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Spectroscopy and Excited States

Overcoming the Failure of Correlation for Out-of-Plane Motions in a Simple Aromatic: Rovibrational Quantum Chemical Analysis of c-CH 3

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Ryan C. Fortenberry, Carlie M. Novak, Joshua P. Layfield, Eduard Matito, and Timothy J Lee J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00164 • Publication Date (Web): 09 Mar 2018 Downloaded from http://pubs.acs.org on March 14, 2018

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Journal of Chemical Theory and Computation

Overcoming the Failure of Correlation for Out-of-Plane Motions in a Simple Aromatic: Rovibrational Quantum Chemical Analysis of c-C3H2 Ryan C. Fortenberry,∗,† Carlie M. Novak,† Joshua P. Layfield,‡ Eduard Matito,¶,§ and Timothy J. Leek †Georgia Southern University, Department of Chemistry & Biochemistry, Statesboro, GA 30460, U.S.A. ‡University of St. Thomas, Department of Chemsitry, St. Paul, MN 55105, U.S.A. ¶Kimika Fakultatea, Euskal Herriko Unibertsitatea, UPV/EHU, and Donostia International Physics Center (DIPC). P.K. 1072, 20080 Donostia, Euskadi, Spain §IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain. kMS 245-3 NASA Ames Research Center, Moffett Field, CA 94035-1000, U.S.A. E-mail: [email protected] Phone: 912-478-7694 Abstract Truncated, correlated, wave function methods either produce imaginary frequencies (in the extreme case) or non-physically low frequencies in out-of-plane motions for carbon and adjacent atoms when the carbon atoms engage in π bonding. Cyclopropenylidene is viewed as the simplest aromatic hydrocarbon, and the present as well

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ACS Paragon Plus Environment

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as previous theoretical studies have shown that this simple molecule exhibits this behavior in the two out-of-plane bends (OPBs). This non-physical behavior has been treated by removing nearly linear dependent basis functions according to eigenvalues of the overlap matrix, by employing basis sets where the spd space saturatation is balanced with higher angular momentum functions, by including basis set superposition/incompleteness error (BSSE/BSIE) corrections, or by combining standard correlation methods with explicitly-correlated methods to produce hybrid potential surfaces. However, this work supports the recently described hypothesis that the OPB problem is both a method and a basis set effect. The correlated wavefunction’s largest higherorder substitution term comes from a π → π ∗ excitation where constructive interference of both orbitals artificially stabilizes OPB. By employing schema to overcome this issue, the symmetric OPB ν9 is the predicted to be the second-brightest transition, and it will be observed very close to 775 cm−1 . However, more work from the community is required to formulate better how carbon atoms interact with their adjacent atoms in π-bonded systems. Such bonds are ubiquitous in all of chemistry and beyond.

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Introduction

Cyclopropenylidene (c-C3 H2 ) is often called the simplest aromatic hydrocarbon since it follows H¨ uckel’s 4n + 2 rule with n = 0, but standard wave function based quantum chemical treatments have been known to fall short in describing the out-of-plane bends (OPBs) for this simple molecule. 1–3 Such behavior is not limited to this molecule, but ethylene, acetylene, benzene, and even nucleobases have been documented as producing imaginary or too-low OPB vibrational frequencies. 4–10 In truth, most wave function-based correlation treatments of π-bonded carbon atoms engaging in OPBs likely exacerbate this problem especially for anharmonic corrections, 3 and c-C3 H2 is no exception. This work will serve to delve more deeply into the problem for this molecule building upon that recently reported 3 and to provide strong predictions for the vibrational frequencies of this molecule for application to

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combustion thermochemistry and astrochemical spectral characterization. Cyclopropenylidene was detected in the interstellar medium (ISM) in 1985 through rotational spectroscopcy, is known to be one of the most abundant molecules in the ISM, and is believed to be a building block for the formation of larger polycyclic aromatic hydrocarbons (PAHs) in the gas phase. 11–14 In fact, c-C3 H2 is so abundant in the ISM that the singlyand doubly-deuterated forms as well as

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C isotopologues have also been detected. 15–17 It is

formed in the gas phase and has been observed towards various types of interstellar objects ranging from stars to galactic centers. 12,18 Hence, its role in the organic chemistry of space is unmatched save for potentially C2 H. The recent reporting of benzonitrile in the Taurus Molecular Cloud 19 highlights how small, aromatic molecules have only just started to be detected in the ISM. While the rotational spectrum of c-C3 H2 is well known, few experimental studies have analyzed the vibrational transitions, and even then only in the condensed phase in an Ar matrix. 20 Such benchmarks are necessary for observation with the newest generation of space telescopes, especially the James Webb Space Telescope (JWST) now set to launch in 2019 or the currently operating Statrospheric Observatory For Infrared Astronomy (SOFIA). Quantum chemical analysis has provided estimates for these values, but the OPB problem has only been addressed in a few modern studies of this ubiquitous molecule. 2 Early insights into the OPB problem for other hydrocarbons appeared to influence most heavily modes where the inversion symmetry is conserved in the OPB. However, other OPB modes without inversion also exhibited this behavior just not as extreme. In any case, computations on the vibrational frequencies of c-C3 H2 possess the same behavior. 1–3 The alternation of the in-plane and out-of-plane motion of the carbon and hydrogen atoms give clues that such behavior may be more systemic and problematic. Previous work on c-C3 H2 3 has shown that this problem is a combined one-partle and n-particle basis set effect. The first clue is that the largest t1 amplitude in the CCSD wave function for c-C3 H2 is the particleon-a-ring n = 2 ← n = 1 transition. The MOs are given in Figure 1. The OPB allows the hydrogen s orbitals to combine with both the π clouds of the occupied and virtual orbitals

