Hydration Energies of Divalent Beryllium and Magnesium Ions: An ab

Feb 29, 1996 - Previous calculations have shown that the potential energy surfaces of divalent beryllium ions surrounded by four, five, and six water ...
2 downloads 0 Views 393KB Size
Download PDF
3488

J. Phys. Chem. 1996, 100, 3488-3497

Hydration Energies of Divalent Beryllium and Magnesium Ions: An ab Initio Molecular Orbital Study George D. Markham,† Jenny P. Glusker,† Cindy L. Bock,†,§ Mendel Trachtman,‡ and Charles W. Bock*,†,‡ The Institute for Cancer Research, The Fox Chase Cancer Center, Philadelphia, PennsylVania 19111, Chemistry Department, The Philadelphia College of Textiles and Science, Philadelphia, PennsylVania 19144, and Biochemistry Department, Elizabethtown College, Elizabethtown, PennsylVania, 17022 ReceiVed: August 29, 1995; In Final Form: NoVember 17, 1995X

Ab initio molecular orbital calculations have been used to investigate contributions of water molecules in the first and second coordination shells to the overall hydration energy of divalent beryllium and magnesium cations. Enthalpy and free energy changes at 298 K have been calculated at a variety of computational levels for the reactions M2+ + [H2O]p f M2+‚nH2O‚mH2O, where M ) Be or Mg, [H2O]p (p ) 2, 4, 6, 8; p ) n + m) are water clusters, and M2+‚nH2O‚mH2O are ion-water complexes with n and m water molecules in the first and second coordination shells, respectively. These reactions involve the disruption of the water cluster and naturally include the competitive effects of ion-water and water-water interactions inherent in the hydration process. At the MP2(FULL)/6-311++G**//RHF/6-31G* computational level, the values of ∆G298 for the reactions which complete the first hydration shells, Be2+ + [H2O]4 f Be2+‚4H2O and Mg2+ + [H2O]6 f Mg2+‚6H2O, are -352.0 and -266.7 kcal/mol, accounting for 61.2% and 60.7% of the experimental free energies of hydration of Be2+ and Mg2+. Reactions that incorporate two additional water molecules into a second hydration shell only change ∆G298 by -43.0 and -24.2 kcal/mol, whereas the values of ∆G298 for the corresponding reactions that incorporate the first two water molecules in the primary hydration shell are -244.6 and -135.2 kcal/mol, respectively. The calculated values of ∆G298 for the formation of the complexes Be2+‚4H2O‚4H2O and Mg2+‚6H2O‚2H2O from eight-water clusters account for approximately 73.2% and 66.2% of the overall free energies for Be2+ and Mg2+, respectively, but convergence toward the experimental hydration energies will be quite slow as additional water molecules are added to the outer hydration shells. This is consistent with the concept of the importance of long-range interactions to the hydration energy.

Introduction Hydration, the formation of a molecular complex between a solute and water, is critical in many areas of biology and chemistry, ranging from protein conformational stability and enzymatic catalysis to the structures and reactivities of both ionic and neutral dipolar molecules.1-3 Ionic hydration phenomena, including the determination of hydration numbers, the rates of exchange of water molecules between outer and inner coordination shells, the interaction energies between ions and water molecules, etc., have attracted considerable attention in recent years.4-11 In particular, the structure of water molecules around metallic cations has been studied extensively, and ion-water distances for the first coordination shell, and in some cases for the second coordination shell, have been established for many common metal ions.11 Such studies have practical significance; for example, the thermodynamic and spectroscopic properties of electrolytic solutions depend to some extent on the ability of water molecules to approach a given ion. Various computational approaches, including ab initio and semiempirical molecular orbital studies of clusters,12 Monte Carlo13 and molecular dynamics simulations of solutions,14 continuum solvation models,15-25 and combined discretecontinuum approaches,26 have proved to be useful in understanding ion-water interactions. The cluster method assumes * Corresponding author: Chemistry Department, Philadelphia College of Textiles and Science, Philadelphia, PA 19144. † Fox Chase Cancer Center. ‡ Philadelphia College of Textiles and Science. § Elizabethtown College. X Abstract published in AdVance ACS Abstracts, February 1, 1996.

0022-3654/96/20100-3488$12.00/0

the formation of a supermolecule consisting of an ion surrounded by a specific number of water molecules. The most exact model would, of course, include a large number of water molecules, but their inclusion has computational limits, particularly if highlevel ab initio molecular orbital or density functional methods are used for the necessary geometry optimizations, frequency analyses, and single-point energy calculations. The hydration energy of a divalent ion is the enthalpy change for the reaction

M2+(g) + H2O(1) f M2+(aq)

(1)

In the present work, we investigate contributions of water molecules in the first and second coordination shells to the enthalpy and free energy changes of this hydration reaction for Be2+ and Mg2+. The cluster approach is used in conjunction with ab initio molecular orbital calculations to compute the enthalpy and free energy changes for the reactions

M2+ + [H2O]p f M2+‚nH2O‚mH2O

(2)

at 298 K, where M ) Be or Mg, [H2O]p (p ) 2, 4, 6, 8; p ) n + m) are water clusters composed of p hydrogen-bonded water molecules, and M2+‚nH2O‚mH2O are ion-water complexes with n and m water molecules in the first (primary, inner) and second (outer) coordination shells, respectively. Reaction 2 involves disruption of the water clusters and thus naturally includes the competitive effects of water-water and ion-water interactions inherent in the hydration process. Although experimental values of the Gibbs free energy, ∆G298, for the overall hydration of © 1996 American Chemical Society

