Ab initio study of NMR shielding constants and spin-rotation constants in N, P and As diatomic molecules

Computational and Theoretical Chemistry 970 (2011) 54–60

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Ab initio study of NMR shielding constants and spin-rotation constants in N, P and As diatomic molecules Andrej Antušek a,⇑, Michał Jaszun´ski b, Małgorzata Olejniczak c,d a

Slovak University of Technology in Bratislava, Faculty of Materials Science and Technology in Trnava, Paulinska 16, 917 24 Trnava, Slovak Republic Institute of Organic Chemistry, Polish Academy of Sciences, Kasprzaka 44, 01224 Warsaw, Poland c Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland d Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Tromsø, N-9037 Tromsø, Norway b

a r t i c l e

i n f o

Article history: Received 4 March 2011 Received in revised form 17 May 2011 Accepted 18 May 2011 Available online 30 May 2011 Keywords: NMR shielding constants Spin-rotation constants Coupled cluster method Relativistic corrections

a b s t r a c t Ab initio calculations of NMR shielding constants and spin-rotation constants of six diatomic molecules XY where X, Y = N, P or As are described. We analyse the dependence of the results on the description of the correlation effects and their convergence with the extension of the basis set. Our best results, obtained at the CCSD(T)/cc-pV5Zs,p-unc level, appear to be sufficiently accurate for the smaller diatomics—N2, PN and P2; for the AsN, AsP and As2 molecules large relativistic effects should be taken into account. Moreover, relativistic four-component density functional calculations of NMR shielding constants for the latter molecules demonstrate that the relativistic corrections cannot be determined at the Hartree–Fock level. The computed spin-rotation constants for PN and AsP can be compared with experimental data, the differences between theory and experiment are below 2% for PN and 10% for AsP. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction The nuclear magnetic resonance (NMR) shielding constant describes the shielding of the nucleus by electrons of the molecule. The shielding constant derived from electronic structure calculations is, in agreement with this definition, given with respect to the bare nucleus. A comparison of computed constants with experiment is complicated, because in a standard NMR spectrum only the chemical shift, that is the shielding with respect to a reference molecule, is measured. To verify experimentally the values of the shielding constants one can use spin-rotation constants provided by high resolution rotational spectroscopy. They are related to the so-called paramagnetic contribution to the shielding constant. Electronic structure calculations of the other, diamagnetic contribution, are much simpler than calculations of the paramagnetic contribution. Therefore, an approach combining the paramagnetic term derived from the spin-rotation constant with an accurate ab initio value of the diamagnetic term often yields the most accurate value of the total shielding constant (see e.g. the review [1]). We begin with a brief summary of the theory, focusing on the relation between NMR shielding constants and spin-rotation constants [2,3]. This relation has been successfully used to establish absolute NMR shielding scales for light nuclei [4]. Recent measurements of spin-rotation constants in molecules which contain ⇑ Corresponding author. E-mail address: [email protected] (A. Antušek). 2210-271X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.comptc.2011.05.026

heavier atoms, such as AsP [5], open the possibility to test the accuracy of corresponding theoretical results. We have studied in this work NMR shielding constants and spin-rotation constants of six diatomic molecules XY where X, Y = N, P or As. In the first part, the performance of standard Dunning’s basis sets [6–8] was assessed in the calculations of NMR shielding constants of the smaller diatomics—N2, PN and P2 molecules. Modified basis sets, leading to better convergence of the results, were then used to examine NMR properties and spin-rotation constants in the AsX series of molecules. We have estimated relativistic corrections to the spin-rotation constants from the four-component calculations of NMR shielding tensor. In the last part, reliability of theoretical prediction is discussed in comparison with available experimental spin-rotation data. 2. Theory In this work we discuss two molecular properties describing different effects due to the nuclear magnetic dipole moment, mK, observed in different spectra. The interaction of the magnetic moment with an external magnetic field B, observed in nuclear magnetic resonance (NMR), is described by the nuclear magnetic shielding tensor rK. Its dependence on the electronic structure is determined from the equation

2 d EðB; mK Þ rK ¼ dBdmK

þ1 B¼0; mK ¼0

ð1Þ

A. Antušek et al. / Computational and Theoretical Chemistry 970 (2011) 54–60

where E is the total energy and 1 accounts for the direct Zeeman interaction between the nucleus and the field. In the non-relativistic approach for closed-shell systems the shielding tensor can be straightforwardly written as a sum of two terms para rK ¼ rdia K þ rK

