The trifluoride anion: a difficult challenge for quantum chemistry

to be thermodynamically more stable than (F2 + P) by about 110 kJ mol'1. An average ... Introduction. Of all the interesting bonding problems posed by...
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J . Phys. Chem. 1992, 96,4359-4366

4359

The Trlfluorlde Anlon: A Dlfficult Challenge for Quantum Chemistry George L. Heard, Colin J. Marsden,* School of Chemistry, The University of Melbourne, Parkville, Victoria 3052, Australia

and Gustavo E. Scuseria* Department of Chemistry, Rice University, Houston, Texas 77251- 1892 (Received: October 4, 1991) The molecular structure, binding energy, and vibrational frequencies of the trifluoride anion, F3-, have been studied by a variety of quantum chemical techniques, ranging from SCF to CCSDT, using a graded selection of basis sets varying in size from DZP to TZ2Pf+ (which includes two sets of d functions and one set of both f functions and diffuse p functions). Our most sophisticated methods are sufficiently reliable to give good accounts of the structure, binding energy, and vibrational frequency of F2and of the electron affinity of the F atom. Both the large value of the CCSD ‘TIdiagnostic and the critical influence on the results of single and triple excitations emphasize that the trifluoride anion has very substantial multireference character, higher than for ozone or molecular fluorine. The F,- anion has remarkably long F-F bonds (1.74 A) and is found to be thermodynamically more stable than (F2 + F) by about 110 kJ mol-I. An average F-F bond energy of some 135 kJ mol-’ is implied. Calculated vibrational frequencies for the trifluoride anion are exceedingly dependent on the method used to obtain them but are less sensitive to basis set, provided that diffuse functions are included. Our best estimates are wI = 440 10, w2 = 260 f 10, and u3 = 535 f 20 cm-I. These match reasonably well the frequencies of 461 cm-l for v I and 550 cm-I for vj which have been observed in inert-gas matrices and assigned to the trifluoride anion.

*

Introduction Of all the interesting bonding problems posed by fluorine and its many varied compounds, the trifluoride anion, F3-, certainly creates some of the most intriguing, for several reasons. Firstly, this is the species in which a fluorine atom can most reasonably be considered to be in a positive formal oxidation state, using the Lewis representation -F-F+-F. In a linear anion of Dmhsymmetry, the central F atom can be regarded as “hypervalent”, having 10 valence electrons, thereby violating the Lewis octet rule. Qualitative aspects of bonding within the trifluoride anion, including the issue of whether the hypervalent description’ and/or the “three-center bonding” approach of Pimente12and Rundle’ are appropriate, have been discussed by Martin and co-workers? Secondly, although the heavier trihalide anions such as IC are very well known and characterized, the experimental evidence concerning Fy is tantalizingly incomplete. It has not so far been isolated, and the only reports of its existence are derived from vibrational spectroscopy in conjunction with matrix isolation. When a mixture of M F vapor (M = Cs, Rb, K) and F2 is codeposited with a large excess of inert gas, two bands are seen in the vibrational spectra which display the Raman/infrared mutual exclusion characteristic of the symmetric and antisymmetric stretching motions of a linear molecule with Dmh~ y m m e t r y . ~ . ~ Since the other X; halogen anions are known to have this symmetry, it is at least plausible to assign these bands to the trifluoride anion. However, there are potential dangers in using observed matrix vibrational features to both identify and characterize a new reactive species, especially when there are fundamental frequencies which remain to be located; these dangers are particularly acute in the present case, since isotopic substitution is not feasible for fluorine-containingcompounds, so it is not possible to be sure that the bands in question are really caused by the trifluoride anion. For example, the vapor above solid M F will contain dimeric and more complex species in addition to monomeric MF, and these other species could in principle also react with Ft. Since the F,- ion is intrinsically unusual, there can be only imperfect precedents to guide our expectations of its vibrational potential surface. Mast observers would therefore probably agree that while the vibrational assignments reported for the __ (1) Musher, J. I. Angew. Chem., Int. Ed. Engl. 1969, 8, 54. (2) Pimentel, G. C. J . Chem. Phys. 1951, 19, 446. (3) Rundle, R. E. Sum.Prog. Chem. 1963, I , 81. (4) Cahill, P. A.; Dykstra, C. E.; Martin, J. C. J . Am. Chem. SOC.1985, 107,6359. ( 5 ) A u k E. S.; Andrews, L. J. Am. Chem. Soc. 1976, 98, 1591. (6) Ault, E. S.; Andrews, L. Inorg. Chem. 1917, 16, 2024. ~

are certainly very plausible, they are not 100% trifluoride secure. In view of these experimental difficulties concerning the characterization of the trifluoride anion, can current quantum chemical techniques predict its vibrational frequencies with sufficient reliability to show that the experimental assignments are secure? A bold assertion was published over 15 years ago in an influential report:’ “With today’s computers, the structure and stability of any molecular compound with up to three first-row atoms can be calculated almost to the best accuracy available through experiment. This capability gives to the chemist many situations not readily accessible to experimental measurement. Short-lived reaction intermediates, excited states, and even saddle points of reaction can now be understood, at least for small polyatomic molecules.” With the dramatic progress which has occurred since the preparation of that report, for both computational hardware and software, one might expect that definitive results will surely now be readily available from electronic structure calculations for a triatomic species which contains only 28 electrons. Rather surprisingly, however, only two theoretical papers have been published in recent years concerning the trifluoride anion,+*and it is fair to say that neither of these can now be regarded as “state-ofthe-art”. For reasons which will emerge below, the trifluoride anion can be expected to pose a particularly difficult challenge for quantum chemical techniques. The third reason for interest in this system is therefore that it provides a timely opportunity to test the performance of various computational techniques in current use on an incompletely characterized species of widespread chemical interest. The current computational literature is full of jargon, so a few introductory remarks are presented here in order to guide the reader who may be unfamiliar with some of the acronyms. All the calculations reported to date on the trifluoride anion, as well as almost all of those described in the present work, are based on a singlereference approach. The effects of electron correlation are incorporated by considering the influence of various excitations out of the single (Hartree-Fock) reference into the virtual orbitals, excitations described as single, double, triple etc., according to the number of electrons involved. These effects may be determined by perturbation theory (referred to in this work as MPn, where MP indicates the Morller-Plesset partition of the Hamiltonian9 (7) Opportunities in Chemistry. National Academy of Sciences Report, G. Pimentel, committee chairman, 1975, p 72. (8) Novoa, J. J.; Mota, F.; Alvarez, S.J . Phys. Chem. 1988, 92, 6561. (9) Msller, C.; Plesset, M.S.Phys. Reu. 1934, 46, 618.

