(1) 2. Thermochemical calculations [2,3]
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1 1. Introduction The exploration of reaction mechanisms and reaction paths that cannot be measured directly during an experiment has nowadays become a daily routine for chemists to support their laboratory work. Among all ab-initio techniques Density Functional Theory (DFT) has become the golden standard due to its wide applicability, ranging from simple organic molecules to complex coordination compounds involving transition metals. For the prediction of thermo-chemical values one has to make a step from a DFT calculation on a single molecule to macroscopic, observable values. With the help of statistical thermodynamics it is possible to describe an ensemble of molecules and close the gap between atomic energy units and units that are more familiar to chemists like for example kj/mol. The properties of such ensembles are described by fundamental thermodynamical variables like enthalpy, entropy, temperature, internal energy etc.. If one finds a way to calculate these values it would be possible to predict the most important values for synthetic working chemists, namely the differences in the heat of enthalpy, the energy of activation and the Gibbs free energy for a chemical reaction. Nevertheless at the beginning one has to accurately calculate single molecule properties with DFT by using an appropriate functional. Which functional should I choose? is the first question that comes into ones mind when dealing with DFT calculations [1]. All functionals currently used are still approximations to the exchange-correlation problem in DFT resulting in the fact that no functional is accurate for all systems and all wanted properties. This fact should be covered by the DFT Jacob s ladder to heaven on which every rung represents a different and more complex approximation, that should have the features of former rungs with additional wider applicability to new problems [2]. Starting from local density approximation functionals (LDA) that are rather useless for calculations with chemical accuracy due to strong overbinding tendency to generalized gradient approximations (GGA) that partially correct this fault, to meta- GGA s and hybrid functionals that are at present the most widely used functionals for molecular DFT calculations. One should also keep in mind that functionals do not perform equally well for molecular systems and crystals with their translational symmetry properties. 1
2 While the local functional LDA performs well for the prediction of for example lattice constants in crystals due to their slow varying electron density, it fails for inhomogeneous systems that are present in molecular structures. By far the most popular functional is the hybrid B3LYP [3,4] which contains about 20% of exact Hartree-Fock exchange to mimic effects of static correlation: (1) Yet today's functionals do not reach chemical accuracies for the prediction of reaction energies where typical errors are in the range of 3-5 kcal/mol [1]. Why are then some functionals more popular than others? The answer is connected to the question which system I want to describe or more precisely in which specific property one is interested! Nevertheless a good starting point are always functionals that are applicable to lots of different questions, which automatically results in a loss of accuracy. Although their results can be crude, they often give at least a qualitatively correct result as a starting point to further calculations. Such functionals that show wide-range applicability are PBE (GGA) and the already mentioned B3LYP (hybrid). 2. Thermochemical calculations [2,3] All calculations assume an ideal gas of non-interacting particles in the rigid rotor, harmonic normal mode approximation. It is assumed that the first and higher electronic excitations are not accessible, a fact which could be troublesome in the case of the existence of low lying electronic excited states. During this exercise the changes in enthalpy and in the Gibbs free energy together with the energy of activation will be calculated for a simple SN 2 reaction at room temperature as shown in equation 2. (2) 2
3 Usually chemists are exclusively interested in stationary points (local extrema), where all forces vanish, whereas the properties of such a point are determined by its Hessian Eigenvalues. To distinguish between a ground state and a transition state or saddle point, one has to calculate the mass-weighted Hessian matrix for a given molecular configuration. Murrel and Laidler defined a transition state as having a single negative Hessian Eigenvalue corresponding to a negative force constant and a single negative imaginary normal mode frequency [5]. Transition states are rather important in terms of chemical reactions because they can be attributed to low energy paths between two ground states (fig.1) leading to the possibility to calculate energy of activations and rate constants from transition state theory (fig.2). Figure 1: An example of a simple PES with two minima (M) connected by a transition state (TS) and a higher order saddle point (S) [6] Figure 2: Energy profile as a function of a reaction coordinate with two minima and a TS [7] 3