ð2Þ

where rdia K is the diamagnetic contribution calculated as an expectation value and rpara is the paramagnetic contribution, which, as K a second-order property, can be calculated as a linear response function [9]. The spin-rotation tensor CK describes the interaction of the nuclear magnetic dipole moment with the effective magnetic field generated by the angular rotational motion of electric charges (electrons and other nuclei in the molecule) around the nucleus under consideration. This interaction leads to a splitting in the rotational spectrum [3,10], which is proportional to the magnetic moment mK and to the rotational angular momentum J. Within the Born–Oppenheimer approximation the total spin-rotation tensor of nucleus K can be written as a sum of a nuclear part and an electronic part

CK ¼ Cnucl þ Cel K K

ð3Þ Cnucl K ,

The nuclear part, is a simple function of molecular geometry and charges of the nuclei [2]. The electronic contribution, Cel K , can be obtained as a linear response function



Cel K ¼ 

lN g K d2 Eel ðmK ; JÞ dmK dJ 2p m

ð4Þ K;

J¼0

where gK is the nuclear g-factor and lN denotes the nuclear magneton. As shown by Flygare [2,3], in non-relativistic theory the elecpara tronic part Cel ðRK Þ—the paramagnetic K is proportional to rK contribution to the NMR shielding constant calculated with the gauge origin at the position of nucleus K. For high accuracy of the results, it is preferable to compute the total values of the shielding using for instance the gauge invariant atomic orbitals (GIAO) approach [11,12], and subtract the diamagnetic contribution derived from a conventional calculation with RK as the gauge origin. Therefore

Cel K ¼

 1 h  lN g K para h lN g K  GIAO r ðRK ÞI1 ¼ r  rdia K ðR K Þ I l B 2p K l B 2p K

ð5Þ

where I is the molecular tensor of inertia and lB is the Bohr magneton. We stress that whenever we discuss the partition of the shielding constant into dia- and paramagnetic contributions, we shall systematically use the definitions corresponding to Eq. (5). However, we should keep in mind that with increasing charge of the nucleus, relativistic effects are becoming more significant and Eq. (5) may become inadequate for comparison of shielding and spin-rotation constants. Although Flygare’s formula was used for analysis of spin-rotation constants for compounds containing third row elements, leading to satisfactory agreement with experimental values (see e.g. [13–15]), the accuracy of a non-relativistic theory should be reconsidered in case of heavier systems. 3. Computational details The molecules under study have strong covalent triple bonds, therefore the quality of computed values of molecular properties is expected to depend significantly on the treatment of electron correlation. We have taken into account the correlation effects on NMR shielding constants and on spin-rotation constants applying the coupled cluster (CC) response theory [16,17]. In all non-relativistic CC calculations all electrons were correlated. We have used Dunning type basis sets: cc-pVXZ series [6] and cc-pCVXZ series

55

(X = T, Q, 5) with additional core-valence correlation functions [7,8], where available. It is known that cc-pVXZ basis sets are not optimal for calculations of NMR properties; one of their drawbacks is slow and often non-monotonous convergence of computed properties with cardinal number X. However, improvements can be introduced by uncontracting s- and p-functions and adding tight functions for a better description of the region near the nucleus [18]. Perturbation-dependent basis sets (GIAO) are systematically used [10,19] in the coupled cluster theory approach implemented in CFOUR program package [20], which we have applied in this work. In addition to correlation effects, which are essential for studied molecules, also relativistic effects must be taken into account— especially for the AsX systems. Relativistic effects on NMR shielding constants are estimated as differences between four-component relativistic and non-relativistic values. In this analysis, we applied two approaches. First, we considered four-component Density Functional Theory method (DFT) related to non-relativistic DFT, both with KT2 functional [21]. This functional was optimized for calculations of NMR shielding and its choice is based on its good performance for such purpose in highly correlated systems [22]. Secondly, we compared four-component Dirac–Hartree–Fock (DHF) results to nonrelativistic Hartree–Fock (HF) values. The latter approach was successfully used for other systems, see e.g. Ref. [15]. Although this method may be inappropriate for highly correlated systems studied in this work, the corresponding results are also shown for completeness. For all relativistic calculations we used an experimental version of the DIRAC package [23] including four-component DHF linear response with GIAO [24] and newly implemented four-component DFT linear response with GIAO [25]. We applied unrestricted kinetic balance for small component basis set generation, and standard gaussian distributions were used to model nuclear charges. The counterpart non-relativistic HF and DFT/KT2 calculations were performed using the DALTON package [26]. In both non-relativistic and relativistic calculations fully uncontracted cartesian cc-pVXZ basis sets were used. All values of NMR shielding constants and spin-rotation constants were calculated for experimental geometries (bond lengths, re, are presented in Table 1) and compared with analogous values obtained for geometries optimized within CCSD(T) method with triple and quadruple-zeta Dunning basis sets. The convergence of the results with the cardinal number towards the experimental values justifies the use of experimental equilibrium geometries for the calculation of molecular properties.

