The Journal of Physical Chemistry, Vol. 83, No. 11,
Local Mode Transitions in C6D,H
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Effect of Bending Vibrations on the High Overtone Spectrum of the C-H Bond Mode in C0D5H Martin L. Sage' Department of Chemistry, Syracuse University, Syracuse, New York 13210 (Received October 20, 1978; Revised Manuscript Received January 22, 1979) Publicatjon costs assisted by Syracuse University
We investigate the effects of bending vibrations on the high overtone spectrum of the C-H bond mode in C,&H using a Morse oscillator to describe the bond mode. We predict appreciable changes in energy and small changes in intensity for the high overtone spectrum. The energies and intensities are well described using an effective potential that differs slightly from the original Morse oscillator.
Introduction In the past few years a great deal of interest has developed in high vibrational states of polyatomic molecules. Knowledge of these llevels is needed for a detailed understanding of a number of problems including overtone spectra, multiphoton photofragmentation, and intramolecular dynamics. Recent measurements of overtone spectra in molecules containing X-13 bonds show that some of the higher vibrational levels are consistent with a local mode modell-5 in which the X-H stretching vibration is described in terms of a Morse oscillator with the true masses of X and H. In this paper we examine the vibrations of an actual polyatomic molecule in more detail and show that the Morse potential derived froim the local mode model is only an effective potential rather than the true potential for the X-H stretch. The effective potential includes appreciable effects from zero point energy of other vibrational modes. In addition intensities in local mode transitions will be lower than those calculated ignoring other vibrations.
Model In the interest of clarity we shall use a simplified model which includes the essentials and avoids computational details. In particular we shall treat the C&H molecule using a simple harmonic oscillator model for P, the in-plane bend, and y,the out-of-plane bend of the CH bond, and a Morse oscillator for s, the C-H bond length. The coordinates and G matrix elements for the molecule are described in detail in Wilson, Decius, and Cross.6 The diagonal G matrix elements are 'Gss
G,, =
(PH
G,, =
(PH
= PH
= G@p,2/2 + Gy&2/2 + G S S P : / ~+ G@sp@s+ k8P2/2 + k,y2/2 + D{exp[-Ba(s - S ) ] - 2 exp[-a(s - $)I) (3)
HCH
where the G matrix elements are given by eq 1. The in-plane and out-of-plane bending force constants are k , and k,, respectively, while D and a are parameters of the Morse potential describing the C-H bond and S is the equilibrium value of the C-H bond length. The Hamiltonian is not separable due to the s dependence of the G matrix elements for the bends and the term involving G,. Nevertheless we shall look for solutions of the form +n,nBny(s,P,Y)
+ PC)/S* + ~ c / s t+ 3yc/4t2 + /lc)/S' + 4 ~ c / s t+ 6 ~ c / t '
(la)
while the only nonvanishing off-diagonal element is
(4)
= +n(s)BnSn(P)cn7n(y)
The wave function +,(s) corresponds to the nth eigenvalue of the C-H stretch Morse os~illator.~ [Gssps2/2 + V(s)l+n(s) = -[AD1"
+ PC
G8, = 31/2/&/4t
mode and the two C-H bending vibrations, H, is the Hamiltonian describing the remaining 27 vibrations, and W are those terms in the potential energy or kinetic energy which couple motions of the C-H with the remaining vibrations. In our model we will find eigenfunctions and eigenvalues of HCH. This model is adequate for determining energy levels and intensities for local mode transitions. However, it will not be sufficient for determining line widths, line shapes, or absorption background^.^,^ A detailed discussion of the partitioning of the Hamiltonian and the effects of W on eigenfunctions of H C H + H, is given by Sage and Jortner.8 The Hamiltonian for the C-H bond is
-
n - 1/2)'/A2I+n(~)
(5) where V(s) is the Morse potential and A = (2/Gs,)1~2/aTz. For each state of the stretching vibration we solve two independent harmonic oscillator equations for the bends with effective G matrix elements
(Ib)
where ,uH and ~c are the reciprocals of the mass of the H and C atoms, respectively, and t is the equilibrium length of the C--C bond. Since we are interested in the high overtone spectrum of the C-H bond mode we can treat all other vibrations as smidl amplitude motions. This fact has already been used in evaluating the G matrix elements. The vibrational Hamiltonian will consist of several parts: H v l b = H c H + H, + W (2) where H C H is the Hamiltonian describing the C-H bond
