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Quartic Force Field Rovibrational Analysis of Protonated Acetylene, C2H+3 , and Its Isotopologues Ryan C. Fortenberry,*,†,∥ Xinchuan Huang,‡ T. Daniel Crawford,¶ and Timothy J. Lee*,∥ †

Department of Chemistry, Georgia Southern University, Statesboro, Georgia 30460, United States SETI Institute, 189 Bernardo Avenue, Suite 100, Mountain View, California 94043, United States ¶ Department of Chemistry, Virginia Tech, Blacksburg, Virginia 24061, United States ∥ NASA Ames Research Center, Moffett Field, California 94035-1000, United States ‡

S Supporting Information *

ABSTRACT: Protonated acetylene, C2H+3 , is among the simplest carbocations. Comprehensive experimental or highly accurate computational spectroscopic data is lacking for this system due to its inherent complexities. Utilizing state-of-the-art quartic force fields (QFFs), the spectroscopic constants and fundamental vibrational frequencies are provided in this work for the nonclassical, bridged, cyclic global minimum. The rotational constants match experiment to better than 0.1%, and the computed ν2 antisymmetric HCCH stretch is less than 3.0 cm−1 different from experiment. Hence, the rovibrational spectroscopic data provided herein for cC2H+3 and its deuterated isotopologues enrich the chemical understanding of this system. Unfortunately, the same rovibrational spectroscopic data is not as trustworthy for the classical, linear form of protonated acetylene due to the shallow well in which it resides on the potential energy surface. However, spectroscopic data are provided for this isomer in the Supporting Information to enhance future studies.

■

INTRODUCTION The complex nature of chemistry can often be exemplified in simple chemical systems. Protonated acetylene (C2H+3 ) is a wonderful instance of such a case. The “vinyl cation”, as it is sometimes referred, was believed for some time to exist in a form very similar to its namesake, the vinyl radical. However, experiments on this molecule, one of the simplest carbocations,1 did not model the assumed “Y”-shaped structure (l-C2H+3 ), which has since been deemed the “classical” isomer. The reactive nature but relatively few electrons in C2H+3 made it a perfect case study for quantum chemical computations from around 30 years ago. Lee and Schaefer2 were among the first to question the linear structural assumption with quantum chemical evidence pointing to a bridged, “non-classical” structure (c-C2H+3 ) lying lower in energy on the potential energy surface (PES). Lindh and coworkers3 corroborated this finding, and it was Curtiss and Pople4 who are often credited with conclusively assigning the lowestenergy isomer of C2H+3 to the bridged or cyclic form. With this and, later, more refined5 theoretical data in hand, the nonclassical isomer of C2H+3 was quickly verified experimentally through various methods6−8 as the lowest-energy structure. To muddy the waters further, Coloumb explosion experiments appear to indicate that a third isomer, where the third hydrogen is loosely bound to the acetylene π cloud, may be present on the PES and may even be lower in energy than the cyclic isomer.7,9 However, this has not been computationally verified, and no other type of experiments have shown such behavior. It is now well-established that C2H+3 is most stable in the bridged, cyclic, nonclassical geometry. © 2014 American Chemical Society

In and of itself, such a story showcases the unexpected, but physically meaningful outcomes that quantum mechanics imparts upon chemistry. Regardless, it is clear that protonated acetylene is not a simple system to examine. Protonated acetylene is significant in organic chemistry as its nature as a carbocation indicates,1 and it has been surmised to be of significance in the hydrocarbon chemistry of the interstellar medium.10 The related cyclopropanylidene (c-C3H2) interstellar molecule11 is believed to be a necessary precursor for the creation of polycyclic aromatic hydrocarbons in space,12 and it is reasonable to assume that c-C2H+3 may be similarly involved in this process. Additionally, larger, related organic complexes based on this protonated structure have been observed in gasphase experiments13 and are certainly relevant to the synthetic organic laboratory and even astrochemical observations. Hence, fundamental understanding of the energetics and rovibrational structure of protonated acetylene is necessary to further characterize this molecule in such environments. Modern quantum chemical computations14 place the classical geometry at nearly 3.7 kcal mol−1 higher in energy than the nonclassical structure with a barrier to the transition state of only 0.07 kcal mol−1 above the classical geometry. This corroborates a previous study using a global C2H+3 PES.15 The latter work generated a PES defined with coupled cluster theory at the singles, doubles, and perturbative triples [CCSD(T)] level16 Received: June 28, 2014 Revised: July 29, 2014 Published: July 30, 2014 7034

dx.doi.org/10.1021/jp506441g | J. Phys. Chem. A 2014, 118, 7034−7043

The Journal of Physical Chemistry A

Article

using Dunning’s aug-cc-pVTZ basis set.17,18 As such, anharmonic vibrational frequencies are produced that promise to be fairly accurate. In the time since, more accurate methods for the computation of vibrational as well as rotational (and rovibrational) spectroscopic data have been developed19−21 with accuracies as compared to experiment for small, closed-shell molecules as good as 1 cm−1 for vibrational frequencies22−24 and better than 0.1% for rotational constants.25−27 These recent methods rely on a quartic force field (QFF; a fourth-order Taylor series expansion of the nuclear Hamiltonian) V=

1 2

∑ FijΔiΔj +

+

1 24

ij

1 6

∑ FikjΔiΔjΔk ijk

∑ FikjlΔiΔjΔk Δl ijkl

(1)

where Fij... are the force constants and Δi are displacements, to describe the nuclear potential. They employ composite schemes that are more complete at each point accounting for the oneparticle basis set limit (C), core correlation (cC), scalar relativity (R), and, in a few cases, higher-order electron correlation (E). The full CcCRE QFF, as it would be called for inclusion of each of these terms,25 is not trivial to develop and compute, but it is not as cumbersome or costly as a global or semiglobal PES. However, because a full description of the PES is not included, QFFs have their limitations in application. Even though a full PES for C2H+3 has produced anharmonic vibrational frequencies, a full set of rovibrational spectroscopic data has not been provided to the community for C2H+3 and certainly not for its five subsequent deuterated isotopologues. Additionally, current state-of-the-art QFF methods are expected to be more accurate than previous work. Hence, the present study provides such spectroscopic data using the energetically descriptive QFF approach to define the PES in order to increase the interplay between theory and experiment such that understanding of this small, complicated, and therefore interesting cation can be enhanced.

