Analysis of Partition Functions for Metallocenes: Ferrocene

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Analysis of Partition Functions for Metallocenes: Ferrocene, Ruthenocene and Osmocene Thiago Ferreira da Cunha, Danilo Calderini, and Dimitrios Skouteris J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b01280 • Publication Date (Web): 22 Mar 2016 Downloaded from http://pubs.acs.org on March 28, 2016

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The Journal of Physical Chemistry

Analysis of Partition Functions for Metallocenes: Ferrocene, Ruthenocene and Osmocene T. Ferreira da Cunha,† D. Calderini,†,‡ and D. Skouteris∗,† Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, Italy, and Istituto Nazionale di Fisica Nucleare (INFN) sezione di Pisa, Largo Bruno Pontecorvo, 3, 56127, Pisa, Italy E-mail: [email protected]

Abstract We present a calculation of the torsional potential of the three metallocenes of the iron group, i.e. ferrocene, ruthenocene and osmocene calculated with the GAUSSIAN program suite. Both a variational method (through computation of the exact energy levels) and our Chebyshev imaginary time propagation method are used to calculate the hindered rotation partition function, demonstrating the efficiency of the Chebyshev scheme. The transition from a semirigid through a hindered rotor to the free rotor regime is demonstrated and the effect of the hindered rotation (as opposed to a harmonic) treatment on the thermodynamics of metallocenes is demonstrated.

INTRODUCTION Metallocenes are a prominent class of organometallic compounds having several applications as catalysts 6,8 and biosensors, 9 besides interesting nonlinear optical and antitumoral properties. 10 ∗

To whom correspondence should be addressed Scuola Normale Superiore ‡ INFN †

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ACS Paragon Plus Environment


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Metallocenes are formed by a pair of cyclopentadienyl (Cp) anions (C5 H5 ) and a transition metal atom in oxidation state II, wherein the metal atom stays between the two Cp units, forming a sandwich-like structure. The earliest studies about metallocenes date back to the discovery of ferrocene. 1,2 A large class of other metallocenes were analyzed by various spectroscopic techniques, obtaining accurate information about their crystallographic structures, molecular geometries and potential energy surfaces. Such studies include X-ray diffraction experiments, 7 as well as vibrational spectroscopy, both Raman and infrared. 3–5 The ferrocene, ruthenocene and osmocene molecules have been recently analyzed from a computational point of view 12 in order to find the best combination of DFT functional and basis set that can describe their infrared spectra in the VPT2 scheme, including combination and overtone bands. The purpose of this paper is to continue this study, investigating the hindered rotational motion between the two Cp anions. Because of the relatively low potential barrier governing this rotation, the motion can be defined as a Large Amplitude Motion (LAM) rather than a (harmonic or anharmonic) oscillation. Obtaining an accurate potential energy curve for this rotation enables us to treat accurately the lowest excited states of the nuclear wavefunction of these compounds, obtaining partition functions and thermodynamic properties which can be of use in further kinetics and thermodynamics studies. Towards this end, we use the Generalized Coordinate (GC) method to find the energy levels pertaining to the hindered rotation and subsequently use a time dependent method, based on the expansion of the Boltzmann operator in a Chebyshev polynomial to compare the results. 13 Using our results, we investigate the effect of treating the hindered rotation as such in the thermodynamics of metallocenes, as opposed to a common harmonic treatment. To this effect, we compare partition functions using a completely harmonic model to the ones obtained with a proper treatment of the hindered rotation. We also illustrate the effect on thermodynamic functions, such as internal energy and entropy. In Section 2 of this paper we present the computational details of our calculation and in Section 3 we present our results and discussion. Section 4 concludes.

