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A Theoretical Study on the Protodeauration Step of the Gold(I)Catalyzed Organic Reactions Rasool BabaAhmadi,† Parisa Ghanbari,† Nasir Ahmad Rajabi,† A. Stephen K. Hashmi,*,‡ Brian F. Yates,*,§ and Alireza Ariafard*,† †

Department of Chemistry, Faculty of Science, Central Tehran Branch, Islamic Azad University, Shahrak Gharb, Tehran, Iran Organisch-Chemisches Institut, Universität Heidelberg, Im Neuenheimer Feld 270, 69120 Heidelberg, Germany § School of Chemistry, University of Tasmania, Private Bag 75, Hobart TAS 7001, Australia ‡

S Supporting Information *

ABSTRACT: Density functional theory was used to investigate the protodeauration of organogold compounds, a process which is thought to be the final step in the gold-catalyzed nucleophilic addition to activated π bonds wherein a proton is added and the gold catalyst is regenerated. In this context, we have studied two important factors which control the effectiveness of this transformation. We find that the nature of the alkenyl group in PMe3Au(alkenyl) affects the reaction barrier through the strength of the Au−C bond; the stronger the Au−C bond, the higher the activation energy. This, in turn, is determined by the π-accepting/donating ability of the substituents on the alkenyl group. We theoretically confirm that, for protodeauration, the reaction should be rapid when π-donating groups are present. In contrast, when πaccepting substituents are present, the intermediate gold complexes may be stable enough to be isolated experimentally. The second important factor controlling the reaction is the nature of the phosphine ligands. We theoretically confirm that electron-rich ligands such as PMe3 or PPh3 accelerate the reaction. We find that this is due to the strong electron-donating nature of these ligands, which strengthens the Au−P bond in the final product and thus provides a thermodynamic driving force for the reaction. Also, it is shown how the protodeauration is affected by the number of molecules solvating the proton. The protodeauration mechanism of some other organogold compounds such as gold−alkyl, gold−alkynyl, and gold−allyl species was investigated as well. The findings of this study can be used to design more effective systems for transformations of organogold compounds.

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INTRODUCTION

of the catalytic reaction is highly reactive toward protodeauration. In contrast, Shi and co-workers synthesized and isolated the gold alkenyl complex VIII through 5-endo-dig cyclization of triazoles (Scheme 3) and demonstrated that VIII is resistant to protodeauration.4 It follows from these results that the stability of the gold alkenyl complexes is determined based on the identity of the alkenyl groups. While examples of gold alkenyl complexes are well-known in stoichiometric organometallic chemistry,5 they could be isolated from catalysis reactions in only rare cases.6 In this context, Xu and co-workers showed that the protodeauration of the gold alkenyl complexes is accelerated by the electron-rich phosphine ligands.7 Although they performed a systematic study to elucidate trends in the protodeauration as a function of the ancillary ligand identities, it is not very obvious why electron-rich ligands enhance the rate of the reaction. They also showed that, for the cases where the protodeauration is the rate-determining step, the additives with the capability of making strong hydrogen bonds increase the rate

The transformation of organic compounds using homogeneous gold catalysts has become recently established as an essential method in the field of organic synthesis.1 In this regard, cationic gold catalysts are able to activate unsaturated C−C bonds toward the addition of nucleophiles via formation of π-complex I_a or I_b (Scheme 1).1 The nucleophilic attack on the activated π bond leads to formation of alkenyl complex II_a or II_b and releasing a proton (step 1). Finally, the alkenyl complex reacts with the proton via protodeauration to give the final product and regenerate the gold catalyst (step 2). The proton transfer can be assisted by solvent, water, counteranion, or other nucleophiles present in the reaction mixture. The validity of the catalytic cycle has been confirmed by various theoretical studies.2 It is well-established that the protodeauration step is fast in many catalytic processes. For example, Hashmi and co-workers developed AuI-catalyzed cycloisomerization of N-propargylcarboxamides III to 5-methylene-4,5-dihydrooxazoles IV and observed a good yield of the product with low catalyst loadings (Scheme 2).3 From this result, it can be concluded that the alkenyl complex V (Scheme 2) that is formed during the course © XXXX American Chemical Society

Received: March 17, 2015

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DOI: 10.1021/acs.organomet.5b00219 Organometallics XXXX, XXX, XXX−XXX

