C-H,20 CH4 (C-H stretch),21 and 0,22. Except for diatomic molecules there are more quadratic force constants than fundamental vibrational frequencies. Thus for the non- linear triatomic system XYX a complete description of the potential energy arising from molecular deformations requires a knowledge of four force constants-those for the X-Y stretch the XYX deformation the interaction between the X-Y stretches and between the stretch and angle motions-whereas there are only three fundamental vibrational frequencies.(The number of vibrational frequencies is 3N - 6 for a non- linear molecule or 3N - 5 for a linear molecule). For molecules with a high degree of symmetry certain of the fundamental vibrations have the l9 A. M. Karo and A. R. Olsen J. Chem. Phys. 1959 30 1232. 2o J. Higuchi J. Chem. Phys. 1954 22 1339. 21 R. G. Parr and A. F. Saturno unpublished data; I. M. Mills Mol. Phys. 1958 22 A. Meckler J. Chem. Phys. 1953 21 1750. 1 99 107. 28 QUARTERLY REVIEWS same frequencies i.e.they are degenerate. As the number of atoms in- creases and the molecular symmetry decreases the situation becomes rapidly more unfavourable. Clearly additional sources of experimental data are needed in order to deduce the force field. By means of isotopic substitution the harmonic vibrational frequencies are altered without affecting the electronic binding. By this expediency it is possible to obtain further sets of experimental data from which to deter- mine the force constants. At first it would appear that M isotopic sub- stitutions would yield Mx sets of data where x is the number of funda- mental frequencies. In practice this is not so. There are several reasons for this. First whilst the molecular vibrational frequencies are properties of the molecule as a whole it is well known that many bonds vibrate almost independently of the remainder of the molecule.Thus >X-H stretching modes interact only very weakly with other modes. This clearly means that isotopic substitution of such atoms will not give significant data on the interaction terms which do not involve the X-H stretching motion. Secondly whilst all the vibrational frequencies may change on isotopic substitution it may be found that all changes are not independent. Vibra- tional theory shows that the product of the vibrational frequencies of any symmetry class are related by the geometry of the molecule and the masses of the atoms to the product of the equivalent vibrations of an isotopically substituted Such internal relationships reduce the number of independent equations relating frequencies to force constants.A third limitation arises from the extremely small changes in the majority of the vibrational frequencies which result from isotopic mass changes of atoms other than hydrogen. This insensitivity of the vibrational frequencies to the isotopic masses drastically restricts the usefulness of the observed data. Frequently all the vibrational frequencies may prove quite insensitive to a particular interaction force constant thus making the uncertainty in the value of that constant large compared with its actual magnitude. An inter- esting case of this type has been discussed by Li~~nett.~* The two stretching vibrations of HCN belong to the C symmetry class and may be considered independently of the angular deformation frequency of the 7~ class.Writing the stretching part of the potential energy as 2v = k + k2(drC$ + 2k, (&d @rcN) and using the values of k and k as determined from HCN and DCN it may be shown that a change in the interaction constant kI2 from 0 to 1 x lo5 dynes/cm. results in a change in the calculated stretching frequencies of HC14N and HC15N of only 1.7 and 1-1 cm.-l whilst for DCN the corresponding changes are 100 and 74 cm.-l. The actual value of k12 is near -0.4 x lo5 dynes/cm. Clearly the constant k12 contributes little to 23 0. Redlich 2. phys. Chem. 1935 b28 371; see also W. R. Angus C. R. Bailey J. B. Hale C . K. Ingold A. H. Leckie C. G. Raisin J. W. Thompson and C. L. Wilson. f. 1936 971. 24 J . W. Linnett A m . Reports 1952 49 8. STEELE INFRARED INTENSITIES 29 the potential energy of HC14N and HC15N and would be difficult to estimate from the vibrational frequencies of those systems.The reason why the frequencies of DCN are so much more sensitive to k12 is that the CD stretching frequency near 2320 cm.-l is much closer to the stretching frequency of the CN group (at 2089 cm.-l in HCN) than is the CH frequency of 3310 cm.-l. If we are to arrive generally at a realistic force field and hence at a good description of the vibrational distortions we must seek additional sources of information about the force constants. Such information can be ob- tained in principle from the magnitudes of vibration-rotation interactions and centrifugal distortions from mean-square vibrational amplitudes as determined by electron diffraction and in certain circumstances from absorption intensity studies.According to the Born-Oppenheimer approxi- mation electronic vibrational and rotational motions are independent of one another for non-degenerate vibrational states. For example the rotational energy levels of a spherical rotor (e.g. CH, SF,) can be written as ( 5 ) where Erot. is the rotational energy h is Planck’s constant and B is the moment of inertia about any axis. Hence since AJ = 1 for the P and R branches of a vibrational band we have Erot./h = BJ (J t I) A Erot.lh = 2 BJ (6) which is independent of the vibrational and electronic states. In the case of a degenerate vibrational band the rotational spacings in the R (AJ = + 1) and the P (AJ = - 