0022-365419212096-4359503.0010 . .. , 0 1992 American Chemical Society I

,

4360 The Journal of Physical Chemistry, Vol. 96, No. 11, 1992

and n gives the order of perturbation theory used), by configuration interaction (CI), or by the coupled-cluster (CC) technique. Within the limits imposed by the basis set used, a full CI calculation (all possible excitations) provides an exact solution to the Schradinger equation, but unfortunately these calculations are far beyond current computing capabilities even for a triatomic molecule like F3- if a large basis set is used. Perturbation theory is by far the “cheapest” correlation technique available, but its use is intrinsically unreliable if the perturbation to be treated is large. CISD methods treat variationally all single (S)and double (D) excitations from the reference configuration and have long been the classic method of describing electron correlation.1° However, while these two classes of excitations provide the majority of the correlation energy, they certainly do not provide all of it. Unfortunately, CISDT, CISDTQ, etc. (where T indicates triple and Q quadruple) calculations are extremely time-consuming for even medium-sized molecular systems and do not seem generally applicable. The coupled-cluster approach” is now widely recognized as the best “all-purpose” philosophy presently available for quantum computations based on a single reference.12 Although it is not variational, its great advantage over the CI technique is that in a CCSD calculation, for example, while the single and double excitations are effectively treated to infinite order of perturbation theory, many of the important effects of higher excitations are also included, albeit approximately. In many cases where electron correlation plays a particularly important role, the CCSD method has been shown to yield results superior to those provided by CISD techniques, and if triple excitations are included, even approximately, the results can be almost as accurate as those yielded by extremely demanding multireference CI methods.I3-I6 If one wishes to determine binding energies, as here for the trifluoride anion, the size-extensive nature of coupled-cluster calculations is a particularly important advantage.17 Cahill, Dykstra, and Martin used both DZP and TZP basis sets in conjunction with SCF and ACCD (approximate coupled-cluster with double substitutions) methods to study F3-.4 They showed that the trifluoride anion is thermodynamically unstable compared to (F, F) at the S C F level but bound by 45 kJ mol-’ (TZP basis) when electron correlation is introduced by the ACCD method. They also calculated vibrational frequencies, and while they obtained very good agreement with the band assigned to v I (calc 472, obs 461 cm-I), the agreement for v3 was so poor (calc 228, obs 550 an-)) that their work failed to establish that the bands assigned to the trifluoride anion are really due to that species. Three limitations may be noted in these calculations, any of which may have caused a poor description of the trifluoride anion. Firstly, the basis set does not contain diffuse functions, which are normally thought to be essential if quantitatively reliable results are sought for an anionic species.I8 The effects of this basis set deficiency are noticeable in their calculated binding energies of F,- compared to (F2 F),which change markedly from 109 to 45 kJ mol-] at the ACCD level as the basis is enlarged from DZP to TZP. Secondly, any CCD implementation of necessity omits all consideration of single and triple excitations from the reference configuration; as it is not possible to obtain accurate data for F2 if these excitations are ignored,IF2’ it seems most likely that they

+

+

(10) Shavitt, I. In Methods of Electronic Structure Theory; Schaefer, H. F. 111, Ed.; Plenum Press: New York, 1977; Vol. 3, p 189. (1 I ) Cizek, J. J . Chem. Phys. 1966,45,4256; Adu. Chem. Phys. 1%9,14, 35. (12) Bartlett, R. J. J . Phys. Chem. 1989,93, 1697. (13) (a) Scuseria, G. E.; Scheiner, A. C.; Lee, T. J.; Rice, J. E.; Schaefer, H. F. I11 J . Chem. Phys. 1986,86,2881. (b) Besler, B. H.; Scuseria, G. E.; Scheiner, A. C.; Schaefer, H. F. I11 J . Chem. Phys. 1988,89, 360. (14) Scuseria, G. E.; Lee, T. J. J . Chem. Phys. 1990,93, 5851. (15) Rendell, A. P.; Lee, T. J.; Taylor, P. R. J . Chem. Phys. 1990, 92, 7050. (16) Scuseria, G. E. J . Chem. Phys. 1991,94, 442. (17) Cizek, J.; Paldus, J. Int. J . Quantum Chem. Symp. 1971,5 , 359. (18) Clark, T.; Chandrasekhar, G.; Spitznagel, G. W.; Schleyer, P. von R. J . Comput. Chem. 1983,4 , 294. (19) Jankowski, K.; Becherer, R.; Scharf, P.; Schiffer, H.; Ahlrichs, R. J . Chem. Phys. 1985,82, 1413. (20) Ahlrichs, R.; Jankowski, K.; Wasilewski, J . Chem. Phys. Lett. 1987, 111, 263.

Heard et al. will also be important for the trifluoride anion. Lastly, the effects of the approximations in the ACCD method, which were adopted to reduce the computational effort, are uncertain. Novoa, Mota, and Alvarez undertook a study of all four trihalide Xc anions (X= F, C1, Br, and I).8 For the trifluoride anion they calculated vibrational frequencies using a DZP basis augmented with diffuse functions, thereby overcoming one of the objections raised above to the work of Martin and co-workers, but used only the MP2 method for the treatment of electron correlation. The results they obtained matched the experimental data for vl (calc 472, obs 461 cm-I) and u3 (calc 526, obs 550 an-)) impressively. However, the MP2 method is the simplest of all possible correlation treatments. It is now realized that, for systems with large multireference character, the results obtained from a perturbation treatment of correlation will almost certainly oscillate seriously from order to order2, and the importance of triple excitations will be badly 0~erestimated.I~ Since correlation effects were shown by Martin and co-workers to be crucially important for the trifluoride anion? producing not just quantitative but qualitative changes to its PE surface, it is therefore at least possible that the MP2 results obtained by Novoa and co-workers were fortuitously successful. These doubts are strengthened when one remembers that large basis sets containing multiple polarization functions are particularly important for electron-rich species and that f functions are also i n f l ~ e n t i a l , ~whereas ~ , ~ ~ *their ~ ~ “DZP+” basis8was perhaps of the minimum size which could be considered acceptable. From the points raised above, it is clear that reliable predictions of vibrational frequencies will be of vital importance if quantum chemical techniques are to provide useful confirmatory evidence concerning the trifluoride anion and its possible p r e p a r a t i ~ n . ~ , ~ Although there is a rule of thumb which states that vibrational frequencies predicted at the DZP/SCF level are a t least semiquantitatively reliable, typically being about 10% too high for ‘normal” molecule^,^^*^^ we cannot expect that the SCF method will perform this well for F c , as the anion has already been shown to be unbound at this theoretical leveL4 There has been much recent interest in the development of sophisticated theoretical methods which are reliable for the calculation of vibrational frequencies, and the ozone molecule has emerged as a particularly severe test for quantum calculations.2632 In this case the antisymmetric stretching frequency is overestimated by 3 1% a t the DZP/SCF and most unusually the DZP/CISD result is in error by more (44%) than the SCF value;26errors comparable to these for Fy would clearly mean that any vibrational frequency calculations would be almost useless for the identification of this species. It has been found possible to obtain rather accurate vibrational frequencies for ozone from standard quantum techn i q u e ~ ,but ~ ) only if very large A N 0 basis sets are used; even g functions(!) produce nontrivial changes of up to 18 cm-I. Levels of theory which are highly successful for “standard” molecules, (21) Langhoff, S. R.; Bauschlicher, C. W.; Taylor, P. R. Chem. Phys. Lett. 1987,135, 543. (22) Gill, P. M . W.; Pople, J. A.; Radom, L.; Nobes, R. H. J . Chem. Phys. 1988,89, 7307 and references therein. (231 Lee. T. J.: Rice. J. E.: Scuseria. G. E.: Schaefer. H. F. 111 Theor. Ch~m.’Acta’1989, 75, 81. ’ (24) Pople, J. A.; Schlegel, H. B.; Krishnan, R.; DeFrees; D. J.; Binkley, J. S.;Frisch, M. J.; Whiteside, R. A.; Hout, R. F.; Hehre, W. J. In!. J . Quantum Chem. Symp. 1981,15, 269. (25) Hout, R. F.; Levi, B. A.; Hehre, W. J. J . Comput. Chem. 1982,3, 234. (26) Lee, T.J.; Allen, W. D.; Schaefer, H. F. 111 J . Chem. Phys. 1987, 87. 7062. (27) Stanton, J. F.; Lipscomb, W. N . ; Magers, D. H.; Bartlett, R. J. J . Chem. Phys. 1989,90, 1077. (28) Scuseria, G. E.; Lee, T. J.; Scheiner, A. C.; Schaefer, H. F. I11 J . Chem. Phys. 1989,90, 5635. (29) Magers, D. H.; Lipscomb, W. N.; Bartlett, R. J.; Stanton, J. F. J . Chem. Phys. 1989,91, 1945. (30) Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Replogle, E. Chem. Phys. Lett. 1989,158, 207. (31) Lee, T. J.; Scuseria, G. E. J . Chem. Phys. 1990,93,489. (32) Watts, J. D.; Stanton, J. F.; Bartlett, R. J. Chem. Phys. Lett. 1991, 178, 471.

