第一性原理整理總結(jié) 英文版適合組會

1、First-principles calculationsXueli Sun ContentsvWhat are first principles calculations? vHow do we do first principles calculations?vWhat is DFT?vHohenberg-Kohn theoremv Kohn-Sham equationvLocal -Density Approximation(LDA)v Pseudopotentials What are first principles calculations? A material is simpl

2、y a collection of atoms that are bound by chemical reactions. Chemical reactions, in turn, are simply interactions between electrons. This means that all material properties (chemical, mechanical, electrical, optical) can, in principle, be predicted from nothing more than the atomic number and mass

3、of the atomic species involved, with the aid of quantum physics. This is precisely what first principles calculations attempt to do.How do we do first principles calculations? Solving the Schrdinger equation by brute mathematical force is extremely demanding computationally and not practical for all

4、 but tiny systems. Instead, we use a combination of two physical approximations: 1. We use density functional theory (DFT). 2. We use first principles pseudopotential theory. What is DFT? DFT maps the original many-electron problem into an equivalent single-electron problem. It does so by lumping al

5、l the many-body quantum phenomena (such as Paulis exclusion principle and electron correlation) into a single additive “exchange-correlation” potential, which is a functional of the charge density alone. In practice, the exact functional is unknown and people use approximate forms for the functional

6、, usually (but not always) derived from properties of a uniform electron gas. 1.Hohenberg-Kohn theorem The first HK theorem demonstrates that the ground state properties of a many-electron system are uniquely determined by an electron density that depends on only 3 spatial coordinates. The second HK

7、 theorem defines an energy functional for the system and proves that the correct ground state electron density minimizes this energy functional. rrdrvVE The original HK theorems held only for non-degenerate ground states in the absence of a magnetic field, although they have since been generalized t

8、o encompass these. 2.Kohn-Sham equation Within the framework of Kohn-Sham DFT (KS DFT), the intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of non-interacting electrons moving in an effective potential . The effective potential

9、includes the external potential and the effects of the Coulomb interactions between the electrons, e.g., the exchange and correlation interactions. 212NiiXCKSiiiKSrrrErrrrdrrVrErrV3.Local -Density Approximation (LDA) The exchange-correlation part of the total-energy functional remains unknown and mu

10、st be approximated. Modeling the latter two interactions becomes the difficulty within KS DFT. The simplest approximation is the local-density approximation (LDA), which is based upon exact exchange energy for a uniform electron. r(r)rrrEVdr)(EXCXCLDAXCLDAXCXCLDAXCr where is the electric density and

11、 xc is the exchange-correlation energy density Pseudopotentials The periodic table tells us that chemical reactivity is governed by valence electrons, with core electrons being chemically inert. Pseudopotentials make use of this basic fact by replacing the inert core electrons with an effective pote

12、ntial. This reduces, sometimes drastically, the number of electrons we need to solve for. Even more importantly, this results in much smoother wave functions for the remaining valence electrons, making the problem much easier to solve numerically. 1.Norm-conserving Pseudopotentials 2.Ultrasolf PseudopotentialsConclusion1.Our approximations are systematic. 2.All “hid