4936
J. Phys. Chem. 1993,97, 4936-4940
Photoelectron Spectrum, Ionization Potential, and Heat of Formation of CC12 Daniel W. Kohn, Eric S. J. Robles, Cameron F. Logan, and Peter Chen'yt Mallinckrodt Chemical Laboratory. Harvard University, Cambridge, Massachusetts 021 38 Received: January 15, 1993
The photoelectron spectrum of dichlorocarbene, CC4, is reported. Simulation of the partially resolved vibrational structure is used to extract the adiabatic ionization potential, IPad[CC12] = 9.27 f 0.04 eV, and the heat of formation, AHof,2~~[CC12] = 5 1.O f 2.0 kcal/mol. A protocol for calculating polyatomic Franck-Condon factors is given and compared to two less general approaches.
We report a photoelectron spectrum of dichlorocarbene, CC12, from which we extract the adiabatic ionization potential of the carbene by a fit to the simulated Franck-Condon envelope of the photoelectron band. The adiabatic ionization potential, IPad[cc12] = 9.27 f 0.04 eV, can be used with the heat of formation of the cation to derive mof,298[cc12] = 51.0 f 2.0 kcal/mol. The heat of formation of dichlorocarbene has been the subject of numerous experimental studies by a variety of techniques,' with results ranging from 39 to 59 kcal/mol. The most recent experiment, by observing the threshold for the CC13CCl2 C1- reaction,2 gave wf,296[cc12] = 52.1 3.4 kcal/ mol. Photoionization and electron impact mass spectrometric determinations of AZf0f,298 [CC12] have been of uncertain reliability, primarily due to problems in the determination of the adiabatic ionization potential of CC12. Mass spectrometric measurements of IP,d[CC12] find a gradual, slowly rising, onset for CC12*+,from which the IP is obtained by extrapolations based on the assumed threshold behavior of the ionization process. Reported values for IPad[cc12] range from 13.2 and 13.1 eV on the high side3 to 9.76 and 9.10 eV on the low side.495 No photoelectron spectra have been reported for this important carbene. Despite the fact that it is only a triatomic molecule, the very large changes in both the C-Cl bond length and Cl-C-Cl bond angle upon ionizationof CCl2 make Franck-Condon factors for ionization near threshold exceedingly small and the IP hard to fix. Having recently used our intense supersonic jet source6 of radicals and carbenes to obtain clean photoelectron spectra of a variety of hydrocarbon reactive intermediates,' we decided to look at CCl2 as a model applicationof Franck-Condon simulations, with the Duschinsky rotation6 expressed in a reduced massweighted Cartesian representation. The large change in equilibrium geometry, which had hindered mass spectrometric determinations of IPad[cc12], is used in this study as the handle by which the spectrum can be made to yield the desired ionization potential. This methodology, previously reported9J0 for the interpretation of UV absorption and resonance Raman spectra, allows us to quantitatively simulate the partially resolved vibrational structure in a photoelectron band for the case of very large geometry changes accompanying ionization. The presentation of the method is aimed at demonstrating, by way of extracting IP,d[cc12], that, having done a vibrational analysis for the molecule (which comes automatically with an ab initio calculation of vibrational frequencies), the additional effort to correctly simulate the Franck-Condon factors is minimal. From the methodological point of view, we are using information on geometries, which are well-modeled by ab initio calculations at a modest level, to help extract absolute thermochemical infor-
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+
*
' NSF Presidential Young Investigator, David and Lucile Packard Fellow, Camille and Henry Dreyfus TeacherScholar, and Alfred P. Sloan Research Fellow. 0022-365419312097-4936$04.00/0
mation, which, even for a triatomic molecule, is a challenging problem for theory.
