Potential Surfaces from Vibration-rotation Data BY IAN M. MILLS Department of Chemistry, University of Reading, Reading, Berkshire RG6 2AD Received 24th May, 1976 The problems of inverting experimental information obtained from vibration-rotation spectro- scopy to determine the potential energy surface of a molecule are discussed, both in relation to semi- rigid molecules like HCN, NOz, H,C=O, etc., and in relation to non-rigid or floppy molecules with large amplitude vibrations like HCNO, C302, and small ring molecules. Although standard methods exist for making the necessary calculations in the former case, they are complex, and they require an abundance of precise data on the spectrum that is rarely available. In the case of floppy molecules there are often data available over many excited states of the large amplitude vibration, but there are difficulties in knowing the precise form of the large amplitude coordinate(s), and in allowing for the vibrational averaging effects of the other modes.In both cases difficulties arise from the curvilinear nature of the vibrational paths which are not adequately handled by our present theories. The object of this Discussion is to bring together workers in the area of molecular physics and molecular chemistry with a common interest in determining potential energy functions. One source of information on the potential energy function (p.e.f.) of a molecule is vibration and vibration-rotation spectroscopy. This is a field which has made considerable progress in the last fifteen years, but it has also become a specialist subject which may not be easy to follow for workers in related fields.My object in this paper is to bring non-specialist readers up to date with develop- ments in this field, and with the strengths and weaknesses of this approach to determining the p.e.f. of a molecule. The prominent weakness of the approach is well known, but it must be stated again at the outset: it is that vibrational spectroscopy only provides information on the p.e.f. close to the minimum, because experimental data are rarely available for more than the first few quanta in each normal mode. Excitation energies of perhaps a few thousand cm-1 from the minimum (i.e., about 0.5 eV E 18 millihartree, or -50 kJ mol-l) represent the upper limit of information from vibrational spectro- scopy, and often the limit is less than 1000 cm-1 (0.12 eV w 4.2 millihartree, or -12 kJ mol-l).This is only 10% of a typical bond dissociation energy. Thus, vibrational and vibration-rotation spectroscopy give detailed information on the shape of the p.e.f. for only a rather smalzpocket in the surface around the minimum, or equilibrium configuration. The detail and precision of the information on the p.e.f. derives from the study of spectra at high resolution, in the gas phase, showing rotational structure and vibration-rotation interaction effects. Both experimentally and theoretically this subject has made great advances. The experimental techniques which have proved particularly fruitful are the developments in conventional high-resolution infrared and Raman spectroscopy, the development of fourier-transform spectrometers based on a Michelson interferometer, the developments in millimetre wave experimental8 POTENTIAL SURFACES FROM VIBRATION-ROTATION DATA techniques, and the application of lasers to high-resolution spectroscopy (for example double-resonance experiments, and laser-Stark and laser-Zeeman spectroscopy).Resolution of 0.02 cm-I is not uncommon today in infrared spectro~copy,l-~ and modern interferometers achieve 0.005 cm-l throughout the mid-infrared region 4 9 5 ; laser experiments can achieve effective resolution of a few MHz (lo4 cm-1).6-8 Gas phase Raman vibration-rotation spectra are now available with resolution of the order 0.25 and pure rotational spectra with much higher resolution.lo Milli- metre wave spectroscopy based on microwave techniques is available in a number of laboratories up to 400 GHz, and, in at least one case, spectra have been observed up to 1000 GHz (30 cm-') with the usual microwave precision and resolution (of less than 1 MHz).ll The analysis of vibration-rotation spectra, and the inversion of the data to give a molecular potential energy function, involve the theory of vibration-rotation spectro- scopy in polyatomic molecules, the theory of force constant calculations, and the application of least-squares refinement techniques to obtain the p.e.f.which gives the best fit to the experimental data. Since I have recently attempted to review these topics in other publication~,'~-'~ and have there given references to other review articles, I shall not go into the detail of the analysis in this paper.I shall discuss some examples in sufficient detail to show where the difficulties arise, first for the usual case of semi-rigid molecules, and then for the particular case of molecules with low- frequency large-amplitude vibration(s) where it is often the case that detailed experi- mental data can be obtained over many excited vibrational states. It is in this last field that the work of my own laboratory has been concentrated in the last few years. 