448
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J. Phys. Chem. 1982, 86, 440-455
Absolute Vibronlc Intensities in the 'A,
'A, Absorption Spectrum of Formaldehyde
S. J. Strlckler' and R. J. Barnhartt Department of Chemistry, University of Colorado, 8ouIder, Colorado 80309 (Received: May 11, 198 1; In Final Form: October 13, 198 1)
Absolute absorption intensities are reported for individual regions within the first singlet-singlet electronic bands of formaldehyde and its deuterated derivatives. Franck-Condon factors for the C-O stretching vibration were calculated by using Morse oscillator wave functions since the harmonic approximation and the method of Doktorov, Malkin, and Man'ko proved to be inadequate. We report a consistent set of intensities of false origins for each molecule which gave a good fit to the spectrum. The oscillator strength of the band is (2.40 f 0.05) X lo-' in H2C0and (1.86 f 0.05) X in D2C0. We find for H2C0that of this intensity about 66% is induced by vibration u4, 26% by u5, 6% by V g , and 0.5% by each of the combinations u4u5 and V4V6; about 1% is due to the magnetic-dipole-allowedcomponent. We discuss a new way of considering vibronic intensity induced by two or more vibrations of the same symmetry. In the case of u5 and u5, we show that the intensity is induced by displacing the H atoms in a direction nearly perpendicular to the C-0 bond, while displacement parallel to that bond induces virtually no intensity.
Introduction In work reported in a separate paper,' the absolute intensities of the symmetry-forbidden 'A2 'Al absorption bands of formaldehyde, formaldehyde-d, and formaldehyde-d2were determined in order to calculate radiative lifetimes of the excited states. Useful byproducta of that investigation were absolute absorption intensities of many individual regions of the spectrum. By selecting regions corresponding to particular vibronic transitions, it is possible to compare the observed intensities to those calculated from a Herzberg-Teller type of analysis? Other workers, for example Y e ~ n g have , ~ made similar comparisons, but not based on the same sort of data as reported here. The term symmetry-forbidden implies that the transition is not allowed by electric dipole selection rules when the molecule is in its equilibrium position. However, the transition may become allowed when the molecule is distorted from the symmetric configuration by vibration along one of the antisymmetric normal coordinates. In the case of formaldehyde, there are three such coordinates: Q4 (out-of-plane bending, bl symmetry), Q5and Q6 (in-plane antisymmetric stretching and bending, both of b2 symmetry, illustrated in Figure 3). Distortion along Q4induces a transition moment along the y direction (in-plane and perpendicular to the symmetry axis); distortion along either Q5or Q6 induces a transition moment in the x direction (out-of-plane). Transition intensities are among the fundamental spectroscopic parameters, and there has been much interest in methods of calculating intensities. Conversely, measurements of transition intensities provide a critical test of the quality of theoretical calculations. Symmetry-forbidden transitions are of particular interest because of the vibrational-electronic interaction involved. One matter of considerable theoretical interest is the afnount of intensity induced by each of the three antisymmetric normal modes, and we wished to determine those values experimentally. A perturbation-theory approach to calculation of vibronic intensities was suggested by Herzberg and Teller in their pioneering paper.2 The effect of nuclear distortions is taken to be a mixing of the various electronic wave functions in the equilibrium configuration, and this mixing
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allows some intensity in the distorted molecule. This approach was first applied to formaldehyde by Pople and Sidman4 They calculated that essentially all of the intensity should be induced by distortion along Q4, with Q5 and Q6contributing less than 1% of the intensity. Lin5 has extended the same type of treatment; although he does not quote figures in quite the same form, his equations 6-10 and 6-11 indicate that the intensities induced by Q4 and Q6should be roughly 45% and 55%,respectively, of the total absorption intensity, with Q5contributing less than 0.1%.
The perturbation method, which expands the electronic wave functions of the distorted confiiation in a basis set of wave functions at the equilibrium configuration, suffers from convergence problems in a truncated expansion. More recently a different approach has become practical, involving complete calculations of molecular-orbital wave functions and transition moments as functions of nuclear configuration. The transition moments are then averaged over the actual vibrational wave functions. CNDO calculations have proved to be successful for aromatic molea* transition of form~ules,~ but J less so for the n aldehyde? However, an ab initio calculation has recently been performed by van Dijk et aL9 Although they also do not quote results in this form, we calculate from their Tables IV and VI the prediction that Q4,Q5,and Q6should induce about 66%, 21 % , and 13 % , respectively, of the band intensity. The calculation is done simply by taking the intensity of some one band induced by a given mode from Table VI and extrapolating to the total intensity induced by that mode using the overlap integrals and/or vibronic factors listed in Table IV. We will show below that these numbers are in fair agreement with experiment. According to the Herzberg-Teller theory, the intensity induced by each mode should show up in the spectrum as bands in which that vibration is excited with an odd number in quanta in the upper state. (This assumes that the transition starts from the u = 0 level in the ground state
-
(1) R. J. Barnhart and S. J. Strickler, to be submitted for publication. (2)G. Henberg and E. Teller, 2.Phys. Chem. Abt. B , 21,410(1933). (3)E.S.Yeung, J . Mol. Spectrosc., 45, 142 (1973). (4)J. A. Pople and J. W. Sidman, J. Chem. Phys., 27, 1270 (1957). (5)S.H. Lin, Proc. R. SOC.London, Ser. A, 362,57 (1976). (6)M.J. Robey, I. G. Ross, R. V. Southwood-Jones,and S. J. Strickler, Chem. Phys., 23,207 (1977). (7)L. Ziegler and A. C. Albrecht, J. Chem. Phys., 60, 3558 (1974). (8)M.J. Robey, Ph.D. Thesis, The Australian National University, Canberra, Australian Capital Territory, 1974. (9)J. M. F. van Dijk, M. J. H. Kemper, J. H. M. Kerp, and H. M. Buck, J. Chem. Phys., 69,2453 (1978).
