Temperature Effects on the Rovibrational Spectra of Pyrene-Based

ARTICLE pubs.acs.org/JPCA

Temperature Effects on the Rovibrational Spectra of Pyrene-Based PAHs F. Calvo,*,† M. Basire,‡ and P. Parneix‡ † ‡

LASIM, Universit
e de Lyon and CNRS UMR 5579, 43 Bd du 11 Novembre 1918, F69622 Villeurbanne Cedex, France ISMO, CNRS and Universit
e Paris-Sud 11, B^at. 210, F91405 Orsay Cedex, France

bS Supporting Information ABSTRACT: Absorption infrared spectra have been computed for a variety of polycyclic aromatic hydrocarbon molecules of the pyrene family, taking into account anharmonicity and temperature effects, rovibrational quantization, and couplings. The energy levels are described by a second-order perturbative expansion of the rovibrational Hamiltonian in the vibrational and rotational quantum numbers, as relevant for a symmetrictop molecule, with ingredients obtained from quantum chemistry calculations. Multicanonical Monte Carlo simulations are carried out to compute bidimensional IR intensity histograms as a function of total energy and vibrational frequency, which then provide the absorption spectrum at arbitrary temperatures via a Laplace transformation. The main spectral features analyzed for neutral, anionic, and cationic pyrene indicate a strong dependence on temperature, in agreement with existing laboratory experiments, and a significant contribution of rotational degrees of freedom to the overall broadenings. The spectral shifts and broadenings reveal some sensitivity of anharmonicities to the charge and protonation states and, in the case of protonated pyrene and pyrenyl cation, on possible isomers and between aromatic and aliphatic C
H bands. Implications of the present work to the general issue of interstellar emission features are discussed.

1. INTRODUCTION The interaction between molecules and light is a primary tool to interrogate physical and chemical properties in the gas and condensed phases. Spectroscopic techniques are especially accurate for isolated compounds, as they can be used to address the complex interplay between rotational, vibrational, and electronic degrees of freedom in the frequency or time domains. This also naturally provides useful benchmark information when treating the same compound in the presence of a complex environment such as a liquid. The nature of the radiative excitation determines which modes can be scrutinized. High-resolution spectroscopy in the far- and mid-infrared ranges, as is of interest in the present work, gives insight into both the global shape of the molecule and the intermolecular vibrations through the rotational and vibrational properties, respectively. Ideally, a low-temperature medium would simplify the spectroscopic analysis by producing a limited number of narrow rovibrational bands resulting from the system being close to its ground state. Unfortunately, at finite temperature the system populates higher rovibrational levels, anharmonicities increase, and for polyatomic molecules the couplings among vibrations or between vibrations and rotations are a further complicating factor.1 As with other experimental techniques, successful assignment of structures by vibrational spectroscopy requires comparison with calculated spectra. So far, the vast majority of theoretical r 2011 American Chemical Society

efforts have relied on first-principles electronic structure calculations at the harmonic level, usually with density-functional theory as the only available tractable method. In this approach, the spectra are corrected a posteriori by empirical scaling factors that account for anharmonicities and basis set incompleteness.2,3 Although undoubtly useful as a starting point, this may be unsatisfactory for several reasons, mainly the neglect of any temperature and vibrational delocalization effects. The role of temperature on the spectral profiles has been experimentally emphasized several times in the past, notably in the astrophysical context, in both IR absorption4 and emission.5
8 In these experiments, anharmonicities could even be effectively probed, posing a case much more challenging to theoreticians than a basic harmonic estimation. In principle, a fully quantum treatment of both electrons and nuclei aimed at solving the vibrational Schr€odinger equation is feasible only for very small systems, and alternative treatments are necessary for larger polyatomic molecules. If the potential energy surface is obtained on the fly by electronic structure methods, then the fully anharmonic IR spectrum can be obtained by molecular dynamics (MD) simulations, most likely in the Born
Oppenheimer or Car
Parrinello schemes but occasionally with Received: March 29, 2011 Revised: July 12, 2011 Published: July 25, 2011 8845

