3060
J. Phys. Chem. 1995,99, 3060-3067
Cation-Water Interactions: The M+(H20), Clusters for Alkali Metals, M = Li, Na, K, Rb, and Cs Eric D. Glendening” and David Feller Environmental Molecular Sciences Laboratory, Pacific Northwest Laboratory, Richland, Washington 99352 Received: September 15, 1994; In Final Form: December 7, 1994@
Gas-phase binding energies and enthalpies are reported for small M+(H20), clusters consisting of an alkali metal cation (Li+, Na+, K+, Rb+, or Cs+) with one to six water molecules. Ab initio molecular orbital calculations were performed at the RHF and MP2 levels of theory with split valence basis sets (3-21G, 6-31+G* with effective core potentials for the heavier alkali metals). Comparison with higher level calculations and with experimentally measured bond dissociation energies suggests that the RHF/6-3 1+G* method provides a reasonable description of cation-water interactions in the smallest (n = 1-3) clusters. Larger clusters, particularly those that involve water-water hydrogen-bonding interactions, require a correlated treatment at the MP2 level. This study serves to calibrate the RHF/6-3 1+G* and MP2/6-3 1+G* methods for applications to cation-ligand interactions in more extended systems (e.g., the ion-selective binding of crown ethers and cryptands) for which calculations at higher levels of theory are not currently feasible.
I. Introduction Increasing interest in the aqueous solvation of metal cations has led to renewed efforts to determine accurate interaction potentials and structures for small cation-water clusters. Bond dissociation energies for gas-phase clusters of the form M+(H20), and M2+(H20), have been determined by a variety of experimental techniques.1-4 Ab initio studies of small cationwater clusters (n = 1-4) employing high levels of theory and extended basis sets yield binding energies that are in excellent agreement with In addition, calculations provide structural data that is hard to obtain experimentally. The cation-water interaction is largely electrostatic, and the clusters therefore favor geometries in which the dipole moment of each water molecule is directed toward the cation. The water molecules are generally organized around the metal cation in highly symmetric arrangements (linear, trigonal planar, tetrahedral) that minimize ligand-ligand repulsions. However, it has been observed that the clusters containing heavier metal cations (particularly those of the fifth and sixth periods of the periodic table) often favor bent or pyramidal structures, stabilized in part by core polarization of the c a t i ~ n . ~ . ’ * - l ~ Our interest in these clusters stems from the influence of cation-water interactions on the ion selectivity of the macrocyclic crown ethers. In an aqueous environment 18-crown-6, (-CH2CH@-)6, is observed to preferentially bind K+ relative to the other alkali metal cations.’5J6 We recently reported an ab initio study of this crown ether and its alkali metal c ~ m p l e x e s . Our ~ ~ calculations revealed that the crown ether favors Li+ in the absence of solvent since the Li+...O distances are shorter and the electrostatic interactions are thereby stronger than those of heavier alkali metals such as Na+ and K+. The K+ selectivity was recovered, however, when the competition of the solvent water molecules for the cation was considered using a simple metal exchange reaction based on the microsolvation of cations in small water clusters (vide infra). Ion selectivity is apparently in part the result of a delicate balance of forces experienced by the cation as the crown ether and solvent molecules compete for the cation in solution. The calculations reported in our crown ether study represent the highest levels of theory (RHF and MP2 with polarized, split @
Abstract published in Advance ACS Abstracts, February 1, 1995.
valence basis sets) applied to date on these systems. However, these levels are rather modest compared to that typically employed in ab initio studies of water clusters with alkali metal c a t i ~ n s . ~To J ~calibrate the binding energies calculated for the cation-crown ether complexes, we examine here the M+(H20), clusters at the same level employed in the crown ether study. We demonstrate that the binding energies and enthalpies calculated at even moderate levels of theory are in good accord with results from higher level calculations and compare favorably with experimentally measured values. The organization of the paper is as follows. In section I1 we present details of the calculations, including