J. Phys. Chem. 1994,98, 8641-8647
8641
Investigation of the Ligand Exchange Reaction for the Aqueous Be2+ Ion M. A. Lee Department of Chemistry, University of California, Davis, California 95616, and Structural Biology Group, Lawrence Livermore National Laboratory, Livermore, California 94550
N. W. Winter' Structural Biology Group, Lawrence Livermore National Laboratory, Livermore, California 94550
W. H. Casey Department of Land, Air, and Water Resources, University of California, Davis, California 95616 Received: February 16, 1994; In Final Form: June 8, 1994'
The geometry and energy of the transition state for the exchange of water-coordinating Be2+in aqueous solution has been investigated by Hartree-Fock (HF) cluster calculations, which include the ion and its near-neighbors. The calculated activation parameters, A P and AH*,are compared to the spectroscopic values. The HF activation volume A P was found to be -0 compared to the experimental value of -13.6 cm3/mol. The discrepancy is attributed to an incomplete description of the second coordination shell, which is needed to accurately define the volume of the transition-state complex. Including a continuum approximation for the solvent did not improve the agreement. The HF activation enthalpy for the exchange reaction, 0.65 eV, is in good agreement with the experimentally determined value of 0.61 eV. A qualitative description of the reaction mechanism has been developed from calculations on 5-fold coordinated Be2+with one B A H 2 distance constrained to the reaction coordinate. The results suggest a reaction scheme where the exchanging water from the second coordination shell is oriented along a 2-fold axis of the tetramer. Optimization of the complex at the saddle point of the reaction surface resulted in a CZ,transition state with approximate trigonal-bipyramidal geometry. The B e H z 0 bonding is dominated by electrostatic interactions with little evidence of dative or covalent bonding between the water lone pairs and the empty 2s and 2p orbitals on Be2+.
I. Introduction Solvation, the formation of a molecular complex between solute and solvent, is important to the study of kinetic processes in many areas of biology, chemistry, and physics, ranging from the folding of proteins to the dissolution of minerals. Insight into solvation kinetics can be obtained from the comparison of models of simple solvent exchange reactions to experiment. Activation parameters such as energy, enthalpy, and volume can be calculated and used in the diagnosis and prediction of reaction mechanisms. This approach is particularly important for reactions at surfaces that cannot be fully characterized spectroscopically. Theoretical studies of the solvent-molecule interaction and the effects of solvation on molecular properties have been carried out by a variety of methods including molecular clusters,' molecular dynamics,2 and Monte Carlo simulations.3 In the molecular cluster approach the whole system is studied with quantum mechanical methods by forming a "supermolecule" consisting of the solute and a selected number of solvent molecules. The model can be improved by including additional solvent molecules, but inclusion of explicit solvent molecules has computational limits. Another limitation is the neglect of the statistical nature of the solvent. Molecular dynamics and Monte Carlo methods incorporate these effects and can make use of quantum mechanical clusters calculations to define the necessary interaction potentials. Recently quantum mechanical methods have been developed that represent the solvent as a polarizable continuous media but do not treat explicit hydrogen bonding and size effects.4-8 In the present work we have used the molecular cluster method to investigate the in vacuo solvent exchange reaction of aquated beryllium ion in order to gain insight into the mechanism for dissolution of bromellite (BeO). This is motivated by the *Abstract published in Aduance ACS Abstracts, July 15, 1994.