Hydration Energies of Be2+ and Mg2+ Be2+ and Mg2+ are known,26,27 very little information is known about the contribution of water molecules in various coordination shells to the overall hydration energy. Recently, the results of several ab initio molecular orbital investigations of the structures of hydrated beryllium and magnesium clusters of the form M2+‚nH2O and M2+‚nH2O‚mH2O (M ) Be, Mg) have been reported in the literature. These calculations have included up to eight water molecules around beryllium4,28-31 and up to six water molecules around magnesium32,33 partitioned between the first and second coordination shells. We have made extensive use of these results to identify the likely global minima on the various potential energy surfaces of these complexes and have augmented these results in several ways: (1) additional lower energy conformers of Be2+ surrounded by eight water molecules have been characterized, (2) conformers with eight water molecules surrounding Mg2+ have been investigated, and (3) the energies of all the complexes have been recalculated at significantly higher computational levels. It has been difficult to assess the contribution of a few water molecules to the overall hydration energies of any ions due to the paucity of reliable data on the necessary water clusters, [H2O]p. Recent calculations by Jordan and co-workers34-37 and by Xantheas and Dunning38,39 have identified the likely global minima for many of the small water clusters required in reaction 2. We have also made extensive use of the structures reported by these authors, although it was necessary to reoptimize all of the clusters at the same levels used for the ion-water complexes. In addition, single-point energy calculations were performed to make it possible to compute the enthalpy and free energy changes for reaction 2 at a consistent level. Of course in a comprehensive treatment of hydration effects an appropriate statistical average over many low-energy conformers would be required. Computational Methods Ab initio molecular orbital calculations were performed at the restricted Hartree-Fock (RHF) and second-order MøllerPlesset (MP2) perturbation levels using the GAUSSIAN 90 and 92 series of programs with the internally stored 6-31G*, 6-311++G**, and 6-311++G(2D,P) basis sets.40-45 The calculations were carried out on the CRAY Y-MP computer at the Advanced Scientific Computing Laboratory at the National Cancer Institute, Frederick, MD, and several Silicon Graphics and DEC Alpha computers located in Philadelphia. Some of the smaller calculations were carried out using the program Spartan46 on a Silicon Graphics computer. RHF/6-31G* optimizations were performed in all instances, and MP2(FULL)/ 6-31G* level optimizations were carried out for many of the smaller complexes to assess the effects of electron correlation on the computed structures. No symmetry constraints were imposed during the optimizations, although in some instances the resulting structure showed various elements of symmetry. Frequency analyses were used to confirm that all the structures presented in this paper are local minima at the RHF/6-31G*// RHF/6-31G* computational level and to estimate the thermal and entropic contributions required to obtain reaction enthalpies and free energies at 298 K.47-50 The energetics for all hydration reactions were calculated at the MP2(FULL)/6-311++G**// RHF/6-31G* level, which includes polarization and diffuse functions on all atoms, to reduce the effects of basis set superposition errors (BSSEs).51-54 MP2(FULL)/6-311++G**/ /MP2(FULL)/6-31G*, MP2(FULL)/6-311++G(2D,P)//MP2(FULL)/6-31G*, and MP4SDQ(FC)/6-31G*//RHF/6-31G* level calculations are also reported for many of the reactions. The computational levels and basis sets used in this study represent

J. Phys. Chem., Vol. 100, No. 9, 1996 3489 a compromise between accuracy and the requirement to include a significant number of water molecules around the beryllium and magnesium ions. Results and Discussion The RHF/6-31G*//RHF/6-31G* and MP2(FULL)/6-31G*// MP2(FULL)/6-31G* optimized geometries of the water clusters [H2O]p (p ) 2, 4, 6, 8) used in the hydration reactions 1 are shown as structures 1-4 in Figure 1, and the total molecular energies, thermal corrections, and entropies are given in Table 1A. In general, the computed geometries of these water clusters are in good agreement with those of Jordan and co-workers34-37 and with those of Xantheas and Dunning.38,39 For comparison, values of ∆H298 for the formation of the water dimer, 1, are calculated to be -5.0, -4.1, and -3.7 kcal/mol at the MP4SDQ(FC)/6-31G*//RHF/6-31G*, MP2(FULL)/6-311++G**//RHF/ 6-31G*, and MP2(FULL)/6-311++G**//MP2(FULL)/6-31G* computational levels, in excellent agreement with the experimental value of -3.6 kcal/mol,55 particularly if diffuse functions are included in the basis set. The RHF/6-31G*//RHF/6-31G* and, where possible, the MP2(FULL)/6-31G**//MP2(FULL)/6-31G* optimized geometries of the divalent beryllium ion-water complexes, Be2+‚nH2O (n ) 2, 4, 6) and Be2+‚4H2O‚mH2O (m ) 2, 4), are shown as structures 5-12 in Figure 2, and the divalent magnesium ionwater complexes, Mg2+‚nH2O (n ) 2, 4, 6) and Mg2+‚nH2O‚ 2H2O (n ) 4, 6), are shown as structures 13-18 in Figure 3.31,32 Total molecular energies at a variety of computational levels and thermal corrections and entropies at the RHF/6-31G*//RHF/ 6-31G* level for all the complexes in Figures 2 and 3 are listed in parts B and C of Table 1, respectively. As the number of water molecules in the complexes increases, the number of possible local minima on the potential energy surface is expected to increase significantly. Consequently, it was necessary to be selective rather than comprehensive in considering various conformers of the larger complexes. In several cases, however, conformations were considered where outer-shell water molecules are hydrogen bonded to either one or two water molecules in the inner shell. Complete geometries of all the structures in Figures 1-3 are available in Table 1S of the supporting information. Vibrational frequency analyses at the RHF/631G*//RHF/6-31G* level show that all the structures in Figures 1-3 are local minima on their respective potential energy surfaces at this computational level; the calculated vibrational frequencies are given in Table 2S of the supporting information. Of course, we cannot be certain that the global minimum has been found for each complex. Calculated values of ∆H298 and ∆G298 for the hydration reaction 2 involving divalent beryllium and magnesium ions surrounded by up to eight water molecules are given in Table 2. A. Hydrated Divalent Beryllium Ions, Be2+‚nH2O‚mH2O. Geometrical parameters from RHF/6-31G* and, in some cases, MP2(FULL)/6-31G* optimizations of selected conformers of divalent beryllium ions hydrated by up to eight water molecules partitioned between the first and second hydration shells are shown in Figure 2. The calculated structures of these complexes for up to six water molecules surrounding Be2+ are in good agreement with the RHF/3-21G//RHF/3-21G level calculations reported originally by Hashimoto et al.28 The total molecular energies, thermal corrections, and entropies are listed in Table 1B. In the case of Be2+‚2H2O, an MP2(FULL)/6-311++G** optimization was carried out to determine the effects on the structural parameters of including polarization functions on the hydrogen atoms and diffuse functions on all the atoms. The Be-O and H-O distances are found to be 1.526 and 0.983 Å,

3490 J. Phys. Chem., Vol. 100, No. 9, 1996

Markham et al.

Figure 1. RHF/6-31G*//RHF/6-31G* and MP2(FULL)/6-31G*//MP2(FULL)/6-31G* optimized structures of the water clusters [H2O]p (p ) 2, 4, 6, 8). Values from the MP2 geometry optimizations are given in square brackets. Bond lengths are in angstrom units, and angles are in degrees. Oxygen atoms are shown in black and hydrogen atoms are in white.

