A potential energy curve illustrating all critical points on where DIIS can converge. Green and red circles, respectively, initial configurations and converged critical points.
Convergence to a given saddle point for C20 using a pure Lanczós scheme (black line) or the mixed Lanczós-DIIS method presented in this paper (red line). Panels (a) and (b) show the energy and the norm of the total force profiles as a function of the cumulative number of force evaluations. Panels (c) and (d) present the lowest eigenvalue and the number of force evaluations as a function of the step number. The arrows in Panel (a) and (b) indicate the transition in the ART nouveau algorithm from steps for leaving the harmonic well to a convergence towards the saddle point.
Exploration of C20's potential energy surface: Energy difference (Δ energy) between each new relaxed structure (an event) and the total energy of the perfect C20 fullerene (zero event). Horizontal full lines and green symbols represent the energy of those configurations accepted by the Metropolis test. Filled squares and open circles represent, respectively, closed and open configurations. For clearness, only the Δ energy of the transition states of accepted events is drawn with dashed arrows. Metropolis temperature for (a) and (b) is 0.5 and 0.9 eV, respectively. In (b) green circles, accepted open configurations.
C20 clusters: (a) Perfect C20 fullerene. (b)–(d) The three lowest energy structures and their energy difference with respect to the perfect C20 fullerene. For clarity, tetragons and hexagons are emphasized by shaded green and yellow colors. (e) A planar structure found along the simulation.
Schematic representation of the total minimum-energy path for the simple diffusion at zero temperature. The intermediate minima is localized at ∼0.04 eV and ∼0.24 Å with respect to the barriers.
Sampling of activated mechanisms related with the vacancy. Energy difference for each event is measured with respect to the energy of the relaxed vacancy. , , and
Top and side view of the SiC () surface with the three possible high-symmetry positions of the adatom (white ball).
Seven independent exploration of 4H-SiC's PES: Energy difference of each event with respect to the total energy of the H 3 symmetry configuration. Green full lines and squares, accepted configurations, red dashed lines, transition states, black square symbols, rejected events. For clearness, only the Δ energy of the transition states of accepted events is drawn with dashed arrows.
Δ of energy with respect of the cumulative displacement between the events E 0, E 2, E 3, and E 4 in Panel (a) of Fig. 8. Green squares, minimum configurations, red dashed lines, transition states.
Schematic representation of the events depicted in Fig. 9. E 0(H 3 H 3), E 2(H 3 T 4), E 3(T 4 H 3), and E 4(H 3 H 3).
Amorphous silicon (a-Si): Statistic of the number of attempted and new (with respect to previous minimum) events obtained from three independent trajectories of 1000 successful events, each one using Lanczós+DIIS, as a function of the number of the Lanczós steps after the minimum in the lowest eigenvalue of the Hessian before launching DIIS (see text for details). Symbol ∞ means that only Lanczós method was used. For the ∞+FIRE and the simulations with DIIS, FIRE was used for relaxing from the saddle to the final minimum. The average number of force evaluation per successful event, 〈f〉 s , is obtained by dividing the total number of force calls needed to generate 1000 successful events by this number.
Comparison between various saddle point converging algorithms. ART nouveau refers to the algorithm presented here and applied to a-Si with Stillinger-Weber potential as well as three systems studied with ab initio method. Results on Pt(111) surface for ARTn (Olsen) and Dimer method are from Ref. 51. The modified dimer methods applied, with various ab initio schemes, to a small organic molecule (Ref. 52) and a peptide in water (Ref. 53). The double-ended growing string method (GSM) is used to find the transition state for H-transfer in methanol oxidation in VO x /SiO2 (Ref. 54). BC refers to boundary condition: bulk, isolated, surface, or solution. DOF is the number of degrees of freedom available for each simulation. 〈f〉 is the number of force evaluations needed to go from a local minimum to a connected saddle point for a successful trajectory, while 〈f〉 s is obtained by dividing the total number of force calls in a simulation by the number of successful events. A/S gives the total number of attempted over the number of successful events in the simulation presented here.
Article metrics loading...
Full text loading...