16538
J. Phys. Chem. 1996, 100, 16538-16544
Harmonic Vibrational Frequencies and Force Constants of M(CO)5CX (M ) Cr, Mo, W; X ) O, S, Se). The Performance of Density Functional Theory and the Influence of Relativistic Effects Attila Be´ rces Steacie Institute for Molecular Sciences, National Research Council, 100 Sussex DriVe, Ottawa, K1A 0R6 Canada ReceiVed: May 14, 1996; In Final Form: August 7, 1996X
We calculated the harmonic force constants and vibrational frequencies of most molecules in the series of M(CO)5CX (M ) Cr, Mo, W; X ) O, S, Se) by local and gradient-corrected density functional theory including relativistic effects. The calculated vibrational frequencies agree well with the experimentally observed ones. In the case of W(CO)6 we studied the contribution of relativistic effects to the reference geometry and the Hessian. These calculations have revealed that relativity has major impact on the geometry of the W complexes studied but has a less significant effect on the force constants comparing calculations at the same reference geometry. The stretching force constants of Cr(CO)5CX (X ) O, S, Se) were compared with empirical force fields. Some of the calculated interaction constants disagree with the empirical ones, but they are consistent with what is expected based on qualitative molecular orbital models.
1. Introduction In recent years several studies have demonstrated that density functional theory (DFT) is very successful for the calculation of molecular geometries and vibrational frequencies. Despite the accumulating evidence about the performance of these methods, the limitations of the method and the importance of the effects of basis sets, electron correlation, reference geometry, and relativity are only partly understood. In a series of investigations we studied the accuracy of force fields and vibrational frequencies of transition metal complexes based on density functional methods.1 We studied various aspects of these calculations, including electron correlation and reference geometry. The extensive experimental data about the compounds in the series of M(CO)5CX (M ) Cr, Mo, W; X ) O, S, Se) provided an attractive opportunity to extend our systematic studies of frequency calculations. The tungsten complexes provide an excellent test case for the significance of relativistic effects. Besides the general performance of density functional theory we are mainly concerned with the effect of relativity in vibrational frequency calculations in this study. We also compare some calculated force constants with those obtained from empirical normal coordinate analysis. For molecules containing heavy elements relativistic effects play an important role in their structural chemistry. The significance of relativistic effects in structural chemistry is well established.2 Schreckenbach et al. have recently implemented the analytic energy gradients based on quasi-relativistic DFT methods.3 On the basis of this implementation, Li et al. investigated the effect of relativity on the geometry and first bond dissociation energies of M(CO)6 (M ) Cr, Mo, W) compounds.4 They found that relativity contracts the W-C bond by 0.07 Å and contributes to first bond dissociation energies by 10 kcal/mol. Therefore, relativity may also have a significant contribution to the force constants and vibrational frequencies as well. In the present paper we study how relativistic effects contribute to vibrational frequencies for transition metal complexes. X
Abstract published in AdVance ACS Abstracts, September 15, 1996.
S0022-3654(96)01383-4 CCC: $12.00
Our objective was to assess the effect of relativity on the gradient and on the Hessian separately. Therefore, we compared calculations in which the frequencies were calculated at both quasi-relativistic and nonrelativistic levels, but kept the same reference geometry. Accordingly, we have done frequency calculations at reference points that were not equilibrium points on the potential energy surface. We have previously implemented a method5 to circumvent the nonzero force dilemma and accounted for this effect in the present investigation.1a Another important goal of this study is to assess the accuracy of quasi-relativistic density functional calculations for vibrational frequencies. Accordingly, we studied molecules for which experimental vibrational frequencies were available. The vibrational frequencies of tungsten hexacarbonyl and W(CO)5CS serve to test the QR-DF method. W(CO)6 is used as an example to study the effects of relativity on gradients and Hessian. The hexacarbonyls of chromium, molybdenum, and tungsten serve to study the effect of relativistic corrections in a series of molecules where relativity is expected to play an increasingly important role. 2. Computational Details The reported calculations employed the Amsterdam Density Functional (ADF 2.0.1) program system developed by Baerends et al.6a and vectorized by Ravenek.6b te Velde7 et al. developed the numerical integration procedure. The atomic orbitals on the metal centers were described by an uncontracted triple-ζ STO basis set with a single polarization function, while a double-ζ single polarized STO basis set was used for C, O, S, and Se.8 A set of auxiliary9 s, p, d, f, and g STO functions, centered on all nuclei, was used to fit the molecular density and represent the Coulomb and exchange potentials in each SCF cycle. We applied three different levels of theory; the energy expression from the local density approximation10a (LDA) in the Vosko-Wilk-Nusair parametrization is the simplest level. In the next level we included gradient corrections to exchange10b of Becke and to correlation of Perdew,10c which were applied self-consistently (generalized gradient approximation or GGA). The most elaborate method also involved quasi-relativistic
