Hydration of MgCC: a quantum DFT and ab initio HF

Journal of Molecular Structure: THEOCHEM 728 (2005) 231–242 www.elsevier.com/locate/theochem

Hydration of MgCC: a quantum DFT and ab initio HF study Martine Adrian-Scottoa, Georges Malletb, Dan Vasilescua,* a

Laboratoire Universitaire d’Astrophysique de Nice, Faculte´ des Sciences, Universite´ de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France b I.Re.T., I.U.T. de Nice-Coˆte d’Azur, 41 Bd Napole´on III, 06041 Nice Cedex, France Received 28 January 2005; revised 10 February 2005; accepted 10 February 2005 Available online 18 July 2005

Abstract The hydration of magnesium dication is studied using the density functional theory BPW91/6-311CCG(3d,3p) and ab initio RHF/631G(d,p) methods. The polarizabilities, enthalpies and Gibbs free energies of MgCC hydration are computed, up to nine added molecules of water. The entropic contribution to the hydration phenomenon is discussed using frequency analyses. The mechanism of hydration is also analyzed using the electronic chemical potential, hardness and electrophilicity concepts. q 2005 Published by Elsevier B.V. Keywords: Hydration; Magnesium dication; Quantum modelling; DFT BPW91/6-311CCG(3d,3p) and HF/6-31G(d,p) methods; Polarizability; Hardness; Electronegativity; Electronic chemical potential; Electrophilicity

1. Introduction Today a great number of studies has been devoted to the hydration of the magnesium dication MgCC. As pointed out by Bock et al. [1], this hard divalent cation is generally found in a hexahydrated octahedral architecture and is implicated in a structural role, when associated with proteins. For instance, it was suggested that two hexahydrated MgCC are involved in the hammerhead ribozyme catalysis [2]. Among the divalent ions of biological interest, the absolute hardness h of magnesium as defined by Pearson (32.55 eV) is inferior to the hardest BeCC (67.84 eV) but superior to CaCC (19.52 eV), ZnCC (10.88 eV), CuCC (8.27 eV) and FeCC (7.24 eV) [3,4]. On another hand, MgCC which is isoelectronic to H2O (i.e. 10 electrons) is an electrophile, much more electronegative (cZ47.59 eV) and hard than the nucleophile H2O (characterized by cZ3.1 eV and hZ9.5 eV). The most recent theoretical studies for understanding the cation–water solvent interactions at the discrete molecular level, of the first and second shells of hydration, involve * Corresponding author. Tel.: C33 492076314; fax: C33 492076321. E-mail addresses: [email protected] (M. Adrian-Scotto), [email protected] (G. Mallet), [email protected] (D. Vasilescu).

0166-1280/$ - see front matter q 2005 Published by Elsevier B.V. doi:10.1016/j.theochem.2005.02.006

essentially the ab initio Hartree–Fock, density functional theory (DFT) and reaction field methods [1,5–8]. The water exchange reactions between the first and the second shells around cations is also of great interest (as seen in the review by Helm and Merbach [9] and Hartmann et al. [6] in the case of ZnCC). As remarked by Markham et al. the cluster approach to compute the enthalpy DH and the Gibbs free energy DG changes seems to be the best when combined with ab initio DFT methods. When considering the reaction MgCC C clusterðH2 OÞn/ cluster½Mg; nH2 OCC

(1)

the computation takes into account both the ligation energies (water–ion interactions) and the water–water interactions. Recently, Bour [10] has demonstrated that computations using the DFT method BPW91/6-31G(d,p), for a cluster of 214 water molecules, reproduce very well the IR spectrum of liquid water. Then, the use of the PW91 exchange correlation functional is preferable, for the dispersion forces are not well treated by the standard DFT procedures. As a continuation of a first investigation [11], we have studied the hydration of MgCC up to nine water molecules, using the cluster approach and the DFT BPW91 at the very large basis set level (6-311CCG(3d,3p)) which includes diffuse functions.

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2. Procedures All computations have been conducted with the GAUSSIAN 98 [12], GAUSSVIEW [13] and HYPERCHEM [14] programs on local PC stations, using ab initio methods SCF Hartree– Fock and DFT BPW91 (Becke’s 1988 exchange functional and Perdew–Wang’s 1991 gradient-corrected correlation functional). The different clusters (of water and hydrated magnesium) were optimized using firstly RHF/6-31G(d,p) computations, then with the DFT Restricted BPW91 method at the 6-311CCG(3d,3p) basis set level. Optimization was done and frequency calculations were performed at the optimized geometry. This procedure allowed to obtain the thermochemical properties and the polarizabilities of both water clusters and MgCC hydrated clusters. The thermal correction to electronic energy (Eel) of a molecule (in hartree/particle) is [15–17] TCE Z Etrans C Erot C Evibr ðTÞ

(2)

where trans, rot and vibr indicate, respectively, the tranlational, rotational and vibrational motions, with Etrans Z ð3=2ÞkT; Erot Z ð3=2ÞkT ðnonlinear moleculeÞ and Evibr ðTÞ Z ð1=2Þk

X

QVi C

i

X fkQVi =½expðQVi =TÞ K 1g i

It is to be noted also that, in our studied supermolecules, there are no internal rotation modes [18]. And in the ideal gas approximation, we obtain: – the total internal energy: E Z Eel C TCE

