J . Phys. Chem. 1992, 96, 2129-2141
ometry drastically. The proton approaches oxygen in the C2, symmetry, while the optimal geometry of OH3+is C3". From the study of geometry variation along the R(N-0) bond the conclusion can be drawn that the proton always belongs to the one subunit or the other, the areas of "sharing" proton are very narrow. Orbital pictures for extreme cases (2.707 and 5.0 A) are significantly different, which may affect the structure of potential surfaces. The correlation between orbital pictures of separated systems and the complex is obvious for the long N - 0 distance. However an intermediate region is recognized also. The orbital pictures are more complicated for shorter distances (2.7-3.2 A). In the case of R(N-0) from 2.707 to 3.2 A the ground state is purely one-determinantal. The gap between ground and excited states is too large to allow interactions. In the case of long distances (5.0 A) the function for the ground state is contaminated by configurations characteristic for the "avoided" region. The gap between ground and excited states is much smaller than for shorter distances, and the interactions are much bigger. The curve in regions around atoms N or 0 resembles the situation in separate molecules. Energy potential curves calculated by coupled cluster method agree well with those from MRD-CI calculations. However, the
2129
multireference character of the wave function a t N-H+ distance of 2.9 A (for N - 0 distance 5.0 A) is manifested by lack of convergence for the excited singlet state, while starting from a single determinantal ground-state function. Mulliken gross atomic population indicates that the reaction proceeds as a proton transfer, while at excited states the proton carries more electron density. Similarly to the geometry optimization, in the ground state the proton belongs to OH2or NH3 and not to a "common" zone. This picture supports the concept of proton transfer as a tuneling between two structures. Electrons in excited states are much more delocalized, and the proton carries an electron acquired partially from ammonia and partially from water. Electrostatic properties of the excited-state systems are completely different from that of the ground state. Acknowledgment. This research was supported by ONR, Power Programs Branch, under Contract NOOO 14-80-C-OOO3 and Grant "14-89-5-1613, U.K. was partly supported by the US-Israel Binational Science Foundation. The calculations were carried out at the NSF San Diego Supercomputer Center on the CRAY YMP. We thank SDSC for the grant of computer time. Registry No. H20, 7732-18-5; NH4, 14798-03-9.
Energetics and Electronic Structure of Chromium Hexacarbonyl Kathryn L. Kunzet and Ernest R. Davidson* Department of Chemistry, Indiana University, Bloomington, Indiana 47405 (Received: October 31, 1991)
We investigate the molecular orbital self-consistent-fieldmodel of bonding in Cr(CO),. The energetics and electron density are examined using a large range of tools. The change in density compared to a promoted 3dt22 'A,, chromium atom and six CO molecules is primarily charge transfer from the tlg orbitals of chromium to the empty 2r*t2, orbital of the (CO), cage. This mixing is counterintuitive, as the largest increase in electron density is in the oxygen p r orbitals. The restricted Hartree-Fock energy is actually repulsive compared to that of ground-state fragments by 111 kcal/mol. This energy change cage formation energy, -272 kcal/mol consists of +266 kcal/mol of fragment promotion energy, +67 kcal/mol of (CO), and finally -329 kcal/mol of orbital of electrostatic attraction, +359 kcal/mol of overlap repulsion between Cr and (CO),, relaxation energy. Most of the relaxation energy is associated with the t2, HOMO-LUMO charge transfer. The u electrons contribute to the bond energy primarily through electrostatic penetration, leading to a large electrostatic attraction between and not through mixing of CO u orbitals with Cr empty valence orbitals. Cr and (CO),,
Introduction The nature of the bonding in low-oxidation-state transitionmetal carbonyl complexes is of considerable importance due to their high catalytic activity, their use as precursors in organometallic chemistry, and their use as models for the chemisorption bond between CO and metal surfaces.'-3 Theoretical studies of the metal-ligand coordinate bond in the metal carbonyls, especially Cr(CO),, Fe(CO)5, and Ni(C0)4, are numerous,"-34 as are qualitative papers "explaining" the b ~ n d i n g . ~ ~ - ~ ~ The main experimental facts about these complexes addressed by qualitative models have been (a) the obedience of the "18electron rule",' (b) the octahedral, trigonal bipyamidal, and tetrahedral shapes of the complexes,' (c) the linear MCO bond with C next to M,"3,4 (d) the relatively constant metal-CO average bond en erg^,^^-$^ (e) the relatively short metal-carbon bond length,', (f) the slight lengthening of the CO bond,43344(g) the decreases in the CO stretching frequency,5842 (h) the decrease (i) the high in the 1s carbon and oxygen ionization potential~,6~*~ d-d transition energy,6446 and (j) the singlet spin ground state.' The very high ligand field transition (d-d) energiesu correspond to a field strength D q / B in excess of 5 . This very high apparent field defies an interpretation in terms of an electrostatic model of fragment charges and multipoles as in ligand field theory.,' 'Deceased, March 28, 1991.
For Cr(CO),, this large value of D q / B suggests that the d6 electrons of the metal can be adequately represented by a single (1) Cotton, F. A.; Wilkinson, G. Advanced Inorganic Chemistry, 5th ed.; Wiley Interscience: New York, 1988. (2) Mingos, D. M. P. Comprehensive Organomeiallic Chemistry; Wilkinson, G . , Stone, F. G. A., Abel, E. W., Eds.; Pergamon Press: New York, 1982; Vol. 3, p 1. (3) (a) Muetterties, E. Science 1976, 194, 1150. (b) Muetterties, E. Science 1977, 196, 839. (4) Arratia-Perez, R.; Yang, C. Y. J . Chem. Phys. 1985, 83, 4005. (5) (a) Baerends, E. J.; Rozendaal, A. Quantum Chemistry: The Challenge of Transition Metals and Coordination Chemistry; Veillard, A., Ed.; D. Reidel: Dordrecht, 1986; p 159. (b) Chornay, D. J.; Coplan, M. A.; Tossell, J. A.; Moore, J. H.; Baerends, E. J.; Rozendaal, A. Inorg. Chem. 1985, 24, 877. (c) Heijser, W.; Baerends, E. J.; Ros, P. J . Mol. Struct. 1980, 63, 109. (6) Barandiaran, Z.; Seijo, L.; Huzinaga, S.; Klobukowski, M. Int. J . Quantum Chem. 1986, 29, 1047. (7) (a) Bauschlicher, C. W.; Bagus, P. S.J. Chem. Phys. 1984,81,5889. (b) Bauschlicher, C. W.; Pettersson, L. G. M.; Siegbahn, P. E. M. J . Chem. Phys. 1987,87, 2129. (c) Bauschlicher, C. W.; Langhoff, S.R.; Barnes, L. A. Chem. Phys. 1989,129,431. (d) Bauschlicher,C. W. J. Chem. Phys. 1986, 84, 260. (e) Bauschlicher, C. W.; Bagus, P. S.;Nelin, C. J.; Roos, B. 0. Chem. Phys. 1986,85, 354. (f) Barnes, L. A.; Rcsi, M.; Bauschlicher, C. W. J. Chem. Phys. 1991, 94, 2031. (g) Barnes, L. A.; Rosi, M.; Bauschlicher, C. W. J . Chem. Phys. 1990,93,609. (h) Barnes, L. A,; Bauschlicher, C. W. J . Chem. Phys. 1989,91,314. (i) Blomberg, M. R.A,; Brandemark, U. B.; Siegbahn, P. E. M.; Wennerberg, J.; Bauschlicher, C. W. J . Am. Chem. Soc. 1988, 110, 6650.
