Hyperfine constants for aluminum hydride and aluminum deuteride

Journal of Molecular Spectroscopy 292 (2013) 8–14

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Hyperfine constants for aluminum hydride and aluminum deuteride Alex Brown ⇑, Roderick E. Wasylishen * Department of Chemistry, University of Alberta, Edmonton, Alberta T6G 2G2, Canada

a r t i c l e

i n f o

Article history: Received 10 June 2013 In revised form 12 August 2013 Available online 20 August 2013 Keywords: Aluminum hydride Aluminum deuteride Nuclear quadrupole coupling constants Spin-rotation constants Magnetic shielding tensors

a b s t r a c t The nuclear quadrupole coupling and spin-rotation constants of aluminum in AlH and AlD have been determined using coupled cluster theory with single and double excitations as well as perturbative inclusion of triples [CCSD(T)] combined with large correlation-consistent basis sets, cc-pCVXZ (X = T, Q and 5) and aug-cc-pCVXZ (X = T, Q). The anharmonic vibrational frequencies have been computed using secondorder vibrational perturbation theory and the effects of vibrational averaging on the hyperfine constants have been determined. The ground state dipole moment has been determined for both isotopologues (AlH and AlD) and shown to depend critically on vibrational averaging. For completeness, the isotropic and anisotropic nuclear magnetic shielding tensors are also reported. All the results agree well with the best available experimental measurements, and in some cases (spin-rotation constants and dipole moments) refine the known data. The present computational results for the vibrationally averaged electric field gradients suggest that the currently accepted nuclear quadruple moment for 27Al of 146:6  1:0 mb may be slightly underestimated. Based on the experimental measurements of the nuclear quadrupole coupling for AlH (AlD) and best computational determinations of the vibrationally averaged electric field gradients, the quadruple moment of 27Al is determined to be 149  2 mb ð148  3 mbÞ. However, this conclusion would be further strengthened with more precise experimental measurement of the 27Al nuclear quadrupole coupling for AlH and AlD. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Aluminum hydride, AlH, the ‘‘simplest’’ aluminum containing molecule, has intrigued spectroscopists and theoreticians for almost 100 years [1–4]. In 1956, Herbig [5] identified AlH as the emitter of bright lines (electronic transitions) observed earlier in the long-period variable star, v Cygni [6]. The first pure rotational spectrum of AlH was reported by Goto and Saito in 1995 [7] and provided the first estimate of the 27Al nuclear quadrupolar coupling constant, C Q (27Al) = 36.72 MHz. In 2001, Gee and Wasylishen [8] pointed out that the analysis of this data did not include perturbations due to the 27Al and 1H nuclear spin-rotation interactions. Their quantum chemistry computations indicated that the 27Al nuclear quadrupolar coupling constant (NQCC) was significantly larger in magnitude, i.e., approximately 49 MHz, and that the 27Al spin rotation constant, C I (27Al), was on the order of 300 kHz. Subsequent laboratory measurements by Halfen and Ziurys [9] determined that C Q (27Al) = 48.59(70) MHz and C I (27Al) = 306(35) kHz for AlH. For the isotopologue, AlD, the corresponding values were determined as C Q (27Al) = 48.48(88) MHz and C I (27Al) = 156(41) kHz. The latter values for AlD have been ⇑ Corresponding authors. Fax: +1 (780) 492 8231. E-mail addresses: [email protected] (A. Brown), [email protected] (R.E. Wasylishen). 0022-2852/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jms.2013.08.003

revised [10] but as explained here the revised spin-rotation value is suspect. One interesting feature concerning the electronic structure of aluminum hydride is the small HOMO–LUMO gap [11]. This property of aluminum hydride makes it challenging to compute parameters such as nuclear spin-rotation tensors and their related magnetic shielding tensors [11,12,8]. The small HOMO–LUMO gap of AlH results in a large paramagnetic contribution to the aluminum magnetic shielding perpendicular to the AlH bond (i.e., significant deshielding of the aluminum nucleus). In fact, the shielding anisotropy for AlH, rk  r?  1000 ppm (vide infra) is comparable to or greater than the range of known aluminum chemical shifts [11]. Furthermore, it is interesting to mention that compared to the aluminum diatomic halides, the aluminum nucleus of AlH is significantly deshielded [12,13]; a result that may be counterintuitive if one incorrectly thinks that magnetic shielding constants are ‘‘in some way’’ related to electron density. The early computations of Gee and Wasylishen [8] are not ‘‘state-of-the-art’’ by today’s standards and clearly need to be replaced. There have been recent ab initio multi-configuration interaction computations focussed on determining the ground state potential energy curve for AlH [14,15] and corresponding vibrational and rotational constants. However, neither of these studies have considered the hyperfine interactions critical for the understanding of the rotational spectrum of AlH (AlD). Therefore,

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the purpose of the present work is to determine, using high-level CCSD(T) computations, spectroscopic constants and the dipole moments which are related to ground electronic state ro-vibrational spectroscopy of AlH and AlD. The work also presents a comprehensive collection of previous theoretical and computational results for these spectroscopic properties in order to facilitate comparison. The harmonic frequency and first-order anharmonic correction are determined to permit the computation of vibrationally-averaged properties in addition to those at the equilibrium geometry. The rotational constant and vibration–rotation coupling are computed. Importantly, both the nuclear quadrupole coupling constants (via electric field gradients) and spin-rotation constants for Al are determined using the highest-level of ab initio electronic structure theory to date. These new computational values allow the assessment of the accuracy of recent experimental measurements of the rotational spectra for AlH and AlD [9,10]. Also, given that the experimental value of C Q (27Al) is known more accurately [9], we are able to combine theory and experiment to propose a revised value of the nuclear quadrupole moment, eQ(27Al), for the 27Al nucleus.

