Article pubs.acs.org/JPCA
Spin-Rotation and NMR Shielding Constants in XF Molecules (X = B, Al, Ga, In, and Tl) Michał Jaszuński,*,† Taye B. Demissie,‡ and Kenneth Ruud‡ †
Institute of Organic Chemistry, Polish Academy of Sciences, Kasprzaka 44, 01-224 Warsaw, Poland Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Tromso̷ −The Arctic University of Norway, N−9037 Tromso̷ , Norway
‡
ABSTRACT: Accurate spin−rotation and absolute shielding constants in a series of XF molecules (X = 11B, 27Al, 69Ga, 115In, and 205Tl) determined using high-level ab initio coupled-cluster and fourcomponent relativistic density-functional theory (DFT) calculations are presented. The accuracy of the results is established by comparing the relativistically and vibrationally corrected calculated values with available experimental data; for spin−rotation and shielding constants for which no experimental data exist, we provide new and reliable values. For both properties, our results can be considered as reference values against which more approximate quantum-chemical methods can be benchmarked.
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INTRODUCTION In rotational spectroscopies such as molecular beam and microwave spectroscopy, the properties of molecules are determined from the measurement of transition energies between the rotational states of molecules. The interaction of nuclear magnetic moments with the rotation of a molecule causes additional observable splittings in the rotational spectra, and the nuclear spin-rotation constants (C) describe this interaction. In quantum chemistry, these properties are obtained from the second derivative of the molecular energy (E) with respect to the nuclear spin of nucleus K, I(K), and the rotational angular momentum J C(K) = −
∂ 2E ∂I(K)∂J
diamagnetic contribution to the shielding constant determined from theoretical calculations σ para(K) = σ SR(K) =
σ(K) = σ para(K) + σ dia(K)
(2) (3)
where mp and me are the proton and electron masses, respectively, B is the molecular rotational constant, and g(K) the nuclear g factor. In accordance with eq 2, there is a direct relation between the paramagnetic part of the shielding and the electronic contribution to C(K). This nonrelativistic approach can be applied for very light atoms but fails when considering heavy atoms. In our recent studies, we discussed these discrepancies for tin compounds,9 heavy-atom hexafluorides,8 as well as for both atoms of HCl,10 and similar numerical evidence has been provided by Aucar et al.11 and Xiao, Zhang, and Liu.12 In quantum chemistry, the NMR shielding tensor σ(K) is identified as the second derivative of the energy with respect to the applied magnetic field and the nuclear magnetic moment. Different quantum-chemical procedures to determine absolute shielding constants are nowadays available. However, the calculated shielding constants, as is the case for the spinrotation constants, may suffer from inaccuracies due to the neglect of relativistic effects (for reviews, see refs 13 and 14). We recently demonstrated these inaccuracies in the electronic
(1)
The accuracy of the rotational constants is of great importance, in particular when they are used according to the so-called Flygare relation,1−3 to determine absolute shielding constants and thereafter the nuclear magnetic moments.4,5 As is well-established, in the absence of accurate experimental data, appropriate quantum-chemical calculations are the best options to determine the spin-rotation constants (see, for example, refs 6−8 and references therein). Nuclear magnetic resonance (NMR) spectroscopy is used to determine and analyze the magnetic properties of nuclei in atoms or molecules. In NMR, the local field experienced by the nucleus of interest differs from the externally applied magnetic field due to the electrons surrounding the nuclei, an effect described by the absolute shielding constant. A common way to determine the absolute shielding constant1−3 combines σSR, the paramagnetic shielding obtained from the electronic contribution to the nuclear spin−rotation constant, with the © 2014 American Chemical Society
1 mp C el(K) 1 2 me B g (K)
Received: July 17, 2014 Revised: September 5, 2014 Published: September 5, 2014 9588
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of Dunning-type basis sets,26−31 focusing on the aug-cc-pCVnZ sequence (n = 2−5). In the nonrelativistic Hartree−Fock (HF) and coupled-cluster calculations, we used (rotational) London atomic orbitals.32−34 The computed spin−rotation constants, shielding constants, and their span are shown in Tables 1, 2, and 3. For all the properties of BF, AlF, and GaF, adding core− valence functions to the basis set has a very large effect: absolute values of C(X) and C(F) increase, σ(X) and σ(F) decrease, and Ω(X) and Ω(F) increase. In the tables, this is illustrated for all the properties by the aug-cc-pVQZ results for AlF and GaF (and even with the largest available, aug-cc-pV5Z, basis set for Ga, we find, for example, C(Ga) = −12.45 kHz at the HF level). We observe good agreement between the results obtained with the largest core−valence type Dunning’s basis sets and the unc-ANO-RCC basis, indicating that the latter basis is also sufficient for InF and TlF. On the other hand, the contracted ANO-RCC basis appears to be useless for the heavier atoms. In TlF at the HF level, it, for instance, gives σ(Tl) = 29768 ppm and σ(F) = 186 ppm, to be compared with the results obtained using the uncontracted basis: 8622 and 287 ppm for σ(Tl) and σ(F), respectively. Thus, we will in the following only discuss the unc-ANO-RCC results. It appears that the CCSD(T) approach gives a good estimate of the electron correlation effects. The differences between the CCSD(T) and CCSD values are much smaller than between the CCSD and HF results. Moreover, whenever we compare the CCSDT and CCSD(T) results, their differences are almost negligible. Thus, although we for the sake of completeness include these differences in the final estimates, CCSD(T) by itself provides a sufficiently accurate description of the properties studied. For comparison with the experiment, we have also computed the zero-point vibrational (ZPV) and temperature corrections (using the CFOUR program, see ref 35 for details). In this approach, a truncated Taylor expansion of the properties in normal coordinates is used, requiring the calculation of first and second derivatives of the properties with respect to the normal coordinates and vibrational expectation values of the normal coordinates and their products, which can be determined from harmonic frequencies and anharmonic force constants.35 The quadratic force constants are computed as analytical second derivatives of the energy, and the cubic force constants and property derivatives are calculated numerically.36,37 The temperature dependence of the properties is determined using thermally averaged values of the normal coordinates (see refs 35, 38, and 39). These corrections are very small and thus their analysis was restricted to the nonrelativistic level. In this analysis, we used the unc-ANO-RCC basis sets and the CCSD(T) approach for BF, AlF, and GaF and CCSD for InF and TlF. To be consistent with the experiment, we shall present the corrections to the equilibrium values of the spin−rotation constants for the lowest rovibrational state (0 K) and for NMR properties the results at 300 K. The temperature effects are in most cases even smaller than the ZPV corrections and do not merit a more detailed analysis. Relativistic Density-Functional Theory Calculations. The four-component relativistic DFT calculations were performed using ReSpect.40 The restricted magnetic balance scheme module was used for the shielding constant calculations,41,42 whereas the restricted kinetic balance scheme was used in the case of the spin-rotation constant calculations.