Published 1996 by the American Chemical Society
Harmonic Vibrational Frequencies of M(CO)5CX
J. Phys. Chem., Vol. 100, No. 41, 1996 16539
corrections to the Hamiltonian introduced by Snijders et al.11 (GGA+QR). The geometry was optimized on the basis of the GDIIS technique.12 The Cartesian force constants and dipole moment derivatives were calculated by numerical differentiation of the energy gradients13 using Cartesian displacements. Fan et al. implemented an automatic scheme for the transformation of symmetry-related Cartesian force constants.13c Therefore, only the symmetry unique displacements had to be made. We used a sufficient number of integration points to ensure a numerical accuracy of 1.0 cm-1 for most vibrational frequencies. For frequencies evaluated at points other than the optimized geometry, the forces or energy gradients are not zero. Therefore, when the force constants are transformed into internal coordinates, q, terms due to the gradient also have to be taken into account.5 The force constants in internal coordinates can be expressed in terms of the Cartesian energy gradients and energy Hessian as 2 ∂2V ∂xk ∂xl ∂V ∂ xl )∑ +∑ ∂qi ∂qj k,l ∂xk ∂xi ∂qi ∂qj l ∂xl ∂qi ∂qj
∂2V
(1)
We have implemented this transformation into a program package developed by Maurer and Wieser,14 based on Schachtschneider's force field program.15 This program was used for the normal coordinate analysis. 3. Results and Discussions We first discuss the influence of relativistic effects on the geometry and on the vibrational frequencies in section 3.1. In 3.1 frequency calculations at various levels are compared for W(CO)6. In 3.2 we discuss the vibrational frequencies of M(CO)5(CX), X ) O, S, Se and M ) Cr, W. In 3.3 we compare experimental and calculated vibrational frequencies of Mo(CO)6. In 3.4 the calculated valence force constants are interpreted in terms of qualitative orbital analysis and compared with the empirical field determined by English,16 Jones,17 and co-workers. Before we present the discussion about specific compounds we make some general comments about the expected accuracy for the harmonic approximation. Most experimental frequencies in our discussion refer to observed fundamental rather than harmonic frequencies, while the theoretical vibrational frequencies are calculated on the basis of the harmonic model. One reason we have chosen the hexacarbonyl series and related compounds is that anharmonic contribution to the observed fundamental frequencies for M(CO)6 compounds is either fairly small or accurately determined.17 The CO stretching vibrations have the most significant anharmonic contribution to the frequencies, which can be easily deduced from the overtone spectra of the CO stretching modes. The most significant anharmonic contribution was 40 cm-1 for the T1u symmetry CO stretches, while it was about 20 cm-1 in the A1g and Eg symmetry combinations. Such experimental corrections are also available for the chalcocarbonyl. For this reason, in all tables we listed the experimentally determined harmonic CO, CS, and CSe stretching frequencies. For the rest of the frequencies the anharmonic component is less accurately determined; however, the magnitude is estimated to be small from the wavenumbers corresponding to combination bands and overtones. Jones et al. found the anharmonic corrections Xkl to be less than 3 cm-1 in most cases, with the exception for k and l both being CO stretches.17 The largest anharmonic contribution, 8.6 cm-1, was found for the ν9 + ν10 combination band. The assignment of this combination band is, however, not certain. Taking all these experimental findings into account, the series of molecules
studied here present an excellent test case for the theoretical harmonic force fields and vibrational frequencies with very few uncertainties due to anharmonic corrections. 3.1. The Vibrational Frequencies of W(CO)6. The vibrational spectrum and force constants of W(CO)6 were first studied by Jones and co-workers.17 In this study, the IR and Raman spectra of hexacarbonyls of Cr, Mo, and W and their 13C- and 18O-substituted derivatives were recorded. As a result of high symmetry, there was enough information to determine the complete set of force constants. The similarity of the spectra of different hexacarbonyls and their isotopomers made it possible to identify vibrational frequencies with confidence. Furthermore, the anharmonic corrections to the CO stretching frequencies have also been deduced. The vibrational assignments for Cr(CO)6 have been confirmed by density functional calculations,18 which can be regarded as a confirmation of assignments for all molecules in the hexacarbonyl series. W(CO)6 is a very attractive subject for a detailed study of the influence of relativistic effects because