2. Computational methods The calculations for AlH were carried out using coupled-cluster theory with single and double excitations as well as perturbative inclusion of triples, i.e., CCSD(T) [16–18]. Since the accurate determination of NQCCs requires a good description of the wavefunction near the nucleus, all computations have been carried out using the cc-pCVXZ and aug-cc-pCVXZ (X = T, Q, or 5) families of basis sets [19]. Some basis sets for aluminum were obtained from the Basis Set Exchange [20,21]. In the CCSD(T) calculations, all electrons have been correlated. The EFGs determined computationally are converted to C Q values using the nuclear quadrupole moments from Ref. [22], i.e., Q ð27 AlÞ ¼ 146:6  1:0 mb and Q ðDÞ ¼ 2:860  0:015 mb. Our primary interest, beyond the EFGs, is in the theoretical determination of the spin-rotation constants. The spin-rotation constants have been computed previously for AlH using HF, MP2, and DFT (B3LYP) methods [8] – theoretical results have not been reported previously for AlD. Therefore, we have determined the spin-rotation constants for both AlH and AlD using CCSD(T) theory [23–25]. The spin-rotation constants are more difficult to determine than the EFGs because they require analytical second derivative techniques [26–29]. The isotropic/anisotropic nuclear magnetic shielding constants, which are available from the same calculations required to determine the spin-rotation constants, are also reported. All the CCSD(T) calculations have been undertaken with the CFOUR program package [30]. The vibrational averaging for molecular properties is carried out using built-in procedures in CFOUR, and its general implementation has been discussed in detail previously [31,32]. Here we present the specific application to the diatomic molecules considered here – a consideration that leads to a much simpler expression. The expectation value of the property X (i.e., the EFG, the spin-rotation constant, or the shielding tensor) is expanded over the vibrational wavefunction in a Taylor series around the equilibrium geometry with respect to bond-coordinate displacement, Q [33]:

hXi ¼ X eq 

1 4x3=2



@X @Q

 /þ Q ¼0

1 @2X 4x @Q 2

! :

ð1Þ

Q¼0

In Eq. (1), x is the harmonic frequency and / is the corresponding cubic force constant, which is computed by numerical differentiation of analytically calculated harmonic force fields with respect to bond-coordinate displacement. The required property derivatives are also obtained via numerical differentiation of the corresponding

property values. The vibrational corrections obtained via Eq. (1) are correct to third order. 3. Results and discussion In the following sections, we detail the results of the present CCSD(T) computations and compare with previous computational determinations and experimental measurements. We report properties at both the equilibrium geometries and based upon vibrational-averaging, see Eq. (1). Since vibrational averaging requires the evaluation of the harmonic frequency, equilibrium properties, and property derivatives evaluated at the equilibrium position, it is first necessary to determine the equilibrium bond length for each basis set. Table 1 reports the equilibrium bond lengths from the present work – these values will be used for all subsequent determinations of equilibrium and vibrationally-averaged properties for a given basis set. While there have been many computational determinations of the AlH equilibrium bond length [14,15,34–49], Table 1 reports selected previous high-level computational results and experimental measurements [50–54]. All the present CCSD(T) results are within 0.005 Å (0.3%) of the most recent experimental measurement [50] of 1.64735188(26) Å. Our best theoretical determination of the equilibrium bond length is within 0.07% of the experimental value. Those utilizing the largest basis set exhibit superior agreement with experiment than recent MRCI results [14,15] and are comparable to the best available ab initio value [35] and that from a composite method based upon high-level computations [34]. Therefore, the current structures provide an accurate basis for determining both equilibrium and vibrationally averaged properties. 3.1. Vibrational and rotational energies In the present work, both equilibrium and vibrationally averaged properties will be determined. Hence, it is important to assess the accuracy of the vibrational averaging by determining the anharmonic vibrational energies for AlH and AlD. The vibrational anharmonicity is treated using second-order perturbation

Table 1 Computationally determined equilibrium bond lengths as compared to previously computed and experimentally measured values. Method

Reference

Re/Å

CCSD(T)/cc-pCVTZ CCSD(T)/cc-pCVQZ CCSD(T)/cc-pCV5Z CCSD(T)/aug-cc-pCVTZ CCSD(T)/aug-cc-pCVQZ

This This This This This

work work work work work

1.6511 1.6458 1.6450 1.6524 1.6465

MRCI/aug-cc-pV6Z MRCI/cc-pV6Z MRCI/aug-cc-pV5Z W4[CCSDTQ] DK3-CCSDTQ/VQZa CCSD(T)/Al[7s7p5d4f]; H[6s4p3d] MRCI/cc-pV5Z FCI/Al[7s,5p,2d]; H[4s,3p]b MRCI/VTZc MRCI/VTZd Experiment Experiment Experiment Experiment Experiment

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

[14] [15] [15] [34] [35] [40] [42] [44] [45] [45] [50] [51] [52] [53] [54]

1.635 1.6707 1.6510 1.6454 1.649 1.6399 1.6508 1.659 1.6517 1.6579 1.64735188(26) 1.645362224046089 (Fit) 1.647395(51) 1.67438(5) 1.6482(2)

a Theoretical best estimate from DK3-CCSD, DK3-CCSDT, DK3-CCSDTQ with DZ, TZ and QZ basis sets. b Included correlation of valence electrons only. c With [He] core. d With [Ne] core.