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COMPUTATIONAL DETAILS Our electronic calculations were performed via Density Functional Theory (DFT) using a development version of the computational quantum chemical package GAUSSIAN 30 that performs the GC analysis. This analysis consists of a relaxed scan along the dihedral angle describing the hindered rotation and a subsequent matrix representation of the relevant Hamiltonian in a DVR basis set. The main idea of this approach is a numerical solution of the Reaction Path Hamiltonian 20 based on the work of Fukui. 21 We point out that, in the present approach, the LAM is separated from the set of all other vibrational and rotational modes. More details about the theoretical background of the method are given in our previous work. 13 It has already been applied on monodimensional hindered rotations and constitutes an alternative to the application of torsional correction factors to the classical partition functions as has been done in many cases . 22–28 In all three cases studied here, the initial geometry of the scan is in the eclipsed conformation (point group D5h ), which corresponds to the global energy minimum in the gas phase, according both to our results and to other computational and experimental work. 11 Following the suggestion of Latouche and coworkers, 12 we have performed our calculations using the B3PW91 functional 14–16 and the LANL2DZ basis set, 17–19 including polarization functions on all atoms (H (p; 0.800), C (d; 0.587), Fe (p; 0.135), Ru (f; 1.235), Os (f; 0.886)). These parameters have previously produced satisfactory agreement with experimental spectra for all three metallocenes considered. In all cases, one Cp unit was held fixed and the other one was rotated during the scan. The torsional dihedral angle itself, varying within the interval [0, 360] degrees, has been scanned in a 72-step sequence of intervals of 5.0 degrees each. We subsequently build a Discrete Variable Representation (DVR) Hamiltonian 29 and find the energy spectrum by diagonalization. In this variational approach, after the relaxed scan, we fit the potential using cubic splines, so the number of DVR functions can be increased to reach the desired convergence in the calculation. The formal reduced mass in this approach is always equal to one, however, the 3



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information about the effective reduced mass is stored in the discrepancy between the GC curve and the dihedral angle curve. 13 This approach reproduces accurately both the quasidegeneracy of the first five states and the twofold degeneracy in the free rotor limit. The partition function has been calculated by direct summation over the eigenfunctions obtained from the diagonalization of the DVR Hamiltonian matrix.

Q(T ) =

∞ X

(1)

e−ǫi /kB T

i=1

where kB is Boltzmann’s constant, T is the temperature and ǫi is an eigenvalue of the Hamiltonian. In parallel, we use the one-dimensional PES obtained from the relaxed scan and expand it as a Fourier series; we compute the Fourier series of the monodimensional PES up to an order high enough to faithfully represent the potential

V (φ) = V0 +

X

ai sin(iφ) +

X

bi cos(iφ)

(2)

i

i

We subsequently perform an imaginary time propagation of a Monte Carlo sampling of random phase wavepackets. The effect of the complex time propagation is to "thermalize" the wavefunction to a specific temperature. In this way we obtain the partition function in an alternative manner as an average of the trace of the Boltzmann operator n

Q(T ) =

1X hψi | e−H/KB T | ψi i n i=1

(3)

where n is the number of initial wavepackets (for more details about the theoretical framework, see 13 ). Having obtained the relevant partition functions, we use the well-known thermodynamic formulae

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U = kB T

2

∂ ln Q ∂T

(4) V

∂ ln Q + ln Q S = kB T ∂T V

(5)

to calculate the internal energy and entropy respectively and assess the effect of the hindered rotation on each.

RESULTS AND DISCUSSION Torsional potentials The computational cost of a DFT relaxed scan is rather demanding for the treatment of LAMs as the accuracy needed in kinetics or dynamic calculations forces us to carefully manage the electronic contribution and the deformation of the molecular structure along the torsion. Interestingly, as shown in Table 1, the potential maximum energies Vmax (Fig.1) (taking 0 as the potential minimum) do not obviously correlate with the distance between ¯ metal and the carbon d(M, C) in the maximum geometry; also the harmonic frequencies, determined by the curvature of the potential, are not directly related to the global behavior of the PES. This confirms the necessity of quantum chemistry calculations to obtain accurate PES for internal rotation. Table 1: Maxima of the torsional potential (in cm−1 ), metal-carbon distance (in Å) and harmonic frequencies (in cm−1 )

Vmax ¯ d(M, C) ωharm

METALLOCENES Ferrocene Ruthenocene Osmocene 293 245 361 2.04 2.18 2.18 30.5061 44.3077 52.3460

In all cases, we use a basis set of 512 DVR functions, that span an energy spectrum that 5



Figure 1: Comparison of the torsional potential in cm−1 of metallocenes. The continuous line refers to ferrocene, the dashed line to ruthenocene and the dotted line to osmocene. 400

300 -1

Torsional Potential (cm )