Figure 2. CcCR equilibrium geometry of l-C2H+3 .

required to describe c-C2H+3 , while 1613 points are required for the classical isomer. In either case, the displacements necessary to describe the Taylor series expansion are defined from previous establishment as 0.005 Å for bond lengths and 0.005 radians for bond angles of all types.19 The nonclassical isomer is described with the following symmetry coordinate system with the atom numbering described in Figure 1 S1(a1) = C1 − C2

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COMPUTATIONAL DETAILS The construction of the QFF for either isomer of C2H+3 first begins with a restricted Hartree−Fock28 CCSD(T)/aug-cc-

(2)

S2(a1) =

1 {(C1 − H1) + (C2 − H1)} 2

(3)

S3(a1) =

1 {(C1 − H 2) + (C2 − H3)} 2

(4)

S4(a1) =

1 {∠(H 2 − C1 − H1) + ∠(H3 − C2 − H1)} 2 (5)

S5(b2) =

1 {(C1 − H1) − (C2 − H1)} 2

(6)

S6(b2) =

1 {(C1 − H 2) − (C2 − H3)} 2

(7)

S7(b2) =

1 {∠(H 2 − C1 − H1) − ∠(H3 − C2 − H1)} 2

Figure 1. CcCR equilibrium geometry of c-C2H+3 .

(8)

pV5Z geometry optimization. This geometry is then modified for the inclusion of core orbitals where the Martin−Taylor (MT) core correlating basis set29 is employed. Structural differences between the CCSD(T)/MT optimized geometry with the core orbitals included in the energy computation and the CCSD(T)/ MT optimized geometry without the core orbitals are appended to the CCSD(T)/aug-cc-pV5Z optimized geometry. From this reference geometry, 1585 symmetry-redundant points are

S8(b1) =

1 {τ(H 2 − C1 − H1 − C2) 2 − τ(H3 − C2 − H1 − C1)}

S9(a 2) =

1 {τ(H 2 − C1 − H1 − C2) 2 + τ(H3 − C2 − H1 − C1)}

7035

(9)

(10)



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Table 1. Symmetry-Internal CcCR Quadratic, Cubic, and Quartic Force Constants (in mdyn/Ån·radm)a for c-C2H+3 F11 F21 F22 F31 F32 F33 F41 F42 F43 F44 F55 F65 F66 F75 F76 F77 F88 F99 F111 F211 F221 F222 F311 F321 F322 F331 F332 F333 F411 F421 F422 F431 F432 F433 F441 F442 F443 F444 F551 F552 F553 F554 F651 F652 F653 F654 a

15.595770 −0.607481 2.012896 −0.218524 0.005921 5.890181 −0.101962 0.082969 0.006175 0.552287 0.718872 −0.029637 5.886684 −0.003656 0.024285 0.393554 0.283944 0.125264 −89.7049 0.6912 1.2774 −7.5704 −0.0809 0.0148 0.0332 0.2143 0.0208 −23.7527 0.5729 −0.0149 −0.0735 0.0522 −0.0667 −0.0126 −0.3348 −0.0567 −0.2299 −0.1113 −3.2629 −7.3684 −0.2667 −1.1537 −0.1922 0.0688 0.1342 −0.0078

F661 F662 F663 F664 F751 F752 F753 F754 F761 F762 F763 F764 F771 F772 F773 F774 F881 F882 F883 F884 F985 F986 F987 F991 F992 F993 F994 F1111 F2111 F2211 F2221 F2222 F3111 F3211 F3221 F3222 F3311 F3321 F3322 F3331 F3332 F3333 F4111 F4211 F4221 F4222


0.2157 0.0397 −23.7492 −0.0271 1.6134 −0.1860 −0.0059 0.1412 0.0512 −0.0472 −0.0290 −0.1793 −0.6823 −0.0323 −0.1729 −0.0654 −0.4628 0.0171 −0.1001 −0.0679 −0.0107 −0.0451 −0.0485 −0.8494 0.0080 −0.0709 0.0267 429.37 0.64 −2.25 −2.73 25.70 −0.29 −0.05 0.02 −0.05 −1.32 0.03 −0.26 −0.98 0.07 85.34 −0.64 −0.24 −0.02 0.25

−0.04 −0.08 0.11 0.04 0.04 0.09 −0.37 0.19 −0.27 0.26 0.04 −0.27 0.00 0.10 0.20 −0.20 7.67 0.79 31.08 1.21 −0.05 −0.35 −0.79 0.02 0.47 −0.50 36.54 0.67 −0.14 −0.07 −0.25 −0.13 −0.30 −0.51 0.18 0.09 −0.03 0.84 −1.37 −0.03 −0.13 −0.94 0.05 85.35 −0.07 0.04


0.15 −0.18 −0.31 −0.34 84.71 −1.94 −0.30 −0.15 −0.57 0.04 0.04 −0.06 −0.30 −0.11 −0.01 −5.72 −0.21 0.17 −0.10 0.10 −0.18 −0.01 0.32 0.04 0.02 0.02 0.49 −0.06 −0.01 −0.40 0.17 −0.20 0.29 0.07 −0.18 0.04 0.14 0.13 0.02 1.75 0.05 −0.06 −0.83 0.05 0.14 0.06


0.11 −0.11 0.26 0.02 −0.14 0.26 −0.00 0.17 −0.19 −0.48 −0.02 −0.16 −0.03 −0.10 −0.15 −0.25 0.19 0.05 0.11 0.11 0.17 −0.02 −0.11 0.02 0.05 0.11 0.02 −0.10 −0.17 0.05 −0.10 0.12 0.07 −0.09 0.55 −0.02 0.12 −0.11 −0.00 −0.03 −0.02 −0.19 0.02 0.08 0.01 0.33

1 mdyn = 10−8 N; n and m are exponents corresponding to the number of units from the type of modes present in the specific force constant.