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Organometallics Scheme 1

carried out at the same level of theory as those for the structural optimization. Transition structures were located using the Berny algorithm. Intrinsic reaction coordinate (IRC)15 calculations were used to confirm the connectivity between transition structures and minima. To further refine the energies obtained from the B3LYP/BS1 calculations, we carried out single-point energy calculations for all of the structures with a larger basis set (BS2) in dichloromethane using the CPCM solvation model16 at the M06 level. BS2 utilizes the quadruple-ζ valence def2-QZVP17 basis set on Au and the 6-311+G(2d,p) basis set on other atoms. Effective-core potentials including scalar relativistic effects were used for the Au atom. We have used the Gibbs free energies obtained from the M06/BS2//B3LYP/BS1 calculations in dichloromethane throughout the paper unless otherwise stated. The atomic orbital populations were calculated on the basis of natural bond orbital (NBO) analyses.18 The AOMix program was also employed in order to compute the molecular orbital compositions.19

Scheme 2

of the catalytic reactions.8 In addition, Roth and Blum investigated protodeauration of various organogold complexes and found that electron-withdrawing substituents on arylgold complexes retard the protodeauration reaction.9 Since the protodeauration is responsible for producing the final product and leads to regeneration of the active catalyst, a better understanding of the mechanistic aspect of this step is necessary and can be useful for the future design of more efficient catalytic reactions. This computational study aims to provide clear guidelines for identifying factors that facilitate or inhibit the protodeauration. In this paper, we will show that, for a given phosphine ligand, the reactivity of the alkenyl complexes is principally determined by the Au−C bond strengths, whereas, for a given reacting alkenyl group, the reactivity of complexes is mainly controlled by the changes in the Au−P bond strength along the reaction coordinate.

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RESULTS AND DISCSSION

We have divided the discussion of the protodeauration process into two parts. The first considers how the identity of the alkenyl group influences the energetics of the protodeauration. The second addresses the question of why the electron-rich phosphine ligands are capable of accelerating the protodeauration process. Effect of the Alkenyl Group Identity on the Protodeauration. To investigate how the identity of the alkenyl group impacts the energetics of the protodeauration, we chose a range of neutral model alkenyl complexes (n), as shown in Table 1. These alkenyl groups are representative of the systems studied experimentally, and some of these alkenyl complexes have been already introduced as isolated or reactive intermediates.1k The protodeauration reaction outlined in Scheme 4 is modeled by DFT calculations. We carried out the calculations by using PMe3 as a phosphine ligand and (H2O)2(H3O+) as a strong electrophile responsible for protodeauration. In this model, the naked LAu+ cation was considered as one of the final products,

EXPERIMENTAL SECTION

Computational Details. Gaussian 0910 was used to fully optimize all the structures reported in this paper at the B3LYP level of density functional theory (DFT).11 The effective-core potential of Hay and Wadt with a double-ξ valence basis set (LANL2DZ)12 was chosen to describe Au. The 6-31G(d) basis set was used for other atoms.13 A polarization function of ξf = 1.050 was also added to Au.14 This basis set combination will be referred to as BS1. Frequency calculations were

Scheme 3

B


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Organometallics

Table 1. Activation and Reaction Energies (ΔG⧧ and ΔGrxn) Calculated for the Protodeauration Reaction of the Model Alkenyl Complexes, n, Where L = PMe3, Based on the Potential Energy Surface Given in Scheme 4, the Dissociation Energies Calculated for the Au−C Bond (D(Au−C)) in n and the C−H Bond (D(C−H)) in n_P, the NBO Charge on the Alkenyl Moieties in n (qalkenyl), and pπ Orbital Population at Cipso (pCipso)a

a

All the energies are given in kcal/mol.

barriers (ΔG⧧) range from 3.1 to 25.8 kcal/mol and reaction energies (ΔGrxn) from −13.3 to 0.2 kcal/mol. These results indicate that, in support of the experimental findings, the identity of the alkenyl group is a strong determinant for the energetics of the protodeauration.

although it should be noted that this cation can be further stabilized by interaction with either the water cluster or the organic product. Table 1 lists the calculated energetics for the protodeauration of the alkenyl complexes. Our calculations show that activation C