0022-3654/92/2096-2129%03.00/0 0 1992 American Chemical Society
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J.; Howell, J. M.; Rossi, A. R. J . Phys. Chem. 1981, 85, 17. (19) (a) Kotzian, M.; Rosch, M.; Schrader, H.; Zerner, M. C. J . Am. Chem. SOC.1989,111,7687. (b) Saddei, D.; Freund, H. J.; Hohlneicher, G. Chem. Phys. 1981, 55, 339. (20) Luthi, H. P.; Siegbahn, P. E. M.; Almlof, J. J . Phys. Chem. 1985,89, 2156. (21) (a) Hay, P. J. J . Am. Chem. SOC.1978,100,241 1. (b) Rohlfing, C. M.; Hay, P. J. J. Chem. Phys. 1985, 85, 4641. (22) (a) Vanquickenborne, L. G.; Verhulst, J. J . Am. Chem. SOC.1983, 105, 1769. (b) Pierfoot, K.; Verhulst, J.; Verbeke, P.; Vanquickenborne, L. G. Inorg. Chem. 1989, 28, 3059. (23) (a) Ziegler, T.; Tschinke, V.; Ursenbach, C. J. Am. Chem. Soc. 1987, 109, 4825. (b) Ziegler, T.; Rauk, A. Inorg. Chem. 1979, 18, 1558. (c) Ziegler, T.; Rauk, A. Inorg. Chem. 1979, 18, 1755. (d) Ziegler, T.; Rauk, A. Theor. Chim. Acta 1977,46, 1. (e) Ziegler, T.; Tschinke, V.; Becke, A. J . Am. Chem. SOC.1987,109, 1351. (f) Ziegler, T.; Tschinke, V.; Versluis, L. Quantum Chemistry: The Challenge of Transition Metals and Coordination Chemistry; Veillard, A., Ed.; D. Reidel: New York, 1986; p 189. (24) (a) Blomberg, M. R. A,; Brandemark. U. B.; Siegbahn, P. E. M.; Mathisen, K. B.; Karlstrijm, G. J . Phys. Chem. 1985,89,2171. (b) Blomberg, M.; Brandemark, U.; Johansson, J.; Siegbahn, P.; Wennerberg, J. J . Chem. Phys. 1988, 88, 4324. (c) Larsson, S.;Braga, M. Int. J . Quantum Chem. 1979, 15, 1. (25) (a) Davidson, E. R. The Challenge of d and f Electrons; Salahub, D. R., Zerner, M. C., Eds.; ACS Symp. 395; 1989; p 153. (b) Frey, R. F.; Davidson, E. R. J . Chem. Phys. 1989, 90, 5541. (c) Frey, R. F.; Davidson, E. R. J. Chem. Phys. 1989, 90, 5555. (26) Spangler, D.;Wendoloski, J. J.; Dupuis, M.; Chen, M. M. L.; Schaefer, H. F. J . Am. Chem. SOC.1981, 103, 3985. (27) Rives, A. B.; Fenske, R. F. J . Chem. Phys. 1981, 75, 1293. (28) (a) Jeung, G. H.; Koutecky, J. Chem. Phys. Lett. 1986,129,569. (b) Koutecky, J.; Pacchioni, G.; Fantucci, P. Chem. Phys. 1985, 99, 87. (29) Barbier, C.; Berthier, G.; Daoudi, A.; Suard, M. Theor. Chim. Acta 1988, 73, 419. (30) (a) Bagus, P. S.;Hermann, K.; Bauschlicher, C. W. J. Chem. Phys. 1984,80,4378. (b) Bagus, P. S.;Hermann, K.; Bauschlicher, C. W. J. Chem. Phys. 1984,81, 1966. (c) Bagus, P. S.;Nelin, C. J.; Bauschlicher, C. W. Phys. Reu. 1983, B28, 5423. (d) Bagus, P. S.;Hermann, K.; Seel, M. J . Vac. Sci. Technol. 1981, 18,435. (e) Bagus, P. S.;Roos, B. 0. J. Chem. Phys. 1981. 75, 5961. (31) Walch, S.P.; Goddard, W. A. J. Am. Chem. SOC.1976, 98, 7908. (32) Ilison, J.; Mavridis, A.; Harrison, J. F. Polyhedron 1988, 7, 1595. (33) Basch, H.; Cohen, D. J . Am. Chem. Soc. 1983, 105, 3856. (34) Mavridis, A.; Harrison, J. F.; Allison, J. J. Am. Chem. Soc. 1989,111, 2482. (35) Hellmann, H. C. R. Acad. Sci. URSS 1939, 24, 549. (36) (a) Pauling, L. Acta Crystallogr. 1968,824,978. (b) Pauling, L. The Nature of the Chemical Bond, Cornel1University Press: Ithaca, 1948; p 251. (c) Pauling, L. J . Chem. SOC.1948, 1461. (37) Dewar, J. S.Bull. Soc. Chim. Fr. 1951, 18, c79. (38) (a) Chatt, J.; Duncanson, L.A. J . Chem. Soc. 1953,2939. (b) Chatt, J. J. Chem. Soc. 1949,3340. (c) Chatt, J.; Wilkins, R. G. J. Chem. Soc. 1952, 2622. (d) Craig, D. P.; Maccoll, R. S.;Nyholm, R. S.;Orgel, L. E.;Sutton, L. E. Trans. Faraday SOC.1954, 333. (39) (a) Craig, D. P.; Maccol, A.; Nyholm, R. S.;Orgel. L. E.; Sutton, L. E. J . Chem. SOC.1954, 332. (b) Sutton, L. E. J. Chem. Educ. 1960,37, 498.
Kunze and Davidson
be the low-field limit for singlet states. Since CO can also be well r e p r m t e d by a single configuration, it is expected that the weight (square of the coefficient) of the Hartree-Fock (RHF) configuration in an expansion of the exact wave function is much larger than the weight for any other configuration. For this reason, past calculations have focused on the R H F wave function. Even so, the weight of the Hartree-Fock configuration in the exact wave function is known to decrease exponentially with the number of electrons and probably is only about 50% for a molecule of this size. Arguments based on the principle of electroneutrality first led to the assignment by Langmuir in 1921 of double-bond character to the metal-carbonyl bond.,* The simple idea of Pauling that the valence bond form for Cr(CO), should be written as (O= C=)3Cr(:CO)3 rather than Cr(:CO), to avoid a high formal charge on the metal explained the short CrC bond lengths, the lengthened CO bond, and the reduced CO force constant.36 One difficulty with this model is that the valence bond fragments from which this complex is formed are formally 'S (ground state) Cr, three ground-state CO molecules, and three n r * jII excited CO molecules. The promotion energy to get three CO molecules into this state is 520 kcal/mol. The qualitative molecular orbital interpretation of bonding of ethylene to a metal, attributed to Dewar,37 was regarded as a natural extension of the Pauling model for carbonyl bonding. In (40) (a) Caulton, K. G.; Fenske, R. F. Inorg. Chem. 1968, 7, 1273. (b) Lichtenberger, D.L.; Fenske, R. F. Inorg. Chem. 1976, 15, 2015. (c) Lichtenberger, D. L.; Kellogg, G. E. Acc. Chem. Res. 1987,20,379. (d) Fenske, A. K.; Cotton, F. A.; Wilkinson, G. J. Am. Chem. Soc. 1957, 79, 2044. (41) (a) Beach, N. A.; Gray, H. B. J . Am. Chem. Soc. 1968,90,5713. (b) Gray, H. B.; Beach, N. A. J. Am. Chem. SOC.1963,85, 2922. (42) (a) Hoffmann, R.; Chen, M. M. L.; Thorn, D. L. Inorg. Chem. 1977, 16,503. (b) Schilling, B. E. R.; Hoffmann, R. J . Am. Chem. Soc. 1979,101, 3456. (43) (a) Whitaker, A.; Jeffery, J. W. Acta Crystallogr. 1967, 23, 977. (b) Whitaker, A.; Jeffery, J. W. Acta Crystallogr. 1967, 23, 984. (44) (a) Jost, A.; Rees, B.; Yelon, W. B. Acta Crystallogr. 1975, 831, 2649. (b) Rees, B.; Mitschler, A. J. Am. Chem. Soc. 1976, 98, 7918. (45) (a) Cotton, F. A.; Fischer, A. K.; Wilkinson, G. J . Am. Chem. SOC. 1956,78,5168. (b) Cotton, F. A.; Fischer, A. K.; Wilkinson, G. J. Am. Chem. SOC.1957, 79, 2044. (c) Cotton, F. A.; Fischer, A. K.; Wilkinson, G. J. Am. Chem. SOC.1956,78, 5168. (46) Bernstein, M.; Simon, J. D.; Peters, K. S . Chem. Phys. Lett. 1983, 100, 241. (47) Junk, G. A,; Svec, H. J. Z . Naturforsch. 1968, 23, 1. (48) Langsam, Y.; Ronn, A. M. Chem. Phys. 1981.54, 277. (49) Lewis, K. E.;Golden, D. M.; Smith, G. P. J . Am. Chem. SOC.1984, 106, 3905. (50) Mikami, N.; Ohki, R.; Kido, H. Chem. Phys. 1988, 127, 161. (51) Pilcher, G.; Ware, M. J.; Pittam, D. A. J. Less-CommonMet. 1975, 42, 223. (52) Smith, G. P. Polyhedron 1988, 7, 1605. (53) (a) Tyndall, G. W.; Jackson, R. L. J. Am. Chem. Soc. 1987,109,582. (b) Tyndall, G. W.; Jackson, R. L. J. Chem. Phys. 1988,89, 1364. (54) Venkataraman, B.; Hou, H.; Zhang, Z.; Chen, S.;Bandukwalla, G.; Vernon, M.J . Chem. Phys. 1990, 92, 5338. (55) Xie, X.;Simon, J. D. J. Phys. Chem. 1989, 93, 4401. (56) Yates, B. W.; von Wald, G. A.; Taylor, J. W.; Grimm, F. A.; Tse, J. S.Chem. Phys. 1990, 147, 431. (57) Conner, J. A. Top. Curr. Chem. 1977, 71, 71. (58) Gerrity, D. P.; Rothberg, L. J.; Valda, V. J. Phys. Chem. 1983,87, 2222. (59) Prasad, P. L.; Singh, S.J . Chem. Phys. 1977, 67, 4384. (60) Pince, R.; Poilblanc, R. Spectrochim. Acta 1972, 28A, 907. (61) (a) Jones, L. H. Inorg. Chem. 1976, 15, 1244. (b) Jones, L. H.; Swanson, B. I. Acc. Chem. Res. 1976, 9, 128. (c) Jones, L. H. J . Mol. Spectrosc. 1970, 36, 398. (62) Horton-Mastin, A,; Poliakoff, M . Chem. Phys. Lett. 1984, 109, 587. (63) (a) Avanzino, S.C.; Baake, A. A.; Chen, H. W.; Donahue, C. J.; Jolly, W. L.; Lee, T. H.; R i m , A. J. Inorg. Chem. 1980, 19, 1931. (b) Jolly, W. L.; Avanzino, S.C.; Reitz, R. R. Inorg. Chem. 1977, 16, 964. (64) Cooper, G.; Sze, K. H.; Brion, C. E. J . Am. Chem. SOC.1990,112, 4121. (65) Joly, A. G.; Nelson, K. A. J. Phys. Chem. 1989, 93, 2876. (66) Wittmann, G. T. W.; Krynauw, G. N.; Lotz, S.;Ludwig, W. J. Organomet. Chem. 1985, 293, C33. (67) (a) Ballhausen, C. J. Introduction to Ligand Field Theory; McGraw-Hill: New York, 1962. (b) Figgis, B. N. Introduction to Ligand Fields; Wiley: New York, 1966. (c) Orgel, L. E. An Introduction to Transition-Metal Chemisrry: Ligand-Field Theory; W h y : New York, 1960. (68) Langmuir, I. Science 1921, 54, 59.
