ARTICLE pubs.acs.org/JPCA
Analysis of the Visible Absorption Spectrum of I2 in Inert Solvents Using a Physical Model Joel Tellinghuisen Department of Chemistry, Vanderbilt University, Nashville, Tennessee 37235, United States ABSTRACT: Absorption spectra of I2 dissolved in n-heptane and CCl4 are analyzed with a quantum gas-phase model, in which spectra at four temperatures between 15 and 50 C are least-squares fitted by bound-free spectral simulations to obtain estimates of the excited-state potential energy curves and transition moment functions for the three component bands—A r X, B r X, and C r X. Compared with a phenomenological band-fitting model used previously on these spectra, the physical model (1) is better statistically, and (2) yields component bands with less variability. The results support the earlier tentative conclusion that most of the ∼20% gain in intensity in solution is attributable to the C r X transition. The Tdependent changes in the spectrum are accounted for by potential energy shifts that are linear in T and negative (giving red shifts in the spectra) and about twice as large for CCl4 as for heptane. The derived upper potentials resemble those in the gas phase, with one major exception: In the statistically best convergence mode, the A potential is much lower and steeper, with a strongly varying transition moment function. This observation leads to the realization that two markedly different potential curves can give nearly identical absorption spectra.
’ INTRODUCTION The I2 visible absorption spectrum is comprised of three overlapping electronic transitions—C 1u (1Π) r X (1Σ+), B 0u+ (3Π) r X, and A 1u (3Π) r X, in order of increasing wavelength.1 In the gas phase, the latter two are very well-known from their rich line spectra,26 while the first (once called B00 X7 but relabeled CX in analogy to the corresponding transitions in Br2 and Cl28) is entirely bound-free and thus has remained less well characterized. However, in a recent reanalysis of lowresolution absorption data over the 400850-nm region, together with data for B f C predissociation and the van der Waals well of the C state, I achieved what should be a significant improvement in the description of the C potential and its absorption and predissociation transitions.9 This analysis employed nonlinear least-squares (LS) simulation of spectra recorded at several temperatures, with numerical evaluation of the quantum mechanical overlap integrals that govern absorption strengths and predissociative decay rates. For I2 dissolved in “inert” solvents like heptane and CCl4, the absorption spectrum resembles that in the gas phase, but is ∼20% stronger and blue-shifted.8,1012 Efforts to resolve such spectra into their components have frequently involved LS fitting to sums of bands of assumed mathematical form, like Gaussians and log-normal functions, sometimes with temperature dependence taken into account.13,14 In a comparative study of such phenomenological methods, my colleagues and I showed that the components and their strengths could vary widely with choice of band form, and that the statistically best resolution was not necessarily the physically best.8 Still, this approach remains popular, as it is relatively easy to use and can give good overall parameterizations of spectra.1519 In the case of I2 in solution, magnetic circular dichroism,11 Raman,20,21 and CARS r 2011 American Chemical Society
spectra22 have been used to augment the simple absorption spectra but have not yielded precise information about component band shapes and intensities. As an alternative to the phenomenological approach of LS fitting to sums of band functions, one can ask how well the physical model that is exact for the gas phase might perform in application to solution spectra. This is the method employed in my recent re-examination of the I2 gas-phase absorption9 and used previously for gaseous Br2 and Cl2.2325 In it, the spectra are assumed to be calculable from potential curves and transition moment functions, both of which are expressed in terms of suitable functional forms containing adjustable parameters. These parameters are then optimized through LS-driven spectral simulations, wherein convergence to a minimum sum of weighted squared residuals (SSR) should yield best estimates of the parameters and hence of the component spectral bands. Such an analysis of the absorption spectrum of I2 in solution, assumed to involve the same three transitions, is the subject of this work. While the use of a gas-phase model to interpret solution spectral properties is commonplace, I believe that this is the first application of such LS simulations to the quantitative analysis of solution spectra. The treatment involves global fitting of spectra recorded at multiple temperatures—an approach that has been found necessary for simultaneous estimation of potential curves and transition moment functions from gas-phase spectra. Before going into the details of the computations, I note several differences required in the model for solution spectra. In the gas phase, there are unique potential curves and transition Received: November 21, 2011 Revised: November 29, 2011 Published: November 30, 2011 391
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I made no refractive index corrections to the spectra before employing them in the LS analysis. As was discussed in ref 8, such corrections worsen rather than improve the gas-phase/solution comparisons. Additional details about the treatment of the data are given in ref 8.