1. SEMI-RIGID MOLECULES By " semi-rigid " I mean molecules without any exceptionally low-frequency large-amplitude vibrations; for the sake of argument " low-frequency " means a wavenumber below about 300 cm-? In this case, the p.e.f.rises rapidly for dis- placements in all directions from the minimum in the potential energy surface. The limitation of vibrational spectroscopy to relatively small displacements from the minimum naturally suggests the representation of the p.e.f. as a power series expansion in displacement coordinates from equilibrium : We may thus talk of the quadratic, cubic, and quartic force field, etc. Most analyses are based on a p.e.f. defined in this way, and vibration-rotation theory is founded on the assumption that this series converges reasonably rapidly, perhaps by a factor of 10 for successive terms in (1) for displacements appropriate to one-quantum excitation. Attempts to determine the coefficients in (1) are described as force constant calculations. The choice of coordinates in (1) requires care.A first requirement seems to be that they should be geometrically defined internal coordinates (so that the force constants are unique and invariant to isotopic substitution); the usual choice is to take valence bond stretching and angle bending coordinates. However, another factor in the choice may be that certain definitions give more rapid convergence than others, a point to which I return later. Also the choice of geometrically defined coordinates leads to a non-linear transformation into normal coordinates, which complicates anharmonic force constant calculations, as also discussed later.IAN M. MILLS 9 The formulation of (1) as a Taylor series ensures that each force constant is equal to the appropriate derivative of Y at equilibrium.For example Equations of this kind allow us to visualise cubic force constants as the rate of change of quadratic constants with respect to one of the coordinates, and thus give a physical interpretation of the higher order constants. Indeed, the signs and magnitudes of cubic and quartic constants can sometimes be estimated by arguments of this kind. 1.1 HARMONIC FORCE FIELD CALCULATIONS The problem in harmonic force field calculations may be described in the following way. The observed vibration frequencies of the fundamentals are essentially the diagonal force constants in the normal coordinates, and the interaction force con- stants in the normal coordinates are by definition zero. Thus the problem is to determine the form of the normaZ coordinates, in terms of which the harmonic force field is known as soon as the vibration frequencies are known.The forms of the normal coordinates are usually expressed in terms of a basis of internal valence coordinates Ri through the L matrix: R = LQ, that is, Ri = ELiQ,. (3) r Thus the elements of the L matrix, L: = (aRJaQr)const Qs etc. give the proportional contributions of the Ri to a unit displacement in Qp Alternatively they may be expressed in a basis of 3N Cartesian displacements 6x subject to the Eckart conditions through the AL matrix: SX = ALQ = M-lBt(L-')+Q. (4) This provides a pictorial representation of the normal coordinates in terms of Cartesian displacements. To determine the p.e.f. in a representation of geometrically defined internal coordinates, as in eqn (I), we construct the transformation F = (L-1)ffi-l (5) 6) (ii) (iii) where A is a diagonal matrix with elements Ar = 47c2c2u),2. The elements of the F matrix are then the quadratic (or harmonic) force constants in eqn (1).The experimental data most sensitive to the forms of the normal coordinates are: Isotopic frequency shifts, because these depend on the amplitude of motion at the substituted atom; centrifugal distortion effects, which are related to the dependence of the moments and products of inertia on displacements along the normal co- ordinates, ar(aB) = (a&B/aQ,) ; coriolis resonance effects, which