J. Phys. Chem. 1993,97, 11625-1 1627
11625
Theoretical Molecular Structures and Vibrational Frequencies for the Dioxodihalides of Chromium(VI) and Molybdenum(VI) Robert J. Deeth Inorganic Computational Chemistry Group, School of Chemistry, University of Bath, Claverton Down, Bath BA2 7AY. U.K. Received: July 1, 1993; In Final Form: August 23, 1993'
Density functional theory calculations are reported for the ground-state geometries and harmonic vibrational energies of MOzX2 (M = Cr, Mo; X = F, Cl). Both local density approximation (LDA) and nonlocal generalized gradient approximation (GGA) functionals were investigated using several different STO basis sets. Both LDA and GGA results using basis sets of double-t plus polarization function quality yield bond lengths and angles within about 0.02 A and 3O of experiment. GGA vibrational energies are systematically predicted to be lower than observed but only by about 5% and 8% for the Cr and Mo species, respectively. The implications of these results for the complete ab initio estimation of zero point energies together with thermodynamic quantities at experimentally accessible temperatures are discussed.
Introduction The description of the complete potential energy (PE) surface for a chemical reaction is a significant problem in computational chemistry.' A more tractable procedure is to concentrate on only a few special points. A reasonably complete description of the mechanism can be derived from the local minima associated withstablemolecules(Le. reactantsandproducts), fromthesaddle points describing transition states along the reaction pathway, and from the energy differences between these various features.2 The first of these is the most straightforward to calculate. Much of the computational chemistry is therefore geared toward computing the molecular geometries corresponding to PE surface minima. Normally, the concomitant energy refers to a molecule in a vacuum at 0 K. Ideally, one would wish to correct the energy to refer to more accessible conditions and this can be achieved by using statistical thermodynamics in conjunction with the molecular vibrational energies.' Computing geometries and vibrational energies from first principles for transition-metal (TM) systems is not easy.4 Electron correlation effects appear to be significant even for the simplest complexes. This makes conventional single-determinantHartreeFock theory a relatively poor approximation.5 In contrast, techniques based on density functional theory (DFT) appear to give results as good as those from correlated HF treatments but at a significantly reduced computational cost.6 This paper illustrates some of these points with reference to the optimized geometriesand vibrational spectra of the do species MOzXz (M = Cr, Mo; X = C1, F). These molecules all have a distorted tetrahedral CZ, geometry. Both the local density approximation (LDA)' and the nonlocal generalized gradient approximation (GGA)8 are examined. Computational Details The DFT calculations use the Amsterdam density functional program system of Baerends et al.9 All the geometry optimizations used analytical energy gradients,IO but two different functionals were considered: the local LDA functional7 (with Stoll correctionll) and the nonlocal exchange12 and correlation'3 corrected GGA approach. Basis set and fit function specifications are as previously described.14 For example, a [ 3d.TZPl Mo basis implies a 3d frozen core with a triple-t valence STO expansion plus a suitable polarization function. *Abstract published in Aduance ACS Abstracts. October 15, 1993.
0022-3654/93/2097- 11625$04.00/0
The program system computesharmonicvibrational frequencies by numerical fitting of analytical first derivatives. Two additional points are used for each degree of freedom.