contribution to the spin-rotation constant and the paramagnetic part of the absolute shielding of heavy atoms, as well as for light atoms in the vicinity of these heavy atoms.8−10 For the molecules considered in this study, there are few available values reported for the shielding and spin−rotation constants; most of them discussed in ref 15 (for ab initio results for the AlF molecule, see also ref 7). Wasylishen et al.15 reported new experimental spin−rotation constants for 69Ga and 19F in GaF and used these experimentally determined values together with ab initio diamagnetic shielding contributions to determine Ω, the span of the shielding tensor. For linear molecules Ω = |σ∥ − σ⊥|, where σ∥ is the shielding experienced during the alignment of the applied magnetic field parallel to the bond axis and σ⊥ when the magnetic field is perpendicular to the bond axis. Hence, for σ∥ > σ⊥, as is the case for all the molecules studied here, the span corresponds to the anisotropy of the tensor. However, additional approximations were used by Wasylishen et al., thus although we shall denote by Ω the values determined from experimental spinrotation constants as well as the computed ab initio σ∥ − σ⊥, these approximations may also lead to some minor differences. In order to estimate the accuracy of the relation between the spin-rotation constant and the span, relativistic effects should also be taken into account. In an attempt to establish accurate results for molecular systems involving heavy atoms, we here present spin−rotation and shielding values that include relativistic corrections. Therefore, considering the importance of both spin−rotation and shielding constants from an experimental and theoretical point-of-view, we use in this study high-level ab initio coupled cluster calculations together with four-component relativistic DFT methods to determine highly accurate values for the spin-rotation and absolute shielding constants of the XF molecules (X = 11B, 27Al, 69Ga, 115 In, and 205Tl).
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COMPUTATIONAL DETAILS For both the relativistic and nonrelativistic calculations, the experimental equilibrium bond distances were used with r e (B−F) = 1.262590 Å, 15 r e (Al−F) = 1.654369 Å, 15 r e (Ga−F) = 1.7743410 Å, 15 r e (In−F) = 3.752 au (1.985473038 Å),16 and re(Tl−F) = 3.93898 au (2.0844186 Å).17 The nuclear g factors used in all the calculations are g(11B) = 1.792433, g(19F) = 5.257736, g(27Al) = 1.456603, g(69Ga) = 1.344393, g(115In) = 1.231289, and g(205Tl) = 3.276429, all taken from ref 18. For the spin-rotation constants, we shall discuss the values of the nonzero tensor component, that is, the component perpendicular to the molecular axis. Nonrelativistic Ab Initio Calculations. The nonrelativistic calculations were performed using the coupled-cluster analytic linear response methods developed by Gauss and coworkers.19−22 The CFOUR program,23 locally modified to incorporate the g factors for the heavy nuclei, was used in the nonrelativistic CCSD, coupled-cluster singles-and-doubles;19 CCSD(T)−CCSD with a perturbative triples correction;20 and CCSDT (singles-doubles-triples) calculations.21 For BF, we also used the MRCC program24 to perform the calculations at the CCSDTQ (-quadruples) level.22 We shall discuss primarily the CCSD(T) results obtained with the uncontracted version of the ANO-RCC25 basis sets (denoted as unc-ANO-RCC); these basis sets were available for all atoms. To check the convergence of the results with the extension of the basis set, we used for BF, AlF, and GaF a series 9589
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Table 1. Ab Initio Spin-Rotation Constants (in kHz) of Nuclei in XF (X = B, Al, Ga, In, Tl) basis set
HF
CCSD
CCSD(T)
CCSDT
HF
CCSD
B aug-cc-pCVDZa aug-cc-pCVTZ aug-cc-pCVQZ aug-cc-pCV5Z unc-ANO-RCC
−15.41 −16.53 −16.75 −16.82 −16.80
−14.01 −15.60 −15.96 −16.07 −16.01
aug-cc-pCVTZ aug-cc-pVQZ aug-cc-pCVQZ aug-cc-pCV5Z unc-ANO-RCC
−8.39 −7.80 −8.40 −8.42 −8.41
−8.52 −7.89 −8.57 −8.60 −8.59
aug-cc-pCVDZ aug-cc-pCVTZ aug-cc-pVQZ aug-cc-pCVQZ unc-ANO-RCC
−12.23 −12.66 −11.70 −12.87 −12.84
−12.95 −13.66 −12.70 −13.95 −14.01
unc-ANO-RCC
−12.54
−13.58
−13.90 −15.48 −15.84 −15.94 −15.88
−13.92 −15.51 − − −15.90
−131.77 −133.29 −135.02 −135.70 −135.69
−124.74 −130.29 −131.88 −132.43 −132.33
−8.59 −7.95 −8.64 −8.68 −8.66
−8.60 − − − −
−33.71 −34.16 −34.36 −34.55 −34.52
−34.76 −34.59 −35.30 −35.48 −35.38
−13.13 −13.92 −12.93 −14.24 −14.30
−13.15 − − − −
−22.64 −23.15 −23.53 −23.49 −23.59
−23.99 −25.23 −25.19 −25.57 −25.77
−13.95
−
−13.76
−15.90
Al
−127.29 −133.01 − − −135.14
−35.99 −35.91 −36.58 −36.76 −36.66
−36.09 − − − −
−25.17 −26.57 −26.58 −26.98 −27.19
−25.27 − − − −
−17.10
−
−
−
F
Tl −45.55
−127.06 −132.77 −134.45 −135.02 −134.92
F
In
unc-ANO-RCC
CCSDT
F
Ga
a
CCSD(T) F
F
−49.27
−
−
−10.73
−12.74
At the CCSDTQ level: C(B) = −13.91 kHz; C(F) = −127.44 kHz.
