Computational details Sample Clauses

Computational details. To include electronic correlation effects for the partially- filled Os 5d band beyond the standard DFT framework we used an local density approximation 1 dynamical mean-field theory (LDA1DMFT) approach25,31. This approach is based on a full-potential linear augmented plane-wave 1 local orbitals technique as implemented in the Wien2k code40 in conjunction with the DMFT implementation provided by the TRIQS package31–34. Our LDA1DMFT framework is fully self-consistent in the charge density. The LDA1DMFT calcu- lations were performed within the scalar-relativistic approximation and using a k-mesh with 32 3 32 3 32 points in the full Brillouin zone. The spin–orbit coup- ling was not included because LDA calculations show that it has a negligible effect on the electronic structure in the vicinity of the Fermi level. The DMFT quantum impurity problem was solved using the numerically-exact imaginary- time hybridization-expansion continuous-time quantum Monte Carlo (CT- QMC) method35. A large number of Monte Carlo cycles, more than 512 million, were performed to obtain a well converged DMFT local self-energy. We adopted a stochastic version of the maximum entropy method36 for the analytical continua- tion of the CT-QMC self-energy to the real frequency axis. For the Coulomb interaction strength U and Xxxx’x coupling constant J we used the values U 5 2.8 eV and J 5 0.55 eV that are estimated in ref. 37. The qualitative results of our LDA1DMFT calculations are not very sensitive to the exact values of U and J. We used the ‘around mean-field form’38 for the double counting correction, which is suitable for weakly correlated metallic systems. In calculations of band structure at the level of DFT39 within the LDA or semi- local generalized gradient approximation (GGA), we used two complementary methods, the full potential (linear) augmented plane waves 1 local orbitals method as implemented in the Wien2k code40 and the electronic-structure method41 RSPt. Both are all-electron methods, which do not impose any approx- imations on the shape of the one-electron potential, and they are known to gen- erate very similar results. The former method allows us to directly compare the LDA and LDA1DMFT results. These methods are particularly suited to high- pressure calculations because the basis functions for any energy, including nom- inally deep core states, can be treated as ‘valence’ states. For calculations with the Wien2k code, we used a k-mesh consisting of 32 3 32 3 32 k-po...

Computational details. The ab-initio molecular dynamics (AIMD) simulations [7] in this work are performed with the CP2K program [8]. The OPBE exchange- correlation functional for the DFT electronic structure calculations is used [9]. The choice of this non-hybrid functional is mainly dictated by the computational efficiency in the AIMD and is justified by previous work where it has been shown to give an accurate description of several transition metal complexes [15-18]. Additional tests were performed to validate the OPBE results using the non-hybrid functional BLYP [10, 11] and the hybrid functionals B3LYP and B3LYP*, with different amounts of exact exchange, 0.2 and 0.15, respectively [11, 12]. The ADF program was used for the calculations with the hybrid functionals [13, 14]. The CP2K program employs a mixed basis set approach with Gaussian type orbitals (GTO) and plane waves (PWs) [15]. GTO functions are used to expand the molecular orbitals and the charge density in real space, whereas PWs are used for the representation of the charge density in reciprocal space. An energy cut-off of 280 Ry is used for the plane-waves basis set. The TZVP-MOLOPT-GTH [16] Gaussian basis set is chosen for all the atoms in the catalyst except ruthenium for which a DZVP-MOLOPT-GTH is used. The water molecules close to the catalyst and involved in the reaction are treated at the TZVP-MOLOPT-GTH level, whereas the DZVP-MOLOPT- GTH is used for all the other water molecules. Pseudopotentials of the GTH form for all the elements [15, 17, 18] are used. The pseudopotential for Ru is generated with 16 valence electrons. In the ADF calculations a TZP Xxxxxx type basis set is applied. Due to the presence of π-cation interactions between the metallic centre and the aromatic ligand, it is crucial to include van der Waals corrections. In all the AIMD simulations the DFT-D2 van der Waals correction by Xxxxxx is applied [19, 20]. Since the OPBE functional does not have its own set of correction parameters, the Xxxxxx parameters for PBE are used instead. Periodic boundary conditions 02 a. u., respectively. In the simulations the collective variables are evolved with one metadynamics step every 20 AIMD time steps. In this way a quick exploration of the reaction pathway can be done, at the expense of accuracy in probing the free-energy surface along the collective variable. For a preliminary estimate of the free-energy, a thermodynamic integration technique with constrained MD [22, 23] can be used instead. Six po...