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Figure 1: The Molecular Orbitals for the largest tai contributor to the CCSD wavefunction for the occupied i orbital (a) and unoccupied a orbital (b).

a)

b)

involved in the wave function substitution. As the atoms move further out of the plane, the s orbitals artificially stabilize the π ∗ orbital. However, in Hartree-Fock computations saturated with spd functions, the erroneous OPB effect is not observed. 3 The question still remains, however, as to what are the correct, gas-phase vibrational frequencies for cyclopropenylidene. If a molecule of such relative simplicity and importance as c-C3 H2 cannot be fully, accurately described, further quantum chemical insights and potential fixes into this behavior need to be addressed. Additionally, if this behavior is present in small hydrocarbons, it will likely be present in larger ones, too. 8,21 This will affect the analysis of PAHs, which are of importance to astrochemistry 13 but also in terrestrial and environmental applications. On Earth, PAHs are known carcinogens making their spectral characterization and that of their precursors for detection and environmental characterization an important application of quantum chemistry. 22,23 The anharmonic vibrational frequencies of cyclopropenylidene and its larger PAH cousins are important beyond mere spectroscopy. These frequencies are necessary for producing highly-accurate thermochemical data for any predictions of internal energy. Hence, quantum chemical descriptions of combustion chemistry will also be affected

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by this error in theoretical predictions of π-bonded hydrocarbons. One of the most accurate methods for computing the potential energy surface involve composite energy schema for quartic force fields (QFFs) based on coupled cluster theory 24,25 at the singles, doubles, and perturbative triples level [CCSD(T)]. 26 These accurate fourthorder Taylor series expansions of the internuclear potential contain energies for complete basis set (CBS) extrapolations, core electron correlation, and scalar relativity to give the so-called CcCR QFF from these pieces, 27–30 which is then coupled with rotational and vibrational perturbation theory at second order (VPT2). 31–33 As a result, this CcCR QFF VPT2 methodology utilized to great effect in the past 29,34–44 has been brought to bear on c-C3 H2 recently, 3 but it was shown to fall short in the OPB frequency computations. This work will more deeply test other approaches and document their OPB behavior in relation to standard CCSD(T) and the CcCR QFF so that an accurate approach can be implemented for computing the vibrational frequencies of π-bonded carbon compounds. Additionally, the present discussion will provide a “best estimate” of currently outstanding spectroscopic constants and vibrational frequencies. Finally, further analysis of the OPB problem in sp & sp2 , carbon−carbon bonded systems must be undertaken in order to provide a path forward for properly treating the correlated wavefunctions of such common and important molecules.

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Computational Details

Multiple methods are utilized for this C2v molecule in order to find a reproducible, computational approach that will more closely corroborate the theoretical spectroscopic constants for the out-of-plane motions, specifically the a2 and b1 modes, to their trusted, corresponding theoretical values by Lee and coworkers 2 that were subsequently compared with Ar-matrix experimental results. 20 c-C3 H2 is a closed-shell molecule and thus a restricted Hartree-Fock reference wavefunction is used. 45 The standard CCSD(T) computations utilize the PSI4

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quantum chemistry program, 46,47 the explicitly correlated methods are computed with MOLPRO 2015.1, 48,49 and the double-harmonic intensities are from Gaussian09. 50 The QFF is defined as

V =

1X 1 X 1X Fij ∆i ∆j + Fijk ∆i ∆j ∆k + Fijkl ∆i ∆j ∆k ∆l 2 ij 6 ijk 24 ijkl

(1)

where Fij define the force constants and ∆i describe the displacements of the ith coordinate defined below. QFFs are accurate for the computational cost and represent well the potential portion of the internuclear Hamiltonian. The QFFs for this molecule was determined using 1585 total grid points as outlined below. Each displacement is 0.005 ˚ A for bond length coordinates and 0.005 radians for the bond angle and torsional coordinates. The symmetry internal coordinates defined for this molecule are:

S1 (a1 ) = C2 − C3

(2)

1 S2 (a1 ) = √ [(C1 − C2 ) + (C1 − C3 )] 2 1 S3 (a1 ) = √ [(C2 − H1 ) + (C3 − H2 )] 2 1 S4 (a1 ) = √ 6 [(H1 − C2 − C1 ) + (H2 − C3 − C1 )] 2 1 S5 (b2 ) = √ [(C1 − C2 ) − (C1 − C3 )] 2 1 S6 (b2 ) = √ [(C2 − H1 ) − (C3 − H2 )] 2 1 S7 (b2 ) = √ 6 [(H1 − C2 − C1 ) − (H2 − C3 − C1 )] 2 1 S8 (b1 ) = √ τ [(H1 − C2 − C1 − C3 ) − (H2 − C3 − C1 − C2 )] 2 1 S9 (a2 ) = √ τ [(H1 − C2 − C1 − C3 ) + (H2 − C3 − C1 − C2 )]. 2

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(3) (4) (5) (6) (7) (8) (9) (10)

