12665
J. Phys. Chem. 1993,97, 12665-12667
Electronic Structure of Palladium Dimer from Density Functional Theory Taketoshi Nakaot and David A. Dixon' DuPont Central Science and Engineering, Experimental Station,? Wilmington, Delaware 19880-0328
Han Chen Cray Research, Inc., 655 Lane Oak Drive, Eagan, Minnesota 551 21 Received: August 30, 1993; I n Final Form: October 14, 1993'
The structure of Pdz has been calculated by density functional theory at a variety of levels. At the best level, relativistic effective core potential with nonlocal correlation and exchange potential corrections, the ground state is predicted to be a triplet formed from a a*(4dZ2)a(5s) occupancy with a bond distance of 2.54 A and a vibrationalfrequency of 185 cin-l. The bond energy with respect to two ground-statePd atoms (4dlOconfiguration) is predicted to be 20.7 kcal/mol, in good agreement with the estimated experimental values.
Introduction The electronic structure of small metal clusters is of interest for a wide range of technologies including understanding the behavior of catalytic systems. Determining the structure of simple transition-metal dimers has proven to be a formidable task both experimentally and theoretically. Metals on the right of the periodic table play an important role in many catalytic systems, notably hydrogenation chemistry. There is significant interest in palladium chemistry because of this, and the palladium dimer has proven to be a challenging molecule. There have been a number of theoretical studies of the electronic structure of the Pd dimer with the most extensive being that of Balasubramanian.' By using relativistic effective core potentials (ECPs) and complete active spaceself-consistentfield (CASSCF) with multireferencesingle and doubles configuration interaction (MRSDCI) calculations,he found that the ground state is a 3Zy+ with a bond length of 2.48 A and ocof 159 cm-l. The lowest singlet state is 1Z8+,and it has a much longer bond length of 2.87 A with we = 121 cm-1. The 3Z8+state is 1257 cm-1 above the 3Zu+state. The dissociation energy of the triplet is calculated to be 2.05 eV with respect to 3D(5s14d9) atoms. Correcting this value to two ground-state Pd atoms with a 4d10configuration by the experimental difference2 of 0.81 eV gives a binding energy of 0.42 eV. Shim and Gingerich3 have performed all electron HF/CI calculationsand concludethat there is no binding between two ground-state Pd atoms but do predict binding between two atoms in the 5s'4d9 excited state. They suggest that the ground state is a singlet. Combining their theoretical data with their experimental mass spectrometric measurements of the Pdz(g) = 2Pd(g) equilibrium (treated by the third-law method) yielded a bond dissociation energy of 23.7 f 3.6 kcal/mol (1.03 f 0.16 eV). This is higher than the value of 16.9 f 6.0 kcal/mol from a third law evaluation but below the value of 26 f 5 kcal/mol from a second law evaluation reported by Lin et a1.4 Kleinman and co-workers5 using relativistic ECPs at the local density functional theory6 (LDFT) level find a ground-state with R = 2.444 A and we = 218 cm-' with a binding energy of 1.61 eV. The same calculations yield a 1Z8+state with R = 2.49 A, we = 178 cm-I, and DO = 1.49 eV. As expected, the LDFT calculation gives a binding energy that is too large as compared to the experimental estimated values. Two important advancements in terms of density functional theory (DFT) calculations are the incorporation of nonlocal f
Contribution No. 6576.
t Permanent address: Matsushita Central ResearchLaboratory, Moriguchi,
Osaka 570, Japan. *Abstract published in Aduonce ACS Absrracfs, November 15, 1993.
0022-3654/93/2097-12665$04.00/0
methods7q8for analytic geometry optimizations and the development of ECPS.~-" ECPs allow one to study the electronic structure of heavier elementsbecause they incorporaterelativistic effects into the core and because the cost of the calculation is reduced due to decreasing the number of atomic orbitals. (The core orbitals are incorporated into the ECP.) We have thus calculated the electronic structure of Pdz with density functional theory at both the local and nonlocal levels and with all electrons included or with just the valence electrons included by using an ECP.