Table 2. Ab Initio NMR Shielding Constants (in ppm) of Nuclei in XF (X = B, Al, Ga, In, Tl) basis set
HF
CCSD
CCSD(T)
CCSDT
HF
CCSD
CCSD(T)
CCSDT
B aug-cc-pCVDZa aug-cc-pCVTZ aug-cc-pCVQZ aug-cc-pCV5Z unc-ANO-RCC
85.51 77.37 75.79 75.23 75.39
96.32 84.76 82.13 81.34 81.83
aug-cc-pCVTZ aug-cc-pVQZ aug-cc-pCVQZ aug-cc-pCV5Z unc-ANO-RCC
576.92 591.99 576.77 576.40 576.53
573.78 589.55 572.67 571.94 572.27
aug-cc-pCVDZ aug-cc-pCVTZ aug-cc-pVQZ aug-cc-pCVQZ unc-ANO-RCC
2121.03 2103.92 2144.20 2094.75 2096.22
2090.20 2061.11 2100.06 2048.77 2046.42
unc-ANO-RCC
4155.68
4089.86
97.13 85.73 83.13 82.36 82.83
97.00 85.54 − − 82.64
133.04 129.38 125.02 123.32 123.33
150.38 136.98 133.11 131.74 131.90
144.38 130.56 126.44 125.02 125.18
143.78 129.93 − − 124.63
572.10 587.95 570.91 570.15 570.53
571.88 − − − −
235.83 232.77 231.35 230.05 230.26
228.33 229.59 224.71 223.56 224.13
219.59 220.21 215.65 214.39 215.04
218.90 − − − −
2082.49 2049.83 2090.31 2036.48 2034.27
2081.57 − − − −
228.39 223.28 219.17 219.67 218.61
213.02 200.49 200.98 197.02 194.81
200.21 185.89 185.87 181.71 179.40
199.08 − − − −
4066.59
−
270.16
238.38
220.40
−
−
−
287.38
252.29
−
−
Al
Ga
In Tl unc-ANO-RCC a
8622.19
8518.03
At the CCSDTQ level: σ(B) = 97.08 and σ(F) = 143.38 ppm.
absolute shielding constants to ensure overall consistency of the results. We assume in the following discussion of all the properties that correlation and relativistic effects are approximately additive. Thus, the computed relativistic corrections are added to the best nonrelativistic coupled-cluster results. Whereas this is an approximation, the rather small electron correlation effects suggest that the errors introduced by this approximation will be small.
We used Dyall’s relativistic augmented cvqz basis sets in uncontracted form for all atoms (unpublished for B, F, and Al;43 see ref 44 for Ga, In, and Tl). The four-component values obtained from the calculations performed using the PBE45 functional are compared in Table 4 with the corresponding nonrelativistic results, and the relativistic effects are estimated from the differences between the relativistic and nonrelativistic values. The common gauge-origin approach was used in the four-component calculations of both spin−rotation and 9590
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Table 3. Ab initio Ω, Anisotropies of Shielding Constants (in ppm) of Nuclei in XF (X = B, Al, Ga, In, Tl) basis set
HF
CCSD
CCSD(T)
CCSDT
HF
CCSD
B aug-cc-pCVDZa aug-cc-pCVTZ aug-cc-pCVQZ aug-cc-pCV5Z unc-ANO-RCC
185.69 197.58 199.92 200.74 200.51
171.27 188.32 192.21 193.37 192.68
aug-cc-pCVTZ aug-cc-pVQZ aug-cc-pCVQZ aug-cc-pCV5Z unc-ANO-RCC
322.47 299.86 322.71 323.27 323.02
328.07 303.69 329.87 331.06 330.36
aug-cc-pCVDZ aug-cc-pCVTZ aug-cc-pVQZ aug-cc-pCVQZ unc-ANO-RCC
776.87 803.45 742.99 817.22 814.95
822.74 867.54 806.97 886.40 889.82
unc-ANO-RCC
1189.47
1288.66
CCSD(T)
CCSDT
F 170.32 187.21 191.06 192.19 191.52
170.53 187.50 − − 191.81
527.05 531.87 538.49 541.07 541.07
499.56 519.95 525.97 528.10 527.71
508.36 529.33 535.71 537.93 537.54
509.26 530.27 − − 538.36
330.77 306.26 332.70 333.94 333.16
331.11 − − − −
375.32 379.96 382.08 384.05 383.74
386.15 384.28 391.73 393.54 392.55
399.05 398.13 405.12 407.08 405.98
400.09 − − − −
834.44 884.55 821.60 904.92 908.13
835.84 − − − −
391.78 399.26 405.49 404.73 406.36
413.44 432.97 432.31 438.35 441.61
432.44 454.62 454.71 461.06 464.48
434.14 − − − −
1323.67
−
333.07
380.28
407.00
−
−
−
Al
Ga
In Tl unc-ANO-RCC a
1907.49
2063.86
F −
−
310.70
362.88
At the CCSDTQ level: Ω(B) = 170.44 and Ω(F) = 509.83 ppm.
