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Vibrational behavior of tetrahedral d0 oxo-compounds: A theoretical study
Chemical Physics Letters 429 (2006) 52–57 www.elsevier.com/locate/cplett
Vibrational behavior of tetrahedral d0 oxo-compounds: A theoretical study Philippe Carbonniere
a,*
, Ilaria Ciofini a, Carlo Adamo b, Claude Pouchan
a,*
a
b
Laboratoire de Chimie The´orique et Physico-Chimie Mole´culaire, CNRS UMR 5624, Universite´ de Pau et des Pays de l’Adour 2, Rue Jules Ferry, F-64000 Pau, France Laboratoire d’Electrochimie et Chimie Analytique, CNRS UMR-7575, Ecole Nationale Supe´rieure de Chimie de Paris, 11 rue P. et M. Curie, F-75231 Paris CEDEX 05, France Received 24 May 2006; in final form 27 July 2006 Available online 5 August 2006
Abstract 2 2 The vibrational behavior of a series of tetrahedral d0 oxo-compounds of general formula MOn 4 , namely CrO4 ,MnO4 ,MoO4 , RuO4, WO2 , ReO and OsO , was investigated by the means of density functional theory. 4 4 4 Both harmonic and anharmonic spectra of such systems were computed, in the gas phase, using a variational treatment to solve the vibrational Schro¨dinger equation considering the rotationless Watson Hamiltonian and including up to anharmonic quartic force fields terms. The effect of basis functions, both for the oxygen and metal atoms, as well as the effect of the pseudo potential used in the description of the metal were discussed in the attempt to propose a easy to handle and cost effective computational approach to compute the vibrational properties of simple d0 compounds. Ó 2006 Elsevier B.V. All rights reserved.
1. Introduction Simple tetrahedral oxoanions of general formula MOn 4 , M being a metal of the 3d, 4d or 5d transition series, have been extensively studied in reason of the role they play in a wide range of domains [1–3]. Here, we briefly mention that for most of them catalytic (such for poly-oxometalates) [4], magnetic [5] and biological activities [6,7] were observed. The numerous fields of application and the consequently large amount of experimental data lead in turn to theoretical works. Initially, most of the theoretical works focused on d0 oxoanions and, in particular, on MnO 4 considered as prototype inorganic molecule in the development of quantum chemical techniques [3]. As a consequence it is not surprising that d0 MOn 4 compounds were used as benchmarks to *
Corresponding authors. E-mail addresses: [email protected] (P. Carbonniere), [email protected] (C. Pouchan). 0009-2614/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.08.010
test the performance of different computational methods for the description of electronic structure both at the ground and at the excited state. In this context, several studies, considering MnO 4 and OsO4 as test molecules for comparative investigations [8– 11], emphasised the necessity to go beyond the Hartree– Fock (HF) approach for the description of the ground and excited states, due to the important role played by electron correlation. Computationally demanding post-HF calculations were performed to analyze the ground and excited states properties of other molecules belonging to the MO4 series [11], up to the massive use of density functional theory models (DFT), much less demanding than their ab initio counterpart. Thus, geometry optimization, harmonic frequencies, bonding description and electronic excitation energies have been reported [1–3,8–13]. In contrast, other molecular properties, such as for instance the magnetic ones, were deserved of less attention, until recently [14]. One reason could be that a reliable modelling of such chemico-physical
P. Carbonniere et al. / Chemical Physics Letters 429 (2006) 52–57
properties, requiring to take into account several effects such as molecular environment, temperature or the zero point vibrational contributions, lead to the need of going beyond the purely electronic model to either vibrationally averaged or ab initio dynamic approaches. Recently, the nuclear magnetic shieldings, of MnO 4 have been computed in gas phase and in solution using an ab initio dynamics approach [14]. This approach is indeed, a viable, even if rather expensive, method to include vibrational and solvent effects, the first being very important in floppy molecules. Another, eventually cheaper, possibility is to describe the dynamic behavior in the gas phase by considering the normal modes of the molecule as uncoupled and solving independent one-dimensional rovibrational Schrodinger equation for each mode [15]. This allowed the computation of the temperature dependence of the property assuming a Boltzmann population of the vibrational levels in agreement with their dynamic counterpart. This latter approach has been applied several times (see for instance Ref. [16]) and it rests on accurate description not only of the electronic surface, but also of its curvature, that is of the corresponding vibrational levels. While nowadays a quite straightforward method is set up to obtain an accurate description of both harmonic and anharmonic effects for organic molecules [17,18], still few studies can be found dealing with the description of the vibrational behavior of compounds containing heavy atoms at this level of theory [19]. In the present Letter, the vibrational spectra of a larger number of oxo-compounds of transition metal, namely MOn 4 , M = Cr, Mn, Mo, Ru, W, Re and Os, will be reported and computed using a variational treatment of the vibrational Schro¨dinger equation and DFT quartic force fields. Here, the elements were chosen with regard to the available experimental works. As by-product, we attempt to propose a general computational approach for the electronic wavefunction of this kind of compounds, easy to handle and cost effective, able to match at best the experimental results. 2. Computational details Anharmonic quartic force fields of CrO2 4 , MnO4 , 2 2 MoO4 , RuO4, WO4 , ReO4 and OsO4 were determined with the GAUSSIAN03 package [20] by finite difference of 6N-11 analytic Hessians (N being the number of atoms) around the optimized geometry. Here, we recall that the best compromise between different error sources is ˚ in the numerical difobtained using a step size of 0.010 A ferentiation of harmonic frequencies, tight geometry optimizations and fine grids (at least 99 radial and 590 angular points) in both size consistent field (SCF) and Coupled-perturbed Kohn–Sham (CPKS) computations. According to vibrational studies investigating semi-rigid organic molecular systems [17], we used as reference the socalled B3LYP Hybrid GGA functional (HGGA) [21] to compute harmonic and anharmonic force constants in rea-
