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RESEARCH ARTICLE (Open Access)
Contents Vol 76(12)

Highly accurate CCSD(T) homolytic Al–H bond dissociation enthalpies – chemical insights and performance of density functional theory

Robert J. O’Reilly https://orcid.org/0000-0002-5000-1920 A * and Amir Karton https://orcid.org/0000-0002-7981-508X A *
+ Author Affiliations
- Author Affiliations

A School of Science and Technology, University of New England, Armidale, NSW 2351, Australia.

* Correspondence to: roreill6@une.edu.au, amir.karton@une.edu.au

Handling Editor: George Koutsantonis

Australian Journal of Chemistry 76(12) 837-846 https://doi.org/10.1071/CH23042
Submitted: 25 February 2023  Accepted: 12 April 2023  Published online: 24 May 2023

© 2023 The Author(s) (or their employer(s)). Published by CSIRO Publishing. This is an open access article distributed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND)

Abstract

We obtain gas-phase homolytic Al–H bond dissociation enthalpies (BDEs) at the CCSD(T)/CBS level for a set of neutral aluminium hydrides (which we refer to as the AlHBDE dataset). The Al–H BDEs in this dataset differ by as much as 79.2 kJ mol−1, with (H2B)2Al–H having the lowest BDE (288.1 kJ mol−1) and (H2N)2Al–H having the largest (367.3 kJ mol−1). These results show that substitution with at least one –AlH2 or –BH2 substituent exerts by far the greatest effect in modifying the Al–H BDEs compared with the BDE of monomeric H2Al–H (354.3 kJ mol−1). To facilitate quantum chemical investigations of large aluminium hydrides, for which the use of rigorous methods such as W2w may not be computationally feasible, we assess the performance of 53 density functional theory (DFT) functionals. We find that the performance of the DFT methods does not strictly improve along the rungs of Jacob’s Ladder. The best-performing methods from each rung of Jacob’s Ladder are (mean absolute deviations are given in parentheses): the GGA B97-D (6.9), the meta-GGA M06-L (2.3), the global hybrid-GGA SOGGA11-X (3.3), the range-separated hybrid-GGA CAM-B3LYP (2.1), the hybrid-meta-GGA ωB97M-V (2.5) and the double-hybrid methods mPW2-PLYP and B2GP-PLYP (4.1 kJ mol−1).

Keywords: aluminium hydrides, bond dissociation energy, CCSD(T), density functional theory, DFT, free radicals, hydrogen storage, W2 theory.

Introduction

Neutral aluminium hydride reagents (i.e. R1R2Al–H) are useful reagents in organic synthesis,[1] and species such as alane (AlH3) have also attracted interest as hydrogen storage materials, demonstrating potential application as a rocket propellant[2] and as a hydrogen source for portable fuel cells.[3,4] Besides the potential technological applications of alane, we note that attention has also been given to ways in which non-polymerised forms of this reagent may be prepared and used in synthetic chemistry. Recently, a 1:2 alane arylphosphane adduct has been synthesised, which was shown to be able to release essentially free non-polymerised AlH3, which could be used in reduction and hydroalumination reactions.[5] Bulkier aluminium hydride species, such as diisobutylaluminium hydride (DIBAl-H), have been employed in effecting a wide range of reduction processes, including the (i) reduction of ketones and aldehydes to the corresponding alcohols, (ii) transforming α,β-unsaturated esters into the corresponding allylic alcohols[6,7] and (iii) epoxide ring opening reactions.[8] While the reactions of alkyl-substituted aluminium hydrides have received considerable attention, other aluminium hydride species have also been used in synthesis, albeit with much more limited scope to date. As one example, we note that Cl2AlH has been employed for the ring-opening of 2-substituted 1,3-dioxolanes.[9] In addition, there has been interest in the synthesis of other Al–H-containing species, and there has been success in generating and characterising species such as ClAlH2.[10]

Owing to the synthetic and technological applications of neutral aluminium hydrides, it would be insightful to have a greater understanding of some of the more salient thermodynamic properties of these compounds. One of the most fundamental thermochemical properties of such species, which has not yet received attention, are the homolytic Al–H bond dissociation enthalpies (BDEs) of such species (i.e. the energies associated with Eqn 1).