Experimental Section Dichlorocarbene was prepared, seeded in a supersonic jet of helium, by the flash pyrolysis of chloroform, which proceeds by the presumably concerted loss of HCl. The pyrolysis nozzle design,6Jl typical operating conditions,' I and the time-of-flight mass and photoelectron spectrometersI2 used in this work have been described previously. The particular flash pyrolysis conditions (1400 IC with a IO-rcs contact time) produced a signal in the mass and photoelectron spectrometers correspondingto CC12 from the 0 OC vapor pressure of HCC13, seeded into 2 atm of helium. Coherent vacuum-UV radiation at 118.2 nm (10.49 eV) for photoionization was produced by tripling the Nd3+-YAG (Spectra-Physics DCR-3G, 20 Hz) third harmonic in 10 Torr of xenon.I2 After optimization of conditions, the photoelectron spectrum was signal-averagedover 2048 laser shots and calibrated against the photoelectron spectrum of nitric oxide. A small retarding potentialofO.183 V wasapplied toimprove theresolution of the spectrum. Ab initio calculations were performed with the Gaussian 90 package" of programs on an IBM RS/6000 workstation.
Results The 10.49-eV photoionization mass spectrum of pyrolyzed HCCl3 is shown in Figure 1. The conversionefficiencyfor HCC13 CC12 HCl is not known because the high ionization potential of chloroform,I4IP[HCClj] = 11.42 eV, precludes its detection at the photon energies used in this experiment. The other product of the pyrolysis, HCl, also does not appear because its ionization potential's is 12.74 eV, well above the 10.49-eV photon energy. Similar spectra were obtained from several other CC12 precursors, but the HCC13 results were the best. Under less-than-optimum conditions of sample concentration, pyrolysis temperature, and nozzle operation, masses and photoelectrons corresponding to CCl, CC13, C2C12, and C2C14are prominent and can even completely obscure the spectrum due to CC12. It is possible to largely suppress the undesired species by adjusting conditions. Because the CC13signal can be reduced to a small fraction of that due to CC12and the two spectra were only partially overlapped, we simply recorded the spectrum of CC13 independently under similar conditionsI6and subtracted it out. A small contribution" from GC12at 10.05and 10.09 eV slightly distorts the falling edge of {he CC12 band but otherwise does not interfere with the spectrum. The resulting time-of-flight photoelectron spectrum of CC12 is shown in Figure 2. Presumably, the facile formation of bimolecular reaction products is why the photoelectron spectrum of CC12 had not been reported until now. We note that the broad Franck-Condon envelope also means that a much higher number density of CC12 is needed to acquire
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0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, NO. 19, 1993 4937
Photoelectron Spectrum of CClz
Calculation of Polyatomic Franck-Condon Factors. To properly extrapolate IP,,j[CC12] from Figure 2, one needs to be able to accurately simulate the Franck-Condon envelope of the photoelectron spectrum. Because of changes in bond lengths, angles, force constants, or symmetry that accompany ionization, the normal coordinates of the neutral and the cation are not the same. Accordingly, a transformation of the coordinate system of the neutral into that of the cation (or visa versa) is needed before one can compute vibrational overlap integrals; i.e., the normal modes of the neutral must be expressed as a linear combination of the normal modes of the cation. This Duschinsky effect, or mode mixing, is usually expressed as
cc12 82
l;;l I' CCI 47
mass Figure 1. 10.49-eV photoionization mass spectrum of the pyrolysate of HCC13. HCl does not appear because of its high ionization potential. The multiple peaks are due to the presence of both 35Cland 37Clisotopes. Monochloro species should show two peaks in a ratio of 3 : l . Dichloro species should show three peaks in a 9:6:1 ratio. Trichloro species should have four peaks in a 27:27:9:1 ratio.
2.4 1.2 I,,,,,I, ,
0.8 I
,
0.4
0.3
I
I
photoelectron kinetic energy (eV)
Figure 2. 10.49-eV vacuum-UV photoelectron spectrum of CC12 (trace a), obtained after subtraction of contributions due to CC13. Electron kinetic energies are given in electronvolts. Two simulated spectra for energy resolutions of 40 meV (trace. b) and 6 meV (trace c) are also shown for comparison. The extrapolated position of the adiabatic ionization potential is marked with an arrow. Subtraction of the electron kinetic energy from the 10.49-eV photon energy gives IP,d[cc12] = 9.27 f 0.04 eV. The two small features marked with asterisks are due to residual CzC12.
