Molecular synthons for accurate structural determinations: the equilibrium geometry of 1-chloro-1-fluoroethene

Alberto Gambi *a, Andrea Pietropolli Charmet b, Paolo Stoppa b, Nicola Tasinato c, Giorgia Ceselin c and Vincenzo Barone c
aUniversità degli Studi di Udine, Dipartimento Politecnico di Ingegneria e Architettura, Via Cotonificio 108, I-33100 Udine, Italy. E-mail: alberto.gambi@uniud.it; Fax: +39 0432 558803; Tel: +39 0432 558856
bUniversità Ca’ Foscari Venezia, Dipartimento di Scienze Molecolari e Nanosistemi, Via Torino 155, I-30172 Mestre (VE), Italy
cScuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy

Received 31st July 2018 , Accepted 27th September 2018

First published on 27th September 2018


The equilibrium structure for 1-chloro-1-fluoroethene is reported. The structure has been obtained by a least-squares fit procedure using the available experimental ground-state rotational constants of eight isotopologues. Vibrational effects have been removed from the rotational constants using the vibration–rotation interaction constants derived from computed quadratic and cubic force fields obtained with the required quantum chemical calculations carried out by using both coupled cluster and density functional theory. The semi-experimental geometry obtained in this way has been also compared with the corresponding theoretical predictions obtained at the CCSD(T) level after extrapolation to the complete basis set limit and inclusion of core-valence corrections. These results allow completion of the molecular geometries of the isomers of chlorofluoroethene in addition to the cis and trans forms of 1-chloro-2-fluoroethene already published.


1 Introduction

The concerns due to the role of halogenated molecules as trace organic pollutants1–4 have motivated many of the theoretical and experimental studies on these compounds that have appeared in the literature. High-resolution spectroscopic techniques are able to detect (and quantify) these species in the atmosphere,5 but they require accurate spectroscopic data (obtained by investigations focused on line positions,6–10 intensities,11,12 and broadening coefficients13) which, in turn, must be obtained from the analysis of highly congested spectra, very often complicated by the effects of anharmonic and Coriolis interactions.14–17 Anyway, recent improvements in both the experimental techniques and the theoretical methods are nowadays able to efficiently support and guide the spectroscopic investigations. At present, high-level ab initio calculations18–20 can provide reliable anharmonic force fields required to take into account all the resonances and also yield the predictions in band positions and intensities, and therefore in recent years they have been successfully applied to the vibrational analysis of many halocarbons.21–29

For a comprehensive understanding of the physical–chemical properties of a molecule, the geometrical structure is mandatory; furthermore it provides fundamental reference data whose determination is the first step toward the challenge of accurate structural determinations for systems of increasing size.30,31 Several methods are available for experimental structure determination, however when accurate geometries are needed, rotational spectroscopy is usually the method of choice.32

However, for polyatomic molecules, the determination of equilibrium structures is not a simple task due to the large number of structural parameters that must be determined by the analysis of the rotational spectra of several isotopically substituted species. Another problem is the consideration of the vibrational effects.

Quantum chemical calculations are of great help also from this perspective because not only they can determine the theoretical equilibrium structure but they are also able to provide the spectroscopic parameters necessary to evaluate the vibrational corrections. Accurate quantum chemical geometries can usually be obtained by using highly correlated wavefunction methods possibly coupled to composite schemes accounting for missing effects such as core-valence correlations and complete basis set extrapolations.31,33–35 Hence, from both experimental and theoretical points of view, obtaining accurate equilibrium geometries becomes prohibitive but for very small molecules. In this respect, a notable improvement is represented by the so-called semi-experimental approach, first introduced by Pulay and co-workers.36 This integrated experimental–theoretical strategy relies on inverting experimental spectroscopic results to equilibrium structures by employing quantum chemical calculations in order to account for the vibrational (and potentially electronic) contributions. Over the last few years the semi-experimental approach has been exploited for several small to medium size molecular systems, leading to the compilation of a database storing accurate molecular structures for about sixty molecules containing up to fifteen atoms.37–39 While such a database provides an invaluable benchmark for the development of new computational methods rooted into quantum or classical mechanics,40–42 one of the challenges for accurate quantum chemical investigations of equilibrium structures is the extension toward systems of increasing size. Here again the availability of reference equilibrium geometries represents the cornerstone to accomplish this target. In an attempt to increase the size of systems amenable to quantum chemical equilibrium structure determination, two approaches, namely the templating molecular approach (TMA) and the linear regression approach (LRA), have been proposed to improve the geometrical parameters optimized by methods rooted into density functional theory (DFT).37,38

The microwave spectrum of 1-chloro-1-fluoroethene was first measured by Stone and Flygare43 and later in the millimeter wave region by Alonso et al.44 by means of two-dimensional Fourier transform spectroscopy. Recently, Leung et al.45 reported the microwave spectra of eight isotopic modifications of ClFC[double bond, length as m-dash]CH2 with the respective rotational constants together with other spectroscopic parameters.

Concerning the geometry, besides the determination of a Kraitchman substitution structure, an empirical structure derived from a fit of the ground state moments of inertia of the eight isotopologues45 was also obtained.

The aim of this work is to determine the accurate equilibrium structure of 1-chloro-1-fluoroethene. Two different techniques have been employed: ab initio geometry optimization at a high level of theory through the application of two basis set extrapolation schemes, and a semi-experimental structure calculation from the rotational constants of eight isotopologues45 corrected with vibrational corrections computed theoretically. These corrections have been obtained from both the ab initio and DFT cubic force fields for the normal modes of the different isotopologues of ClFC[double bond, length as m-dash]CH2.

