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Finite-Temperature IR Spectroscopy of Polyatomic Molecules: A Theoretical Assessment of Scaling Factors† M. Basire,‡ P. Parneix,‡ and F. Calvo*,§ Laboratoire de Photophysique Mole´culaire, | C.N.R.S. Fe´de´ration de recherche Lumie`re Matie`re, Bat 210, UniVersite´ Paris Sud 11, F91405 Orsay Cedex, France, and LASIM, UniVersite´ de Lyon and CNRS UMR 5579, 43 Bd du 11 NoVembre 1918, F69622 Villeurbanne Cedex, France ReceiVed: September 14, 2009; ReVised Manuscript ReceiVed: NoVember 5, 2009
With a recently developed simulation method (Basire, M.; et al. J. Phys. Chem. A 2009, 113, 6947), the infrared vibrational spectra of several polyatomic molecules are calculated over a broad range of temperature, taking into account quantum, anharmonic, and couplings effects. Anharmonic force fields, generated from static first-principle calculations, are sampled in the microcanonical ensemble to provide energy-resolved absorption intensities and their finite temperature analogues after Laplace transformation. Effective anharmonic frequencies are characterized as a continuous function of temperature for vinyl fluoride, the N-acetyl-PheNH2 peptide, and protonated naphthalene. These frequencies generally deviate increasingly from the harmonic value with increasing temperature, although the overestimation due to the harmonic approximation is particularly salient for high-frequency modes. Anharmonicities may also be sufficient to alter structural assignment of experimental spectra with respect to empirically scaled harmonic bands. These results emphasize some possible limitations and inaccuracies inherent to using such static scaling factors for correcting harmonic IR spectra. I. Introduction The structural assignment of polyatomic molecules in the gas phase can be achieved through many experimental techniques such as ion mobility, electron capture, laser- or collision-induced dissociations, electric or magnetic dipole measurements, proton/ deuterium exchange reactions, and NMR, electronic, and vibrational (infrared and Raman) spectroscopies. Most of these methods are however indirect, in that they require comparison with calculated structures. Quantum chemistry, paired with impressive progress in laboratory methods based on Fourier transform infrared, IR depopulation, IR/UV, IR multiphoton dissociation, or messenger tagging, have thus allowed successful structural identification in a broad variety of systems via their vibrational signatures. Ionic molecules,1,2 semiconductor3 or metal4 clusters, hydrogen-bonded complexes,5-8 biomolecules,9-16 and astrochemical compounds17-24 are prominent examples in this respect. From the theoretical perspective, a broad array of very different methods are available for determining vibrational spectra relevant for experimental comparison. The present work aims at addressing some shortcomings of these methods, within a single computational framework in which the anharmonic and finite temperature effects are fully accounted for. In principle, the most accurate way of computing vibrational spectra may be through the Kubo formulas for dipole autocorrelation functions, using dynamical trajectories generated with ab initio potential energy surfaces (PES), together with a quantum description of nuclear motion. Centroid molecular dynamics25 and ring-polymer molecular dynamics26 coupled with first-principles electronic structure methods27-30 have recently †
Part of the “Benoît Soep Festschrift”. * Corresponding author,
[email protected]. C.N.R.S. Fe´de´ration de recherche Lumiere Matiere, Bat 210, Universite´ Paris Sud 11. § Universite´ de Lyon and CNRS UMR 5579. | Laboratoire associe´ a l’Universite´ Paris Sud 11. ‡
been applied to the vibrational spectroscopy of gas-phase molecules. However, these sophisticated approaches are so far only feasible for small systems containing no more than about 10 atoms and, as shown by Witt and co-workers,31 they also suffer from intrinsic problems. Using a classical approximation for nuclei extends these capabilities to larger molecules32-35 through either Car-Parrinello or Born-Oppenheimer propagating schemes. Further approximating the electronic structure description using semiempirical methods36 or even molecular mechanics37,38 still remains the favored way of performing explicit dynamical simulations for systems containing several tens to hundreds of atoms. Besides dipole autocorrelation functions, molecular dynamics trajectories also provide vibrational information through atomic fluctuations.39,40 Such a principal component analysis method can be combined within the Monte Carlo framework.41 The vibrational dynamics of small molecules can also be treated quantum dynamically if the potential energy surface is precalculated and stored on a grid. These methods have been particularly fruitful, e.g., for proton-bound complexes involving amino acids42 or the ammonia dimer.43 Alternatively, very large molecules require more approximate treatments such as the vibrational exciton model.44 Static calculations, on the other hand, are probably the most straightforward (and commonly used) way of estimating vibrational bands and intensities. The static approach benefits from more realistic potential energy surfaces obtained from accurate electronic structure theories, but in practice stops at the harmonic approximation (quadratic PES). Unfortunately, vibrational frequencies calculated at the harmonic level are often in error with respect to measured frequencies by 10-15%. Reasons for these discrepancies include approximations in the treatment of electron correlations, basis set uncompleteness, and anharmonicities. The observation that ab initio harmonic frequencies deviate rather uniformly from the experimentally measured fundamental frequencies has suggested to simply scale the former for