respectively, and the HOH bond angle is 108.2°, in good agreement with the corresponding parameters from the RHF/ 6-31G* and MP2(FULL)/6-31G* optimizations. It should also be noted that Be2+‚2H2O has D2d symmetry, in agreement with the calculations of Hashimoto et al.28 Previous calculations have shown that the potential energy surfaces of divalent beryllium ions surrounded by four, five, and six water molecules have many local minima with different arrangements of the surrounding water molecules.31 In each of these arrangements, however, the lowest energy configuration was found to have four water molecules in the first coordination shell with any remaining water molecules in a second shell hydrogen bonded to waters in the first shell. For example, at the MP2(FC)/6-31G*//RHF/6-31G* level, conformers 7 and 8 of Be2+‚4H2O‚2H2O in Figure 2 are calculated to be 26.4 and 21.5 kcal/mol, respectively, lower in energy than Be2+‚6H2O, 9, at 298 K. If diffuse functions are included, 7 and 8 are still found to be lower in energy than 9, but the differences are somewhat smaller, 21.7 and 18.8 kcal/mol at the MP2(FULL)/ 6-311++G**//RHF/6-31G* level. A molecular dynamics simulation of a 1.1 m aqueous solution of BeCl2 by Probst et al.56 also supports a first hydration shell containing four water molecules, in agreement with diffraction measurements.57 Furthermore, a search of the Cambridge Structural Database (CSD) (1992 version)58 showed no crystal structures in which Be2+ is surrounded by more than four oxygen atoms in the primary shell.31

The two conformers 7 and 8 of Be2+‚4H2O‚2H2O, shown in Figure 2, primarily differ in how the water molecules in the second shell bind to the water molecules in the first shell. Complex 7 has both water molecules in the outer shell hydrogen bonded to two water molecules in the inner shell, forming two identical six-membered rings; the H‚‚‚O distances are 1.872 Å and the O-H‚‚‚O bond angles are 149.4° at the RHF/6-31G*// RHF/6-31G* level. Complex 8 has only a single water molecule in the outer shell hydrogen bonded to a pair of water molecules in the inner shell, forming a six-membered ring. The second outer-shell water in this conformer is hydrogen bonded to only one water in the inner shell; the H‚‚‚O distance for this bond is comparatively short, 1.650 Å, and the O-H‚‚‚O bond angle, 171.7°, is nearly linear. We refer to water molecules in the second shell, which are hydrogen bonded to only one water molecule in the first shell, as “dangling” waters.31 At 298 K, complex 7 is calculated to be 2.9 kcal/mol lower in energy than complex 8 at the MP2(FULL)/6-311++G**//RHF/6-31G* level. Several attempts to find a conformer of Be2+‚4H2O‚2H2O in which both outer-shell water molecules were “dangling” failed to yield a local minimum with this type of configuration. It may be noted that the “dangling” water molecule in 8 is bonded to an inner-shell water molecule that also participates in a hydrogen bond with another water molecule in the outer shell; see Figure 2. Apparently once a water molecule in the second shell binds to two water molecules in the inner shell, the innershell water framework in the neighborhood of the inter-

Hydration Energies of Be2+ and Mg2+

J. Phys. Chem., Vol. 100, No. 9, 1996 3491

TABLE 1: Total Molecular Energies (hartrees), Thermal Corrections (hartrees), and Entropies (cal/(mol‚K)) of the Water Clusters [H2O]n, Hydrated Beryllium Ions, Be2+‚nH2O‚mH2O, and Hydrated Magnesium Ions, Mg2+‚nH2O‚mH2O A. Water Clustersa [H2O]n (n ) 2, 4, 6, 8) level/structure

[H2O]2 1

[H2O]4 2

[H2O]6b 3

MP2(FC)/6-31G*//RHF/6-31G* MP4SDQ(FC)/6-31G*//RHF/6-31G* MP2(FULL)/6-311++G**//RHF/6-31G* MP2(FULL)/6-31G*//MP2(FULL)/6-31G* MP2(FULL)/6-311++G**//MP2(FULL)/6-31G* MP2(FULL)/6-311++G(2D,P)//MP2(FULL)/6-31G* sum of thermal energies (hartrees)d entropy (cal/(mol‚K))e

-152.403 318 -152.420 009 -152.596 611 -152.410 273 -152.596 238 -152.630 948 0.055 347 70.5

-304.844 800 -304.875 860 -305.223 532 -304.860 495 -305.223 239 -305.292 602 0.116 088 94.4

-457.282 279 -457.328 207 -457.843 493 -457.306 616 -457.843 886 -457.948 268 0.176 717 116.8

[H2O]8c 4

-610.477 063 -609.764 460 -610.478 168 0.237 826 136.2

B. Hydrated Beryllium Ions Be2+‚nH2O‚mH2O level/structure

Be2+

Be2+‚2H2O 5

Be2+‚4H2O 6

Be2+‚6H2O 9

MP2(FC)/6-31G*//RHF/6-31G*

-13.609 800 -166.445 035 -319.068 215 -471.530 558

MP4SDQ(FC)/6-31G*//RHF/6-31G*

-13.609 800 -166.460 953 -319.099 995 -471.576 939

MP2(FULL)/6-311++G**//RHF/6-31G*

-13.624 076 -166.630 380 -319.428 261 -472.080 161

MP2(FULL)/6-31G*//MP2(FULL)/6-31G* -13.612 738 -166.456 534 -319.087 416 -471.556 921 MP2(FULL)/6-311++G**//MP2(FULL)/6-31G* -13.624 076 -166.630 044 -319.429 364 -472.081 129 MP2(FULL)/6-311++G(2D,P)//MP2(FULL)/6-31G* -13.624 081 -166.666 819 -319.500 512 0.001 415 0.059 756 0.120 621 0.179 364 sum of thermal energies (hartrees)d entropy (cal/(mol‚K))e

32.5

65.4

89.9

107.4

Be2+‚4H2O‚2H2O Be2+‚4H2O‚4H2O 7, 8 10, 11, 12 -471.574 648f -471.566 282g -471.620 390f -471.612 241g -472.116 811f -472.111 575g -471.602 284f -472.117 834g 0.181 357f 0.180 787g 112.4f 118.0g

-624.780 731h -624.780 700i -624.780 055j

0.240 441h 0.240 413i 0.242 863j 151.2h 149.5i 134.1j

C. Hydrated Magnesium Ions Mg2+‚nH2O‚mH2O level/structure

Mg2+

Mg2+‚2H2O 13

Mg2+‚4H2O 14

Mg2+‚6H2O Mg2+‚4H2O‚2H2O Mg2+‚6H2O‚2H2O 17 15, 16 18

MP2(FC)/6-31G*//RHF/6-31G*

-198.812 105 -351.470 238 -504.055 304 -656.565 895

MP4SDQ(FC)/6-31G*//RHF/6-31G* MP2(FULL)/6-311++G**//RHF/6-31G*

-198.812 105 -351.485 263 -504.086 455 -656.613 000 -198.948 697 -351.776 698 -504.535 789 -657.233 394

MP2(FULL)/6-31G*//MP2(FULL)/6-31G* -198.816 753 -351.483 096 -504.077 394 -656.597 138 MP2(FULL)/6-311++G**//MP2(FULL)/6-31G* -198.948 697 -351.776 556 -504.535 662 -657.233 747 MP2(FULL)/6-311++G(2D,P)//MP2(FULL)/6-31G* -198.949 797 -351.814 299 -504.608 504 -657.336 999 0.001 415 0.058 955 0.118 378 0.177 975 sum of thermal energies (hartrees)d entropy (cal/(mol‚K))e