In Section 2, the ab initio anharmonic force field, the semi-experimental structure and the complete basis set equilibrium geometry are given. Section 3 presents the results obtained and discusses the chemically relevant features of the structures of 1-chloro-1-fluoroethene. In addition, DFT optimized geometries are compared with the semi-experimental structure and it is shown how the LRA can be used to obtain an improved description of DFT geometrical parameters. Concluding, the different structures of the three chloro-fluoroethene isomers, namely ClFC[double bond, length as m-dash]CH2, trans-ClHC[double bond, length as m-dash]CHF and cis-ClHC[double bond, length as m-dash]CHF, will be presented and commented.

2 Computational details and methods

2.1 Harmonic and anharmonic force fields

For each isotopologue, the quadratic and cubic force constants in term of the mass-independent internal coordinates have been transformed into the normal coordinate representation.

The harmonic (quadratic) and the complete cubic anharmonic force fields have been evaluated at the coupled cluster level of theory with single and double excitations augmented by a perturbational estimate of the effects of connected triple excitations,46 CCSD(T). These calculations have been performed with a local version of the CFOUR program package.47

Several different correlation consistent Dunning basis sets have been employed in order to get the best frequency evaluation, as reported in the previous work.48 To be more precise, to account for the electronegative character of the F and Cl atoms, and for an adequate treatment of core and core-valence correlation effects on molecular geometry, the correlation-consistent polarized core-valence triple zeta basis set (cc-pCVTZ) has been employed for H, C, and Cl atoms while for the F atom the aug-cc-pCVTZ basis set has been used in order to improve the overestimated C–F stretching frequency.49–52 For conciseness, these basis sets will be referred to as AFCVTZ in the rest of the text. Spherical harmonics have been used throughout and all the electrons have been correlated excluding the 1s electrons of the chlorine atom.52

At the computed equilibrium geometry, the harmonic force constants have been obtained, in Cartesian coordinates, using analytic second derivatives of the energy.53 The corresponding cubic force field has been determined in a normal-coordinate representation with the use of a finite difference procedure54 involving displacements along reduced normal coordinates55 (step size ΔQ = 0.05 u1/2a0) and the calculation of analytic second derivatives at these displaced geometries.

The quadratic and cubic force constants have been initially obtained for the main isotopologue (H2C[double bond, length as m-dash]C35ClF) and then transformed to the mass-independent internal coordinate representation. The molecule of 1-chloro-1-fluoroethene belongs to the Cs symmetry point group. A chemically intuitive non-redundant set of internal coordinates can be chosen to have 9A′ internal coordinates, which lie in the molecular symmetry plane and correspond to the structural parameters, and 3A′′ out of plane internal coordinates that describe the vibrational modes outside this plane. These internal coordinates are defined in Table 1 where the labeling of the atoms is shown in Fig. 1. In the same table the molecular geometry calculated at the CCSD(T)/AFCVTZ level of theory is also reported.

Table 1 Definition of the internal coordinates used to get the mass-independent force field. In the last column the structural parameters – bond lengths in Å and bond angles in degrees – computed at the CCSD(T)/AFCVTZ level of theory are reported
Symmetry species Internal coordinates Description Geometry
A R 1 C2–F1 1.3344
R 2 C2[double bond, length as m-dash]C3 1.3252
R 3 C3–H4 1.0790
R 4 C2–Cl5 1.7140
R 5 C3–H6 1.0767
R 6 ∠C3[double bond, length as m-dash]C2–F1 122.27
R 7 ∠C3[double bond, length as m-dash]C2–Cl5 125.87
R 8 ∠C2[double bond, length as m-dash]C3–H6 120.12
R 9 ∠C2[double bond, length as m-dash]C3–H4 119.37
A′′ R 10 F1–(Cl5–C2[double bond, length as m-dash]C3)
R 11 H4–(H6–C3[double bond, length as m-dash]C2)
R 12 Cl5–C2[double bond, length as m-dash]C3–H6



image file: c8cp04888f-f1.tif
Fig. 1 Molecular structure of ClFC[double bond, length as m-dash]CH2 and numbering of its atoms.

Since the symmetry of the force constants should belong to the totally symmetric species A′, the force field will be composed as it follows: the quadratic force constants will be 51 detailed as 45AA′ and 6A′′A′′, whereas the cubic force constants are 219 detailed as 165AAA′ and 54A′′A′′A′. Employing the internal coordinates defined in Table 1, the quadratic and cubic force fields reported, respectively, in Tables 2 and 3, have been obtained.

Table 2 Quadratic force constants in internal coordinates, Fij, at the CCSD(T)/AFCVTZ level of theory: the units of the force constants are consistent with energy in aJ, bond lengths in Å and bond angles in rad
i j F ij i j F ij i j F ij
1 1 6.490 6 3 −0.040 8 7 0.001
2 1 0.443 6 4 −0.517 8 8 0.902
2 2 9.615 6 5 0.047 9 1 −0.028
3 1 −0.007 6 6 2.086 9 2 0.226
3 2 0.034 7 1 −0.546 9 3 0.037
3 3 5.756 7 2 0.292 9 4 0.064
4 1 0.567 7 3 0.041 9 5 −0.117
4 2 0.328 7 4 0.010 9 6 −0.039
4 3 −0.001 7 5 −0.033 9 7 0.069
4 4 4.123 7 6 1.098 9 8 0.416
5 1 −0.016 7 7 1.834 9 9 0.892
5 2 0.011 8 1 0.078 10 10 2.186
5 3 0.005 8 2 0.271 11 10 0.448
5 4 −0.008 8 3 −0.119 11 11 0.877
5 5 5.835 8 4 0.002 12 10 0.680
6 1 0.158 8 5 0.024 12 11 0.460
6 2 0.356 8 6 0.091 12 12 0.514


