J. Phys. Chem. 1994, 98, 2038-2043
2038
Octahedral and Prismatic Isomers of CrFs: Energies and Vibrational Frequencies Colin J. Marsden**+ School of Chemistry, The University of Melbourne, Parkville, Victoria 3052, Australia
David Moncrieff Supercomputer Computations Research Institute, B- 186, Florida State University, Tallahassee, Florida 32306-4052
Geoffrey E. Quelch Chemistry Department, Wake Forest University, Winston-Salem, North Carolina 271 09- 7486 Received: September 13, 1993; In Final Form: November 30, 1993”
Detailed ab initio electronic structure calculations are reported for octahedral and prismatic isomers of CrFa. A carefully graded series of basis sets has been used. Correlation effects were calculated at the CCSD(T) level, and some exploratory MR-CISD calculations were performed. Vibrational frequencies calculated at the S C F level are reported. In agreement with other recent work, the octahedral isomer is shown to be the most stable for CrF6; the crucial role played by f functions in the Cr basis is emphasized. Triple excitations preferentially stabilize the octahedral isomer. Our final value for the octahedral-prismatic energy electronic energy separation is 59.3 kJ/mol, which is much lower than the CASPT2 result reported by Pierloot and Roos (Inorg. Chem.
Introduction CrF6 has aroused considerablerecent interest and controversy, both for the question of its preferred shape and for whether it has really been prepared or not. Its preparation was first reported in 1963,and some IR data were presented for the solid materia1.l Some further vibrational data were obtained in 1985, for matrixisolated material; these were interpreted, naturally enough, in terms of an octahedral structure, since only a single band was detected in the stretching region.2 The first ab initio theoretical study of do complexes such as CrF6 was reported in 1989.3 Nonoctahedral geometrieswere predicted for both (trigonalprismatic D3h) and CrH6 (only C3, symmetry), but CrF6 was found to be octahedral at the MP21minimal basis level of theory. A little later, Marsden and Wolynec (MW) carried out a somewhat more elaborate theoretical study of CrFsa4They used a roughly triple-!: basis at the SCF level and found that the prismatic structure was preferred by 11 kJ/mol. Vibrational frequencies were predicted at the SCF level; an imaginary frequency was found for the octahedral isomer, but all frequencies for the prism were real. The spectrum predicted in the stretching region for the prism showed a surprising accidental neardegeneracy, with w4 (az”) and (06 (e’) separated by less than 1 cm-1. The frequency of 818 cm-I for this composite band was in satisfactory agreement with the band reported by Ogden and co-workers (763 cm-1),2 given the tendency for vibrational frequencies to be overestimated at the SCF level. MW also reported that the Mdler-Plesset perturbation series oscillated wildly for CrF6, and they therefore felt that coupled-cluster theory would give more reliable results. They presented CCD (double substitutions only) energies obtained at SCF geometries with a double-{contraction of their basis and found that the preference for the D3h structure had increased to 20 kJ/mol. They commented, “These results strongly suggest that CrF6 is prismatic, but they are probably not definitive...”. Althougha non-octahedral structure for CrF6 might have seemed absurd, given the known structures of MoF6 and WF6,5 there is a clear precedent in the f Present address: Laboratoire de Physique Quantique, IRSAMC, Universitt Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex, France. 8 Abstract published in Aduance ACS Abstracts, February 1, 1994.