35.5

74.5

103.4

115.9

-656.552 029k -656.541 860l -657.213 608k -657.206 381l

-809.894 334

0.179 295k 0.178 402l 121.1k 130.5l

0.238 461 147.6

a For comparison, the total molecular energy of H2O at the computational levels used in the text are as follows: (RHF/6-31G* geometry), MP2(FC)/6-31G* ) -76.195 955; MP4SDQ(FC)/6-31G* ) -76.204 584; MP2(FULL)/6-311++G** ) -76.293 625; (MP2(FULL)/6-31G* geometry) MP2(FULL)/6-31G* ) -76.199 244; MP2(FULL)/6-311++G** ) -76.293 763; MP2(FULL)/6-311++G(2D,P) ) -76.311 027. b Prism.34-36 c D2d symmetry.34-36 d Calculated at the RHF/6-31G*//RHF/6-31G* computational level. e Calculated at the RHF/6-31G*//RHF/631G* computational level. f Structure 7 in Figure 2. g Structure 8 in Figure 2. h Structure 10 in Figure 2. i Structure 11 in Figure 2. j Structure 12 in Figure 2. k Structure 15 in Figure 3. l Structure 16 in Figure 3.

action can become rigid enough to support a second-shell water that interacts with only one water in the first shell. Thus, there appears to be a slight energetic preference for conformers of Be2+‚4H2O‚2H2O in which both water molecules in the second shell are hydrogen bonded to two water molecules in the first shell. As might be expected, the calculated entropy of complex 8 is somewhat greater than that of complex 7; see Table 1B. Nevertheless, the value of ∆G298 for the formation of 7 from Be2+ and [H2O]6 remains more negative than that of 8, although the difference in the ∆G298 values is only 1.3 kcal/ mol at the MP2(FULL)/6-311++G**//RHF/6-31G* level; see Table 2A. Comparing the structure of Be2+‚4H2O with the two conformers of Be2+‚4H2O‚2H2O (see Figure 2) shows that the presence of water molecules in the second shell results in a decrease in the Be-O distances of those water molecules in the first shell that are hydrogen bonded to water molecules in

the second shell. For example, the Be-O distances decrease from 1.654 Å in 6 to 1.645 Å in 7, and the Be-O distance of the water involved in two hydrogen bonds in 8 is only 1.605 Å at the RHF/6-31G*//RHF/6-31G* level. Water molecules in a second shell have been observed to have a similar effect in water clusters around carboxylate groups in the sulfide CH3SCH2CO2and the sulfonium ion (CH3)2S+CH2CO2-, as well as around the sulfur atom in (CH3)3S+.59-61 Marcus et al.26 have studied the nuclear relaxation of the complexes Be2+‚4H2O and Be2+‚6H2O that is induced by a solvent reaction field; the effect of bulk water was studied at the RHF/3-21G*//RHF/3-21G* computational level by embedding these complexes in a spherical cavity surrounded by a reaction field with the dielectric constant of water.62-64 These authors find that the Be-O distances of the water molecules in the first hydration shell lengthen ∼0.02 Å, contrary to the effect produced by including a few discrete water molecules in the second shell.

Figure 2. RHF/6-31G*//RHF/6-31G* and MP2(FULL)/6-31G*//MP2(FULL)/6-31G* optimized structures of the beryllium-water complexes Be2+‚nH2O (n ) 2, 4, 6) and Be2+‚4H2O‚mH2O (m ) 2,4). Values from the MP2 geometry optimizations are given in square brackets. Bond lengths are in angstrom units, and angles are in degrees. Oxygen atoms are shown in black, hydrogen atoms are in white, and the central Be2+ cations, which are also shown in black, have been enlarged for clarity.

3492 J. Phys. Chem., Vol. 100, No. 9, 1996 Markham et al.

Figure 3. RHF/6-31G*//RHF/6-31G* and MP2(FULL)/6-31G*//MP2(FULL)/6-31G* optimized structures of the magnesium-water complexes Mg2+‚nH2O (n ) 2, 4, 6) and Mg2+‚nH2O‚2H2O (n ) 4, 6). Values from the MP2 geometry optimizations are given in square brackets. Bond lengths are in angstrom units, and angles are in degrees. Oxygen atoms are shown in black, hydrogen atoms are in white, and the central Mg2+ cations are in black.

Hydration Energies of Be2+ and Mg2+ J. Phys. Chem., Vol. 100, No. 9, 1996 3493

3494 J. Phys. Chem., Vol. 100, No. 9, 1996

Markham et al.

TABLE 2: Hydration Enthalpies, ∆H298 (kcal/mol), and Free Energies, ∆G298 (kcal/mol), of Divalent Beryllium and Magnesium Ions MP2(FULL)/ MP2(FULL)/ MP2(FC)/6-31G*// MP4SDQ(FC)/6-31G*// MP2(FULL)/6-311++G**// MP2(FULL)/6-31G*// 6-311++G**// 6-311++G(2D,P)// RHF/6-31G* RHF/6-31G* RHF/6-31G* MP2(FULL)/6-31G* MP2(FULL)/6-31G* MP2(FULL)/6-31G* n m 2 4 6 4

0 0 0 2

∆H298

∆G298

∆H298

-269.8 -383.7 -400.5 -426.9a -422.0b

-258.5 -372.7 -388.0 -415.9a -411.0b

-269.3 -384.1 -400.8 -426.8a -422.0b

-159.1 -250.1 -296.6 -287.0f -281.2g

-149.7 -242.1 -285.7 -277.7f -271.9g

-158.1 -250.1 -297.3

4 4

2 4 6 4

0 0 0 2

6 2

∆G298

∆H298 Be2+

A.Reaction Energies for -258.1 -255.8 -373.1 -363.0 -388.3 -384.2 -415.8a -406.0a b -411.0 -403.0b -426.3c -426.3d -424.4e

∆G298

∆H298

+ [H2O]p f -244.6 -352.0 -371.7 -395.0a -393.7b -421.1c -420.6d -414.1e

∆G298

∆H298

Be2+‚nH2O‚mH2O -270.8 -384.0 -399.9 -427.1

(p ) m + n) -259.6 -255.8 -373.0 -363.9 -387.4 -384.6 -416.1 -406.1a

B. Reaction Energies for Mg2+ + [H2O]l f Mg2+‚nH2O‚mH2O (l ) n + m) -148.7 -144.4 -135.0 -159.9 -150.5 -144.6 -242.2 -228.2 -220.3 -251.0 -243.2 -228.3 -286.5 -277.6 -266.7 -298.0 -287.1 -277.5 -264.3f -255.0f -260.3g -253.8g -295.1 -290.9

∆G298

∆H298

∆G298

-244.6 -352.9 -372.1 -395.4a

-257.1 -365.0

-245.9 -354.0

-135.2 -220.4 -266.7

-145.8 -229.8 -276.1

-136.4 -221.9 -265.3

C. Experimental Solvation Energies (kcal/mol) ion

∆G298

Be2+ Mg2+

-575.0 -439.7

a Structure 7 in Figure 2. b Structure 8 in Figure 2. c Structure 10 in Figure 2. d Structure 11 in Figure 2. e Structure 12 in Figure 2. f Structure 15 in Figure 3. g Structure 16 in Figure 3.