Results and Discussion The do oxyhalides of molybdenum and chromium are involved in catalytic olefin metathesis15
c=c + * c = c *
--c
2*c=c
(1) An elegant theoretical study of some of the mechanistic aspects of this reaction has been given by Goddard'6 who used the generalized valence bond configuration interaction (GVB CI) method. While the overall conclusions of these studies are compelling, the authors have not attempted a complete ab initio treatment inasmuch as transition states were not located, not all geometrical parameters were optimized,and vibrational data were often taken from experiment. However, those quantities which were calculated appear to be in good agreement with experiment and these papers provide some benchmarks for comparison with DFT results. Geometries. The optimum theoretical conditions for reproducing molecular geometries were first investigated with respect to Cr02C12. Both DZP and TZP bases were examined with two different frozen cores on the metal and using both LDA and GGA methods. The results are collected in Table I. The computed distances range from 1.548 to 1.603 A for Cr-O (experimental value 1.581 AI7) and from 2.048 to 2.132 A for Cr-Cl (experimental value 2.126 AI7). Changing the basis from DZP to TZP causes only a small change (0.01 A) in Cr-O bond lengths while the Cr-Cl distance can decrease by up to 0.06 A. Changing from LDA to GGA functionals is associated with a lengthening of all bonds by 0.02-0.03 A. The bond angles are not especially sensitive either to basis set or to the choice of the LDAor GGA method. In all cases the 0-0-0 angle is predicted to be less than the Cl-Cr-Cl angle, in agreement with the experimental study of Marsden et aI.I7 They also noted that VSEPR theory18 incorrectly predicts a larger 0-Cr-O angle. The more sophisticatedDFT method corrects this error, although the absolute difference in bond angles is generally underestimated by a few degrees. On the basis of the most reasonable Cr-Cl distance, with everything else reproduced satisfactorily, the [Zp.DZP] GGA results give good agreement with experiment although the LDA results with the same basis set are also satisfactory. The same level of basis (with the GGA functional) was then applied to Cr02F2 and the two Mo molecules. A 3d frozen core 0 1993 American Chemical Society
Deeth
11626 The Journal of Physical Chemistry, Vol. 97, No. 45, 1993
deviation of about 5%. Of course, such good agreement must, in one sense, be fortuitous. The calculations compute harmonic frequencies while the experimental energies are inherently anharmonic. However, if theory consistently gives good agreement, then at least the cancellation of errors is systematic and the computed results, corrected if necessary, can be used with confidence. Systematic errors in vibrational frequenciesare wellknown in HartreeFock theory and are often corrected using simple scale factors.' The calculated GGA harmonic frequencies using DZP basis sets for all four molecules are collected in Table IV. With the exception of two bands for Cr02F2, the calculated values are invariably lower than those experimentally observed.2'-24 Note, however, that a consistenttheoreticaldescriptionof the vibrational spectra of CrOzC12 and Cr02F2 requires three reassignments: A1 ~ M Qbecomes BI P M ~A2 ; 7 becomes AI 6 ~ qBI ; P M O ~becomes A2 T . The latter two assignments had previously been reported to be uncertain.24 The root mean square deviations for the data in Table IV are 16, 32, 60, and 70 cm-l for CrO2Cl2, CrO2F2, Mo02C12, and Mo02F2, respectively. The larger absolute errors for the fluoro species parallel the generally larger energies. If the average percentage errors are computed, then the deviations are 4%,5%, 876, and 8%, respectively. The errors for each band of each molecule are spread fairly evenly indicating a uniformly good treatment of stretching, bending, and torsional modes. With the modificationsfor Cr02F2 described above, the calculated assignments also agree with experiment. (Note that the entries for the two MoO2 stretching vibrations in Table 6 of ref 24 are the wrong way round). The computed variations in band energies from the chloro to fluoro analogues