0022-365419412098-2038%04.50/0
case of W(CH3)6, which has been shown by electron diffraction to have a prismatic structure in the gas phase.6 Meanwhile,there were also experimentaldevelopments. New matrix-isolation IR spectra were reported in 1990 by Jacob and Willner (JW),’ who asserted that the spectra reported earlier by Hope et ala2were actually due to CrFS rather than CrF6; JW concluded that CrF6 had not in fact been prepared. However, Hope, Levason, and Ogden (HLO) analyzed their spectra in greater detail,* defended their vibrational assignment, criticized JW’s work, and repeated their belief that they had prepared CrF6 and that itwasoctahedra1,despitethe predictions by MW. HLO’s strongest evidence in favor of the octahedral structure was the behavior of the Cr isotope shifts. There thus appeared to be a conflict between theory and experiment. Such a situation is almost guaranteed to encourage further work, and three new papers soon appeared, back-to-back. Two of these were theoretical; they both argued that the work by MW was inadequate and that the structure for CrF6 really is octahedral, as HLO had said.2-8 Pierloot and Roos (PR)9 used large primitive basis sets of A N 0 type for Cr (17, 12, 9, 4 contracted to 5, 4, 3, 2), together with a TZP-style basis for F (10,6, 3 contracted to 4, 3, 1). They employed a relatively new method to account for electron correlation, known as CASPT2. A complete active space (CASSCF) calculation is performed, to describe any near-degeneracy effects and to yield suitableorbitals, followed by a multiconfigurational MP2 treatment to account for dynamical correlation effects. Geometriesfor octahedral and prismatic isomers were optimized at this CASPT2 level. The final energies showed a preference for the octahedral isomer of no less than 209 kJ/mol. This figure is much larger than any reported in either of the previous theoretical studies, at any level of theory con~idered.33~ Neuhaus, Frenking, Huber, and Gauss (NFHG)lO used more traditional theoretical methods; they employed effective core potentials for Cr to reduce the number of electrons to be treated explicitly,in combinationwith valence basis sets of either roughly double-or triple-{quality, together with 6-3 lG(d) or TZ2P bases for F. Geometries were optimized at the SCF level only. With these bases, the prismatic isomer is more stable at the SCF level, 0 1994 American Chemical Society
Octahedral and Prismatic Isomers of CrF6 as had already been found by MW,4 by 14 kJ/mol with the DZ basis but only 2.5 kJ/mol with the larger TZ basis. However, a final SCF calculation with the T Z Cr basis to which a set of f functionswere added gave a large change in the relative energies of the two isomers, with the octahedral species favored by 47 kJ/mol. Thus the differential effect o f f functions on the two isomers appears to be astonishingly large, of the order of 50 kJ/ mol. Correlation effects were calculated using the DZ basis with methods increasingin rigor from MP2 to CCSD(T). Oscillations in the relative energies of octahedral and prismatic isomers were again found for MPn energies, following the experienceof MW. NFHG found that coupled-clusterenergies at the CCD level still favored the prismatic isomer, but only by 2.5 kJ/mol, indicating that the correlation contribution due to double excitations was greater for the octahedral system by about 12 kJ/mol, i.e. almost exactly theopposite of theeffect found earlier by MW. Increasing the sophistication of the calculation from CCD to CCSD led to a small further relative stabilization of the octahedral isomer, of 1.2 kJ/mol, but at this level of theory the prismatic isomer is still marginally more stable, by just 1.2 kJ/mol. However, the more traditional CISD and CISD+Q methods, which are generally thought to describe electron corrleation less completely than does the CCSD technique,” gave lower energies for the octahedral isomer, by 7.5 and 11 kJ/mol, respectively. Finally, the CCSD(T) energies favored the octahedral isomer of CrF6 by no less than 47 kJ/mol, implying a large differential effect (about 50 kJ/mol) of connected triple excitations on the relative stabilities of the two isomers. NFHG then added the influence off functions obtained at the SCF level with their T Z Cr basis to the CCSD(T) results obtained with their DZ Cr basis, to estimate that the octahedral isomer of CrF6 must be at least 84 kJ/mol lower in energy than the prismatic form. NFHG also reported vibrational frequencies calculated at the SCF level for both octahedral and prismatic isomers. Unfortunately, they used only the DZ basis for Cr, without any f functions. They found an imaginary frequencyfor the octahedron, as had MW,4 so the relatively close match between the frequencies they predicted for the two IRactive bands of octahedral CrF6 with the matrix results of HL06 and JW7 is of little significance. All their calculated frequencies for the prism were real, though, in contrast to MW, they predicted a readily observable splitting between the two IR-activestretching modes (wq, a p , 910, (45, e’, 901 cm-I). The final recent paper came from Jacobs, Muller, Willner, Jacob, and Burger (JMWJB).IZ They considered the IR spectra of higher Cr fluorides in detail, in both gas and matrix-isolated phases, and repeated the assertion made earlier by JW6 that the spectra observed by HL02.8 are due to CrF5 rather than CrF6. JMWJB were able to detect three bands in the