Computational Methods The density functional theory calculations were done with the programs DGausslZwhich employs Gaussian orbitals and DM0113 which employs numerical orbitals on a Cray YMP computer. We discuss the DGauss calculationsfirst. The all-electron basis set14 onPdisapolarizedvalencedouble-zetasetwith the form (633321/ 5321 1/51) andan [ 11/6/51 fitting basisset. Anorm-conserving p~eudopotential~ was generated for Pd following the work of Troullier and Martins.lo The valence basis set for Pd is (4,2/ 4/3,1) with a fitting basis set of [6/4/4]. The calculations were initially done at the local density functional (LDFT) level with the potential of Vosko, Wilk, and Nusair.I5 Subsequently, the calculations were done at the self-consistent nonlocal level (NLDFT) with the nonlocal exchange potential of Becke7together with the nonlocal correlation functional of Perdew.6 Geometries were optimized by using analyticalgradients12 Second derivatives were calculated by numerical differentiation of the analytic first derivatives. A two-point method with a finite difference of 0.01 au was used. The DMol calculation^^^ were done with a polarized double numerical basis set. The atomic basis functions are given numerically on an atom-centered, spherical-polar mesh. Since the basis sets are numerical, the various integrals arising from the expression for the energy need to be evaluated over a grid. The radial portion of the grid is obtained from the solution of the atomic LDF equations by numerical methods. The number of radial points NR is given as
N~ = 1.2 x i 4 ( z + 2)"'
(1) where Z is the atomic number. The maximum distance for any function is 12.0 au. The angular integration points Ne are generated at the NR radial points to form shells around each nucleus. The value of Ne ranges from 14 to 302 depending on the behavior of the density and the maximum e value for the 0 1993 American Chemical Society
12666 The Journal of Physical Chemistry, Vol. 97, No. 49, 1993
TABLE I: Total Energies (au) for Pdl basis4 calcb suin energy (au) LOC 1 -9869.756 778 Gaussian -9869.754 174 Gaussian LOC 3 LOC 1 -9869.721 854 Gaussian (10 A) LOC 1 -9869.722 027 Gaussian ( m ) -9880.343 553 Gaussian NL 1 NL 3 -9880.342 959 Gaussian NL 1 -9880.322 896 Gaussian (10 A) NL 1 -9880.322 889 Gaussian ( m ) LOC 1 -58.098 303 ECP LOC 3 -58.109 231 ECP LOC 1 -58.060 415 ECP(I0A) LOC 1 -58.060 362 ECP (20 A) LOC 1 -58.060 364 ECP ( m ) -58.390 716 ECP NL 1 -58.401 814 ECP NL 3 ECP (10 A) NL 1 -58.368 778 ECP (20 A) NL 1 -58.368 781 -58.368 782 ECP ( m ) NL 1 LOC 1 -9870.443 844 numerical LOC 3 -9870.437 017 numerical numerical (10 A) LOC 1 -9870.413 705 LOC 1 -9870.413 704 numerical ( m ) 4Gaussian all-electron Gaussian basis set with DGauss; ECP = relativistic effective core potential with DGauss; numerical = numerical basis set from DMol. Values in parenthesesare bond separation between two Pd atoms for bond energy calculations. b L = ~local density calculation; NL = nonlocal calculation. TABLE II: Calculated Properties for Pdz' basis calc spin R (A) we (cm-l) Dob 1 2.703 147 21.81 Gaussian Loc 209 20.17 3 2.474 Gaussian LOC 12.97 1 2.800 123 Gaussian NL 3 2.544 184 12.59 Gaussian NL 160 23.81 LOC 1 2.682 ECP 3 2.480 213 30.66 ECP LOC 1 2.770 134 13.76 ECP NL 3 2.542 185 20.73 ECP NL 147 18.91 numerical LOC 1 2.697 14.63 numerical Loc 3 2.489 211 121 6.29 MRSDCI 1 2.87 ECP 159 9.88 ECP MRSDCI 3 2.48 ECP LOC 1 2.49 178 34.36 3 2.444 218 ECP LOC 37.13 4 See Table I footnotes for abbreviations. In kcal/mol.