(1)R1R2Al–H→R1R2Al˙+H˙

Knowledge of how substituents affect the magnitude of the strength of Al–H bonds toward homolytic dissociation would be insightful, not least because such radical reactions may have industrial applications. For example, in the stabilisation of AlH3 by species such as 2-mercaptobenzothiazole, a key step in the mechanism has been suggested to involve homolytic dissociation of the Al–H bond of AlH3 to form the H2Al˙ radical.[11] While there does not currently exist in the literature any systematic study of Al–H homolytic BDEs, we wish to note that numerous experimental and theoretical studies have been reported which have focussed on the synthesis and characterisation of a number of aluminium(ii) radicals (i.e. of the type R1R2Al˙, which could be formed upon homolytic Al–H bond dissociation). The parent radical, H2Al˙, has been synthesised and characterised by spectroscopic methods.[12,13] A number of substituted aluminium-centred radicals have also been produced including: CH3AlH,[14] HAlNH2,[15–17] Al(NH2)2,17 HAlPH2,[18,19] HAlOH,[20] Al(OH)212 and HAlSH.[21]

The present study addresses the gap in the literature concerning the extent to which substituents affect the strength of Al–H bonds toward homolytic dissociation. To achieve this, we report a high-level quantum chemical investigation (performed using the W2w thermochemical protocol), in which the gas-phase homolytic BDEs of a set of 18 aluminium hydrides (i.e. R1R2Al–H) bearing a diverse range of substituents have been determined (which we refer to as the AlHBDE dataset). In addition, to facilitate future studies of the homolytic BDEs of larger aluminium hydrides, for which the use of the W2w thermochemical protocol is not computationally feasible, we also assess the performance of a wide range of contemporary density functional theory (DFT) methods to determine suitable lower-cost methods that could be applied for such a purpose.

Computational methods

The geometries of all species have been obtained at the B3LYP/AVTZ level of theory (where AVnZ denotes the use of aug-cc-pVnZ basis sets for hydrogen and first-row elements and aug-cc-pV(n + d)Z basis sets for second-row elements).[22,23] The validity of each structure as being a minimum on the potential energy surface was confirmed by all real harmonic vibrational frequencies. Using the geometries obtained at the B3LYP/AVTZ level of theory, we then performed higher-level calculations employing the W2w thermochemical protocol.[24] To compute a W2w energy, several calculations must be performed. First, the underlying SCF/CBS energy is obtained using a two-point extrapolation of the form E(L) = E∞ + A/L5 in conjunction with the AVQZ and AV5Z basis sets. The following corrections were added to the underlying SCF/CBS energy: (i) ΔCCSD (obtained using a two-point extrapolation of the form E(L) = E∞ + A/L3 in conjunction with the AVQZ and AV5Z basis sets), (ii) Δ(T) (a correction for parenthetical triples excitations, obtained using a two-point extrapolation of the form E(L) = E∞ + A/L3 in conjunction with the AVTZ and AVQZ basis sets), (iii) a core-valence correction (ΔCV) obtained as the difference between the all-electron CCSD(T)/MTsmall energies (with the exception of second-row elements, in which the 1s electrons are frozen) and the corresponding frozen core calculations and (iv) a scalar relativistic correction (ΔRel.), which is obtained by way of Douglass–Kroll–Hess (DKH) calculations[25,26] and corresponds to the energy difference between a frozen-core DKH-CCSD(T)/MTsmall and regular CCSD(T)/MTsmall calculations. The final all-electron relativistic, bottom-of-the-well W2w energy is given by the following formula: W2wrel,el = SCF/CBS + ΔCCSD + Δ(T) + ΔCV + ΔRel. To obtain energies at 298 K (i.e. BDE298), the W2w values were amended by the inclusion of scaled ZPVE and Hvib contributions, both of which have been obtained at the B3LYP/AVTZ level of theory, and were scaled by 0.9884 and 0.9987, respectively.[27]