1) branches are different as a result of vibration- rotation interaction. The rotational spacings are now 2B (J + Ci) and 2 4 (J - Ci) respectively.ti is the magnitude of the angular momentum arising from the interaction and may be expressed in terms of the potential constants and the masses. This has been done in algebraic form for many of the more important types of vibration of simple molecules.25 Clearly each ti value gives an extra relationship between the force constants and the result of an observation. Unfortunately Coriolis constants which is the name given to the vibration-rotation coupling parameters can only be determined with reasonable precision for small molecules. In a similar manner the change in rotational spacing with changes in the rotational parameter J can be related to the force constants and molecular para- meters. This effect of Centrifugal distortion has proved of little value owing to the rather large uncertainties in the observed values.In principle it is possible to determine from the electron-diffraction patterns of gases the mean-square amplitudes of bond and angle vibrations as well as the bond and angle values themselves.26 These amplitudes are 23 G. Herzberg “Infrared and Raman Spectra of Polyatomic Molecules,” Van Nostrand New York 1945. 26 J. L. Karle and J. Karle J. Chem. Phys. 1949 17 1052. 30 QUARTERLY REVIEWS readily evaluated from the force field and have been used to test assumed field^.^'^^^ The precision of the experimental data is again low. Even so such calculations have served to show inadequacies of assumed fields. The mathematical difficulties involved in determining the set of force constants which give the most satisfactory fit between observed and cal- culated frequencies distortion constants etc.are very formidable for all but the very simplest of molecules. Electronic computers have been pro- grammed by several groups of research workers to derive optimum sets of force constant^.^^-^^ Early optimism in obtaining good fields in this way was rapidly dispelled when it was discovered that further mathematical problems were of prime importance. The major problem arises from the fact that the solution of a vibrational problem involving n vibrational frequencies involves the solution of an equation of order n to which there are generally n solutions. In the computing procedure a guessed set of constants is used to derive a calculated set of frequencies distortion con- stants etc.The difference between the calculated and observed sets are then used to derive an improved set of force constants by a perturbation tech- nique. It was expected that if a reasonable set of force constants was chosen initially from previous experience with simpler molecules then the pertur- bation would lead to convergence on a unique set of improved constants which would yield a description of the vibrational distortions close to the truth. In practice it was found that the perturbation problem was often unstable and the perturbed constants diverged to impossible values. This usually arises from the differences between the calculated and observed frequencies being too large for a perturbation treatment and can some- times be overcome by taking small fractional improvements i.e.iffi is a typical input constant and J3. is its “improved” value then using fi + (f3 - fi)x where x < 1 in the next cycle may remove the divergence of the constants. Occasionally the cause of the divergence is more deep-seated and arises from an unstable situation in the perturbation solution due to a special case of ill-conditioned beha~iour.~~ Another frequent occurrence is that the convergence of the force constants terminates at a stage of oscillation. This behaviour has been shown to correspond to a set of com- plex solutions. Such complex solutions may arise as a result of simplifying assumptions made in the force field to make the problem tractable. It has been shown however that the oscillations occur about the real compo- nents of the converged set. Finally and perhaps most disturbing of all it has been found that in certain cases slightly different initial guesses lead to a different set of converged solutions.32 Various criteria have been de- scribed to test the validity of the final answers but the uncertainty in the 27 D.A. Long and E. A. Seibold Trans. Faraday. Soc. 1960,56 1105. 28 D. E. Mann T. Shimanouchi J. H. Meal and L. Fano J. Chem. Phys. 1957,27 29 J. Overend and J. R. Scherer J. Chem. Phys. 1960,32,1289. 30 D. A. Long R. B. Gravenor and M. Woodger Spectrochim. Acta 1963 19 937. 31 D. A. Long and R. B. Gravenor Spectrochim. Acta 1963 19 961. 32 Joan Aldous and I. M. Mills Spectrochim. Acta 1962,18 1073. 43. STEELE INFRARED INTENSITIES 31 reliability of the distortional co-ordinates remains a major obstacle to progress in the interpretation of absorption intensities.Once the force field has been deduced and hence the form of the normal co-ordinates determined or approximated the next step is to visualise what the resulting ap/aQj mean. The molecular dipole gradient will generally be difficult to visualise and to utilise. Since physical chemists almost invariably find life a great deal easier if they can translate molecular properties into comparatively simple directed and more-localised pro- perties the next step is to find a suitable set of assumptions that will yield the desired simplifications. The obvious assumptions are those that will decompose the molecular properties into the sum of a set of bond pro- perties and are generally chosen as (a) the stretching of a bond by dr