a spectrum with signal-to-noisecomparable to that in a spectrum with a narrow Franck-Condon envelope, making the suppression of bimolecular chemistry more difficult than usual. The photoelectron spectrum of the much less reactive carbene, CFz, was reported's some time ago. The gross appearance of our spectrum resembles that for CF2, except that the lower frequencyvibrations in CC12 and significant Franck-Condon activity in two (rather than one) modes cause severe congestion of the vibrational structure in our photoelectron band. While there is reproducible vibrational structure in Figure 2, it is insufficiently resolved to use as a basis for conventional spectroscopic analysis. The only way to extract thermochemical information from the spectrum is to simulate the Franck-Condon envelope for the calculated change in geometry upon ionization.
Q=JQ+K (1) where the Q and Q are the normal coordinates for the cation and neutral, respectively, and the matrix J and column vector K define the linear transf~rmation.'~ Having made the transformation, there are several approaches to computing the vibrational overlap integrals, which are more or less equivalent. We use the multidimensionalgenerating function method described by Sharp and RosenstockZ0for the integrals, although our treatment of the Duschinsky effect is different. In practice, there are three ways in which the Duschinsky effect is commonly handled. One may assume that there is no modemixing at all-the parallel mode approximation2'-which is valid only for small changes in geometry. A better approach allows the normal modes of the neutral and cation to differ but assumes that the same set of internal coordinates can be used to describe both. This was used for the Duschinsky effect in the original Sharp and RosenstockderivationZoand also assumes small changes in geometry. The most general approach, which is good for arbitrarily largechanges in geometry (but still assumes a harmonic force field) treats the Duschinsky effect in terms of Cartesian coordinates and displacement^.^ We use this approach below. To obtain the matrix J and vector K in eq 1, in terms of Cartesian displacements, we can write the general linear transformation of an arbitrary distortion, X: X'=ZX+R (2) where X and X' are distortions expressed as Cartesian displacements from the equilibrium geometries of the cation and neutral, respectively. R = Z&-Rtq is the change in equilibriumgeometry between the cation and the neutral in Cartesian coordinates centered on the molecular center of mass, and Z is a rotation matrix,'0q22which is a unit matrix for most molecules of Cz,or higher symmetry. A more detailed discussion of Z is included in Appendix 2. The appendix is included for rigor, even though Z = 1 for CC12. In a conventional normal-mode analysis, the Cartesian displacements, X, are transformed to internal coordinates, S,and then to normal coordinates, Q, and visa versa, by the B and L matrices. The formalism is given in standard texts like Wilson, Decius, and Cross.23 The notation used in this work and, importantly, the definitions of B and L are consistent with those in a series of instructive papers by McIntosh and Michaelian24 and in Sharp and Rosenstock.20 In particular, the usual picture of a normal mode with arrows attached to individual atoms, indicating an unit displacement along that mode, is formalized in the Cartesian atom displacement matrix, adm M-I(L-'B)+, that relates Q to X. M is a diagonal matrix with the atomic masses on the diagonal, once for each Cartesian direction, that gives the proper mass weighting for adm. The atom displacement matrix can then be used, after several substitutions, to relate Q to Q. Q = (L'-'B')[ZM-'(L-'B)+Q where the superscript -7''
+ R]
(3) indicates the transpose of the matrix.
4938 The Journal of Physical Chemistry, Vol. 97, No. 19, 1993 Equation 3, substituted back into eq 1, gives expressions for J and K:
J = (L"B')ZM-'(L-'B)t
and K = (L"B')R (4) Having done vibrational analyses for both the cation and the neutral to obtain B, L, B', and L', one can use eq 4 for a description of the Duschinsky effect that is valid even for large changes in geometry or force constants. The Gaussian 90 package of ab initio electronic structure programs provides, as part of their output, a reduced-massweighted atom displacement matrix representation of the normal modes of the molecule under study. The utility of the formulation of the Duschinsky effect in terms of Cartesian displacements instead of internal coordinates comes not only because it is more rigorous but also because it provides a facile interface to ab initio structure programs. For a molecule with N atoms, there are 3 N - 6 normal modes. The Gaussian 90 output contains a listing of the 3N- 6 normal modes, each as a column vector of length 3N. At the head of each column, among other things, is the reduced mass associated with that mode in atomic mass units. If we define a 3 N X 3N- 6 matrix, g90,containing the normal mode output from Gaussian 90, and a 3N- 6 X 3N- 6 diagonal matrix, V, containing the reduced masses for each mode on the diagonal, then
adm = (g90)V-'/* removes the reduced mass weighting. Substituting eq 5 into eq 4 and using the definition of adm, we get our final working equations for J and K in terms of Gaussian 90 output
J = [M(g90')V'-'/2]tZ(g90)\r'/2 and K = [M(g90')V''/2]tR
(6)
Kohn et al.