0022-3654/94/2098-864I%O4.50/0
observation that the ligand exchange kinetics of metal ions in solution correlates with the dissolution rates of oxide minerals.9 Comparison of the calculated and measured activation parameters for solvent exchange around aquated BeZ+effectively tests the ab initio models for the dissolution reaction. Most oxide minerals exhibit a significant degree of ionic bonding characterized by electrostatic interactions between the constituent atoms. The process of dissolution separates the lattice ions into weakly interacting solvated ions and converts the metal-oxide bonds to metal-water coordinate bonds. Forces responsible for the bond lengths in ionic solids are a combination of both longrange Coulomb (Madelung) and short-range repulsive interactions. For the aquated ion, metal-oxygen distances are determined by shorter range chemical, electrostatic, and steric interactions. The long-range Coulomb forces are less important due todielectric screening by the solvent. Despite these differences, metal-oxygen bond lengths for the first three alkaline earth oxides of the crystalline solid and the aquated cation are quite similar, BeO(s) = 1.65 A, MgO(s) = 2.11 A, and CaO(s) = 2.41 A. These correlate well with the metal-oxygen bond lengths of the cations in aqueous solution, Be2+ = 1.71 A, Mgz+ = 2.10 A, and Ca2+ = 2.42 A. It is reasonable to infer that this implies a similarity in the bondinginteractionsas well. This conjectureiscorroborated by the trends in the rate constants for the dissolution of the crystalline solids (CaO > MgO > BeO) and the rates of ligand exchange for the aquated ions (Ca2+ > Mgz+ > BeZ+) of the first three alkaline earth oxides."J 11. Solvent Exchange
A metal ion with n ligands can undergo ligand exchange by either an associative (A) or dissociative (D) mechanism. For an associative mechanism the incoming ligand coordinates to the metal ion before the original ligands departs, 0 1994 American Chemical Society
8642 The Journal of Physical Chemistry, Vol. 98, No. 35, 1994
ML,
+ L’
-
[L’-ML,]
-
ML,,L’
+L
(1)
Lee et al.
t
and the complex [L’-ML,] defines the transition state for the exchange reaction. If one of the original ligands coordinating the metal ion departs before the new ligand coordinates to the metal, the mechanism is dissociative,
ML,
-
[ML’,,]
+ L + L’
-
ML,,L’
+L
(2)
where the transition state, [ML’,+,], has n- 1 ligands. When the transition state is an activated complex [L’-ML,,-L] formed by the concerted motion of the incoming and outgoing ligands the mechanism is declared an interchange process (I). Associative reactions exhibit a decrease in the molar volume of the transition state relative to that of the reactants, whereas for dissociative reactions the transition state volume is larger than that of the reactants. It follows that the activation volume, the difference between the molar volumes of the transition state and reactants, is a good indicator of the type of mechanism involved in the ligand exchange. A schematic diagram of the relationship of mechanisms, A I D and AV is given in Figure 1, where the transition state for the A mechanism is shown for the limiting case, AV = -Vsolvcnt.For the I mechanism, AV = 0 since the exchanging solvent molecules are each one-half inside the original reactant volume, and for the D mechanism, AV = AV,lvenrsince both exchanging molecules are outside the reactant volume in the transition state. Activation volumes are experimentally observable by highpressure NMR for many ion-solvent systems and are related to the rate constant, k, for the ligand exchange reaction by the following relations:
--
[d(ln k)/aT], = AEJRP
(3)
E, = M*+ RT
(4)
The observed rate of an associative mechanism would increase with applied pressure and decrease for a dissociative mechanism. Typical experiments are carried out at about 2000-3000 atm and involve the measurement of the time-evolution of I7O peaks of bound and unbound water. Measurements for the Be(H20)42+ exchange reaction, Be(H2O)?
+ H,O
-
[Be(H20),]”
-
Be(H20):+
“(9
/
+ H 2 0 (6)
report an activation volume of -13.6 cm3/mol.’0 This strongly suggests an A mechanism with a stable pentacoordinated transition-state complex. The limiting value of AV has been estimated to be between -12.9 and -15 cm3/mol.I1 In order to provide further insight into the relationship between the reaction mechanism and the reaction surface, first principles calculations of AV and AH* as well as for the optimum structure of the transition state have been carried out and are described in the following sections. 111. Details of the Calculations
Previous theoretical work on metal-water clusters includes the alkaline earth and first-row transition elemer1ts.12-I~ Theoretical work on aquated Be2+ has focused on determining the number of primary shell water molcules. Hashimoto et ul.1’3J9 studied models with coordination numbers of 2, 3, 4, and 6 and determined the hydration number to be four. Additional water molecules were found to occupy the second coordination shell. Similarly, Bock and Glusker20 found that the primary hydration
Reaction Coordinate
-
Figure 1. Schematic description of the dissociative (D), interchange (I),
and associative (A) mechanisms for ligand exchange.