53
son of its ability to fit, with a modest basis set, the results obtained with accurate quantum mechanical approaches, such as the CCSD(T)/cc-pVTZ model [22]. Four other approaches, based on density functional theory (DFT), namely the PBE0 (HGGA), the BP86 and the OLYP GGA functionals (GGA), and the TPSS meta GGA functional (MGGA) are used for comparison [23–25]. Different basis sets were considered. The transitions metals have been described using two different effective core-potentials (ECP), and corresponding basis sets: the Stuttgart relativistic small core pseudopotential [26] (SDD basis set) and the Los Alamos ECP (LANL2 basis set) [27]. The large 631+G(3df) and the smaller 6-31+G(d) basis were used for oxygen. Finally, some calculations were carried out with the LANL2DZ basis, including the LANL2 basis set plus the D95 basis for oxygen [28]. The 6-31+G(3df) will be denoted as high basis (HB) and the 6-31+G(d) as low basis (LB), while the LANL2 and SDD basis will be indicated as 1 and 2, respectively. Thus, the LB2 will correspond to the SDD/6-31+G(d) combination. Previous works [17,29] demonstrated that the B3LYP/LB1 level of theory can be successfully applied to solve vibrational problems in the case of first row atomic systems. The Vibrational Schro¨dinger equation was solved by a variational procedure, considering the rotationless Watson Hamiltonian, written as: 1X 1X Hv ¼ xi ðp2i þ q2i Þ þ / qqq 2 i 6 ijk ijk i j k þ
1 X / q q q q þ H Cor 24 ijkl ijkl i j k l
where qi are the dimensionless normal coordinates, pi the conjugated momenta and xi, /ijk, /ijkl the harmonic wavenumbers, cubic and quartic force constants, respectively. HCor is the rotational contribution to anharmonicity (see Ref. [30] and therein). The vibrational Hamiltonian matrix, built-up using a basis set of harmonic oscillators product going up to hexaexcitations of vibrational configurations of interest (see Ref. [31] and therein) was, then, diagonalized. In such a way, the eigenvalues and the eigenvectors obtained correspond to the energetic values of each vibrational state and their description in term of normal mode. 3. Results and discussion Table 1 reports the calculated bond length (dM–O) for the presented d0 MOn 4 species, in their most stable conformation (Td symmetry), in comparison with the available experimental data. These latter were all obtained from crystal structure analysis (X-ray diffraction) of relatively ˚ ), except for the neutral species, poor resolution (0.01 A RuO4 and OsO4, whose geometrical parameters were determined from gas phase measurements [32] and, thus, are more directly comparable to our theoretical results. Generally, we notice an overall good agreement between all the computed values and the experimental data. This is not
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P. Carbonniere et al. / Chemical Physics Letters 429 (2006) 52–57
Table 1 ˚ ) for the MOn species under investigation Computed and experimental M–O distances (in A 4 B3LYP/LB1a CrO2 4 MnO 4 MoO2 4 RuO4 WO2 4 ReO 4 OsO4 MAE a b c d e
1.656 1.597 1.801 1.705 1.801 1.740 1.712 0.007
B3LYP/LB2a 1.653 1.596 1.801 1.699 1.815 1.754 1.720 0.013
PBE0/LB2a 1.649 1.582 1.781 1.689 1.803 1.741 1.706 0.011
B3LYPb b
1.670 1.612a 1.810b 1.735a 1.790b 1.759a 1.741a 0.023
B3LYPc
BP86e
Experimental
1.660 1.610 1.800 1.710 1.810 1.750 1.720
1.684 1.627 1.833 1.734 1.863 1.796 1.762
1.65c –1.66e 1.61–1.63c 1.75–1.78b 1.705e 1.78–1.79b 1.72–1.73c 1.713c
This work. LANL2DZ basis set, see Ref. [1] and therein. CEP pseudo-potential, see Ref. [2] and therein. See Ref [32]. Triple-f Slater type orbitals + polarisation, see Ref. [3].
surprising, since the ground electronic state of these d0 tetra-oxo systems has been already correctly described at DFT level of theory [3]. More in details, all the hybrid functionals, like B3LYP, provide smaller deviations ˚ ) than the pure generalized gradient (MAE 6 0.01 A approximation approaches (GGA), as BP86. Furthermore, the rather constant error along the series suggests a weak solid state effect. Considering the influence of the basis set, the LANL2 ECP performs slightly better than the corresponding SDD pseudo-potentials, when coupled to the 6-31+G(d) basis for the oxygen, as can be easily seen by comparing the B3LYP/LB1 and B3LYP/LB2 data of Table 1 (second and third column). At the same time, the 6-31+G(d) basis provides better results than the D95 basis used in the LANL2DZ set (fifth column of Table 1). On these grounds, the replacement of the LANL2DZ on the oxygen atoms by the 6-31+G(d, p) basis seems preferable for further investigations. The corresponding vibrationally averaged structures were also computed, leading to an increase only of
˚ for the bond length of all the species: more than 0.002 A ten times lower than the discrepancies previously observed and, thus, negligible. Table 2 reports the computed harmonic and anharmonic values of fundamental transitions of the series as well as the description of the vibrational states in term of normal modes. m1 (A1) and m3 (F2) correspond, respectively, to the symmetric and asymmetric stretching modes while m2 (E) and m4 (F2) refer to the bending modes. Here, we do not focus our attention on the accuracy of the computed vibrational frequencies since most of experimental spectra are recorded in aqueous solution where strong solvent effects are expected. Nevertheless, we are confident that a computational tool able to yield reliable bond length and frequencies matching their experimental counterpart by less than 5% in most cases is sufficient to give a fair description of vibrational levels in absence of strong anharmonic couplings like Fermi or Darling–Denisson resonances. Anharmonic effects on heavy atom oxides were firstly studied by Clavague´ra-Sarrio et al. [19] on UO2þ and 2 CUO. Their variational results, reported up to the 7th
Table 2 Harmonic (x) and anharmonic (m) frequencies (cm1) computed at the B3LYP/BL2 level of theory m2 (E)
m4 (F2)
CrO2 4
x m
m1 (A1) 869 859 (95%)
885 872 (99%)
336 334 (99%)
379 377 (99%)
MoO2 4
x m
868 859 (98%)
831 820 (97%)
297 295 (99%)
314 312 (99%)
WO2 4
x m
901 890 (96%)
822 812 (97%)
304 302 (99%)
302 300 (99%)
MnO 4
x m
939 928 (95%)
992 978 (97%)
367 365 (99%)
415 413 (99%)
ReO 4
x m
991 982 (95%)
944 932 (97%)
330 329 (99%)
325 323 (99%)
RuO4
x m
952 940 (96%)
991 977 (97%)
326 323 (99%)
339 337 (99%)
OsO4
x m
1024 1012 (97%)
1016 1002 (98%)
333 331 (99%)
329 327 (99%)
In parenthesis the composition of the anharmonic mode is reported.