0 1982 American Chemical Society
'A2
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The Journal of Physical Chemistry, Vol. 86, NO. 4, 1982 449
'A, Absorption Spectrum of Formaldehyde 6
5
4
1
1
1
250
300
3
1
2
I
I 1
O
I
350
Wavelength, n m Flgure 1. Absorption spectrum of formaldehyde. A number of vibronic origins are labeled. Part of one progression in up is marked for reference. The arrow near the right marks the position of the true origin which is not visible in this spectrum.
and neglects any rotation of normal coordinates between the two states.) In principle, it is only necessary to sum the intensities of all bands having an odd number of quanta excited of each of the three antisymmetric modes to get the total intensity induced by each. In practice, this is not trivial. Figure 1shows the low-resolution absorption spectrum of H2C0 recorded on a Cary Model 17 spectrophotometer. The true band origin, which is forbidden in the electric dipole approximation, cannot be seen in this spectrum; its position is indicated by the arrow near the right-hand side. The spectrum is dominated by progressions in the u2 vibration (the C-0 stretching mode) built upon a number of vibronic origins, some of which are labeled on the spectrum. (The notation 4; for example, means a transition for which vibration u4 has u = 0 in the lower state and u = 1in the upper electronic state.) The assignments are those of Job, Sethuraman, and Innes.lo The strongest bands are those of the progressions built upon the 4; and 4; vibronic origins. These are induced by distortion along Q4,the out-of-plane motion. Normally, Herzberg-Teller theory would suggest that only the 4; progression would have significant intensity. However, in this case, the molecule is nonplanar in the upper state, and this makes the 4: origin and its progression fairly strong. The 5; origin is also fairly strong. This is induced by distortion along Q5,the antisymmetric C-H stretching mode. Because of the nonplanarity of the upper state, the 5142origin should also appear, but this is hidden under the 4020 P S band at about 314 nm. The 6i4: origin is observed; the 6; origin should also appear, but it is hidden under the 4: origin. Two much weaker origins are also observable in Figure 1,the 4: and the 6;4; origins. The former was shown by Callomon and Innes" to be a magnetic dipole transition. The latter is an example of a doubly forbidden transition which requires simultaneous distortion along two normal coordinates, Q4 and Q6,to make the transition allowed. Such transitions are expected to be extremely weak.12 Because of the overlapping of transitions, the intensities of the bands induced by the three vibrations are not directly separable. However, they can be separated if the Franck-Condon (FC) factors for the u2 progressions are known accurately enough. For example, if we can calculate the intensity distribution in the 4i2: progression by the Franck-Condon principle, we can determine what fraction (10)V. A. Job, V. Sethuraman, and K. K. Innes, J.Mol. Spectrosc., 30,365 (1969). (11) J. H.Callomon and K. K. Innes, J . Mol. Spectrosc., 10,166(1963). (12)S.J. Strickler and M. Kasha, "Molecular Orbitals in Chemistry, Physics, and Biology", P.-0. Ldwdin and B. Pullman, Eds., Academic Press, New York, 1964,p 241.
of the 314-nm band is due to the 4;2: transition, and by difference we get the intensity of the 5i4: vibronic origin. The FC factors are also of interest in their own right, as they contain information about the nuclear potential function of the upper state. In this paper we report the experimental intensities of a number of regions of the spectra of H2C0, D2C0, and HDCO. We describe an approach by which FC factors may be calculated and combined with a consistent set of intensities of vibronic origins to reproduce the intensity distributions of the entire bands. We derive from these data values for the intensity induced by each of the three antisymmetric vibrations and compare these with the theoretical calculations referred to above. Finally, we introduce a new way of considering the sort of distortions which induce intensity, and show that x-polarized intensity is induced largely by motion of the hydrogen atoms in a direction perpendicular to the C-0 bond.
Experimental Section All absorption spectra were taken on a Cary Model 17 spectrophotometer. For the intensity measurements the spectra were recorded in digital form. This was done by attaching a 100004 Helipot Model 5711 potentiometer with an independent linearity of f0.1% to the pen drive gear of the Cary 17 recorder. The reference potential was supplied by an Ortec Model 401A power supply, and the center tap potential, which was linearly related to the recorder pen position, was fed to a Canberra Model 6271 digitizer, which converted this voltage to a frequency. This frequency was counted by using the multiscaling mode of a Canberra Model 7010 multichannel analyzer. After subtraction of a base line, the number of counts stored in any one channel is proportional to the integrated absorbance over a small range of wave lengths. In this manner the spectrum could be divided into as many as 1024 segments. The entire system was checked to see whether calculated absorbancies would agree with values measured on the chart paper; in all cases it was found that the agreement was at least as good as the measurements on the chart-better than a tenth of a small division on the chart paper or 0.1% of the full scale reading. The absorption measurements were made by using a 10-cm cell placed in the cell compartment and connected directly to a portable vacuum system used to handle the gases. A Texas Instruments Model 144-01Bourdon gauge was used to measure the pressures to a precision of 0.001 torr. The temperature was measured with a thermometer in contact with the cell inside the cell compartment. Gaseous samples of the various isotopically substituted forms of formaldehyde were obtained by the method of Spence and Wild13 from the appropriate paraformaldehyde. The paraformaldehyde-d2 was obtained from Mdinckrodt and was stated to be 98.8% deuterated. This was confirmed by mass spectrometry, the principal impurity being the monodeuterated molecule. The polymer used for the production of the formaldehyde-d was obtained from Merck Sharp and Dohme; the percentage of deuteration was unstated. Mass spectrometry showed the formaldehydeproduced to be mainly HDCO, and set upper limits to the amounts of H2C0 and DzCO of 5% and 6%, respectively. No corrections were attempted for isotopic impurities. Care was taken that the entire vacuum system and the cell assembly were clean and dry to reduce the amount of polymerization during the time required to record a spectrum. The pressure in the cell was monitored during (13)R.Spence and W. Wild, J. Chem. SOC.,338 (1935).