Table 3 CCSD(T)/AFCVTZ cubic force constants in internal coordinates, Fijk: the units of the force constants are consistent with energy in aJ, bond lengths in Å and bond angles in rad
i j k F ijk i j k F ijk i j k F ijk i j k F ijk i j k F ijk
1 1 1 −41.183 6 4 4 0.976 8 3 2 −0.015 9 5 3 0.223 11 10 3 0.059
2 1 1 −1.486 6 5 1 −0.076 8 3 3 −0.117 9 5 4 0.004 11 10 4 0.091
2 2 1 −1.366 6 5 2 −0.085 8 4 1 0.015 9 5 5 −0.122 11 10 5 −0.106
2 2 2 −55.212 6 5 3 0.010 8 4 2 −0.025 9 6 1 −0.007 11 10 6 −0.222
3 1 1 −0.026 6 5 4 −0.026 8 4 3 0.003 9 6 2 −0.128 11 10 7 0.056
3 2 1 0.095 6 5 5 0.051 8 4 4 −0.019 9 6 3 0.100 11 10 8 0.791
3 2 2 −0.003 6 6 1 −3.440 8 5 1 −0.036 9 6 4 0.017 11 10 9 0.322
3 3 1 0.055 6 6 2 −1.126 8 5 2 −0.237 9 6 5 0.020 11 11 1 0.192
3 3 2 0.175 6 6 3 0.075 8 5 3 0.224 9 6 6 −0.163 11 11 2 −1.450
3 3 3 −32.907 6 6 4 −1.673 8 5 4 0.036 9 7 1 0.002 11 11 3 0.333
4 1 1 −1.470 6 6 5 −0.124 8 5 5 −0.036 9 7 2 0.104 11 11 4 0.003
4 2 1 0.077 6 6 6 0.809 8 6 1 −0.106 9 7 3 −0.021 11 11 5 −0.324
4 2 2 −0.905 7 1 1 0.749 8 6 2 0.115 9 7 4 −0.059 11 11 6 −0.055
4 3 1 −0.021 7 2 1 0.112 8 6 3 −0.024 9 7 5 −0.021 11 11 7 −0.215
4 3 2 −0.037 7 2 2 −0.727 8 6 4 −0.004 9 7 6 −0.009 11 11 8 3.157
4 3 3 0.031 7 3 1 −0.022 8 6 5 −0.043 9 7 7 −0.002 11 11 9 2.821
4 4 1 −1.189 7 3 2 −0.081 8 6 6 −0.016 9 8 1 −0.001 12 10 1 −0.444
4 4 2 −1.013 7 3 3 0.043 8 7 1 0.013 9 8 2 −0.038 12 10 2 −0.769
4 4 3 −0.052 7 4 1 1.282 8 7 2 −0.186 9 8 3 −0.190 12 10 3 −0.062
4 4 4 −21.260 7 4 2 −0.693 8 7 3 0.013 9 8 4 0.002 12 10 4 0.239
5 1 1 −0.036 7 4 3 −0.070 8 7 4 −0.081 9 8 5 −0.193 12 10 5 −0.078
5 2 1 −0.016 7 4 4 0.108 8 7 5 0.108 9 8 6 0.015 12 10 6 −0.330
5 2 2 −0.007 7 5 1 −0.022 8 7 6 0.014 9 8 7 0.018 12 10 7 −0.037
5 3 1 0.003 7 5 2 0.145 8 7 7 −0.206 9 8 8 0.421 12 10 8 −0.086
5 3 2 −0.021 7 5 3 0.009 8 8 1 −0.134 9 9 1 −0.075 12 10 9 −0.052
5 3 3 0.055 7 5 4 0.025 8 8 2 −0.283 9 9 2 −0.231 12 11 1 −0.012
5 4 1 −0.026 7 5 5 0.023 8 8 3 −0.211 9 9 3 −0.449 12 11 2 −0.655
5 4 2 0.069 7 6 1 −1.691 8 8 4 −0.087 9 9 4 −0.112 12 11 3 0.126
5 4 3 0.002 7 6 2 0.137 8 8 5 −0.421 9 9 5 −0.210 12 11 4 −0.054
5 4 4 −0.001 7 6 3 −0.018 8 8 6 −0.173 9 9 6 −0.091 12 11 5 −0.035
5 5 1 0.038 7 6 4 −1.705 8 8 7 −0.124 9 9 7 −0.152 12 11 6 −0.042
5 5 2 0.187 7 6 5 −0.018 8 8 8 −0.264 9 9 8 0.416 12 11 7 −0.123
5 5 3 0.063 7 6 6 2.693 9 1 1 0.059 9 9 9 −0.256 12 11 8 0.725
5 5 4 0.050 7 7 1 −1.583 9 2 1 0.013 10 10 1 −4.493 12 11 9 0.394
5 5 5 −33.348 7 7 2 −0.802 9 2 2 −0.128 10 10 2 −2.276 12 12 1 −0.063
6 1 1 −0.569 7 7 3 −0.119 9 3 1 0.037 10 10 3 −0.132 12 12 2 −0.553
6 2 1 −1.055 7 7 4 −2.905 9 3 2 −0.268 10 10 4 −0.591 12 12 3 −0.053
6 2 2 −0.939 7 7 5 0.117 9 3 3 −0.036 10 10 5 −0.168 12 12 4 −0.014
6 3 1 0.029 7 7 6 2.663 9 4 1 0.004 10 10 6 3.111 12 12 5 −0.057
6 3 2 0.165 7 7 7 0.885 9 4 2 0.018 10 10 7 1.850 12 12 6 −0.099
6 3 3 0.006 8 1 1 −0.079 9 4 3 −0.026 10 10 8 −0.130 12 12 7 −0.072
6 4 1 1.321 8 2 1 −0.016 9 4 4 −0.051 10 10 9 −0.076 12 12 8 −0.043
6 4 2 0.154 8 2 2 −0.252 9 5 1 0.001 11 10 1 −0.229 12 12 9 −0.042
6 4 3 −0.028 8 3 1 0.010 9 5 2 −0.019 11 10 2 −1.136


