Analysis of the Visible Absorption Spectrum of I 2 in Inert Solvents Using a Physical Model Joel Tellinghuisen Department of Chemistry Vanderbilt University Nashville, TN 37235

The visible absorption spectrum of I 2 comprises 3 over- lapped electronic transitions —A  X, B  X, and C  X. Excited-state potential curves in absorption region (gas phase, Es relative to first dissociation limit, dashed regions known from discrete spectroscopy). R eX

Absorption is ~20% stronger in inert solvents than in gas phase. Also slightly blue-shifted in solution. (  10)

How to extract the components? Phenomenological approach: Assume each component can be described by a Gaussian-type function and least-squares fit to a sum of such functions. Widely used for both gas-phase and solution spectra; only method used for the latter to date. Physical model: Nonlinear LS spectral simulations with adjustable parameters representing unknown potential curves and transition moment functions. Used successfully on gas-phase halogen spectra since 1976,* including recent reanalysis for I 2.# * Br 2 — R.J. LeRoy, et al., JCP 65, 1485 (1976). # J. Tellinghuisen, MSS 2011 and JCP 135, (2011). Required: Spectra as a function of temperature T.

Problems with phenomenological method In tests on known component bands in I 2 (g) and Br 2 (g),* Common 3-parameter forms are statistically inadequate for representing single bands. 4- and 5-parameter extensions better but “flaky;” component resolutions can depend strongly on chosen band form. Worst: Among competing functional forms, the statistically best deconvolution is not the physically best. * R.I. Gray, et al., JPC A 105, (2001). Results from this approach on I 2 in n-heptane and CCl 4 :* A  X band blue-shifted but otherwise little changed in shape and intensity. 20% increase in intensity probably due to enhanced C  X band.

Illustration of deficiencies in phenomenological approach

So, what about the physical model? Clearly the way to go on gas-phase spectra: theoretically sound. reproduces spectra within experimental uncertainty. Use on solution spectra? cannot be considered theoretically complete. will require additional parameters to account for T- and solvent-dependent shifts of spectral peaks. Try and see!

Spectra simulated using standard numerical quantum methods as described in previously cited 2011 JCP article on I 2 gas-phase analysis. Spectra treated as purely bound-free — pseudocontinuum model. X potential: Morse curve having  e = (8) cm  1 and  e x e = 0.56 cm  1 (values for I 2 in n-heptane), and R e = Å (gas). Excited-state potentials: U(z) = A 0 + B 0 exp(  a 1 z + a 2 z 2 + …) with z = R  R 0 and R 0 = 2.7 Å. [Results — no need for a 3, a 4, …] Better convergence defining adjustable parameters as U(R 0 ) (= A 0 + B 0 ) and (dU/dR) 0 (=  a 1 B 0 ). T-dependent shifts:  U T = B T exp(  bz), with b state- but not T- dependent. Transition moment functions: |µ e (z)| = µ 0 + µ 1 z Additional: Scale parameters for T  23°C (mostly 1.000±0.001) Nonlinear LS fit model

Data Recorded nm in previously cited 2001 JPCA study; resolution and sampling interval = 1 nm, Shimadzu UV-2101 PC. molar absorptivities,  (l mol  1 cm  1 ), estimated from spectra spanning peak absorbances A = as recorded for 3-5 different concentrations at each T. n-heptane: Three data sets for each of 4 Ts: 15.6°, 22.7°, 40.2°, and 50.0°C (Ts from a calibrated thermistor). CCl 4 : Two data sets, same 4 Ts for one, just 22.7° and 50.0° for the other. wavelength corrections: Following paper.

Results* Preliminary findings: Transition moment slopes µ 1 least-determined (true also for gas-phase analysis). For B–X, within one SE of gas-phase, so fixed at that value (0.72 D/Å). Pursued two models: (1) µ 1 similarly frozen at gas phase for other two bands, and (2) µ 1 freely fitted for these two. Latter statistically better (by ~10% in  2 ) but yielded questionably strong R-dependence for A–X, and unusual U A. Both statistically better than phenomenological model, by % in terms of summed squared residuals, depending on data set and model. But more parameters — vs. 27. * JPC A 116, 391 (2012).

Component resolution for I 2 in CCl 4, (dashed) compared with I 2 (g) (solid). For solution analysis, µ 1 values were fixed at gas- phase results. Error bars are 1-  (too small to see for A  X).

Corresponding potential curves for I 2 (CCl 4 ) (dashed) and I 2 (g) (solid). Small-R downturns in the dashed curves are considered artifacts of the fitting, as this region and that at large R are not well sampled by the data. Energies relative to X-state minimum. (at 23°C)

T-shift parameters for I 2 in CCl 4. Solid curves and points obtained fitting µ 1 for C  X and A  X; open points and dashed lines for all µ 1 frozen at gas-phase values. [All B T  0 at 22.7°C.]

Corresponding results for I 2 in n-heptane. (A state)

Component bands for I 2 in CCl 4, as obtained fitting two µ 1 values (solid) and holding them fixed at gas phase values (dashed)

A-state potential curves and A–X transition moment functions from the two analyses. Solid: two µ 1 values fitted. Dashed: All µ 1 fixed at gas-phase values. Ground- state potential and classical radial probability distribution shown at bottom.

Vibrational components in A  X spectra from two different analyses. Top: µ 1 fitted. Bottom: All µ 1 frozen at gas-phase values.

Summary The physical model is statistically better than phenomenological and it reduces the model-dependent ambiguities in the band composition analysis for I 2 absorption in inert solvents. However, it does require more parameters; and dependence on different functional forms for potentials has not been thoroughly examined. The previous indication that most of the intensity gain in solution occurs in the C  X transition is strongly supported. No obvious need for >3 transitions, nor for >2 dimensions in the potentials. Finding disparate descriptions of the A  X transition from different assumptions in the model shows that the physical model is not foolproof!