TABLE I: Some Franck-Coadon Factors for Cc12'+
-
CCl2
Computed by Using the Cartesian Displacement, Internal Coordinate, and Parallel Mode Methods of Handling the Duschinsky Rotation Cartesian internal band displacement coordinate parallel designation method method mode method 000
3 3 3
72
3 3 02 3
;1 ;2
;1 ;2 0
0
11 22 0 5 1 12 0 0 8 1 32 0
1 5.8 17 35 9.2 43 130 0.030 52 150 232 21991 3487
1 7.0 26 69 8.4 33 82 0.011 57 203 214 2927 15598
1 8 .o 32 87 3.4 6.0 7.0 0.029 28 109 48 424 3469
In the Cartesian displacement method, direct application of eq 6 by using the equilibrium geometries of CC12 and CC12'+ as two column vectors of xyz values in center-of-mass coordinates, two 3 X 3 matrices containing the reduced mass for each normal mode as a diagonal element, and two 9 X 3 matrices, g90 and g9(y, gives J and K, defining the Duschinsky effect of eq 1
Io
0.95 0.31 0 J = -0.31 0.95 0I ]
K=[i:i:]
(7)
where, as usual, unprimed and primed quantities refer to the cation and neutral, respectively. The J matrix is block diagonal Alternatively, J and K could have been produced by using eq 4 by the symmetry of the normal modes. One can choose only the if a conventional vibrational analysis based on a valence force column vectors corresponding to the totally symmetric modes for field (instead of an ab initio calculation) had been done. A brief inclusion in the g90 matrix to reduce the size of the calculation examination of J, which describes the mixing of normal modes, without any loss of generality, if one is confident that even reveal that each one of the two a1 modes in CCl2, the symmetric overtonesof the nontotally symmetric modes will not appear with stretch and the bend, maps onto a linear combination of the two appreciable intensity. Having then compared J and K from the a1 modes in CC12'+ with large mixing coefficients. Note that output of ab initio calculations for the cation and neutral, the each row in J is normalized, the sum of the squares of the mixing Franck-Condon factors are easily produced by using the tables coefficients adding up to unity within rounding errors. Qualifrom Sharp and RosenstockZ0or the more general algebraic tatively, K is a measure of the change in equilibrium geometry expressions given in Appendix 1 . upon ionization along each normal mode of the cation. Large Calculations for Dichlorocarbene. To illustrate the ease with displacements are evident along both V I and u2, the two a1modes. which Franck-Condon factors can be obtained by this methodFor nontotally symmetric modes, e.g., the asymmetric stretch u3, ology, we do thecalculation for CC12, a typical nonlinear triatomic. the corresponding element of K is zero by symmetry. With the A check of the potential surface of CC12'+ by using single-point additionalinput oftwo 3 X 3 matrices with t h e r e d u d frequencies calculations at a series of geometries along the two a1 coordinates on their diagonals, J and K give the predicted Franck-Condon found that the potential is close to harmonic for up to at least 8 factors for photoionization of vibrationally cold CC12. Some quanta of vibration in each mode. In this calculation, only representative values are listed in Table I. Included for comtransitions arising from the vibrationless ground state of CCl2 are included. We had shown in our prior s t ~ d i e s *of~ C3H2 - ~ ~ - ~ ~ parison are the same Franck-Condon factors calculated by using the parallel mode and internal coordinate-based methods. Even and C3H5 that the supersonic jet expansion of species produced for low excitations, both of the less general methods give large by flash photolysis cools them vibrationally to the point that hot errors. While it may be argued that the parallel mode or internal bands are not seen. Geometries and frequencies for CClz and coordinate-basedcalculations are simpler in implementation,given CClz'+ were optimized at the MP2/6-31G* level. Upon ionexperimentally-determined frequencies,one should note that, even ization, the C-Cl bond shortens from 1.7 158 to 1.5883 A and the Cl-C-CIangleincreasesfrom 109.98O to 135.32O. The~nscaled2~ for the parallel mode calculation,a