are related to the forms of the normal co- ordinates through the zeta constants [r,s(a), which give a measure of the vibrational angular momentum generated by a pair of normal coordinates Qr and Qs about the cc axis.It is worth noting that vibration-rotation interactions, which provide data of types (ii) and (iii), provide important information on the p.e.f.; it is for this reason that high resolution gas-phase spectra are so valuable. The calculation of harmonic force-fields from spectroscopic data is usually made by a non-linear least-squares calculation, in which a trial force field is refined to10 POTENTIAL SURFACES FROM VIBRATION-ROTATION DATA achieve agreement between the calculated and observed data. The calculation is complicated by the variety of different types of data involved, by the relatively large number of parameters which have to be simultaneously refined, and by the effects of anharmonicity, for which it is usually difficult to make precise corrections. The uncertainties in the resulting force constants are almost invariably correlated, so that at the end of the day it is hard to assess how reliably the p.e.f.has been determined- even for specialist readers. When there are more than two or three vibrations of the same symmetry species one must always be suspicious of the reliability, i.e., the uniqueness, of a harmonic force field calculation, and it is generally accepted that interaction force constants (FJ # j ) are less reliably determined than diagonal force constants. Despite these difficulties, well-determined harmonic force fields are now available for many simple molecules. In this category I would put most triatomic molecules, many linear four or five-atomic molecules, many simple C,, molecules (XY, or XYZ3), many simple organic molecules like formaldehyde, ethylene, allene, and even benzene, and many simple inorganic molecules for which isotopic shifts associated with substitution at a central metal atom have been studied.Also, chemically related series of molecules have often been studied together, of which the classic example is the series of saturated hydrocarbon molecules; in such cases apparently reliable harmonic force fields have been determined by constraining force constants to remain unchanged between chemically similar groups in different molecules. Many more force constant calculations have been reported in which constraints have been imposed in order to obtain a unique solution for the p.e.f.Lack of data make this unavoidable for larger molecules, but it also makes the reliability of the results even more difficult to assess. Harmonic force field calculations have recently been comprehensively reviewed by Duncan,15 with references to calculations on many molecules and many numerical examples. 1.2 ANHARMONIC FORCE FIELD CALCULATIONS The cubic and quartic anharmonic force constants in eqn (1) become important as the displacements from equilibrium become larger. Their effects appear in higher- order vibration-rotation interactions and in anharmonic spacings of the vibrational energy levels; they are usually calculated from a perturbation treatment of the vibration-rotation hamiltonian. Most anharmonic force constant calculations treat the cubic and quartic terms in the p.e.f.together, because they affect the vibrational energy levels in the same order of magnitude (the cubic terms are larger, but contribute to the energy levels in the 2nd order of perturbation theory; the quartic terms are smaller, but contribute in 1st order). Very few calculations have been attempted in which higher than quartic force constants have been considered. The customary calculation relating cubic and quartic force constants to the spectrum is quite involved. There are two main steps. The first is a non-linear co- ordinate transformation from the geometrically defined internal coordinates Ri to the normal coordinates Qr in terms of which the vibration-rotation hamiltonian is formulated. This transformation has been discussed by Hoy, Mills and Strey.It may be conveniently written in terms of an L tensor, for which eqn (3) is an approxi- mation involving the linear terms only. The exact transformation may be written:IAN M. MILLS 11 The elements of the L tensor must be determined from a preliminary harmonic force field calculation and then used to transform the p.e.f. in eqn (1) to a similar expansion in terms of the normal coordinates Q,, involving the normal coordinate force con- stants, co, (harmonic terms), P)rst (cubic), and qrstu (quartic) etc. : V/hc = 2 3urqr2 + (1/3!) 