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The Journal of Physical Chemistry A quantum path-integral MD.9
12 Using approximate or partly empirical descriptions for the electronic structure alleviates a significant portion of the computational burden, allowing larger systems to be studied on a more satisfactory statistical ground. One difficulty with all molecular dynamics approaches, besides the technicalities associated with an efficient implementation of pathintegral methods, is that the simulations have to be repeated extensively at each new temperature of interest. Alternatively, a perturbative approach can be adopted if the system is likely to remain close to a given equilibrium structure. The quantum mechanical spectroscopic description then rests on well-defined quantum states in model Hamiltonians that expand the rovibrational energy in terms of vibrational and rotational levels.13 Such Hamiltonians contain the standard ingredients of the harmonic calculation, and beyond this zeroth order they also account for anharmonicities through vibrational and rotational couplings. However, the expression of the model Hamiltonian itself does not readily yield the infrared spectrum because the determination of the vibrational populations at thermal equilibrium is a complex combinatorial problem.14 We have recently15 proposed a computational strategy based on Monte Carlo samplings to address this issue, and were able to extract the finite temperature vibrational infrared spectrum of naphthalene, using the accurate anharmonic quartic surfaces computed by Can
e and co-workers.16 Application of this method to other molecules indicated that the difference between harmonic and anharmonic spectra can be sufficient to question conformational assignment,17 but more importantly the method provides a powerful way of quantifying the spectral features (line shifts and broadenings) as a continuous function of temperature. The present paper is aimed at extending this framework to include the rotational degrees of freedom and their couplings with the vibrations. Rotations are expected to contribute mainly to the broadenings, but they also affect the line shifts via the influence of vibrational numbers on the rotational constants. Because of rotation, the spectroscopic analysis must include the possible symmetries of the vibrational modes and their response to parallel or perpendicular excitations. The perspective adopted here bears some similarity with previous work by Mulas and coworkers,18 but is more systematic in that it considers all couplings among vibrational and rotational modes, as well as arbitrary temperatures. Beyond the methodological extensions, our applications deal with larger polycyclic aromatic hydrocarbon (PAH) molecules based on pyrene (C16H10). While naphthalene can be considered as the elementary PAH, pyrene is a significant step toward more realistic compounds. In particular, pyrene is relevant in astrochemistry as one of the candidates possibly responsible for the so-called unidentified bands in the interstellar medium.19
21 In the physical chemistry of combustion processes, pyrene has also been suggested as a key intermediate for the growth of larger carbon and soot particles.22 In the present work, we have carried out a comprehensive characterization of the rovibrational infrared absorption spectrum of pyrene in its neutral and singly charged states, taking temperature effects into account and with a quantum description of the rotational and vibrational modes. The method has been applied to protonated pyrene, as protonated PAHs have been suggested to be better candidates than regular PAHs to explain the aforementioned unidentified bands,23
27 the formation mechanism of protonated PAHs in the interstellar medium conditions having been previously discussed by Herbst and Le

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Page.28 At high temperatures, dissociation may become statistically favored and lead to dehydrogenated cations. Our study has been completed to characterize the IR spectra of these molecules. In addition to their intrinsic interest, the protonated and dehydrogenated compounds also exhibit several competing conformations, which opens some questions regarding the influence of isomerization on their spectroscopic signature. The paper is organized as follows. In section 2 we describe the computational strategy used to calculate rovibrational absorption spectra at finite temperature based on static electronic structure ingredients. In section 3 we present and discuss our applications to pyrene-based molecules, with a particular attention paid to rotational broadenings. Experimental comparison with available data is also reported, and the relevance of the present calculations in astrophysics is emphasized. Finally, we summarize and give some concluding remarks in section 4.

2. METHODS Our goal in this section is to outline the main stages of the computational procedure to determine infrared absorption spectra in a broad range of temperatures. This procedure follows the line of our previous effort on purely vibrational spectra,15 a strategy extended here to account for rovibrational couplings. Briefly, our modeling is based on a rovibrational Hamiltonian extended to second perturbative order, together with an harmonic approximation for the elementary transition intensities. The spectrum at finite temperature is obtained by Laplace transformation of an energy-dependent intensity distribution. The microcanonical entropy needed for this transformation, calculated using advanced Monte Carlo methods,29 is used in turn to achieve a broad sampling of vibrational transitions at various total energies. 2.1. Rovibrational Hamiltonian. Following usual notations, n = (ni), 1 e i e k, stands short for the set of k vibrational quantum numbers, and we denote by J, K, and M the three rotational numbers. In the present work, we approximate the molecular system as a symmetric top with two rotational constants Bz = A (oblate top) or C (prolate top) and Bx = By = B. The rovibrational energy levels depend on a set of harmonic frequencies (νi), second-order anharmonicity coefficients {χij}, on the rotational constants and on the ΔJ, ΔK, and ΔJK quartic centrifugal distortion constants according to30
 1 Eðn, J, KÞ ¼ ∑ hνi ni þ 2 i

 1 1 þ ∑ χij ni þ nj þ þ Bn JðJ þ 1Þ2 2 2 iej þ ðAn
Bn ÞK 2
ΔJ J 2 ðJ þ 1Þ2
ΔK K 4
ΔJK JðJ þ 1ÞK 2

