Infrared and Raman Spectra of C₄H₄Se and C₄D₄Se Isotopomers: A DFT-PT2 Anharmonic Study

Abstract

IR and Raman spectra of selenophene and of its perdeuterated isotopomer have been obtained in gas phase through density-functional theory (DFT) computations. Vibrational wavenumbers have been calculated using harmonic and anharmonic second-order perturbation theory (PT2) procedures with the B3LYP method and the 6-311G^∗∗ basis set. Anharmonic overtones have been determined by means of the PT2 method. The introduction of anharmonic terms decreases the harmonic wavenumbers, giving a significantly better agreement with the experimental data. The most significant anharmonic effects occur for the C–H and C–D stretching modes, the observed H/D isotopic wavenumber redshifts being satisfactorily reproduced by the PT2 computations within 6–20 cm⁻¹ (1–3%). In the spectral region between 500 cm⁻¹ and 1500 cm⁻¹, the IR spectra are dominated by the out-of-plane C–H (C–D) bending transition, whereas the Raman spectra are mainly characterized by a strong peak mainly attributed to the C=C + C–C bonds stretching vibration with the contribution of the in-plane C–H (C–D) bending deformation. The current results confirm that the PT2 approach combined with the B3LYP/6-311G^∗∗ level of calculation is a satisfactory choice for predicting vibrational spectra of cyclic molecules.

1. Introduction

Calculated harmonic vibrational wavenumbers of organic compounds typically deviate from experimental fundamental data, especially overestimating observed wavenumbers of high-energy X–H (X = C, O, N) stretches [1]. Two principal procedures can be employed in practice to correct the shortcomings of the harmonic approximation: (1) scaling methods [2, 3] and (2) anharmonic computations [4–7]. Scaling factors are usually derived for a certain level of theory and basis set by fitting computed harmonic frequencies to experimental data for restricted subsets of molecules. Scaling corrections often work adequately, even if there are specific cases for which scaling factor transferability could originate questionable results [8, 9]. A much more rigorous treatment is furnished by anharmonic computations that are commonly performed through perturbative [4–6] or variational [7] methods. Anharmonic perturbative approaches are generally proven to be less accurate than variational schemes [10], although they are particularly reliable for predicting fundamental wavenumbers of semirigid cyclic compounds [10–17].

In this contribution we report some interesting results on the performances of the anharmonic second-order perturbation theory (PT2) [6] as implemented in the GAUSSIAN 09 program [18] to predict vibrational wavenumbers. We have chosen to study selenophene (C₄H₄Se) and its perdeuterated isotopomer (C₄D₄Se), for which experimental vibrational spectra with complete assignments of all the transitions are available from the literature [19–23]. Theoretically, the vibrational spectra of selenophene were previously computed at ab initio and density-functional theory (DFT) methods under the harmonic approximation [24–27]. In the present study we have used the hybrid functional B3LYP [28, 29] with the fairly flexible 6-311G** basis set [1]. Anharmonic PT2 B3LYP/6-311G** calculations have been recently performed with success on naphthalene, reproducing accurately the experimental wavenumbers [16]. Overtone wavenumbers of C₄H₄Se and C₄D₄Se have been here predicted for the first time. We also will discuss briefly the most intense vibrational transitions of the IR and Raman spectra. Selenophene which is homologue of the furan molecule has long been the subject of many experimental and theoretical studies as promising building block of π-conjugated polymers for nonlinear optical applications [27, 30–34].

2. Computational Details

All calculations were carried out with the GAUSSIAN 09 package. Geometry of selenophene was fully optimized under the C_2v point group symmetry using the B3LYP functional with the 6-311G** basis set. The harmonic frequencies of selenophene and its perdeuterated isotopomer were obtained using an analytical procedure. The anharmonic corrections to wavenumbers were computed through the PT2 treatment. The PT2 procedure in combination with DFT methods is proven to be adequate to predict anharmonic vibrational wavenumbers of cyclic compounds [10–17], including the homologues furan [10, 13] and thiophene [10]. Under the PT2 approximation, third and semidiagonal fourth energy derivatives with respect to normal coordinates were determined using a numerical differentiation scheme implemented in GAUSSIAN 09. Specifically, we used a step-size displacement of 0.025 Å along the normal coordinate as usually adopted in the literature [12–14]. Fundamental frequencies (𝜈_𝑖) were obtained from harmonic (𝜔_𝑖), diagonal (𝜒_𝑖𝑖), and off-diagonal (𝜒_𝑖𝑗) anharmonic constant values as [6]:𝜈_𝑖=𝜔_𝑖+2𝜒_𝑖𝑖+12_𝑗≠𝑖𝜒_𝑖𝑗. (1)