The classical isomer (visually depicted in Figure 2) requires a different set of symmetry coordinates defined as S1(a1) =

1 {(C2 − H 2) + (C2 − H3)} 2

(11)

S5(b2) =

1 {(C2 − H 2) − (C2 − H3)} 2

S6(b2) =

1 {∠(C1 − C2 − H 2) − ∠(C1 − C2 − H3)} 2

(15)

(16)

1 S2(a1) = {∠(C1 − C2 + H 2) + ∠(C1 − C2 − H3)} 2

S7(b2) = LIN1(H 2 − C2 − C1 − x ⃗)

(17)

(12)

S8(b1) = LIN1(H 2 − C2 − C1 − y ⃗ )

(18)

S3(a1) = (C1 − C2)

(13)

S9(b1) = OPB∠(C1 − C2 − H 2 − H3)

(19)

S4(a1) = (C1 − H1)

(14)

where the linear bending coordinates (LIN1) produced by the INTDER program30 have been defined19 and used previ7036



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Table 2. c-C2H+3 CcCR QFF Anharmonic Constant Matrix (in cm−1) HCHCHb

HCDCH

DCHCH

DCDCH

DCHCD

DCDCD

mode

1

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

−26.836 −110.214 −1.959 −11.940 −10.956 −13.877 −12.399 −13.992 −10.280 −26.836 −110.213 −11.938 −1.528a −11.388 −13.185 −12.402 −13.992 −8.229 −53.847 −6.721 −0.915 −7.611 −13.010 −13.433 −18.626 −4.835 −3.831 −53.846 −6.432 −7.591 −0.764a −17.104 −6.768 −18.626 −3.834 −5.030 −12.605 −49.968 −3.692 −17.571 −7.289 −8.848 −9.354 −14.428 −16.156 −12.607 −49.967 −17.334 −3.274 −6.664 −8.615 −9.366 −16.158 −13.833

2

3

4

5

6

7

−28.651 1.507 −6.712 −10.407 −11.546 −9.879 −7.531 −3.407

−35.521 −6.291 52.905a −4.804 −2.470 −7.112 110.827

−6.625 −6.270 −3.822a −1.236 −13.210 −16.063

−27.645 −27.801 −4.137 2.713 −145.588

−1.427 28.974 −0.773 −32.167

−3.617 −3.809 0.087

4.080 26.607

−13.239

−28.651 −6.732 0.221 −9.909 −11.278 −9.884 −7.531 −3.244

−6.624 −4.699 −1.081a −3.924a −1.275 −13.206 −11.977

−18.502 −7.799 −4.241 −1.558 −5.303 69.793

−5.755 −9.828 −1.076 6.519 −43.055

−3.619 −3.815 −1.007

4.075 10.982

−32.890

−22.936 −2.807 −21.729 −5.611 −6.885 −3.276 −12.184 −15.325

−35.514 −4.701 55.765a 17.990 −3.591 83.419 −5.032

−4.941 −7.542 −0.127 −3.325 −8.449 −3.173

−31.559 −31.627 −0.294 −103.836a −1.002

−10.460 25.184 −24.187 0.611

−2.626 1.550 1.163

−10.073 27.221

0.766

−22.932 −21.749 −1.007 −2.989 −9.873 −3.303 −15.319 −9.459

−4.934 −3.714 −0.781 0.973 −3.309 −3.197 −5.562

−18.507 −7.317 8.681a −2.554 −3.656 60.570

−7.039 −4.117 5.741 0.271 −37.509a

−9.822 8.189 1.759 −41.386

−2.631 1.264 0.669

0.812 12.355

−17.907

−15.193 −1.111 −14.151 −7.333 −6.366 −5.316 −3.694 −4.155

−35.504 −4.011 57.556a −3.469 −1.691 83.090 −5.945

−4.102 −7.054 2.903 1.876 −7.616 −3.701

−39.952 −8.193 −3.524 −111.006 −0.847

−1.819 28.362 −22.876 −0.744

−2.136 0.616 −2.448

−7.484 28.832

2.407

−15.194 −14.024 −0.519 −6.084 −6.302 −5.323 −4.154 −2.967

−3.996 −3.764 3.511a 2.383 1.916 −3.597 −0.658

−18.264 33.480a −2.597 −1.163 −4.491 81.721

−11.920 −12.135 −1.805 1.798 −89.358

−1.694 14.750 −0.705 −19.414

−2.131 −2.438 0.223

2.405 13.990

−14.765

−1.302 15.302 −0.682 −6.909a

8

9

a

Constants affected by Fermi resonances. bFor c-C2H+3 , the HCHCH nomenclature follows a left-to-right structural interpretation of Figure 1 as H2C1H1C2H3. This is applicable to the other structures.

ously.19,21,31 The out-of-plane bending angle (OPB∠), also available in the INTDER program,30 in the creation of the QFF points is defined as the minimum angle between the vector

defined from atoms C1 and C2, in this case, and the plane created by atoms C2, H2, and H3; a similar coordinate system has been used by Huang and co-workers for l-C3H+3 .21 7037



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Table 3. CcCR c-C2H+3 Zero-Point (Rα Vibrationally Averaged) and Equilibrium Minimum Energy Structures (in Å and degrees) and Rotational Constants (in cm−1) experimenta HCHCH R(C−C) R(C1−H1) R(C2−H1) R(C1−H2) R(C2−H3) ∠H1−C1−H2 ∠H1−C2−H3 A0/e B0/e C0/e a

zero-point HCDCH

DCHCH

DCDCH

DCHCD

DCDCD

HCHCH

1.22862 1.30414

1.22814 1.29834

1.22664 1.29792

1.22158 1.27790

1.07448

1.22733 1.29865 1.29758 1.07431 1.07638 118.636° 118.663° 6.97050 0.96289 0.84201

