Theoretical study of water-exchange reactions for the divalent ions of

The binding energies of the sixth water ligand of the hexahydrateddivalent first-row ... rate variations occur.6'9 The rate constants for the exchange...
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J . Phys. Chem. 1993, 97, 3165-3114

3765

Theoretical Study of Water-Exchange Reactions for the Divalent Ions of the First Transition Period Ralf Akesson,' Lars G. M. Pettersson,t Magnus Sandstrom,',t Per E. M. Siegbahn,t and Ulf Wahlgrent Department of Inorganic Chemistry, Royal Institute of Technology, S - 100 44 Stockholm, Sweden, and Institute of Theoretical Physics, University of Stockholm, Vanadisvcigen 9, S-113 46 Stockholm, Sweden Received: November 3, 1992; I n Final Form: January 15, 1993

The binding energies of the sixth water ligand of the hexahydrated divalent first-row transition-metal ions from Ca2+ to Zn2+ have been obtained by a b initio S C F calculations. A remarkably accurate linear correlation is obtained between the calculated gas-phase dissociation energies and the logarithm of the experimentally determined reaction rate constants for water exchange in solution, excluding Ca2+ which has a higher hydration number. The result is consistent with a pentahydrated activated complex (except for Ca2+),only weakly interacting with the entering and leaving water ligands in the transition state, Le., an essentially dissociative mechanism for all these ions. This is in conflict with recent interpretations based on experimental activation volumes, which suggest an increasingly associative interchange mechanism to the left in the row. The reason for the discrepancy between the mechanisms for water exchange, proposed on the basis of these theoretical and experimental results, is discussed and analyzed in molecular terms. In cases with weak or no ligand-field stabilization of the pentahydrated complexes, trigonal bipyramidal coordination gives the more stable structures, whereas for some of the ions with strong ligand-field or Jahn-Teller effects, Sc2+, V2+, Cr2+,Ni2+, and Cu2+,square pyramidal structures were favored. An accurate geometry description of the pentahydrated clusters using a large water basis set was found to be important in evaluating the binding energy. The energies of the d orbitals have been studied for an idealized gradual SQP TBP transition (Berry pseudorotation) applied to [Mn(H20)5I2+,in order to investigate their behavior as the geometry is changed.

-

Introduction This study continues our theoretical investigationsof hydrated ions, which have previously encompassed ligand-field effects on the structures and binding energies of the hexahydrated divalent ions of the first transition period, including a comparison with experimental hydration enthalpies,' and also the effect of JahnTeller vibronic couplings in the [Cu(Hz0)6I2+,[Cr(H20)6]2+, and [Mn(H20)6]3+cluster^.^,^ Closely connected to and strongly influenced by these "effects" is the reactivity and reaction rates of the hydrated transition-metal ions. The present study attempts to model the influence of the structure and binding energy on the rate of water exchange for the first-row divalent transition-metal ions. Solvent exchange is a fundamental model reaction for understanding the reactivity of ions in s ~ l u t i o n . ~For - ~ hydrated metal ions without significant ligand-field effects, changes in the waterexchange rates can be related to the electrostatic charge-water dipole interactions,7,*but for the transition-metal ions, much larger rate variations o c c ~ r . ~The . ~ rate constants for the exchange of a water molecule in the first hydration layer have been determined experimentally for all divalent ions of the first transition period, except Sc2+ and Ti2+ (see Table I), with those for Cu2+and Cr2+ being several orders of magnitude faster than for V2+ and Ni2t.6.10.1I Until relatively recently, it was generally believed, from, e+, rates of ligand substitution, that the water-exchange reactions for these hexahydrated ions in solution all followed predominantly dissociative mechanisms. However, after high-pressure NMR techniques became available,allowing measurementsof activation volumes, A V c x p from , the pressure dependence of the reaction rate, the current description favors a gradual interchange mechanism with both the entering and the leaving water molecules simultaneously present in the inner hydration shell.6 Provided +

1

Royal Institute of Technology University of Stockholm.

TABLE I: Experimental Rate Constants k and Activation for Water-Exchange Reactions,' Symmetric Volumes A V.exp M-0 Stretctung Vibrational Frequencies u,~,,,,~and Calculated Mean M-0 Bond Distances R6 for Hexahydrated M ( H Z O ) ~ ~ + Ions' ion Ca2+ sc*+

-

klsl

AVcxD/cm3mol-'

1OXd

Ti?+ V?t Cr2+

Mn2+ Fe2+ co2+ NiZ+

cu-

Zn?+

87 -7 2.1

x 109' x 107 4.4 x 106 3.2 X IO6 3.2 x 104

4.4 x

-4.1

-5.4 +3.8

+6.1 +7.2

i09*

- 2 x 107

Rb/A

2.399 2.307 2.252 2.201 2.225, 2.233 2.185 2.143 2.108 2.1211 2.123

u,,,/cm

-

I

290

-459 352 370 375 39 1 435' 390

Reference6a. * References 15and 16. ' Reference 1,SCFcalculations using themedium-sized basisset. Reference.4. Reference 11. /Average of equatorial and axialdistances in Jahn-Teller distorted structure. 8 This work; broad asymmetric Raman band for 2.3 mol dm-' CrClz solution. * Reference IO. This work; Raman band for saturated Cu(C104)2 solution.