ð1Þ

Whereas the anharmonic matrix {χij} characterizes vibrational couplings, vibration
rotation couplings are included in some dependencies of the rotational constants with the vibrational levels, which we take here at lowest linear order through two sets (A) of coefficients {R(B) i } and {Ri } Bn ¼ B0 þ

∑i RðBÞ i ni

ð2Þ

and a similar equation for An, B0 being the value of B at the equilibrium geometry. The above expansions are valid up to some 8846

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S ðν, EÞ ¼

ðkÞ ∑k σðkÞ n f n þ 1 ∑ FJ , K r J, K δ½hν
ΔEJ, K, J , K  J ,K k

k

0

0

0

0

0

where ΔE(k) J,K,J0 ,K0 is the energy associated with the transition (nk, J,K) f (nk + 1,J0 ,K0 ), leaving all other vibrational quantum numbers unchanged: ! 1 ðkÞ χ ΔEJ, K, J 0 , K 0 ¼ νk þ 2χkk þ 2 i6¼ k ik

∑

∑ χik ni Þ þ Bn J 0 ðJ 0 þ 1Þ 0

i6¼ k


Bn JðJ þ 1Þ þ ðAn0
Bn0 ÞK 0

2


ðAn
Bn ÞK 2
DJ ½J 0 ðJ 0 þ 1Þ2 2


J 2 ðJ þ 1Þ2 
DJK ½J 0 ðJ 0 þ 1ÞK 0
JðJ þ 1ÞK 2 
DK ðK 0
K 4 Þ 4

2

ð4Þ

The expression for S contains an absorption cross section for this transition, which in the harmonic approximation is related to the transition from the ground state as ðkÞ

ðkÞ

σnk f nk þ 1 ¼ ðnk þ 1Þσ0 f 1

ð5Þ

The remaining contribution to S originates from the rotational degrees of freedom, with the oscillator strength being governed by the H€onl
London factor FJ0 ,K0 rJ,K. This procedure is further extended to account for the possible existence of parallel and perpendicular transitions, accumulating two corresponding histograms S (ν,E) and S ^(ν,E). For the pyrene molecule in anionic, neutral, and cationic states (point group D2h), B3u vibrational modes are parallel but B1u and B2u modes are perpendicular. The other molecules considered in this work, although less symmetric, also have parallel and perpendicular transitions. Appropriate selection rules are applied for each vibrational mode, before the two histograms are eventually summed into S (ν,E). In practice, the histograms S , S , and S ^ are calculated by Monte Carlo integration, assuming that all rovibrational states are broadly sampled. Such a broad sampling is achieved from the preliminary determination of the microcanonical )

)

I ðν, EÞ ¼

S ðν, EÞ N ðν, EÞ

ð6Þ

and similar expressions exist for the specific spectra I and I ^ associated with parallel and perpendicular transitions. Finally, the canonical spectrum I (ν, T) at fixed temperature T is recovered by Laplace transformation of I (ν, E) involving the microcanonical density of rovibrational states Ω(E) Z 1 I ðν, EÞΩðEÞe
E=kB T dE ð7Þ I ðν, TÞ ¼ Z in which Z is the partition function Z ZðTÞ ¼ ΩðEÞe
E=kB T dE

ð8Þ

0

ð3Þ

þ ð2χkk nk þ

density of rovibrational states, using for instance Wang
Landau sampling,31,32 but also parallel tempering or kinetic Monte Carlo schemes combined with histogram reweighting.29 Denoting N (ν,E) the number of entries in the multicanonical simulation, the microcanonical absorption intensity is then estimated as

)

finite energy, above which additional higher-order vibrational and rotational terms should be considered. The quartic centrifugal constants are assumed not to depend on the vibrational numbers. All above quantities were obtained from quantum chemistry calculations, whose details are given below. For the neutral pyrene molecule, the Ray parameter χ = (2B0
A0
C0)/(A0
C0) approximately equals
0.4, indicating a moderate level of asymmetry. The neglect of asymmetry in our present framework entails a loss of resolution, especially at low temperatures. 2.2. Infrared Absorption Spectrum. The infrared absorption spectrum at finite temperature is obtained by convolution of its microcanonical equivalent, or the two-dimensional infrared intensity S (ν,E) versus both frequency ν and total energy E. Treating the problem on microcanonical footing provides a more powerful way of accessing the canonical properties, without having to repeat the simulations at each new temperature. This multicanonical procedure was already exploited with success in the case of purely vibrational spectra.15,17 The rovibrational spectrum is thus built by accumulating the quantity