Energetics and Electronic Structure of Cr(C0)6
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energy changes due to sequential (one at a time) relaxation of this model, bonding is attributed to a synergistic u and A delothe orbitals of the free fragments to the optimum M O s of the calization (charge transfer) of electrons to form covalent (shared complex. They concluded that there is negligible 4s and 4p electron) bonds between the filled A and empty A* M O s of ethylene and metal atomic orbitals. Chatt and D u n c a n ~ o n ~ ~ ~contribution to the energy and that the A back-bonding is the dominant contributor to the M-CO bond. Donation of u electrons regarded this Dewar model as a natural extension into MO lanto the empty d orbital also contributed in Fe(CO),. In both cases, guage of the Pauling model for Cr(C0)6. The six 5u (carbon lone the R H F energy was repulsive relative to ground-state metal atoms pair) orbitals of CO are supposed to delocalize into the six possible and CO molecules. The error in the R H F energy, usually called 4s-4p3-3de2 hybrid orbitals of the metal. Simultaneously, the electron correlation, apparently changes between the fragments six electrons in the 3dt2, orbitals of the metal delocalize into the and the complex by more than the bond dissociation energy. 27r*tq CO orbitals. This model has been widely adopted, although not without controversy, because it rationalizes the experimental More complete energy decomposition analyses have been made facts mentioned above as well as trends in the CO bond stretching using X a and modem density functional theory. In these analyses, frequency69 in V(CO),, Cr(CO),, and Mn(C0)6+. It also explains atomic promotion energies, electrostatic interactions between the the trans effect where substitution of NH3 for CO lengthens the fragments, exchange repulsion due to overlap of fragment orbitals, and detailed relaxation energies were r e p ~ r t e d . ~Unlike ~ , ~ ~RHF, ~ Cr-C bonds to the remaining C O ligands.' the best of these studies found the complex bound with about the Like other models, this covalent highest occupied M O correct dissociation energy. These results cannot be compared (HOMO)-lowest unoccupied M O (LUMO) model has some directly with R H F energy terms because they attempt to incorconceptual difficulties. For example, the separated fragments from porate electron correlation into each term. The sum of the which the complex is formed are six ground-state CO molecules, electrostatic and exchange repulsion is usually called the steric, but a highly excited chromium atom in a hypothetical t 2 2 'A, or static, interaction. As with the R H F study, density functionals state. The promotion energy for the chromium is about 150 find the steric interaction repulsive, while the orbital relaxation kcal/mol but is usually not mentioned in the qualitative deis stabilizing and is dominated by A back-bonding. scription. A second problem is the language in which HOMO-LUMO Extensions to linear neutral metal monocarbonyls and to surface descriptions are expressed. Clearly, if the orbitals of C O are chemistry30have also emphasized u donation and A back-bonding. introduced around the metal, with the metal already described In MCO compounds, there is also hybridization of the metal 4s by its exact Hartree-Fock orbitals, no mixing should take place. orbital away from the CO 5u lone pair. Bauschlicher and Bagus If it does, this is called a basis set superposition error and occurs and co-workers7 found the steric interaction to be repulsive and because the free atom was not correctly described. Similarly, the focused on orbital relaxations in an energy decomposition analysis M O s of a CO molecule should not be changed by the introduction for MCO and for CO chemisorption. They found that bonding comes from d r back-bonding and the u interactions are repulsive. of additional orbitals at the site of other fragments. Only when In an EDA study of ScCO, Frey and DavidsonZ5again found the the nuclei and electrons of the other fragments are introduced do the M O s distort. Thus, HOMO-LUMO mixing is a response steric interaction is repulsive, while the orbital relaxation is stabilizing and electron correlation is essential to bonding. to the electrostatic fields from neighboring fragments and not to In this paper we attempt to gain further insight into the bond orbital overlap. energy in Cr(C0)6 using all available theoretical tools for anaA third problem with the HOMO-LUMO model is the tacit lyzing the results. This includes our own approach to an energy assumption that any mixing of this type represents covalent decomposition analysis as well as traditional discussions of debonding, and the energy improvement should be interpreted in formation of charge densities, orbital energy changes, and atomic terms of a covalent bond order or overlap population. This leads populations. Care has been taken to avoid large errors due to basis to the assumption that the bond dissociation energy in these electron pair donor bonds, as in electron pair formation bonds (e.g., set superposition effects. As a consequence, the R H F energy reported here is lower than any previously reported but is still H2or CH4), should be related to lowering of the orbital energy repulsive relative to ground-state fragments. As will be shown, compared with the free fragments. there are several energetic effects which are of the same magnitude For two decades there has been debate about the relative imas the binding energy. Each must be considered in any qualitative portance of the ligand-to-metal u donation and the metal-to-ligand accounting of the strength of the metal-carbonyl bond. A back-donation in this synergistic charge transfer, by both experimentalists and theoreticians. Until recently, the theoretical Preliminaries understanding has been based on changes in M O energies, plots Experimental Background. The crystal structure of Cr(C0)6 of the MOs involved showing nodes characteristic of bonding or was determined in 1967 by Whitaker and Jeffrey43with room antibonding, and population analysis. However, efforts to quantify temperature X-ray diffraction to have Ohsymmetry. After various the u and A contributions have led to theoretical debate on the treatments of the thermal motion and the molecular model of the effect of the quality of the calculation on the results and on structure refinement, it was found to have an atomic configuration technical problems with the Mulliken population analysis and basis Cr-C-O (as opposed to Cr-O-C) with a Cr-C bond length of set superposition error when trying to determine the importance 1.9 16 A, a C-O bond length of 1.17 1 A, and a linear Cr-C-O of 4s and 4p orbitals. bond angle. They noted that the carbon atoms are packed tightly Progress started to be made with the use of energy decomposition analysis (EDA) following concepts introduced by Morokuma70 for studying hydrogen bonds and other van der Waals complexes where no new electron pairs are formed in making the complex. In their R H F study of Fe(CO), and Ni(C0)4, Bauschlicher and B a g u focused ~ ~ ~ on a partitioning of the total (69) (a) Cotton, F. A. Inorg. Chem. 1964, 3, 702. (b) Cotton, F. A.; Krainhanzel, C. S.J . Am. Chem. SOC.1962,84,4432. (70) (a) Morokuma, K. J . Chem. Phys. 1971,55, 1236. (b) Morokuma, K. Acc. Chem. Res. 1977, 10, 294. (c) Morokuma, K.; Kitaura, K. Chemical Applications of Atomic and Molecular Electrostatic Potentials; Politzer, P., Truhlar, D. G., Eds.;Plenum: New York, 1981; p 215. (d) Kitaura, K.; Sakaki, S.;Morokuma, K. Inorg. Chem. 1981, 20, 2292. (e) Kitaura, K.; Morokuma, K. Inr. J . Quantum Chem. 1976,10,325. (0 Nagase, S.;Fueno, T.; Yamabe, S.;Kitaura, K. Theor. Chim. Acta 1978,49,309. (8) Umeyama, H.; Morokuma, K. J . Am. Chem. SOC.1976, 98, 7208. (h) Umeyama, H.; Morokuma, K. J . Am. Chem. SOC.1977, 99, 1316. (i) Yamabe, S.;Morokuma, K. J . Am. Chem. SOC.1975, 97, 4458.
+
around the Cr atom with nearest-neighbor GEM C distances of only 2.7 A. The metal-carbon distances in the metal carbonyls are considered by experimentalists to be fairly short, and the GEM carbon-carbon distance is much less than the sum of the van der Waals radii. In 1975, Jost et al.44reported Cr-C and C-O bond lengths obtained from low-temperature neutron diffraction of 1.915 and 1.140 A, respectively. With a different treatment of thermal
2132 The Journal of Physical Chemistry, Vol. 96, No. 5, 1992
motion, they obtained 1.918 and 1.141 A. They also revised the thermal motion treatment of the Whitaker and Jeffrey data and obtained revised room temperature X-ray values of 1.9 16 and 1.147 A. In 1976, Rees and Mitschler7I combined low-temperature X-ray data with neutron diffraction to obtain 1.915 and 1.140 A. Typical values of the C-O bond length in other metal carbonyls are 1.15 f 0.02 A. These are uniformly longer than the bond length of 1.128 8, in free carbon monoxide. We have used the values of 1.9 16 and 1.17 1 A in the calculations reported here. This leads to a somewhat too large energy to stretch C O but an enhanced r back-bonding. If we had used 1.140 A in place of 1.171 A for the C-O bond length, this would have reduced the SCF energy for stretching six free CO molecules from 34 to only 6 kcal/mol and would have reduced the true stretching energy from 15 to only 1 kcal/mol. Other theoretical calculations on metal carbonyls have used values ranging from the free CO value of 1.128 to 1.15 A. The gas-phase heats of d i ~ s o c i a t i o n ~(to ~ - ~give * ground-state gas-phase atoms) for Cr(C0)6, Fe(CO)5, and Ni(CO), are 154, 140, and 141 kcal/mol. When the average M-CO bond energy is figured relative to atoms in their lowest dnsosinglet state, the bond energies are 58, 61, and 47 kcal/mol. This has been interpreted as indicating that the metal-ligand bonding is qualitatively the same in all three compounds even though nickel has no empty d orbitals. RHF Calculations. The calculations were done with atoms in Oh symmetry positions but only DThsymmetry imposed on the orbitals. With the CO fragments lying along the x, y , and z axes the correspondence between oh and DZhirreducible representations is as follows: Oh
D2h
ab
ag a8 (both x2 - y 2 and 2z2 - x2 - y2) blu, b2u, b3u blU, b2ur b3u bl,, b2gr b3g bl,, b2g. b3g
e8
tl" t2" tk tzs
Hence, the R H F configuration ..8a1,2..5e,2..8t1,6..t2~..tl 6..2t286 'fhe D2h becomes ..l 8a,2..9b1,2..9b2,2..9b3~..3bl~..3b2~..3b~~. orbitals displayed the expected degeneracies, indicating that there was no departure from actual 0,symmetry in the orbitals. The Cr valence orbitals can be classified as u bonding (4s ale, 3d e,), r bonding (3d t2,), or ambiguous (4p ti,) depending on whether they are the same symmetry as the Sa (alg, tlu,e8) or l r and 2 r * (tlu,t2,) orbitals of CO in the octahedral carbonyl cage. Additionally, the r (tl ,t2,) carbonyl cage orbitals are classified as "nonbonding" since tiey have no Cr valence counterpart. Figure 1 shows sketches of the sign patterns in these fragment valence orbitals. The usual MO model for bonding ascribes the bond energy to delocalization of the 50 CO electrons into the 4s, 4p, and 3de, chromium orbitals and "back-donation" of the 3dt2, electrons into the 2r*tZ8orbital of the cage. It is now generally agreed that the back-bonding is d~minant'~ and that the a donation is mainly to the 3de orbital. This conclusion does not readily account for the 18-efectron rule obeyed by Cr(CO),, Fe(CO),, and Ni(C0)4. It has been noticed that there is a mixing between CO 1~ and unoccupied 2 r * orbitals as well, once the Cr 3d orbital is orthogonalized to the C O 1 r orbital. This polarization has been called counterintuitive orbital mixing72 of the three-orbital, four-electron type and can result in a net increase, decrease, or node at carbon in an electron density difference map. This sometimes makes it hard to locate r back-bonding in charge densities or atomic population changes. The details for the construction of the basis set are given in the supplementary material. (Seeparagraph at the end of paper regarding supplementary material.) As shown there, the basis (71) Rees, B.; Mitschler, A. J . Am. Chem. SOC.1976, 98, 7918. (72) Hubbard, J. L.; Lichtenberger, D. L. J . Am. Chem. SOC.1982, 104, 2132.