that changes in the bond distances to the nonexchanging ligands and changes in the solvent structure are negligible at the transition state,I2-l4the volume of activation has been proposed to be a measure of whether the interchange is associative or dissociative. For the divalent ions of the first transition period, AV,,,data (Table I) indicate a gradual change in the interchange process.6 To the left in the period, negative, AV,,, values are obtained, which has been taken as evidence of a mainly associative reaction path, i.e., toward a seven coordination in the transition state. The ions to the right in the period have smaller ionic radii and higher 7d occupations. This leads to an increase in the repulsion of the incoming ligand such that the AV,,,values increase and become positive, suggesting an early release of a water ligand from the 0 1993 American Chemical Society

3766 The Journal of Physical Chemistry, Vol. 97,NO. IS, 1993 complex, giving a transition state more closely corresponding to a five-coordinated species, M(H20)52+.6 Swaddle has used a semiempirical model to estimate limiting values of ~AVexp~ in the extreme cases of completely associative or dissociative ligand exchange.12b.13b For both types of mechanism, the absolute values were proposed to decrease linearly as the ionic radii increase, and for divalent ions of medium size (ri 0.80 A), lAVl,,,,lvalues of 13 cm3 mol-' were obtained. However, for the divalent high-spin aquated metal ions, an inverse correlation were found between AV,,, and the absolute molar volume, VQabs,12,17 and he concluded that "the transition states must resemble each other relatively closely within a series despite some differences in reaction mechanism which AV is thought to reveal".17 The activation volume has thus been used to indicate whether the transition state occurs a t an early stage at the approach of the entering water molecule, in which case a positive volume change indicates a dissociative reaction, or whether it occurs late in which case a negative volume change suggests a penetration of the entering water molecule into the first hydration shell. In the dissociative reaction, the main contribution to the energy barrier of the transition state is expected to be the bond-breaking process taking place when one water ligand leaves the complex. The barrier for water exchange should in that case be close to, or at least related to, the dissociation energy. From the Arrhenius equation, k = A exp(-E,/RT), it is easy to see that a plot of the dissociation energy vs the logarithm of theexperimental reaction rateconstant, log k,for ligand-exchange reactions with similar preexponential factors, A, should be a straight line if either (1) the dissociation energy is close to the Arrhenius activation energy, E,, or (2) the dissociation energy differs from E, by a constant amount. We have calculated the gas-phase dissociation energy for all hexahydrated divalent ions from Ca to Zn relative to their most stable pentahydrated species, initially expecting only the complexes with positive activation volumes to give a linear correlation in such a plot. However, our result is, in fact, that the values for all complexes, except for calcium which is not hexahydrated in solution,l correlate well with a linear Arrhenius plot. We interpret this as showing that at the transition state, the two water molecules undergoing exchange either do not interact strongly with the activated pentacoordinated complex or otherwise give a constant interaction along the period. In either case, this corresponds to a dissociative mechanism in the water exchange for these hexahydrated ions, regardless of whether the activation volume is positive or negative, with a rate-determining step given by the breaking of one metal-water bond. The stereochemistry of five-coordinated transition-metal clustersoffers special features because of the small energy separations between trigonal bipyramidal (TBP) and square pyramidal (SQP) geometries. In some cases, a fluxional behavior or rapid interconversions, "Berry pseudorotations", can be observed.Is Electrostatic calculations of ligand-ligand repulsion show that an elongated TBP structure is slightly preferred over an SQP structure for which the four M-L bonds to the base are longer than the apical o w l 9 In an idealized crystal-field approach (assuming q u a l bond lengths and with the metal atom in the equatorial TBP or base SQP plane), the d-orbital stabilization energy isalwaysequal or more favorable in theSQPconformation than in the TBPconformation for all d-electron configurations.5-9 Bonding interactions,such asvibroniccouplings (first- and secondorder Jahn-Teller effects), ligand-field stabilizations, and *-bonding involving d orbitals, can favor the SQP structure.20,2' Another recent attempt to study the water-exchange reaction theoretically by a b initio methods has been made for the hydrated V2+,Mn2+,Fe2+,and Ni2+ions. In all four cases, the M(H20)72+ species were proposed as the activated complex.22 However, appreciably smaller basis sets were used than in the present study,

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Akesson et al.

n

Figure 1. Schematic view of the pentahydrated ions in square pyramidal (SQP) and trigonal bipyramidal (TBP) geometries,showing the definitions of the Re,,,Rax.and RPYl distances and the coordinate system used. Note that the z axis is along the M-O,,l (shaded H atoms) bond in the TBP structure. In a gradual transition changing the @I and @ 2 angles, the O,,(SQP) atom is converted to the O,,,(TBP) atom.

and for [Fe(H20)6]2+, a low-spin configuration was chosen, contradictory to both experimental observation^^^ and theoretical results.'

Methods Bonding distances have been optimized and binding energies computed a t the SCF level for all M ( H ~ O ) Jclusters ~+ of the first transition period from Ca to Zn. Two limiting stereochemical conformations were considered: SQPand TBP. A planar trigonal coordination of the water molecules has been assumed, corresponding to a C2t.molecular symmetry for both conformations in such a way that the TBP and SQP structures can be easily interconverted; see Figure 1 which also defines the coordinate system used. Close structural relations can be found to the hexahydrated complexes in Th or D2h symmetry.'-3 The internal water geometry was as previously kept at the gas-phase values ( R ~ H0.957 A, H-0-H = 104.5O). Both the axial and equatorial metal-oxygen distances were optimized for each geometrical conformation. For the SQP structure, we also optimized RPY,,which denotes the distance from the metal atom to the basal plane of the pyramid defined by the four equatorial oxygen atoms; see Figure 1. For the TBP geometry, additional calculations were made for an alternative "eclipsed" conformation with the six hydrogen atoms of the equatorial water molecules located in the equatorial plane.