produces a change of dipole moment along the bond of (aplar) dr; (b) the deformation of a bond through an angle dB produces a dipole change (ap/aB)dB perpendicular to the bond and in the plane of movement ; (c) changes in one bond do not result in changes in another bond except when this is geometrically necessary.The test of the validity and usefulness of these assumptions is whether any or all of the following criteria are found to hold and if not whether anything positive can be deduced from the discrepancies. (i) Values of the deduced bond moments and gradients in different molecules are comparable. (ii) Values of given gradients and moments derived from different sym- metry classes of the same molecule are equal. (iii) The perpendicular gradients to any bond are negligible. (iv) Values of the bond dipoles derived are comparable with the static dipoles as measured by other methods.TABLE 2. Efective bond dipole moments and derivatives for C-H bonds. Compound Symmetry Dipole- Effective Measure- Ref. Class moment bond dipole ment derivative moment @/A) (D) CH4 f 2 50.83 'f 0.37 w-w 33 CzH* C2H2Dz b2u 50.26 0-42 w-w 35 } a1,e -0.61 0.33 w-w 34 CHSD CHZD CD3H C2D4 b3U 0.23 =F 0.60 bl u 0.67 33 I. M. Mills Mol. Phys. 1958 1 107. 34 R. E. Hiller and J. W. Straley J. Mol. Spectroscopy 1960 5 24. 35 R. C. Golike I. M. Mills W. B. Person and B. L. Crawford J. Chern. Phys. 1956 25 1266. 32 QUARTERLY REVIEWS TABLE 2.-continued. Compound C2H2 C2D2 C2H2 C2HD C2H6 c6H6 CH,CI CH,Br CH,I NH3* PH3 SiH, SiD Symmetry Class C U + ZU+ n u c u + n u F) a2u e u e1u a2u a1 e a1 e a1 e 01 e a1 e f2 f2 Dipole- moment derivative 0.8 0.78 0.87 0*79(H) 0*78(D) +1-24 &0*75 + 0.45 + 1.00 + 0-24 + 0.98 +0.19 + 0.73 0.61 0.16 1.2 0-8 31-23 5 1-44 4-0.13 Effective bond dipole moment 1 -05 0.89 I -05 0.9qH) 0*92(D) F 0.23 F 0.26 - 0.3 1 - 0.61 +0.17 + 0.27 - 0.48 + 0.42 - 0.46 1-04 0.52 - 0.45 1-58 Me as u r e - ment D w-w w-w w-w w-w w-w w-w w-w W-W w-w w-w w-w w-w Ref.36 37 37 37 40 38 39 39 39 41 41 42 43 W-W Wilson-Wells method. D dispersion method. * Introduction of apu.,,./2a and assumption of complete bond following leads to a, P ~ . ~ . = 0 . 7 4 ~ and ~ N H = -0.650; e ptl.Il. = 0 . 7 0 ~ and ~ N H = - 0 . 6 8 ~ . It can be seen from Tables 2 and 3 that criteria (i) (ii) and hence (iv) certainly do not hold though there is a certain amount of consistency between the gradients and dipoles for similar molecules and some trends are apparent.The situation is particularly bad for carbon-hydrogen bonds. It is from an analysis of these inconsistencies that a great deal of useful information and knowledge of molecular structure has been gleaned. The first step in such an analysis must be a consideration of the four major reasons for the failure of the model. 36 R. L. Kelly R. Rollefson and B. S. Schurin J . Chem. Phys. 1951,19 1595. 37 D. F. Eggers I. C. Hisatsune and I. Van AIten J . Phys. Chem. 1955 59 1124. 38 H. Spedding and D. H. Whiffen Proc. Roy. Soc. 1956 A 238,245. 39 A. D. Dickson I. M. Mills and B. L. Crawford J . Chem. Phys. 1957 27 445. 40 I. M. Nyquist I. M. Mills W. B. Person and B. L. Crawford J . Chem. Phys. 1957 41 D.C. McKean and P. N. Schatz J . Chem. Phys. 1956k 24 316. 42 D. F. Ball and D. C . McKean Spectrochim Acta 1962 18 1019. 43 I. W. Levin and W. T. King J . Chem. Phys. 1962 37 1375. 26 552. STEELE INFRARED INTENSITIES 33 TABLE 3. Eflective bond dipole moments and derivatives for X-F bonds. Compound CF CH3F CF3Br C2F6 C6F6 ;:::: } p-C6H2F4 BF3 NF3* SF6 SiF Symmetry class fi a1 a1 e a2 u eU el U 4 U bl u ef az" a1 e flu f 2 Dipole moment derivative 5.99 or 3.71 4.9 or 3.4 4.0 + 8.1 +4*1 + 3-4 + 3.8 + 5.0 + 6.5 + 5.2 h4.0 or 76.1 + 1.5 to +2-0 or 3.3 ca. $4.7 3.85 3.3 or 7-5 Effective bond dipole moment (4 1.11 or 2.98 1.1 or 2.4 + 2-8 +0*5 ( ~ 5 ) + 2.2 $1.6 + 0.7 1.6 1-3 + 1 * 1 (v6) 12.6 or ~ 0 . 9 1.7 + 0.9 to +1.2 0 to 0-3 2.65 3.3 2.3 Measure- ment D w-w w-w w-w w-w w-w w-w w-w w-w w-w w-w w-w Ref.44 45 46 47 48 49 50 50 51 52 53 53 D dispersion method. W-W Wilson-Wells method. * Introduction of apU.Jaa and assumption of complete bond following for the u1 class leads to pU.p.- 1.7 or 1 . 2 ~ ; p ~ - 1 . 1 ~ . (a) In criterion (i) it is implicitly assumed that the charge distribution in the bond XY is always the same and always alters in the same manner. It is common experience that all XY bonds do not have the same chemical reactivity apart from steric effects and since chemical reactivity is inti- mately related to the valence-shell electronic structure of the bonds this contradicts the above assumption. Furthermore all XY bonds do not have 44 B. Schurin J. Chem. Phys. 1959 30 1. 45 P. N. Schatz and D. F. Hornig J . Chem. Phys. 1953 21 1516. 46 G.M. Barrow and D. C. McKean Proc. Roy. SOC. 1952 A 213,27. 47 W. €3. Person and S. R. Polo Spectrochim. Acta 1961 17 101. 48 1. M. Mills W. €3. Person J. R. Scherer and B. Crawford J. Chem. Phys. 1958,28 49 D. Steele and D. H. Whiffen J. Chem. Phys. 1958 29 1 194. 50 D. Steele and D. H. Whiffen Trans. Furaday SOC. 1960 56 177. 51 D. C. McKean J. Chem. Phys. 1956,24 1002. j2 P. N. Schatz and I. W. Levin J. Chem. Phys. 1958 29 475. 53 P. N. Schatz and D. F. Hornig J. Chem. Phys. 1953 21 1516. 851. L 34 QUARTERLY REVIEWS the same bond length but this in turn depends on the environment of X and Y. Thus the CH bond decreases in length as the local hybridisation at the carbon atoms takes on less p and more s character. Thus this is a further manifestation of the variation of the electronic structure of the bond between two given atoms.