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2.1 2.1.1

Quartic Force Fields CcCR

The first method applied to c-C3 H2 , the CcCR methodology, has been previously utilized in Ref. 3, but only the vibrational frequencies were discussed. In short, the CcCR process, begins with the reference geometry computed by taking the optimized CCSD(T)/aug-ccpV5Z geometry 51,52 and adding the difference of the Martin Taylor (MT) core correlating basis 53 set with and without the inclusion of the core electrons. Then, the displacement geometries that define the QFF grid are determined using a “lazy-Cartesian” approach 54 and computed with a three-point CBS extrapolation 55 for TZ, QZ, and 5Z giving the “C” term. To this, the “cC” term is added for the core correlation energy contribution computed again by taking the difference in the MT with and MT without core electron energies. The “R” term is the scalar relativity added to the composite energy by taking the difference between the energy computed with and without the relativity included at the CCSD(T)/augcc-pVTZ-DK level. 56,57 The CcCR QFF composite energy system is defined as:

ECcCR = ECBS + (EM T (core) − EM T ) + (EDK(Rel.) − EDK ).

(11)

The QFF is then fit using a linear least-squares procedure and produces a sum of the squared residuals (SSR) of less than 2.0 × 10−17 a.u.2 This fitting produces the equilibrium geometry while refitting the points using the equilibrium geometry creates zero gradients and provides the final higher-derivative force constants. These symmetry-internal force constants are then transformed into Cartesian coordinates with the INTDER program 58 in order to provide generalized force constants. The spectroscopic constants as well as the harmonic and anharmonic frequencies from VPT2 are computed using the SPECTRO program. 59

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2.1.2

Explicitly Correlated and Other Methods

In addition to the standard CcCR QFF, explicitly correlated second-order Møller-Plesset purturbation theory (MP2-F12) and CCSD(T)-F12 methods, 60–63 specifically CCSD(T)-F12b, are also employed with various basis sets including aug-cc-pVDZ, aug-cc-pVTZ, and aug-ccpVQZ abbreviated for use in the text as simply DZ, TZ, and QZ. These QFFs are initially begun by optimizing the c-C3 H2 geometry at the respective levels of theory, and then computing the QFFs from the grid of 1585 points based on their respective minimum geometries for the same level of theory unless otherwise noted. In order to fully explore the full surface, standard CCSD(T) QFFs are also computed taking portions of the CcCR methodology based on the CcCR reference geometry. For example, the XZfit QFFs for X = T, Q, and 5 involve CCSD(T)/aug-cc-pVXZ energies including the MT(core) − MT corrections added. CCSD/aug-cc-pVXZ QFFs are also computed with the ∆MT terms included. Additionally, simple Hartree-Fock (HF) QFFs are also computed with the aug-cc-pVDZ basis set as well as the aug-cc-pV5Z basis set but where the latter only contains the s, p, and d orbitals. All methods have SSRs on the order of 10−17 even TZfit where the difference between the reference and equilibrium geometries is greater greater than one displacement in each of coordinates S1 and S2 . Finally, the “Best” results are a combination of the CcCR and MP2-F12/QZ defined later. Hence, multiple many-body perturbation theory methods have been brought to bear to analyze how the OPBs behave anharmonically for this simplest of aromatic hydrocarbons.

2.1.3

Manipulating Force Constants

As discussed below, the out-of-plane CcCR vibrational frequencies of b1 and a2 symmetry raise questions as to their accuracy due to the known OPB issues. In order to overcome these issues, the CcCR force constants (FCs) are manipulated through a series of scaling factors in order to better represent the available experimental data. S8 (b1 ) and S9 (a2 ) are scaled relative to the differences between the CcCR results for the corresponding vibrational 8


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frequencies and the Ar matrix 20 experimental ν9 (b1 ) or the previously computed, 2 reference ν6 (a2 ) since the latter is vibrationally dark. The generic scaling approach follows the below scheme: σ=

νref erence − νCcCR + 1. νCcCR

(12)

Table 1: Nomeclature for the FC Scaling Trials. Trial Coordinate 1A S8 1B S8 S8 2A 2B S8 3 S8 4A S9 4B S9 5A S9 5B S9 6A S8 6B S8 7A S9 7B S9 8A S8 & S9 8B S8 & S9 9A S8 & S9 9B S8 & S9 10A S8 & S9 10B S8 & S9 a n corresponds to the number

Inclusion of nonQFF σ Scalea Coordinate Terms CcCR 1 Yes CcCR 1 No CcCR 2 Yes CcCR 2 No CcCR 1 Yes but No S9 CcCR 1 Yes CcCR 1 No CcCR 2 Yes CcCR 2 No CcCR n Yes CcCR n No CcCR n Yes CcCR n No CcCR 2 Yes CcCR 2 No QZfit 2 Yes QZfit 2 No 5Zfit 2 Yes 5Zfit 2 No of times the given coordinate appears in the FC.