Water-Exchange Reactions for the Divalent Ions The MOLECULE-SWEDEN system of programs has been ~ s e d . 2The ~ basis sets and contraction method have been described previously. I For the geometry optimization, a medium-sized basis set was used for the water molecules, and the energy a t the minimum was then recalculated using a larger water basis set. The latter includes two d functions on oxygen and a p function on hydrogen and gives results for the dipole moment and dipole polarizability of H2O close to the Hartree-Fock limit. This is essential for a reliable estimate of the SCF-level energy barriers, as the bonding is mainly of electrostatic (ion-dipole) character. For the larger basis set, a modified version of the direct S C F program DISC025.26running on a Cray X-MP/416 was used. It is known from the previous study on hexahydrated ions that the optimized geometries corresponding to the larger water basis set give a slight elongation ( Re,. For Cu2+,the singly occupied orbital in SQP geometry is -dX2_,2, whereas for Cr2+the singly occupied orbital is d,z with virtually no contribution from oxygen. For the bigger Cr2+ion, a small RPYr value is obtained, 0.09A, due to the already low Oax-Oeq repulsion. The adiabatic potential surface for the SQP structure is rather insensitiveto variations in the R,, distance (R,, f 0.05 8, corresponds to AE +0.7 kJ mol-I, whereas Re, f 0.05 8, corresponds to AE = +3.8 kJ mol-I). The TBP structures are slightly higher in energy (3.0 and 1 1.2 kJ mol-' for [Cu(H20)512+and [Cr(H2O)5l2+,respectively) with the singly occupied orbitals reversed, dy and d,>-,>,for Cu2+and Cr2+, respectively. In these cases, R,, < Re, and the structure is therefore more closely related to the compressed conformation of the hexahydrated complex. The energy difference to the TBP structure from the compressed octahedral structure would be only 2.6 kJ mol-' higher for Cu2+ (8.4 kJ mol-' for Cr2+) and

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3770 The Journal of Physical Chemistry, Vol. 97, No. 15, 1993

indicates that for Cu2+, this alternative pathway could give a significant contribution to the measured reaction rate. Bersuker has given the following equation for the reduction of the activation energy (AD) due to a distortion of the complex along a Jahn-Teller active symmetry coordinate: with A = ~ E Jand T B = 3AE.28 The theoretical Jahn-Teller energy previously obtained for [CU(H20)6] 2+, E J ~ ( S C F= ) 7.0 kJ mol-',' gives AD = 68 kJ mol-'. Without the Jahn-Teller effect for Cu2+, the "nondistorted activation energy" would thus beAE- 171 kJmol-I. ForCr2+withEJT(SCF)= 10.5 kJmol-I, AD would amount to -75 kJ mol-', giving PE 173 kJ mol-'. Both of these values are much larger than the calculated energy barriers PEB for the nondistorted complexes in the series; see Table V. However, eq 1 is not strictly applicable to the present cases, since the Jahn-Teller distortion coordinate is of E, type, while the distortion toward a loss of a ligand belongs to the TI, representation in O,,symmetry. Also, second-order Jahn-Teller effects may be of importance in the destabilization of the TBP conformation as compared to the SQP conformation for pentacoordination, as previously discussed.20-2' The estimated values above can only be used to indicate the large influence JahnTeller vibronic couplings may have on the activation energies and reaction rates. Eclipsed TBP. In order to investigate how the total energy is affected by the orientation of the water molecules, we also performedgeometry optimizations (with the medium water basis) in an alternative TBP structure with all six H, in the equatorial plane, which is obviously the most unfavorable conformation due to the repulsion between the hydrogen atoms. The He,-H, distances typically are in the range 3.9-4.1 A, which is not large enough to make the interatomic repulsion insignificant. The equilibrium geometries of this "eclipsed" conformation have M-O, bonds -0.02 A longer and M-O,, bonds -0.02 A shorter than the TBP values in Table I1 and, consequently, somewhat longer mean M-O bond lengths. Theenergydifferences compared with the "normal" TBP structure are given in Table 111and were found to be in the range between 10.9 (Cr2+)and 32.1 kJ mol-l (Til+) and in most cases higher than the energy needed for interconversion to a SQP conformation, implying a severely hindered rotation of the water molecules. Mulliken Population Analyses. The results from Mulliken population analyses are given in Table VI for the most stable conformations of the pentahydrates (the higher energy conformation did not show significant differences). As in the previous study on M(H20)62+clusters, the general trend is a reduction of the metal atom charge from left to right in the period, although the highest values found for Sc (+1.96) and Ti (+1.85) both correspond to low third ionization potentials and easily oxidized ions in solution. The increasing degree of charge transfer thus corresponds to the decrease in ionic radii and lower orbital energies. The lowest effective charge is obtained for Zn (TBP), 1.41, and this system therefore has the most covalent bonds. Compared with the corresponding hexahydrated complexes, the metal atom charges are -0.1 e- lower,' consistent with the decrease in the mean M-0 distance, allowing an increased overlap with ligand orbitals. The variation of the charges of the hydrogen atoms, however, displays, as for the hexahydrates, very small changes along the series, cf. Table VI with Table 111, ref 1. The oxygen atoms evidently act as electron buffers, modulating the electron distribution between the positive metal atom and hydrogen centers. As previously found for the hexahydrated complexes,' the metal 3d occupations are very close to the formal occupation numbers, within f0.1 e- for all ions except Sc2+. It should be emphasized, however, that the absolute values of the charges depend strongly on the choice of basis sets and only