2 qrstqrqsqt + (7) r rst Hereg, = ( 2 n ~ ~ ~ / h ) ~ Q , . The effect of the non-linear nature of (6) is that quadratic terms in (1) contribute to quadratic, cubic, quartic, and all higher terms in (7) ; cubic terms in (1) contribute to cubic and all higher terms in (7); and so on.The second step is a perturbation calculation, or contact transformation, in which the vibration-rotation hamiltonian is transformed to an effective hamiltonian in which the coefficients are the spectroscopic constants (such as the arB constants describing the vibrational dependence of rotational constants, and the xr, constants describing the anharmonic contributions to the vibrational energy levels). Although the perturbation calculation is quite complicated, the results are generally applied in the form of analytical formulae for the spectroscopic constants in terms of coefficients in the original hamiltonian, particularly the force constants wry qrsr, yrstu etc.Most of the important formulae are available in the literature.12 Often, further complica- tions arise due to resonances, which require parts of the vibration-rotation hamiltonian to be related to the observed spectrum by matrix diagonalization, or by a mixture of matrix diagonalization and perturbation theory. When both steps of this calculation are made, the spectroscopic observables may be predicted from a trial anharmonic force field. It is finally necessary to reverse the calculation and calculate the force field from observed data. The only general method of achieving this is by a least-squares refinement of the force field to fit the data. All of the difficulties of harmonic force field calculations arise once more in an exaggerated form: too many parameters in the force field, highly correlated results, and unknown errors incurred through the neglect of higher order effects- these are the limiting features of all attempts to determine a general p.e.f. from vibra- tion-rotation data.Our own calculations in this field have been confined to quite simple molecules on which there is an abundance of spectroscopic data (H20, H2S, H2Se, SO3, PF3, AsF,, SbF,, HCN, HCP, HCCH, NCCN and HCCCN).14*16-18 We have also made use of trial calculations as an aid to the analysis and assignment of microwave spectra, in situations where we would hardly claim to have determined an anharmonic force field (e.g., CF3H16 and SiF3H19). Other research groups have made similar calcula- tions with somewhat similar results ; particularly Overend and coworkers 20*21 and Kuchitsu, Morino and coworkers.22 My own feeling is that the greatest success of calculations of this kind lies (i) in the successful interpretation of high-resolution spectra, and (ii) in giving more reliable information on the equilibrium structure and the harmonic force field.I would say that there are relatively few cases where the cubic and quartic anharmonic force constants have been experimentally determined with precision, except for those associated with bond stretching where they can be reliably estimated from assumptions of Morse-like potential curves. Indeed, one general result of anharmonic force field calculations is simply to demonstrate the expected results that bond stretching coordinates give rise to Morse-like sections of the potential surface, and that the " valley-bottom " for an angle bending coordinate tends to follow ;I curvilinear path representing bending at constant bond length.More profound generalizations are hard to make, and I think not very profitable. As a single example to illustrate the quality of calculations of this kind, one may12 POTENTIAL SURFACES FROM VIBRATION-ROTATION DATA consider recent attempts to determine the p.e.f. for the bending vibrations of acetyl- ene,21p23 and their interactions with bond stretching. I choose this example because there has been a lot of precise spectroscopic work on many isotopic species of acetyl- ene; it is hard to imagine that such an abundance of experimental data will become available for any other four-atomic molecule.Also the high symmetry greatly TABLE I.-TERMS IN THE P.E.F. OF ACETYLENE INVOLVING THE ANGLE BENDING COORDINATES. a = HCC bend, z = phase angle between HlClC2 and ClC2H2 bending, Y = CH stretch and R = CC stretch + faat a,cc,cos z V = 3fua (a12 + .2’) simplifies the calculation and reduces the number of force constants involved. In fact, all the signs are favourable for a successful anharmonic force field calculation. The appropriate part of the expansion of the p.e.f. is shown in table 1. The relationship between the force constants and the observed data is shown diagram- matically in table 2 and fig. 1. The best estimates of the force constants to fit the observed data are shown in table 3. The fit to the observed data is good [for this, and other details, see ref.