J . Phys. Chem. 1992, 96, 2448-2455
2448
state to an A, state of higher energy or in the intermediary state. A decision between these alternatives can be based on a comparison of the frequencies of the coupling mode(s) in the three 3p-Rydberg states in cyclopentadiene. If the AI false origins of the B, 3pRydberg state are enabled by coupling between the final state and another electronic state, the frequencies of the bl modes in the B1 electronic state will be shifted from their unperturbed values, as described above. The b, enabling mode vI9 has been assigned in both the B1 and B2 3p-Rydberg states. It is observed to have the same frequency in both states. It cannot, therefore, be coupling the B1 3p-Rydberg state of cyclopentadiene with a state of AI symmetry. T_hus, at least this two-photon-excited AI false origin of the B, X 3p-Rydberg state must be induced by coupling in the intermediate state of the transition. From the above symmetry and intensity considerations, we suggest that it is coupling to an A, state, probably the 3A, valence state, in the intermediary state that gives rise to the vibronic intecsity of the false origins both in the BI X and in the A2 X two-photon-excited transitions in cyclopentadiene. In summary, based on the symmetries of the true and fa@ origins observed in the 2RMPI spectra of the 3p-Rydberg X transitions of cyclopentadiene, the intensities of the one-photonexcited transitions, and the frequencies of the inducing modes in the ground and all three 3pRydberg states, the probable enabling mechanism for the observed vibronically induced transitions has been deduced. The mechanism involves vibronic coupling of the intermediate states of the two-photon transitions with a state of AI symmetry, most probably the 3Al valence state. The results further suggest that this same intermediate state generates th_e experimentally observed intensity in three 3p-Rydberg X transitions.
-
-
-
-
-
V. Summary The polarization-selected two-photon resonant multiphoton ionization spectra of the 3p-Rydberg region of cyclopentadiene
and cyclopentadiene have been measured. The spectra differ from the previously reported one-photon-excited spectra of these isotopomers. The 2RMPI spectra of CP and CPD6 are shown to be composed of transitions from the ground to the B2 3p-Rydberg state and its vibrational substructure and of AI and non-A, false origins of transitions from the ground to the B, and A2 3pRydberg states, respectively, and their vibrational substructures. Transitions from the ground to the true origins of the A2 and BI 3pRydberg states are not observed. Both a , and non-a, vibrations have been assigned in the 3pRydberg states of cyclopentadiene. Some previously reported assignments in the 3pRydberg states of CP and CPD6 have been changed. The analysis of the upper-state vibrational frequency intervals and intensities suggests that the out-of-plane force constants are weaker in the 3pRydberg states than in the ground state but the molecule remains planar in the 3p-Rydberg states. The estimated in-plane distortions are as before. The vibrational substructures of the active transitions are observed not to fully agree with those observed in the optical spectra of the same transitions. Experimental assignments of the three 3p-Rydberg transitions have been deduced. Theoretical calculations of these transition energies agree closely with the experimental energies and assignments. The relative calculated two-photon cross sections to the three 3gRydberg states of cyclopentadiene differ from the relative two-photon intensities of the same transitions. The false origins of the A2 and B1 3p X Rydberg transitions are deduced to arise from mixing of a state of A, symmetry, probably the 3A1 valence state, with the intermediate state of the two-photon transitions. This state probably alto contributes to the intermediate virtual state in the B, 3p X transition.
-
-
+-
Registry No. Cyclopentadiene, 542-92-7; cyclopentadiene-d,, 210216-1.
Iterative Fourier Reconvolution Spectroscopy: van der Waals Broadening of Rydberg Transitions. The 6 k (5p, 6s) Transition of Methyl Iodide +
Arthur M. Halpern Department of Chemistry, Indiana State University, Terre Haute, Indiana 47809 (Received: October 7 , 1991; In Final Form: November 26, 1991)
The pressure-induced shifting and asymmetric broadening of the B (5p, 6s) Rydberg transition of CHJ by the perturbers He, Ar, H2,CH4, and SF6is analyzed using an iterative Fourier reconvolution (IFR) method. Perturbed spectra are analyzed globally, and the full details of the pressure-induced perturbation are accounted for in terms of ground- and excited-state van der Waals complexes between the absorber and perturber. The variance of the observed and calculated perturbed spectra is 0.0123 for an Ar pressure of 81.6 atm. Centrosymmetric potentials are used: 6-12 for the ground state, and exp-6 for the excited state. 6-12 parameters obtained from the IFR analysis are compared with those approximated from simple combining rules. In all cases, the position of the well minimum increases in the excited state whereas the well depth of the van der Waals complex is smaller in the excited state. For the CHJ-He system, repulsive exponential potentials were used for both ground and excited states. The oscillator strength of the CH31transition appears to be conserved between the gas phase and the condensed phase, n-hexane solution (after n correction); at 295 KfgOb = 0.0580 andf,OhF= o.0724. The role of pure homogeneous collisional broadening as an independent, competitive process is discussed. +-
Introduction The very high sensitivity of certain electronic transitions to collision-induced line broadening has long been associated with Rydberg, or extravalence, transitions. In these instances, a molecular electronic transition that may possess many sharp vibronic features under low-pressure conditions will be observed to undergo significant broadening and distortion under the application of 0022-3654/92/2096-2448%03.00/0
moderate pressures (e.g. tens to hundreds of atmospheres).' Indeed, the sensitivity of such transitions to pressure-induced perturbation has been suggested as a tool in assigning a Rydberg transition.* The qualitative rationale for such observations is that (1) (a) Robin, M . B.; Keubler, N. A. J . Mol. Spectrosc. 1970,33,274. (b) Miladi, M.; Falher, J.-P.; Roncin, J.-Y. J . Mol. Spectrosc. 1975, 55, 81.