Besides the PT2 approach, as commonly adopted in the literature, we corrected the computed harmonic wavenumbers by using a scaling factor previously determined by Irikura et al. [36] through a least-mean-squared fitting procedure between 3310 experimental and calculated wavenumbers. In the specific case of the B3LYP/6-311G** level, the scaling factor is 0.9669. The assignments of the vibrational transitions were obtained on the basis of normal modes, as displacements in redundant internal coordinates (in GAUSSIAN 09, option Freq = IntModes) and also through the visualization software Chemcraft [37].

3. Result and Discussion

3.1. Geometry, Rotational Constants, and Dipole Moment

The computed bond lengths, angles, and dipole moment (𝜇) of selenophene (Figure 1) are listed in Table 1, together with the gas phase experimental data for comparison [35]. The B3LYP/6-311G** geometry agree reasonably well with experiment, with a root mean square (rms) deviation {rms=[(1/𝑛)∑^𝑛_𝑖(𝑥_𝑖^exp.−𝑥^calc._𝑖)²]^1/2} where 𝑥_𝑖 is a geometrical parameter value for the bond lengths of 0.008 Å and for bond angles of 1.7°. In Table 1 the vibrationally averaged geometries (𝑟_𝑧 structure) determined using vibration-rotation interaction constant computations [6] are also reported. As can be appreciated from the data in the table, the inclusion of vibrational averaging corrections increases the bond lengths by 0.001–0.007 Å, whereas the bond angles deviate within 0.1°. Table 1 also includes rotational constants (A, B, and C) for C₄H₄Se and C₄D₄Se. The results show that the vibrational averaging corrections little affect the rotational constants which are much more dependent on the H/D isotopic effects, with the values of C₄D₄Se being decreased by ca. 20% with respect to those of C₄H₄Se.

Table 1 

Structural parameters of C₄H₄Se and C₄D₄Se^a.

Figure 1 

Atom numbering for selenophene.

The dipole moment is directed along the C₂ symmetry axis and is here calculated at 0.378 D, in good agreement with the experimental datum of 0.4 D [35] and previous theoretical estimates [25, 30].

3.2. Vibrational Spectra of C₄H₄Se and C₄D₄Se

Experimentally IR and Raman spectra of selenophene were previously investigated by Cataliotti and coworkers [19–23]. Some of theoretical studies restricted to C₄H₄Se were previously carried out under the harmonic approximation [24–27]. To the best of our knowledge anharmonic vibrational wavenumbers of fundamentals and overtones of selenophene and of its perdeuterated isotopomer are lacking to date. Tables 2 and 3 collect the B3LYP/6-311G** harmonic and anharmonic wavenumbers (𝜔 and 𝜈), IR intensities (𝐼_IR), and Raman activities (𝐴_Raman) of C₄H₄Se and C₄D₄Se. These tables also include the available experimental data [21–23] for comparison. Selenophene belongs to the C_2𝑣 symmetry point group with 21 normal modes classified as 8A₁+3A₂+7B₁+3B₂, with all except for the A₂ modes being IR active. The present assignments of the vibrational modes are in good agreement with both experimental [21–23] and previous theoretical investigations [24–27].

Table 2 

Vibrational harmonic, ω, and anharmonic, 𝜈, wavenumbers (cm⁻¹), infrared intensities, 𝐼_IR (km/mol), and Raman activities 𝐴_Raman (Å⁴/amu) of C₄H₄Se.

Table 3 

Vibrational harmonic, 𝜔, and anharmonic, 𝜈, wavenumbers (cm⁻¹), infrared intensities, 𝐼_IR (km/mol), and Raman activities 𝐴_Raman (Å⁴/amu) of C₄D₄Se.

