J. Phys. Chem. 1992, 96, 5783-5789
as a hydrogen bond acceptor. Considering its high molecular weight it therefore seems likely that it will be trapped in PSC particles. W a w System. The absorption spectrum of OClO consists of several bands between 280 and 480 nm with a maximum at 351 nm.23 In the gas phase, OClO is photolyzed to C10 and an oxygen atom.24 With radiation with wavelengths close to 360 nm, atomic chlorine and molecular oxygen may form.25v26In argon matrices, irradiated OClO rearranges to C100.”-’2 The ClOO absorption spectrum is found in the region between 220 and 270 nm with its maximum at 250 nm.44 The first step in the rearrangement process is probably not a simple photodissociation to OC1 and 0,since in that case the oxygen atom should have a finite probability to leave the matrix cage where it was formed. If it did, we should have observed ozone formation when we photolyzed OClO in an oxygen-doped argon matrix. F atoms from the photolysis of F2 become separated even at rather low excess ene r g i e ~ ?and ~ F atoms are approximately of the same size as 0 atoms. Note also that the addition of 0 to O2to form O3has been observed in argon matrices at 20 K during warm up experiments with reactants in thermal equilibrium with the matrix. A possible first step in the OClO photolysis could be the direct formation of 0225,26 followed by the addition of C1 to 02.This type of process should be open also to the water complex, if, as seems reasonable, water is bound to the chlorine atom. CIOCI-N2 System. The observation of N20 as a photoproduct of ClOCl in argon matrices in the presence of N2 was expected since Rochkind and Pimente15got N20in addition to ClClO when they photolyzed ClOCl in nitrogen matricks. N20 can form from the addition of O(ID) to N2, but O(lD) cannot diffuse without losing its excitation energy. The nitrogen molecule therefore has to be trapped in the same cage as the ClOCl molecule which is photolyzed. Our present results do not allow us to say whether N 2 0 forms directly from ClOCl photolysis or via ClClO. References and Notes (1) Chemical Kinetics and Photochemical Data for Use in Stratospheric Modeling Evalution Number 9; JPL Publicatin 87-41; JET Propulsion Laboratory, California Institute of Technology: Pasadena, CA, 1990. (2) Okabe, H. The Photochemistry of Small Molecules; J. Wiley: New York, 1978. (3) Bondybey, V. E.; Fletcher, C. J. Chem. Phys. 1976,64, 3615. (4) Foumier, J.; Salama, F.; Le Roy, R. J. J . Phys. Chem. 1985,89, 3530. (5) Rochkind, M. M.; Pimentel, G. C. J. Chem. Phys. 1967, 46, 4481. (6) Chi, F. K.; Andrews, L. J. Phys. Chem. 1973, 77, 3062. (7) Alcock, W.G.; Pimentel, G. C. J. Chem. Phys. 1968,48, 2373. ( 8 ) Carter, R. 0.; Andrews, L. J . Phys. Chem. 1981, 85, 2351. (9) Cheng, B. M.; Lee, Y. P. J. Chem. Phys. 1989, 90, 5930.
5783
(10) Muller, H. S.P.; Willner, H. Air Pollution Restarch Report 34, Polar Stratospheric Ozone Prccediigs of the First European Workshop 3-5 Oktober; Schliersee: Bavaria, FRG, 1990. (11) Arkell, A,; Schwager, I. J . Am. Chem. Soc. 1967,89, 5999. (12) Bhatia, S.C.; Hall, J. H., Jr. J. Phys. Chem. 1981, 85, 2055. (13) Jansen. M.; Schatte, G.; Tobias, K. M.; Willner, H. Inorg. Chem. 1988,27, 1703. (14) Christe, K.0.;Schack, C. J.; Curtis, E. C. Inorg. Chem. 1971, 10, 1589. (IS) Molina, L. T.; Molina, M. J. J . Phys. Chem. 1978, 82, 2410. (16) Knauth, H. D.; Alberti, H.; Clausen, H. J . Phys. Chem. 1979, 83, 1604. (17) Mishalanie, E. A,; Rutkowski, C. J.; Hutte, R. S.;Birks, J. W. J . Phys. Chem. 1986, 90,5578. (18) Molina, M. J.; Ishiwata, T.; Molina, L. T. J . Phys. Chem. 1980.84, 821. (19) Voght, R.; Schindler, R.N. Air Pollution Research Report 34, Polar Stratospheric Ozone Prodings of the First European Workshop 3-5 O k t o k , Schliersee: Bavaria, FRG, 1990. (20) Bell,A. J.; Pardon, P. R.; Hickman, C. G.; Frey, J. G. J. Chem. Soc., Faraday Trans. 1990,86, 3831. (21) Schwager, I.; Arkell, A. J. Am. Chem. SOC.1967,89,6006. (22) Wahner, A.; Tyndall, G. S.;Ravishankara, A. R. J . Phys. Chem. 1987, 91, 2734. (23) Richard, E. C.; Wickham-Jones, C. T.; Vaida, V. J . Phys. Chem. 1989,93, 6346. (24) Colussi, A. J. J . Phys. Chem. 1990, 94, 8922. (25) Ruhl, E.; Jefferson, A.; Vaida, V. J . Phys. Chem. 1990, 94, 2990. (26) Bishenden, E.; Haddock, J.; Donaldson, D. J. J . Phys. Chem. 1991, 95, 2113. (27) Fredin, L.; Nelander, B. J. Mol. Srrucr. 1973, 16, 217. (28) Enndahl. A.: Nelander. B. J . Chem. Phvs. 1986.84. 1981 (29) No’ble, P.N:; Pimentel, G. C. Spectrolhim. Acta 1968, 24A, 797. (30) Fredin, L. Chem. Scr. 1974,5, 193. (31) Calvert, J. G.; Pitts, J. N. Photochemistry; J. Wiley: New York, 1966. (32) Rochkind, M. M.; Pimentel, G. C. J. Chem. Phys. 1965, 42, 1361. (33) Gmelins Handbuch der Anorganischen Chemie 8 Aujlage; Chlor Erginzungsband Teil B-Liefernung 2; Verlag Chemie: Weinheim/Bergstr., 1969. (34) Barnes, A. J. J. Mol. Strucr. 1983, 100, 259. (35) Maillard, D.; Schriver, A.; Perchard, J. P.; Girardet, C. J. Chem. Phys. 1979, 71, 505. (36) Bohn, R. B.; Hunt, R. D.; Andrews, L. J. Phys. Chem. 1989,93,3979. (37) Jacox, M. E.; Milligan, D. E. J . Mol. Specrosc. 1972, 42, 495. (38) Engdahl, A.; Nelander, B. Unpublished. (39) Gole, J. L. J . Phys. Chem. 1980, 84, 1333. (40) Schack, C. J.; Philipovich, D. Inorg. Chem. 1970, 9, 1387. (41) Andrews, L.; Johnson, G. L. J . Chem. Phys. 1982, 76, 2875. (42) Legon, A. C.; Willoughby, L. C. Chem. Phys. Lett. 1983.95, 449. (43) Mulliken, R. S.;Person, W. B. Molecular Complexes; J. Wiley: New York, 1969. (44) Johnston, H. S.;Moms, E. D., Jr.; van den Bogaerde, J. J . Am. Chem. Soc. 1969, 91, 7712. (45) Feld, J.; Kunttu, H.; Apkarian, V. A. J. Chem. Phys. 1990,93, 1009. (46) Engdahl, A.; Nelander, B. J . Mol. Srruct. 1989, 193, 101.; (47) Nelander, B. J . Phys. Chem. 1988, 92, 5642.