0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 6, 1992 2449
Iterative Fourier Reconvolution Spectroscopy the collisional cross section of a Rydberg excited state is much larger than that of the ground state, owing to the more diffuse nature of the Rydberg, or extravalence, orbital. Thus it is essentially this difference in collisional cross section between the ground and excited (Rydberg) states that gives rise to this form of line broadening. Stated somewhat differently, the difference in the interaction potentials of the ground and excited states of the absorber vis-&-visthe perturber brings about changes in the line shape of the absorber as a function of the collision frequency (Le., pressure) of the perturber. This mechanism of spectral distortion, sometimes referred to as van der Waals broadening, focuses on the differences in the dispersive and repulsive characteristics of the perturber ground-state and the perturber excited-state van der Waals complexes. In certain cases, e.g., the atomic spectra of atoms (alkalis, Hg, etc.), the van der Waals interactions between the atomic absorber and the perturber (often a rare gas atom) are thought to give rise to the appearance of weak satellite bands at relatively low perturber pressures (below 1 atm).3 Under conditions of very high pressures and, in the limit, in the condensed phase, the vibrational structure associated with Rydberg transitions is frequently obliterated, and the (residual) transition often appears as a very broad, blue-shifted band that is nearly Gaussian in shape. Because of the role that Rydberg states might play in photochemical processes initiated at high excitation energies (above ca. 5 eV) where Rydberg transitions are prevalent, it is important to understand the behavior of the excited Rydberg state in the condensed phase, a medium in which the photochemistry of many organic molecules is studied. Many attempts have been made to analyze quantitatively the pressure-broadened atomic and molecular electronic spectra and thus to invert these spectra into pair potential differences. The pioneering investigations by Margenau and co-workers4 and JablonskiS are noteworthy. An interesting approach has been described by Scott6 in which the perturbed spectrum is expanded in a Taylor’s series of the unperturbed spectrum. More recently, Jortner and co-workers7h a y studied the pressure dependence of the Ar-perturbed fi X transition of CH31 using a moment analysis of the origin band. Saxton and Deutschs have analyzed the shift in the Ar-perturbed absorption and emission spectra of Xe using a scaling technique to obtain the interaction potential functions. We recently reported ,a new approach to the inversion of the pressure-perturbed B X spectrum of CH31into pair potentials9 that is based on the dipole correlator method, which has been described by Jortner and co-workers in this context7 and applied by Page and co-workers’O and Champion et al.” to the analysis of Raman excitation profiles. The application described here has the important advantage of analyzing the pressure-perturbed spectra globally, that is, taking into account the complete line shape profile of the entire electronic transition. The dipole correlator formalism operates in the time domain and thus allows the pressure-perturbed spectrum to be expressed as the Fourier convolution of the unperturbed spectrum with the complex (time-dependent) dipole correlator, which contains the interaction pair potentials. In this paper we illustrate the method further by reporting the results of a study of the pressure-induced line shape
-
-
~~~~~~
~
~
-
Experimental Section Experiments and analysis were commenced at Northeastern University (NU) and completed at Indiana State University (ISU). At NU, absorption spectra were acquired using a Varian 2300 spectrophotometer operating with a band pass of 0.08 nm (20 cm-I). At ISU, measurements were carried out using a Varian Cary 5 with a band pass of 0.05 nm (1 3 cm-l). In both cases, the optical path was purged with dry N2 or Ar. The high-pressure cell was fabricated using a stainless steel six-way cube (MDC Mfg. Co., CU-075-6) fitted with “Del-Seal” blank flanges; flanges having 1/2-in. clearings were used on opposite sides of the cube (the optical axis). Suprasil windows (7/8-in. diameter in. thick) were coupled to the cube sides and flanges with Viton gaskets. The effective cell path length was 3 cm. The perturber gas pressure was measured with a transducer (Omega Engineering Co., PX308-4KGlOV) and was converted to molarity using the van der Waals equation of state. At NU, data analysis was carried out with a DEC Mixrovax Ill using appropriate IMSL subroutines (CSINT, FFTCF/B, and UNLSF). At ISU, data analysis was performed on an IBM RS/6000 workstation and code based on standard spline, fast Fourier transform, and nonlinear least-squares routines. Analytical Method Iterative Fourier Reconvolution (IFR)Analysis. We first present the conceptual and computational methods used to invert the van der Waals-broadened spectra to the pair potentials. The observed unperturbed absorption spectrum of the target species can be expressed using the time correlator approach. The unperturbed absorption spectrum, A(?), which corresponds to the electronic transition between the ground and a single excited state of a molecule or atom can be represented by the imaginary part of the linear susceptibility function, F(?), as follows:loa A(?) = CvYF(?) (1) where the constant Cis proportional to the oscillator strength of the transition. The complex function, F(?), is given by the half-range Fourier integral
~
(2) Robin, M. B. Higher Excited States of Polyatomic Molecules; Academic Press: New York, 1974; Vol. 1, pp 8-47. (3) Mahan, G. D. Phys. Reo. A 1972, 6, 1273. (4) (a) Margenau, H.; Watson, W. W. Reu. Mod. Phys. 1936, 8, 22. (b) Margenau, H. Phys. Reo. 1951, 82, 188. (c) Klein, L.; Margenau, H . J . Chem. Phys. 1959, 30, 1556. (5) Jablonski, A. Phys. Reo. 1945,68, 78. (6) Scott, J. D. In Photophysics and Photochemistry in the Vacuum Ultrauiolet; McGlynn, S. P., Findley, G . L., Heubner, R. H., Eds.; Reidel: Dortrecht, The Netherlands, 1985; pp 729 ff. (7) (a) Messing, I.; Raz, B.; Jortner, J. J . Chem. Phys. 1977,66,4577. (b) Messing, I.; Raz, B.; Jortner, J. Chem. Phys. 1977, 25, 55. (c) Messing, I.; Raz, B.;Jortner, J. Chem. Phys. 1977, 23, 351. (2) Saxton, M. J.; Deutsch, J. M. J . Chem. Phys. 1974, 60, 2800. (9) Halpern, A. M.; Ziegler, L. D. Chem. Phys. Lett. 1989, 161, 1. (10) (a) Page, J. B.; Tonks, D. L. J. Chem. Phys. 1981, 75, 5694. (b) Chan, C. K.; Page, J. B. Chem. Phys. Lett. 1984,104,609. (c) Chan, C. K.; Page, J. B. J. Chem. Phys. 1983, 77, 5234. ( 1 1) Schomacker, K. T.; Srajer, V.;Champion, P. M. J . Chem. Phys. 1987, 86, 1796.