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200

100

0

1.57

3.14 Dihedral Angle (rad)

4.71

6.28

reaches approximately 5000 cm− 1, corresponding to an equivalent temperature of 7000 K. This allows us to safely calculate partition functions up to 1000 K. The eigenvalue spectrum shows a relatively fast transition between a five-fold quasidegenerate harmonic oscillator to a free rotor limit when the PES maximum is surpassed. The only significant pure quantum effect of the potential to the hindered rotor spectrum is a slow decrease of the energy gap between the quasi-degenerate quintets, as expected. From the topology of the PES it is clear that the only relevant terms in the Fourier expansion of the potential are the ones whose order is a multiple of 5, as the potential shows a five-fold periodicity. Moreover, the vertical reflection symmetry of the potential permits to discard all sine terms in the expansion, keeping only cosine terms as shown in Table 2. We have chosen to truncate the Fourier expansion at an order of 15 as the next coefficient B20 is already negligible (of the order of 1 · 10−2 cm−1 ). In this case, we use the reduced mass calculated following the approach of Kilpatrick and Pitzer. 31 After this fitting procedure, we are able to use any number of wavepackets to perform a statistical sampling of the phase space. We have found that a small set of seven

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Table 2: Coefficients of Fourier expansion of the torsional potential (cm−1 ).

V0 B5 B10 B15

METALLOCENES Ferrocene Ruthenocene Osmocene 149.72 124.06 00 182.48 145.86 123.01 181.10 2.62 1.11 1.32 0.20 4.66 · 10−2 0.24

initial wavepackets permits to reproduce well the partition functions.

Symmetry considerations As the energy increases, the eigenfunctions of the Hamiltonian are expected to switch from a localized nature among the five potential minima to the delocalized nature of the free rotor levels. However, at all energies (and for all kinds of fivefold symmetric potentials) a 5-cycle permutation of the carbon and hydrogen atoms in a Cp unit is a symmetry operation. Briefly, this implies that, in all cases, the energy levels can be classified into three categories: 1. Totally symmetric nondegenerate wavefunctions, A 2. Doubly degenerate wavefunctions, whose character under a 5-cycle is 2 cos(2π/5), E1 . 3. Doubly degenerate wavefunctions, whose character under a 5-cycle is 2 cos(4π/5), E2 . In the free rotor case, these correspond respectively to K being a multiple of 5, congruent to ±1 modulo 5 and congruent to ±2 modulo 5 (in this case there is also the additional degeneracy for K being a nonzero multiple of 5). On the other hand, for an infinitely high potential, all three cases merge into a 5-fold degeneracy. In Table3 are shown the first 15 eigenvalues for all three cases and the transition of regimes can be seen. The lowest 5 eigenvalues have approximately the same energy (although the totally symmetric among them are already seen to be splitting away). This is the case also for the highest eigenvalues shown apart from the case of ruthenocene, where the 1 − 2 − 2 splitting pattern starts being visible. 7



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Table 3: First 15 eigenvalues in the Variational Approach(cm−1 ).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

METALLOCENES Ferrocene Ruthenocene Osmocene 22.579 19.864 23.982 24.615 21.005 25.523 24.615 21.006 25.525 24.616 21.007 25.525 24.616 21.008 25.525 67.229 59.540 72.252 70.258 61.795 75.339 70.260 61.796 75.343 70.266 61.811 75.343 70.267 61.812 75.343 110.087 97.944 119.498 113.400 100.637 123.168 113.401 100.639 123.174 113.402 100.666 123.174 113.403 100.668 123.174

Partition functions and thermodynamical properties In the following part of the paper, we show the partition functions for all metallocenes and how the inclusion of the hindered rotation as such makes it different from the one calculated using a purely harmonic behavior. In order to compare our results with other methods for the inclusion of the hindered rotation, we compute the partition function using the equation proposed by McClurg and all 22

Q(T ) = Qc (T )

Qhq (T ) (¯hω)2 /kT (2¯hω+16V0 ) e Qhc (T )

(6)

where V0 is half the potential maximum and Qc (T ) is the classical hindered rotor partition function 2πIkT Qc (T ) = h

Z

2π

e−V (φ)/kT dφ

(7)

0

Qhc and Qhq are the classical and quantum mechanical partition functions of the harmonic oscillator, respectively 8


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Qhc =

Qhq

=

∞ X

kT h ¯ω

(8)

(9)

e−(n+1/2)¯hω/kT

n=0

We refer to the partition function obtained by this equation as the ’corrected’ result. Methods based on the vibrational perturbation theory, such as SPT, 32 cannot be applied for the same reason that forces us to use the GC method: the Taylor series does not converge fast enough and therefore the anharmonic contribution suffers from numerical instabilities. Figure 2: Ferrocene partition functions: the continuous line refers to the time dependent Chebyshev expansion and the dotted line refers to the variational results, while the dashed and dash-dotted are the harmonic and the corrected one. 80 Variational Chebyshev Harmonic McClurg