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theory[31,32] using the built-in procedures in CFOUR [30]. Table 2 reports the harmonic vibrational frequencies and first order anharmonic corrections for AlH as determined at the CCSD(T)/cc-pCVXZ (T, Q and 5) and aug-cc-pCVXZ (X = T, Q) levels of theory. Also included in the table are previously computed and experimentally measured values. Table 3 reports the corresponding data for the AlD isotopologue. The present results are in excellent agreement with the best available experimental measurements. For AlH, the fundamental vibrational transition is predicted within 0.3% (5 cm1) of the experimental measurements [50,55,56,51,57] – except for the aug-cc-pCVTZ basis where the predicted fundamental frequency of 1606.9 cm1 underestimates the experimental result by 18.2 cm1. For AlD, not surprisingly, the theoretical predictions are equally good. Clearly, since the vibrational frequencies are well-predicted, one expects the vibrational averaging of the molecular properties to be well-described. The corresponding rotational and vibration–rotation coupling constants for AlH and AlD are given in Table 4. As expected from the agreement between the experimental and theoretical bond lengths, see Table 1, the equilibrium rotational constants are also in excellent accord. For the two largest basis sets, cc-pCV5Z and aug-cc-pCVQZ, the predicted vibrational–rotational coupling constants are both within 0.003 cm1 of the most recent experimental determination [50,55]. Table 2 Harmonic vibrational frequencies and first order anharmonic corrections for AlH as compared to previous computed and experimentally measured values. Second-order corrections are also provided if they were determined. All values reported with the same significant figures provided in the original work.

a

xe ye

Method

xe =cm1

xe xe =cm1

CCSD(T)/cc-pCVTZ CCSD(T)/cc-pCVQZ CCSD(T)/cc-pCV5Z CCSD(T)/aug-ccpCVTZ CCSD(T)/aug-ccpCVQZ MRCI/aug-cc-pV5Za W4[CCSDTQ]b DK3-CCSDTQ/VQZc MRCI/cc-pV5Zd MRCI/VTZe MRCI/VTZf CASSCFg CIPSI/DZPh PNO-CIi PNO-CEPAi Experimentj Experimentk Experimentl Experimentm Experimentn Experimento Experimentp

1677.03 1689.27 1689.83 1662.89

27.07 29.46 33.69 27.99

— — — —

1685.51

28.54

—

1683.37 1682.14 1690 1681.3 1727.9 1694.7 1678.6 1662.7 1703.6 1691.7 1682.37499(17) 1682.3694(22) 1682.37474(31) 1683.7590(52) 1683.7419(25) 1682.435 1653.65(3)

29.3786 28.61 30 28.3 27.27 26.53 29.22 30.2 28.7 29.6 29.05111(15) 29.0466(15) 29.050978(287) 28.2510(43) 28.2402(14) 29.106 29.09(6)

— 0.032 — — 0.133 0.206 0.24 — — — 0.247654(44) 0.24631(46) 0.247615(115) 0.2501(19) 0.2461(37) 0.2665 —

Ref. [15]. Ref. [34]. c Ref. [35]. Theoretical best estimate from DK3-CCSD, DK3-CCSDT,DK3-CCSDTQ with DZ, TZ and QZ basis sets. d Ref. [42]. e Ref. [45] using [He] core. f Ref. [45] using [Ne] core. g Ref. [47] with basis H (12s,6p,4d/8s,4p,2d) and Al (16s,10p,5d,2f/12s,7p,4d,1f). h Ref. [48]. i Ref. [49] with basis H (4s,2p,1d) and (valence only) Al (4s,4p,2d,1f). j Refs. [50,55]. k Ref. [56]. l Ref. [51]. m Ref. [76]. As determined from the mass-independent Dunham parameters. n Ref. [52]. As determined from the mass-independent Dunham parameters. o Ref. [53]. p Ref. [54]. b

Table 3 Harmonic vibrational frequencies and first order anharmonic corrections for AlD as compared to experimentally measured values. Second-order corrections are also provided if they were determined. All values reported with the same significant figures provided in the original work.

a b

Method

xe =cm1

xe xe =cm1

xe ye =cm1

CCSD(T)/cc-pCVTZ CCSD(T)/cc-pCVQZ CCSD(T)/cc-pCV5Z CCSD(T)/aug-ccpCVTZ CCSD(T)/aug-ccpCVQZ Experimenta Experimentb

1207.43 1216.24 1216.65 1197.25

14.02 15.36 18.39 14.71

— — — —

1213.53

14.88

—

1211.77402(15) 1211.7821(18)

15.064765(114) 15.07302(87)

0.0924425(369) 0.09534(16)

Ref.[51]. Ref. [52].

3.2. Dipole moments Knowledge of the dipole moments of AlH and AlD is important for determining the intensity of rotational (and ro-vibrational) transitions [58]. The dipole moments at the equilibrium bond length and as determined by vibrational averaging are given in Table 5. Wells and Lane [14] have determined the permanent dipole moment as a function of internuclear separation at the MRCI/augcc-pcV6Z level of theory and it clearly exhibits a strong dependence on bond length. Hence, an accurate determination requires incorporating the effects of vibrational averaging. This is also apparent from the previous calculations of the dipole moment which have accounted for vibrational averaging [46,47,49] and where the effects increase the dipole by 50–90% from its equilibrium value. Previously, the most accurate determinations of the vibrationally averaged dipole moments for AlH have been at the MBPT(4) ðhli ¼ 0:300 DÞ [46], CASSCF (0.206 D) [47], and PNOCEPA [49] (0.185 D) levels of theory. There is only one previous prediction of the vibrationally-averaged dipole moment for AlD: 0.160 D from PNO-CEPA computations [49]. The present results based on CCSD(T) are broadly in keeping with these previous computations both in terms of the effects of vibrational-averaging leading to an increase in the dipole moment compared to the equilibrium values and the overall magnitudes. For example, the CCSD(T)/aug-cc-pCVQZ vibrationally-averaged dipole moments are 0.233 D and 0.208 D for AlH and AlD, respectively. These results suggest that the rotational transitions for AlD will be weaker than those for AlH – in addition to the significant differences due to relative isotopic abundance (0.0115 versus 99.9885). 3.3. Nuclear quadrupole couplings and spin-rotation constants The NQCCs, C Q (27Al), as determined from the calculated EFGs are given in Table 6. Also provided are the previous computational results of Gee and Wasylishen [8] and the recent experimental measurements of Ziurys and co-workers [9,10]. For completeness, we note that C Q (D) is determined to be 50 kHz for all basis sets used in the present work. As discussed previously [8], the experimental analysis for determining C Q (27Al) by Goto and Saito [7] did not account for the spin-rotation interaction and, hence, resulted in a value that was too low. Therefore, that result is not reported here. Note that the theoretical results from Ref. [8] have been corrected to reflect the most recent value of the nuclear quadrupole moment for Al of 146.6 ± 1.0 mb [22] – the previous recommended value was 140.3 mb. Gee and Wasylishen used several theoretical methods and both the 6-311++G (3df,3pd) and cc-pVQZ basis sets. Their cc-pVQZ results are not provided here due to known deficiencies in the basis for second row atoms that were

