Curved-line search algorithm for ab initio atomic structure relaxation

Ab initio atomic relaxations often take large numbers of steps and long times to converge, especially when the initial atomic configurations are far from the local minimum or there are curved and narrow valleys in the multidimensional potentials. An atomic relaxation method based on on-the-flight force learning and a corresponding curved-line search algorithm is presented to accelerate this process. Results demonstrate the superior performance of this method for metal and magnetic clusters when compared with the conventional conjugate-gradient method.

Figure 1

The flowchart of the CLS algorithm. OTFF indicates on-the-flight fitting.

Figure 2

(a),(b) Deviation of atomic forces for ${\mathrm{Pt}}_{100}$ between DFT calculations and approximate models: (a) conventional unfitted Gupta potential; (b) force-fitted Gupta potential. Their $x$ axis is DFT force $F_{\mathrm{DFT}}$, while the $y$ axis is the deviation $|F_{\mathrm{Gupta}}-F_{\mathrm{DFT}}|$ (in units of eV/Å). (c),(d) Comparisons of the values of force difference $F-F_0$ between DFT and approximate models: (c) unfitted potential; (d) force-fitted potential, where $F_0$ is the force at the atomic structures ($R_0$) used for force fitting and $F$ is the one at randomly displaced structures ($R$) around $R_0$. Their $x$ axis is the DFT force difference ${(F-F_0)}_{\mathrm{DFT}}$, while the $y$ axis is the approximate one (in units of eV/Å). The color bar for (c) and (d) indicates the distance (in units of Å) of $R$ from $R_0$. Note that the plot contains several different groups of $R_0$.

Figure 3

The relaxation process of five random ${\mathrm{Pt}}_{20}$ clusters with different initial structures and corresponding different initial energies by the CLS and CG methods, respectively. The $x$ axis is the number ($N$) of ab initio force evaluation, while the $y$ axis is $E-E_f$ (in unit of eV), where $E$ is the energy of the current step and $E_f$ is the energy of the finally sought structure.

Figure 4

The mean energy evolution in the ab initio relaxation of five prerelaxed ${\mathrm{Pt}}_{20}$ clusters and the relaxation of a ${\mathrm{Pt}}_{100}$ cluster. The $x$ axis is the number ($N$) of ab initio force evaluation, while the $y$ axis is $E-E_f$ (in unit of eV), where $E$ is the energy of the current step and $E_f$ is the energy of the finally sought structure.

Figure 5

The relaxation process of ${\mathrm{Co}}_{120}$ and ${\mathrm{Cu}}_{20}{\mathrm{Au}}_{18}$ clusters with the initial geometry from the global minimum of conventional unfitted Gupta force field. The $x$ axis is the number ($N$) of ab initio force evaluation while the $y$ axis is $E-E_f$ (in unit of eV), where $E$ is the energy of the current step and $E_f$ is the energy of the finally sought structure.