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Organometallics

We found a linear free energy relationship between ΔG⧧ and ΔGrxn values with a correlation coefficient R2 of 0.94 (Figure 1a); the more thermodynamically favorable the reaction, the lower the activation barrier. The origin of this trend is mainly traced back to the Au−C bond strengths; a plot of the ΔG⧧ values versus the homolytic Au−C bond dissociation energies (D(Au−C), Table 1) gives a satisfactory linear correlation with an R2 value of 0.85 (Figure 1b). This correlation suggests that the Au−C bond which is being weakened in the transition structure contributes to the barrier height; the stronger the Au−C bond, the higher the activation barrier. Although perhaps not unexpected, the significance of this result is that it indicates that the reaction is thermodynamically controlled. To probe the reason why the Au−C bond strength is reliant on the nature of the reacting alkenyl group, the NBO charges carried by the alkenyl moiety (sum of the NBO charges of all the atoms of alkenyl) in complexes n were calculated (Table 1). We found a good correlation between the NBO charge of the alkenyl moiety and the Au−C bond strength (R2 = 0.86) (Figure 1c).20 The corollary is that a more negative charge on the alkenyl group gives a stronger Au−C bond, a result which suggests that the ionicity of the Au−C bond is the dominant factor controlling the Au−C bond strength. Indeed, according to the Pauling electronegativity concept,21 M−R bond energy increases with bond polarity; the stronger the M−R bond, the more negative the charge on R. This behavior is not unprecedented and is welldocumented in the literature.22,23 For instance, Eisenstein found a similar correlation between ionicity and strength of M−C

Scheme 4

Figure 1. Plots of (a) ΔG⧧ vs ΔGrxn, (b) ΔG⧧ vs D(Au−C), (c) qalkenyl vs D(Au−C), and (d) qalkenyl vs pCipso based on the values given in Table 1. D


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bonds in studies on the C−H oxidative addition of fluorinated benzenes to transition-metal complexes.22a,c The calculations reveal that the population of the Cipso p(π) orbital plays an important role in the thermodynamic stability of the alkenyl complexes; a less populated p(π) orbital gives a stronger Au−C bond. The NBO population of the Cipso p(π) orbital in the alkenyl complexes is shown in Table 1. We found a reasonable correlation between the Au−C bond strength and the population of the p(π) orbital (R2 = 0.86) (Figure 1d).24 The trend can be explained as follows: the presence of π-accepting groups on the alkenyl ring tends to depopulate the Cipso p(π) orbital through π-conjugation, increasing the electronegativity of the Cipso atom, thereby enhancing the ionicity and strength of the Au−C bond. In short, the reactivity of the gold alkenyl complexes toward protodeauration is dictated by the π-accepting/donating ability of the groups on the alkenyl ring. The strong π-accepting groups give the most stable complexes, whereas the strong π-donating groups give the least stable complexes. This result agrees well with the experimental observations by Roth and Blum, who established that electron-withdrawing substituents on gold aryl complexes retard the protodeauration reaction.9 From a computational perspective, the gold alkenyl complexes with electron-withdrawing groups should be the most promising targets for isolation. The calculations predict that these alkenyl complexes would be even more stable than the cationic complex isolated by Shi and co-workers (Scheme 3);4 the activation energy for protodeauration of cationic complex 15 (15.2 kcal/ mol) (Scheme 5) is calculated to be lower than that for protodeauration of species 1−5 (Table 1). At this point, it is worthwhile to note that the π-conjugation effect also modulates the strength of the newly formed C−H bonds of the products. A strong π-conjugation that removes the electron density away from the Cipso atom results in a stronger Cipso−H bond. The homolytic C−H bond dissociation energies for the products (n_P) are listed in Table 1. There exists a reasonable correlation between the bond energies of Au−C and C−H (R2 = 0.66) (Figure S1, Supporting Information). A similar correlation between the strength of the M−C and C−H bonds has been already reported in the literature.25 The results presented in Table 1 indicate that the C−H bond energies follow the same trend as the Au−C bond energies. The slope of this plot is found to be 0.27, suggesting that the Au−C bond energies are more sensitive to the π-conjugation effect. This result explains the trend obtained for the ΔGrxn values in Table 1 and the reason that the protodeauration activation energy of the alkenyl complexes is principally determined by the Au−C bond strengths. However, it is worth noting that the ease of protodeauration of other organogold compounds relative to the alkenyl complexes cannot be interpreted in terms of the Au−C bond strength. For example, although the Au−C bond strength in the gold alkynyl complex PMe3Au(CCMe) is estimated to be as strong as 120.6 kcal/mol (about 12 kcal/mol stronger than that in species 1), the activation energy for protodeauration of this complex is only 3.6