m3 (F2)
P. Carbonniere et al. / Chemical Physics Letters 429 (2006) 52–57
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Table 3 1 Most relevant computed anharmonic force constants in MOn 4 species (in cm ) and their contribution to the anharmonic coupling (in percentage) /111 2m1/m1a /331 m1 + m3/m3b a b
CrO2 4
MoO2 4
WO2 4
MnO 4
ReO 4
RuO4
OsO4
122 2.7% 130 2%
113 1.9% 114 1.3%
115 1.8% 115 1.3%
132 2.8% 142 2.2%
124 1.7% 125 1.3%
132 2.7 % 134 1.1%
135 2% 137 1.2%
2m1 contribution in the m1 fundamental transition. m1 + m3 contribution in the m3 fundamental transition.
order combination bands, revealed a weak anharmonicity decreasing the harmonic wavenumbers by less than 5 cm1. The three eigenvectors of the first excited vibrational states represented more than 98% of the corresponding normal modes. The picture observed in Table 2 is quite similar: m2 and m4 bending modes are not affected by cubic and quartic force constants since they decrease by about 2 cm1 for all the species. As a consequence, they can be considered as pure harmonic frequencies and uncoupled normal mode. For m3 and m1 stretching modes we note an anharmonicity of 10–15 cm1, more marked for the m3 mode. Here, the anharmonic coupling is essentially due to the /111 and /331 cubic force constants of about 130 ± 25 cm1 in the series (see Table 3) yielding a slight 2m1 and m1 + m3 character in m1 and m3, respectively, of 1.5–2.5%. Note also that m3 and m1 modes are very close in energy to F2 and A1 sub-levels of m2 + 2m4, m4 + 2m2, 3m4 and 3m2 ( A1), depending on the nature of the metal. Nevertheless, no Darling–Dennisson couplings are observed since the A1 components of /1244, /2344, /2234, /1444, /3444, and /1222 type quartic derivatives are negligible (less than 2 cm1). Therefore, we can conclude that for these systems anharmonic effects are very small.
The aim of our analysis is also to select of the protocol more suitable for further investigation on these d0 oxo compounds. To this end, the mean signed errors (MSE) for the stretching and bending harmonic frequencies of oxo-compounds computed for different combinations of model theory/basis sets are reported in Figs. 1 and 2, respectively. All the transitions are given in Supplementary Materials (Table S1–S7). The MSE’s are evaluated using the experimental values (mexp) removing the anharmonicity contribution calculated at the B3LYP/BL1 level of theory in order to estimate their harmonic counterpart (xexp). In such a way, we assume that: (1) the anharmonic force field is basically the same for all the presented theoretical approaches and (2) that anharmonicity is additive in reason of the very weak anharmonic couplings observed. Furthermore, solvent effects on vibrational transitions, another feature which could be even more important than anharmonicity in the description of their IR and Raman Spectra of oxo-compounds, are not taken into account. In this respect, we can safely assume that the m2 and m4 bending modes are not strongly affected in aqueous solution, as observed for M(CO)6 species in several solvents [33], for which the bending MCO mode is
stretching modes 135 120 105 90 75 60 45 30 15 0 -15 -30 -45 -60 -75 -90 -105 -120 -135
B3LYP LB1 B3LYP LB2 B3LYP HB2 BP86 LB1 PBE0 LB2 OLYP LB2 TPSS LB2
Cr
Mn
Mo
Ru
W
Re
Os
LB : 6-31+G(d) ; HB : 6-31+G(3df) ; for oygen. 1 : LANL2DZ ; 2 : SDD ; for d0 metal. Fig. 1. Mean signed error for stretching mode (MSE, cm1) of the oxo-compounds analyzed obtained with selected functionals and basis sets.
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P. Carbonniere et al. / Chemical Physics Letters 429 (2006) 52–57 bending modes 45
30 B3LYP LB1 B3LYP LB2 B3LYP HB2 BP86 LB1
15
0
PBE0 LB2 OLYP LB2 TPSS LB2
-15
-30
-45 Cr
Mn
Mo
Ru
W
Re
Os
LB : 6-31+G(d) ; HB : 6-31+G(3df) ; for oygen. 1 : LANL2DZ ; 2 : SDD ; for d0 metal. Fig. 2. Mean signed error for the bending mode (MSE, cm1) the oxo-compounds analyzed obtained with selected functionals and basis sets.
shifted by 5 cm1 while the C@O stretching mode decreases by about 25 cm1. Indeed, the interaction of oxygen atoms with the solvent increases the reduced mass relative to the oxygen which explains, at least partially, the decrease of the corresponding frequency. This ‘mass effect’, as important as the strength of the intermolecular bond, is certainly greater for charged species like MOn systems. These 4 assumptions suggest that a fair DFT model in the gas phase must overestimate the derived xexp values of the stretching modes observed in liquid phase, at least for the totally symmetric m1 frequency, while the bending modes have to be correctly reproduced.