(b) The effect of lone-pair electrons on the dipole change during a vibration is ignored. As the molecule vibrates the hydridisation of the orbitals will generally change and consequently affect the infrared ab- sorption. Burnelle and Coulson62 have shown using wave-mechanical TABLE 4. Efective bond dipole moments and derivativesfor various bonds. Com- Symmetry Bond pound class C6H6 el u c160180 c ClCN c n BrCN c n CH3CN a e HC1 c c 17 SO2 b a1 c-c c=o c=o c=o c=o c=o C r N CEN 12CrN 13CrN C r N C=N C r N CGN C-N H-Cl C=N C f N s=o s=o Dipole- moment derivative 0.0 5.79 6.0 5.85 (D/& 0.585 0-595 0.605 0-3 to 0.7 0.5 to 0.8 1.21 0.66 4.17 f 2.0 or &4.3 Effective Measure- moment bond dipole ment ( D ) w-w w-w w-w w-w 1 *33 w-w 1.33 C-O-G w-w 1.2 w-w w-w 1.4 1.3 W-W w-w w-w w-w w-w 1.3 1.8 1.3 f l .2 W-W Wilson-Wells method. C-o-G curve-of-growth method. 54 D. F. Eggers and C. B. Arends J. Chem. Phys. 1957 27 1405. 55 T. Miyazawa J. Chem. Phys. 1958,29 421. 56 J. W. Schultz and D. F. Eggers J. Mol. Spectroscopy 1958 2 113. 57 R. W. Hendricks and D. F. Hornig see ref. 58. 58 D. F. Hornig and D. C. McKean J . Phys. Chem. 1955 59 1133. 59 A. V. Golton D.Phi1. Thesis Oxford 1953. 6 o S. S. Penner and D. Weber J. Chem. Phys. 1953 21 649. 61 G. E. Hyde and D. F. Hornig J. Chem. Phys. 1952,20 647. 62 L. Burnelle and C. A. Coulson Trans. Furaduy SOC. 1957 53,403. Ref. 38 54 16 17 13 12 55 56 57 58 59 60 61 33 STEELE INFRARED INTENSITIES 35 calculations of Ellison and Shu1P3 and Higu~hi,~* that the lone-pair contribution to the molecular-dipole change associated with the bending vibrations of H,O and NH3 is nearly as large as the contribution resulting from the bond deformations.Thus for HzO apL/aa - 1 . 4 1 ~ and apB/aa - -2.13~ where pL and pB are the lone-pair and bonding con- tributions respectively to the molecular dipole. That this must be the case can be seen from the fact that the lone-pair contribution to the molecular dipole is calculated to be 1.69~ for H,O. In the deformed state where the oxygen and hydrogen atoms are collinear the lone-pair contribution must be zero by symmetry. Several of the discrepancies between the effective bond moments given in Tables 2 3 and 4 can be explained by similar reasoning to the above. Thus in boron trifluoride which in its equilibrium position is planar there is a vacant p orbital associated with the boron atom and perpendicular to the plane of the molecule.During the symmetrical out-of-plane bending vibration rehybridisation of the B-F bonding electrons at the boron atom I -\ I \ ‘ I ‘ - - I FIG. 3. Electron rehybridisations in the p a orbitals during the transitions of symmetry (a) aN2 of boron trifluoride; (b) a21 of benzene. results in electron-flow into an orbital on the opposite side of the original atomic plane to the fluorine atoms [see Fig. 3(a)]. This makes the fluorine atoms appear to carry less negative charge than they actually do. In the e’ class the F-B-F angular deformation cannot result in electron flow into the vacant p z orbital. In agreement with this reasoning the effective BF dipole as deduced from the a,” class is only 1 .7 ~ compared with 2 . 6 ~ as deduced from the e’ class. In the out-of-plane CH deformation of benzene a similar effect occurs in the a, out-of-plane deformation. The p z orbital is fully occupied in this case but rehybridisation at the carbon atom takes place in the form of s character being introduced in the p z orbital (Fig. 3(b)]. That this must be so is readily apparent from the fact that when the HCH angles are deformed to 108” the carbon hybridisation must be sp3 compared with sp2 for the planar configuration. The effect 6s F. 0. Ellison and H. Shull J . Chem. Phys. 1953 23,2348. 64 J. Higuchi J. Chem. 1956 24 535. 36 QUARTERLY REVIEWS once again is to make the substituent appear less negative. Since the hydrogen atoms in benzene are known to be at the positive end of the CH dipole,65 the numerical value of the effective dipole should be greater for the a 2 class than for the in-plane el class.The observed values are 0 . 6 1 ~ and 0 - 3 1 ~ respectively. Clearly an adequate theory of infrared intensities must incorporate terms of the nature of 8pPz/&. Coulson and Stephen66 have shown that the variations in the deduced effective dipoles in benzene acetylene and ethylene are compatible with reasonable degrees of rehybridisation and bond following (see below). However they were unable to deduce the relative contributions of the two effects. (c) Hybridisation changes can also occur as a bond stretch. If we con- sider the CH stretching in methane and assume that in the extreme case we have a CH radical and an H atom then it can be seen that rehybridisa- tion of the orbitals around the carbon atom must have occurred.The configuration of the CH radical is still uncertain but it seems probable that it is planar. In this case the hybridisation at the carbon atom will be sp2 as compared with sp3 in CH,. Though in both the extreme cases i.e. those of the unperturbed CH molecule and of a CH radical and an H atom there is no total dipole moment it seems necessary to assume that there is a dipole gradient during the vibrational motion. This gradient can be expected on the above grounds to be different from that in say the vibrational motion in which the stretching of one CH bond is out of phase with the remaining two (v of symmetry classf,). (d) A bond is usually thought of as being directed along the line con- necting the bonded atoms.This is frequently a false picture particularly for vibrationally distorted configurations. Wave-functions are not usually