Article pubs.acs.org/JPCA

Twisted Triplet Ethylene: Anharmonic Frequencies and Spectroscopic Parameters for C2H4, C2D4, and 13C2H4 Xiao Wang, Walter E. Turner, II, Jay Agarwal, and Henry F. Schaefer, III* Center for Computational Quantum Chemistry, University of Georgia, Athens, Georgia 30602, United States ABSTRACT: Ethylene is an exceptional example of a stable closed-shell singlet molecule with a low-lying triplet state of very different symmetry. Triplet C2H4 (the ã 3A1 state), which has a twisted D2d geometry, is studied herein with high-level theoretical methods, namely, CCSD(T) (coupled cluster theory with single, double, and perturbative triple excitations) and Dunning’s correlation-consistent quadruple-ζ basis set (cc-pVQZ). Geometric parameters, including equilibrium (re) and vibrationally corrected (rg) values, are reported for C2H4, C2D4, and 13C2H4. Harmonic and anharmonic vibrational frequencies are also predicted using second-order vibrational perturbation theory (VPT2). Challenges encountered for the wagging vibrational features are discussed.

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INTRODUCTION Small hydrocarbons are ideal species for research in that they have a limited number of degrees of freedom and few lowenergy conformers, while also exhibiting the complexity of larger molecules, including hyperconjugation and multiple bonding.1 As a prominent member of this set, ethylene gains distinction as the simplest π-bonding molecule and as a prototype system for understanding larger unsaturated systems.2,3 The stability of ethylene makes it useful for monitoring reactions, especially in kinetics studies where it is used to elucidate the rate constants of radical processes. Fundamental properties of ethylene, such as its rotational barrier, have been studied by experimentalists and theoreticians for decades;4−11 Crawford and co-workers reported the vibrational spectra of ethylene’s 1Ag ground state in the 1940s,12,13 and J. L. Duncan and co-workers carefully studied its vibrational modes,14−23 reporting the general harmonic force field of ethylene in 1973.24 Studies regarding the lowest-energy triplet state of ethylene are scarce. This is due, in part, to its transient nature and the difficulty associated with synthesis in appreciable concentrations. Study of the ã 3A1 state is further inhibited by its low ultraviolet cross section, necessitating the use of spectroscopic techniques involving greater sensitivity, such as hemispherical electron spectrometry, high-resolution electron monochromatry, or cavity ring down spectroscopy.25−36 Despite the aforementioned difficulties associated with the ã 3A1 state, it has garnered interest partially due to its twisted geometry. As the methylene groups twist, the movement toward a perpendicular structure decreases the π-orbital overlap, transforming the planar D2h structure to the 90° D2d structure. For ethylene, the planar excited-state geometry (3B1u) is obtained from vertical excitation; the equilibrium triplet geometry (3A1) is realized after the structure twists to a lower-energy D2d © 2014 American Chemical Society

configuration. Further, because studies suggest synthesizing triplet ground states via schemes where significant steric hindrance will result in near-90° structures, the 3A1 state also has usefulness as a benchmarking compound for these endeavors.37 Notwithstanding the interest in the ã 3A1 state, studies of its fundamental properties are not abundant. A. G. Suits and coworkers reported an important measurement on the ethylene adiabatic (X̃ 1Ag → ã 3A1) energy difference using tunable synchrotron radiation to probe the dissociation dynamics of ethylene sulfide.6 The groups of Dixon38 and earlier Peyerimhoff4 computed both adiabatic and vertical singlet−triplet excitation energies for ethylene. As the 3A1 state is likely a relevant species in excited electronic state pathways, knowledge of its vibrational frequencies is paramount. A theoretical study by Kim and co-workers39 predicted the harmonic vibrational frequencies of the ã 3A1 state, but there is still no report of its anharmonic vibrational frequencies. Fortunately, theory is directly applicable to this system because it is both rigid and small.39−43 In this work, we present the anharmonic vibrational frequencies for the ã 3A1 state (1a12 1b22 2a12 2b22 1ex2 1ey2 3a12 2ex1 2ey1) of ethylene, C2H4, and two of its isotopologues, 13 C2H4 and C2D4, computed at the CCSD(T) level of theory with the cc-pVQZ basis set. No experimental or theoretical reports of the anharmonic vibrational frequencies for C2H4 and its isotopologues have been published to date. Our research thus gives the first complete set of anharmonic frequencies for Special Issue: Kenneth D. Jordan Festschrift Received: March 5, 2014 Revised: March 28, 2014 Published: April 2, 2014 7560