In searching for local minima with Be2+ surrounded by eight water molecules, only conformations with four water molecules in the first shell were considered; the remaining four water molecules were placed in the second shell. The second shell could accommodate up to eight water molecules if each of the waters interacts with only a single hydrogen atom of the four water molecules in the first shell. Computational limitations, however, restricted consideration to a maximum of four waters in the second shell. The first eight-water complex studied was the C2V conformer first obtained by Lee et al.;4 see structure 12 in Figure 2. Each of the four water molecules in the second shell forms hydrogen bonds with two waters from the first shell, effectively completing the second shell with the smallest possible number of water molecules; the H‚‚‚O distances are 1.934 Å, more than 0.06 Å longer than in conformer 7 of Be2+‚4H2O‚ 2H2O, and the O-H‚‚‚O bond angles are 149.7° at the RHF/ 6-31G*//RHF/6-31G* level. A frequency analysis shows that this structure is indeed a local minimum at this computational level. As can be seen in Figure 2, the addition of these water molecules to the second hydration shell shortens the Be-O distances in the first shell by ∼0.006 Å at the RHF/6-31G*// RHF/6-31G* level. Two additional local minima on the potential energy surface were found and are shown as structures 10 and 11 in Figure 2. In these conformations only two of the four water molecules in the outer shell interact with two waters in the primary shell, while the remaining two water molecules are “dangling”; the H‚‚‚O bond distances for the “dangling” water molecules in conformers 10 and 11 are relatively short, 1.682 and 1.681 Å, respectively, and the O-H‚‚‚O angles are nearly linear, 178.7° and 178.4°. It should be noted that the “dangling” water molecules in both 10 and 11 are again bonded to inner-shell water molecules that participate in other hydrogen bonds; see Figure 2. These two conformers have almost the same total molecular energy, and at the MP2(FULL)/6311++G**//RHF/6-31G* level, both are found to be 1.9 kcal/ mol lower in energy than conformer 12 at 298 K, suggesting some strain in the C2V conformer. Thus, there appears to be a slight energetic preference for conformers of Be2+‚4H2O‚4H2O in which at least two water molecules in the second shell are

“dangling”. Furthermore, the calculated entropies of conformers 10 and 11 are significantly greater than that of 12 (see Table 1B), leading to a 7.0 kcal/mol difference in the values of ∆G298 for the formation of conformers 11 and 12. We did not search for conformers in which all four waters are “dangling”. There is some evidence that the coordination number of the second shell around Be2+ is six.4,65 In view of the structure of complexes 10 and 11, it may be that a conformer with six water molecules in the second shellstwo hydrogen bonded to two water molecules in the inner shell and four in “dangling” configurationssand two water molecules in a third hydration shell may be competitive in energy with a conformer where all eight water molecules are in “dangling” configurations in the second shell. The enthalpies and free energies, ∆H298 and ∆G298, for the reactions Be2+ + [H2O]p f Be2+‚nH2O‚mH2O (p ) n + m) are given in Table 2A at a variety of computational levels. It may be noted that these calculated values do not depend significantly on whether the RHF/6-31G* or MP2(FULL)/631G* geometries are employed in the computations. This is important since it is not yet practical to determine the geometries of the larger complexes at the MP2(FULL)/6-31G* level. However, the calculated hydration energies do depend more significantly on the inclusion of diffuse functions in the basis set, lowering calculated reaction enthalpies or free energies by as much as 20 kcal/mol; see Table 2A. On the other hand, including a second set of d-type polarization functions on the heavy atoms does not alter the calculated enthalpy changes or free energy changes for the smaller two- and four-water complexes by more than ∼1.3 kcal/mol. The values of ∆G298 for the reactions 1 are plotted as a function of the number of water molecules, p ) n + m, in the Be2+‚nH2O‚mH2O complexes in Figure 4 using the values from the MP2(FULL)/6-311++G**//RHF/6-31G* computational level; the lowest energy complex at each p value was used in the graph. The values of ∆G298 for the reaction Be2+ + [H2O]4 f Be2+‚4H2O, which completes the first coordination shell of Be2+, are calculated to be very similar at the MP2(FULL)/6311++G**//RHF/6-31G*, MP2(FULL)/6-311++G**//MP2-

Hydration Energies of Be2+ and Mg2+

J. Phys. Chem., Vol. 100, No. 9, 1996 3495

Figure 4. Values of ∆G298 (kcal/mol) for the reactions Be2+ + [H2O]p f Be2+‚nH2O‚mH2O (p ) n + m) plotted as a function of the total number of water molecules p in the complex. At each value of p, the lowest energy arrangement of water surrounding the beryllium was used; see Table 2. Black circles indicate all the water molecules are in the first shell; white squares indicate the water molecules are in the first and second coordination shells.

TABLE 3: Percentage of the Experimental Free Energies of Hydration (∆G298)a for Be2+ and Mg2+ Accounted for by the Reaction M2+ + [H2O]p ) M2+‚nH2O‚mH2O (p ) n + m; M ) Be, Mg) at the MP2(FULL)/6-311++G**//RHF/6-31G* Computational Level Be2+‚nH2O‚mH2Ob n

m

2 4

0 0

4 4

2 4

%

Mg2+‚nH2O‚mH2Ob n

First Hydration Shell 42.5% 2 61.2% 4 6 Second Hydration Shell 68.7% 6 73.2%

m

%

0 0 0

30.7% 50.1% 60.7%

2

66.2%

a Experimental bulk hydration free energies: Be2+ ∆G298 ) -575.0 kcal/mol; Mg2+ ∆G298 ) -439.7 kcal/mol.21 b Lowest energy conformers are used throughout for the percentage calculation.