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Table 4 Rotational and vibration–rotation coupling constants for AlH and AlD as determined in the present work and from previous experimental studies. All values reported with the same significant figures provided in the original work. Method AlH CCSD(T)/cc-pCVTZ CCSD(T)/cc-pCVQZ CCSD(T)/cc-pCV5Z CCSD(T)/aug-cc-pCVTZ CCSD(T)/aug-cc-pCVQZ Experimenta Experimentb Experimentc Experimentd Experimente Experimentf Experimentg Experimenth AlD CCSD(T)/cc-pCVTZ CCSD(T)/cc-pCVQZ CCSD(T)/cc-pCV5Z CCSD(T)/aug-cc-pCVTZ CCSD(T)/aug-cc-pCVQZ Experimenth Experimenti Experimentb a b c d e f g h i

Be =cm1

ae =cm1

B0 =cm1

6.365 6.406 6.412 6.354 6.401 6.3937898(20) — — — 6.2978 6.3937(4) 6.39378418(169)

0.178 0.182 0.186 0.181 0.184 0.1870560(30) — — 0.179496(11) 0.179602(39) — 0.1868(3) 0.18705266(148)

6.276 6.315 6.319 6.264 6.309 6.30071641 6.300706(15) 6.300072554(90) 6.226219(4) — — — —

3.299 3.321 3.324 3.292 3.318 3.31839293(80) — —

0.066 0.068 0.069 0.067 0.068 0.069877296(436) — —

3.266 3.287 3.289 3.260 3.284 — 3.2835751(12) 3.2835763(7)

Refs. [50,55]. Ref. [9]. Ref. [57]. Ref. [52]. Ref. [76]. Ref. [77]. Ref. [53]. Ref. [51]. Ref. [10].

Table 5 Dipole moments (in Debye) for AlH and AlD as determined using CCSD(T) methods and from previous computations. Current values are given at the corresponding optimized geometry and as determined from vibrational averaging, see text for details. All orbitals are correlated for CCSD(T) computations.

a b c d e f

Method

l at Re

hli for AlH

hli for AlD

CCSD(T)/cc-pCVTZ CCSD(T)/cc-pCVQZ CCSD(T)/cc-pCV5Z CCSD(T)/aug-cc-pCVTZ CCSD(T)/aug-cc-pCVQZ MP2/6-311++G (3df,3pd)a B3LYP/6-311++G (3df,3pd)a CCSD(T)b FCIc SDCIc ACPFc QDVPTc MBPT(4)d CASSCFe PNO-CEPAf

0.133 0.123 0.128 0.172 0.145 0.11 0.23 0.033 0.066 0.071 0.004 0.000 0.193 0.119 0.098

0.217 0.209 0.211 0.260 0.233 — — — — — — — 0.300 0.206 0.185

0.193 0.185 0.192 0.236 0.208 — — — — — — — — — 0.160

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

[36]. [40]. [78]. [46]. [47]. [49].

Basis Basis Basis Basis Basis

H is H H H

(6s,4p,3d) and Al (7s,7p,5d,4f). Al (7s,5p,2d) and H (4s,3p) and determined at R = 3.1 Bohr. (8s,4p,2d) and Al (12s,7p,4d,1f). (8s,4p,2d) and Al (12s,7p,4d,1f). (4s,2p,1d) and (valence) Al (4s,4p,2d,1f).

subsequently corrected [59]. The previous (non-vibrationally averaged) MP4(SDQ)/6-311++G (3df,3pd), CID/6-311++G (3df,3pd), and CISD/6-311++G (3df,3pd) values, i.e., 48.49 MHz, 48.34 MHz, and 48.24 MHz, respectively, as determined at the experimental bond length are in excellent agreement with the recent experimental measurement for AlH [9]. Similarly, the new CCSD(T)/cc-

pCVQZ, CCSD(T)/cc-pCV5Z, and CCSD(T)/aug-cc-pVQZ values for C Q (27Al) as determined at the corresponding equilibrium bond lengths are also in good accord with the experimental measurement for AlH and AlD, i.e., they agree to within the experimental error bars. Surprisingly, upon correcting for the effects of vibrational averaging, the present CCSD(T) results consistently underestimate the experimental results. While they are all within the experimental error plus uncertainty of the 27Al quadrupole moment [22], i.e., 146:6  1:0 mb, except for the cc-pCVTZ and augcc-pCVTZ values, these results suggest that the nuclear quadrupole moment for 27Al may be slightly underestimated. Based on the experimental measurements of the nuclear quadrupole coupling for AlH (AlD) and CCSD(T)/cc-pV5Z, and CCSD(T)/aug-cc-pVQZ determinations of the vibrationally averaged electric field gradients, the quadruple moment of 27Al is determined to be 149  2 mb ð148  3 mbÞ.The present recommended value for the quadrupole moment of 27Al comes from experimentally measured values for C Q (27Al) combined with accurate CCSD(T) computations of the EFGs in AlCl and AlF, including vibrational corrections using the Buckingham formula [60] at the MBPT2 level of theory, plus MCHF calculations on atomic Al (3p,2P3/2) [61]. For AlF and AlCl, the vibrational contribution was estimated to be only 1.5% of the EFG at equilibrium and a similar contribution is found for AlH in the present work – for AlD, the vibrational contribution is only 1%. In the previous work, the core 1s orbitals were left uncorrelated, but this effect is expected to be insignificant, e.g., the NQCC at the CCSD(T)/ cc-pCVQZ level of theory with the 1s orbital frozen is 48.53 MHz, which is a change of only 0.01 MHz. The experimental values of C Q (27Al) for AlF and Al35Cl (Al37Cl) of 37.53 ± 0.12 MHz [62] and 30.4081 ± 0.0027 MHz (30.4112 ± 0.0028 MHz) [63], respectively, have much smaller error bars than the present measurements for AlH and AlD. Therefore, before definitively re-assigning