Different trials correspond to different means of employing this scaling factor for both S8 and S9 CcCR FCs and are collected in Table 1. Trial 1 applies Eq. 12 to S8 . This is broken into 2 subsets. Trial 1A is for every FC occurrence of S8 . Trial 1B employs Eq. 12 for those FCs composed of only S8 i.e. the F88 and F8888 FCs. Trials 2A and 2B are the same as the corresponding approach in Trials 1A and 1B except that the scaling factor in Eq. 12 has been doubled for closer empirical correlation. Trial 3 is the same as Trial 2A except when there is a dependence upon S9 (instead of S8 ) in the FC, the scaling factor is neglected. For instance, F882 is manipulated but F985 is not in Trial 3. Trials 4A, 4B, 5A, and 5B are the

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same approach as that for 1A-2B except that they are applied to S9 . Trial 6A has the scaling factor is multiplied by the number of instances of S8 in a given FC. For instance F8881 will have the scaling factor tripled. Trial 6B is the same approach but only applies, again, to those FCs that are exclusively comprised of S8 . Trials 7A and 7B have the same construction as 6A and 6B but are defined for S9 . Trial 8A is a combination of Trial 2A and 4A where both S8 and S9 are utilized. Trial 8B is only for those FCs comprised exclusively of their respective coordinates. Trial 9A is the QZfit QFF but with the Trial 8A scaling factor. Trial 9B is the same but only for exclusive S8 and S9 terms. Trials 10A and 10B follow 9A and 9B but are for the 5Zfit QFF.

2.2

Other Metrics

In order to gauge how well the OPBs are treated, CCSD(T)/aug-cc-pVTZ computations are employed to scan the S8 and S9 coordinates from 0.0◦ to 90.0◦ in 1.0◦ increments. Additionally, this same scan is undertaken for cyclopropylidene (c-C3 H4 ), another carbene where the ring is maintained, but the π-bond is removed. Differences in energy for the scan between c-C3 H4 and c-C3 H2 + H2 showcase how the π bond interacts with the atoms as they move out-of-plane and is similar to that done in Ref. 3. The H2 is the static energy of the optimized CCSD/aug-cc-pVTZ geometry, which is a complete description for this twoelectron system. This same scan and analysis is also done for ethylene versus acetylene and hydrogen as well as ethane versus ethylene and hydrogen. Finally, explicit computations of the aromaticity within c-C3 H2 , c-C3 H4 , and the same above scans of these cyclopropyl carbenes are also undertaken utilizing the multicenter index (MCI) 64 in order to explore the change of this property along the OPBs in these molecules. Aromaticity calculations employed the quantum theory of atoms in molecules (QTAIM) 65 to define the atomic boundaries in the Cartesian space, using the AIMall program 66 to obtain the atom overlap matrices (AOMs). The AOMs were input into ESI-3D code 67–69 to compute 10


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the MCI indices, 64 which despite its limitations in large rings 70 provides a reliable account of aromaticity. 71,72 The MCI indices were calculated from the energy-derivative natural orbital occupancies obtained from Gaussian 50 employing the formulation suggested in Refs. 73,74.

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Results

3.1

QFF Analysis of c-C3 H2

The harmonic/quadratic FCs for the major QFFs utilized in this work are given in Table 2 where the numbers correspond to the Sx values given earlier. For now, the most notable item is that the second-order CcCR FCs are in marked agreement with those from Ref. 2. Hence, the CcCR QFF is behaving well thus far which is further evidenced in the correlation between methods in the harmonic frequenices of Table 3. Additionally, the F11 value is in line with that expected for a C=C bond while F22 is in line with that expected for a lower bond order C−C value. Unfortunately, once the anharmonic FCs are added (given in Table 1 of the supplemental information, SI) to construct the QFF, the VPT2-computed ν6 (a2 ) and ν9 (b1 ) OPB modes begin to behave differently for CcCR as compared to both previous theory 2 and Ar matrix data 20 as given in Table 3. Table 3 gives the vibrational frequencies for c-C3 H2 with several different QFFs as well as argon-matrix experimental results. 20 Previous, high-level results did not see evidence for the known OPB bending issue 2 and compare well to experiment. Hence, the Lee, Huang, and Dateo (LHD) results from Ref. 2 are considered the standard for this study. Clearly the symmetric OPB, ν9 of b1 symmetry, is well described by the LHD QFF since it produces a 776.0 cm−1 fundamental frequency with the experimental results at 787.4 cm−1 where a matrix shift compared to gas phase results is likely. Granted, the other vibrational frequencies for which there are experimental data (ν4 , ν5 , and ν7 ) differ by 1-3 cm−1 , but the 776.0 cm−1 value from LHD is closer to 787.4 cm−1 than the earlier CCSD(T)/aug-cc-pVTZ 768.5 cm−1 results from Ref. 1. The LHD antisymmetric OPB, ν6 of a2 symmetry, is likely similarly 11



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Table 2: The Harmonic Symmetry-Internal Force Constants For Various Methods Utilized for c-C3 H2 (in mdyn/˚ An ·radm ). F11 F21 F22 F31 F32 F33 F41 F42 F43 F44 F55 F65 F66 F75 F76 F77 F88 F99 F11 F21 F22 F31 F32 F33 F41 F42 F43 F44 F55 F65 F66 F75 F76 F77 F88 F99

Ref. 1 9.60495 -0.46727 5.02355 0.04168 -0.03883 5.78698 -0.34432 0.07337 0.00286 0.58541 5.58939 0.24381 5.78360 0.26784 -0.04307 0.46889 0.09906 0.12819 MP2-F12/DZ 9.62930 -0.53066 5.30129 0.02616 -0.03760 5.88486 -0.33532 0.07821 0.00765 0.58433 5.88695 0.24024 5.89791 -0.23162 -0.04072 0.46309 0.10379 0.13101