(FULL)/6-31G*, and MP2(FULL)/6-311++G(2D,P)//MP2(FULL)/6-31G* levels, -352.0, -352.9, and -354.0 kcal/mol, respectively, and account for approximately 61-62% of the total experimental free energy of hydration, -575.0 kcal/mol;27 see Table 3. The value of ∆G298 for the reaction Be2+ + [H2O]6 f Be2+‚4H2O‚2H2O, using complex 7, is found to be -395.0 kcal/mol at the MP2(FULL)/6-311++G**//RHF/6-31G* level. Thus, the first two water molecules in the second shell account for approximately 19.3% of the hydration free energy not already taken into account by Be2+‚4H2O, and the six water molecules account for about 68.7% of the experimental free hydration energy. Finally, the value of ∆G298 for the reaction Be2+ + [H2O]8 f Be2+‚4H2O‚4H2O is -421.1 kcal/mol for complex 10 at the MP2(FULL)/6-311++G**//RHF/6-31G* level. The second pair of water molecules in the outer shell provides only about 14.5% of the residual free energy not accounted for by Be2+‚4H2O‚2H2O, and the eight waters account for approximately 73.2% of the experimental free energy. While it was not possible to perform the calculations with more than a total of eight water molecules, it seems likely that completing the first and second hydration shells surrounding

Be2+, with four waters in the primary shell and with either six or eight waters in the second shell, will account for more than 75% of the total hydration free energy. However, since each successive pair of water molecules accounts for a smaller percentage of the residual free energy, and the second pair of water molecules in the second shell already accounts for less than 15% of this residual energy, it is clear that a large number of water molecules would be required to approach the experimental value of the free energy of hydration for divalent beryllium ions. B. Hydrated Divalent Magnesium Ions, Mg2+‚nH2O‚ mH2O. Geometrical parameters from RHF/6-31G* and, in some cases, MP2(FULL)/6-31G* optimizations of selected conformers of divalent magnesium ions hydrated with up to eight water molecules partitioned between the first and second hydration shells are shown in Figure 3, and the total molecular energies, thermal corrections, and entropies are given in Table 1C. Comparing the magnesium complexes with the corresponding beryllium complexes shows that Mg-O distances are significantly longer than Be-O distances, in accord with the much larger ionic radius of divalent magnesium;66 the H‚‚‚O bond lengths are also longer in the magnesium compounds; compare Figures 2 and 3. For Mg2+‚2H2O the O-Mg-O configuration is found to be linear, in agreement with the calculations of Bauschlicher et al.33 Previous calculations have shown that the potential energy surface of a divalent magnesium cation surrounded by six water molecules has several local minima with different arrangements of the surrounding water molecules.32 The lowest energy arrangement, however, was found to have all six water molecules in the inner shell. For example, at the MP2(FC)/631G**//RHF/6-31G* computational level, conformers 15 and 16 of Mg2+‚4H2O‚2H2O in Figure 3 are 9.5 and 15.4 kcal/mol higher in energy at 298 K than Mg2+‚6H2O, 17. If diffuse functions are included in the basis set, 15 and 16 remain higher in energy than 17, and the energy separation is increased to 13.2 and 17.3 kcal/mol, respectively, at the MP2(FULL)/6311++G**//RHF/6-31G* level. A molecular dynamics simulation of a 1.1 m aqueous solution of MgCl2 by Dietz et al.67 also supports a first hydration shell containing six water molecules. Although a search of the CSD58 found crystal structures that contain divalent magnesium ions with coordination numbers from three to eight, the largest number of entries had magnesium with coordination number six.32 Furthermore, all the structures in which the magnesium ion was surrounded solely by water molecules were hexahydrates, with all six waters in the primary shell. Conformer 15 of Mg2+‚4H2O‚2H2O has both water molecules in the outer shell hydrogen bonded to two waters in the inner shell and resembles the corresponding beryllium structure, 7. At 298 K, this conformer of Mg2+‚4H2O‚2H2O is calculated to be 4.0 kcal/mol lower in energy than conformer 16 at the MP2(FULL)/6-311++G**//RHF/6-31G* level. Conformer 16, which has one “dangling” water molecule with a short H‚‚‚O distance of 1.693 Å and a nearly linear O-H‚‚‚O bond angle, resembles the beryllium structure 8. It may be noted that in conformer 16 the “dangling” water molecule is again bound to an innershell water molecule which is part of a second hydrogen bond; see Figure 3. Comparing the structures of Mg2+‚4H2O, 14, and Mg2+‚4H2O‚2H2O, 15 and 16, shows that water molecules in the second shell surrounding divalent magnesium ions tighten up the first shell; for example, the Mg-O distance is reduced from 2.011 Å in 14 to 1.995 Å in 16 at the RHF/6-31G*//RHF/ 6-31G* level, similar to the situation observed for divalent beryllium ions.

3496 J. Phys. Chem., Vol. 100, No. 9, 1996

Figure 5. Values of ∆G298 (kcal/mol) for the reactions Mg2+ + [H2O]p f Mg2+‚nH2O‚mH2O (p ) n + m) plotted as a function of the total number of water molecules p in the complex. At each value of p, the lowest energy arrangement of water surrounding the magnesium was used; see Table 2. Black circles indicate all the water molecules are in the first shell; the white square indicates the water molecules are in the first and second coordination shells.

Only one complex with eight water molecules was considered; see structure 18 in Figure 3. (An attempt to find a local minimum on the potential energy surface with eight waters in the first coordination shell around Mg2+ failed; two water molecules migrated into a second coordination shell during the optimization.) In conformer 18, both water molecules in the outer shell are hydrogen bonded to two water molecules in the inner shell, and the remaining two inner-shell waters do not participate in any hydrogen bonds. The six-membered rings in structures 15 and 18 are quite similar, although the Mg-O and H‚‚‚O distances are slightly longer in 18. As can be seen in Figure 3, the water molecules in the second shell reduce the Mg-O bond distances associated with the water molecules in the first shell which are involved in hydrogen bonds, similar to what is observed in the case of beryllium ions. Marcus et al.26 also studied the nuclear relaxation of Mg2+‚6H2O induced by the reaction field of bulk water using a cavity model. They found that the Mg-O distances of the water molecules in the first hydration shell lengthen more than 0.03 Å contrary to the effect produced by including a few discrete water molecules in the second shell. When calculations on larger complexes become possible, it will be interesting to see how water molecules in the third and higher hydration spheres change the Be-O and Mg-O distances of waters in the first and second hydration spheres. The enthalpies and free energies, ∆H298 and ∆G298, for the reactions Mg2+ + [H2O]p f Mg2+‚nH2O‚mH2O (p ) n + m) are given in Table 2B at a variety of computational levels. Where comparisons are possible, it is clear that the calculated enthalpies and free energies do not depend significantly on whether the RHF/6-31G* or MP2(FULL)/6-31G* geometries are used or whether a second set of d-functions is employed on the heavy atoms. However, the calculated enthalpies and free energies do depend on the inclusion of diffuse functions. In Figure 5, the values of ∆G298 are plotted as a function of the number of water molecules, p ) n + m, in Mg2+‚nH2O‚mH2O at the MP2(FULL)/6-311++G**//RHF/6-31G* level; the lowest energy complex at each value of p was used in the graph. The values of ∆G298 for the reaction Mg2+ + [H2O]6 f Mg2+‚6H2O,