A Theoretical Study of the Reactivity of Pd Clusters with Methane Margareta R. A. Blomberg,* Per E. M. Siegbahn, and Mats Svensson Institute of Theoretical Physics, University of Stockholm, Vanadisviigen 9, S - I 1346 Stockholm, Sweden (Received: January 2, 1992; In Final Form: February 25, 1992) Correlated, size-consistent calculations have been performed for the reaction between Pdz and CHI. Some calculations have also been done for Pd3. The results are used to interpret the experimental observations for the reactivity of palladium clusters of different sizes with CH4. The calculations indicate that the C-H bond is broken with only a small barrier of 4-6 kcal/mol and thus confirm the experimental interpretation of dissociative chemisorption. This result is in contrast to the case of N2, which was found in calculations to be only molecularly bound. There are two dominating effects for the size of the barrier in the methane reaction. The first factor is the repulsion in the entrance channel, and the other factor is the capability of strong bond formation in the product channel. For palladium clusters of different sizes these two effects compete, and this is the origin for the maximum in the curve for the reactivity versus cluster size found experimentally a t about 10 atoms.
Blomberg et al.
5184 The Journal of Physical Chemistry, Vol. 96, No. 14, 199‘2
standards. From a quantum chemical point of view, there are two approaches. The first approach is to actually treat the large problems, but then rather low-accuracy methods have to be used. The second approach is to try to find representative results for the smallest possible clusters and treat these systems with highaccuracy methods. This latter approach has been taken, for example, in a recent treatment of the reactivity between palladium clusters and N2 where the Pd2N2system was studied in detail., This is also the approach taken here where the particular reactivity of palladium clusters toward methane3 is explicitly investigated only for Pd,. An attempt is then made to extrapolate the Pdz result to larger clusters to try to explain the experimental reactivity trend. In a recent paper by Fayet et al.,, palladium clusters with up to 25 atoms were studied for reactions with seven different compounds including methane. One of the most interesting results of that study was the finding that most palladium clusters were found to activate methane. The only other metal, for which methane activation has been noticed, is platinum! Other metals which had been investigated previously were, for example, iron,5 rhodium: and aluminum.’ It should in this context also be noted that infinite palladium metal surfaces are not particularly active toward methane, so the observed reactivity is an interesting property which is particular for rather small clusters ( n < 25). The activation of C-H bonds in alkanes is a topic of high interest also in homogeneous catalysis (see for example ref 8). The first observations of C-H activation of saturated hydrocarbons were reported in 198293’0for iridium complexes. Since then, a few other complexes have been found to be active including rhodium,”,12 iron, rhenium, and osmium comple~es.’~No active palladium complex has yet been reported. Theoretically, the reactivity of the entire sequence of second-row transition-metal atoms toward methane has recently been studied.I4 The palladium atom was found to be the atom after rhodium with the lowest activation barrier for breaking the C-H bond. The barrier for the palladium atom is 16 kcallmol and for rhodium, 14 kcal/mol. However, the product PdHCH, complex is, in contrast to the corresponding rhodium complex, only in a local minimum with a negative binding energy with respect to the atom and methane of 9 kcal/mol. Poor binding energies of the product complexes could thus be one reason palladium complexes have not been found to activate alkanes as well a rhodium complexes do. In a recent study, very similar to the present one, the reaction between N2 and Pd, was studied? In this case the conclusion from the experiments was that N, was dissociatively chemisorbed. This conclusion was based on the formation of a Pd2N2complex which was stable under the experimental conditions. The results of the calculations did not support this conclusion, and the experiment was instead interpreted in terms of a formation of an unusually stable moderately bound Pd2N2complex. It was also suggested that nondissociative N2 chemisorption is probably observed also for the larger clusters studied experimentally. A theoretical study, related to the present one, has previously been done by Nakatsuji et al.15for the reaction between Pd, and H,. It was found that the H2 bond was broken without any bamer and that the moderately bound Pd2H2complex has nearly the same energy as the dissociatively bound H-Pd2-H system. The reaction was found to proceed with the H-H axis parallel to the Pd-Pd axis. This is a quite unexpected reaction pathway, and we therefore decided to reinvestigate this system following other pathways. 11. Computational Details
