Anharmonic Vibrational Analysis of the Infrared ... - ACS Publications

Article pubs.acs.org/JPCA

Anharmonic Vibrational Analysis of the Infrared and Raman GasPhase Spectra of s‑trans- and s‑gauche-1,3-Butadiene Sergey V. Krasnoshchekov,*,† Norman C. Craig,‡ Praveenkumar Boopalachandran,§ Jaan Laane,§ and Nikolay F. Stepanov† †

Lomonosov Moscow State University, Leninskiye Gory, 119991, Moscow, Russian Federation Department of Chemistry and Biochemistry, Oberlin College, Oberlin, Ohio 44074, United States § Department of Chemistry, Texas A&M University, College Station, Texas 77843-3255, United States ‡

S Supporting Information *

ABSTRACT: A quantum-mechanical (hybrid MP2/cc-pVTZ and CCSD(T)/cc-pVTZ) full quartic potential energy surface (PES) in rectilinear normal coordinates and the second-order operator canonical Van Vleck perturbation theory (CVPT2) are employed to predict the anharmonic vibrational spectra of strans- and s-gauche-butadiene (BDE). These predictions are used to interpret their infrared and Raman scattering spectra. New high-temperature Raman spectra in the gas phase are presented in support of assignments for the gauche conformer. The CVPT2 solution is based on a PES and electro-optical properties (EOP; dipole moment and polarizability) expanded in Taylor series. Higher terms than those routinely available from Gaussian09 software were calculated by numerical differentiation of quadratic force fields and EOP using the MP2/cc-pVTZ model. The integer coefficients of the polyad quantum numbers were derived for both conformers of BDE. Replacement of harmonic frequencies by their counterparts from the CCSD(T)/cc-pVTZ model significantly improved the agreement with experimental data for s-trans-BDE (root-mean-square deviation ≈ 5.5 cm−1). The accuracy in predicting the rather well-studied spectrum of fundamentals of s-trans-BDE assures good predictions of the spectrum of s-gauche-BDE. A nearly complete assignment of fundamentals was obtained for the gauche conformer. Many nonfundamental transitions of the BDE conformers were interpreted as well. The predictions of multiple Fermi resonances in the complex CH-stretching region correlate well with experiment. It is shown that solving a vibrational anharmonic problem through a numerical-analytic implementation of CVPT2 is a straightforward and computationally advantageous approach for medium-size molecules in comparison with the standard second-order vibrational perturbation theory (VPT2) based on analytic expressions.

1. INTRODUCTION

The traditional form of second-order vibrational perturbation theory (VPT2) is based on analytic expressions for spectroscopic quantities such as anharmonic constants (x) and anharmonic intensities and is followed by a numerical diagonalization of the quasi-diagonal Hamiltonian matrix containing Fermi resonance (FR) and Darling-Dennison (DDR) resonance couplings (W and K parameters, accordingly).21−34 We call this approach AVPT2+WK, where the prefix “A” stands for “analytic”. A complication in applying AVPT2+WK is that the number and complexity of the analytical expressions needed for full-scale implementation make this method error-prone. Another difficulty is associated with the treatment of resonance terms. Within AVPT2+WK, all expressions are deduced for a resonance-free case. Accounting for resonance effects is performed post factum by empirically

The purpose of this article is twofold. We demonstrate the power of canonical Van Vleck perturbation theory at the second-order level augmented by the variational step (CVPT2+WK), 1−20 and we apply this theory to the anharmonic vibrational analysis of the s-trans and s-gauche conformers of butadiene (further denoted as trans- and cisBDE). Figure 1 depicts the structures of the two conformers of BDE. A proper treatment of anharmonicity, resonances, and combination transitions for the two conformers of BDE is provided for the first time. For the trans conformer a principal weakness is addressed in existing assignments of vibrational fundamentals in the CH stretching region where resonances play a major role. For the less abundant gauche conformer, gaps in the assignments of fundamentals are closed. Expanded and improved gas-phase Raman scattering (RS) spectra are reported, as are gas-phase infrared (IR) spectra at medium resolution. © XXXX American Chemical Society

Received: August 6, 2015 Revised: October 3, 2015

A

DOI: 10.1021/acs.jpca.5b07650 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A

approach helps check on the consistency of the solution by comparing calculations with observations for the frequencies of nonfundamental transitions and polyad components connected strongly to fundamental states. The second step in this part of the study is to apply the same methodology to the gauche conformer of BDE to verify earlier assignments and extend them. Many experimental investigations have been made of the vibrational spectroscopy of BDE. In the following review, references focus on recent literature that reports improved resolution and accuracy. References to extensive earlier literature can be found, for instance, in refs 36 and 37. IR spectra have been observed at medium resolution and at high resolution. Examples of medium-resolution IR studies are reported in the work of Wiberg and Rosenberg,38 and Craig et al.39 Halonen et al. did a high-resolution IR investigation in the high end of the CH-stretching region.40 The two other highresolution studies were the analysis of rotational structure in two bands of the dominant trans conformer by Craig et al.39 and the search for bands due to the gauche conformer, which included analysis of the rotational structure in four bands of the trans conformer.41 RS spectra have been observed in the liquid and gas phases. Liquid-phase RS spectra were reported by Craig et al.39 Gasphase RS spectra were observed by Wiberg and Rosenberg,38 by McKean et al.,42 and by Boopalachandran et al.43 The latter work included spectra at temperatures as high as 250 °C for the purpose of increasing the relative amount of the higher energy gauche conformer, which goes from ∼2% at room temperature to ∼20% at 250 °C.41,44 Gas-phase RS spectra were also recorded in the region below 350 cm−1 by Durig et al.,45 Carreira,46 and Engeln et al.47 with a focus on the torsional motion around the C−C bond. IR and RS spectra in argon matrices were recorded by several groups. In all of these investigations a very hot beam of dilute BDE in argon was frozen on an IR-transparent window or on a plate for RS observation. An enhanced content of the gauche conformer present at high temperature was secured in the quick-freezing process. An early report of an IR investigation was in a note by Squillacote et al.48 Following a note,49 HuberWälchli and Günthard provided a full report of a similar study.50 Additional IR matrix-isolation studies were by Furukawa et al.51 and Arnold et al.,52 followed by a matrixisolation RS study by Choi et al.53 Furukawa et al. also did an RS study.51 High-level ab initio calculations have been done to map out the potential function for internal rotation around the C−C bond in BDE.54 These calculations gave 12.6 kJ/mol (min to min.) for the electronic energy of the gauche conformer relative to the trans conformer. The cis conformer corresponds to a saddle point of 2.0 kJ/mol above the gauche conformer. This potential function was consistent with an experimental study by Boopalachandran et al.55 that improved on the earlier work. Predictions of anharmonic frequencies were made for s-transBDE by Feller and Craig with the CCSD(T)/aug-cc-pVTZ model and core−valence electron-correlation corrections for the harmonic frequencies and with an MP2(fc)/aug-cc-pVTZ model (fc for frozen core) for the anharmonic corrections.54 In recent examples, approximate predictions of anharmonic frequencies have been made with scaling of frequencies43 or with scaling of force constants.37,42 Scaling of force constants to anharmonic frequencies is useful for predicting fundamental frequencies but is not a proper treatment of anharmonicity.

Figure 1. Structures of the trans (upper) and gauche (lower) conformers of 1,3-BDE.

spotting terms with “vanishing” denominators. Also, the energy expansion becomes intractable beyond the second order. Our alternative methodology, useful for circumventing the aforementioned problems, is based on the operator implementation of vibrational perturbation theory. In brief, this approach involves solving the vibrational Schrödinger equation every time f rom scratch when a new set of input data is provided and uses the canonical Van Vleck perturbation theory (CVPT). In this methodology, the original vibrational Watson Hamiltonian, expressed in dimensionless normal coordinates, is subjected to operator transformations within orders of perturbation theory until the Hamiltonian gains the desired quasi-diagonal form. In this method, the representation of the Hamiltonian is kept in analytic operator form while the numerical coefficients of operators are consolidated and retained with high numerical precision. Any order of perturbation theory is treated uniformly; handling of resonances is accomplished in a standard form. In addition, transformations can be seamlessly extended to higher order than the second, although doing so requires additional terms in the Taylor expansion of the potential energy surface (PES). As was shown for 1,1difluoroethylene,18 the fourth-order corrections are important; they are of the order of 3−4 cm−1 for CH-stretches. Similar to the AVPT2+WK approach, CVPT requires a variational “+WK” step. Thus, we shall further abbreviate our main method while restricting it to the second order as CVPT2+WK. As the first step of our study, we demonstrate for the trans conformer how well the accepted experimental values are predicted by CVPT2+WK and a hybrid MP2/cc-pVTZ// CCSD(T)/cc-pVTZ PES, with an advanced higher-level calculation of the harmonic part of the PES. In addition, we properly analyze multiple resonances and make assignments of combination bands with greater confidence than in the past. A new, well-supported analysis of the CH-stretching region is performed, permitting assignment of additional polyad peaks. As a further test, we apply the Handy method35 by varying the harmonic f requencies of the trans conformer to achieve the best fit of fundamental anharmonic vibrations. These derived harmonic frequencies are used in further analysis. This B


Article

The Journal of Physical Chemistry A Predictions of vibration fundamentals for the gauche conformer, which is of lower symmetry than the trans conformer, have been made by Feller and Craig with a simpler model than for the trans conformer. The harmonic frequencies were predicted with the CCSD(T)/aug-cc-pVDZ model and core− valence electron-correlation corrections.54 Anharmonic corrections were made at the same level of theory as for the trans conformer.54 Boopalachandran et al. computed anharmonic frequencies for the gauche conformer by scaling predicted frequencies.43 Panchenko and De Maré used the force field scaled for the trans conformer to predict anharmonic frequencies for the gauche conformer.37

(k) Ĥk = Hk(k − 1) + i[Sk , H0]

(3.1)

Here Sk is a specially chosen anti-Hermitian operator Sk = −S†k , ⟨Sk⟩diag = 0 that warrants the unitary property of the canonical Van Vleck transformation Ĥ (k) = UkĤ (k−1)U−1 k , Uk = exp(Sk). The lower index in eq 3.1 designates the order of VPT in partitioning of the Hamiltonian, and the upper index designates the resulting form after the k-th canonical transformation. The S-operator becomes cumbersome in the second order;67−71 its knowledge is necessary for transforming dipole moment components, for instance.72 However, Wigner’s theorem, stating that “the block-diagonal part of the k-times transformed Hamiltonian coincides with the effective Hamiltonian up to order 2k + 1” greatly simplifies the problem when only energy levels are needed.62 After applying a single transformation (eq 3.1) and putting all parts of the Hamiltonian together, the energy levels expression as a function of the vibrational quantum numbers ν̅ = {ν1, ν2, ..., νM} is given by4 1 (1) ⟨Ψ(ν ̅ )|Ĥ |Ψ(ν ̅ )⟩ = ⟨Ψ(ν ̅ )|H0 + H2 + i[S1 , H1]|Ψ(ν ̅ )⟩ 2