of its high symmetry and high influence of relativistic effects on the structural chemistry of tungsten complexes. The high symmetry considerably reduces the cost of computation of force constants, making it feasible to compare calculations at various levels. In comparing different levels of theories, the effect of reference geometry has to be carefully taken into consideration. The derivatives of the nuclear part of the potential energy depend only on the geometry of the molecule. Therefore, when two sets of calculations are compared with different reference geometries, it is difficult to judge if the frequency shifts are the result of the different nuclear contributions to the force constants, electron correlation, or other effects. We had previously adopted a scheme in which the effect of reference geometry can be separated from other effects by comparing vibrational frequencies obtained by calculations in which either the reference geometry or the applied level of theory was varied. These types of calculations showed that the effect of reference geometry is probably the most important error source for calculated vibrational frequencies by density functional methods.1a Even the simple local density approximation (LDA) was remarkably successful in reproducing harmonic vibrational frequencies when experimental geometries were used as a reference point for carefully selected molecules with highly accurate and reliable experimental data.1a A higher level of electron correlation treatment by gradient corrections did not affect the results significantly at that geometry. The effect of reference geometry naturally becomes important when dealing with relativistic effects since relativity is know to contribute to bond shrinkage of 0.01-0.1 Å in tungsten complexes.3 Therefore, the question arises to what extent relativity contributes to the gradient and to the Hessian. Another question to be addressed is how well quasi-relativistic methods reproduce the vibrational frequencies of heavy element complexes. In the present study we compare three levels of theory: the simple local density approximations (LDA), the generalized gradient approximation (GGA) or nonlocal DFT,19 and inclusion of quasi-relativistic corrections (GGA+QR). Details of different approaches are given in the Computational Details section. We optimized the geometry of W(CO)6 at these three levels of theory and calculated the force field at each geometry by the method that corresponds to the optimized geometry and by all lower levels of theory. The nonstationary force fields were transformed into a set of nonredundant internal coordinates by eq 1, and the vibrational frequencies were determined in the Wilson formalism.20 The calculated vibrational frequencies at
16540 J. Phys. Chem., Vol. 100, No. 41, 1996
Be´rces
TABLE 1: Vibrational Frequenciesa of Tungsten Hexacarbonyl
A1g E T1g T1u
T2g T2u a
expt
ref. geom. Hessian
2.059b 1.148b
rWCd rCOd
2153.2c 426c 2037.6c 410c 361.6c 2037.6c 586.6c 374.4c 82.0c 482.0c 81.4c 521.3c 61.4c
CO stretch WC stretch CO stretch WC stretch CO stretch WC stretch
GGA+QR GGA+QR
GGA+QR GGA
GGQ+QR LDA
GGA GGA
GGA LDA
LDA LDA
2.053 1.155
2.053 1.155
2.053 1.155
2.102 1.153
2.102 1.153
2.054 1.146
2081.2 436.0 1987.4 420.1 339.0 1953.8 603.3 331.4 85.9 460.9 80.8 543.9 63.6
2089.8 452.0 1996.4 434.9 329.9 1965.5 601.6 371.4 77.7 473.5 73.4 539.9 62.6
2090.1 447.3 2000.2 431.5 316.1 1968.6 590.5 364.1 76.3 458.3 69.9 527.1 60.9
2085.1 405.6 1995.6 387.5 323.2 1969.6 578.0 310.1 74.7 459.1 75.2 521.3 69.4
2086.6 400.6 2000.3 383.9 309.7 1973.4 568.3 302.5 74.6 444.0 67.2 508.9 65.2
2155.1 446.6 2065.7 432.4 319.3 2035.3 594.8 365.0 70.2 461.8 77.1 529.1 64.1
Frequencies in cm-1. b Reference 21. c Reference 17. d Distances in angstroms.
various reference geometries and by various levels of force constant calculations are listed in Table 1. It is apparent from Table 1 that the CO stretching frequencies are not affected much by electron correlation or relativistic effects. When comparing CO stretching frequencies calculated by different levels of theory but at the same reference geometry, the frequencies hardly differ at all. The only significant effect on CO stretching frequencies is caused by the change in reference geometries. Usually the calculated CO bond length increases by about 0.01 Å when gradient corrections are added to the exchange correlation functional. This general observation is supported by calculations on a variety of transition metal carbonyls. Unfortunately, the higher level of electron correlation results in a less accurate CO bond length. Previously we found that the LDA method gave a CO bond length very close to the experimental CO equilibrium distance for transition metal carbonyls.1a,b Accordingly, this reference geometry usually yields the most accurate CO stretching vibrational frequencies. W(CO)6 is another example that supports this general observation. Relativistic effects most significantly influence the WC stretching frequencies, which is expected since the relativistic bond contraction of the WC bond in W(CO)6 based on our calculations is 0.05 Å. The data in Table 1 suggest that relativistic effects contribute significantly to the Hessian, not only to the gradient. This is an important difference between gradient corrections