Computational details. The ab-initio molecular dynamics (AIMD) simulations in this work were performed with the CPMD program [24]. Nuclear forces in AIMD are derived from the electronic structure using DFT with the BLYP functional [25, 26]. The choice of a non-hybrid functional is mostly dictated by computational efficiency in the AIMD. However, single point calculations using the hybrid functionals OPBE0, B3LYP, B3LYP* [27], that differ in the amount of exact exchange (25%, 20% and 15%, respectively), have been performed with the ADF program [28-30] to check the relative energy of different complexes in order to validate the BLYP results. Further tests have been performed with the ADF program to check the effect of empirical dispersion corrections in the form proposed by Xxxxxx [31]. Norm-conserving pseudopotentials of xxx Xxxxxxx–Xxxxxxxxx [32] form are used for all the atomic elements except for Mn, for which the Xxxxxxxxx form is used instead [33]. The Xxxx-Sham orbitals are expanded on a plane-waves basis set with an energy cutoff of 100 Ry, which provides a good convergence in relative energies. The Car-Xxxxxxxxxx AIMD simulations are performed with a time step Δt = 5 a.u. and a fictitious electronic mass μ = 400 a.u. In order to efficiently explore possible reaction pathways, the CPMD code with the metadynamics approach is used [23]. The metadynamics is a coarse- grained dynamics on the free-energy surface (FES) defined by a few collective variables, such as distances, angles, and coordination numbers. The method uses an adaptive bias potential to escape from local minima. In these simulations the evolution of the collective variables takes one metadynamics step every ten AIMD time steps. At each metadynamics step the evolution of the collective variables is guided by a generalized force that combines the action of the thermodynamic force, which would trap the system in a free energy minimum, and a history-dependent force that disfavors configurations already visited. This history-dependent potential is built as a sum of Gaussian functions centered in the explored values of the collective variables [23]. The width and height of the Gaussian are parameters that can be tuned to find the best compromise between accuracy in the FES estimate and speed in crossing energy barriers to sample the whole collective variable space. In this work these values are 0.1 a.u. and

Computational details. The details of the numerical method how to calculate the BF were described in details earlier [10, 11]. Here, we will list only the main calculations steps focusing on the differences to the previous studies. 2.1 Bias factor a) since the point defects are absorbed [19] within the dislocation core radius (at r=r0), then (r0 )  C1  0 . The previously used [9] definition of dislocation core radius based on an interaction energy gradient threshold, was applied in this work. The radius r0 for the particular defect (vacancy or SIA) is defined as the distance from the dislocation where the thermal energy becomes comparable with the gradient of the interaction energy scaled by the Burgers vector, i.e. b E  kBT , where b is the Burgers vector. b) far from the dislocation (at the dislocation capture range r=R, where the dislocation defect interaction is negligible and the defect concentration C is a constant) the drift term (βDC∇E) is zeroed and the second boundary condition takes the form (R)  C2 1. According to the model used [7], the choice of C1 and C2 constants was arbitrary except for the fact that they should satisfy the condition C1 < C2 so that the concentration difference drives point defect flow to the center, i.e. to the dislocation core. The total current of the point defects towards the dislocation can be evaluated as follows:

Computational details. The details of the numerical method how to calculate the BF were described in details earlier [10, 11]. Here, we will list only the main calculations steps focusing on the differences to the previous studies. 2.1 Bias factor a) since the point defects are absorbed [19] within the dislocation core radius (at r=r0), then Ψ(r0 ) = C1 = 0 . The previously used [9] definition of dislocation core radius based on an interaction energy gradient threshold, was applied in this work. The radius r0 for the particular defect (vacancy or SIA) is defined as the distance from the dislocation where the thermal energy becomes comparable with the gradient of the interaction energy scaled by the Burgers vector, i.e. b∇ E = kBT , where b is the Burgers vector. b) far from the dislocation (at the dislocation capture range r=R, where the dislocation defect interaction is negligible and the defect concentration C is a constant) the drift term (βDC∇E) is zeroed and the second boundary condition takes the form Ψ(R) = C2 =1. According to the model used [7], the choice of C1 and C2 constants was arbitrary except for the fact that they should satisfy the condition C1 < C2 so that the concentration difference drives point defect flow to the center, i.e. to the dislocation core. The total current of the point defects towards the dislocation can be evaluated as follows: 2π J tot = ro ∫ J r (r0 ,θ )dθ , (4) where r0 is the dislocation core radius vector and Jr(r0,θ) is the current of the point defects towards the core in the certain position. Finally, the capture efficiency of a point defect can be evaluated as: in the bulk iron (the crystal without a dislocation). , where J0 is the flux The general strategy of the numerical calculation of the BF following the above mentioned methodology: Firstly, the atomistic simulations are performed to get the interaction energy between dislocation and point defect in the vicinity of the dislocation core, i.e. in the region where the elasticity theory considerations are not sufficient to describe the interaction in an adequate way. Later on, the interaction energy map is extended outside the zone studied by atomistic simulations up to the desired dislocation density using the analytical prediction of the interaction energy. The final step consists in the numerical integration of the bias factor using the finite element approach following the Eqs. 2-4. 2.2 Atomistic calculations towards the construction of the point defect-dislocation interaction energ...