2. EXPERIMENT The gas-phase IR spectra were recorded with 400 scans at 25 °C with a Nicolet Magna 760 FT infrared spectrometer at 0.1 cm−1 resolution. The gas cell had an internal length of 10 cm and windows made of potassium bromide. The detector was a DTGS device. Two pressures were used: 32 Torr (Figure S1a,b) and 151 Torr (Figure S2a,b). Aldrich supplied the BDE for all the experiments. The gas-phase RS spectra were observed by Boopalachandran et al.43 Additional experimental results are presented in this study. Details of these experiments were reported in the reference. Briefly, the frequency-doubled Nd:YAG laser gave 6 W of 532 nm green light. Resolution for the spectra was 0.7 cm−1. The pressure of the sample at room temperature was 1 atm. The liquid-phase RS spectrum (Figure S3a,b) was recorded at room temperature with the Nicolet Magna 760 FT infrared spectrometer at 2 cm−1 resolution and an associated Raman module. Excitation was with 180° optics and 0.5 W of Nd:YVO4 laser light at 1064 nm. 10 000 scans were accumulated and observed with a cooled (77 K) germanium detector.

(3.2)

Evaluating the commutator [S1, H1] and assembling coefficients in powers of vibrational quantum numbers plus half, the effective Dunham-type energy expression can be obtained, where E0 is a zero-point energy correction term, ωr denotes harmonic frequencies, and xrs denotes anharmonic constants:4 M

−1

E(ν ̅ )(hc)

= E0 +

∑ ωr ⎛⎝vr + ⎜

r

M

+

∑ xrs⎛⎝vr + ⎜

r≤s

1 ⎞⎟ 2⎠

1 ⎞⎟⎜⎛ 1⎞ vs + ⎟ ⎠ ⎝ 2 2⎠

(3.3)

In addition, VPT2 enables evaluation of expressions for the resonance constants W and K, zero-point vibrational energy (ZPVE), and transition strengths in IR and RS spectra. All AVPT2+WK expressions are derived for the general case when no a priori information about the values of different Hamiltonian constants is available. Hence, resonant terms cannot be removed at the stage of deducing the formulas. The resonances reveal themselves in the form of abnormally large resonance denominators.1,4,14,24,27,28,31,32 The commonly accepted scheme of treating FRs has two steps. First, all algebraic expressions for spectroscopic constants and transition moments (x, K, etc.) are converted into expanded forms with terms containing denominators with harmonic frequencies only in the first power, (±aωr ± ··· ± bωs ...)−1. Second, if the fraction containing potentially vanishing denominators is recognized as abnormally large, the whole “resonant” term must be removed from the expression and accounted for during the variational +WK step, as the corresponding energy levels are considered explicitly via evaluation of the matrix elements and a numerical diagonalization. It was recently shown that in certain situations,32 when the expressions for DDR constants contain signature denominators linked to recognized FRs, the value of the K-constant may be wrong. In brief, if the first-order part of the Hamiltonian is subdivided into nonresonant (*) and purely resonant (†) parts, H1 = H1* + H†1, then the S1* operator cancels only the H1* terms. As was shown in ref 32, if the commutator [S1*, H†1] includes the same operator term as a DDR operator under

3. THEORY For a better understanding of the theoretical conclusions drawn and to point out some new elements in the applied theoretical methodology, we discuss briefly the essentials of VPT2 in both the AVPT2+WK and CVPT2+WK formulations. We limit our consideration to a nonlinear asymmetric top N-atomic molecule with M = 3 × N − 6 degrees of vibrational freedom. The AVPT2+WK implementation of the VPT21−17,21−34 rests on general analytical expressions derived from the vibrational (rotational quantum number J = 0) Watson Hamiltonian4 expressed in dimensionless rectilinear normal coordinates and conjugate momenta. The Hamiltonian includes some numerical constants: ϕrst and ϕrstu as the cubic and quartic force constants (in cm−1), along with Bαe and ζαrs being the rotational constants (cm−1) and Coriolis coupling (zeta) constants, respectively. The VPT2 solution of the Schrödinger equation with the chosen Hamiltonian can be implemented in various forms56 by employing its basic Rayleigh−Schrödinger version (RSPT)3 or by the canonical Van Vleck operator perturbation theory (CVPT).16,56−66 The latter is our method of choice due to its intuitive transparency and generality. In the nondegenerate case both approaches (RSPT2 and CVPT2) can be applied.3 The universally accepted term uniting these approaches is VPT2. It can be shown that at each k-th order of CVPT the following equation must be used:4 C


Article

The Journal of Physical Chemistry A consideration, the corresponding resonance constant K will be calculated incorrectly using the usual VPT2 expressions. In the framework of an automated procedure of the variational +WK step, one can include, for instance, all states Ψ0(ν)̅ with total vibrational excitation limited to Nmax quanta for each state,

a† =

1 1 (q − ip), a = (q + ip) 2 2

(3.5)

In the framework of this method, the Hamiltonian or any original or canonically transformed operator is expressed through CAO, preserving so-called normal ordering,56 M

M

Ψ0(ν ̅ ) =

M

∏ ψk(0)(νk), ∑ νk ≤ Νmax k=1

k=1

H=

∑ hj ∏ (al†)mjl (al)njl (3.6)

l=1

j

(3.4)

Here the summation on j is performed over all operator terms of the Hamiltonian with a scalar multiplier hj, and the integers mjl and njl are powers of the corresponding CAO products. The CAO-representation greatly simplifies obtaining the S operator needed for cancellation of the nonresonance off-diagonal terms in the Hamiltonian expressed in the form of eq 3.6,56,61

where ψ(0) k are individual normal mode wave functions. The parameter Nmax can have a minimal value of two and possibly increase to three or four, molecular size permitting. In theory, a larger value of Nmax produces a more accurate result for energy levels, but in practice an increase of Nmax can be counterproductive, as it can become difficult to identify relevant states. The picture of mixed states can also be very sensitive to the input data (force constants), especially for larger basis sets. For the BDE molecule, the choice of Nmax = 4 produces 20 474 basis functions, which is not prohibitive for the calculation even without symmetry considerations. However, we universally employed the setting Nmax = 2, which results in 324 basis functions in total. This setting takes into account an important type of DDR (11−11), requiring the quartic force constants ϕrstu, often set to zero for the “semi-diagonal” force field approximation. There is an alternative polyad technique for a more targeted assembling of relevant basis functions to account efficiently for important couplings between states.17,73−76 A tremendous advantage of using the polyad quantum number concept is that far fewer basis functions are required to account for resonance couplings, which permits a more accurate treatment of much larger molecules. This concept requires knowledge of a set of positive integer polyad coefficients associated with individual normal modes. Any resonance vector composed of powers n,m of the creation and annihilation operators (a†r )n and (ar)m in any “admissible” resonance operator (positive numbers for a†r and negative for ar) must be orthogonal to the vector of polyad coefficients. With an increase of molecular size, it becomes rather difficult to find the polyad vector in a “blackbox” manner. The trouble lies in the necessity of a “state of the art” selection of a set of exactly M − 1 linearly independent vectors corresponding to the strongest resonances that must be orthogonal to the vector of polyad coefficients.17,73−76 The aforementioned shortcomings of AVPT2+WK led us to an alternative implementation of VPT2. We found a solution from the studies of Sibert et al.,64 who suggested applying a general operator form of CVPT to a vibrational problem with the Meyer−Günthard−Pickett Hamiltonian for general curvilinear coordinates.77−79 The approach we employed is similar to the one of Sibert, but the main feature of our approach is use of the Watson Hamiltonian that, being less accurate for highly excited vibrational states (over ∼10 000 cm−1), is easier to implement and is fully compatible with the AVPT2+WK method. The CVPT2+WK preserves the analytic operator representation of the Hamiltonian during the process of canonical transformations until the stage when the Hamiltonian gains the desired quasi-diagonal form. Following Birss and Choi,61 the original Hamiltonian is converted into a new representation in creation and annihilation operators (CAO) of vibrational quanta (nondegenerate case),61

M

M

S = −i ∑ hj(∑ (mjl − njl)ωl)−1 ∏ (al†)mjl (al)njl j

l=1

l=1

(3.7)

The evaluation of (nested) commutators of the general form [SK, HL] in eq 3.1 can be performed through transformation of the operator products into an equivalent representation:16,64 (a†)k al(a†)m (a)n = (a†)k + m (a)l + n min(l , m) ⎡⎛ i−1 1 + ∑ ⎢⎜⎜ ∏ (l − j)(m − ⎢ i = 1 ⎣⎝ i! j = 0

⎤ ⎞ j)⎟⎟(a†)k + m − i (a)l + n − i ⎥ ⎥ ⎠ ⎦ (3.8)

After two canonical transformations, the Hamiltonian is reduced to a quasi-diagonal form. The resonant operators are recognized in accordance to the following general criterion that is applied for all orders of CVPT,16,24 M

Ωk = |hk (∑ (mkl − nkl)ωl)−1| > Ω† l=1

(3.9)

The threshold value of the cutoff parameter Ω† (resonance index) can be determined by examination of the values of Ωk for experimentally proven resonances or by using certain physical conditions.32 In addition, the “resonance denominator” (the value of sum in parentheses in eq 3.9) can be required not to exceed a certain threshold value Δ†.32 The matrix representation of the transformed quasi-diagonal Hamiltonian is obtained by integration with a chosen set of harmonic oscillator basis functions; this matrix is subject to the final numerical diagonalization (+WK step). The process of canonical transformations by CVPT2 produces an explicit form of the S-operators, so that any molecular property (such as a dipole moment vector or polarizability tensor components) that is expanded in a power series of normal coordinates qi can also be subjected to same canonical transformations for evaluation of the corresponding anharmonic properties. This scheme is applicable for the evaluation of intensities of infrared and/or Raman bands.80−83 Note that the preferred way of presenting RS intensities is evaluation of normalized absolute differential cross sections,84 expressed in units of 1 × 10−48 cm6/sr (steradian), which we will further abbreviate as RSU. In the framework of AVPT2+WK the evaluation of anharmonic intensities of fundamental transitions is a rather difficult task.85−87 Willetts et al.72 obtained a cumbersome expression for IR fundamental transitions using an explicit form D


Article

The Journal of Physical Chemistry A of S2. Later, Vázquez and Stanton88 applied Rayleigh− Schrödinger perturbation theory in the second order to obtain an expression for the transition moment of vibrational fundamentals. Recently, Barone et al.89,82 published the corrected form of the formula obtained by Vázquez and Stanton and extended the treatment to other molecular properties.

