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2.9 First‐principles Simulations of Glass‐formers

       Walter Kob and Simona Ispas

       Laboratoire Charles Coulomb (L2C), University of Montpellier, CNRS, Montpellier, France

1 Introduction

      In their early days, i.e. in 1940–1950, computer simulations were mainly done to address questions in statistical physics, such as the properties of hard‐sphere systems or the dynamics of simple crystals [1]. When a few decades later computers became more powerful and more accessible, it became possible to study the properties of real materials. For this, researchers used an approach that today is referred to as classical molecular dynamics (MD), i.e. the interactions between the particles that constitute the material are described by an effective potential with a form chosen in a rather ad hoc manner, e.g. Lennard‐Jones. The parameters of the potential (e.g. the depth and the position of the well in the Lennard‐Jones potential) are selected such that the density, the melting temperature for a crystal, or other macroscopic properties of the simulated system matches experimental data. Once these interactions are known, one solves numerically Newton’s equations of motion and hence obtains the trajectories of all the particles (Chapter 2.8, [1]). From these trajectories, one then can determine physical properties of the system such as the radial distribution function, diffusion coefficients, or elastic constants.

      Although simulation studies with effective potentials are very valuable to gain qualitative insight into the properties of liquids and solids, they usually do not allow to obtain a good quantitative description. The reason for this failure is that most properties depend in a quite sensitive manner on the potential that describes the interactions between the constituting particles of the material. Since normally these interactions depend not only on the type of atoms considered but also on the microscopic environment of the particles (e.g. the bond strength between an oxygen and a silicon will change if a hydrogen is approached to this pair), it is basically impossible to come up with a simple potential‐energy expression that could describe faithfully all these different environments. This problem is particularly pronounced for glasses where, in contrast to crystals, each atomic species has multiple local environments. At present, the only method that can address this problem in a systematic manner is the ab initio formalism (also called first‐principles). In this approach one does not rely on a fixed functional form for the interaction potential, but uses instead the electronic degrees of freedom of all atoms to compute the forces acting on them in the system. Once these forces are known, one solves Newton’s equation of motion to determine the trajectory of the particles and, subsequently, the properties of the system. Thus a priori the only input needed for these simulations are the species of the particles, a feature that has allowed ab initio simulations to become a highly attractive method to gain a detailed understanding of the properties of glass‐forming liquids and glasses.

      The goal of this review is to give a brief introduction to the method, discuss its advantages but also its problems, and then to present some specific examples showing that this type of simulation is indeed very useful to improve our understanding of glasses at the microscopic scale. We will first focus on structural properties in view of the major problems caused by atomic disorder, especially in complex glasses. Vibrational properties will then be examined as the heat capacity, thermal conductivity, or transparency directly or indirectly depends on them. Because solid‐state nuclear magnetic resonance (NMR) yields detailed insights into the local structure of materials (Chapters 2.2 and 2.4), especially when long‐range symmetry is lacking, we will conclude this chapter with a brief description of the calculation of NMR spectra.

2 Ab Initio Simulations

      2.1 General Principles

      In the ab initio approach, the forces on the atoms are obtained from the electronic degrees of freedom of the system as determined from the Schrödinger equation,

(1)

      where the (complex) many‐electron wavefunction Ψ({r _i}; {R _I}) depends on the positions of the n electrons, {r _i}, as well as the positions of the N nuclei, {R _I} [2], E is the energy of the electronic degrees of freedom, and the operator ℋ_e is given by

      (2)

with Z_I the charge of ion I. This equation is written in atomic units, i.e. Planck's constant, the electronic charge and mass are set to unity, and the Laplacian ∇² is given by ∇² = ∂ ²/∂x ² + ∂ ²/∂y ² + ∂ ²/∂z ². In Eq. (1), only the electronic degrees of freedom are treated quantum mechanically whereas those of the nuclei are not. This approximation, often called “Born–Oppenheimer” or “adiabatic,” is quite accurate since the mass of the electron is almost 2000 times smaller than the one of the lightest nucleus, i.e. hydrogen. Hence, the degrees of freedom of the electrons are basically decoupled from those of the nuclei, i.e. the heavy nuclei move more slowly than the light electrons which thus adapt instantaneously to the changes of the nuclear positions. As a consequence, one assumes for the electronic structure calculations that the nuclei are clamped at fixed positions and that hence the electronic wavefunction Ψ depends on {R _I} only parametrically [2].

      To reduce the computational complexity, even further one considers only the ground‐state solution of Eq. (1), Ψ₀, i.e. the electronic state with the lowest energy, which is in fact the only one that matters even for melts at high temperatures. The interaction potential between the nuclei is then given by and the force on particle I is given by F_I = −∇_IΦ({R _J}) [2].

      Finding the wavefunction Ψ₀ that is the solution of the Schrödinger equation (1) is a formidable task and there are two main approaches to solve it. The first one is a wavefunction‐based method, often known as the quantum‐chemistry approach, which starts from the Hartree–Fock method. In a first‐order approximation, it factorizes the many‐body electronic wavefunction into one‐particle wavefunctions whose ground states are then searched numerically. Although at present the solution is found in a reasonable amount of time for several tens