and kB the Boltzmann constant. Once the infrared intensity I (ν,T) is calculated, it can be further analyzed in terms of its various moments near relevant spectral bands. 2.3. Monte Carlo Simulations. The auxiliary use of microcanonical quantities is not only convenient for postprocessing the histograms as a continuous function of temperature, it is also a key ingredient for achieving extensive sampling of the space of rovibrational states through multicanonical simulations. The microcanonical density of rovibrational states, Ω(E), was obtained here by Wang
Landau Monte Carlo integration31,32,29 for all systems. Once Ω is estimated, its inverse provides the statistical weight of a second Monte Carlo simulation aimed to sample the configuration space uniformly, during which the histograms S are recorded. Both simulations consist of the same set of elementary moves, either purely vibrational (nk f nk + δn for random k and random δn in the interval [
Δn;Δn]), or purely rotational (J f J + δJ or K f K + δK, again with random δJ and δK in the intervals [
ΔJ;ΔJ] and [
ΔK;ΔK]). Due to the symmetric-top approximation, all rovibrational states must be further weighted by 2J + 1 to account for degeneracy in the rotational number M. This weight is incorporated in all evaluations of Metropolis probabilities and in the accumulated histograms. 2.4. Computational Details. All molecular properties were obtained from density-functional theory calculations using the very standard hybrid functional B3LYP and a moderate basis set (4-31G) for the anharmonic coefficients. For the purpose of energetic comparison (especially between isomers), the structures were further tightly optimized using the larger 6-311++G** basis set. The ingredients for the anharmonic analyses were provided by second-order perturbation theory, as incorporated in the Gaussian03 software package.33 All these ingredients are given as Supporting Information. We should emphasize here that the relatively small basis set 4-31G was necessary in order to carry out the anharmonic calculations for molecules as large as pyrene. Larger basis sets can be used for smaller molecules, but the effects on the infrared spectra are not very significant. For instance, in the case of naphthalene, calculations performed with the 6-31 +G(d) basis set mostly alter some vibrational shifts with respect 8847

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The Journal of Physical Chemistry A to a 4-31G calculation, leaving the broadenings essentially unaffected (see Supporting Information). The Wang
Landau simulations consisted of 20 iterations of 107 Monte Carlo cycles each, one cycle being a set of moves for each of the k + 2 degrees of freedom. The penalty factor f of the Wang
Landau procedure, initially taken as f = 2, was decreased by taking its square root after each iteration. The accuracy of the density of states Ω(E) was checked against alternative kinetic Monte Carlo simulations for the same system.29 The multicanonical simulations of the infrared absorption intensity S (ν,E) were carried out using the same elementary Monte Carlo moves, with a total of 108 MC cycles after an equilibration period of 2  107 cycles. The histograms in total energy used a bin of 25 wavenumbers, and the spectral resolution was taken as 0.5 cm
1. For the molecules of interest here, the temperature was chosen to be varied in the range 0
1500 K, which is sufficient to activate even the hardest C
H stretching modes. This temperature also roughly corresponds to excitations close to 10 eV, typically found in the interstellar medium within the UV domain.

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Figure 1. Rovibrational absorption spectra of neutral pyrene in the singlet electronic state. The vertical lines correspond to static calculations with harmonic and anharmonic treatments. The continuous spectrum corresponds to a T = 300 K anharmonic calculation.

3. RESULTS AND DISCUSSION The aforedescribed methods have been used to calculate the anharmonic infrared spectra of an archetypal polycyclic aromatic molecule, namely pyrene, in various charge states, as well as protonated and dehydrogenated species. Although these molecules are relatively small for standard, harmonic characterizations, the determination of anharmonic coefficients is more challenging and, as aforementioned, could be achieved only with the modest 4-31G basis set. For all molecules studied, the first task consisted in locally optimizing the geometries in several spin states, which was greatly facilitated by inspection of the literature.23,24,42
44 In the case of protonated pyrene and the pyrenyl cation, several isomers had to be considered, and we also located the lowest transition states connecting these isomers. 3.1. Neutral Pyrene: Vibrational versus Rovibrational Spectra. The static spectra of neutral pyrene in its lowest spin state