We have additionally assessed a diverse array of different DFT functionals for their ability to compute gas-phase homolytic Al–H BDEs (in conjunction with the AVTZ and AVQZ basis sets), using the W2w non-relativistic bottom-of-the-well BDEs as reference values. The DFT exchange-correlation functionals considered in this study, ordered by their rung on Jacob’s ladder, are the generalised gradient approximation (GGA) functionals: BLYP,[28,29] B97-D,[30] HCTH407,[31] PBE,[32] revPBE,[33] PB86,[29,34] and BPW91,[29,35] the meta-GGA (MGGA) functionals: M06-L,[36] TPSS,[37] τ-HCTH,[38] VSXC,[39] M11-L,[40] MN12-L,[41] MN15-L,[42] r2SCAN,[43] and B97M-V,[44] the hybrid-GGAs (HGGA): BH&HLYP,[45] B3LYP,[28,46,47] B3P86,[34,46] B3PW91,[35,46] PBE0,[48] B97-1,[49] X3LYP,[50] SOGGA11-X,[51] APF,[52] and the range-separated functionals ωB97,[53] ωB97X,[53] ωB97X-V,[54] N12-SX,[55] CAM-B3LYP,[56] the hybrid-meta GGAs (HMGGA): M05,[57] M05-2X,[58] M06,[59] M06-2X,[59] M08-HX,[60] MN15,[42] BMK,[61] TPSSh,[62] τ-HCTHh,[38] PW6B95,[63] and the range separated functionals MN12-SX,[55] M11,[64] ωB97M-V,[65] and the double hybrid (DH) functionals: B2-PLYP,[66] mPW2-PLYP,[67] B2GP-PLYP,[68] DSD-BLYP,[69] PWPB95,[70] DSD-PBEP86,[71,72] DSD-PBEB95,[71] PBE0-DH,[73] PBEQI-DH.[74] Empirical D3 dispersion corrections[75,76] were included using the Becke–Johnson[77] damping potential (denoted by the suffix -D3). All calculations have been performed using the Gaussian 16 (rev. C.01) and ORCA 5.0 programs.[78–81]

Results and discussion

General overview of the AlHBDE dataset

The full set of gas-phase homolytic Al–H BDEs (Eqn 1), which have been obtained in conjunction with the W2w thermochemical protocol, are provided in Table 1. In addition to reporting the final homolytic gas-phase Al–H BDEs at 298 K (i.e. BDE298), we have also included the various contributions that lead to these values. These contributions include the underlying BDEs obtained using Hartree–Fock theory (ΔSCF), a correction for single and double excitations (ΔCCSD), a correction for the inclusion of quasi-perturbative triple excitations (Δ(T)), a core-valence correction (ΔCV), a scalar relativistic correction obtained within the Douglass–Kroll–Hess approximation (ΔRel.), the zero-point vibrational energy contribution (ΔZPVE) and finally, an enthalpy correction (ΔHvib) at 298 K. The species have been selected such that there is a reasonable selection of electron-donating and electron-withdrawing substituents. From a general perspective, we note that the Al–H BDEs of the molecules in this dataset (at 298 K) differ by up to 79.2 kJ mol−1, with (H2B)2Al–H having the lowest BDE (288.1 kJ mol−1), while (H2N)2Al–H is associated with the largest BDE (367.3 kJ mol−1). We note that the simplest molecule within this family, namely H2Al–H, has a BDE of 354.3 kJ mol−1. Prior to embarking on a more specific discussion concerning the effect of substituents in governing the magnitude of the Al–H BDEs, we make a few points to: (i) address the likely accuracy of the reported BDEs obtained at the W2w level by using an energy-based diagnostic for the importance of post-CCSD(T) contributions and (ii) examine the performance of the lower cost W1w thermochemical protocol, which may be applied to the high-level study of larger aluminium hydride species, for which the use of the more rigorous W2w protocol might be computationally prohibitive.

Table 1. Component breakdown and final W2w gas-phase homolytic Al–H BDEs (all components and energies are reported in kJ mol−1).

MoleculeΔSCFΔCCSDΔ(T)ΔCVΔRel.ΔZPVEΔHvibBDE298
(H2B)2Al–H (1)266.443.1−7.2−1.6−0.5−17.04.9288.1
(H2Al)2Al–H (2)246.861.7−3.0−0.7−0.3−15.84.6293.3
(H2B)HAl–H (3)296.742.5−5.6−1.8−0.6−20.66.3317.0
(H2Al)HAl–H (4)277.164.6−1.8−0.9−0.4−18.95.1324.8
(H2N)(H2B)Al–H (5)305.843.7−5.5−1.6−0.5−20.15.7327.4
(H3Si)HAl–H (6)282.379.1−0.3−0.4−0.4−20.05.4345.7
(PH2)HAl–H (7)285.579.2−0.7−0.4−0.5−20.15.4348.5
(SH)HAl–H (8)287.180.6−0.7−0.3−0.4−20.05.4351.6
Cl(H)Al–H (9)288.181.1−0.6−0.5−0.5−20.85.5352.4
Cl2Al–H (10)286.982.3−1.0−0.2−0.6−18.95.3353.8
H2Al–H (11)288.582.40.0−0.3−0.3−21.75.7354.3
(OH)HAl–H (12)290.980.9−0.9−0.3−0.3−21.15.6354.9
F(H)Al–H (13)290.382.0−0.7−0.4−0.4−21.25.6355.2
CH3(H)Al–H (14)289.982.0−0.3−0.2−0.3−20.15.0356.0
(CN)(H)Al–H (15)290.382.9−0.5−0.2−0.3−20.75.6357.1
(NH2)HAl–H (16)293.582.8−0.4+0.1−0.2−21.05.6360.3
F2Al–H (17)290.285.8−1.1+0.1−0.3−19.45.4360.7
(H2N)2Al–H (18)298.284.2−0.7+0.6−0.2−21.16.2367.3