450
The Journal of Physical Chemistry, Vol. 86,No. 4, 1982
TABLE I: Relative Intensities in the 4:2: Progression for D,CO calcd harmonic oscillatoP
n obsd d = 3.782 d = 4.547 d = 5.236 0 1 2 3 4 5 6 7 8 a
1.00 2.10 2.67 252 1.71 1.39 0.84 0.51 0.18
1.00 2.10 1.84 0.84 0.20 0.02 0.00 0.00 0.00
1.00 3.04 4.07 3.13 1.50 0.44 0.07 0.00 0.00
1.00 4.03 7.38 5.90 2.94 1.00 0.22 0.03 0.00
d is the displacement in Q, in units of
Doktorov formula Morse 1.00 2.15 2.68 2.51 1.95 1.32 0.81 0.45 0.24
lo-”
1.00 2.18 2.66 2.40 1.79 1.18 0.71 0.40 0.22
g”’ A.
spectral measurement so as to follow the remaining slow pressure decrease as polymerization occurred. The program used to compute intensities corrected for this pressure decrease.
Results and Discussion Harmonic Oscillator Franck-Condon Factors. We will discuss the problem of fitting FC factors to the u2 progressions in terms of the 4$2: progression of D&O, which is the best-resolved progression in all of our spectra. Actually, part of the intensity is due to 6k2: progression, but this does not affect the relative intensities since all evidence indicates that the same FC progression is built on each vibronic origin. The second column of Table I shows the experimental intensities in this progression. The application of the FC principle to polyatomic molecules was first discussed by Herzberg and Teller.* Under some reasonable assumptions the intensity of the nth band in any progression is expected to be proportional to ( @”2,01W2,n)2 where @’’2,0 represents the state u = 0 for vibration v2 in the ground electronic state, and @’2,n represents the state u = n for this vibration in the upper electronic state. This general method was first applied successfully to the forbidden absorption spectrum of benzene by Craig.’* He determined the increase in C-C bond length in the excited state by using the observed vibration frequencies in the two states and varying the displacement of the normal coordinate to fit the observed intensities in the progression in the symmetric stretching frequency. Parmenter and co-workers15 have applied the same technique to the fluorescence spectrum. A similar treatment was applied by Coon and co-workerP6to triatomic molecules in which two vibrations form progressions. Using the method of Coon et al., we have attempted to fit the intensity distribution in the u2 progressions of formaldehyde. We assumed that harmonic oscillator wave functions were adequate, used the observed frequencies of this vibration in the two states, and calculated the intensity distribution for various displacements of the coordinate. These calculations fail completely to explain the observed intensity distribution. The third, fourth, and fifth columns of Table I, headed “harmonic oscillator” show three intensity distributions calculated by using different displacements. The first displacement, d = (3.782 X 10-21)g1/2A,was chosen to fit the intensity of the 2; band, but in that ~~~
~
(14)D. P.Craig, J. Chem. SOC.,2146 (1950). (15)A. E.W. Knight, C. S.Parmenter, and M. W. Schuyler, J. Am. Chem. Soc., 97,1993 (1975). (16)J. B. Coon, R. E. DeWames, and C. M. Loyd, J . Mol. Spectrosc., 8,285 (1962).
Strickler and Barnhart column all higher bands are calculated to be much weaker than observed. The calculated intensities of higher bands can be increased by using larger values of d ; two other calculations are shown in the next two columns, but the agreement with experiment is still very poor. Other well-resolved progressions in H2C0, D2C0 and HDCO absorption spectra also cannot be fitted. Apparently additional factors must be considered to explain these intensities. Neglected factors which could be important in determining intensities include large second derivatives of the electronic transition moment such as d2Md/ (dQ2dQ4),a rotation or mixing of normal coordinates upon going from one electronic state to the other, and anharmonicity in the potential function for displacement along Q2. We considered and eliminated the first of these, since no reasonable values for the second derivative would produce the observed effect. Furthermore, our analysis for each molecule indicates that the same intensity distribution applies to the progression built on every vibronic origin, and it would require a coincidence of second derivatives for this to be true for bands induced by both the in-plane and out-of-plane vibrations. We then turned to consideration of rotation of the excited-state normal coordinates relative to the ground-state coordinates, or the Duschinsky effect.” A method of treating such a rotation has been given by Doktorov, Malkin, and Man’ko.1&20 It is based on harmonic oscillator wave functions for all vibrations and uses parameters obtained by comparing the intensities of a few specific bands. In the case of a single progression such as the present case, one can use the intensities of the u = 1 and u = 2 bands relative to the u = 0 band to obtain two parameters. A recursion relationship then permits the calculation of the intensities of all other bands in the progression. Parameters can be chosen to give an excellent fit to the intensities in the isotopic formaldehydes. The sixth column in Table I, headed “Doktorov formula”, shows the fit for the D2C0 progression. Comparable fits are obtained for the other isotopic molecules with slightly different parameters. The agreement between experiment and the values calculated by the Doktorov formula led us at first to believe that a Duschinsky effect was, in fact, important in formaldehyde. However, when we tried to use known vibration frequencies and numerical integration to discover just what type of rotation was involved, it gradually became clear that no possible rotation among the existing modes could explain the experimental intensities. Anharmonic Franck-Condon Factors. We then turned to consideration of anharmonicity of the u2 vibration. Since this mode is primarily a C-0 stretching motion, we decided to try a Morse potential for motion along Q2 and to calculate overlap integrals numerically by using the known exact solutions to the Morse o~cillator.~l-~~ This treatment also gives a good fit to the experimental intensities. We write the Morse potential
(17)F.Duschinsky, Acta Physicochim. URSS, 7,551 (1937). (18)E.V. Doktorov, I. A. M&, and V. I. Man’ko, J. Mol. Spectrosc., 56, 1 (1975). (19)E.V. Doktorov, I. A. Mallrin, and V . I. Man’ko,Chem. Phys. Lett., 46,183 (1977). (20)E.V . Doktorov, I. A. Malkin, and V. I. Man’ko, J.Mol. Spectrosc., 64,302 (1977). (21)P.M. Morse, Phys. reu., 34,57 (1929). (22)J. L. Dunham, Phys. Reo., 34,438 (1929). (23)H.S. Heaps and G. Herzberg, 2. Phys., 133,48 (1952).
