Rate constant calculations Sample Clauses

Rate constant calculations.Β The general expression used for calculating rate constants is taken from standard transition-state theory (TST): π‘˜B𝑇 𝑄‑ βˆ’Ξ”πΈ/π‘˜ 𝑇 π‘„π‘Ÿ οΏ½οΏ½ B where Q‑ and Qr are the partition functions of the transition state and reactant state, respectively, and Ξ”E is the electronic energy difference between a transition state and a reactant state. For surface reactions the transition state is characterized by a saddle point on the potential energy surface and the reactant state is found at a local minimum. Since electronic contributions to the partition functions are neglected, only vibrational contributions contribute to Q in this case. For more background information see for instance the book by NΓΈrskov et al.2 For the surface reactions we consider three levels of approximation to the TST expression: 1. Total neglect of partition functions, since these are generally more time consuming to calculate than the energy difference between two configurations, the former requiring vibrational frequencies and the latter only optimized geometries and the corresponding electronic energies. The rate constant expression then becomes: π‘˜ = πœˆπ‘’βˆ’Ξ”πΈ/π‘˜B𝑇 where the prefactor Ξ½ is either set to a default value (1013 s-1) or specified by the user. 2. Classical vibrational partition functions calculated from a set of 1-dimensional harmonical oscillators (one for each vibrational mode). The expression is then: βˆπ›Ό πœˆπ‘Ÿ βˆπ›Όβˆ’1 πœˆβ€‘ π‘’βˆ’Ξ”πΈ/π‘˜B𝑇 where the Ξ½i are the vibrational frequencies of the reactant state (r) and the transition state (‑). There is one less vibrational frequency for the transition state, since the normal mode frequency corresponding to the unstable mode at the saddle point is imaginary (because of negative curvature of the potential). 3. Quantum vibrational partition functions calculated from a set of 1-dimensional harmonical oscillators (one for each vibrational mode). The expression can be written: 𝑖=1 𝑖 π‘˜B𝑇 βˆπ›Ό sinh(β„Žπœˆπ‘Ÿβ„2π‘˜B𝑇) π‘’βˆ’Ξ”πΈ/π‘˜B𝑇 β„Ž βˆπ›Όβˆ’1 sinh(β„Žπœˆβ€‘οΏ½2π‘˜B𝑇) Note that the vibrational partition functions may be calculated in other equivalent ways as well.
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