3.1. NMR shielding constants 3.1.1. Basis set dependence In Table 2, we collected Hartree–Fock and coupled cluster results for isotropic NMR shielding constants of light molecules – N2, PN and P2. These systems were chosen to investigate the convergence of Dunning’s cc-pVXZ and cc-pCVXZ basis set series with the cardinal number X. Although the cc-pVXZ basis sets are not appropriate for all-electron correlated calculations in case of N2 no significant difference of the computed shielding constants between cc-pVXZ and cc-pCVXZ series was found. The convergence with X is monotonous and the convergence trends indicate that the basis set limit is the same for both series. However, the situation is different in case of PN and P2. For standard cc-pVXZ basis sets, non-monotonous behavior of phosphorus shielding constants is encountered, with off-trend cc-pVQZ values obtained at the Hartree–Fock level as well as at CCSD and CCSD(T) levels (see all the PN results in Table 2). This confirms that standard contracted ccpVXZ basis sets are not sufficiently flexible to compute accurate shielding constants for third row atoms (similar strong basis set

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Table 1 Comparison of CCSD(T) optimized and experimental equilibrium geometries (Å).

cc-pVTZ cc-pVQZ exp. a b c d e f

Ref. Ref. Ref. Ref. Ref. Ref.

N2

PN

P2

AsN

AsP

As2

1.10059 1.09809 1.09768(5)a

1.50327 1.49517 1.490867(3)b

1.91194 1.90275 1.8934c

1.59893 1.61476 1.618d

2.01320 2.00486 1.9995440(2)e

2.11069 2.10487 2.103f

HF

CCSDa

CCSD(T)a

345.04 348.96 352.43 346.61 350.26 352.47 351.56 353.78 354.12

326.47 330.73 334.12 327.62 331.62 333.99 333.02 335.41 335.59

[34]. [35]. [36]. [37]. [5]. [38].

Table 2 Isotropic NMR shielding constants in N2, PN and P2 molecules [ppm]. Basis set

HF

N2

N

cc-pVTZs,p-unc cc-pVQZs,p-unc cc-pV5Zs,p-unc cc-pCVTZ cc-pCVQZ cc-pCV5Z

104.69 109.54 112.29 107.13 110.82 112.30

PN

P

CCSD(T)a

56.09 61.31 64.18 59.71 63.79 65.10

50.71 55.71 58.43 54.20 58.10 59.30

66.16 17.38 92.16 96.36 89.76 84.88 89.44 83.86 83.54

75.29 106.28 35.86 48.60 43.56 41.28 45.90 42.81 40.00

99.46 128.40 59.45 73.91 67.66 64.87 70.70 66.53 63.51

P2

P

cc-pVTZs, p-unc cc-pVQZs,p-unc cc-pV5Zs,p-unc

387.10 379.55 374.33

217.51 219.86 223.17

184.20 189.01 191.77

cc-pVTZ cc-pVQZ cc-pV5Z cc-pVTZs,p-unc cc-pVQZs,p-unc cc-pV5Zs,p-unc cc-pCVTZ cc-pCVQZ cc-pwCV5Z

a

CCSDa

N 485.31 486.69 488.41 484.63 486.77 487.85 482.65 484.96 486.82

All electrons were correlated in the coupled cluster calculations.