10.1021/jp9088639 2010 American Chemical Society Published on Web 12/04/2009
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improving the agreement.45-48 Appropriate scaling factors for use with different levels of electronic structure methods and different basis sets have been extensively tabulated in a number of studies,49-53 especially in the context of density-functional theory (DFT) methods.54-57 Because of a much heavier computational burden, the impact of anharmonicities on calculated frequencies could be quantified only in a few cases.58-61 Still, these static methods are unable to account for any temperature effect naturally included in the aforementioned dynamical methods. Several experimental studies have highlighted the importance of temperature on the vibrational spectra, particularly the vibrational shifts and band profiles.17-19,62,63 Recently, several models have been developed to simulate the effects of a finite temperature on the vibrational spectrum based on pure static ingredients.64-67 These models considered the influence of anharmonicities on transition energies but neglected the thermal effects on the equilibrium vibrational populations. In a recent investigation,68 we have shown how to include both contributions in the calculation of finite temperature infrared spectra. The model used in this work, initially tested on the neutral naphthalene molecule, is here applied to a broader variety of systems. Effective anharmonic frequencies can be characterized from the vibrational spectrum obtained at a finite temperature, and their deviation with respect to the harmonic value provides a natural comparison with the predictions of applying simple scaling factors. Recommended values for these factors are shown to work reasonably well, even though temperature effects can be significant and increasingly shift the calculated frequencies from their harmonic reference. The article is organized as follows. In the next section, we briefly summarize the model, starting with its main assumptions and subsequent stages. Several applications are then presented and discussed for the cases of vinyl fluoride, the N-Ac-Phe-NH2 peptide (simply denoted as NAPA below), and protonated naphthalene for which experimental results are available.24,61,69 Some summarizing remarks and perspectives are finally detailed. II. Method We consider a molecular system at its equilibrium geometric structure, as predicted from a standard quantum chemical calculation. The vibrational energy levels are described by a quadratic force field E({n}), the total energy E of the system being expressed as a second-order Dunham expansion with respect to m quantum numbers {n} ) n1, ..., nm as m
E({n}) )
∑ hνi(h)(ni + 21 ) + ∑ i)1
(
χij ni +
1eiejem
1 1 nj + 2 2 (1)
)(
)
The set of harmonic frequencies νi(h) as well as anharmonic coefficients χij are assumed to be also supplied by quantum chemical calculations.70 At thermal equilibrium and temperature T, the infrared absorption intensity I(ν,T) at frequency ν of this molecule can be expressed from the microcanonical absorption intensity ˜I(ν,E) at fixed excess energy E through a Laplace transformation
I(ν, T) )
1 Z
∫ I˜(ν, E)Ω(E)e-E/k T dE B
(2)
in which kB is the Boltzmann constant and Z denotes the canonical partition function
Z(T) )
∫ Ω(E)e-E/k T dE B
(3)
The two previous equations involve the microcanonical density of vibrational states Ω(E), which is determined from the
expression of the vibrational levels by a dedicated Monte Carlo simulation based on the Wang-Landau algorithm.71 Compared to other available methods,72-78 the Wang-Landau approach provides numerically accurate densities of states for fully coupled anharmonic oscillators without resort to approximations. The detailed description of the Wang-Landau method, as applied to the present problem, is given elsewhere.79 Once the density of states Ω(E) is estimated in the energy range 0 e E e Emax, a second Monte Carlo simulation is carried out in the multicanonical ensemble, using Ω(E) as a bias to achieve flat-histogram sampling. During this second simulation, the two-dimensional histogram H(ν,E) of the intensity of transitions at frequency ν and excess energy E is accumulated. Here, combination bands and overtones are omitted for simplicity, and the intensity of an elementary, single-quantum transition involving mode k from nk to nk + 1 is taken proportional to the Einstein coefficient σ(k) nkfnk+1. Hence the microcanonical intensity I˜(ν,E) reads
I˜(ν, E) ∝ H(ν, E) )
∑ ∑ σn(k)fn +1δ[hν - ∆E(k)]
MC steps
k
k
k
(4) (k)
∆E being the energy difference associated with this transition. ∆E(k) straightforwardly follows from eq 1 as
∆E(k)({n}) ) νk(h) + 2χkk +
1 2
∑ χik + 2χkknk + ∑ χikni i*k
i*k