Markham et al. which completes the first coordination shell of Mg2+, are calculated to be very similar at the MP2(FULL)/6-311++G**// RHF/6-31G*, MP2(FULL)/6-311++G**//MP2(FULL)/6-31G*, and MP2(FULL)/6-311++G(2D,P)//MP2(FULL)/6-31G* levels, -266.7, -266.7, and -265.3 kcal/mol, respectively, and account for 60-61% of the experimental free energy of hydration for Mg2+, -439.7 kcal/mol;27 see Table 3. Thus, although the first hydration shells of divalent beryllium and magnesium ions contain different numbers of water molecules, this shell accounts for nearly the same fraction of the total hydration energy for both ions. Including two additional water molecules in the second coordination shell gives a value of ∆G298 for complex 18 at the MP2(FULL)/6-311++G**//RHF/ 6-31G* level of -290.9 kcal/mol. This first pair of water molecules in the second shell accounts for only about 14.0% of the residual free energy not taken into account by the first hydration sphere; the eight water molecules account for about 66.2% of the experimental free hydration energy. It is again clear that a much larger number of water molecules would be required to approach the experimental value of the free energy of hydration. Conclusions The reactions M2+ + [H2O]p f M2+‚nH2O‚mH2O, where M ) Be and Mg, have been studied using ab initio molecular orbital calculations to determine the contributions of water molecules in the first and second hydration shells to the overall free hydration energy. These reactions naturally include the competitive effects of ion-water and water-water interactions inherent in the hydration process. The value of ∆G298 for the reaction Be2+ + [H2O]4 f Be2+‚4H2O, which completes the first hydration shell of Be2+, is calculated to be -352.0 kcal/ mol at the MP2(FULL)/6-311++G**//RHF/6-31G* computational level; comparable values are found at other computational levels as long as diffuse functions are included in the basis set. Similarly, the value of ∆G298 for the reaction Mg2+ + [H2O]6 f Mg2+‚6H2O, which completes the first hydration shell of Mg2+, is calculated to be -266.7 kcal/mol at the same computational level, some 85 kcal/mol less negative than for the completion of the first shell surrounding beryllium, even though two additional water molecules are involved. Despite these differences, the inner hydration shells surrounding divalent beryllium and magnesium cations account for approximately 61% of their respective experimental free energies of hydration, -575.0 and -439.7 kcal/mol. Reactions that incorporate two additional water molecules into a second hydration shell account for only 19.3% and 14.0% of the residual free energies not accounted for by the first shells. Incorporating a second pair of water molecules in the second hydration sphere around Be2+ picks up only 14.5% of the residual free energy not accounted for by Be2+‚4H2O‚2H2O, suggesting that successive pairs of water molecules account for a progressively smaller percentage of the remaining free energy. The calculated values of ∆G298 for the formation of the complexes Be2+‚4H2O‚4H2O and Mg2+‚6H2O‚2H2O from eight-water clusters already account for a large portion of the overall free hydration energies 73.2% and 66.2%, respectively, but convergence toward the experimental values will be quite slow as additional water molecules are added to the outer hydration spheres. Acknowledgment. We thank Dr. Ken Jordan for providing initial, x, y, and z coordinates for the prism form of the water hexamer and the low-energy D2d form of the water octamer. We also thank the Advanced Scientific Computing Laboratory, NCI-FCRF, for providing time on the CRAY Y-MP supercom-

Hydration Energies of Be2+ and Mg2+ puter. This work was supported by Grants CA-10925, CA06927, and GM-31186 from the National Institutes of Health, by an appropriation from the Commonwealth of Pennsylvania, and by an Undergraduate Summer Fellowship Program from the Bristol Meyers Oncology Division (to C.L.B.). The contents of this paper are solely the responsibility of the authors and do not necessarily represent the official views of the National Cancer Institute, or any other sponsoring organization. Supporting Information Available: Complete geometries of all structures in Figures 1-3 and calculated vibrational frequencies (22 pages). Ordering information is given on any current masthead page. References and Notes (1) Ladanyi, B. M.; Skaf, M. S. Annu. ReV. Phys. Chem. 1993, 44, 335. Leikin, S.; Parsegian, V. A.; Rau, D. C. Annu. ReV. Phys. Chem. 1993, 44, 369. (2) Rose, G. D. Annu. ReV. Biophys. Biomol. Struct. 1993, 22, 381. Teeter, M. M. Annu. ReV. Biophys. Biomol. Struct. 1993, 22, 577. (3) Ben-Naim, A. Curr. Opin. Struct. Biol. 1994, 4, 264. (4) Lee, M. A.; Winder, N. W.; Casey, W. H. J. Phys. Chem. 1994, 98, 8641. (5) Sanekata, M.; Misaizu, F.; Fuke, K.; Iwata, S.; Hashimoto, K. J. Am. Chem. Soc. 1995, 117, 747. (6) Feller, D.; Glendening, E. D.; Kendall, R. A.; Peterson, K. A. J. Chem. Phys. 1994, 100, 4981. (7) Hertel, I. V.; Huglin, C.; Nitsch, C.; Schultz, C. P. Phys. ReV. Lett. 1991, 67, 1767. (8) Barnett, R. N.; Landman, U. Phys. ReV. Lett. 1993, 70, 1775. (9) Kebarle, P. Annu. ReV. Phys. Chem. 1977, 28, 445. (10) Keesee, R. G.; Castleman, A. W., Jr. J. Phys. Chem. Ref. Data 1986, 15, 1011. (11) Marcus, Y. Ion SolVation; Wiley: Chichester, 1986. (12) Tunon, I.; Sila, E.; Bertran, J. J. Phys. Chem. 1993, 97, 5547. (13) Cieplak, P.; Kollman, P. A. J. Chem. Phys. 1990, 92, 6761. (14) Dann, L. X.; Rice, J. E.; Caldwell, J.; Kollman, P. A. J. Am. Chem. Soc. 1991, 113, 2481. (15) Kirkwood, J. G. J. Chem. Phys. 1934, 2, 351. (16) Onsager, L. J. Am. Chem. Soc. 1936, 58, 1486. (17) Miertus, S.; Scrocco, E.; Tomasi, J. Chem. Phys. 1981, 55, 117. (18) Cramer, C. J.; Truhlar, D. G. Science 1992, 256, 213. (19) Cramer, C. J.; Truhlar, D. G. J. Am. Chem. Soc. 1991, 113, 8305. (20) Wong, M. W.; Frisch, M. J.; Wiberg, K. B. J. Am. Chem. Soc. 1991, 113, 4776. (21) Honig, B.; Sharp, K. A.; Yang, A.-S. J. Phys. Chem. 1993, 97, 1101. (22) Tannor, D. J.; Marten, B.; Murphy, R.; Friesner, R. A.; Sitkoff, D.; Nicholls, A.; Ringnalda, M.; Goddard, W. A., III; Honig, B. J. Am. Chem. Soc. 1994, 116, 11875. (23) Klapper, I.; Hagstrom, R.; Fine, R.; Sharp, K.; Honig, B. Proteins: Struct., Funct., Genet. 1986, 1, 47. (24) Grant, J. A.; Williams, R. L.; Scheraga, H. A. Biopolymers 1990, 30, 929. (25) Bajorath, J.; Kraut, J.; Li, Z.; Kitson, D. H.; Hagler, A. T. Proc. Natl. Acad. Sci. U.S.A. 1991, 88, 6423. (26) Marcos, E. S.; Pappalardo, R. R.; Rinaldi, D. J. Phys. Chem. 1991, 95, 8928. (27) Noyes, R. M. J. Am. Chem. Soc. 1962, 84, 513. Noyes, R. M. J. Am. Chem. Soc. 1964, 86, 971. Gomer, R.; Tryson, G. J. Chem. Phys. 1977, 66, 4413. Madden, W. G.; Gomer, R.; Mandell, M. J. J. Phys. Chem. 1977, 81, 2652. (28) Hashimoto, K.; Yoda, N.; Iwata, S. Chem. Phys. 1987, 116, 193. (29) Hashimoto, K.; Iwata, S. J. Phys. Chem. 1989, 93, 2165. (30) Hashimoto, K.; Yoda, N.; Osamura, Y.; Iwata, S. J. Am. Chem. Soc. 1990, 112, 7189.

