Appendix A Ab initio Computational Methods The objective of ab initio approaches based on applied quantum theory is to calculate stationary states for electrons in the electrostatic field of nuclei,i.e., the electronic structure (ES).The energy of this ground state can then serve as a potential energy for displacements of nuclei.From the point of view of ideal-strength (IS)calculations,the total energy of the system is the most important output of ab initio methods. The main advantage of ab initio methods is their independence of exper- imental data.Unlike in the case of empirical and semi-empirical methods, there is no need for calibration parameters.Thus,they can also be used for calculation of some structural and mechanical characteristics of hypothetical systems(prediction of properties of materials that have not yet been devel- oped)or study of materials behaviour under large deformations(far from the equilibrium state)that can give a better understanding of micromechanisms of materials failure. First attempts to develop applicable theories were made in the late 1920s [421,422,a few years after the foundations of modern quantum the- ory were laid(derivation of the Schrodinger equation).A very successful step forward was the Hartree-Fock method [422,423].This method yields very accurate bond lengths in molecules.On the other hand,the binding energies are generally not in good agreement with experimentally obtained energies. Moreover,for solids,the Hartree-Fock method has problems with a descrip- tion of band structures.The density functional theory (DFT)37,38]was invented to include correlation effects without using the very costly wave function methods.All the methods used within this book are based on the DFT. In DFT the energy is not obtained as eigenvalues of a wave function,but rather as a functional of the electron density.The complex problem of many interacting electrons is transformed into a much simpler study of single elec- tron interactions with an effective potential Uef created by other electrons and all nuclei.This is expressed by the Kohn-Sham equation (one-electron Schrodinger equation) 249Appendix A Ab initio Computational Methods The objective of ab initio approaches based on applied quantum theory is to calculate stationary states for electrons in the electrostatic field of nuclei, i.e., the electronic structure (ES). The energy of this ground state can then serve as a potential energy for displacements of nuclei. From the point of view of ideal-strength (IS) calculations, the total energy of the system is the most important output of ab initio methods. The main advantage of ab initio methods is their independence of exper￾imental data. Unlike in the case of empirical and semi-empirical methods, there is no need for calibration parameters. Thus, they can also be used for calculation of some structural and mechanical characteristics of hypothetical systems (prediction of properties of materials that have not yet been devel￾oped) or study of materials behaviour under large deformations (far from the equilibrium state) that can give a better understanding of micromechanisms of materials failure. First attempts to develop applicable theories were made in the late 1920s [421, 422], a few years after the foundations of modern quantum the￾ory were laid (derivation of the Schr¨odinger equation). A very successful step forward was the Hartree–Fock method [422, 423]. This method yields very accurate bond lengths in molecules. On the other hand, the binding energies are generally not in good agreement with experimentally obtained energies. Moreover, for solids, the Hartree–Fock method has problems with a descrip￾tion of band structures. The density functional theory (DFT) [37, 38] was invented to include correlation effects without using the very costly wave function methods. All the methods used within this book are based on the DFT. In DFT the energy is not obtained as eigenvalues of a wave function, but rather as a functional of the electron density. The complex problem of many interacting electrons is transformed into a much simpler study of single elec￾tron interactions with an effective potential Ueff created by other electrons and all nuclei. This is expressed by the Kohn–Sham equation (one-electron Schr¨odinger equation) 249