60 Partition Function Q(T)

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40

20

0

0

500 Temperature (K)

1000

The potential maximum of ferrocene (Table 1) is equivalent to a temperature of about 420K. On the other hand, the derivative of the partition function (Fig. 2) does not show a significant change around that value, starting to decrease after 500K. The energy gap between the quasidegenerate quintets slowly decreases from 40 to 20 cm− 1, while the E1 , E2 doublets are separated from each other at the beginning of the free rotor region (around 300cm− 1),

9



by a difference of 10 cm− 1. For this reason the derivative of the partition function starts to decrease significantly when the spectrum becomes similar to the free rotor one. Figure 3: Ruthenocene partition functions: the continuous line refers to the time dependent Chebyshev expansion and the dotted line refers to the variational results, while the dashed and dash-dotted are the harmonic and the corrected one. 80 Variational Chebyshev Harmonic McClurg


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40

20

0

0

500 Temperature (K)

1000

Ruthenocene (Fig. 3) shows a transition to the free rotor limit at an equivalent temperature of 352 K, which makes it an interesting molecule because it exhibits this transition at a temperature near the ambient one. As can be seen, this molecule has the highest partition function (see also Table 3), even though at high temperatures this difference becomes less significant. Osmocene, which has the highest potential variation, has the lowest partition function (Fig. 4) . This trend should be confirmed by further investigations on the field of organometallic molecules. In all three cases, it can be seen that, when anharmonicity is taken into account by the McClurg method, the partition function is much closer (the deviation is typically reduced by a factor of 4) to the variational value than in the harmonic case, as would be expected. Nevertheless, in all three cases, the Chebyshev expansion method provides a much more 10


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Figure 4: Osmocene partition functions: the continuous line refers to the time dependent Chebyshev expansion and the dotted line refers to the variational results, while the dashed and dash-dotted are the harmonic and the corrected one. 70 Variational Chebyshev Harmonic McClurg

60


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40

30

20

10

0

0

500 Temperature (K)

1000

accurate answer, yielding a deviation itself by a factor of 4 less than the McClurg value. In Fig.5 can be seen the thermodynamic entropy of the three metallocenes, calculated either using a harmonic model for all degrees of freedom or using a harmonic model for all vibrations, treating the hindered rotation as such using our approach. There are various effects at play here. In particular: • In the harmonic model, the residual entropy at 0K is positive as the five-fold degeneracy is exact (the potential is assumed infinite). On the other hand, in the hindered rotor case, the degeneracy is lifted (if only slightly) and the residual entropy is zero. • As the energy is increased, the quasi-degenerate quintets move closer together as the potential peak is reached and therefore the entropy tends to be higher in the hindered rotor case. • In the free rotor limit, the density of states decreases with increasing energy while that of the harmonic oscillator remains constant. This would again tend to increase the 11



entropy of the harmonic oscillator at high enough energies. In the case of ferrocene, the HR entropy remains constantly below the harmonic one, while for ruthenocene a transition is observed whereby the harmonic entropy is lower for up to 350K, after which temperature it becomes higher. This ties with our previous observation of ruthenocene reaching the free rotor regime fast. In the case of osmocene, the two curves remain close together. The highest effect observed is the case of ferrocene, where the difference between the two curves can be of the order of 2 percent, i.e. having observable results. Figure 5: Comparison of the entropy of the three metallocenes, obtained both with a hindered rotor and a harmonic treatment. 40

30

Entropy S/k

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20

H-Ferrocene Ferrocene H-Ruthenocene Ruthenocene H-Osmocene Osmocene

10

0

0

200

400 600 Temperature (K)

800

1000

In Fig.6 can be seen the internal energy of the three metallocenes scaled by the temperature. Again, this is calculated using a harmonic model for all degrees of freedom or using a hindered rotor model for the torsional motion. At higher temperatures the equipartition region is seen to be reached, where the ratio of internal energy to temperature becomes constant. A transition is observed around 220K for both ferrocene and ruthenocene, where the harmonic model starts having a higher internal energy than the free rotor model. The same transition is seen for osmocene at around 320K. The same factors mentioned previously 12


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which influence the entropy do so also for the internal energy and it is not easy to predict which will prevail at given conditions. Also for the case of internal energy, hindered rotor vs. harmonic model can cause a variation of the order of 2 percent, indicating the need of taking account of such degrees of freedom properly (even though such errors would probably tend to cancel out in the case of chemical reactions where both reactants and products have similar characteristics). Figure 6: Comparison of the internal energy of the three metallocenes, obtained both with a hindered rotor and a harmonic treatment. 20