the geometries of the clusters, levels of theory employed, and basis sets. Section IIIpresents the binding energies and enthalpies and a comparison of the present values with those calculated at higher levels and with experiment. Section IV discusses the geometry of the clusters, and we conclude with a brief summary of the work in section V. 11. Methods Figure 1 displays a representative set of the M+(H20), clusters examined in this work. Each structure is identified by a label of the form n m and its point group symmetry. The value n indicates the number of primary shell water molecules directly coordinating the metal cation, and m gives the number of secondary shell water molecules, i.e. those that are separated from the cation by one intervening water. For example, we examined two configurations for M+(H20)2, one labeled 2, in which both water molecules directly coordinate the cation, and another labeled 1 1, in which only the proton donor molecule of the water dimer coordinates the cation. Note that the n rn labels alone cannot uniquely identify all cluster geometries. For instance, we optimized three distinct structures labeled 2, one “linear” structure of Du symmetry and two “bent” structures of C2 and C, symmetry. Two basis set levels were employed throughout this study. Preliminary calculations were performed with the standard 3-21G basis that are readily available for all atoms but cesium. For the latter, we developed a 3-21G-type contraction (specifically, a (18s12p6d)/[7s6p2d] contraction) of Huzinaga’s MIDI set.21 Higher level calculations were per-
+
+
0022-3654/95/2099-3060$09.00~0 0 1995 American Chemical Society
+
J. Phys. Chem., Vol. 99, No. 10, 1995 3061
Cation-Water Interactions
(H2O) clusters. The outermost (n - 1) shell of core electrons for these atoms was treated explicitly. The field of the remaining core electrons was described by the ECP. The rubidium and cesium ECPs directly incorporate the massvelocity and one-electron Darwin effects into the potentials and, hence, should approximately treat the dominant relativistic corrections that may contribute importantly to the description of these atoms. Ab initio wave functions were calculated with the GAUSSIAN 9224and GAMESSz5programs at the restricted-HartreeFock (RHF) and second-order Mprller-Plesset perturbation (MP2) levels of theory.22 Correlation involving the inner shell 1s electrons of lithium and oxygen and Is, 2s, and 2p electrons of sodium was neglected in the frozen-core MP2 treatment. All electrons of potassium, rubidium, and cesium (except those described by the ECP) were correlated in the MP2 calculations. Failure to correlate the (n - 1) shell of core electrons in the latter cases resulted in significant overestimation of the M+- -0 distances and, as might be expected, underestimation of the corresponding binding energies. Full geometry optimizations were performed at the RHF level using standard analytic gradient techniques with the “tight” gradient convergence threshold of GAUSSIAN 92 or a threshold of O.OOOO3 au for GAMESS. These low thresholds were required to converge the geometrical parameters to an acceptable level since the M+(H20), clusters are rather floppy. The geometry optimizations of the larger (n = 5 , 6) clusters were quite time-consuming, typically requiring 100 optimization steps or more. Electron correlation effects were generally examined by calculating MP2 energies at the RHF-optimized geometries. Geometry optimization at the MP2 level was, however, performed for several of the smaller clusters. Calculations were therefore performed at four levels of theory: (i) RHF/3-21G with full geometry optimization, (ii) RHF/6-31+G* with full geometry optimization, (iii) MP2/6-31+G* at the RHF/631+G*-optimized geometry, and (iv) MP2/6-31+G* with full geometry optimization. These levels are specified in the tables and figures using the standard labeling convention (energy level/ /geometry level). For example, (iii) is denoted MP2/6-31+G*/ /RHF/6-3 1+G* . Binding energies and enthalpies corresponding to the reactions M+
+ nH20 - M+(H20),
(1) were evaluated for each of the M+(H20), clusters, where the reactant fragments are infinitely separated and at their calculated equilibrium geometries. We discuss these values either in terms of the raw quantities of eq 1 or, where convenient, in terms of the incremental binding energies of the reactions 5+1(C1)
6(Sd
6P3)
Figure 1. Optimized geometries of the M+(H20), clusters.