1.22707 1.30374

1.07434

1.22778 1.30467 1.30318 1.07420 1.07635 118.701° 118.727° 13.33024 0.96195 0.89397

1.07609

1.07612

1.07608

118.754° 13.33966 1.14204 1.04645

equilibrium

HCHCH

13.35575 1.14216 1.04760

118.685° 6.99090 1.14376 0.97741

118.679° 13.29783 0.82260 0.77241

118.618° 6.94405 0.82351 0.73322

118.221° 13.80093 1.14763 1.05953

Gas discharge experimental data from ref 8.

interpretation of Figure 1 as H2C1H1C2H3. For instance, deuteration of the hydrogen in the nonclassical, bridged position is called HCDCH herein. Each isotopologue requires a different set of resonances to be included in the SPECTRO computations. HCHCH possesses a 2ν6 ≈ ν9+ν5 ≈ ν4 resonance polyad, a 2ν5 ≈ ν3 type-1 Fermi resonance, and a ν6 + ν3 ≈ ν1 type-2 Fermi resonance. HCDCH requires a 2ν6 ≈ 2ν5 ≈ ν3 resonance polyad, 2ν4 ≈ ν1 and 2ν9 ≈ ν6 type-1 Fermi resonances, and a ν9 + ν5 ≈ ν4 type-2 Fermi resonance. DCHCH possesses 2ν5 ≈ ν8 + ν4 ≈ ν3 and 2ν8 ≈ ν8 + ν6 ≈ ν5 resonance polyads and ν6 + ν3 ≈ ν1, ν6 + ν4 ≈ ν2, and ν8 + ν5 ≈ ν4 type-2 Fermi resonances. DCDCH has a 2ν6 ≈ ν6 + ν5 ≈ ν9 + ν5 ≈ ν4 polyad, 2ν4 ≈ ν1 and 2ν9 ≈ ν5 type-1 Fermi resonances, and ν6 + ν3 ≈ ν2 and ν6 + ν5 ≈ ν3 type-2 Fermi resonances. DCHCD necessitates the input of a 2ν5 ≈ ν6 + ν4 ≈ ν3 polyad and ν8 + ν5 ≈ ν4 and ν8 + ν6 ≈ ν5 type-2 Fermi resonances. The fully deuterated form has the fewest Fermi resonances of the c-C2H+3 protonated acetylene with a 2ν5 ≈ ν3 ≈ ν4 polyad and a ν9+ν4 ≈ ν2 type-2 Fermi resonance. All isotopologues require ν7/ν6 and ν9/ν8 A-type Coriolis couplings. Most of the c-C2H+3 VCI computations reported herein are comprised of a five-mode representation (5MR) of the CI expansion with 12 737 a1, 12 233 b2, and 9291 b1/a2 vibrational basis functions or 14 348 a′ and 10 292 a″ basis functions for the two Cs isotopologues. 4MR computations and those with fewer basis functions were also undertaken, and the reported values are deemed to be converged because the 4MR − 5MR difference is less than 1 cm−1, and an increase in 1000 or more vibrational basis functions does not alter the reported fundamental frequency again by more than 1 cm−1. The nature of DCHCD requires slightly fewer basis functions to exhibit convergence and fully describe the vibrational space, 7390 a1, 6958 b2, and 5146 b1/a2, in this case. The equilibrium and zero-point (Rα vibrationally averaged) geometries and primary rotational constants are given in Table 3. Due to the use of the Born−Oppenheimer approximation, the equilibrium geometry will remain unchanged for the various isotopologues. The standard isotopologue has A0, B0, and C0 values that are in agreement with experiment8 to better than 0.1%. Hence, the produced rotational constants for the deuterated isotopologues should be similarly (or more) accurate for comparison to laboratory experimentation or even interstellar observation. Deuteration reduces the zero-point rotational constants by expected amounts based upon where the standard hydrogen atom is replaced. Once the symmetry is broken by deuteration, as in the cases of DCHCH and DCDCH, the newly

A three-point complete basis set (CBS) limit extrapolation scheme32 is employed from CCSD(T)/aug-cc-pVTZ, aug-ccpVQZ, and aug-cc-pV5Z energies computed at each point. To the energy at each point, the differences in CCSD(T) energies for core correlation from the MT basis set and energy differences for scalar relativity33 from the cc-pVTZ-DK basis set are added to the CBS energy. This composite procedure for the determination of highly accurate energies19 has previously been defined as the CcCR QFF.25 Further inclusion of higher-order electron correlation (E) is not included in the composite energies. The T1 diagnostics are 0.012, less than 0.02 (ref 34), and recent experimental work has shown that the CcCR QFF employed for c-C3H+3 (ref 21) is accurate to better than 1.0 cm−1 for the asymmetric C−H stretch35 with the E terms excluded. The MOLPRO 2010.1 quantum chemistry package36 is utilized for these electronic structure calculations. A least-squares fitting with a sum of residuals squared on the order of 10−17 au2 produces the equilibrium geometry for both isomers. A refitting to this new minimum forces the gradients to zero and gives the other force constants given in Table 1 for cC2H+3 and in Table 1 of the Supporting Information for l-C2H+3 . Transformation of the force constants into Cartesian coordinates from symmetry coordinates via the INTDER program30 allows for the use of second-order perturbation theory (VPT2)37−39 for the computation of vibrationally averaged values, rotational constants, and anharmonic vibrational frequencies utilizing the SPECTRO40 program. Concurrently, transformation of the force constants into Morse-cosine coordinates41,42 gives the proper limiting behavior in the symmetry coordinates such that vibrational configuration interaction (VCI) theory from the MULTIMODE program43,44 may also be employed to produce anharmonic vibrational frequencies. The single-reference (SR) version of MULTIMODE is necessary for this work because the QFF is built around the respective local minimum geometry for each isomer. Comparison of the VPT2 and VCI anharmonic frequencies is one metric by which the reliability of the QFF can be tested.25