Kunze and Davidson
t1g
12"
3d ta
2x* 12g
n
0
4P t i u
0 5 0 alg 4s alg Figure 1. Sign pattern in the valence molecular orbitals of the (CO), cage and the chromium atom. Only one member of each degenerate set is shown.
TABLE I: Previous Hartree-Fock Results for Cr(CO)n HillierQ Hallb HillierC Vanquickenborned Baerends' Hall' present work
year 1971 1983 1983 1983 1985 1987 1991
energy -1702.6129 -1708.5768 -1717.2739 -1719.4277 -1708.3061 -1714.3016 -1749.8760
Bauschlicherg
1984
Ni(C0)4
WCOh
-4 -80 ?
-103 ? ? ? +111 +58 +lo4
'Reference 14a. *Reference 13b. 'Reference 14h. dReference 22a. 'Reference 5b. /Reference 13e. g Reference 7a.
gives an energy which is 4 kcal/mol above the S C F limit for CO and 0.2 kcal/mol above the SCF result for Cr. From calculations on each fragment in the full Cr(CO), basis, the total basis set superposition error is estimated to be -7 kcal/mol. Table I shows a comparison of our R H F energy with previous literature values. In spite of our low absolute energy, we find the molecule to be unbound relative to ground-state Cr and CO by 104 kcal/mol (1 11 kcal/mol after correction for BSSE). Because of the high quality of this basis set, the conclusion that MO theory predicts no bonding energy should remain true for a calculation in a complete basis set. We will not deal with the correlation energy in this paper. From the experimental interaction energy of -1 54 kcal/mol compared with the R H F interaction energy of +111 kcal/mol, it is clear that -265 kcal/mol of additional energy must come from some effect. Approximately -80 kcal/mol of this could come from the correlation energy error in the Hartree-Fock promotion energy of the Cr atom. Another -30 kcal/mol could come from an
Energetics and Electronic Structure of Cr(C0)6 improved estimate of the energy needed to prepare ‘stretched” CO molecules at the bond length in the complex. (This is a combination of correlation energy and use of a better estimate of the bond length.) No more than -20 kcal/mol is expected from use of a more flexible basis set for the complex. This leaves approximately -1 30 kcal/mol for the “extramolecular” correlation energy, i.e., the correlation energy change upon molecule formation. This extramolecular correlation energy is not expected to be due to a large mixing between a few configurations. In fact, the free atom is expected to show a much larger near-degeneracy mixing of configurations (in Dul symmetry) than does the complex. Rather, the dominant effect is expected to be “dispersion energy” which is used here to include all the correlation energy arising from induced-dipole-induced-dipole interactions (double excitations involving electrons from different fragments in configuration interaction or many-body perturbation theory language). While a number as large as -20 to -25 kcal/mol of bonds for the dispersion energy contribution to M-CO bonds may seem large, there is precedent in the literature for such a large effect. Frey and DavidsonZ5 estimated the effect at -25 kcal/mol in ScCO, Moncrieff et al.14hobtained -16 kcal/mol of Cr-CO bonds in Cr(C0)6, and Demuynck” estimated -10 kcal/mol for Mo-Kr (at a 3-A bond length). While dispersion energy is normally small in van der Waals complexes where it is the dominant attractive force, the rapid change with bond length ( “ P can ) cause it to become large at normal chemical bond lengths. Bauschlicher and Bagus also found the R H F energy for Ni(C0)4 and Fe(CO)S unbound by 15 and 20 kcal/mol of M-CO bonds relative to free ground-state atoms and CO molecules (even without BSSE corrections which would increase these numbers).
Energy Decomposition Analysis Morokuma Analysis. In this paper, we attempt to break the R H F energy into physically meaningful components. These components, together with the -265 kcal/mol of correlation energy change, will account for the -154 kcal/mol of binding energy in Cr(CO)6. As we will show, there are, unfortunately, several components to the energy which are each as large as the net binding. Hence, the binding energy cannot be understood as simply due to one effect like A back-bonding. The first energy step we consider is the preparation of the monomers for the formation of a lAl, state of the complex. From our own and previous calculations, we know the R H F wave function for the complex is best described as ta6 IA, Cr complexed to carbonyls in their ground state. In order to be able to compute smooth potential curves for symmetric dissociation, we adopt for our reference states the prepared fragments with the Cr atom described by this hypothetical configuration and the CO molecules already stretched to the experimental bond length they have in the complex. The R H F energy of the prepared monomers compared to that of the monomers in their ground state a t their free monomer experimental equilibrium geometry will be called the promotion energy. For the calculations reported here, the total promotion energy is +266 kcal/mol. Assembling the fragments into a larger cluster will be done in two stages. First, we will consider the energetics of placing the six CO molecules into their final octahedral cage positions with the cage M O s allowed to fully relax using all of the basis functions of the complex (including the Cr basis functions). Then we consider the energy involved in inserting the prepared Cr atom into the cage center. Following M ~ r o k u m a , ’we ~ break the energy change for each of these stages into three pieces. First we consider the classical electrostatic interaction, ES, between the unperturbed charge distributions of the fragments being assembled. The electrostatic energy would be the interaction energy between fragments if the wave function were simply the product of the unmodified fragment wave functions with no antisymmetry between fragments. Then we consider the actual energy, E l , of the wave function formed by placing all of the unmodified nonorthogonal monomer MO’s into a single Slater determinant 3 (with the energy defined as
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2133 (+lHl@)/(@l@)). The “exchange interaction”, EX, is then defined as EX = E1 - Eo - ES
where Eo is the sum of the R H F energies of the prepared fragments. This exchange interaction contains attractive terms due to the actual exchange integrals between electrons of the same spin on different fragments and repulsive terms due to the nonorthogonality of the orbitals. This energy is easily evaluated because 0 is equivalent to a Slater determinant in which the fragment orbitals have been orthogonalized. The two terms ES and EX together73are usually called the “steric interaction”, ST. The third piece of the energy in a Morokuma EDA is the orbital relaxation energy, RLX. In this step, the nuclei are held in place and the energy change is computed while the orbitals change from the free monomer orbitals into the final MO’s of the complex. Then RLX is defined by RLX = ERHF - El
so the total interaction energy, INT, is just I N T = ERHF - Eo = E S + EX + RLX In a traditional Morokuma analysis one tries to further decompose this energy into polarization, PL (mixing of occupied and empty orbitals of the same fragment), and charge transfer, CT (mixing of occupied orbitals of one fragment with empty orbitals of another fragment). Clearly, the qualitative description of bonding in this complex as due to H O M e L U M O mixing of the filled 3dt2, of Cr with the empty A* of CO, and/or mixing of the filled 50 of C O with the empty 4s, 4p, and 3de, orbitals of Cr, is ascribing the energy to a charge-transfer term. A large number of variations of the Morokuma procedure have been proposed for separating RLX into PL and CT. The original Morokuma definition considered PL first and then ascribed the balance of RLX to CT. This produced reasonable results for hydrogen-bonded complexes with small basis sets. Later, Kitaura and Morokuma considered schemes for computing CT and PL separately with the difference between CT + PL and RLX described as a mixing term, MX. All of their schemes for PL involved wave functions in which the polarized orbitals of one fragment were not orthogonal to the occupied orbitals of the other fragment, and the energy formula did not fully include all nonorthogonality effects (see Gutowski and Piela74 or Frey and DavidsonZ5). The Morokuma scheme corresponded exactly to the method for computing the polarizability of a molecule in an external field,’5,76 but interaction with another molecule must take into account that the field is actually generated by other electrons which are subject to a Pauli exclusion effect. Apart from this easily solved technical problem, there is a serious conceptual problem with PL and CT. Like a Mulliken population analysis, separation of RLX into PL and C T assumes that basis functions can be meaningfully assigned to individual fragments. When very large sets of diffuse basis functions are used, then the whole orbital relaxation can be equally well described as either PL or CT. If PL and C T are evaluated separately, the MX term will tend to cancel one of them. If they are evaluated sequentially, then the first will give the whole effect while the second will give zero. Consequently, in this paper we will not attempt to separate orbital relaxation into CT or PL. For the same reason, we will demonstrate that a meaningful Mulliken population analysis cannot be made for the basis set used in this calculation. We will, however, show plots of the change in the charge density which will allow a visual assessment of the actual relaxation process. Vinal Partitioning of Orbital Relaxation. One problem with interpreting orbital relaxation is that the orbital energy is not a direct measure of the total energy. Suppose 3’is an initial (73) Stone, A.; Erskine, R. W. J . Am. Chem. SOC.1980, 102, 7185. (74) (a) Gutowski, M.; Piela, L. Mol. Phys. 1988, 88, 943. (b) Chalasinski, G.; Gutowski, M. Chem. Reo. 1988, 88, 943. (75) Dykstra, C. G.Acc. Chem. Res. 1988, 21, 3 5 5 . (76) Buckingham, A. D. Adu. Chem. Phys. 1967, 12, 107.
2134 The Journal of Physical Chemistry, Vol. 96, No. 5, 1992
Kunze and Davidson
closed-shell Slater determinant wave function with orthonormal orbitals 4ki and is the final wave function with orbitals $ k f . From each wave function, a density matrix p can be formed as P =
2c’$k$k* and the Coulomb and exchange operators, J(p) and K(p), computed. Then, even if the Fock operator F=T+VeN+J-l/zK is not diagonal, an average orbital energy can be defined as ck = ( 4 k l q $ k ) The energy change AE = Ef - E’ is not given by the sum of the changes in the orbital energies. The HOMO-LUMO emphasis on mechanisms for stabilizing orbitals may fail to account for factors which stabilize the total energy. The energy change may, however be expressed as
.
