J. Phys. Chem. 1994,98, 4551-4563
4551
A Microwave Study and Centrifugal Distortion Analysis of the Pyrrole-CO Complex Ryan P. A. Bettens, Sonja R. Huber, and A. Bauder' Luboratorium f i r Physikalische Chemie, Eidgenossische Technische Hochschule, CH-8092 Zurich, Switzerland Received: December 9, 1993; In Final Form: February 14, 1994'
The rotational spectrum of the pyrrole...CO complex has been measured between 7.8 and 18.1 G H z with a pulsed nozzle Fourier transform microwave spectrometer. In addition to the parent species, seven I3C, 15N, and '80isotopic variants have been analyzed. Rotational constants, quartic centrifugal distortion constants, and, where necessary, quadrupole coupling constants have been fitted to the measured transition frequencies. The determination of ro and r, structures of the complex has been affected seriously by the van der Waals vibrations. The structure is compatible with CO lying above the pyrrole ring in the symmetry plane of pyrrole. In order to correct for the effects of the van der Waals vibrations, an anharmonic potential function for the five van der Waals modes has been adjusted from the centrifugal distortion constants and rotational constants. The frequencies of the van der Waals modes and the equilibrium dissociation energy of the complex have been estimated from the potential. An improved structure followed from the rotational constants corrected for the low-frequency zero-point vibrational contributions. However, a second possible orientation of CO above the pyrrole plane could not be ruled out.
1. Introduction We report here the first high-resolution study of a complex between CO and an aromatic molecule, pyrrole. An understanding of the binding of CO to pyrrole is of interest for a number of reasons. First, one would like to determine whether CO binds preferentially to the hydrogen attached to the nitrogen of pyrrole, thus forming a H-bond, or whether the preferred configuration is a van der Waals bonded CO, or both. Considering the counterintuitive, small, and opposite-directed dipole of COI, it would also be interesting to determine which atom is closest to pyrrole. Second, pyrrole is the basic building block of porphyrins which are found in biologically significant molecules, e.g., chlorophyll and hemoglobin.2 When one considers that the binding of 0 2 in hemoglobin is not well understood and that CO is unlikely to bind in the same way as 02, then the work presented here may furnish some enlightening insights into the nature of possible interactions between CO and hemoglobin. There have been a number of high-resolution infrared and microwave studies of complexes containing CO. These complexes are CO--HF,3*4HCl,4g5HBr,4,6HI,7 HCN,* H20,9NH3,l0C12,11 HCCH,l2 CO2,13 and the trimer CO-HCN-HF.14 Most recently, CO-.Ar has also been measured.15 As for pyrrole, there has been, to our knowledge, only one complex studied under high resolution and that was the complex with Ar.I6 Apart from the "trimer", all of the above complexes have had the stretching force constant estimated from the centrifugal distortion constant, DJ or AJ. No attempt has been made in the above works to fit the obtained centrifugaldistortion constantsto a force field. However, several centrifugal distortion analyses have been performed in other complexes (see, for instance, Xu et a1.17 and Kukolich and Pauleyls), but to our knowledge these analyses have considered only the centrifugal distortion constants in the fitting procedure. In this work we have simultaneously fitted theobserved A-reduced rotational and quartic centrifugal distortion constants from a number of isotopically substituted species to a potential function. The function used for the radial part of the potential is valid for the full range of R while the angular part of the potential is only accurate about the minimum. In section 2 of this paper we give the experimental details for the measurement of the microwave spectrum of the complex. Next (section 3) we discuss the assignment of the spectrum, and *Abstract published in Advance ACS Abstracts, April 1, 1994.
0022-3654/94/2098-455 lW4.50 10
then we determine the ro and r, structures (section 4) from the observed rotational constants. We discuss the difficulties encountered in these structural determinations. We then present the evidence for substantial vibrational effects incorporated within the observed rotational constants and go on to give full details of our normal mode/centrifugal distortion analysis (section 5 ) . In section 6 the results of our analysis are given and discussed, and finally, in section 7, we present our conclusions.