to exchange correlation and relativistic corrections. If one compares two sets of calculations where the reference geometry is the same, the inclusion of gradient corrections does not influence the frequencies significantly. On the other hand relativistic corrections make a significant difference in the WC stretching frequencies when comparing two sets of calculations at the same reference geometry. For example, the T1u W-C stretching frequency is shifted from 371.4 to 331.4 cm-1 as a result of including relativistic corrections. Unfortunately, the nonrelativistic result matches better with the experimental frequency of 374.4 cm-1. Relativistic corrections decrease the WC stretching frequencies at a given reference geometry, but the relativistic bond contractions result in an increased stretching frequency. Overall, relativity slightly increases the WC stretching frequency. In this particular molecule, W(CO)6, the gradient corrections to the exchange correlation functional and the relativistic corrections almost cancel each other for the WC bond. Therefore, the geometry calculated by the simple LDA method is in excellent agreement with the experimental geometry obtained by Arnesen and Seip21 and with the gradient-corrected
relativistic results. Furthermore, the LDA CO bond length is the most accurate of all calculations. Therefore, by cancelation of errors the LDA method provides the most accurate reference geometry. Consequently, it is not surprising that the simple LDA method provides vibrational frequencies with accuracy similar to the most sophisticated GGA+QR calculations. For the CO stretching frequencies, the LDA method gives the best agreement with experiment. This finding also underlines the importance of reference geometry in vibrational frequency calculations. 3.2. The Vibrational Frequencies of M(CO)5CX, Where M ) Cr, W and X ) S, Se. The vibrational spectra and force field of pentacarbonyl-(chalcocarbonyl)-metal complexes were studied by English et al.16 English et al. have determined the valence force constants of these complexes and compared them to the force constants of the hexacarbonyls determined by Jones et al.17 English and co-workers used compliance constants rather then force constants for the normal coordinate analysis to ensure better transferability of potential constants from one molecule to another. The compliance matrix is the inverse of the force constant matrix, and the values of its elements Cij depend only on the definition of coordinates i and j, but are independent of the other coordinates. This property does not hold for force constants. Although English et al. carried out the normal coordinate analysis using the compliant field, the final compliance matrix was inverted to obtain the more familiar force constants. We have not found previous theoretical force field calculations for these compounds. The vibrational frequencies of the chromium and the tungsten complexes are summarized in Table 2 and 3, respectively. All these calculations were done by gradient-corrected quasirelativistic calculations. For the chromium complexes it was not necessary to do the calculations at quasi-relativistic level. However, these calculations give us an opportunity to test the method for molecules where one does not expect strong relativistic effects. The computational cost of including relativistic corrections is very modest. In Tables 2 and 3 we have included the experimental vibrational frequencies where they were available. In these tables we also included the calculated vibrational frequencies of the corresponding hexacarbonyls correlated to the C4V symmetry for easier comparison. For Cr(CO)6, comparison with experimental frequencies was given in refs 1a and 1b. The difference in the present calculations is that now relativistic corrections to the Hamiltonian are included. These values are virtually identical to our previous calculations by the GGA method.1a This finding was
Harmonic Vibrational Frequencies of M(CO)5CX
J. Phys. Chem., Vol. 100, No. 41, 1996 16541
TABLE 2: Vibrational Frequenciesa of Cr(CO)6, Cr(CO)5CS, and Cr(CO)5CSe Calculated with Relativistic Corrections Cr(CO)6 Oh rCrCeqc rCrCax rCrCtrans rCOeq rCX rCOtrans
GGA+ QR
C4V
expb
1.917
2091.8 1977.0 1996.7 692.9 446.6 387.6 384.8 103.6 364.0 1996.7 522.9 387.6 63.5 548.8 90.2 1977.0 692.9 522.9 548.8 446.6 364.0 103.6 90.2 63.5
GGA+ QR
Cr(CO)5CSe expb
1.906 1.871 1.925 1.153 1.567 1.153
1.153
A1g T1u E T1u T1u E A1g T1u T1g E T2u E T2u T2g T2g T1u T1u T2u T2g T1u T1g T1u T2g T2u a
Cr(CO)5CS
A1 A1 A1 A1 A1 A1 A1 A1 A2 B1 B1 B1 B1 B2 B2 E E E E E E E E E
2118.1 2060.7 1287.3 650.4 421.2 376.0 346.6 98 364 2052.4 512 390 525 85 2044.9 650.4 525 488 424.6 340.7 95 56
2079.5 1994.6 1273.9 685.7 437.3 396.8 353.6 100.9 353.5 2001.6 513.3 400.1 31.9 543.9 92.4 1982.4 682.0 526.9 494.5 444.2 329.8 92.8 82.6 53.6
GGA+ QR 1.907 1.864 1.930 1.153 1.702 1.153
2116.0 2064.6 1101.1 643.1 406.4 370.2 280.1 86 363 2054.3 505.5 389 68 528 85 2044.6 643.1 528 480.5 419.8 328.3 95.5 45.5
2077.6 1994.2 1105.9 678.5 426.2 396.7 277.3 94.4 353.2 2002.0 511.7 401.5 33.2 542.7 92.0 1983.0 676.3 522.9 487.4 439.6 319.0 93.3 79.8 43.3
-1 b
Frequencies in cm . Reference 16. CO, CS, and CSe stretching frequencies are corrected for anharmonicity. c Distances in angstroms.