The applicability of the polyad technique for molecules of the size of BDE has not been studied previously. We attempted to find out if a unique polyad vector with reasonable values of coefficients can be obtained for the BDE conformers. Different strategies can be applied for finding the optimum form of the polyad vector on the basis of a given set of resonance vectors.17 In Section 5 we describe the results of determination of the polyad vector. The variational (+WK) step was performed using harmonic oscillator basis functions with a total sum of quantum numbers less or equal to Nmax = 2, 3, or 4. For the BDE molecule, these values of Nmax correspond to 324, 2924, and 20 474 basis functions, respectively. The CPU time on a desktop computer for a single CVPT2+WK calculation with Nmax = 3 is approximately an hour, while the setting Nmax = 4 takes ∼1 d and requires more RAM. We used the Chebyshev supercomputer resources at Moscow State University in such cases.

4. THE METHOD OF CALCULATION In our study of anharmonic vibrations of BDE conformers, we employed CVPT2+WK as implemented in our upgraded software package ANCO (acronym for Analysis of Normal COordinates). The current version permits working with molecules containing up to 12 atoms.15,16,32 The traditional form of VPT2 (AVPT2+WK) is also implemented in ANCO for development purposes and for vibrational calculations of bigger size molecules (up to 40 atoms in the current version), although not in a comprehensive form. The Gaussian09 program package (G09)90 was utilized for the geometry optimizations of both conformers of BDE and evaluations of the Hessians (Cartesian harmonic force constants) and electro-optical properties (EOP; equilibrium value and derivatives of dipole moment vector and polarizability tensor) at the point of equilibrium and displaced configurations in normal coordinates. The quantum-mechanical second-order Møller−Plesset electronic perturbation theory model (MP2) with the Dunning correlation consistent basis set (cc-pVTZ) was employed for these calculations. In addition, systematic errors of the MP2 method were compensated by substituting the harmonic frequencies ωr with their counterparts, as computed by the higher-level CCSD(T)/cc-pVTZ model. This hybrid approach has proved its efficacy in several studies.91−94 The full quartic PESs were obtained by the one- and twodimensional numerical differentiation of Hessians in normal coordinates, using the 9 and 3 × 3 equidistant grids, accordingly, and a 0.02 Å × (a.m.u.)1/2 step size. Calculation of the cubic surfaces of the dipole moment and the quadratic surfaces of the polarizability tensor components were performed by the single and double differentiation of the analytic dipole moment first derivatives and the polarizability, respectively. In the framework of the CVPT2+WK approach, a universal resonance criterion can be applied for both FRs and DDRs; it is outlined by eq 3.9 above in Section 2. According to our experience,32 suitable threshold values of the “resonance index” Ω† lie in the range of 0.01−0.5 (dimensionless), and values of a resonance denominator Δ† are ∼200−600 cm−1. In theory, variation of these criteria should not severely affect the final CVPT2+WK solution, because sufficiently weak couplings are correctly accounted for either perturbatively or variationally. However, variations in the criteria Ω† and Δ† affect the number of resonance operators, which can be used for determination of the polyad vector. In the ideal case, which holds reasonably well for small molecules, the number of strong “independent” resonances [in the vector sense, treating the powers of the resonance operator CAO as coordinates in M = 3 × NATOM − 6(5) space] is equal to the number of vibrational degrees of freedom less one.17,73−76 Thus, the unique vector of integer polyad coefficients is orthogonal to the subspace of resonance vectors. This construction ensures block-diagonal structure of the Hamiltonian and simplifies both the classification of energy levels and assembling the complete sets of basis functions.

5. RESULTS AND DISCUSSION 5.1. Reference Geometry. The numbering of atoms for both s-trans- and s-gauche-1,3-BDEs is presented in Figure 1. The results of BDE geometry optimizations with two quantummechanical models (MP2/cc-pVTZ and CCSD(T)/cc-pVTZ) along with the comparison with the semiexperimental values appear in Table S1 in the Supporting Information. As shown by Allen and co-workers,95 the accuracy of the reference geometry is of prime importance for a good reproduction of the harmonic part of the potential. Table S1 demonstrates a good agreement between the MP2/cc-pVTZ reference geometry and the semiexperimental re structure.96 For all vibrational calculations in this study, the MP2/cc-pVTZ geometry was used. 5.2. Resonance Effects in s-trans-1,3-Butadiene. As explained above, the CVPT2+WK methodology leads to the dimensionless criterion Ω† for detecting resonances in a straightforward way. To verify the stability of the solution (in particular, the values of fundamentals) upon the choice of the resonance index threshold Ω† and to find a suitable form of the vector of polyad coefficients, a series of calculations was undertaken. The setting of Ω† was changed from 0.01 to 0.10 in steps of 0.01 and further to 0.50 in steps of 0.05. The value of Δ† was fixed at 300 cm−1, and the basis set was restricted to Nmax ≤ 3. The RMDS is quite stable for Ω† = 0.01 to 0.10, varying by ±0.1 cm−1 near the value of ∼5.5 cm−1. A large number of resonances was detected (between 575 and 125 for Ω† = 0.01 and 0.10, accordingly), the dominant contribution being from DDRs of type 11−11. This finding demonstrates the necessity of using the full quartic force field with all ϕrstu force constants. Increasing Ω† above 0.10 gradually degrades the fit of three CH fundamentals ν3, ν18, and ν19. Therefore, for further calculations we chose the standard value Ω† = 0.05. However, for finding the form of the polyad vector it was necessary to decrease the number of resonance vectors by an increase of Ω†. To find a suitable form of the polyad vector, the dimension of the resonance vector space must be equal to M − 1.17,73−76 This target is achieved by setting Ω† = 0.40. This choice produces 58 Fermi and Darling−Dennison resonances, of which 23 = M − 1 are linearly independent. Although the setting Ω† = 0.40 rules out certain significant resonances, this restriction is an inevitable condition for finding the form of the polyad vector for a molecule of this size. The polyad vector is given by E


Article


cm−1, K(3+118+12−119−1) = 116 cm−1, K(3+119+11−117−1) = 176 cm−1, K(3+119+12−118−1) = 115 cm−1, K(17+119+11−12−1) = −127 cm−1, K(17+119+11−13−1) = 161 cm−1, K(18+119+12−13−1) = 115 cm−1. The level repulsion effect ε for all these resonances was >10 cm−1. For handling the 11−11 resonances the inclusion of the ϕrstu force constants is compulsory. The 2−2 Darling−Dennison resonances are most commonly recognized. There are at least four significant resonances of this kind in BDE: K(2+23−2) = −44 cm−1, K(2+218−2) = −66 cm−1, K(1+219−2) = −54 cm−1, K(17+219−2) = −54 cm−1. It should be added that although 2−11 resonances are generally less frequently of significance, it is not difficult to include them. There is only one of significance for BDE: K(18+22−13−1) = −95 cm−1. The least important resonances have the total quanta of excitation of the second state equal to three (1−3, 1−21, 1− 111). The values of the corresponding K values in our case were 0.05. The total number of basis functions in 32 lower polyads is less than 8000 with the restriction that the maximum excitation for each mode Nmax ≤ 3. The biggest polyad with P = 32 has 1400 basis functions, and this block is further factored into four symmetry sub-blocks. This outcome demonstrates that using the polyad technique significantly reduces the number of basis functions. However, the comparison of the predictions of fundamental frequencies of BDE made by using the polyad technique and without it shows that it is sufficient to employ a less elaborate automated technique with resonance selection criteria settings Ω† = 0.05, Δ† = 300 cm−1, and Nmax ≤ 2. We note that this work is the first time a meaningful form of the polyad vector for a 10-atomic molecule has been obtained with a systematic procedure and the polyad technique has been applied to molecules of this size and complexity. All subsequent calculations were performed using the universal settings mentioned above (Ω† = 0.05, Δ† = 300 cm−1, Nmax ≤ 2). The total number of resonances was equal to 219, of which the number of FRs is 56 (1−2 type: 10). The analysis shows that modes 3−5 and 18−22 are most involved in FRs. To mention a few, FRs 3+14−15−1 and 19+15−120−1 correspond to values Ω ≈ 1.0 and Δ ≈ 30.0 cm−1. Of greater interest is the analysis of the results for Darling− Dennison resonances. The correct AVPT2 expressions for all seven types of D−D resonances were derived very recently.31,32 The literature data on the extent of occurrence of various types of D−D resonances is rather scarce. In addition, there is no well-established criterion for including them. From our analysis of D−D resonances that were detected using CVPT2 with aforementioned settings, we can recommend the following approximate parameters for selecting D−D resonances for AVPT2+WK: (1) the value of the D−D constant: K ≥ 5.0 cm−1; (2) the value of resonance denominator: ΔDD ≤ 200 cm−1; and (3) the minimum 2 × 2 effect ε of “repulsion” of the deperturbed diagonal energy levels E1 and E2 after diagonalization with the coupling K: ε ≥ 0.1 cm−1: ε1 − 2

⎤ |E − E1| ⎡ 4K 2 ⎢ 1+ ⎥ 1 = 2 − 2 ⎢⎣ ⎥⎦ (E2 − E1)2

P = 26 × (ν1 + ν2 + ν3 + ν14 + ν15 + ν16) + 14 × (ν4 + ν17) + 12 × (ν5 + ν18) + 11 × (ν6 + ν19) + 9 × (ν7 + ν20) + 8 × (ν8 + ν9 + ν21 + ν22) + 7 × ν10 + 6 × ν11 + 5 × ν23 + 4 × ν24 + 2 × ν12 + ν13

(4.2)

(4.3)

This result shows that for a molecule of a rather large size as two conformers of BDE it is possible to find a sensible form of the polyad vector. However, we found that using the polyad technique does not significantly improve the mean value of deviations, while its application requires expert knowledge. We

Concerning the occurrence of different types of D−D resonances, we found that the most common contributors are the 11−11 resonances. The strongest of them are the following: K(2+117+11−119−1) = −127 cm−1, K(3+117+11−119−1) = 161 F