(5) To proceed, the Einstein coefficient is estimated using an (k) harmonic approximation, namely σn(k)kfnk+1 ) (nk + 1)σ0f1 . The are taken from the static IR intensities obtained coefficients σ(k) 0f1 from the same quantum chemical calculations used to locate the vibrational ground state structure. Finally, the histogram H(ν,E) obtained from the multicanonical simulation is converted into a microcanonical, wavenumberdependent intensity ˜I(ν,E) by simply normalizing by the number of entries and then into the finite-temperature infrared spectrum after applying eq 2. III. Results and Discussion The computational methods outlined in the previous section must be fed with several static ingredients that characterize the anharmonic potential energy surface at the ground electronic state. For vinyl fluoride, these ingredients were taken from the literature.61 For the larger organic systems studied here, the NAPA peptide and protonated naphthalene, we had to perform additional electronic structure calculations to estimate the anharmonic surfaces. These calculations also provided absorption intensities, allowing in turn some comparison with experimentally measured IR spectra. A. Computational Details. For each molecule, the WangLandau and multicanonical Monte Carlo simulations are carried out on a finite energy range 0 e E e Emax, relative to the zeropoint reference. Emax is a fixed upper range chosen in order to contain the thermal distribution at the highest temperature of interest. The values of Emax are 2, 3.5, and 4 eV for vinyl fluoride, protonated naphthalene, and the NAPA peptide, respectively. These values correspond to approximate upper temperatures of 1500 K (vinyl fluoride) and 1000 K (protonated naphthalene and NAPA). The energy histograms were discretized into 161, 282, and 644 bins, respectively. The elementary random moves of both simulations involve the shift of all quantum numbers ni by +1 or -1 with a fixed probability p. An optimal value of p is estimated in
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TABLE 1: Harmonic Frequencies of the H2CdCHF Molecule Obtained by Stoppa et al.61 at the CCSD(T)/ cc-pVTZ Levela mode 1 (A′) 2 (A′) 3 (A′) 4 (A′) 5 (A′) 6 (A′) 7 (A′) 8 (A′) 9 (A′) 10 (A′′) 11 (A′′) 12 (A′′)
ν(h) (cm 3280 3217 3178 1703 1425 1335 1186 946 481 956 871 725
-1
)
β0
βexp
0.956 0.958 0.957 0.975 0.972 0.979 0.980 0.984 0.999 0.975 0.981 0.984
0.958 0.962 0.964 0.972 0.968 0.978 0.974 0.981 1.004 0.972 0.991 0.983
a Fundamental and experimental frequencies relative to these harmonic values are denoted by β0 and βexp, respectively.
order to reproduce the harmonic density of states exactly known for the harmonic system.72 The Wang-Landau simulations were initiated with a modification factor of f ) e, and consisted of several million MC steps at each iteration. A 90% flatness criterion of the energy distributions was used to monitor convergence. The density of states was finally averaged from the results of 100 independent runs. The subsequent multicanonical simulations were carried out over a finite wavelength range taken as 0 < ν e 4000 cm-1 for all molecules, with a 0.1 cm-1 bin size. Canonical absorption spectra I(ν,T) were obtained after Laplace transformation of the energy-resolved intensities H(ν,E) every 50 K. Peaks in the spectra were analyzed into their first moments, providing in particular the average frequency 〈ν〉 as a function of temperature. The deviations of these frequencies with respect to harmonic references, which are the main focus of the present work, were quantified by defining appropriate scaling factors. Depending on the context, care should be taken to distinguish between several such quantities. When full theoretical details are available at both harmonic and anharmonic levels, β ) 〈ν〉/ν(h) denotes the ratio between the effective (anharmonic) and harmonic frequencies both originating from the model. β implicitly depends on temperature, and the specific 0 K value related to the fundamental and harmonic frequencies is noted β(T)0) ) β0. It also seems natural to compare these scaling factors to the generally accepted values βref obtained from the computational literature. These latter factors carry some dependence with the type of electronic structure method (in the more specific case of DFT, on the functional) and with the basis set. Finally, an experimental scaling factor βexp ) νexp/ν(h) can be defined from the measured frequency, but again this quantity carries some dependence on the type of calculation through the harmonic value. B. Vinyl Fluoride. We first apply the method to a rather small molecule, H2CdCHF, for which both experimental and theoretical investigations can be carried out with substantial accuracy. Vinyl fluoride belongs to a class of atmospherically relevant molecules that may react with ozone.80 For this first and the molecule, the harmonic vibrational frequencies ν(h) i anharmonic couplings χij were borrowed from coupled cluster CCSD(T) calculations performed by Stoppa and co-workers61 with the cc-pVTZ basis set. These authors also conducted FTIR measurements at medium resolution in the 400-5000 cm-1 wavelength range at room temperature. The results of β0 and βexp, deduced from the data of Stoppa et al.,61 are compared to each other in Table 1 for all 12 modes.