'Ap
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TABLE 11: Morse Function Parameters Used to Fit Spectra D $0 H,CO lower upper lower upper state state state state D , c m - ' 63100 34900 57100 28800 w , , c m - ' 1764 1200 1770 1200 xZz,cm-' 12.3
k pa
The Journal of Physical Chemistty, Vol. 86, No. 4, 1982 451
'A, Absorption Spectrum of Formaldehyde
10.3 116 6.08 4.44
143 6.64
13.7 129 7.00
12.5 96 6.68 4.44
TABLE 111: Relative Intensities for the 2: Progressions Calculated from Morse Wave Functions and Used To Fit the Spectra
HDCO
lower state
upper state
61400 30000 1780 1200 12.9 12.0 138 100 6.79 6.55 4.44
displacementb a In l O I 9 g-", cm-'. In g l / ' cm. The displacements were not forced to be equal.
where Q is the displacement of the normal coordinate Q2 from its equilibrium position. We take Q to have units of g'l2 cm, which makes our definition slightly different from the usual application to diatomic molecules where internuclear distance is used. D is the dissociation energy; for a Morse oscillator it is related to the harmonic frequency w2 and the anharmonicity constant x22 by
D = W22/4x22
(2)
0 is given by
The parameter
The Morse wave functions in the form given by Dunhamn are expressed in terms of a parameter k = w 2 / x 2 2 . For simplicity in numerical evaluation, we take k to be an integer. The wave functions are calculated by the following formulas for any given vibrational quantum number u: z = (4)
]
P(k - 2u - 1) A , , = [ u!(k - u - l ) !
(5)
where
L k u ( Z ) = zu - u(k - u - 1)Zv-1 + 1
-u(u
2!
- l)(k - u - l)(k - u - 2)z*2 - ... (7)
Each Morse potential function is defined by two parameters which can be taken to be any two of the quantities w2, x22, D, 0, or k. To calculate FC factors, we also need the displacement along Q2 between the two states, so there is a total of five parameters. We have chosen the values of w2 to agree with the known frequencies in the two states and have made the reasonable assumption that the ground and excited states dissociate to the same state of CH2 + 0, so that the two values of D for each molecule differ by the excitation energy. This leaves two parameters to be determined for each molecule. In the case of H2C0, we chose the value of D for the ground state to agree with thermodynamic data24on the enthalpies of formation of H2C0, CH2, and 0 and then varied the displacement to give the best fit to the observed spectrum. The resulting fit is excellent. In the cases of D2C0 and HDCO, use of (24) D. D. Wagman, W. H.Evans, V. B. Parker, I. Halow, S. M. Bailey, and R. H.Schu", "Selected Values of Chemical Thermodynamics Properties", US Government Printing Office, Washington, DC, 1968, NBS Tech. Note (U.S.), No.270-3.
n
H,CO
D ,CO
HDCO
0 1 2 3 4 5 6 7 8 9 10 11 12
1.00
1.00 2.26 2.82 2.58 1.94 1.27 0.75 0.42 0.22 0.11 0.053 0.025 0.012
1.00 2.29 2.88 2.63 1.97 1.28 0.75 0.41 0.21 0.10 0.048 0.022 0.010
2.30 2.89 2.61 1.90 1.20 0.67 0.35 0.17 0.075 0.033 0.014 0.005
the same dissociation energy did not give good fits, and it was necessary to use Morse parameters corresponding to a lower dissociation energy, i.e., more anharmonicity. While we have not attempted to optimize all of the parameters, we have chosen values which give reasonable fits to the spectrum of each molecule. The last column of Table I shows a Morse oscillator fit to the 4!2$ progression in D2C0 for comparison with the other methods. Fitting the Entire Spectra. Table I1 lists the Morse parameters used for each molecule. Table I11 gives the complete set of intensities calculated from Morse wave functions and used for each molecule; values are taken up to u = 12 for the fit to the complete spectrum. The progression for D2C0 is slightly different from the one in Table I because it is chosen to give the best fit to the whole spectrum. Table IV shows the intensities of vibronic origins which, combined with the progression intensities in Table 111, were used to fit the spectra. The intensities chosen for the vibronic origins are in some cases fairly definite and in others somewhat arbitrary. Because of overlapping with adjacent bands, the values for the weakest origins are only approximate. The choices were made by observing peak heights and areas and attempting to make the values internally consistent within each molecule and consistent between H2C0 and D2C0. For example, values were chosen so that
1(4$5a)/1(56) = 1(4!66)/1(6h) for each molecule. Also, the intensity induced by Q5and Qsrelative to that induced by Q4 should be nearly the same for H2C0 and D2C0. We also considered FC factors (@"4,0(W4,n)2 and vibronic intensity factors ( @"4,0(Q4(@'4,n)2 calculated from the excited-state vibrational wave functions of Jones and Coon.25 However, we found that those wave functions consistently overestimated the intensities of the bands of higher vlaues of n compared to the bands with n = 0 or 1, so they could be used only in a qualitative sense. The most important criterion was always a good fit of the entire spectrum. We have used some origins which were not identified by Job et al.'O because their presence seems reasonable and they contribute to the good fit of the observed spectra. Finally, Tables V-VI1 give the observed intensities in individual regions of the spectra (generally corresponding to intervals between absorption minima), the bands expected to lie in these regions, and the intensities calculated for those bands. To determine which bands should, lie within each region, we used anharmonicity only in the u2 progression. The upper-state vibrational energies for this vibration were calculated from the formula E,, = (u + l/Z)w2 (25) V. T. Jones and J. B. Coon, J. Mol. Spectrosc., 31, 137 (1969).