(singlet) are first shown in Figure 1, where the anharmonic results are compared with the harmonic reference. The general spectral features fall into the range of 0
4000 cm
1, with a series of intense peaks corresponding to C
H stretchings (near 3000 cm
1), C
H out-of-plane bendings (near 800 cm
1), C
C stretchings, and C
H in-plane bendings (1000
1500 cm
1). At T = 0, rotational effects are negligible for such a relatively large compound and only the vibrational bands are seen. Anharmonicities contribute to significant shifts of these bands, an effect that can be roughly be accounted for by a scaling of 0.96. Anharmonic effects are, however, much clearer as soon as we assume a finite temperature, and we have superimposed in Figure 1 the rovibrational spectrum obtained at 300 K. The bands exhibit some slight additional shift, but more importantly they become much broader with clear rotational features on both sides of the central peak. Those features are absent from the static spectra, simply because they are due to rotational populations. More insight into the role of rotational degrees of freedom, and how they couple with the vibrations, can be obtained by comparing the rovibrational spectra with the purely vibrational spectra. For neutral pyrene, such spectra are illustrated in Figure 2 at the two temperatures of T = 300 K and T = 1500 K. As expected, the peaks in the rovibrational spectrum are always broader, but they do not display additional line shifts. Increasing

Figure 2. Vibrational and rovibrational absorption spectra of neutral pyrene in the singlet electronic state: (a) T = 300 K; (b) T = 1500 K. The insets show a narrow spectral range where rotational broadenings are best seen.

the temperature further broadens the spectral lines, in an especially asymmetric way for the tightest C
H stretching modes. 8848

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Table 1. Rates of the Linear Variations Obtained for the Average Vibrational Frequencies Æνæ of C
H Stretching and Out-of-Plane Bending Modes and the Line Broadenings Δν, in the 500
1000 K Rangea Æνæ slope (expt) vibrational mode C
H

Æνæ slope (calcd)

(10
2 cm
1 (10
2 cm
1
1


1

Δν slope (expt)

Δν slope (calcd)

(10
2 cm
1

(10
2 cm
1


1

K )

K )

K )

K
1)


2.84


2.59

3.40

2.55


1.35

not reported

0.81

stretchings
2.22 out-of-plane C
H bending


1.43
1.67

a

The corresponding experimental slopes from Joblin and co-workers4 are given when available. Figure 3. Variations with temperature of the spectral shifts (upper panels) and spectral widths (lower panels) for three bands of the neutral pyrene molecule near (a and d) 210 cm
1; (b and e) 850 cm
1; and (c and f) 3050 cm
1. The results are shown for pure vibrational and rovibrational spectra, as well as for vibrational spectra convoluted by a Gaussian with 4(BkBT)1/2 width at midheight.

A much better view of rotational broadenings is obtained by looking at specific spectral bands, and we have chosen to focus on the 460
540 cm
1 region hosting two active modes (see insets of Figure 2). These two transitions are characterized by harmonic frequencies of 512.1 cm
1 (B3u parallel transition) and 522.1 cm
1 (B1u perpendicular transition). At 300 K, the P, Q, and R rotational branches are clearly resolved for both modes. The two modes, although well separated in the pure vibrational spectrum, slightly overlap with each other once rotational broadening is included. At 1500 K, the rotational structure is no longer resolved and the two rovibrational bands are essentially merged into a single broad peak with a red shoulder. On a more quantitative footing, the average frequency Æνæ and spectral width Δν have been calculated from the first and second moments of I (ν,T), for three active IR bands as a continuous function of temperature in the range 100
1500 K. The bands considered here cover the softest out-of-plane bending mode (near 212 cm
1), C
H bendings (near 870 cm
1) and stretchings (near 3060 cm
1), which for the latter can be compared with experimental measurements by Joblin and co-workers.4 The results of these calculations are shown in Figure 3, together with available experimental data. For the three bands, the average frequency varies very smoothly with temperature, eventually linearly in the high-temperature classical regime. The onset of this classical regime strongly depends on the mode considered, tighter modes behaving more quantum mechanically. The rotational degrees of freedom barely affect the average frequency, hence there is no additional shift due to the rovibrational couplings. The experimental data reported for C
H stretching band, superimposed in Figure 3, also decrease linearly with increasing temperature in the 600
900 K range.4 The agreement with the present calculations is very satisfactory both for the average frequency and its slope, as further seen in Table 1 where the rates of these linear variations are detailed. In the case of the line shifts, two experimental values are indicated based on different ways of analyzing the measurements in ref 4.