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Table 6 Nuclear quadrupole coupling constants (in MHz) of 27Al in AlH and AlD as determined using CCSD(T) methods. Values are given at the optimized geometry and as determined from vibrational averaging, see text for details. All orbitals are correlated for CCSD(T) computations. C Q determined using Q = 146.6 ± 1.0 mb from Ref. [22], and, therefore, theoretical error bars reflect the uncertainty in the theoretical quadrupole moment. Also, reported are the experimental measurements and the previous computational results [8] determined at the experimental equilibrium geometry [51] of r e ¼ 1:6453622 Å. The previous computational values have been multiplied by 146.6/140.3 as the value of Q Al has been revised [22].

a b

Method

C Q ðAlÞ at Re

hC Q ðAlÞi

hC Q ðAlÞi

CCSD(T)/cc-pCVTZ CCSD(T)/cc-pCVQZ CCSD(T)/cc-pCV5Z CCSD(T)/aug-cc-pCVTZ CCSD(T)/aug-cc-pCVQZ HF/6-311++G (3df,3pd) MP2/6-311++G (3df,3pd) MP3/6-311++G (3df,3pd) MP4(SDQ)/6-311++G (3df,3pd) CID/6-311++G (3df,3pd) CISD/6-311++G (3df,3pd) CCD/6-311++G (3df,3pd) B3LYP/6-311++G (3df,3pd) Experimenta Experimentb

AlH 47.64 ± 0.32 48.54 ± 0.33 48.39 ± 0.33 47.47 ± 0.32 48.46 ± 0.33 51.14 ± 0.35 49.79 ± 0.34 49.00 ± 0.33 48.49 ± 0.33 48.34 ± 0.33 48.24 ± 0.33 47.85 ± 0.33 52.83 ± 0.37 — —

AlH 47.02 ± 0.32 47.89 ± 0.33 47.73 ± 0.33 46.83 ± 0.32 47.80 ± 0.33 — — — — — — — — 48.59 ± 0.70 —

AlD 47.19 ± 0.32 48.07 ± 0.33 47.91 ± 0.33 47.01 ± 0.32 47.98 ± 0.33 — — — — — — — — 48.48 ± 0.88 48.64 ± 0.37

Ref. [9]. Ref. [10].

the nuclear quadrupole moment for Al, the NQCCs for AlH and AlD should be remeasured to improve the precision or it may be worthwhile to try to improve the previous computational results for AlF and/or AlCl. In addition to the C Q values, the spin-rotation constant plays an important role in the rotational spectra of AlH and AlD, e.g., see [8]. Table 7 provides the spin-rotation constants for aluminum C I (Al) in AlH and AlD as determined in the present work at the CCSD(T) level of theory as well as from previous computations [8] and experimental measurements [9,10]. The previous MP2 values were computed directly while all other values were calculated from nuclear magnetic shielding data. The corresponding computational values of the spin-rotation constants for H and D of (approximately) 4.5 kHz and 0.4 kHz, respectively, suggest that this interaction can be safely neglected from the analysis of the rotational spectrum. The spin-rotation constant is related to the isotropic nuclear magnetic shielding constant, riso , as follows in the Flygare approximation [64,65,23,66]:

riso 

mp C I þ rfree atom : 3mg N B

Table 7 Spin-rotation constants (in kHz) of 27Al in AlH (AlD) as determined using CCSD(T) methods. Values are given at the optimized geometry and as determined from vibrational averaging, see text for details. All orbitals are correlated for CCSD(T) computations. Also, reported are the experimental measurements and the previous computational (HF, MP2, and B3LYP) results [8] determined at the experimental equilibrium geometry [51] of r e ¼ 1:6453622 Å. Method

C I ðAlÞ at Re

hC I ðAlÞi

hC I ðAlÞia

CCSD(T)/cc-pCVTZ CCSD(T)/cc-pCVQZ CCSD(T)/cc-pCV5Z CCSD(T)/aug-cc-pCVTZ CCSD(T)/aug-cc-pCVQZ HF/6-311++G (3df,3pd) HF/cc-pV6Z MP2/6-311++G (3df,3pd) MP2/cc-pV5Z B3LYP/6-311++G (3df,3pd) B3LYP/cc-pV6Z Experimentb

283.4 290.7 292.7 282.5 290.5 298.7 299.7 297.7 301.1 324.8 330.0 —

279.9 286.9 288.8 278.9 286.6 — — — — — — 306 ± 35

282.8 288.0 289.7 281.9 288.0 — — — — — — —

146.9 150.7 151.7 146.4 150.6 — —

145.6 149.3 150.3 145.1 149.1 156 ± 41 108 ± 22

145.5 148.2 149.0 145.2 148.2 — —

AlH

AlD

ð2Þ

In Eq. (2), mp and m are the proton and electron masses, respectively. For the values reported in Ref. [8], g N is the nuclear g-factor, 1.4565 for 27Al [67], B is the rotational constant of 191680(5) MHz at the equilibrium bond length [51], and the non-relativistic nuclear magnetic shielding of the free atom, rfree atom ¼ 789:88 ppm for aluminum [68]; the relativistic value of the nuclear magnetic shielding for aluminum is nearly the same, rfree atom ¼ 790:7 ppm [69]. We have used Eq. (2), these same constants, and newly computed vibrationally averaged isotropic magnetic shielding constants, see Table 8, to determine the approximate CCSD(T) spin-rotation constants for comparison. For determining the spin-rotation constants for AlD, we have used the recent experimental measurement of the rotational constant [10] of 98439.108(32) MHz. Clearly, all the results from Eq. (2) are within 3 kHz (1.5%) of the directly computed CCSD(T) values. The complete set of isotropic and anisotropic nuclear magnetic shielding tensors at the CCSD(T) level of theory along with the previous computational results [8] are given in Table 8. The computationally determined spin-rotation constants for AlH agree with the experimental measurement within the experimental error bars – although the computations suggest that the experimental

