2044
J. Phys. Chem. 1994,98, 2044-2049
Na+/K+-(Hz)1 , ~Clusters: Binding Energies from Theory and Experiment John E. Bushnell, Paul R. Kemper, and Michael T. Bowers' Department of Chemistry, University of California, Santa Barbara, California 931 06 Received: October 6, 1993; In Final Form: December 7, 1993"
Dissociation energies for H2 loss from Na+.(H2) 1.2 and K+.(H2) 1,2 clusters have been determined via temperaturedependent equilibrium measurements. DO= - M o o = 2.45 f 0.2 and 2.25 f 0.2 kcal/mol for Na+-H2 and HyNa+-H2, respectively, and 1.45 f 0.2 and 1.26 f 0.4 kcal/mol for K+-H2 and HyK+-H2, respectively. Also, a b initio calculations on Na+.H2 were carried out at the HF and MP2 levels with an extended basis set and compared with previous calculations on this system as well as with experiment. By extrapolating these purely electrostatic results to the first-row transition series, one can conclude that covalent interactions dominate the bonding in V+-H2 and Co+-H2 systems.
Introduction The coordination of dihydrogen with transition metals has become an active area of research since the pioneering studies of Kubas and co-workers nearly a decade ago.' Initial studies have indicated d6 metals are the best candidates for dihydrogen addition2 while de metals oxidatively add H2 to form dihydride~.~ The oxidation state of the metal appears to strongly influence the way hydrogen adds. As electrons are withdrawn from the metal, it becomes more difficult to oxidize H2 by donation into the u* orbital, and dihydrogen addition becomes favored. Consequently, bare transition metal ions become superb candidates for nonclassical dihydrogen addition. In our laboratory, we have explored these systems by forming state-selected transition metal ions4 and sequentially clustering them with up to seven H2 molecules. By measuring equilibrium constants as a function of temperature, we have been able to determine accurate binding energies for each successive H2 ligand.s.6 Binding energies between metal ions and rare gas neutrals have also been investigated.' This work has stimulated substantialtheoretical interest in an effort to understand the trends observed experimentally.8-10 In the first-row transition series, 4s and 3d electrons dominate the bonding, which is due to a mix of electrostatic and covalent interactions. There is some question as to the relative importance of these two effects, with Perry et a1.10 maintaining that the majority of the interaction is electrostatic, while Bauschlicher and c o - w ~ r k e r scontend ~ - ~ that covalent effects play a significant role. In an attempt to simplify the problem, we felt it would be useful tocarefully study a system in which the electrostatic results clearly dominate. The closed shell alkali metal ions were chosen since charge quadrupole and charge-induced dipole electrostatic interactions shoulddominatetheir bonding with H2. Furthermore, the Na+.H2 system has received considerable theoretical attenti0n.~JI-l3 All investigators agree on the general aspects of the interaction. The H2 approaches side on (to maximize the charge-uadrupole interaction), the Na+ ion and the Hz ligand are largely unperturbed, and the resulting bond is weak (De 3 kcallmol). Significant disagreement exists over the best value for De, however. Recently, Falcetta et al.13 have reviewed the existing theory and resolved most of the disagreement. A question remains, however, as to the relative importance of electron correlation and basis set superposition error (BSSE) in the calculations. In an attempt to resolve this question and to gauge the absolute accuracy of the calculations, we present here a careful experimental determination of the Na+.H2 bond energy as well as a theoretical investigation of the importance of electron
-
0
Abstract published in Aduance ACS Absfracrs, January 15, 1994.