Figure 1. Ratio between the effective line position 〈ν(T)〉 and the harmonic frequency ν(h) for several modes of the vinyl fluoride molecule, as a function of temperature. The horizontal dashed lines, when shown, represent the corresponding experimental frequencies obtained by Stoppa et al.61 also normalized by the calculated ν(h). Note the different vertical scale for the mode at 871 cm-1.
The corresponding harmonic frequencies are also reported. These experimental and fundamental ratios are in rather good agreement, both ratios increasing from 0.96 to 1.00 with decreasing mode frequency. This trend has long been reported based on analysis of different systems and at different levels of computational chemistry. For example, in the seminal paper by Scott and Radom,49 the use of different scaling factors was recommended depending on the vibrational range of interest. This procedure, which relies on the allocation of the vibrational frequencies into clearly distinct groups, seems quite hard to apply here except for the tightest stretching modes. The three frequencies above 3000 cm-1 can be correctly reproduced with a scaling factor close to 0.96, whereas β spans the entire range 0.97-1.0 for softer modes. If a single scaling factor were used for the entire set of frequencies, as is probably the most common practice in assigning experimental spectra, our results should be compared with the predictions of Thomas and co-workers,85 who recommended the value βref ) 0.978 for the present CCSD(T)/ccpVTZ treatment of electronic structure. This factor is remarkably j 0 of β0 over all frequencies, close to the arithmetic average β j which equals β0 ) 0.975. The previous comparisons are relevant only for fundamental frequencies. The spectral shifts have also been characterized as a function of temperature for four specific bands characterized by harmonic frequencies of 481, 871, 1186, and 1703 cm-1, respectively. The corresponding scaling factors β are represented in Figure 1, together with the experimentally assigned value βexp. They generally show significant variations with temperature. Except for the softest mode at 481 wavenumbers, for which β(T) ≈ 1 and steadily increases, β is generally below 0.98 and further decreases as temperature increases. One special case is the 871 cm-1 mode, which remains almost unchanged in the 0-1500 K temperature range. The small blue shift found below 1000 K is interpreted from the details of the anharmonicity matrix as due to small positive couplings with the lowest modes,61 which are populated first at low temperatures. Because the couplings with the higher frequency modes are negative,61 the reverse effect takes place at higher temperature. Similar considerations allow explanation of the blue shift of the softest mode at 481 cm-1 seen in Figure 1 above 300 K, with a nearly constant slope of about 1.3 × 10-5 K-1. Five
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Basire et al. TABLE 2: Harmonic (νh) and Fundamental (ν) Frequencies of the N-Ac-Phe-NH2 Peptide Obtained at The B3LYP/4-31G Level, and Corresponding Scaling Ratio β0 ) ν/ν(h) a mode
νh (cm-1) ν (cm-1)
β0
IR N-H (Phe)
3562
3375
0.948
sym NH2
3571
3408
0.954
antisym NH2
3704
3520
0.950
βexp
{ } { } 0.964 0.959
or
0.962 0.962
0.956
a The experimental frequencies of Chin and co-workers at 3426, 3434, and 3543 wavenumbers (ref 69), also normalized by the calculated harmonic frequencies ν(h) are given, including two possible assignments for the two former lines.