expected since relativistic effects should not play a significant role in Cr structural chemistry. The comparison between calculated and observed frequencies for Cr(CO)5CS and Cr(CO)5CSe is another example of excellent performance of density functional theory for the vibrational frequencies of transition metal complexes. Most vibrational frequencies agree within 10-20 cm-1 with the experimental values. The CO stretching frequencies are too low by as much as 40-60 cm-1, which is the consequence of the too long CO bond length predicted by gradient-corrected methods. The quantum mechanical calculations confirmed most experimental assignments by English et al.16 There is minor ambiguity about the assignments and values of the CCrC bending frequencies. These frequencies were identified by solid state Raman measurements and IR combination bands. The calculated frequencies in this region usually agree with their experimental counterparts within a few wavenumbers, as demonstrated in this and previous calculations. Therefore, it is unusual that the A1 CCrC bending frequency of Cr(CO)5CSe is calculated to be 94.4 cm-1 while experimentally it is 86 cm-1, determined from the combination band. Similar ambiguity can be raised about the B2 CCrC bending frequency of both Cr(CO)5CS and Cr(CO)5CSe. These deviations are in the range of uncertainties in the absolute values of the experimental frequencies. The B1 CCrC bending frequency of Cr(CO)5CSe is probably incorrectly assigned, since the experimentally suggested value of 68 cm-1 is more than twice the theoretical value of 33.2 cm-1. The experimental frequency may be the overtone and not a fundamental frequency of the corresponding mode. The vibrational frequencies of W(CO)6, W(CO)5CS, and W(CO)5CSe are listed in Table 3, where experimental frequen-
TABLE 3: Vibrational Frequenciesa of W(CO)6, W(CO)5CS, and W(CO)5CSe Calculated with Relativistic Corrections W(CO)6 ref. geom.
Oh
W(CO)5CS
GGA+QR C4V
expb
rWCeqc rWCax rWCtrans rCOeq rCX rCOtrans A1g T1u E T1u E T1u A1g T1u T1g E T2u E T2u T2g T2g T1u T1u T2u T2g T1g T1u T1u T2g T2u
2081.2 1953.8 1987.4 603.3 420.1 331.4 436.0 85.9 339.0 1987.4 543.9 420.1 63.6 460.9 80.8 1953.8 603.3 543.9 460.9 339.0 331.4 85.9 80.8 63.6
A1 A1 A1 A1 A1 A1 A1 A1 A2 B1 B1 B1 B1 B2 B2 E E E E E E E E E
2123 2045 1294 569 426 380 346 2054 518 412 479 80 2039 569 491 463 375 332 50
W(CO)5CSE
GGA+QR
GGA+QR
2.058 1.990 2.111 1.155 1.556 1.153
2.058 1.986 2.118 1.155 1.671 1.154
2058.9 1977.4 1285.5 591.0 427.4 373.4 319.6 65.8 339.8 1985.6 536.0 414.8 48.3 458.2 79.5 1953.0 584.8 507.2 440.7 328.2 313.8 87.3 71.3 48.6
2056.4 1973.7 1125.5 591.0 434.0 329.9 243.9 42.7 339.5 1984.7 534.3 422.3 49.9 457.5 79.6 1953.2 581.6 495.6 436.6 341.7 306.5 73.6 64.3 31.5
Frequencies in cm-1. b Reference 16. CO and CS stretching frequencies are corrected for anharmonicity. c Distances in angstroms. a
cies of W(CO)5CS are also included. Unfortunately experimental vibrational frequencies for W(CO)5CSe are not available. The calculated frequencies of W(CO)5CS are also in good agreement with experimental results. The experimental frequency at 80 cm-1 was originally assigned to the A1 CMC bending mode. These frequencies are usually predicted with only a couple of cm-1 error for W(CO)6 and for all other molecules in this study. Therefore, this frequency more likely belongs to the B2 symmetry, where the corresponding calculated frequency is 79.5 cm-1. The calculated T1g MCO bending frequency of W(CO)6 (calcd 339.0; exptl 361 cm-1) and the correlating E frequency of W(CO)5CS (calcd 340.7; exptl 375cm-1) are in poor agreement with the experimental values. The corresponding frequencies for both Cr and Mo carbonyls were predicted within a few wavenumbers compared to experiment. It is hard to tell if this disagreement for W complexes is due to improper treatment of relativistic effects or incorrectly assigned frequencies. The calculations at different levels of theory for W(CO)6 suggest that the contribution of relativistic effects to this frequency is modest. Therefore, we expect this frequency to be fairly accurately predicted by DFT. Other than these minor ambiguities, we have confirmed the assignments for all frequencies by English et al.16 It is also interesting to compare the vibrational frequencies of W(CO)5CS and W(CO)5CSe. Most corresponding vibrational frequencies of these two complexes are remarkably close to one another. The CWC bending frequencies of the A1 and E symmetry are somewhat lower for W(CO)5CSe. Naturally, the WC(X) and CX stretching frequencies are different. All frequencies of A2 and B1 symmetry and all CO stretching frequencies differ by less then 1-2 cm-1 for these two systems. The same holds for the Cr analogues, for which such close
16542 J. Phys. Chem., Vol. 100, No. 41, 1996
Be´rces
TABLE 4: Vibrational Frequenciesa of Mo(CO)6 exptb 2.063 1.145 A1g E T1g T1u
T2g T2u
2144.2 406.8 2043.3 392.2 341.6 2043.1 595.6 367.2 81.6 477.4 79.2 507.2
ref. geom. c
rWC rCO
GGA+QR GGA+QR 2.085 1.152 2089.2 405.8 1996.3 391.1 337.9 1979.5 593.9 377.5 90.4 468.0 78.0 504.5 53.4
a Frequencies in cm-1. b Reference 17. CO frequencies are corrected for anharmonicity. c Distances in angstroms.