Article

The Journal of Physical Chemistry A conclude that finding and applying the polyad technique for BDE is possible, but by the balance of pros and cons it is not of much value for a molecule of this size. Introduction of many additional states to the matrix representation of the Hamiltonian makes the picture more “blurred” and difficult to interpret. Therefore, as in the case of the trans conformer, we chose the same universal settings (Ω† = 0.05, Δ† = 300 cm−1, Nmax ≤ 2) for the gauche conformer. 5.4. Theoretical Analysis of the Observed Infrared and Raman Scattering Spectra for s-trans- and s-gauche-1,3Butadiene. The previous studies, except the AVPT2+WK study with the DFT-based quartic PES by Krasnoshchekov et al.,81 did not include a full-scale theoretical modeling of anharmonic vibrational states of the trans and gauche conformers of BDE. In addition, they did not take into account frequency shifts due to multiple FRs and DDRs, especially in the CH-stretching region. In the following discussion of the vibrational assignments, we refer to various vibrational spectra of BDE. Two scans of the gas-phase IR spectrum at 0.1 cm−1 resolution are in Figure S1a,b at 32 Torr and in Figure S2a,b at 151 Torr in the Supporting Information. A liquid-phase RS spectrum at 2 cm−1 resolution and high S/N is in Figure S3a,b. Polarization information for the RS spectrum of the liquid phase was confirmed in the present study. Previously unpublished parts of gas-phase RS spectra for 250 °C compared with 25 °C are supplied as figures in the current text. Detailed compilations of the observed bands, theoretical predictions, and assignments are in Table S2 in the Supporting Information. Because spectra were recorded at different temperatures, intensity descriptions vary. For gas-phase IR spectra and liquid-phase RS spectra relative intensities are for room temperature. For gas-phase Raman spectra intensities for the two conformers are for 250 °C; these intensity estimates were revised since the earlier report in Boopalachandran et al.43 For matrix-isolation spectra the relative intensities are as observed in the matrices. In the interpretation of the matrix-isolation spectra, authors were ambivalent about the structure of higher-energy form of BDE.50−53 Was it the gauche conformer or the cis conformer? Recent high-level calculations cited in the Introduction give strong support to the gauche conformer being the species observed in matrix-isolation. Thus, we regard the species in the matrix spectra as being the gauche conformer. In citing matrix spectra, we emphasize the IR investigation of Huber-Wälchli and Günthard because they gave details of their observations and not merely assignments of fundamentals.50 To organize the interpretation of the observed IR and RS spectra of the whole range of 100−4000 cm−1, we split the discussion into eight subranges. They span the following intervals of wavenumbers in ascending order: 100−800 cm−1 (range 1; see Section 5.5.1 below, etc.), 800−1250 cm−1 (range 2), 1250−1350 cm−1 (range 3), 1350−1550 cm−1 (range 4), 1550−1800 cm−1 (range 5), 1800−2800 cm−1 (range 6), 2800−3200 cm−1 (range 7), and 3200−4000 cm−1 (range 8). In the Discussion below, we compare the anharmonic frequencies and intensities (IR and RS) of fundamental and nonfundamental transitions, predicted without any fitting of spectroscopic constants to experimental data, with observed spectral features and analyze assignments of observed band peak positions to vibrational quantum numbers. The new hightemperature, gas-phase RS data receive special attention for assignments of the gauche conformer. The discussion emphasizes resolving uncertainties in the assignments for the

less abundant gauche rotamer as well as clarifying CHstretching assignments for the trans conformer. The majority of the assignments are drawn from gas-phase spectra. However, observations in matrix-isolation reinforce some assignments and are depended upon in a few cases. 5.5.1. Range 100−800 cm−1. This range contains ν9 (Ag, RS-active), ν12, ν13 (Au, IR-active), ν16 (Bg, RS-active), and ν24 (Bu, IR-active) for the trans conformer and ν11, ν12, ν13 (A) and ν23, ν24 (B) for the gauche conformer, as well as some observed nonfundamental transitions. Annotated gas-phase RS (250 °C compared with 25 °C) spectra for 100−800 cm−1 are presented in Figure 2.

Figure 2. Gas-phase RS spectra of 1,3-BDE at 25 °C (blue) and 250 °C (red) in the range of 800−100 cm−1.

The trans conformer reveals itself in the RS spectrum range of 100−800 cm−1 with the two resonance-free fundamentals, ν9 [512 cm−1 (vs)] and ν16 [748 cm−1 (vvw)]. These values have reasonable agreement with the theoretical wavenumber values of 506 cm−1 (A = 8.4 RSU) and 749 cm−1 (A = 3.1 RSU) even though the theoretical scattering intensities in parentheses at the MP2/cc-pVTZ level are poor descriptions of the observations, the value of A for ν9 being too small and the value for ν16 being much too large. For the scattering intensity values from the MP2/cc-pVTZ model in Table S2, we must be cautious when using intensity predictions to support experimental assignments. In the liquid phase, the band at 513 cm−1 is polarized, in accord with Ag symmetry for this transition. In addition to the allowed band for ν16(Bg) in the RS spectrum at 748 cm1 (vvw), this transition may possibly appear weakly in the IR spectrum. A faint Q branch occurs in the gasphase IR spectra at 751 cm−1 at 32 and 151 Torr with a greater relative intensity at 151 Torr. This feature is absent from the high-resolution spectrum recorded at 5 Torr.41 The IR frequency of 750.6 cm−1 is in agreement within experimental uncertainty with the observation of the vvw band at 748 cm−1 in the gas-phase Raman spectrum. The unusual IR observation may be caused by collisional effects and requires further study. In the RS spectrum, there are two nonfundamental bands for the trans conformer in the region up to 800 cm−1. They lie at 322 cm−1 (w) (2ν13, theoretical values 314 cm−1, A = 1.6 RSU) and at 683 cm−1 (w) (ν12 + ν13, theoretical values 676 cm−1, A = 0.32 RSU). In the room-temperature IR spectrum for the 100−800 cm−1 range, we consider peaks attributable to the trans conformer. Fundamentals ν12 and ν13 of Au symmetry and ν24 of Bu G


Article

The Journal of Physical Chemistry A symmetry are all resonance-free and are predicted at 520 cm−1 (I = 11.7 km/mol), at 158 cm−1 (I = 0.44 km/mol) and at 292 cm−1 (I = 2.4 km/mol), respectively. Their correspondence with observed bands is free of ambiguities. Craig and Sams41 performed a high-resolution study of the band for ν12 and determined its center as 524.5739 cm−1 (m, C-type). Craig and Sams41 also analyzed the rotational structure of the band for ν13 at 162.42 cm−1 (vw, C-type), and Craig et al.39 observed the band for ν24 at 299.1 cm−1 (w, B-type). The IR intensity predictions have greater consistency with the observations than was found for intensities of the Raman bands. There is a weak observed band (see Figure S2b) at 675 cm−1 (vw, C-type) that we assign to the combination band ν9+ν13, since its predicted value and intensity are 667 cm−1 and 0.19 km/mol for this Au transition. For the gauche conformer in the range of 100−300 cm−1 the calculation predicts the presence of three rather strong bands in the RS spectrum: 134 cm−1 (ν13, A = 33 RSU) and an FR doublet of ν12 and 2ν13: 284 cm−1 (77% ν12, 22% 2ν13, A = 16 RSU); and 228 cm−1 (23% ν12, 76% 2ν13, A = 4.0 RSU). The IR counterparts are predicted to have negligible intensity. Boopalachandran et al.43 did not observe frequencies below 200 cm−1 in the gas-phase RS spectrum. However, Boopalachandran et al.55 observed a double-jump torsion frequency 215 cm−1 (vw) for the (0+−2+) transition (gauche), which correlates fairly well with our calculated value of 134 cm−1 for ν13 and better with 106 cm−1 predicted from the experimental potential function.43 The absence of the ν13 band in the RS spectrum may indicate that the predicted activity is much too large, in conjunction with the poor prediction of the frequency. However, spectral observations close to the exciting line are difficult. Concerning the prediction of the ν12 and 2ν13 FR doublet, we must be cautious due to the poor prediction for ν13. The band at 282 cm−1 (w) is assigned to ν12 with the possibility of some mixing with 2ν13. There are two more fundamentals (ν24 and ν11) of the gauche conformer with reasonable intensities in the RS spectrum in the region up to 800 cm−1. These transitions also are predicted to have significant IR intensities. A third fundamental (ν23) of the gauche conformer in this spectral range has negligible RS intensity predicted but significant predicted IR intensity. The predicted resonance-free peaks for the RS-observable transitions are 457 cm−1 (ν24, I = 12 km/ mol, A = 2.0 RSU) and 727 cm−1 (ν11, I = 3.7 km/mol, A = 3.6 RSU). Boopalachandran et al.55 reported a peak at 734 cm−1 in the high-temperature (250 °C) spectrum, now revised to 729 cm−1, and attributed it to ν11, which agrees with our prediction. A band in this location was also observed in several matrixisolation IR investigations (Table S2). A weak band for ν11 in the gas-phase IR spectrum recorded at room temperature would be obscured by the difference band [ν15(Bg) − ν13(Au)] at 746 cm−1 for the trans conformer.41 Earlier, De Maré et al.97 had mistakenly attributed this band to the gauche conformer. In Figure 2, a very weak band at 460 cm−1 in the RS spectrum may have a higher relative intensity at 250 °C. A corresponding band is seen in various matrix-isolation IR spectra. In addition, Craig and Sams44 reported weak features in the high-resolution IR spectrum at 462 and 464 cm−1. We assign these features to ν24 (gauche). The remaining transition for the gauche conformer in the region below 800 cm−1 is ν23. The prediction for this band of B symmetry is 600 cm−1 (I = 8.5 km/mol, A = 0.02 RSU). A very weak Q-branch feature appears at 601.7 cm−1 in the gas-phase

IR spectrum, we assign it to ν23 (gauche). Corresponding features occur in the matrix-isolation IR spectra (Table S2). A close-lying RS quite weak peak at 603 cm−1 is attributed to 2ν24 of the trans conformer; its predicted intensity is 0.12 RSU. 5.5.2. 800−1250 cm−1. The next range, 800 to 1250 cm−1, is crowded with seven bands for each conformer. The trans conformer has the following fundamental transitions in this region: ν7, ν8 (Ag, Raman-active), ν10, ν11 (Au, infrared-active), ν14, ν15 (Bg, Raman-active), and ν23 (Bu, infrared-active). The gauche conformer has four IR- and Raman-active fundamentals for the symmetry species of A-type: ν7, ν8, ν9, ν10 and three for the symmetry species of B-type: ν20, ν21, ν22. Figure 3 compares the gas-phase Raman spectra at 25 and 250 °C in the range of 800−1250 cm−1.