Figure 2. Absorption spectra of the N-Ac-Phe-NH2 peptide calculated at 300, 500, and 700 K using the anharmonic force field determined at the DFT/B3LYP level with the 4-31G basis set. The three vertical arrows show the experimental line positions of the N-H stretches (ref 69).
anharmonic couplings between this mode and other soft modes are found to be positive, other couplings being unsufficiently negative to compensate this effect. For the two tighter modes at 1186 and 1703 cm-1, the effective frequency drops linearly when T > 600 K with corresponding slopes of -1.7 × 10-5 K-1 and - 2.4 × 10-5 K-1, respectively. Deviations of the estimated frequency due to temperature effects appear to have the same magnitude as pure static anharmonic effects above 1000 K, which provides an upper limit of validity for using these factors. From a practical point of view, the difference between experimental and anharmonic frequencies is minimal when temperature lies in the approximate range 500-700 K but is only within a few wavenumbers of the measured values at 0 K. Hence, unless a high resolution is necessary or the experimental temperature exceeds about 1000 K, fundamental frequencies already provide a very satisfactory answer. C. The N-Ac-Phe-NH2 Peptide. Small peptides containing only a few amino acids have been extensively studied by means of IR spectroscopy for improving our understanding of intramolecular hydrogen bonding in biomolecules, in the perspective of the emergence of secondary structures. The infrared signature of different backbone conformations has been investigated by Chin and co-workers69 for the N-acetylphenylalaninylamide model peptide. Using density functional theory calculations, these authors found evidence for two sets of conformations based on either a γ-turn type or the β-sheet manifold.69 In the present work, we have chosen to study more specifically the β-sheet structure, which is lower in energy than all γ conformers. The geometry was reoptimized using a similar electronic structure method as in ref 69, namely, density functional theory with the B3LYP hybrid functional; it is depicted as an inset in Figure 2. The anharmonic force field was obtained using the 4-31G basis set with the Gaussian03 suite of programs.81 The IR region of interest is here related to N-H stretching modes and lies experimentally near the 3200-3600 wavenumbers range. We checked that the harmonic frequencies of the three relevant vibrations were barely affected by the relatively small basis set chosen. For this system, the softest mode (associated to the rotation of methyl group) and some other very soft modes (with harmonic frequencies lower than 100 cm-1) turned out to be so anharmonic that their fundamental frequencies become very low or even negative. These values, in addition to being unphysical, appear problematic for the Wang-Landau simula-
tion. This was circumvented by neglecting the couplings between the six softest modes with modes other than the three of interest in the calculation of the vibrational density of states. This procedure ensures that all important anharmonicities are correctly included in the absorption spectra, at least for the tightest vibrational modes we are interested in. The harmonic and fundamental frequencies, together with their ratio β0, are reported in Table 2 for the three N-H modes. Experimental frequencies measured by Chin and co-workers,69 relative to the harmonic values, are also indicated by the βexp factors. In the experiment, two of these stretching modes are very close and differ from each other by less than 10 cm-1. Interestingly, visualization of the normal mode vectors at the harmonic approximation reveals that the frequencies of these modes are reversed upon increasing the basis set from 4-31G to 6-31+G(d). Due to this inversion, the assignment of the frequencies to specific modes is ambiguous, hence the two possible values for βexp in Table 2. Fortunately, as seen in Table 2, the same ordering is found for the harmonic and fundamental frequencies in a given basis set. Experimental and fundamental scaling factors have comparable magnitudes for the three N-H stretchings, β0 being slightly underestimated with respect to βexp. A good agreement is also found with the value βref ) 0.96 recommended by Langhoff and co-workers,86 even though it should be mentioned that these authors did not include nitrogeneous compounds in their database. The variations of the absorption intensity calculated at several temperatures are represented in Figure 2 in the 3200-3600 cm-1 spectral range. The experimental frequencies measured by resonant IR-UV ion dip spectroscopy (IR spectra in the ground electronic state with T < 100 K) are also indicated in this figure. Due to both a larger number of degrees of freedom and the stronger couplings, numerical convergence of the two-dimensional histogram H(ν,E) turned out to be somewhat slower than that for vinyl fluoride. By doubling the statistics, we could however get reproducible spectra even at temperatures as low as 50 K. From Figure 2, the frequencies computed using the present method are slightly underestimated compared to the measured values. The two softer stretching modes also differ much more from each other when anharmonicity is taken into account: they are shifted by about 40 cm-1, whereas their harmonic value differ by 9 wavenumbers only. The 40 cm-1 shift can be partitioned into the static anharmonicity effects, with a 33 cm-1 contribution from fundamental frequencies, and the remaining 7 cm-1 contribution from pure thermal effects. This thermal contribution results from the strongly anharmonic character of the N-H stretching band associated with the phenyl group. Inspecting Figure 2, the three bands exhibit very different shifts, broadenings, and even asymmetries when temperature increases. These effects are best evidenced on the variations of
IR Spectroscopy of Polyatomic Molecules
Figure 3. Ratio between the effective line position 〈ν(T)〉 and the harmonic frequency ν(h) for the three N-H stretches of the N-Ac-PheNH2 peptide, as a function of temperature. Note the different vertical scale of the upper panel.
