Assessing the efficiency of first-principles basin-hopping sampling

We present a systematic performance analysis of first-principles basin-hopping (BH) runs, with the target to identify all low-energy isomers of small Si and Cu clusters described within density-functional theory. As representative and widely employed move classes we focus on single-particle and collective moves, in which one or all atoms in the cluster at once are displaced in a random direction by some prescribed move distance, respectively. The analysis provides detailed insights into the bottlenecks and governing factors for the sampling efficiency, as well as simple rules of thumb for near-optimum move settings that are intriguingly independent of the distinctly different chemistry of Si and Cu. At corresponding settings, the observed performance of the BH algorithm employing two simple general-purpose move classes is already very good and for the small systems studied essentially limited by frequent revisits to a few dominant isomers.

Figure 1

(Color online) Schematic representation of the original and transformed potential-energy surfaces $E\left(\left\{{\mathbf{R}}_m\right\}\right)$ and $\tilde{E}\left(\left\{{\mathbf{R}}_m\right\}\right)$, respectively, as well as of a basin-hopping trial move (see text).

Figure 2

(Color online) Schematic illustration of successful, unsuccessful, and high-energy trial moves in the BH scheme. The horizontal dashed (red) lines indicate the targeted energy window entering the acceptance criterion.

Figure 3

(Color online) Histograms of the probability with which trial moves end up in the lowest-energy isomers of ${\text{Si}}_7$, ${\text{Si}}_{10}$, and ${\text{Cu}}_7$. The identified isomers are numbered with decreasing stability, with isomer 1 corresponding to the identified ground state and those isomers shown with bracketed numbers revealed as unstable by an a posteriori vibrational analysis (see text). The histograms comprise all isomers found in an energy range up to 2 eV above the ground state isomer, as obtained from long BH hopping runs using single-particle moves and normally distributed move distances around the average values $r_{\text{o}}=1.5$, 2.0, and 2.5. The geometric structures behind the truly stable isomers in an energy range up to 1 eV above the identified ground state are summarized in Figs. 5, 6, 7.

Figure 4

(Color online) Probabilities for the lowest-energy isomers of ${\text{Si}}_7$ as in Fig. 3. Shown is the evolution when binning the histogram entries over consecutive sampling periods containing 50 moves each using single-particle moves and normally distributed move distances around the average value $r_{\text{o}}=2.5$. Entries for all isomers higher in energy than isomer 4 are bundled into one entry labeled “$>\#4$.”

Figure 5

(Color online) Identified stable ${\text{Si}}_7$ isomers in the energy range up to 1 eV above the ground state. The isomer numbering follows the one of Fig. 3 and reflects the decreasing cluster stability as indicated by the stated energies relative to the ground-state isomer 1.

Figure 6

(Color online) Identified stable ${\text{Si}}_{10}$ isomers in the energy range up to 1 eV above the ground state. The isomer numbering follows the one of Fig. 3 and reflects the decreasing cluster stability as indicated by the stated energies relative to the ground-state isomer 1.

Figure 7

(Color online) Identified stable ${\text{Cu}}_7$ isomers in the energy range up to 1.1 eV above the ground state. The isomer numbering follows the one of Fig. 3 and reflects the decreasing cluster stability as indicated by the stated energies relative to the ground-state isomer 1. Note that isomer 4 exhibits small imaginary eigenmodes but is nevertheless retained in the performance analysis (see text).

Figure 8

(Color online) Performance analysis of BH runs for ${\text{Si}}_7$, ${\text{Si}}_{10}$, and ${\text{Cu}}_7$ using collective moves and a normal distribution for the atomic displacements. Upper panel: variation in the average number of moves $N_{\text{av}}$ required to determine the low-energy dominant isomers with the average move distance $r_{\text{o}}$ (see text). Lower panel: corresponding variation in the fraction of unsuccessful moves ${\alpha }_{\text{unsucc},\text{av}}$ of moves into high-energy isomers ${\alpha }_{\text{high}E,\text{av}}$ and of successful moves ${\alpha }_{\text{succ},\text{av}}$ [cf. Eqs. (4, 5, 6)]. The error bars reflect the uncertainty due to the employed approximate hopping matrix procedure (see text).