Starting from the comparison of the basis sets, the inclusion of 3 d and 1f functions on the 6-31+G(d, p) for oxygen poorly affects the B3LYP results: globally less than 5 cm1, particularly observed for the heavier transition metals. On the other hand, we want to notice that the computational time is increased by a factor 6 in going from the 631G(d) to the larger 6-31+G(3df) basis set, which induce to discard this last choice for larger systems. Concerning the choice of the pseudo potential for the metal atom, a direct comparison between the theoretical results stemmed from LANL2 and SDD pseudo-potentials indicates an important discrepancy for stretching m1 mode
50 45 40
LB1-LB2 (cm-1)
35 30 B3LYP
25
PBE0
20 15 10 5 0 Cr
Mn
Mo
Ru
W
Re
Os
element 1
Fig. 3. Differences (in cm ) along the
MOn 4
series between the results obtained at LB2 and LB1 levels, using the B3LYP and PBE0 functionals.
P. Carbonniere et al. / Chemical Physics Letters 429 (2006) 52–57
as reported in Fig. 3. While it is negligible for Cr, it slightly increases with the atomic number of the central metal to reach the value of 45 cm1 for Os. This progression suggests that the SDD pseudo-potential is more suitable for vibrational computation of this kind of systems, at least when 5d type metals are involved. Among the five functionals tested, the BP86 model which seemed very promising according to the results of Andrews et al. [32], tend to systematically underestimate the experimental data, namely those issuing from spectra recorded in aqueous solution (see Figs. 1 and 2). This is particularly true for bending modes, where unlike the stretching frequencies, no plagued results are found. Note that OLYP and TPSS functionals follow the same trend. B3LYP and PBE0 results are less contrasted. We invoke here the gas phase data i.e. the four fundamental transitions of OsO4, the m3 frequency of ReO 4 and RuO4, as well as the value of the transitions observed in liquid phase. Starting from OsO4, a mean difference of 6 cm1 is reported for B3LYP/BL2 computations with respect to experimental results, much better than their PBE0 counterpart (22 cm1). Conversely, they similarly fit the m2, m3 and m4 observed frequencies of ReO 4 . Besides, the picture is basically the same concerning the bending modes for 2 RuO4, MnO and WO2 (DxB3 xexp ¼ 18 cm1 , 4 , MoO4 4 DxPBE0 xexp ¼ 20 cm1 . As a consequence, B3LYP and PBE0 can both be used despite their non negligible difference above all on the m1 and m3 stretching mode for which the computed values suffer from a serious mistake. Although a solvent effect of some dozens of cm1 is expected, the discrepancies reported on M@O stretching mode can overtake a hundred cm1. This reminds the substantial difference observed on double bond stretching of organic compounds (40 cm1 for mC@O ; 30 cm1 for the ke´kule´ mode of benzene ; 70 cm1 for the aka-ke´kule´ mode of s-tetrazyne) [17] since a correct description of the curvature along these vibrations is essentially a two reference problem at large amplitude [34]. 4. Conclusion We have analyzed, the vibrational spectra of a series of d0 MOn 4 (M = Cr, Mn, Mo, Ru, W, Re, Os) compounds, using DFT calculations and an accurate variational procedure, including up to anharmonic quartic force fields terms, to solve the corresponding vibrational Schro¨dinger equation. In particular, we have calculated the harmonic and anharmonic spectra of such compounds and analyzed their dependence upon basis functions, pseudo potential and exchange–correlation functionals. Our study clearly shows that vibrational approaches, applied up to now mainly to organic systems, are also appropriate for transition-metal compounds, providing that a correct description of the curvature of the potential energy surface is used.
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Acknowledgement The authors thank the COST action D26/0013/02 for financial support. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.cplett.2006. 08.010. References [1] P. Gili, P. Martin-Zarza, P.A. Lorenzo-Luis, Inorg. Chim. Acta 357 (2004) 3397. [2] A.J. Bridgeman, G. Cavigliasso, Polyhedron 20 (2001) 2269. [3] A.C. Stu¨ckl, C.A. Daul, H.U. Gu¨del, J. Chem. Phys. 107 (1997) 4606. [4] M.T. Pope, Heteropoly and Isopoly Oxometalates, Springer, Heidelberg, 1983. [5] G. Liu, J.E. Greedan, W. Gong, J. Solid State Chem. 105 (1993) 78. [6] M. Cieslak-Golonka, Polyhedron 15 (1996) 3667. [7] J.J.R. Frausto da Silva, R.J.P. Williams, The Biological Chemistry of the Elements. The Inorganic Chemistry of Life, Clarendon Press, Oxford, 1991, pp. 423, 432, and 538. [8] M.A. Buijse, E.J. Baerends, J. Chem. Phys. 93 (1990) 4129. [9] H. Johansen, S. Rettrup, Chem. Phys. 74 (1983) 77. [10] G. Ujaque, F. Maseras, A. Lledos, Int. J. Quantum Chem. 77 (2000) 544. [11] S. Jitsuhiro, H. Nakai, M. Hada, H. Nakatsuji, J. Chem. Phys. 101 (1994) 1029. [12] C.J.M. Coremans, J.H. Van der Waals, J. Konijnenberg, A.H. Huizer, C.A.G.O. Varma, Chem. Phys. Lett. 125 (1986) 514. [13] R.M. Dicksson, T. Ziegler, Int. J. Quantum Chem. 58 (1996) 681. [14] M. Bu¨hl, J. Phys. Chem. A 106 (2002) 10505. [15] I. Ciofini, C. Adamo, J. Mol. Struct. (Theochem) 762 (2006) 133. [16] V. Barone, C. Adamo, Y. Brunel et, R. Subra, J. Chem. Phys. 105 (1996) 3168. [17] P. Carbonniere, T. Lucca, C. Pouchan, N. Rega, V. Barone, J. Comp. Chem. 26 (2005) 384. [18] P. Carbonniere, V. Barone, Chem. Phys. Lett. 399 (2004) 226. [19] C. Clavague´ra-Sarrio, N. Ismail, C.J. Marsden, D. Begue, C. Pouchan, Chem. Phys. 302 (2004) 1. [20] M.J. Frisch et al., GAUSSIAN03, Revision