-

+

TABLE VI: Mulliken Population Analyses of the M(H20)s2+ Complexes (a) Atomic Charges

Ca2+ Ti2+ Mn*+ Fe2+ coz+ Znz+

+ 1.94 + 1.96 +1.77 + 1.70 + 1.62

sc2+

+ 1.94

V*+ Cr*+ Ni2+ cu*+

+1.91 +1.86 +1.50 1.42

+1.42

+

Trigonal Bipyramids -0.40 -0.39 -0.36 -0.38 -0.34 -0.34 -0.32 -0.33 -0.30 -0.32 -0.29 -0.29

+0.20 +O. 19 +0.19 +0.19 +0.19 +0.20

+0.20

Square Pyramids -0.39 -0.38 -0.37 -0.37 -0.35 -0.36 -0.29 -0.30 -0.30 -0.29

+O. 19 +0.19 +0.19 +0.20 +0.20

+0.20 +0.19 +0.19 +0.20 +0.20

+O. I9 +0.20 +0.19 +0.19 +0.2 1

(b) Orbital Populations M*+ Ca2+ Ti2+ Mn2+ Fe2+ co2+ Zn2+

sc*+ V*+ Cr2+ Ni2+ cu2+

S

P

d

Trigonal Bipyramids 6.00 1 1.96 6.00 11.99 12.14 6.1 I 6.14 12.20 6.17 12.24 6.23 12.35

0.10 2.06 4.96 5.96 6.97 10.00

Square Pyramids 5.99 1 1.93 6.04 12.02 6.10 12.05 6.23 12.27 6.26 12.30

1.13 3.03 3.98 8.00 9.03

relative comparisons are meaningful. In particular, the diffuse metal s and p functions exert a strong influence on the calculated charge distributions as noted in the previous study.' Correlation with Experimental Water-Exchange Rates. The smallest system which can be used to describe a water-exchange process, assuming a concerted process, should contain seven water molecules. However, if the transition state occurs early in the entrance pathway of the incoming water molecule, the activated complex will be an essentially pentahydrated complex with the two exchanging water molecules at a relatively large distance from the metal ion. In such a case, the barrier to water exchange will be largely determined by the energy needed to remove one water molecule from the hexahydrated complex, and the dissociation energy AEBmight provide a good estimate of the barrier. From the Arrhenius rate equation, it follows that a plot of the dissociation energy vs the logarithm of the observed rate constant k should then essentially show a straight line (see below) for processes with similar preexponential factors. A linear plot is alsoobtained if thedissociation energy differsfrom the true barrier by a constant amount for all complexes. For complexes where the transition state takes place late along the entrance pathway ("associative reactions"), the bonding energy contribution to the incoming ligand should become more important. If this would be the case, such points in a PEBvs log k plot would deviate from a straight line. In the present calculations, we have only considered a "dissociative" mechanism for the water-exchange process; i.e., we have used the gas-phase dissociation energy as a measure of the barrier to water exchange. Contrary to our expectations, this approach appears to be valid for all the divalent first-row transition-metal ions. The energy differences, A& = E{M(HzO)s2+)+ E(H20) E{M(H20)62+), corresponding to the gas-phase dissociation energy of the sixth water molecule, have been calculated using the large water basis set and are given in Table V. The values of E{M(H20)62+) were taken from a previous study' and generally correspond to optimized geometries in Th symmetry (all M-O

The Journal of Physical Chemistry, Vol. 97, No. 15, 1993 3771

Water-Exchange Reactions for the Divalent Ions

100 ll01 1

2

I

I

4

1

I

6

I

I

8

1

I

log A

10

Figure 4. Theoretical bond dissociation values, A E B / ~ Jmol-', plotted against experimental water-exchange rate constants, log k, in a linear Arrhenius plot.