Ref. 2 Best 9.72876 -0.47718 5.16443 0.05300 -0.0239 5.80469 -0.33803 0.07642 0.00541 0.58653 5.78257 0.24289 5.81407 -0.25731 -0.04072 0.47198 0.10097 0.12989 MP2-F12/TZ 9.64920 -0.53929 5.33857 0.03033 -0.03453 5.88326 -0.33462 0.08082 0.00786 0.58538 5.95167 0.24202 5.89831 -0.22923 -0.04174 0.46513 0.10532 0.13289

Ref. 3 CcCR 9.73341 -0.47391 5.18854 0.05138 -0.02196 5.80465 -0.33580 0.07701 0.00580 0.58557 5.83618 0.24120 5.82028 -0.25610 -0.03968 0.47347 0.10098 0.13049 MP2-F12/QZ 9.62390 -0.46311 5.11681 0.04901 -0.02617 5.76554 -0.33755 0.07605 0.00444 0.58434 5.74133 0.24117 5.77505 -0.25309 -0.04183 0.46835 0.10074 0.13006

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QZfit 9.68607 -0.46627 5.16130 0.05141 -0.02441 5.80205 -0.33681 0.07752 0.00564 0.58553 5.78780 0.24091 5.81071 -0.25440 -0.04086 0.47062 0.10072 0.12964 CCSD(T)-F12/TZ 9.62396 -0.46310 5.11666 0.04902 -0.02618 5.76555 -0.33757 0.07605 0.00443 0.58433 5.74123 0.24117 5.77506 -0.25313 -0.04183 0.46835 0.10072 0.13006


5Zfit 9.70902 -0.47014 5.17829 0.05043 -0.02399 5.79914 -0.33584 0.07706 0.00523 0.58546 5.81524 0.24133 5.81009 -0.25506 -0.04057 0.47219 0.10077 0.13007 CCSD(T)-F12/QZ 9.64965 -0.46559 5.13116 0.04851 -0.02637 5.77516 -0.33858 0.07603 -0.00426 0.57144 5.76085 0.24154 5.78554 -0.25409 -0.04114 0.46430 0.09619 0.12560

Trial 8 9.73341 -0.47391 5.18854 0.05138 -0.02196 5.80465 -0.33580 0.07701 0.00580 0.58557 5.83618 0.24119 5.82028 -0.25610 -0.03968 0.47347 0.11015 0.13713 CCSD/TZ+∆MT 9.99188 -0.44062 5.38377 0.05312 -0.02598 5.87580 -0.35130 0.08854 0.00232 0.60231 6.01858 0.24626 5.86810 -0.25595 -0.04355 -0.48525 0.10430 0.13304

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Table 3: The Harmonic and Anharmonic Vibrational Frequencies (in cm−1 ) and Rotational Constants (in MHz) of c-C3 H2 for Various Levels of Theory. Exp.a 35092.6 32212.9 16749.3 – – – – – – – – – – – – 1278.8 1061.5 – 886.4 – 787.4