■

RESULTS AND DISCUSSION c-C2H+3 . The force constants for nonclassical protonated acetylene are given in Table 1 numbered in the same fashion as the symmetry coordinates defined previously. The anharmonic constant matrices for c-C2H+3 and each of the five different deuterated isotopologues produced from VPT2 are given in Table 2. For ease of discussion of the isotopologues, the HCHCH nomenclature follows a left-to-right structural 7038



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Table 4. c-C2H+3 CcCR QFF Vibration−Rotation Interaction Constants and S-Reduced Hamiltonian Terms vib−rot constants (MHz) mode

α

α

α

1 2 3 4 5 6 7 8 9

−182.3 −109.2 8234.7 −651.6 −2468.6 −93.5 459.5 9664.0 11839.5

212.9 183.3 52.1 174.9 −119.2 −93.6 9.2 41.1 −131.6

177.7 151.7 64.3 143.0 177.6 22.0 −101.1 −89.2 168.1

HCDCH

1 2 3 4 5 6 7 8 9

−96.0 −58.0 −330.6 2985.5 −5979.0 −137.0 366.4 7210.4 6364.8

212.9 183.3 174.9 35.2 −136.2 −91.3 9.3 41.1 −196.1

DCHCH

1 2 3 4 5 6 7 8 9

−155.6 −270.4 8325.4 −510.1 −1237.8 −5693.0 1702.5 15618.2 9732.6

151.7 206.9 40.5 122.6 −86.4 −75.0 12.6 −118.8 −6.8

DCDCH

1 2 3 4 5 6 7 8 9

−81.0 −150.2 −255.9 3063.0 −8323.2 −5681.7 6992.3 7456.2 7719.7

DCHCD

1 2 3 4 5 6 7 8 9

DCDCD

1 2 3 4 5 6 7 8 9

B

Watson S reduction

distortion constants

HCHCH

A

C

(MHz)

(Hz)

′ τaaaa τ′bbbb τ′cccc τ′aabb τ′aacc τ′bbcc

−245.245 −0.185 −0.142 −17.650 −1.332 −0.157

τ′aaaa τbbbb ′ τ′cccc τ′aabb τ′aacc τ′bbcc

−66.389 −0.185 −0.124 −15.430 −1.213 −0.136

129.6 173.2 54.5 102.4 138.8 40.4 −73.4 77.8 −113.8

τ′aaaa τ′bbbb τ′cccc τ′aabb τ′aacc τ′bbcc

−251.648 −0.130 −0.103 −11.814 −0.979 −0.113

151.5 207.0 122.5 28.2 −101.7 −90.7 12.6 −7.3 −145.4

114.6 156.0 87.1 46.1 83.6 60.6 −65.1 −100.5 149.2

τaaaa ′ τbbbb ′ τ′cccc τ′aabb τ′aacc τ′bbcc

−68.213 −0.129 −0.090 −10.530 −0.916 −0.099

−421.5 −63.8 8463.5 −489.2 5392.8 −214.1 454.0 11184.7 4208.4

181.6 140.4 32.1 90.8 −64.0 −77.6 −10.1 −94.3 −11.3

159.1 89.9 73.6 75.8 125.3 4.5 −89.3 51.7 −90.7

τ′aaaa τbbbb ′ τ′cccc τ′aabb τ′aacc τ′bbcc

−259.843 −0.091 −0.074 −8.214 −0.741 −0.080

−228.2 −38.8 −96.5 3017.0 −1913.0 −218.3 273.7 3962.3 6464.4

181.7 140.4 89.6 23.5 −94.1 −76.6 −10.1 −11.3 −110.0

142.5 108.8 69.6 35.8 100.5 7.2 −80.8 −81.1 105.3

τ′aaaa τ′bbbb τ′cccc τ′aabb τ′aacc τ′bbcc

−70.480 −0.091 −0.066 −7.451 −0.715 −0.072

154.09 132.15 122.14 52.46 117.08 31.99 −88.30 −76.81 258.89

7039

10−3Φaaa Φbbb Φccc Φaab Φabb Φaac Φbbc Φacc Φbcc Φabc Φaaa Φbbb Φccc Φaab Φabb Φaac Φbbc Φacc Φbcc Φabc Φaaa Φbbb Φccc Φaab Φabb Φaac Φbbc Φacc Φbcc Φabc Φaaa Φbbb Φccc Φaab Φabb Φaac Φbbc Φacc Φbcc Φabc Φaaa Φbbb Φccc Φaab Φabb Φaac Φbbc Φacc Φbcc Φabc Φaaa Φbbb Φccc Φaab Φabb Φaac Φbbc Φacc Φbcc

(MHz)

10.626 0.035 0.022 −7196.161 −5825.258 −0.266 3.897 1.288 0.013 −3.625 1745.304 0.035 0.020 −2648.824 110.489 −1815.661 −1.151 −0.302 0.051 −22.897 7268.265 0.024 0.015 −4792.065 34.459 −3816.267 −0.073 2.051 0.017 −2.043 1538.521 0.023 0.014 −1814.453 51.991 −1241.163 −0.390 0.012 0.027 −12.683 4104.964 0.015 0.010 −3399.700 17.978 −2629.348 −0.019 1.158 0.014 −1.015 1471.245 0.015 0.010 −1332.704 27.316 −883.341 −0.142 0.065 0.018

(Hz)