bonds equal) except for the hexahydrated Cr2+and Cu2+ions for which the geometry was optimized in Dzh symmetry (tetragonally elongated octahedron) in order to account for Jahn-Teller effects. The value of E(H20) = -76.050 761 hartree (R*H = 0.957 A; H-O-H = 104.5') was taken from the same study.' For Ti2+ and Co2+,the CASSCF corrections (see the Methods section) for the pentahydrated complexes were 3 1.81 (Co*+;TBP), 26.28 (Co2+;SQP), 19.16(Ti2+;TBP), and 13.79 (Ti2+;SQP) kJ mol-' and for the hexahydratedcomplexes 30.01 (Cd+)and 16.34(Ti2+) kJ mol-'. The preliminary A& values obtained with the medium-sized basis set, which gives an exaggerated water dipole moment and thus increased binding energies, displayed a rather constant increase, +22 f 3 kJ mol-', as compared with the large basis set values, recalculated for the refined geometry (Table V). This previously utilized procedure, which is based on the assumption that the energy improvement by performing the geometry optimization with the larger basis set would be negligible,' was tested by a reoptimization of [Mn(H20)sl2+(TBP) in the large basis, resulting in a A& value 0.41 kJ mol-' lower than in Table V, a minor difference in this context. The relation between the calculated energy barriers and experimental water-exchangerate constants is illustrated in Figure 4,where MB is plotted vs log k. The plot is satisfactorily linear (correlation coefficient 0.990excluding Ca2+),in agreement with the Arrhenius equation, k = A exp(-E,/RT), using the calculated A& as the Arrhenius activation energy E,. The deviating value for Ca2+,and the motif for excluding it from the fitting procedure, is related to its larger radius, giving it a hydration number in solution which most probably is greater than six, as discussed previously. The high correlation coefficient allows us to predict the rate constants for water exchange of the Sc2+and Ti2+ions, which are unstable in aqueous solution. The computed value of 114.7 kJ mol-' for Sc2+would correspond to a rate constant of k lo7s-I, while the higher dissociation energy of Ti2+ at 123.7 kJ mol-! leads to a predicted value of k 10s s-1. The linear plot shows that the calculated energy for the bond breaking in the gas phase gives a good account of the variations of the activation energy barrier in solution and is consistent with a pentacoordinated complex with only weak M-O interactions to other water molecules as the transition state for all divalent ions (except calcium). Seven water molecules are involved in the process in all cases, and the difference between an Yassociative" and a "dissociative" water interchange,as labeled by theactivation volumes, would thusbe the degreeof penetration into the hydration

-

-

Figure 5. Illustration of the initial stage in the water exchange. The plot shows six (shadowed) water oxygen atoms octahedrally surrounding a metal atom ( M - 0 distances 2.233 A as in Mn(H20)b2+). The entering (dotted) water oxygen atom is at a M-O distance of 3.4 A, in contact with the oxygen atoms of a triangular face of the octahedron (with radius 1.4 A). Note the remainingspace between the coordinated water ligands (plotted with a radius estimated to 1.34 A for coordinated water).z9

sphere of the metal ion by the two "exchanging" water molecules in the transition state and not their amount of M-O bond formation. Mechanism of Water Exchange. At the initial stage of the water-exchange process, one can envisage a water molecule approaching a triangular surface of an M(H20)62+octahedron, attracted by the metal ion (Figure 5). Estimationsof the M-O,,,, distance at contact with the three bound oxygen atoms of a face in a regular octahedron reveal very small variations with metal ion size RH^^ = 1.40A gives M-Omt, 3.43A for a Ni-O distance of 2.108 A, and 3.41 A for Mn-O 2.233 A, Table I). The mean M-O distances to the hydrogen-bonded water molecules in the second shell are about 4.1-4.2 A already for the small Ni2+ and hydrogen bonding from the first shell to the entering water molecule would be highly unfavorable and can safely be ignored at the contact distance. However, in the M a 6 octahedron, the nearest O--O distances increase by 0.18 A from Ni2+ to Mn2+. This will allow the water molecules around the bigger metal ions to be pushed further back by the entering water molecule, which then can penetrate more deeply into the first hydration shell. In addition, the increasing filling and contraction of the tZgdorbitals, which are pointing out between the oxygen ligands through the edges of the octahedron, will further reduce the possibility to distort the octahedral coordination for the ions to the right in the period. This is also reflected by the increasing frequencies of the symmetric stretching M-O vibration, corresponding todecreasing vibrational amplitudes and flexibility of the oxygen atom movements, for the smaller non-Jahn-Teller distorted hexahydrated ions, Table I. Thus, the repulsive force acting on a water molecule entering the first hydration shell will be substantial already at a fairly large M - O distance for the smaller ions, while the bigger and more deformable hexahydrated ions to the left in the period can allow a partial penetration before the crowding becomes so severe that an M-O bond breaks. The point of bond breaking will then largely be given by steric and electronic repulsion effects, Le., the size of the metal ion determining the O--O distances in the first shell and the degree

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The Journal of Physical Chemistry, Vol. 97,No. 15, 1993

of filling of the d orbitals, respectively, but will in any case take place with the entering water molecule rather far from the metal ion, giving only a relatively weak M.-O,,,,, interaction even for the bigger ions (Figure 5). The principle of microscopic reversibility requires the transition state to be symmetrical, with the same interaction to the leaving ligand.9s12 This model would also be consistent with results from a computer simulation which indicate that at an exchange process, the activated complex persists for a relatively long period on the collisional time scale.31 Thus, the pentahydrated species in the activated state will have time to find the energetically most favorable conformation after the bond breaking, and the process will proceed according to an essentially dissociative mechanism regardless of whether the activation volume is positive or negative. The intrinsic (corrected) activation volume will therefore only reflect the degree of penetration of the entering water into the hydration shell (see the discussion below) and cannot be used as a measure of bond making in the interchange. The equation of the fitted line in Figure 4 is AEB = -5.14 log k 150.43 (kJ mol-'). The slope equals -RT In 10, which corresponds to a T value of ca. 268 K, in reasonable agreement with the experimental temperature of 298 K. The mean "preexponential factor", A = 1.86X s-I, obtained from the intercept of the fitted line, is related to the activation entropy of the individual complexes536bbut will in this comparison also include any constant differences between the AEBand E, quantities, e.g., from hydration of the gas-phase species (see below). This is born out by a comparison between the calculated bond dissociation energy AEB and the Gibbs standard free energy of activation AG*, obtained from the Eyring equation for a secondorder processs