0022-365417912083-1455$0 1 ,0010
where j = /3 or y. The approximate energy levels are
En,+pnr= -[(AD1/'
-
n - 1/2)2/A2]+ hvp,,(n'
+ Yz) +
hv,,n(nY +
72)
(7)
where v, = (klGll(n))1~2/2x. Transitions that are attributed to the excitation of the bond stretch mode from the 0th to the nth level will occur a t AE,/hc = nZs - n26,+
0 1979 American
Chemical Society
+ AZ,,J/2
(8)
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The Journal of Physical Chemistry, Vol. 83, No. 11, 1979
TABLE I: Effective G Matrix Elements Divided by Equilibrium G Matrix Elements for the C-H Bending Vibrations. Exact Value and Various Series Expansion Approximations Gpp ("'lGpp
Martin L. Sage
TABLE 11: Morse Parameters for C-H Stretch Effective Parameters for C,H, and C,D,H, and True Parameters for C,D,H
Gyy(n)/Gyy
n
exact
1 term
5 term
exact
1 term
5 term
0 1 2 3 4 5 6 7 8 9 10
0.9870 0.9606 0.9339 0.9069 0.8796 0.8519 0.8239 0.7955 0.7667 0.7376 0.7081
0.9743 0.9245 0.8765 0.8304 0.7860 0.7432 0.7020 0.6623 0.6241 0.5872 0.5516
0.9870 0.9605 0.9335 0.9058 0.8774 0.8483 0.8183 0.7876 0.7561 0.7240 0.6912
0.9895 0.9682 0.9467 0.9249 0.9028 0.8804 0.8577 0.8348 0.8115 0.7879 0.7640
0.9798 0.9405 0.9026 0.8661 0.8308 0.7968 0.7639 0.7321 0.7014 0.6717 0.6429
0.9895 0.9681 0.9464 0.9241 0.9012 0.8777 0.8536 0.8289 0.8036 0.7777 0.7613
where 5, = (2AD1/2- l)/hcA2,6, = l/hcA2, and = (u,,, - v,,o)/c. The two A? are negative, decreasing functions of n. All the terms in eq 8 must be used in finding the best Morse potential for describing the C-H stretch.
Calculations Wave functions for the Morse oscillator, in the usual approximation, do not satisfy the correct boundary conditions of vanishing a t zero but vanish at --. For Morse potentials used to describe molecules, these functions can be used to give accurate values of many matrix element^.^ However, matrix elements of s-l and s - ~needed in our computations will diverge since the wave functions do not vanish a t zero but are only very small. Exact, analytic methodsg cannot be used. Two obvious methods remain. In method one we expand l / s 2 as a series in (s - s,)/s, where s, is the expectation value of s in the nth level of the Morse oscillator. The expansion is accurate whenever the wave function is appreciable. For s I 0 the expansion breaks down but the wave function is small and does not make a significant contribution to the integrals. Likewise when s 1 2s, the expansion fails but again the wave function is small. Had we expanded about the equilibrium position, S , the contribution for s 1 2s could be large and lead to highly inaccurate values of G,,(,). All necessary integrals can be found using the analytical procedures of ref 9. In method two we use the approximate wave functions directly in a numerical integration. However, we must cut off the integration a t a small but finite value of s. The exact results should be found using the correct confluent hypergeometric wave functions which vanish at s = 0, but the cutoff procedure works because the approximate wave function is small near the cutoff. Method two requires much more computer time than method one, especially for highly excited states. Table I gives G,l(n)/GIJ, the ratio of effective G matrix elements to equilibrium G matrix elements for the two bending vibrations calculated using both methods. The parameters of the oscillator are listed in Table 11. Exact results are obtained using method two. Series expansion results obtained with a five-term expansion are accurate t o within a few percent for all the levels. Likewise the one-term expansion is qualitatively correct. Estimates of the bending force constants are obtained from published analyses of the normal modes of benzene. The constants for in-plane bendlo and out-of-plane bend'l are k,= 8.4 X erg and ha = 4.4 X 10l2erg, respectively, 8-l cm-2 and G,, = 8.54 X and G,, = 6.19 X 8-l cm-2. The two vibrational frequencies are 1210 and 1030 cm-'. Due to uncertainties in the force constants these frequencies are only qualitatively correct. However, they
effective parameters C6H6 a
A-', cm-' D , cm-' a, A-'
57.7 4.33 X 1.S6
lo4
true parameters
C,D,Ha
C,D,Ha
58.7 4.31 x l o 4 1.87
58.5 4.33 x l o 4 1.87
a The equilibrium bond length for C-H and C-C was taken t o be 1.07 and 1.40 A , respectively.
