Lecture3DensityFunctionMethod
VASP lecture_2
Lec 2 (2/8/02) Survey of Simulation TechniquesThe Multiscale Materials Modeling ConceptDuring the past fifteen or so years, materials research has gained increasingly wide-spread recognition in the scientific community for its relevance to technological innovation and competitiveness. The advent of another powerful driver -high performance computing -has given rise, essentially in parallel, to intense interest in the computational approach to advanced materials research. The term multiscale materials modeling (MMM) has now taken on the meaning of theory and simulation of materials properties and behavior across length and time scales from the atomistic to the macroscopic. Although sometimes the importance of experiments is not explicitly acknowledged, it is nevertheless understood, at least by the more informed practitioners of materials simulation that selected experimental information will be indispensable to successful modeling, for providing appropriate database to determine the theoretical parameters and for validating the critical model assumptions.The relevance of multiscale modeling is fundamentally predicated on the belief that such analysis and prediction will bring about better understanding and control of materials microstructures, which in turn are essential in the application areas of processing, performance evaluation, and ultimately the design of new materials. Because physics, chemistry, and various fields of engineering are all disciplines important to materials modeling, this emerging field has broad appeal to students with a diverse range of backgrounds. For basically the same reason, there is considerable impetus for different groups of research investigators at universities and research labs to team up to tackle grand-challenge type problems that lie beyond the expertise and capabilities of any single group.As materials research expands in both breadth and depth, molecular engineering of materials becomes more of a reality. This is the long-held dream of every materials scientist and engineer where new materials can be created with such benefits as enhanced performance, extended service life, and acceptable environmental impact, not to mention cost reduction. Even though computer-aided materials design has yet to achieve the success of computer-aided molecular (drug) design, there is impressive progress being made, especially in the area of functional materials for microelectronics, optical and magnetic applications. In contrast, for structural materials the complexities of mechanical, thermal, chemical (alloying, corrosion, etc.) phenomena continue to pose formidable challenges to reliable and predictive modeling. As a result, it is blieved that the most promising approach to understanding and control of these phenomena is to effectively combine several simulation techniques, each one being suited for a particular length and time scale.One can think of MMM as an open environment in which materials phenomena are described over a continuous distribution of length and time scales. The spirit of MMM is outward looking, intuitively logical, and flexible that it invites the investigtor to come up with his own interpretation. Yet despite the different ways of saying what is MMM, no general concensus on how to define the underlying concept has been established. Much like a good painting, it can have different meanings for different people, and in this way it maintains a certain freshness and contemporary appeal.Simulation Techniques in Multiscale Materials ModelingReturning to Fig. 1-1, shown at the last lecture, one sees that in materials processing the length scales span the range from 1 angstrom to about 1 m. The 10 decades go from nanotechnologies at the short end to system components and structures we normally associate with the macroscopic world. The order in which the topics themselves are arranged is interesting in that it begins with atomistic modeling which is one of the 4 characteristic length scales in MMM (see below) but not the smallest length scale. Several entries in this listing could be relevant to what we will be discussing later in the term, such as molecular dynamics/diffusion, phase equilibria/transformation, elasticity-plasticity, thermodynamics, thermomechanics, and deformation processing. Just these alone already give one the impression that materials processing involves a very wide range of physical phenomena, and this is only a partial list. We can regard this viewgraph as a reminder of the vastness of the material modeling landscape and the need to have a sense of direction in one's own research.In many fields of scientific inquiry, it is commonly recognized that a single physical phenomenon can be examined at several levels or length (time) scales. For example, the complicated motion of a ocean wave as it washes onto a beach can be observed by seeing it in