Calculations have been performed for the systems Pd2CH4, Pd2H2,and Ni2CH4using standard basis sets and size-consistent correlation methods where all valence electrons were correlated. To obtain more reliable relative energies, some calculationswere also performed for the Pd2CH4system using a somewhat larger basis set including f functions on palladium. Finally, a few calculations were made on a Pd3 cluster. The standard basis set used in this paper is for palladium the Huzinaga (17s,l lp,8d) primitive basis,I6 augmented with one
TABLE I: Geometrical Parameters Used in the Calculntions on the Pd2CH4Systed (Bond Distpnces in .o)
Pd-Pd Pd2C Pd2-H P d C Pd-H product complex square planar trans cis transition state q2 complex M2
+ RlR2
4.88 4.97 4.97 5.10 5.43 4.81b
3.36
1.80
3.36 4.69
1.80
4.15 3.68 3.68 4.22
3.03 2.87 2.87 3.12
bend angle, deg 180 78
For the methyl group the C-H bond lengths are kept at 2.08 a. and the HC-H angle at 107.8’. bFor the ’2: ground state. The Pd-Pd state is 5.44 a,. distance in the
diffuse d function and two p functions in the 5p region, yielding a (17s,13p,9d) primitive basis. A generalized contraction scheme” is used, leading to a [8s,7p,4d] contraction. In a few calculations a set of three f exponents were added on palladium and contracted to one function leading to a [8s,7p,4d,lfl contracted basis set. The basis set used for nickel is of similar quality as the standard basis set for palladium. A (14s,9p,5d) primitive basis set was contracted to [6s,5p,2d] using a generalized contraction scheme with exponents and contraction coefficients taken from Wachters.IB This basis set was augmented with one diffuse d function leading to a final [6s,5p,3d] contracted basis. For carbon the Huzinaga (9s,5p) set’9 was generally contracted to a [3s,2p] set. One d function, with exponent 0.63, was added, leading to a [3s,2p,ld] contraction. For the hydrogen in the C-H (or H-H) bond to be broken the primitive (5s) basis from ref 19 was used, augmented with one p function with exponent 0.8 and contracted to [3s,lp]. The inactive methyl hydrogens were described by the (4s) basis from ref 19 contracted to [2s] and with the exponents scaled by a factor of 1.2. The zeroth-order wave function is generated at the SCF level, and electron correlation is accounted for using the size-consistent modified coupled pair functional (MCPF) method.20 All valence electrons on both palladium (nickel) and CH4are correlated, Le., 28 electrons for Pd2CH4(Ni2CH4). The MCPF method is based on one reference state, and it can be questioned whether this treatment of the correlation effects is accurate enough. To investigate the multireference effects, average coupled pair functional (ACPq2l calculations were performed on Pd2. Using the standard basis set, the ’2: ground state of Pd2 was found to be bound by 17.0 kcal/mol in a two reference state ACPF calculation, as compared to 14.8 kcal/mol in the MCPF calculation. The multireference effects on the relative energies of the CH4 reaction are therefore expected to be only a few kcal/mol. The Pd3calculations were performed using a newly developed relativistic effective core potential (RECP).22 The valence basis set used in these calculationsdid not contain f functions. MCPF calculations were performed correlating all 30 valence electrons on Pd,. The accuracy of the RECP was tested on Pd2. Two reference state ACPF calculations on the ground state gave a binding energy of 17.6 kcal/mol. This value is very close to the corresponding all-electron result of 17.0 kcal/mol, and the difference is due to a larger basis set superposition error in the RECP calculations. It is therefore expected that the RECP results for Pd, are reliable to within a few kcal/mol. The relativistic contribution to the energy in the all-electron calculations is obtained by means of first-order perturbation theory where the mass-velocity and the Darwin terms are retained in the perturbation operator.23 Calculations were performed for several different structures of the Pd2CH4system, which are depicted in Figure 1, and the most important geometrical parameters used are given in Table I. These parameters were chosen in the following way. At long distance the methane geometry is taken from experiment, and the Pd, distance is optimized at the MCPF level using the basis set including f functions. For the molecularly bound q2 complex (Figure 1a) an undistorted methane molecule was assumed, and
Reactivity of Pd Clusters with Methane
The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 5785 TABLE Ik Calculated Energies for the Reaction of CH4 with Pd2 and NiZand H2 with Pd2" Pd2CH4 Ni2CH4 Pd2H2 state AE state AE state AE product complex square planar 'A' -3.5 (-6.2) 'A'' 19.6 'Al -35.1 1.6 'A' trans 'A' 21.0 'A1 -13.9 cis 'A' 7.2 3A' 29.0 'A1 -9.1 transition state 'A' 10.4 (6.7) 3A'' 65.4 v2 complex 'A' 7.1 (2.6) M 2 + R1R2 '2; 0 32; 0 32; 0 '2; 11.2 (7.0)
a
AE
state M2
32; 12;
M -f M
'S(d'O)
14.8 (10.8) 3.6 (3.8)
0
state '2;
'D(d9 s')
AE 44.4
0
"The energies (in kcal/mol) are given relative to ground-state metal dimer and free CH4 or H2. The calculated binding energies of the dimers are also given, relative to ground-state atoms. Results with f functions on Pd are given within parentheses.
b
C
d
e
Figure 1. Different structures investigated for the Pd2CH4system: (a) molecularly bound T~ complex, (b) pseudo-square-planar product complex, (c) transition-state, (d) cis form of the product complex, and (e) trans form of the product complex.