(23). However, this fit has been obtained with a quartic force field in which two of the four TABLE 2.-DIAGRAMMATIC REPRESENTATION OF THE OBSERVED SPECTROSCOPIC CONSTANTS AND THEIR MLATION TO THE FORCE CONSTANTS IN TABLE 1 a 4 - F 4 4 = fa, - faat 05 - F55 = faa + faanIAN M. MILLS 13 oo 2* oo 2O pl'l l"llF1 22 oo 2O oo oo l1 l1 oo oo oo Cr) 0 rn l- ? e4 I4 (0 FIG. 1 .-The low-lying bending vibrations of acetylene, to illustrate the constants x1414, x1415, x.1515 and Y ~ ~ . Some of the observed transitions are marked with approximate vibrational origins for C2Hz in cm-l; those around 612 cm-l have been observed in the Raman spectrum, and those around 730 cm-l in the infrared spectrum. TABLE 3.-BEST ESTIMATES OF THE FORCE CONSTANTS IN TABLE 1, WITH STANDARD ERRORS FROM THE LEAST SQUARES CALCULATION IN PARENTHESES, OBTAINED BY STREY AND MILLS^^ AND BY SUZUKI AND OVEREND.*= force force constant units Sand M S and 0 constant units Sand M Sand 0 f a a aJ 0.2510(5) 0.258 8 frua aJA-l -0.67(4) -0.39 frah -0.52(4) 0.01 faaaa aJ 1.418(43) 0.515 2 fRucc -0.34(3) -0.76 fad 0.0925(5) 0.100 1 h a a s -0.02(1) -0.09 Luau' (a) (b) fRaa 0.27(1) 0.36 faaci~a' + 0.1 60( 17) &a,*,, - ) (a) (b) (a) constrained; (6) S and 0 ' s results ar0 not comparable14 POTENTIAL SURFACES FROM VIBRATION-ROTATION DATA angle-bending force constants are constrained to zero.Moreover, the uncertainties in the cubic force field, and the comparison between the two independent calculations, suggest that the cubic and quartic force constants have not been well determined.The correlation matrix shows that the force constants are highly correlated, and the standard errors are probably an optimistic estimate of uncertainty by a factor of 5. However, in this case (and in all our experience) one can say that the harmonic force field is much more precisely determined as a result of including the cubic and quartic terms in the analysis. The difficulties which these results illustrate are partly inherent in attempts to determine potential surfaces from vibration-rotation spectroscopy, arising from the fact that higher order force constants relate to the shape of the surface further and further from equilibrium, which is the part of the surface to which the data are least sensitive. They also arise from the fact that the convergence of the power-series expansion of Vin normal coordinates, in terms of which the hamiltonian is formu- lated, is not sufficiently rapid.The difficulty is that the zero’th-order model of vibration-rotation theory, used to establish the basis functions, is based on harmonic vibrations in rectilinear normal coordinates ; but even for one-quantum excitation many bending vibrations follow a significantly curvilinear path, to such an extent that the deviations cannot be comfortably handled by perturbation theory. The procedure described above involves using a hamiltonian formulated in recti- linear normal coordinates, truncated at quartic terms in the potential. If it were possible to formulate the hamiltonian directly in curvilinear normal coordinates, the equivalent truncation would give a much better representation of the p.e.f.for large displacement in the angle bending coordinates, and recent work has shown that the use of (drlr) as a bond stretching coordinate, rather than the displacement 6r itself, also gives improved convergence of the potential energy expansion.24 The use of curvilinear coordinates, however, greatly complicates the kinetic energy terms in the h a m i l t ~ n i a n , ~ ~ . ~ ~ and there is as yet no widely accepted development of the yerturba- tion treatment in this format. However Handy and Kern and co- workers,28 and Whiffen and have all developed variational methods of handling the vibration-rotation hamiltonian which allow one to use a potential energy expansion directly in curvilinear coordinates, as discussed by Handy in a later paper presented at this Discussion.2. LOW FREQUENCY VIBRATIONS When a molecule has one or more low-frequency mode of vibration, the presence of hot bands often make it possible to determine the vibrational energy levels over many quanta in the low-frequency mode. Such vibrations are also often observed in combination bands in near-infrared and Raman spectra; again in microwave