Deb ref
21.6 19.9 12.8 12.3 23.6 30.4 13.6 20.5 18.7 14.3 6.1 9.6 34.1 36.8
a a a a
a a
a a a a
1
1 6
6
spherical harmonics (Lx = 29). The local potential of von Barth and Hedini6is used in DMol. Results and Discussion The total energies of the optimized Pdz structures are given in Table I together with the energies of two Pd atoms in their dl0 ground state separated by 10 A and a t infinity.17 At the ECP level, we also calculated the energy at a separation distance of 20 A. The difference in energies between the 10 A and infinite separation is always small except for the all-electron local calculation where it is 0.1 1 kcal/mol in the wrong direction. The calculated molecular properties are summarized in Table I1 for the lowest-energy singlet and triplet states. The bond distance of the singlet is significantly longer than that of the triplet as expected from the population analysis discussed below. Thedifference in the bond distanceof thesinglet isquitedependent on the inclusion of nonlocal corrections as might be expected since the singlet is nominally formed from two Pd(4dlo) atoms interacting and LDFT is known to predict overbinding. The effect of nonlocal corrections is to increase r(Pd-Pd) by about 0.1 A. There is a much smaller dependence on the form of the basis set or the use of ECPs. The ECP leads to a shortening of the bond distance for the singlet of 0.02-0.03 A. The triplet geometries also show a dependence on the inclusion of nonlocal corrections.
Letters TABLE III: Pd2 Nonlocal (Full) Populations (Electrot1.9)~ orbital singlet total triplet total triplet A(spin)* 5s 0.46 0.11 0.50 0.00 0.00 0.01 5Px 0.00 0.01 0.00 5PY 0.01 0.02 0.02 5P* 1.62 0.13 1.57 4dxx 1.62 0.13 1.57 4dYY 1.64 0.27 1.35 4dzz 2.00 0.00 2.00 4dxy 1.99 0.00 2.00 4dxz 1.99 0.00 2.00 4dyz Bond axis is z. Difference in alpha and beta spin populations. The bond length again lengthens when nonlocal corrections are included by 0.06-0.07 A. For the triplet, the replacement of the all-electron basis set by ECPs has very little effect. The frequencies show the same dependence on spin state and on the computational level as do the bond distances. The local frequencies are higher than the nonlocal ones and the triplet frequencies are higher than the singlets. The higher values of the local frequencies are consistent with the overbinding at the local level. The bond energies display a somewhat different behavior. With the Gaussian basis sets, the singlet is slightly more stable than the triplet (AE= 1.6 kcal/mol) and the bond energy is about 22 kcal/mol (0.95 eV). When nonlocal corrections are included, the singlet is still more stable than the triplet, but the splitting is reduced to 0.4 kcal/mol and the binding energy to 13kcal/mol (0.56 eV). At the nonlocal level with numerical basis sets, the singlet is 4.3 kcal/mol more stable than the triplet and the bond energy is almost 19 kcal/mol (0.82 eV). When a relativistic ECP is used, the triplet is more stable than the singlet at both the local and nonlocal levels. At the LDFT level, the triplet is 6.9 kcal/mol more stable than the singlet and the dissociation energy is -31 kcal/mol (1.33 eV). At the NLDFT level, the triplet is 7.0 kcal/mol more stable than the singlet just as found at the local level but the binding energy is reduced to -21 kcal/ mol (0.90 eV). The difference between the local and nonlocal levels of calculation is well-understood as the local density approximation usually leads to overbinding. What is surprising is that the relativistic ECPs favor the triplet over the singlet as compared to the all-electron calculations which favor the singlet. The population analysis (Table 111) shows the differences in the electronic structures of the singlet and triplet states. The population analysis is independent as to whether ECPs are used or not. The singlet is composed of two interacting 4dl0 atoms. In order to get a stabilizing interaction, there is a promotion of 0.14 electron into the 5s and 5p orbitals from the 4d. Most of this is from the d,Z u* orbital. The triplet is formed by promoting an electron out of the d,2 u* into the 5s u orbital to give a u b * l occupancy. Although one is promoting an electron which in the atom costs about 0.8 eV, it is into a bonding orbital. This reduces an antibonding interaction and forms a bonding one. The spin populations show that the electron is promoted from the 4d,* to the 5s for the triplet. The population analysis shows why there may be a dependence on the use of ECPs. Palladium is heavy enough that relativistic effects could become important. If the size of the 5s orbital changes, then it could lead to the triplet being more stable as it should contract when relativistic effects are included.18 In order to test this, we calculated a new pseudopotential without relativistic effects. The results are shown in Table IV. The effect of not including relativistic effects in the ECP is to lengthen the Pd-Pd bond a t the local level but to shorten it at the nonlocal level. The triplet bond length is predicted to decrease at the local level by more than a t the nonlocal when relativistic effects arenot included in the ECP. The order of the singlet and triplet states reverses at the local level with the singlet now more stable than the triplet
The Journal of Physical Chemistry, Vol. 97, No. 49, 1993 12667
Letters
TABLE I V Nonrelativistic ECP Results for Pdf basis calc spin energy (au) R (A) we (cm-I) DO(kcal/mol) 141 17.07 ECP(NR) Loc 1 -58.547 192 2.727 ECP(NR) ECP(NR) ECP(NR) ECP(NR) ECP(NR) 0
LOC 3 LOC 1 NL 1 NL 3 NL 1
-58.538975 2.495 -58.519875 20.0 -58.848262 2.740 -58.849979 2.527 -58.830911 20.0
206
11.99
105 142
10.89 11.96
SccTable I footnotes for abbreviations. (NR) = nonrelativistic ECP.