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Table 1. Harmonic Vibrational Frequencies (in cm−1) for Triplet Ethylene Isotopologues with IR Intensities (in km/mol, listed parenthetically) Predicted by the Frozen-Core Method and the All-Electron Correlated Method 13

C2H4 mode ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9 ν10 ν11 ν12

frozen 3115.4 1465.7 1135.1 693.7 3119.8 1439.7 3207.6 3207.6 939.7 939.7 407.6 407.6

a

(0.0) (0.0) (0.0) (0.0) (18.5) (0.1) (5.3) (5.3) (1.2) (1.2) (53.8) (53.8)

C2H4

all-electron 3121.2 1468.3 1139.4 696.5 3125.7 1442.5 3213.4 3213.4 942.3 942.3 414.1 414.1

b

(0.0) (0.0) (0.0) (0.0) (17.7) (0.2) (5.0) (5.0) (1.2) (1.2) (54.4) (54.4)

frozen 3110.1 1456.1 1099.5 693.7 3114.9 1434.0 3194.5 3194.5 930.5 930.5 404.6 404.6

a

C2D4 all-electronb

(0.0) (0.0) (0.0) (0.0) (18.4) (0.1) (5.2) (5.2) (1.0) (1.0) (53.1) (53.1)

3115.8 1458.5 1103.8 696.5 3120.8 1436.8 3200.3 3200.3 933.1 933.1 411.1 411.1

(0.0) (0.0) (0.0) (0.0) (17.7) (0.2) (4.9) (4.9) (1.0) (1.0) (53.7) (53.7)

frozena 2261.9 1220.8 939.3 490.7 2252.1 1067.2 2387.4 2387.4 739.4 739.4 311.7 311.7

(0.0) (0.0) (0.0) (0.0) (9.5) (0.1) (3.5) (3.5) (2.6) (2.6) (31.1) (31.1)

all-electronb 2266.1 1224.7 941.4 492.7 2256.4 1069.2 2391.7 2391.7 741.8 741.8 316.5 316.5

(0.0) (0.0) (0.0) (0.0) (9.1) (0.1) (3.3) (3.3) (2.6) (2.6) (31.5) (31.5)

a Computed at the CCSD(T)/cc-pVQZ level of theory with frozen core. bComputed at the CCSD(T)/cc-pCVQZ level of theory with core correlation.

at the CCSD(T)/cc-pVQZ level of theory were utilized. During VPT2 analysis, corrections were made for the C2D4 species in order to treat Fermi−Dennison resonances.7 Note, discussion of the problematic anharmonic corrections for the wagging modes are analyzed in the final section of this paper.

twisted triplet ethylene. Additionally, we also offer parameters pertinent to other spectroscopic measurements, including the optimized geometries (re and rg) and rotational constants (Be and B0) for the three isotopologues.

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METHODS The geometric parameters of the twisted structure of triplet ethylene were fully optimized here using coupled cluster theory with single, double, and perturbative triple excitations [CCSD(T)].44−49 We used Dunning’s correlation-consistent cc-pVQZ basis set,50 which contains 230 contracted spherical-harmonic Gaussian functions. All computations were carried out using the CFOUR51 program package. An unrestricted Hartree−Fock (UHF) reference was adopted because of the open-shell character of the triplet state. A small amount of spin contamination (⟨Ŝ2⟩ ≤ 2.015) was encountered, but we note that it has been previously shown that spin contamination in the reference wave function is often diminished by subsequent coupled-cluster computations.52,53 Also, no instabilities were detected in the UHF wave function. Analytic second derivatives54,55 were utilized in frequency computations. Because the cc-pVQZ basis set does not include a description of core electron correlation, the lowest-energy 1s-like molecular orbitals of carbon are frozen in the correlation computations. In order to gauge the error introduced by using this “frozen core” approximation, we computed the harmonic vibrational frequencies of the three isotopologues using the CCSD(T)/ cc-pCVQZ method (288 contracted spherical-harmonic Gaussian functions) and compared it to the CCSD(T)/ccpVQZ computations. The results are collated in Table 1. We find that the addition of core correlation effects resulted in an at most +0.5% difference in the vibrational frequencies of triplet ethylene. Additionally, our choice to use a basis set without core correlation is supported by research that has shown that the addition of core electron correlation without the inclusion of quadruple excitations results in a misleading blue shift of the harmonic vibrational frequencies.56−58 Second-order vibrational perturbation theory (VPT2)59 was used in the frequency computations in order to take into account the anharmonic nature of the potential energy surface. Numerical differentiation of second derivatives at 15 nuclear displacements was used to generate the requisite cubic and semidiagonal quartic force fields. Analytic derivatives computed