452
Strickler and Barnhart
The Journal of Physical Chemistty, Vol. 86, No. 4, 7982
TABLE IV: Vibronic Origins in Formaldehyde with the Intensities Used To Fit the Spectra H ,CO D,CO designation transition 3, em-' P transition V , cm-' la transition A B C D E F G H I J K L M N P
4
R S T U
4; 4: 4; 4: 6; 4: 4;6: 4,26: 4i6: 4: 5: lb4: 4;5: 4:5; 1:6: 1:4,3 4; 5; 1:4,26: lA5: 1:4:5;
I = !e d In V in units of
+
27 021 28188 28313 28 730 29092 29136 29217 29634 30 047 30 127 31 156 31 160 3 1 284 3 1 698 31 939 31 983 3 2 104 3 2 483 34 003 34545
0.03 0.08 4.06 0.05 0.41 3.05 0.03 0.27 0.03 0.22 1.72 0.81 0.03 1.14 0.08 0.61 0.05 0.34 0.23
27367 28301 28369 28688 29006 28970 29075 29393 29678 29699 30534 30448 30602 30922 31 085 31046 31550 31 472 32613 33001
0.08 0.07 2.74 0.06 0.19 2.74 0.03 0.18 0.03 0.56 1.22 0.19 0.03 1.16
40
HDCO -
em-'
Ia
27185 28 330 28 708 29 039 29 117 29 460 29 800 29 761 30 408 3 1 167 30 488 30 862 3 1 607 31 030
0.06 3.50 0.15 3.20 0.05 0.20 0.10 0.15 0.20 1.70 0.60 0.15 0.40 0.50
u,
0.19 0.12 0.09 0.08
L mol-' cm-'.
+ ( u 1/2)2x22.The values of w2 and x22 used were 1195 and -7.8 cm-' for H2C0, 1185 and -7.0 cm-' for D2C0, and 1210 and -10.0 cm-' for HDCO. Energy levels cannot be expected to fit such a simple formula very exactly-indeed the accurate band energies quoted by Job et al.'O fit only with a standard deviation of some 20 cm-'-but these numbers give reasonable fits to our spectra. We consider the agreement between calculated and observed intensities in Tables V-VI1 to be excellent. Perfect agreement cannot be expected, as the minima between peaks do not go to the base line, so there is overlapping of bands in adjacent regions. Furthermore, anharmonicities will give some mixing of close-lying vibrational states and therefore some redistribution of intensity. However, the overall patterns of intensities are certainly well reproduced by the calculations. We believe that the agreement between observed and calculated intensity distributions means that the use of Morse wave functions is a satisfactory way of taking account of anharmonicity for a long progression in a vibration like u2 of formaldehyde, which is primarily a bond stretching. It is gratifying that in H2C0 the parameters which fit thermodynamic data also fit the spectrum. In DzCO and HDCO this is not true, and to get a good fit one must use a larger degree of anharmonicity than indicated by the dissociation energies. However, it is not too surprising that the simple Morse calculation should not fit exactly. Even in a diatomic molecule a Morse function is not an exact representation of a true potential curve. In formaldehyde, the small amplitude motion is a normal coordinate involving both C-H and C-0 motions, but for larger amplitudes the coordinate must change character to become more exclusively a C-0 motion. Our simple use of a Morse oscillator cannot reflect the details of changes in the nature of the coordinate. Such difficulties may become more serious as the mass of the light atoms is increased. A check on the consistency of the treatment can be made by converting the values of AQ2 and AQl to changes in bond lengths. We have used a ground-state normal-coordinate analysis for H2C0 and D2C0,values of AQz from Table 11, and values of AQl calculated from the observed intensities of the 1; origins by the use of the harmonic oscillator formulas of Coon et al.16 We find the change in C-0 bond length to be 0.121 A in both isotopic molecules.
