Читать книгу Encyclopedia of Glass Science, Technology, History, and Culture - Группа авторов - Страница 324

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 of atoms, new methods have been proposed in this field with very promising results for systems ten times bigger. The second method, which will now be described in some detail, is the density functional theory (DFT).