formed H ith a hybrid basis set that we shall, for brevity, denote 6-31+G*. This set consisted of the standard 6-31+G* set for hydrogen, lithium, oxygen, and sodiumZ2and Hay and Wadt’s effective core potential (ECP) with split valence basis for potassium, rubidium, and cesium.23 The 6-31+G* basis includes six-term, d-type polarization functions and a diffuse sp shell for all atoms but hydrogen. Previous work on the Li+(H20), clusters with highly extended basis sets clearly demonstrated that diffuse functions, particularly those on oxygen, are required to accurately describe cation-water interactions.10For potassium, rubidium, and cesium, we used a (5s5p)/[3s2p] contraction of Hay and Wadt’s valence basis set23augmented by six-term, d-type polarization functions. The exponents of the polarization functions (@(K) = 0.48, @(Rb) = 0.24, and @(Cs) = 0.19) were chosen to minimize the energy of the M+-
M+(H20),-,
+ H 2 0 - Mf(H20),
(2)
The counterpoise correction (CP) of Boys and Bernardi26was applied to each binding energy to approximately account for basis set superposition error (BSSE). Enthalpy corrections at 298 K were calculated using standard expressions with the RHF vibrational frequencies scaled by the usual factor, 0.9.22-27 111. Binding Energies and Enthalpies
Tables 1 and 2 list the calculated binding energies and enthalpies for the M+(H20), clusters shown in Figure 1. Binding energies for the smaller clusters (n = 1-3) are dominated by cation- water interactions that are principally electrostatic and are known to be well described at the RHF We find that correlation at the MP2 level tends to weaken the binding interaction for clusters containing the lighter alkali metals and to strengthen the interaction for those with
3062 J. Phys. Chem., Vol. 99, No. 10, 1995
Glendening and Feller
TABLE 1: Counte oise-Corrected Binding Energies (in kcal mol-') of the M'p(HzO). Complexes at Various Levels of Theorp ~~
structure
Li
1(CZ") l+l(Cs) W2d) 2+1(C2) 3P3) 3(C3) 3+1(c2) 4(S4) 4(C4) 4+1(c2) 5(G) s(C2v) 4+2(c2) 4+2(Cs) 4+2(c2,)
-48.1 -72.5 -90.0 -108.4 -120.0 b -139.0 -139.5 b -156.1
W 3 ) 6@6)
l(C2") l+l(Cs) W2d) 2(CJ 2+1(C2) 3P3) 3(C3) 3+1(cz) 4(S4) 4(C4) 4+1(c2) 5(G) ~(CZV) 4+2(c2) 4+2(CJ 4+2(c2") 5+1(Ci) W 3 )
6@6)
Na
~~~~
K
RHF/3-21G//RHF/3-21G -31.6 -20.4 -51.9 -34.9 -60.0 -39.2 -77.5 -55.5 -82.0 -54.8 b -50.2 -100.1 -71.3 -99.6 -68.0 b -72.0 -115.5 -83.0 -88.3 -77.2 -167.3 -131.2 -169.2 -155.5 -149.2 -119.3 -92.8 -149.5 -119.4 -92.8
Rb
cs
-17.5 -33.4 -33.6 -49.3 -47.2 -45.2 -63.0 -58.9 -67.2
-15.0 -30.0 -28.9 -43.7 -40.9 -41.0 -55.7 -51.2 -62.8
-24.3 -38.9 -45.5
-80.2 -87.5 b -103.1 -104.1 b -118.5
-60.6 -63.5 b -78.0 -77.5 b -91.0
-129.6 -130.9 (- 123.9)
-103.6
-13.7 -24.3 (-25.9) -26.0 (-26.0) -37.6 -37.1 -36.3 -48.0 -46.8 -48.1
l(C2") 1+1(C,) 2(D2d)
-46.7 -69.2 -86.6 -102.6 -114.7 -131.2 -132.9 -146.9 -154.7 -157.5 -143.3 -137.9 -138.2
Na
K
RHF/3-21G//RHF/3-21G -30.5 -19.6 -48.9 -32.0 -57.2 -36.7 -72.1 -50.4 -77.5 -50.8 -44.7 -93.2 -64.7 -93.6 -62.5 -64.3 -107.1 -75.1 -79.0 -70.4 -119.1 -108.0 -108.1
-16.1 -28.2 (-30.6) -30.7 (-30.7) -44.2 -43.7 -43.6 -56.6 -55.1 -57.4
6(s6)
-14.1 -25.5 (-26.6) -26.8 (-26.8) -39.8 -38.2 -39.5 -50.6 -48.4 -53.0
UCZJ 1+1(CJ 2(D2d) 2(Cd 2(CJ 2+1(Cz) 3P3) 3(C3) 3+1(cz) 4V4) 4(C4) 4+1(c2) 5(G) ~(CZV) 4+2(c2) 4+2(C,) 4-t2(Czv) W 3 )
6(s6)
-88.1 -99.8
MP2/6-3 l+G*//MP2/6-31+G* -18.8 -24.2 -16.1 -34.4 -31.6 -38.8 -28.0 -51.4 (-35.7) -45.6 (-30.5) -64.3
-34.7 -49.0 -63.7
-24.4 -36.4 -45.2
-76.1 -85.6
-56.5 -62.5
-97.6 -100.4
-73.4 -75.8
-111.2
-85.9
-117.1 -120.2 (- 113.9)
-92.1
Rb
cs
-16.7 -30.6 -31.2 -44.4 -43.3 -39.7 -56.6 -53.6 -59.6
-14.3 -27.2 -26.7 -38.9 -37.2 -35.5 -49.5 -46.1 -55.3
-81.1 -81.1
-14.0 -25.4 (-26.5)
Binding energies evaluated relative to infinitely separated M+ and HzO fragments. Values in parentheses correspond to saddle point structures having at least one imaginary vibrational frequency. Unable to locate an equilibrium structure of this symmetry.
UCZY) l+l(Cs) 2(D2d)
-18.0 -28.2 (-33.2) -33.2 -33.2 -43.5 -46.4 -56.3 -57.5 -53.4 -66.8 -65.2 (-70.4)
-15.3 -24.7 (-28.1) -28.1 (-28.1) -38.0 -39.5 -36.8 -48.9 -49.2 -47.0
-13.2 -22.0 (-24.0) -24.1 (-24.1) -33.6 -33.8 -32.2 -42.8 -42.2 -42.1
-93.3 -73.1
W 3 )
-101.4 -122.2
Li
RHF/6-31+G*//RHF/6-31+G* l(C2") 1+1(CJ 2(D2d) 2(C2) 2(CJ 2+1(C2) 3(D3) 3(C3) 3+1(cz) 4G4) 4(C4) 4+1(Cz) S(C2) ~(CZY) 4+2(c2) 4+2(CJ 4+2(c2") 5+1(CI)
5+1(C1)
W 3 )
6@6)
-18.9 -31.7 (-35.8) -35.8 -35.8 -49.8 -50.9 b -64.1 -63.7 -63.2 -76.4 -76.9 (-73.8)
l(C2") 1+1(C,) 2(D2d) 2+1(C2) 3P3) 3(C3) 3+1(c2) 4(S4) 4(C4) 4+1(c2) S(C2) 5(czv) 4+2(c2) 4+2(C,) 4+2(c2,) W 3 )