First, we wish to point out that the W2w thermochemical protocol, which in effect affords an energy at the all-electron CCSD(T) basis-set-limit level, does not include post-CCSD(T) corrections. In some systems, for example those that exhibit high degrees of non-dynamical correlation, post-CCSD(T) corrections can be of considerable magnitude and their exclusion can render any computed thermodynamic quantity substantially less accurate. To address the likely reliability of the CCSD(T) method in this context, an energy-based diagnostic, namely the percentage of the atomisation energy accounted for by parenthetical connected triple excitations, %TAE[(T)] has been developed.[24,82] This diagnostic has been used previously for the purposes of validation of datasets of the BDEs of a range of other chemical bonds.[83–87] It has been shown that for species with %TAE[(T)] ≤ 5%, post-CCSD(T) contributions are unlikely to exceed 2 kJ mol−1. An analysis by Chan also supported the adoption of this recommended cut-off of ≤ 5%.[88] As the %TAE[(T)] diagnostics of all molecules considered in this study are well below the 5% threshold (ranging from 0.1% in the case of AlH3 and AlH2˙ to 3.0% in the case of (CN)HAl˙), inclusion of post-CCSD(T) contributions are therefore unlikely to affect the BDEs to any significant extent. On this basis, it stands to reason that the W2w BDEs reported in this study are expected to be within chemical accuracy (i.e. with deviations below 1 kcal mol−1) from reference values obtained at the full configuration interaction (FCI) infinite basis-set limit.[89]

We have also sought to consider whether the W1w thermochemical protocol, which is less computationally demanding than W2w, and hence could be applied to the study of larger aluminium hydride species, would also result in reliable Al–H BDEs. W1w theory employs the AVTZ and AVQZ (rather than AVQZ and AV5Z) basis sets for the computation of the ΔSCF and ∆CCSD components, and the AVDZ and AVTZ (rather than AVTZ and AVQZ) basis sets are employed for the computation of the Δ(T) correction. Having performed this analysis (see the Supporting material for the non-relativistic bottom-of-the-well valence W1w BDEs and the values of the various contributions giving rise to these BDEs), we note that the differences between W2w and W1w are relatively small. First, we note that the W1w protocol systematically overestimates the Al–H BDEs compared with those obtained using the W2w protocol with an MD and MAD of +0.5 kJ mol−1. Second, we note that the largest deviation, which amounts to 1.0 kJ mol−1, was observed in the case of the BDE of (H2B)2Al–H. As a result of breaking down the BDE of this molecule into the individual components, we find that the deviation of 1.0 kJ mol−1 between the W1w and W2w value arises predominantly because of a larger difference in the ΔCCSD correction (0.7 kJ mol−1), with only a 0.3 kJ mol−1 difference in the underlying SCF energy and a difference of –0.1 kJ mol−1 for the Δ(T) contribution.

Substituent effects in governing the magnitude of Al–H BDEs

Upon inspection of the AlHBDE dataset (Table 1), we note that, although the BDEs span a range of 79.2 kJ mol−1, this relatively wide range belies the fact that upon exclusion of those molecules bearing at least one –BH2 or –AlH2 substituent (i.e. molecules 1–5) the resulting variation in BDEs amounts to 21.6 kJ mol−1. Prior to discussing the BDEs of molecules 1–5, we note that, whereas substitution by third-period elements (i.e. Si, P, S, and Cl) results in molecules that have lower Al–H BDEs than that of H2Al–H, attachment of second-period elements (i.e. C, N, O, and F) serves to increase the Al–H BDEs. Of the species considered in this study, we note that (H2N)2Al–H is associated with the largest BDE (i.e. 13.0 kJ mol−1 higher than that of H2Al–H). The finding that aluminium hydrides substituted by electron-donating substituents are not associated with significantly lower Al–H BDEs (and in the case of electron-donating groups belonging to the second period actually serve to increase the Al–H BDEs compared with that of H2Al–H), can be attributed, in part, to the fact that the resulting aluminium-centred radicals adopt electronic states that are σ rather than π. By performing NBO calculations at the B3LYP/AVTZ level of theory, we note that overlap of the lone-pairs (of the π-electron-donating substituents) and the formally vacant 3p orbitals of the central Al atoms in Al–H-containing precursor molecules are conserved in the resulting radicals, and consequently the radicals do not benefit from the stabilisation that might be expected to arise if overlap between the half-filled orbital on the aluminium and the lone pair of the donor substituents (for example, which has been noted previously in the case of C–H BDEs)[90] were to occur.