C.02, Gaussian Inc., Wallingford, CT, 2004. [21] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [22] K. Raghavachri, G.W. Trucks, J.A. Pople, M. Head-Gordon, Chem. Phys. Lett. 157 (1989) 479. [23] A.D. Becke, Phys. Rev. A 38 (1988) 3098. [24] J.P. Perdew, Phys. Rev. B 33 (1986) 8822. [25] C. Adamo, V. Barone, J. Chem. Phys. 110 (1999) 6158. [26] M. Dolg, H. Stoll, H. Preuss, R.M. Pitzer, J. Phys. Chem. 97 (1993) 5852. [27] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 270. [28] T.H. Dunning Jr., P.J. Hay, in: H.F. Schaefer (Ed.), Modern Theoretical Chemistry, vol. 3, Plenum Press, New York, 1976, p. 1. [29] P. Carbonniere, H. Hagemann, J. Phys. Chem. A 110 (2006) 9927. [30] P. Carbonniere, V. Barone, Chem. Phys. Lett. 392 (2004) 365. [31] D. Begue, P. Carbonniere, C. Pouchan, J. Phys. Chem. A 109 (2005) 4611. [32] M. Zhou, A. Citra, B. Liang, L. Andrews, J. Phys. Chem. A 104 (2000) 3457. [33] J.H. Robin Clark, B. Crociani, Inorg. Chim. Acta 1 (1967) 12. [34] J.M. L Martin, P.R. Taylor, T.J. Lee, Chem. Phys. Lett. 275 (1997) 414.
Vibrational behavior of tetrahedral d0 oxo-compounds: A theoretical study Philippe Carbonniere
a,*
, Ilaria Ciofini a, Carlo Adamo b, Claude Pouchan
a,*
a
b
Laboratoire de Chimie The´orique et Physico-Chimie Mole´culaire, CNRS UMR 5624, Universite´ de Pau et des Pays de l’Adour 2, Rue Jules Ferry, F-64000 Pau, France Laboratoire d’Electrochimie et Chimie Analytique, CNRS UMR-7575, Ecole Nationale Supe´rieure de Chimie de Paris, 11 rue P. et M. Curie, F-75231 Paris CEDEX 05, France Received 24 May 2006; in final form 27 July 2006 Available online 5 August 2006
Abstract 2 2 The vibrational behavior of a series of tetrahedral d0 oxo-compounds of general formula MOn 4 , namely CrO4 ,MnO4 ,MoO4 , RuO4, WO2 , ReO and OsO , was investigated by the means of density functional theory. 4 4 4 Both harmonic and anharmonic spectra of such systems were computed, in the gas phase, using a variational treatment to solve the vibrational Schro¨dinger equation considering the rotationless Watson Hamiltonian and including up to anharmonic quartic force fields terms. The effect of basis functions, both for the oxygen and metal atoms, as well as the effect of the pseudo potential used in the description of the metal were discussed in the attempt to propose a easy to handle and cost effective computational approach to compute the vibrational properties of simple d0 compounds. Ó 2006 Elsevier B.V. All rights reserved.
1. Introduction Simple tetrahedral oxoanions of general formula MOn 4 , M being a metal of the 3d, 4d or 5d transition series, have been extensively studied in reason of the role they play in a wide range of domains [1–3]. Here, we briefly mention that for most of them catalytic (such for poly-oxometalates) [4], magnetic [5] and biological activities [6,7] were observed. The numerous fields of application and the consequently large amount of experimental data lead in turn to theoretical works. Initially, most of the theoretical works focused on d0 oxoanions and, in particular, on MnO 4 considered as prototype inorganic molecule in the development of quantum chemical techniques [3]. As a consequence it is not surprising that d0 MOn 4 compounds were used as benchmarks to *
Corresponding authors. E-mail addresses: [email protected] (P. Carbonniere), [email protected] (C. Pouchan). 0009-2614/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.08.010
test the performance of different computational methods for the description of electronic structure both at the ground and at the excited state. In this context, several studies, considering MnO 4 and OsO4 as test molecules for comparative investigations [8– 11], emphasised the necessity to go beyond the Hartree– Fock (HF) approach for the description of the ground and excited states, due to the important role played by electron correlation. Computationally demanding post-HF calculations were performed to analyze the ground and excited states properties of other molecules belonging to the MO4 series [11], up to the massive use of density functional theory models (DFT), much less demanding than their ab initio counterpart. Thus, geometry optimization, harmonic frequencies, bonding description and electronic excitation energies have been reported [1–3,8–13]. In contrast, other molecular properties, such as for instance the magnetic ones, were deserved of less attention, until recently [14]. One reason could be that a reliable modelling of such chemico-physical
P. Carbonniere et al. / Chemical Physics Letters 429 (2006) 52–57
properties, requiring to take into account several effects such as molecular environment, temperature or the zero point vibrational contributions, lead to the need of going beyond the purely electronic model to either vibrationally averaged or ab initio dynamic approaches. Recently, the nuclear magnetic shieldings, of MnO 4 have been computed in gas phase and in solution using an ab initio dynamics approach [14]. This approach is indeed, a viable, even if rather expensive, method to include vibrational and solvent effects, the first being very important in floppy molecules. Another, eventually cheaper, possibility is to describe the dynamic behavior in the gas phase by considering the normal modes of the molecule as uncoupled and solving independent one-dimensional rovibrational Schrodinger equation for each mode [15]. This allowed the computation of the temperature dependence of the property assuming a Boltzmann population of the vibrational levels in agreement with their dynamic