Journal of Molecular Structure 930 (2009) 21–25 Contents lists available at ScienceDirect Journal of Molecular Structure journal homepage: www.elsevier.com/locate/molstruc Equilibrium CAs and CSb bond lengths Jean Demaison a,*, Harald Møllendal b, Jean-Claude Guillemin c a b c Laboratoire de Physique des Lasers, Atomes, et Molécules, UMR CNRS 8523, Université de Lille I, F-59655 Villeneuve d’Ascq Cédex, France Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, P.O. Box 1033, Blindern, N-0315 Oslo, Norway Sciences Chimiques de Rennes, Ecole Nationale Supérieure de Chimie de Rennes-CNRS, 35708 Rennes, France a r t i c l e i n f o Article history: Received 31 March 2009 Accepted 17 April 2009 Available online 3 May 2009 Keywords: Equilibrium structure Carbon–arsenic bond length Carbon–antimony bond length a b s t r a c t The equilibrium structures of some small molecules containing the CAs or the CSb bond are determined using the method of predicate observations. The input data are the semi-experimental equilibrium rotational constants (experimental ground state rotational constants corrected for the rovibrational contribution calculated from the ab initio cubic force field) and the bond lengths and bond angles calculated ab initio using the CCSD(T) method and a relativistic effective core potential of quadruple zeta quality. It is shown that this method, avoiding difficult isotopic substitutions, allows us to obtain reliable equilibrium structures. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction The equilibrium CX bond lengths are accurately known for different atoms X in many molecules where X = H [1], C [2], N [3], O [3], F [4], and Hal (where Hal = Cl, Br, and I) [5]. On the other hand, data for the CAs and CSb bond lengths are extremely scarse. For the CAs bond, the microwave spectra of difluoromethylarsine, CH3AsF2 [6], ethylidynearsine, CH3CAs [7,8] and trimethylarsine, (CH3)3As [9] have been measured and the structure of arsabenzene, cC5H5As, has been investigated by microwave spectroscopy [10] and by electron diffraction [11]. For all these molecules, an approximate value of the CAs bond length has been estimated, but, with the exception of CH3CAs [8], this estimation was based on many assumptions and it is thus rather inaccurate, see Table 1. For the CSb bond length, the data are still more scarse, only the microwave spectra of vinylstibine, H2C@CHSbH2 [12], and stibabenzene, cC5H5Sb [13] have been measured leading to a very approximate value of the CSb bond length (see Table 1). In addition to that, the microwave spectra of vinylarsine, H2C@CHAsH2 [14] ethynylarsine, HC„CAsH2, and ethylarsine, CH3CH2AsH2 [15] have also been recently measured in our laboratories allowing us to obtain very accurate ground state rotational constants. The microwave spectrum of methylarsine, CH3AsH2, has also been measured but never published [16]. These experimental ground state rotational constants do not permit to calculate an equilibrium structure but they can be corrected using ab initio rotation–vibration interaction constants (a-constants) in order to obtain semi-experimental equilibrium rotational constants [17]. * Corresponding author. Fax: +33 3 20 33 70 20. E-mail address: jean.demaison@univ-lille1.fr (J. Demaison). 