103DJ DJK DK 103d1 103d2 exp.a 103DJ exp.a DJK exp.a DK exp.a 103d1

36.986 4.686 56.589 −2.721 −1.967 37.65 6.0393 43.1 3.139

HJ HJK 10−3HKJ 10−3HK h1 h2 h3

−0.046 47.804 −13.104 23.683 −0.013 0.037 0.016

103DJ DJK DK 103d1 103d2

26.054 4.154 12.416 −3.811 −6.343


−0.259 66.219 −4.580 6.268 −0.076 0.144 0.080

103DJ DJK DK 103d1 103d2

26.954 3.150 59.734 −1.673 −0.938


−0.072 23.216 −8.650 15.895 −0.003 0.013 0.005

103DJ DJK DK 103d1 103d2

21.211 3.841 14.191 −2.439 −3.088


−0.086 31.551 −3.112 4.619 −0.025 0.052 0.027

103DJ DJK DK 103d1 103d2

19.607 2.203 62.738 −1.059 −0.480


−0.002 12.625 −6.052 10.145 −0.001 0.005 0.002

103DJ DJK DK 103d1 103d2

16.336 2.020 15.583 −1.558 −1.604


−0.031 17.156 −2.248 3.702 −0.009 0.022 0.010



Article

Table 4. continued vib−rot constants (MHz) mode

α

A

α

B

α

(MHz)

(Hz) Φabc

a

Watson S reduction

distortion constants C

(MHz)

(Hz)

−7.124

Gas discharge experimental data from ref 8.

substantial amount of mixing is present in the VPT2 eigenfunctions for the computations of these modes. The ν3 + ν6 combination band is in a type-2 Fermi resonance, with ν1 creating a blending of states at this frequency. Similarly, the substantial 95.4 cm−1 VPT2 − VCI difference for ν3 is caused by over 66% mixing in the polyad matrix with the anharmonic 2ν5 mode at 2235.72 cm−1. Because these combinations of states are treated with more complete coupling within the VCI wave function expansion, the VCI computations should be more reliable. The VPT2 − VCI difference in the ν5 frequencies is present in each of the other five deuterated isotopologues, as well. This is lessened for the inclusion of deuterium in the bridging position and is minimized for full deuteration, where the VPT2 frequency is 846.6 cm−1 and the VCI frequency is 860.0 cm−1. Such behavior is expected for full deuteration46 because the vibrational wave function has been more tightly bounded as compared to the standard isotopologue. Other examples of mixing in the VPT2 computations for the isotopologues are noted with footnote c in Table 5. Similarly, VCI frequencies marked in the same fashion are those where the primary mode contributor coefficient for the VCI expansion is less than 0.90 or at least one other contributing coefficient is greater than 0.50. A combination of coefficients such as these indicate an impure state that involves significant resonances between the normal coordinated-based VCI basis functions.47 As such, the large VPT2 − VCI differences are attributable in most cases to problems brought about by imprecisions either in the lack of VPT2 properly to treat coupling or by the difficulty that VCI has exhibited in effectively handling some torsional motions. Finally, the VCI wave function behaves very well for the fully deuterated isotopologue. Even the mixed VPT2 ν2 and ν4 modes of c-C2D3+ are within 5 cm−1 of the VCI modes. The ν3 CDC stretch is better described by VCI, again due to a more complete treatment of mode coupling within the CI expansion. l-C2H+3 . The force constants for the classical isomer are given in Table 1 in the Supporting Information. They follow the same ordering, once more, as the symmetry coordinates provided previously. The HCCHH description of this molecule follows the top-to-bottom flow of the atoms for H1C1C2H2H3, as laid out in Figure 2. For the SPECTRO computations, HCCHH uses 2ν4 ≈ ν9 + ν2 ≈ ν1 2ν7 ≈ ν8 + ν7 ≈ ν4 resonance polyads and ν5 + ν4 ≈ ν3, ν9 + ν3 ≈ ν2, ν9 + ν6 ≈ ν5, and ν9 + ν8 ≈ ν7 type-2 Fermi resonances. DCCHH requires 2ν5 ≈ ν5 + ν4 ≈ ν2 and 2ν8 ≈ ν9 + ν7 ≈ ν5 polyads, a 2ν7 ≈ ν4 type-1 Fermi resonance, and a ν9 + ν2 ≈ ν1 type-2 Fermi resonance. HCCDH possesses 2ν4 ≈ ν9 + ν2 ≈ ν1 and 2ν5 ≈ ν6 + ν5 ≈ ν3 Fermi resonance polyads, a 2ν7 ≈ ν4 type-1 Fermi resonance, and a ν9 + ν8 ≈ ν7 type-2 Fermi resonance. HCCDD requires a 2ν6 ≈ 2ν7 ≈ ν4 polyad, 2ν4 ≈ ν1 and 2ν5 ≈ ν3 type-1 Fermi resonances, and ν9 + ν3 ≈ ν2, ν9 + ν6 ≈ ν5, and ν9 + ν8 ≈ ν7 type-2 Fermi resonances. DCCDH requires 2ν6 ≈ 2ν7 ≈ ν4, ν5 + ν4 ≈ ν9 + ν3 ≈ ν2, and 2ν8 ≈ ν9 + ν6 ≈ ν5 polyads and a ν6 + ν3 ≈ ν1 type-2 Fermi resonance. VPT2 computations of DCCDD necessitate a 2ν5 ≈ 2ν6 ≈ 2ν7 ≈ ν4 polyad as well as ν9 + ν3 ≈ ν2 and ν9 + ν2 ≈ ν1 type-2 Fermi resonances. The 5MR VCI computations reported here all make