are sufficiently accurate to determine the significance of zero-point energy changes described by eq 8. Intensities calculated using the single anharmonic oscillator will be incorrect since the bending wave functions change. Franck-Condon-like factors must be included. For ground state harmonic oscillator functions the Franck-Condon factor needed for the n n ' local mode transition is
-
JBo,n(P)Bo,n,(P) dPJCO,n(y)CO,n,(Y) dy = [2b,bn!/(b,2 + bn~2)]1/2[2C,C,+,2 + cn?)]1/2 (9) where b, = (h2Gpii(n'/ka)1~4 and c, = (h2Gyy(n)/hy)1/4. The overlap depends on the product of the ratios G,p(n)/G,p(no and Gyy(n)/Gyy(n'). Our functions given by eq 4 are only approximate eigenfunctions of HCH. The functions with quantum numbers n, n,, and nY are coupled to other functions by the Gp,pop, term and through the dependence of G , and G,, on s. For the overtone spectrum of the C-H bond mode we are interested in states in which no = nY = 0 so we can focus on these states. In addition we will only look a t states with n I 10. The term G,,p,p, couples n, 0, 0 to n', 1, 0. Matrix elements with n'= n f 1 are at least an order of magnitude larger than other matrix elements. They range in magnitude from 50 cm-l for the coupling of 0, 0, 0 to 1, 1, 0 to 200 cm-l for 10, 0, 0 to 11, 1, 0. Second-order perturbation theory leads to energy shifts ranging from -1 cm-' for 0, 0, 0 to +16 cm-' for 10, 0, 0. These shifts are small compared to those calculated using eq 8. The dependence of G,, on s leads to the coupling of n, 0,O with n', 0,O and n', 2,O while G,, leads to the coupling of n, 0, 0 with n', 0, 0 and n', 0, 2. Again the significant matrix elements involve n ' = n f 1. We calculate the matrix elements by expanding G in a Taylor series in s and retaining only the linear term. Matrix elements diagonal in no and n, range from approximately 130 cm-l for n = 0, n ' = 1 to 370 cm-' for n = 10, n ' = 11. The overall contribution of these terms to the energy of n, 0, 0 is an almost constant value of -6 to -8 cm-'. Matrix elements off diagonal in n8 and nY are smaller but can make appreciable contributions to the energy for n 1 8 since the unperturbed energy of n - 1 , 2 , 0 may approach that of n, 0, 0. Nevertheless, for states with n I 6 our original procedure is adequate for a description of the energy levels.
Results and Conclusions Table I1 lists the parameters which describe the true and the effective Morse oscillators for C6D5H. The effective Morse oscillator parameters are obtained from the observed 0 1 , 0 5 , and 0 6 C-H stretching transitions in C6D5H. The parameters of the true oscillator differ but slightly from those of the effective oscillator. When the zero-point shifts shown in eq 8 are applied to the true oscillator, energy levels are found which agree with those of the effective oscillator to within 1 cm-' for n I 10.
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The Journal of Physical Chemistry, Vol. 83, No. 11, 1979
Internal Rotation in Isopropyl Alcohol
TABLE 111: C-H Stretching Energy Levels, Zero-Point Shifts, Squares of Franck-Condon Factors, and Bending Vibration Frequencies Morse aiscillator, cm- '
--__
n 0 1
2 3 4 5 6 7 8 9 10
effective
0
3052 5987 8805 11505 14088 16553 18901 21132 23245 25242
sq.
zero F-C point factor shift, 0 n cm-' transn
true 0 3052 5987 8805 11505 14088 16554 18902 21132 23245 25241
0 -14 -28 -42 -57 -72 -87 -103 -120 -137 -154
h
up,:, cm1.000 1202 1.000 1186 1.000 1169 1.000 1152 0.999 1135 0.999 1117 0.998 1098 0.997 1079 0.996 1059 0,995 1039 0.994 1018 -+
N
y9p,,
cm
1025 1013 1002 991 979 966 954 941 928 914 900
Includes zero-point shifts.