(6)

– the enthalpy: H Z Eel C TCH ðfrom H Z E C PVÞ

(7)

– the Gibbs free energy: G Z Eel C TCG ðfrom G Z H K TSÞ

(8)

The standard procedures for computing the energies of hydration are based on the reaction MgCC C nH2 O/ ðMg; nH2 OÞCC C DH1 ðor DG1 Þ

(9)

In the cluster approach, a cluster of water containing nH2O is symbolized by (H2O)n, and a cluster of hydrated MgCC containing nH2O is symbolized by (Mg, nH2O)CC. Taking into account that MgCC is hexahydrated, the structure of this cluster is [(Mg, mH2O)pH2O]CC with m the number of the water molecules in the first hydration shell (mZ1.6), p the number of water molecules in the second hydration shell, and mCpZn. In our modelling, pZ1, 2, 3. Then, the hydration reactions of MgCC in the cluster procedure are: MgCC C ðH2 OÞn/ ðMg; nH2 OÞCC C DH2 ðor DG2 Þ

with iZ1, 2,. m, where

(10)

– m is the number of normal modes of harmonic vibrations (mZ3nK6; n number of atoms) – QVi are the vibrational temperatures, related to the harmonic vibrational wave number 6i (cmK1) by: QViZ(hc/k)6i.

Reaction of successive binding water enthalpies or Gibbs free energy:

In this harmonic oscillator treatment, the zero-point vibrational energy is: X ZPE Z ð1=2Þk QVi (3)

The resulting enthalpies (or Gibbs free energies) for (9)– (11) are

i

The resulting thermodynamic relations are (in hartree/particle):

½Mg; ðn 1ÞH2 OCC C H2 O/ ðMg; nH2 OÞCC C DH3 ðor DG3 Þ ð11Þ

DH1 Z HðMg; nH2 OÞCC K HðMgCCÞ K nHðH2 OÞ ðfor the standard procedureÞ DH2 Z HðMg; nH2 OÞCC K HðMgCCÞ K HðH2 OÞn ðfor the clusterÞ

ð12Þ

ð13Þ

– thermal correction to enthalpy: TCH Z TCE C kT

(4)

– thermal correction to Gibbs free energy: TCG Z TCH K TS

DH3 Z HðMg; nH2 OÞCC K HðH2 OÞ K Hð½Mg; ðn 1ÞH2 OCCÞ ðapproachÞ

(14)

(5)

where S, the computed entropy from the canonical partition functions, involves the three contributions: SZStransCSrotCSvibr(T)

with the same procedure for obtaining DG1, DG2, DG3. For the water clusters we define DH1 Z HðH2 OÞn K nHðH2 OÞ

(15)


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3. Results and discussion 3.1. Structure, polarizabilities and thermochemistry of (Mg, nH2O)CC clusters

Fig. 1. Ellipsoı¨d of polarizability of one optimized water molecule using BPW91/6-311CCG(3d,3p) computation. The principal axes of the tensor of polarizability are figurated with the dipole moment along the z axis. The equation of the ellipsoı¨d of polarizability is: x2 =a2xx C y2 =a2yy C z2 =a2zz Z 1.

DH2 Z HðH2 OÞn K HðH2 OÞ K HðH2 OÞnK1

(16)

and the same definitions for DG1 and DG2. Electron affinities, ionization energies, chemical potentials and hardnesses were obtained from two methods: the DFT way [3,19–22] using BPW91/6-311CCG(3d,3p) computations for (Mg, nH2O)CC clusters, the frontier orbital way [4] using HF/6-31G(d,p) computations for both H2O clusters and hydrated MgCC clusters.

The performance of the DFT method BPW91 at the 6-311CCG(3d,3p) level with regard to the experimental tensor of polarizability of one molecule of water (see Fig. 1) is illustrated in Table 1. This table shows that for the same level of basis set 6-311CCG(3d,3p), DFT methods are preferable to the ab initio HF computations, while BPW91 gives better results for polarizabilities and vibrational frequencies than the B3LYP procedure. This remark remains valid in the case of MgCC, for which the value of polarizability axx Z ayy Z azz Z haiZ 0:454a30 is nearer to the experimental one 0:51 a30 [23] when computed with BPW91, when B3LYP gives 0:446 a30 . Such a comparison was also made for the hydrogen atom: the experimental polarizability is given to be aH Z 4:5 a30 [24], while we obtain aH Z 4:117 a30 using U BPW91/ 6-311CC G(3d,3p), and aH Z 4:031 a30 using U B3LYP/6-311CC G(3d,3p). Our calculated optimized structures of water clusters correspond to local minima (with no symmetry constraint) controlled by frequency analyses (no imaginary frequencies). The obtained n-mers are: 1. A dimer with a structure very close to those found by Xantheas and Dunning [26] using an HF/MP2/aug-ccpVDZ computation with a distance between the two

Table 1 Systematic Hartree–Fock and DFT study of H2O (R computations)