Confirmation of the assignments of the trans conformer bands for ν7 and ν8 in the Raman spectrum is immediate. These bands have sufficient intensity (especially ν7), and no bands of significant intensity fall nearby. In addition, FR does not affect these modes. Literature assignments agree well with our theoretical predictions: Boopalachandran et al.43 assigned the band at 1204 cm−1 (s) to ν7 (theoretical value 1198 cm−1 and A = 14 RSU) and the band at 889 cm−1 (w) to ν8 (theoretical value 883 cm−1 and A = 0.47 RSU), assignments that we confirm. Both transitions are polarized in the liquid-phase RS spectrum, as required for modes of Ag symmetry. The predicted RS intensities are closer to the observations than was so for ν9(Ag) and ν16(Bg). The prediction of the intensity for the ν16(Bg) mode was much too large. The other two Raman-active fundamentals in this frequency range, ν14 and ν15 of Bg symmetry, are predicted to be weak. The prediction for ν14 is 972 cm−1 (A = 0.79 RSU), and for ν15 the prediction is 903 cm−1 (A = 3.2 RSU). Armed with these predictions, we assign ν14 to the very very weak RS band in the gas-phase at 972 cm−1 and ν15 to the very weak RS band at 910 cm−1 (Figure 3). Corresponding bands were observed in the liquid-phase RS spectrum at 967 and 908 cm−1 (Figure S3b). Making depolarization measurements for very weak bands is unreliable. The gas-phase assignment for ν14 differs from the 977 cm−1 value of Boopalachandran et al.43 Confirmation of the assignments of the three IR-active fundamentals of the trans conformer in the 800−1250 cm−1 range is certain (Figure S1b). Strong bands for the two Au fundamentals, ν10 and ν11, with C-type band shapes are at 1014 cm−1 (s) and 908 cm−1 (vs). They agree with the predictions, H


Article

The Journal of Physical Chemistry A which are 1011 cm−1 (I = 35 km/mol) and 901 cm−1 (I = 74 km/mol). The third IR-active fundamental in this frequency range is ν10 with Bu symmetry. The predictions give 982 cm−1 (I = 1.4 km/mol). Despite the prediction of a low intensity, a band appears as a shoulder with medium intensity and an appropriate B-type shape at 990 cm−1. None of these IR-active fundamentals of the trans conformer is involved in FR. We turn to the gauche conformer in the 800−1250 cm−1 range. The lowest vibrational band for this conformer in this range belongs to ν10 of A symmetry. According to the theoretical prediction, ν10 = 873 cm−1 is slightly perturbed by FR with the combination state ν11 + ν13. The resonance shift is small (2.7 cm−1), and the component with the major contribution (84%) from ν10 is four times more intense (A = 1.1 RSU) than the other component in the RS spectrum. The observed RS band at 869 cm−1 (vw) (Figure 3) fits the calculation and supports the assignment. Unfortunately, additional support does not appear in the matrix-isolation RS or IR spectra. The predicted IR intensity of only 0.39 km/mol makes this band too weak to observe. The spectral range from 890 to 920 cm−1 is compromised for the gauche conformer both in the IR spectrum and in the RS spectrum. The RS spectrum contains the weak band for ν15 of the trans conformer at 910 cm−1 (Figure 3), while the closelying ν9 transition of the gauche conformer is heavily mixed with 2ν24 due to a strong FR. Theoretical values of the doublet are 905 cm−1 (A = 1.0, 52% ν9 and 47% 2ν24) and 910 cm−1 (A = 0.83, 47% ν9 and 52% 2ν24). The weak FR doublet, belonging to the gauche conformer, can be experimentally observed as a single peak, because the theoretical zero-order states lie very close to each other by coincidence (907.4 and 907.5 cm−1), while the FR constant W is only 9.5 cm−1. As a new assignment, we attribute ν9 for the gauche-conformer to the RS band at 922 cm−1 (vw), as shown in Figure 3. This gauche conformer fundamental ν9 should be also observable in the IR spectrum. Indeed, Arnold et al.52 found a close-lying peak at 920 cm−1 in the matrix-isolation IR spectrum, as was reflected by Choi et al.,53 in accord with our assignment. The gauche conformer fundamental ν22 also lies in this vicinity with a theoretical value of 910 cm−1 and a strong IR intensity of I = 65 km/mol. Its predicted RS intensity is very low (A = 0.12 RSU). In the gasphase IR spectrum in this region, the intense R branch for the ν11 band of the trans conformer hides any band for ν22 (gauche). However, a band for ν22 for the gauche conformer appears at 914 cm−1 in the matrix-IR spectrum.52 A possible feature for this band is near 915 cm−1 in the gas-phase RS spectrum (Figure 3). All in all, ν22 is assigned at 915 cm−1. Two fundamentals, namely, ν8 and ν21, for the gauche conformer are predicted to lie in the 970−1030 cm−1 region. The band for ν21 is predicted to be strong in the IR spectrum (I = 33 km/mol) at 993 cm−1, but it is, nonetheless, overshadowed by the two bands of the trans conformer in this region. Huber-Wälchli and Günthard50 observed clear evidence for this transition in a matrix at 996 cm−1 (s). For ν8 of the gauche rotamer, the predictions are 975 cm−1 (I = 3.3 km/mol, A = 8.0 RSU). A band at 978 cm−1 (vvw) in the gas-phase RS spectrum (Figure 3) is a good candidate for this fundamental. It has a counterpart in the matrix-IR spectrum. Two remaining fundamentals, ν7 and ν20, of the gauche conformer lie in the region between 1000 and 1100 cm−1. Predictions for these transitions are 1046 cm−1 (I = 0.25 km/ mol, A = 0.42 RSU) for ν7 and 1082 cm−1 (I = 3.0 km/mol, A = 1.9 RSU) for ν20. The very weak RS band at 1051 cm−1 (Figure

3) is assigned to ν7, and the certain matrix-IR feature of medium intensity at 1087 cm−1 is assigned to ν20.50 A band at 1043 cm−1 (w) in the IR matrix spectrum agrees with the assignment for ν7.50 The ν7 mode has a significant FR with ν9 + ν13, predicted at 1141 cm−1 with low intensity that implies this combination transition is too weak to be observed in either spectrum. 5.5.3. 1250−1350 cm−1. The range of 1250−1350 cm−1 contains the Raman-only band for the ν6(Ag) fundamental and the IR-only band for the ν22(Bu) fundamental of the trans conformer. This spectral region also contains bands for ν6(A) and ν19(B) of the gauche conformer, involving transitions active in both RS and IR spectra. For both conformers, the ν6 state is strongly perturbed by FR. The gas-phase RS spectrum at 250 and 25 °C for this region appears in Figure 4.


For the trans conformer, the calculation predicts a strong FR between ν6 and the combination state ν23 + ν24(Ag) with frequency shifts of ±8 cm−1. As a result of the FR, the computed RS intensity of ν6 (A = 15 RSU) is redistributed, and the “dark” state ν23 + ν24 gains ∼30% of the intensity. Predicted frequencies are 1291 (63% ν6, 36% ν23 + ν24) and 1268 (36% ν6, 63% ν23 + ν24). With the Handy method (see below), the contributions of the two modes are exchanged: 51% ν6, 48% ν23 + ν24 for the lower-frequency component and 51% ν23 + ν24, 48% ν6 for the higher-frequency component. As seen for the gas phase in Figure 4, there are two RS bands, 1298 and 1278 cm−1, that reasonably match the theoretical predictions. The spacing between the observed bands of 20 cm−1 is calculated to be 23 cm−1. The higher-frequency band (1298 cm−1) is the first member of a hot band series, arising from ν13 states, that extends upward to 1311 cm−1.98 The integral intensity for this band, which is the sum of the intensities for the hot band series, is approximately equal to the intensity of the lower-frequency band at 1278 cm−1, the peak height of which is higher. In the liquid-phase RS spectrum, the lower-frequency component of the FR doublet is decidedly more intense (Figure S3b). Because the intensity estimates in the liquid phase are more certain than in the gas phase, we conclude that the higher-intensity band at 1278 cm−1 in the liquid phase comes from the state with the larger component of ν6. It is unlikely that the order of the deperturbed levels is swapped when going from the gas to liquid state. Furthermore, the frequencies for the two components agree within 2 cm−1 for the liquid and gas phases. Guided by these considerations, we choose the lower frequency I


Article

The Journal of Physical Chemistry A more intense component at 1278 cm−1 as the principal feature for ν6. McKean et al. estimated the “deperturbed” value of ν6 as 1285 cm−1.42 The CVPT2+WK model predicts that the deperturbed values of the ν6 FR dyad as shifted by ±8 cm−1, so that the pre-FR experimental value of the lower more intense component can be estimated as 1278 + 8 = 1284 cm−1 in good agreement with the estimate in ref 42. The anharmonic frequency of 1298 cm−1 for ν6 computed by Feller and Craig is unreliable because FR was not taken into account.54 The difficulty with the assignment of this FR dyad shows that a highly accurate estimation of the zero-order frequencies can be very important for spectral interpretations. The region includes the band for the ν22(Bu) fundamental of the trans conformer, active only in the IR spectrum. There is a weak FR (W = 17 cm−1) between ν22 and ν12 + ν16. Figure 5