the distance between the carbon and the Pd2 midpoint was optimized together with the Pd-Pd distance, at the MCPF level. In all calculations on the q2 complex the larger hydrogen basis set, including one p function, was used on the two hydrogens pointing toward the palladium dimer. In all geometry optimizations on the Pd2CH4.systemthe standard basis on palladium, without f functions, was used. For the pseudo-square-planar Pd2HCH3
product complex (Figure lb) the Pd-Pd, Pd-C, and Pd-H distances were optimized at the SCF level. These Pd-C and Pd-H bond distances were kept at the transition state (Figure IC),and the bend angle determining the bond distance of the C-H bond to be broken was optimized two dimensionally together with the methyl tilt angle. Finally, the Pd-Pd distance was optimized. All optimizationsat the transition state were performed at the SCF level. Two more structures of the Pd2HCH3product complex, shown in Figure 1 and referred to as the trans (Figure Id) and the cis (Figure le) structures, were finally investigated. For these structures the Pd-Pd distance was taken from ref 15, and the Pd-C and Pd-H distances were taken from the PdHCH3 complex of ref 24. In the calculations performed for the Pd2H2system, the Pd-Pd and Pd-H distances for the different product structures (square planar, trans, and cis) were taken from the corresponding Pd2HCH3structure. In the calculations on the Ni2CH4system the geometrical parameters were taken from the Pd2CH4system. For the Pd2CH4and the Pd2H2systems the closed-shell singlet state is the lowest for all structures, while for the Ni2CH4system triplet states were found to be lowest. As a reference point for the results of the calculations presented in the next section, it is useful to know the splitting between the two lowest states of the palladium atom calculated with the present basis sets and methods. The experimental energy difference between the ' S (d'O) ground state and the 3D(d9s1)excited state is 0.95 eV, whereas the calculated result using the standard basis set is 0.65 eV. The main discrepancy compared to experiment is due to the lack of f functions in the basis set. Inclusion of these functions leads to an improved value of the splitting of 0.88 eV. For nickel the most relevant splitting is instead between the 3F(d8s2)and the 3D(d9s1)states, which are practically degenerate experimentally. The standard basis set makes the 3Fstate the ground state by 1.54 eV at the SCF level and by 0.66 eV at the correlated level. The calculated values include relativistic effects.
In. Results and Discussion The results of the present calculations for selected points of the potential energy surface of Pd2CH4are given in Table I1 and displayed in Figure 2. The most interesting result is perhaps that on the singlet surface the transition state for breaking the C-H bond is below the Pd2 + CH4 asymptote. The energy difference to the asymptotic value is 0.3 kcal/mol for the basis set including f functions (0.8 kcal/mol for the standard basis set). The product complex, where a C-H bond has been broken, is the lowest point on the singlet surface with a binding energy with respect to the singlet asymptote of 13.2 kcal/mol using the basis set with f functions. There is also a substantially bound q2 complex, with a nearly undistorted methane on this surface. The binding energy with respect to the singlet asymptote is 4.4 kcal/mol using the
5786 The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 AE [kcollmol]
t
-5
\
\t
'A'
t1 Pd,+
CH, F i i
7,complex
Transltlon state
PdzHCH,
2. Calculated potential energy surface for the dissociation of CHI
on Pd2.
basis set with f functions. If it were not for the fact that the ground state of Pd2 is not a singlet, the present results would easily explain the experimental observation that Pdz activates methane. As seen in Figure 2, the situation is more complicated when the transition state is compared to the lowest triplet asymptote of Pd2. In fact, the actual barrier height counted from the ground-state asymptote will be nearly equal to the singlet-triplet splitting of Pdz. Since the actual value of the barrier height for the C-H insertion reaction depends directly on the singlet-triplet splitting of Pdz, several larger calculations were performed for this quantity. The standard basis set gives a binding energy for the ground 32: state of 14.8 kcal/mol relative to two 'S Pd atoms and for the excited '2: state of 3.6 kcal/mol. The splitting between the states is thus 11.2 kcal/mol. Adding the f function described in section I1 decreases the binding energy for the ground triplet state to 10.8 kcal/mol, whereas for the excited singlet state there is a small increase of the binding energy to 3.8 kcal/mol. The addition of a single f function has thus reduced the splitting between the states by 4.2 kcal/mol, and the barrier for the Pdz insertion reaction is reduced by almost the same amount, 3.8 kcal/mol. The splitting changed from 11.2 kcal/mol down to 7.0 kcal/mol. The decrease in the splitting when the basis set was i n d was not unexpected since the excited '2' state has more 4d electrons and thereby more correlation energy tkan the 32: state. With this line of reasoning, it might also be expected that a further increase in the basis set should reduce the splittng and the barrier height even further, but this does not turn out to be the case. Going to a very large Pd basis consisting of [8~,7p,Sd,3flANO-contracted functions gives results very close to those obtained with the basis set containing only one f function. The splitting actually increases slightly to 7.2 kcal/mol. The binding energy for the 'Z: state is with this basis set 11.2 kcal/mol and for the 'Zistate, 4.0 kcal/mol. The effects on the splitting between the Pdz states from adding f functions to the basis set can be traced to changes in the splitting between the IS and 3Dstates of the palladium atom. When the single f function is added, this splitting is changed from 0.65 to 0.88 eV, which is already close to the experimental value of 0.95 eV. The largest basis set, including three f functions, improves the splitting further to 0.92 eV. With this interpretation, further increases of the basis set should not influence the barrier height very much, since the atomic splitting is now so close to the experimental value. A best estimate of the barrier height for the breaking of the C-H bond of methane by the ground-state palladium dimer should therefore be in the range 4-6 kcal/mol. In the most straightforward interpretation of the experimental result that Pdz activates methane, the triplet Pd2 molecule has sufficient amount of kinetic energy to surmount the 4-6 kcal/mol barrier. Since one also has to remember that in surmounting the barrier the Pdz molecule has to switch spin state, which requires