0022-3654/94/2098-2044%04.50/0
correlation and BSSE in this system. We also present results on N ~ + - ( H Zsince ) ~ both Falcetta et al.13 and Bauschlicher et ale8 have done calculations on this second cluster. The K+-(H2),,2clusters were also investigated. If electrostatic forces dominate, these clusters should be more weakly bound than the analogous Na+ system, a result our experiments confirm. On the basis of the Na+.H2 and K+.H2 bond strengths, we then estimate the dissociation energies for Rb+.HZ and Cs+.Hz by extrapolation and comment on the relative importance of electrostatic and covalent interactions of Hz with first-row transition metals. Experimental Technique and Data Analysis Both the instrument14 and data analysis methods5*' have been described previously. Sources of error have also been discussed extensively.5 A brief overview is given here with emphasis on the differences encountered in these experiments. The Instrument. The Na+ and K+ ions are formed via surface ionization on a hot filament (Re ribbon T 2500 K), the source of sodium and potassium being a small tube of alumina ceramic through which the filament was threaded. The nascent ions are accelerated to 5 keV, mass selected with a double-focusing reverse geometry mass spectrometer, decelerated to approximately 3-5 eV, and injected into a reaction cell containing H2 a t about 1 X 10" molecule/cm3 (3 Torr at 300 K.) The ions are quickly thermalized via collisions with the H2 and move through the 4-cmlong reaction cell under the influence of a small, uniform electric field. The electric field is kept small enough that the ion thermal energy is not significantly perturbed. The Hz pressure in the reaction cell is monitored directly with a capacitance manometer. Cell temperatures are varied using a flow of heated or cooled Nz, and temperatures are measured using a thin-film platinum resistor suspended in the bath gas. Ions exiting the cell are accelerated slightly (2-5 eV), quadrupole mass analyzed, and collected using standard ion-counting techniques. The quadrupole is computer scanned over the mass range of interest, and the baseline-resolved peaks are then integrated to give the relative ion intensities. Equilibrium Experiment. After the temperature and H2 pressure are observed to be stable within the reaction cell, product/ parent ion ratios are measured as a function of reaction time. This time is varied by changing the drift voltage across the cell. As the drift time is increased, the product/parent ion ratios eventually become constant, indicating that equilibrium has been reached. The reactions were probed out to 3.5 ms to ensure equilibrium had been established. This measurement also ensures that the drift field is not significantly perturbing the ion thermal kinetic energy. For selected experiments, the pressure of H2 was varied by a factor of 2 with no significant change observed in
-
AGO T.
0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 8, 1994 2045
Na+/K+-(H2)1,2 Clusters
TABLE 1: Summary of Experimental Results -A&o T -AHTO Na+ + H2- Na+.H2 Na+H2+ H2- Na+.(H2)2 K+ + Hz K+-H2 -+
K+H2
+ H2
+
TABLE 2 property re (H2), A
r ~ nA,
R h 9A De, kcal/mol v
(Hz), cm-'
V I , cm-I --
v2.
I
0
100
200
300
Temperature (K) Figure 1. Equilibrium data for Na+ and Na+-H2clusteringwith H2. The data are plotted as free energy (AGO) us temperature. The ion ratios are converted to equilibrium constants using eql
where P H is~ the hydrogen pressure in Torr and M+.(Hz), and M+.(H2),l are the measured intensities of the cluster ions of interest. The standard free energy change is calculated using eq 2 AGO = -RT In K O , (2) where R is the gas constant and T i s the temperature. A plot of AGO us temperature gives a straight line with intercept A H O T and a slope of &TOT, where T is the average experimental temperature. This is functionally equivalent to a van't Hoff plot but is more convenient for our data analysis. Since we are forming closed shell ions with noble gas configurations and using relatively mild ionizing conditions (surface ionization), there are no interfering excited states which would require deactivation (this was a problem in our study of Co+-H25). In order to obtain the desired cluster bond dissociation energies (BDE), the A H O T must be converted to A H 0 o (I-BDE). This is done by calculating AGO as a function of temperature using statistical mechanics. The bond lengths, frequencies, and dissociation energy used in the calculation are varied until the experimental and calculated functions agree. In all cases, the vibrational frequencies were varied over a wide range to see the effect on the resulting A H 0 o (DO), and these uncertainties are included in the error limits. A number of potential sources of error are present in these experiments, e.g., pressure and temperature inaccuracies as well as mass discrimination and resolution. The effect of these factors on our dissociation energies and entropies is discussed in refs 5 and 7. The net result is that the bond dissociation energies are essentially unaffected by these uncertainties. The entropies are affected to a small extent by any mass discrimination in the quadrupole mass analyzer, but in these experiments such mass discrimination shourd be very small since the parent/product mass difference is only 2 amu. One additional source of error is collision-induced dissociation (CID) of the clusters as they are accelerated into the quadrupole. This process most likely occurs to a small extent for weakly bound clusters, and its importance has previously been di~cussed.~ Such processes can have a minor effect on measured values of A S O T
cm-1
~ 3 cm-1 ,
-Woo
(cal mol-' K-I) (K) (kcal/mol) (kcal/mol)
reaction
K+*(H2)2
13.2 12.4 13.5 11.2
190 130 170 120
*
2.93 2.41 1.86 1.47
2.45 0.2 2.25 f 0.2 1 . 4 5 f 0.2 1.35 t 0.4
Summary of ab Initio Calculations on Na+.H2 previous ab initio studies" this work 11
12
8
13
HF
MP2
0.730 0.732 2.541 3.25b 4649 260 4577 439 0.80 2.45a
0.734 0.739 2.458 2.87 4582 306 4518 558 1.15 1.72
0.742 0.750 2.469 3.1 -
0.735 0.740 2.475 2.95 4594 286 4522 502 1.03 1.92
0.734 0.739 2.475 2.92 4590 296 4514 554
0.737 0.741 2.463 2.85 4528 304 4458 533 0.99 1.86
AZPE, kcal/mol 1.oo DO,kcal/mol 1.92 Numbers are references in this paper. b Too large due to BSSE,see text. (1
but have essentially no effect on M O T or M o o . The reason for this is the fraction of clusters undergoing CID is largely independent of temperature since the pressure outside the cell is very nearly temperature independent. Consequently, any loss of clusters via CID appears as a constant factor in the equilibrium constant. That this fraction of CID is small is apparent from experiments where the cell pressure was changed by a factor of 2, and the measured value of AGO showed little or no change. This result argues against significant CID (which should scale with cell pressure).