0022-2860/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.molstruc.2009.04.029 With the exception of CH3CAs, the number of rotational constants is not large enough to determine an equilibrium structure. However, each molecule may be divided in three parts: (i) the alkyne (hydrocarbon) group, CmHn, whose structure may be accurately computed using standard ab initio methods; (ii) the arsine (resp. stibine) group, AsH2 (resp. SbH2) whose ab initio structure may be scaled using the known equilibrium structure of AsH3 [18] (resp. SbH3 [19]); (iii) the CX (X = As, Sb) bond length and the \(CCX) bond angle whose values may be crudely estimated using ab initio methods. The goal of this paper is to show that it is possible to obtain a reliable equilibrium structure for these molecules by combining the ab initio equilibrium structure and the semi-experimental rotational constants in the method of predicate observations. In this method, the rotational constants and the ab initio structure are used as input in a least-squares fit, each datum being weighted according to the value of its estimated uncertainty [20,21] . The paper is organized as follows. Section 2 describes the ab initio calculations. Section 3 is dedicated to the equilibrium structure of CH3CAs. Section 4 is devoted to the structure of HC„CAsH2. This section is more detailed than others because it explains the methodology used which is then applied in the subsequent sections to the other molecules, CH2@CHAsH2, CH3CH2AsH2, and CH2@CHSbH2. 2. Ab initio calculations Geometry optimizations have been carried out at two levels of electronic structure theory, second-order Møller–Plesset perturbation theory (MP2) [22] and coupled-cluster (CC) method with single and double excitations (CCSD) [23] augmented by a perturbational estimate of the effects of connected triple excitations [CCSD(T)] [24]. For the H and C atoms, Dunning’s 22 J. Demaison et al. / Journal of Molecular Structure 930 (2009) 21–25 Table 1 Previous determinations of the CAs or the CSb bond length (values in Å). Molecule Bond Structurea value Ref. (CH3)3As CH3AsF2 c-C5H5As CH3CAs c-C5H5Sb CH2@CHSbH2 C–As r0 r0 rg r eb r0 rs 1.959(10) 1.92(10) 1.850(3) 1.658(2) 2.050(5) 2.13(3) [9] [6] [11] [8] [13] [12] C–Sb a r0 = effective structure from ground state rotational constants, rg = average internuclear distance at thermal equilibrium from electron diffraction, re = equilibrium structure, rs = substitution structure. b Extrapolated from the average structure (rz). correlation-consistent polarized valence n-uple zeta basis sets, ccpVnZ [25] with n {T, Q} were employed. For the As and Sb atoms, the correlation-consistent valence basis sets with the StuttgartDresden-Bonn relativistic effective core potential, SDB-cc-pVnZ [26] (abbreviated as SDB-VTZ) were used. In all calculations, the frozen core approximation was used, unless otherwise stated. The anharmonic force field was calculated at the MP2 level of theory using the SDB-cc-pVTZ basis set. There is indeed a large documented evidence which shows that, in most cases, the CCSD(T) force field only provides a negligible improvement over the MP2 force field in determining the semi-experimental equilibrium structure [27,28]. This will be further checked here in the particular case of CH3CAs. The reference geometry was first determined via optimization prior to computing the force field at the same level of theory. Then, the associated harmonic force field was evaluated analytically in Cartesian coordinates. The cubic (/ijk) and semidiagonal quartic (/ijkk) normal coordinates force constants were determined with the use of a finite difference procedure involving displacements [29] along reduced normal coordinates and the calculation of analytic second derivatives at these displaced geometries. This procedure was repeated for all isotopologues. The semi-experimental equilibrium rotational constants needed to determine a semi-experimental equilibrium structure are then obtained by correcting the experimental ground state rotational constants with the a-constants, computed from the ab initio cubic force field via the expression Bne ¼ Bn0 þ X i ani di 2 ð1Þ where n = a, b, c, referring to the principal axes of the