First-principles Method Md Mehidi Hasan Dept. of MSE, KUET Reference: Computational Materials Science, An Introduction by June Gunn Lee 2nd Edition Chapter 4 Basic forces acting on matter • Electromagnetic Force • Gravitational Force • Strong nuclear force • Weak nuclear force Only this force is important, since studies on materials rarely encounter the other three forces The electromagnetic force is the very governing force of nuclei and electrons The governing law for these small particles is known as quantum mechanics Calculations are based solely on the basic laws of physics. Do not use any fitting parameters from the experimental data as used in MD Describes the behavior of microscopic bodies such as subatomic particles, atoms, and other small bodies. They are named as first-principles (or ab initio) methods. In principle, the only information the methods require is the atomic numbers of the constituent atoms. Then, the electromagnetic properties and bond-forming/-breaking that are unobtainable in MD (Molecular Dynamics) can be calculated with these firstprinciples methods. Nuclei and electrons are considered as point-like particles when we treat them as a particle in terms of masses, charges, positions, and so on The subject is especially difficult for materials people since we have accustomed ourselves to relying on instruments to amplify the subatomic phenomena into some form of signal that they can observe macroscopically. The quantum world is a magical world in the sense that it can make an object disappear at the first floor and reappear at the third floor at the same time, or make it penetrate a high wall and place it outside the wall. Note that quantum acts are real things, whereas magic is all well-designed and manipulated illusions. Furthermore, it is fascinating that the quantum world is seamlessly interwoven with the classical world. “Anyone who is not shocked by quantum theory has not understood it”. Niels Bohr (1885–1962) “I think I can safely say that nobody understands quantum mechanics”. Richard Feynman (1918–1988) Classical mechanics was found to be inadequate for explaining phenomena in the subatomic scale Describes the behavior of macroscopic bodies Four theories in the development of the quantum mechanics • • • • Niels Bohr and the quantum nature of electrons De Broglie and the dual nature of electrons Schrödinger and the wave equation Heisenberg and the uncertain nature of electrons Schrödinger wave equation Simplifying the problem All entities including nuclei, electrons, and even time variable are active participants in this model via the wave functions. This system is in fact an extremely complex one that can be solved only for hydrogen-like systems. Unless we do some drastic measures on the model, the equation has no use in practice. A very schematic of the Schrödinger system for an n-electron system with all quantum interactions indicated by black arrows Simplifying the problem ❖ Forget about gravity, relativity and time The first two simplifications—forgetting about gravity and relativity—are obvious, considering that an electron’s mass is so small, and its speed is much slower than that of light. For heavy atoms, however, there are significant relativistic effects. ❖ Forget about nuclei and spin Nuclei are far more massive than electrons and thus, whenever a nucleus moves, the electrons respond instantaneously to nuclear motion and always occupy the ground state of that nuclear configuration. This means that the positions of nuclei are considered to be “frozen” and become not variables but only parameters from the electron’s view. ❖ Forget about the excited states It is limited only within the ground states of electrons at 0 K, which provides a most practical solution for the problems encountered in materials science. Since the ground-state energy is independent of time, we can use a much simpler timeindependent wave equation. ❖ Use of atomic units Unless otherwise written, convenient atomic units are used. These units make many quantum quantities to 1 and thus make equations simpler. Energy Operator: Hamiltonian The Hamiltonian operator, Hoperator, is a sum of all energy terms involved: kinetic and potential energies. • Kinetic energies are of the nuclei and electrons. • Potential energies are of the Coulomb interactions by nucleus–electron, electron-electron and nucleus-nucleus. The terms connected with nuclei can be skipped and therefore Summing all together, the Hamiltonian operator, H , is now given by For example, for hydrogen molecule H2, Early first-principles calculations The early first-principles calculations, the Hartree method and the Hartree–Fock (HF) method, that are based on the Schrödinger wave equation ❖ n-electron problem The underlying physics of the first-principles calculations is not very complex. It works nicely for simple systems such as an electron in a well, hydrogen, or helium. It becomes complex due to: • Materials made of up to several hundred atoms that easily contain several thousand electrons. Calculations for these n-electron systems are truly complex. This is the so-called many-body problem that is discussed for the N-atom systems in MD. • The size of the system. Dealing with n electrons that interact with all other electrons at the same time is just too complex to solve even numerically. Early first-principles calculations ❖ Hartree method: One-electron model To bring the problem down to the tractable level it is assumed that each body is independent and interacts with others in an averaged way. This means that, for an n-electron system, each electron does not recognize others as single entities but as a mean field. Hence, an n-electron system becomes a set of noninteracting one-electrons where each electron moves in the average density of the rest. Schematic of the Hartree model with the mean field approximation for electrons. With this simplified model, Hartree (1928) treated one electron at a time and introduced a procedure he called the self-consistent field method to solve the wave equation: Uext is the attractive interaction between electrons and nuclei UH is the Hartree potential coming from the classical Coulomb repulsive interaction between each electron and mean field Since electrons are independent, the total energy is the sum of n numbers of one-electron energies Then, the n-electron wave function can be simply approximated as the product of n numbers of one-electron wave functions Limitations of Hartree method due to these oversimplifications: ❖It does not follow two basic principles of quantum mechanics: the antisymmetry principle and thus Pauli’s exclusion principle ❖It does not count the exchange and correlation energies coming from the nelectron nature of the actual systems Early first-principles calculations ❖ Hartree Fock method: One-electron model and mean-field approach In the HF method, the true n-electron wave function is approximated as a linear combination of noninteracting one-electron wave functions in the form of a Slater determinant. For example, the wave function of an He atom with two electrons is It exactly expresses the actual electron that changes its sign whenever the coordinates of any two electrons are exchanged This is called the antisymmetry principle, and the determinant follows it by changing the sign when two rows or columns are exchanged. The two wave functions, have additional spin variables (one up-spin and one downspin). Then, the more exact expression for the Slater determinant is a combination of Now, notice that any determinant with two identical rows or columns is equal to zero. This implies that, if two electrons occupy the same spin wave functions (≡ spin orbital), such wave function just does not exist, and thus the Pauli’s principle is satisfied The general expression of the Slater determinant for an n-electron system, if we neglect the spin variable, is Born-Oppenheimer Approximation The Born-Oppenheimer approximation is the assumption that the electronic motion and nuclear motion in molecules can be separated. It leads to a molecular wave function in terms of electron positions and nuclear positions. Due to BO approximation Energy level diagrams can be constructed like: Nuclei remain fixed during much faster electronic transition - transitions are vertical.