shell of Be2+contained four water molecules with additional waters taking positions in the second shell. Intermediate structures appropriate to describing dissociative exchange of water between the first and second coordination shells were also studied, for example Be(H2O)32+ withadditional water molecules in thesecond shell. They also investigated the equilibrium structure of Be(H20)s2+ with one water molecule in the second shell, which is a possible starting point for the search for the associative exchange transition structure. In the present work HF calculations of the optimum structures for the Be(H20)42+ and Be(Hz0)s2+complexes have been carried out by using a 6-3 lG* basis set. This basis includes 3d polarization functions of the heavy atoms but does not include hydrogen 2p functions. In order to more accurately describe the inter- and intrashell hydrogen bonding interactions, pentahydrate calculations were repeated with a 6-31G** basis set that includes hydrogen 2p polarization functions. Because the 2s and 2p orbitals of Be2+ are unoccupied, bonding electrons are provided by the 2p lone pairs on the oxygen atoms of the coordinating water molecules. Formation of the pentahydrate transition state by addition of water to the tetrahydrate complex and the subsequent dissociation to reform the tetrahydratecomplexdo not involve the pairing or unpairing of electrons. Consequently, the restricted HF approximation remains valid along the reaction path for the ligand exchange. Calculations were carried out with the Spartanz1Gaussian integral and direct self-consistent field program with gradient searches carried out in Cartesian coordinates employing the E F algorithm of Baker and Hehre.22
N. Results The initial calculations were carried out to determine the positions of waters in the first coordination shell and to compare to the results of Hashimoto et ul.18Jgand Bock and Glusker.20 The bond distances and angles for the resulting 4-fold coordinated complex, Be(H20)42+, having approximate tetrahedral symmetry and bond lengths of R m = 1.654 A and ROH= 0.960 A, are shown in Figure 2. Our results are in good agreement with those of Bock and Gluskerm who also carried out their study at the 6-31G* level. Calculations carried out by Hashimoto et al.19 a t the 3-21G* level found the tetramer to have S d symmetry with shorter B e 4 bond lengths ( 1.648 A) and longer 0-H bond lengths (0.974 A) than those reported here. The calculations were repeated with the 6-31G** basis set, which predicted R m = 1.650 A and ROH= 0.954 A, slightly shorter than the 6-31G* basis. In order to test the effects of electron correlation, the tetrahedral complex was reoptimized a t the MP2 level, giving
The Journal of Physical Chemistry, Vol. 98, No. 35, 1994 8643
Investigation of Ligand Exchange 108.5'
08.2" 108.3') 3.4A
Figure 2. Structure of the tetrahedrally coordinated Be(H~0)4~+ complex showing the bond distances and angles. All H-0-H angles and OH bond distancesare equivalent. The 01-Be+ and 04-Be-05 angles are equal, as are the 01-Be-04, 01-Be-05, 03-Be+, and 03-Be-04 angles. All B e 4 distances are the same. The values in parenthesis were determined by an MP2 optimization of the geometry.
-
3.OA
-
-
2.5A
-
2.2A
2.0A
Figure 4. Formation of the pentahydrate transition state from a second shell water as a function of the B e 4 distance to the exchanging water molecule. The intermediate structures were determined by optimizing all degrees of freedom for the pentahydrate complex except the Be-O distance for the incoming water molecule. This was constrained to the values given below each figure.
'
/&
0.956(0.951)A
109 l(lO95)"
(105.7O)
1
A
(1 653)A
0.955(0.950)A
Figure 5. Pentahydrate transition state for the ligand exchange reaction of tetrahedrally coordinated Be(H20)42+. The unlabeled bond distances and angles can be determined from the CZ,symmetry of the complex. In each case the first value given was determined with the 6-3 lG* basis set and the value in parenthesis with the 6-31G** basis set.
Figure3. Optimized structure of the Be(H20)s2+pentahydrate complex with the fifth water in the second coordination shell. The unlabeled bond distances and angles can be determined from the C2 symmetry of the complex. In each case the first value given was determined with the 6-3 lG* basis set and the value in parenthesis with the 6-3 1G** basis set.
