The Journal of
Physical Chemistry
0 Copyright, 1983, by the American Chemical Society
VOLUME 87, NUMBER 5
MARCH 3,1983
LETTERS A Theoretical Investigation of the Bond Length of Dichromium Randall A. Kok and Ylchael B. Hall' Deparlment of Chemistry, Texas A & M University, College Station, Texas 77843 (Received November 8, 1982; In Final Form: January 17, 1983)
Generalized molecular orbital (GMO) and configuration interaction (CI) calculations of dichromium predict a bond length of 1.73 A. This is in close agreement with recent resonant two-photon ionization spectroscopy results which yielded a bond length of 1.68 A.
Introduction Recently, there has been considerable interest in the bonding of dichromium. Early experimental work on this system was done in 1974 by Efremov et al.' who examined the spectrum of chromium hexacarbonyl by pulsed photolysis. They studied the rotational structure of the 4600-8, band and concluded that the observed spectrum was due to Cr2although they also admitted that it might have been due to CrOz or CrCz. Their analysis yielded a bond distance of 1.685 8, for CrP2 In 1975, Kundig, Moskovits, and Ozh3 published results of quantitative atom concentration studies in low-temperature matrices. Their results on chromium also predicted that the 4600-A band was due to Crz. Theoretical calculations on dichromium have predicted a wide range of bond Early calculation^,^^ (1) Efremov, Yu. M.; Samoilova, A. N.; Gurvich, L. V. Opt. Spectrosc. 1974. .36. 381. ---7
- - 7
(2) Efremov, Yu. M.; Samoilova, A. N.; Kozhukhovsky, V. B.; Gurvich, L. V. J.Mol. Spectrosc. 1978, 73, 430. (3) Kundig, E. P.;Moskovits, M.; Ozin, G. A. Nature (London) 1975, 254, 503. (4) Cooper, W. F.;Clarke, G. A.; Hare, C. R. J.Phys. Chem. 1972, 76, 2268. This paper reported a calculated bond length of 1.9 A. (5) Klotzbucher, W.; Ozin, G. A. Inorg. Chem. 1977, 16, 984. This paper reported a calculated bond length of 1.8 A. 0022-3654/83/2087-0715$01.50/0
mainly extended Huckel, gave bond lengths in the region of 1.7-1.9 8,and predicted a strong chromium-chromium bond with a high formal bond order. Later more sophisticated calculations, however, cast doubt on the possibility of an extremely short Cr-Cr bond. In 1979, Harris and Jones' used the density functional formalism of Hohenberg, Kohn, and Sham: with the local spin density approximation for the exchange-correlation energy, to calculate the bond lengths of the iron-series dimers. Their calculation yielded a bond length of 2.84 8, and a stretching frequency of 200 cm-' for the ll&+ state. Multiconfiguration self-consistent-field(MCSCF) calculations by Wood (6) Klotzbucher, W.; O h , G. A.; Norman, Jr., J. G.; Kolari, H. J. Inorg. Chem. 1977, 16, 2871. This paper reported a calculated bond length of 1.7 A. (7) Harris, J.; Jones, R. 0. J. Chem. Phys. 1979, 70, 830. (8) (a) Hohenberg, P.;Kohn, W. Phys. Reu. B 1964, 136, 864. (b) Kohn, W.; Sham, L. J. Phys. Rev. A 1965, 140, 1133. (9) Wood, C.; Doran, M.; Hillier, I. H. Symp. Faraday SOC.1980,14, 159. (10) Bursten, B. E.;Cotton, F. A. Symp. Faraday Soc. 1980,14, 180. (11)Goodgame, M. M.; Goddard, 111, W. A. J. Phys. Chem. 1981,85, 215. (12) Goodgame, M. M.; Goddard, 111, W. A. Phys. Reu. Lett. 1982,48, 135. (13) Atha, P.M.; Hillier, I. H. Mol. Phys. 1982, 45, 285.
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The Journal of Physical Chemistry, Vol. 87, No. 5, 1983
et ala9predicted an equilibrium bond length of 1.9 A but they estimated that the sextuple bond only contributed about 19% to the multiconfigurational wave function. At about the same time, Bursten and Cottonloreported results of SCF-Xa-SW calculations on Cr2 at 1.685 A and they concluded that the bond length was probably incorrect and the experimental work should be checked. In early 1982, Goodgame and Goddard"J2 published a 6000-configuration MCSCF calculation predicting that the ground state of Cr2 would best be described as an antiferromagnetic dimer with a bond length of 3.06 A rather than a sextuply bonded dimer with a very short interatomic distance. Their calculation gave a stretching frequency of 110 cm-l. Recently, a 3196 configuration MCSCF calculation by Atha and Hillier13yielded a bond length of 3.14 A and a stretching frequency of 92 cm-'. They also concluded that a better description of the bonding in Crz would involve the weak interaction of two ' S Cr atoms, rather than an MO description involving a Cr-Cr sextuple bond. Thus, the early experimental work was largely disregarded by those who felt that 1.685 A was simply too short for dichromium. However, a new experimental study of Cr, has been completed. Michalopoulos et al.14 used mass-selected resonance two-photon photoionization probes of the spectrum of jet-cooled Cr, to confirm that indeed Cr2 was the carrier of the 4600-A spectrum and that the bond length in the u = 0 level of the lZg+ground state is 1.68 f 0.01 A. Recently, resonance Raman spectra from Cr, in a cold argon matrix showed the X IZg+vibrational frequency to be 427.5 cm-l.15 We believed that a new theoretical calculation of the potential energy curve of dichromium might shed light on the wide discrepancy between recent experimental and theoretical work. Having completed a theoretical calculation of the potential energy curve of dimolybdenum16 that gave an equilibrium bond length very close to the experimental value, we felt that perhaps we might have equally good success with Cr,. We now report the results of calculations on the lZg+ state of Cr, by the GMO method with CI. The technique usually provides an accurate determination of the potential curve near the equilibrium internuclear distance but is not appropriate at the dissociation limit. Our predicted spectroscopic constants, re = 1.73 A and W , = 396 cm-', provide excellent support for the experimental values.