created C−D bond lengths are shortened, while the bond angles are increased slightly. The spectroscopic constants for each isotopologue of the nonclassical form of C2H+3 are reported in Table 4. Even though these values are not vibrationally averaged, comparison to experimental results8 for the quartic distortion constants is still quite good, especially for DJ. Experimentally, this value is 37.65 kHz, and the CcCR computations put DJ at 36.986 kHz, a difference of less than 0.7 kHz. Once more, similar accuracies are expected for the spectroscopic constants of the five subsequent deuterated isotopologues. The vibrational frequencies for c-C2H+3 and its deuterated isotopologues are listed in Table 5 from highest frequency to lowest. A previous, full PES produced by Sharma and coworkers15 has generated both harmonic and 5MR VCI anharmonic vibrational frequencies available for comparison. It can be seen in Table 5 that the CcCR harmonic frequencies parallel the previous15 RCCSD(T)/aug-cc-pVTZ full PES harmonic frequencies quite well for the most part. The notable exceptions are the ν8 and ν9 low-frequency antisymmetric HCCH in-plane and out-of-plane bending modes. This difference is rectified for the ν9 b2 in-plane mode when anharmonicity is accounted for because the CcCR frequency is 539.4 cm−1 and the previous full PES is 535.0 cm−1. The ν8 a2 out-of-plane modes still differ by 66.8 cm−1, with the CcCR frequency higher in energy at 628.9 cm−1. The CcCR VPT2 anharmonic frequency of 620.1 cm−1 indicates that this difference in frequency between the present and previous studies is probably attributable to the difference in nature of the two potential surfaces. For the other anharmonic fundamental frequencies, there are differences of as much as 24.5 cm−1 (ν5) between the present CcCR VCI frequencies and those from Sharma and coworkers.15 It can be interpreted that the present study gives the more accurate ν8 frequency, and this may be a systematic trend. The ν2 b2 antisymmetric HCCH stretch is a strong indicator of the improved accuracy for the CcCR QFF over the RCCSD(T)/aug-cc-pVTZ PES because the experimental frequency reported by Gabrys and co-workers8 is 3142.2 cm−1. This ν2 frequency is only 2.3 cm−1 lower than the CcCR value, while it is 22.6 cm−1 higher than the previous PES15 result. Additionally, the current QFF includes corrections for extrapolation to the one-particle basis set limit, core correlation, and scalar relativity, whereas the previous PES does not, and, as mentioned previously, similar approaches as those employed here have proven to be exceptionally accurate, especially for C− H stretching modes.23,24 A further metric of the present predictive performance for the CcCR QFF with c-C2H+3 is the VPT2 − VCI difference. Many fundamental frequencies differ between the two vibrational methods by around 5 cm−1 or less, with ν1, ν3, and ν5 being the notable exceptions. The large VPT2 − VCI ν5 difference is related to similar differences observed in the torsional motion of Cs tetra-atomics,25−27 where this difference can be as large as 15− 25 cm−1. It has been shown that the VPT2 torsional fundamental is more accurate in certain cases,45 and such is probably correct here, as well. The ν1 and ν3 VPT2 differences are different cases entirely, with the VCI frequency being more trustworthy. A 7040



Article

Table 5. CcCR VPT2 and VCI Fundamental Vibrational Frequencies (in cm−1) for c-C2H+3 and Its Deuterated Isotopologoues harmonic HCHCH

HCDCH

DCHCH

DCDCH

DCHCD

DCDCD

anharmonic

mode

description

this work

previousa

VPT2

VCI

previous VCIa

ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9 ν1 ν2 ν4 ν3 ν5 ν6 ν7 ν8 ν9 ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9 ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9 ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9 ν1 ν2 ν4 ν3 ν5 ν6 ν7 ν8 ν9

a1 HCCH symm. stretch b2 HCCH antisym. stretch a1 CHC stretch a1 CC stretch b2 CHC ipb a1 HCCH symm. ipb b1 HCCH symm. opb a2 HCCH antisym. opb b2 HCCH antisym. ipb a1 HCCH symm. stretch b2 HCCH antisym. stretch a1 CC stretch a1 CDC stretch b2 CDC ipb a1 HCCH symm. ipb b1 HCCH symm. opb a2 HCCH antisym. opb b2 HCCH antisym. ipb a1 DCCH symm. stretch b2 DCCH antisym. stretch a1 CHC stretch a1 CC stretch b2 CHC ipb a1 DCCH symm. ipb b1 DCCH symm. opb a2 DCCH antisym. opb b2 DCCH antisym. ipb a1 DCCH symm. stretch b2 DCCH antisym. stretch a1 CDC stretch a1 CC stretch b2 CDC ipb a1 DCCH symm. ipb b1 DCCH syms. opb a2 DCCH antisym. opb b2 DCCH antisym. ipb a1 DCCD symm. stretch b2 DCCD antisym. stretch a1 CHC stretch a1 CC stretch b2 CHC ipb a1 DCCD symm. ipb b1 DCCD symm. opb a2 DCCD antisym. opb b2 DCCD antisym. ipb a1 DCCD symm. stretch b2 DCCD antisym. stretch a1 CC stretch a1 CDC stretch b2 CDC ipb a1 DCCD symm. ipb b1 DCCD symm. opb a2 DCCD antisym. opb b2 DCCD antisym. ipb

3378.3 3278.7 2365.6 1951.1 1263.6 913.9 759.0 620.5 604.7 3378.0 3278.7 1951.1 1720.5 1037.1 905.3 759.0 620.5 549.4 3334.3 2564.9 2364.1 1822.3 1228.9 802.6 700.5 552.5 528.2 3334.3 2564.8 1822.2 1717.2 1004.4 762.5 700.5 528.2 523.1 2697.1 2408.6 2362.7 1728.3 1177.4 671.2 534.4 557.1 517.4 2697.1 2408.4 1728.4 1713.9 909.1 665.7 557.1 517.4 511.0

3355 3245 2358 1929 1258 928 775 594 534

3225.4c 3142.3 2416.8c 1913.3 1133.6 878.2 749.4 620.1 543.3 3232.8 3142.1 1913.9 1726.6 952.2c 816.7 743.9 615.1 486.8 3195.0 2482.0 2402.1c 1788.6 1097.8c 765.4 694.6 511.7 530.0 3193.4 2483.0 1791.9 1737.6 971.7c 721.6 689.2 524.6 474.6 2608.2 2332.4 2386.7c 1699.7 1053.5 657.9 495.9 557.1 519.7 2609.3 2341.7c 1700.0c 1753.7c 846.6 646.0 551.2 514.4 466.3