spectra hot bands may be observed over many quanta, so that in general more detailed information is available on that section of the potential surface associated with the low-frequency mode. Interpreting such data, however, poses its own problems. By their nature low frequencies imply large amplitudes of vibration, for which curvilinear paths tend to play a more important role; as discussed above, the customary theory of molecular vibrations is based on rectilinear normal coordinates and is thus ill-adapted to large- amplitude vibrations.Large amplitudes also tend to involve highly anharmonic potential surfaces, for which a power series expansion as in eqn (1) may not give a satisfactory representation; yet it may not be easy to find an analytical function that gives a suitable parameterization of the surface. Finally it may be difficult to deter-IAN M. MILLS 15 mine both the precise form of the coordinate involved in a large amplitude vibration, and the zero-point averaging effect of the remaining vibrations. 2.1 RING PUCKERING VIBRATIONS Four membered ring molecules show a low frequency out-of-plane puckering vibration which illustrates many of these problems.Ring puckering vibrations are usually very anharmonic, and in many cases vibrate in a double minimum potential with two equivalent non-planar equilibrium configurations. The subject has beel\ recently reviewed by L a f f e r t ~ . ~ ~ Cyclobutane (C4H8) is the parent molecule of the series, and provides a good ex- ample. The puckering vibrational energy levels have been determined over the first 10 quanta from the Raman spectrum3' and from near-infrared combination band^.^^*^^ By treating this vibration separately from the rest, and solving the one- dimensional anharmonic oscillator problem exactly (usually by expanding the hamiltonian in a harmonic basis set), it is found that the observed puckering energy levels can be fitted to an anharmonic double-minimum potential of the form V/hic = -Ax2 + BX4. (8) Here x represents the puckering coordinate (usually taken as half the separation of the ring diagonals).Since this two-parameter function gives a good fit to the first 10 vibrational intervals, one might think that the potential surface in this coordinate is well determined. In the case of cyclobutane the barrier to inversion is found in this way to be 515 5 5 cm-I (see Lafferty3' for other examples). However, although the barrier height may be quite reliably determined, the exact form of the puckering normal coordinate (e.g., the amount of CH2 rocking vibration mixed with ring puckering) is not known, so that the correct reduced mass associated with ring puckering is uncertain.This in turn introduces uncertainty into the horizontal scale of the puckering potential function. Furthermore, the path of the coordinate is almost certainly curvilinear in the sense that the C-C bond lengths remain constant as the ring puckers : this implies a coordinate-dependent effective mass, which is a complication that is omitted from most ring-puckering potential function calculations. Mal10y~~ has discussed these problems at length. In prin- ciple, the vibrational energy levels of an isotopically substituted species (C4D8 in this case) give information on the form of the coordinate, but even this is complicated by the fact that the form of the puckering coordinate may change on isotopic substitution. Malloy concludes that barrier heights are generally more reliable than are the other features of the puckering potential function.Four membered ring molecules of lower symmetry, such as oxetane (trimethylene oxide, C3H60), thietane (C3H6S), and silacyclobutane (C3H6SiH2), exhibit far infrared puckering spectra, and microwave spectra which may be analysed to give precise rotational constants in many excited vibrational states. These give further informa- tion on the form of the puckering potential, because it is found that the effective rotational constants show an anomalous vibrational dependence on the puckering quantum number.30 This may be interpreted by equations of the form where q is a scaled (usually dimensionless) puckering coordinate, and the averages are calculated using the exact anharmonic wavefunctions in the puckering coordinate.The observed vibrational dependence of the rotational constants A,, B, and C, proves to be a sensitive measure of the anharmonic wavefunction, and hence of the16 POTENTIAL SURFACES FROM VIBRATION-ROTATION DATA puckering potential function, as shown by the example of oxetane which has been most comprehensively In this case, the microwave spectrum has also been analysed in sufficient detail to give the vibrational dependence of the distortion constant^.