HF

6-31 G 3d3p 6-31C G 3d3p 6-31CC G 3d3p 6-311 G 3d3p 6-311C G 3d3p 6-311CC G 3d3p DFTB3LYP 6-31 G 3d3p 6-31C G 3d3p 6-31CC G 3d3p 6-311 G 3d3p 6-311C G 3d3p 6-311CC G 3d3p DFTBPW91 6-31 G 3d3p 6-31C G 3d3p 6-31CC G 3d3p 6-311 G 3d3p 6-311C G 3d3p 6-311CC G 3d3p Experimental values

n bending (cmK1)

n s-stretch (cmK1)

n a-stretch (cmK1)

m (D)

axx ða30 Þ

ayy ða30 Þ

azz ða30 Þ

haiða30 Þ

Daða30 Þ

1775.5264 1753.2593 1752.8400 1775.4886 1757.9391 1757.8452 1667.1170 1629.4550 1629.0116 1659.2502 1636.3047 1635.5517 1638.3620 1603.3457 1602.8594 1630.9752 1608.0905 1607.7299 1595

4132.4427 4115.5222 4114.6313 4129.2904 4127.9022 4127.9522 3809.7695 3789.5030 3788.0092 3814.0757 3807.4312 3809.5756 3712.6121 3693.1358 3691.3772 3714.8510 3713.3276 3712.9605 3657

4236.1295 4222.9935 4222.5333 4223.5426 4227.5307 4227.4764 3912.5304 3898.5575 3897.6848 3907.4508 3908.9501 3911.0729 3818.0113 3804.5382 3803.4866 3811.9983 3817.4059 3816.9544 3756

1.9216 1.9601 1.9552 1.8742 1.9760 1.9754 1.7758 1.8394 1.8286 1.7015 1.8549 1.8524 1.7340 1.7976 1.7843 1.6609 1.8165 1.8134 1.8500

6.2570 7.4850 7.4920 6.288 7.394 7.404 6.7680 8.9810 9.0170 7.0600 8.8880 8.9810 7.0880 9.5100 9.5630 7.4130 9.3880 9.5050 9.55G0.09

8.3230 8.4500 8.5290 8.395 8.525 8.549 9.2980 9.4910 9.6810 9.5070 9.7150 9.7630 9.5990 9.8080 10.0290 9.8330 10.0340 10.1010 10.31G0.09

7.4170 7.8320 7.8840 7.089 7.741 7.764 8.4410 9.0830 9.2100 8.0820 9.0370 9.1150 8.8260 9.5010 9.6590 8.4730 9.4700 9.5600 9.91G0.09

7.3323 7.9223 7.9683 7.2573 7.8867 7.9057 8.1690 9.1850 9.3027 8.2163 9.2133 9.2863 8.5043 9.6063 9.7503 8.5730 9.6307 9.7220 9.9233

1.7937 0.8466 0.9069 1.8421 1.0036 1.0141 2.2287 0.4674 0.5916 2.1287 0.7635 0.7244 2.2275 0.3026 0.4262 2.1011 0.6092 0.5705 0.6585

The mean polarizability is: haiZ ð1=3Þðaxx C ayy C azz Þ and the polarizability anisotropy is defined by: DaZ fð1=2Þ½ðaxx K ayy Þ2 C ðaxx K azz Þ2C ðayy K azz Þ2 g1=2 .

234


Table 2 Thermochemistry of clusters (H2O)n (R BPW91/6-311GCC(3d,3p) computations) Number of water molecules

H (hartree)

G (hartree)

Eel (hartree)

S (cal molK1KK1)

m (D)

DH1 (kcal molK1)

DH2 (kcal molK1)

DG1 (kcal molK1)

DG2 (kcal molK1)

1 H2 O 2 H2 O 3 H2 O 4 H2 O 5 H2 O 6 H2 O 7 H2 O 8 H2 O 9 H2 O

K76.429580 K152.862933 K229.303916 K305.748930 K382.186522 K458.621936 K535.062774 K611.508195 K687.944835

K76.451025 K152.895798 K229.341041 K305.791697 K382.237724 K458.679982 K535.124663 K611.571714 K688.014824

K76.454172 K152.914741 K229.383406 K305.855737 K382.320286 K458.783454 K535.251377 K611.724744 K688.189234

45.13 69.17 78.14 90.01 107.76 122.17 130.26 133.69 147.31

1.8135 2.4348 1.2575 0 2.6899 1.4348 3.4204 4.7661 2.8204

0 K2.37 K9.52 K19.21 K24.24 K27.90 K34.96 K44.90 K49.33

0 K2.37 K7.16 K9.68 K5.03 K3.66 K7.06 K9.94 K4.43

0 3.92 7.55 7.78 10.92 16.42 20.40 22.90 27.86

0 3.92 3.63 0.23 3.14 5.50 3.98 2.49 4.97

TZ298.15 K, PZ1 atm.

