Abstract
In several domains of physics, including first principle simulations and classical models for polarizable systems, the minimization of an energy function with respect to a set of auxiliary variables must be performed to define the dynamics of physical degrees of freedom. In this paper, we discuss a recent algorithm proposed to efficiently and rigorously simulate this type of systems: the Mass-Zero (MaZe) Constrained Dynamics. In MaZe, the minimum condition is imposed as a constraint on the auxiliary variables treated as degrees of freedom of zero inertia driven by the physical system. The method is formulated in the Lagrangian framework, enabling the properties of the approach to emerge naturally from a fully consistent dynamical and statistical viewpoint. We begin by presenting MaZe for typical minimization problems where the imposed constraints are holonomic and summarizing its key formal properties, notably the exact Born–Oppenheimer dynamics followed by the physical variables and the exact sampling of the corresponding physical probability density. We then generalize the approach to the case of conditions on the auxiliary variables that linearly involve their velocities. Such conditions occur, for example, when describing systems in external magnetic field and they require to adapt MaZe to integrate semiholonomic constraints. The new development is presented in the second part of this paper and illustrated via a proof-of-principle calculation of the charge transport properties of a simple classical polarizable model of NaCl.
Highlights
The method of mass-zero constraints was originally introduced in the early 1980s to study the rotational– translational coupling in diatomic molecules [1]
Current methods adopted for the Molecular Dynamics (MD) simulation of such systems combine standard propagation schemes for the slow variables—that we shall indicate as the ions for simplicity—with algorithms to find, or approximate, the minimum of the potential with respect to the fast dofs at each ionic configuration
The always stable predictor–corrector scheme is only approximately time-reversible [13,14] leading again to energy drifts that are usually quenched via a Berendsen thermostat, and it contains system-dependent parameters that can only be determined by trial and error
Summary
The method of mass-zero constraints was originally introduced in the early 1980s to study the rotational– translational coupling in diatomic molecules [1] It has undergone a new set of developments when adiabatic systems were identified as an important area where MaZe dynamics can provide an original formal approach and an effective integration algorithm [2,3,4]. In Molecular Dynamics (MD), a typical example of adiabatic dynamics is the evolution of ionic (slow) and electronic (fast) degrees of freedom in first principle calculations based on Kohn–Sham or orbital-free Density Functional Theory Another important example is given by classical models of polarization in which electrons do not appear directly, but the dynamical system is extended to include sets of classical auxiliary variables of null mass mimicking different polarization effects.
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