Quantum Chemistry of the Trifluoride Anion

The Journal of Physical Chemistry, Vol. 96, NO. 11, 1992 4361

such as DZP/CCSD, produce rather disappointing results for 0zone,2’.~~ especially for the antisymmetric stretching frequency, but certainly perform far more successfully than the less elaborate DZP/CISD technique. Analysis of the CC wave function for ozone, both in its equilibrium C, geometry and when distorted along the antisymmetric stretching coordinate (C, symmetry), has revealed the origin of the problem;27not only is there strong multireference character in the ground state, but in addition there are two further singly excited configurationswhich become important as the molecule is distorted in the antisymmetric stretching motion though which is forbidden by symmetry from mixing into the C, ground state. Triple substitutions from a single reference therefore have a very large influence on the antisymmetric stretching frequency of ozone. In principle, precisely analogous effects can arise for the antisymmetric stretching motion of the trifluoride anion, so it is not surprising that the ACCD frequency prediction by Martin and co-workers4was so unsuccessful. The DZP/MP2 method gives an error of over loo%(!) for o3for so the doubts expressed above about the MP2 result for F y are reinforced. Since the SCF method provides a much poorer description of the antisymmetric stretching motion for F3- than for ozone (the SCF calculated then one might expect that the frequency is at least real for 03), variation with level of correlated theory will be at least as great, if not greater, for F3- than for ozone. Now the computational effort required for the CCSDT method, in which single, double, and triple excitations are all treated completely, increases nominally as the eighth power of the number of correlated molecular orbitals, whereas for the CCSD technique the effort increases (only) as the sixth power. There is therefore a strong incentive to estimate the effects of triple substitutions rather than calculating them rigorously. Various approximate ways of estimating these effects have been proposed, such as CCSD+T(CCSD), CCSD(T), or CCSDT-la, but unfortunately, in the case of ozone, they produce astonishingly and depressingly different results from the full CCSDT p r o c e d ~ r e . * ~ J It ~ ~is~thus ’ * ~ of ~ great interest to examine the performance of several methods used in current quantum chemistry on another molecule of high multireference character, to see whether the vibrational results already obtained for ozone are typical. The QCISD method3) is now used quite widely, since it is available in the GAUSSIAN 90 series of programs;34 note that despite the appearance of CI in its name, it is probably better thought of as a slightly truncated version of the CCSD technique.35 In earlier versions of the GAUSSIAN programs, the most elaborate methods available were CCD36and derivatives thereof known as CCD(S) or CCD(ST), in which the effects of both single and triple excitations were treated a p p r ~ x i m a t e l y . ~ ~ These latter methods are now superseded by the QCISD procedures, and it is valuable to examine the extent of the improvements brought about by the newer methods for a “difficult” case. The precise relationships between the various different computational methods mentioned above have been clearly analyzed.38 Results obtained at CCSD, QCISD, CCSD(T), and QCISD(T) levels of theory have recently been compared39 for a wide variety of molecules; it was found in that study that the CCSD and QCISD methods give rather similar results in general, though the focus there was on binding energies and equilibrium geometries rather than on vibrational frequencies. (33) Pople, J. A.; Head-Gordon, M.; Raghavachari, K. J . Chem. Phys.

In view of the importance of the trifluoride anion in fluorine chemistry and the uncertainty surrounding its properties, we have undertaken a thorough theoretical study with the aim of providing clear answers to the questions raised above. The great difficulty experienced in calculating accurate values of v3 for the closely related hydrogen bifluoride anion, HFt,40adds to the interest of the present work. We have used a variety of basis sets and theoretical methods to probe the sensitivity of our results to the level of theory used, so as to gain an indication of the likely reliability of our results. We have also performed parallel calculations on F and Fzto provide some calibration information for the trifluoride data.

Computational Procedures A carefully graded series of basis sets has been used in this work. The smallest is denoted DZP and is a 4s2p contraction41of Huzinaga’s 9s5p augmented with a set of d polarization functions whose exponent was chosen as 1.O. Variationally, a d exponent of about 1.5 bohr2 is optimum for F or F2;19however, a function with such a tight exponent has its maximum amplitude at only 0.4 A from the nucleus, and we felt that a rather more diffuse function was more appropriate for the trifluoride anion, given its long bonds and extended charge distribution. The presence of diffuse p functions (exponent 0.074) in the basis is indicated by +, which is a standard notation.I8 Dunning’s 5s3p c o n t r a ~ t i o nof~ Huzinaga’s ~ 1Os6p primitives4 was chosen for the s,p part of our TZ bases. Two sets of d functions, exponents 2.0 and O S , were used in bases denoted 2P, while a set off functions (exponent 1.O) was also added to give our largest basis denoted TZZPf+, which contains 34 contracted functions per F atom. We have found that use of a d function whose exponent is 1.5, in conjunction with the TZ s,p basis, gives a bond energy for Fz at the CCD level of 83 kJ mol-’; an exponent of 0.8 gives the maximum bond energy of 95 kJ mol-’. If the ability to reproduce bond energies is taken as important, this single d exponent performs as well as the 2d set with exponents of 0.5 and 2.0. Only the pure spherical harmonic components of polarization functions were used. At the suggestion of a reviewer, a few calculations were performed to check the possible importance of diffuse s functions, whose exponent was taken as 0.09. The changes found when these functions were added to the TZP+ basis, which were slight, are described in the appropriate sections below. Ab initio molecular orbital calculations were performed in Melbourne using the GAUSSIAN 8645 and programs and the SCF, MPn,46 CCD,36 CCD(S), CCD(ST),37 QCISD, and QCISD(T)33techniques internal to those programs. Geometries were optimized by gradient methods for SCF calculations but by interpolation of energies for correlated calculations. At least four closely spaced points (not more than 0.03-A separation) were fitted by a polynomial (order at least three) in these interpolations, and checks with gradients for MP2 calculations showed that geometries optimized in this way for F3- are reliable to about 0.0002 A. Harmonic vibrational frequencies were obtained from analytical SCF second derivatives, but for correlated calculations, either from the quadratic force constant obtained from geometry optimization for the symmetric stretching mode or from energy changes caused by finite displacements (fO.O1 A for the antisymmetric stretching mode, two degrees for the bending mode). Numerical checks with gradients where these were available indicated that the precision in vibrational frequencies obtained following these procedures is

1981, 87, 5968.