the scaling factors β with temperature, shown in Figure 3. The softest N-H mode displays a strong, essentially linear red shift over the entire temperature range with a slope of -2.8 × 10-5 K-1. This particular mode is strongly coupled with very low frequency modes, which explains why β varies down to low temperatures. The two other N-H stretchings, which both involve the N-terminus, are also slightly perturbed by the couplings with low frequency modes, as manifested by a small blue shift below 300 K. Above 400 K, these two NH2 modes linearly shift to the red with approximate slopes of -2.6 × 10-6 and -5.1 × 10-6 K-1, respectively, or about 1 order of magnitude below the effects found for the N-H mode at the phenyl side chain. This example shows that a single scaling factor does not satisfactorily perform for these three bands except at low temperature. Nevertheless, as far as T remains below 300 K, using the average value of β = 0.95-0.96 successfully accounts for most anharmonic effects. One remarkable prediction of our calculations is the major difference between the locations of the anharmonic bands and the scaled harmonic frequencies. Experimental results have also been resolved for conformations other than the β-sheet structure,69 with an inverse γ-turn (γL) character of the side chain. These measurements show vibrational bands in closer agreement with the spectra of Figure 2, especially for the gauche conformer in ref 69. This further demonstrates the difficulty denoted as γ(g+) L in assigning vibrational spectra based on static frequencies only and suggests that similar anharmonic calculations should be performed on other conformers to disentangle and clarify this assignment. These theoretical results have been obtained within a rather limited basis set and should probably be confirmed with a larger basis set. Due to computational difficulties in extracting anharmonic potential energy surfaces for the present peptide, these aspects have been investigated on the slightly smaller aromatic hydrocarbon system. D. Protonated Naphthalene. Polycyclic aromatic hydrocarbons (PAHs) have been proposed as likely sources of carbon in different environments of the interstellar medium.82,83 The so-called unidentified infrared emission bands, in particular, may be explained by IR fluorescence of such molecules after UV excitation. In addition to being more convenient for mass spectrometry studies in the laboratory, ionized PAHs appear to match astronomical observations better than neutrals.18,22 The possibility that protonated PAHs are formed in the interstellar
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Figure 4. Ratio β0 between the fundamental (ν) and harmonic (ν(h)) frequencies at T ) 0, as a function of ν(h), obtained for protonated naphthalene using anharmonic DFT force fields with the 4-31G and 6-31+G(d) basis sets. The horizontal dashed line highlights the value βref recommended by Bauschlicher et al.86 for the 4-31G basis set.