real life, or it can be visualized (in our minds eye) in terms of the individual movements of the water molecules which make up the wave. Depending on the scale of interest, the relevant dynamics requires quite different ways of analysis -in this case, continuum fluid dynamics to describe waves breaking on a beach and discrete-particle molecular dynamics to describe the atomic motions of the molecules.Fig. 2-1For many materials problems one can identify four distinct length scales where different aspects of a physical phenomenon can be analyzed. As shown in Fig. 2-1 these four regions maybe referred to as electronic structure, atomistic, microstructure, and continuum. Imagine a piece of material, say a crystalline solid. The smallest length scale of interest is about a few angstroms (10-8 cm). On this scale one can deal with the individual electrons of an atom. The appropriate method for modeling relevant processes is called density functional theory or first-principles (ab initio) calculation. Being fully quantum mechanical, it is the most computationally demanding of all the techniques, and as a result it can be applied only to small simulation systems, the current limit being about 300-400 atoms. The system being modeled at the electronic structure level is therefore a collection of atoms in the form of nuclei (or ions) and electrons, see Fig. 2-2.Fig. 2-2Sometimes the electrons are further separated into core and valence electrons. The ions are described classically using Newton's equations of motion, F = ma , while the electrons are governed by the Schroedinger equation, H y = E y . The computationally intensive part of the calculation lies in the solving the Schroedinger equation. Once the wave function y is determined, we then know the electron charge distribution in the system, r = y 2 , which tells us about the bonding between atoms, a very fundamental and useful piece of information.The next scale in Fig. 2-1, spanning hundreds of angstroms, is called atomistic. Here the system is represented a collection of atoms. The appropriate simulation techniques are called molecular dynamics (MD) and Monte Carlo (MC); they are quite well developed and will be discussed in detail in Weeks 3 and 4 respectively. Such methods require the knowledge of an interatomic potential function U(r 1,..., r N ) (recall Lec 1) which are often obtained empirically byfitting a functional form with several parameters to experimental data. By not treating the electrons explicitly, atomistic simulation is computationally much simpler than electronic structure calculations, which means that one can treat a much larger number of particles, the limit being about 109 at the present time. The equations to be solved in the case of molecular dynamics is again Newton's equation of motion, while in the case of Monte Carlo one does not solve any equations of motion but instead sample a certain distribution of positions. With either simulation the essential input is the same, the potential function U. The goodness of a simulation therefore depends on how accurately do we know U for the material of interest. Because the electronic structure effects are ignored, atomistic simulations are not as reliable as ab initio calculations. This is especially true in situations where chemical bonds are broken as in fracture or rearranged as in chemical reactions. On the other hand, one has the advantage of being able to simulate much larger systems over much longer times. This in turn means one can study more physical properties and behavior of the material.The length scale above atomistic is sometimes called the mesoscale, the characteristic length being a micron (104 angstrom). Here the simulation technique commonly in use is the finite-element method (FEM), where the system is represented by a grid of elements, triangles in 2 dimensions, cubes or tetrahedra in 3 dimensions, for example. The equations to be solved describe force balance on each element, the output being the nodal displacements of the specified grid. Typically, the grid is chosen to be fine enough to represent idealized microstructural features of micron size, which makes this method well suited for the simulation of materials microstructure in many applications. Notice, however, for FEM calculation one needs to specify the force acting on an element due to its neighbor, and this kind of mterials property has to be provided either from experiment, if available, or from studies at the atomistic or ab initio level.Beyond the mesoscale one has the macroscale where the material is represented as a continuum described by distributions which vary rather smoothly on the scale of mm and larger. As an example we can imagine the temperature distribution in a rod that is heated at one end. The equation describing this system is the time-dependent heat conduction equation (Fig. 2-2), with k denoting the thermal conductivity of the