CCSD(T)/cc-pCVTZ CCSD(T)/cc-pCVQZ CCSD(T)/cc-pCV5Z CCSD(T)/aug-cc-pCVTZ CCSD(T)/aug-cc-pCVQZ Experimentb Experimentc a b c

As estimated using Eq. 2 and hriso i, see Table 8. Ref. [9]. Ref. [10].

measurement may be slightly overestimated. For AlD, the computational results agree with the previous experimental measurement [9]. However, the revised experimental value of 108  22 kHz determined in Ref. [10] is too small by  50%. One can also note this potential discrepancy, since according to Eq. (2) [70], the ratio of the spin-rotation constants for AlH and AlD should be (approximately) equivalent to the ratio of their rotational constants, i.e., C I (AlH)/ C I (AlD) = B (AlH)/B (AlD) = 1.947. Clearly, the earlier result [9] of 156 ± 41 kHz should be considered more reliable. Finally, two other points regarding our computations in Tables 7 and 8 deserve comment. Firstly, the computed vibrationally aver-

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Table 8 Isotropic and anisotropic nuclear magnetic shielding tensors (in ppm) of Al in AlH (AlD) as determined using CCSD(T) methods. Values are given at the optimized geometry and as determined from vibrational averaging, see text for details. All orbitals are correlated for CCSD(T) computations. Also, reported are the previous computational (HF, MP2, and B3LYP) results[8] determined at the experimental equilibrium geometry [51] of r e ¼ 1:6453622 Å. Method

riso AlH/AlD

CCSD(T)/cc-pCVTZ CCSD(T)/cc-pCVQZ CCSD(T)/cc-pCV5Z CCSD(T)/aug-cc-pCVTZ CCSD(T)/aug-cc-pCVQZ HF/6-311++G (3df,3pd) HF/cc-pV6Z MP2/6-311++G (3df,3pd) MP2/cc-pV5Z B3LYP/6-311++G (3df,3pd) B3LYP/cc-pV6Z

166.1 154.3 150.5 167.0 154.2 136.8 132.9 137.3 129.8 77.7 66.3

hriso i

hriso i

AlH

AlD

169.9 158.4 154.8 171.0 158.5 — — — — — —

168.8 157.2 153.6 169.9 157.3 — — — — — —

aged isotropic shielding constant for AlH is approximately 1.2 ppm greater than that for AlD. That is, the derivative of the isotropic shielding with respect to bond extension is positive, which is unusual (e.g., see Ref. [71]) but predicted on the basis of early calculaþ tions for AlH3 and AlH4 by Chesnut [72]. Invariably, experimental NMR studies demonstrate that the heavier isotopologue is more shielded than the lighter isotopologue (e.g., Ref. [73]); however, this is not the case for aluminum hydride. Secondly, since the anisotropy of the magnetic shielding tensor is related to the spin-rotation constant [64,65,23,66,74], one can use the spin-rotation data in Table 7 (C I values at Re ) to estimate the shielding anisotropy, raniso ¼ rk  r? , i.e.,

raniso 

mp C I : 2mg N B

ð3Þ

For example, substitution of the C I value, 283.8 kHz, from the CCSD(T)/cc-pCVTZ into the above equation yields a shielding anisotropy, raniso , of 931.9 ppm while direct calculation gives 932.7 ppm. Similarly, using the CCSD(T)/aug-cc-pCVQZ value of C I gives raniso ¼ 955:3 ppm in good agreement with the value calculated directly given in Table 8, 950.7 ppm. The experimental value of C I , 306  35 kHz and the experimental value [9] of B0 (188890.4 MHz) yields hraniso i  1020  120 ppm. It is important to recognize that the isotropic magnetic shielding constant for aluminum in AlH is very small, i.e., approximately 160 ppm (Table 8), versus the free-atom value of approximately 790 ppm. As well, the aluminum shielding anisotropy,  1000 ppm, is much larger than the typical range of observed aluminum chemical shifts, 300 ppm, e.g., see Ref. [75]. Again, qualitatively this is related to the small HOMO–LUMO gap in AlH as pointed out by several researchers earlier [11,12,8].

raniso AlH/AlD 932.7 950.6 956.4 931.4 950.7 975.8 981.8 975.3 986.2 1065.5 1082.3

hraniso i

hraniso i

AlH

AlD

927.0 944.4 949.9 925.3 944.3 — — — — — —

928.6 946.2 951.7 927.0 946.1 — — — — — —

AlH. The present results confirm the very large effect (50–60% increase) in the dipole moment upon vibrational averaging that has been shown previously [46,47,49]. The present work represents the first determination of the C Q for AlH and AlD incorporating the effects of vibrational averaging, which are shown to decrease the C Q values by approximately 1.5%. While the values of the computationally determined NQCCs are within the experimental (and, due to the uncertainty in the nuclear quadrupole moment, computational) errors, they are systematically lower than the experimentally measured values of 48:59  0:70 MHz ð48:48  0:88 MHzÞ for AlH [9,10] and 48:64  0:37 MHz [10] for AlD. The computational results correctly predict that the value for AlH is slightly smaller than that for AlD. However, the systematic difference from experiment suggests that the value of the nuclear quadruple moment for 27Al of 146:6  1:0 mb [22] should be revised – however, any revision should be based upon more precise experimental determination of C Q for AlH. The spin-rotation constants have also been determined and confirm the original experimental measurements for both AlH and AlD [9]. Based on the present computational results, the experimental measurements for AlH [9], and the known theoretical relationship between the spin-rotation constants for AlH and AlD isotopologues, see Eq. (2) [64,65,23,66], the revised value of 108  22 kHz for AlD [10] is clearly incorrect and should not be used in analysis of hyperfine spectra for AlD. Overall the present work provides a complete summary of the rotational and ro-vibrational spectroscopy of the important molecule AlH (and its deuterated isotopologue) and suggests that new more precise experimental measurements of C Q may be warranted. Acknowledgments