15 Internal Energy U/kT

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10

H-Ferrocene Ferrocene H-Ruthenocene Ruthenocene H-Osmocene Osmocene

5

0

0

200

400 600 Temperature (T)

800

1000

CONCLUSIONS The combination of a small rotational constant and low torsional potential peaks in metallocenes leads to a dense Hamiltonian spectrum. The Boltzmann averaging of the eigenvalues make them individually less important; nevertheless, the study of high degenerate system, which has to obey specific symmetry restrictions, is fundamental to checking the accuracy of the GC method, which can be also used to describe rate constants within the frame of transition state theory. 13



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On the other hand, the generality of the present approach can be used to find partition functions of molecules with irregular shape and unsymmetrical internal rotations, increasing the field of applicability of computational algorithm to predict kinetics and dynamics of medium-sized molecules: in this sense, the catalytic properties of metallocenes can be described with higher lever of accuracy. It is important that large amplitude motions, such as hindered rotations, be taken into account accurately. As we show, observable effects can be caused both at the entropy and at the internal energy level. Using a simple harmonic model where such is not applicable can cause deviations from observed results.

Acknowledgement The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. [320951]. T.F.C is grateful for the support given by CAPES (Brazil).

References (1) Long, N. J. Metallocenes: An Introduction to Sandwich Complexes, 1st Edition; WileyBlackwell: New York, 1998. (2) Astruc, D. Organometallic Chemistry and Catalysis; Springer: Berlin, Heidelberg, Germany, 2007. (3) Bodenheimer, J.; Loewenthal, E.; Low, W. The Raman Spectra of Ferrocene. Chem. Phys. Lett. 1969, 3, 715-716. (4) Bodenheimer, J. The Raman Spectra of Ruthenocene. Chem. Phys. Lett. 1970, 6, 519-520.

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(24) Katzer, G.; Sax, A. F.; Identification and Thermodynamic Treatment of Several Types of Large-Amplitude Motions. J. Comp. Chem., 2005, 26, 1438-1451 DOI 10.1002/jcc.20280 (25) Vansteenkiste, P.; Van Neck, D.; Van Speybroeck, V.; Waroquier, M. An Extended Hindered-Rotor Model with Incorporation of Coriolis and Vibrational-Rotational Coupling for Calculating Partition Functions and Derived Quantities J. Chem. Phys. 2006 124 044314 DOI 10.1063/1.2161218 (26) Ellingson B. A.; Lynch, V. A.; Mielke, S. L.; Truhlar, D.G. Statistical Thermodynamics of Bond Torsional Modes: Tests of Separable, Almost-Separable, and Improved PitzerGwinn Approximations J. Chem. Phys. 2006 125 084305 DOI 10.1063/1.2219441 (27) Reinisch, G.; Leyssale, J.M.; Vignoles, G. L., Hindered Rotor Models with Variable Kinetic Functions for Accurate Thermodynamic and Kinetic Predictions J. Chem. Phys. 2010 133 154112 DOI 10.1063/1.3504614" (28) Zheng, J.; Yu, T.; Papajak, E.; Alecu, I. M.; Mielke, S. L.; Truhlar, D. G. Practical Methods for Including Torsional Anharmonicity in Thermochemical Calculations on Complex Molecules: The Internal-Coordinate Multi-Structural Approximation Phys. Chem. Chem. Phys. 2011 13 10885-10907 RSC DOI 10.1039/C0CP02644 (29) Colbert, D. T.; Miller, W. H. A Novel Discrete Variable Representation for Quantum Mechanical Reactive Scattering via the S-matrix Kohn Method. J. Chem. Phys., 1992, 96, 1982-1991. (30) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson G. A. et al., Gaussian 09, Revision E.01, Gaussian, Inc., Wallingford CT, 2009. (31) Kilpatrick J. E.; Pitzer K. S. Energy Levels and Thermodynamic Functions for

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Molecules with Internal Rotation. III. Compound Rotation. J. Chem. Phys. 1949, 17, 1064-1075. (32) Truhlar, D. G.; Isaacson, A. D. Simple Perturbation Theory Estimates of Equilibrium Constants from Force Fields J. Chem Phys., 1991, 94, 357-359 DOI 10.1063/1.460350

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400 Fe Ru Os

300 -1

Torsional Potential (cm )

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200

100

0

0

90

180 Angle (degrees)

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270


360

Analysis of Partition Functions for Metallocenes: Ferrocene

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