2. Experimental Section 2.1. Chemicals. The chemicals used in the microwave measurements were obtained from the following sources. Normal pyrrole was purchased from Aldrich and further purified by pumping off the low boiling point impurities. IzC160,purity 4.7, was obtained from Union Carbide. The isotopically enriched CO, which contained 99% and approximately 10% l8O,was obtained from Cambridge Isotope Laboratories as was pyrrole15N. The enriched pyrrole was further purified by preparative gas chromatography. 2.2. Spectrometersand Experimental Conditions. Rotational spectra were recorded with a pulsed nozzle beam, Fabry-Perot cavity Fourier transform microwave (FTMW) spectrometer. The design of this spectrometer is similar to the instrument of Balle and Flygare.19 Only a short description of this spectrometer is given here as details have been reported elsewhere.20 Theconfocal Fabry-Perot cavity was formed of two mirrors with 40-cm diameters. The microwave energy was coupled through a circular irisintoandoutofthecavity. Gaspulsesofless than 1-msduration crossed the cavity perpendicular to its axis at a repetition rate of approximately 2 Hz through a nozzle of 500-pm diameter. This was achieved using an electromechanical valve (General Valve Corp., series 9). The gas mixture contained approximately 0.5% pyrrole and 4% CO in He. The optimal backing pressure was around 2.5 atm. Molecules from the jet plume in the beam waist of the cavity were polarized with microwave pulses of 1-ps duration and an optimal power of 1 mW. After the energy stored in the cavity had decayed sufficiently (i.e.,after 3-20 ps, depending on conditions), the molecular emission signal was amplified and first converted down to 30 MHz. It was then mixed with a coherently derived 28.75-MHz signal down to the G2.5-MHz frequency band. This signal was sampled at a rate of 5 MHz with an 8-bit AID converter and, in most cases, for 512 channels. In a few instances, in order to observe significantly weaker transitions at 0 1994 American Chemical Society
4552 The Journal of Physical Chemistry, Vol. 98, No. 17, 1994
9271.5
9272.5
9273.5
8274.5
FrequencylMHz
Figure 1. Observed hyperfine multiplet for the 21,rlo.l transition of pyrrole-CO. This figure is a composite of five separate scans each made with 50 gas pulses at eight microwave pulses per gas pulse. The vertical lines indicate the places where each scan was joined. The relative amplitudes of these hyperfine components are on the same scale. Identificationof this pattern to the above transition led to the assignment of the spectrum.
adequate signal-to-noise ratios (Le., high J, high K,, for the 14N isotopes and some transitions of the 13Cspecies measured in natural abundance) under practical conditions, the number of channels was reduced to 256. Up to eight microwave pulses sampled the molecules in a single gas pulse. From 25 gas pulses for the strongest transitions up to 1600 gas pulses for the weakest, that is, the data from a total of 200-12 800 individual microwave pulses, were added and Fourier-transformed to produce the power spectrum. The resulting line widths were of the order of 50 kHz. We found that we were able to reproduce the center frequencies under these conditions for the strongest transitions to within 3 kHz. It was found that when one microwave pulse per gas was used, the observed line widths could not be reduced below 30-40 kHz. Under these conditions no Doppler doublets were observed for transitions due to the complex whereas transitions due to the pyrrole monomer were observed to posses a Doppler splitting. No further structure was resolved with a second spectrometer having higher resolving power. 3. Assignment 3.1. Parent Species. Initially, the frequency regions 9.0959.700 and 10.774-10.922 GHz were scanned in an attempt to find lines due to the hydrogen-bonded complex of pyrrole*-CO. Within the former region, among other transitions, a quintet was observed (Figure 1). It was soon realized that the spacing and relative intensities of this distinctive multiplet matched that of the transition 21.2-10,l for a complex where C O lies above the pyrrole plane, assuming the quadrupole coupling constants of the pyrrole monomer. With this tentative assignment, double resonance wasused to locate the transition 32.1-212. Theexpected triplet was quickly found to occur a t 18.021 GHz and matched the predicted quadrupole splittings. This transition is the upper frequency member of a K-doublet. The other member, also a triplet, was located at 17.898 GHz. Using the above assigned multiplets, an initial set of rotational constants was determined. These constants were then used to predict further low-J, rb-type R-branch transitions (bR). These transitions were then included in the fit, and the constants thus obtained were used to further predict higher J , rb-type R-branch transitions, rb-type Q-branch transitions and pa-type R-branch transitions (aR). Tables with listings of all measured transition frequencies are available as supplementary material. The rotational constants, centrifugal distortion, and quadrupole coupling constants were simultaneously fitted to the observed transitions. The final set of fitted parameters is given in Table 1 along with their associated uncertainties. Included in the fit were 97 hyperfine components made up of 12 bR,10 bQ, and 15 *R rotational transitions. It was N
-
Bettens et al.