dependence of the shielding constants in PN has been observed previously [14,27]). The problem was eliminated by uncontracting s- and p-type functions in the standard cc-pVXZ basis sets. The new results, ccpVXZs,p-unc in our notation, show very similar convergence patterns to those obtained with cc-pCVXZ basis sets but the results of similar quality are reached with much smaller number of basis set functions. We note that the usual basis set modification aimed at improving the performance of NMR property calculations— uncontracting of s-type functions only—did not bring any improvement; the results (not included in the table) differed only by a fraction of ppm from the cc-pVXZ values. We have analysed further improvements of basis sets, such as the extension of cc-pVXZs,punc by additional tight s- and p-functions or the addition of standard augmentation functions of Dunning’s basis sets. In both cases the change of the results was negligible. In view of the behavior of Dunning type basis sets for light systems, we analysed NMR shielding constants of AsX (X = N, P, As) primarily for cc-pVXZs,p-unc basis sets. The results are presented in Table 3. The influence of uncontracting on r(N) in AsN was only a few ppm for all basis set used. More pronounced effect of uncontraction was encountered in case of r(P) in AsP, where the change varies up to about hundred ppm in case of quadruple-zeta basis set results, similarly to r(P) in PN and P2 molecules. Uncontracting of s- and p-functions had significant effect on all the r(As) already at Hartree–Fock level—the change was hundreds of ppm (as shown in

Table 3 for AsP). In case of r(As) constants uncontracting of basis set functions did not lead to monotonous convergence with the cardinal number, but the oscillation was decreased by an order of magnitude in comparison to contracted basis sets. As expected, electron correlation has very strong influence on NMR shielding constants in AsX series, shifting each of the constants by hundreds of ppm towards positive values. In case of r(As) in AsX, electron correlation changes deshielding at Hartree–Fock level to positive shielding at coupled cluster level, although for r(As) in As2 the effect is visible at CCSD(T) level only (however, the convergence trend indicates small deshielding as final result for r(As) in As2). The contribution of non-iterative triple excitations is relatively large in all cases, indicating that higher coupled cluster expansion contributions may also be non-negligible. In each case the convergence with the coupled cluster expansion is in opposite direction to the convergence with the basis set, therefore partial cancellation of errors can be expected. Taking into consideration both trends, we do not attempt any extrapolation and we treat CCSD(T)/cc-pV5Zs,p-unc shielding constants as our best non-relativistic results. 3.1.2. NMR shielding constants—overall trends To understand the behavior of NMR shielding constants we have plotted (see Fig. 1) r(N) in NX series, r(P) in PX series and r(As) in AsX series as functions of the atomic number of the X atom, Z(X). This presentation of data reveals common trends in

57

A. Antušek et al. / Computational and Theoretical Chemistry 970 (2011) 54–60 Table 3 Isotropic NMR shielding constants in AsX series (X = N, P, As) [ppm]. basis set

HF

AsN

As

cc-pVTZs,p-unc cc-pVQZs,p-unc cc-pV5Zs,p-unc

a

CCSDa

CCSD(T)a

HF

CCSDa

CCSD(T)a

477.70 487.32 492.37

432.06 440.48 444.89

281.88 243.47 348.46 325.80 334.48 340.07

236.57 201.39 300.25 277.44 287.21 291.69

N

30.21 23.74 25.15

AsP

As

cc-pVTZ cc-pVQZ cc-pV5Z cc-pVTZs,p-unc cc-pVQZs,p-unc cc-pV5Zs,p-unc

170.69 101.36 250.17 339.51 331.63 330.65

As2

As

cc-pVTZs,p-unc cc-pVQZs,p-unc cc-pV5Zs,p-unc

709.04 703.44 706.23

585.03 552.60 538.87

695.41 664.28 651.27

295.62 323.54 174.23 132.71 109.14 95.72

384.01 407.76 268.52 230.04 204.86 193.99

80.20 118.15 136.02

49.14 13.96 0.16

762.27 768.55 773.32 P 517.58 454.60 569.43 568.19 561.99 557.87

All electrons were correlated in the coupled cluster calculations.