(34) Gaussian 90, Revision F Frisch, M. J.; Head-Gordon, M.; Trucks, G. W.; Foreman, J. 8.; Schlegel, H. B.; Raghavachari, K.; Robb, M.; Binkley, J. S.; Gonzalez, C.; Defrees, D. J.; Fox, D. J.; Whiteside, R. A,; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.;Kahn, L.R.; Stewart, J. J. P.; Topiol, S.;Pople, J. A. Gaussian, Inc., Pittsburgh, PA, 1990. (35) (a) Paldus, J.; Cizek, J.; Jeziorski, B. J . Chem. Phys. 1989,90, 4356. (b) Scuseria, G. E.; Schaefer, H. F. J . Chem. Phys. 1989, 90, 3700. (36) Pople, J. A.; Krishnan, R.; Schlegel, H. B.; Binkley, J. S. Inr. J . Quantum Chem. 1918, 14, 545. (37) Raghavachari, K. J. Chem. Phys. 1985, 82, 4607. (38) Raghavachari, K.; Trucks, G. W.; Pople, J. A,; Head-Gordon, M. Chem. Phys. Letr. 1989, 157,479, (39) Lee, T. J.; Rendell, A. P.; Taylor, P. R. J . Phys. Chem. 1990, 94, 5463.

(40) Janssen, C. L.;Allen, W. D.; Schaefer, H. F. 111; Bowman, J. M. Chem. Phys. Letr. 1986, 131, 352. (41) Dunning, T. H. J . Chem. Phys. 1970, 53, 2823. (42) Huzinaga, S.Approximate Atomic Wauefunctions;Chemistry Department, University of Alberta: Alberta, 1971. (43) Dunning, T. H. J . Chem. Phys. 1971, 55, 716. (44) Huzinaga, S. J . Chem. Phys. 1965, 42, 1293. (452 Gaussian 86: Frisch, M. J.; Binkley, J. S.; Schlegel, H. B.; Raghavachari, K.; Melius, C. F.; Martin, R. L.; Stewart, J. J. P.; Bobrowicz, F. W.; Rohlfing, C. M.; Khan, L.R.; Defrees, D. J.; Seeger, R.; Whiteside, R. A.; Fox, D. J.; Fleuder, E. M.; Pople, J. A. Carnegie-Mellon Quantum Chemistry Publishing Unit, Pittsburgh, PA, 1986. (46) Krishnan, R.; Frisch, M. J.; Pople, J. A. J . Chem. Phys. 1980, 72, 4244.

4362 The Journal of Physical Chemistry, Vol. 96, No. 11, 1992

TABLE I: Calculated Results for F2,F, and F a basis and theory

Heard et al.

SCF MP2 MP3 MP4DQ MP4SDQ MP4SDTQ CCD CCD(S) CCD(ST)

E -0.738 46 -1.095 88 -1.096 66 -1.099 36 -1.105 15 -1 . I 14 70 -1.098 88 -1.104 01 -1.1 I3 70

r 1.3402 1.4241 1.4150 1.4177 1.4269 1.4392 1.4160 1.4241 1.4398

DZP+

SCF MP2 MP3 MP4DQ MP4SDQ MP4SDTQ CCD CCD(S) CCD(ST) QCISD CCSD QCISD(T) CCSD(T)

-0.740 21 -1 .100 86 -1.101 40 -1,104 22 -1.11063 -1.12065 -1.103 70 -1 .lo9 25 -1.1 I9 26 -1.111 34 -1.10958 -1.12006 -1.1 19 42

TZP+

SCF MP2 MP3 MP4DQ MP4SDQ MP4SDTQ CCD CCD(S) CCD(ST) QCISD CCSD QCISD(T) CCSD(T)*

TZ2P+

TZ2Pf+

DZP

experimentale

D

w

-1 52

109 125 97 103 119

1240 969 982 974 937 914 983 948 874

EA 0.46 2.12 1.86 1.88 1.89 1.89 1.88 1.88 1.87

1.3384 1.4256 1.4149 1.4178 1.4283 1.443 1 1.4160 1.4246 1.4412 1.4278 1.4275 1.4452 1.4465

-156 140 95 98 107 123 94 101 117 104 104 120 I21

1245 959 979 971 929 868 980 944 867 933 932 857 852

1.19 3.30 2.81 2.90 3.02 3.14 2.87 3.02 3.OO 2.93 2.91 2.97 2.96

-0.762 38 -1.173 95 -1. I68 69 -1.17256 -1. I79 23 -1.192 90 -1.171 92 -1.177 36 -1.190 36 -1.179 05 -1.177 78 -1.191 33 -1 .I90 7 1

1.3386 1.4201 1.4066 1.4108 1.4220 1.4418 1.4089 1.4176 1.4370 1.4204 1.4191 1.4406

-133 148 92 98 109 135 94 102 126 105 104 128 (128)

1257 974 1009 995 946 856 1006 970 885 96 1 962 865

1.27 3.34 2.78 2.88 3.00 3.16 2.85 2.91 3.02 2.95 2.90 3.02 3.00

SCF MP2 MP3 MP4DQ CCD CCD(S) CCD(ST)

-0.762 60 -1.22445 -1.22074 -1.223 18 -1.222 48 -1.228 27 -1.244 24

1.3376 1.4233 1.4085 1.4112 1.409 1 1.4181 1.4407

-150 151 93 97 92 93 129

1238 955 985 981 1002 957 865

1.18 3.46 2.89 2.98 2.96 2.91 3.15

SCF MP2 MP3 MP4DQ CCD CCD(S) CCD(ST) QCISD CCSD QCISD(T) CCSD(T)*

-0.767 73 -1.266 75 -1.259 99 -1.261 68 -1.26095 -1.26621 -1.287 55 -1.267 77 -1.251 82 -1.284 47 -1.268 76

1.3308 1.4083 1.3949 1.3969 1.3950 1.4029 1.4323 1.4054 1.4049 1.4266

-142 188 111 113 109 118 155 121 119 147 (146) 160

1258 993 1021 1018 1028 99 1 91 1 990 988 903

1.15 3.52 2.95 3.04 3.02 3.06 3.20 3.10 3.07 3.20 3.20 3.40

1.4119

143 98 100

916

"Absolute energy (E) of F2, in au below -198.0, optimum bond length ( r ) for F2 in A, dissociation energy ( D ) of F2 in kJ mol-', harmonic vibrational frequency ( w ) for F2 in cm-I, and electron affinity (EA) for F in eV. bResults obtained at CCSD-optimizedgeometries, 'From refs 52 and 53. not worse than f3 cm-I, a limitation which was felt tolerable in Geometries were optimized and vibrational frequencies obtained view of the size of many of the calculations and the substantial in Houston mostly from gradients, but the polynomial fitting and variations found with level of theory (see below). finite-displacement procedure outlined above was adopted for the Calculations were performed in Houston at the SCF, CCSD, CCSDT results. In all correlated calculations in this work one CCSD(T), and CCSDT levels using codes developed in Berkeley, occupied MO per F atom was frozen, and excitations into the Georgia, and H o ~ ~ t o n . Very l ~ careful ~ ~ ~checks ~ ~ were ~ ~ ~ ~corresponding ~ - ~ ~ virtuals were excluded. performed to ensure that the same bases were used, and the same results obtained at SCF levels, in both Melbourne and Houston! Results and Discussion Calibration Chlculatiom on F2and F.In order to provide some (47) Scheiner, A. C.; Scuseria, G. E.; Rice, J. E.; Lee, T. J.; Schaefer, H. indication of the quality of the basis sets we have used and the F. 111 J . Chem. Phys. 1987, 87, 5361. reliability of the various quantum chemical techniques employed, (48) Scuseria, G. E.; Janssen, C. L.;Schaefer, H.F. 111 J . Chem. Phys. we provide in Table I values calculated for the optimum bond 1988,89, 7382. length, the dissociation energy, and the harl'l'ionic vibrational (49) Scuseria, G. E.; Schaefer, H. F. 111 Chem. Phys. Lerr. 1988,152,382, (50) Scuseria, G. E. Chem. Phys. Lett. 1991, 176, 27. frequency of F2,together with the electron affinity of the F atom.