-
distortions of the B 9 (5p, 6s) Rydberg transition of methyl iodide brought about by rare gases He and Ar and by the nonpolar molecules, H2,CH4, and SF,. This transition is an excellent model for quantitatively analyzing the effects of van der Waals perturbations on a Rydberg transition because of the relative simplicity of the spectrum. The transition is characterized by an intense origin at 49726 cm-I (emn = 2.00 X IOS M-Ism-’), which is relatively narrow ( a . 25 cm-I bandwidth). The most prominent vibrational structure consists of two members of a weakly coupled vibrational progression in v i (the symmetric C-H bending mode). Other vibronic features, which are even more weakly allowe, hav! been assigned in that region.I2 Thus, the origin of the B X transition of CH31, being narrow and well isolated, and highly susceptible to pressure broadening, serves as an excellent *marker” for collisional perturbation. Density effects by several rare gases on the high-n molecular Rydberg states of CH31have been studied by KBhler et al.13
F(?) = i i mexp(rvt)G(t)fo(r) dt
(2)
in which G ( 7 ) is the relaxation that accounts for the dephasing of the initially prepared levels of the excited state. Although G ( t ) is generally represented as a simple exponential damping function (that produces the Lorenzian line shape in the frequency domain), more complicated forms of G ( t ) ,such as stretched exponentials, have been suggested to describe distributed relaxation in heterogeneous systems.” In the application reported here, however, G(t) is regarded as phenomenological. This point is significant because the dephasing function for a polyatomic molecule may (12) Felps, S.;Hochmann, P.; Brint, P.; McGlynn, S.P. J. Mol. Spectrosc. 1976, 59, 355. (13) Kohler, A. M.; Reininger, R.; Saile, V.; Findley, G . L. Phys. Reo. A 1987, 35, 79.
2450 The Journal of Physical Chemistry, Vol. 96, No. 6, 1992
not be expressible in analytical form. fo(t) is the dipole correlation function, or time correlator, which has been expressed in analytical form in application to several model cases, e.g., multimode excitation at finite temperature.I4 In the method described here, however, f o ( t ) is also considered to be a phenomenological entity that represents the full FranckCondon activity of the mode@) that is(are) coupled into the electronic transition. The important point here is that the product of the two time-dependent functions, G ( t ) andf,(t), defined as Z(t),viz.
= G(t)fo(t)
(3)
is the full time domain representation of the observed unperturbed absorption spectrum. We can rewrite eq 2 in terms of the time domain representation of the unperturbed spectrum, namely F(5) = i x mexp(iiU)Z(t) dt
(4)
Halpern
,
-
Figure 1. Schematic diagram of the IFR method. The observed, unperturbed absorption spectrum, A(?), is Fourier transformed to Z ( t ) , which is convolved with the dipole correlator,fp(t). fp(t) is constructed from the pair potentials V,(r) and V,(r) and a perturber number density M at a temperature T. The calculated spectrum, I=,#), obtained from the Fourier backtransform, is compared with the experimental perturbed
using a nonlinear regression procedure, which furnishes spectrum, lobs(?), the optimized potential parameters. sorption spectrum can be expressed mathematically by taking the product of the two (complex) time domain functions, Z ( t ) and fp(t). Thus if P(t) is defined as P(t)
and it can now be seen from eq 4 that Z(t) is the Fourier transform of A(:). Thus Z(t) can be obtained from the experimental, unperturbed spectrum. The computational details for obtaining Z(r) are outlined in the Appendix. We can incorporate the effects of inhomogeneousvan der Waals broadening on the absorption spectrum in the time domain through the use of the dipole correlator. If the perturber number density is sufficiently low such that only two-body (i.e., absorber-perturber) interactions need be considered, one can show that the dipole correlator that accounts for the dephasing of the initially prepared excited state caused by a statistical distribution of perturbers (surrounding an absorber) has the form7c
where D is the perturber number density, r is the absorber-perturber separation, gI2is the pair distribution function of perturbing species about an absorber, and AV(r) is V,(r) - V8(r), the difference in the pair potentials of the excited- and ground-state absorber and perturber. We assume at this point that these potentials are centrosymmetric; hence, glz(r) is the pairwise radial distribution function (RDF). Although the use of such potentials obviously represents a simplification, in view of the anisotropy of CHJ, we present the results here at this level of approximation to keep the number of parameters to a minimum (four or five). It may be henceforth possible to use a potential involving an expansion in spherical harmonics to represent anisotropy. A very useful simplification that stems from the assumption of pair interactions (low perturber number densities) is that the RDF can be approximated by the Boltzmann factor of the ground-state interaction potential, viz. g12m
exp[-vg(r)/wl
(61
It should be noted here that the method is applicable to a system under equilibrium conditions; hence, the system temperature and pressure are well-defined. Although spectroscopic measurements provide direct information about differences in state properties, the interaction potential of the ground electronic state of the absorber with the perturber appears independently through eq 6. In principle, the method described here can uncover the absolute forms of V,(r) and V,(r). However, as will be shown below, the differences in interaction distances and well depths between the ground and excited electronic state van der Waals complexes are reliably obtained using the technique described here. Nevertheless, the use of eq 6 for the RDF imposes a degree of self-consistency on the analysis. The main idea of the method is that the imposition of the pressure-induced van der Waals broadening on the observed ab(14) Schomacker, K. T.; Champion, P.M . J . Chem. Phys. 1986,84, 5314.
= a t ) XfP(f)
(7)
then we can associate the line shape function of the observed perturbed absorption spectrum as the Fourier transform of P(t), i.e.