As mentioned previously, the most dramatic substituent effects were observed in the case of molecules containing at least one –BH2 or –AlH2 substituent(s), with the magnitude of such effects being more pronounced in the case of the former. In this regard, we note that introduction of one –BH2 substituent (as in (H2B)HAl–H) serves to reduce the resulting Al–H BDE by 37.3 kJ mol−1 compared with that of H2Al–H, while introduction of a single –AlH2 substituent reduces the BDE by a smaller magnitude (29.5 kJ mol−1). The results of our computation for such species also indicate a sizable additive effect concerning the introduction of two such substituents, with the BDE of (H2B)2Al–H being 66.2 kJ mol−1 lower than that of H2Al–H (BDE = 354.3 kJ mol−1) and the BDE of (H2Al)2Al–H being reduced by 61.0 kJ mol−1 compared with that of H2Al–H. The significantly lower Al–H BDEs of molecules containing either –BH2 or –AlH2 substituent(s) may be attributed, at least in part, to the unpaired electron in each of the resulting radicals being delocalised onto either the substituent –BH2 or –AlH2 groups (refer to the Supporting material for images of the SOMOs of the radicals resulting from the homolytic dissociation of the Al–H bonds of molecules 1–4). The extent of such delocalisation effects can be probed by way of Mulliken spin density calculations. For example, in (H2B)HAl˙ (which adopts an almost planar structure, having a H–Al–B angle of 173.1° at the B3LYP/AVTZ level), we compute spin densities (at the ROHF/AVTZ level) of 0.437 for the boron atom and 0.531 for the aluminium atom. In a similar vein, we note that the (H2B)2Al˙ radical, which adopts a planar structure with D2h symmetry, is associated with spin densities of 0.194 on each of the substituent boron atoms and 0.607 on the central aluminium atom. Analogous effects, although of smaller magnitude, were also noted in the case of radicals containing at least one –AlH2 substituent. In this regard, we compute a spin density of 0.140 on the –AlH2 group of (H2Al)HAl˙, while in the case of (H2Al)2Al˙, the substituent aluminium atoms are each associated with spin densities of 0.102. The reduced extent of delocalisation in the aluminium-substituted radicals may account, in part, for why these species have higher Al–H BDEs than the comparable –BH2 substituted molecules.

Assessment of DFT methods for the computation of Al–H BDEs

Attention is now turned to considering the performance of a diverse array of lower-cost DFT methods for their ability to compute gas-phase homolytic Al–H BDEs (against the AlHBDE dataset). To assess these methods, we have utilised electronic non-relativistic bottom-of-the-well BDEs (i.e. those obtained according to BDENR,el = ΔSCF + ΔCCSD + Δ(T) + ΔCV). Table 2 gives the mean absolute deviations (MADs), mean deviations (MDs), largest deviations (LDs; the species that correspond to the largest deviation is given in bold) and the number of outliers (NOs, which constitute the number of species with an absolute deviation from the W2w reference value of ≥10 kJ mol−1), in conjunction with both the AVTZ and AVQZ basis sets.

Table 2. Performance of various DFT procedures (in conjunction with the AVTZ and AVQZ basis sets) for the calculation of gas-phase homolytic Al–H bond dissociation energies relative to W2w values (in kJ mol−1).