’ COMPUTATIONAL METHOD Theory Summary. The LS fit algorithm employed methods nearly identical to those used in my recent analysis of the gasphase spectrum.9 All transitions are treated as bound-free, for which the absorption cross section from level (υ00 ,J00 ) to continuum level at energy ε0 is as follows:
σ ν, ab ðυ00 , J 00 Þ ¼
ð2Þ
where I have replaced the sum over R, P, (Q) branches by the single Q-branch transition (hypothetical for BX in the gas phase). Here ν is the transition wavenumber, and Gab accounts for electronic degeneracy and is unity for BX (00 in Hund’s case c) and 2 for AX and CX (both 10). For absorption from I2 in thermal equilibrium at temperature T, this expression must be averaged over the Boltzmann distribution of absorbing υ00 and J00 levels at that T. The computations included the first 10 υ00 levels, accounting for over 99.9% of the population at the highest T (50 C). Nuclear-spin degeneracy effects average out and can be ignored. The bound and free wave functions needed for the matrix element in eq 2 are obtained by numerical methods that may be considered exact.27 Converting the absorption cross section to decadic molar absorptivity εν and expressing the transition moment function in debye (1 D = 3.3456 C m), we have,
Figure 1. Absorption spectra (molar absorptivity) of I2 in CCl4 and in n-heptane at two temperatures. The differences are displayed at bottom (scale to right).
moment functions, and the T-dependence in the spectra is the result of changes in the equilibrium rotational and vibrational population distributions in the ground state. For the halogen absorption spectra, the primary effect of increasing T is a broadening of each component band in the absorption spectrum. Such thermal broadening can be approximately accommodated by the band-fitting model;13,14 however, in our earlier phenomenological study of the solution spectrum,8 there were also changes in the peak positions with T that required additional peak shift parameters in the fit model (see Figure 1). With the ground-state potential fixed, there will be corresponding shifts required in the upper potentials in the physical model; and the slopes in the FranckCondon region also change with T, placing further demands on the solution fit model.
εν ¼ 108:861νGab ½jÆε0 J 00 jμe ðRÞjυ00 J 00 æj2 avg
ð3Þ
In solution as in gas phase, much of the BX absorption and some of AX is boundbound rather than bound-free. However, there is no perceivable vibrational structure in the spectra; and intensities are determined entirely by the shapes of the upper potentials in the FranckCondon region.27 These properties justify treating the transitions as bound-free. Of course, the determination can say nothing about the attractive branches of the A and B states in their bound wells. In the gas phase, the rotational averaging can be done by taking a small number of representative J levels for each T, computing the corresponding wave functions from effective potentials having a centrifugal term proportional to J(J + 1)/R2. Even a single J (the thermal average) yields good results, and an average over five is fully adequate.23 While small molecules dissolved in inert solvents appear to undergo essentially free rotation,28 the gas-phase centrifugal term must be an approximation for the solution. In the course of the calculations I considered this form with a single J (average), three representative Js, and simply J = 0 (no centrifugal term). Comparing J = 0 with the average J for the three I2/heptane data sets, there was a significant statistical difference for only one of these, and that in favor of the average J. There was essentially no statistical difference between the 1-J and 3-J calculations. Since the latter are more time-consuming, and since the gas-phase model is anyway not rigorous for the solution, I have chosen to report results for just the single, average J. This choice, rather than just J = 0, also preserves the gas-phase analogy, making gas-phase/solution comparisons more meaningful.