AE = 2CA($kIT + VeN + Y2CJ - f / 2 K ) I $ k ) which suggests that it would be interesting to look at changes in the expectation values of T , VeN,and (J - ‘ / 2 K ) . It will be important later to note that, if only one orbital is allowed to vary in this equation, the energy contribution from every orbital is changed through the change in J - 1 / 2 K . An alternative equation was noted by Ziegler23d
-
: -240-
Clossicoi Coulomb ,nsertion
I
’
0;
‘ Distortion
:”
insertion of Cr*
AE = 2CA($k(FTS1$k) where F is the Fock operator for the “transition state“, defined as the average of F’ and F‘. This equation is appealing because the RLX energy is expressed directly as a sum of changes in orbital energies, albeit with an average Fock operator. Also, if only one orbital changes, then AE is just‘the change in that one orbital energy. If this definition of orbital energies is used, or if relaxation does not change p (and hence J - 1/2K) too much, then the HOMO-LUMO emphasis on orbital stabilization is justified. These two ways of assigning the energy change due to changes in an orbital differ only in the way the change in electron repulsion between electrons in different orbitals is assigned. In the first definition this change is partitioned equally between the orbitals involved, while in the second definition it is ascribed entirely to the orbital which changed. In both of these equations it is essential that the orbitals be orthonormal. In the case of full orbital relaxation, the final orbitals will be chosen so that P is diagonal in the M O basis. In order that no orbital energy contributions appear just due to use of a different definition for the initial orbitals, we choose the orthogonalizing transformation of the fragment MO’s so that F‘ is diagonal within the occupied M O basis. It is also possible with our programs to consider the relaxation of one MO with the others fixed at their nonorthogonal freefragment values, since this is exactly equivalent to using sequentially orthogonalized orbitals with the ones to be frozen placed fmt. In order to explore the effect of HOMO relaxation, we report the results of constrained orbital relaxation for the 3dt2, set alone, the 5u set alone, and both together, as well as the all-electron relaxation. This approach, which was inspired by an error in Baerends’ paper,58leads to a deeper insight into the stabilizing effects of orbital relaxation and synergistic interactions (see also B a ~ s c h l i c h e rW , ~ ~~ l f e Ber~~adi’~). ,~~ Results Partitioning of the Interaction Energy. The total interaction energy of six stretched C O S and a t22 prepared Cr* atom to form Cr(CO), is shown in Figure 2 as a function of the symmetric stretch Cr-C bond length. As expected, the total interaction has its minimum at a bond length slightly longer than the experi(77) Wolfe, S.;Mitchell, D. J.; Whangbo, M. H. J . Am. Chem. SOC.1978, 100. 1936. (78) Bernadi, F.; Bottoni, A,; Mangini, A.; Tonachini, G. J . Mol. Strucr. 1981, 86, 163.
R(Cr-C)
A
Figure 2. Pieces of the interaction energy for Cr and six CO molecules, cage, as a function of the Cr-C distance arranged in an octahedral (0,) for symmetrical stretch.
mental. The cluster is bound, relative to the prepared fragments, by -162 kcal/mol but is unbound relative to the ground-state fragments by +I04 kcal/mol. This becomes + 111 kcal/mol when a +7 kcal/mol correction for BSSE overbinding is included. The total repulsive interaction for assembling the CO molecules to form an empty (CO), octahedral cage is also shown in Figure 2. This can be analyzed into electrostatic, exchange, and orbital distortion energies. The repulsive interaction, at the experimental Cr-C distance, is dominated by exchange repulsion (+I 11 kcal/mol) with small attractive electrostatic (-27 kcal/mol) and relaxation (-24 kcal/mol in the full molecular basis including Cr basis function) contributions. This gives a net repulsion of 60 kcal/mol in building the cage (+67 kcal/mol if the BSSE correction is applied here). The GEM carbons, at a C-C distance of 2.710 A, are quite close together. The individual energy components for inserting Cr* into the cage are also shown in Figure 2. At the experimental Cr-C distance, the dominant term is exchange repulsion (+359 kcal/ mol). There is a very large electrostatic term (-272 kcal/mol) and an even larger orbital relaxation term (-309 kcal/mol). This gives a net attractive interaction between Cr* and (CO)6 of -222 kcal/mol in the RHF model. These numbers are quite comparable to the results of Bauschlicher and Bagus for Fe(CO)5 and Ni(CO),. Because the Cr* atom and the cage both have very high symmetry, the long-range electrostatic interaction is expected to decrease like Rnwith n 2 5. The large magnitude of ES, shown in Figure 2, was an unexpected contribution to the bond energy because both C O and Cr are usually regarded as “nonpolar” fragments. Also, this ES energy has a predominantly exponential dependence on R over the limited range of R shown. Because of the sp hybrid nature of the 5a lone pair, however, carbon has the possibility of a significant “penetration” component to the charge electrostatic energy. Later figures will show that the (CO),
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2135
Energetics and Electronic Structure of Cr(CO),
S
"#I: ;,;a'
2n'
I
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:"
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,
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Figure 4. Contour map of the total electron density of the (CO)6 cage at the equilibrium Cr-C distance.
chromium. The energies shown here are "canonical HartreeFock", so for occupied orbitals they are approximate ionization energies and for empty orbitals they are electron affinities. Differences in these orbital energies do not give excitation energies. In fact, the Cr* 3dt$ and ezgare essentially the same orbital and ,- ell the orbital energies differ just because occupied and empty orbital , energies are defineddifferently. These orbital energies are, ; So #I .', however, the differences in the diagonal matrix elements of the , Fock operator when the fragments are far apart; so they are the (1" In 'i correct differences to consider when using perturbation theory c: =,' :' ; ; , z to estimate the amount of orbital mixing. 1,)m ', &I" *-In the absence of the Cr, the Sa orbitals of the GEM CO's *. overlap to form a bonding al MO, a nonbonding tl, set of MOs, illl -20 and an antibonding eg set of MO's in accordance with the sign CO' (CO), Cr(CO), Cr' pattern shown in Figure 1. The orbital energies split by 5.4eV Figure 3. Canonical Hartree-Fock orbital energies. due to this interaction. By comparison, the 1r orbital energies split by only 0.5 eV. The order is the one expected from the sign density penetrates the Cr* charge cloud significantly. This reduces pattern in Figure 1, but the GEM overlap is small because the the electron-electron repulsion below the value included in the lr MO is shifted to the oxygen end of CO. The 2r* orbital multipole expansion of the electrostatic energy and gives an enenergies also split in the same pattern, but by a larger amount hanced ES energy with an exponential dependence on R.79-81 because this orbital is shifted to the carbon end of CO and gives The exchange interaction is also quite large considering that larger GEM overlaps. it comes from overlap between filled fragment orbitals, in this case The orbital energy changes during insertion of Cr* into the mainly 5a of CO and the 3s and 3p core of Cr. Near the (CO), cage, shown in Figure 3, appear to agree with elementary equilibrium distance, this large exchange effect would also indicate textbook arguments about H O M G L U M O mixing lowering the a large penetration contribution to the electrostatic energy. At orbital energy of 3dt2, and 5ue,. The situation is a little more longer Cr-C distances, the exchange term decreases faster than complex than this, however. Considering only the electrostatic the ES term. At large R,the electrostatic term dominates, so the energy would cause the sum of orbital energies, C(nktk), to change steric interaction shows a shallow attractive well with a minimum by AE A( V,) - A( V , , ) , or -1422 kcal/mol. Most of this near 2.4 A. appears as stabilization of each of the six 5a (CO), cage MO's The orbital relaxation energy is nearly equal to the electrostatic by an average of -100 kcal/mol. The exchange repulsion similarly energy over the range of Cr-C distances considered. The shapes changes the sum of the orbital energies by +1800 kcal/mol with of the two curves are the same. With a good basis set, the orbital most of the change appearing as an average 100 kcal/mol shift relaxation can equally well be represented by mixing with empty in the 5u algand t,, (CO), cage and the 3s algand 3p t,, Cr atom orbitals on the same fragment or with empty orbitals on the other orbital energies. The final relaxation step produces a further fragment. The exponential shape of the RLX energy, however, change of -5260 kcal/mol in the orbital energy sum. Most of can be taken as an indication that this is primarily charge transfer this is not apparent at all in Figure 3 because the 18 core electrons rather than polarization. of Cr are stabilized by about 250 kcal/mol each. This stabilizing Canonical Orbital Energies. Figure 3 shows the changes in shift in orbital energy is nearly twice the shift in the 3dt2, orbital orbital energies for formation of the cage and insertion of the energy and is associated with the reduction in core-3d electron repulsion due to the expansion of the size of the 3d orbital during the orbital relaxation step. (79) Hoinkis, J.; Ahlrichs, R.; BBhm, H. J. Int. J. Quantum Chem. 1983, The large magnitude of the orbital energy changes, and the fact 23, 821. (80) (a) Sokalski, W. A.; Hariharan, P. C.; Kaufman, J. J. J . Comput. that the energy changes even when the orbital does not, makes Chem. 1983, 4, 506. (b) Sokalski, W. A.; Chojnacki, H . Int. J . Quantum interpretation of the self-consistent-field orbital energies difficult. Chem. 1978, 13,679. Electron Densities. Figure 4 shows the total electron density (81) van Lenthe, J. H.; van Duijneveldt-van de Rijcht, J. G. C.; van of the (CO), cage. This figure makes clear the large electron Duijneveldt, F. B. Adu. Chem. Phys. 1987, 69, 521. (82) (a) Partridge, H. J . Chem. Phys. 1987, 87, 6643. (b) Partridge, H. density in the interior of the cage before the Cr is inserted which NASTA Technical Memorandum 101044; Ames Research Center: Moffett leads to a large electrostatic penetration energy and to a large Field, CA, 1989. overlap repulsion with the Cr 3s and 3p core orbitals. Figure 5 (83) Sundholm, D.; Pyykko, P.; Laahonen, L. Mol. Phys. 1985,56, 141 1 . shows the charge density of the Cr atom in the t22configuration. (84) Hay, P. J. J . Chem. Phys. 1977, 66, 4311.
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Kunze and Davidson
2136 The Journal of Physical Chemistry, Vol. 96, No. 5, 1992
---.
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Figure 5. Contour map of the total electron density of a t 2 t 'A,, chromium atom.