normal mode analysis is needed prior to calculation of Franck-Condon factors. Having done the harmonic frequencies and reduced masses, in the order, V I normal-mode analysis, eq 4 would give J and K for the Duschinsky (symmetric stretch), u2 (bend), u3 (asymmetric stretch), are 771 effect and a proper simulation of the spectrum without much (14.21), 355 (29.80), 794 (13.85) cm-I (amu) for CCl2 and 807 additional work. (17.18), 371 (21.89), 1297 (13.48) cm-l (amu) for CC12*+. Vibrational analysis were done with the UMAT package of Discussion programs:* available from the Quantum Chemistry Program Exchange. An empirically determined energy resolution of 40 With the equilibrium geometries and frequencies from the ab meV was used in simulation. initio calculations, we calculated the Franck-Condon factors for
Photoelectron Spectrum of CC12 the photoelectron spectrum of CC12. We used J and K from eq 6, to produce a stick spectrum that is then convoluted with a linewidth. Finally, the linear-in-energy simulation is transformed to a linear-in-flighttime simulation with the integrated area under each peak held constant. No adjustments of the ab initio geometries were done. The frequencies were scaled to match the one experimentally known fundamental.29 v3 = 1195 cm-I, in CCl2*+by multiplying the abscissa of the linear-in-frequency simulationby 0.9214 prior to transformation to a linear-in-flighttime spectrum. Simulations for two energy resolutions, 40 and 6 meV, are shown in Figure 2. The deceptively simple structure in Figure 2c arises because V I = 2~2,at least within the resolution of the simulation, creating the false impression that there is only one long progression. Matching of the overall band contour of Figure 2b to Figure 2a (rather than any single line, although the partially resolved structure is well-reproduced) gives the adiabatic ionization potential, IPad[CC12]= 9.27 f 0.04 eV. We note that the largest Franck-Condon factor, for the 5vl + 7v2 line (see Table I), is almost 22 000 relative to that for the origin. It is clear that, even if sensitivity issues were resolved, limits on the dynamic range of detection would have made location of the adiabatic ionization potential exceedingly difficult without some sort of extrapolation procedure. Our value for IPad[cc12] compares well with the best photoionization mass spectrometric value of 9.10 f 0.10 eV by Rademann, Jochims, and Baumglrtel: which was extracted by a semilogarithmic extrapolation of the photoionization efficiency curve for CCl2 to its origin. Our slightly higher IP, along with AWf,298[CC12'+] = 264.8 f 1.8 kcal/mol,5 gives a value for the heat of formation of CC12, AWf,298[CC12] = 51.0 f 2.0 kcal/mol, in excellent agreement with the 52.1 f 3.4 kcal/mol result from collision-induced dissociation threshold measurement2 for CC&-, which relies on a wholly independent set of auxiliary thermochemical data. Having used the fit to a simulated Franck-Condon envelope to extract IPad[cc12] from the photoelectron spectrum, one may legitimately question the extent to which the different ways of handling the Duschinsky effect change the result. A crude estimate may be made from the differencein energy between the most intense single line in the simulation and the band origin (which does not correspond to the apparent IP,,,, in Figure 2 because of the transformation to a linear-in-flight time spectrum and thesuperpositionof many lines). These differences are 0.758, 0.864, and 0.864 eV for Cartesian displacement, internal coordinate, and parallel mode treatments, respectively, with the maximum occurring at 8 ~ + 1 3v2 in the latter two. The error in AZf'r.298 [CC12] introduced by either the internal coordinate or parallel mode treatments of the Duschinsky effect is at least 2.4 kcal/mol. This number assumes that the error in the identification of the most intense line propagates through to wf,298[CC12]. The different simulations, convoluted with a linewidth, gave Franck-Condon envelopes of different FWHM and shape,which would make fitting of the experimental spectrum problematic and perhaps introduce additional errors.