stretching region in matrix-isolated spectra, an observation clearly incompatible with an octahedral species. The situation in early 1993 could thus be summarized as follows: a measure of consensus has been achieved by theoreticians concerning the structure of CrF6, as the two most recent studiesgJO both favor the octahedron, but quantitative convergence in the predictions is still far from achieved, as the energy differences between octahedral and prismatic forms are very different in the two cases. It is of considerable interest to discover which value predicted for the energy separation is more reliable, slightly over 200 kJ/mol (PR)9 or at least 84 kJ/mol (NFHG).lO Although the value by NFHG was presented as a lower limit, they gave no indication that the real value could be as large as 200 kJ/mol. There are features in both the recent theoretical papers by PR and by NHFG which merit further attention. While the CASPT2 method used by PR should be relatively reliable, we note that it is difficult to use any multiconfigurational method to compare energies in a balanced manner of two systems with different electronic structures. This point is particularlyacute in the present case, since their active space is quite small.9 Only 10 electrons
The Journal of Physical Chemistry, Vol. 98, No. 8, 1994 2039 in 10 orbitals were considered, whereas there are 48 valence electrons for CrF6. PR acknowledge, in a note added in proof, that if a larger active space is used, the energy difference between the octahedral and prismatic isomers is “somewhat smaller”than the 209 kJ/mol they report. We note that the Cr-F bond lengths predicted by PR at the CASPT2 level are substantially longer than those found by other worker~.~JOWhile we could anticipate some lengtheningdue to correlation effects, the SCF method has in fact been found to be rather accurate for the prediction of bond lengths in high oxidation-statefluorides for early transition metals; we note the results of MW4for CrF4 (calculated 1.705, observed13 1.701(2) A) and VF5 (calculated 1.695 and 1.734, observed14 1.704(5) and 1.732(7) A for equatorial and axial bonds, respectively). In their communication, PR report only CASPT2 energies. Although these are of course more complete than SCF results, it is still useful to have separate SCF and correlated data available, to investigate the trends in calculated properties at different levels of theory. The communication by NFHGlO was intended primarily to establish the qualitative point that CrF6 is indeed predicted to be octahedral by quantum chemical methods. While that has been the most crucial point at issue in recent work, we may note that, in severaltechnical aspects, their paper does not reach thecurrent “stateof theart” and that their quantitative results must therefore be regarded as somewhat unreliable. First, the basis they used for their correlated calculations is not balanced; it contains polarization functions on F but not on Cr. Second, their F basis used in the post-HF calculations (6-31G*) is at the lower limit of what could be regarded as sufficiently flexible for F in a fairly electron-rich environment. Third, they estimate a final energy difference between isomers by adding basis extension effects obtained at the SCF level to the correlated results obtained with an unbalanced basis. Fourth, their calculated vibrational frequencies are unreliable, since they were obtained with a basis which did not include f functions on Cr, and their own work showed just how important those functions are. Indeed, their reported frequencies for octahedral CrF6 predict that species to be a transition state. Unfortunately,there are some typographical errors in their paper; while they report shorter axial (1.685 A) than equatorial bonds (1.717 A) in VF5, those entries in their Table I1 should be interchanged to bring them into much better agreement with both the experimental results and the theoretical results of MW.4 As a comment, but not as a criticism, we note that in some cases NFHG’s results contradict those obtained earlier by MW; for example, does the influence of correlation stabilize or destabilize the prismatic isomer compared to the octahedron? This analysis of the recent literature shows that further work on CrF6 is not only justified but also urgently needed. While the quantitative value of the energy difference between two isomers of CrF6 is an interesting question, the larger issue of which combinations of basis and theoretical method may reliably be used for transition metal compounds is very important indeed for future applications of quantum chemistry. It is also important to obtain reliable calculated values of the vibrational frequencies for CrF6, since these should be helpful in the continuing debate as to whether the compound has ever actually been prepared.2.7.8J2
Computational Results
SCF Level. In order to help resolve the questions which still remain concerning quantum chemical predictions for the molecular structure of CrF6, we have used a carefully graded series of basis sets of progressively increasing size, studying their effects on the molecular energies, geometries,and vibrationalfrequencies of the octahedral and prismatic isomers. These basis calibration tests were performed at the SCF level. Once a basis had been found which gave apparentlyconverged results for these properties of interest, we used the most elaborate treatment of electron
Marsden et al.