kcal/mol. The justification for this low activation energy might be attributed to the relatively strong C−H bond in MeCC−H (D(C−H) = 136.6 kcal/mol). For the sake of completeness, we also compared the energetics of protodeauration for PMe3Au(Me) and PMe3Au(vinyl). The calculations show that the activation energy for protodeauration of PMe3Au(vinyl) (ΔG⧧ = 6.3 kcal/mol) is only 1.5 kcal/mol lower than that of PMe3Au(Me) (ΔG⧧ = 7.8 kcal/mol). This result finds support from the work of Roth and Blum;9 the protodeauration of the vinyl gold complex PPh3Au(vinyl) was found to be slightly faster than that of the methyl gold complex PPh3Au(Me) by a factor of 4.3. It follows from these findings that, in contrast to other organometallic systems in which the mixing of the metal−Cipso σ bond and the Cipso π system in the transition structure results in a dramatic lowering of the activation barrier,26 the effect of this orbital mixing on the protodeauration of organogold complexes is not very significant. Effect of the Phosphine Ligand Identity on the Protodeauration. As stated in the Introduction, the protodeauration is accelerated by electron-rich phosphine ligands.7 To understand the role of the phosphine ligands, we modeled the protodeauration reaction given in Scheme 6 with the aid of DFT calculations for L = PMe3, PPh3, P(OMe)3, PH3, PF3, and PCl3. The detailed structures of the representative stationary points 4, 4TS, and PMe3Au+ are depicted in Figure 2. Three dimensional ball-and-stick of other stationary points are shown in Figure S2 (Supporting Information). The calculations show that, consistent with the experimental observations, the electron-rich Scheme 6

E


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Figure 2. Optimized structures with selected structural parameters (bond lengths in Å) for 4, 4_TS, and PMe3Au+. For clarity, the hydrogen atoms of PMe3 and the alkyl ligand were omitted.

Table 2. Activation and Reaction Energies (ΔG⧧ and ΔGrxn) Calculated for the Protodeauration Reaction Based on the Potential Energy Surface Given in Scheme 6, the Dissociation Energies Calculated for the Au−C and Au−P Bonds in n (D(Au−C) and D(Au−P)1) and for the Au−P Bond in AuL+ (D(Au−P)2), and the Energy Difference between D(Au−P)1 and D(Au−P)2 (ΔD(Au−P))a

a

n

L

ΔG⧧

ΔGrxn

D(Au−C)

D(Au−P)1

D(Au−P)2

ΔD(Au−P)

4 16 17 18 19 20

PMe3 PPh3 P(OMe)3 PH3 PF3 PCl3

17.2 17.4 19.9 20.2 22.1 23.7

−7.2 −6.7 −1.7 −0.1 6.4 7.0

97.8 95.3 97.7 96.0 93.6 87.1

43.9 41.3 39.9 34.0 24.4 22.9

64.6 60.0 55.3 48.3 30.2 30.2

20.7 18.7 15.4 14.3 5.8 7.3

All the energies are given in kcal/mol.

Figure 3. Plots of (a) ΔG⧧ vs ΔGrxn and (b) ΔG⧧ vs ΔD(Au−P) based on the values given in Table 2.

phosphine ligand is a strong electron-donating ligand such as PMe3 and PPh3, the stability of the alkenyl complex falls off noticeably due to a considerable weakening of the Au−P bond. In such a case, the system becomes more reactive toward the protodeauration reaction. For the strong electron-donating ligands, the Au−P bond is strengthened more rapidly upon going from n to AuL+ due to cancellation of the trans influence of the alkenyl group. The calculations, for instance, show that, from 4 to PMe3Au+, the Au−P bond is strengthened by 20.7 kcal/mol, whereas, for the case of PCl3 (a weak electron-donating ligand), the Au−P bond is strengthened by only 7.3 kcal/mol. Table 2 gives the Au−P bond dissociation energies for species n (D(Au−P)1) and AuL+ (D(Au−P)2). In this table, ΔD(Au−P) stands for the energy difference between D(Au−P)1 and D(Au−P)2. According to the energy values given in Table 2, we find an excellent correlation between the ΔG⧧’s and the ΔD(Au−P)’s with an R2 value of 0.90 (Figure 3b). This linear correlation strongly supports the claim that the reactivity of alkenyl complexes with a given reacting alkenyl group but with different phosphine ligands