RHF/6-3 l+G*//RHF/6-31+G* -25.1 -18.6 -15.8 -35.6 -38.8 -30.5 -27.0 -51.6 (-35.2) (-30.0) -47.3 -66.3 -35.2 -30.1 -35.2 (-30.1) -80.6 -60.7 -47.6 -42.0 -89.7 -66.1 -49.9 -42.8 b b b -40.8 -103.7 -79.1 -61.7 -54.2 -105.9 -80.8 -62.3 -53.8 b b -59.2 -52.9 -118.7 -92.9 -73.6 -72.3 (- 7 1.8) -126.9 -101.6 -129.6 (-123.3) -102.0 -82.1 -119.7 -100.2 -34.5 -51.5 -64.4
structure
6&)
MP2/6-31+G*//RHF/6-31+G* l(C2") 1+1(C,) 2(D2d) 2(C2) 2(CS) 2+1(C2) 3P3) 3(C3) 3+1(c2) 4@4) 4(C4) 4+1(c2) ~(CZ) s(c2v) 4+2(cz) 4+2(C,) 4+2(Cz,) 5+1(C1)
TABLE 2: Binding Enthalpies (298 K, in kcal mol-') of the M+(&O), Complexes at Various Levels of Theory"
-110.9
-91.7
MP2/6-3 1+G*//RHF/6-3 1+G* -33.6 -23.6 -18.3 -15.6 -48.9 -36.5 -29.4 -25.9 -61.8 -43.4 (-33.8) (-28.7) -33.8 -28.7 -33.8 (-28.7) -75.6 -56.4 -45.7 -40.2 -83.4 -59.9 -47.5 -40.4 -39.6 -97.0 -72.3 -58.7 -51.2 -98.6 -72.5 -58.9 -50.5 -57.5 -51.5 -110.9 -84.0 -69.7 -69.8 (-67.8) -119.8 -94.1 -121.5 (-114.5) -92.7 -79.2 -113.4 -91.2 MP2/6-31+G*//MP2/6-31+G* -33.4 -23.5 -18.2 -15.5 -48.5 -36.2 -29.1 -25.5 -61.5 -43.2 (-33.6) (-28.4)
-13.6 -23.3 (-24.7) -24.9 (-24.9) -35.9 -35.0 -35.5 -45.4 -43.8 -47.1
-13.4 -22.9 (-24.5)
Binding energy contributions taken from Table 1. RHF harmonic frequencies are scaled by 0.9. MP2 frequencies are unscaled. Enthalpy corrections at the MP2/6-31+G*//RHF/6-3l+G* level are taken from the RHF/6-31+G*//RHF/6-3 1+G* calculations. Values in parentheses correspond to saddle point structures having at least one imaginary vibrational frequency.
J. Phys. Chem., Vol. 99, No. 10, 1995 3063
Cation-Water Interactions 01
- 10 ~~
h
E
I
I
I
I
I
1
MP2 / 6-31+GI/ / M p 2 /6-31 +G*
-20 -
1
-30 -
-40
RHF/3-21G/ /RHF/3-21G
-50 Li
Na
1
I
I
K
Rb
Cs
-
(-
RHF/6-31+G* RHF/3-21G
Na
Li
Rb
K
Cs
Figure 2. CP-corrected binding energies for the M+(H20) clusters as
Figure 3. Me .O distances in the M+(H20) clusters as a function of
a function of cation type.
cation type.
the heavier metals. But this effect is rather small, typically less than 1 kcal mol-' per water molecule or only a few percent of the total binding energy. The geometries of 1, 1 1, and 2(&) were optimized at both the RHF and MP2 levels. The MP2 optimizations had only marginal influence on the stability of these clusters. Comparison of the MP2 binding energies at the RHF- and MP2-optimized geometries reveals differences of about 0.1 kcal mol-' per water molecule. Correlation effects are generally more significant in the larger clusters (n = 4-6), where hydrogen-bonding interactions and associated dispersion effects play a larger role. The binding energies and enthalpies determined at the 3-21G basis set level differ significantly from the corresponding 6-31+G* values, particularly for the clusters with Li+, Na+, and K+. Figure 2 shows the binding energies of the M+(H20) clusters as a function of cation type. The 3-21G binding energies for these clusters are stronger than the 6-3 1+G* values by 2-13 kcal mol-', and differences are even more pronounced for the clusters containing several water molecules. Compared to the 6-31+G* basis, 3-21G favors geometries for the larger clusters that have hydrogen-bonding interactions. For example, for Cs+(H20)4, the 4(C4) structure, which has four hydrogen bonds, is 11.6 kcal mol-' more stable than 4(S4), which has no hydrogen bonds. In contrast, this energy difference is only 1.3 0 and 4.6 (MP2) kcal mol-' at the 6-31+G* level. Waterwater interactions are poorly described by the 3-21G basis. This level strongly overestimates the strength of the water dimer interaction, giving a binding energy of nearly -1 1 kcal mol-' compared to about -5.1 kcal mol-' for large basis sets and correlated methods.1° BSSE effects are substantial at the RHF/3-21G level. BSSE tends to spuriously stabilize the clusters relative to the separated fragments and shorten the cation-water distances. The latter effect is clearly seen in Figure 3, where the equilibrium M+* Q distances in the M+(H20) clusters are plotted as a function of cation type for several levels of theory. The distances calculated at the 3-21G level are consistently shorter (by roughly 0.1 A) than those determined with the larger 6-31+G* basis set. We evaluated CP corrections at the 3-21G level of 7-12 kcal mol-' per water molecule for the range of Li+(H20), clusters. The corresponding corrections are much smaller for the 6-31+G* basis set, only 1.0 and 2.5 kcal mol-' per water molecule at the RHF and MP2 levels, respectively. Somewhat smaller CP corrections were calculated at the 3-21G level for the heavier cations.