˚ and a bend hydrogen bond oxygens R(O–O)Z2.93 A dO–H–OZ172.38, with a deviation from linearity of sZ5.148. 2. A cyclic trimer close to a perfect equilateral triangle with ˚ (corresponding to two hydrogen R(O–O)Z2.79 A bonded water molecules) and three bended-hydrogen bonds with a d angle which lies between 151.5 and 153.78. 3. A cyclic tetramer close to a square defined by R(O–O)Z ˚ and four bended hydrogen bonds with dZ169.58. 2.73 A This structure is remarkable from the electrostatic point of view because of the zero value of its dipole moment (see Table 2).These trimer and tetramer structures are close to those described in Refs. [25,26], respectively. 4. The conformation of our cluster (H2O)5 is close to the cyclic pentamer defined by Xantheas and Dunning [26], while the (H2O)6 differs from their S6 cyclic structure; the (H2O)8 is like the octamer with D2d structure described by Tsai and Jordan [27], Dang [28] and Rodriguez et al. [29], and also the cage structure obtained by Markham et al. [5]. For the (H2O)5 cluster we obtain a R(O–O) varying from ˚. 2.70 to 2.72 A For the (H2O)6 cluster the R(O–O) distances vary from ˚. 2.72 to 2.80 A For the (H2O)7 cluster we obtain a variation from 2.58 to ˚. 2.86 A For the (H2O)8 cluster we observe that the distance ˚. R(O–O) varies from 2.58 to 3.07 A For the (H2O)9 cluster we obtain a cage-like structure ˚. (H2O)8CH2O with R(O–O) varying from 2.64 to 3.00 A The extra H2O forms a basket with (H2O)8. This structure was also obtained by Dang [28]. The distances R(O–O) can be compared with those obtained by Silvestrelli and Parinello [30] for clusters of ˚ , close to the average 32–64 water molecules: 2.75–2.78 A ˚ distance of 2.80 A in liquid water. The thermochemistry and the dipole moments for these water clusters are listed in Table 2. We can note that the binding enthalpy for the water dimer DH1ZDH2ZK2.37 kcal molK1 is less accurate than the

MP2 aug-cc-PVDZ (K3.21 kcal molK1) given in Ref. [26], and to the experimental value (K3.6G0.5 kcal molK1). The value of 2.43 D for the dipole moment is closer to the experimental value (2.60 D) given by Yu et al. [31] versus that given by Silvestrelli and Parinello [30]: 2.15 D. The dimer electronic binding energy DE2ZE(2H2O)K2E(H2O)Z K0.009127HZK5.72 kcal molK1 is also in agreement with DE2 expZK5.26 kcal molK1 (i.e. K22 kJ molK1) given by Sprik et al. [32]. Our water clusters present a dipole moment close to 3 D, except for the (H2O)7. The obtained magnesium–water clusters, which correspond to global minima (without any imaginary frequency) are represented in Fig. 2. The structures of dihydrated, tetrahydrated and hexahydrated magnesium are equivalent to those computed by Markham et al. [5]. Concerning the first coordination shell, our computed distances Mg–O are close but slightly longer than those obtained by Pavlov et al. [7] who have used the B3LYP DFT method with double z basis sets. We obtain for the different ˚ , dihydrated 1.96 A ˚, clusters: monohydrated 1.93 A ˚ , quadrihydrated 2.02 A ˚ , quintahydrated trihydrated 1.99 A ˚ , hexahydrated 2.12 A ˚. 2.06–2.12 A The second shell of hydration was initiated with the structures [(Mg, 6H2O)H2O]CC, [(Mg, 6H2O)2H2O]CC and [(Mg, 6H2O)3H2O]CC. We can observe in Fig. 2(g) and (h) that the water molecules belonging to the second shell are hydrogen bounded to two water molecules of the first shell. This structure was obtained also by Markham et al. [5] for the [(Mg, 6H2O)2H2O]CC cluster. Fig. 3 represents the structure of the cluster with 3H2O in the second shell. We observe also in this case the presence of double hydrogen bonds to three water molecules of the first shell. It is to be noted that the distances between the oxygen atoms of the water molecules belonging to the first shell and the oxygen atoms of water molecules of the second shell are ˚ , in agreement with those reported by Pavlov et al. w2.79 A ˚ ). [7] and very closed to the experimental value (2.76 A Figs. 2 and 3 show the evolution of the Mulliken net charge on magnesium cation. At our basis set level, we observe a continuous decrease of the charge with


235

Fig. 2. Structures of optimized Mg–water clusters: [Mg, nH2O]CC with BPW91/6-311CCG(3d,3p) computation. (a) nZ1 to (h) nZ8. Bond lengths are in angstroms and angles in degrees. Mulliken charge of magnesium is given in brackets.

the number of water molecules around the magnesium; in the case of the hexahydrated structure (Fig. 2(f)) about 0.65 of the cationic charge was transferred to water molecules. This type of magnesium charge variation was also observed

by Pavlov et al. [7] in their B3LYP/6-311CG(2d,2p) computations. The polarizabilities of our obtained [Mg, nH2O]CC clusters are given in Table 3. As suggested by Ghanty

236


Fig. 3. Structure of optimized Mg–water cluster [(Mg, 6H2O)3H2O]CC with BPW91/6-311CCG(3d,3p) computation. Bond lengths are in angstroms and angles in degrees. Mulliken charge of magnesium is given in brackets.