Subsequently, for each isotopologue for which experimental ground-state rotational constants are available,45 the cubic force fields have been used to compute the spectroscopic parameters. Table 4 reports the spectroscopic parameters computed for six additional isotopologues of 1-chloro-1-fluoroethene, and to be more precise H2C[double bond, length as m-dash]13C35ClF, H2C[double bond, length as m-dash]13C37ClF, H213C[double bond, length as m-dash]C35ClF, H213C[double bond, length as m-dash]C37ClF, (E)-HDC[double bond, length as m-dash]C35ClF, and (Z)-HDC[double bond, length as m-dash]C37ClF, in addition to those reported for H2C[double bond, length as m-dash]C35ClF and H2C[double bond, length as m-dash]C37ClF in the previous work48 on the vibrational spectra and absorption cross sections of this molecule.

Table 4 CCSD(T)/AFCVTZ computed values (upper line) of spectroscopic parameters for six isotopomers of H2C[double bond, length as m-dash]CClF and their comparison with the experimental (lower line) data.45 For the sextic centrifugal distortion constants, no experimental data are available for these isotopologues
CH213C35ClF 13CH2C35ClF CH213C37ClF 13CH2C37ClF (E)-CHDC35ClF (Z)-CHDC35ClF
A 0/MHz 10635.063 10316.682 10634.706 10315.754 10369.817 9669.620
10679.35951 10358.24279 10679.03348 10357.36802 10409.47886 9710.13722
B 0/MHz 5053.180 4990.536 4906.649 4845.719 4808.470 5036.998
5090.81635 5028.33230 4943.14860 4882.36362 4844.58835 5075.59772
C 0/MHz 3420.988 3359.031 3353.134 3292.676 3280.949 3307.467
3442.89104 3380.62815 3374.65403 3313.90292 3301.79819 3328.91615
Δ J /kHz 1.380 1.338 1.308 1.268 1.186 1.386
1.354 1.313 1.310 1.253 1.188 1.372
Δ JK /kHz 4.930 4.757 4.965 4.628 4.216 4.840
4.92 4.93 4.54 5.10 4.24 4.944
Δ K /kHz 5.108 4.852 5.139 5.037 7.198 2.151
5.10 4.98 5.47 4.89 7.246 2.237
δ J /kHz 0.472 0.462 0.441 0.432 0.401 0.494
0.462 0.466 0.439 0.424 0.4042 0.5022
δ K /kHz 5.373 5.085 5.22 4.950 4.592 5.035
5.71 4.92 5.85 4.36 4.55 4.874
Φ J /Hz 0.000820 0.000832 0.000745 0.000755 0.000670 0.000932
Φ JK /Hz 0.0148 0.0133 0.0140 0.0126 0.0104 0.0144
Φ KJ /Hz −0.0143 −0.0141 −0.0130 −0.0128 −0.0107 −0.0182
Φ K /Hz 0.0257 0.0270 0.0252 0.0263 0.0345 0.0195
ϕ J /Hz 0.000428 0.000432 0.000390 0.000393 0.000350 0.000478
ϕ JK /Hz 0.00893 0.00825 0.00844 0.00780 0.00665 0.00876
ϕ K /Hz 0.0950 0.0861 0.0944 0.0856 0.0838 0.0736
χ aa (Cl)/MHz −72.1423 −72.4036 −56.8453 −57.0517 −72.49066 −72.42782
−72.9957 −73.2380 −57.5169 −57.7101 −73.30626 −73.25541
χ bb (Cl)/MHz 38.0177 38.2790 29.9511 30.1575 38.36610 38.30326
38.6977 38.9472 30.4870 30.6864 39.01017 38.93886
χ cc (Cl)/MHz 34.1246 34.1246 26.8942 26.8942 34.12456 34.12456
34.2981 34.2909 27.0299 27.0237 34.29609 34.31654
χ aa (D)/MHz 0.2053 −0.0421
0.1882 −0.0374
χ bb (D)/MHz −0.0942 0.1547
−0.0844 0.1388
χ cc (D)/MHz −0.1111 −0.1126
−0.1039 −0.1013


Table 4 lists also the diagonal elements of the inertial chlorine and deuterium quadrupole tensor, χij, obtained at the same level of theory. The elements of this tensor, evaluated from an applied field gradient qij, have been calculated with the formula

 
χij (MHz) = 234.9647·Q(bqij (a.u.) i, j = a, b, c(1)
where Q is the nuclear electric quadrupole moment of chlorine or deuterium nucleus in units of barn56 and a, b, and c are the inertial axes. Due to the molecular planarity, the inertial nuclear quadrupole tensor has only the χab off-diagonal element.

In addition the cubic force field required to compute the vibrational corrections to rotational constants has been evaluated from density functional theory calculations for all the required isotopologues. Both the B3LYP57,58 functional in conjunction with the SNSD59 basis set and the double hybrid B2PLYP60 functional coupled to the cc-pVTZ49,50 basis set have been employed. All DFT calculations have been performed by using the Gaussian16 quantum chemical package,61 with the vibrational corrections to rotational constants obtained within the generalized vibrational perturbative engine.62

2.2 Semi-experimental equilibrium geometry

Subsequently, the vibration–rotation interaction constants αir (where r denotes the vibrational normal mode and i = a, b, or c the inertial axis) have been computed using expressions from vibrational second-order perturbation theory (VPT2), which has the advantage of having analytical formulas for obtaining the spectroscopic parameters.63,64

In VPT2, the equilibrium molecular rotational constants Bie is related to the vibrational ground state rotational constants Bi0 as

 
image file: c8cp04888f-t1.tif(2)
where the summation is over the r = 3N − 6 vibrational modes, N being the number of nuclei.