4. Conclusions In the present work, the important parameters related to the rotational and ro-vibrational spectroscopy of AlH and its deuterated analog AlD have been determined at the CCSD(T) level of theory combined with large basis sets (i.e., cc-pCVXZ and augcc-pCVXZ, X = T, Q, and 5). As expected, the equilibrium bond lengths and, hence, rotational constants are in excellent accord with the experimentally measured results. The fundamental vibrational frequencies are determined (incorporating anharmonicity) and are shown to be within 0.5% of the best available experimental measurements. Therefore, the vibrational-averaging performed provides an accurate representation of this effect. The CCSD(T)/ aug-cc-pCVQZ vibrationally averaged dipole moments of 0.233 D and 0.208 D for AlH and AlD, respectively, represent the best available determinations to date, and suggest that the intensities of lines observed for AlH will be approximately 25% larger than for

This work was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grants for AB and REW). REW holds a Canada Research Chair in physical chemistry at the University of Alberta. AB thanks the Canadian Foundation for Innovation (New Opportunities Fund) for support for computational infrastructure and Ms. Chen Liang for performing preliminary computational work on this project. References [1] [2] [3] [4] [5] [6] [7] [8]

E. Bengtsson, Z. Phys. 51 (1928) 889–894. W. Holst, E. Hulthén, Nature 133 (1934) 496. W. Holst, E. Hulthén, Nature 133 (1934) 796. R.S. Mulliken, Phys. Rev. 33 (1929) 730. G.H. Herbig, Pub. Astron. Soc. Pacific 68 (1956) 204–210. P.W. Merrill, Astrophys. J. 118 (1953) 453. M. Goto, S. Saito, Astrophys. J. 452 (1995) L147–L148. M. Gee, R.E. Wasylishen, J. Mol. Spectrosc. 207 (2001) 153–160.

14

A. Brown, R.E. Wasylishen / Journal of Molecular Spectroscopy 292 (2013) 8–14

[9] D.T. Halfen, L.M. Ziurys, Astrophys. J. 607 (2004) L63–L66. [10] D.T. Halfen, L.M. Ziurys, Astrophys. J. 713 (2010) 520–523. [11] J. Gauss, U. Schneider, R. Ahlrichs, C. Dohnmeier, H. Schöckel, J. Am. Chem. Soc. 115 (1993) 2402–2408. [12] R.W. Schurko, R.E. Wasylishen, H. Foerster, J. Phys. Chem. A 102 (1998) 9750– 9760. [13] M. Gee, R.E. Wasylishen, in: J.C. Facelli, A.C. de Dios (Eds.), Modeling NMR Chemical Shifts, ACS Symposium Series, vol. 732, 1999, pp. 259–276. [14] N. Wells, I.C. Lane, Phys. Chem. Chem. Phys. 13 (2011) 19018–19025. [15] D.-H. Shi, H. Liu, X.-N. Zhang, J.-F. Sun, Y.-F. Liu, Z.-L. Zhu, Int. J. Quant. Chem. 111 (2009) 554–562. [16] K. Raghavachari, G.W. Trucks, J.A. Pople, M. Head-Gordon, Chem. Phys. Lett. 157 (1989) 479–483. [17] R.J. Bartlett, J.D. Watts, S.A. Kucharski, J. Noga, Chem. Phys. Lett. 165 (1990) 513–522. [18] J.F. Stanton, Chem. Phys. Lett. 281 (1997) 130–134. [19] D.E. Woon, T.H. Dunning, J. Chem. Phys. 103 (1995) 4572–4585. [20] D. Feller, J. Comput. Chem. 17 (1996) 1571–1586. [21] K.L. Schuchardt, B.T. Didier, T. Elsethagen, L. Sun, V. Gurumoorthi, J. Chase, J. Li, T.L. Windus, J. Chem. Inf. Model 47 (2007) 1045–1052. [22] P. Pyykkö, Mol. Phys. 106 (2008) 1965–1974. [23] W.H. Flygare, Chem. Rev. 74 (1974) 653–687. [24] J. Gauss, D. Sundholm, Mol. Phys. 91 (1997) 449–458. [25] J. Gauss, K. Ruud, T. Helgaker, J. Chem. Phys. 105 (1996) 2804–2812. [26] J. Gauss, J. Chem. Phys. 116 (2002) 4773–4776. [27] M. Kállay, J. Gauss, J. Chem. Phys. 120 (2004) 6841–6848. [28] J. Gauss, J.F. Stanton, J. Chem. Phys. 104 (1996) 2574–2583. [29] J. Gauss, J. Stanton, Chem. Phys. Lett. 276 (1997) 70–77. [30] J.F. Stanton, J. Gauss, M.E. Harding, P.G. Szalay, Cfour, coupled-cluster techniques for computational chemistry, a quantum-chemical program package, 2010. With contributions from A.A. Auer, R.J. Bartlett, U. Benedikt, C. Berger, D.E. Bernholdt, Y.J. Bomble, L. Cheng, O. Christiansen, M. Heckert, O. Heun, C. Huber, T.-C. Jagau, D. Jonsson, J. Jusélius, K. Klein, W.J. Lauderdale, D.A. Matthews, T. Metzroth, L.A. Mück, D.P. O’Neill, D.R. Price, E. Prochnow, C. Puzzarini, K. Ruud, F. Schiffmann, W. Schwalbach, S. Stopkowicz, A. Tajti, J. Vázquez, F. Wang, J.D. Watts and the integral packages MOLECULE (J. Almlöf and P.R. Taylor), PROPS (P.R. Taylor), ABACUS (T. Helgaker, H.J. Aa. Jensen, P. Jørgensen, and J. Olsen), and ECP routines by A.V. Mitin and C. van Wüllen. For the current version . [31] C. Puzzarini, J.F. Stanton, J. Gauss, Int. Rev. Phys. Chem. 29 (2010) 273–367. [32] A.A. Auer, J. Gauss, J.F. Stanton, J. Chem. Phys. 118 (2003) 10407–10417. [33] A.D. Buckingham, W. Urland, Chem. Rev. 75 (1975) 113–117. [34] A. Karton, J.M.L. Martin, J. Chem. Phys. 133 (2010) 144102. [35] S. Hirata, T. Yanai, W.A. de Jong, T. Nakajima, K. Hirao, J. Chem. Phys. 120 (2004) 3297–3310. [36] L. Andrews, X. Wang, J. Phys. Chem. A 108 (2004) 4202–4210. [37] X. Wang, L. Andrews, S. Tam, M.E. DeRose, M.E. Fajardo, J. Am. Chem. Soc. 125 (2003) 9218–9228. [38] C.J. Cobos, J. Mol. Struct. (Theochem) 581 (2002) 17–29. [39] M.T. Swihart, L. Catoire, J. Phys. Chem. A 105 (2001) 264–273. [40] G.L. Gutsev, P. Jena, R.J. Bartlett, J. Chem. Phys. 110 (1999) 2928–2935. [41] P. Schwerdtfeger, J. Ischtwan, J. Mol. Struct. (Theochem) 306 (1994) 9–19.