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(−)17.89(15) kHz, whereas we obtain −16.90 kHz). Overall, the results show satisfying agreement between the experimental and calculated spin−rotation constants. Moreover, the results confirm that when dealing with spin−rotation constants of heavy nuclear centers, as well as light atoms in the vicinity of heavy atoms, the neglect of relativistic effects may lead to wrong conclusions. NMR Shielding Constants. The absolute shielding constants are listed in Table 6. As expected, relativistic effects increase going down the group, and the same trend is also observed for the vibrational effects, the largest vibrational correction being obtained for TlF. The vibrational corrections to σ(19F) are relatively larger compared to the other nuclei. Since there are no experimental absolute shielding constants for all the molecules studied here, the results presented in Table 6 obtained from high-level computational procedures can be considered as accurate new results for all the nuclei. Let us discuss in some detail the role of the relativistic effects in InF. The paramagnetic contribution to the absolute shielding (σpara) of 115In determined from the experimental spin−rotation constant is −1153 ppm.47 The corresponding result obtained in this work from the four-component relativistic calculation is −495.56 ppm, while the nonrelativistic value is −1123.93 ppm, the latter being in good agreement with the experimental value. On the other hand, for 19F in InF, σpara determined from experiment is −510 ppm,47 whereas the relativistically calculated value is −408.09 ppm and the nonrelativistic value is −390.16 ppm. These results show that determining σpara from the electronic contribution to the spin−rotation constant leads to large errors both for this contribution and the total absolute shielding, as also observed in previous studies.9,12 It is noteworthy, for example, that the nonrelativistic σSR of 115In is −1124.01 ppm and of 205Tl is −1846.87 ppm, whereas σSR calculated from relativistic spin-rotation constants are −1408.34 ppm and −4742.28 ppm, respectively, with the relativistic effect amounting to 20% for 115In and 61% for 205Tl, respectively.
RESULTS AND DISCUSSION Spin−Rotation Constants. Theoretical and experimental values of spin−rotation constants are compared in Table 5 using the notation (−) for the sign of the experimental results as different conventions are applied for spin-rotation constants. The total calculated values, including both relativistic and vibrational corrections, are in satisfactory agreement with the corresponding available experimental results. For instance, the calculated C(11B) is −15.77 kHz, whereas the experimental value is (−)16.6(2) kHz.46 The relativistic effect on the spinrotation constants is not significant for C(11B) and C(27Al) but contributes 8.6% for C(69Ga) and increases to ∼25% for the spin-rotation constant C(115In) and 68% for C(205Tl). On the other hand, rather small vibrational corrections are obtained for all the spin-rotation constants of 11B, 27Al, 69Ga, 115In, and 205 Tl. The calculated values for C(19F) are in good agreement with available experimental data (for instance, with errors of 5% and 3%, respectively, for C(19F) in BF and InF). There is no experimental spin-rotation constant for C(F) in AlF. However, considering the level of approximation, our nonrelativistic value for C(F) in AlF is likely to be more accurate than that previously reported in ref 15. Our results for C(Al) and C(F) in AlF differ from the very accurate CCSD(T) of Teale et al.,7 8.36 and 36.23 kHz. This difference appears to be due to the fact that their optimized geometry of 1.674942 Å, obtained at the CCSD(T)/cc-pVTZ level, is significantly longer than the experimental bond length, 1.654369 Å, as well as the geometry we get by optimizing at the CCSD(T) level with the unc-ANORCC basis set, 1.658988 Å. The relativistic effects on the fluorine spin−rotation constants are small compared to the other nuclei. For example, in InF the relativistic effect contributes only 8% to C(19F), whereas it is ∼25% for C(115In). Even though the calculated and the old experimental values of C(205Tl) differ noticeably, there is satisfactory agreement for C(F) in TlF (the experimental value is 9591
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Table 4. Four-Component Relativistic and Nonrelativistic Spin-Rotation Constants (C in kHz), Shielding and Span (σ and Ω in ppm) in XF (X = B, Al, Ga, In, and Tl) Calculated at the DFT Level (PBE/Dyall-aug-CVQZ)a σ(X)
C(X) NR
ΔC
rel.
NR
Δσ
rel.
NR
ΔΩ
BF 27 AlF 69 GaF 115 InF 205 TlF
−17.68 −9.91 −18.32 −21.70 −168.51
−17.70 −9.84 −17.06 −17.20 −64.81 C(F)
0.02 −0.07 −1.27 −4.50 −103.69
69.96 552.58 2079.11 4553.04 11752.96
69.09 540.52 1917.60 3861.29 8083.98 σ(F)
0.87 12.07 161.52 691.75 3668.98
212.56 382.72 1176.09 2121.45 7799.87
212.50 378.57 1083.59 1632.33 2715.22 Ω(F)
0.06 4.15 92.49 489.13 5084.65
rel.