R w = 1.658 A and ROH= 0.98 1 A, somewhat longer than either H F result. Solvating this structure in a continuum dielectric did not significantly affect the predicted geometry. A fifth water molecule was added to the original complex and the atomic positions of the new complex, Be(H20)s2+, were optimized without constraints. The resulting structure has C2 symmetry along the Be-0 axis to the second sphere water, which is at a distance of 3.4 A from the central Be2+ion. This is consistent with the second shell water distances of 3.2 to 3.9 A, determined by a molecular mechanics calculation for solvated Be(H20)42+. The calculated bond distances and angles for the C2 structure at the 6-31G* and 6-31G** basis sets are shown in Figure 3. The bond length of the oxygen of the fifth water to the first coordination shell hydrogens, 1.85 A, is in the range of a typical hydrogen bond.23 The fifth water is bound to the Be(H20)42+cluster by 1.39eV, considerably more than the typical hydrogen bond energy. A subsequent series of calculations was carried out at fixed Be-Odistances of 3.0,2.5,and 2.2 A, where all degrees of freedom were optimized except the distance between the oxygen of the reactant water and the Be2+ ion. As the distance between the oxygen atom of the fifth water and the Be2+ion is decreased, the three nearest waters move into the equatorial plane and the remaining water increases its distance from Be2+as it moves into an opposing axial position. This is illustrated in Figure 4 where the optimized structures of the pentamer at selected points along the reaction coordinate are shown. The final Be(H20)s2+ transition state was optimized by constraining the two axial bonds of the approximate trigonal-bipyramidalstructure shown in Figure 5 to be equal and parametrically varying them while optimizing all other degrees of freedom. The optimized bond distances and angles are also given in Figure 5 for both the 6-3 lG* and 6-3 lG** basis sets. The resulting symmetric stretch potential energy curve
I
I
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
Be-0 Distance, Angstroms
Figure 6. Potential energy for the symmetric stretch of the two axial water molecules of the pentahydrate transition state.
TABLE 1: Definition of the Reaction Coordinate and the Calculated Energies for the Exchange of HzO(6) by HzO(5) in Be(H20h2+a Be-05, A Be-06, A displacement from saddle point, A energy, aub ~
~
3.43 3.0 2.5 2.2 2.0
~~
1.661 1.673 1.726 1.813 2.0
1.4 1.o 0.5
0.2 0.0
-394.361 1 -394.3 555 -394.3440 -394.3382 -394.3364
The numbering of the atoms is given in Figure 3. RHF/6-31G*.
is shown in Figure 6, where the optimum axial bond distances are 2.05 A. The optimized transition-state complex has C, symmetry with a plane of reflection containing the three oxygens of the equatorial waters and a second reflection plane passing through the oxygen atoms of the two axial waters and one of the equatorial waters. The trend of average bond lengths in the complexes is C, > Td > C2. The total energies along the reaction coordinate corresponding to the intermediate ligand exchange structures in Figure 4 are listed in Table 1 and plotted in Figure 7. The difference in energy of the transition state relative to the C2 pentamer is 0.67 eV. Inclusion of the zero-point energy to this
8644
The Journal of Physical Chemistry, Vol. 98, No. 35, 1994
08t
I
-2
I -15
-1
-05
0
0.5
I
15
2
Reactron Coordinate. Angstroms
Figure 7. Total energy of the pentahydrate complex along the ligand
exchange reaction coordinate. value results in AH* = 0.65 eV, in good agreement with the experimental value of 0.61 eV." Rearrangement of the remainder of the solvent along the reaction path also contributes to the calculated activation parameters. Preliminary calculations using the Born-Onsager reaction field approximation24q25implemented in the Gaussian 92 program,2s which determines the solute wave function selfconsistently in the presence of a continuum dielectric, show the change in solvation free energy, PAC,,, = 0.02 eV, is small relative to E, (0.67 eV). This suggests that solvent rearrangements do not significantly affect the calculated energies. However, the Born-Onsager method approximates the solute as a dipole in a spherical cavity, neglecting higher order moments needed to accurately reproduce the solute charge distribution. Therefore the change in the solvation free energy calculated with this model is underestimated. In order to quantify this, further calculations are being carried out that employ more accurate solvation models.