phosphine ligands such as PMe3 or PPh3 significantly decrease the protodeauration barriers and favor the reaction thermodynamically (Table 2). According to the energy values given in Table 2, we obtained a good correlation between the ΔG⧧’s and the ΔGrxn’s (R2 = 0.97) (Figure 3a). This trend again shows that the thermodynamic factors principally affect the ease of the protodeauration process. In contrast to the complexes with different alkenyl groups (Table 1), there is a reverse correlation between the Au−C bond strength and the ΔG⧧ values; the weaker the Au−C bond, the higher the activation barrier! It follows from this result that the difference in the reactivity between the alkenyl complexes with different phosphine ligands is not explainable in terms of the Au− C bond strength. We find that the change in the Au−P bond strength along the reaction coordinate is the most crucial factor in controlling the difference in the reactivity. In the complexes, both the phophine and the alkenyl ligands are trans to each other. Since the alkenyl group is a strong trans influencing ligand, the stability of the complex depends on the electron-donating property of the phosphine ligand. If the F


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is principally controlled by the changes in the Au−P bond strength along the reaction coordinate. The argument about the strengthening of the Au−P bond is also supported by the decrease of the Au−P distance upon going from n to nTS to LAu+. For example, the Au−P bond is shortened from 2.382 Å in 4 to 2.341 Å in 4TS and then to 2.285 Å in PMe3Au+ (Figure 2). To further explore the effect of spectator ligands, the protodeaurations of N-heterocyclic carbene (NHC) complexes 21 and 22 were also investigated (Scheme 7). The protodeauration barriers for the NHC complexes are calculated to be lower than those for the phosphine complexes, a result which is in accordance with the experimental observation.9 This result can be explained to some extent by the relatively large energy values derived for ΔD(Au−NHC) (Scheme 7). Effect of Electrophilicity of Protonating Agent on the Protodeauration. As expected, when the number of the water molecules solvating the proton increases, the electrophilicity of the proton decreases. A decrease in the electrophilicity is reflected by an increase in the LUMO energy of the protonating agents. The calculations at the M06/BS2//B3LYP/BS1 level of theory show that the Kohn−Sham LUMO energies for H3O+, (H2O)(H3O+), (H2O)2(H3O+), and (H2O)3(H3O+) are −1.71, −0.94, −0.77, and −0.56 eV, respectively. The protodeauration is also found to have a strong dependence on the electrophilicity of the protonating agent. For example, the protodeauration of 4 by a proton solvated by one, two, three, and four water molecules was investigated, and it was found that the activation energies increase in the order H3O+ (2.1) < (H2O)(H3O+) (14.9) < (H2O)2(H3O+) (17.2) < (H2O)3(H3O+) (20.0 kcal/mol). It follows that the less electrophilic the protonating agent, the higher is the activation energy for protodeauration. The LUMO energies have an excellent linear correlation with respect to the activation energies (R2 = 1.00) (see Figures S3 and S4, Supporting Information, for details). Protodeauration of a Gold Allyl Complex. We completed our study by investigating the protodeauration of the gold allyl complex 23 (Figure 4). Two pathways are found for protodeauration of the complex 23; pathway (I) protonates directly the α-carbon of the allyl group, while, in pathway (II), the interaction of the Au−Cα σ orbital with the C−C π* orbital promotes the protonation of the γ-carbon. Although both

Figure 4. Energy profile calculated for protodeauration of the allyl complex 23 via pathways (I) and (II). The relative free energies obtained from the M06/BS2//B3LYP/BS1 calculations in dichloromethane are given in kcal/mol.

pathways give the same product, pathway (II) is found to be 9.3 kcal/mol more favorable than pathway (I) (Figure 4). This result corroborates findings reported by Lee and co-workers that the proposed allyl intermediate formed in the course of the gold(I)catalyzed addition of indoles to cyclopropenes is protonated only at the γ-carbon and not the α-carbon.27 The higher reactivity of the γ-carbon over the α-carbon of η1-allyl complexes has also been discussed by previous theoretical studies.28 G