localised in one bond. Indeed the orthogonality of the wave-functions generally forbids this. By their delocalisation these wave-functions are not directly related to the usual conception of chemical bonds. Burnelle and Cou1son62 transformed the accurate SCF-LCAO wave-functions for H20 and NH so that the resulting wave-functions satisfied the following conditions (a) that they were orthogonal; (b) that the orbitals associated with identical bonds should be equivalent; and (c) that the lone-pair orbitals should contain only orbitals of the central atom. The resulting picture of the molecular orbitals showed that the hybrid orbitals at the central atom are not in general directed along the bond direction.This means that the bonds may be considered to be bent and to have dipole gradients perpendicular to the internuclear axis. The individual contributions to the dipole moment of H,O are for an inter- bond angle of 105" pL = 1 . 6 9 ~ pB = 0.20D and pBy = - 0 - 3 7 ~ where pL pB and pBY are the contributions due to the lone-pair electrons the 65 (a) R. P. Bell H. W. Thomspon and E. E. Vago Proc. Roy. SOC. A 192 498; (6) A. R. H. Cole and H. W. Thompson Proc. Roy. Suc. 1951 A 208,341. 66 C. A. Coulson and M. J. Stephen Trans. Furuduy SOC. 1957,53 272. STEELE INFRARED INTENSITIES 37 bond electrons in the direction of the internuclear axis and the bond electrons in a direction perpendicular to the internuclear axis respectively.If the HOH angle is deformed the resulting molecular dipole gradient of -@72~/radian is derived from the following contributions = 1-41 D/Radian; apBZ/aa = -0.20 D/Radian; and apBy/&t = 1-93 D/Radian. The transverse moment is very sensitive to the inter-bond angle and its derivative is the major contributing term to the absorption in- tensity. An analysis of the absolute absorption intensities of the infrared bands of 1 ,2,4,5-tetrafluorobenzene50 showed that the dipole gradient ap/ar,, lies along the internuclear axis within the experimental error. However the asymmetry of the CF bond in this compound is not very pronounced and it is not unreasonable to expect the above result. The Polarity of the Dipole Gradients.-According to equation (3) the absorption intensity is proportional to the square of the dipole gradient.In the preceding section it has been assumed that in solving for the gradient the sign of the square root of the intensity is known. In fact the question of these signs introduces what is frequently a serious uncertainty into the interpretations. The seriousness of this problem can be ap- preciated from the fact that if there are n fundamental vibrations in a given symmetry class then there are 2" possible ways of choosing the sign combinations leading to 2" distinct solutions for the bond parameters. Isotopic substitution should leave the dipole gradients unchanged whilst changing the form of the vibrational modes. The sets of dipole gradients with respect to bond co-ordinates-or with respect to combinations of the co-ordinates in the form of symmetry co-ordinates-will not all be con- sistent with the observed intensities of the other isotopic system.Generally there are rarely more than three sets which yield acceptable values for the intensities of the other isotopic system. A choice between sets acceptable by the above criterion is usually made on the basis of lack of credulity of the authors to certain of the derived gradients. When no isotopic data existed sign choices have often been made on the basis of one set giving bond gradients and dipoles which were in line with values for similar molecules. Clearly this is not an acceptable criterion especially as the aim has usually been to discover if the gradients and dipoles were indeed com- parable for bonds of a given type in similar molecules.Blatant cases of this practice have been excluded from the Tables. As an example of the above technique the CN dipole gradient for cyanogen derived from the infrared-active CEN stretching vibration of 12CN12CN is compared in Table 5 with those derived from the ''almost symmetric" CN and CC stretching vibrations of 13CN12CN.56 Owing to the presence of normal cyanogen in the heavy-isotopic system the intensity of the antisymmetric mode was not measured. In this case it is very clear that the ap/aQ values must have the same sign. Occasionally the sign of the square root in equation (3) can be deter- 38 QUARTERLY REVIEWS TABLE 5. Dipole-moment derivatives for cyanogen D/A Dipole-moment (12CN) Relative signs for 12CN13CN derivative ap/aQ values aplar(cN)l &0.585 same k0.595 different 0-41 6 aPlar,c,l ‘f 0.585 same ‘f 0.607 different &0*718 aPlar(c-,) zero by symmetry same ~ 0 4 0 0 3 different TO-107 mined from vibration-rotation interaction studies.Thus Hermann and WalW7 were able to show that for a diatomic molecule the ratio of intensities of corresponding rotational absorption lines in P and R branches is proportional to (1 + 4y8J)J where y = 2B,/ve 8 = [po/((?p/ar)]l/re and J is the rotational quantum number of the initial state. B is the rota- tional constant for the equilibrium bond length re; po is the molecular dipole; and ap/& is the dynamic dipole gradient. If the sign of p0 is known then the sign of the dipole gradient can be deduced. In this way they were able to show that ap/ar for HCl has the same sign as po.Clearly it is also possible to use the above technique to evaluate the ratio of the magnitude of the gradient to the static dipole. Thus p/(+/ar)r for LiH has been determined as -1.8 & 0 . 