TABLE 3: Selected Harmonic Frequencies Obtained for Protonated Naphthalene at the B3LYP/4-31G and B3LYP/ 6-31+G(d) Levels, and Corresponding Fundamental Frequencies β0 ) ν/ν(h) Relative to These Harmonic Values B3LYP/4-31G mode 8 (A′) 9 (A′) 12 (A′) 13 (A′) 18 (A′) 43 (A′′) 44 (A′′) 49 (A′′)
ν
(h)
-1
(cm )
2978 1670 1566 1517 1395 810 762 265
β0 0.952 0.975 0.978 0.980 0.974 0.979 0.985 1.012
B3LYP/6-31+G(d) ν
(h)
(cm-1)
2988 1664 1553 1499 1374 789 737 252
β0 0.953 0.984 0.978 0.978 0.970 0.986 1.003 1.001
medium after hydrogen attachment on these ionized species84 has recently led Lorenz and co-workers24 to measure the infrared spectrum of protonated naphthalene. These authors have shown that the best assignment with quantum chemistry calculations at the DFT/B3LYP/6-311G(2df,2pd) level was found with the lowest energy isomer, depicted as an inset in Figure 4 and also chosen in the present work. The anharmonic force field for protonated naphthalene was also obtained using density functional theory with the B3LYP functional; however the basis sets had to be restricted to 4-31G and 6-31+G(d) for computational efficiency. To our knowledge, the only experimental work currently available for gas phase protonated naphthalene is the IRMPD experiment from Lorenz et al.24 Unfortunately, spectroscopic data deduced from such a multiphotonic absorption experiment cannot be directly compared to our simulations. The theoretical study for this protonated system will thus mainly focus on the influence of the basis sets on the scaling factors at zero and finite temperature. Table 3 summarizes some harmonic frequencies and the scaling factors β0 obtained from the fundamental frequencies with the two density functional calculations. These bands were selected according to their absorption intensity. For all the selected modes, the harmonic frequencies do not differ significantly when the basis set is enlarged: at most, mode 18 has its frequency dropped by 21 wavenumbers. Looking more closely at the entire set of harmonic frequencies, we find that the values obtained with the 4-31G basis set differ from those at the 6-31+G(d) basis set by about 2.14% in average. Turning now to fundamentals, the values of β0 are quite similar for the two basis sets, and increase from about 0.95 up
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Figure 5. Absorption spectra of protonated naphthalene calculated at 300, 500, and 700 K using the anharmonic force fields determined at the DFT/B3LYP level with the 4-31G and 6-31+G(d) basis sets.
to 1.00 as the frequency decreases from the tightest C-H and C-C stretchings down to soft out-of-plane collective modes. This trend, previously noted for vinyl fluoride, is further magnified in Figure 4 where the scaling factors β0 are depicted against the harmonic frequency for the two basis sets. On this figure we also highlighted the reference scaling factor of βref ) 0.9578 recommended by Bauschlicher and Langhoff for the DFT/B3LYP/4-31G computational method.86 Discrepancies between the two basis sets are usually small, except for a few modes. In average, β0 is close to 0.978 when the frequencies are calculated with the 4-31G basis set, and 0.980 with the larger 6-31+G(d) basis set. From the above results, the recommended value βref is suitable only for the tightest stretching frequencies but performs poorly for soft modes. If we analyze more precisely the C-H stretching spectral region, the two C-H aliphatic modes (with harmonic frequencies close to 2980 cm-1) appear much more red shifted with β0 values of about 0.95, i.e., a significant 150 cm-1 difference. For the seven C-H aromatic stretchings, β0 is 0.958 in average. This finding is important because these modes may be responsible for the 3.4 µm feature of spectra in the interstellar medium.87-89 Of course, further evidence that protonated PAHs may be involved in these bands would have to be searched on other molecules as well. On the basis of the above study, we could recommend the use of two separate scaling factors to approximately account for anharmonicities, namely, β ≈ 0.96 for all frequencies higher than 1800 wavenumbers (including C-H stretchings) and β ≈ 0.98 for lower frequencies. This last value is remarkably close to the scaling factor recommended by Scott and Radom49 for the comparable 6-31(d) basis set with the B3LYP functional. It is also consistent with the recent experimental and theoretical investigation on protonated 1,2-dihydronaphthalene, where the authors used β ) 0.955 for C-H stretching modes and β ) 0.975 for the other modes.90 In Figure 5, the IR absorption spectrum in the spectral region of mode number 9 is displayed at three temperatures and for the two basis sets. The DFT calculations carried out for this molecule yield significant 0 K intensities of 79 and 88 km/mol for the 4-31G and 6-31+G(d) basis sets, respectively. When anharmonicities are included, the frequency of this mode becomes compatible with one emission band detected at 6.2 µm in the interstellar medium.91 For this mode, the basis set effects on the absorption spectrum are found to be more than only quantitative. The harmonic
Basire et al.
Figure 6. Ratio between the effective line position 〈ν(T)〉 and the harmonic frequency ν(h) for selected modes of protonated naphthalene, as a function of temperature. The results obtained with anharmonic force fields at the DFT/B3LYP level with both 4-31G and 6-31+G(d) basis sets are shown.