-
Intensities for the 0 n overtone transition are reduced by the square of the Franck-Condon factor 9. The reduction is less than 1%for n 5 10. For the n n+1 transition the reductions are even smaller. These numerial
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results are summarized in Table 111. The effective potential reproduces the energy levels of the true Morse oscillator with zero point shifts to within 1 cm-l for n 5 10 and to 10 cm-l for n I20 despite shifts of hundreds of wavenumbers. Likewise intensities are accurately predicted. On the other hand coupling terms that have not been treated quantitatively will lead to a breakdown of the Morse energy level structure for n L 8 since nearly degenerate levels can interact.
References and Notes (1) R. Wallace, Chem. Phys., 11, 189 (1975). (2) R. L. Swofford, M. E. Long, and A. C. Albrecht, J . Chem. fhys., 65, 179 (1976). (3) R. L. Swofford, M. E. Long, M. S. Burberry, and A. C. Albrecht, J . Chem. fhys., 66, 664 (1977). (4) R. L. Swofford, M. S. Burberry, J. A. Morrell, and A. C. Albrecht, J . Chem. fhys., 66, 5245 (1977). (5) R. Bray and M. J. Berry, to be published. (6) E. B. Wilson, J. C. Decius, and P. C. Cross, "Molecular Vibrations", McGraw-Hill, New York, 1955. (7) D. F. Heller and S. Mukamel, J . Chem. fhys., 70, 463 (1979). (8) M. L. Sage and J. Jortner, Chem. Phys. Lett., in press. (9) M. L. Sage, Chem. fhys., 35, 375 (1978). (10) J. C. Duinker and I. M. Mills, Spectrochim. Acta, Part A , 24, 417 (1968). (11) E. J. O'Reilly, J . Chem. fhys., 51, 2206 (1969).
Internal Rotation in Isopropyl Alcohol Studied by Microwave Spectroscopy Eizl Hirotat' Depaflment of Chemistry, Faculty of Science, Kyushu University, Fukuoka 8 12, Japan and Institute for Molecular Science, Okaraki 444, Japan (Received October 10, 1978) Publication costs assisted by the Institute for Molecular Science
The microwave spectrum of isopropyl alcohol was reinvestigated in much more detail than in a previous paper. The trans spectra were analyzed in terms of a rigid-rotor model modified by first-order centrifugal distortion effects. For the gauche form, the spectrq of which were complicated by a tunneling effect between the two equivalent minima, the b-type and C-type transitions in both the symmetric and antisymmetric sublevels were observed in addition to a-type ones which occurred between the symmetric and antisymmetric levels. The observed transition frequencies were analyzed by using an effective 2 X 2 Hamiltonian. The tunneling splitting was determined to be 46798.50 f 0.11 (AQ)MHz. An analysis of the Stark effects gave I(sIpLala)l= 1.114 f 0.015 D, pb = 0.737f 0.025 D, and pLc= 0.8129 f 0.0049 (&to) D. The trans spectra of (CH&CHOD were analyzed similarly to those of the normal species, but a definite assignment could not be made for the gauche form.
Introduction Until 1960 rotational isomerism in molecules was investigated mainly by infrared and Raman spectroscopy, electron diffraction, and dielectric constant measurements? all these methods established the existence of rotameric forms in many molecules and also provided approximate structural parameters of rotamers, including the dihedral angles and their relative stabilities. In 1960 a systematic study of rotational isomerism was started in Professor Wilson's laboratory by using microwave spectroscopy, which was expected to make possible more detailed comparison of molecular properties of different isomers such as structures, dipole moments, quadrupole coupling constants, internal-rotation barriers of methyl groups, torsional frequencies, relative stabilities, and so on. Wilson summarized the results obtained from earlier studies in 0022-3654/79/2083-1457$01 .OO/O
this field in a review articlea3 One interesting feature of microwave investigations is a direct observation of torsional splittings. As is well known, torsional splitting is a very sensitive function of the potential barrier through which tunneling occurs. Therefore, the observed torsional splitting is important in determining the potential function for internal rotation. It should be noted that the rotamer is a static concept; in other words, it only means the presence of a minimum in the potential function. On the other hand, we can derive any information on rotational isomerism from the potential function, and therefore its precise determination is the final goal of experimental approaches t o rotational isomerism. The observation of torsional splittings is possible only when two equivalent rotamers are present. This is the case
0 1979 American
Chemical Society