Quantum Chemistry of the Trifluoride Anion These systems have been studied very intensively by previous workers, and they are known to provide severe tests of any ab initio method. Experimental values are also provided for comparison. It is not necessary to discuss all the entries in exhaustive detail, but we note the following points: (i) The SCF method provides a very poor description of F,; the bond is substantially too short, the molecule is unbound by approaching 150 kJ mol-I, and the stretching frequency is too high by some 35%. These deficiences have been known for a long time5’ and are not caused by the limitations of the bases used, as the TZ2Pf+ set performs no better than the much smaller DZP set. The SCF method is also poor for F-,giving an electron affinity which is only about one-third of the correct value. Once the basis contains diffuse functions, further extension has little effect a t the SCF level. It is therefore clear that the methods used to describe electron correlation must perform very well if good results are desired for these systems, as the SCF starting point is so poor. (ii) Perturbation theory provides results rather similar to the more exact coupled-cluster-based methods for the bond length and dissociation energy of F,; the influence of triple substitutions on these quantities is large but not overwhelming (about 0.02 A and 25 kJ mol-’, respectively) and is not greatly overestimated by perturbation theory. At the MP2 level, the electron affinity of F is clearly too large, by more than 0.1 eV with our largest basis; presumably, the error would increase still further with more complete bases. Perturbation theory also overestimates the effect of triple substitutions on the electron affinity of F, by about 0.15 eV compared to the coupled-cluster methods. Recent CCSD and CCSD(T) calculations have shown that it is not possible to obtain an accurate value for the electron affinity without including the triple excitation^.^^ QCISD and CCSD results with a given basis are very similar for the bond length, dissociation energy, and electron affinity. QCISD absolute energies are lower than those provided by the CCSD technique, by almost 16 mhartrees with our TZ2Pf+ basis, whereas the estimate of the energy improvement due to triple excitations is slightly larger for the CCSD(T) method than for QCISD(T). The addition of diffuse s functions to the diffuse p set was found to increase the electron affinity and binding energy and decrease the F-F bond length, but the changes amounted to only a few hundredths of an electronvolt, 2 kJ mol-’, or two thousandths of an angstrom, respectively, and so are of no real significance. (iii) The vibrational frequency provides a more demanding test of the various methods. All the methods which include only double excitations overestimate the frequency by an amount which increases as the basis is improved, reaching about 100 cm-l (or 11%) with our largest TZ2Pf+ basis. Single and triple excitations both decrease the frequency, by about 35 and 90 cm-I, respectively. Perturbation theory estimates of these effects are slightly greater than those obtained from coupled-cluster methods. Most of the CCD(S) results compare tolerably well with those obtained using the more exact QCISD theory; the vibrational frequency is always higher at the CCD(S) level than from QCISD data, but not by more than 20 cm-’. CCD(ST) frequencies are higher than those obtained at the QCISD(T) level, by comparable amounts. Where they can be compared, QCISD and CCSD frequencies differ by no more than 2 cm-I, which is essentially the precision of our data, and our QCISD(T) and CCSD(T) results are also very similar. These observations confirm the close correspondence between QCISD and CCSD results already noted in a recent comparison.39 CCSDT and CCSD(T) values for the vibrational frequency differ by only 2 cm-’ with the DZP+ basis. (iv) Our best calculated properties of the bond length, dissociation energy, vibrational frequency, and electron affinity are in quite reasonable agreement with the experimental value^,^^^^^ differing by only 0.015 A, 13 kJ mol-’, 13 cm-I, and 0.19 eV, respectively, or 1.1, 8.1, 1.4, and 5.8%. It is possible to obtain (51) Wahl, A. C. J . Chem. Phys. 1964, 41, 2600. (52) Scuseria, G. E. J . Chem. Phys. 1991, 95, 7426. (53) Huber, K. P.; Herzberg, G. Constants of Diatomic Molecules; van Nostrand Reinhold: New York, 1979. (54) Hotop, H.; Lineberger, W. C. J . Phys. Chem. Ref: Dara 1975,4,539.

The Journal of Physical Chemistry, Vol. 96, NO. 11, 1992 4363 better agreement than we have done,’%z1,52~55 but the convergence in these properties with increase in size of basis is frustratingly slow. S t ” e and Dimciition Energy of Fy. In Table I1 we provide values of the absolute energy, the optimized bond length, the dissociation energy compared to (F, F),and the harmonic vibrational frequencies calculated for the trifluoride anion with several different basis sets and methods. At the SCF level the F-F distance is already long at about 1.64 A, and it is substantially increased by correlation effects, by a full 0.1 A with our most elaborate treatment, to near 1.74 A, some 23% longer than in molecular Fz. Since the formal F-F bond order is only one-half in the trifluoride anion, long bonds are of course e ~ p e c t e d .The ~ binding energy is negative, Le., the anion is unbound by nearly 50 kJ mol-’, a t the S C F level. It is noticeable that the addition of diffuse functions reduces the stability of F3- relative to (F, F);this apparent anomaly may be understood when it is realized that although the absolute energy of F3- is of course lowered substantially by the addition of diffuse functions, the energy of F is reduced even more, presumably because the negative charge is more concentrated in the fluoride anion. Perturbation theory is certainly not without some merit for calculations on F3-, as the major effects of double and quadruple excitations appear to be well described at fourth order. It is notable that the CCD optimum value for the bond length is consistently just 0.008 A greater than the MP4DQ result, and the CCD dissociation energy just a few U mol-’ higher. However, the MP2 dissociation energies are much larger than the MP4DQ values, by an amount which decreases as the basis is enlarged but is still almost 70 kJ mol-I with the TZ2Pf+ basis. Similarly, the MP2 bond distance is about 0.05 A greater than the MP4DQ value, a difference which is not sensitive to the size of basis. Therefore, low orders of perturbation theory are clearly unreliable for the trifluoride anion. Perhaps the most informative and sensitive way of assessing the multireference character of a molecular system is from the magnitude of the CCSD 7, d i a g n o ~ t i c . For ~ ~ the trifluoride anion this is 0.034 with the DZP+ basis, which indicates very substantial multireference character; the value for ozone at its DZP/CCSD equilibrium geometry is 0.029,31 and for the “problem molecule” FOOF is 0.033,23while for a molecule which is well described by a single reference, such as water, the value is about 0.01 or less.23 To obtain perhaps semiquantitatively useful information about the origin of the multireference character of F3-, we have undertaken a few MCSCF (MC indicates multiconfigurational) and MR-CISD (MR indicates multireference) calculations with the TZP+ basis. These show that there are only three important configurations in all. The coefficient of the Hartree-Fock reference configuration in MR-CI calculations is 0.93; for the other two configurations, which both involve only a orbitals, we find coefficients of 0.25 for the HOMO-LUMO double excitation (a,-.,*) and 0.14 for a a,-LUMO double excitation. These are the three a orbitals discussed by PimenteP and Rundle) in the “three-center” approach; they are of bonding (a,), essentially nonbonding (a,), and antibonding character (a,,*), respectively. For purposes of comparison, the comparable figures for Fz are 0.96 for the reference configuration and 0.18 for the ag-ae* double excitation. Orbital occupations for F3- are 1.93 (a,,), 1.82 (a,), and 0.24 (a,*), while for F, they are 1.91 (a,) and 0.091 (a,,*). The HOMO-LUMO gap in F3-is smaller than in Fz (SCF orbital energies are -0.749 and +0.078 au for F2 but -0.237 and +0.227 au for F3-), and therefore the corresponding double excitation has greater importance. The smaller gap in F3is caused by the longer, weaker bonds with corresponding reduced orbital overlap. Both single and triple substitutions increase the F-F distance by about 0.02 and 0.035 A, respectively, when coupled-cluster methods are used, but perturbation theory grossly overestimates the importance of these effects. With the DZP basis, the change in bond length from MP4SDQ to MP4SDTQ is 0.08 A, while