The rotational constants Bie is inversely proportional to the principal moment of inertia Ii at the equilibrium geometry, and within the Born–Oppenheimer approximation, the equilibrium geometry is the same for all isotopologues.

The semi-experimental equilibrium structure rSEe has been obtained by a least-squares fit of the molecular structural parameters, corresponding to the planar internal coordinates of symmetry species A′, listed in Table 1, to the given experimental rotational constants corrected for the vibrational contributions computed theoretically. Actually, the fit requires at least as many independent rotational constants as there are structural degrees of freedom: for 1-chloro-1-fluoroethene, 5 bond lengths and 4 bond angles are needed as listed in Table 1. However, using more rotational constants, when available, is preferred since in this way the consistency of the experimental data can be checked.

It should also be mentioned that 1-chloro-1-fluoroethene being a planar molecule, only two of the three rotational constants are independent and can be included in the fit, since the relation

1/C = 1/A + 1/B
which holds for all planar molecules where the c axis is perpendicular to the molecular plane containing the a and b axes. Therefore at least the constants of 5 isotopologues are required to obtain the complete structure given by 9 parameters.

The fit has been carried out using the same weight for all the rotational constants, Bie, and employing the recently proposed Molecular Structure Refinement (MSR) program.65,66 In addition to the availability of different optimization algorithms, this program presents a flexible choice of coordinates (ranging from the common Z-matrix to delocalized internal coordinates of A′ symmetry) and an extended error analysis providing t-Student distribution confidence intervals besides the more common standard deviation. Furthermore, it has been equipped with the method of predicate observations67 to augment the dataset when the number of experimental points is low due to the lack of some isotopic substitutions.

2.3 Complete basis set limit equilibrium geometry

The equilibrium geometry of a molecule can be calculated with accuracy if an adequate electron correlation and basis-set convergence are taken into account. The cost-effective coupled cluster theory approach with single and double excitations and including treatments of triple excitations with a perturbative, non-iterative method, CCSD(T) has become the standard for highly accurate theoretical structure computations.68–71

These CCSD(T) calculations with sufficiently large basis-sets as the correlated molecular wave functions are able to deliver an accuracy of about 0.2 to 0.3 pm for bond distances.72 However, increasing the number of basis functions to reach a near complete basis, the computational cost becomes the main obstacle to perform accurate calculations of molecular parameters.

The problem can be solved using an extrapolation scheme which should speed up the systematic convergence to the complete basis set (CBS) limit.34 After the development of the family of correlation consistent polarized basis sets with an hierarchical structure, cc-pVnZ (n = D(2), T(3), Q(4), 5,… are the cardinal numbers of the basis sets) by Dunning and coworkers,49,50 several basis set extrapolation (BSE) methods have been proposed.51,73–78

The convergence in the computed structural parameters of the molecular geometry is assumed to have the same functional form of the energy.79 To be more precise, extrapolation has been carried out on geometric parameters instead of on gradients as would be more justified. Indicating the structural parameter with P, the basis-set extrapolation contributions are

 
P = P(HF-SCF/A) + ΔP(CCSD(T)/B) + ΔP(core/C)(3)
with large basis sets A, smaller basis sets B and a basis set C well suited for the treatment of the core correlation. The Hartree–Fock basis set limit is obtained from the consolidated, though empirical, extrapolation formula80
 
P(HF-SCF/A) = P(HF-SCF/cc-pVnZ) + a[thin space (1/6-em)]exp(−bn)(4)
with n being the cardinal number of the corresponding members of Dunning's hierarchy basis sets. P is obtained from three different basis set calculations as
 
image file: c8cp04888f-t2.tif(5)
where Pn = P(HF-SCF/cc-pVnZ).

The extrapolation due to the correlation to the basis-set limit is obtained from the following two-parameter correction,79

 
image file: c8cp04888f-t3.tif(6)

Applying the previous equation with two sequential basis sets, the correction at the CBS limit is given by

 
image file: c8cp04888f-t4.tif(7)
where
 
ΔPn = ΔP(CCSD(T)/cc-pVnZ) = P(CCSD(T)/cc-pVnZ) − P(HF-SCF/cc-pVnZ)(8)

Since the correlation contributions have been obtained in the frozen-core approximation in order to minimize the computational cost, the inclusion of eqn (9) is then necessary. The effect of the core-valence electron correlation is obtained taking into account the difference between all-electron and frozen-core calculations:

 
ΔP(core/C) = Pae(CCSD(T)/cc-pwCVnZ) − Pfc(CCSD(T)/cc-pwCVnZ)(9)

In addition to this composite procedure another extrapolation scheme has been considered based on a mixed exponential and gaussian function81,82

 
Pn = P + de−(n−1) + fe−(n−1)2(10)

The structural parameter at the CBS limit is calculated from three quantum chemistry calculations and is given by

 
image file: c8cp04888f-t5.tif(11)
where
 
Pn = P(CCSD(T)/cc-pVnZ)(12)

3 Results and discussion

In Table 4, the computed ground-state constants and quartic centrifugal distortion constants are compared with the experimental values given by Leung et al.45 At present, no experimental data are available for the sextic centrifugal distortion terms of the 6 isotopologues of 1-chloro-1-fluoroethene reported in this table.

As far as the ground-state rotational constants are concerned, the calculated values are all slightly underestimated with respect to the experimental data. The values of the quartic centrifugal distortion constants are all very close to the experimental ones. Moving to the diagonal elements of the inertial chlorine and deuterium atoms quadrupole coupling tensor χii, the calculated values are also in good agreement with the observed values.

The overall agreement between the experimental and theoretical spectroscopic parameters should be considered more than satisfactory, thus confirming the validity of the chosen level of theory.

The semi-experimental equilibrium structure of 1-chloro-1-fluoroethene has been obtained, as explained in Section 2.2, using the experimental ground-state rotational constants together with the theoretical α-constants deduced from the ab initio force field.