RESULTS AND DISCUSSION Structures and Energetics. Figure 1 shows a depiction of the equilibrium structure of the lowest triplet state of C2H4 (ã

Figure 1. The equilibrium geometry (in Å and degrees) of 90°-twisted triplet ethylene, possessing D2d symmetry, computed at the CCSD(T)/cc-pVQZ level of theory. 3

A1, D2d symmetry) computed at the CCSD(T)/cc-pVQZ level of theory. The optimized equilibrium geometric parameters (re) are shown in Table 2 along with the rg values, discussed below, that include vibrational corrections.60−62 We predict C2H4 to have an equilibrium C−C bond length of 1.453 Å. Compared to the C−C single bond length in ethane (1.533 Å)63 and the C−C double bond length in singlet ethylene (1.334 Å), the computed C−C bond length in triplet ethylene is approximately halfway between them, though closer to the former, which suggests some vestige of a π orbital. Wu and Schleyer64 proposed that hyperconjugation exists between the σ orbitals of the C−H bond and the p orbitals on the opposite carbon atom, which could probably compensate for the substantial, but not complete, loss of π conjugation. The rough equivalence of the equilibrium C−H bond in triplet ethylene (1.084 Å) and singlet ethylene (1.082 Å) reveals that the drastic change in C−C π bonding has a small influence on the C−H σ bond. Similarly, the ∠H−C−C bond angle (121.5°) shows minor distortion from the geometry of singlet 7561

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Table 2. Structural Parameters (in Å and degrees) for Selected Isotopologues of Triplet Ethylene Molecules zero-point-correcteda 13

parameter

re

C2H4

r(C−C) r(C−H) ∠H−C−C

1.4528 1.0838 121.49

1.4620 1.1045 120.18

C2H4

1.4617 1.1044 120.19

C2D4

ref 5b

1.4614 1.0990 120.45

1.4556 1.0820 121.48

a

Computed at the CCSD(T)/cc-pVQZ level of theory. bVariational Monte Carlo with a smooth relativistic norm-conserving pseudopotential; C2H4 isotopologue.

ethylene (121.4°). We note, for reference, that the parameters of our equilibrium structure agree well with the quantum Monte Carlo work of Barborini and co-workers.5 The equilibrium geometries (re) for the 13C- and deuteriumsubstituted isotopologues are equivalent to those of the parent species. Corrections for thermal vibrations, which are dependent on the atomic masses, yield an rg structure that differs among the three species (see Kuchitsu, ref 60, for definitions). Using the normal coordinates, rg may be expressed as61,62 rg = re +

∑ γs⟨Q s⟩ + s

≈ re +

∑ γs⟨Q s⟩ + s

1 2

∑ γst⟨Q sQ t ⟩ + ...

1 2

∑ γss⟨Q s2⟩

Table 3. Descriptions and Symmetries for the Vibrational Modes of Triplet C2H4

st

s

(1)

where γs and γst are the first and second derivatives of the internuclear distances with respect to the corresponding normal coordinates. At 0 K, the linear average ⟨Qr⟩ and quadratic average ⟨Q2s ⟩ on the right-hand side of 1 are computed using the formulas ⎛ ℏ ⎞ ⎟ ⟨Q r ⟩ = −⎜ 3 ⎝ 2πcωr ⎠ ℏ 4πcωs

description

symmetry

ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9 ν10 ν11 ν12

CH2 symmetric stretching CH2 in-plane scissoring CC stretching CH2 out-of-plane twisting CH2 symmetric stretching CH2 in-plane scissoring CH2 antisymmetric stretching CH2 antisymmetric stretching CH2 in-plane rocking CH2 in-plane rocking CH2 out-of-plane wagging CH2 out-of-plane wagging

a1 a1 a1 b1 b2 b2 e e e e e e

Table 4. Harmonic and Anharmonic Vibrational Frequencies (in cm−1) for Triplet Twisted C2H4 with IR Intensities (in km/mol, listed parenthetically) this researcha