V. Analysis of the Bonding in the Be(H20),,2+ Complexes The Be2+ ion would be expected to participate in dative bonding with its near-neighbor water molecules, since the unoccupied 2s and 2p orbitals are available to accept electrons from the oxygen lone pair orbitals. In addition, the formation of sp3hybrids from the beryllium 2s and 2p orbitals is consistent with the observed 4-fold coordination and near tetrahedral geometry of Be(H20)d2+. However, the small ionic radius of Be2+ (0.31 A) prevents water molecules from approaching the central metal ion close enough
Lee et al. to achieve good orbital overlap for charge transfer and dative bond formation. This is supported by the natural orbital populations for Be(2s,2p) andO(2p) orbitalsof H20, B e ( H ~ 0 ) 4 ~ + , C2 Be(H20)++, and Cb Be(Hz0)S2+ given in Table 2. Results for the tetramer and pentamers show that a relatively small amount of charge is transferred to the accepting orbitals of the Be2+ ion (-0.24-0.26 electrons), with most going into the Be(2s) orbital. The oxygen atom electron population also increases (-0.090.19 electrons), while the hydrogen atoms lose electrons in each case relative to the isolated water molecule. In contrast, the Mulliken analysis listed in Table 3 shows that a significant amount of charge (-0.85-0.95 electrons) is transferred to the Be(2s,2p) orbitals from the nearest-neighbor water molecules, with greater occupation of Be(2p) than Be(2s) orbitals. The oxygen atom populations remain approximately the same as for the isolated water molecule, with the net electron transfer being from the hydrogens to the BeZ+ion. These results are consistent with those of Hashimoto et ~ l . , 1who ~ reported gross Mulliken orbital populations for Be(2s) of 0.273 and Be(2p) of 0.725 for Be(H20)42+using a smaller 3-21G basis set without polarization functions on the hydrogens or heavy atoms. The total atomic populations and predicted charges are given in Tables 4 and 5 . The two charge partitioning schemes suggest somewhat different bonding pictures for the hydrated complexes. The natural orbital population analysis indicates that electron charge is not only retained on the oxygen atoms but is increased upon coordinating the Be2+ cation. The relatively small amount of charge transferred to the empty Be(2s,2p) orbitals occurs at the expense of the hydrogens and goes mostly into the s orbitals. The trend in Be(2s) character for the complexes is CZ> T d > C,. The Mulliken population analysis shows that a more significant amount of charge (- 1 electron) is transferred to the accepting orbitals of Be2+ and goes mostly into the p orbitals. The trend in Be(2p) character for the complexes is the same as for the Be(2s) The natural orbital population analysis indicates there is only minor participation of the Be(2p) orbitals in the coordinate bonding. Because the natural orbital population analysis is based on localized orbitals intrinsic to the wave function, it is well suited to ionic systems and less sensitive to the size of the basis set used in the ~ a l c u l a t i o n . ~This ~ ~ ~is8a distinct advantage over Mulliken analysis, which can lead to nonphysical populations that violate the Pauli exclusion principle as the basis set becomes larger and more diffuse. The natural orbital picture is supported by the charges derived from the fit to the electrostatic potential. An alternative method for estimating the atomic charges is to fit the electrostatic potentia129330of the complex produced by the
TABLE 2: Natural Orbital Populations for Be(2s,2p), 0(2p), and H(1s) Orbitals for HzO, Be(HzO)43+, C3 Be(HzO)s3+, and Cf. Be(H30)s2+;RHF/6-31G* H20
OI(2P) 2Px 2PY 2P* H1,2( 1 S)
5.195 1.464 1.998 I. 733 0.523
