Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов

results for systems ten times bigger. The second method, which will now be described in some detail, is the density functional theory (DFT).

      2.2 Density Functional Theory

      The DFT formalism exploits certain ground‐state properties of many‐electron systems in an external field, thus in our case the Coulomb field of the nuclei. Following up ideas proposed by Thomas and Fermi in the 1930s, it has been rigorously established by Hohenberg, Kohn, and Sham in the early 1960s [2], who showed that the total energy of the system is a functional of the electronic density. DFT gets rid of the many‐body wavefunction, which depends on 3n electronic spatial coordinates, and replaces it by the simpler electronic density ρ(r) that only depends on three spatial coordinates:

      (3)

      Kohn and Sham [3] showed that this density can be written as a sum of the density of noninteracting particles, , where ϕ_i are the fictitious one‐particle Kohn–Sham (KS) orbitals, and thus the ground‐state density ρ₀(r) is given by the sum of the ground states of these particles. Hence, the total energy of the system can be expressed as

      (4)

where T[ρ] is the kinetic energy of a system of n noninteracting electrons having the density ρ, , and the second term is the Coulomb interactions between electron–electron and electron–nuclei. The third term is the so‐called exchange and correlation energy which accounts for all quantum many‐body effects due to the Pauli exclusion principle which correlates electrons. Since this term cannot be evaluated exactly, approximations have to be made. Even though E_xc[ρ] is usually substantially smaller than the two other terms, the manner it is approximated may become crucial for chemically complex systems [2]. The simplest one, proposed originally by Kohn and Sham [3], relies on the assumption that at each point the exchange‐correlation energy density corresponds to that of a homogeneous gas of electrons. This approximation is called the “local density approximation” (LDA) and is given by:

      (5)

      Even for such a simple model system, the expression of the correlation energy has to be calculated numerically with Monte‐Carlo methods. A more advanced approximation, the so‐called “generalized gradient approximation” or GGA, is based on a more complex operator making use of the density gradient of mth order:

      (6)

      However, this higher approximation does not necessarily give more reliable results so that it is a priori not clear which exchange‐correlation functional should be used [2]. Despite this problem one can say that the KS approach and reasonable (simple) approximations for the exchange‐correlation term have opened the door to the calculations of the electronic structure for many‐atom systems relevant to the study of real materials.

      Although expressing the full quantum mechanical problem in the language of DFT leads to a significant reduction of the computational effort for calculating the forces on the nuclei, in practice this task is still extremely demanding when dealing with more than a few tens of atoms. Therefore, one usually makes the further approximation that all the core electrons of an atom are lumped together so that their effect is replaced by an effective potential, the so‐called “pseudo‐potential,” for the remaining valence electrons which are described by a pseudo‐wavefunction [2]. The physical motivation for this approximation is that the chemical bonds between two atoms are usually related to the outer valence electrons and thus depend only weakly on the inner core electrons.

      2.3 Computational Limitations

      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