+

+

(55) Bauschlicher, C. W.; Taylor, P. R. J . Chem. Phys. 1986,85, 2779. ( 5 6 ) Lee, T. J.; Taylor, P. R. Int. J . Quantum Chem. Symp. 1989, 23, 199.

4364 The Journal of Physical Chemistry, Vol. 96, No. 11, 1992

Heard et al.

TABLE II: Calculrted Results for FJ-O basis and theory

r 1.6500 1.7356 1.6974 1.6871 1.7264 1.8051 1.6943 1.7115 1.7333

ob

WI

w2

w1

-9 166 94 86 129 192 92 114 144

567 426 488 512 442 292 496 468 430

349 178 299 315 282 23 1 302 290 270

412i 761 50 1 305 439 628 415 686 660

DZP

SCF MP2 MP3 MP4DQ MP4SDQ MP4SDTQ CCD CCD(S) CCD(ST)

E -0.15676 -0.758 04 -0.730 34 -0.13206 -0.755 17 -0.790 69 -0.733 80 -0.748 61 -0.771 35

DZP+

SCF MP2 MP3 MP4DQ MP4SDQ CCD CCD(S) CCD(ST) QCISD CCSD QCISD( T) CCSD(T) CCSDT

-0.16990 -0.787 74 -0.752 50 -0.756 11 -0,786 26 -0.758 14 -0.776 14 -0.802 64 -0.78491 -0.177 47 -0.809 84 -0.808 70 -0,80821

1.6537 1.7459 1.6995 1.6926 1.7423 1.7002 1.7202 1.7474 1.7307 1.7226 1.7653 1.7727 1.7648

-36 110 40 33 93 39 72 94 75 68 114 115 111

558 403 490 507 41 1 488 456 433 445 455 388 373 393

341 263 298 306 27 1 311 284 260 278 277 250 245 248

5741’ 725 262 3291’ 253 64 373 573 296 380 496 569 546

TZP+

SCF MP2 MP3 MP4DQ CCD CCD(S) CCD(ST) QCISD CCSD QCISD(T) CCSD(T)‘

-0.19974 -0.89403 -0.848 56 -0.854 82 -0.85591 -0.873 39 -0.904 15 -0.881 78 -0.875 16 -0.91 3 02 -0.91086

1.6841 1.7395 1.6893 1.6843 1.6917 1.7115 1.7409 1.7222 1.7133 1.1594

-42 102 29 23 29 52 85 65 57 104 (103)

564 407 496 510 493 460 410 446 459 387

345

5981

302 290

154i 338

280 249

245 339 477

SCF MP2 MP3 MP4DQ CCD CCD(S) CCD(ST)

-0.201 49 -0.976 75 -0.933 70 -0.937 06 -0.938 67 -0.956 43 -0.993 03

1.6478 1.7372 1.6887 1.6822 1.6903 1.7090 1.7378

-38 109 37 33 36 65 93

566 415 500 518 499 469 420

346

5931’

307 293

120i 352

SCF MP2 MP3 MP4DQ CCD CCD(S) CCD(ST) QCISD CCSD QCISD(T) CCSD(T)‘

-0.203 33 -1.026 98 -0.989 03 -0.990 79 -0.992 48 -1.009 38 -1.048 14 -1.016 49 -0.989 66 -1.053 92 -1.03209

1.645 1 1.7240 1.6790 1.6723 1.6802 1.6980 1.7252 1.7079 1.7008 1.7425

-46 86 31 21 29 53 75 67 57 110 (103)

566 42 1 509 526 506 477 432 464

346

618i

293

237

409

262

486

TZ2P+

TZ2Pf+

-

‘Energy (E) in au below -298.0, optimized bond length ( r ) in A, dissociation energy (D)in kJ mol-I, and harmonic vibrational frequencies ( w ) in cm-I. bEnergy for the reaction F3- F2 + F.‘Results obtained at CCSD-optimized geometries.

with the DZP+ basis it is at least 0.4 A! The energ was still decreasing as the bond distance was increased to 2.1 without any indication that a minimum would be found, and the full MP4 calculations were therefore abandoned at this point and not performed with any basis larger than DZP+. Perturbation theory also appears to indicate very large increases of up to 60 kJ mol-’ in the dissociation energy of the trifluoride anion due to single and to triple excitations. With coupledcluster methods the binding energy is certainly increased by these effects, but by more modest amounts of the order of 30 kJ mol-’. These observations demonstrate that even orders of perturbation theory as high as the fourth should not be used for F3- if quantitatively useful results are desired. It is of course the large multireference character which renders perturbation theory unsuitable for the trifluoride anion. Whereas QCISD(T) and CCSD(T) bond lengths for F2are almost identical, they differ by as much as 0.008A for F3-with the DZP+ basis; the QCISD(T) values are the shorter and very close to the

X,

full CCSDT result. However, when the effects of triple substitutions are excluded, the CCSD bond lengths are about 0.008 A less than the QCISD values. As the size of basis is increased from DZP+, competing effects are seen on the dissociation energy of the trifluoride anion. Expansion of the s,p basis reduces the binding slightly, as does the use of two sets of d functions rather than just one. Presumably these results are due to the greater difficulty of describing F,with its very concentrated negative charge distribution, than F3-. On the other hand, addition of f functions to the basis increases the binding energy slightly, so the net effect of increasing the basis from DZP+ to TZ2Pf+ is almost zero at the QCISD(T) level. We were able to use the complete CCSDT method with the DZP+ basis, which showed that the much less demanding CCSD(T) and QCISD(T) techniques give results which differ by no more than 4 kJ mol-’ from the full CCSDT value (and trivially from each other). We therefore feel that extrapolation to the “very large