Pyrrole-CO Complex
+ ;......
"
Jp., *...
The Journal of Physical Chemistry, Vol. 98, No. 17, 1994 4553
"-
...............................
TABLE 2 Fitted Structural Parameters for the r, and Modified ro Structures I and 11 structure I structure II uarameter mod ro r. mod ro r. L/Pm
elk3
scale rcolpm ufita/amu pm2
349.86(9) -2.6(5) 21.7(9) 0.9966(4) 109.0(9) 540
348.27(9) -3.4(1) 21.3(2) 109.9(5) 110
349.7(1) -3.9(7) 150(l) 0.9966(6) llO(1) 730
348.1(4) -4.7(5) 147(1) 109(2) 480
Unit weight was given to all quantities fitted. Figure 2. Definition of the internal coordinates &, 6, and p in the pyrrole.40 complex. Also indicated (upper left) are the principal inertial axes of the complex.
possible, due to the variety of transitions measured, to determine all the quartic centrifugal distortion constants. However, it will be noticed by examining Table 1 that the least determined parameter is 6 ~ .This parameter has an uncertainty of w 1 / 3 of its value. From the F-test it was concluded that the fit was indeed improved at the 95% confidence level by the inclusion of 8 ~ . 3.2. Isotopic Species. Seven other isotopic species were measured, and the rotational transitions assigned. Those species included 13C in natural abundance substituted in the a and j3 positions of pyrrole (pyrrole-2-13C-CO and pyrrole-313C-.CO), pyrroleJ3C0, pyrrole-J3CI*O, pyrr~le-~~N-.CO, pyrrole-ISNJ3C0, and p y r r ~ l e - ~ ~ N . . J ~ CFor ~ ~ Othe . three species, pyrrole.-I3CO, pyrr~le-~~N.-CO,and pyrroleISN..J3CO, we were able to obtain all the quartic centrifugal distortion constants. For the other four species 8~ was constrained at the parent species value of -3.6 kHz. Tables with listings of all measured transition frequencies are available as supplementary material. The fitted rotational, centrifugal distortion, and quadrupole coupling constants as well as the standard deviation of the fits are given in Table 1.
TABLE 3 Observed and Differences with the Calculated Planar Moments of Inertia between the Parent Species and the Isotopically Substituted Species for Structures I and 11 diff in ob-talc/ ob-calc/ planar obs/amu amu AZ, amu A2, species momenta A2 struct I struct II 1.0950 0.0885 1.2330 0.8494 1.0814 0.4279 4.6999 0.4023 -0.0026 18.4188 0.6466 -0.0043 1.2687 1.1471 -0.0129 6.0020 1.5523 -0.0153 19.8756 1.7864 -0.0165
-0.0228 -0.0159 0.0106 -0.0189 -0.0028 -0.0028 0.0029 -0.0060 -0.0026 0.0009 0.0029 -0,0043 -0.0049 0.0026 -0.0129 -0.0013 -0.0049 -0,0153 0.0013 0.0025 -0.0165
-0.0198 -0.0394 0.0159 -0.0272 0.0541 -0.0020 0.0160 0.1270 -0.0026 -0.0157 -0.0051 4,0043 0.0060 -0.0619 -0.0129 -0.01 16 0.0717 -0.0153 0.0161 -0.0820 -0.0165
4. Structures of the Complex
4.1. Least-Squares Fit of the Structure. It was obvious from geometric models for possible structures of the pyrrole-CO complex that the observed rotational constants and quadrupole splittings were only compatible with C O located above the plane of pyrrole. The location of CO with respect to pyrrole requires, in general, five structural parameters corresponding to the degrees of freedom lost when the complex is formed. When Kraitchman's equations21 were applied to this complex, using pyrrole.J3CO as the parent, we obtained imaginary c coordinates for carbon and oxygen of CO and nitrogen. This proves that CO lies in the symmetry plane of pyrrole. Thus the structural parameters for the position of CO are reduced to R,, the distance between the center-of-mass of the monomers, 8, the angle between & and the c'principal axis of the pyrrole monomer, and p, the tilt angle of CO with respect to the inertial plane a%' of the pyrrole monomer. The parameters are defined in Figure 2. They always refer to the parent species of the complex (pyrrole.-12C160). A simple least-squares fit of the ro structure, Le., R,,, 8, and p, assuming no change in the structure of the monomers while the complex was formed, did not give satisfactory results including all isotopic species. The bond length, rco, was held fixed at its rovalue determined from its rotational constant.22 The structural parameters of pyrrole were constrained to the rovalues determined by us from directly fitting the observed rotational constants.23 The failure of the structure determination of the complex was attributed to large amplitude vibrations between the monomers. In an attempt to consider theeffects of largeamplitude motions, an adjustable scale factor was introduced by which the Cartesian coordinates of the nuclei in pyrrole were multiplied. In addition, the effective bond length, rco, was included as an adjustable