are well reproduced but the subtler variations between different metals are not. For example, the MO2 symmetricstretch energies are observed to vary in the sequence Cr02F2 (1006) > Mo02C12 (994) > M002F2 (987) > Cr02C12(984) while they are computed to change in the order CrO2F2 (965) > CrOzClz (955) > Mo02C12 (914) > Mo02F2 (908). This is probably to be expected given that for CrO2Clz this energy can change by up to 130 cm-l merely as a function of basis set. (See Table 111.) Nevertheless, the overall agreement between observed and calculated vibrational energies is satisfactory. Zero Point Energies and AG. Accurate vibrational energies would be required to estimate zero point corrections. The latter are not always considered when modeling reaction energetics partly because they might be relatively small and partly because, in the absence of experimental data, computing the necessary vibrational energies can be time-consuming. However, such corrections can be important and should not be ignored. The present DFT results suggest that good ab initio estimates of zero point energies can be obtained. In addition, computed total energies normally refer to 0 K and are not corrected to relate to more reasonable temperatures. However, providing the energies are accurate in themselves, the vibrationalenergiescan be used in conjunction with the techniques of statistical thermodynamics3to estimate M a n d the corrections
TABLE I: Calculated and Observed Geometrical Parameters for CrOzcl2 (Distances in Angles in d e d basisa
method
[3p.DZP] [3p.TZP] [3p.DZP] [3p.TZP] [Zp.DZP] [Zp.TZP] [Zp.DZP] [Ip.TZP]
LDA LDA GGA GGA LDA LDA GGA GGA
Obsdb
Cr-0 1.548 1.559 1.564 1.573 1.584 1.576 1.603 1.591
Cr-CI 2.064 2.035 2.090 2.063 2.104 2.048 2.132 2.074
0-Cr-O 108.8 108.5 109.3 108.8 109.5 108.9 109.8 109.1
Cl-Cr-CI 110.8 111.8 109.9
1.581
2.126
108.5
113.3
111.1
110.8 113.4 110.0 110.0
Metal basis specified. Ligand bases use comparable valence STO expansions. Reference 17.
TABLE II: Calculated and Observed Geometrical Parameters for M o a 2 Species. (GCA) and [Zp.DZP]/ [3d.DZP] Cr/Mo Metal Basis Sets, Distances in A, Andes in des) molecule Cr02C12 Cr02F2 M002C12 M002Fz
M-O 1.603
M-X 2.132
0-M-O 109.8
X-M-X 110.0
1.581
2.126
108.5
1.603
1.740
108.5
1.575
1.720
107.8
113.3 111.3 111.9
1.728
2.288
107.2
110.4
obsd
1.75
2.28
106
113
calcd obsda
1.727
1.889
106.7
112.4
(1.71)
(1.94)
calcd obsd calcd obsd calcd
ref 17 19 2Oa
2Ob
#Estimated data. (See text and refs 16 and 24.)
TABLE III: Calculated and Observed Vibrational Energies for Cr02C12 (All Values in cm-l) AI basis' method vm% vmx2 dm% DZP LDA io34 486 350 TZP LDA 1121 521 364 DZP GGA 955 449 340 TZP GGA 1082 475 353 obsdb 984 465 357 0
BI
b," A27 140 200 151 220 143 193 151 214 144 216
B2
vmo,
pm% vmxl PM%
1067 1155 986 1116
215
265 sa4 272 49s 258 525 273 538
226 211 224
994 230
497
263
*
2p frozen core on metal. Reference 21.
was employed on Mo. The data in Table I1 demonstrate that the good accuracy found for Cr02C12carries over to the other three molecules. Bond lengths are reproduced to about 0.02 A and bond angles to within 2 or 3O, although definitive experimental data appear to be lacking for Mo02F2. VibrationalEnergies. Thevibrational data for CrO2C12in Table I11 show that the GGA-corrected [Zp.DZP] calculation gives the best agreement with experiment.z1 However, as found for geometries, the LDA results with the [2p.DZP] basis are also reasonable, although the error in the CrO2 symmetric and asymmetric stretching modes is larger for the LDA functional. Comparing the data in Tables I and 111, it appears that all the stretchingfrequenciesroughly parallel the calculated bond lengths with shorter contacts associated with higher energies. For the [2p.DZP] GGA results, the average root mean square error is a remarkably small 16 cm-l, which corresponds to a TABLE I V Calculated and Observed Vibrational Energies of All Value8 in cm-1)
M02X2
(GGA and [Zp.DZP]/[M.DZP] Cr/Mo Metal Basis Sets,
Calcd
vmol 955
obsd 984 calcd 965 obsd' 1006 M00zC12 calcd 914 obsd 994 M002F2 calcd 925 obsd 1009 a Some bands reassigned. See text. Cr02F2
Bz
BI
Ai
molecule CrOzClZ
VMOl
PMOa
WX2
143
A17 193
994
21 1
495
~. .