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Figure 6. Change in electron density caused by antisymmetrizing the (CO), and Cr fragment orbitals.
The preference for this configuration is correctly interpreted by ligand field theory as due to the channeling of 3d electron density between the ligands which leads to lower electron-electron repulsion and lower overlap repulsion. Figure 6 shows the difference between the density of Cr(CO), computed from the wave function built from unrelaxed fragment orbitals and the sum of the fragment Cr* and (CO), cage fragment densities. This difference shows the effect of the Pauli exclusion principle on the charge density. It leads to a buildup of density in the oxygen p r orbital and an increase near the Cr and C nucleus. There is a general depletion of density in the region of overlap of 5a with 3s and 3p. Figure 7 shows the change of total density upon orbital relaxation. Figure 8 shows the change in density within the tZgset of orbitals, and Figure 9 shows the total change in ag, ti", and eg orbitals. The most obvious feature in Figure 8 is the loss of density in the 3dt2 orbital and the increase in density in the oxygen ?r orbital. Tiere is a smaller increase near the carbon atom and a general increase in size of the MO. This corresponds to counterintuitive orbital mixing as 2r* has its largest density near carbon. In the u space, there is an increase in electron density in the region of the 3de, orbital and a shift of u electrons from oxygen to carbon. The key 5u eg and 3dt2, orbitals before and after orbital relaxation are shown in Figure 10. The u relaxation effects in the total density are evident in the e orbital and come,from admixing of 3degcharacter. The t2gorbita! displays predominantly a general increase in size and concentration in a pr pattern around 0 which could be represented as mixing with the lr and 2r* orbital of CO. There are minor changes in the other orbitals. In particular, the Cr 3s and 3p show a slight contraction consistent with their
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deshielding as the 3dt2, orbitals expand. We also tried many schemes for assigning Mulliken populations to atomic orbitals and fragment molecular orbitals. All failed to produce sensible values. The main difficulty is that the overlap between the 5 0 a l gand t,, cage orbitals and the 4s and 4p Cr orbitals is greater than 0.9. This makes partitioning of charge between these orbitals meaningless. In fact, a calculation on the (CO), cage using the full molecular basis set results in -0.26 electron in the 4s Cr orbital and a total of -0.61 electron on the Cr center, according to the Mulliken population scheme. Less than no electrons on an atom is complete nonsense. When the
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2137
Energetics and Electronic Structure of Cr(CO),
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d ~ " " " " " " ' " i ' " ' " " " " " ' ~
Figure 10. The Sue, and 3dt2, orbitals after orthogonalization and after relaxation of all orbitals: (a) Sue, before relaxation, (b) Sue, after relaxation, (c) 3dt2, before relaxation, (d) 3dt2, after relaxation.
Cr atom is actually included, the net electron deficiency on the Cr center is -0.1 electron, which is a net gain of 0.5 electron. Hence, the usual Mulliken population analysis result would be to report the result Cr+O.'. If we make a BSSE correction to the population, however, the result could be reported as Cr-'".5. The 3d orbitals have less of a problem with large overlap so one might believe the BSSE corrected electron population changes of -1.1 in t2! and +0.4 eg. Virial Characterization of Changes. In an effort to better understand the energy changes in each step, we have computed the changes in quantities related to the virial theorem. If the system had changed from one R H F minimum (free fragments) to another (complex), the kinetic energy would have increased and the potential energy decreased. Because we are using experimental coordinates, this will not be true. The analysis summarized in Table I1 shows, first of all, that the changes in the virial quantities are orders of magnitude larger than the net energy change. Beyond that, one sees that the kinetic energy contribution, A( T ) , to the exchange energy is large and repulsive because of introduction of additional nodes in the orbitals from orthogonalization to other fragments. The net change in the electronelectron repulsion, A( V,), due to exchange is also repulsive even though this includes the additional attractive exchange integrals between orbitals on different fragments. The change in the electron-nuclear attraction, A( V,,), in the exchange part of step 2 is stabilizing because orthogonalization of valence orbitals on CO to core 3s and 3p orbitals on Cr introduces tails on these functions which intrude closer to the Cr nucleus. The electrostatic energy is slightly stabilizing due to the fact that the enormous attraction of the electrons on one fragment for the nuclei of the other fragment slightly exceeds the sum of the repulsion between the nuclei of the fragments, A ( V,,), and the repulsion between electrons on separate fragments. The fact that A ( V , , ) > A(-1/2Vne) > A( V,) is consistent with ES arising from penetration. Orbital relaxation in step 2 reduces the kinetic energy. This is different from the usual situation in covalent bonding where orbital relaxation is accompanied by orbital contraction and an increase in kinetic energy. As discussed earlier, the dominant orbital relaxation is an increase in size of the 3dt2, orbital (including delocalization into 27r* of CO) which leads to a decrease in kinetic energy. This same increase in size also reduces (V,) by moving the 3d electrons away from the other Cr electrons but loses energy in (V,,) because the 3d electrons move further from the Cr nucleus. Changing the 3d orbital would be unfavorable in the absence of the CO cage, and increased shielding from penetration of Cr by the 5u orbitals of CO is the dominant cause of the distortion. Subspace Relaxation. Although the orbital relaxation energy during insertion of Cr into the (CO), cage is only one piece of the total binding energy, it is the exclusive focus of semiempirical models and has dominated the discussion of past a b initio calculations. In this section, we will consider simultaneous relaxation of all orbitals of a given symmetry, relaxation of individual orbitals, and possible synergistic effects from simultaneous relaxation of two kinds of orbitals. This latter possibility is important since the synergistic model of bonding is taught as reality in textbooks. An advantage to considering all orbitals of a given symmetry together is that it gives the same answer no matter how they are defined during orthogonalization. In the transition-state method of ascribing energy changes to orbitals, all the change in (V,) is ascribed to the orbital which changes. As seen in the preceding discussion of orbital energy changes, this avoids a great deal of confusion. Figure 1 1 shows the transition-state partitioning of the total relaxation energy into contributions from orbitals grouped by symmetry. The t2grelaxation at the experimental Cr-C distance is -224 while the egis -76 out of a total of -309 kcal/mol. For comparison, B a e r e n d ~using , ~ ~ density functional theory which gets the complex overbound by -100 kcal/mol, obtained -353 for tzgand -141 for eg out of a total of -549 kcal/mol. Both of these calculations agree that the dominant effect is due to t2gorbitals.
2138 The Journal of Physical Chemistry, Vol. 96, No. 5, 1992
Kunze and Davidson
TABLE II: Analysis of Morokuma Partitioning of the RHF Interaction Energy; Changes in ( T ) , ( Va), ( V - ) , and ( V,,,,), in kcal/mol, Accompanying Morokuma Partitioning of the RHF Interaction Energy, Starting with Promoted Fragments A( T) A(Ven) A( V A A(Vnm) AE 523 871 total interaction E 582 -1 053 596 528 982 -162 254871 41 1 -510545 255 323 60 interaction E, step 1“ 171 -222 269 000 interaction E, step 2* -543 047 273 659 255 21 1 813 -51 1263 255 323 84 steric E, step 1 -402 718 -340 -24 0 distortion E, step 1 273951 273 659 2678 -550 196 steric E, step 2 87 -4 98 1 0 -2507 7 149 -309 distortion E, step 2 254 860 0 255 323 electrostatic E, step 1 -510210 -26 813 0 351 -1 053 110 exchange E, step 1 0 272 509 -546 440 -272 273 659 electrostatic E, step 2 2678 0 1441 -3 756 exchange E, step 2 359 “Step 1 is 6CO
-
(CO),. bStep 2 is (CO),
+ Cr*
-
Cr(C0)6. Distortion f o r Insertion o f Cr* i n t o Oh
+
( C O ) 6 cage, Cr* 8000
c-
6
-320
-4001
-480
1.6
i
;
I I
Derived Tronsition ,Stote Theory , ,
,
1
F ‘1.
4000-
Y
2000-
v
cr(C0)~
I Differences in S u m
p., x
->
I Q>
P
(CO)6
Over S y m m e t r y of
Total Energies
x
E?
Total
~
W
2.0
2.4
2.8
R(Cr-C) A
Figure 11. Total relaxation energy within each symmetry when all orbitals are relaxed partitioned according to the change in ( @ l p I b ) .
Bauschlicher and B a g u ~ using , ~ ~ a sequential limited subspace relaxation method for Ni(C0)4 and Fe(CO),, came to the same conclusion for those molecules. When the total energy change is partitioned using A ( T + V,, + 1/2(Jas the definition of the orbital energy contribution to the total energy, a quite different picture emerges. As shown in Figure 12 and Table 111, the t2* relaxation is now repulsive, while all the rest are attractive. That is, when the changes in electron repulsion are partitioned equally between the orbitals involved, regardless of which one changes, then the relaxation of the tZgorbital is seen to destabilize that orbital while lowering the total energy of the molecule. In Table IV and Figure 13, relaxations of individual molecular orbitals are considered keeping the rest frozen at their nonorthogonal free fragment values. As shown in Figure 13, the change in the total energy from relaxing just the 3dt, orbitals accounts for half of the total RLX. Relaxation of just the 5u eg orbitals of (CO), gives about one-third as much improvement. The synergistic effect of letting both sets of orbitals relax together contributes another -61 kcal/mol, which is spread evenly among the t2gand eg electrons. There is another -42 kcal/mol from simultaneous relaxation of all the other orbitals. Figure 13 shows that this relaxation mechanism extends to large Cr-C distances where the overlap between 3d and A* should be smaller. Hall had concluded, based on a population analysis, that the r back-bonding should occur at shorter range than 5u mixing with 3der Our results find the opposite conclusion. Figures 11 and 13 show that at 2.5 8, almost all the RLX is due to tZg. Inspection of the MO coefficients at this distance shows that relaxation is even more localized in charge transfer to oxygen with much less general enlargement of the 3d orbital or charge transfer to carbon. Table IV shows the partitioning of these individual orbital relaxation energies according to the scheme of equal sharing of
‘/,a)
I Experimental R ( C r - C )
-4000,
2.0
16
2
2.4
R(Cr-C)
a
A
Figure 12. Total relaxation energy within each symmetry when all orbitals are relaxed partitioned according to the change in 1/2(bIF hi&).
