Reliable low-level seeding of nanocluster global-minimum searches: transferable DFTB repulsive refits

The structure of a nanocluster is predicted by searching for the global minimum (GM) of its potential-energy surface, but global optimization (GO) requires so many energy evaluations that it must run at an inexpensive electronic-structure level, with a few low-energy candidates then locally refined at a benchmark level such as CCSD(T) at the complete-basis-set limit. This two-stage protocol is reliable only if the inexpensive surface ranks the true benchmark GM among its own low-lying minima. Galvão, Viegas, Salahub and Lourenço (J. Mol. Model. 2020) showed that standard semiempirical Hamiltonians (AM1, PM3) and standard DFTB2 parametrizations fail this test for period-3 element clusters, that DFTB3 is partially reliable, and that refitting only the pairwise repulsive term of DFTB2 (FASP refits for Na–Na, Mg–Mg, Al–Al) recovers benchmark GMs and transfers out of sample to Na$_{20}$ — though fitted isomerization-energy deviations still reach about 7.7 kcal mol$^{-1}$. The open problem is to turn that demonstration into a reliable methodology: establishing whether general-purpose and transferable repulsive refits — or comparably inexpensive surrogates — can be made to satisfy this prerequisite for other elements, mixed-element (alloy) clusters and larger sizes across alkali, metallic and covalent bonding types, and identifying the cases in which the prerequisite fails for such surrogates.

Background

Nanoclusters — aggregates of a few to a few hundred atoms such as Na$_{10}$, Mg$_{10}$, or Al$_{12}$ — have potential-energy surfaces (PES) whose number of local minima (structural isomers) grows roughly exponentially with size. The global minimum (GM), the lowest-energy isomer, is the structure most likely realized experimentally and the starting point for predicting cluster properties, so locating it is the central step of computational structure prediction for nanoparticles. Global optimization (GO) algorithms such as basin hopping (repeated local minimization with stochastic jumps) and genetic algorithms (population-based search that recombines and mutates candidate structures) must evaluate energies thousands of times or more, which forces the use of inexpensive energy models: direct GO at a benchmark quantum-chemical level — coupled cluster with perturbative triples, CCSD(T), extrapolated to the complete-basis-set (CBS) limit — is feasible only for the smallest systems.

The standard workaround is a two-stage (hierarchical) protocol: run the GO with an inexpensive surrogate Hamiltonian, collect its low-lying minima, and locally reoptimize a small subset at the benchmark level. Its reliability hinges on an unproven prerequisite: the surrogate PES must rank the true benchmark GM among its own low-energy minima, so that refining a handful of candidates recovers it. Galvão and Viegas (J. Phys. Chem. A 2019) tested this prerequisite for common density-functional theory (DFT) functionals and MP2 (second-order Møller–Plesset perturbation theory) variants against CCSD(T)/CBS isomerization energies (energy differences between isomers) of period-3 clusters, concluding that when the low-level search also reaches a modest number of low-lying structures, several DFT functionals are reliable choices — but DFT-level GO remains too costly for larger clusters, motivating still cheaper surrogates.

Density-functional tight-binding (DFTB) approximates DFT at roughly two orders of magnitude lower cost: a fixed electronic Hamiltonian built from tabulated Slater–Koster integrals (precomputed two-center tight-binding matrix elements) is solved self-consistently with fluctuating atomic charges (self-consistent charge, SCC), and a short-range pairwise repulsive energy term is added on top. The repulsive terms are supplied by parameter sets such as 3OB (fitted mostly to organic and biological molecules) or matsci-0-3 (bulk materials); a precedent for systematically redesigning SCC-DFTB repulsive terms to reproduce reference data is the iterative Boltzmann inversion refit (an iterative scheme that adjusts a repulsive potential until simulation reproduces a reference radial distribution function) for bulk water of Lourenço, dos Santos, Pettersson and Duarte (J. Chem. Theory Comput. 2020). Older semiempirical Hamiltonians of the NDDO type (neglect of diatomic differential overlap, a semiempirical simplification of the electron-repulsion integrals) such as AM1 and PM3 are cheaper still but were parameterized on small organic molecules.

Galvão, Viegas, Salahub and Lourenço (J. Mol. Model. 2020) benchmarked AM1, PM3, DFTB2/matsci and third-order DFTB3/3OB against CCSD(T)/CBS reference data for Na$_{10}$, Mg$_{10}$, Al$_{12}$, P$_{11}^{+}$ and the alloy Al$_4$Si$_7$. The AM1/PM3 surfaces contain many spurious minima with poor energetic ordering; standard DFTB2 misplaces the benchmark GM; DFTB3 is reasonable, but its lowest isomer is not the benchmark GM for Na$_{10}$ and P$_{11}^{+}$. The pivotal finding: keeping the electronic Slater–Koster terms unchanged and refitting only the pairwise repulsive term — with the FASP workflow (a framework for automating Slater–Koster file parameterization), trained on benchmark GM geometries, representative isomers, and their isomerization energies — yields DFTB2-FASP parametrizations for Na–Na, Mg–Mg and Al–Al whose low-energy minima include the benchmark GMs, and the Na-fitted repulsion transfers out of sample to Na$_{20}$ (a database of 1500 isomers), recovering the reported CCSD(T) honeycomb-like GM. Remaining fitted isomerization-energy deviations nevertheless reach about 7.7 kcal mol$^{-1}$, and the refits were tested on only a few homonuclear pairs and one larger cluster.

Problem Statement

The problem is to establish whether inexpensive surrogates — above all DFTB2 in which only the short-range pairwise repulsive term is refitted against benchmark data (FASP refits), or comparably inexpensive semiempirical Hamiltonians — can be made to satisfy the prerequisite of the two-stage protocol beyond the demonstrated scope: that their lowest-energy structures serve as starting points for a subsequent local optimization at the CCSD(T)/CBS benchmark level that yields the true global minimum. The demonstration covered Na–Na, Mg–Mg and Al–Al refits trained on benchmark data for Na$_{10}$, Mg$_{10}$ and Al$_{12}$, tested on P$_{11}^{+}$ and the alloy Al$_4$Si$_7$, with out-of-sample transfer of the Na-fitted repulsion to Na$_{20}$. The open question is whether the prerequisite can be secured for other elements, mixed-element (alloy) clusters and larger sizes, across alkali, metallic and covalent bonding types — including systematic, general-purpose (ideally automated) procedures for constructing refits that simultaneously balance geometric fidelity of low-energy minima, correct energetic ordering, and transferability where such refits are achievable, and identification of the cases in which the prerequisite fails for pairwise-repulsive refits and can be met only by a different class of inexpensive surrogate. What resolves the problem is a demonstration for a representative subset of this scope beyond the demonstrated Na, Mg and Al cases — at least one system spanning a bonding type or element pair not covered there — that the prerequisite is either secured or shown to fail; exhaustive coverage of every listed element, alloy and size is not required, and the general-purpose refit-construction procedure and the identification of failure cases are goals pursued within such a demonstration rather than separately mandated deliverables.

References

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