The vibrational corrections, BieBi0, for all the eight isotopologues of 1-chloro-1-fluoroethene computed at different levels of theory and used in the fits are reported in Table 5. In general, CCSD(T) vibrational corrections are larger than B2PLYP/cc-pVTZ ones that are in turn larger than the B3LYP values.

Table 5 Vibrational corrections, BieBi0, in MHz, for the eight isotopologues obtained at different levels of theory and used to fit the molecular structurea
A eA0 B eB0 C eC0
a Each set of lines for each isotopologue refers, from top to bottom, to the CCSD(T)/AFCVTZ, B2PLYP/cc-pVTZ and B3LYP/SNSD levels of theory, respectively.
CH2C35ClF 56.605 20.448 19.854
54.473 20.083 19.429
53.014 19.576 18.975
CH2C37ClF 54.519 19.931 19.150
54.530 19.438 18.932
53.104 18.769 18.407
CH213C35ClF 53.482 20.009 19.307
53.481 19.539 19.055
52.031 19.052 18.611
13CH2C35ClF 51.189 20.760 19.337
51.410 20.168 19.058
50.057 19.673 18.622
CH213C37ClF 53.553 19.295 18.781
53.537 18.904 18.561
52.278 18.305 18.087
13CH2C37ClF 51.286 20.014 18.818
51.481 19.503 18.570
50.175 18.824 18.055
(E)-CHDC35ClF 47.019 20.693 18.579
47.400 20.032 18.283
46.014 19.499 17.834
(Z)-CHDC35ClF 47.704 20.333 18.751
47.726 19.997 18.577
46.689 19.390 18.126


The semi-experimental equilibrium geometry has been obtained from a non-linear least squares fitting procedure based on eqn (2) since the equilibrium rotational constants Bie depend on the moments of inertia at the equilibrium geometry which, in the Born–Oppenheimer approximation, is the same for all the isotopic modifications of 1-chloro-1-fluoroethene. The differences in Bie are therefore due to differences in isotope masses. In particular, the semi-experimental equilibrium structure of 1-chloro-1-fluoroethene has been obtained according to three different fits in which CCSD(T), B2PLYP, and B3LYP vibrational corrections have been employed and the results are collected in Table 6 labeled as rSEe(CC), rSEe(B2), and rSEe(B3), respectively. As it can be seen the three fits provide the same equilibrium geometry of ClFC[double bond, length as m-dash]CH2 with differences in bond lengths and angles well within 0.002 Å and 0.2 degrees, respectively, thus supporting the consistency of the vibrational corrections computed by using the cost-effective B2PLYP/cc-pVTZ and B3LYP/SNSD model chemistries.

Table 6 Semi-experimental structure of 1-chloro-1-fluoroethene obtained by using vibrational corrections computed at the coupled cluster and DFT levels (bond lengths in Å and bond angles in degrees)
r SEe(CC)a,b r SEe(B2)a,c r SEe(B3)a,d
a Figures in parentheses represent one time the standard deviation. The numbers below the values of the structural parameters are 95% confidence intervals from t-Student distribution. b Vibrational corrections at CCSD(T)/AFCVTZ. c Vibrational corrections at the B2PLYP/cc-pVTZ level. d Vibrational corrections at the B3LYP/SNSD level.
C2–F1 1.3287(52) 1.3294(8) 1.3282(1)
0.012 0.0018 0.0042
C2[double bond, length as m-dash]C3 1.3233(37) 1.3220(6) 1.3224(1)
0.0089 0.0013 0.0030
C3–H4 1.0780(13) 1.0780(2) 1.0777(4)
0.0030 0.00045 0.0010
C2–Cl5 1.7081(31) 1.7089(5) 1.7099(1)
0.0073 0.0011 0.0025
C3–H6 1.0750(16) 1.0765(2) 1.0762(5)
0.0037 0.00055 0.0013
∠C3C2F1 125.44(30) 122.48(4) 122.56(10)
0.70 0.10 0.24
∠C3C2Cl5 125.52(32) 125.61(5) 125.54(1)
0.75 0.11 0.26
∠C2C3H6 120.02(23) 119.85(3) 119.90(8)
0.53 0.08 0.18
∠C2C3H4 119.30(20) 119.33(3) 119.30(7)
0.48 0.08 0.16


As far as the complete basis set limit equilibrium geometry is concerned, the contributions to the structural parameters of eqn (3) have been computed as described below. The Hartree–Fock basis-set limit and the unknown parameters a and b of eqn (4) have been determined from three calculations employing sequential basis sets. Choosing n = 6, P can be easily obtained from eqn (5) employing the results of Hartree–Fock calculations with cc-pVQZ, cc-pV5Z, and cc-pV6Z basis sets. The corrections due to the correlation have been calculated from eqn (7) and n = 5. Therefore, at the CCSD(T) level of theory the quantum chemical computations have been carried out with cc-pVQZ and cc-pV5Z basis sets. Although the TZ and QZ basis sets could be sufficient for this contribution because they require less computational time, the larger basis sets have been used here since they are also necessary for the other extrapolation method as explained below. For the last contribution, the inner-shell correlations have been computed with the weighted core-valence cc-pwCVQZ set52 in eqn (9).

For the extrapolation scheme described in eqn (10), three calculations are necessary in order to get the structural parameter at the CBS limit with eqn (11). Using n = 3, the calculations have been carried out employing cc-pVTZ, cc-pVQZ, and cc-pV5Z basis sets at the CCSD(T) level of theory.

The molecular geometries of 1-chloro-1-fluoroethene obtained from these calculations are reported in Table 7.