4 The enthalpy E at a temperature of 298K can be calculated from equation 3 [8]: (3) Whereas can be calculated from the following equation 4 [8]: (4) = Difference in energy between educts and reactants at 0K = Difference in the zero point energy between educts and reactants at 0 K = Change in the vibrational energy by going from 0K to 298K = Difference in the rotational energy between products and reactants = Difference in the translational energy between products and reactants dn with dn beeing the difference in the number of molecules of reactants and products The free Gibbs energy can be calculated from equation 5 as: (5) where (see at ref.9): contains all contributions to the total partition function corrected to 298K (6) The program will calculate and print all corrections to the thermal energy, the enthalpy and the Gibbs free energy at the end of every *.log file but one has to take care and check for different units used (convert them!). 4
5 Remember that the values corrected to 298K are already containing the zero point energy (ZPE)! Never add the ZPE except for calculations at 0K, in that case it must be added manually. To calculate the reaction enthalpy and the Gibbs energy one simple has to use equation (7) and (8) by using the corrections to enthalpy and Gibbs free energy and the printed total energies for every molecule [9]: (7) (8) It is recommended to print a table containing all the calculated total energies, enthalpies and Gibbs free enthalpies for every species involved in the reaction. How is the energy of activation calculated then? 3. Instructions Optimize the geometries for all educts and products with DFT(B3LYP) as programmed in the g03[11] software and a 6-31+G* basis set [10] and compare them with published results. Use point groups for every species involved to speed up all calculations. What are the geometries for [GS1] -, [GS2] - and [TS1] -? Do they correspond with published results? To find the geometry [TS] - start with a good guess based on your chemical intuition together with the proper point group. Verify that every structure is indeed a ground- or transition state by running a mass-weighted normal coordinate analysis at the same level of theory. Calculate the change in enthalpy, Gibbs free energy and the energy of activation (for educt -> TS1 -> product) for every reaction step at 298 K and print a table with all results and a diagram of Gibbs free energy vs. reaction coordinate (enthalpies in brackets!). 5
6 Calculate the reaction rate with the Eyring Polanyi equation and compare with results from the literature (most likely an MP2 calculation). 4. Manual Open a terminal and start the gaussview program by typing gv. Use the elements fragment table and the modify bond/angle/dihedral buttons to draw an initial structure. Assign the correct point group symmetry to the model using the edit Point group submenu. Switch to the calculations tab and set the option button there to Opt+Freq and optimize to a minimum. At the Method tab choose the Ground state calculation, the DFT method together with the B3LYP functional and as we are considering closed shell molecules in this exercise do not forget to set the Restricted flag option as well. Basis set should be set to the already mentioned 6-31G using one additional diffuse and one additional polarization function on all atoms except hydrogen (6-31+G(d)) and depending on the species the correct charge has to be set as well. To take advance of the multicore installation of g03 we use two shared processors and choose an appropriate folder for the Chkpoint File location. To initialize our calculation we start from molecular orbitals previously obtained from an extended Hückel calculation. After the calculation has finished, the *.log file can be opened by any text editor (vim,gedit, notepad...) and at the end of the file below the Thermochemistry record, the necessary results at standard conditions (298K, 1 atm) for the calculations of the gibbs free energy and enthalpy can be extracted. Structural results including an animation of the vibrations can be visualized by using gaussview or any other program like avogadro, gabedit, chemcraft. The location of the TS is more complicated and some chemical intuition is needed to create a proper structural starting model (recall the general SN2 reaction mechanism!). It is recommended to use lower level methods like the semi empirical PM6 model or molecular mechanics to create a structure, that is close to the actual transition state with a minimum in computational effort. On the Job Type tab choose Optimize to a TS(Berny) and Calculate Force constants always, then switch from the DFT method to Semi-empirical and check the PM6 method. Additionally add opt=noeigentest in the keywords field and start the calculation. After it has finished, check that the TS has been identified (one and only one imaginary frequency!) and use that structural model as a starting point for the DFT calculation. 6
7 5. Sources [1] Rappoport,D.; Crawford,N.M.R.; Furche, F.; Burke,K. Computational Inorganic and Bioinorganic Chemistry ; E. I. Solomon, R. B. King, and R. A. Scott, Eds.,Wiley, Chichester (2009) [2] Perdew,J.P.; Schmidt, K.; AIP Conf. Proc. 577, 1 (2001) [3] Becke,A.D. J.Chem.Phys. 98 (1993) [4] Lee, C.W. Yang; Parr,R.G. Phys. Rev. B 37 (1988) [5] Murrel,J.N.; Laidler,K.J.; Trans. Faraday Soc. 64 (1968) 371 [6] Wales, D.J.; in Atomic Clusters and Nanoparticles (2001) [7] [8] Foresman,J.B; Frisch,A.; Exploring Chemistry with Electronic Structure Methods, Sec.Ed. Gaussian Inc. (1996) [9] Ochterski,J.W.; Thermochemistry in Gaussian, Gaussian Inc. (2000) [10] Hehre, W. J.; Random, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, [11] Gaussian 03, Revision C.02, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez, and J. A. Pople, Gaussian, Inc., Wallingford CT,
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