Be(H20)d2+ C2 Be(H20)s2+ Be(2s,2~) Be(2s92p) 0.258 2s 0.220 2s 2P 0.039 2P 2Px 0.013 2Px 2PY 0.013 2PY 2PZ 0.013 2PZ ~ I , ~ z , ~ ~ , ~ 5.358 ~ ( ~ P ) 01,03(2P) 2Px I.739 2P* 2PY 1.843 2PY 2PZ 1.776 2PZ Hts-H13( 1s) 0.417 05,06(2p) 2Px 2PY 2PZ 05(2P) 2Px 2PY 2PZ Hi~,i6,9,10(1S) Hii,i2,7,~(1~) Hi3.141~)
0.262 0.222 0.039 0.013 0.013 0.013 5.386 1.911 1.767 1.709 5.350 1.845 1.755 1.749 5.280 1.594 1.909 1.777 0.419 0.422 0.454
C2" Be(Hz0)P Be(2s92p) 2s 2P 2Px 2PY 2PZ 01,03(2P) 2PX 2PY 2PZ O&P) 2Px 2PY 2PZ 05,06(2p) 2Px 2PY 2PZ HI5.16.9.10(1 S) Hi 1 . d 1s) H13,14,7,d 1 S)
0.237 0.201 0.036 0.013 0.009 0.014 5.351 1.933 1.571 1.847 5.350 1.872 1.571 1.908 5.3 15 I .649 1.779 1.887 0.426 0.438 0.427
Investigation
of Ligand Exchange
The Journal of Physical Chemistry, Vol. 98, No. 35, 1994 8645
TABLE 3: Mulliken Orbital Populations for Be(2s,2p), 0(2p), and H(1s) orbitals for HzO, Be(Hz0)d2+, C2 Be(HzO)Z+, and Cg Be(H*OV+:RHF/6-31G* Be(H~0)4~+ CZBe(H20)s2+ CZ Be(Hz0)S2+ 0.875 0.208 0.668 0.220 0.220 0.227 2.991
0.882 0.205 0.677 0.210 0.230 0.238 2.994 1.085 0.974 0.934 2.988 1.026 I .047 0.915 2.935 0.884 1.078 0.973 0.437 0.453 0.488
1.001
1.001 0.989 0.447
TABLE 4
Natural Atomic Populations and Charges for HzO, Be(H20h2+,C2 Be(H20)2+, and CzVBe(H2O)s2+;RHF/6-31C* Be(H~0)4~+ CZBe(Hz0)s2+ Ch Be(HzO)Sz+ H20
occupancy
0 HI,^
0.786 0.206 0.580 0.220 0.122 0.237 2.988 1.092 0.879 1.017 2.987 0.939 1.154 0.894 2.951 0.921 0.956 1.074 0.461 0.455 0.472
8.955 0.523
charge -0.955 0.477
Be 0~,0~,04,0s
occupancy
charge
2.260 9.101 0.4 17
1.740 -1.101 0.583
occupancy Be
2.263 01,03 9.121 04.06 9.092 05 9.038 Hi5.16,9,10 0.418 H I I J Z , ~ , ~ 0.422 Hi3.14 0.454
charge 1.737 -1.121 -1.092 -1.038 0.582 0.578 0.554
occupancy
charge
2.238 9.090 9.093 9.068 0.422 0.422 0.438
1.762 -1.090 -1.093 -1.068 0.578 0.578 0.561
Be 01,03 0 4 0 5 0 6
His.16.9.10 Hii,iz H13,14,7,8
TABLE 5: Mulliken Atomic Populations and Charges for HzO, B ~ ( H Z O ) ~CZ ~ +Be(H20)s2+, , and C2, Be(H20h2+;RHF/631G* HzO Be(H~0)4~+ CZBe(HzO)Sz+ CZ Be(HzO)sz+
0 Hl,z
occupancy
charge
8.869 0.566
-0.869 0.434
occupancy Be 0.597 0 ~ , 0 ~ , 0 4 , 0 ~ 8.809 Hb13 0.595
charge 0.403 -0.809 0.405
Be
Oi,o, 04,06
os
Hi 5,16,9,10 Hii,i2,7.8 H13.14
occupancy
charge
0.609 8.859 8.802 8.869 0.533 0.600 0.546
0.391 -0.859 -0.802 -0.869 0.467 0.400 0.454
occupancy Be 0.605 01,Oa 8.797 O4 8.798 O5,Os 8.800 H I S , I ~ . ~ , I O 0.561 Hii.12 0.599 H13,14,7,8 0.601
charge 0.395 -0.797 -0.798 -0.800 0.439 0.401 0.399
TABLE 6 Charges from Fit to the Electrostatic Potential for HzO, Be(Hz0)42+,CZBe(HP)a2+, and C2* Be(H20)s2+; RHF/631C* Hz0
0 HI,Z
4.808 0.404
Be 01,0~~04~0s H6-13
Be(H~0)4~+ 1.609 -1.102 0.600
Be 01~03 04906
OS H9,ls Hio.16 Hi 1.12,7.8 H13,14
HF electron density. The predictions for each of the complexes by this approach are listed in Table 6. The fitted atomic charges for the free water molecule are smaller than those derived from either of the population analysis methods by 7-15%. The overall trends for the fitted charges are in better agreement with the natural orbital charges than with the Mulliken charges. The latter method assigns more of the oxygen population to the BeZ+ ion, reducing their net charge. The charges on the hydrogen atoms, as predicted by all three methods, are in reasonable agreement. Water molecules in the first coordination shells of the complexes are tightly bound by electrostatic forces and are highly polarized relative toneutral water. This results in increased positive charge on the first shell hydrogen atoms, which in turn strengthens the
C2 Be(HzO)sZ+ 1.380 -1.013 -0.978 -0.961 0.576 0.565 0.565 0.51 1
Be 0 1 ~ 0 3 0 4
oS,06 His,16,9,io HIIJZ H13,14.7.8
C a Be(Hz0)P 1.645 -1.081 -1.047 -1.041 0.583 0.574 0.539