a
0
+
t
I
,
,
,
,
,
2
3
4
5
6
I
4
-40
1
1
2
3
4
5
6
Figure 4. Comparison of the incremental binding enthalpies for the Li+(HzO), clusters calculated with the 6-31+G*and aug-cc-pVDZ basis sets at the (a) RHF and (b) MP2 levels of theory. The solid lines correspond to the aug-cc-pVDZ values taken from Feller er al. (ref 10). Despite CP corrections, the RHF/3-21G description of the cation-water clusters remains poor, and the energies and enthalpies listed for this level in Tables 1 and 2 are not reliable. However, we report these values as we were previously able to demonstrate that calculations at the 3-21G level on extended systems are relatively inexpensive and provide qualitative trends that are potentially quite valuable." For example, we used the cation exchange reaction
K+(18-crown-6)
+ M+(H,O), - M+( 18-crown-6) +
K+(H,O), (3) together with the binding energies and enthalpies reported here to demonstrate that the observed K+ selectivity of 18-crown-6 in aqueous solution is qualitatively reproduced by cluster calculations at the 3-21G as well as at higher levels of theory. The 6-31+G* energies and enthalpies compare quite favorably with those calculated at higher levels of theory. Feller et aZ.l0 recently examined the Li+(H20), clusters using a variety of computational techniques including MP2, MP4, and QCISD(T) with correlation consistent basis sets of double- through quadruple-5 quality. At the MP2 level, they estimated an energy and enthalpy for Li+(H20) at the complete basis set limit of -35.2 and -34.2 kcal mol-', respectively, values that are each only 0.8 kcal mol-' larger than the MP2/6-31+G* quantities reported here. Figure 4 compares the 6-31+G* incremental binding enthalpies for the clusters up to n = 6 to those calculated with the more extended aug-cc-pVDZ basis set. At the RHF level (Figure 4a) these values differ by less than 0.8 kcal mol-'. The differences are somewhat larger at the MP2 level (Figure
Glendening and Feller
3064 J. Phys. Chem., Vol. 99, No. IO, 1995
-a fi
a
t o
T
1
t t
-5
i
n
0 RHF/3-21G//RHF/3-21G 0 RHF/6-31tG'//RHF/6-31+GW
+ X
-0-
1
2
3
4
5
MP2/6-31+G'//RHF/6-31+G* MP2/ 6-31+G*/ / M P 2 / 6-31tG' Expt
6
n
Figure 5. Comparison of the calculated incremental binding enthalpies for the M+(H20), clusters with the experimental values of Dzidic and
Kebarle (ref 1). 4b), up to 1.6 kcal mol-', but the overall agreement remains satisfactory. Feller et u1.l0noted that lithium-water interactions were rather insensitive to electron correlation effects but that diffuse functions on oxygen were necessary to obtain accurate structures and energies. The 6-31+G* basis set appears to describe the lithium-water interactions quite well even though it is significantly smaller than the aug-cc-pVDZ set (157 vs 260 basis functions for Li+(H20)6). The 6-31+G* basis also appears to adequately describe the sodium-water interaction. Bauschlicher and co-workers5 calculated binding energies for the Na+(H20), clusters (n = 1-4) with highly correlated methods and polarized double- and triple-5 basis sets. Again, the agreement between our 6-31+G* binding energies and their triple-g values is satisfactory. Root mean square deviations of 0.3 and 1.0 kcal mol-' are calculated for the RHF and MP2 levels, respectively. The 6-3 1+G* enthalpies compare favorably with experimentally determined values. Figure 5 shows calculated incremental binding enthalpies together with the measurements of Dzidic and Kebarle.' In preparing the calculated values for this figure, we only considered the clusters (for fixed n) of lowest energy. In effect, this assumes that the gas-phase clusters only populate the lowest energy structures and that structures of higher energy contribute negligibly to the enthalpies. The 6-3 1+G* values
generally reproduce the experimental trend quite well. The root mean square deviations at the MP2 level are 1.5, 0.7, 0.5, 0.4, and 0.8 kcal mol-' for the Li+, Na+, K+, Rb+, and Cs+ clusters, respectively. Since the incremental binding enthalpies typically range in magnitude from 10 to 20 kcal mol-' (or even larger for the small clusters with Li+ and Na+), these deviations represent an average error of only 5-lo%, an acceptable level considering the uncertainties in the experimental measurements and the rather modest levels of theory employed here. The comparison at the RHF level is rather poor, as revealed by the corresponding deviations of 2.4,2.2, 3.1, 1.0, and 1.4 kcal mol-'. If one neglects, however, the largest clusters (n = 5, 6), the root mean square deviations for Li+, Na+, and K+ improve somewhat to 1.9, 1.0, and 0.6 kcal mol-', respectively, emphasizing the importance of a correlated treatment in the calculation of the large clusters. This is seen quite clearly for the K+(H20), clusters in Figure 5. While the RHF enthalpies agree well with the experimental values for the small clusters, for n = 5, 6 they deviate strongly (by up to 7 kcal mol-'). In contrast, the Mp2 enthalpies compare favorably with experiment over the full range of cluster sizes. Although Dzidic and Kebarle reported enthalpies at 298 K, the actual measurements were performed over a range of temperatures that depended on the size of the cluster.' No
J. Phys. Chem., Vol. 99,No. IO, I995 3065
Cation-Water Interactions TABLE 3: Temperature Dependence of the Incremental Binding Enthalpies for the Na+(H20), Cluster@ n 1 2 3 4 5 6 r m s dev"
AH290
AHT
T
AHexptb
-23.6 -19.8 -16.5 -12.6 -11.5 -10.1
-23.5 -18.9 -16.0 -12.4 -11.5 -10.1 0.8
670 550 450 375 300 290
-24.0 -19.8 -15.8 -13.8 -12.3 -10.7
0.7
1 -
9
4,,,
0-1
-
-2
-
-3
-
-4
-
r
.a fi
yd
MP2/6-31+G*//RHF/6-3l+G* values. Enthalpies in kcal mo1-I. Temperaturesin degrees Kelvin. Dzidic and Kebarle, ref 1. Deviation from AHexpt. (1
extrapolation was applied to the reported values since the van't Hoff analysis of the data revealed good linearity over the temperature range examined. However, a recent analysis of the Na+(H20), clusters with the OPLS model potential revealed that neglecting the temperature extrapolation is unjustified for large clusters.28 To determine whether the temperature dependence might account in part for differences between the calculated and measured values, we evaluated enthalpies AHT for the Na+(H20), clusters at temperatures within the range employed in the experiment. These are listed in Table 3 together with the standard A H 2 9 8 and experimental AHexptenthalpies. Temperature extrapolation has little influence on the calculated enthalpies and, in fact, slightly worsens the agreement between calculated and experimental values.