NR
ΔC
rel.
NR
Δσ
rel.
NR
ΔΩ
BF AlF GaF InF TlF
−154.65 −45.58 −36.58 −25.60 −24.69
−154.22 −45.36 −35.77 −24.13 −20.50
−0.44 −0.22 −0.81 −1.47 −4.19
78.98 156.69 82.30 98.35 40.52
75.58 153.70 86.51 116.17 116.88
3.40 2.99 −4.21 −17.82 −76.36
613.62 500.93 620.50 606.42 712.45
611.35 497.63 603.41 562.87 565.27
2.27 3.30 17.09 43.55 147.18
11
a
Ω(X)
rel.
ΔC, Δσ, and ΔΩ are the differences between relativistic (rel.) and nonrelativistic (NR) values.
Table 5. Comparison of Calculated and Experimental Results for Spin-Rotation Constants (in kHz) of Nuclei in XF (X = B, Al, Ga, In, and Tl) BF
AlF
GaF
InF
TlF
C(X) CCSD(T)/unc-ANO-RCCa full triplesb relativistic correctionc rovibrational correction (0 K)d total exp.
−15.88 −0.03 0.02 0.12 −15.77 (−)16.6(2)e
−8.66 −0.01 −0.07 0.05 −8.69 (−)8.2(13)f
CCSD(T)/unc-ANO-RCCa full triplesb relativistic correctionc rovibrational correction (0 K)d total exp.
−134.92 −0.22 −0.44 0.21 −135.36 (−)143(14)e
C(F) −36.66 −0.10 −0.22 0.09 −36.89 −
−14.30 −0.02 −1.27 0.08 −15.51 (−)14.86(56)g (−)15.3(18)h (−)14(5)i
−13.95 − −4.50 0.07 −18.38 (−)17.47(10)j (−)17.50(1)k
−49.27 − −103.69 0.25 −152.71 (−)126.03(12)l
−27.19 −0.10 −0.81 0.06 −28.03 (−)32.0(21)g
−17.10 − −1.47 0.04 −18.53 (−)18.3(10)j (−)18.77(10)k
−12.74 − −4.19 0.03 −16.90 (−)17.89(15)l
a
CCSD(T)/unc-ANO-RCC values (CCSD for TlF) at the equilibrium geometries. bFull triples correction: CCSDT-CCSD(T), see the text and Table 1. cRelativistic correction: PBE, uncontracted Dyall aug-cvqz basis set, see the text. dNonrelativistic results, unc-ANO-RCC basis set; BF, AlF, GaF−CCSD(T), InF, TlF−CCSD. eRef 46. fRef 47. gRef 15. hRef 48. iRef 49. jRef 47. kRef 50. lRef 51.
As mentioned in the introduction, for linear molecules, Ω corresponds to the anisotropy of the shielding tensor. Among the molecules studied here, the experimental value for Ω(69Ga) determined by Wasylishen et al.15 directly from the spin− rotation constant is 949(35) ppm and our calculated value including relativistic and vibrational corrections for Ω(69Ga) is 997.48 ppm. Wasylishen et al. also estimated the total absolute shielding constant (based on the electronic contribution to the spin−rotation constant) as 2005 ppm, whereas we obtain 2197.86 ppm, showing that the previously reported value is off by 9% from the value which includes relativistic corrections. The previously reported approximate values of the shielding spans for 115In in InF and 205Tl in TlF are 1660 and 5300 ppm, respectively, both obtained without relativistic corrections.15 For Ω(19F), there is good agreement between our calculated value and the available experimental datum, 543.83 and 549 ppm, respectively. On the other hand, the previously reported calculated value of Ω(19F) in AlF is 443 ppm,15 whereas we
obtain 410.54 ppm. The relativistic effects play an increasingly important role as one proceeds to the heavier elements in the periodic table (see Table 4), and our calculated values for Ω(115In) and Ω(205Tl) are accordingly 1805 and 7136 ppm, respectively, showing the importance of the relativistic effects in determining the shielding spans. For instance, the previously reported Ω(205 Tl) based on experimental spin−rotation constant15 is smaller by 26% than our relativistically corrected value. Overall, for the light nuclei there is good agreement between the calculated values and those derived from experimental data, which is not the case of the heavier nuclear centers (see Table 6).
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CONCLUSIONS In this contribution, we have investigated the spin−rotation and absolute shielding constants of the monofluorides of group IIIA elements. We have presented results obtained from high-level ab initio nonrelativistic and four-component relativistic DFT 9592
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having well-established spin-rotation constants, the approach we have followed reproduced the experimental values (in most cases within ±5%, corresponding to 2−3 times the experimental standard deviation). An exception is C(205Tl), with the error bar reported being much smaller than for the other molecules and the computed spin-rotation constant dominated by the huge relativistic correction. On the other hand, for the molecules we studied, there are no experimental absolute shielding constants and only a few shielding spans for some of the nuclei. Therefore, considering the importance of both the spin−rotation and absolute shielding constants from the theoretical and experimental point-of-view, accurate results for all the nuclei of the studied molecules should be very useful, especially when there are no available experimental values.