hydrogen bonding to the second shell water molecules. The nature of the first and second shell bonding is illustrated in Figure 3, where the second shell oxygen atom strongly interacts with two hydrogens from different first shell water molecules. The CZ cluster predicts the distance between the oxygen of the fifth water and the beryllium to be 3.4 A. This is expected to increase as more second shell waters are included in the calculation due to steric interactions.
VI. Discussion of the Activation Parameters The calculated activation parameters, energy (A&), enthalpy volume (AV), were determined for the ligand exchange reaction (6). Inclusion of zero-point vibrational energies
(AH*), and
8646 The Journal of Physical Chemistry, Vol. 98, No. 35, 1994
Lee et al.
results in the calculated value of AH*of 0.65 eV, which is in good agreement with the experimental value of 0.61 eV.” Using eq 4 and T = 273.15 K, thecalculated AEawas 0.67 eV. Molecular volume was calculated by using the Monte-Carlo method in the Gaussian 92 program26and the activationvolume, AVI, was found to be -0.803 cmg/mol. Although the sign is in agreement with the experimentally determined value of -13.6 cm3/mol and is consistent with an associative (A) reaction mechanism, the magnitude is indicative of an I mechanism. The saddle point character of the reaction surface in the region of the transition state is also consistent with the intercharge reaction mechanism. Since MP2 calculations of the reactant and transition-state energies gave essentially the same A& as the HF calculations, electron correlation should have a small effect on the shape of thereaction surface. It is more likely that including the complete second solvation shell in the calculation will have the most significant effect on the calculated activation volume. For a binary gas phase reaction with an associative (A) Figure 8. Calculated structure for the Be(HzO)d2+:4H20 cluster. Be is mechanism, the transition state would be expected to be at a local white, 0 is gray, and H is black. The geometry was optimized without minimum on the reaction surface. In the gas phase the shape of constraint, resulting in Cb symmetry. the potential energy surface shown in Figure 7 would suggest an interchange (I) mechanism for the ligand exchange. However, mination of the geometry and energy of the Be(H20)52+:3H20 in the liquid phase rather than the reactants undergoing binary transition state corresponding to ligand exchange in Be(Hz0)d2+: collisions the exchange reaction is analogous to a unimolecular 4H20 is in progress, as well as analogous calculations for Berearrangement. The “molecule” undergoing rearrangement is (H20)42+:8H20. the Be+2 cation and its first and second shells. There are two second shell clusters corresponding to two extremes of hydrogen bonding between the first and second shells. Be(H~0)4~+:4H20 W.Conclusions or Be(H20)42+:8H20. In the first case, Be(H20)42+:4H20,the Starting with a simplified model for the reactants, the transition oxygen from each of the four second shell waters interacts with state for the exchange of water coordinating to Be2+ has been two hydrogens from the first shell. Making the maximum number found to be a distorted trigonal bipyramid with overall C b of hydrogen bonds between the two shells results in the cluster symmetry. Hartree-Fock calculations along the proposed reacBe(H20)d2+:8H20, where each of the eight second shell waters tion coordinate indicate that the exchange reaction proceeds interacts with a single first-shell hydrogen. Taking into account through a saddle point lying 0.67 eVabove reactants and products. the dynamic behavior of water molecules in the outer coordination The calculated activation enthalpy of 0.65 eV is within 0.04 eV shell, it is reasonable to expect that on the average the actual of the measured value.11 Preliminary