molecule; the sum is running over all normal modes and di is a degeneracy factor (1 for non-degenerate vibrations and 2 for doubly-degenerate vibrations). The evaluation of the a-constants as well as the other spectroscopic parameters was based on second-order perturbation theory [30]. For the calculation of the magnetic g-tensor, the Kohn-Sham density functional theory [31] using Becke’s three-parameter hybrid exchange functional [32] and the Lee-Yang-Parr correlation functional [33] together denoted as B3LYP, was also employed together with the split-valence basis set 6-311G(3df,2pd), as implemented in GAUSSIAN03 (g03) [34] for the H,C and As atoms. For molecules containing Sb, the DGDZVP basis set, as implemented in g03, was used. The diagonal elements of the g-tensor permit to calculate the electronic correction to the rotational constants using the relation Bncorr ¼ Bnexp 1 þ ðm=Mp Þg nn ð2Þ where gnn is expressed in units of the nuclear magneton, m is the electron mass, and Mp the proton mass. The calculated values of the g-tensor are given in Table S1 of the Supplementary material. This correction was found negligible for all molecules of this work. All computations were performed with g03. 3. Ethylidynearsine Ethylidynearsine, CH3CAs, is the most promising molecule from the point of view of structure determination because it has already been thoroughly studied. The ground state rotational constants of eight isotopologues have been determined, its infrared spectrum has been measured, an average structure (rz) has been calculated and an approximate equilibrium structure has been deduced [8]. The anharmonic force field, calculated at the MP2/SDB-VTZ level of theory, was used to calculate the values of the spectroscopic parameters for which experimental values are known: quartic centrifugal distortion constants and band centers. The results are given in Table S2 of the Supplementary material. The good agreement between the calculated and experimental parameters indicates that the force field is reliable. The semi-experimental (se) equilibrium structure was calculated from a weighted least-squares fit of the related equilibrium moments of inertia and is given in Table 2. The system of normal equations is moderately well conditioned (condition number j = 264) [35]. This is mainly due to the fact that there is no isotopic substitution available for As. However, the standard deviation of the derived parameters is extremely small, indicating that the re(se) structure should be accurate. The semi-experimental structure is in very good agreement with the previous equilibrium structure (extrapolated from the rz structure), but it is much more accurate. The ab initio structures are also given in Table 2. The CCSD(T)/SDB-VQZ values with all electrons correlated are in very good agreement with the semi-experimental structure. For the methyl part, it was expected [36] but for the CAs bond length, it might be accidental. The CC bond length, at 1.463 Å, is longer than in methyl cyanide (1.459 Å) [27] indicating a smaller degree of double bond character. Finally, it as to be noted that the CH bond length at 1.0910(3) Å is longer than in CH3CN where the value is 1.0865(1) Å [27]. This is in agreement with the fact that the CH stretching frequencies are larger in CH3CN than in CH3CAs. For instance, the CH3 symmetric stretch, m1, is at 2840 cm1 in CH3CAs [8] and 2954 cm1 in CH3CN [27]. 