and Ghosh [33,34] in case of standard reactions ACB/ ABCDH!0 , we can observe that for all our clusters the cube root of the mean polarizability hai obey the equation: hai1=3 ðMgCCÞ C hai1=3 ððH2 OÞn ÞO hai1=3 ð½Mg; ðH2 OÞn CCÞ ð17Þ We have also reported in Fig. 4 the variation of the volume corresponding to the electronic spatial extent versus the mean polarizability hai of these clusters. We can observe that the slope changes when penetrating the second hydration sphere; in the first hydration shell, the correspondence between hR2i3/2 and hai is quasilinear. Fig. 5(a)–(d) represents the polarizability ellipsoid for MgCC hydrated by 1, 2, 6 and 9 water molecules, respectively. The thermochemistry of our n-hydrated divalent magnesium structures is reported in Table 4. We have reported there the energies of magnesium hydration obtained with the standard procedure (DH1 and DG1) and with the cluster procedure (DH2 and DG2). DH3 and DG3 are the values for

the successive binding water enthalpies or Gibbs free energies, in a cluster procedure. The first line (0 H2 O) corresponds to the Mg CC energetic and entropic properties. In all cases, the negative signs of DH and DG indicate that the hydration process of MgCC is exothermic and exotropic, with Dhai1/3!0, where Dhai1=3Z hai1=3 ð½Mg; ðH2 OÞn CCÞK hai1=3 ðMgCCÞK hai1=3 ððH2 OÞn Þ. The results presented by Pavlov et al. [7] concern only the standard procedure and only internal energy variations during the magnesium hydration. Thus, a direct comparison with our results is not possible. For instance, in the case of the hexahydrated magnesium, they found with our notations: DE 1ZK303.9 kcal mol K1 and DE 3Z K24.5 kcal molK1. Our results may be compared with those obtained by Markham et al. [5] for a number of water molecules nZ2, 4, 6, 8, using ab initio Hartree–Fock MP2 computations. The correspondence between their results and ours is DH298ZDH2 and DG298ZDG2. For the hexahydrated magnesium, they found DG 298Z K266.7 kcal molK1, using a MP2 (FULL)/6-311CCG** computation and DG298Z K286.5 kcal molK1 when using the MP4 SDQ(FC)/ 6-31G* procedure. Our DG2 value is K251.5 kcal molK1. An another fact emphasized by Markham et al. [5] is to compare the DG2 values as n increases with the experimental one: DGexpZK439.7 kcal molK1. They observe that for an MP2 (FULL)/6-311CCG** computation, in the case of hexahydrated magnesium, the first shell of hydration accounts for 61% of the DG2 experimental value (in our case, this value is 57.2%). Then, when two water molecules are added in the second shell, they account for: 2 OÞ ½DGðMg;8H 2

CC

2 OÞ 2 OÞ K DGðMg;6H =DGexp K DGðMg;6H %: 2 2 CC

CC

We found in our case 14.2%, a result in agreement with those obtained by Markham et al. (14%). We observe also that when n increases, DG3 tends towards zero, which is the necessary condition for an easy exchange of a water molecule between the second shell of hydration and the water environment [9]. Because the Gibbs free energy involves the entropic contribution TS, it is interesting to understand the entropy

Table 3 Polarizabilities of clusters [Mg, nH2O]CC (R BPW91/6-311CCG(3d,3p) computations) Number of water molecules

axx ða30 Þ

ayy ða30 Þ

azz ða30 Þ

hai ða30 Þ

Da ða30 Þ

Electronic spatial extent hR2 i ða20 Þ

0 H2 O 1 H2 O 2 H2 O 3 H2 O 4 H2 O 5 H2 O 6 H2 O 7 H2 O 8 H2 O 9 H2 O

0.454 9.155 22.561 29.083 36.394 46.896 54.532 69.785 81.633 89.518

0.454 11.670 16.718 29.065 36.050 45.523 54.551 61.895 73.888 89.431

0.454 8.455 16.704 23.930 36.041 43.484 54.565 63.731 71.640 80.341

0.454 9.760 18.661 27.359 36.162 45.301 54.549 65.137 75.720 86.430

0 2.928 5.850 5.144 0.349 2.974 0.029 7.151 9.080 9.134

4.783 93.29 326.16 501.97 691.57 911.97 1135.38 1660.52 2266.67 2957.71

M. Adrian-Scotto et al. / Journal of Molecular Structure: THEOCHEM 728 (2005) 231–242 180000 9

160000 140000

3/2 (a03)

120000

8

100000 80000

7

60000 6

40000

5

20000 0

0

1

2

3

4

0.45 9.76 18.7 27.4 36.2 45.3 54.6 65.1 75.7 86.4 (a03)

Fig. 4. Variation of the volume corresponding to the electronic spatial extent expressed in a30 versus the mean polarizabilities of the [Mg, nH2O]CC clusters. The labelling 0–9 indicates the value of n.

Fig. 5. Ellipsoı¨d of polarizability for the clusters: (a) [Mg, H2O]CC, (b) [Mg, 2H2O]CC, (c) [Mg, 6H2O]CC, (d) [(Mg, 6H2O)3H2O]CC.