Computational Method All a b initio calculations were done using the Gaussian 92 packagewitha6-31 lG+(3df,2p) basisset.15 Optimizationswere done at the Hartree-Fock (HF) level as well as with secondorder Moller-Plesset perturbation theory (MP2.) Only the two valence (Hz) electrons were correlated in the MP2 calculations. Basis set superposition errors (BSSEs) were estimated using the counterpoise approximation.16.17
Results and Discussion
Na+.H2. Figure 1 shows our experimental data plotted as AGO us T. The straight lines through the points are linear least-squares
fits. According to statistical mechanics, AGO is not an entirely linear function of temperature. Over our limited temperature range, however, the deviation from linearity is imperceptible. As noted, the slopes and intercepts correspond to -MOTand M O T , respectively, where T is the midpoint of the experimental temperature range. Table 1 gives the experimental values of AHo T and ASoT as well as the - M o o (DO) values obtained from our statistical mechanical fitting procedure (see refs 5 and 7 for details). We find the Na+-H2 bond dissociation energy to be 2.45 f 0.2 kcal/mol. To our knowledge, there are no other experimental determinations with which to compare. The DOfor Li+-H2 has been measured to be 6.5 f 4.6 kcal/moll* and computed to be 5.64 kcal/mol.lg Although the Li+.H2 system should be more strongly bound due to the smaller Li+ radius, the lack of measurement precision makes quantitative comparison impossible. Table 2 lists results from the present calculations as well as those done previously. There is general agreement in all calculations as to the structure of Na+.H2. The Ha binds side-on to the Na+ (CZ, symmetry, to maximize the chargequadrupole attraction) with a Na+ to Hz bond midpoint (BMP) distancevery close to 2.47 A (Rmin). The H2 ligand is largely unperturbed with
The Journal of Physical Chemistry, Vol. 98,No.8. 1994
Bushnell et al.
the H-H distance (rmin)about equal to 0.735 %, (0.005 8,longer than that calculated for isolated H2). The theoretical DC’s range from 2.85 to 3.25 kcal/mol. Falcetta et al.13 have extensively discussed and critiqued the calculations of Curtiss and Pople,” Bauschlicher et a1.,8 and Dixon et al.12 Falcetta et aL13used their calculations (large basis set, basis set superpositionerror (BSSE) correction, no electron correlation, De = 2.95 kcal/mol) as a basis for comparison. They concluded that the calculations of Bauschlicher et a1.8 (large basis set, no BSSE correction, electron correlation included, De = 3.1 kcal/mol) were the most reliable. They questioned the lack of BSSE, however, and thought their smaller De might be more reliable. Falcetta et al.13 found the bond energy calculated by Curtiss and Poplell (smaller basis set, no BSSE correction, electron correlation included, De = 3.25 kcal/mol) to be 0.32 kcal/mol too large due to neglect of BSSE. Thus, Curtiss and Pople’s De should be -2.93 kcal/mol. The low De value calculated by Dixon et a1.12 (small basis set, no BSSE correction, no correlation, De = 2.87 kcal/mol) is due to the small basis set which failed to reproduce the HZpolarizability (aI).Further, the lack of BSSE correction in Dixon’s work spuriously increased the calculated De. In summary, Falcetta et aL13concluded that the best theoretical value for De was about 3.0 kcal/mol. In an attempt to gauge the relative importance of BSSE and electron correlation in the calculation of the Na+.H2 binding energy, we did both Hartree-Fock (HF) and second-order perturbation theory (MP2) calculations using large basis sets and including counterpoise approximations of BSSE. Our H F results are very close to those of Falcetta et al.13 BSSE was calculated to be only 0.05 kcal/mol at the H F level. Correlating the two valence (Hz) electrons using MP2 actually decreases our De by 0.07 kcal/mol when the slightly larger BSSE of 0.09 kcal/ mol is included. The large BSSE of Falcetta et al. (-0.2 kcal/ mol) is probably the result of a larger basis set for H2. They also find a (very small) decrease in attraction at the MP2 level, though they have estimated the dispersion attraction to be -0.36 kcal/ mol (at R = 2.5 A) using combination rules.20 Bauschlicher et a1.8 show an increase in binding energy of about 0.2 kcal/mol due to correlation, which may be due to their large A N 0 basis sets and MCPF technique, though BSSE may account for part of this difference. The best theoretical estimate for De is then 3.0-3.1 kcal/mol. In order to compare the results of the various theoretical calculations (0;s) with our present experimental finding (DO), the change in zero point energies (AZPE) must be determined. Although our statistical mechanical calculation of AGO us Tdoes give some indication of the vibrational frequencies present, it is insensitive to high-frequency modes, and the results are not sufficiently accurate for the AZPE calculation. Thus, we must rely on theoretically calculated frequencies. The accuracy of these frequenciesthen becomes important since any error directly affects the comparison between experiment and theory. In general, the agreement between the frequencies from different calculations is good for the Na+*H2 system. However, these calculated frequencies are generally expected to be somewhat high. The Na+.Hzstretch frequencies(vl and v3) from the present calculations were scaled by a factor of 0.9 1 as recently suggested by Grev, Janssen, and Schaefer.21 This factor differs from the common practice of scaling by 0.89, which was originally intended to give good agreement between ab initio frequencies and experimentally observed fundamental frequencies. The H-H stretch frequency ( 4 was scaled by 0.97, the ratio of the experimental and calculated Y (H2) frequencies (4400 and 4528 cm-l, respectively). This ratio of Y (H2) is essentially the same in all the theoretical calculations (Table 2). Curtiss and Pople11 and Dixon et al.12both scaled all frequencies by 0.89. Falcetta et al.I3 scaled only the H-H stretch (v2 and Y (H2)) by 0.89. The resulting AZPEs range from 0.80 to 1.2 kcal/mol (Table 2). We