material. Like the FEM simulation, the materials properties needed for simulation at the continuum level have to be supplied externally. In this case the conductivity is a quantity that one measure experimentally, or calculate by means of atomistic simulation.We can see why it is important to couple the different length scales. For practical design calculations, either continuum or finite-element methods are the most useful. However, these methods require parameter or property specifications that cannot be generated internally, nor can they provide the electronic structure or atomic-level understanding of mechanisms that are essential for prediction and design. It is only when the different methods are effectively integrated that one can expect materials modeling to give fundamental insight, or reliable predictions on the appropriate length scale. Once this is achieved, then materials modeling can justifiably claim to be a valuable complement to experiment and theory.In Fig. 2-1 we also indicate the regions where nanoscience and technology would operate, in contrast what is previously the microscale. With regard to biological science at the cellular levl, the range spans from the nanoscale up to tens or even hundreds of microns. The presnetlength scale classification has counterparts in time scales. Moreover, one can discuss various physical properties that manifest on the different length-time scales, and finally one can even roughly associate different disciplines with each level of scales. For example, electronic structure would be the domain of quantum chemistry and condensed matter physics, atomistic and microstructural levels would be materials science and engineering, and continuum level would be the domain of engineering.In as much as the advantages of linking simulation and modeling techniques across different length scales are quite apparent, actual implementation of coupling the different levels of description is highly nontrivial. It is fair to say that no single research group thus far has fully succeeded in demonstrating such a capability, so there are still many opportunities for original research. To emphasize this point we return to Fig. 1-3 which describes a research program currently on-going at the Lawrence Livermore National Laboratory with the goal of understanding dislocation dynamics for applications in high strain-rate deformation. The idea is to take a bottoms-up approach of linking atomistic simulation of dislocation interactions and mobility with mesoscale simulation of single crystal and polycrystal plastic deformation to arrive at appropriate constitutive relations which then may be incorporated into design codes at the continuum level. This is presently one of the most comprehensive efforts to solve a critical materials problems. Its success will go a long way in demonstrating the unique power of the MMM approach to materals research.Fig. 2-3Fig. 2-3 shows several examples of FEM simulations in materials processing. One sees (counter clockwise) the grain microstructure in a turbine blade and a block undergoing directional solidification, the flow pattern or velocity plot in a casting where solid and melt zones begin toseparate, and the thermal profile or temperature zones in a directionally chilled continuous casting. The results demonstrate the usefulness of modeling microstructure in materials technology.Image RemovedFig. 2-4Fig. 2-4 is a cover of a journal devoted to computer-aided materials design, showing a composite of simulation results at different length scales. Starting at the bottom one has the charge distributions in crystalline silicon given by electronic-structure method of density functional theory. As we have mentioned, such information cannot be obtained by any of the other methods in Fig. 2-1. Going up on the left side, one has a molecular dynamics simulation of a crack tip in a two-dimensional crystal under uniaxial tension. The crack (dark region), while moving upward, has started to branch out to take on a jagged appearance which is also seen in laboratory measurement. In Fig. 2-5 we see the same simulation in greater details, the individual atoms in the simulation are now visible. The sequence of six snapshots begins on the left from the top down. The two colors denote the two opposing directions of strain. The fact that the strain directions start to mix as the simulation proceeds means that the state of strain is no longer simpleonce the crack moves further into the crystal. This simulation involves about 30 million atoms, quite large by present standards. If one were to show all the particles in some kind of colorFig. 2-5Fig. 2-6coding scheme, as in Fig. 2-6, one would lose the discreteness of the simulation and the system starts to look like a continuum, including the stress waves that are reflected from the fixed borders of the simulation cell (reminiscent of ripples on the surface of a lake). This is an illustration of how one level of simulation can merge into the next level. The impression we would like to leave here is that atomistic simulation can reveal very local details of how the material is tearing apart, a level of microscopic information that experiments at present cannot provide. The challenge is to understand this kind of complexity and use the knowledge to design materials that would bemore resistant to failure.。