Conclusions We have used information from ab initio calculations on the geometry and frequencies of CC12 and CC12*+ to extract the adiabatic ionization potential, and hence the heat of formation, of CC12 from a partially resolved photoelectron spectrum of the carbene. Franck-Condon factors were computed in a massweighted Cartesian representation to represent properly the coordinate transformations for large changes in equilibrium geometry. Comparison to less sophisticatedmethods finds that, although they do not require less prior information,extrapolations to theadiabatic ionization potential of CC12 by internal coordinatebased or parallel mode methods would be significantly in error. Acknowledgment. This work was funded by the Department of Energy and the Petroleum Research Fund, administered by
The Journal of Physical Chemistry, Vol. 97, No. 19, 1993 4939 the AmericanChemicalSociety. WealsoacknowledgeDr. David Yaron and Professor William Klemperer for assistance in the coordinate transformations.
Appendix 1 EquationsAl.1 andA1.2aretakenfromref 20. Thederivation is not given here. Only the resulting working equationsare listed. The input for the equations are the J and K matrices from the Duschinsky effect and a diagonal matrix r that contains the harmonic frequencies as diagonal elements. If mass are given in amu and distances in angstroms, then the'l matrix elements are the frequencies, in cm-l, multiplied by 0.029 660. Sharp and Rosenstock then define
C = 21'1/2[JtI"J
+ 1']-11'1/2- 1 + l"]-lJtl"K
(Al.l)
D = -21"/*[JtI"J (Al.2) The elements of these arrays are oridinarily used with the tables in ref 20 to compute the Franck-Condon factors. The major difficulty in using the multidimensional generating function method of ref 20 is the algebraic complexity of the expressions for the Franck-Condon factors as the number of vibrational quanta increase. Errors in collecting and reducing terms in the power series expansion of the generating function are difficult to avoid. Sharp and Rosenstock give an analytic expression for the Franck-Condon factor for an arbitrary number of quanta of excitation in one mode that is well-suited to coding as a computer program. They, however, do not extend their expression beyond a single mode and rely instead on a tabulation that becomes increasingly cumbersome as the number of quanta and number of modes increase. By inspection of the form of the entries in Table I or ref 20 and by analogy to a comparable expression in work by Hutchisson,3° we generalize the analytic expression to the two-mode case needed for the al modes of CC12. The FranckCondon factor for the transition from the vibrationless ground state of CC12 to CC12'+, with m quanta in mode i and n quanta in mode j , is m!n! FCF(m,nl X . . _= 2m+n
The limit for the outermost summation is the lesser of the two numbers m and n. The bracketed expressions for the limits of the remaining two summations are taken to mean the greatest integer less thanor equal to(m-p)/2or (n-p)/2. Generalization to an arbitrary number of modes, and inclusion of hot bands, is straightforward, but tedious. For example, the three-mode expression, where there are m quanta in mode i, n quanta in mode j , and o quanta in mode k, is m!n!o!
FCF(m,n,o) = -X 2m+n+0
/
~ 2 ' - l - p D ~ " - 4 - p C ~ j ~[p!q!r!s!r!U! k ~ j ~ ~x ~ k ]
(m-2u-r - q)!(n- 2t - r - p ) ! ( o- 2s - q -p)!]
I'
(A1.4)
where the limits are defined as above. The multiple, nested
Kohn et al.