3 ~ . ~ * Clearly a prerequisite of this tech- nique is the existence of an equilibrium molecular dipole. The Hermann- Wallis theory has been extended to the case of linear polyatomic mole- c u l e ~ ~ ~ (group C,,) but no application to intensity interpretation has yet been made. that the intensities of the out-of-plane deformations in various alkyl- and halogen-substituted benzenes indicate that the effective CH dipole is of opposite sign to the effective halogen dipole. As it is reasonably certain that the halogen atom is at the negative end then this implies that the hydrogen atom is at the positive end of the CH dipole in benzenes in the out-of-plane deformations.Since out-of-plane deformations have the effect of making the dipoles appear less positive the hydrogen atom must also be at the positive pole of the static bond. Interaction between two vibration-rotation bands can occur if the sym- metry of the vibrations and of a principal rotation axis bear a certain simple relationship to one another. Specifically the Coriolis force on each atom is of magnitude 2ma Va w sin + where YlZa is the mass of the atom w is the angular velocity of the co-ordinate system with respect to a fixed co-ordinate system Va is the velocity of atom “a” and+ is the angle between the axis of rotation and the direction of Va. The force is directed at right- angles to the direction of Va and to the axis of rotation.If the forces on the T. C. James W. G. Norris and W. Klemperer J. Chem. Phys. 1960,32,728. Bell Thompson and Vago and Cole and Thompson have 67 R. Hermann and R. F. Wallis J. Chem. Phys. 1955 23 637. 69 G. A. Gallup J. Chem. Phys. 1957 27 1338. STEELE INFRARED INTENSITIES 39 atoms constituting the molecule which arise from a vibration j and from rotation about a specific axis m have the same symmetry as a second vibration k which is not too far removed in frequency fromj then the two vibrations will interact. The effect of this interaction on the vibrational frequencies is to make the P and R rotational band spacings different from one another (apart from the small effect arising from centrifugal distor- tion). This Coriolis interaction has been mentioned previously in connec- tion with the dependence of the 5 constants on the potential constants.In addition to the frequency effect the intensities are also altered. In the approximation of no vibration-rotation interaction the lines in the P and R branches arising from transitions from the same initial state have equal strengths. The effect of the Coriolis interaction is to increase (or decrease) the R-line intensities of the higher-frequency band and also the P-line intensities of the lower band whilst decreasing (or increasing) the intensi- ties of the other AJ = &- 1 lines. Which in fact occurs depends on the sing of Cjkz (ap/aQj)z(ap/aQk)g where the j and k subscripts specify the two vibrations. The sign of cjk is generally determinable and hence from the intensity asymmetry of the bands the relative signs of the two dipole transitions can be deduced.70 This new technique has considerable promise.The Effect of Change of State on Intensities.-Recently there has been a renewed interest in the intensities of absorption bands in condensed phases. Until the last two or three years it was believed that intensity changes arising from intermolecular interactions would be small except where structural changes took place or where definite bonds were formed be- tween molecules. Examples of these special cases are the crystallisation of trans-dichloroethane from a mixture of the trdnS- and gauche-forms and the formation of intermolecular hydrogen bonds in hydroxyl-containing compounds. Considerable theoretical work has been carried out to evaluate the magnitude of the intensity changes in the absence of ap- preciable intermolecular interaction~.~l-~~ The basic theory is encompassed in the e q ~ a t i o n ~ ~ y ~ ~ (7) rliq (n2 + 2)2 r g a s 9rr ' which is derived for an oscillating dipole in a spherical cavity in a medium of refractive index n.This equation implies that the intensities ought always to be greater in a condensed phase than in the gaseous phase. Experi- mental work until very recently was restricted to isolated absorption bands-isolated in the sense of a single band of a certain molecule. Equation (7) and its refined forms have met with very limited success. - - - 7 0 I. M. Mills unpublished work. 71 N. Q. Chako f. Chem. Phys. 1934,2 644. 72 S. R. Polo and M. K. Wilson f. Chem. Phys. 1955 23 2376. 73 J. van Kranendonk Physica 1957 23 825.74 A. D. Buckingham Proc. Roy. Sac. 1960 A 255 32. 75 L. Onsager f. Amer. Chem. Soc. 1936 58 1486. 40 QUARTERLY REVIEWS Intensity data now exist for all infrared active bands of in vapour liquid and solid phases. The results are very disturbing. The in- tensity changes are far greater than can be accounted for by dielectric changes or by the expected magnitudes of intermolecular perturbations. Similar results have been reported for ethylene.79 The results for benzene are shown in Table 6. These results pose a fascinating theoretical problem TABLE 6. Absolute intensities of benzene trdnSitiOnS in various phuses ( 103cm2/mole). Band (crn.-l) r.cl rt r s 3060 1 -95 1 *45 0.65 1480 0.878 1 -29 2.56 1036 0.850 1 . 1 1 1 *69 673 12.95 14.1 13.6 on which no headway has been made at the present time and which is certain to attract considerable attention from theoretical chemists and physicists in the future.Applications of Absolute Intensity Measurements to Determination of Chemical Structure.