counterpart. This latter approach has been applied several times (see for instance Ref. [16]) and it rests on accurate description not only of the electronic surface, but also of its curvature, that is of the corresponding vibrational levels. While nowadays a quite straightforward method is set up to obtain an accurate description of both harmonic and anharmonic effects for organic molecules [17,18], still few studies can be found dealing with the description of the vibrational behavior of compounds containing heavy atoms at this level of theory [19]. In the present Letter, the vibrational spectra of a larger number of oxo-compounds of transition metal, namely MOn 4 , M = Cr, Mn, Mo, Ru, W, Re and Os, will be reported and computed using a variational treatment of the vibrational Schro¨dinger equation and DFT quartic force fields. Here, the elements were chosen with regard to the available experimental works. As by-product, we attempt to propose a general computational approach for the electronic wavefunction of this kind of compounds, easy to handle and cost effective, able to match at best the experimental results. 2. Computational details Anharmonic quartic force fields of CrO2 4 , MnO4 , 2 2 MoO4 , RuO4, WO4 , ReO4 and OsO4 were determined with the GAUSSIAN03 package [20] by finite difference of 6N-11 analytic Hessians (N being the number of atoms) around the optimized geometry. Here, we recall that the best compromise between different error sources is ˚ in the numerical difobtained using a step size of 0.010 A ferentiation of harmonic frequencies, tight geometry optimizations and fine grids (at least 99 radial and 590 angular points) in both size consistent field (SCF) and Coupled-perturbed Kohn–Sham (CPKS) computations. According to vibrational studies investigating semi-rigid organic molecular systems [17], we used as reference the socalled B3LYP Hybrid GGA functional (HGGA) [21] to compute harmonic and anharmonic force constants in rea-
53
son of its ability to fit, with a modest basis set, the results obtained with accurate quantum mechanical approaches, such as the CCSD(T)/cc-pVTZ model [22]. Four other approaches, based on density functional theory (DFT), namely the PBE0 (HGGA), the BP86 and the OLYP GGA functionals (GGA), and the TPSS meta GGA functional (MGGA) are used for comparison [23–25]. Different basis sets were considered. The transitions metals have been described using two different effective core-potentials (ECP), and corresponding basis sets: the Stuttgart relativistic small core pseudopotential [26] (SDD basis set) and the Los Alamos ECP (LANL2 basis set) [27]. The large 631+G(3df) and the smaller 6-31+G(d) basis were used for oxygen. Finally, some calculations were carried out with the LANL2DZ basis, including the LANL2 basis set plus the D95 basis for oxygen [28]. The 6-31+G(3df) will be denoted as high basis (HB) and the 6-31+G(d) as low basis (LB), while the LANL2 and SDD basis will be indicated as 1 and 2, respectively. Thus, the LB2 will correspond to the SDD/6-31+G(d) combination. Previous works [17,29] demonstrated that the B3LYP/LB1 level of theory can be successfully applied to solve vibrational problems in the case of first row atomic systems. The Vibrational Schro¨dinger equation was solved by a variational procedure, considering the rotationless Watson Hamiltonian, written as: 1X 1X Hv ¼ xi ðp2i þ q2i Þ þ / qqq 2 i 6 ijk ijk i j k þ
1 X / q q q q þ H Cor 24 ijkl ijkl i j k l
where qi are the dimensionless normal coordinates, pi the conjugated momenta and xi, /ijk, /ijkl the harmonic wavenumbers, cubic and quartic force constants, respectively. HCor is the rotational contribution to anharmonicity (see Ref. [30] and therein). The vibrational Hamiltonian matrix, built-up using a basis set of harmonic oscillators product going up to hexaexcitations of vibrational configurations of interest (see Ref. [31] and therein) was, then, diagonalized. In such a way, the eigenvalues and the eigenvectors obtained correspond to the energetic values of each vibrational state and their description in term of normal mode. 3. Results and discussion Table 1 reports the calculated bond length (dM–O) for the presented d0 MOn 4 species, in their most stable conformation (Td symmetry), in comparison with the available experimental data. These latter were all obtained from crystal structure analysis (X-ray diffraction) of relatively ˚ ), except for the neutral species, poor resolution (0.01 A RuO4 and OsO4, whose geometrical parameters were determined from gas phase measurements [32] and, thus, are more directly comparable to our theoretical results. Generally, we notice an overall good agreement between all the computed values and the experimental data. This is not
54
P. Carbonniere et al. / Chemical Physics Letters 429 (2006) 52–57
Table 1 ˚ ) for the MOn species under investigation Computed and experimental M–O distances (in A 4 B3LYP/LB1a CrO2 4 MnO 4 MoO2 4 RuO4 WO2 4 ReO 4 OsO4 MAE a b c d e
1.656 1.597 1.801 1.705 1.801 1.740 1.712 0.007
B3LYP/LB2a 1.653 1.596 1.801 1.699 1.815 1.754 1.720 0.013
PBE0/LB2a 1.649 1.582 1.781 1.689 1.803 1.741 1.706 0.011
B3LYPb b
1.670 1.612a 1.810b 1.735a 1.790b 1.759a 1.741a 0.023
B3LYPc
BP86e
Experimental
1.660 1.610 1.800 1.710 1.810 1.750 1.720
1.684 1.627 1.833 1.734 1.863 1.796 1.762
1.65c –1.66e 1.61–1.63c 1.75–1.78b 1.705e 1.78–1.79b 1.72–1.73c 1.713c
This work. LANL2DZ basis set, see Ref. [1] and therein. CEP pseudo-potential, see Ref. [2] and therein. See Ref [32]. Triple-f Slater type orbitals + polarisation, see Ref. [3].