take the best estimate for AZPE to be the average of our AZPE and the AZPE calculated from the frequenciesof Falcetta et al.13 This gives a AZPE of 0.97 kcal/mol and a De derived from experiment of 3.42 f 0.3 kcal/mol for Na+.H2, including both the experimental and zero point uncertainties. As discussed, the best theoretical value for De(Na+.H2)appears to be that of Bauschlicher et a1.* with De = 3.1 kcal/mol. Thus, theory is able to account for 90 f 8% of the experimental binding energy. While this may not seem quite as good as expected for this simple closed shell system, it is actually very good agreement for such a weakly bound complex where dispersion forces are (relatively) significant. One possible explanation for the discrepancy could be that the cluster vibrational frequencies are in fact somewhat lower than those from theory, due to large anharmonicities known to be present in this very shallow potential well (this would lower the AZPE correction which equals 1/2. (ae- aexe/2)). Our statistical mechanical analysis of the experimental data did require lower frequencies if we assumed that no mass discrimination was present. Although this is not definitive, it does at least support the possibility that the true cluster frequencies may be lower than those calculated. Another possible source of error is BSSE. Bauschlicher et a1.* do not include any estimate of BSSE in their calculations but instead rely on their A N 0 basis sets’ normally very small BSSE. Our calculations as well as those of Falcetta et al. utilize the counterpoise correction for BSSE. Though it is often used as an estimate of BSSE, it has not been confirmed as a valid technique in all cases.16J7 Significanterrors in this correctionseem unlikely, however, given the excellent agreement between our H F calculation and those of Falcetta et al. and Bauschlicher et al. Na+.(H&. The experimental results are shown in Figure 1, with the resulting -AH” T,- S O T , and -WO (DO)listed in Table 1. The DOfor the second H2 is 2.25 f 0.2 kcal/mol, or 0.20 kcal/mol smaller than the first. Since the first and second bond dissociation energies are so close, we used identical frequencies for the stretching modes of the first and second hydrogens in our analysis. This left only the degenerate bend and the internal rotation undetermined. Falcetta et ai.” determined the barrier to internal rotation to be less than 0.001 kcal/mol, and this mode was treated as a free rotation. Because of the low experimental temperatures, the rotational partition functions were determined by direct summation. The effect of ortho/para type species in the rotational states was also included (see the Appendix). This left only the frequency of the HrNa+-H2 degenerate bend as an adjustable parameter in fitting the experimental and statistical mechanical AGO us T curves. The bend frequency was found to be very low (20 f 5 cm-l), in keeping with the weak Na+-Hz interaction (and even weaker H2-H2 interaction). The resulting AZPE for the second Na+-H2 bond should be very close to that for the first (since the high frequencies are all about the same and the additional bending frequency is very low). Thus, based on experiment, we expect the De for HrNa+-H2 to be -0.2 f 0.2 kcal/mol less than the De for Na+-H2 or 3.22 f 0.3 kcal/mol. Both Bauschlicheret a1.* and Falcetta et aLl3have investigated the bonding in the Na+.(H2)2 cluster. Both agree the second H2 is bound less strongly than the first, the weaker bond being due to induced dipole-induced dipole repulsion rather than steric interference. Falcetta et al. calculate the difference in D i s to be 0.1 kcal/mol, while Bauschlicher et al. find a difference of 0.5 kcal/mol. Neither calculation has been corrected for BSSE. The experimental change in De agrees well with the value of Falcetta et al. (0.1 kcal/mol), but the rather large difference reported by Bauschlicher et al. is probably outside expected experimental (and theoretical) error. Interestingly, both calculations find the same decrease in De (0.2 kcal/mol) in going from the second to the third cluster. We expect a larger change in De for the third H2 addition (relative to the second) since the third H2 is repelled by two other hydrogen molecules (instead of only one), and the
2046
The Journal of Physical Chemistry, Vol. 98, No. 8, 1994 2047
Na+/K+.(H2)1,2 Clusters
I
1
’
io
1o ;
1;o
1o;
l&l
li0
2;o
2;o
2;o
Temperature (K) Figure 2. Equilibrium data for K+ and K+.H2 clustering with H2. The data are plotted as free energy us temperature.