Blomberg et al. spin-orbit coupling, it is perhaps worthwhile to suggest also another possible mechanism for breaking the C-H bond which might be present in the experiments. In this mechanism the Pd2 molecule exists also in its excited singlet state in the cluster beam, which would then lead to a barrierless dissociation reaction with methane (see Figure 2). One factor which could cause this to m u r is that the Pd atom has a singlet (IS(4d'O)) ground state. Two Pd atoms would thus form singlet Pdz with a high probability, and only afterward will the Pd2 molecule fall down into its ground triplet state. Which of these two dissociation mechanisms is most important in the experiments must be a complicated function of the time scale of these processes and also of the temperature of the clusters, and a detailed analysis is beyond our present capabilities. However, quite generally the present calculations support the experimental conclusions that palladium clusters have unusually low barriers for breaking the C-H bond in methane, much lower than for any well-defined transition-metal surface or any transition-metal atom, for example. In order to gain insight into the origin of the unusual capabilities of palladium clusters to dissociate methane, a few comparative calculations on the corresponding nickel dimer reaction have also been performed with results also given in Table 11. The difference between Pdz and NiZis perhaps even more striking than might have been expected. Standard basis sets give a barrier height of Niz for breaking the C-H bond, which is as high as 65.4 kcal/mol compared to the correspondingvalue for Pdz of only 10.4 kcal/mol. It should be noted that the present calculations are less accurate for the nickel than for the palladium case, due to the presence of relatively large near-degeneracy effects for nickel, but this can only explain a t most 10-15 kcal/mol of the difference between the barrier heights. For the product complexes the binding energy difference is smaller, 23.1 kcal/mol, between PdzCH4and NizCH4. One major part of the explanation for the difference between palladium and nickel can be found in the bond strengths of the dimers. Using the standard basis sets Pdz is only bound by 14.8 kcal/mol whereas Nil is bound by 44.4 kcal/mol. In forming the bonds to CH3and H the intermetallic bond is essentially broken so that most of this binding energy is lost in the final products. One should thus expect a difference in binding energy for the final products of about 30 kcal/mol. Since the actual difference is somewhat smaller, 23.1 kcal/mol, also other effects are present such as differences in repulsion to nonbonding d electrons, differences in overlap, etc. The reason Pdz is so much weaker bound than Niz is that Pd, in contrast to Ni, has to be promoted to an excited state (d9s1)to form the bond in the dimer. Using the standard basis set, this promotion effect is 30 kcal/mol and thus explains the whole binding energy difference. The same promotion effect should prevail also for larger clusters, and it will probably lead to the fact that some of the Pd-Pd bonds are weaker than they are for an infinite Pd surface. As seen in Table 11, the difference between the palladium and nickel reactions is even larger at the transition state, 55.0 kcal/mol, than for the final products, 23.1 kcal/mol. One of the reasons for the larger difference is simply the accuracy of the calculations, which is (as already mentioned) lower for the transition state of the nickel reaction. However, this would amount to only up to 10-15 kcal/mol of the difference. The remaining difference leads us to one of the main effects which makes palladium clusters active in breaking the C-H bond in methane. In a recent study of the methane reaction for all second-row transition-metal atoms, it was found that the repulsion in the entrance channel is of major importance for the size of the barrier.I4 Therefore, for the atoms it is important to have a low-lying dI0 ("so") state since the repulsion is largely determined by the number of s electrons. The same repulsion effect should be present for metal clusters. At the transition state for the Pdz reaction the 4d population is as high as 9.7 (see Table HI), whereas for the Niz reaction the 3d population is only 8.8, implying larger s and p populations for the nickel case since the metal charges are about the same. The smaller number of s,p electrons for Pdz should have a large effect on the barrier height and will thus explain why the difference between the energies for the palladium and nickel reactions is so
The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 5181
Reactivity of Pd Clusters with Methane TABLE III: Mulliken Population Analysis for tbe Reaction of CH, witb Pdz lad Nii, lad HZwitb Pd2, Using tbe Standard Basis, and chrrge, qM, and Valeace d Poplllations on b c b Metal Atom
P d-X R. 4Pd
P d_ d 2_
Ni,CH, . qNi
3d
4W
+0.3 +0.3 +0.1 +0.3
8.9 8.8 8.7 8.1 8.6 8.8
+0.1
9.4
+0.1
9.2
+O.O
9.4
product complex
"square planar" trans (Mc) trans (MH) cis (Mc) cis (MH) transition state n2 complex 32:
'2;
+0.2 +0.2 +0.1 +0.2 +0.1
+0.1 0.0
9.3 9.2 9.2 9.2 9.2 9.7 10.0
0.0 +0.1
Pdi 4d
Ni2 3d
9.4 9.9
8.8
much larger at the transition state than for the product complexes. It should be added that palladium is the only one of the first- and second-row transition-metal atoms which has an so ground state. The populations for Pd2CH4and Ni2CH4are given in Table 111. The most striking difference between these two systems has already been mentioned: the d population at the transition state. For the product complexes the difference is less pronounced with 9.3 4d electrons for palladium and 8.9 3d electrons for nickel. These populations clearly show that the d9s1state is the optimal state for forming the two covalent bonds required for the product complexes. This state is also the leading state for the metal atoms in the bulk metals and for most atoms in the clusters. However, due to the gain in atomic energy there will be a tendency for some palladium atoms in a cluster toward adopting an so (d'O) state which will not be present for nickel. As an interesting comparison, for cobalt and iron clusters there will be a corresponding tendency toward adopting s2 states instead, which could explain the irregularities observed for different cluster sizes of these atoms in their reactivities toward adsorbate^.^^,^^ On the basis of the present results, it is tempting to speculate over the origin of the trend of the results obtained experimentally for the reactivities of the larger palladium clusters toward methane.3 These results can be described schematically in the following way. The Pdl, Pd3, and Pd4 clusters are totally unreactive in contrast to Pdz. From Pd5 and upward the reactivity increases with a maximum at about 10 atoms and then decreases down to a minimum at about 18-20 atoms, where again an increase in reactivity starts. The Pd9 cluster is a notable exception to this general trend with a very low reactivity. Finally, the largest clusters studied and also the infinite palladium surface have a rather low reactivity compared to the smaller clusters. Most of the results can be rationalized by considering two of the main factors determining the reactivities, described above. The first of these factors is the presence of a state which can bind the final products of the dissociation. It is expected that the capability to bind the final products will in general increase as the clusters get larger since the d9s' state will be more and more dominating. This state will provide at least one covalent bond which can be used to bind the final products of the dissociation. This explains the general increase of the reactivities up to 10 atoms. To understand why the reactivities decrease for clusters larger than 10 atoms, the other factor determining the barrier has to be considered. This factor is tha local accessibility of states with few s electrons, which is important in order to reduce the repulsion in the incoming channel. The least repulsive atomic state is the so state, as mentioned above. It is expected that the energy required to reach such a state should increase at about the same rate as the d9s1 state becomes more and more important for the larger clusters. Since the latter state is important for forming a strongly bound final product, there are thus two competing factors which determine the barrier height. A maximum in the reactivity as a function of cluster size is therefore expected, and from the experiments it appears that this maximum is at about 10 atoms. The detailed reason for the minima at 9 atoms and at 18-20 atoms is more difficult to give, but the shell-closing effect is the most