-The intensities of some absorption bands character- istic of certain groupings such as those generally associated with the stretching of -C= N -C = 0 -C-H ( in hydrocarbons) and -OH bonds have been correlated in an empirical manner with the structure of the attached group. In many cases it is possible to deduce the number of absorbing centres in an unknown molecule. This aspect of intensity measurement has been thoroughly reviewed by T. L. Brown.80 It is inter- esting that such empirical correlations actually do exist. The so-called characteristic group vibrations are rarely confined to one bond and the mode may change quite drastically for small structure changes without the frequency being much affected.It is likely that the main reason for the success of these empirical correlations is that the groups studied usually have very large dipole gradients associated with their stretching motions so that participation in the vibration by other parts of the molecule is relatively unimportant. Even so these correlations imply that the stretching gradients of these groups are reasonably constant. Very few detailed studies of molecules involving these groupings have been carried out at present. As a consequence of the interpretational difficulties of intensity measure- ments absorption intensities have been little used in elucidating molecular structure apart from the empirical approach mentioned above.The following examples show that in special circumstances absolute in- tensity measurements can be used in structure determinations. 76 D. A. Dows and A. L. Pratt Spectrochim. Acta 1962 18 433. 77 I. S. Hisatsune and E. S. Jayadevappa J. Chem. Phys. 1960 32 565. 78 J. L. Hollenberg and D. A. Dows J. Chem. Phys. 1962 37 1300. 7 9 G. M. Wieder and D. A. Dows J. Chem. Phys. 1962,37 2990. 8o T. L. Brown Chem. Rev. 1958,58 581. STEELE XNFRARED INTENSITIES 41 Until recently magnesium dicyclopentadienyl was believed to be an ionic salt. However its solubility in benzene was inconsistent with this belief. An analysis of its vibrational spectrum indicated a marked simil- arity in its vibrational modes to ferrocene.81 If it is principally ionic then the absorption intensity of the antisymmetric ring-metal stretching vibra- tion ought to be well represented by the ionic model Fig.4 where the rings + 2E - € - € CP M9 C P FIG. 4. An ionic model for the antisymmetric stretching vibration of magnesium di- and the metal atom are represented by point charges. The absolute in- tensity calculated for this model was approximately seventy times as large as the observed value.81 This can only be explained on the basis of princip- ally covalent bonding. Magnesium dicyclopentadienyl or magnacene is the first established covalently bonded sandwich compound of a non-transition element. The vibrational stretching frequencies of cyanide bonds are outstanding in their insensitivity to the presence of other groups.Furthermore there is little coupling between the stretching vibrations of cyanide groups attached to a common atom. Thus in maleonitrile only one C N stretching frequency is observed in the infrared and Raman spectra,82 and in sulphur dicyanide the symmetric and antisymmetric modes are at 2184 and 2179 cm.-l respectively only 5 cm.-l apart.83 This indicates that the electronic wave-functions are highly localised in the bonds themselves. The stretching gradients for cyanides are all of the same order of magnitude. The only reported absolute intensity measurements on a dicyanide are those on ~ y a n o g e n . ~ ~ Rather surprisingly the stretching gradients derived from the symmetric stretching modes of 12CN13CN and from the antisymmetric mode of 12CN12CN are in excellent agreement.These facts have been used as the basis for a determination of the angle between the cyanide groups in the dicyanamide ion N(CN)2-.84 Assuming that the bond-moment hypothesis holds for this ion then it is easy to show that the ratio of the intensities of the antisymmetric to the symmetric stretching modes is given by ras sin28/2 where 8 is the inter-bond angle. A fortuitous splitting of the antisymmetric E. R. Lippincott J. Xavier and D. Steele J. Amer. Chem. SOC. 1961 83 2262. 82 F. Halverson and R. J. Francel J. Chem. Phys. 1948 17 694; K. W. F. Kohl- 83 D. A. Long and D. Steele Spectrochim. Acta 1963 19 1731. 84 D. A. Long J. Y. H. Chau and D. Steele unpublished results. cyclopentadienyl (MgCp,) - rs cos2e/2’ rausch and G. P. Ypsilanti Z . phys. Chem. 1934 b29 274. 42 QUARTERLY REVIEWS mode occurs as a result of Fermi resonance.This allowed the relative intensities of the two cyanide bands to be measured with reasonable precision. The value derived for the inter-bond angle was 145". The original measurements were made in potassium bromide media. Accord- ing to the dielectric theories of the effects of phase on intensities the rela- tive intensities should be unaffected by phase changes. However in view of the previously noted influences on the bands of benzene and ethylene the measurements have been repeated in aqueous The two results agree well. Other Aspects and Present Trends.-A certain amount of research has been devoted towards the evaluation of higher-order derivatives of the dipole moment in suitable systems-generally diatomic ~ y s t e m s .~ ~ - ~ ~ An interesting special case was the interpretation of the intensities of certain combination bands of benzene arising from the out-of-plane motions of the C-H These bands which are characteristic of benzene and sub- stituted benzenes occur in the region 2000-1500 cm.