4940 The Journal of Physical Chemistry, Vol. 97, No. 19, I993
summations are easily coded3’and evaluated by computer. This, along with the formulation of the Duschinsky rotation in terms of Cartesian displacements, makes the evaluation of multidimensional vibrational overlap integrals for polyatomic FranckCondon factors very easy. Appendix 2
The matrix Z in eqs 2-4 and eq 6 is a rotation matrix that removes contributionsto the Franck-Condon factors due tooverall rotation of the molecule. The need to remove contributions from overall translation or rotation can be justified qualitatively. In the calculation of Franck-Condon factors in terms of Cartesian displacements, there are nonvertical transitions with nonzero intensity when any atom in the cation is displaced relative to its position in the neutral. If one were to take a neutral and a cation with exactly the same geometry and either displace the origins of the coordinate systems or rotate one coordinate system with respect to the other, then there would be artifactual atom displacements giving incorrect Franck-Condon factors. The contribution from overall translation can be removed by setting the origins of both the cation and neutral Cartesian coordinate systems at the respective centers of mass. The contribution from overall rotation can be removed, as described in refs 9, 10, and 22, by a matrix Z, derived by using a static analogue to the Eckhart condition^,^^ which, in the harmonic limit, setsvibrationrotation interactions to zero. The cation and neutral coordinate systems are first set up with as many symmetry elements in common as possible. If the subgroup defined by the symmetry elements common to both the cation and neutral is D,, C,,,Dnh, , D,d, Td, oh,C,,, D.+ or I h , then Z is a unit matrix, and no rotation is needed. If the subgroup common to the cation and neutral is C,,Ci, C,,Cnh, or S,,then a rotation is needed. The equation to be satisfied is analogous to that used for axis-switching by Hougen and Watson:33 N
(A2.1) where mi are the masses of the N atoms and S i s a 3 X 3 rotation matrix. Z will then be a 3N X 3N block diagonal matrix with Sas the block repeating N times down the diagonal. A closed form of the solution for S,satisfying eq A2.1, is given by Yaron.22 The notation follows ref 22, which contains the derivation of the working equations used below. Note that the italicized matrices S,C, and R a r e not the S , C, and R matrices used earlier in this report. To find the rotation matrix S, first define a matrix C with elements N
In eq A2.2, a,and 0 denote x or y for planar molecules (cation and neutral) and x, y , or z for nonpolar molecules. For planar molecules, the element C,,is set to unity. This assumes that the planar molecule has been placed in the xy plane and a rotation about the z axis is needed to orient the neutral and cation. Diagonalizing the product O C yields a matrix R (eigenvectors of QCas rows) and a diagonal matrix X (eigenvalues of O C o n thediagonal) that satisfyROclpI = A. With theadditional matrix A, a 3 X 3 diagonal matrix with + 1’s or -1’s for diagonal elements,
the solution for a nonplanar molecules is
s = R?AX‘ f 2 ~ c ‘ (A2.3) The only ambiguity is the matrix A, for which there are eight distinct possibilities. Four can be discarded because, S being a rotation matrix, 1 4must be equal to 1. The choiceof the correct A can be made from among the remaining four possibilities by inspection after superimposingthe cation and neutral structures.
+