TABLE V
Calculated Energy Levels (eV) of Pd and Pd2
orbital
nonrelativistic
relativistic
~
Pd 5s 5P
3.10 0.27
4d
4.38 Pd2
HOMO@) HOMO(T) 4 5 s ) HOMO(T) u*(4dZ2)
4.01 4.56 5.34
by 5.1 kcal/mol. The bond energies at the local level are significantly lower with the singlet about 7 kcal/mol lower and the triplet 19 kcal/mol lower in energy than the relativistic ECP results. The results of the local nonrelativistic ECP calculations are very similar to the LDFT calculations with numerical basis sets. At the nonlocal level the nonrelativistic ECP has the triplet below the singlet. For this ECP, there is essentially no nonlocal correction to the triplet, but there is a significant nonlocal correction to the energy of the singlet. Both structures at this level are the most weakly bound as shown by the low bond energies and frequencies. In order to provide more information about the relativistic effects, we calculated the energies of thevalence states of Pd with or without relativistic corrections by using a numerical atomic code. As shown in Table V, the energy of the 5s level is lowered by 0.5 eV if relativistic effects are included, which reduces the energy difference between the 5s and 4d orbitals. Thus, it is easier to promote an electron to 5s from 4d if relativistic effects are included. This is consistent with the stabilization of the triplet as we have predicted. In fact, the singlet-triplet splitting is -0.5 of the calculated 4d-5s promotional energy difference due to relativistic effects. It is useful to note that the most pronounced effect of including relativistic corrections on the energy levels is for the 5s orbital. The energy difference between the 5s and the average of the multiplets for the 4d is 0.78 eV, comparable to the experimental energy differencebetween the 4d10and 4d95s1states. The HOMO energies (Table V) of the singlet and triplet states are consistent with this. The HOMO of the triplet shows only a small effect if relativistic effects are included. Although the orbital energies for open-shell species are not as precisely defined as those for closed-shell species, we will discuss them for the qualitative features. The triplet exhibits a much larger lowering of the energy of the singly-occupied 4 5 s ) orbital of almost 0.4 eV, and the u*(d,Z) orbital is only lowered by about 0.2 eV if relativistic effects are included, consistent with the atomic energy level calculations. The nonlocal relativistic ECP calculations can be compared to the results of Ba1asubramanian.l The geometries and frequencies are similar although we find a shorter bond length by 0.07 A for the singlet. The ECP/MRSDCI calculations find a singlet/ triplet splitting that is about one-half of our value. The binding energy to two ground-state atoms is not reported so we corrected the reported binding energy' to two 5sl4d9 atoms by twice the experimental energy splitting.2 This yields a bond energy about one-half of our value. Our calculated bond energy for Pdl at the nonlocal ECP level is within the experimental error bars of the
-
combined theoretical/experimental result of Shim and Gingerichn3 However, they used the wrong geometry and frequency in their data analysis as they used the results of their singlet calculations. We would suggest that thedata be reanalyzed withour parameters for the triplet. Our value for DOis in good agreement with the average of the two values of Lin et a1.,4 which is 21.5 kcal/mol and falls essentially within the error limits of the two individual measurements. We note that our triplet would actually dissociate to a ground-state Pd(4dIo) atom and an excited Pd(4d95sl) atom or to two excited Pd(4d95s1) atoms unless curve crossings occur. For such heavy atoms, this is likely. In conclusion, we have calculated the bond dissociation energy of Pdzand find that twoeffects must be included in thecalculation. There is a significant relativistic effect on this value at the density functional level. Nonlocal corrections are also required if one is going to predict the bond energy with any accuracy. The interplay between nonlocal and relativistic corrections is not straightforward. The ground state is predicted to be a triplet formed from a ~ * ( 4 d 2 ) ~ a ( 5occupancy s)~ with a bond distance of 2.54 A and a vibrational frequency of 185 cm-I.