1/2

⟨Q s2⟩ =

mode

mode

∑ ϕrss

ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9 ν10 ν11 ν12

s

(2)

where ωr is the harmonic frequency of the rth mode and ϕrss is a cubic force constant. Note that both ω and ϕ in eq 2 depend on the nuclear masses, yielding different zero-point-corrected geometries for the three isotopologues. Relative to the re structure, C−C bonds are increased by 0.0086 Å or more and the C−H bonds by 0.0152 Å or more. The latter bonds are strongly affected due to the high degree of anharmonicity along those coordinates (vide infra). Table 2 also shows that the bond lengths of the rg structures are always longer than those of the equilibrium geometry, which is consistent with the general features of vibrationally averaged structures. Vibrational Frequencies. The 12 normal vibrational modes of triplet ethylene and their symmetries are described in Table 3. The D2d symmetry makes all vibrational modes Raman-active, with ν5−ν12 being infrared (IR)-active. Due to the fleeting nature of triplet C2H4, neither Raman nor IR spectroscopy has been observed experimentally. Therefore, our predicted vibrational frequencies may provide assistance to future experimental work. In Tables 4−6, we list our computed harmonic and anharmonic vibrational frequencies for triplet C2H4 and two isotopologues. These results show that the anharmonic frequencies are generally lower than the harmonic predictions by about 3.2%, except for the two wagging modes with lowest frequencies (ν11 and ν12). More discussion of the large

harmonic 3115.4 1465.7 1135.1 693.7 3119.8 1439.7 3207.6 3207.6 939.7 939.7 407.6 407.6

(0.0) (0.0) (0.0) (0.0) (18.5) (0.1) (5.3) (5.3) (1.2) (1.2) (53.8) (53.8)

previous research

anharmonic 2978.2 1426.8 1103.7 651.2 2982.4 1403.8 3050.0 3050.0 925.7 925.7 516.0 516.0

(0.0) (0.0) (0.0) (0.0) (16.8) (0.5) (6.2) (6.2) (2.0) (2.0) (36.8) (36.8)

ref 39b 3060.6 1449 1109.6 662.2 3062.7 1424.3 3150 3150 925.6 925.6 339.4 339.8

a Computed at the CCSD(T)/cc-pVQZ level of theory. bComputed at the QCISD/6-311G(d,p) level of theory and scaled by the factor 0.9776.

anharmonic corrections for ν11 and ν12, which we consider to be unphysical, is given in a later section. C2H4. Vibrational frequencies and IR intensities for the parent molecule are reported in Table 4. Compared to the scaled harmonic vibrational frequencies obtained theoretically by Kim and co-workers, most differencies are approximately 2.0% except for ν11 and ν12, which differ by around 20%, as mentioned. Note that Kim and co-workers report harmonic frequencies scaled by a factor of 0.9776. For most vibrational modes, such scaled frequencies lie between our corresponding harmonic and anharmonic frequencies, closer to the latter for ν3, ν4, ν9, and ν10 while slightly closer to the former for the rest. The IR intensity of the CH2 in-plane scissoring mode, ν6, is predicted to be relatively low, which may prohibit observation 7562

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Table 5. Harmonic and Anharmonic Vibrational Frequencies (in cm−1) for Triplet Twisted 13C2H4 with IR Intensities (in km/mol, listed parenthetically)

Table 7. Triplet Ethylene Isotopic Vibrational Shifts (in cm−1) with Respect to the Parent Isotopologue C2H4

mode ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9 ν10 ν11 ν12 a

harmonic 3110.1 1456.1 1099.5 693.7 3114.9 1434.0 3194.5 3194.5 930.5 930.5 404.6 404.6

ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9 ν10 ν11 ν12

anharmonic

(0.0) (0.0) (0.0) (0.0) (18.4) (0.1) (5.2) (5.2) (1.0) (1.0) (53.1) (53.1)

2972.6 1418.8 1070.8 651.0 2977.3 1397.3 3040.2 3040.1 916.2 916.2 513.4 513.3

(0.0) (0.0) (0.0) (0.0) (16.6) (0.5) (6.0) (6.0) (1.7) (1.7) (35.8) (35.9)

Computed at the CCSD(T)/cc-pVQZ level of theory.