surprising, since the ground electronic state of these d0 tetra-oxo systems has been already correctly described at DFT level of theory [3]. More in details, all the hybrid functionals, like B3LYP, provide smaller deviations ˚ ) than the pure generalized gradient (MAE 6 0.01 A approximation approaches (GGA), as BP86. Furthermore, the rather constant error along the series suggests a weak solid state effect. Considering the influence of the basis set, the LANL2 ECP performs slightly better than the corresponding SDD pseudo-potentials, when coupled to the 6-31+G(d) basis for the oxygen, as can be easily seen by comparing the B3LYP/LB1 and B3LYP/LB2 data of Table 1 (second and third column). At the same time, the 6-31+G(d) basis provides better results than the D95 basis used in the LANL2DZ set (fifth column of Table 1). On these grounds, the replacement of the LANL2DZ on the oxygen atoms by the 6-31+G(d, p) basis seems preferable for further investigations. The corresponding vibrationally averaged structures were also computed, leading to an increase only of
˚ for the bond length of all the species: more than 0.002 A ten times lower than the discrepancies previously observed and, thus, negligible. Table 2 reports the computed harmonic and anharmonic values of fundamental transitions of the series as well as the description of the vibrational states in term of normal modes. m1 (A1) and m3 (F2) correspond, respectively, to the symmetric and asymmetric stretching modes while m2 (E) and m4 (F2) refer to the bending modes. Here, we do not focus our attention on the accuracy of the computed vibrational frequencies since most of experimental spectra are recorded in aqueous solution where strong solvent effects are expected. Nevertheless, we are confident that a computational tool able to yield reliable bond length and frequencies matching their experimental counterpart by less than 5% in most cases is sufficient to give a fair description of vibrational levels in absence of strong anharmonic couplings like Fermi or Darling–Denisson resonances. Anharmonic effects on heavy atom oxides were firstly studied by Clavague´ra-Sarrio et al. [19] on UO2þ and 2 CUO. Their variational results, reported up to the 7th
Table 2 Harmonic (x) and anharmonic (m) frequencies (cm1) computed at the B3LYP/BL2 level of theory m2 (E)
m4 (F2)
CrO2 4
x m
m1 (A1) 869 859 (95%)
885 872 (99%)
336 334 (99%)
379 377 (99%)
MoO2 4
x m
868 859 (98%)
831 820 (97%)
297 295 (99%)
314 312 (99%)
WO2 4
x m
901 890 (96%)
822 812 (97%)
304 302 (99%)
302 300 (99%)
MnO 4
x m
939 928 (95%)
992 978 (97%)
367 365 (99%)
415 413 (99%)
ReO 4
x m
991 982 (95%)
944 932 (97%)
330 329 (99%)
325 323 (99%)
RuO4
x m
952 940 (96%)
991 977 (97%)
326 323 (99%)
339 337 (99%)
OsO4
x m
1024 1012 (97%)
1016 1002 (98%)
333 331 (99%)
329 327 (99%)
In parenthesis the composition of the anharmonic mode is reported.
m3 (F2)
P. Carbonniere et al. / Chemical Physics Letters 429 (2006) 52–57
55
Table 3 1 Most relevant computed anharmonic force constants in MOn 4 species (in cm ) and their contribution to the anharmonic coupling (in percentage) /111 2m1/m1a /331 m1 + m3/m3b a b
CrO2 4
MoO2 4
WO2 4
MnO 4
ReO 4
RuO4
OsO4
122 2.7% 130 2%
113 1.9% 114 1.3%
115 1.8% 115 1.3%
132 2.8% 142 2.2%
124 1.7% 125 1.3%
132 2.7 % 134 1.1%
135 2% 137 1.2%
2m1 contribution in the m1 fundamental transition. m1 + m3 contribution in the m3 fundamental transition.
order combination bands, revealed a weak anharmonicity decreasing the harmonic wavenumbers by less than 5 cm1. The three eigenvectors of the first excited vibrational states represented more than 98% of the corresponding normal modes. The picture observed in Table 2 is quite similar: m2 and m4 bending modes are not affected by cubic and quartic force constants since they decrease by about 2 cm1 for all the species. As a consequence, they can be considered as pure harmonic frequencies and uncoupled normal mode. For m3 and m1 stretching modes we note an anharmonicity of 10–15 cm1, more marked for the m3 mode. Here, the anharmonic coupling is essentially due to the /111 and /331 cubic force constants of about 130 ± 25 cm1 in the series (see Table 3) yielding a slight 2m1 and m1 + m3 character in m1 and m3, respectively, of 1.5–2.5%. Note also that m3 and m1 modes are very close in energy to F2 and A1 sub-levels of m2 + 2m4, m4 + 2m2, 3m4 and 3m2 ( A1), depending on the nature of the metal. Nevertheless, no Darling–Dennisson couplings are observed since the A1 components of /1244, /2344, /2234, /1444, /3444, and /1222 type quartic derivatives are negligible (less than 2 cm1). Therefore, we can conclude that for these systems anharmonic effects are very small.