Theory and Results The basis functions employed in this study were linear combinations of Gaussian-type orbitals (GTOs), obtained from a least-squares fit of near Hartree-Fock quality Slater atomic orbitals.17 In this study, the number of Gaussians used for each function was increased until the integral error of the fit was less than 2 X for valence functions and 5 x for core functions. It was found that three Gaussians per atomic orbital were sufficient except for the Cr 3d where five Gaussians were required. The two most diffuse components of the 3d orbital were split off to form a triple-{ representation. Also two extra s functions with exponents of 0.15 and 0.05 and two extra p functions with the same exponents were added to the basis, resulting in a set of (lls8p5d) primitive GTOs contracted to [5s4p3d] on each atom. Finally, the basis set included single s, p, (14)Michalopoulos, D. L.;Geusic, M. E.; Hansen, S. G.; Powers, D. E.; Smalley, R. E. J . Phys. Chem. 1982,86,3914. (15)Dilella, D. P.; Lipson, R. H.; Moskovits, M.; Taylor, K. 'Resonance Raman Studies of Metal Dimers and Metal Clusters". Proceedings of the 8th Raman Conference, Bordeaux, France, 1982. (16)Bursten, B. E.;Cotton, F. A.; Hall, M. B. J.Am. Chem.Sac. 1980, 102,6348. (17)Roetti, C.; Clementi, E. J . Chem. Phys. 1974,60, 3342.
Letters
t
I
1.5
1.6
1.7
1.8
1.9
r(cr-m) A Figure 1. Potential energy curve for Cr . The curve represents the The scale on the left side C I energy and has its minimum at. 1.73 plus 2068 hartrees represents the absolute value of the C I energy.
1.
TABLE I : Experimental and Theoretical Molecular Constants for Dichromium re,
a
w e , cm-'
ref
Experiment 1.685
2 14
1.68 427.5
15
Theory
1.9 3.06 3.14 1.73
110 92
396
9 12 13 this work
and d GTOs, all with exponents of 0.7, at the midpoint of the bond. The GMO methodla was used to obtain an optimized set of strongly and weakly occupied valence orbitals. The metal-metal bonding 6ug, 7ug, 3aU, and 16, orbitals (strongly occupied) and their corresponding antibonding orbitals (weakly occupied) were used to define a configuration space for the CI calculation. A full CI calculation on lZ,+ Cr, within this space of 12 orbitals would require more than 35 000 spin-adapted configurations and is computationally infeasible. Initially, a configuration space was chosen to be all paired excitations from the bonding orbitals to their corresponding antibonding orbitals. This set of 64 configurations contains those necessary for proper dissociation into neutral but not 7S Cr atoms. Within the orbital optimization afforded by the GMO method, this is the equivalent of a perfect pairing GVB c a l c ~ l a t i o n . ~ ~ All single and double excitations out of these 64 configurations were included for a total of 3520 spin-adapted configurations. The potential energy curve calculated from a fourthorder polynomial fit of calculations at r = 1.5, 1.6, 1.7, 1.8, and 1.9 8, is shown in Figure 1. The polynomial functions (18)(a) Hall, M. B. Chem. Phys. Lett. 1979,61,467. (b) Hall, M. B. Int. J . Quantum Chem. 1978,14,613.(c) Hall, M. B. Int. J. Quantum Chem. Symp. 1979,13, 195. (19)Goddard, 111, W.A.; Dunning, Jr., T. H.; Hunt, W. J.; Hay, P. J. Acc. Chem. Res. 1973,6,368.
The Journal of phvsical Chemistry, Vol. 87,No. 5, 1983 717
Letters
were analyzed by the method of Dunham.20 The calculated molecular constants are listed in Table I, along with previously reported theoretical and experimental values. The final electronic configuration a t 1.7 A is The sextuply bonded leading configuration comprises 50% of the total wave function a t this bond length. Analysis of the wave function indicates that the dominant excited configurations in the CI wave function are very similar to those discussed previously for Mo2.16 Again the most important excited configurations are the result of excitations from the 6, to the 6, orbitals indicating strong right-left correlation of the 6 electrons. Also the u, to a, correlation is much smaller than 6, to 6, correlation implying that in Crz the second u bond again is far more important than -
(20) Dunham, J. L. Phys. Reo. 1932,41, 721.
the second 6 bond in making the bond length of Cr2 shorter than the quadruple Cr-Cr bonds. We are not sure why the other ab initio calculations predict such long Cr-Cr bond lengths. In all these calculations the two major sources of error are the basis set error and the correlation error. It is well-known that HartreeFock-Roothaan calculations with very good basis sets tend to predict bond lengths shorter than experiment. On the other hand, calculations with highly correlated wave function but inadequate basis sets tend to predict bond lengths longer than experiment. Perhaps the latter problem has influenced the previously reported calculations. We feel that in our calculation the basis set errors and correlation errors balance each other resulting in the close agreement between our calculated bond length and the experimental one.
Acknowledgment. This work was supported by the National Science Foundation (CHE79-20993).