resemblance of vibrational spectra can be confirmed by the experimentally observed frequencies. This observation suggests that the error in the vibrational frequencies is systematic and reproducible. Therefore, one may use a small number of empirical factors as suggested by Pulay et al.22 to correct for the systematic error in the force constants if highly accurate vibrational frequencies and eigenvectors are needed. We previously found that eigenvectors and eigenvector related quantities improve significantly by correcting for systematic errors, although the frequencies change less dramatically.1e,f 3.3. The Vibrational Frequencies of Mo(CO)6. The geometry of Mo(CO)6 was experimentally determined by Arnesen and Seip,21 which differs somewhat from the optimized geometry at the GGA+QR level. The calculated MoC bond length is too long by 0.018 Å. Usually, one expects better agreement between experiment and theory for metal-carbon bond distances at this level of theory. If our reference bond length is too long, than the corresponding calculated frequencies should be too low. The MoC stretching frequencies are, however, in excellent agreement with experimental values (Table 4). This observation suggests some uncertainty of the experimental value of the MoC bond length. The vibrational frequencies of Mo(CO)6 calculated by quasirelativistic gradient corrected density functional theory are compared to experimental frequencies in Table 4. The experimental frequencies were measured by Jones et al.17 With the exception of the CO stretching frequencies all frequencies agree within a few wavenumbers with the experimental values. The CO stretching frequencies are systematically too low by about 50-60 cm-1, as we found for other metal carbonyls. This error reflects that the CO bond length is too long at the GGA (and GGA+QR) level of calculations. The CS- and CSe-substituted Mo carbonyls were not studied experimentally; therefore we have not calculated the vibrational frequencies of these systems. 3.4. Valence Force Constants. The shift in the CO stretching vibrational frequencies and force constants upon substitution of a ligand with a better σ donor or π acceptor is a textbook example for the explanation of the strength of π bonding between a transition metal and CO.23 One of the motivations for empirical normal coordinate analysis was to explain the electronic structure based on force constants and vibrational frequencies. Cotton and Kraihanzel suggested some rules for correlating bond strength with π bonding and how to approximate force constants.24 Early simplified MO calculations attempted to quantify the correlation between the empirically derived force constants and the occupancies of 5σ and 2π
TABLE 5: Stretching Force Constants of Cr(CO)6 and WCO)6 Cr(CO)6 expt FD FR FDDc FDDt FRRc FRRt FRD
1a
17.04 2.10 0.17 0.08 -0.20 0.47 0.69
W(CO)6
2b
theoryc
expt
17.24 2.08 0.21 0.02 -0.19 0.44 0.68
16.28 2.16 0.18 0.14 -0.01 0.35 0.63
17.02 2.32 0.19 0.11 0.05 0.61 0.84
expt
1a
expt 2b theoryd 17.22 2.36 0.22 0.00 0.05 0.56 0.79
17.26 2.48 0.16 0.11 0.05 0.76 0.74
CO str. MC str.
a Reference 17, gas. b Reference 17, solution. c GGA+QR calculations at optimized geometry. d LDA calculations at optimized geometry.