3239.4 3144.5 2321.4 1915.2 1158.7 865.5 754.8 628.9 539.4 3237.2 3144.7 1911.4 1721.6 999.6 823.3c 748.8 622.8 486.5 3194.5c 2483.8 2326.2c 1792.1 1128.2 764.5 701.3 506.8 536.5 3194.5 2485.2 1791.5 1710.6 986.4 721.3 695.2 530.3 472.1 2608.5 2334.4 2330.0c 1704.5 1035.8c 657.5 491.8 560.4 526.3 2614.2 2337.1 1695.1 1707.8 860.0 644.0 554.0 520.0 463.2

3219.2 3119.6 2308.4 1897.5 1134.2 859.3 767.8 562.1 535.0

experimentb 3142.2

a CCSD(T)/aug-cc-pVTZ PES MULTIMODE VCI computations from ref 15. bGas discharge experiment from ref 6. cImpurities in the eigenfunctions (VPT2) or VCI wave funtions describing each state. More discussion is given in the text.

use of 6801 a1, 6206 b2/b1, and 5427 a2 basis functions for the C2v isotopologues, while the Cs structures are computed with 14 348

a′ and 10 292 a″ basis functions. HCCDD, however, actually requires 4069 a1, 3331 b1, 2506 b1, and 2008 a2 basis functions 7041



Article

the issues causing a larger difference are rectifiable such that the predicted values indicate this QFF as the most accurate PES developed to date for protonated acetylene in its nonclassical form. Hence, the spectroscopic constants and fundamental vibrational frequencies provided in this work for the standard and five associated deuterated isotopologues of c-C2H+3 will remove another layer of complexity in this simple, yet tricky molecule that has proved to be very challenging. With the growth of astronomical telescopic power from the Stratospheric Observatory for Infrared Astronomy (SOFIA), the Atacama Large Millimeter/submillimeter Array (ALMA), and the upcoming James Webb Space Telescope (JWST), such accurate spectroscopic data are absolutely essential to resolve the wealth of spectral data generated by the newest generation of modern instruments.

due to convergence issues for larger basis sets in the VCI formulation. Convergence of the basis space is actually not robustly achieved for the VCI computations of l-C2H+3 in each case. This is largely due to the small barrier above the local minimum (the classical structure minimum), which is less than any of its possible harmonic vibrational frequencies.14 Even though the CcCR QFF equilibrium geometries put the energy difference at 3.94 kcal/mol (1376.6 cm−1), the classical isomer’s well is as shallow as 25 cm−1 but probably no higher than 150 cm−1.14,15 Hence, the SR formulation of MULTIMODE as well as the VPT2 treatment will behave well for the global minimum, cyclic isomer because the barrier is high enough to treat this isomer as an isolated molecule. VCI or VPT2 computations of the classical isomer cannot function in the same way because the barrier is not large enough for this minimum-energy structure to be considered a similar isolated system. The equilibrium and zero-point geometries for the six deuterated isotopologues are given in Table 2 of the Supporting Information. In Table 3 of the Supporting Information, correspondence between the CcCR QFF harmonic frequencies and those computed previously15 is within the same frequency spread as that observed in the nonclassical isomer. However, the ν7 mode of l-C2H+3 is in substantial disagreement between the present and earlier works. The culprit for this behavior is coming, once more, from the difference in the descriptions of the PESs, the more complete electronic structure method and QFF used here versus a less complete electronic structure approach but with a larger PES used previously. It is also clear from this table that the VPT2 − VCI agreement is good in some cases but often very poor for l-C2H+3 and its deuterated isotopologues. Hence, the CcCR QFF, which is local by construction, cannot treat this system that needs a larger, more global PES. The VCI computations may be more correct than their VPT2 counterparts because the transformation into Morse-cosine coordinates gives proper limiting behavior and proper behavior far away from the local minimum.41,42 For instance, the double well for ammonia has been produced from a QFF.42 However, the low transition barrier relative to the local minimum for this isomer precludes any strong theoretical assignment of vibrational frequencies using a QFF, and the moreaccurate vibrational frequencies of this isomer are probably those from ref 15, where a full six-dimensional PES was computed even though it is at a lower level of electronic structure theory.

■

ASSOCIATED CONTENT

S Supporting Information *

The tables related to the classical isomer of protonated acetylene are contained in the Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (R.C.F). *E-mail: [email protected] (T.J.L.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Georgia Southern University provided start-up funds, and the NASA Postdoctoral Program administered by Oak Ridge Associated Universities through a contract with NASA financially supported the work performed by R.C.F. T.D.C. was supported by National Science Foundation (NSF) Award CHE-1058420 and by NSF Multi-User Chemistry Research Instrumentation and Facility (CRIF:MU) Award CHE-0741927, which provided the employed computer hardware. T.J.L., R.C.F., and X.H. gratefully acknowledge funding from NASA Grant 12-APRA120107. X.H. was funded by NASA/SETI Institute Cooperative Agreement NNX12AG96A. Support from NASA’s Laboratory Astrophysics “Carbon in the Galaxy” Consortium Grant (NNH10ZDA001N) is acknowledged by T.J.L., R.C.F., and X.H. The figures were created using the CheMVP program developed at the University of Georgia’s Center for Computational Quantum Chemistry.

■

CONCLUSIONS Cyclic C2H+3 is confirmed here to be the lowest-energy isomer of protonated acetylene at 3.94 kcal mol−1 lower in energy than the classical structure from the minimum fitted CcCR QFF energies of both forms. As such, the MULTIMODE results should not require treatment beyond SR for the VCI fundamentals of the nonclassical isomer, and the VPT2 computations are based on the QFF isolated to the global well. Hence, the global minimum, nonclassical, bridged form of C2H+3 is well-described by this CcCR QFF procedure even though the use of this established method of QFFs encounters instabilities in the PES description of the classical form of protonated acetylene. The predicted vibrationally averaged rotational constants are in very close agreement (

Quartic Force Field Rovibrational Analysis of ... - ACS Publications

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