Table 6. Comparison of Calculated and Experimental Results (in ppm) for Absolute Shielding (σ) and Shielding Span (Ω) of Nuclei in XF (X = B, Al, Ga, In, and Tl)
CCSD(T)/uncANO-RCCa full triplesb relativistic correctionc rovibrational correction (300 K)d total CCSD(T)/uncANO-RCCa full triplesb relativistic correctionc rovibrational correction (300 K)d total
BF
AlF
GaF
InF
TlF
82.83
σ(X) 570.53
2034.27
4066.59
8518.03
−0.19 0.87
−0.22 12.07
−0.93 161.52
− 691.75
− 3668.98
−0.03
0.84
3.13
5.01
8.31
83.48
2197.99
4763.35
12195.32
125.18
583.20 σ(F) 215.04
179.40
220.40
252.29
−0.55 3.40
−0.70 2.99
−1.13 −4.21
− −17.82
− −76.36
−3.04
−1.09
−1.12
−0.67
−0.60
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Tel: +48-223432333. Fax: +48-226326681. Notes
The authors declare no competing financial interest. 124.99
216.25 Ω(X) 333.16
172.94
201.91
175.33
908.13
1323.67
2063.86
0.29 0.06
0.34 4.15
1.40 92.50
− 489.13
− 5084.65
−0.15
−1.33
−4.76
−7.56
−12.50
191.72 −
336.32 997.26 − 949(35) Ω(F) 405.98 464.48
1805.24 1660
7136.01 5300
407.00
362.88
CCSD(T)/uncANO-RCCa full triplesb relativistic correctionc rovibrational correction (300 K)d total exp.e
191.52
CCSD(T)/uncANO-RCCa full triplesb relativistic correctionc rovibrational correction (300 K)d total exp.e
537.54 0.83 2.27
1.04 3.30
1.69 17.09
− 43.55
− 147.18
4.51
1.57
1.57
0.87
0.74
545.14 549
411.89 443
484.83 518
451.42 419
■
ACKNOWLEDGMENTS We are grateful to Dr. Kenneth G. Dyall for providing us the basis sets for B, F, and Al prior to publication. We are indebted to Mr. Piotr Kuszaj for his help in the nonrelativistic calculations. This work has received support from the Research Council of Norway through a Centre of Excellence grant (Grant 179568) and a research grant (Grant 177558). Support from the European Research Council through a Starting Grant is also gratefully acknowledged (Grant 279619). This work has benefited from computer time provided by the Norwegian supercomputing program NOTUR (Grant NN4654K). T.B.D. acknowledges the Centre for Theoretical and Computational Chemistry, The Norwegian Centre of Excellence, for financial support.
■
REFERENCES
(1) Ramsey, N. F. Magnetic Shielding of Nuclei in Molecules. Phys. Rev. 1950, 78, 699−703. (2) Flygare, W. H. Spin-Rotation Interactions and Magnetic Shielding in Molecules. J. Chem. Phys. 1964, 41, 793−800. (3) Flygare, W. H. Magnetic Interactions in Molecules and an Analysis of Molecular Electronic Charge Distribution from Magnetic Parameters. Chem. Rev. 1974, 74, 653−687. (4) Bryce, D. L.; Wasylishen, R. E. Microwave Spectroscopy and Nuclear Magnetic Resonance Spectroscopy: What Is the Connection? Acc. Chem. Res. 2003, 36, 327−334. (5) Jaszuński, M.; Jackowski, K. Nuclear Magnetic Dipole Moments from NMR Spectra: Quantum Chemistry and Experiment. In Lecture Notes in Physics; Karshenboim, S. G., Ed.; Springer: New York, 2008; Vol. 745, pp 233−260. (6) Puzzarini, C.; Stanton, J. F.; Gauss, J. Quantum-Chemical Calculation of Spectroscopic Parameters for Rotational Spectroscopy. Int. Rev. Phys. Chem. 2010, 29, 273−367. (7) Teale, A. M.; Lutnæs, O. B.; Helgaker, T.; Tozer, D. J.; Gauss, J. Benchmarking Density-Functional Theory Calculations of NMR Shielding Constants and Spin-Rotation Constants using Accurate Coupled-Cluster Calculations. J. Chem. Phys. 2013, 138, 024111. (8) Ruud, K.; Demissie, T. B.; Jaszuński, M. Ab Initio and Relativistic DFT Study of Spin-Rotation and NMR Shielding Constants in XF6 Molecules, X = S, Se, Te, Mo and W. J. Chem. Phys. 2014, 140, 194308. (9) Malkin, E.; Komorovsky, S.; Repisky, M.; Demissie, T. B.; Ruud, K. The Absolute Shielding Constants of Heavy Nuclei: Resolving the
510.80 467
a
CCSD(T)/unc-ANO-RCC values (CCSD for TlF) at the equilibrium geometries. bFull triples correction: CCSDT-CCSD(T), see the text, Tables 2 and 3. cRelativistic correction: PBE, uncontracted Dyall aug-cvqz basis set, see the text. dNonrelativistic results, unc-ANORCC basis set; BF, AlF, GaF−CCSD(T), InF, TlF−CCSD. e Approximate values derived using experimental spin-rotation constants.15 See also refs 46 (BF), 49 (AlF), 47 (InF), and 51 (TlF).