repulsion is stronger due to the smaller H2-H2 distance. On the basis of this argument and the experimental results, the change in De between adding the first and second hydrogen ligands appears to be about 0.2 kcal/mol. K+-(H2)1,2. Our experimental data are shown in Figure 2, and the results are listed in Table 1. The larger uncertainty reflects both the smaller number of data points and the more limited temperature range in these experiments. The observed binding energies are smaller than those in the Na+ clusters, presumably because of the larger K+ ionic radius (1.33 A us 0.95-0.98 A).22 The K+-(H2)1,2 bond lengths and frequencies needed to determine M o o with our statistical mechanical analysis had to be estimated from the corresponding Na+.H2 clusters since no theoretical investigations of K+.Hz have been made. The M+H2 bond length was assumed to vary roughly as to De-1/3 basedon either a 4-10 potential (charge-induddipoleattraction/ hard sphere repulsion) or a 3-10 potential (chargequadrupole attraction/hard sphere repulsion).’ The K+-Hz bond length (R,”) was thus set at (1.17 f 0.04)Rmin(Na+.H2) or 2.89 f 0.1 A. This increase in Rmin(0.42 A) also corresponds fairly well to the increase in ionic radius between Na+ and K+ (0.35-0.38 A).22 In order to estimate the K+.H2 vibrational frequencies, we note that the symmetric stretch frequency of a charge induced-dipole bound complex varies as De3/4.7 A plot of experimental frequencies us De was found to vary roughly as De0.65. On this basis, we expect vl(K+.H2) to be -0.75vl(Na+.Hz) or 200 cm-l. The asymmetric stretch was also scaled by 0.75 (to 350 cm-l), although this is more uncertain. The H-H stretch in K+.H2 should be even closer to that of neutral H2 due to the weaker interaction. As noted, the APovalues determined in our analysis are not sensitive toerrorsin thesemolecular parameters. The resulting DO(K+.H2) is 1.45 f 0.2 kcaI/mol. The AZPE is about 0.75 kcal/mol, giving a De of 2.20 f 0.4 kcal/mol. As with Na+-(H&, frequencies and bond lengths in K+.(H2)2 were set equal to those in the first cluster. Free rotation of the two hydrogens was again assumed. The H2-K+-H2 bend frequency from the data fit was 15 5 cm-l. The resulting Do was 1.35 kcal/mol, corresponding to a De of about 2.1 kcal/mol. The K + . ( H Z ) ~binding ,~ energies (D,’s) are both 60% of those in the corresponding Na+.(H2) 1.2 clusters. This is expected since all the clusters are governed by nearly identical attractive potentials with the only difference being the size of the ion. The difference in De between the first and second clusters is slightly smaller in the K+ system, however, and this reflects the reduced induced dipole (H2-Hz) repulsion with the longer K+-H2 bond lengths.