Zcalc(:) can be compared with the experimentally observed line shape function, I,,&). The transformation of the experimental absorption spectrum to the line shape function is outlined in the Appendix. From eqs 7 and 8, it is evident that the calculated van der Waals-broadened line shape function is obtained from the Fourier convolution of the experimental, unperturbed line shape function with the time correlator,fp(t). The latter function depends on the analytical forms chosen for the interaction potentials, V&r) and Ve(r), and the system conditions: the perturber number density and the temperature. An implicit assumption in this method is that the empirical relaxation function, G ( t ) ,and the dipole correlator,fo(t), of the unperturbed spectrum are independent of perturber pressure. In other words, line broadening is accounted for solely in terms of van der Waals interactions. The relaxation of this assumption results in the inclusion of homogeneous collisional (or interruption) broadening. These two points will be discussed below. In principle, the interaction potentials, V&r) and Ve(r), can be obtained in numerical form using Fourier deconvolution methods. However, this approach is fraught with mathematical difficulties. It is interesting to note that this type of problem also pertains to the extraction of the “true” decay function in time domain luminescence experiments in which the excitation pulse distorts the observed decay curve.I5 In the approach described here, analytical forms of the interaction potentials will be used to analyze the pressure-perturbed spectrum of an electronic transition through eq 5 and 6 and the use of appropriate Fourier transforms. Interaction potential parameters contained in Vg(r) and VJr) will be found that optimize the fit between the calculated and observed perturbed spectra. This procedure is thus an iterative Fourier reconvolution technique in which optimization is controlled by a nonlinear least-squares routine. Although Fourier transforms are repeatedly performed (often many hundreds of times) in an analysis, this method is practical because of the availability of highly efficient Fourier transform algorithms. The method is portrayed schematically in Figure 1.
-
Results and Discussion The use of the IFR method is demonstrated using the B 3 transition of methyl iodide as perturbed by Ar, He, H,, CH4, and SF6. It is interesting to note that with all of :he pefiurbers studied here no satellite bands associated with the B X of CH31 were
-
(15) Halpern, A. M.; Frye, 40, 5 5 5 .
S. L.;KO,J.-J. Pholochem. Photobiol. 1984,
The Journal of Physical Chemistry, Vol. 96, No. 6, 1992 2451
Iterative Fourier Reconvolution Spectroscopy
TABLE I: Optimized Pair Potential Parameters for the Ground- and Excited-State CHJ-X van der Waals Complexes Obtained Using the Iterative Fourier Reconvolution MethodL perturber rmg/A emg/cm-' rme/A c,,,Jcm-' a X Arm Ac Ar (3.64 M) 4.944 209 5.330 166 12.63 0.0123 0.39 -43 Ar (3.64 M) 4.990 264 5.238 232 b 0.0147 0.25 -32 Ar (4.90 M) 4.967 223 5.335 184 12.83 0.0165 0.37 -39 Ar
4.55SC 6.79 3.59c 4.970 5.530' 5.057 4.740'
H2 (3.59 M) H2 SF, (1.32 M) SF6
CH4 (1.71 M) CH4
149' 65.6 122' 335 293' 307 240'
7.43
52.0
15.34
0.0216
0.64
-14
5.169
334
12.293
0.01 50
0.20
-1
5.302
29 1
14.33
0.0117
0.25
-16
"The Lennard-Jones 6-12 potential, eq 9, was used for the ground state, and a modified Buckingham exp-6 potential, eq 10, was used for the excited state (except for ( b ) ) . b6-12 potential used for Ve(r).'Calculated from rg = (1/2)(r, + r,); data from ref 20. dCalculatedfrom e,, = (qc,)'/2; data from ref 20.
w
0
1
z
4
m
U 0 u)
s 0
0.5 H
Q
J
a
I
a 0 z
0.-
\
Vg ( R )
-
-200-
0.0
49500
50000
50500
WAVENUMBERS.
51000
51500
CM-1
Figure 2. Absorption spectra of CHJ vapor alone (-) and in the presence of 81.6 atm (3.64 M) Ar ( - - - ) at 295 K. The IFR-fitted corresponds to the potential parameters shown in Table I. spectrum Spectra are maximum normalized to 1.O. (-e)
observed. This is in contrast with the situation in the resonance transitions of Xe (which is formally isoelectronic with Xe), in which satellite bands are observed with several rare gas perturb e r ~ . ~The ~ Jabsence ~ of such bands may be associated with the fact that the highest perturber pressure used in this study is 136 atm (for Ar). It is in fact an objective in future studies employing the IFR method to determine whether it can account for the formation of satellite bands. Perturbation by Argon. Figure 2 shows the unperturbed absorption spectrum of methyl iodide and the spectrum produced in the presence of 81.6 atm (3.64 M) Ar. The two spectra are normalized at the respective frequencies of maximum absorbance. We do not have sufficient data a t this time to ascertain the accurate dependence of the oscillator strength of this transition with respect to Ar pressure. Preliminary results indicate, however, that this dependence is small (see below). As is evident in Figure 2, the origin band head, and that of the v2 fundamental, are red shifted, the former by ca. 57 cm-I. In addition, the Ar-induced perturbation results in a dramatic asymmetric broadening to the blue, the result of which is a significant increase in bandwidth. For example, the origin, which has a full width at half-maximum (fwhm) of ca. 25 cm-l in the unperturbed spectrum, has a fwhm of 180 cm-' in the presence of 81.6 atm (3.64 M) Ar. The optimized fit using the I F R method is also shown in Figure 2. This analysis, which is based on a Lennard-Jones (6-12) potential for VgW
(16) Castex, M. C.; Granier, R.; Romand, J. Comptes Rendue, Acad. Sci. (Paris) Ser. B, 1969, 268, 5 5 2 .
for the excited-state interaction, closely matches both the red shift of the band maxima and also accounts well for the contours of the asymmetrically blue-shifted bands. The quality of this fiveparameter fit is evidenced by its statistical variance, x, which is 0.0123. In eqs 9 and 10, tgand t, are the well depths of the ground and excited van der Waals complexes, respectively, and rm and r,, are the pair separations at the potential minima. a is a stijfness parameter that describes the repulsive wall. Figure 3 displays the interaction potentials uncovered by the IFR method, Vg(r),V&), and AV(r) (see eqs 9 and 10). A somewhat poorer fit is obtained (x = 0.0147) if 6-12 potentials are used for both the ground and excited states. This is not unexpected in part because of the reduction of the problem to a four-parameter analysis. Moreover, the 6-1 2 parameters returned for the CHJ-Ar ground-state complex differ somewhat from those obtained using the exp-6 potential for the excited-state complex. It is not possible to determine at this point whether the exp-6 potential is intrinsically superior to the 6-12 potential in depicting the excited CH31-Ar complex. Nevertheless, this difference in the V&r) parameters can be understood in view of the fact that it is the difference between the upper and lower state potentials, AV(r), that primarily affects the position and line shape of the van der Waals broadening. As mentioned above, the separation (17) Hirschfelder, J . 0.;Curtiss, C. F.; Bird, R. B. Molecular Theory OJ Gases and Liquids; John Wiley & Sons, Inc.: New York, 1964; pp 33-34.