DFT的14个问题
Fourteen Easy Lessons in Density Functional TheoryJOHN P.PERDEW,ADRIENN RUZSINSZKYDepartment of Physics and Quantum Theory Group,Tulane University,New Orleans,LA70118Received1April2010;accepted5May2010Published online in Wiley InterScience().DOI10.1002/qua.22829ABSTRACT:Density functional theory(DFT)is now the most commonly usedmethod of electronic structure calculation in both condensed matter physics andquantum chemistry,thanks in part to the focus it has received over the first50years ofthe Sanibel Symposium.We present a short history,and review fourteen short and easybut important lessons about nonrelativistic DFT,with some partiality but with aminimum of technical complication.V C2010Wiley Periodicals,Inc.Int J Quantum Chem000:000–000,2010Key words:density functional theory;exchange–correlation energy;electronicstructure theory;Kohn–Sham theoryIntroduction and Short History of Density Functional TheoryT his article summarizes a talk given at the 50th anniversary Sanibel Symposium on Quantum Chemistry.Its original title,‘‘Some Things We have Learned About Exchange and Correlation in the Last Fifty Years,’’was some-what too restrictive in both subject matter and time frame.It presents14easy lessons in nonrela-tivistic density functional theory(DFT)at a quali-tative level.The selection of lessons is partial in both senses of the word:it is incomplete,and it reflects our own biases about what is most impor-tant and interesting(or at least most familiar). The same can be said of the short history we present.Our subject is appropriate to the occa-sion,since the Sanibel Symposium has played a major role in the development of modern elec-tronic structure theory,including but not limited to DFT.Possible companion pieces to this article are our short discussion[1]of‘‘perplexing’’issues in DFT and a detailed review[2]of the exact theory and its approximations.Often we need to predict the ground-state properties of an atom,molecule,solid,nanostruc-ture,or other system.We might need the equilib-rium geometry or structure,vibrational frequen-cies,electron density,total energy,and various total energy differences such as atomization and surface energies,as well as the linear and nonlin-ear responses to external static probes.One way to find the needed properties is to solve theCorrespondence to:J.P.Perdew;e-mail:*****************Contract grant sponsor:National Science Foundation.Contract grant number:DMR-0854769.International Journal of Quantum Chemistry,Vol.000,000–000(2010) V C2010Wiley Periodicals,Inc.N -electron Schroedinger equation for the N -elec-tron ground-state wavefunction.This approach is potentially accurate and complete (providing all the information that can be known),but it is com-putationally inefficient and impractical for large N .A second way is to solve for more limited in-formation,as provided by a Green’s function,density matrix,or electron density.This approach may be less accurate and complete,but it can achieve a useful computational efficiency even for large N .Kohn–Sham DFT [2,3]uses spin orbitals to predict the ground-state electron density,total energy,and related important properties.It pro-vides an often useful and improvable compromise between accuracy and computational efficiency.Thus,it is now the most widely used method of electronic structure calculation in both condensed matter physics and quantum chemistry,two sub-jects that have long been intertwined at the Sani-bel Symposia.However,other methods [4],including full wavefunction methods,continue to improve in accuracy and efficiency,providing for selected systems a benchmark of accuracy.Orbital-free DFT began in the 1920s with the Thomas–Fermi theory [5,6],which expresses the total energy E approximately in terms of the elec-tron density n ð~r Þ,where n ð~r Þd 3r is the average number of electrons in volume element d 3r at position ~r ,using the simplest density functional that makes sense.The density is then varied atfixed electron number N ¼R d 3rn ð~r Þto minimize the energy functional.This approach gives simple and useful estimates for the density and total energy of an atom (defined as minus the mini-mum work to strip all the electrons from the nu-cleus)but it is far too crude for chemistry.In fact,Teller [7]proved that in Thomas–Fermi theory,atoms do not bind together to form molecules and solids.Without the exchange–correlation energy,‘‘nature’s glue’’[8],chemical bonds are ei-ther absent or far too long