^^^^^ These have also been successfully interpreted by using the exact anharmonic wavefunctions in the puckering coordinate. The analysis also gives information on the degree of mixing of CH2 rocking into the puckering coordinate, as shown by Mallinson and Mills.40 However, despite the fact that five different partially deuterated isotopic species of oxetane have been studied in detail, there still remain small inconsistencies in the observed isotopic dependence of the various data.2.2 LOW-FREQUENCY BENDING VIBRATIONS OF A LINEAR CHAIN In my own laboratory we have recently been studying three examples of quasi-linear molecules, involving a low wavenumber, large amplitude anharmonic bending vibra- tion. These are fulminic acid (HCN0),41 silylisocyanate (SiH3NC0),42 and carbon- suboxide These examples involve a two-dimensional anharmonic oscillator (in contrast to the one-dimensional example of the four-membered ring molecules). Once again, information on the potential function can be obtained from both the vibrational energy levels and the vibrational dependence of rotational constants, and the experimental data available on these molecules are summarized in table 4.The TABLE 4.-sOTJRCES OF INFORMATION ON THE LOW FREQUENCY BENDING POTENTIAL FUNCTION OF HCNO, SiH,NCO, AND C302. THE BODY OF THE TABLE GIVES REFERENCE NUMBERS TO PAPERS ON THE APPROPRIATE SPECTRA (ABSENCE OF A REFERENCE INDICATES ABSENCE OF THAT KIND OF DATA) HCNO, SiH,NCO C302 v5 v 10 V l molecule and vibration: DCNO vibrational energy levels from: 1. near i.r. combination and hot bands 47 2. far i.r. spectra 44 46 3. Raman spectra 46 1. microwave spectra 45 42 rotational constants from: 2. high resolution i.r. 44 47 diffraction 48 49 vibrationally averaged distances from gas electron analysis of the data has been made in a similar manner to that described above for ring puckering vibrations, and the combination of vibrational and rotational data leads to the potential surfaces illustrated in fig.2. With SiH3NC0 there is no direct observation of the vibrational energy levels, and the potential function and energy levels shown in the figure have been inferred from rotational structure. However for both C302 and SiH3NC0 a gas electron diffrac- tion study has been made (as indicated in the table); this essentially confirms the other evidence on the potential surface. Details of the analysis are given in ref. (41), (42) and (43). Although we regard the main features of these surfaces as well-determined (partic- ularly the height of the central potential hump in GO2 and SiH,NCO), the exact form of the bending coordinate is less well determined; this reflects uncertainty on theIAN M.MILLS 80. 60. 40. 17 33 0 - A 1 1 2' c - A t 22 A - T 2od 01 -20 201 ' I *--[ i -20 O W , > , I , 1 I 0' 10' 20° 30° 40. 0 10. 20' 3O0 40° 0. 10' 20' 30. 40. 's i 1": O=C&crC=O 3- -c=o .-.. ,--.I . . H-!-C&N=O FIG. 2.-The potential energy surface for the large amplitude bending vibrations of HCNO, GO2, and SiH3NC0. In each case the coordinate involved is bending of the linear chain, the major component of the bending being as indicated below each figure. The bending coordinate is doubly degenerate, and the potential surface should be visualized in three dimensions, with a circular valley and a central hump.The figure shows a radial section of this surface. appropriate reduced mass and hence on the horizontal scale in the figures. Also, the curvilinear nature of the bending coordinate has not been properly accounted for in relating the potential surface to the observed spectrum, as in the example of ring molecules. Another feature of these potential surfaces is illustrated by fig. 3 for C,O,. For this molecule fairly high resolution infrared spectra of the fundamentals v4 and v2 are available, and these show hot bands and combination bands involving the large amplitude bend v7. From the resulting vibrational and rotational constants we have been able to determine43 the effective bending potential function in v7, in both the 820 810 780 775 760 J.. -r ground stole 30 40{ 2 0 4 ---I # Go loo ?CO JC" 40' FIG.3.