*
Using the present results, we can estimate bond strengths and lengths in Rb+.H2 and Cs+-H*. The ionic radii of Na+, K+, Rb+, and Cs+ are -0.97, 1.33, 1.48, and 1.68 A, respectively.22 The expected Rmin (Rb+.H2) and Rmin (Cs+.H2) are then about 3.0 and 3.2 A, respectively, based on Na+/K+.H2. As noted, the bond strengths are expected to vary as Rmin4or Rmin3.Based on theNa+/K+.H2values, theexpected Defer Rb+.H2is then between 1.6 and 1.9 kcal/mol, while that for Cs+.H2 is between 1.2 and 1.6 kcal/mol. Since as Rd,, increases the charge-quadrupole interaction is expected to dominate, the larger De values are probably better estimates. One further point is worth making. For V+ and Co+,the derived binding energies with H2 are 10.2 f 0.5 and 18.2 f 1.0 kcal/ m 0 1 , ~respectively, ~~ which is substantially larger than those for Na+ and K+. Barnes et aLZ3calculated the orbital radii for the first and second transition series of the singly charged ions. For V+(3d4)and Co+(3d8),these values are approximately 0.71 and 0.61 A, respectively. If one extrapolates the Na+.H2 properties to obtain the “pure electrostatic” contributions to bonding in V+.H2 and Co+.Hzclusters,oneobtains 3.8 and4.6 kcal/mol (for moo), respectively. Using this simple model, one would conclude that covalent effects dominate the bonding of transition metal centers with H2. This conclusion must be tempered, however. The lowlying s and d orbitals on transition metal centers allow two effects to occur that cannot be observed in Na+ and K+. First, charge transfer can occur between the H-H u-bond orbital and the empty s-orbital on M+. Also, back donation into the antibonding u* orbital from the filled 3dyz-orbitalon M+ can occur. Second, the approaching H2 ligand can cause reorganization of the M+ d-orbital populations to minimize repulsion (i.e., polarization of the charge on the metal center). Perry et al.10,24 calculate a bond length for Co+-H2 that is substantially shorter than expected from ionic radii considerations discussed above. They indicate this shortening is due primarily to electron correlation which is the primary theoretical expression of charge transfer and M+ charge polarization. From this shortened bond length they calculate an “electrostatic” bond strength of 20.9 kcal/mol. The question then arises, how do you determine”e1ectrostatic” and “covalent” contributions to a bond? In our view, “electrostatic” contributions to bonds are probably best described by charge distributions characteristicof the isolated bonding partners. Effects induced by charge transfer and/or redistribution (hybridization) of metal orbitals by the H2 partner are probably best classified as “covalent” bonding contributions. Nonetheless, it is primarily a semantic question, and both the simple pictures we present here or complex pictures derived from detailed quantum chemical calculations have value.
Conclusions We have made experimental determinations of binding energies and entropies for Na+.(H2)1,z. The agreement with theory is good. The main difficulty in comparing theoretical and experimental binding energies is the uncertainty in the zero point energies. The theoretical results presented here agree well with previous high-level calculations. Our results confirm that the bonding is dominated by electrostatic interactions and is only very slightly covalent. Bond energies and entropies were also determined for K+-(H2)13. The K+ ion binds less strongly to H2 than Na+ due to its larger size. Estimates were made for the binding energy in Rb+.H2 and CS+.H~. Finally, by extrapolating the purely electrostatic binding energies of Na+.H2 and K+.H2 to the first-row transition series, it was concluded that covalent interactions dominate the bonding in V+.H2 and Co+.H2 systems, although ambiguity exists in how
2048
Bushnell et al.
The Journal of Physical Chemistry, Vol. 98, No. 8, 1994 U
to quantify "electrostatic" and "covalent" contributions to a chemical bond.
(A-6a)
Acknowledgment. We gratefully acknowledge the support of the National Science Foundation under Grant CHE 91-19752.