Table 7 Equilibrium geometries of 1-chloro-1-fluoroethene as computed at different levels of theory employing different basis sets (see the text). Distances in Å and angles in degrees
HF-SCF/cc-pVnZ CCSD(T)/cc-pVnZ CCSD(T)/cc-pwCVQZ
Q 5 6 T Q 5 fc ae
C2–F1 1.3051 1.3053 1.3053 1.3304 1.3294 1.3299 1.3295 1.3275
C2[double bond, length as m-dash]C3 1.3034 1.3035 1.3035 1.3286 1.3255 1.3251 1.3253 1.3224
C3–H4 1.0709 1.0708 1.0707 1.0800 1.0793 1.0792 1.0793 1.0778
C2–Cl5 1.7146 1.7123 1.7121 1.7245 1.7183 1.7140 1.7145 1.7108
C3–H6 1.0679 1.0678 1.0678 1.0775 1.0769 1.0768 1.0769 1.0755
∠C3C2F1 123.01 122.93 122.93 122.70 122.67 122.54 122.59 122.59
∠C3C2Cl5 125.39 125.44 125.44 125.24 125.37 125.51 125.40 125.40
∠C2C3H6 120.53 120.52 120.52 120.17 120.03 119.98 120.02 120.06
∠C2C3H4 119.52 119.55 119.56 119.30 119.33 119.33 119.34 119.36


Below the line reporting the theoretical method, the cardinal number of Dunning's hierarchical basis-set sequences is also indicated. To be more precise, three values (Q, 5, and 6) are for HF-SCF calculations, and other three values (T, Q, and 5) for the CCSD(T) method. The last two columns are relative to the core-valence contribution and report the frozen core (fc) calculation as well as the computation correlating all the electrons (ae).

As far as the Hartree–Fock self-consistent-field computations are concerned, it is confirmed – as often assumed – that the basis set limit is reached with the HF-SCF/cc-pV6Z level of theory. Concerning the coupled cluster calculations, from the data shown in Table 7, we can see a progressive smooth increase or decrease, with the exception of the C–F bond length (probably due to the lack of the diffuse functions which are present in the augmented basis set), in the value of the structural parameter when we change from n = 3 (T) to n = 5.

Applying eqn (5), (7), and (9) with the data reported in Table 7, all the contributions necessary to employ eqn (3) could be determined. For the sake of completeness, in the first and in the second column of Table 8 the coupled cluster contribution from eqn (7) and the core-valence correlation of eqn (9), respectively, are listed. The complete basis set structure obtained is reported in Table 8 labeled as CBS-I. Using the mixed exponential and gaussian function of eqn (11) the molecular structure, indicated in Table 8 as CBS-II, is obtained. The two molecular geometries, CBS-I and CBS-II, are almost equivalent, with a maximum difference in the bonding distance of 0.003 Å while the bond angles differ at most by 0.03°. It should however be pointed out that CBS-II requires 3 calculations against the 7 required by CBS-I. On the opposite, DFT geometries, reported in Table 9, show significant differences with respect to the semi-experimental structure. At the B2PLYP/cc-pVTZ level of theory, bond lengths and angles are reproduced quite well with a mean absolute deviation (MAD) of 0.005 Å and 0.2°, however with a maximum deviation of 0.02 Å for the C–Cl bond length. As to the B3LYP/SNSD level of theory, the errors are larger and up to 0.03 Å for the C–Cl bond length and 0.5° for the CCCl bond angle. As noted, B3LYP/SNSD and B2PLYP/cc-pVTZ predictions of 1-chloro-1-fluoroethene geometrical parameters suffer from large inaccuracy with respect to the semi-experimental parameters. A possible strategy to improve the DFT results is given by the LRA. This is based on the observation that DFT errors are systematic and follow a linear trend and hence it is possible to obtain a linear transformation for back-correcting structural parameters:

 
rLRAe = rDFTe + Δr with Δr = ArDFTe + B(13)
where A and B are, respectively, (1-slope) and the intercept obtained by performing a linear regression of a semi-experimental structural parameter as a function of the corresponding DFT value. For C–H, C[double bond, length as m-dash]C, C–F, and C–Cl bond lengths, the LRA has been applied by adopting the regression parameters A and B determined in previous works by Barone and co-workers38,66 and derived from a set of 100, 45, 6 and 5 items, respectively. The interested reader is referred to the cited literature and to the SNS structural database39 for further details on the molecules used to set up both the LRA and the TMA.37,38,66 In the present work, it has been possible to obtain a new LRA correction for the CCF bond angle (A = −0.13136, B = 15.63107 for B2PLYP/cc-pVTZ; A = −0.12032, B = 14.40384 for B3LYP/SNSD), by using in addition to the CCF parameter of 1-chloro-1-fluoroethene those of vinyl-fluoride, cis-CHFCHCl, 2-fluoropyridine and 3-fluoropyridine already present in the SMART database of molecular structures.39Table 9 reports the results of this procedure.

Table 8 Complete basis set equilibrium molecular structures of 1-chloro-1-fluoroethene obtained by applying eqn (3) (CBS-I) and eqn (11) (CBS-II) (bond lengths in Å and bond angles in degrees). ΔCC is the coupled cluster correlation contributed from eqn (7) and ΔCV, from eqn (9), is the core-valence correction
Δ CC Δ CV CBS-I CBS-II
C2–F1 0.0250 −0.0020 1.3283 1.3282
C2[double bond, length as m-dash]C3 0.0210 −0.0029 1.3216 1.3219
C3–H4 0.0083 −0.0014 1.0777 1.0776
C2–Cl5 −0.0003 −0.0037 1.7081 1.7078
C3–H6 0.0091 −0.0014 1.0754 1.0754
∠C3C2F1 −0.44 0.00 122.48 122.47
∠C3C2Cl5 0.16 −0.01 125.59 125.58
∠C2C3H6 −0.59 0.03 119.96 119.98
∠C2C3H4 −0.25 0.01 119.32 119.35