2452 The Journal of Physical Chemistry, Vol. 96,No.6,1992
of V&r) from AV(r) is possible because it is explicitly expressed in the pair distribution function, g12;see eq 6. The dependence of the calculated perturbed spectrum on Vg(r), however, is not very strong; hence, the parameters associated with the individual potentials are softer than are their differences, e.g., Armand Ac. The optimized potential parameters obtained for the CH31-Ar system are summarized in Table I. The two V,(r) potentials used indicate very similar values for rmgand r,, but differ considerably in the well depths. Differences in these parameters are reasonably consistent, however, and indicate an increase of 0.25-0.39 A for the Ar-CH,I “bond” length in the CH31-Ar van der Waals complex in the excited state vis-&vis the ground state. It is significant that the analysis indicates :he excited-state complex to be less strongly bound in the CH31B state by ca. 32-43 cm-I relative to the ground state. This reduction isperhaps a result of the lower electron density of CHJ in the B state, which is characterized by the more extensive IbSorbital. As a test of the IFR method, it is instructive to compare the well depth and the collision diameter of the ground-state van der Waals complex as extracted from the spectra with the respective values obtained from physical properties of CHJ and Ar by using the simple combining rules’* rm,, =
(y2)(ri + rj)
Ci,
=
(EiC,)’”
(11)
As can be seen in Table I, these approximations give values of rmgand cg that are smaller than those obtained from the IFR analysis. A slight refinement can be made in these approximations using the angle-averaged Stockmayer potential,I9 which accounts for the interaction between a polar molecule and a nonpolar species. This correction takes the dipole moment of CH31 and the polarizability of the nonpolar perturber into account. With such modifications, the values of rmgand cg become 4.600 A and 152 cm-I, respectively, for the CH,I-Ar complex. Given the crudeness of the approximations made in the combining rules, and especially the variation in the self-association parameters for CH31 that exists in the literature, we present the data in Table I based on the simple combining rules and parameters taken from a single sourceZofor comparative purposes. At higher Ar pressure (e.g., 100-120 atm), the van der Waals distortion of the CH31spectrum increases; the red shift increases, as does the extent of the asymmetric broadening to the blue. For example, the IFR analysis of the spectrum acquired with 109 atm (4.9 M) Ar using 6-12 and exp6 potentials produces a satisfactory fit with x = 0.0165, a value slightly larger than that obtained at 81.6 atm. The five parameters uncovered from these data are also listed in Table I. It is likely that the slightly poorer fit and different parameters returned for the higher pressure system indicate that many-body interactions between Ar and CHJ are more significant at the higher Ar pressure. Nevertheless, Ar and ACvalues obtained with 3.64 and 4.90 M Ar are in close agreement. The ground-state and excited-state pair potential thus uncovered for the CH31-Ar system can be used to calculate the Ar-perturbed CH31 spectrum at any arbitrary Ar number density. Such calculations show that the red shift in the band heads observed at moderate pressures gives rise to a blue shift at much higher Ar number densities. This is in accordance with the qualitative predictions based on the shape of the AVvs r curve extracted from the data at moderate Ar pressure (see Figure 3). Although we do not have direct experimental data on systems at Ar pressures higher than 137 atm, the pressure at which the red-shifted band head “turns around” and moves to the blue is determined from simulations of pressure-perturbed spectra to be ca. 6.2 M, or 135 atm. We now discuss the behavior of the CH31 (5p, 6s) transition in the limit of very high perturber number density, the condensed phase. For convenience, we used n-hexane as the nonpolar solvent at 295 K; the quantitative absorption spectrum is shown in Figure
Halpern *
250000
000
* 0 I
..
U
* 200000 \
I-
t
f~150000
H
0 H LL
=u 100000
0
0
2 0 5 + X W
50000
0
48000
IX
49000
50000 51000 52000 WAVENUMBERS.
53000
54000
w
CM-I
Figure 4. Absorption spectra of CH31 in the gas phase (left-hand axis) and in n-hexane solution (right-hand axis) at 295 K.
4. As can be seen,the condensed-phase spectrum is considerably blue shifted and Gaussian in appearance (FWHM N 2530 cm-I). For example, t,, is 51 500 cm-I, as compared with the gas-phase origin, which is at 49 726 cm-I. It is interesting to compare the oscillator strength of the CH31 (5p, 6s) transition in the absence of van der Waals broadening with that in the condensed phase. For experimental reasons, it was not possible to obtain reliable values of the oscillator strength of the CH31transition in the high pressure perturbed systems. Nevertheless, it is valuable to learn whether the oscillator strength is conserved between the limiting cases portrayed in Figure 4,since the IFR method assumes that fo(t) is pressure independent (see eq 2). In order to make this comparison quantitative, however, the calculation of the oscillator strength in n-hexane solution must take into account at least the dispersive property of that medium. We use for that purpose a relation that attempts to account for the local field correction that is based on the Onsager cavity model as follows:21,22 uC/fg) = L2/n
(12)
where f, and fg denote the oscillator strength of a transition in the condensed phase and gas phase, respectively. L2 is a function of n, the refractive index, that accounts for the dispersive property of the medium. With L2 = ((n2
+ 2)/3)2
(13)
and n = 1.482 at 200 nm,23the quotient fc/fg = 1.32. Although this model does not take the structural anisotropy of the solute into account, and may not adequately represent the solvation of Rydberg, or extravalence states, in a continuous medium, it represents the minimal necessary correction that is required here. Unfortunately, the accuracy of experimental value off, is compromised by the fact that the condensed-phase spectrum extends above 54 000 cm-l, which is the high-energy limit of our instrument. We estimated the value off, by smoothly truncating the absorption spectrum at higher energy, and obtained thereby a value of 0.0724 at 295 K. In having to take this measure, we estimate the error in fcto be less than 10%. The experimental value of fg, which is more precisely determined, is found to be 0.0580 at 295 K. The experimental ratiof,/f, is thus 1.25, which compares favorably with 1.32, as found above. Although this comparison is too imprecise to permit a rigorous test of the validity of the expressions in eqs 11 and 13, it does indicate that there is reasonably close conservation of transition strength in the CH31(5p, 6s) Rydberg between the low-pressure gas phase to n-hexane solution. (a) Vibrational Spectrum. It was considered worthwhile to calculate the one-dimensional vibrational spectrum of the binary CHJ-Ar van der Waals complex. This was accomplished by ~~~~~~~
(18) Reference 17, p 168. (19) Reference 17, pp 597-600. (20) Mourits, F. M.; Rumens, H . A. Can. J . Chem. 1977, 55, 3007.