parameter. The structural parameters R,, 8, p, rco, and the scale factor were fitted to the differences between the planar moments of inertia, Pa = I/2(-Za + Zp + ZJ, a,8, y = x, y, z, and cyclic permutations (Zx, Z,, Zz are the principal moments of inertia of the complex), of the isotopically substituted species and the parent species as well as the moments of inertia of the parent species. Thus the ro structure reported here is not determined in the usual manner and is termed a modified ro structure in this work. It was possible to fit two different structures of the pyrrole...CO complex compatible with the observations. The residual errors of the two structures do not differ much, as can be seen from the results of the fits in Table 2. Both structures have CO lying above the pyrrole plane with C O oriented along the symmetry plane of pyrrole. CO is tilted with the carbon end toward the nitrogen of pyrrole in structure I. Structure 11had the carbon tilted to the opposite end of pyrrole. In both structures, carbon of CO lies closer to the pyrrole ring than oxygen. As an alternative, an r, structure was determined in the same manner as the modified ro structure except that the moments of inertia of the parent were not included in the fit. The structural parameters &, 8, p, and rco were adjusted whereas the structural parameters of pyrrole were held fixed at the previously determined r, values of the monomer.23 The results are listed in Table 2. Again, two structures similar to the ro structures I and I1 were obtained. Structure1 withcarbonof CO tilted toward the nitrogen of pyrrole exhibited smaller residual deviations between differences of observed and calculated planar moments of inertia of all isotopic species than structure 11, as demonstrated in Table 3. 4.2. Discussion of the ro and r, Structures. It is clear upon examination of Table 2 that the r, structures lead to smaller
4554
The Journal of Physical Chemistry, Vol. 98, No. 17, 1994
residual errors than the modified ro structures despite the adjustment of only four structural parameters. There is substantial cancellation of vibrational effects for the r, structures. It is not possible on the basis of the rather small variations of the residual errors for the two structures to eliminate structure I1 from further consideration. However, there is no doubt that the observed rotational constants and quadrupole coupling constants are fully consistent with a complex where C O lies in the symmetry plane of pyrrole and is positioned above the pyrrole ring with carbon of C O lying closer to the ring than oxygen. The origin of the failure to determine an unambiguous structure of the complex is clear. Because of the large separation between C O and pyrrole, the a axis of the complex passes nearly through the center of mass of CO. This can be verified from eqs 16h, 18a, 18b, and 18d (4 = e), which are given later. Thus, the b substitution coordinates of C and 0 in C O must be opposite in sign, otherwise an unrealistic rco distance is obtained, but whether the b coordinate of C is positive or negative, i.e., the relative orientation of CO, cannot be easily established. Intuitively, with the knowledge that the negatively charged end of CO is centered a t carbon' and that the positively charged side of pyrrole is a t the nitrogen,*3 structure I would be expected on the basis of permanent dipole-dipole interaction. Furthermore, an induced dipole in CO may be estimated from the results of the dipole moment measurements of the pyrrole-Ar complex.I6 If we assume that the back-polarization of pyrrole is negligible, then the induced dipole in Ar is calculated to be approximately 0.07 D parallel to the C2 axis of pyrrole. C O has a 20% higher polarizability than Ar, so the size of the induced dipole moment in CO parallel to the pyrrole ring is near 0.1 D. This value adds roughly to the permanent dipole ofO.1 D for structure[. However, it would nearly cancel the permanent dipole of CO for structure II,rendering the dipole-dipole electrostatic interaction small and making the formation of a stable complex less likely. The fitted bond length rco for both structures is considerably smaller than the equilibrium rco in the