-l. It was shown that the intensities of these bands of C6H6 C6D6 and p-C6H,D2 could be interpreted satisfactorily using only one parameter. This parameter which is the second derivative of the dipole moment with respect to the out-of- plane deformation of thejth bond 82p/ay2j has the value 1.10~. Several important theorems concerning the vibrational frequencies and intensities of isotopically related systems have been advanced by B. Craw- f0rd.9~9~3 The most important of these shows that the function ;I'/va is invariant to isotopic substitution when the summation is over all vibra- tional bands in a given symmetry class.ra is the absorption intensity of the band centred at a frequency va. This theorem when applicable provides a very useful test on measured intensities. A typical set of results is shown below for C2H6 and C2DG40 C2H6 C2D6 class 2.810 0.024 class 0-737 & 0.008 2.942 0.140 ~rn.~/mole. 0.775 r f i 0.037 ~m.~/mole. Le Fkvre and Rao,94995 Whiffen,96 and Illinger and SmythS7 have shown 85 D. Stele unpublished results. 86 W. S. Benedict R. Hermann G. E. Moore and S. Silvermann J. Chem. Phys. 88 S . S . Penner and D. Weber J. Chem. Phys. 1953 21 649. 89 B. Schurin and R. Rollefson J. Chem. Phys. 1957 26 1089. 91 F. E. Dunstan and D. H. Whiffen J. 1960 5221. 92 B. Crawford J.Chem. Phys. 1952 20 977. 93 I. M. Mills and D. H. Whiffen J. Chem. Phys. 1959,30 1619. 94 D. A. A. S. N. Rao Trans. Faraday SOC. 1963 59 43. 95 R. J. W. Le Fkvre and D. A. A. S. N. Rao Austral. J. Chem. 1955 8 39. 96 D. H. Whiffen Trans. Faraday Soc. 1958,54 327. g7 K. H. Illinger and C. P. Smyth J. Chem. Phys. 1960 32 787. 1957,26. 1671. G. A. Kuipers J . Mol. Spectroscopy 1958 2 75. E. K. Plyler W. S. Benedict and S. Silvermann J. Chem. Phys. 1952 20 175. STEELE INFRARED INTENSITIES 43 that the atomic polarisation of a molecule is related to its infrared absorp- tion by the relationship p * = - - p . Nc 3T2 j vj The significance of the atomic polarisation can be understood by con- sidering the effect of an applied oscillating electric field on a molecular gas. The induced and the permanent dipole moments align as far as permitted by the thermal motions against the applied field thus reducing the effective field.The molecular polarisation is defined as the total dipole moment per unit volume parallel to the field arising from the above contributions. As the field frequency is decreased the total molecular polarisation de- creases in certain frequency ranges. At very high oscillating frequencies only the electrons are mobile enough to follow the field changes. In the infrared frequency range the nuclei become able to follow the field and finally at still lower frequencies the molecular dipoles are able to align with the field (see Fig. 5). Clearly it is reasonable to consider the molecular FIG. 5. polarisation as I I lnfro U l t r o - -red vlolot Y- The variation of molecular polarisability with frequency.consisting of three parts the electron and the atomic induced polarisations and that arising from the permanent dipoles. Values for the atomic polarisation of non-polar molecules as deter- mined from refractive-index and dielectric-constant studies (see e.g. ref. 94) agree very well with those deduced from intensity measurements. Some typical results are shown in Table 7. In the case of polar molecules the discrepancies are generally much greater but it is very likely that these discrepancies arise from difficulties in determining the total polarisability at frequencies such that only Po is measured. It is clear from preceding sections that any successful theory of absorp- tion intensities must incorporate terms involving derivatives of the dipole (PE P A and Po).44 QUARTERLY REVIEWS Compound PA Infrared studies ( ~ m . ~ ) BF3 2.19 CF4 2-89 C2H2 1 -28 CH4 0.1 1 TABLE 7. PA Compound Dielectric studies ( ~ m . ~ ) 0.08 SiF4 1 a27 SF6 2.8 1 C2H6 2-86 C6H6 p* 1 nfrared studies ( ~ r n . ~ ) 0.12 0.73 4-82 5.07 PA Dielectric studies ( c ~ I . ~ ) 0.09 0.80 5.46 5.20 contribution of lone-pair electrons rehybridisable orbitals conjugated systems etc. with respect to bond deformations. McKean and Schatz41 and Hornig and M ~ K e a n ~ ~ have utilised terms involving the lone-pair electrons. Sverdlovg8 has developed a complete second-order bond- moment theory which incorporates terms such as api/aRj where the i and j subscripts refer to different bonds. In this way instead of single terms being determinable only combinations of terms such as (ap/aO,),,) - (ap/M,) (4) can be evaluated.An appreciation of these results necessitates a judicious assessment of the relative importance of the additional terms. Much of the present experimental data cannot be satisfactorily treated owing to the sensitivity of the modes to uncertainties in the force fields. Further developments in gas-phase intensities must come through the deduction of satisfactory force fields and the use of treatments such as those of Hornig and McKean and of Sverdlov. It is usually the unexpected results which prove to be the most fascinating and the most rewarding. The changes of intensities with phase changes are certainly the most surprising results during recent years of the field re- viewed. It is too early to even surmise the importance of an interpretation but it must modify the present concepts of condensed phases. In conclusion much has been achieved in the interpretation of infrared intensities but many important problems remain to be solved. I sincerely thank Dr. I. M. Mills for describing his work on the determination of the sign of dipole gradients from Coriolis interaction studies prior to its publication. 98 L. M. Sverdlov Optics and Spectroscopy 1959 6 477; 1959 7 1 1 ; 1960 8 316; 1960 8,96.