2040 The Journal of Physical Chemistry, Vol. 98, No. 8, 1994
TABLE 1: Theoretically Predicted Geometries and Energies for Octahedral (Ob)and hismatic
A B C D E F F‘ G H F’ F’ F’ F’ F’ a
8,6,2 8,6,3 8,6,3 8,6,2,1 8,6,2,1 10,8,3, 1 10,8,3, 1 10,8, 3, 1 10,8,3, 1
SCF SCF SCF SCF SCF SCF SCF SCF SCF CISD CISD + size MR-CISD CCSD CCSD(T)
1.706 1.702 1.698 1.683 1.678 1.679 1.673 1.679 1.678 (1.673) (1.673) (1.673) (1.673) (1.673)
-1639.54087 -1639.54303 -1639.555 58 -1639.62238 -1639.693 93 -1639.738 09 -1639.725 32 -1639.742 29 -1639.799 76 -1640.894 80 -1641.216 31 -1640.910 34 -1641.207 40 -1641.327 50
1.724 1.720 1.716 1.699 1.693 1.694 1.689 1.695 1.693 (1.689) ( 1.689) (1.689) (1.689) (1.689)
CrFs Isomers
50.5 50.5 50.5 50.6 50.5 50.5 50.5 50.5 50.5 (50.5) (50.5) (50.5) (50.5) (50.5)
-1 2.0 -14.4 -10.9 +22.3 +23.2 +22.5 +21.4 +21.2 +21.1 +31.7 +27.9 +53.7 +18.1 +59.3
Angle between a threefold axis and a Cr-F bond.
TABLE 2 SCF Predictions of Vibrational Wavenumbers (cm-’) for Octahedral and Prismatic CrF6 Isomers. oh Dah basis A basis D basis F basis A basis D w1 al‘ 744 779 813 829 w1 alp 772 546 635 642 608 w2 520 w2 e, 67i w3 a]’’ 52 850 (1734) 827 ( 1466) 847 (1788) U3 tl” 837 (303) 324 (1 87) 329 (165) 04 a2/1 811 (174) w4 261 (185) 361 (24) 408 394 406 W5 330 (13) U5 t28 836 (456) w6 e’ 811 (281) 98 100 w6 41i 462 (1 8) a7 437 (10) 401 (3) w8 380 (3) w9 e’’ 582 599 247 w10 235 a Non-zero IR intensities in parentheses, in km/mol. correlation which was available to us, namely the CCSD(T) method. This method has been shown to perform very well in a variety of “difficult” cases involving substantial multireference character, such as ozone,ls small Be clusters,16 or F0OF.l’ The SCF and CCSD(T) calculations were performed using the Gaussian 90 and 92 programs.I8J9 We also performed multirefererence CISD calculations to obtain some insight into the apparent importance of triple substitutions; the DIRECT-CI program20 was used for this purpose. Throughout this paper, the pure spherical-harmonic components of d-type (five) and f-type functions (seven) were used. We started with the 14s, 9p primitives developed for Cr by Wachters.21 They were contracted reasonably flexibly, especially in the valence region; the contraction schemes were 6,2,6*1 for s functions and 4, 2, 3*1 for p functions. The two most diffuse s-type functions in Wachters’ original set (exponents 0.10 and 0.04) were replaced by two others (exponents 0.22 and 0.076) so as to make a more even-tempered set, which we felt would be more appropriate for molecular calculations. An extra, more diffuse p function was added (exponent 0.15) to give an 8s, 6p contracted set. Since the d functions as optimized for atoms by Wachters are known not to be optimal for use in molecular calculations,22 we chose instead the more diffuse 5d set due to Goddard and co-workers,23contracted 4 , l in double-{style. This 8s, 6p, 2d basis, in combination with the 4s,2p contraction24of Huzinaga’s 9s, 5p primitives25for F, is denoted A in what follows. Geometries and energies obtained for the octahedral and prismatic isomers are reported in Table 1. The angle 0 for the prismatic isomer is the angle between a threefold axis and a Cr-F bond. If the same bond angles are maintained in the prism as in the octahedron, 0 would be half the tetrahedral angle, or 54.7’. Basis A leads to the prismatic form of CrF6 being more stable than the octahedral by 12.0 kJ/mol at the SCF level. Vibrational frequencies obtained at the SCFlevel for both isomers are reported in Table 2. The most important feature of the results obtained with basis A is the imaginary frequency &f found for the octahedral
basis F 795 547 63i 849 (310) 369 (20) 849 (445) 465 (14) 405 (1) 604 25 1
isomer. This triply-degenerate motion is effectively a torsion of one triangular F3 face about its threefold axis, and it transforms the octahedron into the trigonal prism. All fequencies for the prismatic isomer, however, are real. Thus a t this level of theory, the predictions are quite clearly in favor of a prismatic structure for CrF6. It is noticeable that although there are two IR-active stretching modes for a trigonal prism, of a*’’ and e’ symmetries, these two are almost exactly (accidentally) degenerate (frequency separation less than 1 cm-l) when basis A is used at the SCF level. , The contraction scheme used for basis A was relaxed in two different ways, to test the influence of further flexibility in the valencefunctionsforCrandF. First,a triple-{(3,l,l)contraction of the d functions was adopted for Cr, giving basis B. The results in Table 1show that this extension produces only marginal changes to the structure of the two isomers, with the Cr-F bond length decreasing by 0.004 A in each case, while the energy separation between the isomers increases slightly to 14.9 kJ/mol. To obtain basis C from B, we adopted a 5, 3 contraction,24 which is of triple-{quality at least for the valence space, of the 9,5 primitive basis for F. This extension gave a further modest