monomer CO, for which a value of 112.82 pm was established from the Be rotational constant.24 This fact was observed for the r, and modified ro structures. Typically, rq is smaller than ro for diatomic molecules with r, (req+ r0)/2. The short value for rco obtained in the fits for the structure of the pyrrole..CO complex is a clear indication of large amplitude vibrations. If a torsional motion of C O around its center-of-mass either in the symmetry plane of pyrrole or perpendicular to it is considered, then an effective bond length of C O = cos ( T ) is expected where (7)is the vibrationally averaged torsional angle. Using ro = 113.09 pm, determined from BO of CO?* an average angle of ( T ) = 15.5O was obtained. A trend for the R , distance of the r, and modified ro structures similar to that for diatomic molecules is observed. For the r, structures I and 11,R,is 1.6 pm shorter than in the corresponding modified ro structures. The bending vibrations of C O above the pyrrole plane as represented by its center-of-mass motion also affect the structural determination. Similar effects were noticed in the pyridine-Ar complex by Klots et al.25 These authors introduced vibrationally averaged bending angles to compensate for the deviations of the planar moment of inertia Pc from the corresponding value of the pyridine monomer. The scale factor introduced for the fit of the modified ro structure may be interpreted as a vibrationally averaged bending angle. Indeed, if one takes the arc cosine of the scale factor as a measure of the rms displacement of the pyrrole monomer out of the a'b'inertial plane (cf. Figure 2), then a value of approximately 5O is obtained for both structures I and II. Similar values were observed for several aromatic molecule-Ar complexes.26 For the p y r r o l e 4 0 complexes which possess C, symmetry, the planar moment of inertia Pc of a complex should be equal to
-
$3
Bettens et al.
TABLE 4 Differences between the Planar Moments of Inertia Poin the Comslex and R in Pvrrole species @lamu A2 parent -0.4128 pyrrole- W O -0.4154 pyrrole... W s O -0.4171 pyrrole-15N-.CO -0.4228 pyrrole-ISN-J3CO -0.4252 ~~
pyrrole-l5N-J3ClsO
-0.4264
pyrrole parent pyrroleJ5N 0 AP = P~""PlCx)- Pimu. pyrrole-I5Nfrom ref 23.
PJamu
A2
-0.0083 -0.0083
Rotational constants of pyrrole and
Pb' of the corresponding pyrrole monomer. Only the same nuclei in the complex or in the monomer contribute to this planar moment of inertia. It should be noted that the complex and the pyrrole monomer share a common inertial plane, but the inertial axes are labeled differently. The differences Pc - PV for the appropriate complexes are listed in Table 4. It is obvious that they are not approaching zero, indicating again therms displacement of pyrrole out of the a%' inertial plane. 4.3. Evidence for Vibrational Effects. The first evidence for substantial vibrational motions came from the centrifugal distortion constants of the pyrrole..CO complex. When the magnitudes of the fitted constants were compared to those of the pyrrole monomer, the centrifugal distortion constants of the complex were significantly larger. One would expect the distortion constants of a "rigid" molecule to scale approximately with the magnitude of the rotational constants. For the complex, with much smaller rotational constants than the pyrrole monomer, the distortion constants thus should be smaller. The shortening of the bond length rco in the fitted structures of the complex compared to the CO monomer indicated large amplitude torsional motions of CO. The scale factor necessary for a satisfactory fit of the ro structure pointed to large amplitude bending vibrations. In an attempt to better quantify the vibrational effects, we adopted a new approach to model the intermolecular dynamics of a complex between a planar molecule and a diatomic molecule. In this model we considered the fivevan der Waals modes. They should be responsible almost exclusively for the effects observed and discussed above. 5. Intermolecular Vibrations In the treatment which follows we consider only the five intermolecular vibrations and the three equilibrium structural parameters for the complex formed between C O and pyrrole. Without any prior knowledge of the intermolecular potential function, we were forced to assume its particular form. The potential and a full description of the procedure adopted to fit the observed spectroscopic constants are given below. 5.1. The Potential. The assumed potential had the following form:
v=
.[
l - ( y ] ' + 4
AS,ASi
+ 2 Ci k m , AR ASi
1
(1)
where
and Si refers only to the angular internal coordinates 8, p , x, and
The Journal of Physical Chemistry, Vol. 98, No. 17. 1994 4555
Pyrrole-CO Complex y of the parent species of the complex. No internal coordinates of either monomer are considered in eq 1. Hence the sum appearing in the potential is over the five internal coordinates defining the position and relative orientation of CO with respect to pyrrole. The angles x and y refer to out of symmetry plane motions of CO. x is the angular displacement of the centerof-mass of CO perpendicular tp the symmetry plane of pyrrole and measured in a plane parallel to the c’b’plane (see Figure 2). y is the angle of rotation of C O measured around the c’axis. R,, is the equilibrium center-of-mass distance between CO and pyrrole. AR = R - R,, and ASi = Si - S,(w), where S,(q) is the equilibrium value of internal coordinate Si, and for x and y this corresponds to Oo. At equilibrium, V = 0 and as R -, V c and is thus the equilibrium dissociation energy. Equation 1 is a combination of a Lennard-Jones potential function, used for the radial part of the interaction between CO and pyrrole defined for the full range of R, and a modified harmonic force field for ASi and AR. The modification made to the harmonic force field is seen from eq 1 to be the premultiplier, AR). This factor has been included, somewhat arbitrarily, so that as R -, V-. t. It was necessary to include to ensure that the angular dependence of the potential diminishes the further C O is from pyrrole. The two terms were chosen on the basis of the form of the Lennard-Jones potential. That is, the angular part of the potential goes to 0, at large values of R, as l/R6, and for small values of R, the angular part increases rapidly as 1/R12. At equilibrium the premultiplier is unity. It should be noted that the anharmonicity in this potential arises from the LennardJones function and the premultiplier only; no explicit cubic or higher force constants have been included in eq 1. For clarity, the coordinates of carbon and oxygen of CO in the axis system of pyrrole shown in Figure 2 are given below, where r is the distance from the center-of-mass of CO to carbon along the CO axis.
-
-
+ rcos p cosy b’, = R cos 0 sin x + r cos p sin y a‘, = R sin 0
c,’
= R cos 8 cos x - r sin p
(2a) (2b)
with a # 8. The derivatives in eq 3 have been evaluated using the formulaz7 (5)
where
and
a/,81,and y~ are the equilibrium coordinates of atom 1 in the
ca
principal axis system and are to be taken in cyclic order. is the moment of inertia about axis a a t equilibrium. The 6a/,etc., are the set of small increments which produce an increment SSI in the lth internal coordinate and leave the other internal coordinates at their equilibrium values. Using eqs 5 4 b , the 6a/, etc.,are permitted to be arbitrary although they must be measured with respect to the principal axis system. Equations 5-6b have been derived by converting an unallowed set of displacements, Sa/,etc., into an allowed set, Sa,,etc., which satisfy the six c0nditions,2~
(2c)
ab = R sin 0 - (rco - r ) cos p cos y
(2d)
bb = R cos 0 sin x - (rco - r ) cos p sin y cb = R cos 0 cos x + (rco - r ) sin p
(2e) (20
5.2. Centrifugal Distortion Analysis. Below we briefly outline the approach taken in our centrifugal distortion analysis. The quartic centrifugal distortion constants can be obtained from a perturbation treatment of one of the terms in the rotation-vibration Hamiltonian. It is implicit in this treatment that the amplitudes of vibration are small. With this assumption the quartic centrifugal distortion constants can be written in terms of the internal coordinates of the molecule rather than normal coordinates. The quartic centrifugal distortion constants are related to the force field of the molecule in the following way:27
where the derivatives are evaluated with respect to the internal coordinates and at the equilibrium position (Le.,all 6Si = 0). The Si here refer to any internal coordinate of the molecule including R. The matrix F-1 is the inverted force constant matrix with respect to internal coordinates. It is convenient at this point to define another set of T ’ S (in units of hertz) to be used in later expressions.
