Orbital-Free Molecular Dynamics Simulations of Melting in Na8 and Na20: Melting in Steps

The melting-like transitions of Na8 and Na20 are investigated by ab initio constant energy molecular dynamics simulations, using a variant of the Car-Parrinello method which employs an explicit electronic kinetic energy functional of the density, thus avoiding the use of one-particle orbitals. Several melting indicators are evaluated in order to determine the nature of the various transitions, and compared with other simulations. Both Na8 and Na20 melt over a wide temperature range. For Na8, a transition is observed to begin at approx. 110 K, between a rigid phase and a phase involving isomerizations between the different permutational isomers of the ground state structure. The ``liquid'' phase is completely established at approx. 220 K. For Na20, two transitions are observed: the first, at approx. 110 K, is associated with isomerization transitions between those permutational isomers of the ground state structure which are obtained by interchanging the positions of the surface-like atoms; the second, at approx. 160 K, involves a structural transition from the ground state isomer to a new set of isomers with the surface molten. The cluster is completely ``liquid'' at approx. 220 K.Comment: Revised version, accepted for publication in J. Chem. Phys. The changes include longer simulations for the Na20 microcluster, a more complete comparison to previous theoretical results, and the discussion of some technical details of the method applie

Weighted bounds for multilinear operators with non-smooth kernels

Let $T$ be a multilinear integral operator which is bounded on certain products of Lebesgue spaces on $\mathbb R^n$. We assume that its associated kernel satisfies some mild regularity condition which is weaker than the usual H\"older continuity of those in the class of multilinear Calder\'on-Zygmund singular integral operators. In this paper, given a suitable multiple weight $\vec{w}$, we obtain the bound for the weighted norm of multilinear operators $T$ in terms of $\vec{w}$. As applications, we exploit this result to obtain the weighted bounds for certain singular integral operators such as linear and multilinear Fourier multipliers and the Riesz transforms associated to Schr\"odinger operators on $\mathbb{R}^n$. Our results are new even in the linear case