this worka ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9 ν10 ν11 ν12 a

harmonic 2261.9 1220.8 939.3 490.7 2252.1 1067.2 2387.4 2387.4 739.4 739.4 311.7 311.7

(0.0) (0.0) (0.0) (0.0) (9.5) (0.1) (3.5) (3.5) (2.6) (2.6) (31.1) (31.1)

anharmonic 2181.8 1191.9 919.9 469.1 2173.4 1051.0 2300.0 2300.0 733.7 733.7 376.3 376.3

C2H4

−5.6 −8.1 −32.9 −0.2 −5.0 −6.5 −9.8 −9.8 −9.6 −9.6 −2.6 −2.6

C2 D 4 −804.8 −234.9 −183.8 −182.2 −800.6 −352.7 −750.0 −750.0 −192.1 −192.1 −139.6 −139.6

was the mode least affected by 13C isotopic substitution in the case of 13C2H4. Our VPT2 analyses were generally free from problematic Fermi−Dennison resonances; however, the ν5 vibrational mode of C2D4 suffers from a Fermi type-II resonance when perturbed by the modes ν2 and ν6 (ω2 + ω6 ≈ ω5). As is the standard procedure, first proposed by Nielsen,65 we remove the contributions with small denominators from the summations in the VPT2 analyses and estimate the energetic effect of neglecting such items via the first-order couplings

Table 6. Harmonic and Anharmonic Vibrational Frequencies (in cm−1) for Triplet Twisted C2D4 with IR Intensities (in km/mol, listed parenthetically) mode

13

mode

this worka

(0.0) (0.0) (0.0) (0.0) (9.0) (0.1) (3.9) (3.9) (3.2) (3.2) (23.0) (23.0)

⎛ ϕ ⎞ ⎜ ω2 + ω6 2,5,6 ⎟ 8 ⎟ ⎜ ⎜ ⎟ ⎜ ϕ2,5,6 ⎟ ω5 ⎟ ⎜ ⎝ ⎠ 8

(3)

where ϕijk is the cubic force constant with respect to the modes i, j, and k. It is straightforward to consider the diagonal terms to be the zero-order states and the off-diagonal terms to represent a first-order perturbation. Denoting the separation between ω2 + ω6 and ω5 by Δ, it may be shown that the eigenvalues of 3 are given by

Computed at the CCSD(T)/cc-pVQZ level of theory.

of this particular vibration in future experiments. The most intense vibrational modes are found to be the out-of-plane wagging vibration of the two CH2 terminal groups, ν11 and ν12. 13 C2H4. Vibrational frequencies and IR intensities for 13C2H4 are reported in Table 5. As for the parent isotopologue, the anharmonic corrections and IR intensities for 13C2H4 reveal a distinct pattern. From Table 7, isotopic shifts of the anharmonic frequencies with respect to the parent molecule for all vibrational modes may be inspected. All frequencies are reduced in magnitude, with the C−C stretching mode ν3 affected most (3.1%) by 13C isotopic substitution. The CH2 in-plane rocking modes, ν9 and ν10, shift 1.0% because they also involve vibration of the carbon atoms. As expected, the CH2 out-of-plane twisting mode, ν4, remains nearly unchanged (0.01%) by 13C substitution. C2D4. Vibrational frequencies and IR intensities for C2D4 are reported in Table 6. Deuterium substitution results in a considerable shift (>17%) for all vibrational modes, with respect to the parent molecule. This is, of course, due to the relative mass change arising from deuterium substitution. The CH2 out-of-plane twisting mode, ν4, which involves only vibrations of hydrogen atoms, suffers the most significant reduction (28%) in magnitude upon deuteration. Note that ν4

⎛ Δ⎜ ν = ω5 + 1± 2⎜ ⎝

⎞ 2 ϕ2,5,6 ⎟ 1+ 2Δ2 ⎟ ⎠

(4)

Our final ν5 prediction is obtained by adding a correction, which is the difference between the desired eigenvalue computed from eq 4 and ω5, compared to the deperturbed ν5. This eigenvalue of eq 3 corresponds to an eigenvector containing the maximal ω5 content. The resulting value is 2156.5 cm−1, 16.9 cm−1 lower than the deperturbed frequency. Rotational Constants. Rotational constants for the equilibrium and vibrationally averaged structures of the species under study are shown in Table 8. Such corrections arise from two sources, vibrational zero-point effects and quartic centrifugal distortion effects. Including vibrational zero-point effects alone yields A0, B0, and C0, while the inclusion of both effects gives A′, B′, and C′. To second order, the molecular rotational constant B0 for the vibrational ground state is related to the equilibrium rotational constant Be by66 7563