’ EXPERIMENTAL SECTION The analyzed spectra were those recorded in the earlier study,8 at a resolution and sampling interval of 1 nm on a Shimadzu UV-2101PC UVvisible spectrophotometer equipped with a temperature-controlled cuvette compartment, at temperatures between 15 and 50 C. Each spectrum used in the fitting was in the form of molar absorptivity, as estimated from 3 to 5 different concentrations spanning an absorbance range of 0.3 1.5. [Absorbance, A log10 (I0/I), where I0 and I are incident and transmitted intensity, respectively.] There were three such data sets for I2 in heptane, each having spectra at 15.6, 22.7, 40.2, and 50.0 C. There were two for CCl4—one at the same four Ts, the other at just 22.7 and 50.0 C. Two of these data sets were among the Supporting Information provided with ref 8. The nominal wavelengths of the spectra were corrected in accord with a prior wavelength calibration of the instrument,26 Δλ ¼ ½0:041 þ 0:025 sin2 ð1:623λ 550Þ sin ð180x=10Þ þ 0:188 þ 0:00055x þ 8:4 107 x2
2π2 ν Gab jÆε0 J 00 jμe ðRÞjυ00 J 00 æj2 3ε0 hc
ð1Þ
with the correction (nm) being additive (true apparent) and the argument x = (λ λ0), the phase λ0 being 638.0 nm. (In ref 8, this equation incorrectly had the 00 200 of sin2 after the argument, and I failed to note that the arguments of the sine functions were in degrees.) 392
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Fit Model. In n-heptane and CCl4, the I2 vibrational frequencies— 212.51(10) and 212.59(8) cm1, respectively29—are slightly reduced from that in the gas phase (214.58 cm1). The first anharmonicity constants are also smaller than gas-phase and statistically equivalent for these solvents. For all of the computations I represented the X potential by a Morse curve defined by the heptane ωe and ωexe (0.56 cm1) and fixed at the gas-phase internuclear distance of 2.6664 Å. The three excited-state potentials were taken as exponential polynomials,
UðzÞ ¼ A0 þ B0 expð a1 z þ a2 z2 þ :::Þ; z R R0 ð4Þ centered at R0 = 2.7 Å, with energies relative to the minimum of the ground state. However, I found better convergence properties fitting not to A0 and B0 but to the energy and slope at R0, U0 ¼ UðR0 Þ ¼ A0 þ B0 ; ðdU=dRÞ0 ¼ a1 B0
ð5Þ
Figure 2. Absorption component bands at 23 C for I2 in the gas phase (solid, ref 9) and dissolved in CCl4 (dashed), with error bars shown for the latter (1 σ, too small to show in this display for A r X, even with 10 scaling).
Checks indicated that terms beyond quadratic in the exponent in eq 4 were not statistically significant, so results below are for 4 adjustable parameters for each potential. (A 4-parameter inverse power construct — a + b/(R + c)d — was tried on one data set and gave SSR values more than a factor of 2 larger than the exponential polynomial, so was not pursued further.) The transition moment functions were taken as follows: jμe ðRÞj ¼ μ0 þ μ1 ðR R0 Þ
ð6Þ
it governs most of the intensity in the main peak, making the SSR very sensitive to it.
’ RESULTS AND DISCUSSION
To accommodate the T-dependent shifts mentioned earlier, the potentials for 50 C, 40 and 16 C were modified by addition of a term, ΔUT ¼ BT expð bzÞ
Observations. In initial studies, spectra at a single T were fitted to an appropriately simplified version of the model described above, then were fitted two Ts at a time and three at a time. In this way, the 50 spectrum for one of the heptane data sets was judged anomalous and was greatly downweighted in all subsequent fits. All other data were accepted as-is for the multispectrum fitting. In the analogous fitting of gas-phase spectra, the physical parameters most difficult to determine were the transition moment slopes μ1, so in initial fitting, I examined the precision and variability of these parameters in the three heptane data sets. The estimates of μ1 for BX were roughly within one SE (typically 0.15 D/Å) of the gas-phase value of 0.72 D/Å.9 The estimates for CX were less precise (SE ≈ 0.5 D/Å) but were also generally consistent with the gas-phase value, 0.16 D/Å. However, the smallest-SSR solutions for the A r X band preferred a strongly negative μ1, typically 2 D/Å and precise within a few percent. Accordingly, in all final fitting I froze μ1 for BX at the gas-phase value. I then obtained solutions for two cases: μ1 for AX and CX fixed at the gas phase values or fitted. The present physical model yields SSR values significantly below the lowest from the phenomenological approach of ref 8. For reasons discussed there, the point-by-point σε values for the spectra (and hence weights 1/σε2) were not absolutely reliable, so I present no χ2 values. However, since the same weights are used here (except for the single 50 spectrum mentioned above), the SSR values give a valid comparison of the statistical qualities of the two methods. For the four data sets that can be compared in this way, the SSR ratios are 0.54, 0.64, 0.77, and 0.80 for the statistically best present fits, with μ1 for AX and CX fitted. With μ1 fixed at the gas-phase estimates, the corresponding ratios rise to 0.70, 0.67, 0.81, and 0.86, respectively. As was noted earlier, the number of adjustable parameters in the present model
ð7Þ
with b specific to each potential but a single value used for all T, for a total of 12 correction parameters for four spectra. This form forces the slope corrections to have the same dependence on T as the shifts; there was no statistical indication of more complex behavior. As was true also in the phenomenological fitting of ref 8, it was necessary to include scale factors for all spectra beyond the first (23 C). These factors account for slight concentration errors in the spectra at different T; they were usually within 0.001 of unity but had standard errors (SEs) typically