and weak.However,orbital-free methods continue to improve;see recent work by Trickey and others [9–11].Spin orbitals or fictitious one-electron wave-functions w i ð~r ;r Þwere introduced in the 1930s by Hartree,Fock,and Slater [12].The energy (includ-ing sometimes exchange but not Coulomb correla-tion)was expressed in terms of these orbitals and their occupation numbers f i (1or 0,since electrons are fermions),and minimized with respect to them.This approach binds atoms into molecules and solids,although usually too weakly [8],andgenerally improves the total energy E and elec-tron density n ð~r Þ¼P i r f i r w i r ð~r Þj j 2over orbital-free methods.The Hartree and Hartree-Fock methods were not easy to implement on early computers and omitted important correlation effects.In the 1950s,Slater combined the orbital and density functional approaches by creating a local density approxima-tion (LDA)for the exchange–correlation energy and potential (which he called the X a approxima-tion [12]).He found that it could be easily imple-mented self-consistently on the computer and that it included a rough but useful estimate of correla-tion.After retiring from MIT in 1966,Slater joined the Quantum Theory Project at the University of Florida,Gainesville,which held the first Sanibel Symposium in 1961.In 1964–1965,the Hohenberg–Kohn [13]and Kohn–Sham [3]theorems,the twin pillars of mod-ern DFT,were published.These theorems showed that,given the right density functionals,one can find the exact ground-state density and energy of an N -electron system in an external scalar poten-tial using either the total density or the orbitals as variational objects.Kohn and Sham also proposed an LDA for the exchange–correlation energy,which,unlike Slater’s,is exact for an electron gas of uniform or slowly varying density.Kohn–Sham theory was not widely known until around 1970,when condensed matter physi-cists started to find that this theory in the LDA gives a remarkably realistic description of bulk solids and their surfaces.Since then,DFT has dominated electronic structure calculations for solids.For the surface energy [14,15]of a solid,density functionals proved to be more accurate than early correlated wavefunction calculations.The densities of many solids,especially simple metals,are sufficiently like those of uniform elec-tron gases for the LDA to work well.However,it did not work so well for atoms and molecules.In particular,the atomization energies of molecules were strongly overestimated (although they were still better than those of Hartree-Fock theory).In the period 1970–1986,DFT in chemistry had only a few prophets,including notably Parr [16],Jones and Gunnarsson [17],and Levy [18].This was however a time when theoretical work by Langreth and Per-dew [19,20],and by Gunnarsson and Lundqvist [21],explained why the local approximation worked as well as it did and suggested approaches to improve it.The new approaches included generalized gradient approximations and hybridPERDEW AND RUZSINSZKY2INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL.000,NO.000functionals,which greatly increased the accuracy of atomization energies and thus the relevance of the theory to chemistry.Since about1992,DFT has swept chemistry,as it earlier swept condensed matter physics.This development was led notably by Becke[22],who proposed many creative approaches to functional construction,and by John Pople and Nick Handy.Since the1920s,many important lessons have been learned about DFT.We summarize14of these lessons below.Lesson1:Density vs.Correlated WavefunctionThe electron density nð~rÞhas a lower dimen-sionality than the N-electron wavefunction Wð~r1;r1;…;~r N;r NÞ.Using n instead of W as thebasic variational object makes electronic structure calculations much faster but typically much less accurate.This lesson comes from Thomas[5]and Fermi [6].In his lecture for the1998Nobel Prize in Chemistry[23],Walter Kohn explained this fact in a clear if oversimplified way,which we para-phrase here:suppose we use a mesh of points in real space,with10points along each of the x,y, and z axes.Then we can compute and store the density on the mesh as103numbers.A one-elec-tron wavefunction can also be represented by 103numbers,but a10-electron correlated wave-function must be represented by(103)10¼1030 numbers.There are of course cleverer ways[4] to compute a10-electron wavefunction,but the fact remains that the effort to compute an N-electron wavefunction scales up very rapidly with N.Lesson2:Orbitals vs.Correlated WavefunctionWhile too much accuracy can be lost by using the density nð~rÞas the basic variational object, considerable accuracy can be restored by using also the occupied orbitals or fictitious one-electron wavefunctions w1ð~r;rÞ…w Nð~r;rÞ.This lesson comes from Hartree and Fock in the1920s and from Slater in the1950s[12].In our example from Lesson1,an orbital description of