TypeAFunctionalAVTZAVQZ
MADMDLDNOMADMDLDNO
GGArevPBE34.3−34.346.3 (1)1833.6−33.645.1 (1)18
BPW9131.5−31.544.5 (1)1830.8−30.843.4 (1)18
PBE30.9−30.946.9 (1)1830.2−30.245.6 (1)18
HCTH40727.4−27.438.8 (1)1826.6−26.637.5 (1)18
BLYP17.6−17.630.2 (1)1816.9−16.928.6 (1)18
BP8615.5−15.530.3 (1)1815.0−15.029.1 (1)18
B97-D7.6−7.418.7 (1)36.9−6.617.7 (1)3
MGGAr2SCAN21.4−21.431.6 (1)1820.5−20.530.3 (1)18
τ-HCTH12.8−12.823.1 (1)1412.0−12.022.2 (1)14
MN12-L12.2+8.418.6 (1)1511.9+8.421.8 (14)13
B97M-V7.1+1.419.3 (1)37.6+3.815.5 (1)3
VSXC6.9−5.025.2 (1)55.9−2.622.1 (1)4
MN15-L6.2+0.818.5 (1)25.8+0.718.7 (1)2
TPSS5.8−5.811.7 (1)25.4−5.411.1 (1)1
M06-L2.0−1.67.1 (1)02.3−2.27.9 (1)0
HGGAPBE024.5−24.531.5 (5)1823.7−23.730.5 (5)18
APF22.6−22.629.2 (1)1821.9−21.928.2 (5)18
APF-D21.7−21.728.4 (1)1821.0−21.027.3 (1)18
B3PW9119.9−19.926.5 (1)1819.3−19.325.4 (1)18
B97-113.2−13.220.6 (1)1612.4−12.419.4 (1)16
ωB9710.4+10.416.5 (18)711.6+11.617.2 (18)11
N12-SXB10.9−10.623.7 (1)79.2−8.821.2 (1)5
X3LYP8.1−8.114.3 (1)27.4−7.412.8 (1)1
B3LYP7.9−7.914.2 (1)37.2−7.212.8 (1)1
B3P865.4−5.413.6 (1)24.8−4.812.5 (1)2
ωB97XB3.1+3.18.5 (1)04.2+4.210.4 (1)1
BH&HLYP4.1−3.25.8 (10)03.8−2.45.8 (3)0
SOGGA11-X3.6−2.58.7 (1)03.3−2.18.0 (1)0
ωB97X-DB3.2−3.15.1 (10)02.7−2.14.1 (12)0
ωB97X-VB3.3+3.310.4 (1)14.1+4.111.7 (1)1
CAM-B3LYPB2.5−2.34.6 (13)02.1−1.54.3 (13)0
HMGGAM06-2X15.3−15.320.7 (10)1717.2−17.222.4 (10)17
M11B9.6+9.636.6 (2)510.9+10.935.5 (2)5
τ-HCTHh11.4−11.420.8 (1)1410.6−10.619.6 (1)11
PW6B9510.7−10.719.3 (1)710.0−10.018.0 (1)4
PW6B95-D310.6−10.619.2 (1)69.9−9.917.9 (1)4
M05-2X8.0−7.711.3 (5)68.5−8.411.6 (5)7
M08-HX6.4+6.411.8 (2)17.8+7.811.5 (2)3
MN158.0−0.223.8 (1)45.9−1.921.8 (1)3
BMK4.4−4.49.3 (5)04.1−4.18.8 (5)0
TPSSh4.2−4.27.3 (1)03.7−3.76.7 (1)0
M064.2+4.09.7 (18)03.4+3.28.4 (18)0
ωB97M-VB3.1+2.78.2 (18)02.5+2.27.1 (18)0
DHPBE0-DH18.4−18.421.2 (5)1817.2−17.219.8 (15)18
PBEQI-DH15.0−15.016.1 (18)1813.1−13.114.6 (17)18
DSD-PBEP869.1−9.111.1 (10)26.5−6.58.4 (10)0
DSD-PBEB958.4−8.49.6 (10)05.8−5.87.8 (17)0
B2-PLYP7.3−7.310.1 (1)15.5−5.57.8 (1)0
DSD-BLYP7.8−7.89.7 (17)05.4−5.48.0 (17)0
PWPB957.1−7.19.7 (1)05.4−5.47.6 (1)0
B2GP-PLYP6.2−6.28.1 (10)04.1−4.16.3 (17)0
mPW2-PLYP5.7−5.77.5 (10)04.1−4.15.7 (10)0

AGGA, generalised gradient approximation; MGGA, meta-GGA; HGGA, hybrid-GGA; HMGGA, hybrid-meta-GGA; and DH, double hybrid.

BRange separated XC functional.