overshadowed by the band for the trans conformer at 1298 cm−1. The gauche conformer’s resonance-free band for ν19(B) has a significant, predicted RS intensity of 3.8 RSU, and its predicted frequency is 1276 cm−1. There is shoulder at 1274 cm−1 on the low-frequency side of the 1277.8 cm−1 trans conformer band that increases in strength at 250 °C (Figure 4). We therefore assign the feature at 1274 cm−1 to ν19(B). The IR intensity of ν19 is only 0.39 km/mol, and thus this band is unlikely to be observable in the IR spectrum in competition with the ν6 band from the trans conformer. 5.5.4. 1350−1550 cm−1. The range of 1350−1550 cm−1 (see Figures 4 and S1b) contains the Raman-active ν5(Ag) fundamental and the IR-active ν21(Bu) fundamental of the trans conformer, while the gauche conformer is represented by ν5(A) and ν18(B), active in both types of spectra. The Ag fundamental ν5 for the trans conformer with a predicted frequency of 1437 cm−1 and a strong RS intensity of 12 RSU is very slightly coupled through FR with a combination state ν8 + ν9 (Ag); the share of the latter is ∼5%. The observed main state ν5 is the strong band at 1442 cm−1 in Figure 4, which matches well with the predicted value and is polarized the liquid-phase RS spectrum. We discuss the assignment of the FR partner ν8 + ν9 below. As for the Bu fundamental ν21 of the trans conformer, the predicted frequency is 1378 cm−1 (Figure 5) with an intensity 4.9 km/mol. The corresponding IR peak is at 1381 cm−1 with weak-medium intensity and a type-A/B shape. For the gauche conformer the predicted frequencies and intensities for the resonance-free ν5(A) and ν18(B) fundamentals are 1424 cm−1 (8.6 km/mol and 4.6 RSU) and 1398 cm−1 (1.4 km/mol and 1.9 RSU), respectively. In the RS spectra in Figure 4 there are three bands that are close to the predicted values: 1432 cm−1 (w), 1428 cm−1 (w), and 1398 cm−1 (w). The band at 1398 cm−1 in the gas-phase RS spectrum does not seem to increase in strength at 250 °C and thus is assigned to ν8 + ν9 (Ag) of the trans conformer; the calculation predicts that this combination state gains intensity through a weak FR with ν5(Ag). Matrix IR spectra provide strong evidence that the transition near 1424 cm−1 and one near 1401 cm−1 arise from the gauche conformer. Consequently, we assign the matrix band at 1424 cm−1 to ν5(A) and the matrix band at 1401 cm−1 to ν18(B). For ν5(A) the gas-phase RS band counterpart is at 1428 cm−1; it shows some intensity gain at 250 °C. The RS band at 1432 cm−1 is a band for the trans rotamer, presumably the combination state ν11 + ν12 (Ag), although the predicted value is 1420.5 cm−1. 5.5.5. 1550−1800 cm−1. The next range, 1550−1800 cm−1, contains the Raman-active ν4(Ag) fundamental and the IRactive ν20(Bu) fundamental of the trans conformer, while the gauche conformer is represented by the ν4(A) and ν17(B) fundamentals, active in both types of spectra. In this range, the trans conformer appears in the gas-phase RS spectrum as a Fermi-resonance doublet with predicted frequencies and activities of 1642 (37 RSU; 80% ν4, 15% ν15 + ν16) and 1653.47 (6.0 RSU; 14% ν4, 85% ν15 + ν16). Their observed counterparts, as seen in Figure 4, are the very strong band at 1644 cm−1 and the medium intensity band at 1661 cm−1 for ν15 + ν16. This latter band has a series of hot band components, arising from ν13 states, spilling off to low frequency. All the peaks in the hot-band sequence contribute to the overall intensity for this transition. In this range of the IR spectrum, there is a strong peak at 1596 cm−1 (A-type; Figure S1b),

Figure 5. Fermi resonance doublet ν22, ν12 + ν16 and fundamental band ν21 of trans-1,3-BDE in the gas-phase IR spectrum (range of 1420−1220 cm−1). Resolution 0.1 cm−1, pressure 151 Torr.

shows this FR doublet for the trans conformer in the IR spectrum. Predicted frequencies, intensities (km/mol), and contributions of the zero-order states are 1287 cm−1 (1.8; 91% ν22, 8% ν12 + ν16) and 1265 cm−1 (0.56; 8% ν22, 91% ν12 + ν16). Observed peaks with weak intensities are at 1294 and 1268 cm−1 with supporting type-A/B contours (boldface indicates the stronger component), see also ref 99 and Table S2. These peaks are assigned to ν22 and ν12 + ν16 of the trans conformer, respectively. The higher component of the dyad is only slightly shifted (∼1 cm−1) after FR as per the CVPT2+WK prediction, so that the “deperturbed” experimental value of ν22 = 1281 cm−1 estimated by McKean et al.42 lies too far from our calculation. The intrinsic intensity of the FR partner state ν12 + ν16 is not nil, a feature that can partly explain the failure of the simpler model. This example demonstrates that use of isolated effective Hamiltonians for estimation of the FR perturbation effects on fundamental states requires more knowledge, such as W-constants and deperturbed intensities of all states, than is often used in the literature. In the calculations for the 1250−1350 cm−1 region, the gauche conformer state ν6(A) is perturbed by an interaction with the combination state of ν7 + ν12 with a frequency shift of ∼11 cm−1 for ν6. The zero-order RS intensity of ν6 (A = 6.3 RSU) is evenly redistributed between the two states, and the theoretical frequency values after diagonalization are 1298 cm−1 (58% ν6, 41% ν7 + ν12) and 1324 cm−1 (40% ν6, 59% ν7 + ν12). The lower-frequency component is 1284 cm−1, as seen in Figure 4; the higher-frequency component is presumably J


Article

The Journal of Physical Chemistry A corresponding to the ν20(Bu) fundamental of the trans conformer. Our theoretically predicted value for the resonance-free ν20 fundamental is 1594 cm−1 (I = 12 km/mol), in good agreement with the experiment. For the nonfundamental transitions of the trans conformer in this range, a weak band observed in the RS spectrum at 1583 cm−1 in Figure 4 matches with the predicted value of 1580 cm−1 (A = 0.087 RSU) for the combination band ν22 + ν24 of Ag symmetry. Another peak of medium intensity in the Raman spectrum is observed at 1680 cm−1 and is assigned to the ν21 + ν24 transition. This assignment corresponds well to the theoretical value of 1671 cm−1 (A = 1.7 RSU). Similarly, a theoretical value for ν7 + ν9 equals 1705 cm−1 (A = 0.087 RSU) and agrees with the observed peak of weak intensity at 1716 cm−1. In the IR spectrum (Figure S2b), an observed very weak peak at 1655 cm−1 (A/B-type) can be attributed to the combination band ν11 + ν16, the theoretical value of which is 1650 cm−1 (I = 0.28 km/mol). For bands for the gauche conformer in the 1550−1800 cm−1 range, the two fundamental transitions that fall in this range, ν4(A) and ν17(B), are perturbed by Fermi-resonances. As seen in Figure 4, ν4 in the gas-phase Raman spectrum is a weak peak at 1614 cm−1. This transition was also attributed to the gauche rotamer in the matrix investigations. According to our calculation, ν4 is affected by a weak Fermi resonance with ν9 + ν11; the frequency shift is ∼5 cm−1. The calculation predicts ν4 as a strong Raman band at 1614 cm−1 (22 RSU, 86% ν4 and 12% ν9 + ν11). The doublet partner, predicted at 1635 cm−1, is ∼10 times weaker (2.8 RSU) and may be a shoulder at 1633 cm−1. Good support for this band is found in the matrix spectra. According to the theoretical prediction, the transition, ν17, appears in the IR as a doublet at 1627 (1.6 km/mol; 61% ν17, 38% ν11 + ν22) and at 1640 cm−1 (1.4 km/mol; 33% ν17, 60% ν11 + ν22). In the RS spectra the intensities are too low to be observed. In the gas-phase IR spectrum at 151 Torr, weak Qbranch features appear at 1636.0, 1633.8, and 1632.7 cm−1. Thus, it seems that there is possible IR evidence for ν17 near 1634 cm−1, which needs to be investigated with high-resolution methods. Panchenko and De Maré37 assembled previous data and quoted 1632 and 1633 cm−1 for ν17 from several sources, including matrix data of Furukawa.51 Other matrix spectra show evidence for a band of the gauche rotamer in this region. However, that band seems to be part of the FR doublet for ν4, as indicated above. We tentatively conclude that ν17 for the gauche rotamer is at 1634 cm−1, almost coincident with the band for ν9 + ν11. The RS band at 1594 cm−1 may be ν10 + ν11. There is a vvw peak at 1740 cm−1 visible on Figure 4, which corresponds to the overtone 2ν10 of the gauche conformer that has a predicted value of 1740 cm−1 (0.79 RSU). This peak seems to grow in intensity in the spectrum between 25 and 250 °C, as is appropriate for the gauche conformer. 5.5.6. 1800−2800 cm−1. The wide range of 1800−2800 cm−1 does not contain any fundamental transitions of either conformer. The gas-phase RS spectra (25 and 250 °C) are presented in Figure 6. There are many overtone and combination transitions in this region. On the basis of our predicted spectrum, we make the following Ag assignments (in cm−1) for the trans conformer, where in square brackets are the predicted frequencies and RS intensities in RSU: 1776 (w) 2ν8 [1764 (0.9)], 1791 (vw) ν6 + ν9 [1790 (0.044)], 1822 (vvw) 2ν15 [1810 (0.23)], 1896 (vvw) ν20 + ν24 [1886 (0.047)], 2088 (w) ν7 + ν8 [2076 (0.38)], 2114 (vvw) ν7 + ν15 [2113.9


(0.009)], 2152 (vvw) ν4 + ν9 [2153 (0.015)], 2328 (vw) ν5 + ν8 [2315 (0.34)], 2403 (vw) 2ν7 [2392 (0.18)], 2478 (vvw) ν6 + ν7 [2478 (0.052)], 2524 (vw) ν4 + ν8 [2526 (0.25)], 2642 (vvw) ν5 + ν7 [2632 (0.068)], and 2752 (vvw) 2ν21 [2753 (0.14)]. There are two peaks that can be attributed to A transitions of the gauche conformer because they grow in intensity at 250 °C: 2582 (vvw) ν4 + ν8 [2591 (0.11)] and 2663 (vvw) ν4 + ν7 [2654 (0.31)]. There was just one previous attempt81 to interpret IR spectra in this region. The summary of our assignments according to our new calculation is given in Table S2. 5.5.7. 2800−3200 cm−1. The region of 2800−3200 cm−1 includes mainly C−H stretching fundamental states as well as some “dark” states that gain transition intensity from the “bright” fundamental states through multiple resonances. For the trans conformer, this region contains six fundamentals [half in the Raman spectrum (Ag), ν1, ν2, ν3, and half in the IR spectrum, ν17, ν18, ν19 (Bu)]. Different from the lower-energy regions, modeling of anharmonic states in C−H region generally cannot be properly accomplished with a pairwise resonance interaction model because of strong multiple resonance couplings. See Table S2 in the Supporting Information for details of the assignments. As an illustration, we present (see Figure 7) a computer-simulated IR spectrum of the two conformers of BDE in this region with annotations.