calculations, including also rovibrational corrections. As expected, the relativistic effects on the spin-rotation constants increase as one goes down the group; for instance it is insignificant for C(11B) but increases to 68% for C(205Tl) (see Table 5). The same is true for the absolute shielding constants. We included also the rovibrational corrections, which are very small compared to the relativistic effects. The overall analysis of the results shows that there is good agreement between the available experimental and our calculated values. The results also show the importance of considering relativistic corrections in determining calculated spin−rotation as well as absolute shielding constants, derived experimentally in most cases from measured spin−rotation constants. In general, for the nuclei 9593
dx.doi.org/10.1021/jp507129y | J. Phys. Chem. A 2014, 118, 9588−9595
The Journal of Physical Chemistry A
Article
Enigma of the 119Sn Absolute Shielding. J. Phys. Chem. Lett. 2013, 4, 459−463. (10) Jaszuński, M.; Repisky, M.; Demissie, T. B.; Komorovsky, S.; Malkin, E.; Ruud, K.; Garbacz, P.; Jackowski, K.; Makulski, W. SpinRotation and NMR Shielding Constants in HCl. J. Chem. Phys. 2013, 139, 234302. (11) Aucar, I. A.; Gómez, S. S.; Melo, J. I.; Giribet, C. C.; Ruiz de Azúa, M. C. Theoretical Study of the Nuclear Spin-Molecular Rotation Coupling for Relativistic Electrons and Non-Relativistic Nuclei. II. Quantitative Results in HX (X=H,F,Cl,Br,I) Compounds. J. Chem. Phys. 2013, 138, 134107. (12) Xiao, Y.; Zhang, Y.; Liu, W. New Experimental NMR Shielding Scales Mapped Relativistically from NSR: Theory and Application. J. Chem. Theory Comp. 2014, 10, 600−608. (13) Kaupp, M. Relativistic Effects on NMR Chemical Shifts. In Relativistic Electronic Structure Theory. Part 2. Applications; Schwerdtfeger, P., Ed.; Elsevier: Amsterdam, 2004; pp 552−597. (14) Autschbach, J. Relativistic Calculations of Magnetic Resonance Parameters: Background and some Recent Developments. Philos. Trans. R. Soc., A 2014, 372, 20120489. (15) Wasylishen, R. E.; Bryce, D. L.; Evans, C. J.; Gerry, M. C. L. Hyperfine Structure in the Rotational Spectrum of GaF: A Comparison of Experimental and Calculated Spin-Rotation and Electric Field Gradient Tensors. J. Mol. Spectrosc. 2000, 204, 184−194. (16) Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules; Van Nostrand Reinhold: New York, 1979. (17) Coveney, P. V.; Sandars, P. G. H. Parity- and Time-Violating Interactions in Thallium Fluoride. J. Phys. B: At. Mol. Phys. 1983, 16, 3727−3740. (18) Raghavan, P. Table of Nuclear Moments. At. Data Nucl. Data Tables 1989, 42, 189−291. (19) Gauss, J.; Stanton, J. F. Gauge-Invariant Calculation of Nuclear Magnetic Shielding Constants at the Coupled-Cluster Singles and Doubles Level. J. Chem. Phys. 1995, 102, 251−253. (20) Gauss, J.; Stanton, J. F. Perturbative Treatment of Triple Excitations in Coupled-Cluster Calculations of Nuclear Magnetic Shielding Constants. J. Chem. Phys. 1996, 104, 2574−2583. (21) Gauss, J. Analytic Second Derivatives for the Full CoupledCluster Singles, Doubles, and Triples Model: Nuclear Magnetic Shielding Constants for BH, HF, CO, N2, N2O, O3. J. Chem. Phys. 2002, 116, 4773−4776. (22) Kállay, M.; Gauss, J. Analytic Second Derivatives for General Coupled-Cluster and Configuration-Interation Models. J. Chem. Phys. 2004, 120, 6841−6848. (23) CFOUR, A. Quantum Chemical Program Package written by Stanton, J. F.; Gauss, J.; Harding, M. E.; Szalay, P. G. with Contributions from Auer, A. A.; Bartlett, R. J.; Benedikt, U.; Berger, C.; Bernholdt, D. E.; Bomble, J. et al. and the Integral Packages MOLECULE (Almlöf, J. Taylor, P. R.), PROPS (Taylor, P. R.), ABACUS (Helgaker, T.; Jensen, H. J. Aa.; Jørgensen, P. Olsen, J.), and ECP Routines by Mitin, A. V. van Wüllen, C. (for the current version, see http://www.cfour.de). (24) MRCC, A Quantum Chemical Program Suite written by Kállay, M.; Rolik, Z.; Ladjánszki, I.; Szegedy, L.; Ladóczki, B.; Csontos, J.; Kornis, B. See also Rolik, Z.; Szegedy, L.; Ladjánszki, I.; Ladóczki, B. Kállay, M. J. Chem. Phys. 139, 094105, 2013 (and www.mrcc.hu). (25) Roos, B. O.; Lindh, R.; Malmqvist, P.-Å.; Veryazov, V.; Widmark, P.-O. Main Group Atoms and Dimers Studied with a New Relativistic ANO Basis Set. J. Phys. Chem. A 2004, 108, 2851−2858. (26) Kendall, R. A.; Dunning, T. H., Jr.; Harrison, R. J. Electron Affinities of the First-Row Atoms Revisited. Systematic Basis Sets and Wave Functions. J. Chem. Phys. 1992, 96, 6796−6806. (27) Woon, D. E.; Dunning, T. H., Jr. Gaussian Basis Sets for use in Correlated Molecular Calculations. III. The Atoms Aluminium through Argon. J. Chem. Phys. 1993, 98, 1358−1371. (28) Woon, D. E.; Dunning, T. H., Jr. Gaussian Basis sets for use in Correlated Molecular Calculations. IV. Calculation of Static Electrical Response Properties. J. Chem. Phys. 1994, 100, 2975−2988.