+

k = (kT/h) exp(-AG*/RT) = (kT/h) exp(-AH*/RT) exp(AS*/R) (2) where k, h, and R are the Boltzmann, Planck, and gas constants, respectively. A rather constant difference ranging from 78 to 85 kJ mol-l is obtained between AEBand AG*, Table V. There are several contributions to this discrepancy. The calculated A E B value gives an energy barrier for a molecular gas-phase process at 0 K, while AG* refers to a statistical mean in solution at 298 K. In a comparison of energies between the transition state and the reactants for the gas-phase and solution processes, differences in zero-point energies, heat capacity, activation entropies AS*, solvation, and hydrogen-bonding energies should be accounted for, and also differences in medium effects on the experimental rate constantss should be considered. Thus, neglecting the temperature-dependent corrections needed to obtain AEBvalues at 298 K, the activation enthalpy is

AH* = AG* + TU* = h ~- ,AH^^^^^ + AHF'n - A H H 2 ? P

(3)

according to the Born-Haber cycle below.

2M(H~o),z'(~) +

M(H~O)C(~)

H,o(~)

The major contribution to the difference between the theoretical and experimental energies is the energy needed for vaporization of water, AHH20VaP = 44.02kJ mol-' at 298 K.32 For six of the ions, AS*values have been obtained fromNMR studies,6although the inclusion of the rather small TAS' terms to account for their different activation entropies does not seem to reduce the variations in thecorrelation (Table V). Other contributions come from the effects of further solvation of the penta- and hexacoordinated

species, which should not differ very much because of the similar sizes and shapes of these clusters. The absolute values of the hydration enthalpy u 6 ' O i n of the gas-phase clusters, according to M(H20)62+(g) M(H20)62+(aq),have previously been estimated to be in the range from about -926 to -1020 kJ mol-'.] An estimate of the increase in solvation energy due to the contraction of the M-0 bonds for the pentahydrates can be made by approximating the complexes with spheres and use of the Born equation. For the radii defined in Table V (R6 + RH*O or RS + RH*&the energy difference between AH* and AEB is reduced by about 2 kJ mol-' for a 0.01-Acontraction of the M - O distance. The calculated mean contractions range from 0.015 (V+) to 0.067 A (Cr2+), and the remaining discrepancies between the experimental activation enthalpies and the calculated energy barrier after these approximative solvation corrections are then not more than about 30 kJ mol-l, an acceptable agreement in view of the neglected difference AH5soln- AH6"'" and the other terms above not corrected for here. It is notable that nocorrelation exists between the total binding energiesof the hexahydrates' (instead of thesixth binding energy) and log k, which means that the successive binding energies for the ions along the period do not follow a regular pattern. Also the use of, e.g., only SQP geometries for all pentahydrated ions gives a clearly worse correlation. Comparisons with Crystal-Field and Angular Overlap Method (AOM) Models. By means of the previously shown crystal-field stabilization energies, Table IV, relative crystal-field activation energies can be estimated for a ML6 MLs(SQP) high-spin reaction ~ a t h . Using ~ , ~ Dq units (assumed to be equal for five and six coordination), the different d-electron configurations lead to d4 -3.14,d6 -0.57, d7 -1.14, ds + 2.00,and d9 -3.14. This corresponds qualitatively to the large difference in reaction rates between the fast exchange of Cu2+ (d9) and Cr2+ (d4) ions and the slow Nil+ (d8) ion. However, the model cannot correctly account for the observed relative reaction rates for the sequence Mn2+-Fe2+-Co2+-Ni2+. This is proposed to be due to the omission of *-bonding in the model, leading to an exaggerated splitting between the d, and d,,, d,, orbitals.33 Application of the AOM model gives the following stabilization energies for a-type interactions involving the d orbitals in an idealized regular SQP geometry ( R , = R,,, Rpyr= 0) in units of (proportional to the square of the overlap integral): d,, d,,, and d,,, 0; d,i, 2;d,2_,2, 3.33 By subtracting the d-orbital SQP u-stabilization energies from those of octahedral geometries and adding a linearly increasing energy contribution from the metal s and p orbitals, which become gradually lower in energy across the period, the relative trend of the observed reaction rates can be approximately accounted for.33 Solvent Effects on Activation Volumes. The hydration of the solvated species, i.e., the effect of the second and possibly also the third hydration layer, constitutes the main difference between the quantum chemical model and the real solution. The hydrogen bonds are of primary importance for the structure and energy of the hydration shells. As discussed above, the solvation should to a first approximation only make constant contributions to the estimated activation energy values and would, therefore, not significantly influence the slope. of the straight line used todescribe the correlation with the reaction rates. Similar inherent assumptions are the basis for the conclusions made from the measuredactivationvolumes, AVVtexp= AV,ntr+ AVncont+ A V W l V , where the intrinsic activation volume, AV",,,,, reflects the volume change due to the entering and leaving solvent molecules in the transition state, and where the corrections for the contraction of the activated complex, A V c o n tand , the solvent contraction, AVWlv, often are neglected.6.13,14 Our results show that in some cases, the contraction of the M-0 distances is appreciable in a dissociative process (Table V), with substantial reductions of the volume of the transition-state

-

-.