Ref. 1 Ref. 2 Best CcCRb QZfit 5QZfit A0 34436 35000 35157.8 35158.1 35154.7 31939 32189 32242.4 32243.8 32244.5 B0 C0 16520 16717 16767.8 16768.0 16767.2 ω1 (a1 ) C−H Symm. Stretch 3286.5 3290.7 3290.9 3290.1 3289.5 ω2 (b2 ) C−H Antisymm. Stretch 3247.8 3256.5 3258.4 3255.7 3255.6 ω3 (a1 ) C−C−C Bend/C=C Stretch 1616.2 1629.9 1631.6 1628.0 1629.8 1289.3 1312.4 1314.2 1310.7 1312.8 ω4 (a1 ) C−C Symm. Stretch ω5 (b2 ) C−H Antisymm. Stretch 1080.3 1093.1 1097.3 1093.0 1095.4 990.0 995.7 997.5 994.2 995.8 ω6 (a2 ) C3 OPB Twist ω7 (a1 ) H−C−C Symm. Bend 901.0 911.4 914.8 911.8 913.6 899.4 904.4 904.3 904.0 904.1 ω8 (b2 ) H−C−C Antisymm. Bend ω9 (b1 ) Wing Flap 783.4 789.4 788.8 787.8 788.0 ν1 (a1 ) C−H Symm. Stretch 3131.6 3145.0 3142.0 3144.2 3139.2 ν2 (b2 ) C−H Antisymm. Stretch 3112.9 3123.5 3119.7 3122.1 3118.0 ν3 (a1 ) C−C−C Bend/C=C Stretch 1586.0 1600.2 1599.2 1587.4 1596.0 ν4 (a1 ) C−C Symm. Stretch 1265.3 1279.4 1277.5 1278.7 1276.0 ν5 (b2 ) C−H Antisymm. Stretch 1050.2 1063.0 1064.6 1063.6 1062.5 968.6 976.3 952.1 992.3 935.0 ν6 (a2 ) C3 OPB Twist ν7 (a1 ) H−C−C Symm. Bend 879.6 889.2 887.2 892.8 882.6 ν8 (b2 ) H−C−C Antisymm. Bend 877.8 882.2 873.9 881.3 875.7 ν9 (b1 ) Wing Flap 768.5 776.0 753.2 794.9 739.1 MP2-F12 CCSD(T)-F12 CCSD/QZ DZb TZb QZ TZ QZ +∆MT A0 MHz 35155.9 35296.0 34850.4 34849.8 35098.7 35193.0 35155.9 32027.7 32043.9 32044.1 31929.1 32281.2 B0 MHz C0 MHz 16684.0 16740.9 16643.7 16643.6 16669.6 16787.1 ω1 (a1 ) C−H Symm. Stretch 3314.9 3314.2 3279.9 3279.9 3283.0 3312.3 ω2 (b2 ) C−H Antisymm. Stretch 3280.3 3280.4 3245.7 3245.7 3248.6 3271.9 ω3 (a1 ) C−C−C Bend/C=C Stretch 1622.0 1624.3 1622.1 1622.1 1623.5 1658.8 ω4 (a1 ) C−C Symm. Stretch 1323.8 1327.5 1305.7 1305.7 1304.1 1330.8 ω5 (b2 ) C−C Antisymm. Stretch 1095.5 1100.6 1088.4 1088.4 1089.5 1112.3 ω6 (a2 ) C3 OPB Twist 996.0 1003.7 994.6 994.6 977.6 1008.3 ω7 (a1 ) H−C−C Symm. Bend 911.5 916.1 907.4 907.4 904.4 928.9 ω8 (b2 ) H−C−C Antisymm. Bend 903.5 906.0 901.2 901.2 893.9 919.4 ω9 (b1 ) Wing Flap 797.1 803.3 787.2 787.2 769.4 802.7 ν1 (a1 ) C−H Symm. Stretch 3198.6 3194.4 3132.8 3129.0 3168.3 3044.7 ν2 (b2 ) C−H Antisymm. Stretch 3151.0 3148.2 3112.7 3110.9 3115.7 3007.5 ν3 (a1 ) C−C−C Bend/C=C Stretch 1594.3 1590.5 1592.3 1589.7 1567.5 1607.9 ν4 (a1 ) C−C Symm. Stretch 1288.9 1287.3 1271.9 1271.7 1258.2 1292.7 ν5 (b2 ) C−C Antisymm. Stretch 1066.8 1069.0 1058.8 1057.4 1027.6 1067.7 ν6 (a2 ) C3 OPB Twist 974.8 922.6 968.8 946.6 698.3 994.4 ν7 (a1 ) H−C−C Symm. Bend 889.1 878.3 884.9 879.8 841.4 911.8 ν8 (b2 ) H−C−C Antisymm. Bend 878.9 872.3 879.0 879.3 900.2 913.6 ν9 (b1 ) Wing Flap 782.4 731.2 772.0 746.2 483.1 790.5 a Vibrational frequencies are Ar-matrix data from Ref. 20 while the rotational constants are gas-phase from Ref. 11. b Previously reported in Ref. 3.

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Trial 8 35166.5 32229.4 16765.5 3290.9 3258.4 1631.6 1314.2 1097.3 1022.5 914.8 904.3 823.9 3141.8 3119.5 1599.7 1277.9 1065.1 973.8 888.3 875.2 784.4 Best 35156.6 32243.1 16767.7 3290.9 3258.4 1631.6 1314.2 1097.3 995.8 914.8 904.3 787.8 3142.6 3120.8 1600.9 1278.8 1065.1 970.2 888.6 878.6 772.8


well described at 968.6 cm−1 but will have no intensity for experiment to be observed due to symmetry. Unfortunately, the CcCR results 3 fall short of these marks at 753.2 cm−1 and 952.1 cm−1 even though the other modes, especially those involving the heavy atoms, coincide nicely between the CcCR and LHD frequencies. While the frequency of the a2 mode is not as bad as that from the b1 mode and will be spectroscopically dark, highly-accurate thermochemical computations require proper descriptions of all frequencies. The LHD results are based upon a CCSD(T)/aug-cc-pVQZ QFF corrected for core correlation at the zeroth-order (geometry) and the second-order (quadratic FCs), and the CcCR approach has the one-particle CBS extrapolation giving the latter a more complete physical description. As a result, the CcCR OPBs have a decrease in frequency compared to what they likely should be, but at least they are not imaginary. As mentioned previously, 3 the harmonic frequencies do not exhibit such large discrepancies between the LHD and CcCR ω6 and ω9 values. The ω9 values, in particular, are within 1.0 cm−1 of each other. However, there appears to be an overcorrection in the anharmonic terms. Analysis with the QZfit and 5Zfit QFFs reveals similar correspondence with the ω6 and ω9 harmonic frequencies, but ν6 and ν9 are clearly in error. The QZfit QFF actually has a minimum within one displacement of the totally symmetric coordinates but, from Table 3, produces positive anharmonicities for these OPB modes while 5Zfit overcorrects the anharmonicity. This give and take does not balance nicely in the CcCR QFF indicating that clearly, the basis set affects the energies as the ∆i terms in Eq. 1 are increased beyond two displacement terms. The CCSD level of theory, instead of CCSD(T), from the CcCR reference geometry for an aug-cc-pVQZ QFF also including ∆MT terms (CCSD/QZ+MT in the bottom half of Table 3) reduces the positive anharmoncity for ν9 but not ν6 . Clearly, the correlation and the basis set are at play in these erroneous descriptions of the OPBs. Employing the host of scaling factors (with results given in Table 2 of the SI) produces varying results, but the Trial 8 scaling generates terms in excellent agreement with the Ar-