Figure 7. Simulated superposition of IR spectra of trans- and gauche1,3-BDE in the range of 3150−2850 cm−1, 25% intensity for gauche conformer in red. K


Article


the assignment of the gas phase peak 2877 cm−1 (2870 cm−1 in liquid-phase) to 2ν542 is corroborated by our calculation. At least five distinct band origins can be observed in the gasphase IR spectrum of BDE in the C−H stretching region (Figures 7 and 9) for the trans conformer. The high-resolution spectral studies by Craig and Sams41 yielded the following bands of A/B shape: 2984.0, 3011.4, 3031.4, 3055.2, and 3100.63 cm−1. The frequency of the last is from Halonen et al.40 Figure 9 is at 0.1 cm−1 resolution for a better display of the overall band structure. Figure S4 shows the overall structure of this region at 0.003 cm−1 resolution. The highest wavenumber band ν17 was found to be in weak FR with an unknown state νx (see Discussion below).40 The state ν18 is approximately equally mixed by FR with the state ν4 + ν21; they both are shifted by ∼12 cm−1. The resulting transitions are observed at 3011.4 cm−1 (m) and 3031.4 cm−1 (m), respectively, in good agreement with the predictions (see Table S2). This resonance was missed in previous studies. The state ν19 is strongly perturbed (shift ≈ 30 cm−1) by FR with the combination state of ν5 + ν20. The result is two observed peaks 2984.0 cm−1 (m) and 3055.2 cm−1 (m), the lower of which can be assigned mostly to ν19. McKean et al.42 attempted to estimate the unperturbed origin of ν18 from the intensity redistribution. They incorrectly assumed that FR involving ν18 produces only the dyad 2984 and 3055 and obtained the “deperturbed” value of ν18 = 3026 cm−1, which was later quoted as an “experimental” value of ν18 by Panchenko and De Maré.37 This example shows that using deperturbed values of experimental frequencies should be avoided unless the parameters of all the contributors to FR are known with sufficient confidence. In a high-resolution IR slit-jet study of the trans conformer of BDE, Halonen et al.40 determined a precise value for ν17(Bu) of 3100.6328 cm−1, and thus they discovered an additional nearby “dark” state νx at 3096.14 cm−1 that was shown to belong to the trans conformer by a rotational analysis. However, the identity of this state remained unclear. The measured 1:4 intensity ratio and a 4.5 cm−1 origin difference yielded the FR constant value of W = 1.8 cm−1 and deperturbed band centers separated by 2.7 cm−1.40 A CVPT2+WK calculation with an enlarged basis set (Nmax ≤ 4) predicts a picture very similar to the one obtained experimentally, with the exception that the predicted order of the levels is reversed (ν17 < νx). The DDR (type 1−111) coupling of deperturbed states ν17 and ν6 + ν11 + ν15 has a DDR constant K = 7.1 cm−1 (matrix element: K/4 = 1.78 cm−1) and a separation of levels of 3.4 cm−1 that results in final redistributed IR intensities of 9.6 and 2.1 km/mol, the ratio of which is 4.6. The striking similarity between the observed and predicted parameters of the resonance implies that the unknown “dark” state may indeed belong to ν6 + ν11 + ν15, but this conclusion must be verified by higher-level calculations. The calculation with adjusted harmonic frequencies shows that another possibility is a ternary state ν5 + ν11 + ν16. The predicted value of 3095.0 cm−1 is close to the observed one. For the lower symmetry gauche conformer, all six CH stretching modes are, in principle, active in both the RS spectrum and the IR spectrum. For assignments for the gauche conformer in this complex spectral region, we focus on the gasphase RS spectrum, where bands from the gauche species grow in intensity in going from 25 to 250 °C. Counterparts of some of these gas-phase features were observed in matrix-isolation Raman spectra and IR spectra.48−53 The gauche conformer is represented in the A symmetry block by three C−H

Figure 8 is the gas-phase Raman spectra at 25 and 250 °C. Figure 9 is the gas-phase IR spectrum in this region.


Figure 9. IR spectrum of 1,3-BDE in the CH-stretching region. Resolution 0.1 cm−1; pressure 32 Torr.

In the Raman-active Ag symmetry block, the highest-energy state for trans conformer ν1(Ag) is only slightly perturbed by resonances (weak FR with the state of ν20 + ν22), as is reflected in its separate polyad coefficient P = 32, and is observed as a strong peak at 3099 cm−1. The states ν2 and ν3, belonging to the polyad with P = 30, are slightly coupled through DDR (1− 1) with the constant K = −4.6 cm−1. In addition, they are in FR with a few more two-quanta states, the most important being ν3/ν4 + ν5 (W = −86.5 cm−1) and ν3/ν20 + ν21 (W = −73.8 cm−1). The vvs RS band at 3012 cm−1 is largely ν2. The predicted FR between ν3 and ν20 + ν21 causes a significant repulsion of the order of 10 cm−1 between the “deperturbed” states and an intensity redistribution. Therefore, the band at 2961.5 cm−1 (ms) marked “unassigned” with an asterisk on Figure 7 of Boopalachandran et al.43 can now be confidently assigned to ν20 + ν21. This band appears (w) in our Figure 8 and Table S2. The vs RS band, which is a shoulder at 3010 cm−1, is the principal feature for ν3. The calculation confirms some additional trans conformer assignments made in previous studies. McKean et al.42 assigned the liquid-phase Raman band at 3180 cm−1 [gas-phase value 3187.7 cm−1 (s)] to 2ν20. They assigned the band 2953 cm−1 (2961.5 cm−1 in gas phase) to ν20 + ν21 as specified above. Also, L


Article

The Journal of Physical Chemistry A Table 1. Fundamentals of trans-1,3-Butadienea mode Ag 1 2 3 4 5 6 7 8 9 Au 10 11 12 13 Bg 14 15 16 Bu 17 18 19 20 21 22 23 24

harm, MP2b

harm, CCc

harm, fitted

anharm, Nmax = 2

anharm, Nmax = 3

intensity, IR/RSd

observed, this work or hi-res IR

calcd, ref 54

3282.6 3189.5 3178.3 1706.7 1482.9 1314.1 1231.0 907.6 511.7

3237.9 3154.7 3142.0 1695.7 1476.9 1307.2 1221.9 896.7 506.8

3239.5 3154.7e 3139.8 1696.0 1481.2 1313.1 1228.0 901.8 513.2

3096.4 3017.5 3011.4 1642.4 1437.3 1267.6 1198.1 883.3 505.7

3097.3 3017.0 3009.8 1642.5 1437.3 1267.6 1198.1 883.3 505.7

25 60 6.0 37 12 4.3 14 0.47 8.4

3099. s 3012.5 vvs 3010. vs 1644.3 vvs 1442.2 s 1277.8 sf 1204.2 s 888.8 w 512.2 vs

3082.2 2997.3 2977.5 1657.8 1439.6 1298.4 1205.7 889.9 517.3

1057.8 931.9 541.6 168.0

1037.5 918.4 532.0 163.7

1039.9 925.3 536.7 167.5

1010.9 900.9 519.6 158.3

1011.1 900.9 519.6 158.3

35 74 12 0.44

1013.8 908.07 524.57 162.42

1026.4 914.4 520.8 163.5

996.4 932.4 778.9

981.3 919.5 761.8

985.3 925.6 760.5

962.0 903.2 749.3

962.0 903.2 749.3

0.79 3.2 3.1

3282.9 3194.0 3180.7 1643.6 1420.0 1312.8 997.9 290.1

3238.2 3159.3 3144.5 1635.8 1412.5 1308.1 994.0 288.2

3242.1 3146.6 3155.0 1638.2 1414.8 1317.9 1000.4 295.1

3096.3 3015.7 2978.2 1593.7 1377.9 1286.9 982.2 292.3

3096.7 3012.5 2982.5 1594.0 1377.9 1286.6 982.2 292.3

12 3.0 5.4 11 4.9 1.8 1.4 2.4

s vsg mg vwg

972. vvw 909.5 vw 748. vvw 3100.63 sh 3011.4 m 2984. vw 1596.45 sg 1380.6 wm 1296.2 w 990.3 m 299.1 w

973.1 917.5 758.4 3082.4 3019.3 3003.6 1611.9 1386.5 1299.4 995.4 301.2

a Calculated using CVPT2+WK and a hybrid PES (MP2/cc-pVTZ//CCSD(T)/cc-pVTZ) and the following settings (see text): Ω† = 0.05, Δ† = 300 cm−1, Nmax = 2, 3). Intensity is given in units of km/mol for IR spectra or 1 × 10−48 cm6/sr (RSU) for RS spectra. bHarmonic frequencies calculated using QM model MP2/cc-pVTZ. cHarmonic frequencies calculated using QM model CCSD(T)/cc-pVTZ. dIntensities are presented for the setting Nmax = 2. eExcluded from fitting to avoid oscillations. fThe FR dyad partner band is 1298 cm−1 (m). gHigh-resolution studies, refs 39 and 41. hHighresolution study, ref 40.

In the B-symmetry block three C−H fundamentals, ν14, ν15, ν16 produce a similar picture to those of A symmetry: ν14 is essentially FR free, while ν15 and ν16 participate in a polyad with six two-quanta states, the most important interactions are with ν4 + ν18 and ν5 + ν17. The B-symmetry fundamentals ν14 and ν16 have low predictions for RS intensity and are questionable observations in the gas-phase Raman spectrum due to overlaps with stronger A-symmetry fundamentals. The calculation predicts the frequency of ν14 at 3097 cm−1 being practically equal to ν1 (Figure 7), while the intensity of ν14 is of ∼20× smaller. An intensity of 9.6 km/mol is predicted for ν14 in the IR spectrum, and distinct features are observed in matrix IR spectra at 3103 cm−1, as reported in Table S2. For the ν15 transition the frequency is predicted to be 3018 cm−1 and the IR intensity to be 7.1 km/mol. Bands are found at 3014 and 3010 cm−1 in matrix IR spectra (Table S2). Despite an intensity prediction of 9.6 RSU, an RS band is overlapped by an intense band of the trans conformer. Thus, the observation in the matrix-IR becomes the basis for the assignment of ν15. The predicted intensities for ν16 seem too weak to be observable in the IR or the RS spectra. The predicted bands for ν16 at 2978 cm−1 would be overlapped by the stronger bands for ν3 2988 cm−1. The band at 2978 cm−1 is predicted to be 56% ν16 and 33% ν4 + ν18. In sum, we take a band for ν16 to be overlapped in the gas-phase RS by the stronger band for ν3 and thus not directly observable.