(A-6b) (A-6~)
Appendix. Rotational Energies and Partition Functions in K+/Na+W2)1,2 Na+/K+.H2. Although these clusters are technically asymmetric tops, two of the moments of inertia are almost equal, and they can be treated as symmetric tops.25 Thus
J(J + 1)RdB
E,
+ K2R(eA- 6,)
(A-1)
Again, E11 (the ground-state energy) must be subtracted from Edd. The rotational energy and entropy of neutral H2 are treated similarly. K + / N ~ + S ( H ~ )The ~ . calculation of statistical mechanical quantities for the K+/Na+-(H2)2 clusters is greatly complicated at low temperatures by the almost completely free internal rotation that is present. In this ca~e25.2~,28 EjKKi = J(J + l)RdB
where R is the gas constant. The rotational temperatures are defined by (A-2)
+ K2R(dA- 6,) + K?RdIR (A-7)
where Ki is the internal rotation quantum number (Kiis even for K = even and odd for K = odd).25 The rotational temperature is defined by27928
where h = Planck's constant, k = the Boltzmann constant, I is the moment of inertia, and 6A always refers to the symmetry axis. The corresponding classical partition function is (A-3) where u is the classical symmetry number. This integral form of qrotcan only be used at temperatures greater than 36,,. Since Omax in Na+.H2 is 86.5(6~)and our experimental temperatures range down to 100 K, a direct summation of qrotmust be done. The problem of ortho and para rotational species now arises for both H2 and K+/Na+.H2. One-fourth of the H2 and Na+.H2 molecules will have hydrogen nuclei with paired nuclear spins and even J (in H2) or K (in Na+.H2) rotational levels. Threefourths will have parallel spins and odd J (or K ) levels. The H2 case is discussed in several place^.^^^^^ For Na+.H2, the treatment is completely analogous and the following formulas result: m
m
even LD
m
odd
(A-4~) where gK is the K degeneracy (1 for K = 0, 2 for K > 0). The zero of energy for the odd rotational states is E1 1. The rotational energy and entropy are given by26 m
Here (A-9) where I l and 1 2 are the moments of inertia of the individual internal rotations. In the case of K+/Na+.(H2)2, I1 = 1 2 = ( 1 / 2 ) 1 ~and dlR = 4dA. The classical partition function for completely free internal rotation is then25.27+28 (A-10) Equation A- 10 is of limited use, however, since 6lR is 170°. In order to calculate qmtby direct summation, we must determine the rotational groups (ortho/para type species) which are present. This is a four-step process: first, the rotational point group for the molecule (including internal rotation) must be determined. Second, the irreducible representations of the nuclear spin species present must be found. Third, the irreducible representations of the rotational levels are calculated as a function of Ki,K,and J. Fourth, the various nuclear spin representations are paired with the proper rotational representation to give a molecular wave function of the correct overall symmetry. This process is discussed by Hertzberg25 and has been applied by Wils0n2~to the C2H6 molecule. The results for the Na+/K+.(H2)2 moleculeare outlined below. The point group is a modified D2 group (02 X internal rotation). The number of proper rotations is eight, corresponding to the classical symmetry number. DZIR A+ A-
m
B+ BE m
m
Erot
1 3 = zEeven + z E o d d
(A-5~)
E
CU
1 1
1 1 1 1
1
1 2
-2
2c21
2C21R
2CZIRC21
1 1 -1 -1
1 -1 1
0
0
1 -1 -1 1 O
-1
The+ and-refertothesymmetrywithrespect tointernalrotation. The CzA is a rotation about the symmetry axis (HrNa+-H2 bond axis). The C ~ Iare R the two internal rotations. The 16 possible nuclear spin functions (each hydrogen nucleus has INS = &1/2; 24 = 16) are readily seen to have a reducible representation in the D ~ Igroup R composed of 6A+ 1A- 3B+
+
+
Nat/K+.(H2)1,2 Clusters
The Journal of Physical Chemistry, Vol. 98, No. 8,1994 2049
+
3E. Thus, there are four noninterconverting nuclear spin species present, their populations being equal to their functional abundance (Le., 6/16 of the ions have an At nuclear spin symmetry, etc.) Finding the rotation species present requires determining the character of ( J , K,Ki) under the five symmetry operations in D21~.This gives the reducible representations for different J , K and Ki. These are decomposed into the irreducible representations. This is also discussed by Wilson,29and the results for Na+.(H2)2 are as follows: J
Ki
K
even odd
0 0 0 0 even even odd
0 0 even even even even odd
K/2 ~~
-
-
-
0 0 even odd even odd -
irreducible reDresentation ~~
~~
A+ B+ A+ + B+ A- 32A+ 2B+ 2A- 2B2E
+ + +
These \kmtsymmetries must be paired with the 9"s symmetries above to give an A- symmetry for \ktoUI. Only in this case is *total symmetric with respect to molecular rotations (where two pairs of identical spin 1/2 particles are exchanged) and antisymmetric with respect to internal rotation (where only two identical spin 1/2 particles are exchanged). This symmetry with respect to molecular rotations is also found in CzH4.*5 Thus, the A+ rotational levels are paired with the A- nuclear spin functions. Since 1/16 of the clusters have A- nuclear spin functions, 1/ 16 of the clusters have A+ rotational levels. The resulting partition functions are
even
even
(A-1
The factor of 1/2 in 44 arises from the product of rPNS X I?%, where only 1/4 of the resulting symmetries are A-(E X E = A+ + A- + B+ B-), but the odd rotational levels make up 2E representations. The corresponding values of ErOtand SI, are easily found by analogy with eqs A-4, A-5, and A-6.