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Table 8. Twisted Triplet Ethylene Rotational Constants (in MHz) from the Equilibrium Geometry and with Vibrational Zero-Point Corrections (in MHz) parameter

C2H4

Ae Be Ce A0 B0 C0 A′ B′ C′

146761.9 23928.8 23928.8 144580.7 23806.3 23806.3 144581.4 23806.1 23806.1

B0 ≈ Be −

1 2

13

C2H2

146761.9 22786.4 22786.4 144559.8 22669.2 22669.2 144560.4 22669.0 22669.0

Table 9. Harmonic and Anharmonic Vibrational Frequencies (in cm−1) of Wagging Modes for the C2H4, C2H3H′, and C2H3D Isotopomersa of Twisted Triplet Ethylene

C2 D 4 73437.4 17094.9 17094.9 72741.2 17036.7 17036.7 72741.5 17036.6 17036.6

mode

harm.

anharm.

harm.

anharm.

harm.

anharm.

ν11 ν12

407.6 407.6

516.0 516.0

401.3 402.0

512.1 512.9

352.7 397.6

439.9 507.0

Table 10. Theoretical Harmonic and Fundamental Frequencies (in cm−1) of Allene C3H4 Compared with Experiment

∑ αrB + ... r

1 (3τcaca − 2τabab − 2τbcbc) 4

C2H3D

a The results for C2H3H′ and C2H3D are both computed at the CCSD(T)/cc-pVTZ level of theory. bThe mass of H′ is 1% greater than that of hydrogen.

(5)

mode

symmetry

ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9 ν10 ν11 ν12 ν13 ν14 ν15

e e e e b1 e e a1 b2 a1 b2 a1 b2 e e

where αxr (x = A, B, or C) denotes the vibration−rotation interaction constants, which describe the coupling of rotations about the principal axis x with the normal mode r. Further taking into account the quartic centrifugal distortion constants ταβγδ, one may obtain the effective rotational constant B′ (which corresponds to the Hamiltonian used to fit the observed energy levels) via67 B′ = B0 +

C2H3H′b

C2H4

(6)

harmonica fundamentala 348.0 348.0 856.7 856.7 870.3 1017.8 1017.8 1080.8 1438.4 1488.6 2012.2 3142.8 3144.3 3226.4 3226.4

348.2 348.2 837.8 837.8 849.3 997.9 997.9 1067.2 1395.2 1441.5 1953.5 3011.3 3015.7 3078.7 3078.7

Auer and Gaussb 351 351 851 851 857 1005 1005 1078 1411 1457 1985 3040 3044 3096 3096

experimentc 352.73 840.93 848.59 999.00 1072.22 1359.0 1442.55 1959.1 3006.7

The zero-point effects (eq 5) provide the most significant corrections, lowering {Ae, Be, Ce} by {2181.7, 122.6, 122.6} MHz, respectively, for the parent isotopologue, C2H4. The quartic centrifugal distortion effects (eq 6) contribute minimally, further reducing {A0, B0, C0} by {−0.70, 0.17, 0.17} MHz. The corrected rotational constants are lower than their equilibrium counterparts by {1.5, 0.5, 0.5}%, consistent with the bond elongation observed in the vibrationally averaged structure, relative to the equilibrium structure. Similar trends are observed for the other two isotopologues.

a

CHALLENGES FROM THE ANHARMONICITY OF THE WAGGING MODES As reported in Tables 4−6, the wagging modes, ν11 and ν12, of triplet C2H4, 13C2H4, and C2D4 involve large anharmonic corrections to the vibrational frequencies (+24.9% on average) compared to those of other modes (−3.2% on average). There may be several possible explanations for this behavior. First, as a symmetric top molecule, twisted triplet ethylene with a D2d point group allows for degenerate frequencies whose complexity68 might not be treated properly by the VPT2 module in CFOUR. To evaluate this possibility, the molecular symmetry may be lowered by either replacing one hydrogen with deuterium or by changing the mass of one hydrogen by a small amount, yielding an asymmetric top molecule. Table 9 compares harmonic and anharmonic frequencies of the wagging modes for triplet C2H4, C2H3H′, and C2H3D, where C2H3H′ is a molecule containing a hypothetical atom, H′, that has 1% larger mass than hydrogen. Here, we can see that the results of two symmetry-broken molecules have similar anharmonic corrections as those found for the parent molecule, which refutes our first hypothesis. In 2001, Auer and Gauss69 reported anharmonic frequencies for allene, which also has a D2d symmetry and is a symmetric top, using the VPT2 method implemented in CFOUR. We obtained a set of allene