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In order for the protodeauration to occur, the proton should interact with the HOMO of 23. The molecular orbital decomposition analysis shows that the γ-carbon has a greater contribution to the HOMO than the α-carbon (Figure S5, Supporting Information). In addition, the energy decomposition analysis outlined in Scheme 8 for transition structures 23TS and 23′TS indicates that the allyl complex 23 requires a larger structural deformation to reach transition structure 23TS. This argument is supported by the much lower stability of 23_def versus 23′_def, as reflected by the larger deformation energy ΔE1d versus ΔE2d (Scheme 8); the species 23_def and 23′_def in Scheme 8 are the fragments derived from the deletion of the acid from 23TS and 23′TS, respectively, without any geometry relaxation. The α-carbon nucleophilicity is relatively low due to its interaction with the gold metal center. In order for the αcarbon to start interacting with the proton, the Au−Cα bond should be significantly weakened. The longer Au−Cα bond distance in 23TS (2.248 Å) as compared to the shorter Au−Cα bond distance in 23′TS (2.138 Å) supports this argument (Scheme 8). The less availability of the α-carbon for interaction with the proton renders 23TS a later transition structure than 23′TS, as evidenced by the 0.214 Å shorter C−H bond distance being formed and the 0.145 Å longer O−H bond distance being broken (Scheme 8).

protodeauration is dictated by the exergonicity of the reaction; a more exergonic reaction has a lower activation energy. We also explored the origin of the barrier from the point of view of the phosphine ligand bound to the gold and found that, in agreement with the experimental findings,7 electron-releasing ligands lower the barrier. This result was explained by the fact that the strong electron-donating nature of these ligands strengthens the Au−P bond in the final product and thus provides a thermodynamic driving force for the reaction. The effect of the number of water molecules solvating the proton on the protodeauration was also investigated. It was found that, as the number of the water molecules increases, the electrophilicity of the proton decreases, which, in turn, results in an increase in the activation energy of the protodeauration. This explains why the protodeauration is sensitive to the strength and concentration of the acid. Finally, a brief investigation of the protodeauration step for a simple allyl complex shows that the protonation mechanism of this species is different from the others and occurs at the least-hindered carbon furthest away from the gold (γ-carbon). Overall, this study has emphasized the importance of taking into account the energetics of this final step in the overall catalytic cycle involving gold, and we have confirmed that the groups on the alkenyl substrate and the ligands on the metal can both contribute to the overall barrier. These results can be used in the design of catalytic reactions of unsaturated organic compounds. In addition, we have identified specifically several gold complexes that might be isolated experimentally due to their large barriers for the protodeauration reaction.

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CONCLUSIONS We have explored protodeauration, the final step, for example, in the gold-catalyzed nucleophilic addition process in which a proton is added and the gold catalyst is regenerated, from a variety of different perspectives. For a range of (mostly cyclic) alkenyl complexes, the barrier for this step can be traced back to the π-accepting/donating ability of the groups on the alkenyl ring, a result which is supported by the experimental data.9 The greater the π-accepting ability, the greater the barrier and the slower the reaction. Indeed, we predict that some of these gold complexes have such large reaction barriers that they could be isolated experimentally. In determining the overall reactivity, there is a subtle interplay of Au−C and C−H bond strengths. In the cyclic alkenyl compounds studied here, the C−H bond strengths are all fairly similar and so the Au−C bond strength determines the outcome. However, this is not true in other cases, such as some simple gold−alkyl and gold−alkynyl species. Indeed, the reactivity of the organogold compounds toward the

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ASSOCIATED CONTENT

S Supporting Information *

Additional supporting information, which includes an xyz file giving the Cartesian coordinates of all optimized structures along with energies and Figures S1−S7. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.organomet.5b00219.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (A.A.). *E-mail: [email protected] (B.F.Y.). *E-mail: [email protected] (A.S.K.H.). H


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Organometallics Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the support of the University of Tasmania for a Visiting Scholarship (to A.A.) and the generous allocation of computing time from the Australian National Computational Infrastructure and the University of Tasmania.

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