shrinking of the Cr-F bond lengths for both isomers and a decrease in the relative stability of the prismatic isomer to 10.9 kJ/mol below the octahedral species. Since the changes in total energy produced by extending basis A to B then C were so slight (only 2 m H for B and a further 12 mH for C), it did not seem necessary to calculate vibrational frequencies with basis B. Results obtained with set C are already a ~ a i l a b l ethey ; ~ differ only by a few reciprocal centimeters from those yielded with basis A. The next basis extension was the addition of a set off functions on Cr to basis A, to give set D. The optimum exponent was found to be 0.47,inSCFcalculationson theoctahedral isomer. Addition of these polarization functions had a profound effect on the energies and character of the PE surface calculated for CrF6 isomers. First, the energy of the octahedral structure decreased by no less than 214 kJ/mol, though the bond length changed by
Octahedral and Prismatic Isomers of CrF6 only 0.023 A. More importantly, the vibrational frequencies calculated for the octahedron are now all real, with the value for being 98 cm-’, and it is now the prismatic isomer which has an imaginary frequency (wj, 67i cm-1, al”symmetry, which is the “torsional”mode rotating one triangular face relative to the other, thereby transforming the prism to the octahedron). There are also substantial changes to other vibrational frequencies in octahedral CrF6 caused by the addition of f-type polarization functionsto the Cr basis; both algand egstretching modes increase by about 5%, and the increase of 24%in the tl, bending mode from 261 to 324 cm-’ is particularly noteworthy, though the tzg bending mode is increased by only 12 cm-’ (3%). With these changes reflected in vibrational character, the octahedral isomer is now 22.3 kJ/mol more stable than the prismatic structure. Thus the presence of a set of polarization functions on Cr has reversed the relative energies of the two isomers, completely inverted the nature of the two stationary points, and made quantitatively important changes to several (but not all) vibrational frequencies. It is interesting to compre the energy gain achieved here from a set of Cr f functions with that reported by NFHG;IO in their case, an improvement of only 99 kJ/mol was obtained, with an exponent of 1.14, though they added Cr f functions to a basis which already contained d functions on the F atoms, and it is well-known that energy gains from polarization functions on different atoms are not strictly additive.26 As one set off functions on Cr had made such a large improvement to the total energy of CrF6. we carried out a few tests to discover whether two sets would give a further substantial improvement. Rather to our surprise, the use of two sets with exponents greater and smaller by a factor of 2 than the optimum single-set value gave energies marginally worse than those of the single set. Several combinations of two sets were tried, but as the greatest improvement in energy obtained was only 4.7 mH (with exponents 0.3 and 0.74, which lead to a bond length for the octahedral isomer of 1.681 A), we felt it was not worth pursuing basis enhancements of this type. It was clearly important to consider the addition of polarization functions to the F atoms. Basis E was obtained from D by optimizing a set of fluorine d-type functions for the octahedral isomer. The optimum exponent was found to be 0.67;this change produced a marginal decrease in the Cr-F bond length of just 0.005 A but a substantial improvement in energy of 188 kJ/mol. Most importantly, however, the effects on the energy difference between the octahedral and prismatic isomers were insignificant, at less than 1 kJ/mol. Since Cr is known to be one of the most difficult elements to study by quantum chemistry,l7we wanted to be sure that our Cr basis was adequate, even though the preliminary indications from comparisons of bases A and B suggested that it might be. To obtain basis F, we first decontracted the s set slightly, in the style 5, 9*1, then replaced the additional p function whose exponent was 0.15 by two others (exponents 0.22 and 0.076), and made thepcontractionslightlymore flexible (4,7*1). Wealsoenlarged the d basis by using a triple-contractiono of the larger 6d primitive set due to Goddard et al. (6/3).23 Basis F is thus of 10, 8, 3, 1 quality for Cr and 4, 2, 1 for F. It did not seem necessary to check the effects of these three enlargements individually. As shown in Table 1, the total energy for CrF6 is 116 kJ/mol lower with basis F than E, the Cr-F bond lengths are reduced by 0.005 A for the octahedral isomer or 0.006 A for the prism, but the quantity of primary interest to us, namely the energy difference between the octahedron and prism, is changed by only 0.7 kJ/ mol. We tested the 6/3 contraction of Wachters’ d primitives for Cr21 and found it slightly inferior (1.2 mH) to Goddard’s set. Vibrational frequencies for both the octahedral and prismatic structures of CrF6 were calculated at the SCF level using basis F and are reported in Table 2. Although the frequencies for the