the10-electron system requires computing and storing only10Â103¼104numbers.Lesson3:Something Proved to Exist That Cannot be FoundIn principle,we can find the exact ground-state energy and density of N electrons in an external potential vð~rÞa.by solving an Euler equation for the densitynð~rÞas proved by Hohenberg and Kohn[13],orb.by solving self-consistent one-electronSchroedinger equations for the orbitals asproved by Kohn and Sham[3].In approach(b),the functional derivative d E xc=d nð~rÞserves as the exchange–correlation contribu-tion v xcð~rÞto the effective one-electron orKohn–Sham potential v KSð~rÞ.In practice,the exact density functional E v[n] for the total energy needed for approach(a),and the exact density functional E xc[n]for the exchange–correlation energy needed for approach (b),are not accessible in any practical way and must be approximated.Although the Hohenberg–Kohn–Sham theo-rems are only existence theorems,the knowledge that exact density functionals exist has strongly driven the quest for better and more accurate approximations.Lesson4:From Uniform ElectronGas to Atoms,Molecules,and SolidsThe local density approximation(LDA)E LDAxc½n ¼Zd3rnð~rÞe unifxcðnð~rÞÞ;(1)or better the local spin density approximation (LSDA)E LSDAxc½n";n# ¼Zd3rnð~rÞe unifxcðn"ð rÞ;n#ð~rÞÞ;(2)which start from e unifxc,the exchange–correlation energy per particle of a uniform electron gas,areFOURTEEN EASY LESSONS IN DENSITY FUNCTIONAL THEORY VOL.000,NO.000DOI10.1002/qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY3accurate enough for useful calculations on solids and solid surfaces but not accurate enough for atoms and molecules.LDA comes from Kohn and Sham[3]and LSDA from von Barth and Hedin[24].The accu-racy of these local approximations for solids and surfaces was established by Lang and Kohn[25] and by Moruzzi et al.[26].Jones and Gunnarsson [17]found good structures but overestimated atomization energies for molecules.For open-shell or magnetic systems,in the ab-sence of an external magnetic field,one can use either the total density or the separate spin den-sities in principle,but in practice LSDA is more accurate than LDA because it inputs more infor-mation about the system[27].This was the first indication of a possible ladder of density func-tional approximations for the exchange–correla-tion energy to be discussed in Lessons9and10 below.All rungs of the ladder are actually imple-mented on the spin densities,but we will use the total density below to simplify the notation. Lesson5:The Electron Digs a HoleThe exact exchange–correlation energy is[19–21]the electrostatic interaction between the elec-tron density nð~rÞand the density n xcð~r;r0Þat r0of the exchange–correlation hole around an electronat~r:E xc¼ð1=2ÞZd3rnð~rÞZd3r0n xcð~r;~r0Þ=~r0À~rj j:(3)The exchange–correlation hole n xc¼n xþn c is the sum of the separate exchange and correlation holes.The exchange hole is the same as in Har-tree–Fock theory,with the Hartree–Fock orbitals replaced by Kohn–Sham orbitals,and the correla-tion hole is an average of an expectation value over the coupling constant for the electron–elec-tron interaction at fixed electron density.The exact holes satisfy the constraintsn x<0;Zd3r0n xð~r;~r0Þ¼À1;Zd3r0n cð~r;~r0Þ¼0:(4)These constraints are also satisfied by the LDA hole(that of a uniform gas)and this fact explains why LDA works as well as it does.The con-straints can also be imposed to develop other approximations that can improve over LDA by satisfying additional exact constraints.This lesson comes from Langreth and Perdew [19,20]and from Gunnarsson and Lundqvist [21].Lesson6:Searching Over Wavefunctions for theDensity FunctionalThe original proof of the Hohenberg–Kohn the-orem[13]was by reductio ad absurdum and thus not a constructive proof.Alternatively,the vari-ous density functionals,including E xc[n],can be defined by searches over N-electron wavefunc-tions constrained to yield a given electron density [18].Thus,the functionals can be intuitively understood and their exact properties can be derived.This lesson comes from Levy[18].It is an im-portant one because the derived exact properties (e.g.,Refs.[28,29])can be used to constrain the needed approximations.Lesson7:After the Local Density Comes its GradientSecond-order gradient expansionsE GEAxc½n ¼E LDAxc½n þZd3rC xcðnÞr nj j2=n4=3(5)improve on LDA for very slowly varying den-sities but are typically less accurate than LDA for realistic densities[30,31],because[31]the sec-ond-order gradient expansion of the hole violates the exact hole constraints of Lesson5.However, generalized gradient approximations or GGAs [30–35]E GGAxc½n ¼Zd3rn e GGAxcðn;r nÞ(6)can be derived with or without nonempirical parameters,and are more accurate than LDA for the separate exchange and correlation energies of atoms and for the atomization energies of mole-cules and solids.Nonempirical constructions often impose exact constraints on(1)the hole[33],or(2)the energyPERDEW AND RUZSINSZKY4INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI10.1002/qua VOL.000,NO.000functional[35],or(3)the coupling constant de-pendence[36].This lesson comes from Ma and Brueckner[30], Langreth and Perdew[31],Langreth and Mehl [32],Perdew and Wang[33],Becke[34,22],Per-dew et al.