14


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matrix and LHD benchmarks. Such a scaling should perform this way since the answer is known, but the fact that the scaling term in Eq. 12 has to be doubled for no apparent reason is somewhat concerning. Increasing the scaling factor as a product of the number of FCs (Trials 6 and 7) overcorrects. One hope was that this aribtrary scaling factor could be utilized in all further OPB FCs for hydrocarbons, but, again, this would have to be fully benchmarked. Another workaround for this problem in addition to such scaling or the equally distateful and seemingly acroamatic clipping of the basis set is to use explicitly correlated methods. Previous work 3 showed that the MP2-F12/DZ heavy atom frequencies given in Table 3 are moderately well described with this method, and the MP2-F12/QZ describes the hydride stretches better but with similar behavior in the OPB modes. However, such behavior may simply be coincidence. Moving from MP2-F12 to CCSD(T)-F12 here allows the improper OPB behavior to reemerge. Most notably, the CCSD(T)-F12/QZ results in Table 3 have very large discrepancies for both ν6 and ν9 with errors to the red on the order of multiple hundreds of cm−1 relative to all other methods as well as the benchmarked LHD and Armatrix results. While CCSD(T)-F12/TZ is slightly better, the OPB error is still present for these explicitly correlated methods in spite of the promise that MP2-F12 had given. Although CCSD(T)-F12 performed well for C2 H2 in Ref. 63, explicit correlation, except potentially in the MP2-F12/DZ and QZ Pauling points, 75 is not a clear solution.

3.2

Exploring the OPB

The hypothesis of this emerging set of studies beginning with its explicit statement in Ref. 3 is: “the very act of adding correlation is the culprit from a combined one- and n-particle basis set effect,” making this issue a combined basis set and method effect. By taking away the π cloud in saturated c-C3 H4 and comparing to c-C3 H2 + H2 , the aromatic ring becomes stabilized more quickly, 3 plotted in Figure 2 as both the S8 symmetric OPB (b1 ) and S9 antisymmetric OPB (a2 ) coordinates are scanned as defined in Figure 3. The difference in 15



the “product” (c-C3 H4 ) and “reactants” (c-C3 H2 and H2 ) yields the value plotted in Figure 2. The point at 0.0 kcal/mol is where the two are equal; any value above this point is where the unsaturated hydrocarbon is more favored. Figure 2: The S8 symmetric (red) and S9 antisymmetric (blue) OPB scans of c-C3 H2 (with H2 ) versus c-C3 H4 .

Scanning the aromaticity as computed from the MCI with CCSD/aug-cc-pVTZ wavefunctions in both c-C3 H2 and c-C3 H4 over the same range results in little change for either molecule (see Fig. 4). Note that in Fig. 4 the y-axis description for c-C3 H4 is on the left and c-C3 H2 is on the right. The c-C3 H4 molecule has an MCI value of 0.066 at equilibrium, and it grows, albeit very slightly to 0.070 at 90◦ along the motion of S8 . S9 has almost no change. Cycloprenylidene, however, has a much greater MCI value of 0.327 at equilibrium, 16


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Figure 3: Depictions of S8 (a) and S9 (b).

a)

C

C

C

C

C

C

b)

Figure 4: The MCI values for the S8 -type symmetric (red) and S9 -type antisymmetric (blue) OPB scans of c-C3 H4 (solid line) and c-C3 H2 (dotted line).

17



and this value decreases somewhat as S8 and S9 twist to 90.0◦ . The former reduces to 0.314 and the latter to 0.307. These represent changes of less than 6% indicating that while the aromaticity is being decreased, it is not being greatly affected in the OPB motion of c-C3 H2 . Neither the energy of the molecular orbitals involved in the largest tai nor the T1 diagnostic, which is a comfortable 0.012, 76 of c-C3 H2 are also affected to any meaningful percentage in the scans. Figure 5: The S8 -type symmetric (red) and S9 -type antisymmetric (blue) OPB scans of C2 H2 (with H2 ) versus C2 H4 .

The same energy scan is also applied to acetylene and H2 versus ethylene with the ethylene carbon atoms coming out of the plane of the molecule. This scan highlights that anytime π electrons are involved in bonding, this phenomenon is present in addition to the evidence 18


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Figure 6: The S8 -type symmetric (red) and S9 -type antisymmetric (blue) OPB scans of C2 H4 (with H2 ) versus C2 H6 .

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from previous computations. 4–6 The ethylene is favored at the minimum. However, a swap to favoring of acetylene (+ H2 ) shown in blue in Fig. 5 occurs at 48◦ in the antisymmetric bend (equivalent to S9 in Fig. 3b), but the symmetric bend of the S8 type always favors ethylene over acetylene plus H2 . The swapping points for ethylene with H2 versus ethane are shown in Fig. 6 and take place at 21◦ in the symmetric bend and 59◦ for the antisymmetric, S9 /a2 -type bend. A scan of the in-plane C2 rotation with respect to the four hydrogen atoms in ethylene versus acetylene with H2 yields no change with the ethylene always lower than the acetylene. The opposite behavior is present for fluorine replacement of the hydrogens as produced in difluoroacetylene. 77 The 2p fluorine orbitals already are interacting with the π cloud leading to exaggerated electron repulsion. This will be left for future study.