(21) Sension, R. J.; S t r a w , H . L. J . Chem. Phys. 1986, 8 5 , 3739. (22) See also: Myers, A. 8.; Birge, R. R. J . Chem. Phys. 1980, 73, 5314. (23) MacRae, R. A.; Arakawa, E. T.; Williams, M. W. J . Chem. Eng. Data 1978, 23, 189.
The Journal of Physical Chemistry, Vol. 96, No. 6,1992 2453
Iterative Fourier Reconvolution Spectroscopy
5 0.5
1
-0.0
-.--
-
49500
50000
q-M-MJk-1
50500
WAVENUMBERS.
WAVENUMBERS, CM-I
Figure 5. Calculated Franck-Condonprofile of the absorption spectrum of the CHJ-Ar van der Waals complex based on the V&r) and V J r ) potentials shown in Figure 3. A vibrational temperature of 30 K and a Gaussian line width of 3 cm-l were used in the calculation. The vibrationless transition is shown at 0 wavenumbers.
diagonalizing the one-dimensional Schrijdinger equations in Vg(r) and V,(r) using a finite element method and obtaining the respective eigenfunctions and eigenvalue^.^^ The reduced mass of the pseudo-diatomic CH31-Ar pair was used in this calculation. The Franck-Condon matrix was thus assembled, and the absorption spectrum was calculated from a set of Gaussian-broadened sticks whose intensities are given by ZVr,,,,
0:
B,,,( T)F~,,,~?
(14)
where B,,,( T ) is the Boltzmann factor, exp(-E,,,/kT), and Fvtt,v, is the overlap of the u"th ground-state and u'th excited-state wave function. The calculation indicates ground- and excited-state fundamental van der Waals stretching frequencies of 23.6 and 18.8 cm-', respectively. Figure 5 shows this spectrum calculated from the optimized 6-1 2 and exp-6 potentials discussed earlier and a vibrational temperature of 30 K. For presentational purposes, the lines are given a Gaussian line width of 3 cm-l, which roughly corresponds to the full rotational line width estimated at that temperature for the CH31-Ar complex with B = 0.028 cm-l, based on the separation of CH31 and Ar point masses by 4.944 A, the value of rmg(see Table I). (b) Collisiooal Broadening. At this point, we discuss an attempt to incorporate interruption, or collisional broadening, into the scheme of pressure-induced perturbation. Although this process might be expected to contribute a small component to the total spectral perturbation, its inclusion may result in a more complete description of the line broadening observed. Mukamel has recently considered the problem of homogeneous and inhomogeneous line broadening using a stochastic model and has discussed the interpolation between these limiting dephasing mechanism^.^^ Accordingly, a pressure-dependent exponential damping factor was added to the phenomenological zero-pressure relaxation function, Z ( t ) (see eq 3), such that Z ( t ) reads
a t ) = G(t)fdt)GP(t) (15) where the function G,(t) accounts for the purely collision-induced dephasing, which can be taken to occur with gas kinetic cross section. Thus G,(t) = exp(-rt)
(16)
where I.( is the reduced mass of the absorber-perturber pair, kB (24) Halpern, A. M.; Ziegler, L. D.; Ondrechen, M. 0.J . Am. Chem. Soc. 1986, 108, 3907. (25) Mukamel, S. J . Chem. Phys. 1985, 82, 5398. (26) He data taken from ref 17, p 11 10.
-.. 51000
51500
CM- 1
Figure 6. Absorption spectra of CHJ alone (-) and in the presence of 110 atm (4.10 M) He at 295 K. The IFR-fitted spectrum (---) was determined from ground- and excited-state potentials represented by eq 18. The optimized parameters are contained in the text. Spectra are maximum normalized to 1.0. (e-)
is the Boltzmann constant, and the other terms are defined above. The incorporation of r from eq 17, which expresses the collisional frequency of excited-state molecules with perturbing species based on simple gas kinetic theory, into eq 15 assumes that purely homogeneous dephasing is strictly additive with inhomogeneous van der Waals broadening. When the IFR procedure was applied to the CH31-Ar system discussed earlier ([Ar] = 3.64 M), and eq 15 was used to obtain Z ( t ) ,a poorer fit to the observed pressure-perturbed spectrum was obtained (i.e., x = 0.0247; cf. 0.0123; see Table I). Moreover, although not surprisingly, different parameters for the groundand excited-state pair potentials were obtained. Thus, when collisional broadening was additionally incorporated, the following results were obtained (cf. Table I): rmg= 5.180 A; eg = 306 cm-'; rme = 5.414 A; t, = 283 cm-'; CY = 14.05. The main difference between these sets of parameters is in the well depths, which are significantly larger when collisional broadening is included in Z(t),Le., by 94 cm-' for the ground state. The difference in potential well minima as determined using Z(t) from eq 15, i.e., Ar = 0.23 A, is comparable with that obtained without this mechanism (see Table I). The difference in well depths, however, is smaller, -0.23 cm-' as compared with -43 cm-I. It should be noted that the value of r in eq 17 is calculated to be ca. 18 cm-I for the CH31-Ar system described above. This compares with the fwhm of the Ar-perturbed spectrum of CH31 at 3.64 M Ar, which is 180 cm-'. Because of the order of magnitude difference in these line widths (and hence in the respective time scales), and also because it is not clear whether collision broadening should be included as an independent, competitive process vis-&vis van der Waals broadening, as implied in eq 15, we did not include collision broadening in the remainder of the IFR applications described here. A further rationalization of this approach is provided by the facts that the quality of fit is poorer and that the larger well depth determined above (i.e., e = 306 cm-') is at further variance with the well depth of the CH,-Ar pair estimated from combining rules relative to the value reported in Table I (Le., 209 cm-I). Perturbation by Helium. The IFR methcd was applied to an analysis of the pressure broadening of the B X transition of CH31by He. The results are shown in Figure 6 for a He pressure of 110 atm (4.10 M). It is evident that, as in the case of Ar perturbation, the spectrum is asymmetrically broadened to higher energy; however, unlike that system, the maximum of the Heperturbed spectrum is blue shifted (by ca. 23 cm-') relative to the "zero-pressure" spectrum. It should be noted that over the range of He pressures studied (103-164 atm; 3.87-5.86 M), only blue-shifted spectra were observed. In analyzing these spectra in terms of a 6-12 ground-state potential and a exp-6 or 6-12 excited-state potential, we consistently obtained unreasonable values for the ground-state parameters. For example rmgexceeded 10 8, and tmgconverged at values less than 0.1 cm-l. For com-
-
2454 The Journal of Physical Chemistry, Vol. 96, No. 6, 1992
Halpern
I w 1 .o
u z a m a 0 cn m U cl
p
.5
-I
a L
\
,500 000~ 3.'5
0 0
4
4.'5
z
5
5.'5
6
6.'5
7
0 .o
49500
50000
R/ANGSTROMS
51000
WAVENUMBERS. CM-1
Figure 7. Potential curves, V8(r),Ve(r),and AV(r), for the CH31-He pair obtained from the IFR analysis using the potential in eq 18. The parameters are contained in the text.