Prior to considering the performance of the functionals within each class, we make the following general points concerning the performance of these methods overall. First, the best-performing methods overall are CAM-B3LYP, M06-L, ωB97M-V and ωB97X-D, which in conjunction with the AVQZ basis set, are associated with MADs of 2.1, 2.3, 2.5 and 2.7 kJ mol−1, respectively. Both CAM-B3LYP and ωB97X-D attain similar LDs of 4.1 and 4.3 kJ mol−1, respectively, whereas ωB97M-V attains a slightly higher LD of 7.1 kJ mol−1 (Table 2). Second, of the 53 functionals considered in this study, we note that in most of the cases, the use of the larger AVQZ basis set affords better performance than with the smaller AVTZ basis set. Having said that, the magnitude of these performance improvements is generally small for the conventional DFT methods (i.e. for the most part, being less than 1.0 kJ mol−1). As expected, these differences can become more significant for the double-hybrid functional; for example, for the DSD-PBEP86 method, the difference reaches 2.6 kJ mol−1. Of the seven functionals in which we note that using the smaller AVTZ basis set was found to offer better performance, the largest performance improvement was noted in the case of M06-2X (1.9 kJ mol−1). Third, the overwhelming majority of the functionals systematically underestimate the BDEs. In this regard, only ten out of the 53 functionals assessed are associated with positive MDs. This finding, that the selected functionals tend to underestimate the Al–H BDEs, is consistent with previous studies that have demonstrated that DFT methods generally underestimate the BDEs of other bonds, for example, in the case of C–Cl,[84] B–Cl,[85] N–Br[91] and S–F[83] bonds. Fourth, for approximately 60% of the functionals considered in this study (in conjunction with the AVQZ basis set), their largest deviation was attributed to the computation of the Al–H BDE of molecule 1 (i.e. (BH2)2B–H).

We now turn our attention to considering the performance of the functionals within each family. These results will be discussed in the context of those values obtained in conjunction with the AVQZ basis set, not only as for the vast majority of the functionals use of this basis set affords better performance, but also because these values are closer to the basis-set-limit values for each functional. This is particularly true for the double-hybrid methods, which exhibit a substantially slower basis set convergence due to the MP2-like correlation term.[92]

With the exception of B97-D, the GGA functionals show very poor performance with MADs between 15.0 (BP86) and 33.6 (revPBE) kJ mol−1. B97-D performs considerably better but still results in a large MAD of 6.9 kJ mol−1. The inclusion of the kinetic energy density in the functional form considerably improves the performance. With the exception of three functionals (r2SCAN, τ-HCTH and MN12-L) which attain MADs between 11.9 and 20.5 kJ mol−1, the MGGAs attain MADs between 2.3 (M06-L) and 7.6 (B97M-V) kJ mol−1. Of the considered HGGA methods, only one functional provides better performance than M06-L, namely the range-separated hybrid CAM-B3LYP with a MAD of merely 2.1 kJ mol−1. This performance is followed by another range-separated hybrid, ωB97X-D, which attains a MAD of 2.7 kJ mol−1, respectively. We note that these long-range-corrected functionals involve 65–100% exact exchange at long-range. The best-performing global hybrids are SOGGA11-X and BH&HLYP, with MADs of 3.3 and 3.8 kJ mol−1, respectively. We note that these two methods involve 40–50% of exact exchange. The other global hybrids show relatively poor performance, with MADs ranging between 4.8 (B3P86) and 23.7 (PBE0) kJ mol−1. The HMGGA methods do not result in better performance relative to the best-performing MGGA and HGGA methods. The best-performing HMGGA methods attain MADs of 2.5 (ωB97M-V), 3.4 (M06), 3.7 (TPSSh) and 4.1 (BMK) kJ mol−1. Thus, again we find that a range-separated HMGGA provides better performance than the global HMGGAs. Interestingly, the global HMGGA functionals involve a wide range of exact exchange amounts, ranging between 10% (TPSSh) and 42% (BMK). Somewhat surprisingly, the considered DHDFT methods also do not provide better performance relative to the best-performing MGGA and HGGA methods. The best-performing DHDFT methods are mPW2-PLYP and B2GP-PLYP with the same MAD of 4.1 kJ mol−1. These results suggest that the performance for the Al–H BDEs does not strictly improve along the rungs of Jacob’s Ladder.