orbitals of CO.25 The Cotton-Kraihanzel interpretation of CO force constants has been criticized by others.26 These empirical and early theoretical calculations were important contributions to the understanding of the electronic and molecular structures of metal carbonyls. A detailed theoretical investigation of the π acceptor and donor effects based on density functional theory has been given by Ziegler and Rauk.27 Kraatz and co-workers studied the π acidity of thioethers and selenoethers in quasioctahedral (CO)5Cr-L complexes by density functional methods.28 Most recently Ehlers and co-workers used high-level ab initio methods to study the structure and bonding of (CO)5Cr-L compounds with a variety of ligands.29 These density functional and ab initio calculations took electron correlation into account and therefore gave a solid basis for quantitative study of π bonding. Since there are already many extensive studies on π bonding, we concentrate here on the quantitative differences between theoretical and empirical force constants and present electronic structure arguments to substantiate the calculated values only if necessary. The significance of sophisticated electronic structure calculations in the present study is that they can provide highly accurate values of potential energy constants of these systems subject to some small systematic error. Empirical normal coordinate analysis was always based on simplified models for polyatomic molecules, since the number of force constants is significantly larger that the number of vibrational frequencies and other information to be used in the determination of force constants. Some of the interaction constants had to be neglected in order to reduce the number of unknowns in the equations. It is very difficult to judge which interaction constants are important without elaborate quantum mechanical calculations. The simplifying approximations have introduced uncertainty especially about the values of interaction constants. Previous experience with elaborate quantum mechanical force fields shows that the sign and relative magnitude of interaction constants are predicted fairly accurately by these calculations.1e.f Table 5 includes the calculated and empirical stretching force constants for Cr(CO)6 and W(CO)6. The Cr(CO)6 force constants represent GGA+QR calculations of the present study. For W(CO)6, we listed the force constants at the LDA level, since these force constants reproduce the CO stretching frequencies better than other fields. The experimental values were taken from Jones et al.17 We listed the empirical and theoretical CO, CX, and CrC stretching force constants of Cr(CO)5CS and Cr(CO)5CSe in Table 6. The theoretical constants are those of the present GGA+QR calculations, while the empirical values are taken from ref 16. There are significant discrepancies between the CO stretching force constants for all Cr complexes (Table 5 and Table 6). The theoretical constants are 1 mdyn/Å lower than the experimental values. This discrepancy is the consequence of the too long CO bond obtained by the gradient corrected methods. At
Harmonic Vibrational Frequencies of M(CO)5CX
J. Phys. Chem., Vol. 100, No. 41, 1996 16543
TABLE 6: Stretching Force Constants of Cr(CO)5CS and Cr(CO)5CSea Cr(CO)5CS expt FD FDa FDx FR FRa FRx FDDc FDDt FDxDa FDxDe FRRc FRRt FRxRa FRD FRxDx FRtDt
b
17.28 17.42 7.68 2.04 2.03 2.45 0.18 0.04 0.13 0.13 -0.04 0.42
Cr(CO)5CSe
theoryc
exptb
theoryc
16.32 16.33 7.40 2.32 2.17 2.80 0.20 0.17 0.11 0.14 0.00 0.36 0.41 0.70 0.41 0.68
17.28 17.47 5.77 2.00 1.90 2.66 0.16 0.05 0.12
16.33 16.31 5.83 2.33 2.12 2.87 0.20 0.17 0.11 0.13 0.00 0.37 0.43 0.71 0.35 0.67
-0.06 0.41
CO equat. CO axial CX MC equat. MC axial MC(X)
a
Internal coordinates are defined in reference 16. b Reference 16. c GGA+QR calculations at optimized geometry.
the experimental reference geometry, the calculated CO stretching constant of Cr(CO)6 (by LDA calculations) is 17.10 mdyn/ Å, which is much closer to the experimental value and reproduces the experimental frequency quite well.1a The calculated CO stretching constant of W(CO)6 is in good agreement with the experimental values since we listed the value obtained by the LDA method in this case. The better agreement is due to the more accurate CO bond length at the LDA level, which demonstrates the importance of reference geometry at the level of force constants. The CS and CSe stretching constants (Table 6) are in fairly good agreement with the empirical values, while the theoretical CrC stretching constants are systematically higher than the experimental ones. The Cr-C constants corresponding to the substituted ligands (FRx) and of those in the axial (trans) position (FRa) are significantly different from those obtained by the calculations. The corresponding vibrational frequencies are well reproduced by the theoretical field, so the discrepancy is probably related to the assumptions made in the empirical normal coordinate analysis. The normal coordinate analysis was done using compliance constants rather than the more familiar force constants. Therefore, it is difficult to trace back what assumptions may be responsible for this discrepancy. The interaction force constants between equatorial CO stretchings of all