The experimentalz5change is 0.1174 A. The difference of only 3% represents good agreement. (The observed change in the C-H bond length is only -0.02 A. The changes calculated from the normal coordinates are -0.04 A for H2C0 and +0.03 A for D2C0, neither as consistent nor in as good agreement with experiment as the C-0 changes.) Although we believe that anharmonicity is the correct explanation for the failure of the simple harmonic oscillator treatment, a comparable fit can be obtained by using FC factors calculated by the method of Doktorov et al.lpz0 For reasons indicated earlier, we do not believe that a rotation of normal coordinates, for which the Doktorov formula is designed, is actually important in formaldehyde. The question then arises, why does the method of Doktorov et al. work so well? In answer to this question, we note that it uses a two-parameter harmonic oscillator formula. When applied to the specific case of a single progression starting from u = 0, it appears to be mathematically equivalent to treating a single harmonic mode and allowing both the displacement and the ratio of vibration frequencies in the two states to be adjustable parameters. However, the ratio of observed frequencies obtained is not physically meaningful. If we use the observed ground-state frequency of about 1746 cm-' in H2C0, the Doktorov formula is equivalent to using a higher excited-state frequency of about 2440 cm-l instead of the observed 1190 cm-'. Alternatively, accepting the excited-state frequency, the ground-state frequency would have to be about 850 cm-', lower than any actual frequency in the ground state. In fact, a transformation of coordinates in which Q2 was rotated in the direction of a mode with a lower frequency would give FC factors corresponding to a lower effective ground-state frequency, but it could not become lower than the frequency of the other mode. Figure 2 is designed to illustrate why the Doktorov method works so well. The lower solid curve is a fragment of a potential curve around the u = 0 level in the ground state of H2C0. (Anharmonicity has little effect this close to the bottom of the well.) The upper solid curve is the Morse function for the excited state, and the horizontal lines indicate the energies of the fist few vibrational levels. The dashed curve is a harmonic potential curve for the excited state with displacement and frequency which give FC factors the same as the Doktorov method. The spacings between the vibrational levels indicated by the hor-
'Ap
+
The Journal of Physical Chemistry, Vol. 86, No. 4, 1982 453
'A, Absorption Spectrum of Formaldehyde
TABLE V: Calculated and Observed Spectrum
TABLE VI: Calculated and Observed Spectrum of Formaldehyde-d
of Formaldehyde
wavelength region, nm 363.4 357.8 351.5 342.4 337.5 336.1 332.6 328.4 324.2 322.8 320.2 316.5 313.7 31 1.7 308.1 302.5 300.4 297.6 292.7 290.6 287.8 282.9 278.0 274.5 270.3 266.1
236.7 258*4
1
1 0 Z l ed In a, L mol-' cm-I
--
assumed vibronic bandsa
calcd obsdb
A0 0.03 A l , BO, CO 4.2 DO, EO, FO, GO 3.5 A2, B1, C1 9.6 HO 0.3 0.1 D1, IO 8.3 E l , F1,G1, JO A3, B2, C2 12.0 H1 0.6 3.3 D2, 11, J1, KO, LO, MO E2, F2, G 2 10.1 A4, B3, C3, NO 12.0 H2, PO, QO 1.5 D3, 12, K1, L l 6.1 A5, B4, C4, E3, F3, G3, 52, M1, 20.4 N1, SO H3, P1, Q1 2.3 D4, 13, K2, L2 7.5 A6, B5, C5, E4, F4,G4,53,M2, 15.7 N2, S1, TO H4, P2, Q2 2.5 D5, 14, K3, L3, UO 6.7 A7, B6, C6, E5, F5, G5, 54, M3, 10.6 N3, S2, T1 D6, H5,15,55, K4, L4,M4, P3, 7.3 Q3, U1 A8, B7, C7, E 6 , F 6 , G 6 , N4,S3, 6.1 T2 D7, H6,16, 56, K5, L5,P4, Q4, 4.7 u2 A9, B8, C8, E7, F 7 , G 7 , H7, M5, 3.5 N5, S4, T3 A10, B9, C9, D8, E8, F8, G8, 4.4 H8, 17, 57, K6, L6, M6, N6, P5, Q5, 55, T4,U3 most remaining bands
3.9
-0.03 4.0 3.2 9.4 0.2 0.4 8.0 12.4 0.4 2.9 10.3 11.5 1.7 6.8 20.9 2.4 7.3 15.3 2.6 6.8 11.2 7.4 5.8 4.5 3.6 4.5
3.7
The letter designates the origin from Table IV; the number gives the vibrational quantum number in the u 2 progression. These are the band heads calculated to lie within the wavelength region. The values have an uncertainty of about 0.2.
I
I I
1
/
W
I
Q, Figure 2. Morse and harmonic oscillator potential curves drawn to scale to Illustrate why the Doktorov et al. method works. See text for details.
izontal dashed lines bear no resemblance to the actual spacings in the u2 progressions. However, as illustrated elegantly by Anderson, Crosley, and Men,%FC fadora are
-
102je d In J,
wavelength region, nm 361.3 358.5 351.5 348.0 342.4 334.7 330.5 3 24.9 321.4 317.9 313.0 310.2 307.4 302.5 299.0 296.9 292.7 289.9 287.1 283.6 280.8 2'78.7 274.5 272.4 270.3 266.8 264.0 262.6 259.1 236.7
L mol-I cm-' assumed vibronic bandsa
A0 BO, CO A l , DO EO, FO, GO A2, Bl; C1, D1, HO, IO, JO E l , F1, G 1 A3, B2, C2, H1, KO, LO, MO D2,11, J1, NO, QO E2, F2, G 2 A4, B3, C3, H2, 12, K1, L1, M1. RO D3, J2, N1, Q1 E3, F3, G 3 A5,B4, C4, H3, K2, L2,M2, R1, TO, UO D4, 13, 53, N2, Q2 E4, F4, G4 A6, B5, C5, H4, K3, L3,M3, R2. T 1 D5, 14, 54, N3, Q3, U1 E5,F5,G5 B6, C6, H5, K4, L4, M4, R3, T2 A7, D6,15, 55, N4, Q4, U2 E6, F6, G6 B7, C7, H6, K5, L 5 , M 5 , R 4 ,
calcd
obsdb
0.08 -0.09 2.5 2.8 0.4 0.2 3.0 3.0 7.5 6.9 6.7 6.3 10.0 10.5 3.1 2.8 8.0 8.3 11.4 11.8 4.8 7.6 10.5
4.6 7.6 11.2
5.4 5.7 8.2
6.3 5.1 8.1
4.9 3.8 5.7
5.1 4.2 5.5
3.7 2.2 3.6
3.3 2.5 3.7
A 8 , b 7 , 16, J6, N5, Q5, U3 2.4 E7, F7, G7 1.2 A9, B8, C8, K6, L6, M6, R5, T 4 2.1 D8, H7, N6, Q6, U4 1.4 E8, F8, G8, 17, 57 0.6 A10, B9, C9, K7, L7, M7, R6, 1.2 T5 most remaining bands 3.2
2.0 1.5 1.9 1.5 0.5 1.0
T.?