216
986
497
263
264
987
745
296
vMXl
8M4
6~x2
449
340
465
357
144
679
377
191
727
422
182
259
1016
230 27 1 274
397
314
110
178
897
165
630
334
164
225
908
22 1
437 692
972 987
P M ~
258
789
304
420
218
453 66 1 710
25 1
ref 21 22 23 23
Geometries and Spectra of M02Xz to Lw which are required in order to compute AG at any temperature. Goddard16employs these procedures to good effect in his studies on Cr02C12. Several workers have indicated the success of the GGA method with regard to molecular energies. This in conjunction with the good DFT vibrational data should yield reasonable corrections from which estimates of, say, the Gibbs free energy at 300 K, AG~w,can be obtained. Comparisons with Other Theoretical Treatments. Sosa et ale6' compare HartreeFock (HF) and density functional theory treatments of the geometries of Cr02F2 and Cr02C12 plus the vibrational frequencies for the latter. DFT appears to be significantly more accurate than HF theory. The present [2p,DZP] STO LDA results for Cr02C12 closely parallel their D M o P and DGaussz6 LDA data and also tend to give slightly too short metal-ligand distances. The LDA vibrational energies for Cr02C12 are within about 30 cm-l or less of each other although it appears that DMol cannot predict the correct ordering of the Cr-0 stretching modes. Given that the basis sets are all fairly comparable at about the polarized double-t level, the similarity of the LDA treatments is reassuring. The GGA correction lengthens the chromium-ligand bonds as discussed above and as noted by Sosa et al., albeit in the context of metal-carbonyl and metal-nitrosyl contacts. These authors do not report calculations of the Mo02X2 species but do give LDA data for MoOX4, X = C1 and F. The bond lengths are predicted to be slightly longer than observed. Conclusions Density functional theory calculations using LDA and nonlocal GGA functionals give accurate geometries and vibrationalenergies for the molecules M02X2 (M = Cr, Mo; X = C1, F). Bond lengths and angles for both methods are reproduced to about 0.02 A and 3O, respectively, while the average error between observed and calculated vibrational energies is about 5% for the chromium species and about 8% for the molybdenum species. The vibrational data are systematically predicted to be lower than observed. These data should provide an excellent basis for computing zero point energies and room temperature thermodynamic quantities. Work is in progress to extend these studies to the modeling of reactions of the MO2X2 with H2 and ethylene for comparison with Goddard's AG3m GVB CI results.l6 Acknowledgment. The author acknowledges the financial support of the Royal Society and the Science and Engineering Research Council Computational Science Initiative for provision of computing equipment and Professor Evert Jan Baerends for making the ADF code available.