800

nonrelativistic σ(N) in NX series nonrelativistic σ(P) in PX series nonrelativistic σ(As) in AsX series relativistic σ(N) in NX series relativistic σ(P) in PX series relativistic σ(As) in AsX series

600

σ [ppm]

400

200

0

-200

-400

-600

5

10

15

20 Z(X)

25

30

35

Fig. 1. Trends in NMR shielding constants.

all the series. We observe first a steep decrease of the shielding when nitrogen atom is replaced by phosphorus atom. Further – much smaller – decrease of the shielding follows due to replacement of phosphorus atom by arsenic atom. Separation of total shielding constants into paramagnetic and diamagnetic parts shows that these trends are fully driven by the paramagnetic shielding, whereas the diamagnetic part is almost constant. Diamagnetic shielding can be considered as an atomic property not much affected by chemical bonds. Indeed, in the AsX series diamagnetic shielding of As nucleus is almost independent of X—the relative differences are less than 0.1%, and variations of diamagnetic shielding of P in PX series and N in NX series are below 0.2% and below 3%, respectively. The changes of NMR shielding constants are in qualitative agreement with the trend of Pauling electronegativities of N, P and As atoms—3.04, 2.19 and 2.18, respectively (similar correlations were observed in other systems, see e.g. [28]). In the AsX series, As2 molecule is non-polar and the paramagnetic shielding is 2878.26 ppm. A replacement of one As atom by a P atom makes the bond slightly polar (AsP dipole moment is 0.15 a.u.) and

paramagnetic shielding of As becomes 2683.16 ppm. In AsN the bond becomes strongly polar, the dipole moment is 1.11 a.u. and the absolute value of As paramagnetic shielding is significantly reduced, to 2224.60 ppm. This analysis suggests that removing (bringing) electron density due to chemical bonding from (to) the atom leads to decrease (increase) of paramagnetic shielding of the nucleus under consideration. The same model works for the PX and NX series. Paramagnetic shielding of P is 1260.29 ppm in AsP, 1159.99 ppm in P2 and 902.09 ppm in PN; these changes are qualitatively consistent with the shift in electron density in chemical bond. Moreover, the magnitudes of the changes correlate with magnitudes of dipole moments, 0.15 a.u. in AsP versus 1.07 a.u. in PN. Finally, paramagnetic shielding of N in AsN is 784.20 ppm. Replacing the As atom by P or N leads to decreased charge on N atom and paramagnetic shielding of N is significantly decreased, to 669.54 ppm and 386.02 ppm in PN and N2, respectively. We note also that correlation effects systematically reduce paramagnetic shielding, leading to smaller deshielding, or—when the diamagnetic part begins to dominate—to a change from

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A. Antušek et al. / Computational and Theoretical Chemistry 970 (2011) 54–60

deshielding to shielding. This is consistent with the fact that correlation effects reduce the dipole moments in these molecules (the values given above were calculated at CCSD(T)/cc-pVQZs,p-unc level). Although it is not possible to separate precisely contributions to NMR shielding from electron density localized on the atoms in the molecule, a description relating the observed effects to electronegativities summarizes the results consistently with a qualitative chemical bond picture and may be applied to predict the properties of heavier systems in the series.

3.1.3. Relativistic corrections The approach assuming additivity of correlation effects and relativistic corrections calculated at Hartree–Fock level was successful for NMR shielding constants and spin–spin coupling constants in hydrides (see e.g. [15,29]), with such an approximation justified either by perturbation treatment [30] or by comparison with experiment [29]. In systems like hydrides the correlation and relativistic effects are small in comparison to HF values; group 15 diatomics are different. The correlation effects are very significant, changing HF value by hundreds of ppm, and adding these contributions and relativistic corrections calculated at HF level is not expected to be an accurate approximation. For this reason we utilized the newly developed relativistic four-component DFT approach [25], which should provide more reliable estimate of relativistic corrections. We have chosen, at both non-relativistic and relativistic levels, the KT2 functional for studies of NMR shielding constants [21]. This choice is justified in our calculations, showing that non-relativistic KT2 density functional results (Table 4) recover at least 70% of the CCSD correlation contribution to the shielding constants in AsX series. Relativistic corrections to isotropic shielding constants for fully uncontracted cc-pVQZ (cc-pVQZunc) basis sets, calculated using

Table 4 Relativistic and non-relativistic shielding constants for cc-pVQZunc basis set and relativistic corrections D to the shielding [ppm]. basis set

rel.

N2

N

non-rel.

rel.