The Journal of Physical Chemistry, Vol. 98, No. 8, 1994 2041 stretching modes have increased slightly compared to those obtained with basis D, due to the shorter bonds predicted by the larger basis, the overall changes are very minor. In particular, the lowest frequency for the octahedral geometry has changed by only 2 cm-1, and that for the prism by only 4i cm-1. It therefore seems very likely that the pattern of frequencies obtained with basis F would not be altered in any significant way even if a very much larger basis were used at the SCF level. It is interesting to note that the two stretching frequencies predicted for the prismatic isomer are still separated by less than 1 cm-1. Two further extensions were considered to the fluorine basis beyond the 4, 2, 1 set used in bases E and F. First we added diffuses and p functions(exponents0.095 and 0.074,respectively) to yield basis G. The results in Table 1 show that these extra functions lowered the total energy by only 1 1 kJ/mol at the SCF level and led to only a slight change (1.3 kJ/mol) in the octahedralprismatic energy separation. Finally, we used a triple-c representation of the s, p functions (Huzinaga’s 10,6 basis25contracted 5 , 3 27) to obtain basis H. This improved description, especially of the core orbitals, gave a much larger improvement in total energy (1 62 kJ/mol) but again led to only marginal changes in geometry and the difference in energy between the octahedral and prismatic isomers. Bases G and H both contain 170 contracted functions. At the SCF level, they appear to give converged results for the energy difference between octahedral and prismatic isomers, and only minimal differences from the resulted obtained with basis F. Correlated Level. For largely pragmatic reasons, we carried out correlated calculations at geometries optimized at the SCF level. This choice can be justified since we are mostly interested here in energy differences between two isomers, and we may anticipate that any geometrical changes due to correlation will be at least similar for the two structures, so that any effects on energy differences will tend to cancel. It has already been shown that low-leveltreatments of correlation are unreliable for crF6$,10 so if correlatedmethods were to be used for geometry optimization, at least the CCSD level of theory would need to be employed, and we did not think that the necessary computational effort (see below) could be justified. We also observe, as noted in the Introduction, that the SCF method performs surprisingly well for the prediction of geometries for early transition metal fluorides, so the geometries we have used are probably reliable. The largest basis which was feasible (CrF6 has 78 electrons) for high-level correlated studies was set F, which contains 146 contracted functions. The d-type function on fluorine in basis F is too diffuse for use in correlated calculations; although we optimized its exponent at the SCF level to be 0.67, it is well established that a value of about 1.4 is optimal in post-HF calculations.28 The difference arises because the primary role of polarization functions at the SCF level is truly to describe polarization in the bonding region, whereas in correlated calculations on F-containing compounds their principal role is to permit angular correlation; since the typical pair-pair separation is much less than the bond distances, a tighter d-type function is needed for correlated than for SCF calculations.26 Basis F’ therefore differs from F only in the tighter d-type function used on fluorine (exponent 1.4). Results in Table 1 show that use of the tighter d function gave a loss in energy of 34 kJ/mol at the SCF level, a slight shortening of the Cr-F bonds, but a change of less than 1 kJ/mol in the energy separation between octahedral and prismatic isomers. CISD calculations for both isomers were carried out in C, symmetry which generated around 1.3 M configuration state functions (CSF) in the CI expansion. The low-symmetry point groupwas used because a symmetry-adaptedbasis was not possible in the program which generated the atomic integrals. The direct CI method used in the CI calculations restricts the maximum number of electrons in the secondary orbital space to two in any
2042
The Journal of Physical Chemistry, Vol. 98, No. 8, 1994