likely explanation. It can be noted that for singly valent metals shell closings generally do occur in these regions, at 8, 18, and 20 atoms. It is possible that palladium is effectively monovalent for these clusters, but this speculation needs to be supported by detailed calculations. A few preliminary calculations were also performed for Pd3 in order to get an understanding of the nonreactivity of this cluster. Correlated calculations for this cluster have been presented previously by Balasubramanian?' who has also done similar calculations for Pd2.28 The present calculations agree reasonably well at a qualitative level with those of ref 28 for Pd2,even though there are notable quantitative differences. For example, our binding energy for the '2: state is 4.0 kcal/mol whereas about 12 kcal/mol is obtained in ref 28. For Pd3 the differences between the present results and those of ref 27 are very large. Our calculations used a newly developed relativistic effective core potential (RECP)22 which was tested against all-electron calculations for several systems including Pd2 and gave very accurate results (see Computational Details). In ref 27 an RECP developed by LaJohn et was used, and it appears that the two different RECP's give very different results. For example, for the 'Al state, which is composed of three dl0 atoms, we obtain a binding energy of 16 kcal/mol with respect to ground-state atoms, whereas about 60 kcal/mol is obtained in ref 27. Our ground state is 3B2(not the same as in ref 27), which can be described as a triangular molecule with a Pd2 unit in the '2: state bridge binding a d'O Pd atom. If CH4 approaches this Pd3 cluster, one likely pathway would be similar to the one for Pd2, Le., with a C-H bond perpendicular to the Pd-Pd axis of the Pd2 unit of the molecule. Again, in analogy to the Pdz case, to break the C-H bond this Pd2 unit has to be promoted to a '2; state with two dI0 atoms. This promotion energy is only 7 kcal/mol for Pd2, but for Pd3 it is as large as 19 kcal/mol. This result would at least rule out a similar type of dissociation for Pd3 as for Pd2 and would be one part of an explanation for the nonreactivity of Pd3. The geometries at the transition state and for the final product of the Pd2CH4system are given in Table I and Figure 1. The reaction path follows a route where the C-H bond to be broken is essentially perpendicular to the Pd-Pd axis and the product is a pseudo-square-planar complex. Two other structures have been investigated for the product complex. The trans isomer (Figure Id), where methyl binds to only one palladium atom and hydrogen to the other, is found to be 5.1 kcal/mol higher in energy at the MCPF level (standard basis sets) than the optimal square-planar complex where the ligands are bridging Pd2. The cis isomer (Figure le) was found to be 5.6 kcal/mol higher than the trans isomer at the MCPF level. A structure where CPdPdH is linear is quite unfavorable with an energy that is 40 kcal/mol higher than the trans isomer at the SCF level. The latter result is easy to understand. The bonding Pd atoms are in a d9s' state with one singly occupied d orbital and one singly occupied s orbital. When these orbitals are hybridized, they will form lobes at right angles to each other so that bonds with 180° angle, as in the linear type structure, will be unfavorable. The slightly lower energy for the trans than for the cis isomer is probably due to weak steric interactions. The main reason the bridging structure is optimal is that the spherical hydrogen atom can simultaneously bind to both palladium atoms in this structure. There are also ionic contributions to the bonding in this structure which are important. As seen in Table 111, the palladium atoms are slightly positive whereas the carbon atom is strongly negative, which will both lead to ionic bonding and facilitate methyl bonding to both palladium atoms. With a pure covalent bond to methyl there would be a strain introduced since methyl has a rather directional bond in contrast to hydrogen. As an interesting comparison, it should be noted that there is a clear difference between the present Pd2CH4system and the recently studied Pd2N2system.2 The side-on structure of the latter system was found to be 17 kcal/mol more stable with N2 parallel to the Pd-Pd bond than with the two bonds perpendicularly bridging. The reason for this difference between N2 and CH4 must be due to the retention of most of the intramolecular N-N bonding in contrast to the C-H bond which is broken at
Blomberg et al.
5788 The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 the transition state. The positions of the nitrogen atoms in PdzNz are thus constrained to rather short N-N distances and cannot find as optimal positions as H and CH3 in PdzHCH3. There are also clear differences in the repulsion between the nonbonding electrons in the two systems. The reaction between Pdz and H2 has previously been studied theoretically by Nakatsuji et a1.I5 as a model for Hz dissociation on a palladium surface. High-accuracy correlated methods were used, but the results of that study differ markedly from some of the results obtained in the present study for PdzCH4. The major difference concerns the reaction pathway which we have found to have the C-H bond perpendicularly bridging the Pd-Pd axis, whereas a parallel orientation between the H-H axis and the Pd-Pd axis is described in ref 15. This difference could of course be due to real differences between the systems, and to be able to make a more direct comparison to ref 15, calculations were performed for the final product of PdzH2with results given in Table 11. As can be seen in this table a square-planar ground state, similar to the one for Pd2CH4,is found also for the PdzH2system. In fact, the energy of this configuration is found to be as much as 26 kcal/mol lower than for the cis configuration described in ref 15. This is larger than the difference between the corresponding Pd2CH4geometries of about 11 kcal/mol, which is not unexpected since both hydrogens can form bonds to both palladium atoms in the square-planar structure of Pd2Hz. The CH3 group, with its directional bond, is under some strain in the squareplanar form of PdzCH4. As a final comment, it is clear that the cis form of PdzHzhas definite limitations as a model for the Hz dissociation on a palladium surface, since it is far away from the optimal reaction pathway for H2 both for Pd2 and for the infinite surface.
