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1、First-principles calculationsXueli Sun,Contents,What are first principles calculations?How do we do first principles calculations?What is DFT?Hohenberg-Kohn theorem Kohn-Sham equationLocal-Density Approximation(LDA)Pseudopotentials,What are first principles calculations?,A material is simply a colle
2、ction 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 of the atomic speci
3、es 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 but tiny systems.,I
4、nstead,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 all the many-body quantum phe
5、nomena(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,usually(but not always)derived f
6、rom properties of a uniform electron gas.,1.Hohenberg-Kohn theorem,The first HK theorem demonstrates that theground stateproperties of a many-electron system are uniquely determined by an electron densitythat depends on only 3 spatial coordinates.,The second HK theorem defines an energy functional f
7、or the system and proves that the correct ground state electron density minimizes this energy functional.,The original HK theorems held only for non-degenerate ground states in the absence of amagnetic field,although they have since been generalized to encompass these.,2.Kohn-Sham equation,Within th
8、e framework of Kohn-ShamDFT(KS DFT),the intractablemany-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 includes the external potential and the effects of the
9、Coulomb interactionsbetween the electrons,e.g.,the exchangeand correlationinteractions.,3.Local-Density Approximation(LDA),The exchange-correlation part of the total-energy functional remains unknown and must be approximated.Modeling the latter two interactions becomes the difficulty within KS DFT.T
10、he simplest approximation is the local-density approximation(LDA),which is based upon exact exchange energy for a uniform electron.,whereis theelectric densityandxc is the exchange-correlation energy density,Pseudopotentials,The periodic table tells us that chemical reactivity is governed by valence
11、 electrons,with core electrons being chemically inert.Pseudopotentials make use of this basic fact by replacing the inert core electrons with an effective potential.This reduces,sometimes drastically,the number of electrons we need to solve for.,Even more importantly,this results in much smoother wa
12、ve functions for the remaining valence electrons,making the problem much easier to solve numerically.1.Norm-conserving Pseudopotentials 2.Ultrasolf Pseudopotentials,Conclusion,1.Our approximations are systematic.2.All“hidden parameters”within them are determined objectively from theoretical results for properties of the electron gas(DFT)or the isolated atom(pseudopotentials),and not by fitting experiments.,