+
Distortion for Insertion of Cr* into o h (C0)E coge, Cr* + ( c o ) ~-> Cr(C0)C 0, I &--------4
-‘“I
I
-480
1.6
Bonding
I Erperimentol R(Cr-C) 2.0
2.4
8
R(Cr-C) A
Figure 13. Energy change upon relaxation of tlg and e8 HOMO orbitals separately and synergetically.
electron-electron repulsion improvements. When only the t, HOMO is relaxed, the total energy improves by -154 kcal/mo! but the energy contribution of the tzgorbital is repulsive by 2373
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2139
Energetics and Electronic Structure of Cr(CO),
-
TABLE III: Contribution to the Relaxation Energy (kcrl/mol) from Relaxing All Orbitals Simultaneously during the Insertion Step Cr* Cr(CO),
MO lalg 2% 1tlU 3a1, 2tlU 1% 4a1, 34, 2% 5a1, 4t1, 6alg 5t1, 3% 7a1, 6tl" 4% 7hU 1t2g 1t2u Itl, 8% 8tlU 5% 2t2g
SUM"
A( T) -6 10
9 -10
0 -6 -19 -3 -1 1 144 85 11 -7 3 -174 -26 -32 20 -36 -6 -3 195 64 45 -57 1 -2509
A(V,,) 6 -12 -1 1 11 0 6 21 3 12 -241 -147 -24 -3 -1 3 -848 -233 -288 -626 -347 -25 -2 1 590 476 -767 3012 7147
A L / 2 ( J -'/2K) -147 -152 -151 -3 -3 -3 2 3 3 -5 3 -6 3 7 5 5 505 129 159 300 188 15 11 -394 -269 347 -1269 -4947
AL/2(F+ h ) b -147 -153 -153 -3 -3 -2 4 3 4 -151 -1 25 -6 -5 -5 -516 -1 30 -161 -306 -195 -16 -13 39 1 270 -375 1173 -309
A ( P ) 0 0
+ (CO)'
Cr 1s Cr 2s Cr 2p 0 1s 0 1s 0 1s c Is c 1s c 1s Cr 3s Cr 3p co 3a co 30 co 30 co 4 a co 4a co 40 co 1 r co 1 r co l r
0 0 0 0 0 0 0 0 -1 0 0 0 -5 -1 -2 -3 -3 0 0 4 3 -17 -34
co lr co 5a co 50 co 5a Cr 3d
-309
'SUM = ~nkgkA(@klol@k) where nk is the orbital occupancy and gk is the orbital degeneracy. The sum gives the total change in each energy quantity. The sum of AI/2(J - 1/2K)gives A(V,). The sum of 1/2(F+ h ) and the sum of (Ps) both give AE. b 1 / 2 ( F h ) = ( T V,, l/*(J -'/2K)) = (F-'/z(J-'/&); ( F ) ( T + V,,+ J - ' / & ) ; ( h ) = ( T + Vcn).
+
kcal/mol. The energy improvement comes in the a, and tl, core orbitals which now feel less electron repulsion from 3d. By contrast, relaxing the e, orbital works in the expected way since the e, orbital contribution is attractive by -903 kcal/mol. In this case, it is moving closer to the Cr core and the core orbital contributions are repulsive. The very small relaxation effects in the a, and tl, symmetries suggest that the 4s and 4p orbitals of the Cr are not very important. In fact, elimination of the 4s and 4p Hartree-Fock valence atomic orbitals from the basis set (i.e., working in a projected basis orthogonal to these orbitals) raises the energy by only 4.9 kcal/mol. A similar result was reported by Baerends and by Bagus and Bauschlicher. Hall has phrased this result differently by stating that the 4s orbital is contracted in the molecule. This is equivalent to saying that the 4s orbital is unimportant and that other changes in the a, space such as 3s contraction can make better use of an "s" basii function of a different size. Since the Sa ag (CO), orbital has an overlap of 0.96 with the Cr 4s orbital, the 4s orbital represents little additional freedom in the wave function. Thus, it is certainly incorrect to describe the IJ bonding as donation into sp3d2metal hybrid orbitals. At the same time, it is essential that all these orbitals be empty on the metal since an electron in 4s or 4p would cause very large overlap repulsion. Thus, the origin of the "18-electron rule" is not the use of metal hybrid orbitals for accepting electron pairs but the huge overlap repulsion from the Pauli principle if ligand lone pairs overlap filled metal 4s or 4p orbitals. Comparison with Density Functional Results. Baerends and RozendaalSBreported a basis set Xa calculation with the free electron gas correlation functional and a = 0.7. Their binding energy was -239 kcal/mol(-208 kcal/mol after BSSE correction) compared to the experimental -155 kcal/mol and the R H F result discussed above of +111 kcal/mol. This occurred mainly because the interaction between the seven prepared fragments was -458 kcal/mol compared to the R H F value of -162 kcal/mol. Ziegler et al.23aincorporated modern gradient corrections and a better correlation energy functional. Their calculation gave a binding energy of -1 53 kcal/mol, in excellent agreement with experiment. In addition, their Cr promotion energy, +144 kcal/mol, is close to the value, 150 kcal/mol, we estimate in the supplemental material to be correct. Their total steric repulsion
+
+
for assembling the seven fragments into the complex in one step, +204 kcal/mol, is close to the R H F value of + 160 kcal/mol we calculate for this direct process. Their orbital relaxation energy, -461 kcal/mol is much greater than our total R H F relaxation energy (relative to Cr* 6CO) of -322 kcal/mol. Our estimate of -1 30 kcal/mol for the extramolecular correlation energy should be built into the Ziegler calculation, but it is unclear whether it should appear in the steric or orbital relaxation energy. Importance of Radical Pair Configurations. An alternative description which could be considered is ionic bonding between Cr+ and a (CO),- cage. At the equilibrium bond length, the Mulliken population in fact shows a loss of 1.1 t2, electrons. Hence, a reference configuration for the fragments might be chosen to be t285 2T2,Cr+ and tZgl2T2g(CO),-. This would be only slightly higher in energy because the R H F ionization energy of d6 '1 Cr to give d5 21Cr+ is only 21 kcal/mol(37 kcal/mol for ionization from t 2 t Cr). The large relaxation energy of the d electrons upon ionization makes Koopmanns' theorem not valid, so the IP is not close to the negative of the d orbital energy. While CO is not a good electron acceptor, the (CO), cage has a stabilized t2, empty orbital as shown in Figure 2. With this basis set, COis above C O by 53 kcal/mol but (CO),- is above (CO), by only 18 kcal/mol. This stabilization comes from the oxygens on the outside of the cage. Hence, Cr+ + (CO),- is only 55 kcal/mol higher than Cr (CO), in the Hartree-Fock model. Insertion of Cr+ into the (CO),- cage can then take place with greater electrostatic energy, and less orbital relaxation energy, than starting with neutral fragments. Closely coupled diradical ion pairs of this type, (A+)(B'-), are often suggested as responsible for bonding in exciplexes. If the molecule were actually a singlet-coupled diradical, the R H F description would be inadequate. To describe insertion of Cr'+ into a (CO)6'- cage to form an A,, state, we would need the (unnormalized) wave function
+
+
+I
= A[ ...3dx~3dx,23dyz12?r*tYz1(aP - @a)]
42 = A[ ...3dx~3dy,23dxy'2?r*txy'(aj3 - @a)] c $ ~ = A[ ...3dx~3dyz23dx,'2~*txz1(a@ - Pa)]
9 = 9,
+ 42 + 43
Kunze and Davidson
2140 The Journal of Physical Chemistry, Vol. 96,No. 5. 1992
-
TABLE IV: Contribution to the Relaxation Energy (kcal/mol) from Relaxing Selected Orbitals during the Insertion Step Cr* + (CO), Cr(CO), relax. A1/2(J - ‘ / 2 W MO t28 e8 both la,. -167 40 -170 Cr 1s -164 40 Cr 2s -164 Cr 2p 40 -3 3 0 1s 3 -3 0 Is 3 -3 0 Is 2 -3 c 1s 2 -3 c 1s 2 -3 c 1s -105 33 Cr 3s -98 32 Cr 3p 1 -1 co 3u 1 co 3a -1 1 -1 co 3u -3 2 eo 40 1 0 co 4u 1 0 eo 4u 1 -2 co 1n -1 0 co In 0 0 co In 0 -1 co In -6 6 co 5a -5 co 50 6 222 482 co 5a -848 26 Cr 3d
1733 -120
-1664 -52
-5656 -2816 8265
-267
“SUM = Cnkg,A1/2($kJJ- 1/2q$k) where n is the orbital occupancy and g, is the orbital degeneracy. This sum gives A ( Vw). bFor nkgkelectrons in this degenerate orbital. For tlg or e, relaxation, only the average value for the relaxed orbital changes. Also n , g , A ( p ) is AE for the orbital which changes and zero for the rest of the orbitals. When both orbitals relax, 6A(t2,1qt2,) is -2904 kcal/mol and 4A(e,lqeg) is 28 kcal/mol. Also, 6A(t2,(Ve,(t2,) is 12342 kcal/mol and 4A(e,lVe,Jeg)is -4076 kcal/mol. This gives 6A(t2,1plt2g) to be -194 kcal/mol and 4A(eglple,) to be -73 kcal/mol. when A denotes antisymmetrization. This wave function cannot be expressed as a simple RHF wave function, as it always has one, and only one, electron in the cage 217* orbital. By contrast, the R H F wave function with a t2%orbital of the form X(3d) + p(2?r*) contains components with (2r*)”, n = 0, 1, ..., 6 electrons. If \k were truly ionic, the R H F wave function would be very inadequate. While we do not have programs to investigate this form for \k, we can compare $q with the S C F result. In DZhnotation, we consider replacing one b3g electron pair by a split-valence description, b3%b3;. This lowers the energy by 21 kcal/mol and gives an overlap (b3Jb3gl) of 0.62. If this wave function is written in the alternative form, c , ( X ) ~- C ~ ( Y )where ~ , X is b3g+ b3g) and Y is b3g- b3gl, then c2/c1 is 0.24. Thus, the exact \k probably contains other configurations with coefficients 0.2 times as large as the S C F coefficient. A coefficient of this size is intermediate between typical correlation coefficients, 0.05, and true diradicals, c2/cI 1 0.5. However, when the free Cr atom is computed with the same b3 b 3 i form for the wave function, its energy is improved by 13 kcalrmol and c2/q is 0.17. This describes radial correlation in the free atom and is not very different from the result in Cr(CO),+ Thus, diradical ion pair configurations do not seem to be of major importance. If each of the bilowered the energy by the same 8 kcal/mol relative to the free atom, this would account for only 24 kcal/mol of extra molecular correlation energy. As a footnote, it should be noted that it is often claimed that electron correlation’” increases the 3d to 2?r* charge transfer based on excitations of the X2 Y2type. From the results, we have obtained, these past claims must be interpreted cautiously since they do not report the corresponding result for the free atom in
-
the full molecular basis set. Our split-valence atomic results are the same in the atomic basis and full molecular basis, but literature results have not reported any possible basis set superposition errors in the electron correlation energy. Conclusion