Line broadenings differ significantly upon including rotational effects, which add up to the conventional thermal broadening effects. This difference is particularly strong for the lowest frequency bands of out-of-plane modes, but in the case of the tighter C
H stretchings rotational degrees of freedom have a minor influence. This shows that rotational broadening itself is smaller than the anharmonic contribution to the set of five active modes in this spectral range. It should also be noted that the five individual modes of this band become resolved below approximately 500 K, making the line width obtained from the second moment somewhat ill-defined. As was found for the spectral shifts, the variations of the spectral broadenings are linear with increasing temperature in the range 500
1500 K, a feature also found experimentally.4 The slopes of the linear adjustments, as reported in Table 1, are in semiquantitative agreement with the measured value for the C
H stretching band.4 The slight underestimation may be possibly due to the clearly nonlinear behavior in our calculation near 500 K (see Figure 3). For this band, the broadening itself can be compared favorably to the measurements, as our calculations for the rovibrational spectra give Δν = 31 cm
1 versus 36 wavenumbers in the experiment.4 In the limit where rovibrational couplings are negligible, and at sufficiently high temperature, the rotational broadenings can be approximately represented by a convolution of the pure vibrational spectrum with a Gaussian having a width at midheight of 4(BkBT)1/2, as would be expected from the Boltzmann rotational populations.34 This convolution was applied to the pure vibrational spectra, and the results superimposed in Figure 3 show that this procedure is actually nearly quantitative. A more stringent test can be realized if we temporarily assume an harmonic system (χij = 0 for all i, j), thereby suppressing the intrinsic thermal broadenings. Repeating the calculations of vibrational and rovibrational spectra for the two bands near 210 and 850 cm
1 leads to variations in Δν with temperature that are represented in Figure 4. These variations are now purely indicative of the rotational broadening and can be successfully compared to the Gaussian width 4(BkBT)1/2, deviations being manifested only above 1000 K. This very good agreement on a simplified system indicates that the computational procedure is correct and was correctly implemented. In a general compound, line widths will not be represented by a simple Gaussian width, but instead they will result from a more complex interplay between thermal and rotational broadenings. 8849

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Figure 4. Rotational broadenings obtained after removing vibrational anharmonicities, for two modes of the neutral pyrene molecule near 210 and 850 cm
1. The broadening for a pure Gaussian with 4(BkBT)1/2 width at midheight is also shown.

3.2. Charge Effect. Ionized PAHs have been shown to be much better candidates than neutrals as possible carriers of the UIBs,6,35 and previous quantum chemistry studies actually indicate that the (harmonic) infrared spectrum of polycyclic aromatic molecules is rather sensitive to their charge state.36,37 Briefly, ionization strongly activates the C
C stretching modes in the 1200
1600 cm
1 range with respect to the spectrum in the neutral molecule. However, differences between the ionization states are best seen on the C
H stretching modes, which are redshifted in the anion but blue-shifted in the cation. Their intensities in the cation are also reduced by at least an order of magnitude, although they are otherwise similar to those in the anion below about 1600 cm
1. We have extended the previous calculations of anharmonic spectra on neutral pyrene to the cationic and anionic cases. In both cases, the doublet spin state is more stable than the quartet state by about 3 eV, in agreement with the systematic survey carried out by Malloci and co-workers.38 The results of these calculations at 300, 1000, and 1500 K are depicted in Figure 5, where the two spectral ranges 0
2000 and 2800
3200 cm
1 are considered separately. Again, temperature effects are mainly manifested by the increasing red shift and broadening exhibited by most active lines. Interestingly, the tightest C
C stretching frequencies are found to be near 1500 cm
1 at 1500 K in both the cation and anion, about 100 cm
1 below the value found in neutral pyrene. This could contribute to explaining the 6.2 μm feature observed in emission spectra of such vibrationally excited molecules in the interstellar medium.39,40 Unfortunately, the very low intensity of this band in the neutral case makes it a rather unlikely candidate as a carrier of these emission bands at this particular wavelength. The role of temperature on the line shifts and broadenings has been characterized for the three charged states, focusing on three well-resolved active bands. More specifically, we have chosen a very soft, out-of-plane deformation mode near 210 cm
1, some out-of-plane C
H bendings near 850 cm
1, and the C
H stretchings band near 3050 cm
1, the temperature of interest spanning the 0
1500 K range. The variations of the average frequencies Æνæ are represented in Figure 6. At T = 0, the fundamental frequencies of the three bands clearly exhibit some significant shift as the charge states goes

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Figure 5. Absolute rovibrational spectra obtained for neutral, cationic, and anionic pyrene in their most stable electronic state, at 300, 1000, and 1500 K. Two spectral ranges are highlighted with different vertical scales. Note that the spectra have been vertically shifted for clarity.