HF KT2 PN

112.24 63.84 P

109.00 59.15

3.24 4.69

HF KT2 P2

76.66 54.67

87.40 47.95

10.74 6.72

P

HF KT2 AsN

376.80 236.42 As

376.10 237.11

0.70 0.69

HF KT2 AsP

55.60 580.74 As

23.92 491.06

31.68 89.68

HF KT2 As2

418.98 16.23 As

330.81 35.17

88.17 51.40

HF KT2

768.42 281.87

704.49 273.70

63.92 8.17

D

non-rel.

D

485.44 364.06

14.31 3.12

859.88 495.29 P

768.32 489.50

91.57 5.79

541.05 369.66

559.49 353.66

18.44 16.00

N 499.75 367.18

N

both approaches: as the difference of relativistic and non-relativistic KT2 results at a correlated level and as the difference of DHF and non-relativistic Hartree–Fock results, are shown in Table 4. Comparison of these two approaches shows no significant differences for N2, PN and P2, but for AsX systems the overall relativistic correction changes dramatically, often even changes its sign, when correlation effects described at the DFT level are simultaneously included. The quality of cc-pVQZunc basis set for calculations of relativistic corrections at HF level was tested against analogous results obtained with cc-pV5Zunc basis set (and, for light systems, ccpCVQZunc basis set). These tests showed that the cc-pVQZunc results are reasonably converged with basis set, the differences between quadruple-zeta and pentuple-zeta relativistic corrections are in most cases <3 ppm (except As2 where the difference was 8 ppm) and the differences between valence and core-valence basis set results were 1 ppm. Our final results for the NMR shielding constants are obtained as the sum of CCSD(T)/cc-pV5Zs,p-unc non-relativistic values and the KT2/cc-pVQZ-unc values of relativistic corrections. These results are collected in Table 5 and plotted, together with non-relativistic shielding constants, in Fig. 1. 3.2. Spin-rotation constants For each nucleus in a diatomic molecule the spin-rotation tensor is diagonal, with two equal components Cxx = Cyy and Czz = 0 (where z is the molecular axis). In Tables 6 and 7 we report ab initio values of equilibrium Cxx components of spin-rotation constants calculated at HF and coupled cluster level of theory, and compare the resuls with available experimental data. Although the corresponding NMR shielding constants strongly depend on the basis set and on the coupled cluster approximation used, the final nonrelativistic CCSD(T)/cc-pV5Zs,p-unc values of spin-rotation constants are well converged. The convergence patterns indicate uncertainties of few per cent in these values. For comparison with experimental spin-rotation constants, zero-point vibrational (ZPV) corrections were calculated for PN and AsP. We used second-order vibrational perturbation theory with harmonic frequencies and cubic force constants calculated at CCSD(T)/cc-pVQZ level and derivatives of spin-rotation constants at CCSD(T)/cc-pVQZs,p-unc level. This level of approximation was tested on N2, and for 15N2 we obtain 0.024 vs. 0.025 kHz in Ref. [19]. For PN molecule we obtain 0.01 kHz for C(P) and 0.001 kHz for C(N), and for AsP we found 0.005 kHz and 0.003 kHz for C(As) and C(P), respectively. ZPV corrections for AsP estimated from experimental data are much larger; it appears that the difference between v = 0 and v = 1 values of both spinrotation constants is overestimated in experiment [5]. The computed ZPV corrections are generally very small in comparison with equilibrium values, due to weak dependence of spin-rotation constants on internuclear distance, and it is very unlikely that a change to a higher level of approximation would increase ZPV corrections by orders of magnitude. We note here that analogous ZPV corrections to NMR shielding constants are also very small. For AsP they are 6.7 ppm for rpara(As) and 3.4 ppm for rpara(P), with both corrections to rdia below 0.02 ppm. These ZPV corrections to the

Table 5 Total isotropic NMR shielding constantsa [ppm]. N2 N a

PN 63.12

P N

P2 71.59 337.11

P

AsN 191.08

As N

AsP 740.95 450.68

CCSD(T)/cc-pV5Zs,p-unc non-relativistic values with KT2/cc-pVQZ-unc values of relativistic corrections.