The Journal of Physical Chemistry, Vol. 97, No. 45, I993
11627
References and Notes (1) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; John Wiley and Sons: New York, 1986. (2) (a) Andzelm, J.; Sosa, C.; Eades, R. A. J. Phys. Chem. 1993, 97, 4664. (b) Versluis, L.; Ziegler, T.; Baerends, E. J.; Ravenek, W.J. Am. Chem. Soc. 1989, 111, 2018. (c) Ziegler, T.; Tschinke, V.; Fan, L.; k k e , A. D. J. Am. Chem. Soc. 1989, I l l , 9177. (d) Masters, A. P.; Sorensen, T. S.; Ziegler, T. Organometallics 1989,8, 1088. ( e ) Upton, T. H.; Rappe, A. K. J. Am. Chem. Soc. 1985,107, 1206. (3) Benson, S.W. Thermochemical Kinetics;John Wiley and Sons: New York, 1976. (4) Veillard, A., Ed. Quantum Chemistry: The Challenge of Transition Metals and Coordination Chemistry; NATO AS1 Series C; Reidel, Holland, 1986; Vol. 176. ( 5 ) Buijse, M. A.; Baerends, E. J. Theoret. Chim. Act4 1991, 79, 389. (6) (a) Ziegler, T. Chem. Rev. 1991,91,651, and references therein. (b) Deeth, R. J. J . Chem. Soc., Dalton Trans. 1993, 1061. (c) Woo, T.; Folga, E.; Ziegler, T. Orgunometallics 1993, 12, 1289. (d) Ziegler, T.; Tschinke, V.;Becke, A.D. Polyhedron 1987,6,685. (e)Ziegler,T.;Cheng, W.; Baerends, W.; Ravenek, W. Inorg. Chem. 1988, 27, 3458. (f) Sosa, C.; Andzelm, J.; Elkin, B. C.; Wimmer, E.; Dobbs, K. D.; Dixon, D. A. J . Phys. Chem. 1992, 96, 6630. (7) Slater, J. C. Ado. Quantum Chem. 1972, 6, 1. (8) Perdew, J. P. Physica 1991, 8172, 1. (9) Baerends, E. J.; Ellis, D. E.; Ros, P. Chem. Phys. 1973, 2, 41. (10) Versluis, L.; Ziegler, T. J. Phys. Chem. 1988, 88, 322. (1 1) (a) Stoll, H.; Golka, E.; Preuss, H. Theor. Chim. Acra 1980,55,29. (b) Stoll, H.; Pavlidou, C. M. E.; Preuss,H. Theor. Chim. Acra 1978,49,143. (12) Becke, A. J. Chem. Phys. 1986,84,4524. (1 3) (a) Perdew, J. P. Phys. Rev. 1986,833,8822. (b) Perdew, J. P. Phys. Rev. 1987, 834, 7406 (erratum). (14) Snijders, J. G.; Vernwijs, P.; Baerends, E. J. At. Data Nucl. Data Tables 1981,26,483. (b) Vernooijs, P.; Snijders, G. P.; Baerends, E.J. Slater Type Busis Functionsfor the Whole Periodic System; Internal Report; Free University: Amsterdam, The Netherlands, 1981. (c) Baerends, E. J.; Ellis, D. E.; Ros, P. Theor. Chim. Acra 1972, 27, 339. (1 5) (a) Grubbs, R. H. Prog. Inorg. Chem. 1978,24,1. (b) Calderon, N.; Lawrence, J. P.; Ofstead, E. A. Adv. Organomet. Chem. 1979, 17, 449. (16) (a) Rappe, A. K.; Goddard, W. A. J . Am. Chem. Soc. 1980, 102, 51 14. (b) Rappe, A. K.; Goddard, W. A,; J. Am. Chem. Soc. 1982,104,448. (c) Rappe, A. K.; Goddard, W. A. J. Am. Chem. SOC.1982, 104, 3287. (17) Marsden, C. J.; Hedberg, L.; Hedberg, K. Inorg. Chem. 1982, 21. 1115. (18) Gillespie, R. J.; Nyholm, R. S.Q. Rev. Chem. Soc. 1957, ! I , 339. (19) French, R. J.; Hedberg, L.; Hedberg, K.; Gard, G. L.; Johnson, B. M. Inorg. Chem. 1983, 22,892. (20) (a) Skinner, H. A. quoted in Chem. Soc. Spec. Pub/. 1958,II. (b) Estimated data for M002F2 derived from refs 16 and 24. (21) Stammreich, H.; Kawai, K.;Tavares, Y. Spectrochim. Acta 1959, 15, 438. (22) Hobbs, W. E. J . Chem. Phys. 1958, 28, 1220. (23) Ward, B. G.; Stafford, F. E. Inorg. Chem. 1968, 7, 2569. (24) Griffith, W. P. Coord. Chem. Rev. 1970,5,459. (25) Delley, B. J. Chem. Phys. 1990, 92, 508. (26) Andzelm, J. In Density Functional Methods in Chemistry; Labanowski, J., Andzelm, J., Eds.; Springer-Verlag: New York, 1991; p 155.