calculations using the number of second shell waters is between these two extremes. In Onsager reaction field which places the cluster in a fact the average of the two clusters (6H20) is consistent with our continuous medium with a uniform dielectric constant, show an molecular mechanics simulations and with results from diffusion increase in the activation volume by a factor of about 4 and experiments.” demonstrate the sensitivity of the results to more accurate Calculation of the positions of the four second shell waters hydration models. shows them to be positionedon the edges of the tetrahedron formed by the four first shell waters. The reaction proceeds as one of Acknowledgment. We thank Drs. D. A. McQuarrie, J. H. the second shell water molecules joins the first shell, giving the Swinehart, M. E. Colvin, and W. H. Fink for helpful discussions. transition state Be(Hz0)s2+:3H20, similar to the mechanism This work was performed under the auspices of the Department shown Figure 4. If the waters entering and leaving the first shell of Energy by Lawrence Livermore National Laboratory under are each one-half in the first shell (as is the case for the BeContract W-7405-ENG-48 and supported by the Structural (H20)52+ transition state in Figure 4), then AV 0 and the Biology Initiative funded by the Laboratory Directed Research mechanism is I, as illustrated in Figure 1. However, if the second and Development Office of the Lawrence Livermore National shell water enters the first shell before the leaving water departs, Laboratory, NSF Grant EAR-93-02069, and by the University the mechanism is A and AV* V(H20) as is also shown in of California Research Mentorship Fellowship Fund. Figure 1 . The experimental value of AV = -13.6 cm3/mol, which is close to the electrostricted molar volume of a water molecule in the second shell, suggests an A mechanism. It is References and Notes likely that our Be(Hz0)4*+:H20cluster, which has only one second (1) Tunon, I.; Sila, E.;Bertran, J. J. Phys. Chem. 1993, 97, 5547. shell water, predicts A P 0, because the leaving water is not (2) Dang, L. X.;Rice, J. E.; Caldwell, J.; Kollman, P. A. J. Am. Chem. constrained by proper interactions with the remainder of the Soc. 1991, 113, 7, 2481. second coordination shell. In order to better illustrate this, the (3) Cieplak, P.; Kollman, P. A. J. Chem. Phys. 1990, 92, 6761. (4) Kirkwood, J. G. J . Chem. Phys. 1934, 2, 351. optimized structure for the Be(H2O)d2+:4H20 cluster is shown (5) Onsager, L. J. Am. Chem. Soc. 1936, 58, 1486. in Figure 8. The second-shell waters are a t a distance of 3.496 (6) Miertus, S.; Scrocco, E.; Tomasi, J. Chem. Phys. 1981, 55, 117. A from Be2+,slightly longer than for the Be(H20)42+:H20cluster. (7) Cramer, C. J.; Truhlar, D. G. Science 1992,256, 213. The optimized inner shell Be-0 distance is reduced to 1.644 A ( 8 ) Cramer, C. J.; Truhlar, D. G. J . Am. Chem. Soc. 1991, 113, 8305. (9) Casey, W. H. J. Colloid Inrerfoce Sci. 1991, 146, 2, 586. by a more complete outer shell. As the exchanging second shell (10) Merbach, A. E.; Akitt, J. W. High Resolution Variable Pressure water moves into the first shell, the departing first shell water is NMR for Chemical Kinetics, NMR Basic Principles and Progress; considerably more confined than in the calculations with only a Springer-Verlag: Berlin, 1990; Vol. 24, p 202. (1 1) Pittet, P-A.; Elbaze, G.; Helm, L.; Merbach, A. E. Inorg. Chem. single water in the second shell as previously described. Presum1990, 29, 10, 1936. ably this will result in a smaller transition-state volume and a (12) Bauschlicher, C. W.;Sodupe, M.;Partridge, H.J. Chem. Phys. 1992, more negativevalue for AVI, in agreement with experiment. Deter96, 6 , 4453.
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