fundamentals, ν1, ν2, and ν3. The first one (ν1) is FR free and has a predicted frequency of 3096.8 cm−1 and a Raman intensity of 23 RSU. It is seen in the gas phase as an uncertain frequency of ∼3088 cm−1 in a broad shoulder in Figure 8. The fundamentals ν2 and ν3 are coupled through DDR (1− 1) and FR (1−2), including states ν4 + ν5, ν4 + ν6, ν17 + ν18 and ν17 + ν19. Intrinsic RS activities of ν2 and ν3 are equal to 236 and 82 Å4 × amu−1, and their redistribution into close-lying dark states appears to be sensitive to the number of basis functions included for consideration. Deperturbed values of ν2 and ν3 (before diagonalization) equal to 3019.5 and 3007.2 cm−1. After the diagonalization, the value of ν2 becomes 3028.6 cm−1, while ν3 is shifted to 2981.4 cm−1 due to a complex interaction with dark states. The resulting positions of energy levels are very sensitive to underlying values of the harmonic frequencies. The calculations point to the peak 3023 cm−1 (w) being largely attributable to ν2 and to 2988 cm−1 being significantly ν3 (vw). The weak gauche band at 2988 cm−1 is predicted to have 24% ν17 + ν18 character. Another band at 2924 cm−1 (w) that is not observed at 25 °C but is observed at 250 °C can be attributed mostly to ν4 + ν6. A gas-phase gauche band at 2851 cm−1 is attributable to 2ν5; the calculation predicts 2849.5 cm−1. The problems with the assignment of the CH-stretch polyad for the gauche conformer demonstrate the importance of very accurate theoretical predictions of the underlying harmonic frequencies. M


Article

The Journal of Physical Chemistry A Table 2. Fundamentals of gauche-1,3-Butadienea mode A 1 2 3 4 5 6 7 8 9 10 11 12 13 B 14 15 16 17 18 19 20 21 22 23 24

harm, MP2b

harm, CCc

anharm, Nmax = 2

anharm, Nmax = 3

IR inten, km/mold

RS inten RSUd

3284.0 3200.6 3182.2 1674.9 1474.2 1337.9 1067.1 1014.5 940.9 899.2 757.7 273.0 182.2

3239.2 3165.0 3145.8 1666.8 1468.1 1332.8 1058.0 998.2 927.3 886.5 742.2 270.7 161.8

3096.8 3028.6 2981.4 1613.8 1423.6 1298.3 1046.1 975.1 910.2 872.8 726.8 284.3 133.7

3096.7 3029.0 2986.2 1613.9 1423.6 1303.7 1046.6 975.1 910.2 872.8 726.8 283.8 133.7

3.7 1.9 8.4 2.4 8.6 0.005 0.25 3.3 2.3 0.39 3.7 0.001 0.11

23 52 5.8 22 4.6 3.5 0.42 8.0 0.83 1.2 3.6 16 33

3282.5 3189.2 3179.6 1680.6 1440.7 1306.7 1101.5 1036.4 941.8 620.0 470.0

3237.6 3152.3 3143.3 1674.9 1436.9 1300.8 1098.9 1017.4 929.3 609.0 466.8

3096.7 3018.5 2978.5 1627.3 1398.4 1275.8 1082.2 992.6 909.8 599.8 456.7

3097.0 3019.7 2977.8 1626.9 1398.4 1275.7 1082.2 992.6 909.8 599.8 456.7

9.6 7.1 0.74 1.6 1.5 0.39 3.0 32 65 8.6 13

1.2 9.6 0.95 0.035 1.9 3.8 1.9 0.2 0.44 0.018 2.0

observed, gas-phase RS intens

calcd, ref 54

3088 vvw 3023 w 2988 vw 1614 w 1428 w 1283.6 m 1051 vvw 978 vw 922 w 869 vw 729 vw 271 w (106)e

3092.9 3019.4 3023.3 1605.2 1408.5 1285.1 1034.5 950.6 890.5 863.5 709.1 263.6 142.8

3088 vvw 3014f 2988 vw 1634 vvwf,g 1401f 1274 sh, m 1087f 993 vw 915 vw 601.7 vwh 463.8i

3091.7 3004.4 3026.3 1613.0 1381.8 1261.3 1065.7 971.7 891.8 596.4 446.9

Calculated using CVPT2+WK and a hybrid PES (MP2/cc-pVTZ//CCSD(T)/cc-pVTZ) and the following settings (see text): Ω† = 0.05, Δ† = 300 cm−1, Nmax = 2, 3). Intensity is given in units of km/mol for IR spectra or 1 × 10−48 cm6/sr (RSU) for RS spectra. bHarmonic frequencies calculated using QM model MP2/cc-pVTZ. cHarmonic frequencies calculated using QM model CCSD(T)/cc-pVTZ. dIntensities are presented for the setting of Nmax = 2. eν13 is not observed directly. It was calculated to be 106 cm−1 from the potential function for internal rotation; ref 43. fIR observation in argon matrix. gAn uncertain feature in the gas-phase IR spectrum. hIR observation in gas phase, this work. iPossible IR observation; ref 41. a

5.5.8. Region 3200−4000 cm−1. In this region in the gasphase RS spectra at 25 and 250 °C, which are not shown, only two peaks were observed: 3277 (vw) and 3282 (vw) cm−1 for the trans conformer. They can be explained as a resonance dyad of the overtone 2ν4 and a combination state of ν19 + ν24. A few features observed in this part of the IR spectrum (P = 151 Torr) were interpreted as binary combinations (see Table S2): ν4 + ν20 (Bu) = 3235.7 cm−1 (vw, A), ν1 + ν24 (Bu) = 3398.9 cm−1 (vw, B), ν1 + ν12 (Au) = 3623.0 cm−1 (vw, C?), ν3 + ν11 (Au) = 3917.7 cm−1 (vvw, B). Tables 1 and 2 provide summaries of the predictions and assignments of fundamentals for the trans and gauche conformers, respectively. Predictions made with two- and three-wave function basis sets are shown in data columns 4 and 5. For the most part, the differences in the two sets of predictions are small; the exceptions are only as large as 5 cm−1. For comparison the last data columns report the results of the best previous calculations. They are the anharmonic calculations of Feller and Craig.54 For the trans conformer, they used the CCSD(T)/aug-cc-pVTZ model for harmonic frequencies and the MP2/aug-cc-pVDZ for the anharmonic corrections. For the less symmetric gauche conformer, they used the CCSD(T)/aug-cc-pVDZ model for harmonic frequencies and the MP2/aug-cc-pVDZ for the anharmonic corrections. Core/ valence corrections were made. FR was not taken into account. The assignments for fundamentals of the trans conformer of butadiene in Table 1 rest on sound theoretical and experimental bases. All the experimental wavenumber values

are from the gas phase. Only the higher-frequency component of the strong FR pair is given for ν6. In other instances of FR, especially in the CH stretching region, the stronger component is also given. Comparison of the anharmonic predictions from Feller and Craig54 shows that a number of these are somewhat closer to the experimental values than to those in the present work, a consequence of a higher-level model. Table 2 gives the assignments of the fundamentals for the gauche conformer of BDE. Because of low intensities from bands of the gauche conformer, these assignments are less secure than for the trans conformer. Matrix-isolation spectra are important in making some of the assignments for the gauche confomrer. For all but ν13(A), ν16(B), and possibly ν17(B) credible observations support the assignments. No feature for ν13 was observed. A value of 106 cm−1 for the 0+ − 1+ transition of ν13 was computed from the experimental potential function for internal rotation.55 This value is quite different from the predicted value of 134 cm−1, which does not, however, take tunneling into account. A possible band for ν16 is presumed to be overlapped by the stronger band for ν3. The assignment for ν17 comes from an uncertain observation in the gas-phase IR spectrum. As for the trans conformer, the strongest components of FR multiplets are listed in Table 2. Outside the CH stretching region, the biggest effect of FR is on ν6. With a few exceptions the anharmonic frequencies predicted by Feller and Craig agree less well for the gauche conformer than for the trans conformer.54 For the gauche conformer, the N


Article

The Journal of Physical Chemistry A assignments were put on a firmer experimental footing and were extended to more fundamentals. 5.5. Data Consistency Test: Fitting Calculated Fundamental Frequencies to Observed Ones. As we have seen from the discussion above, the CVPT2+WK model of anharmonic vibrations with the hybrid MP2/cc-pVTZ// CCSD(T)/cc-pVTZ model ensures a good reproduction of fundamental frequencies with a systematic underestimation of non-CH stretch frequencies by ca. −5 cm−1. This result arises from a complex superposition of effects, such as the deviation of the predicted equilibrium geometry from the actual one, deficiencies in the QM models at the MP2 and CCSD(T) levels, incompleteness of basis set for the electronic problem, and the cutoff the perturbative vibrational solution at the second order. Such effects may compensate each other, but it is evident that the outcome for predicted frequency shifts is systematic. To perform an additional test of the quality of the calculated anharmonic force constants and to assess the adequacy of the CVPT2+WK methodology and its implementation, we attempted to fit the predicted f undamental frequencies to observed ones by varying the harmonic frequencies only. Since the number of observed vibrational transitions (including resonance splittings, overtones, and combinations bands) is much larger than the number of normal modes, such a test verifies assignments of fundamental and nonfundamental transitions. Adjusted harmonic frequencies can be subsequently compared with predictions from different high-level QM models to find the “best hybrid” model. The technique of varying harmonic frequencies within the anharmonic VPT2 model was first proposed by Handy et al.35 It is an iterative procedure, which corrects the values of harmonic frequencies using the differences between calculated and target fundamental frequencies at each step. This procedure may lead to oscillations and lack of convergence, especially if certain states are involved in strong resonances. In such cases, applied changes can be damped by a multiplier smaller than unity, or, certain normal modes can be excluded from consideration. Results of our fitting for the trans conformer are presented in Table 1 as “harm, fitted”. To reach the convergence (root-mean-square deviation < 0.3 cm−1), it was necessary to exclude ω2 from the fit. Once a convergence was achieved, we compared values of nonfundamental transitions with experimental data and assignments made earlier. The analysis of the results (see Table S2) shows that for the majority of modes a very good level of agreement between predicted and observed overtones and combination bands is obtained. Although CH-stretching modes form only highfrequency combination bands, we verified good fit of some combination bands with modes ν1, ν3, and ν19. For non-CH stretches, a typical match between predicted combination bands or overtone is of the order of

Anharmonic Vibrational Analysis of the Infrared ... - ACS Publications

Recommend Documents