3.1
a The letter designates the origin from Table IV; the number gives the vibrational quantum number in the u 2 progression. These are the band heads calculated to lie within the wavelength region. The values have an uncertainty of about 0.2.
largely determined by the overlap with the ground-state wave functions of the innermost maximum of the higher vibrational wave function, and this occurs near the classical turning point of the vibration. The vertical lines in Figure 2 indicate that the classical turning points of the vibrational levels of this harmonic oscillator (which gives the same FC factors as the Doktorov formula) agree closely with the classical turning points of the Morse levels. It is this agreement which makes the calculated FC factors very similar. In fact, if the only aim is to get FC factors to fit the spectrum, the Doktorov formula can be used just as well in this case as the Morse calculations, even though the former does not correspond to a true picture of the vibrations of this molecule. Vibronic Intensities. The intensities of vibronic origins given in Table IV make it possible to divide up the total intensity of the electronic transition into fractions induced by each antisymmetric vibration. Table VI11 lists the results for H,CO, including the small contributions of the magnetic dipole and doubly forbidden bands. The values for DzCO are similar. For comparison we have listed the values derived from the calculations of van Dijk et al.9 The (26) W. R. Anderson, D. R. Crosley, and J. E. Allen, Jr., J. Chem. Phys., 71, 821 (1979).
454
The Journal of Physical chemistry, Vol. 86, No. 4, 1982
TABLE VII: Calculated and Observed Spectrum of Formaldehyde-d
1 0 Z l e d In V , L mol-' cm-'
wavelength region, nm 368.3 356.4 350.8 347.3 342.4 337.5 334.0 3 29.1 324.2 320.0 317.2 313.0 311.6 306.0 302.5 3 00.4 295.5 292.7 290.6 286.4 283.6 280.8 277.3 274.5 269.6 266.1 264.7 26 2.6 259.1 256.3 252.1 245.1 240.9 225.5
assumed vibronic bands'
calcd obsdb
0.06 -0.2 3.9 3.8 co 0.2 0.4 3.2 DO, EO 3.4 8.1 8.7 B1, FO A2, C1, HO, GO 0.9 0.7 D1, E l 7.9 7.9 A3, B2, F1, IO, KO 12.2 12.0 C2, G1, H1, 50, LO 3.0 3.9 D2, E2 9.9 8.8 A4, B3, F2, 11, K1, MO 12.8 12.4 L1, NO 0.9 1.1 14.4 14.5 C3, D3, E3, G2, H2, J1 A5, B4, F3,12, K2, M1 11.3 11.3 1.7 2.0 L2, N1 C4, D4, E4, G3, H3, 52 13.0 12.5 7.8 8.6 A6, B5, F4, 13, K3, M2 2.1 L3, N2 1.9 9.0 9.9 C5, D5, E5, G4, H4, J3 6.3 5.9 A7, B6, F5, 14, K4, M3 2.2 C6, G5, H5, L4, N3 2.2 D6, E6, 54 6.1 5.9 3.5 A8, B7, F6, 15, K5, M4 3.6 4.3 C7, D7, E7, G6, H6, 55, L5, N4 5.2 2.8 A9, B8, F7, 16, K6, M5 2.0 0.6 C8, G7, H7, L6 0.3 D8, E8, J6, N 5 2.7 1.3 2.2 A10, B9, C9, F8, 17, K7, M6 1.1 G8, H8, L7, N6 0.5 0.8 1.8 A l l , B10. D9. E9. F9. 18. K8, 1.6 577 M7 1.1 1.6 0.5 0.8 0.4 2.2
A0 A l , BO
a The letter designates the origin from Table IV; the number gives the vibrational quantum number in the u 2 progression. These are the band heads calculated to lie The values have an within the wavelength region. uncertainty of about 0.2.
*
TABLE VIII: Fraction of the Absorption Intensity Induced by Various Vibrations
-vi bration
Q, QE Q, QdQ,, Q,Q, magnetic dipole
obsd 66 26 6 1 1
H,CO, % van Dijk et al.
obsd D,CO, %
66 21 13
66 27 4 1 2
agreement is quite good. It illustrates the power of ab initio SCF-CI calculations to predict at least relative intensities. We find the total oscillator strength of the transition in H2C0 to be (2.40 f 0.05) X We estimate the total oscillator strength predicted by the calculation of van Dijk et al. to be 2.1 x slightly low, but also in remarkable agreement with the experiment, so their method gives good absolute intensity as well as relative vibronic intensities. The experimental oscillator strength of D2C0 is (1.86 f 0.05) X lo4. This is lower than that of HzCO by a factor of 0.78 f 0.02. This ratio can be compared to that estimated by Herzberg-Teller theory. We make the assumption that for Q4, Q5,and Q6 it is sufficient to consider only the first derivatives of the transition moment with
Strickler and Barnhart
respect to each coordinate. Then the total intensity induced by each mode is proportional to the mean square displacement along that mode. If we also assume that the actual form of the distortion for each normal mode is unaffected by isotopic substitution, it can be shown that, for a harmonic oscillator, this mean square distortion decreases by the same factor as the frequency. The magnetic-dipole-allowed component should be unaffected by isotopic substitution, and the doubly forbidden bands changed by the product of the frequency ratios of the two vibrations. Using the harmonic frequencies of Duncan and Mallins~n,~' we estimate that the ratio of intensities of D2C0 and HzCO should be 0.788, in good agreement with the observation. This shows that the first-order Herzberg-Teller theory is adequate for this symmetry-forbidden transition. We now propose a new and useful way to view the effect of distortions when two or more vibrations can induce a transition moment in the same direction, as is the case for Q5 and Q6 of formaldehyde. Since for small-amplitude vibrations we need only consider a linear variation of M, with the orthogonal normal coordinates, there must be a unique direction in normal-coordinatespace which induces all of the x-polarized intensity, while small displacements perpendicular to that unique direction contribute nothing. We will call the unique direction Q,. The effectiveness of Q5 and Q6 in inducing intensity then depends only on the component of Q, in those directions. To determine the direction Q, we require the ratio of the derivatives (a M/aQ& and (a M/d&6)0.Each derivative is related to the intensity of the corresponding vibronic bands by
1-
8a3N dM S e d l n D = 3000hc In 10 dQi
'Q?