4. Ethynylarsine, HCCAsH2 The experimental ground state rotational constants of HC„CAsH2, B0 = 3666.1654(6) MHz and C0 = 3659.6748(6) MHz [15] are corrected with the a-constants from the MP2/SDB-VTZ force field to give the semi-experimental equilibrium rotational constants, Be = 3659.3(2) MHz and Ce = 3652.8(2) MHz where the quoted uncertainty takes into account the uncertainty on the ab initio a-constants. The ab initio structure has been optimized at Table 2 Structure of ethylidynearsine, CH3CAs (distances in Å, angles in degrees). Method Basis set a r0 reb MP2 fcc CCSD(T) aec re(se)d a SDB-VTZ SDB-VQZ r(CH) r(CC) r(CAs) \(HCC) 1.105(7) 1.091(3) 1.090 1.090 1.0910(3) 1.465(2) 1.465(2) 1.466 1.465 1.4634(2) 1.660(1) 1.658(2) 1.677 1.659 1.6585(2) 110.6(2) 110.6(2) 110.71 110.71 110.575(10) r0 = effective structure from a fit of the ground state rotational constants, Ref. [8]. b c d Extrapolated from the average structure (rz), Ref. [8]. fc = frozen core approximation, ae = all electrons correlated. Semi-experimental equilibrium structure. 23 J. Demaison et al. / Journal of Molecular Structure 930 (2009) 21–25 Table 3 Structure of ethynylarsine, HC„CAsH2 (distances in Å, angles in degrees). MP2a r(H–C) r(CC) r(C–As) r(As–H) \(CCAs) \(CAsH) \(HAsH) \(HCC) a b CCSD(T)a SDB-VTZ SDB-VQZ SDB-VTZ SDB-VQZ 1.0623 1.2201 1.8922 1.5121 173.145 94.727 92.252 178.608 1.0624 1.2174 1.8842 1.5091 172.928 94.682 92.013 178.554 1.0642 1.2167 1.9025 1.5207 173.388 94.663 92.132 178.611 1.0643 1.214 1.8945 1.5177 173.171 94.618 91.894 178.557 Predicateb Uncertainty re 1.0628 1.2105 1.8945 1.5120 173.171 94.618 91.894 178.557 0.0020 0.0020 0.020 0.002 0.3 0.5 0.4 0.3 1.063 1.211 1.894 1.512 173.25 94.85 92.24 178.55 Frozen core approximation. See text. the MP2 and CCSD(T) levels with triple and quadruple zeta basis sets (see Table 3). The CCSD(T)/SDB-VQZ bond lengths of the ethynyl group are corrected for the core correlation which is estimated to be 0.0015 Å for the CH bonds and 0.0035 Å for the C„C bonds [5]. The uncertainty of the CH and CC bond lengths was assumed to be 0.002 Å. For the AsH bond length and the \(HAsH) bond angle, the MP2/SDB-VTZ values were preferred because, in the case of AsH3, the MP2/SDB-VTZ structure was found to be the closest one of the equilibrium structure (see Table 4). The uncertainty of the MP2/SDB-VTZ AsH bond length was assumed to be 0.002 Å and that of the \(HAsH) bond angle, 0.4 degree. For the remaining parameters, the CCSD(T)/SDB-VQZ values were used with an uncertainty of 0.020 Å for the r(CAs) bond length, 0.3 degree for the \(HCC) and \(CCAs) bond angles and 0.5 degree for the \(CAsH) bond angle. We also made a sensitivity analysis. The most sensitive parameters for the B and C rotational constants are r(C-As) and \(CCAs), and to a lesser extent \(CAsH), i.e. the parameters whose ab initio values are difficult to calculate accurately. This is a very favourable case for the use of the method of predicate observations. The ab initio equilibrium parameters (column predicate of Table 3) were fitted together with the two semi-experimental equilibrium rotational constants giving a structure reported in the last column (re) of Table 3. The results of the fit are worth to be commented. The CH and CC and AsH bond lengths remain unchanged, as expected. The CAs bond length is also unchanged which was not expected. Only two bond angles are significantly changed, the \(HAsH) angle being larger by 0.35 degree and the \(CAsH) angle being also larger by 0.23 degree but these changes are rather small indicating that the predicate (ab initio) structure is likely to be close to the true equilibrium structure and confirming that the fitted structure is probably reliable. Another interesting conclusion is that the HCCAs moiety is not linear, the \(CCAs) angle being as large as 6.8 degree and the ethynyl H being in trans position with \(HCC) = 1.45 degree. 