Table 9 Equilibrium geometries of 1-chloro-1-fluoroethene computed at B2PLYP/cc-pVTZ and B3LYP/SNSD levels. Distances in Å and angles in degrees
B2PLYP B2PLYP+LRAa B3LYP B3LYP+LRAb
a B2PLYP/cc-pVTZ equilibrium geometry corrected through the LRA approach (see the text). b B3LYP/SNSD equilibrium geometry corrected through the LRA approach (see the text). c Uncorrected value.
C2–F1 1.3306 1.3281 1.3372 1.3278
C2[double bond, length as m-dash]C3 1.3206 1.3218 1.3247 1.3220
C3–H4 1.0776 1.0778 1.0838 1.0780
C2–Cl5 1.7236 1.7115 1.7369 1.7094
C3–H6 1.0785 1.0786 1.0811 1.0755
∠C3C2F1 122.83 122.33 122.80 122.43
∠C3C2Cl5 125.39 125.39c 125.70 125.39c
∠C2C3H6 120.38 120.38c 120.49 120.38c
∠C2C3H4 119.32 119.32c 119.5 119.32c


Even if the number of items employed to obtain the A and B parameters for the CCF bond angle is limited, the linear regressions present an R2 = 0.995. Moreover, it can be observed how the LRA improves the description of this angle that at B2PLYP and B3LYP levels is predicted with an error of 0.4° that reduces to −0.1° and 0.01° when the LRA is applied to the B2PLYP/cc-pVTZ and B3LYP/SNSD values, respectively. Very notably, when the LRA correction is applied to the C–Cl bond length, that was badly predicted at both B2PLYP and B3LYP levels, the error is reduced by about one order of magnitude. LRA corrected bond lengths present a MAD around 0.001 Å thus showing the effectiveness of this scheme, whose accuracy competes with that of highly correlated wavefunction methods however with a significantly lower computational cost.

Taking as a reference the CBS-I theoretical equilibrium structure reported in the third column of Table 8, chemically relevant structural comparison with the best theoretical equilibrium structures of cis-1-chloro-2-fluoroethene and trans-1-chloro-2-fluoroethene reported by Puzzarini et al.83 can be discussed.

Fig. 2 features these data showing the molecular geometry of the three isomers of chlorofluoroethene. Although not all parameters can be compared appropriately some considerations can still be carried out. At first we note that the CH2 group is very close to the trigonal geometry, the angles being almost 120°. The C–H distances remain rather unchanged upon isomerization. The largest effects are observed for the C–Cl and C–F distances. The C–Cl distance in the gem compound is shortened by 0.0082 Å with respect to the trans form and by 0.0026 Å compared to the cis form. A similar behavior is observed for the C–F distance which is shortened by 0.0093 Å and 0.0027 Å, respectively, when compared to the trans- and cis-isomer. The changes in the CCF and CCCl bond angles in the cis- and trans-isomers were explained invoking the steric effects, as the fluorine and chlorine atoms move away from each other. In the gem isomer the decrease of the FCCl angle to 111.93° of course cannot be explained by steric hindrance and should therefore be due to the electronic arrangement of the molecular orbitals. It should also be mentioned that in 1,1-dichloroethene and 1,1-difluoroethene the ClCCl and FCF angles are, respectively, 117.8° and 115.5°, as determined in the rs molecular structure.84 Finally, the C[double bond, length as m-dash]C bond length must be considered; we note that this bond distance is shortened further than the cis and trans values, confirming the well-known effect induced by the substitution of one or more hydrogens by halogens.85


image file: c8cp04888f-f2.tif
Fig. 2 Comparison of the best theoretical estimate for the equilibrium structure of (a) trans-1-chloro-2-fluoroethene,83 (b) cis-1-chloro-2-fluoroethene,83 and (c) 1-chloro-1-fluoroethene. Distances in Å and angles in deg.

4 Conclusions

One of the present challenges of quantum chemistry is the accurate prediction of equilibrium geometries for systems of increasing size, this being a fundamental prerequisite for the reliable modeling of spectroscopic properties. In this context, the knowledge of the semi-experimental equilibrium structure of small molecules serves as the cornerstone in building up new cost-effective methodologies or theoretical approaches. In the present work, the accurate equilibrium geometry of 1-chloro-1-fluoroethene has been first obtained by exploiting the semi-experimental approach and using vibrational corrections evaluated at three different levels of theory, namely CCSD(T), B2PLYP and B3LYP. These have led to the same geometrical parameters, thus confirming the reliability of vibrational contributions evaluated at the DFT level. The obtained semi-experimental equilibrium structure has then been compared with those obtained by exploiting two different composite schemes based on the CCSD(T) theory and accounting for extrapolation to the CBS and inclusion of the core-valence correlation. The two approaches have led to the same equilibrium geometry whose agreement with the semi-experimental structure is within 0.002 Å and 0.1° for bond lengths and angles, respectively. Conversely, geometries optimized at B2PLYP and B3LYP levels have shown significant errors. The structural parameters have been improved by applying the LRA, thus providing DFT-based structures with an accuracy rivalling that of highly correlated wavefunction methods. Hence, the LRA appears as a cost-effective strategy for determining accurate equilibrium geometries through DFT thus disclosing the route toward medium sized molecular systems. However, the parametrization of the linear transformations behind the LRA for different geometrical parameters requires the careful investigation of the semi-experimental equilibrium geometries for a variety of molecules thus also improving the available database of accurate equilibrium structures.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work has been supported by MIUR “PRIN 2015” funds (Grant Number 2015F59J3R), Scuola Normale Superiore (GR16_B_TASINATO) and University Ca’ Foscari Venezia (ADiR funds). The SMART@SNS Laboratory (http://smart.sns.it) is acknowledged for providing high-performance computer facilities. GC thanks SNS for her research fellowship.

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