References and Notes (1) A tabulation of experimental results may be found in: Lias, S.G.; Karpas, Z.; Liebman, J. F. J. Am. Chem. SOC.1985,107, 6089. (2) Paulino, J. A.; Squires, R. R. J . Am. Chem. Soc. 1991,113,5573. (3) Blanchard, L. P.; LeGoff, P. Can. J . Chem. 1957,35,89.Reed, R. I.; Snedden, W. Trans. Faraday SOC.1958,54, 301. (4) Shapiro, J. S.;Losing, F. P. J . Phys. Chem. 1968,72, 1552. (5) Rademann, K.; Jochims, H.-W.; Baumglrtel, H. J. Phys. Chem. 1985,89,3459. ( 6 ) Kohn, D.W.; Clauberg, H.; Chen, P. Rev. Sci. Instrum. 1992,63, 4003. (7) Blush, J. A.; Clauberg, H.;Kohn, D. W.; Minsek, D. W.; Zhang, X.; Chen, P. Acc. Chem. Res. 1992, 25, 385. (8) Duschinsky, F. Acta Physicochim. URSS 1937,7, 551. (9) Warshel, A.; Karplus, M. Chem. Phys. Lett. 1972,17, 7. Warshel, A.; Karplus, M. J. Am. Chem. SOC.1974,96,5677. Warshel, A. J . Chem. Phys. 1975,62, 214. (10) Ozkan, I. J . Mol. Specrrosc. 1990, 139, 147. (1 1) Clauberg, H.; Minsek, D. W.; Chen, P. J . Am. Chem. SOC.1992,114, 99. (12) Minsek, D. W.; Chen, P. J. Phys. Chem. 1990,94,8399. (13) Frisch, M. J.; Head-Gordon, M.; Trucks, G.W.; Foreman, J. B.; Schlegel, H. B.; Raghavachari, K.; Robb, M. A.; Binkley, J. S.;Gonzalez, C.; DeFrees, D. J.; Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Topiol, S.;Pople, J. A. Gaussian 90; Gaussian Inc.: Pittsburgh PA, 1990. (14) Watanabe, K.; Nakayama, T.; Mottl, J. J . Quant. Spec. Radiat. Transfer 1962,2, 369. (15) Kimura, K.; Katsumata, S.; Achiba, Y.; Yamazaki, T.; Iwata, S. Handbook of HeI Photoelec t ron Spec t ra of Fundamental Organic Molecules; Halsted Press: New York, 1981. (16)The photoelectron spectrum of CCI, will be the subject of a separate studv. (i7) Allan, M.; Kloster-Jensen, E.; Maier, J. P. J . Chem. Soo., Faraday Trans. 2, 1977,73, 1417. (18) Dyke, J. M.; Golob, L.; Jonathan, N.; Morris, A.; Okuda, M. J. Chem. SOC.Faraday Trans. 2 1974. 70. 1828. (19) Within the. harmonic approximation, the transformation is linear. For the small corrections in the more general case, see ref 10. (20) Sharp, T. E.;Rosenstock, H. M. J . Chem. Phys. 1964,41, 3453. Botter, R.; Dibeler, V. H.; Walker, J. A.; Rosenstock, H. M. J . Chem. Phys. 1966,44,1271. Rosenstock, H.M.Int. J . Mass Spectrom. Ion Phys. 1971, 7,33. Botter, R.; Rosenstock, H. M. J . Res. NBS 1969,73A, 313. (21) Recent examples of the use of parallel modes may be found in: Ohno, K. Chem. Phys. 1979,37,63.Giles, M. K.; Polak, M. L.; Lineberger, W. C. J. Chem. Phys. 1992,96,8012. (22) Yaron, D. Ph.D. Dissertation, Harvard University, 1991. (23) Wilson, E. B., Jr.; Decius, J. C.; Cross, P. C. Molecular Vibrations, McGraw-Hill: New York, 1955. (24) McIntosh, D. F.;Michaelian, K. H. Can. J. Spectrosc. 1979,24, 1. McIntosh,D. F.;Michaelian,K. H.Can.J.Spectrosc. 1979,24,35.McIntosh, D.F.; Michaelian, K. H. Can. J. Spectrosc. 1979, 24, 65. (25) Clauberg, H.; Chen, P. J . Phys. Chem. 1992,96, 5676. (26) Minsek, D. W.; Blush, J. A,; Chen, P. J . Phys. Chem. 1992,96,2025. Blush, J. A.; Minsek, D. W.; Chen, P. J . Phys. Chem. 1992,96, 10150. (27) The frequencies are scaled to match the one known fundamental of CC12’+ at a later p i n t in the simulation: (28) QCPE Program No. QCMP 067, available from the Quantum Chemistry Program Exchange, Indiana University, Bloomington IN. ( 2 9 ) Jacox, M. E. J. Phys. Chem. ReJ Data 1984,13, 945. (30) See eq 1 1 in: Hutchisson, E. Phys. Rev. 1930,36,410. (31) We havecoded thegeneralizationof thesharpand Rosenstock tables as a FORTRAN program that can handle an arbitrary number of quanta distributed among up to eight normal modes in the cation and eight normal modes in the neutral (hot bands). (32) Bunker, P. R. Molecular Symmetry and Spectroscopy; Academic Press: Orlando, FL, 1979;pp 137-153. (33) Hougen, J. T.; Watson, J. K. G.Can. J . Phys. 1965,43,298.