[35],and Seidl et al.[36].It was the main development that made DFT of interest to chemists.Lesson8:What is a Fractionof an Electron?The exact density functional E v[n]for the energy can be defined for an open system with noninteger average electron number N[37]by extending the constrained search of Lesson6from wavefunctions to ensembles.The energy E varies linearly between adjacent integer electron num-bers,with derivative discontinuities at the inte-gers.This explains why neutral molecules dissoci-ate to integer-charged atoms,and why the fundamental band gap[38,39]in the exact Kohn–Sham band structure is unphysical.This lesson comes from Perdew et al.[37], Perdew and Levy[38],and Sham and Schlueter [39].Lesson9:The More Information Input,the More Accuratethe OutputWe can make more accurate functionals by adding ingredients to the exchange–correlation energy density beyond n and!n:a.Meta-GGAs add the orbital kinetic energydensitysð~rÞ¼ð1=2ÞXi rf i r r w i rð~rÞj j2;(7)as proposed by Becke[40]with other non-empirical constructions by Perdew et al.[41–43].b.Hyper-GGAs or hybrid functionals addexact exchange information,such as the exact exchange energy or the exact exchange hole,as proposed by Becke[44]and refined by Savin[45],Vydrov and Scuseria[46],etc.Most hyper-GGAs do not properly scale to exact exchange under uniform density scal-ing to the high-density limit but some do[47].c.Random phase approximation(RPA)-likefunctionals add the unoccupied orbitals asdeveloped by Furche[48,49],Harl andKresse[50],etc.Lesson10:We are ClimbingJacob’s LadderThere is a five-rung Jacob’s ladder of common density functional approximations(LSDA,GGA, meta-GGA,hyper-GGA,and RPA-like functionals), as proposed by Perdew and Schmidt[51]and explained in Lesson9.All rungs except the hyper-GGA rung now have nonempirical constructions. Accuracy tends to increase up the puta-tional cost increases modestly from LSDA to GGA to meta-GGA(the three semilocal rungs)but can increase considerably on ascent to higher rungs.The Perdew-Burke-Ernzerhof(PBE)GGA[35] provides a moderately accurate description of atoms,molecules,and solids but is too simple to achieve high accuracy for all three kinds of sys-tems.The optimum GGA for solids has a weaker gradient dependence than the optimum GGA for atoms and molecules,as argued by Perdew and coworkers[52–54].However,a single nonempirical meta-GGA(revTPSS)can work well for the equi-librium properties of atoms,molecules,and solids, as shown by Perdew et al.[43].A meta-GGA can provide different GGA descriptions for solids(especially metals),which have important regions of strong orbital overlap where sð~rÞ)s Wð~rÞ r nj j2=ð8nÞ,and for atoms and mol-ecules,which have important regions where a sin-gle orbital shape dominates the density(as in an electron-pair bond),making sð~rÞ%s Wð~rÞ.Lesson11:Where Semilocal Functionals Fall DownSemilocal functionals(LSDA,GGA,and meta-GGA)necessarily fail(and thus full nonlocality is needed)when the exact exchange–correlation hole has a long-range tail,because semilocal function-als know nothing about the electron density far from an electron.This failure occurs when long-FOURTEEN EASY LESSONS IN DENSITY FUNCTIONAL THEORY VOL.000,NO.000DOI10.1002/qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY5range van der Waals interactions between sepa-rating systems are important;the needed correc-tion is present in RPA-like functionals[48–50]but neither in semilocal functionals nor in hyper-GGAs.The failure also occurs when electrons are shared across stretched bonds between distant centers(as in certain transition states and dissoci-ation limits).Semilocal functionals cannot describe such stretched bonds,but hyper-GGAs and RPA-like functionals can do so by displaying more or less the right derivative discontinuities of Lesson8.Dramatic stretched-bond effects have been dis-cussed by Ruzsinszky et al.[55,56],Vydrov et al.[57],Tsuchimochi and Scuseria[58],and Yang and coworkers[59].Lesson12:Elaborate Functionals Need Corrections TooThe RPA,which uses the unoccupied orbitals in the simplest way that makes sense for all sys-tems,is not quite good enough for chemistry.It underestimates atomization energies of molecules by about the same amount that the PBE GGA overestimates them as discovered by Furche[48]. For this property,RPA is far less accurate than meta-and hyper-GGAs.RPA uses exact exchange but makes the correlation energy too negative by roughly0.02hartree/electron.Thus,a correction to RPA is needed.The needed correction to RPA is semilocal(de-scribable by LSDA or GGA)in atoms and at solid surfaces[60]but it is fully nonlocal in molecules [61].An accurate nonempirical nonlocal correction remains to be found.Lesson13:The Kohn–ShamPotential Shapes UpStarting from an orbital functional,the exact Kohn–Sham potential v KSð~rÞof Lesson3can be constructed by the optimized effective potential method of Talman and Shadwick[62,63].Starting from the correlated N-electron wave-function,we can directly construct the energy and electron density.However,it is also computation-ally practical to construct therefrom the exact Kohn–Sham potential and so to study its features. This can be done in two different ways:(1)The Zhao et al.