frequencies differ by 10 cm-1 or an equivalent 0.6%. The spectral widths are quite similar, but an additional low intensity feature occurs on the blue side, close to 1640 cm-1, for the larger basis set only. This feature is not caused by fundamental bands but is due to some positive anharmonic couplings of the ninth mode with vibrations close to 500 and 600 wavenumbers. The anharmonic couplings of these two modes with the mode of interest change from -1.0 and -0.5 wavenumbers to +7.0 and +4.5 wavenumbers as the basis set enlarges from 4-31G to 6-31+G(d). These differences, though not exceedingly large, are sufficient to alter the shape of the band already at low temperatures. The consequences in terms of vibrational shift and broadening are better depicted in Figure 6 where the ratios between the effective frequency and the harmonic value are represented as a function of temperature for the two basis sets. These ratios are also shown for the three modes numbered as 12, 43, and 49 in Table 3. Besides the aforementioned 0 K difference in the β0 values, the variations with temperature generally remain parallel in the 0-1000 K temperature range. Only the softest of the four modes exhibits some blue shift with increasing temperature and varies by more than 2% with respect to the fundamental frequency at room temperature. Between 500 and 1000 K, β varies almost linearly with temperature for all modes, with a slightly higher rate for the 4-31G basis set reaching about twice those for modes 9 and 43. Figure 6 confirms the validity limit of static scaling factors in reproducing experimentally measured frequencies and suggests that these factors should be revised at temperatures exceeding about 500 K for the present systems, by about 1% per 500 K. Beyond protonated naphthalene, we anticipate that the results obtained here will also be representative of a broader class of aromatic hydrocarbons. IV. Summary and Outlook Assignment of molecular structure based on vibrational spectroscopy techniques is becoming more and more reliable with complementary advances in both experimental and theoretical methods. However, one major issue remains the role of anharmonicities on fundamental frequencies and its subsequent influence on finite-temperature spectra. Simple scaling of harmonic frequencies with appropriate factors that depend on the level of the electronic structure calculation and the basis set provides a first, simple correction for the static anharmonic
IR Spectroscopy of Polyatomic Molecules effects. This procedure, though commonly used for larger polyatomic systems, may lack safeguards if the measurements are carried out at significant temperatures, as is often the case for gas-phase species or molecules produced by electrospray. Explicit anharmonic force fields, on the other hand, though increasingly accurate and available for large molecules, do not capture temperature effects alone. In the present work, we have further tested a systematic computational approach to compute finite-temperature vibrational spectra, incorporating anharmonicity effects both in the evaluation of transition energies and in the statistical weighting of vibrational states. The method is based on a quadratic Dunham expansion of the energy surface in terms of the vibrational quantum numbers and also requires an estimate of the absorption intensities for all bands. The finitetemperature spectra are obtained in two steps, starting with the Wang-Landau calculation of the density of vibrational states, followed by a multicanonical simulation of the two-dimensional histogram resolved in frequency and total energy. Because these simulation stages can be undertaken without much difficulties, the limiting step of the method mostly lies in the determination of the anharmonic force field. With this approach, the finite-temperature vibrational spectra were computed for three different systems for which experimental measurements or other reference calculations are available. In general, we found that static anharmonicities play a major role on the frequency shifts, which of course is the result expected at lower temperatures. Under those circumstances, scaling factors perform rather well in bringing the band frequencies much closer to the experimental fundamental bands. However, consistent with previous studies,49,90 for the present hydrocarbon-based molecules different scaling factors should apply to soft (below 1000 cm-1) and tight (above 3000 cm-1) modes. Temperature also contributes to shifting the effective frequency with respect to experimental bands, but the effects become significant above 300-500 K depending on the system. The vibrational shifts can be captured by new temperaturedependent scaling factors with respect to the harmonic value, and we have shown that the specific temperature contribution to band shifting usually adds up to the static contribution. In vinyl fluoride and protonated naphthalene, for instance, the underestimation of soft frequency modes (scaling factor β > 1) and the overestimation of tight frequency modes (scaling factor β < 1) by harmonic calculation are both amplified linearly for temperatures below about 1000 K. The results obtained in the present work provide some estimation of the validity range of the simple scaling approach used in most works aimed at characterizing experimental structures through comparison with the spectra computed for selected conformations. They also suggest how to account for temperature effects, which might further amplify the discrepancy between the harmonic and measured frequencies and even exceed the pure static shifting effect. Future improvements of the method could be useful for a more accurate testing on experimental grounds. Those include taking into account rotational degrees of freedom and rotational/vibrational couplings, as well as combination bands and overtones. The method can also be adapted to determine excitation spectra, which would pave the way for modeling more dynamical multiphoton experiments. Work along these lines is currently in progress. Acknowledgment. Some calculations have been carried out at the regional Poˆle Scientifique de Mode´lisation Nume´rique, which is gratefully acknowledged. We also thank GDR 2758 for financial support.
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