References and Notes (1) Balasubramanian, K. J. Chem. Phys. 1988, 89,6310. See also: J . Phys. Chem. 1989, 93,6585. (2) Moore, C. E. Atomic Energy Levels; Natl. Bur. Stand. (US) Circ. No. 467; U S . GPO. Washington, D.C., 1952; Vol. 111, p 38. (3) Shim, I.; Gingerich, K. A. J . Chem. Phys. 1984,80, 5107. (4) Lin, S.-S.; Strauss, B.; Kant, A. J . Chem. Phys. 1969, 51, 2282. ( 5 ) Lee, S.;Bylander, D. M.; Kleinman, L. Phys. Rev. B 1989,39,4916. ( 6 ) (a) Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University Press: New York, 1989. (b) Salahub, D. R. In Ab Initio Methods in Quantum Chemistry-II; Lawley, K. P., Ed.; J. Wiley &Sons: New York, 1987; p 447. (c) Wimmer, E.; Freeman, A. J.; Fu, C.-L.; Cao, P.-L.; Chou, S.-H.; Delley, B. In Supercomputer Research in Chemistry and Chemical Engineering, Jensen, K. F., Truhlar, D. G.,Eds.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987; p 49. (d) Jones, R. 0.;Gunnarsson, 0. Rev. Mod. Phys. 1989,61,689. (e) Zeigler, T. Chem. Rev. 1991, 91, 651. (7) (a) Becke, A. D. Phys. Rev. A 1988,38,3098. (b) Becke, A. D. In The Challenge of d and f Electrons: Theory and Computation; Salahub. D. R., Zerner, M. C., Eds.;ACS SymposiumSeries No. 394; American Chemical Society: Washington, DC, 1989; p 166. (c) Becke, A. D. Int. J. Quantum Chem., Quantum. Chem. Symp. 1989, 23, 599. (8) Perdew, J. P. Phys. Rev. B 1986, 33, 8822. (9) Chen, H.; Kraskowski, M.; Fitzgerald, G., Hay, P. J.; Martin, R. L. J . Chem. Phys. 1993,98, 8710. (10) Troullier, N.; Martins, J. L. Phys. Reu. B 1991, 13, 1993. (11) Bylander, D. M.; Kleinman, L. Phys. Rev. B 1984, 29, 2274. (12) Andzelm, J.; Wimmer, E.; Salahub, D. R. In The Challenge of dand f Electrons: Theory and Computation; Salahub, D. R., Zerner, M. C., Eds.; ACS Symposium Series No. 394; American Chemical Society: Washington, DC, 1989;p 228. (b) Andzelm, J. In Density Functional Merhodsin Chemistry; Labanowski, J., Andzelm, J., Eds.; Springer-Verlag: New York, 1991;p 101. Andzelm, J. W.;Wimmer, E. J. Chem. Phys. 1992,96, 1280. DGaua is a density functional program available via the Cray Research Unichem Project. (13) Delley, D. J . Chem. Phys. 1990, 92, 508. DMol is available commercially from BIOSYM Technologies,San Diego, CA. The multipolar fitting functions for the model density used to evaluate the effective potential have angular momentum numbers of 3 for Pd. The FINE parameter was used for generating the mesh. (14) Godbout,N.;Salahub,D. R.;Andzelm. J.; Wimmer,E. Can.J. Chem. 1992, 70. 560. (15) Vosko, S.J.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200. (16) von Barth, U.; Hedin, L. J . Phys. C 1972, 5, 1629. (17) The differences in the total energies for the all-electron calculations at the local level is due to differences in the basis sets (numerical vs Gaussian) and in the exchange-correlation potentials. (18) (a) Balasubramanian, K.; Pitzer, K. S. In Ab Initio Methods in Quantum Chemistry-I; Lawley, K. P.,Eds.; John Wiley and Sons: New York, 1987; p 287. (b) PykkB, P. Adu. Quantum Chem. 1978, 11, 353. (c) Pykko. P.; Desclaux, J. P. Acc. Chem. Res. 1979, 12, 276. (d) Pitzer, K. S. Aec. Chem. Res. 1979,12,27 1. (e) Culberson, J. C.; Knappe, P.; Rbch, N.; Zerner, M. C. Theor. Chim. Acta 1987, 71, 21.
