Metadynamics is a powerful sampling technique that uses a nonequilibrium history-dependent process to reconstruct the free-energy surface as a function of the relevant collective variables s . In Bussi [Phys. Rev. Lett. 96, 090601 (2006)] it is proved that, in a Langevin process, metadynamics provides an unbiased estimate of the free energy F(s) . We here study the convergence properties of this approach in a multidimensional system, with a Hamiltonian depending on several variables. Specifically, we show that in a Monte Carlo metadynamics simulation of an Ising model the time average of the history-dependent potential converge to F(s) with the same law of an umbrella sampling performed in optimal conditions (i.e., with a bias exactly equal to the negative of the free energy). Remarkably, after a short transient, the error becomes approximately independent on the filling speed, showing that even in out-of-equilibrium conditions metadynamics allows recovering an accurate estimate of F(s) . These results have been obtained introducing a functional form of the history-dependent potential that avoids the onset of systematic errors near the boundaries of the free-energy landscape.

Copper-modified titania is a system of interest for its potential for photocatalytic applications in the production of solar fuels. Still, the role of copper in the process is unclear. In this work, small copper clusters on the (101) and (100) surfaces of anatase have been investigated by first-principles simulations based on density functional theory, to shed light on their atomic and electronic structure, and to understand their effect on the photocatalytic process. The main effects of copper on the electronic structure are to provide states above the edge of the valence band of titania and to lead to the formation of midgap states. There are two types of midgap states, respectively, associated with direct Cu−Ti bonds and to Ti 3+ polarons. The latter are the result of charge donation from copper and lie in the vicinity of the surface. Moreover, the copper tetramer (Cu 4 ) displays empty states at the bottom of the conduction band that play a key role in accommodating excess electrons. We discuss how these features should enhance the photoresponse of TiO 2 , contribute to increase the lifetime of the photogenerated electron−hole pairs and contribute to increase the activity of this material for CO 2 reduction, a key step in the photoproduction of hydrocarbons.

We introduce a simple lattice model with Ising spins to explain recent experimental results on spin freezing in a hollandite-type structure. We argue that geometrical frustration of the lattice in combination with nearest-neighbour antiferromagnetic (AFM) interactions is responsible for the appearance of a spin-glass phase in presence of disorder. We investigate this system numerically using parallel tempering. The model reproduces the magnetic behaviour of oxides with hollandite structure, such as α − MnO2 and presents a rich phenomenology: in absence of disorder three types of ground states are possible, depending on the relative strength of the interactions, namely AFM ordered and two different disordered, macroscopically degenerate families of ground states. Remarkably, for sets of AFM couplings having an AFM ground state in the clean system, there exists a critical value of the disorder for which the ground state is replaced by a spin-glass phase while maintaining all couplings AFM. To the best of our knowledge this is the only existing model that presents this kind of transition with short-range AFM interactions. We argue that this model could be useful to understand the relation between AFM coupling, disorder and the appearance of a spin-glass phase.

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