parison, the algebraic mean of the CH31and He collision diameters is 3.61 A, and the geometric mean of the self-association well depths is 64.4 cm-1.26These results indicate that the ground-state CH31-He pair is very weakly bound, and possibly repulsive. We found that the CHJ-He system could be successfully analyzed, however, using a repulsive potential for both ground and excited states. The potential function we used has the form V(r) = A exp(Br)
50500
Figure 8. Absorption spectra of CHJ alone (-) and in the presence of 93.0 atm (3.59 M) H2 at 295 K. The IFR-optimized spectrum (---) corresponds to the parameters shown in Table I. The leading edge artifact produced in the calculated spectrum (below ca. 49 700 cm-I) is rejected from the x 2 calculation (see text). Spectra are maximum normalized to 1.0. (-a)
(18)
The optimized parameters for the data shown in Figure 6 are as follows: A , = 2.750 X lo* cm-I; B, = 2.944 A-l; A, = 3.071 X lo6 cm-l; Be = 1.751 A-'. These potentials, and their difference, are displayed in Figure 7 . The repulsive wall is steeper for the excited-state complex relative to the ground state. AV vs r is negative, indicating that blue-shifted spectra should be observed, as is the case. The maximum in the AVvs r curve that appears at low r values, which corresponds to the very high pressure regime, may not be quantitatively significant. It should be noted here that the leading edge of the calculated spectrum shown in Figure 6 (below a.49 700 cm-') is an example of the "wraparound effect" produced in the Fourier transform. This is caused by the fact that the observed perturbed spectrum (and hence also the calculated spectrum) does not approach zero at the high-energy limit (see Figure 6). The R procedure, which produces a continuous spectrum between the initial and final points, forces the nonzero leading edge onto the calculated spectrum. In order to avoid computational artifacts that may arise from this complication, a discrimination threshold was imposed on the observed and calculated spectral arrays such that values less than ca. 3% of the maximum values were rejected from the x2 calculation. The x2 cutoff threshold was chosen according to the requirements of each individual data set. Perturbation by H2. The application of 93.0 atm (3.59 M) H2 causes only a very small red shift (ca. 10 cm-I) in the origin of the methyl iodide spectrum. Accompanying this shift is the characteristic asymmetric broadening to higher energy. The unperturbed, H2-perturbed, and IFR-analyzed spectra are shown in Figure 8. These data were difficult to fit unambiguously; evidently there are several local minima in the x2 surface. Using a 6-1 2 ground-state and exp-6 excited-state potential for the CH31-H2 pair, we were able to locate a global minimum (with x = 0.0216) that indicates a loosely bound complex for both states. The data are summarized in Table I. The results indicate a relatively large CHJ-H2 separation, i.e., 6.79 A for the ground state with an increase of 0.64 8,in the excited state. Also, as in the case with the Ar complex, the IFR analysis suggests that the well depth of the complex decreases (by ca. 13.6 cm-I) upon electronic excitation of CH31. For comparative purposes, we note that the mean self-association rmgvalues of H2and CHJ is 4.27 A, which is considerably smaller than rmgfound for the complex.
J
n n
-.-
-
.....
.....
49500
50000
b
b
.....
50500
5iOOO
51500
WAVENUMBERS. CM-1
Figure 9. Absorption spectra of CHJ alone (-) and in the presence of 22.8 atm (1.32 M) SF6 at 295 K. The optimized spectrum using the IFR method (- -) is based on potential parameters shown in Table I. Spectra are maximum normalized to 1.0.
-
(e-)
Also, the geometric mean of the self-association well depths is 84.7 cm-l, which compares with 65.6 found for the complex. To ensure that the data reported above for the CH31-H2system corresponded to the global minimum, a grid search was conducted of the five-dimensional x2 surface. This procedure, which required the calculation of about 6000 6, arrays (see eq 8), spanned preset ranges of each of the five fitting parameters. Once the global minimum was found, it was used as the starting point in the more sensitive least-squares algorithm. Not surprisingly, we found that a fit based on repulsive potentials (eq 18) could reproduce the data, but with a somewhat larger x value, 0.071. Perturbation by SF6. Finally, we report the results of the pressure-perturbed spectrum of CHJ in the presence of SF6,a sperical, nonpolar species that has a relatively large collision diameter (5.2 A).2o With this perturber, however, we were limited by its vapor pressure at 295 K, Le., 22.8 atm (1.32 M). The perturbed and IFR-fit spectra are displayed in Figure 9. The quality of the fit is very good, with x = 0.0105. The ground-state 6-1 2 and excited-state exp-6 parameters that are returned are summarized in Table I. These results show that rmgfor the CHJ-SF, complex is 4.97 A, which compares closely with the mean of the self-association rm values of the species, 4.93 A. The ground-state well depth of the complex as determined from the IFR analysis is 319 cm-l, which is larger than the geometric mean of the self-association well depths, which is 204 cm-I. It should
J. Phys. Chem. 1992, 96, 2455-2463
be noted, however, that the well depth data used for this comparison are calculated from critical point data. The excited-state parameters interestingly show little difference relative to the respective ground-state values. For example, r,, is 5.17 A, which is only 0.20 A larger than the ground-state value. Also, the well depth of the excited-state complex is nearly identical to the ground-state value.