Water-Exchange Reactions for the Divalent Ions

The Journal of Physical Chemistry, Vol. 97, No. 15, 1993 3773

complex, AV,,,,. The calculated gas-phase volume changes obtained from the mean radii of the penta- and hexahydrated complexes are given in Table V. The Cr2+and Cu2+ions show the largest contractions and V2+the smallest in the transition state for a dissociative reaction. The contraction of the M-0 distances in the pentahydrates should increase the polarization of the bound water molecules and increase their hydrogen-bonding ability, although the calculated increase in the charge of the hydrogen atoms from the Mulliken population analyses (cf. Table VI and ref 1, Table 111) is small. The enhanced hydrogen bonding should cause a contraction of the water structure around the activated complexes and thuscontribute tothenegativesolvent term, AVsolv. However, as discussed by Hahn and S~addle,3~.12 the compression of a bulk water molecule due to electrostriction effects in the second hydration sphere around divalent ions is relatively small (they estimate values of -1.1 and -2.5 cm3 mol-’, respectively). The limited decrease of the M-0 distances in the transition state (giving rise to AV,,,,, see Table V) would thus only cause minor additional compressions in the second sphere, and even if the number of hydrogen-bonded water molecules is large (>12),30 their contribution to the AVsolv term would (in particular for V2+) be small. A more important contribution is certainly the contraction and increased hydrogen bonding of the water molecule entering the first hydration sphere.6bs1ZParticularly for the bigger ions, for which such penetration issterically possible, the electrostriction of the entering water molecule in the field of the central divalent metal is expected to make a large negative contribution to AVsOlv (values between -2.5 and -9 cm3 mol-’ depending on the degree of penetration,6b-5.7 cm3 mol-’ for an ion with 1-A radius,34and up to -7 cm3 mol-’ l 2 have been estimated). In order to extract and compare the intrinsic volume changes, AV,,,,,due to the entering and leaving water molecules in the transition state, the experimental activation volumes, A p e x , , should be corrected for these different contraction terms; i.e., AVln,r AT,,, - AV,,,, - AVsolv.Values of AV,,, - AV,,,, (sAV,,,~ + AVsolv) aregiven inTableV. Alreadyaftercorrection for AV,,,, the activation volume (PPI”,, AV,,I~) value is more negative for V2+ than for Mn2+, which would indicate a somewhat deeper penetration of the entering water into the first hydration shell provided that the AVs,lvterms are similar. Thus, the increase in repulsion of the entering water molecule due to the increased filling of the d orbitals of Mn2+seemingly outweighs the increased steric repulsion from the bound water molecules for the slightly smaller V2+ion (Table I). The solvent contraction terms, AVsolv, for the entering water are, however, expected to be large for both ions (see above), which means that the AV,,,, values should be well into the positive region. Concluding Remarks. The energy differences between the SQP and TBP conformations of the pentahydrates of the divalent firstrow transition-metal ions are quite small. Because of the dominant electrostatic nature of the bonding, large basis sets, in particular for the water molecules to adequately describe their dipole moments, are necessary in order to obtain reliable energy differences. Due to the atomic-like character of the 3d orbitals and the absenceof s-d mixing (as for the hexahydrates),’ electron correlation effects are not important for relative comparisons, and qualitative agreement with predictions from the crystal-field model of, e.g., the d-orbital occupation order is obtained. Strong ligand-field effects are generally found to favor the SQP conformation relative to the TBP conformation, although the pentahydrates of Co2+and Ti2+deviate from the idealized scheme of ligand-field stabilization. Also, an intermediate conformation along a Berry pseudorotation path may have still lower energy, as was found for [ M I I ( H ~ O ) ~ ] ~ + . The smallest gas-phase energy differences for the dissociation of the sixth water molecule from all the divalent hexahydrated

+

ions (excluding Ca2+which has a higher hydration number) were found to agree remarkably well with the experimental rate constants for water exchange in an Arrhenius plot, including the Jahn-Teller distorted hexahydrated Cu2+ and Cr2+ ions. We have interpreted this in terms of a dissociative mechanism, with pentahydrated activated complexes and bond breaking to the leaving water ligand as the rate-determining step, regardless of whether the activation volumes have positive or negative values. A mechanism is discussed, where the intrinsic activation volume depends on how far the entering water molecule has to penetrate into the first hydration shell by pushing the bound water molecules in an octahedral triangular face aside before the force exerted on the crowded ligands is strong enough to cause a bond breaking. This is consistent with Swaddle’s conclusion for this system that ‘the molal volumes of the transition states within a given series of solvent exchange reactions are effectively the same regardless of the identity of the central ion”.l2b Corrections of energy and activation-volume values to account for the contractions of (1) the entering water molecule, (2) the activated complex, and (3) the surrounding second sphere of the hydrogen-bonded water molecules have been discussed and will in many cases make important contributions. For the hydrated trivalent first-row transition-metal ions, the reaction rates of water exchange span an even greater range, with rate constants from k = 2.4 X 10“ (Cr3+) to 1.8 X lo5s-I (Ti’+), both with highly “associative” activation volumes.6a A theoretical study is in progress for this series.36 As could be expected for these more highly charged ions, a similar dissociative reaction mechanism, based on the bond dissociation energy of the sixth ligand, does not explain these rate variations equally well as for the divalent ions, and an associative mechanism is also being investigated.