237

variation versus the n added water molecules in a water cluster or around the magnesium bication. Tables 5 and 6 show the results obtained for the (H2O)n and [Mg, nH2O]CC clusters, with the translational, rotational and vibrational contributions (with all vibrational modes taken into account) to the total entropy S. For all n, we observe that: Sð½Mg; nH2 OCCÞ! SðMgCCÞC SððH2 OÞnÞ, and consequently DS!0. The variation of these entropic contributions, plotted as a function of n added molecules in the clusters, is shown in Fig. 5. It is to be noted that our results for the total entropy values and for nZ2, 4, 6, 8, are comparable to those obtained by Markham et al. [5] using an ab initio calculation at the RHF/6-31G* level; the only discrepancy concerns the hexahydrated magnesium for which these authors have found SZ115.9 kcal molK1. Fig. 6(a) and (b) shows clearly the weight of the three contributions to the total entropy Stot. For our water clusters, the vibrational entropy becomes a predominant contribution when nR6 molecules. In the case of the hydrated magnesium clusters, we obtain the same result for nR5; and as soon as nZ3 we observe that SvibzSrot and when nZ 4, SvibzStrans. To explain this behaviour, one is reminded that Strans depends only on the mass of the cluster, while Srot depends on the three rotation temperatures which are related to the structure of the cluster (three principal moments of inertia and rotational symmetry number). For all n, we have StransOSrot. The vibrational contribution Svib depends on the vibrational modes in the clusters and in particular the most important contribution comes from the low frequency vibration modes which correspond to the intermolecular movements (liberation of water molecules and translations of MgCC along the O–O axis of two ligated H2O) with vibrational temperatures inferior to 900 K. Fig. 7(a) and (b) shows the vibrational spectra obtained, respectively, for (H2O)9 and [Mg, 9H2O]CC clusters. We observe that in the case of the water cluster the IR spectrum is composed of large bands corresponding to the stretching (2861–3606 cmK1) and the bending (1605–1696 cmK1) of water, and in the low frequency domain to the intermolecular movements (109–1128 cmK1). This spectrum is, even with

Table 4 Thermochemistry of clusters [Mg, nH2O]CC (R BPW91/6-311CCG(3d,3p) computations) Number of water molecules

H (hartree)

G (hartree)

Eel (hartree)

S (cal molK1 KK1)

DH1 (kcal molK1)

DH2 (kcal molK1)

DH3 (kcal molK1)

DG1 (kcal molK1)

DG2 (kcal molK1)

DG3 (kcal molK1)


K199.232187 K275.786929 K352.324238 K428.837840 K505.332988 K581.803337 K658.269512 K734.727351 K811.185995 K887.644248

K199.249036 K275.814530 K352.360961 K428.881769 K505.384512 K581.859112 K658.329868 K734.795077 K811.258570 K887.721753

K199.234547 K275.815113 K352.379832 K428.920975 K505.443513 K581.941350 K658.434852 K734.919980 K811.406288 K887.892144

35.46 58.09 77.29 92.46 108.44 117.39 127.03 142.54 152.75 163.12

– K78.54 K146.14 K198.87 K240.01 K265.59 K288.56 K306.29 K324.53 K342.52

–

– K78.54 K67.60 K52.72 K41.14 K25.58 K22.96 K17.73 K18.24 K17.99

–

– K71.83 K135.62 K183.04 K215.72 K233.65 K251.54 K264.42 K278.23 K287.33

– K71.83 K59.87 K43.79 K32.45 K14.79 K12.38 K8.90 K7.82 K7.63


K78.54 K143.77 K189.34 K220.80 K241.36 K260.66 K271.33 K287.20 K293.19

K71.83 K131.70 K175.49 K207.94 K222.73 K 235.12 K244.02 K251.84 K259.47

238


Table 5 Entropy of clusters (H2O)n (R BPW91/6-311CCG(3d,3p) computations) Number of water molecules

S (cal molK1 KK1)

Strans (cal molK1 KK1)

Srot (cal molK1 KK1)

S vib (cal molK1 KK1)

Vibrational modes

Stretching and bending vibrations

Intermolecular vibrations

1 H2 O 2 H2 O 3 H2 O 4 H2 O 5 H2 O 6 H2 O 7 H2 O 8 H2 O 9 H2 O

45.135 69.171 78.137 90.012 107.764 122.167 130.257 133.687 147.305

34.609 36.675 37.883 38.741 39.406 39.949 40.409 40.807 41.158

10.519 21.185 25.069 26.960 28.620 29.921 30.272 30.555 31.383

0.007 11.312 15.185 24.312 39.738 52.296 59.577 62.325 74.764

3 12 21 30 39 48 57 66 75

3 6 9 12 15 18 21 24 27

0 6 12 18 24 30 36 42 48

TZ298.15 K, PZ1 atm. Table 6 Entropy of clusters [Mg, nH2O]CC (R BPW91/6-311G(3d,3p) computations) Number of water molecules

S (cal molK1 KK1)

Strans (cal molK1KK1)

Srot (cal molK1KK1)

Svib (cal molK1KK1)