IV. Structure We now turn to a brief discussion of the optimal geometries for the M+(H20), clusters. Full specifications of these geometries are available as supplementary material in the form of GAUSSIAN 92 and GAMESS input decks. We were unable to locate equilibrium structures in several cases, such as the 3(C3) clusters for Li+, Na+, and K+. Instead, these structures generally reverted to one of higher symmetry (e.g. C3 0 3 ) during geometry optimization. All clusters optimized at the 3-21G basis set level correspond to energy minima according to the vibrational normal mode analysis. Several of the clusters optimized with the 6-31+G* basis, those with binding energies in parentheses in Table 1, are saddle point structures having at least one imaginary vibrational frequency. The "linear" ~ ( D u complexes ) containing K+, Rb+, and Cs+ are perhaps the most interesting of the saddle point structures. One would generally anticipate that two water molecules would interact with a cation in a linear arrangement with the water dipoles directed toward the cation. However, as demonstrated by Kaupp and Schleyer, heavy alkali metals often favor bent or pyramidal structures that are stabilized in part by the polarization of the cation core.' At the MP2/6-31+G* level, the linear M+(H20), structures have a degenerate pair of bending modes with imaginary frequencies 18i, l l i , and 1Oi cm-' for K+, Rb+, and Cs+, respectively. We optimized two bent structures of lower energy having C2 and C, symmetry (cf. Figure 1). The MP2/6-31+G* binding energies reported in Table 1 suggest that these structures are essentially isoenergetic and are only slightly (0.03-0.12 kcal mol-') more stable than the DZd form. The bent structures are significantly distorted from Du,having optimized 0-M-0 angles of 145', 134', and 123" for K+, Rb+, and Cs+, respectively. Kaupp and Schleyer calculated angles that were even smaller for the Rb+ (126') and Cs+ (1 13') clusters using somewhat larger basis sets than employed here.7 The bending potential for these complexes is clearly flat and sensitive to the level of theory and basis set employed. More recent MP2 calculations of the Rb+(H20)2
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-5
'
I
Li
1
I
K
Na
Figure 6. Binding energies of the 3 4(&) as a function of cation type.
I
I
Rb
Cs
I
+ 1 and 4(C4) clusters relative to
complex with an ECP and highly extended valence basis set suggest, in fact, that the linear form is a stable s t r u c t ~ r e . ~ ~ Small cation-water clusters (n = 1-3) generally favor geometries such as 1,2, and 3(D3) in which each of the n water molecules directly coordinates the cation in a highly symmetric arrangement. The alternative 1 1 and 2 1 structures having one water molecule hydrogen bonded to the cluster are somewhat less stable. This suggests that cation-water rather than water-water interactions dominate the binding energy of these small clusters. The preference for structures such as 3(D3) rather than 2 1 becomes less clear as the size of the cation increases. For example, for Rb+(H20)3 we identified three minimum energy structures, 2 1, 3(D3), and 3(C3), that are separated by 2.0 kcal mol-' at the RHF/6-31+G* level and by only 0.6 kcal mol-' at the MP2/6-31+G* level. Considering the accuracy of these methods discussed in the previous section, we cannot unequivocally specify which of these structures is most stable. Hydrogen-bonded structures are increasingly favored as the size of the cation increases. Figure 6 shows the MP2/6-31+G* binding energies of the M+(H20)4 clusters relative to the 4(S4) form. Except for Li+, this level of theory suggests that either 3 1 or 4(C4) is the most stable form. For Cs+, 4(C4) is 2.4 kcal mol-' more stable than 3 1, which is in turn 2.2 kcal mol-' more stable than 4(S4). The 4(C4) structure is particularly interesting, as it resembles a cation interacting with a cyclic water tetramer, simultaneously stabilized by four hydrogen bonds and four cation-water interactions. These eight interactions are somewhat strained since the O...H-O linkages are nonlinear and the water dipole moments are not optimally directed toward the cation. Nevertheless, the 4(C4) structure