The aim of our analysis is also to select of the protocol more suitable for further investigation on these d0 oxo compounds. To this end, the mean signed errors (MSE) for the stretching and bending harmonic frequencies of oxo-compounds computed for different combinations of model theory/basis sets are reported in Figs. 1 and 2, respectively. All the transitions are given in Supplementary Materials (Table S1–S7). The MSE’s are evaluated using the experimental values (mexp) removing the anharmonicity contribution calculated at the B3LYP/BL1 level of theory in order to estimate their harmonic counterpart (xexp). In such a way, we assume that: (1) the anharmonic force field is basically the same for all the presented theoretical approaches and (2) that anharmonicity is additive in reason of the very weak anharmonic couplings observed. Furthermore, solvent effects on vibrational transitions, another feature which could be even more important than anharmonicity in the description of their IR and Raman Spectra of oxo-compounds, are not taken into account. In this respect, we can safely assume that the m2 and m4 bending modes are not strongly affected in aqueous solution, as observed for M(CO)6 species in several solvents [33], for which the bending MCO mode is
stretching modes 135 120 105 90 75 60 45 30 15 0 -15 -30 -45 -60 -75 -90 -105 -120 -135
B3LYP LB1 B3LYP LB2 B3LYP HB2 BP86 LB1 PBE0 LB2 OLYP LB2 TPSS LB2
Cr
Mn
Mo
Ru
W
Re
Os
LB : 6-31+G(d) ; HB : 6-31+G(3df) ; for oygen. 1 : LANL2DZ ; 2 : SDD ; for d0 metal. Fig. 1. Mean signed error for stretching mode (MSE, cm1) of the oxo-compounds analyzed obtained with selected functionals and basis sets.
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P. Carbonniere et al. / Chemical Physics Letters 429 (2006) 52–57 bending modes 45
30 B3LYP LB1 B3LYP LB2 B3LYP HB2 BP86 LB1
15
0
PBE0 LB2 OLYP LB2 TPSS LB2
-15
-30
-45 Cr
Mn
Mo
Ru
W
Re
Os
LB : 6-31+G(d) ; HB : 6-31+G(3df) ; for oygen. 1 : LANL2DZ ; 2 : SDD ; for d0 metal. Fig. 2. Mean signed error for the bending mode (MSE, cm1) the oxo-compounds analyzed obtained with selected functionals and basis sets.
shifted by 5 cm1 while the C@O stretching mode decreases by about 25 cm1. Indeed, the interaction of oxygen atoms with the solvent increases the reduced mass relative to the oxygen which explains, at least partially, the decrease of the corresponding frequency. This ‘mass effect’, as important as the strength of the intermolecular bond, is certainly greater for charged species like MOn systems. These 4 assumptions suggest that a fair DFT model in the gas phase must overestimate the derived xexp values of the stretching modes observed in liquid phase, at least for the totally symmetric m1 frequency, while the bending modes have to be correctly reproduced.
Starting from the comparison of the basis sets, the inclusion of 3 d and 1f functions on the 6-31+G(d, p) for oxygen poorly affects the B3LYP results: globally less than 5 cm1, particularly observed for the heavier transition metals. On the other hand, we want to notice that the computational time is increased by a factor 6 in going from the 631G(d) to the larger 6-31+G(3df) basis set, which induce to discard this last choice for larger systems. Concerning the choice of the pseudo potential for the metal atom, a direct comparison between the theoretical results stemmed from LANL2 and SDD pseudo-potentials indicates an important discrepancy for stretching m1 mode
50 45 40
LB1-LB2 (cm-1)
35 30 B3LYP
25
PBE0
20 15 10 5 0 Cr
Mn
Mo
Ru
W
Re
Os
element 1
Fig. 3. Differences (in cm ) along the
MOn 4
series between the results obtained at LB2 and LB1 levels, using the B3LYP and PBE0 functionals.
P. Carbonniere et al. / Chemical Physics Letters 429 (2006) 52–57
as reported in Fig. 3. While it is negligible for Cr, it slightly increases with the atomic number of the central metal to reach the value of 45 cm1 for Os. This progression suggests that the SDD pseudo-potential is more suitable for vibrational computation of this kind of systems, at least when 5d type metals are involved. Among the five functionals tested, the BP86 model which seemed very promising according to the results of Andrews et al. [32], tend to systematically underestimate the experimental data, namely those issuing from spectra recorded in aqueous solution (see Figs. 1 and 2). This is particularly true for bending modes, where unlike the stretching frequencies, no plagued results are found. Note that OLYP and TPSS functionals follow the same trend. B3LYP and PBE0 results are less contrasted. We invoke here the gas phase data i.e. the four fundamental transitions of OsO4, the m3 frequency of ReO 4 and RuO4, as well as the value of the transitions observed in liquid phase. Starting from OsO4, a mean difference of 6 cm1 is reported for B3LYP/BL2 computations with respect to experimental results, much better than their PBE0 counterpart (22 cm1). Conversely, they similarly fit the m2, m3 and m4 observed frequencies of ReO 4 . Besides, the picture is basically the same concerning the bending modes for 2 RuO4, MnO and WO2 (DxB3 xexp ¼ 18 cm1 , 4 , MoO4 4 DxPBE0 xexp ¼ 20 cm1 . As a consequence, B3LYP and PBE0 can both be used despite their non negligible difference above all on the m1 and m3 stretching mode for which the computed values suffer from a serious mistake. Although a solvent effect of some dozens of cm1 is expected, the discrepancies reported on M@O stretching mode can overtake a hundred cm1. This reminds the substantial difference observed on double bond stretching of organic compounds (40 cm1 for mC@O ; 30 cm1 for the ke´kule´ mode of benzene ; 70 cm1 for the aka-ke´kule´ mode of s-tetrazyne) [17] since a correct description of the curvature along these vibrations is essentially a two reference problem at large amplitude [34]. 4. Conclusion We have analyzed, the vibrational spectra of a series of d0 MOn 4 (M = Cr, Mn, Mo, Ru, W, Re, Os) compounds, using DFT calculations and an accurate variational procedure, including up to anharmonic quartic force fields terms, to solve the corresponding vibrational Schro¨dinger equation. In particular, we have calculated the harmonic and anharmonic spectra of such compounds and analyzed their dependence upon basis functions, pseudo potential and exchange–correlation functionals. Our study clearly shows that vibrational approaches, applied up to now mainly to organic systems, are also appropriate for transition-metal compounds, providing that a correct description of the curvature of the potential energy surface is used.
57
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