Figure 6. Variations with temperature of the spectral shifts (upper panels) and spectral widths (lower panels) for three bands of the neutral (0, green), anionic (
, red), and cationic (+, black) pyrene molecules near (a and d) 210 cm
1; (b and e) 850 cm
1; and (c and f) 3050 cm
1.

from
1 to 0 and +1, the two successive shifts being of comparable magnitude. Temperature effects are very similar for the three charge states, with variations in the effective frequencies that are essentially similar. This indicates that, contrary to frequencies, vibrational anharmonicities do not appreciably vary with the charge state for these molecules. In the case of C
H stretchings, the red shift appears reduced for the cation. This could have some consequences on the emission spectrum resulting from the IR emission cascade after excitation, which according to the larger variations in Æνæ(T) would be more spread out in the neutral and ionic cases. The spectral width in emission spectra would thus be an adequate quantity to probe the difference in anharmonicities of the C
H stretching modes. The variations of the line broadenings Δν(T) with increasing temperature, also represented in Figure 6, sometimes show larger differences that can be explained based on static considerations. 8850

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Figure 7. Most stable isomers of protonated pyrene in their lowest (singlet) electronic state, and the corresponding transition states. The relative DFT energies are given including harmonic zero-point corrections. All energies are expressed in kcal/mol.
1 The width of the softest √ out-of-plane mode near 210 cm varies approximately as T as the combined result of thermal and rotational broadenings. Both types of broadening have a comparable magnitude for this mode, notably due to the very similar rotational constants shared by the molecules in their different ionic states. However, for the two tighter modes the widths are significant already at T = 0, which marks the splitting of the band into several but close-by active modes. This is particularly problematic for the C
H stretchings, due to the numerous B1u and B2u bands merging above about 500 K only. 3.3. Protonated Pyrene. Having validated the computational methodology on neutral and ionized pyrene, we turn to more complex but related compounds differing by a single hydrogen atom. Protonated PAHs have gained interest as promising candidates for the UIBs,23
27 and they are also relevant in combustion processes.41 Protonation can take place on several carbon sites, and we summarize in Figure 7 the lowest part of the potential energy surface obtained from our DFT calculations. Again, the singlet spin state was found to be always lower by a few electronvolts, and we do not show the results for the triplet state. Similarly to previous calculations on protonated PAHs,23,24,42
44 the preferred protonation site in pyrene appears to be on the hydrogenated ring, as a peripheral, nonsymmetric carbon site in the 1-hydropyrene cation. All other isomers are significantly higher, by more than 10 kcal/mol. In addition to the three natural peripheral protonation sites, protonating the intermediate carbon in the 3-hydropyrene isomer produces a fourth, higher-lying structure. The transition states connecting these four isomers are also quite high, and again in quantitative agreement with previous results on related polyaromatic compounds.23,24,42
44 This indicates very strong thermodynamic and kinetic stabilities of the most stable structure. The lowest decay channel, hydrogen loss, has a dissociation energy of 2.62 eV after including harmonic zero-point energy at the same DFT/B3LYP/6-311++G** level. This is in agreement with the value of 2.60 eV reported by Hudgins et al. using the same method.43 The anharmonic IR absorption spectra of the three lowestenergy isomers of protonated pyrene are represented in Figure 8 at the relatively low temperature of 300 K. The most salient difference with the spectra of pure sp2 pyrene is the appearance of aliphatic C
H stretching modes near 2800 wavenumbers that are even more active than the aromatic C
H stretchings. These modes also show a significant dependence on the protonation site, and they exhibit a progressive red shift as the extra proton binds to higher lying isomers. The intensity ratio

Figure 8. Rovibrational spectra of the three lowest-energy isomers of protonated pyrene at 300 K, using the same vertical scale. The spectra have been shifted for clarity.

Table 2. Isomer Distribution in Protonated Pyrene C16H11+ and Pyrenyl Cation C16H9+, Based on Harmonic Partition Functions at 500, 1000, and 1500 K C16H11+

C16H9+

temp (K)

[1]

[2]

[3]

[1]

[2]

[3]

500

100.0

0.0

0.0

70.0

28.0

2.0

1000

99.5

0.5

0.0

60.0

34.0

6.0

1500

96.0

3.5

0.5

58.0

34.0

8.0

between the aliphatic and aromatic bands could thus be used as a probe of the protonation site. Compared to planar pyrene, lowering the symmetry in protonated pyrene results in many additional bands active in the 1200
1600 cm
1 range of C
C stretchings, with intense rovibrational bands near 1600 wavenumbers. Hence the present results further support the hypothesis that protonated PAHs are better candidates to explain the 6.2 μm feature in the ISM,23,25
27 which encourages us to look more closely into anharmonic effects. For the present system with competing isomers, it is necessary to estimate the statistical weight of the various minima before carrying the analysis of infrared spectra. A simple harmonic estimation of quantum partition functions, including the degeneracy factor for permutation-inversion isomers, yields the probabilities given in Table 2. Those readily show that the lowest-energy isomer of protonated pyrene contributes dominantly (more than 96%) up to temperatures as high as 1500 K. Even at this temperature, the fourth isomer never significantly contributes to the overall population. However, it should be kept in mind that 1500 K is a rather high temperature for protonated aromatics, as statistical hydrogen dissociation becomes more likely. A rough estimate of the dissociation rate kd at canonical temperature T is provided by a simple Arrhenius expression
 ΔE kd ðTÞ ≈ k0 exp
ð9Þ kB T where the prefactor k0 is related to the Gspann parameter45 G and to the experimental time window τ through k0 = eG/τ. Even though the value of the Gspann parameter is not known very accurately, it 8851