Finally, it is of interest to examine the effect of adding an empirical dispersion correction on the performance of the DFT methods. For this purpose we consider the pairwise D3 dispersion correction. Table 3 gives an overview of the effect of adding the dispersion corrections for a selection of DFT methods. The tabulated values are the difference in MAD between the dispersion-uncorrected and dispersion-corrected functionals, namely ∆MAD = MAD(DFT) − MAD(DFT-D3). Therefore, a positive ∆MAD value indicates that adding the dispersion correction improves the overall performance of the functional. Inspection of Table 3 reveals that this is indeed the case for practically all of the considered DFT methods. With the exception of revPBE, the improvements are relatively small and range between 0.1 (PW6B95) and 1.0 (BLYP) kJ mol−1. Nevertheless, for functionals that are capable of sub-chemical accuracy, these corrections are statistically significant.

Table 3. Overview of the effect of adding a D3 dispersion correction on the performance of the DFT methods.

TypeAFunctionalAVTZAVQZ
GGArevPBE1.41.4
PBE0.60.6
BLYP1.01.0
BP860.60.7
MGGATPSS0.70.7
HGGAPBE00.50.5
B3PW910.80.9
B3LYP0.70.7
HMGGAPW6B950.10.1
BMK0.0–0.1
ωB97M-V0.20.2
DHB2-PLYP0.30.2

The tabulated values are ∆MAD = MAD(DFT) – MAD(DFT-D3) (in kJ mol−1). A positive ∆MAD value indicates overall improvement in performance upon addition of the D3 dispersion correction.

AGGA, generalised gradient approximation; MGGA, meta-GGA; HGGA, hybrid-GGA; HMGGA, hybrid-meta-GGA; and DH, double hybrid.

Conclusion

Using the high-level W2w thermochemical protocol, we have computed a dataset of gas-phase homolytic Al–H BDEs for a set of 18 neutral aluminium hydrides (which we refer to as the AlHBDE dataset). The intention of this study was two-fold. First, we wanted to investigate the magnitude by which substituents can induce variations in the Al–H BDEs. Second, we assess the performance of DFT functionals for their ability to compute accurate Al–H BDEs. In addressing the first aim of this study, we note that the Al–H BDEs of the species in this dataset span a range of 79.2 kJ mol−1, with (H2B)2Al–H having the lowest BDE (288.1 kJ mol−1) and (H2N)2Al–H having the largest BDE (367.3 kJ mol−1). Of the selected substituents, both –AlH2 and –BH2 have by far the greatest effect in terms of altering the Al–H BDEs compared with that of the parent molecule H2Al–H. In fact, when molecules containing at least one of these two substituents are removed from the set, the Al–H BDEs of the remaining 13 molecules span a smaller range of 21.6 kJ mol−1. In terms of the second part of the study, our assessment of a broad range of DFT methods for the computation of Al–H BDEs (relative to the non-relativistic bottom-of-the-well W2w reference values) reveals that M06-L and CAM-B3LYP result in the best overall performance with MADs of 2.3 and 2.1 kJ mol−1, respectively. However, it should be noted that CAM-B3LYP is preferable since it results in a significantly smaller deviation of 4.3 kJ mol−1 relative to the largest deviation of M06-L of 7.9 kJ mol−1.

Supplementary material

Structural formulae for all aluminium hydrides investigated in the present study are provided in Supplementary Table S1. The geometries (in Cartesian coordinates) of all molecules investigated in this study (obtained at the B3LYP/AVTZ level of theory) are provided in Supplementary Table S2. In addition, the non-relativistic bottom-of-the-well valence W1w Al–H BDEs, as well as the individual components leading to these values, are provided in Supplementary Table S3. Images of the SOMOs (obtained at the ROHF/AVTZ level of theory) for the radicals arising via the homolytic Al–H dissociation of molecules 1–4 are provided in Supplementary Table S4. Supplementary material is available online.

Data availability

The data that support the findings of this study are available in the supplementary material of this article and from the corresponding authors upon reasonable request.

Conflicts of interest

Amir Karton is an Associate Editor of the Australian Journal of Chemistry but was blinded from the peer-review process for this paper.

Declaration of funding

This research was undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI, project dv9), which is supported by the Australian Government.

Acknowledgements

This work is dedicated to Professor Brian Yates in honour of his outstanding contributions to computational organometallic and carbene chemistry, as well as lifelong contributions to chemistry and science education in Australia. The authors gratefully acknowledge the generous allocation of computing time from the National Computational Infrastructure (NCI) National Facility and system administration support provided by the Faculty of Science, Agriculture, Business and Law at the University of New England to the Linux cluster of the Karton group.

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