Equations 6a and 6b are valid only if the increments are small. There are various ways of reducing the effective centrifugal distortion Hamiltonian. We have chosen in this work to use the A-reduction.28 With this reduction, as with others, there is a folding of some centrifugal distortion into the experimentally determined rotational constants. Because we have a potential function for this complex, we are able to calculate the centrifugal distortion contribution to the experimentally determined constants. We summarize these correction terms below2*
where S A ) (X = A, B, C) is an experimentally determined A-reduced rotational constant and XOis a vibrationally averaged rotational constant. Thecalculation of the latter quantity is given in the next section.
4556 The Journal of Physical Chemistry, Vol. 98, No. 17, 1994
The expressions used for calculating the A-reduced centrifugal distortion constants are those given by Watson28(Irrepresentation) and are reproduced below, but with ground-state rotational constants
- cO> (Bo - CO)
7bk(2A0
- BO
The 7's are calculated for the equilibrium configuration which are assumed to be not significantly different from the corresponding ground-state quantities. 5.3. Vibrationally Averaged Rotational Constants. The vibrationally averaged rotational constants are given by the usual expression,
where /3 is A, B, or C and u describes the vibrational state of the effective constants. The ai? can be calculated from a perturbation treatment of various terms in the rotation-vibration Hamiltonian. The results of this perturbation treatment yield eq 11,28 where the ai6) includes the contributions made by the harmonic, anharmonic, and Coriolis interactions. Equation 11 is the expression used in this work to obtain aiB).
Bettens et al. of pyrrole. Thus the usual Wilson vector method for determining the C matrix is not applicable. We have, however, derived the elements of C-1 from equations27 6a,= 66, - 67, pi
+ 67, y i + 6a;
for the parent species of thecomplex and then inverted the matrix to obtain G. The ai, 6a/,etc. are the same as those given in eqs 6a and 6b. The small rigid translations, baa, etc., and rotations, bv,, etc., of the complex are chosen such that the six conditions (7a, 7b) are satisfied. That is, one converts a set of unallowed displacements, 6a/,etc., into an allowed set, 6ai, etc.,via eqs 12-14. 6aa is a rigid translation of the complex along the a principal inertial axis, and 6or is a rigid rotation of the complex about the y axis. The general expression for the elements of C-1 obtained from the above equations is given by
where
((
ask
e
'si
1
+ ( a-b ' : ) (ab':) - + ( 6- ~ ' : ) (ad:)
- (6a':) 6a':) e
ask
e
"1
e
"k
e
"1
e
(15b) Using the above general expression for (3-1, we have derived the elements of G for this complex. The non-zero elements for the parent species of the complex are given below.
1
G,=-
1 Gee= -
(16a)
CC
+ -1 4'
zd
(1 1) Here, B, is the rotational constant a = x,y, z,okis the fundamental frequency for mode k , and k k k j is a cubic anharmonicity constant. These constants are related to the cubic force constants through the L matrix obtained from a normal mode analysis. g:) is the Coriolis coupling constant coupling modes k and 1 about axis /3.29 The is a dimensionless coupling parameter which is related to the derivatives 5-6b through the L matrix. Equation 11 and further relevant formulas can be found in the work of Gordy and Cook.30 The use of perturbation theory for centrifugal distortion and the vibrational correction to the rotational constants requires that the spacing of the rotational energy levels be small compared to thevibrational spacings and that no near-degeneratevibrational energy levels are connected by the perturbation. 5.4. The G Matrix. To evaluate eq 11 requires the results of a normal mode analysis. To perform such an analysis requires the C matrix. However, in this complex the internal coordinates, R , 8, p , x, and 7 , are not defined between atoms in the complex but from the principal inertial axis system of the parent species
(12)
+-1
G, = IC0
lp
(zc0 + + 2(cy+ cy))sin(p - 6')) where
(16g)
P y r r o l e 4 0 Complex
The Journal of Physical Chemistry, Vol. 98, No. 17, 1994 4551
(lap$) = 2(1y (lap;”,co)
COS 2(e - 24) + COS 28)
= 2(CY - zp)(cos 2(p - 24) + cos 2p)
E,, = sin(p + 0 - 44) + sin(p + e) [d
= sin(p
+ 319- 44)
cy