Quantum Chemistry of the Trifluoride Anion basis"/CCSDT limit would give an electronic binding energy of 110 f 15 kJ mol-'. In comparison with uncertainities of this magnitude, the differences between De and Do, or between AI3 and AH values, may safely be ignored. When diffuse s functions were added to the TZP+ basis, the F-F bond length was decreased by 0.005 A at the SCF level or by 0.007 A at both QCISD and QCISD(T) levels. These changes are not trivial but are still of only minor quantitative importance. The extra diffuse s functions caused the binding energy to decrease by 3 kJ mol-'. Dissociation of the trifluoride anion to F and two F atoms requires either 257 or 270 kJ mol-', depending on whether one adopts our best calculated or the experimental value for the dissociation energy of Fz. The "average energy per F-F bond" in F3- is therefore about 135 kJ mol-', surprisingly similar to the value of 160 kJ mol-' in F s3 In view of their substantial length (calculated above as 1.74 compared to 1.41 A for F2) and low formal order of only 0.5: one would surely have expected weaker bonds in F). We draw attention to the fact that even a semiquantitatively correct indication of the relative energies of the two "half-bonds" in F3- compared to the single bond in Fz emerges only when electron correlation effects are considered to fairly high order, i.e., with inclusion of triple excitations from a CC wave function if a single-reference approach is used. This may appear surprising, as the comparison of F3-with (F2 F) involves no bond of order higher than unity, and there is no change in the number of unpaired electrons. While the analogy should probably not be overemphasized, a similar pattern is found for dib~rane,~' in which the substantially greater strength of two bridging B-H-B bonds (formal order 0.5 each) compared to one regular single B-H bond only appears correctly at correlated levels of theory. Vibrational Frequencies for Ff. We note firstly that the antisymmetric stretching frequency is strongly imaginary at the SCF level, with its value tending to increase in magnitude as the basis is enlarged, exceeding 600i cm-I with the TZ2Pf+ basis. There might therefore be some concern that SCF orbitals would be qualitatively unsuitable for correlated treatments of the molecule if C,, distortions are applied, as for example in the calculation of the antisymmetric stretching frequency. To investigate the seriousness of any possible symmetry-breaking problems, we first ensured that the Dmhstructure is a smooth col at the SCF level rather than a cusp, at least for small distortions of the magnitudes which concerned us directly. We also verified that if converged SCF orbitals from a calculation carried out in C, symmetry were used as the initial guess for the optimized Dmhgeometry, then the same SCF energy was obtained as that from a calculation in which Dmhsymmetry was also imposed on the orbitals. We have a fairly complete selection of predicted frequencies available with the DZP and DZP+ bases. In particular, it was possible to obtain CCSDT results with the DZP+ basis, but these calculations were not feasible with larger bases, as the single energy point needed to produce the bending frequency required almost 250 CPU hours on a relatively fast (4 Mflops) workstation. The primary use of the CCSDT data will therefore be to calibrate the probable reliability of results obtained at lower levels of theory, levels which could be extended to larger bases. In view of our experience gained on the differences in CCSD(T) and QCISD(T) results, compared with the differences found with a given method but different basis sets, it was not judged worthwhile to obtain frequencies with the largest bases at both CCSD(T) and QCISD(T) levels; each QCISD(T) calculation with the TZ2Pf+ basis required over 2 CPU hours on a Cray Y-MP. The symmetric stretching frequency is fairly sensitive to the level of theory used for its calculation. With the DZP+ basis, the CCSD(T) value is only 74% of the MP4DQ result, the latter being the highest of all the correlated levels. As could be expected from the work of Pulay and c o - w o r k e r ~there , ~ ~ is a pronounced inverse relationship between the optimum bond length at various levels of theory and the value predicted for wl,being about 8 times

x

+

(57) Curtiss, L. A,; Pople, J. A. J . Chem. Phys. 1988, 89, 4875 and references therein. (58) h l a y , P.; Lee, J.-G.; Boggs, J. E.J . Chem. Phys. 1983, 79, 3382.

The Journal of Physical Chemistry, Vol. 96, No. 11, 1992 4365 greater than that in r. They have shownS6that a change in bond distance Ar should lead to a proportional change in harmonic force constant of about -3aAr, where a is the Morse anharmonicity. This relationship therefore predicts that if the only influence on the force constant arises from the change in bond length, a change in wI of about 116 cm-' will follow a change in re from 1.69 (DZP/MP4DQ) to 1.77 A (DZP+/CCSD(T)), taking a to be 2.2 A-' and the average value of wIto be 440 cm-I. Inspection of Table I1 shows the actual change to be 139 cm-I; although the prediction is not exact, the similarity of this figure to that predicted by such a simple relationship emphasizes the importance of the reference geometry in calculations of vibrational frequency. While both single and triple substitutions lower wI appreciably, the different methods are far from unananimous in their estimates of these effects. The difference between CCSD(T) and QCISD(T) results is not negligible, amounting to 15 cm-I with the DZP+ basis; for this particular aspect of the PE surface, the QCISD(T) value is closer to the CCSDT result than is the CCSD(T) figure. It is clear from the data in Table I1 that the value predicted for wlwith a given method tends to increase as the basis is increased in size, following the decrease in optimum bond length which also results from an increase in basis (see above). We do not imagine that we have reached the "infinite basis limit" with our TZ2Pf+ basis, so we extrapolate. Frequencies obtained with the various bases at the QCISD(T) level are as follows: DZP+, 388; TZP+, 387; TZ2Pf+, 409 cm-'. We also have a more complete set of CCD(S) results: DZP+, 456; TZP+, 460, TZZP+, 469; TZ2Pf+, 477 cm-I. These progressions show that basis extension from DZ to T Z gives only minor changes in wl,so further enlargement beyond TZ should give insignificant changes. Increasing the primary polarization space from P to 2P increases olby 9 cm-l at the CCD(S) level. We suppose that similar changes would be seen using QCISD(T) methods, so we estimate that further increases of the d basis might increase wl by perhaps 5-10 cm-'. Adding one set o f f functions to the TZ2P+ basis increases wI by 8 cm-' at the CCD(S) level. Again we suppose that similar changes would be found using QCISD(T) energies, and we allow for possible further increases of about 5 cm-' for extra f functions. Finally, we consider that g and higher functions should increase wI by no more than 5-10 cm-I, given the rather small effect of f functions. When we combine all these possible basis extensions, we obtain an estimated value of wl at the "very large basis/ QCISD(T) limit" in the range 420-430 cm-I. Since with the DZP+ basis the CCSDT frequency is only slightly higher (just 5 m-')than the QCISD(T), we suppose that a similar relationship would obtain with a very large basis, to give an estimated "very large basis"/CCSDT value for wI of some 430-450 cm-I; we believe these uncertainties are conservative. Vibrational anharmonicity will probably (though not inevitably, given the behavior of the related HF2- ion40) make uI slightly lower than wl,but probably by only a few wavenumbers. The Raman peak observed in rare-gas matrices and attributed to u1 of the trifluoride anion is found at 461 cm-1.56In assessing the level of agreement between this experimental value and our extrapolated estimate, the possible influence of "matrix effects" must not be overlooked. These are quite likely to shift vibrational peaks by as much as a few tens of wavenumbers (though it is not at all obvious to us in which direction!); in particular, in the case of F), the anion is surely coupled to some extent to the alkali-metal cation in the matrix, so the "matrix shifts" are probably greater than for neutral molecules. In view of these uncertainties, we regard the level of agreement as quite encouraging. The bending frequency w2 is less sensitive to the level of theory used for its calculation than is wI. It varies by "only" 66 cm-l over the whole range of 12 different correlated methods used with the DZP+ basis, and with a particular method changes by only a few wavenumbers as the basis is enlarged from DZP to TZ2Pf+. Triple substitutions lower this frequency, but not dramatically. As the DZP+/CCSDT value is almost exactly between the CCSD(T) and QCISD(T) results, which themselves differ by only 5 cm-l, we may extrapolate fairly confidently to the "very large basis"/CCSDT limit, in the manner described above the wl, to