is more stable than either 3 1, which has three cation-water interactions and two hydrogen bonds, or 4(S4), which has just four cation-water interactions. Hashimoto and Morokuma'' recently reported calculations of S4 and D4 symmetry structures of Na+(H20)4 at the same levels of theory employed here. Since the D4 structure was a second-order saddle point, they concluded that the S4 form corresponds to the only minimum on the potential energy surface. Our calculations clearly show, however, that 3 l is also an equilibrium structure that is nearly isoenergetic to 4(S4). For Li+, Na+, and K+, we optimized several clusters containing five and six water molecules. Several attempts failed to locate an equilibrium structure for the Li+(H20)5 and Na+(HzO)~clusters in which each water molecule directly coordinates the metal (labeled 5 according to the n m convention). Beginning with an initial geometry of form 5, the optimization procedure moved one water molecule from the primary shell
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3066 J. Phys. Chem., Vol. 99, No. 10, 1995
Glendening and Feller
into a hydrogen-bonding position of the secondary shell to give 4 1. For Kf(H20),, we located an equilibrium structure S(C2) that resembles a cation resting on a cyclic water tetramer with the fifth water coordinating the cation from above. The saddle point structure 5(CzV)is similar except that the four water molecules that form the tetramer of 5(c2)are not oriented for hydrogen bonding but rather have their dipoles directed toward the cation to enhance the cation-water interactions. Although we were able to identify structures of the form 5 for K+(HzO)5, the hydrogen bonded 4 1 geometry remains the most stable. We optimized a few structures of the form 4 2 , s 1, and 6 for the M+(H2O)6 clusters. For Li+ and Na+ we generally found the 4 2 structures to be most stable by several kcal mol-’. The only exception is the 5 1 structure for Na+ that is essentially isoelectronic to the 4 2(C2) structure at the RHF/ 6-31+G* level. However, the latter structure is favored again at the correlated MP2 level. The 4 2 structures each have the four primary shell water molecules coordinating the metal in a tetrahedral arrangement with the secondary shell waters participating in two or three hydrogen-bonding interactions. These water molecules could be arranged in many hydrogenbonding patterns, but we only examined the three structures shown in Figure 1. For Li+, Na+, and K+ we optimized the structures 6(&) and 6(D3) at the 3-21G level. Both of these structures resemble a sandwich complex with the cation interacting with two cyclic water trimers. The 0 3 form is similar to 6(&) except that the OH bonds of the water trimers are oriented in counter-rotating cycles. The 6(&) form is slightly favored over 6(D3) for Li+ and Na+, but the two structures are essentially degenerate for K+. Hydrogen-bonded clusters such as 4 1 and 4 2 have been observed in molecular dynamics simulations of gas-phase Na+(H20), clusters. Kollman and c o - w ~ r k e r have s ~ ~ shown ~~~ with their polarizable POL1 model potential that the Na+(H20)6 cluster prefers to have a “4 2” structure rather than an octahedrally coordinated one. Using similar methods, Perera and B e r k o ~ i t zobserved ~ ~ , ~ ~a “3 1” structure for Na+(Hz0)4 at high temperature and found the most stable cluster of Na+(H2O)5 to be “4 1”. It is interesting to note the degree of hydration n at which the hydrogen-bonded forms with at least one water molecule in the secondary shell become more stable than the clusters in which all n waters directly coordinate the cation in the primary shell. The MP2/6-31+G* data of Table 1 suggest that this crossover occurs at n = 5, 4, 4, 3, 3 for the respective cations Li+ through Cs+. The incremental binding energy for adding the nth water in each case is about 12-14 kcal mol-’, or roughly twice the strength of a single hydrogen bond. This suggests that the hydrogen-bonded forms are favored when the addition of a water molecule to the primary solvation shell stabilizes the cluster by less than two hydrogen bonds. In such cases, the formation of the clusters like 2 1 or 3 1 is favored.