-The bending potential surface of c302 in the doubly degenerate large amplitude vibration v7, and the effect on this surface of exciting one quantum of v2 and one quantum of v4. The form of v2 and v4 is illustrated in the diagram.18 POTENTIAL SURFACES FROM VIBRATION-ROTATION DATA ground vibrational state of all other modes, and in the excited states v4 = 1 and v2 = 1. The resulting surfaces are shown in fig. 3. The interesting feature is that in both cases there is a significant change in the shape of the v7 potential surface on exciting another vibration; the hump at the linear configuration is doubled in the v4 = 1 state but almost removed in the v2 = 1 state. This draws attention to the point that the potential surface in any low-frequency large-amplitude vibration is really an efective potential obtained by averaging over all the remaining high frequency vibrations, in a manner analogous to the adiabatic separation of electronic from nuclear degrees of freedom.A similar comment applies to all of the previous examples: in each case the section of the potential surface which is determined should be thought of as vibration- all) averaged over the remaining normal modes. 3. DISCUSSION The limitation of vibration-rotation spectroscopy to a small region around the equilibrium configuration in the potential surface suggests that the most fruitful comparison with other methods of study will be with ab initio calculations. This, indeed, is proving to be true, since there are now many papers appearing which report the ab initio calculation of both harmonic and anharmonic force constants for small molecules, and it is clear that sufficiently careful calculations give good results. There have been fewer attempts to calculate potential surfaces for large amplitude vibra- tions, such as those described in section 2 of this paper, perhaps because the energy changes involved are smaller, implying the need for an even greater degree of precision in the calculation.In this paper I have tried to emphasize two features of the difficulties in determining potential surfaces from vibration-rotation data. The first is the difficulty of knowing the exact form of the coordinate in terms of which some section of the potential surface may have been determined.I began by describing this as the fundamental difficulty of harmonic force field calculations, and I finished by emphasizing that this is the least certain feature of large-amplitude potential surfaces such as those in fig. 2 and 3. The second is the importance of vibrational averaging effects. Kuchitsu, in a following paper, will emphasize the importance of these effects on molecular structure determinations; fig. 3 shows that they can make an important contribution to the effective potential surface. If we were to compare an ab initio calculation of the bending potential in C302 with the experimental surface, we would have to face these two problems in making a comparison. From the ab initio end, it would be necessary not only to recalculate the energy for each value of the large amplitude coordinate, but also to recalculate the local potential surface with respect to all other vibrational coordinates.It would then be necessary to determine the vibrational wavefunctions (including anharmonic effects) in the remaining coordinates, holding the large amplitude coordinate frozen, and use them to average the potential energy function. In this way an effective poten- tial in the large amplitude mode might be determined. From the experimental end, it is clearly desirable to determine the effective potential as a function of as many other parameters as possible (e.g., isotopic substitution, vibrational excitation) in order to provide the information necessary to extrapolate back to the true potential surface. At present there are no examples where an analysis of this kind has been made from either the experimental or the ab initio approach.I am grateful to my colleagues for numerous discussions of these problems, andIAN M . MILLS 19 particularly to Dr. A. G. Kobiette and Mr. J. A. Duckett for permission to quote results from our forthcoming paper on C301. I also thank Dr. Robiette for a critical reading of this manuscript. A. G. Maki, J. Mol. Spectr., 1973, 47, 2 17. D. Van Lerberghe, J. J. Wright and J. L. Duncan, J. Mol. Spectr., 1972, 42, 251. C. Camy-Peyret, J. M. Flaud, G. Guelachvili and C. Aniiot, hfol. Phys., 1973, 26, 825. G. Guelachvili, Nouv. Rev. Opt. Appl., 1972, 3, 317. S. M. Freund, G. Duxbury, M. Romheld, J. T. Tiedje and T. Oka, J. Mol. Spectr., 1974,52, 38.H. E. Radford, K. M. Evenson and C. 3. Howard, J. Chem. Phys., 1974, 60, 3178. T. H. Edwards and S. Brodersen, J . Mol. Spectr., 1975, 54, 121. ’ A. Cabana, L. Lambert and C . Pepin, J. Mol. Spectr., 1972, 43,429. ’ A. 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