As P

As2 245.39 307.69

As

8.01

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A. Antušek et al. / Computational and Theoretical Chemistry 970 (2011) 54–60 Table 6 Spin-rotation constants in N2, PN and P2 [kHz].a basis set

a b c d

HF

CCSD

CCSD(T)

15.06 15.27 15.38

14.85 15.05 15.16 15.14

HF

CCSD

CCSD(T)

10.46 10.52 10.56

10.17 10.23 10.27 10.33c 10.4(5)c,d

14

N2

C( N)

cc-pVTZs,p-unc cc-pVQZs,p-unc cc-pV5Zs,p-unc Ref. [19]b

16.96 17.15 17.26

PN

C(31P)

cc-pVTZs,p-unc cc-pVQZs,p-unc cc-pV5Zs,p-unc Ref. [14] Exp.

92.16 91.58 91.16

P2

C(31P)

cc-pVTZs,p-unc cc-pVQZs,p-unc cc-pV5Zs,p-unc

45.38 45.12 44.95

C(14N) 79.55 79.99 80.17

77.35 77.89 78.12 79.61c 78.2(5)c,d

39.68 39.76 39.87

38.56 38.72 38.81

12.56 12.63 12.65

Cxx component calculated at the experimental equilibrium geometry. All electrons were correlated in the coupled cluster calculations. Converted from the 15N2 values. Different sign conventions are in use in theory and experiment; moreover theory indicates that both values have the same sign (see the comments in Ref. [14]). Experimental values for vibrational state m = 0 [31].

Table 7 Spin-rotation constants in AsX series [kHz].a Basis set

a b c

HF

CCSD

CCSD(T)

75

AsN

C( As)

cc-pVTZs,p-unc cc-pVQZs,p-unc cc-pV5Zs,p-unc rel., Eq. (6)c rel., Eq. (7)c

74.58 74.41 74.45

AsP

C(75As)

cc-pVTZs,p-unc cc-pVQZs,p-unc cc-pV5Zs,p-unc rel., Eq. (6)c rel., Eq. (7)c Exp.

29.09 29.02 29.01

As2

C(75As)

cc-pVTZs,p-unc cc-pVQZs,p-unc cc-pV5Zs,p-unc rel., Eq. (6)c rel., Eq. (7)c

17.15 17.13 17.14

HF

CCSD

CCSD(T)

14

C( N) 58.77 59.60 59.96

55.94 56.74 57.07 65.15 65.30

11.74 11.81 11.86

8.68 8.78 8.83

8.18 8.27 8.32 9.47 7.11

C(31P) 24.81 25.02 25.15

23.93 24.16 24.26 26.20 27.99 25.6(7)b

14.14 14.32 14.41

13.52 13.69 13.76 15.71 16.05

32.62 32.48 32.40

27.45 27.64 27.75

26.42 26.63 26.72 27.59 26.16 23.6(2)b

Cxx component calculated at the experimental equilibrium geometry. All electrons were correlated in the coupled cluster calculations. Experimental values [5] for equilibrium geometry. Different sign conventions are in use in theory and experiment. Eq. (6): CCSD(T)/cc-pV5Zs,p-unc non-relativistic values with relativistic KT2/cc-pVQZunc corrections; Eq. (7); relativistic KT2/cc-pVQZunc values of tensor components.

shielding constants are clearly much smaller than the errors due to basis set incompleteness, truncation of the coupled cluster expansion and approximate treatment of relativistic effects. For N2, PN and P2 molecules one can expect accurate prediction of spin-rotation constants from non-relativistic theory. The accuracy of such calculations is confirmed by the comparison of the CCSD(T)/cc-pV5Zs,p-unc spin-rotation constants for PN with experimental data [31], where the agreement is very satisfying— the differences between theoretical and measured values of C(31P) and C(14N) are 1%. In the AsX series, the non-relativistic spin-rotation constants are presumably less accurate. Experimental spin-rotation constants are known only for AsP [5]. Comparing our best non-relativistic CCSD(T)/cc-pV5Zs,p-unc results for AsP with experimental ones, we observe 2–3 kHz differences, exceeding the experimental error bars.