given CSF; this orbital set is usually spanned by the virtual orbital space generated by the zero-order wave function. The primary orbital space usually contains the H F occupied orbitals, with no restrictions placed on occupation patterns except for spin and space conservation. The core orbitals on Cr (Is, 2s, 2p, 3s, and 3p) and F (1s) were frozen at the transformation phase and did not participate in any of the correlated calculations described in this paper. CISD results lead to an increase in the relative stability of the octahedral isomer, by 10.3 kJ/mol, though when the sizeconsistency correction19 (similar to the Davidson correction for quadruple excitations29) is applied, that increase becomes only 6.5 kJ/mol. Although the oscillations already noted4J0 in the MPn energies suggest appreciable multireference character in CrF6, the CISD calculations show that there is no single particularly important additional configuration, as the largest coefficient after that for the H F reference was less than 0.05, for both isomers. CrF6 therefore resembles FOOF; correlation is important, but because of the large influence of many small contributions, rather than from any speciallysignificant additional config~ration.~~ TheHFreferenceconfigurationcontributes 81% to the total CISD wave function in both octahedral and prismatic isomers of CrF6. This percentage is typical for transition-metal compounds, either “simple” dimers or complex systems, where the HF configuration frequently gives a poor representation of bonding in the molecule.32 CCSD calculations with basis F’ produced a small decrease (3.3 kJ/mol) in the energy separation between octahedral and prismatic isomers compared to the result obtained at the SCF level. These results are therefore consistent in direction with, but smaller in magnitude than, those obtained earlier at the CCD level with a much smaller basis by MW.4 The CCSD calculations were large with the programs available to us,which unfortunately do not exploit symmetry for correlated calculations; just under 19 CPU hours were needed for the octahedral isomer, and over 26 hours for the prism, on a Cray Y-MP, largely because the CCSD coefficients converged painfully slowly, with 42 cycles being needed for the octahedron with the program Gaussian 92. QCISD calculations failed to converge with Gaussian 90. Triple excitations were found to give a large contribution to the total energy of the octahedral isomer of 315 kJ/mol at the CCSD(T) level of theory. This figure is astonishinglysimilar to thevalueof 3 17 kJ/mol reported by NHFG.10 Almost nine hours of additional CPU time were required to calculate the triples’ contribution for the octahedral isomer. Even though coupledcluster calculations are not strictly variational, it is noteworthy that our final energy for octahedral CrF6 at the CCSD(T) level is 291 kJ/mol below that reported by RP using the CASPT2 method? despite their use of larger primitive atomic basis sets. It thereforeappears that the CCSD(T) method describes electron correlation more completely than does the CASPT2 approach, at least when only a small active space is used in the latter. Triple excitations lead to a large preferential stabilization of the octahedral isomer, by 41.2 kJ/mol with our basis F’; NHFG reportedlOasomewhat largervalue of 49.4 kJ/mol. In an attempt to gain some insight into the origin of this stabilization effect, we performed MR-CISD calculations on the octahedraland prismatic isomers, using basis F’, at the geometries optimized at the SCF level. Close examination of the CISD wave function for the octahedral isomer shows that the next dominant configuration after the H F reference has a coefficient of around 0.04. In fact there are three similar configurations with nearly identical coefficients. These configurations are generated via a single excitationfrom the H F referenceconfigurationto the three lowest virtual orbitals. The nature of this excitation is an F-F intracorrelationprocess. This dominant single-excitationprocess could be regarded as a second-ordereffect, similar to that found in the ozone CI wave function.33 The CISD wave function for
Marsden et al. the prismatic isomer reveals a similar dominant single-excitation process, whose coefficient is 0.039, slightly less than for the octahedral case. Unlike the octahedral isomer, the prismatic species does not have three identical single-excitation configurations with equal coefficients based on the H F reference. The other two configurations can be found with a coefficient value of around 0.031 in the prismatic CISD wave function. The MRCISD references consisted of the H F reference and the three configurationsdescribed above. Although in the prismaticisomer these three configurationsare not quite those with highest weight in theCISD wave function, theywereadopted eoensure a balanced treatment of both isomers. The primary space for the MR-CISD consisted of the H F orbitals and the three lowest virtual orbitals, while the secondary space contained the remainder of the HF virtual orbitals. The MR-CISD calculations were performed in C, symmetry, giving about 13.1 M CSF in the CI expansion. Although the results cannot be definitive, in view of the severely limited number of configurationsin the reference space, the data in Table 1 do show that the MR-CISD energy differencebetween the two isomers of 53.7 kJ/mol is almost as large as that found at the CCSD(T) level of theory (59.3 kJ/mol). Therefore, the origin of the greater importance of triple substitutions for the octahedral isomer is shown to be its greater multireference character.