Is it possible to extract from these pictures the “driving force” for bond formation? For covalent bonds, the driving force is said to be the valence bond exchange effect from a@ Pa “spin-exchange resonance”. For ionic bonds, the driving force is ascribed to electron transfer a t large R followed by Coulomb attraction between the ions. For hydrogen-bonded complexes the driving force is electrostatic attraction between polar groups, and for nonpolar van der Waals complexes it is induced-dipole-induced-dipole dispersion (another name for extra-molecular correlation energy). In all of these statements what has been identified is a dominant attractive interaction in the range of maximum force (not minimum energy) before the short-range repulsive forces have become large. In each case, the term identified contributes a major part, but not all, of the attractive force. Can such terms be identified for Cr(CO),? Near 2.5 a where the force is maximum, the electrostatic penetration energy and orbital relaxation energy are dominant and nearly equal in importance. The repulsive cage formation energies are small. The exchange energy for inserting an atom, already prepared in the t2: configuration, is also small. Examination of the orbital relaxation energy and coefficients a t this distance shows that the relaxation is almost entirely due to tzr with little change in any other orbitals. Further, even at this distance, the relaxation is counterintuitive with most of the transferred charge appearing in the oxygen ?r orbitals. The extra-molecular electron correlation, which has not been computed here, generally has a very sharp bond length dependence25and does not produce a large force at longer distances. Do these results confirm the usual model of a large 3d-2?r* covalent mixing as the primary source of bonding? In our opinion (with which many people can reasonably disagree) the answer is no. Carbon monoxide is a very special ligand. The 5u lone pair is sp hybridized and extends far from the carbon nucleus. The oxygen end of the molecule is a good electron acceptor. This acceptor ability is enhanced in a CO cage with the oxygen on the outside. When a chromium atom is placed in a CO cage, there is a large electrostatic attraction because of the interpenetration of the 5u lone pairs of CO and the metal 3s, 3p, and 3d orbitals. We regard this as the primary effect. The 3d orbital is much more polarizable than the 3s and 3p orbitals, so it responds to penetration by the 5u electrons by becoming more diffuse. This results in considerable additional lowering of the energy but is not primarily a covalency effect. In fact, the Cr-C ?r bond order remains small because most of the charge appears in the oxygen ?r orbital. Arguments about 3d,2?r* overlap are irrelevant because no mixing will take place in the absence of some driving potential-in this case the increased shielding of the 3d electrons by the penetrating 56 electrons. The existence of the empty 21r* orbital is important, since blocking of this part of function space would effectively prevent the 3d orbital expansion, but the mere existence of this orbital does not cause the relaxation. Correct description of the electron correlation between the electrons in these interpenetrating orbitals is also important for a quantitative estimate of the energy. At shorter distances there is some relaxation of the 5a orbitals, as well, into the empty 3degorbitals. There is also some synergistic effect as this increased penetration further shields the 3dt2, electrons and causes those orbitals to expand even more. We tend to use the word “bond” to describe the situation where new intermolecular electron pairs are formed and the forces are large and the word “complex” to describe the situation where no new electron pairs are formed and the forces are weaker. Often, as in the case of the carbon atom, the initial promotion of the atom in preparation for bonding is regarded as spin-unpairing to provide a greater number of unpaired electrons. For transition metals the situation is different from either of these cases. The initial promotion of the atom is regarded as emptying the 4s shell and
-
2141
J. Phys. Chem. 1992, 96, 2141-2146 pairing the electrons in the 3d shell. Usually this is said to be done to provide empty orbitals to accomodate the incoming lone pairs of the ligands in forming a complex. Calculations indicate, however, that the ligand makes very little use of these empty metal orbitals. Even so, it is essential that they be empty because they have a very high overlap with the ligand orbitals and would give rise to large repulsive effects if occupied. Emptying these orbitals and pairing the electrons of the metal has a high energy cost, but the repulsive energy from leaving them filled would be even higher. At first sight paying a promotion energy price to form a weak coordinate bond does not seem possible. In fact, most nonpolar ligands will not stick to a transition metal in oxidation state zero. Carbon monoxide is exceptional. Previous explanations for this have focused on covalent bonding with the empty 27r* orbital of CO. On the basis of the calculations presented here, we prefer an explanation which emphasizes the electrostatic energy made possible by the penetration of the polar 5u lone pair of CO into the metal charge distribution. The 27r* orbital remains very important as a place for the 3d electrons to go in response to this perturbation. As far as the detailed energetics of the bond energy of Cr(Co), is concerned, one can see that the atomic promotion energy is actually about 150 kcal/mol(232 kcal/mol SCF, -80 kcal/mol correlation correction) and the CO stretching promotion is small (34 kcal/mol in the present work because of the too long bond length used in this paper). The (CO), cage formation is about 60 kcal/mol. The electrostatic attraction between the cage and the chromium is about -270 kcal/mol, and the exchange repulsion is 360 kcal/mol. The basis set limit orbital relaxation is estimated to be about -330 kcal/mol (the present basis gives -309 kcal/mol),
and the extramolecular correlation energy is estimated to be about -130 kcal/mol. All numbers should be reliable to within f 1 0 % and are in good relative agreement with similar numbers25derived for ScCO. Clearly several numbers are large compared to the net bond energy of -150 kcal/mol and must be included in a quantitative accounting of the bond energy. Semiempirical MO theory neglects all effects except the orbital relaxation and interprets it as covalent bonding. Ab initio MO theory misses the extra correlation energy and overestimates the atomic promotion energy by 80 kcal/mol, so it is hopeless as a quantitative tool. No qualitative changes occurred in the ScCO wave function at the equilibrium bond length as a result of including electron correlation, and none are expected for Cr(CO),; so the present interpretation of the source of bonding is unlikely to be changed by improved calculations. Acknowledgment. This work was supported in part by the National Science Foundation. This research was mostly conducted using the Cornell National Supercomputer Facility, a resource for the Center for Theory and Simulation in Science and Engineering a t Cornell University, which is funded in part by the National Science Foundation, New York State taxpayers, and the IBM Corp. Registry No. Cr(C0)6, 13007-92-6.
Supplementary Material Available: Details concerning the choice of orbital exponents, the contracted orbitals, and the energies of the fragments with this basis (4 pages). Ordering information is given on any current masthead page.
Interferences in HgH Spectral Line Intensities Related to Permanent Electric Dipole Moments Odette NUBlec* and Jean Dufayard Laboratoire de Spectrometric Physique, UniversitB Grenoble 1, B.P. 87, 38402 Saint Martin d’HZres, France (Received: July 17, 1991; In Final Form: September 24, 1991)
-
In the HgH A 211142u’- X 2Z+u” bands, the measured relative intensities of the lines originating from a J’level depend on the 2111/2 2Z spin-orbit interaction. It gives an ”interference effect” on R and P lines, between transitions parallel and perpendicular to the molecular axis. In the u’ = 0 v” = 0, 1, 2, and 3 bands, the sign of the interference reverses + - p(21Xl/2v’), the progressively as v” increases. This change is attributed essentially to the variation of p l y = M ( ~ Zvf’) difference between the permanent electric dipole moments of the two levels of the observed band. Associated with the radiative lifetime and the relative band intensities which give p I U y ‘ , the relative line intensities provide pli(Yd’= k(+0.45, +0.2, -0.35, 0.6) D for u” = 0, 1, 2, and 3.
-
1. Introduction
In a HgH A 2111 v f - X 2Z+v” band, the relative intensities of the P, Q, and lines originating from a J’ e/f level are measured. Such intensities in CdH agree with those predicted with 2111/2 2113/2 mixing by rotational interaction. For HgH, there is a deviation from these formulas, which appears on R and P lines and depends on the parity index.’ Our interpretation rests on the fact that spin-orbit interaction, large in the Hg atom, mixes 211 and 2Z+basis states. As a result, transitions between A 2111 an& 2 ~ energy + levels are superpositions of 2nIl2. 2 ~ + 2 , d+ 2Z+,and 2111/2 2111/2 transitions. Then, they involve transition moments between different basis states, and also permanent electric dipole moments, as the same basis states appear in both
-
-
k
-
-
(1) NWlec, 0.; Majournat, B.; Dufayard, J. Chem. Phys. 1989,134, 137.
0022-36S4/92/2096-2141$03.00/0
A and X levels. Relative phases are found to be opposite for R and P lines, which gives an interference effect and accounts for the observed intensities.’ Precise spectroscopic measurements2 provide molecular constant^^,^ and state mixing coefficients. In the present line intensities, the interference amplitude is related to the values of the permanent electric dipole moments of the A 2111,2u’and X 2Z+ v” levels of the observed bands. In our previous work,’ the measurements were restricted to the HgH A 2111/2 X 2Z+,v’ = 0 v f f= 0 band. In the present work, they are extended to v ‘ ~= 1, 2, and 3, nearly up to the dissociation of the ground state. The interference amplitudes exhibit an important variation with v”, which is interpreted as
-
-
(2) Phillips, L. G.; Davis, S. P. Berkeley Analysis of Molecular Spectra; University of California Press: Berkeley, CA, 1968. Eakin, D. M.; Davis, S.P. J. Mol. Spectrosc. 1970, 35, 27. (3) Veseth, L. J . Mol. Spectrosc. 1972, 44, 251.
0 1992 American Chemical Society