IV. Conclusions The reactivity of palladium clusters with methane has been studied for the particular case of the palladium dimer. The reaction is shown to occur on the lowest singlet potential energy surface. On this surface methane is first bound in a molecular precursor state with a binding energy of 4.4 kcal/mol. Methane then approaches Pdz with a C-H bond perpendicular to the Pd-Pd axis. When the C-H bond starts to break, the energy goes up until a transition state is reached, which is still 0.3 kcal/mol lower than the singlet asymptote. The final product complex has a pseudo-square-planar geometry and is bound by 13.2 kcal/mol with respect to the singlet asymptote. A trans configuration is found to be 5.1 kcal/mol higher in energy and a cis configuration still 5.6 kcal/mol higher in energy. Since Pdz has a triplet ground state, the reaction between methane and Pdz has to proceed by an excitation to the singlet state. Since there is no barrier on the singlet surface, the singlet-triplet splitting in Pdz will be close to the actual bamer height for the reaction. The best value for this splitting obtained here is 7.2 kcal/mol, and the barrier height is therefore estimated to be in the 4-6 kcal/mol range. A few preliminary calculations on Pd3indicate that a similar reaction mechanism as for Pdz would lead to a barrier which is 12 kcal/mol higher than for Pd2 and can thus explain why Pd3 is found to be unreactive in the cluster experiments. The main factors responsible for the barrier height of the methane reaction are very similar to the ones recently described for the reaction between methane and single second-row transition-metal atom.14 For these atoms it was found that to obtain a low barrier it was important to have both a low repulsion in the entrance channel and fairly strong bond formation in the product region. The atomic state with the lowest repulsion is the state with the lowest number of s,p electrons, and this is the so state (the dIo state for palladium). For Pd2this corresponds to the '2; state built from two dIo atoms. To form a strongly bound product state, which has two covalent bonds to hydrogen and methyl, a low-lying state with two open shells is required. This requirement is fulfilled by the ground '2: state of Pd2, and for the atoms the best state of this type is the s1 state (the d9s1state for palladium). For the second-row transition-metal atoms these two requirements with low-lying so and s1 states are best fulfilled by the rhodium
atom, which therefore has the lowest barrier for the methane reaction. For palladium clusters, these two requirements can explain the general shape of the curve for the reactivity versus cluster size. The dominating atomic state for an infinite palladium system is the s1 state, and this state will therefore dominate the larger clusters. This state can form strong, covalent bonds which is necessary to get stable products but is rather repulsive in the entrance channel. Since the palladium atom has an so ground state, there will be a tendency for the smaller clusters to have low-lying states with large contributions from this state, which is less repulsive than the s1 state but cannot form covalent bonds for the product. There are thus two competing effects which will lead to a reactivity maximum for some optimal cluster. From experiments it appears that this cluster has about 10 atoms. The two reactivity minima found for clusters with nine atoms and with 16-18 atoms should be due to shell-closing effects. A few general comments can finally be made concerning the reactivity of palladium clusters toward different adsorbates. These comments will be based on the present results for methane and the previously obtained results for NZs2It has been found for the Pdz cluster that bridge bonding is an absolute requirement for strong bond formation for both N2 and CH4. The reactivity curves as a function of cluster size for these adsorbates are as a whole very similar, with minima and maxima at roughly the same positions. It is therefore likely that bridge bonding will continue to be important also for the larger clusters. The same appears to be tme for HZ. The reactivity curve for CzH4 has a maximum at about the same place as these other adsorbates, but the deep minima at nine atoms and in the 16-18 range are missing. This can be interpreted as a similar dominating binding mechanism but that for CzH4 there is also another binding site with strong bond formation available. From the relatively strong binding energy known for ethylene to one palladium atom, this binding site is likely to be an on-top site. As a final speculation, since PdCO also has a strong bond, the reactivity curve for CO should be more similar to C2H4 than to NZ,but this remains to be shown by experiments. R-try
NO. CH4, 74-82-8;Pdz, 12596-93-9.
References and Notes (1)Small Particles and Inorganic Clusters; Echt, O., Recknagel, E., Eds.; Springer-Verlag: Berlin, 1991. (2)Blombcrg, M. R. A.; Siegbahn, P. E. M. Chem. Phys. Lett. 1991,179, 524. (3) Fayet, P.;Kaldor, A.; Cox, D. M. J . Chem. Phys. 1990,92, 254. (4)Trevor, D. J.; Cox, D. M.; Kaldor, A. J . Am. Chem. SOC.1990,112, 3142. ( 5 ) Whetten, R. L.; Cox, D. M.; Trevor, D. J.; Kaldor, A. J. Phys. Chem. 1985,89, 566. (6)Cox, D. M.; Trevor, D. J.; Whetten, R. L.; Kaldor, A. J . Phys. Chem. 1988,92,421. (7)Zakin, M. R.; Cox, D. M.; Kaldor, A. J . Chem. Phys. 1988,89,1201. ( 8 ) Selective Hydrocarbon Activation: Principles and Progress; Davies, J . A., Watson, P. L., Greenberg, A., Liebman, J. F., Eds.; VCH Publishers: New York, 1990. (9)(a) Janowicz, A. H.; Bergman, R. G. J . Am. Chem. Soc. 1982,104, 352. (b) Janowicz, A. H.; Bergman, R. G. J . Am. Chem. SOC.1983,105, 3929. (10)(a) Hoyano, J. K.; Graham, W. A. G. J. Am. Chem. Soc. 1982,104, 3723. (b) Hoyano, J. K.;McMaster, A. D.; Graham, W. A. G. J . Am. Chem. Soc. 1983,105, 7190. (11)Jones, W. D.; Feher, F. J. J . Am. Chem. SOC.1982, 104, 4240. (12)Sakakura, T.; Sodeyama, T.; Sasaki, K.;Wada, K.; Tanaka, M. J . Am. Chem. Soc. 1990, 112, 7221. (13) Perspective in the Selecrive Activation of C-Hand C-C Bonds in Saturated Hydrocarbons; Meunier, B.,Chaudret, B., Eds.; Scientific Affairs Division-NATO: Brussels, 1988. (14)Svenason, M.; Blombcrg, M. R. A.; Siegbahn, P. E. M. J . Am. Chem. SOC.1991, 113, 7076. (15) Nakatsuji, H.; Hada, M.; Yonezawa, T. J . Am. Chem. SOC.1987, 109, 1902. (16) Huzinaga, S.J . Chem. Phys. 1977,66,4245. (17) Raffenetti, R. C. J. Chem. Phys. 1973,58,4452.Almltif, J.; Taylor, P. R.;J . Chem. Phys. 1981,86,4070. (18)Wachters, A. J. H. J . Chem. Phys. 1970,52, 1033. (19)Huzinaga, S.J . Chem. Phys. 1965,42, 1293. (20)Chong, D.P.;Langhoff, S.R. J . Chem. Phys. 1986,84,5606. (21)Gdanitz, R. J.; Ahlrichs, R. Chem. Phys. Leu. 1988,143, 413.