5. Other molecules The structure of the other molecules, syn-vinylarsine, transethylarsine, and syn-vinylstibine is calculated in the same way as for ethynylarsine. The experimental ground state rotational constants and the derived semi-experimental equilibrium rotational constants are given in Table 5. The calculated structures are given in Tables 6–8. The only difference is that, for the predicate structure of syn-vinylstibine, an offset of 0.0040 Å for the SbH bond was estimated at the MP2/DSDB-VTZ level from the equilibrium structure of stibine (bottom of Table 8) and the predicate value of the CSb bond length was assumed to be equal to the MP2/ DSDB-VTZ value. It is important to try to estimate the accuracy of the derived structures. However, the number of input data is not large enough to try a statistical estimation. Nevertheless, a comparison with many other molecules indicates that the accuracy of the bond an- Table 5 Experimental ground state and semi-experimental equilibrium rotational constants (in MHz). Basis set r(As–H) \(HAsH) This work VTZ-PPe SDB-VTZ SDB-VQZ SDB-VTZ SDB-VQZ SDB-VTZ SDB-VQZ 1.5111 1.513(2) 1.5109 1.4896 1.5106 1.5078 1.5201 1.5168 1.5200 1.5188 92.069 92.08(7) 92.189 92.3874 92.0892 91.8824 92.110 91.729 91.9694 91.881 b re r ec r ed MP2 MP2 MP2 CCSD(T) ae CCSD(T) ae CCSD(T) CCSD(T) Ref.a B C 3666.16536(55) 3659.67482(55) 3659.34(20) 3652.77(20) 15 syn-H2C@CHAsH2 A B C 36322.959(29) 3941.36086(89) 3678.56301(99) 36634.7(31) 3964.01(23) 3701.56(23) 14 trans-CH3CH2AsH2 A B C 24410.16(34) 3537.6052(18) 3313.1621(17) 24582.4(31) 3575.95(23) 3349.43(23) 15 syn-H2C@CH121SbH2 A B C 32773.1831(3) 3116.0358(1) 2944.7042(1) 33017.7(30) 3133.18(20) 2961.57(20) 12 syn-H2C@CH123SbH2 A B C 32766.0381(5) 3107.9617(2) 2937.4356(2) 33010.3(30) 3125.11(20) 2954.31(20) syn-H2C@13CH121SbH2 A B C 32107(1) 3068.7252(5) 2897.1315(5) 32344.2(10) 3085.35(20) 2913.56(20) syn-H2C@13CH123SbH2 A B C 32117(1) 3060.525(1) 2889.763(1) 32354.2(10) 3076.91(20) 2906.15(20) a Frozen core approximation unless otherwise specified, ae = all electrons correlated. b Ref. [18]. c Ref. [42]. d This work, using the a-constants of Ref. [43]. e Correlation consistent-like basis set with a small-core relativistic pseudopotential, Ref. [44]. Equilibrium HC„CAsH2 Table 4 Structure of arsine, AsH3 (distances in Å, angles in degrees). Methoda Ground state a Reference for experimental ground state rotational constants. 24 J. Demaison et al. / Journal of Molecular Structure 930 (2009) 21–25 Table 6 Structure of syn-vinylarsine, H2C@CHAsH2 (distances in Å, angles in degrees). Parametera r(CC) r(CHc) r(CHt) r(CAs) r(CHg) r(AsH) \(CCHc) \(CCHt) \(CCAs) \(HgCAs) \(CAsH) \(HAsH) a b c d MP2 CCSD(T) Predicatec Uncertainty re 1.3340d 1.0828 1.0823 1.9427 1.0827 1.5142 121.407 121.584 120.205 120.264 95.572 91.354 0.002 0.002 0.002 0.020 0.002 0.003 0.3 0.3 0.5 0.3 0.5 0.3 1.334 1.083 1.082 1.939 1.083 1.516 121.41 121.59 120.12 120.27 95.57 91.53 b SDB-VTZ SDB-VQZ SDB-VTZ SDB-VQZ 1.3361 1.0825 1.0824 1.9427 1.0826 1.5142 121.315 121.501 119.893 120.256 95.562 91.354 1.3338 1.0818 1.0814 1.9339 1.0821 1.5112 121.278 121.489 119.903 120.264 95.613 91.303 1.3397 1.0850 1.0848 1.9524 1.0847 1.5231 121.444 121.595 120.195 120.256 95.520 91.284 1.3374 1.0843 1.0838 1.9436 1.0842 1.5201 121.407 121.584 120.205 120.264 95.572 91.233 c = cis to As; t = trans to As; g = gem. CCSD(T)/SDB-VTZ + MP2/SDB-VQZ – MP2/SDB-VTZ, this approximation is expected to be accurate for the CC and CH bonds lengths as well as the \(HCC) angles. See text. Offset, 0.0034 Å, Ref. [2]. Table 7 Structure of trans-ethylarsine, CH3CH2AsH2 (distances in Å, angles in degrees). Parametera r(CC) r(CHs) r(CHa) r(CAs) r(CH)b r(AsH) \(CCHs) \(CCHa) \(HaCHa) \(CCAs) \(HCAs) \(HCH)b \(CAsH) \(HAsH) a b c d MP2 CCSD(T) SDB-VTZ SDB-VQZ SDB-VTZ SDB-VQZc 1.5213 1.0906 1.0889 1.9724 1.0906 1.5146 111.334 110.856 107.570 115.557 105.987 106.535 94.770 