systems show a different picture based on the empirical and theoretical fields. The corresponding cis and trans interaction constants, FDDc and FDDt, both have significant magnitudes in the theoretical field, the cis constant being somewhat larger. On the other hand, the empirical field suggests that the trans interaction constant is almost negligible or even zero on the basis of the gas phase force constants of W(CO)6. This is unexpected on the basis of the qualitative orbital picture. The stretching of the CO bond lowers the π-π* gap, resulting in increased dπ f pπ* donation. This effect is unfavorable for the π bonding of the other ligands and increases CO bond strength in both the cis and trans position. By simplified qualitative bonding arguments Cotton and Kraihanzel even expected the trans interaction constant to be about twice as large as the cis constant.23 Accordingly, the theoretical field better fulfils the expectations based on orbital arguments. For the hexacarbonyls two sets of force constants are available, and trans CO interaction constants are strikingly different in the two sets. The two sets were derived from CO frequencies in the gas phase and solution, which are not significantly different. Therefore, the high sensitivity of this constant to the input frequencies
suggests its value to be uncertain. This is probably true for the related complexes as well. There is also qualitative disagreement between theory and experiment for the cis CrC stretching interaction constant FRRc for the Cr complexes. The interaction constant between the two CrC stretches in the cis position is almost negligible on the basis of our study (FRRc ), while it is a significantly large negative value in the empirical field for all systems, especially for Cr(CO)6. The small calculated value is the result of cancelation of σ and π bonding contributions to this interaction constant. Only on the basis of qualitative arguments, it is difficult to assess which effect is more significant. This particular force constant is not sensitive to the input frequencies, since it is about the same in the two sets of empirical constants for both Cr(CO)6 and W(CO)6. 4. Conclusions In this study we have shown that quasi-relativistic gradientcorrected density functional methods provide accurate force fields and vibrational frequencies of transition metal complexes including compounds of heavy elements. On the example of W(CO)6 we have shown that accurate reference geometry is one of the most important requirements for accurate calculation of vibrational frequencies, even for molecules where relativistic corrections are important. The reference geometry was found to be more important than either the gradient correction or relativistic effects. In fact one of the best agreements between experimental and empirical frequencies of W(CO)6 was obtained at the simple local density functional level, which can be explained by the most accurate reference geometry obtained due to cancelation of neglected effects. The vibrational frequencies of pentacarbonyl-chalcocarbonyl-metal complexes were also in excellent agreement with experiment. The comparison between empirical and theoretical force constants has shown that the calculated interaction constants often disagree with empirically determined values. The theoretical values are more consistent with the expectation based on qualitative molecular orbital arguments. Acknowledgment. Financial support from the National Research Council (Canada) is gratefully acknowledged. This paper is issued as NRCC # 39117. References and Notes (1) (a) Be´rces, A.; Ziegler, T. J. Phys. Chem. 1995, 99, 11417. (b) Be´rces, A.; Ziegler, T. J. Phys. Chem. 1994, 98, 13233. (c) Be´rces, A.; Ziegler, T. Fan, L. J. Phys. Chem. 1994, 98, 1584. (d) Ziegler, T.; Cavallo, L. Be´rces, A. Organometallics 1993, 12, 3586. (e) Be´rces, A.; Ziegler, T. J. Chem. Phys. 1993, 98, 4793. (f) Be´rces, A.; Ziegler, T. J. Chem. Phys. Lett. 1993, 203, 592. (2) (a) Pyykko¨, P. Chem. ReV. 1988, 88, 563. (b) Pyykko¨, P. The Effect of RelatiVity on Atoms, Molecules, and the Solid State; Wilson, S., Ed.; Plenum Press: New York, 1991. (c) Ziegler, T.; Snijders, J. G.; Baerends, E. J. J. Chem. Phys. 1981, 74, 1271. (3) Schreckenbach, G.; Ziegler, T.; Li, J. Int. J. Quantum Chem., Quantum Chem. 1995, 56, 477. (4) (a) Li, J.; Schreckenbach, G.; Ziegler, T. J. Phys. Chem. 1994, 98, 4838.(b) Li, J.; Schreckenbach, G.; Ziegler, T. J. Am. Chem. Soc. 1995, 117, 486. (5) Allen, W. D.; Csa´sza´r, A. J. Chem. Phys. 1993, 98, 2983. (6) (a) Baerends, E. J.; Ellis, D. E.; Ros, P. Chem. Phys. 1973, 2, 41. (b) Ravenek, W. In Algorithms and Applications on Vector and Parallel Computers; te Riele, H. J. J., Dekker, Th. J., van de Vorst, H. A., Eds.; Elsevier: Amsterdam, 1987. (7) (a) Boerrigter, P. M.; te Velde, G.; Baerends, E. J. Int. J. Quantum Chem. 1988, 33, 87. (b) Velde, G. te; Baerends, E. J. J. Comput. Phys. 1992, 99, 84. (8) (a) Snijders, G. J.; Baerends, E. J.; Vernooijs, P. At. Nucl. Data Tables 1982, 26, 483. (b)Vernooijs, P.; Snijders, G. J.; Baerends, E. J. Slater Type Basis Functions for the Whole Periodic System. Internal Report; Free University of Amsterdam, The Netherlands, 1981.
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