Acknowledgment. The continuing financial support and CPU time on a CRAY-XMP/416 obtained through the Swedish National Science Research Council is gratefully acknowledged. Professor Dr. Isaac B. Bersuker and Professor Ingmar Grenthe are thanked for helpful discussions. References and Notes ( I ) Akesson, R.; Pettersson, L. G . M.; Sandstram, M.; Siegbahn, P.; Wahlgren, U . J . Phys. Chem. 1992, 96, 10773. (2) Beagley, B.; Eriksson, A.; Lindgren, J.; Persson, I.; Pettersson, L. G. M.; Sandstram, M.; Wahlgren, U.; White, E. W . J . Phys.: Condens. Matter 1989, I, 2395. (3) Akason, R.; Pettersson, L. G . M.; Sandstram, M.; Wahlgren, U . J . fhys. Chem. 1992, 96, 150. (4) Eigen, M. Pure Appl. Chem. 1963, 6, 105. (5) Basolo, F.; Pearson, R. G . Mechanisms of Inorganic Reactions; Wiley: New York, 1967; Chapter 2. (6) (a) Ducommun, Y.; Merbach, A . E. In Inorganic High Pressure Chemistry, Kinetics and Mechanisms; van Eldik, R., Ed.;Elsevier: Amsterdam, 1986; Chapter 2.2. (b) van Eldik, R. Ibid. Chapters 1 and 8. (7) Crumbliss, A . L.;Garrison, J . M . Comments Inorg. Chem. 1988,8, I. ( 8 ) Katakis, D.; Gordon, G . Mechanisms of Inorganic Reactions; Wiley: New York, 1987; Chapters 3.6, 3.8, and 6.4. (9) Jordan, R.B. Reaction Mechanismsoflnorganicand Organometallic Systems; Oxford University Press: New York, 1991; Chapters 2.4 and 3.7. (IO) Powell, D. H.; Helm, L.; Merbach, A . E. J . Chem. fhys. 1991. 95, 9258. (I I ) van Eldik, R.; Gaede, W.; Cohen, H.; Meyerstein, D. Inorg. Chem. 1992, 31, 3695. ( I 2) Swaddle, T.W. In Mechanistic Aspects oflnorganic Reactions, ACS Symposium Series 198; Rorabacher, D. B., Endicott, J . F., Eds.; American Chemical Society: Washington, D.C., 1982; Chapter 2. ( 1 3) (a) Swaddle, T.W . Inorg. Chem. 1980,19,3203; (b) 1983,22,2663. (14) Newman, K. E.; Merbach, A . E. Inorg. Chem. 1980, 19, 2481. (15) Kanno, H.J . Phys. Chem. 1988, 92,4232. (16) Brooker, M . H . In The Chemical Physics of Solvation; Dogonadtc, R. R., Kdlmdn, E., Kronyshev, A . A,, Ulstrup, J., Eds.; Elsevier: Amsterdam, 1986; Part B, Chapter 4. (17) Swaddle, T.W.; Mak, M. K. S . Can. J . Chem. 1983, 61, 473 and references therein. (18) Berry, R. S. J . Chem. fhys. 1960, 32, 933. (19) Kepert, D. L.Inorganic Chemistry Concepts; Springer: New York, 1982: Vol. 6, Chapter 4 .

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(20) Reinen, D.; Atanasov, M. Chem. Phys. 1989, 136, 27. (21) Bacci, M. Chem. Phys. 1986, 104, 191. (22) Kang, S.-K.;Lam. B.; Albright, T. A,; OBrien, J. F. New J . Chem. 1991, IS, 757. (23) Lever, A. B. P. Inorganic ElecrronicSpectroscopy, 2nd ed.; Elsevier: Amsterdam. 1984: Chanter 6. D 458. (24) MOLECULE-SWEDEN is an electronic structure program written

by J. AlmlBf, C. W. Bauschlicher, M. R. A. Blomberg, D. P. Chong, A. Heiberg, S. R. Langhoff, P.-A. Malmqvist, A. P. Rendell, B. 0. Rms, P. E. M. Siegbahn, and P. R. Taylor. (25) Almldf, J.; Faegri, K., Jr.; Forsell, K. J . Comput. Chem. 1982,3,385. (26) Saebe, S.; Almlof, J. Chem. Phys. .Lett. 1987, 154, 521. (27) Pearson, R. G. Symmetry Rules for Chemical Reactions; Wiley: New York. 1976: Chaaters 1 and 3. (28) Bersuker, 1. B.*The John-Teller Effect and Vibronic Interactions in Modern Chemistry; Plenum: New York, 1984; Chapter 6.

Akesson et al. (29) Beattie, J. K.; Best,S. P.;Skelton, B. W.; White.A. H. J . Chem.Soc.. Dalton Trans. 1981, 2105. (30) Powell, D. H.; Neilson, G. W.; Enderby, J. E. J . Phys.: Condens. Matter 1989* I * 8721' (31) Swaddle, T. W. Comments Inora. - Chem. 1991. 12. 237. (32) American Institute o j Physics Handbook; Gray, D. E., Ed.; McGraw-Hill: New York, 1972. (33) Burdett, J. K. Adu. Inorg. Chem. Radiochem. 1978,21, 113. (34) Hahn, R. L. J . Phys. Chem. 1988, 92, 1668. (35) Conway, B. E. Ionic Hydration in Chemistry and Biophysics; Elsevier: Amsterdam, 1981; Chapter 25. (36) Akesson, R.; Pettersson, L. G.M.; S a n d s t r h , M.; Siegbahn, P. E. M.; Wahlgren, U. To be published.