+
References and Notes (1) Kubas, G. J.; Ryan, R. R.; Vergamini, P. T.; Wasserman, H. J. J. Am. Chem. SOC.1984,106,451. Wasserman, H. J.; Kubas, G. J.; Ryan, R. R. Ibid. 1986, 108, 2294. Kubas, G. J.; Ryan, R. R.; Wrableski, D. Ibid. 1986, 1239. Kubas, G. J.; Unkefer, C. J.; Swanson, B. I.; Funkushima, E. Ibid. 1986, 7000. (2) Crabtree, R. H.; Hamilton, D. G. Adv. Organomet. Chem. 1988,28, 299. (3) Unpublished data of Crabtree, R. H.; Hamilton, D. J. quoted in ref 2, p 312. (4) Kemper, P. R.; Bowers, M. T. J . Phys. Chem. 1991, 95, 5134. (5) Kemper, P. R.; Bushnell, J.; von Helden, G.; Bowers, M. T. J . Phys. Chem. 1993, 97, 52. (6) Bushnell, J.; Kemper, P. R.; Bowers, M. T. J . Phys. Chem. 1993.97, 11628. (7) Kemper, P. R.; Hsu, M-T.; Bowers, M. T. J . Phys. Chem. 1991,95, 10600. (8) Bauschlicher, C. W., Jr.; Partridge, H.; Langhoff, S. R. J . Phys. Chem. 1992, 96, 2475. (9) (a) Maitre, P.; Bauschlicher, C. W., Jr. J . Phys. Chem. 1993, 97, 11912 (describes calculations on V+/CO+/CU+.H~/H~O/CH~). (b) Maitre, P.; Bauschlicher,C. W., Jr. J.Phys. Chem.,submittedforpublication(describes calculations on V+*(H2)14). (c) Bauschlicher, C. W., Jr.; Langhoff, S. R.; Partridge, H. Modern Electronic Structure Theory; in press. (10) Perry, J. K.; Ohanessian, G.; Goddard, W. A., 111 J . Phys. Chem. 1993, 97, 5238. (11) Curtiss, L. A.; Pople, J. A. J . Phys. Chem. 1988, 92, 894. (12) Dixon, D. A.; Gole, J. L.; Komornicki, A. J . Phys. Chem. 1988,92, 1378. (13) Falcetta, M. F.; Pazun, J. L.; Dorko, M. J.; Kitchen, D.; Siska, P. E. J . Phys. Chem. 1993, 97, 1011. (14) Kemper, P. R.; Bowers, M. T. J . Am. SOC.MassSpec. 1990, 2, 197. (15) Frisch, M. J.; Trucks, G. W.; Head-Gordon, M.; Gill, P. M. W.; Wona. M. W.: Foresman. J. B.; Johnson, B. G.: Schleael. H. B.: Robb. M. A.; Gplogle, E. S.; Gomperts, R.; Andres, J. L.; RaghGachari, K.; Binkley, J. S.; Gonzalez, C.; Martin, R. L.; Fox, D. J.; Defrees, D. J.; Baker, J.; Stewar, J. J. P.; Pople, J. A. Gaussian 92, Revision A; Guassian, Inc.: Pittsburgh, PA, 1992. (16) Frisch, M. J.; Del Bene, J. E.; Binkley, J. S.;Schaefer 111, H. F. J . Chem. Phys. 1986, 84, 2279. Schwenke, D. W.; Truhlar, D. G. J . Chem. Phys. 1985,82, 2418. (17) Schwenke, D. W.; Truhlar, D. G. J . Chem. Phys. 1985, 82, 2418. (18) Wu, C. H. J. Chem. Phys. 1979, 72,783. (19) Lester, W. A., Jr. J . Chem. Phys. 1970, 53, 1511; 1971, 54, 3171. (20) Kramer, H. L.; Herschbach, D. R. J. Chem. Phys. 1970,53, 2792. (21) Grev, R. S.;Janssen, C. L.; Schaefer 111, H. F. J . Chem. Phys. 1991, 95, 5128. (22) Cotton, A. F.; Wilkinson, G. Advanced Inorganic Chemistry, 2nd ed.; Wiley Interscience: New York, 1966. (23) Barnes, L. A.; Rossi, M.; Bauschlicher, C. W. J. Chem. Phys. 1990, 93, 609. (24) Perry, J. K. Private communication, 1993. (25) Herzberg, G. Molecular Spectra and Molecular Structure I I . Infrared and Raman Spectra of Polyatomiic Molecules; Van Nostrand Reinhold: New York, 1945. (26) McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976. (27) Davidson,N. Statistical Mechanics; McGraw-Hill: New York, 1962. (28) Lewis, G. N.; Randall, M. Thermodynamics (revised by Pitzer, K. S.; Brewer, L.); McGraw-Hill: New York, 1961; p 438. (29) Wilson Jr., E. B. J . Chem. Phys. 1938,6, 740; 1935, 3, 276.
