Читать книгу Encyclopedia of Glass Science, Technology, History, and Culture - Группа авторов - Страница 326

So far, we have discussed how DFT allows one to obtain the forces between the nuclei due to their surrounding electrons. These forces can now be used to solve the equations of motion for the nuclei:

(7)

where the energy E_KS[{ϕ_i}, R] can be calculated from the KS orbitals within the KS scheme of the DFT. In the above equations the nuclei are considered as classical particles and Eq. (7) is solved using the same methods as in classical MD (see Chapter 2.8 or [1] for details). Because of the resulting motion of the ions, the electronic structure changes and hence one has in principle to recalculate the total energy of the electronic ground state, a procedure that is computationally very costly even within the DFT approach. One possibility to avoid this problem was proposed in a seminal paper by Car and Parrinello in 1985 [4]. The idea behind this so‐called Car–Parrinello molecular dynamics (CPMD) is to introduce a fictive dynamics to the electronic degrees of freedom and thus to recast the quantum mechanical problem into a classical one with the electronic wavefunctions as new effective degrees of freedom.

Although the CPMD approach allows to obtain the correct equilibrium properties of a system, the introduction of the fictive dynamics for the electronic degrees of freedom makes that the motion of the system in configuration space is not completely realistic [2]. Despite this shortcoming, CPMD simulations are still widely used to study complex systems by means of computer simulations (cf. software available at http://www.psi‐k.org/codes.shtml). This problem can, however, be avoided with the so‐called Born–Oppenheimer molecular dynamics (BOMD) in which one solves at each time step the electronic problem. This approach thus allows to give a correct dynamics but at the cost of an increased computational load. Only in the last few years the numerical algorithms have been improved to such an extent that today it is possible to simulate within BOMD several hundreds of particles [5, 6].

This brief description of the ab initio simulations should make it clear that in practice the codes for such simulations must be extremely optimized in order to keep the necessary computer time within a reasonable limit. This is why, in contrast to simulations performed with effective potentials, first‐principles calculations are not made with “homemade” codes but with one of the very sophisticated and highly optimized packages that have been developed over the years. Various groups use different approaches to maintain and develop these packages, the most popular ones being CPMD, VASP, Quantum Espresso, CP2K, Siesta, CASTEP, etc. (see on http://www.psi‐k.org/codes.shtml for a more extended list). Each of them has advantages and disadvantages regarding the scaling of computational effort with system size, accuracy, ensembles that can be simulated (microcanonical, canonical, constant pressure, etc.), quantities that can be calculated, etc., and therefore the best choice will depend on the application.

To give an idea of what can be done at present with ab initio simulations, we compare in Table 1 ab initio and classical simulations. Since the computational load for the former does depend on the system considered (owing to the different electronic structure for the atoms), we will consider the example of an oxide glass. We note that, for a given computing time, there is trade‐off between the size of the system and the time span covered by the calculation. For a classical system the relevant number is basically the product of the two quantities so that doubling the system size increases the computer time by a factor of two. For ab initio simulations the situation is not that favorable since doubling of the system size usually leads to an increase of the computational load by a factor of 2^α , where α is about 3. As a consequence, system sizes are smaller and accessible timescales are shorter in ab initio than in classical simulations. Therefore, the quench rates used to cool a system from its liquid to its glass state are usually on the order of 10¹³–10¹⁵ K/s. Such rates are thus significantly larger than those of classical simulations (10¹⁰–10¹⁴ K/s) and of course ways higher than experimental values (10⁻² − 10⁶ K/s). Despite these huge differences, the resulting glasses are surprisingly similar since many of their properties depend only in a logarithmic way on these rates. Therefore, it does make sense to use ab initio simulations to investigate the properties of glasses on the microscopic scale.

Table 1 Comparison of various features of large‐scale computer simulations carried out with classical and ab initio methods.

	Classical MD	Ab Initio MD
Size	1 000–500 000 atoms	100–500 atoms
Box size	∼100 Å	∼15–20 Å
Trajectory length	∼1 ns–10 μs	∼20–100 ps
Transferability	No	Yes