(8)
where the integral is over the bands induced by vibration i, N is Avogardo's number, and 8;2 = h/(8?r2cwi)for the u = 0 level of a harmonic oscillator. From the experimental ratio of intensities induced by Q5 and Q6, 0.2610.06, we obtain the absolute value of the ratio of derivatives
(aM/aQsl/ld M/dQ6l = [4.3(~5/(~6)]'/~ = 0.95/0.30 This corresponds to a rotation of about 18' between Q, and Q5. However, the direction of rotation is undetermined by the experiment, since only the absolute value of the ratio is determined. To find the direction of rotation, we assume that the relative sign of the derivatives is given correctly by the theoretical calculations of van Dijk et aL9 We have done a normal-coordinate analysis using the general harmonic force field of Duncan and Mallin~on.'~ (Properly, we should use normal coordinates of the excited state, but for simplicity we use those of the ground state which can be calculated more accurately.) It should be noted that in a normal-coordinate analysis it is always arbitrary which direction of nuclear displacement is taken to be a positive Qi; we chose the phases in accord with the schematic drawings in Figure 1of van Dijk et al. With that choice, we use the results from their Figure 2 indicating that a M/aQ5 and d have opposite sign, meaning that to go from positive Q5 toward Q, requires a rotation away from positive Q6. Taking the appropriate linear combination, 0.95Q5- 0.30Q6, of our normal coordinates, we can determine the form of the displacement Q,. Of course, we can also determine the form of the orthogonal displacement in the two-dimensional subspace of the b2 (27) J. L. Duncan and P. D. Mallinson, Chem. Phys. Lett., 23, 597 (1973).
455
J. Phys. Chem. 1982, 86, 455-459
P
1
Flgure 3. Normal coordinates Q 5 and Q e ,and the displacement di, and Om of b, symmetry. For small amplitudes, x-porections 0 , , ?C larized intensity is induced only by displacement in the direction while displacement in the direction of C?& induces no intensity In the 'A, 'A, transition. +
vibrations which induces no intensity. We call that Qdn, and it is given by -0.30Q5 - 0.95Q6. Figure 3 shows the actual nuclear displacements of Q, and Qmin in H2C0 with the forms of Q5 and Q6 for comparison. Because the hydrogen atoms are so light, they do mwt of the moving in each case. It is interesting to note that, to induce x-polarized intensity, it is most favorable to displace the hydrogen atoms in a direction nearly perpendicular to the C-0 bond. Displacement of the hydrogen atoms parallel to the C-0 bond has essentially no effect on the transition moment. It is tempting to speculate qualitatively about the effects of displacements along Q, and Qminon the electronic wave function and the resulting effect on the transition moment. However, it would be more to the point to examine the SCF wave functions under such displacements, and these are presently unavailable to us. We have emphasized the concept of Q, here because we believe it will prove to be very useful in more complicated molecules
where there may be many more vibrations of the proper symmetry to induce intensity.6 In such cases a considerrather than the ation of the unique displacement Q, many normal-coordinate directions could be very helpful in gaining an understanding of the vibronic intensity. A few other points may be noted. Comparing D2C0with H2C0,we find that Q6 seems to induce a smaller fraction of the intensity in the former, while Q5 induces a bit more. A normal-coordinate analysis of DzCO shows that Q6 involves a displacement of the hydrogen atoms which is more nearly parallel to the C-O bond than is the case in H&O, Le., Qs is closer to being along Qmin,and this presumably explains the lower intensity of the Q6 bands. We also note in Table VI11 that the calculations of van Dijk et a1.9 somewhat overestimate the intensity induced by Q6 and underestimate that induced by Q5,but that the s u m of the two is very close to our experimental value. It seems possible that the difference lies in the form of the normal coordinates used.
Conclusions In this paper we have discussed the vibrational structure of the 'A2 'A, electronic transition of formaldehyde and its deuterated derivatives. We have shown that it is possible to fit that vibrational structure well by using Herzberg-Teller theory, including Franck-Condon factors which take account of anharmonicity in the C-O stretching vibration. The vibronic effects have been treated in considerable detail, including intensity induced by different vibrations, isotope effects, and the concept of a unique displacement Q, which is responsible for all of the intensity induced with a given polarization. The resulting picture is self-consistent and in good agreement with high-quality calculations.
-
Acknowledgment. We thank the National Science Foundation for the support of this work. We also thank Drs. V. T. Jones and J. B. Coon for sending us their vibrational wave functions. R.J.B. was the recipient of a fellowship from the Graduate School of the University of Colorado during part of this work.
Contribution of Second and Subsequent Water Shells to the Potential Energy of Guest-Host Interactions in Clathrate Hydrates V. T. John Department of Chemical Engineering, Columbia University in the City of New York, New York, New York 10027
and 0. D. Holder' Chemical and Petroleum Engineering Department, UniversRy of Pittsburgh, Pittsburgh, Pennsylvania 1526 1 (Received: May 22, 198 1; In Final Form: September 1, 1981)
Dispersion interactions between a guest molecule and the second and third neighbor water molecules have been calculated using quasi-sphericalKihara potentials. The magnitudes of these secondary interactions are significant and can influence hydrate equilibrium predictions to a large extent, changing the calculated Langmuir constants (for a given set of Kihara parameters) by several orders of magnitude.
Introduction In predicting hydrate equilibrium, the statistical mec h a n i d theory advanced by van der Waals and Platteeuw' (1) J. W.van der Waals and J. C.Plath (1959).
is currently used. A final expression resulting from the theory is pB- P H =
k T C vi In (1 + Ccijfj) i
N,
Adu. Phys. Chem., 2 , l
0022-3654/82/2086-0455$01.25/0
I
(1)
where p@- pH is the difference in chemical potential be0 1982 American Chemical Society