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V. Summary We have calculated gas-phase binding energies and enthalpies for small M+(H~O),,clusters consisting of an alkali metal cation with one to six water molecules. The energy and enthalpy quantities evaluated at the RHF and MP2 levels of theory with the 6-3 1+G* basis set (including effective core potentials for the heavier alkali metals) compare favorably with values determined at higher levels of theory and with the experimental measurements of Dzidic and Kebarle.’ In contrast, the RHF/ 3-21G level performs rather poorly, strongly overestimating the binding energies of most clusters and underestimating the M+.*.O distances. The small clusters (n = 1-3) favor
structures in which each of the water molecules resides in the primary solvation shell, directly coordinating the cation in a highly symmetric arrangement. Since cation-water interactions are principally electrostatic, the RHF/6-3 1+G* level treatment provides an adequate description of these clusters. The larger clusters (n = 4-6) often favor geometries in which one or two waters occupy the secondary solvation shell, each forming two hydrogen bonds to water molecules in the primary shell. Dispersion effects associated with hydrogen bonding require a correlated (MP2) approach to obtain reliable binding energies for these clusters. Comparison with experiment suggests that the MP2/6-3 l+G* method is capable of determining binding enthalpies with an average error of 5-lo%, or roughly 1 kcal mol-’ per water molecule. Acknowledgment. This research was supported by the U.S. Department of Energy under Contract No. DE-AC06-76RLO 1830. The authors acknowledge the support of the Division of Chemical Sciences, Office of Basic Energy Sciences. E.D.G. also acknowledges the support of Associated Western Universities, Inc. (on behalf of Washington State University) under Grant No. DE-FG06-89ER-75522 with the US. Department of Energy. Portions of this work were completed on the computer resources at the National Energy Research Supercomputer Center with a grant provided by the Scientific Computing Staff, Office of Energy Research, U.S.Department of Energy. The Pacific Northwest Laboratory is a multiprogram national laboratory operated by Battelle Memorial Institute. Supplementary Material Available: Optimized geometries for the M+(H20), clusters are available in the form of GAUSSIAN 92 and GAMESS input decks (53 pages). Ordering information is given on any current masthead page. References and Notes (1) Dzidic, I.; Kebarle, P. J . Phys. Chem. 1970, 74, 1466. f2) Blades. A. T.: Javaweera., P.:, Ikonomou. M. G.: Kebarle. P. J. Chem. Phys. 1990, 92, 5900. I
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(3) Willev. K. F.: Yeh. C. S.: Robbins. D. L.: Pilmim. J. S.: Duncan. ” M. A.’J. Chem. Phys. 1992, 97, 8886. (4) Dalleska, N. F.; Honma, K.; Sunderlin, L. S.; Armentrout, P. B. J . Am. Chem. Sac. 1994, 116, 3519. ( 5 ) Bauschlicher, C. W., Jr.; Langhoff, S. R.; Partridge, H.; Rice, J. E.; Komornicki, A. J . Chem. Phys. 1991, 95, 5142. (6) Bauschlicher, C. W., Jr.; Sodupe, M.; Partridge, H. J . Chem. Phys. 1992, 96, 4453. (7) Kaupp, M.; Schleyer, P. v. R. J . Phys. Chem. 1992, 96, 7316. (8) Klobukowski, M. Can. J . Chem. 1992, 70, 589. (9) Sodupe, M.; Bauschlicher, C. W., Jr. Chem. Phys. Lett. 1992,195, 494. (10) Feller, D.; Glendening, E. D.; Kendall, R. A,; Peterson, K. A. J. Chem. Phys. 1994, 100, 4981. (11) Hashimoto, K.; Morokuma, K. Chem. Phys. Lett. 1994, 223, 423. (12) Kaupp, M.; Schleyer, P. v. R.; Stoll, H.; Preuss, H. J . Chem. Phys. 1991, 94, 1360. (13) Kaupp, M.; Schleyer, P. v. R.; Stoll, H.; Preuss, H. J. Am. Chem. SOC.1991, 113, 6012. (14) Kaupp, M.; Schleyer, P. v. R. J . Am. Chem. SOC.1992, 114,491. (15) De Jong, F.; Reinhoudt, D. N. Adv. Phys. Org. Chem. 1980, 17, 279. (16) Pedersen, C. J. Angew. Chem., Znt. Ed. Engl. 1988, 27, 1021. (17) Glendening, E. D.; Feller, D.; Thompson, M. A. J . Am. Chem. SOC. 1994, 116, 10657. (18) Binkley, J. S.; Pople, J. A.; Hehre, W. J. J. Am. Chem. SOC.1980, 102, 939. (19) Gordon, M. S.; Binkley, J. S.; Pople, J. A,; Pietro, W. J.; Hehre, W. J. J. Am. Chem. SOC.1982, 104, 2791. (20) Dobbs, K. D.; Hehre, W. J. J . Comput. Chem. 1986, 7, 359. (21) Huzinaga, S.; Andzelm, J.; Klobukowski, M.; Radio-Andzelm, E.; Sakai, Y.; Tatewaki, H. Gaussian Basis Sets for Molecular Calculations; Elsevier: Amsterdam, 1984. (22) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986. (23) Hay, P. J.; Wadt, W. R. J. Chem. Phys. 1985, 82, 299.
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Cation-Water Interactions (24) Frisch, M. J.; Trucks, G. W.; Head-Gordon, M.; Gill, P. M. W.; Wong, M. W.; Foreman, J. B.; Johnson, B. G.; Schlegel, H. B.; Robb, M. A.; 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, Rev. A; Gaussian, Inc.: Pittsburgh, PA, 1992: (25) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A,; Elbert, S. T.; Gordon. M. S.: Jensen. J. H.: Koseki. S.; Matsunaga, N.: Nguven, K. A.; Su,S.; Windus, T. L.; Dupuis, M.; Montgomery, J. i., Jr. J. C>mput. Chem. 1993, 14, 1347. (26) Boys, S. F.; Bemardi, F. Mol. Phys. 1970, 19, 553. (27) Del Bene, J. E.; Mettee, H. D.; Frisch, M. J.; Luke, B. T.; Pople,
J. A. J . Phys. Chem. 1983, 87, 3219. (28) Jorgensen, W. L.; Severance, D. L. J. Chem. Phys. 1993,99,4233. (29) Feller, D.; Glendening, E. D.; Woon, D. E.; Feyereisen, M. W. In preparation. (30) Caldwell. J.:. Dane. -. L. X.: Kollman. P. A. J . Am. Chem. SOC. 1990. ,
112,9144.
(31) Dane. L. X.:. Rice.. J. E.:. Caldwell. J.: Kollman. P. A. J . Am. Chem. Soc. 1991, 173, 2481.
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