91.835 1.5188 1.0894 1.0879 1.9629 1.0897 1.5115 111.349 110.842 107.546 115.400 106.073 106.512 94.891 91.742 1.5271 1.0937 1.0921 1.9827 1.0930 1.5238 111.226 110.884 107.610 115.508 106.011 106.678 94.883 91.769 1.5246 1.0925 1.0911 1.9732 1.0921 1.5207 111.241 110.871 107.586 115.351 106.097 106.655 95.004 91.677 Predicate Uncertainty re 1.5213d 1.0910 1.0896 1.9732 1.0906 1.5146 111.241 110.871 107.586 115.351 106.097 106.655 95.004 91.677 0.002 0.002 0.002 0.020 0.002 0.002 0.3 0.3 0.3 0.5 0.3 0.3 0.3 0.3 1.521 1.091 1.090 1.970 1.091 1.516 111.25 110.87 107.68 115.38 106.11 106.74 95.05 91.85 s = H in symmetry plane; a = H out of symmetry plane. CH2 group. CCSD(T)/SDB-VTZ + MP2/SDB-VQZ – MP2/SDB-VTZ, this approximation is expected to be accurate for the CC and CH bonds lengths as well as the \(HCC) angles. Offset, 0.0033 Å, Ref. [2]. Table 8 Structure of syn-vinylstibine, H2C@CHSbH2 (distances in Å, angles in degrees) Parametera MP2 CCSD(T) SDB-VTZ SDB-VQZ SDB-VTZ SDB-VQZb r(C@C) r(C-Sb) r(CHc) r(CHt) r(CHg) r(SbH) \(CCSb) \(CCHc) \(CCHt) \(CSbH) \(HgCSb) \(HSbH) 1.3375 2.1403 1.0829 1.0834 1.0832 1.7079 120.5416 121.7448 121.5725 94.6510 120.1275 91.1136 1.3354 2.1323 1.0822 1.0825 1.0829 1.7025 120.6989 121.7926 121.5314 94.7083 120.1004 91.2947 1.3409 2.1505 1.0855 1.0858 1.0854 1.7189 120.7617 121.7950 121.6427 94.5655 119.8221 91.0197 1.3388 2.1425 1.0848 1.0849 1.0851 1.7135 120.919 121.843 121.602 94.623 119.795 91.201 SbH3 r(SbH) \(HSbH) 1.7040 92.012 a b c 1.7153 91.818 Predicate Uncertainty re 1.3355c 2.1403 1.0833 1.0834 1.0836 1.7039 120.919 121.843 121.602 94.623 119.795 91.201 0.002 0.020 0.002 0.002 0.002 0.003 0.5 0.3 0.3 0.5 0.3 0.3 1.335 2.131 1.083 1.084 1.084 1.703 120.65 121.89 121.62 94.72 119.82 91.14 1.7000 91.56 c = cis to Sb; t = trans to Sb; g = gem. CCSD(T)/SDB-VTZ + MP2/SDB-VQZ – MP2/SDB-VTZ, this approximation is expected to be accurate for the CC and CH bonds lengths as well as the \(HCC) angles. Offset, 0.0034 Å, Ref. [2]. J. Demaison et al. / Journal of Molecular Structure 930 (2009) 21–25 gles is not worse than a few tenths of a degree [37]. Likewise, the accuracy of the CH and CC bond lengths is probably not worse than a few thousandths of an Angstrøm [2]. The accuracy of the CAs and CSb bond lengths is still more difficult to estimate. However, a sensitivity analysis indicates that it is also probably not worse than a few thousandths of an Angstrøm, a very conservative estimate being 0.005 Å. 6. Discussion The ethynyl CH bond length in HC„CAsH2 is only faintly longer than in acetylene: 1.0628 Å instead of 1.0618 Å [38]. On the other hand, the C„C bond length, at 1.211 Å is much longer than in acetylene (1.203 Å) [38]. It is also larger than in cyanoacetylene (HC„C–C„N, 1.206 Å) [39]. The structure of the vinyl group is almost identical in H2C@CHAsH2 and H2C@CHSbH2 (Tables 6 and 8). The C@C bond length is slightly longer than in ethylene (1.331 Å) [2] but slightly smaller than in vinylamine (1.336 Å) [40]. The CH bond lengths are up to 0.005 Å longer than in vinylamine but the CH2 group is much less asymmetric (cis and trans CH bond lengths and \(HCC) bond angles are almost identical). The C–C bond length in ethylarsine is not significantly different to the value found for ethane (1.522 Å) [41] but the CH bond lengths are again 0.001–0.002 Å longer. As expected, the C–As bond length increases from ethynylarsine to ethylarsine going through vinylarsine. The As–H bond length in ethynylarsine, at 1.512 Å, is almost identical to the value found in arsine (1.511 Å, see Table 4) but it is longer in both vinylarsine and ethylarsine where the value is 1.516 Å. The \(HAsH) remains close to the value found for arsine. 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