J. Phys. Chem. 1992'96, 5789-5793 (22) Wahlgren, U.; Siegbahn, P. E. M. To be published. (23) Martin, R. L. J . Phys. Chem. 1983,87, 750. See also: Cowan, R. D.; Griffin, D. C. J . Opt. Soc. Am. 1976, 66, 1010. (24) Low, J. J.; Goddard 111, W. A. Organometallics 1986, 5, 609. 125) Morse. M. D.: Geusic. M. E.: Heath. J. R.: Smallev. R. E. J. Chem. Phjs. 1985,83,2293. Geusic, M. E.; Morse,M. D.; Smaky, R. E. J . Chem. Phys. 1985, 82, 590.
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(26) Panas, I.; Siegbahn, P.; Wahlgren, U. In Computational Chemistry-The Challenge of d and f Electrons; Salahub, D. R., &mer, M. C., Eds.; Toronto, 1988. (27) Balasubramanian, K. J . Chem. Phys. 1989, 91, 307. (28) Balasubramanian. K. J . Chem. Phvs. 1988. 89. 6310. (29) LaJohn, L.; Christiansen,P. A.; Rks, R. B.; Atashroo, T.; Ermler, W. C. J . Chem. Phys. 1987,87, 2812.
Theory of Magnetic Resonance of Higher Spin Nuclei: Some Closed Expressions for Eigenvalues of Spin 1 and Spin Systems T. H. Siddall III* Chemistry Department, University of New Orleans, New Orleans, Louisiana 70148
and J. J. Sullivan Physics Department, University of New Orleans, New Orleans, Louisiana 70148 (Received: January 9, 1992)
After some discussion of theory, closed expressions are presented for calculating the eigenvalues of the two-body interaction within clusters of equivalent nuclei with either spin 1 or spin 3/2. While specifically discussed in the context of nuclear magnetic resonance spectroscopy, the expressions will have utility for clusters of any particles with intrinsic angular momentum of but still one or of three halves. The closed expressions are insufficient for calculating eigenvalues for spins greater than have considerable utility in dealing with systems of such higher spin.
Introduction In an earlier publication' one of us treated AzBz nuclear magnetic resonance (NMR) systems where the nuclei have spins higher than spin By exploitation of some equations of Racah2 from another context, closed expressions could be written down for the spinspin (two-body interactions) in these systems. However, this development is applicable only to clusters of just two nuclei. In this present publication we report closed expressions for the coupling eigenvalues within spin 1 and spin 3 / 2 clusters of arbitrary size with the proviso that the nuclei within a cluster are equivalent and all coupling constantsof a given rank are equal. For static configurations of nuclei such equivalence is possible only when the number of nuclei in a cluster, n, is less than five. However, in the limit of fast exchange, coupling constants of a given rank can be equal for any value of n. The Zeeman effect (chemical shift) is not discussed here since it would not be qualitatively different from that in spin systems. While the discussion in this publication centers on application to NMR spectroscopy, the basic mathematics applies to clusters of particles of any sort so long as the particles are taken to have intrinsic angular momentum of one or of three halves. For example, there is a direct application to the magnetic properties of clusters of transition metal ions. This publication is organized around four main headings with subheadings under the first and fourth main headings. The first main heading is Computer Output for a Cluster of Three Nuclei with Spin 3/z. This computer output comes from a program for calculating the spinspin coupling eigenvalues for this cluster. This is the simplest situation to provide background for the rest of the discussion. The algorithm is the most direct and straightforward that can be used for such a calculation. It is just an extension to higher spin from algorithms for spin sy~tems.~-~ The subheads describe the Hamiltonian used and the basis set that was used, Table I, which contains the computer output and some additional information, and, under the fourth subheading, conclusions drawn from Table I and supplementary information. The idea behind including the material in the fmt main heading is that a guided study of such computer output can evolve intuitively into the theoretical background for treating the coupling in higher spin clusters. The theoretical background would be
developed conventionally through a discussion in arcane and unfamiliar (to chemists and to many physicists as well) terms and propositions in the theory of higher groups. We have taken here the unconventional approach of presenting computer output and then finding a theoretical framework to match the output. The second main heading is Symmetry. It extends the conclusions from Table I to a more general examination of the symmetry and groups appropriate to spin-spin coupling. Cotton's classic book6 covers the same sort of material for the application of point groups. It is just that for present purposes groups higher than point groups are needed. These higher groups cannot be examined by direct geometric visualization in the way that point groups allow. It is not necessary to delve very deeply into these groups here, but texts are a ~ a i l a b l e for ~ - ~those who wish to do so.
The third main heading is The Closed Expressions. It covers the closed expressions needed to calculate eigenvalues for spin 1 and spin 3 / z clusters which is the main result of this publication. The first subheading covers a specific numerical example of using the expressions. The second subheading covers the utility that these expressions can have in application to clusters of spin higher than spin 3/2. For this still higher spin there is not enough information in the expressions to allow the general calculation of spinspin coupling eigenvalues. Nevertheless the expressions can be useful in dealing with still higher spin. The fourth main heading is: Future Investigations. It covers the course that future investigations might take.
Computer Output for a Cluster of Three Nuclei with Spin 3/2 Table I is the computer output that gives the eigenvalues and other data for the effect of spinspin coupling in a cluster of three nuclei with spin 'I2. The Zeeman effect, the chemical shift contribution, was omitted from the calculation. The three spin 3/2 cluster was chosen as the simplest system that can provide sufficient examples to facilitate further discussion. The Hamiltonian and basis set that were used in the algorithm are described immediately below. The Hamiltonian. The Hamiltonian, as given by C ~ r i ofor ,~ spin '/znuclei (with notation slightly modified) is H = -&Iz, - ~ ~ J i j ~ y ~ J (1) J