Vibrational modes

H2O stretching and bending vibrations

Intermolecular vibrations


35.462 58.092 77.289 92.456 108.442 117.388 127.030 142.541 152.747 163.123

35.462 37.132 38.196 38.978 39.597 40.110 40.547 40.928 41.266 41.569

0 19.253 22.445 26.473 27.593 28.406 29.077 30.162 31.101 31.973

0 1.707 16.648 27.005 41.252 48.873 57.406 71.450 80.380 89.581

– 6 15 24 33 42 51 60 69 78

– 3 6 9 12 15 18 21 24 27

– 3 9 15 21 27 33 39 45 51


only nine molecules of water, similar to the IR spectrum of liquid water. The corresponding spectrum for the nanohydrated-magnesium is not composed of large bands but of sharp peaks: H2O stretching (w3404 and 3450 cmK1), H2O bending (around w1610 cmK1) and librations of water molecules and translations of Mg along an O–O axis (from 342 to 762 cmK1). This drastic difference between the IR spectra of water clusters and Mg-hydrated complexes is due to the strong structuration of water molecules around the bication, in the inner shell of hydration. In particular, the two peaks of maximal intensities at 3450.0 and 3450.5 cmK1 are among the six a-stretches corresponding to the three double hydrogen bonds between two water molecules from the first hydration shell and one water molecule belonging to the second hydration shell. 3.2. Hardness and electronic chemical potential considerations Concerning the monohydrated metallic cations, Chang [35] has recently related linear correlations between the water binding electronic energy and the jE(HOMOK H 2O)KE(LUMOKmetal Z C)j computed with DFT (BPW) and coupled cluster (CCSD(T)) methods. It seems that these methods are not very well adapted for reproducing the frontier orbitals of magnesium bication or H2O.

Fig. 6. Entropy contributions for (a) water clusters, (b) hydrated magnesium clusters.


239

Fig. 7. Vibrational line spectra for clusters with nine water molecules: (a) water cluster (H2O)9, (b) hydrated magnesium cluster [(Mg, 6H2O) 3H2O]CC.

When observing the results, the HOMO of water is situated at w7.2 eV; this value is far from the ionization potential of water which is 12.6 eV [3]. The same remark is valid in the case of metallic cations LUMO’s, because the ionization potential and the hardness of these species (hwjELUMOK EHOMOj/2) are not very well computed using the DFT methods. An ab initio Hartree–Fock method is better to reproduce the frontier orbital properties. We have computed for our (Mg, nH2O)CC clusters the ionization potential I, the electron affinity, the electronic chemical potential m (opposite of the electronegativity c) and the hardness h, using the DFT way [19].

Using the finite difference approach, the electronic energy E(N) was calculated for the mono, di and tricationic clusters, from the conformation of the dicationic clusters. This procedure is used to determine the vertical values of the electron affinities and ionization potentials. Pearson [4] suggest that the vertical values are the only ones that satisfy the density functional computations which require ‘no change in nuclear positions’: – [Mg, nH2O]C containing (NC1) electronsZ11C10n – [Mg, nH2O]CC containing N electronsZ10C10n – [Mg, nH2O]CCC containing (NK1) electronsZ9C10n

240


Table 7 Vertical ionization potential (I), electron affinity (A) and their derivatives (c, h, u) for the clusters [Mg, nH2O]CC (BPW91/6-311CCG(3d,3p) computations) Number of water molecules

I (eV)

A (eV)

cZKm (eV)

h (eV)

u (eV)

Mg Mulliken charge


80.3593 26.2768 27.9036 25.8245 22.5122 25.2535 23.8135 20.3876 22.1118 21.5977

15.2196 13.0232 10.7181 9.088 8.0439 7.44 7.1132 6.7413 6.3918 6.0445

47.79 19.65 19.31 17.46 15.28 16.35 15.46 13.56 14.25 13.82

32.57 6.63 8.59 8.37 7.23 8.91 8.35 6.82 7.86 7.78

35.06 29.12 21.70 18.21 16.15 15.00 14.31 13.48 12.92 12.27

2 1.667 1.485 1.428 1.443 1.380 1.346 1.317 1.308 1.363

Then the ionization potential and the electron affinity, for the cluster [Mg, nH2O]CC, were computed as:

at the 6-311CCG(3df,2p) level of basis: c Z 47:98 eV and h Z 32:52 eV

I Z EðN K 1Þ K EðNÞ

ðwith the functional B3LYPÞ

A Z EðNÞ K EðN C 1Þ

c Z 47:84 eV and h Z 32:63 eV

From the valence state parabola model [20], we deduce: c Z Km Z ðI C AÞ=2 h Z ðI K AÞ=2 The obtained results are presented in Table 7, which shows also the values of the electrophilicity index defined by Parr et al. [20] in the valence state scheme: uVs Z m2 =2h This index seems to be the most appropriate for potent electrophiles like MgCC [21]. We can observe on the line corresponding to 0 H2O the values we have obtained for MgCC (cZ47.79 eV and hZ 32.57 eV), in agreement with those compiled by Pearson [4] (cZ47.59 eV and hZ32.55 eV). De Proft and Geerlings [19] have determined the following using the DFT method Table 8 Vertical ionization potential (I), electron affinity (A) and their derivatives (c, h, u) for the clusters [Mg, nH2O]CC (OPT BPW91 6-311CCG(3d,3p) Koopmans procedure with SP RHF 6-31G(d,p) computations) Number of water molecules

EHOMO (eV)

ELUMO (eV)

cZKm (eV)

h (eV)


K81.86 K27.25 K26.34 K25.35 K24.39 K23.48 K22.80 K21.81 K21.39 K20.94

K14.70 K12.41 K9.66 K7.66 K6.19 K5.32 K4.50 K4.01 K3.50 K2.91

48.28 19.83 18.00 16.50 15.29 14.40 13.65 12.79 12.44 11.92

33.58 7.42 8.34 8.85 9.10 9.08 9.15 8.78 8.95 9.02

u (eV)