4366

The Journal of Physical Chemistry, Vol. 96, No. 11, 1992

obtain a predicted harmonic value of 260 f 10 cm-I. This band has not been observed experimentally. Its intensity predicted at the SCF level varies rather narrowly in the range 28-24 km/mol, as the basis is enlarged from DZP+ to TZ2Pf+. It therefore appears as though this vibration should be fairly easily observable; perhaps it lies just at or beyond the limit of the spectral range accessible in the experimental studies, which appears from the published spectra5v6to be 250 cm-l, though the background is clearly sloping downward below about 300 cm-I. Of all aspects of the present calculations, it is the predicted value for the antisymmetric stretching mode w3 which is the most variable. This could be expected by analogy with results already reported for as mentioned above. With the DZP+ basis, for example, it changes from 574i to 725 cm-I (real) comparing SCF with MP2 data. Then as the order of perturbation theory is increased from MP2, the frequency drops precipitously to 262 cm-I (MP3) and becomes strongly imaginary (3293 cm-l) at the MP4DQ level, only to rise to 253 (real) cm-I when single excitations are included by the MP4SDQ formalism. These enormous oscillations surely indicate that any value predicted by perturbation theory of orders 2-4 is quite unreliable, and we believe that the quite close agreement between the value reported for w3 by Novoa and co-workersa (526 cm-') and the frequency attributed to u3 of the trifluoride anion in an inert matrix (550 cm-1)536must be regarded as fortuitous. We do not understand the substantial difference (199 cm-I) between our MP2 result and that reported by Novoa and co-workers.* Their DZP+ basis is not quite identical to ours, as they used a d exponent of 0.9 compared to our 1.0, and their diffuse functions were of both s and p type whereas we used only p functions. But these differences are very minor and would certainly not be expected to cause a change in frequency of some 200 cm-'; in particular, the addition of diffuse s functions to our TZP+ basis changed the values of w1 and w 3 by only +3 cm-l, at both QCISD and QCISD(T) levels. Triple substitutions raise the value of w 3 quite substantially, but the various methods of incorporating these substitutions produce markedly disparate results; compare the CCD(ST), CCSD(T), CCSDT, and QCISD(T) values with the DZP+ basis of 573,569, 546, and 496 cm-'. Fortunately, however, the frequency changes only slightly for a given theoretical method with the size of basis, decreasing by just 10 cm-' at the QCISD(T) level as the basis is increased from DZP+ to TZ2F'f+. We assume that the CCSDT values with DZP+ and TZ2Pf+ bases would differ by about the same margin as the QCISD(T) results and can therefore extrapolate to obtain a "very large basis"/CCSDT prediction of 535 f 20 cm-I, which encompasses the matrix feature observed at 550 cm-I. CCD(S) and CCSD results for the bond length, binding energy, and vibrational frequencies of the trifluoride anion are very similar in all cases where we can make direct comparisons. Performance of the CCD(ST) method is rather variable; it underestimates the bond lengthening due to triple excitations by almost half, compared to the CCSDT or CCSD(T) values, and correspondinglyprovides a lower estimate of the increase in binding energy due to these excitations. However, the CCD(ST) antisymmetric stretching frequency obtained with the DZP+ basis agrees very well with the CCSD(T) result, whereas the symmetric stretching frequency is substantially too high, due to the underestimate of the bond length. It is of interest to compare the major findings from a computational study40 of the vibrational frequencies of the hydrogen trifluoride anion, HF,, with those obtained here for the closely related trifluoride anion. Calculated values of both w1 and w2 for that system are remarkably insensitive to the level of theory used, whereas we have seen that uIvaries appreciably for the trifluoride anion. The most striking results for the bifluoride anion concern the antisymmetric stretching mode. Rather unusually, its magnitude is substantially increased by correlation effects (though these effects are not as dramatic as for F3-, since the vibration is real for the bifluoride anion at the SCF level). It displays an enormous sensitivity to the details of the basis used, being greatly

Heard et al. lowered by the addition of diffuse functions on F and raised by d functions on H. It is also notable and unusual that the fundamental frequency u1 for HF2- is substantially higher than the harmonic value; we have no direct information on anharmonic effects for the trifluoride anion but expect them to be much less pronounced than for the bifluoride case, in view of the absence of a very light atom. Triple excitations appear to have rather little effect on the antisymmetric stretching mode for HF,, but as their influence was evaluated only with a rather limited DZP basis, this conclusion may be premature. The best calculated frequency of w1 for HF, is over 1200 cm-1,40much higher than the 535 cm-I predicted here for the trifluoride anion, but it is more meaningful to compare force constants than frequencies for systems of different atoms. As Fj3values are about 0.45 and 1.07 mdyn A-', it is clear that the trifluoride anion has a markedly greater resistance to antisymmetric stretching than the bifluoride. The CCSD(T)-QCISD(T) differences found here for the antisymmetric stretching frequency of the trifluoride anion are more pronounced, especially on a proportional basis, than those reported for 0zone,~O3~~ where the DZP+ values are 976 and 934 cm-I, respectively; however, in that case, the CCSDT value of 1077 cm-I is much higher than either of the others,32whereas in the present case it lies between them, closer to the CCSD(T) result than to the QCISD(T) figure. These observations suggest, rather disappointingly, that data on the reliability of various different computational techniques from one particular molecule of high multireference character may provide quite imperfect precedents for other molecules. However, in view of possible diversities in electronic structure, such differences in performance from molecule to molecule should perhaps not be unexpected. In the trifluoride anion the most important multireference character is associated with a u-u* excitation, as discussed above, while in ozone the s t y p e electrons are involved.

Conclusions We have carried out a theoretical study of the structure, binding energy, and vibrational frequencies of the trifluoride anion, F3-. Large basis sets (triple-{ for the valence space, augmented with diffuse p functions, two sets of d functions, and one set of f functions) have been used in combination with coupled-cluster methods which incorporate the influence of triple substitutions (QCISD(T), CCSD(T) and CCSDT, though this latter was feasible only with a DZP+ basis). Very pronounced multireference character is indicated for Fq, by the CCSD 7ldiagnostic, by preliminary MR-CI calculations and by the important influence of single and triple excitations on the calculated results. Perturbation theory and approximate methods for including triple excitations such as CCD(ST) have been shown to be unsuitable for quantitative studies of this system. The trifluoride anion is bound by about 110 kJ mol-' compared to (F + F),and while its bond distance is remarkably long at 1.74 its average F-F bond energy of about 135 kJ mol-' is rather higher than might be expected. Our best estimates of the harmonic vibrational frequencies of F3- are wI= 440 f 10, w2 = 260 f 10, and w3 = 535 f 20 cm-'. These match reasonably well the frequencies of 461 cm-I for vI and 550 cm-' for u3 which have been observed in inert-gas matrices5V6and attributed to the trifluoride anion. We therefore feel that the difficult challenge posed to quantum chemistry by the trifluoride anion has probably been met, and the coupled-cluster method has again shown its usefulness and reliability.

1,

Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. C.J.M. thanks the University of Melbourne's Information Technology Service for programming assistance, Cray Research for generous acccss to Cray X-MP and Y-MP computers,and the Australian Research Council for financial support. We thank the referees for helpful and perceptive comments. Registry No. FF, 25730-99-8.