Discussion It is convenient to summarize the main results found here. First, we agree with PR9 and NFHGlO that the most stable structure for CrF6 is an octahedron. Second, we have shown that the earlier erroneous prediction by M W4 of a prismatic structure was caused by the limitations of the basis they used; in particular, we have shown that f-type (polarization) functions on Cr are of vital importance, not just for the total energy but also for the energy separation between the octahedral and prismatic isomers, even at the SCF level. Third, we have confirmed the results obtained by NFHG that correlation effects have a substantial influence on that energy difference, but only if triple excitations are considered; our most completeresult for the energy separation is about 60 kJ/mol. We do not feel courageous enough to put quantitative uncertainty limits on our predicted value at present, in view of its sensitivity to the level of theory used. It is noticeable that our result for the octahedral-prismatic energy separation is much lower than the value reported by PR (209 kJ/mol) and somewhat lower than that estimated by NFHG (at least 84 kJ/ mol). We suspect that the limited active space used by PR is responsible for their anomalous result; direct confirmation of this suggestion would be most valuable. There is a clear need for a set of MR-CISD calculationsbased on optimizedvalence orbitals (obtained by either MCSCF or CASSCF techniques); investigations are underway in this area.33 Fourth, we have &set of calculated vibrational frequencies for CrF6 which should be reasonably reliable. It is hoped that these frequencies and IR intensities will be helpful in the continuing debate as to whether CrF6 has in fact been prepared.2g7.*J2We note that the harmonic frequencies we predict at the SCF level for the IR-active modes of isolated octahedral CrF6 (850 and 329 cm-l for basis F) agree about as well as could be expected with the older matrix results for “CrF6” 2.’ (763 and 332 cm-l), though the most recent experimental paper asserts that there are three IR-activestretching modes12 rather than just one. Intrigued by the profound influence of Cr f functions on the calculatedenergy of octahedralCrF6, weinvestigated, for purposes of comparison, the effect at the SCF level of adding f functions to the S basis in SF6. It is already known that the effect of d functions on the energy and structure is large;*6 when using standard double-{ basis ~ets,21.3~ we find a gain of 825 kJ/mol and a decrease in bond length of 0.107 A from a single set of d functions on S ({OSS, optimized). A set off functions on S ({
Octahedral and Prismatic Isomers of CrF6 0.53, optimized) then produces a further gain of 239 kJ/mol, Le. a slightly greater improvement than that found for CrFs! The bond length decrease due to f functions in SF6 is 0.022 A, also marginally greater than the change produced in CrF6. These observations seem to indicate that the particular importance of f functions in these hexafluorides is due to symmetry factors. As an octahedronhas a center of symmetry,d functionson the central atom are unable to make any contribution to the MO of u symmetry, for which orbitals of p or f type are needed. However, the prismatic isomer of CrF6 has D3h symmetry, no center of symmetry, and no u/g distinction, so we can understand why Cr f functions are less important for that geometry. In a related way, it was learned some time ago that if a large set of s and p functions, but no d-type functions, are used in the N basis, NH3 is predicted to be planar rather than pyramidal by quantum chemical calculations.3~ However, this symmetry argument has limited scope; while one of the tZgvibrational frequencies (06) in octahedral CrF6 is dramatically influenced by the presence off functions on Cr, the other is only marginally affected. One of the t,, modes is changed substantially, but the other only slightly. It could be argued that the test calculations on SF6 described above are unbalanced, in that a set o f f functions was added to the S basis before any possible d contribution on F was considered. Indeed, if f functions on S are added to a basis containing d functions on both S and F (l0.68,optimized at the SCF level), the resulting energy lowering is only 108 kJ/mol, or less than half the gain achieved in the absence of any polarization functions on F. However, the importance of f functions on Cr in CrF6 was probed using bases without any polarization functions on F (compare results obtained with bases D and A in Table l), so the comparison of results for SF6 and CrF6 should be reasonably valid. The important qualitative point to which we wish to draw attention is that, even at the SCF level, f-type functions can be quantitatively important in some cases of particularly high symmetry. Acknowledgment. We thank Professor G . E. Scuseria for helpful discussions. C.I.M. thanks the University of Melbourne for generousaccess to supercomputerfacilities and the Australian Research Council for financial support. D.M. acknowledges the support of the U S . Department of Energy through Contract DE-FCOS-85ER2500000. G.E.O. acknowledges the support of the US.National Aeronauticsand Space Administrationthrough cooperative agreement NCCI-5 5. References and Notes (1) Hellberg, K.H.; Miiller, A,; Glemser, 0.Z . Naturforsch. 1963,218, 188.
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