dx.doi.org/10.1021/jp202935p |J. Phys. Chem. A 2011, 115, 8845–8854

The Journal of Physical Chemistry A

ARTICLE

Figure 10. Most stable isomers of pyrenyl cation in their lowest (triplet) electronic state, and the corresponding transition states. The relative DFT energies are given including harmonic zero-point corrections, and the numbers between brackets indicate the optimized energies in the singlet state. All energies are given in kcal/mol. Figure 9. Absolute rovibrational spectra of neutral and protonated pyrene in the 1400
1800 and 2800
3200 cm
1 ranges, at 300 and 1000 K.

can be reasonably well approximated as G ∼ 23.5 in the time range of 10 μs relevant in mass spectrometry time-of-flight experiments.46,47 This value for G leads to kd(T=1000 K) = 103 s
1 and kd(T=1500 K) = 2.5  107 s
1, indicating that the absorption spectrum will hardly be measurable at such temperatures. However, in the situation where the molecule is excited in a higher electronic state by a visible/UV photon, internal conversion within a subpicosecond time scale could produce high vibrational temperatures exceeding 1000 K. Dissociation will then compete with radiative IR emission, and all rovibrational anharmonicities will be manifested on the emission spectrum as well, even though they cannot be seen on the absorption spectrum. The influence of temperature on the anharmonic absorption spectrum of protonated pyrene is represented in Figure 9, only the lowest-energy isomer being considered accordingly with the weights of Table 2. In the same figure, the rovibrational spectra for neutral pyrene are depicted as well for comparison. While the aromatic C
H stretching bands are resolved at 300 K, they are merged at 1000 K, with a width (44 cm
1) only slightly higher than the aliphatic band (33 cm
1). The spectral shifts show contrasted dependencies on temperature, with approximate slopes of 2.3  10
2 cm
1 K
1 and only 6  10
3 cm
1 K
1 for the aromatic and aliphatic bands, respectively. The slopes for the aromatic lines are of comparable magnitude as their values in both pyrene and its cation. The present work indicates that anharmonicities associated to the aliphatic and aromatic modes are quite different from each other. It would be interesting to confirm this prediction on other aromatic molecules, and of course through any experimental measurement. As stated previously, the lower symmetry of protonated pyrene gives rise to many additional bands in the mid-IR region, and it becomes more difficult to characterize individual peaks. In particular, intense IR absorption bands are present near 1600
1640 cm
1 at 300 K. Increasing temperature to 1000 K induces a rather small red shift of about 20 cm
1 in this spectral range. In the astrophysical context, IR photons are emitted from out-of-equilibrium, vibrationally hot molecules subsequently to a visible-UV electronic excitation. The emission spectral profile for these C
C stretching modes should thus be characterized by an intense emission band in the 1600
1640 cm
1 range resulting from the IR emission cascade. This is consistent with

a possible contribution of protonated PAHs to the 6.2 μm emission band. 3.4. Pyrenyl Cation. The energy landscape has been explored for the pyrenyl cation (C16H9+), and Figure 10 shows the lowest part of the potential energy surface obtained from our DFT calculations. As was found in previous quantum chemical studies,48
50 the triplet state is now more stable than the singlet state for all isomers (and transition states), the energy gap approaching or exceeding 10 kcal/mol. The three main isomers are the counterparts of protonated pyrene, in the sense that in order of increasing stability the dehydrogenation site is the same as the protonation site. However, the isomerization energies are now much lower than in the protonated case, being close to a few kcal/mol. As seen in Table 2, this leads to much closer statistical weights and some actual competition between the two lowest-energy isomers already at 300 K. Interconversion between isomers, though, is not very likely at low temperatures due to the rather high barriers of about 70 kcal/ mol. These values are similar to the hydride shift barrier rearrangement in the phenyl51 and tolyl52 cations, which confirms the reliability of the present computations. On the basis of these static results, we now illustrate the possible effects of isomerization on infrared spectra. In a superposition approximation,53,54 the spectrum at a given temperature is obtained as the weighted average of individual spectra, and the associated kinetics can be modeled by solving a master equation in which the state-to-state isomerization rates are estimated by RRKM theory. We have computed the partition functions for the reactants and transition states using anharmonic densities of states, and found that the isomerization rates are sufficiently low (