(29) Woon, D. E.; Dunning, T. H., Jr. Gaussian Basis Sets for use in Correlated Molecular Calculations. V. Core-Valence Basis Sets for Boron through Neon. J. Chem. Phys. 1995, 103, 4572−4585. (30) Wilson, A. K.; Woon, D. E.; Peterson, K. A.; Dunning, T. H., Jr. Gaussian Basis Sets for use in Correlated Molecular Calculations. IX. The Atoms Gallium through Krypton. J. Chem. Phys. 1999, 110, 7667− 7676. (31) DeYonker, N. J.; Peterson, K. A.; Wilson, A. K. Systematically Convergent Correlation Consistent Basis Sets for Molecular CoreValence Correlation Effects: The Third-Row Atoms Gallium through Krypton. J. Phys. Chem. A 2007, 111, 11383−11393. (32) London, F. Théorie Quantique des Courants Interatomiques dans les Combinaisons Aromatiques. J. Phys. Radium 1937, 8, 397− 409. (33) Wolinski, K.; Hinton, J. F.; Pulay, P. Efficient Implementation of the Gauge-Independent Atomic Orbital Method for NMR Chemical Shift Calculations. J. Am. Chem. Soc. 1990, 112, 8251−8260. (34) Gauss, J.; Ruud, K.; Helgaker, T. Perturbation-Dependent Atomic Orbitals for the Calculation of Spin-Rotation Constants and Rotational g Tensors. J. Chem. Phys. 1996, 105, 2804−2812. (35) Auer, A. A.; Gauss, J.; Stanton, J. F. Quantitative Prediction of Gas-Phase 13C Nuclear Magnetic Shielding Constants. J. Chem. Phys. 2003, 118, 10407−10417. (36) Gauss, J.; Stanton, J. F. Analytic CCSD(T) Second Derivatives. Chem. Phys. Lett. 1997, 276, 70−77. (37) Stanton, J. F.; Gauss, J. Analytic Second Derivatives in HighOrder Many-Body Perturbation and Coupled-Cluster Theories: Computational Considerations and Applications. Int. Rev. Phys. Chem. 2000, 19, 61−95. (38) Harding, M. E.; Lenhart, M.; Auer, A. A.; Gauss, J. Quantitative Prediction of Gas-Phase 19F Nuclear Magnetic Shielding Constants. J. Chem. Phys. 2008, 128, 244111. (39) Ruden, T. A.; Ruud, K. Ro-Vibrational Corrections to NMR Parameters. In Calculation of NMR and EPR parameters. Theory and Applications; Kaupp, M., Bühl, M., Malkin, V. G., Eds, Wiley-VCH: Weinheim, 2004; pp 153−173. (40) ReSpect, version 3.2, 2013; Relativistic Spectroscopy DFT Program of Authors Repisky, M.; Komorovsky, S.; Malkin, V. G.; Malkina, O. L.; Kaupp, M.; Ruud, K. with contributions from Bast, R.; Ekstrom, U.; Knecht, S.; Malkin Ondik, I.; Malkin, E. (41) Komorovsky, S.; Repisky, M.; Malkina, O. L.; Malkin, V. G.; Malkin Ondik, I.; Kaupp, M. A Fully Relativistic Method for Calculation of Nuclear Magnetic Shielding Tensors with a Restricted Magnetically Balanced Basis in the Framework of the Matrix DiracKohn-Sham Equation. J. Chem. Phys. 2008, 128, 104101. (42) Komorovsky, S.; Repisky, M.; Malkina, O. L.; Malkin, V. G. Fully Relativistic Calculations of NMR Shielding Tensors using Restricted Magnetically Balanced Basis and Gauge Including Atomic Orbitals. J. Chem. Phys. 2010, 132, 154101. (43) Dyall, K. G. Basis Sets Available from the Dirac Web Site, http://dirac.chem.sdu.dk (unpublished). (44) Dyall, K. G. Relativistic Quadruple-Zeta and Revised Triple-Zeta and Double-Zeta Basis Sets for the 4p, 5p, and 6p Elements. Theor. Chem. Acc. 2006, 115, 441−447, (Basis Sets Available from the Dirac Web site, http://dirac.chem.sdu.dk). (45) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (46) Cazzoli, G.; Cludi, L.; Degli Esposti, C.; Dore, L. The Millimeter and Submillimeter-Wave Spectrum of Boron Monofluoride: Equilibrium Structure. J. Mol. Spectrosc. 1989, 134, 159−167. (47) Hensel, K. D.; Gerry, M. C. L. Fourier-Transform Microwave Spectroscopy of InF, InCl, and InBr, Prepared using Laser Ablation. J. Chem. Soc., Faraday Trans. 1997, 93, 1053−1059. (48) Honerjäger, R.; Tischer, R. Magnetische Konstanten der Molekeln CuF und GaF. Z. Naturforsch., A 1974, 29, 1919−1921. (49) Hoeft, J.; Lovas, F. J.; Tiemann, E.; Törring, T. Microwave Absorption Spectra of AlF, GaF, InF, and TlF. Z. Naturforsch., A 1970, 25, 1029−1035. 9594
dx.doi.org/10.1021/jp507129y | J. Phys. Chem. A 2014, 118, 9588−9595
The Journal of Physical Chemistry A
Article
(50) Hammerle, R. H.; Van Ausdal, R.; Zorn, J. C. Hyperfine Structure of Indium Fluoride. J. Chem. Phys. 1972, 57, 4068−4070. (51) Boeckh, R. V.; Gräff, G.; Ley, R. Die Abhängigkeit Innerer und Ä usserer Wechselwirkungen des TlF-Moleküls von der Schwingung, Rotation und Isotopie. Z. Phys. 1964, 179, 285−313.
9595
dx.doi.org/10.1021/jp507129y | J. Phys. Chem. A 2014, 118, 9588−9595