Relativistic effects on NMR shielding constants for AsX systems are significant and one can expect non-negligible effects on spinrotation constants. We shall follow here two approaches [5,32,33] which have been used to relate the total, experimental spin-rotation constants, to the total shielding constants (although, strictly speaking, application of non-relativistic Flygare’s formula, Eq. (5), is not justified when relativistic effects are included). In the first approach, diamagnetic shielding is treated as a property of the atom, not much affected by chemical bonding and we assume that the relativistic effects on rdia for N, P and As can be approximated by relativistic corrections to diamagnetic shielding of closed-shell ions N3+, P3+, As3+. Therefore, we estimate the electronic part of spin-rotation constant as para dia C el / ðrGIAO nonrel þ DÞ  ðrnonrel þ Dion Þ ¼ rnonrel þ D  Dion

ð6Þ

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where D is the relativistic correction to the total shielding constant in the molecule and Dion is the correction for the relevant closedshell ion. The values we use for N3+, P3+, As3+are 1.75, 19.67 and 253.91 ppm, respectively, calculated as differences of relativistic and non-relativistic KT2/cc-pVQZ-unc results. Similar approximations have been used to analyse the results obtained for properties of group 13 fluorides [32] and SbN and SbP [33]. They are based on the equation para dia C el / rk  r? ¼ rdia  rpara ? k  r?  r?

ð7Þ

relating Cel to the span of the shielding tensor. For diatomics, the span is equal to the anisotropy, and practically equal to the paramagnetic part—the contributions of the diamagnetic part to the parallel and perpendicular component are almost identical and cancel out. For instance, in the non-relativistic approach for all the AsX series the differences between the span and the paramagnetic part are below 1%. We have shown in Table 7 relativistically corrected spin-rotation constants obtained applying Eqs. (6) and (7). Both procedures increase electronic part of As spin-rotation constants in all the AsX systems by 10%; for the light nuclei—N in AsN and P in AsP—the corrections derived using Eqs. (6) and (7) differ in sign. In case of AsP, neither the relativistic correction to C(As) nor the correction to C(P) explain the discrepancies between theory and experiment. The accuracy of spin-rotation constants obtained for light molecules is confirmed by good agreement between theoretical and experimental values for PN. To our knowledge, experimental spin-rotation constant for P2 molecule is not available; the predicted CCSD(T)/cc-pV5Zs,p-unc value is 38.8 kHz, with estimated uncertainty of ±2 kHz. The accuracy of spin-rotation constants in AsX series is difficult to assess. Considering the discussed approximate treatment of relativistic effects we may assume that the accuracy of calculated values is 10 kHz for AsN and similar for AsP and As2. Such an estimate is supported by comparison of theoretical and experimental results for AsP.

for theoretical predictions of spin-rotation constants for AsN and As2 systems. Acknowledgements We are indebted to Dr. Magdalena Pecul for reading and commenting on the manuscript. We acknowledge support of the Slovak Grant VEGA 1/0520/10, Polish Ministry of Science and Higher Education research Grant N N204 244134 (2008–2011) and a grant of computer time from the Norwegian Supercomputing Program (Notur). A.A. acknowledges financial support from CTCC during his stay in Tromsø. M.O. was supported by Foundation for Polish Science (Project operated within the Foundation for Polish Science MPD Programme co-financed by the EU European Regional Development Fund) and by PhD Grant, number N N204 116539. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

4. Conclusions We have studied NMR shielding constants and spin-rotation constants in N2, PN, P2 and AsN, AsP, As2 molecules. First, the dependence of the computed properties on the correlation treatment and on the basis set was analysed for the light systems. We applied modifications of valence type Dunning basis sets, which lead to monotonous convergence of shielding constants with the cardinal number of the basis set—similar to that of corresponding core-valence basis sets, but with much smaller number of basis set functions. Partial cancellation of errors due to basis set incompleteness with those due to truncation of the coupled cluster expansion presumably improves the accuracy of the CCSD(T)/ccpV5Zs,p-unc shielding constants, the final non-relativistic values. For the spin-rotation constants we used the same approach, observing noticeably better convergence with the correlation treatment and basis set extension. The main contribution to the uncertainty of the results originates in significant relativistic corrections. For the shielding constants in AsX series these corrections have to be estimated at a correlated level, and we have applied for this purpose DFT approach with KT2 functional, which yields reasonable non-relativistic results. The corresponding effect on the calculated spin-rotation constants is much more difficult to estimate, due to non-relativistic nature of Flygare’s formula. However, a comparison of theoretical predictions with recent experimental data for AsP shows agreement up to several kHz, what can be used as estimate of accuracy

[21] [22] [23]

[24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

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