anharmonic frequencies in good agreement with the experimental frequencies70 and the theoretical results from Auer and Gauss. Table 10 compares the results obtained by the different methods. For the wagging modes of allene, ν3 and ν4, our anharmonic corrections at the CCSD(T)/cc-pVTZ level of theory diminish the deviation from experimental frequencies to 3 cm−1. The computations on allene, along with our results for C2H3H′ and C2H3D, preclude the degenerate vibrations of the wagging modes as a possible problem. Other errors could arise from the finite difference technique that is utilized to obtain cubic and quartic force constants from second derivatives computations at displacements from the equilibrium geometry. Values for the displacement size along the normal coordinates must be large enough to capture the vibrational motion but small enough to be local to the equilibrium region. As such, it is possible that to capture the nature of the potential energy surface for the wagging modes, larger or smaller displacement values are needed. Table 11 shows the anharmonic vibrational corrections of the wagging modes with respect to the displacement size. We find that varying the displacement value does not noticeably affect the anharmonic corrections of the wagging modes. In fact, the displacements used in the finite difference method are less crucial to the wagging modes than to the stretching or bending

Harmonic anharmonic b Harmonic anharmonic ref 70.

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7564

3085.43

frequencies computed at CCSD(T)/cc-pVTZ with correction computed at the same level of theory as well. frequencies computed at CCSD(T)/cc-pVTZ with correction computed at MP2/cc-pVTZ. See ref 69. cSee

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Table 11. Anharmonic Vibrational Corrections (in cm−1) of Wagging Modes for Triplet C2H3H′ with Respect to Values of the Displacement (in terms of reduced normal coordinates)a

a

0.050

0.060

0.075

0.100

ν11 ν12

+112.5 +107.2

+109.8 +110.4

+111.0 +110.6

+110.8 +110.9

Computed at the CCSD(T)/cc-pVTZ level of theory.

because the region of the potential energy surface for the former appears to be much flatter than that of the latter. Third and perhaps most problematically, the quartic force constants of the wagging modes are arrestingly larger than those for the other modes. With inspection of the contribution to the anharmonic frequencies and constants (the relevant equations are detailed in ref 7), the largest positive component of the anharmonic corrections is derived from the self-couplings (χ11,11 and χ12,12) and intercoupling (χ11,12) of the two wagging modes. For these, the largest positive contributor comes from the quartic force constants ϕ11,11,11,11, ϕ11,11,12,12, ϕ12,12,11,11, and ϕ12,12,12,12. The VPT2 method, which starts from the harmonic oscillator rigid rotator approximation, cannot be reliably applied to modes where the quadratic terms do not dominate the potential energy surface. To summarize this section, we have analyzed issues arising from the symmetric top nature of the molecule and the influence of displacement increments as possible explanations for the unphysical anharmonic corrections to the wagging modes. A rational candidate among possible reasons for the failure of the VPT2 method when applied to the wagging modes of triplet ethylene appears to be the absence of a dominant harmonic term for the potential.

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CONCLUSIONS The lowest triplet state (ã 3A1) of C2H4 is of both theoretical and experimental interest. In the present research, we have focused on the structure and vibrational modes of this electronic state of twisted C2H4 and its two isotopologues using the CCSD(T)/cc-pVQZ level of theory. With VPT2 theory, we have also studied the zero-point-corrected structures and fundamental vibrational frequencies. This is the first time that the anharmonic frequencies of the 10 vibrational modes for triplet ethylene (and two of its isotopologues) have been predicted. We hope that this research will provide guidance for future experimental investigations of triplet ethylene.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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ACKNOWLEDGMENTS

This work was funded by the Department of Energy, Office of Basic Energy Sciences (Grant No. DE-FG02-97-ER14748). X.W. thanks Dr. Y. Yamaguchi and Dr. D. S. Hollman for insightful discussions. 7565

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