In Figures 2 and 3 we plot the percentage deviations (((𝜔^calc._𝑖−𝜔_𝑖^exp.)/(𝜔_𝑖^exp.))×100) of the B3LYP/6-311G** harmonic and anharmonic vibrational wavenumber values from the experimental data of C₄H₄Se and C₄D₄Se, respectively. Not surprisingly, the harmonic wavenumbers of C₄H₄Se and C₄D₄Se systematically overestimate the experimental values: in fact the percentage deviations from the observed data are within 4.6% and 6.5%, respectively, whereas the rms deviations for all the modes are calculated to be 63 and 38 cm⁻¹, respectively. When excluding the C–H stretches (νC–H), these rms deviations are reduced to 23 cm⁻¹ and 22 cm⁻¹, respectively. For C₄H₄Se, when the harmonic wavenumbers are corrected by the scaling factor of 0.9669, the rms deviation for all the modes decreases to 18 cm⁻¹, whereas the percentage deviations from the experimental data are within 3.7%. As can be appreciated from Figures 2 and 3, in comparison to both the harmonic and scaled harmonic values, the anharmonic wavenumbers are in better agreement with experiment (with the exceptions of mode no. 10 for C₄H₄Se and modes no. is 11 and 18 for C₄D₄Se, which are better reproduced by the harmonic computations), with a percentage error within 2.2% for C₄H₄Se and 5.3% for C₄D₄Se. In line with previous results found for other cyclic compounds [10–17], the most significant anharmonic corrections occur for the νC–H vibrations, which reduce the harmonic wavenumber values by 122–126 cm⁻¹ (ca. −4%), improving noticeably the agreement with the experimental data (within 4–19 cm⁻¹, 0.1–0.6%). It is worth noting that, for a certain 𝑖th C–H (or C–D) stretching mode, the largest vibrational anharmonic constants (𝜒_𝑖,𝑗) involve the remaining C–H (or C–D) stretches as well as the diagonal term. In Figure 4 we report the calculated 𝜒_𝑖,𝑗 (𝑖=1,    𝑗= 1–21) values for mode no. 1, taken as an example. The results show that the most significant contribution originates from the coupling with the mode no. 15, with the 𝜒_1,15 value being calculated to be −108 cm⁻¹ for C₄H₄Se and −47 cm⁻¹ for C₄D₄Se. Through (1), these anharmonic terms determine ca. 90% (C₄H₄Se) and 60% (C₄D₄Se) of the total anharmonic corrections. Thus, on going from the harmonic to the anharmonic data of C₄H₄Se, the rms deviation from the observed values for all the modes is decreased by ca. one order of magnitude (from 63 to 9 cm⁻¹). Note that for the perdeuterated isotopomer the rms deviations are much more closer to each other, being 38 cm⁻¹ for the harmonic and 11 cm⁻¹ for the anharmonic values.

Figure 2 

Percentage deviation relative to experiment [21] of the B3LYP/6-311G** wavenumbers of C4H4Se.

Figure 3 

Percentage deviation relative to experiment [22, 23] of the B3LYP/6-311G** wavenumbers of C4D4Se.

Figure 4 

Anharmonic vibrational constants for the C–H stretching mode no. 1 with the vibrational modes (𝜒1,𝑗,  𝑗=1--21) of C4H4Se and C4D4Se. B3LYP/6-311G** results. For the mode numbering see Tables 2 and 3.

The high-energy IR and Raman spectral regions for C₄H₄Se (C₄D₄Se) are exclusively characterized by the νC–H (νC–D) peaks, predicted in the 3058–3119 cm⁻¹ (2264–2335 cm⁻¹) wavenumber range by the anharmonic computations, in good agreement with experiment [21–23]. Figures 5 and 6 display the anharmonic B3LYP/6-311G** vibrational spectra in the 1500–500 cm⁻¹ wavenumber range for C₄H₄Se and C₄D₄Se, respectively. The spectral profiles were constructed with pure Lorentzian band shapes with a full width at half maximum of 10 cm⁻¹. In these figures we also show graphical representations of the atomic displacement vectors of the most interesting vibrations. As can be appreciated from Figure 5, the IR spectrum of C₄H₄Se is dominated by an absorption (𝐼_IR = 131.5 km/mol) located at 703 cm⁻¹ (707 and 684 cm⁻¹ under the harmonic and scaled harmonic approximation, resp.). This mode is attributed to a pure out-of-plane C–H bending deformation. It is worth noting that the anharmonic calculations are in satisfactory agreement with experiment, which gives a very strong peak at 700 cm⁻¹ (0.4%). The corresponding absorption in the calculated spectrum of the perdeuterated isotopomer (Figure 6) is placed at 522 cm⁻¹ (𝐼_IR = 74.9 km/mol), which well reproduces the observed datum of 520 cm⁻¹.