34.71 26.50 19.42 15.38 12.85 11.42 10.18 9.34 8.65 7.88

ðwith the functional B3PW91Þ For comparison, we have reported in Table 8 the results obtained using the frontier orbitals (HOMO and LUMO) method for the same clusters, by computing a single point of energy at the level RHF/6-31G(d,p). In this case, using the Koopmans’ theorem [36], Pearson [4] has defined: I yKEHOMO and A yKELUMO The obtained values for c and h are close to those of Table 7 but the values of hardness and electronegativity of MgCC are better in the case of DFT computations. For the (H2O)n clusters, the DFT method with three points of electronic energy is not applicable because of the intervention of instable anionic clusters [37,38]. Table 9 presents the computed results with the frontier orbitals method at the level RHF/6-31G(d,p). The values obtained for 1 H2O: cZ3.90 eV and hZ9.61 eV are comparable to those reported by Pearson [4] (cZ3.1 eV and hZ9.5 eV). Table 9 Vertical ionization potential (I), electron affinity (A) and their derivatives (c, h, u) for the clusters (H2O)n (OPT BPW91 6-311CCG(3d,3p) Koopmans procedure with SP RHF 6-31G(d,p) computations) Number of water molecules

IZKEHOMO (eV)

AZKELUMO (eV)

cZKm (eV)

h (eV)

u (eV)

1 2 3 4 5 6 7 8 9

13.51 12.66 13.38 13.35 13.20 13.23 12.80 12.68 13.01

K5.70 K5.05 K5.48 K5.46 K5.31 K5.35 K4.84 K4.57 4.76

3.90 3.80 3.95 3.94 3.94 3.94 3.98 4.05 4.12

9.61 8.86 9.43 9.41 9.26 9.29 8.82 8.63 8.89

0.79 0.81 0.83 0.82 0.84 0.84 0.90 0.95 0.95

H2O H2O H2O H2O H2O H2O H2O H2O H2O


241

hydrated by nZ1–9 molecules of water. We can see that for liquid water (clusters (H2O)n), c increases slightly with n, while h decreases with n. For the hydrated magnesium, the decrease of c and h is dramatic when nZ 1, then when n increases, the hardness tends to reach the hardness of water and the behaviour of the c is to reduce the gap of electronic chemical potential between the [Mg, nH2O]CC clusters and liquid water. This tendency is stabilized when reaching the second shell of hydration, with the increase of the number of water molecules, with a stabilization of the electronic flow (H2O)n/MgCC as indicated by the values of the magnesium charge in the [Mg, nH2O]CC clusters.

4. Conclusion Fig. 8. Frontier orbital energy diagram showing the electronic potentials (m)—as introduced by Pearson et al. [4]—for MgCC, H2O and [Mg, H2O]CC cluster.

Fig. 8 shows the frontier orbital energy diagram for MgCC, 1 H2O and the cluster [Mg,H2O]CC corresponding to the values of Tables 8 and 9. It can be seen clearly that when one molecule of water is associated with MgCC, the flow of electrons moves from the highest chemical potential mH2 O to the lowest chemical potential mMgCC; then the charge of Mg is lowered. The obtained chemical potential of the resulting cluster (mZK19.83 eV) is close to the medium value (K22.19 eV) between mH2 O ZK3:90 eV and mMgCCZK48.28 eV. Fig. 9 shows the ensemble of obtained results in Tables 7–9 for the c and h parameters, when MgCC is

Fig. 9. Variation of the electronegativity (c) and hardness (h) of the [Mg, nH2O]CC clusters versus n varying from 0 to 9.

This study presents an investigation for the magnesium dication hydration up to nine added molecules of water, using the DFT computations (OPTCFREQ)-RBPW91/ 6-311CCG(3d,3p), which seems to be the best for comparison of polarizability and vibrational frequencies of a water molecule, and for polarizability of MgCC.

Fig. 10. Electrophilicity index u of the [Mg, nH2O]CC clusters: (a) versus n varying from 0 to 9, (b) versus the Mulliken charge Q on Mg.

242


Our results show that the cluster procedure for obtaining the enthalpies or the Gibbs free energies of hydration is preferable to the standard one. Our results are close to those obtained by Markham et al. [5] in their ab initio Hartree– Fock MP2 (FULL) 6-311CCG** and MP4SDQ(FC) 6-31G* computations. Our obtained vibrational spectra for [Mg, nH2O]CC and (H2O)n clusters show a difference in the low frequency domain which concerns the intermolecular movements, that become important as n increases. The entropy contributions may be an important factor for stabilizing the hexahydrated form of MgCC [7]. It is interesting to note that for the hydration process: MgCC C ðH2 OÞn/ ðMg; nH2 OÞCC C DH ðor DG or DS or Dhai1=3 Þ are negative in all cases: Finally, the hydration of MgCC was interpreted using the hardness and the electronic chemical potential concepts. Clearly, as the number n of water molecules increases around the magnesium dication, the hardness of the cluster decreases and tends to the hardness of water. The decrease of the Mulliken net charge of magnesium with n can easily be explained by the electronic flow directed from water (high m) to Mg (low m). This effect is also related to the correlation observed between the electrophilicity index uVsZm2/2h defined by Parr et al. [20] as a ‘propensity to soak up electrons’, as illustrated through the behaviour of this index versus the charge on magnesium (Fig. 10(a) and (b)).

Acknowledgements We would like to thank Dominique Toupin and Olivier Brousseau from the Colle`ge Shawinigan (Que., Canada) for their technical support.

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