J. Phys. Chem., Vol. 100, No. 9, 1996 3497 (31) Bock, C. W.; Glusker, J. P. Inorg. Chem. 1993, 32, 1242. (32) Bock, C. W.; Kaufman, A.; Glusker, J. P. Inorg. Chem. 1994, 33, 419. (33) Bauschlicher, C. W.; Sodupe, M.; Partridge, H. J. Chem. Phys. 1992, 96, 4453. (34) Tsai, C. J.; Jordan, K. D. J. Chem. Phys. 1991, 95, 3850. (35) Tsai, C. J.; Jordan, K. D. J. Phys. Chem. 1993, 97, 5208. (36) Tsai, C. J.; Jordan, K. D. Chem. Phys. Lett. 1993, 213, 181. (37) Kim, K.; Jordan, D. D.; Zwier, T. S. J. Am. Chem. Soc. 1994, 116, 11558. (38) Xantheas, S. S.; Dunning, T. H. J. Chem. Phys. 1993, 98, 8037. (39) Xantheas, S. S.; Dunning, T. H. J. Chem. Phys. 1993, 99, 8774. (40) Møller, C.; Plesset, M. S. Phys. ReV. 1934, 46, 618. (41) Frisch, M. J.; Head-Gordon, M.; Trucks, G. W.; Foresman, J. B.; Schlegel, H. B.; Raghavachari, K.; Robb, M.; Binkley, J. S.; Gonzalez, C.; Defrees, D. J.; Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Topiol, S.; Pople, J. A. GAUSSIAN 90, Revision F.; Gaussian Inc.: Pittsburgh, PA, 1990. (42) Frisch, M. J.; Trucks, G. W.; Head-Gordon, M.; Gill, P. M.-W.; Wong, M. W.; Foresman, J. B.; Johnson, B. G.; Schlegel, H. B.; Robb, M.; Replogle, E. S.; Gomperts, R.; Andres, J. L.; Raghavachari, K.; Binkley, J. S.; Gonzalez, C.; Martin, R. L.; Fox, D. J.; Defrees, D. J.; Baker, J.; Stewart, J. J. P.; Pople, J. A. GAUSSIAN 92, Revision C; Gaussian Inc.: Pittsburgh, PA, 1992. (43) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973, 28, 213. (44) Clark, T.; Chandrasekhar, J.; Spitznagel, G. W.; Schleyer, P. v. R. J. Comput. Chem. 1983, 4, 294. (45) Frisch, M. J.; Pople, J. A.; Binkley, J. S. J. Chem. Phys. 1994, 80, 3265. (46) SPARTAN; Wavefunction, Inc.: 18401 Von Karman Ave., #370, Irvine, CA 92715, 1995. (47) Binkley, J. S.; Pople, J. A. Int. J. Quantum Chem. 1975, 9, 229. (48) Pople, J. A.; Binkley, J. S.; Seeger, R. Int. J. Quantum Chem. 1976, 10S, 1. (49) McIver, J. W., Jr.; Komornicki, A. J. Am. Chem. Soc. 1972, 94, 2625. (50) Pople, J. A.; Krishnan, R.; Schlegel, H. B.; Binkley, J. S. Int. J. Quantum Chem., Quantum Chem. Symp. 1979, 13, 225. (51) Kim, K. S.; Mhin, B. J.; Choi, U. S.; Lee, K. J. Chem. Phys. 1992, 97, 6649-6662. (52) Feller, D. J. Chem. Phys. 1992, 96, 6104-6114. (53) Szalewicz, K.; Cole, S. J.; Kolos, W.; Bartlett, R. J. J. Chem. Phys. 1988, 89, 3662-3673. (54) Frisch, M. J.; Del Bene, J. E.; Binkley, J. S.; Schaefer, H. F., III. J. Chem. Phys. 1986, 48, 2279-2289. (55) Curtiss, L. A.; Frurip, D. J.; Blander, M. J. Chem. Phys. 1979, 71, 2703. (56) Probst, M. M.; Spohr, E.; Heinzinger, K. Chem. Phys. Lett. 1989, 161, 405. (57) Yamaguchi, T.; Ohtaki, H.; Spohr, E.; Palinkas, G.; Heinzinger, K.; Probst, M. M. Z. Naturforsch. 1986, 41a, 1175. (58) Allen, F. H.; Bellard, S.; Brice, M. D.; Cartwright, B. A.; Doubleday, A.; Higgs, H.; Hummelink, T.; Hummelink-Peters, G. G.; Kennard, O.; Motherwell, W. D. S.; Rodgers, J. R.; Watson, D. G. Acta Crystallogr. 1979, B35, 2331-2339. (59) Markham, G. D.; Bock, C. W. J. Phys. Chem. 1993, 97, 5562. (60) Markham, G. D.; Bock, C. W. J. Phys. Chem. 1995, 99, 10188. (61) Markham, G. D.; Bock, C. W. Structural Chem., in press. (62) Rivail, J. L.; Rinaldi, D. Chem. Phys. 1976, 18, 233. (63) Rinaldi, D.; Ruiz-Lopez, M. F.; Rivail, J. L. J. Chem. Phys. 1983, 78, 834. (64) Rivail, J. L.; Terryn, B.; Rinaldi, D.; Ruiz-Lopez, M. F. J. Mol. Struct. (THEOCHEM) 1985, 120, 387. (65) Spandau, H.; Spandau, C. Z. Phys. Chem. 1943, 192, 211. (66) Cotton, F. A.; Wilkinson, G. AdVanced Inorganic Chemistry, 4th ed.; Wiley; New York, 1980; p 14. (67) Dietz, W.; Riede, W. O.; Heinzinger, K. Z. Naturforsch. 1982, 37a, 1038.

JP952531T