Abstract
Finding the optimal alignment between two structures is important for identifying the minimum root-mean-square distance (RMSD) between them and as a starting point for calculating pathways. Most current algorithms for aligning structures are stochastic, scale exponentially with the size of structure, and the performance can be unreliable. We present two complementary methods for aligning structures corresponding to isolated clusters of atoms and to condensed matter described by a periodic cubic supercell. The first method (Go-PERMDIST), a branch and bound algorithm, locates the global minimum RMSD deterministically in polynomial time. The run time increases for larger RMSDs. The second method (FASTOVERLAP) is a heuristic algorithm that aligns structures by finding the global maximum kernel correlation between them using fast Fourier transforms (FFTs) and fast SO(3) transforms (SOFTs). For periodic systems, FASTOVERLAP scales with the square of the number of identical atoms in the system, reliably finds the best alignment between structures that are not too distant, and shows significantly better performance than existing algorithms. The expected run time for Go-PERMDIST is longer than FASTOVERLAP for periodic systems. For finite clusters, the FASTOVERLAP algorithm is competitive with existing algorithms. The expected run time for Go-PERMDIST to find the global RMSD between two structures deterministically is generally longer than for existing stochastic algorithms. However, with an earlier exit condition, Go-PERMDIST exhibits similar or better performance.
Highlights
Quantifying the difference or similarity between two structures is of broad relevance
We have shown that it is possible to estimate the rootmean-square distance (RMSD) between structures by calculating the maximum kernel correlation, so long as the interatomic separation is relatively large compared to the kernel size
We demonstrated that the FASTOVERLAP algorithm can find the maximum value of the overlap in periodic and isolated systems efficiently and deterministically using fast Fourier transforms (FFT) and fast special orthogonal transforms (SOFT)
Summary
Quantifying the difference or similarity between two structures is of broad relevance. Calculating the RMSD is a global optimization problem, requiring the identification of the relative lattice vectors and/or rotation, permutation and translation that defines the global minimum Locating this global minimum is equivalent to finding the optimal alignment of two structures, where the total squared displacement between them is minimized. Sadeghi et al.[13] developed a Monte Carlo algorithm for calculating the global minimum RMSD for both clusters and periodic systems.[27] In this method an initial permutational alignment is performed by either matching the principal axes of the moment of inertia or by matching atoms with similar local environments. The number of permutation subsets with a lower bound below a given distance scales approximately exponentially with the distance, which means that the computational complexity scales approximately with the exponential of the minimum RMSD
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