[42] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 99 (1993) 1914–1929. [43] P. Schwerdtfeger, G.A. Heath, M. Dolg, M.A. Bennett, J. Am. Chem. Soc. 114 (1992) 7518–7527. [44] C.W. Bauschlicher, S.R. Langhoff, J. Chem. Phys. 89 (1988) 2116–2125. [45] V.J. Barclay, J.S. Wright, Chem. Phys. 121 (1988) 381–391. [46] J.M.O. Matos, B.O. Roos, A.J. Sadlej, G.H.F. Diercksen, Chem. Phys. 119 (1988) 71–77. [47] J.M.O. Matos, P.A. Malmqvist, B.O. Roos, J. Chem. Phys. 86 (1987) 5032–5042. [48] M. Pelissier, J.P. Malrieu, J. Chem. Phys. 67 (1977) 5963–5965. [49] W. Meyer, P. Rosmus, J. Chem. Phys. 63 (1975) 2356–2375. [50] W. Szajna, M. Zachwieja, R. Hakalla, R. Keßpa, Acta Phys. Polon. A 120 (2011) 417–423. [51] J.B. White, M. Dulick, P.F. Bernath, J. Chem. Phys. 99 (1993) 8371–8378. [52] R.-D. Urban, H. Jones, Chem. Phys. Lett. 190 (1992) 609–613. [53] J.L. Deutsch, W.S. Neil, D.A. Ramsay, J. Mol. Spectrosc. 125 (1987) 115–121. [54] P.B. Zeeman, G.J. Ritter, Can. J. Phys. 32 (1954) 555–561. [55] W. Szajna, M. Zachwieja, Eur. Phys. J. D 55 (2009) 549–555. [56] F. Ito, T. Nakanaga, H. Takeo, H. Jones, J. Mol. Spectrosc. 164 (1994) 379–389. [57] R.S. Ram, P.F. Bernath, Appl. Opt. 35 (1996) 2879–2883. [58] J.M. Brown, A. Carrington, Rotational Spectroscopy of Diatomic Molecules, Cambridge University Press, Cambridge, UK, 2003. [59] T.H. Dunning, K.A. Peterson, A.K. Wilson, J. Chem. Phys. 114 (2001) 9244–9253. [60] A.D. Buckingham, J. Chem. Phys. 36 (1962) 3096. [61] V. Kellö, A.J. Sadlej, P. Pyykkö, D. Sundholm, M. Tokman, Chem. Phys. Lett. 304 (1999) 414–422. [62] J. Hoeft, F.J. Lovas, E. Tiemann, T. Törring, Z. Naturforsch. A 25 (1970) 1029– 1035. [63] K.D. Hensel, C. Styger, W. Jäger, A.J. Merer, M.C.L. Gerry, J. Chem. Phys. 99 (1993) 3320–3328. [64] N.F. Ramsey, Phys. Rev. 78 (1950) 699–703. [65] W.H. Flygare, J. Chem. Phys. 41 (1964) 793–800. [66] W.H. Flygare, Molecular Structure and Dynamics, Prentice Hall, Englewood Cliffs, NJ, 1978. [67] R.K. Harris, E.D. Becker, S.M. Cabral de Menezes, R. Goodfellow, P. Granger, Solid State NMR 22 (2002) 458–483. [68] G. Malli, C. Froese, Int. J. Quant. Chem. IS (1967) 95–98. [69] F.D. Feiock, W.R. Johnson, Phys. Rev. 187 (1969) 39–50. [70] N.F. Ramsey, Am. Sci. 49 (1961) 509–523. [71] C.J. Jameson, in: R.K. Harris, R.E. Wasylishen (Eds.), Encyclopedia of Magnetic Resonance, John Wiley and Sons Ltd., New York, 2011, pp. 2197–2214. [72] D.B. Chesnut, Chem. Phys. 110 (1986) 415–420. [73] C.J. Jameson, in: E. Buncel, J.R. Jones (Eds.), Isotopes in the Physical and Biomedical Sciences, vol. 2, Elsevier Science Publishers, Amsterdam, 1991, pp. 1–54. [74] C.J. Jameson, in: R.K. Harris, R.E. Wasylishen (Eds.), Encyclopedia of Magnetic Resonance, John Wiley and Sons Ltd., New York, 2011, pp. 493–502. [75] J.W. Akitt, Prog. NMR Spectrosc. 21 (1989) 1–149. [76] C. Yamada, E. Hirota, Chem. Phys. Lett. 197 (1992) 461–466. [77] Y.F. Zhu, R. Shehadeh, E.R. Grant, J. Chem. Phys. 97 (1992) 883–893. [78] R.J. Cave, J.L. Johnson, M.A. Anderson, Int. J. Quant. Chem. 50 (1994) 135–149.