Conclusion The dipole correlator method provides a successful route to the Fourier reconvolution of an unperturbed electronic absorption to generate the spectrum broadened by van der Waals interactions between the absorber and perturber. Using nonlinear optimization methods, the potential parameters describing the ground- and excited-state van der Waals complexes can be inverted from the spectra. The iterative Fourier reconvolution (IFR) method is thus capable of accurately accounting for the complete line shape of a perturbed spectrum without having to resort to cruder approaches, such as monomer(s) analysis. Although a t this point only centrosymmetric potentials have been used, the data uncovered from pressureperturbed spectra using IFR provide insights to molecular interactions that may not easily yield to analysis with anisotropic potentials. Moreover, that data are complementary to results obtained from other studies, such as scattering experiments. It is not clear whether inhomogeneous van der Waals interactions ("sticky" collisions), involving a distribution of perturbed absorbers, operates independently of homogeneous, or purely collisional dephasing. In all cases studied here of the pressure perturbation of the (5p, 6s) transition of CH31,the IFR analysis indicates an increase in the van der Waals complex well minimum in the excited state relative to the ground state. This increases ranges from about
2455
0.20A for SF6 to ca. 0.64 A for H2. In addition, all systems reported here reveal that the van der Waals complex is less bound in the excited state than in the ground state. These changes range from ca. 0 for SF6 to 43 cm-I for Ar. The ability of the IFR method to extract potential parameters from pressure-perturbed spectra of CH31with such relatively high precision suggests that it can be applied to other, more complex molecular systems. Comparisons between molecular Rydberg and intravalence transitions can be made using IFR in terms of the behavior of these types of excited states toward perturbers. Work along these lines is continuing. Acknowledgment. Profs. L. D. Ziegler and P. M. Champion are acknowledged for many helpful discussions. Ms. V. Srajer provided help with the FT and Hilbert transform code. Messrs. A. Taaghol and B. King assisted with preliminary data acquisition.
Appendix The normalized line shape function, I(?), which is produced in the calculation from eq 8, is obtained from the observed absorption spectrum, A(?) through
The imaginary part of the complex susceptability function, F(I), is proportional to the line shape function, I(?) (see eq 1). The real component of F(?), which is proportional to d?' I(?')/(?' - ?), is obtained as the Hilbert transform of I(?). An analytical expression for obtaining the Hilbert transform is presented in ref 1Oc. The complex array constructed from the experimental unperturbed absorption spectrum that forms the input to the F R program to obtain Z ( t ) (see eq 7) is (H(?), I(?)), where H(F) is the Hilbert transform of I(?).
Changes in the Structure of Alkali-Metal Silicate Glasses with the Type of Network Modifier Cation: An ab Initio Molecular Orbital Study Takas& Uchino,* Tetsuo Sakka, Yukio Ogata, and Matae Iwasaki Institute of Atomic Energy, Kyoto University, Uji,Kyoto-Fu 61 I , Japan (Received: August 12, 1991)
The structure of alkali-metal silicate glasses has been investigated by ab initio molecular orbital calculations on clusters H$iz07 and H4Si207MZ(M = Li, Na, K). The optimized geometries at the 3-21G* level are in good agreement with the experimental values obtained by diffraction techniques. Our calculations have further revealed that for a given alkali-metal concentration the silicon-bridging oxygen bond length increases while the silicon-nonbridging oxygen bond length decreases in the order Li, Na, K. Spectroscopic data such as photoelectron, X-ray emission, and UV excitation energies are satisfactorilyreproduced by the calculations, and 29SiNMR chemical shifts are interpreted in terms of charge distributions calculated for the clusters. Both Si and 0 d functions contain substantial populations although the 0 d orbitals hardly contribute to the Si-0 overlap populations. Activation energies for conduction of alkali-metal cations are estimated on the basis of the calculated charges and optimized geometries. We also discuss the mechanism of the charge transfer from alkali-metal to network_formingatoms in the course of the reaction between alkali-metal oxides and the Si-0-Si network.
1. Introduction
Spectroscopic analyses such as X-ray photoelectron (XPS), nuclear magnetic resonance (NMR), X-ray emission, ultraviolet (UV), infrared (IR), and Raman spectroscopies have been extensively used to characterize the nature of the network-forming and network-modifying atoms in silica and alkali-metal silicate glasses. XPS studies, for example, have demonstrated that two types of oxygen atoms exist in alkali-metal silicate glasses:'i2 H.-U.; Goretzki, H. Glasrech. Ber. 1978, SI, H.-U.; Goretzki, H.;Sammet, M. J. Non-Cryst.
(1) (a) Brkkner, R.; Chun, 1. (b) Briickner, R.; Chun, Solids 1980, 42, 49.
0022-3654/92/2096-2455$03.00/0
oxygens bonded to two silicons, Le., bridging oxygen (0,)atoms, and oxygens bonded to only one silicon, Le., nonbridging oxygen (Onb)atoms. Alkali metals in these glasses are considered to sit near the nonbridging oxygen atoms in order to maintain the local charge balance. Also, recent high-resolution 29SiN M R studies have investigated the distribution of Q" units in alkali-metal silicate glasses through deconvolution of the 29Siline where Q (2) Smets, B. M. J.; Lommen, T. P. A. J . Non-Cryst. Solids 1981, 46, 21. (3) (a) Lippmaa, E.; Magi, M.; Samonson, A,; Engelhardt, G.;Grimmer, A. R. J . Am. Chem. SOC.1980, 102, 4889. (b) Grimmer, A.-R.; MHgi, M.; Hahnert, M.; Stade, H.; Samonson, A.; Wieker, W.; Lippmaa, E. Phys. Chem. Glasses 1984, 25, 105.
0 1992 American Chemical Society