Figure 5 

B3LYP/6-311G** anharmonic IR (bottom) and Raman (top) spectra of C4H4Se in the 500–1500 cm−1 wavenumber range. Lorentz line shapes with half-width of 10 cm−1 were used.

Figure 6 

B3LYP/6-311G** anharmonic IR (bottom) and Raman (top) spectra of C4D4Se in the 500–1500 cm−1 wavenumber range. Lorentz line shapes with half-width of 10 cm−1 were used.

In the Raman spectra of selenophene and its perdeuterated isotopomer the A₁ symmetry 𝜈C–H and 𝜈C–D vibrations (modes no. 1 and 2) are characterized by the highest 𝐴_Raman values (Tables 2 and 3). As can be seen from Figures 5 and 6, for wavenumbers <1500 cm⁻¹, the strongest Raman peak is placed at 1424 cm⁻¹ (𝐴_Raman=38.0Å⁴/amu) for C₄H₄Se and at 1403 cm⁻¹ (𝐴_Raman=40.4Å⁴/amu) for C₄D₄Se. This vibrational transition (mode no. 3) is mainly assigned to the C=C + C–C bonds stretching with the nonnegligible contribution from the in-plane C–H (C–D) bending motions. The abovecalculated wavenumbers can be compared with the experimental values of 1419 cm⁻¹ (+0.4%) and 1398 cm⁻¹ (+0.4%), respectively.

In Figure 7 we compare the computed (harmonic and anharmonic) and experimental H/D isotopic wavenumber shifts for all the νC–H vibrations (modes 1, 2, 12, and 13). Following the experimental results, upon deuteration the νC–H wavenumbers are downward shifted by 765–787 cm⁻¹. Under the harmonic approximation the νC–H-νC–D wavenumber shifts are calculated to be in the 834–840 cm⁻¹ wavenumber range, overestimating the observed shifts by 50–75 cm⁻¹ (6–10%). The introduction of the anharmonic corrections noticeably improves the accordance with experiment within 6–20 cm⁻¹ (1–3%). For the remaining vibrations the magnitudes of the anharmonic corrections for the H/D isotopic shifts are little relevant.

Figure 7 

H/D isotopic wavenumber shifts for the C–H stretching modes between C4H4Se and C4D4Se. Comparison between experimental [21–23], harmonic, and anharmonic B3LYP/6-311G** data.

Table 4 lists the wavenumber values for the overtone transitions of C₄H₄Se and C₄D₄Se here calculated under the harmonic and anharmonic treatments. Similarly to the results of the fundamental wavenumbers, the largest anharmonic effects are found for the νC–H (νC–D) vibrations, with the harmonic data being decreased by 5% (3-4%).

Table 4 

B3LYP/6-311G** first overtone wavenumbers (cm⁻¹) of selenophene isotopomers.

4. Conclusions

The gas-phase equilibrium structure and harmonic and anharmonic IR and Raman spectra of selenophene and of its perdeuterated isotopomer have been determined and analyzed at the B3LYP/6-311G** level of theory. The overtone transitions have been here computed under the harmonic and anharmonic procedures. On the whole the harmonic wavenumbers overestimate the experimental data of the fundamental transitions with an rms deviation of 63 cm⁻¹ for C₄H₄Se and 38 cm⁻¹ for C₄D₄Se. The corresponding rms deviations for the anharmonic calculations are reduced to 9 and 11 cm⁻¹, respectively. In particular, the C–H and C–D stretches are strongly affected by the anharmonic corrections, which reproduce the experimental H/D isotopic frequency shifts within 6–20 cm⁻¹ (1–3%). The present results suggest that the anharmonic PT2 scheme in combination with the B3LYP functional and the 6-311G** basis set can be considered a promising method to calculate the vibrational spectra of cyclic compounds for which observed data are incomplete or not yet available in the literature.