is the moment of inertia of the parent species of pyrrole around its principal axis a. We have dropped the subscripts “eq” on the angles 0 and p for compactness. Thus, it should be noted that the expressions 16a - g give the elements of G a t equilibrium and are not general expressions for this matrix as a function of internal coordinates. 4 is the angle which rotates an axis system with an origin at the center-of-mass of the parent species of the complex and parallel to the principal inertial axis system of the parent species of pyrrole (axis system x’, y’, z’), into the principal inertial axis system of the complex. 4 is given by
multiplied by a convenient distance. Note that there is no kinetic coupling between the stretch and any other motion, and thus any mixing between stretch and bend occurs entirely through the F matrix. Even this mixing is limited because the potential energy function must transform as the totally symmetric irreducible representation of the permutation inversion group, in this case A’of C,(M). This group is used because the inversion operation and the permutation of the symmetric hydrogens and carbons of pyrrole are not considered feasible operations if one assumes small amplitudes of displacement. The internal displacement coordinates, AI?, AB, and Ap, belong to the symmetry species A’, and Ax and AT belong to A”. Thus the stretch can only mix with the bend AB and the torsion Ap. It can be seen from the expressions for G that there is no need to introduce any structural parameters of the monomers. All elements are defined in terms of the five internal coordinates and the vibrationally averaged moments of inertia of the respective monomers. The latter quantities areused because the vibrational modes of the monomers have been ignored in this work; thus the obtained structure is termed “pseudo-r,”. The expressions 16a-g, are valid for all isotopic species, but because of the definitions of the internal coordinates of the complex, the expressions can only be used for one isotopic species. We have chosen the parent species of the complex to be the species to which the internal coordinates refer. With this choice, more complex expressions arise for isotopically substituted species if one wishes to refer all quantities to the parent species. Rather than derive more general expressions, we have numerically calculated the extended moment of inertia matrix ( p - l ) for each complex using eqs 12-14,19, and 20. In the latter two equations, given below, the sk refer to any of the five internal coordinates of the complex. These two equations plus the definitions for the moments of inertia furnish the matrix elements for the extended moment of inertia tensor, an 8 X 8 matrix, with ( F - I ) ~ (a ~ = x, y, z ) being the diagonal I defined above.
The rotation matrix, R, takes the moment of inertia tensor from the translated a’, ,b‘, c’axis system into the principal inertial axis system, i.e., I = RI’R where
[
]
cos 4 -sin 4 0 R = s i n 4 cos 4 0 (17) 0 0 1 The coordinates of atom i, given by the position vector r‘ in the axis system a’, c’, are rotated to r in the principal inertial axis system by r = Rr’. Here the transpose of r’ is [c’, a’, b l where a’, b’, c’ are the coordinates of atom i in the complex of the translated principal inertial axis system of the parent species of pyrrole. The transpose of r is [a, b, c] where a, b, c denote the coordinates of atom i in the principal inertial axis system of the complex. Thus, the b’axis is the c axis of the complex, and the c’ and a‘axes are approximately the a and b axes, respectively, in the complex. The moment of inertia tensor is now given by
i’,
za;
=
cy + zc0 cos’ p +
Id
sin2 e
(18a)
(2) (%)} (20) ask
(18 4 Equations 2a-f, 17, and 18a-d were necessary for the analytical calculation of G . For the expressions 16a-g, the angular internal displacement coordinates are given in units of radians and have not been
e
A structure is required for both monomers in the numerical calculation, so we have fitted an ro structure for pyrrole from the rotational constants of Nygaard et al.23 and determined the ro bondlengthof 12C160fromitsBo rotationalconstant.22 To obtain the G matrix, we inverted the extended p-I and extracted the purely vibrational part. Expressions 16a-h were used as a check on the numerical accuracy, as was the psS. The latter elements should be zero in the equilibrium configuration if the Eckart conditions are properly fulfilled; Le., eqs 7a and 7b should be satisfied given that
(3) =(”) e
Ia