3.3

Spectroscopic Data for c-C3 H2

Regardless of these problems, actual data, especially gas-phase vibrational frequencies, are still needed for cyclopropenylidene. Fortunately for the present analysis, the gas-phase experimental rotational constants 11 for c-C3 H2 are in good agreement with the Rα vibrationallyaveraged rotational constants for each QFF tested. Most are actually in excellent agreement. The standard CcCR QFF, for instance, produces an A value of 35157.8 MHz while experiment is 35092.6 MHz, a difference of 65.2 MHz. The B and C constants are within 29.5 MHz and 17.5 MHz, respectively. The OPBs have no affect on these values since they are not totally symmetric which indicates that the other vibrational modes predicted by CcCR, especially the a1 modes but likely also the b2 modes, should be quite close to the actual values as they are to the LHD results and those available from Ar-matrix experiments. As such, the “Best” results produced for the vibrational frequencies of c-C3 H2 are a combination of the CcCR and MP2-F12/QZ. These “Best” values reported in Table 3 are the result of taking the FCs from the CcCR QFF except those that possess at least one S8 or S9 term. Those are chosen from MP2-F12/QZ. While MP2-F12/DZ actually correlates better 20


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to the LHD and Ar-matrix benchmarks, the larger basis sets are typically less susceptible to OPB error in the first place. In any case, the brightest frequency from double harmonic MP2/6-31+G∗ intensities is the ν4 C−C symmetric stretch, and it will occur in the 1279 cm−1 range as predicted by our “Best” results, the CcCR value, and that from LHD as well as the Ar-matrix experiments. Additionally, the ν9 “Wing Flap” is the second-brightest transition at half the intensity of ν4 and will likely be observable whether in the laboratory or in space with JWST or SOFIA with the EXES instrument in the region of 773 cm−1 to 776 cm−1 in the gas phase. Further spectroscopic data for the “Best” QFF VPT2 results are given in the SI for the quartic and sextic distortion constants as well as the vibrationally-excited rotational constants for standard c-C3 H2 and for the singly- and doubly-deuterated species. However likely trustworthy, this hybrid QFF approach may only be applicable for c-C3 H2 and related systems. The underlying problems of the erroneous OPB behavior, even if it only appears in the anharmonic vibrational frequencies, must be addressed before further insights can be provided from quantum chemistry for π-bonded hydrocarbons.

4

Conclusions

The “Best” predictions for the fundamental vibrational frequencies of c-C3 H2 are from a combination of the CcCR QFF and the MP2-F12/QZ FCs for S8 and S9 . These are in excellent agreement with previous benchmarks and available Ar-matrix data. Such an approach yields trustworthy predictions for the vibrational frequencies and the unknown spectroscopic constants, notably the vibrationally-excited rotational constants for c-C3 H2 . This hybrid QFF should be employed in the very least for future studies of hydrocarbons where OPBs are treated beyond the harmonic approximation until more complete basis set treatments of the carbon atom can be constructed. Additionally, the problems with OPBs appear not to affect the rotational constants in any way. Unfortunately, most large, aromatic hydrocarbons

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are largely comprised of bonded, sp2 -hybridized carbon atoms. While MP2-F12 is promising for addressing this method and basis set issue, expclicit correlation is not a savior for this issue. CCSD(T)-F12 failed to correct these issues as effectively as MP2-F12 does. As a result, more work within the quantum chemistry community is required to develop basis sets or even methods that can properly represent the orbital space contained once carbon atoms engage in multiple bonds. Such are a must before pure ab initio data can be fully trusted for OPB mode frequencies for pure hydrocarbons with π electrons. Such a formulation is applicable across all areas of chemisty where quantum chemistry can be brought to bear since carbon is the atom upon which most “interesting” chemisty is based. Only when the OPBs can be treated properly can computational predictions for spectroscopy, combustion, and environmental science be routinely and robustly trusted.

Acknowledgement RCF and JPL acknowledge funding from NASA grant NNX17AH15G that supported this work. RCF and CMM a grateful for funding from the Georgia Space Grant Consortium (NASA grant NNX15P85H, subaward RF964-G5) as well the Georgia Southern University Office of the Vice President for Research. Additionally, TJL is supported by the National Aeronautics and Space Administration through the NASA Astrobiology Institute under Cooperative Agreement Notice NNH13ZDA017C issued through the Science Mission Directorate as well as support from the NASA 16-PDART16 2-0080 grant. The authors also acknowledge Dr. Julia Rice of the IBM Almaden Research Lab for useful insights into this work. Figure 1 was constructed with the WebMO graphical user interface, 78 and Figure 3 was constructed with the CheMVP program developed at the Center for Computational and Quantum Chemistry at the University of Georgia. EM acknowledges financial support from the Spanish MINECO/FEDER Projects CTQ2014-52525-P and EUIN2017-88605, the

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Basque Country Consolidated Group Project No. IT588-13, as well as the computational resources and technical and human support provided by the DIPC and the SGI/IZO-SGIker UPV/EHU. Finally, the authors would like to thank the organizers of the 2017 International Conference on Chemical Bonding (specifically Lai-Sheng Wang of Brown University and Alexander I. Boldyrev of Utah State University) for invitations to attend this meeting where TJL and RCF were able to meet and discuss these things with EM.

Supplemental Information This document contains SI material including tables for the CcCR cubic and quartic force constants as well as the vibrational frequencies and rotational constants for c-C3 H2 determined from each of the scaled, trial force constants.

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