[64]approach first constructs the den-sity and then numerically varies the Kohn–Shampotential to reproduce this density in a system of N noninteracting electrons.(2)Another approach due to Sahni[65]and others constructs the Kohn–Sham potential from expectation values computed from the correlated N-electron wavefunction.Lesson14:After Statics Comes DynamicsDFT can be extended(exactly in principle and approximately in practice)to time-dependent and excited states.The essential idea is to solve time-dependent one-electron Schroedinger equations in which the time-dependent exchange–correlation potential v xcð~r;tÞis a functional of the electron density nð~r0;t0Þ.Poles of the frequency-dependent linear density response function then yield the ex-citation energies.This lesson comes from Runge and Gross[66] and from others[67].ACKNOWLEDGMENTSThe authors thank Sam Trickey for the invita-tion to speak at the50th Sanibel Symposium,and Kieron Burke and Jianwei Sun for suggestions on the manuscript.References1.Perdew,J.P.;Ruzsinszky,A.;Constantin,L.A.;Sun,J.;Csonka,G.I.J Chem Theory Comput2009,5,902.2.Perdew,J.P.;Kurth,S.In A Primer in Density FunctionalTheory;Fiolhais,C.,Nogueira,F.,Marques,M.,Eds.;Lecture Notes in Physics;Springer:Berlin,2003;Vol.620,p1.3.Kohn,W.;Sham,L.J.Phys Rev1965,140,A1133.4.Shavitt,I.;Bartlett,R.J.Many Body Methods in Chemistryand Physics:MBPT and Coupled-Cluster Theory;Cambridge University Press:Cambridge,2009.5.Thomas,L.H.Proc Cambridge Philos Soc1926,23,542.6.Fermi,E.Zeit fuer Physik1928,48,73.7.Teller,E.Rev Mod Phys1962,34,627.8.Kurth,S.;Perdew,J.P.Int J Quantum Chem2000,77,819.9.Perdew,J.P.;Constantin,L.A.Phys Rev B2007,75,155109.10.Karasiev,V.V.;Jones,R.S.;Trickey,S.B.;Harris,F.E.Phys Rev B2009,80,245120.11.Huang,C.;Carter,E.A.Phys Rev B2010,81,045206.12.Slater,J. C.The Self-Consistent Field for Molecules andSolids;McGraw-Hill:New York,1974.13.Hohenberg,P.;Kohn,W.Phys Rev1964,136,B864.PERDEW AND RUZSINSZKY6INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI10.1002/qua VOL.000,NO.00014.Fiolhais,C.;Perdew,J.P.Phys Rev B1992,45,6207.15.Constantin,L. A.;Pitarke,J.M.;Dobson,J. F.;Garcia-Lekue,A.;Perdew,J.P.Phys Rev Lett2008,100,036401. 16.Parr,R.G.;Yang,W.Density Functional Theory of Atomsand Molecules;Oxford University Press:Oxford,1989.17.Jones,R.O.;Gunnarsson,O.Rev Mod Phys1989,61,689.18.Levy,M.Proc Nat Acad Sci USA1979,76,6062.ngreth,D.C.;Perdew,J.P.Solid State Commun1975,17,1425.ngreth,D.C.;Perdew,J.P.Phys Rev B1977,15,2884.21.Gunnarsson,O.;Lundqvist,B.I.Phys Rev B1976,13,4274.22.Becke,A.D.Phys Rev A1988,38,3098.23.Kohn,W.Rev Mod Phys1999,71,1253.24.von Barth,U.;Hedin,L.J Phys C:Solid State Phys1972,5,1629.ng,N.D.;Kohn,W.Phys Rev B1970,1,4555.26.Moruzzi,V.L.;Janak,J.F.;Williams,A.R.Calculated Elec-tronic Properties of Metals;Pergamon:New York,1978. 27.Gunnarsson,O.;Lundqvist,B.I.;Wilkins,J.W.Phys Rev B1974,10,1319.28.Levy,M.;Perdew,J.P.Phys Rev A1985,32,2010.29.Oliver,G.L.;Perdew,J.P.Phys Rev A1979,20,397.30.Ma,S.-K.;Brueckner,K.A.Phys Rev1968,165,18.ngreth,D.C.;Perdew,J.P.Phys Rev B1980,21,5469.ngreth,D.C.;Mehl,M.J.Phys Rev B1983,20,1809.33.Perdew,J.P.;Wang,Y.Phys Rev B1986,33,8800;Erratum:Perdew,J.P.;Wang,Y.Phys Rev B1989,40, 3399.34.Becke,A.D.J Chem Phys1986,84,4524.35.Perdew,J.P.;Burke,K.;Ernzerhof,M.Phys Rev Lett1996,77,3865;Erratum:Perdew,J.P.;Burke,K.;Ernzerhof,M.Phys Rev Lett1997,78,1396.36.Seidl,M.;Perdew,J.P.;Kurth,S.Phys Rev Lett2000,84,5070.37.Perdew,J.P.;Parr,R.G.;Levy,M.;Balduz,J.L.Phys RevLett1982,49,1691.38.Perdew,J.P.;Levy,M.Phys Rev Lett1983,51,1884.39.Sham,L.J.;Schlueter,M.J.Phys Rev Lett1983,51,1888.40.Becke,A.D.;Roussel,M.R.Phys Rev A1989,39,3761,and references therein.41.Perdew,J.P.;Kurth,S.;Zupan,A.;Blaha,P.Phys Rev Lett1999,82,2544;Erratum:Perdew,J.P.;Kurth,S.;Zupan,A.;Blaha,P.Phys Rev Lett1999,82,5179.42.Tao,J.;Perdew,J.P.;Staroverov,V.N.;Scuseria,G.E.Phys Rev Lett2003,91,146401.43.Perdew,J.P.;Ruzsinszky,A.;Csonka,G.I.;Constantin,L.A.;Sun,J.Phys Rev Lett2009,103,026403.44.Becke,A.D.J Chem Phys1993,98,5648.45.Savin, A.In Recent Developments and Applications ofModern Density Functional Theory;Seminario,J.M.,Ed.;Elsevier:Amsterdam,1996;p327.46.Vydrov,O.A.;Scuseria,G.E.J Chem Phys2006,125,234109.47.Perdew,J.P.;Staroverov,V.N.;Tao,J.;Scuseria,G.E.Phys Rev A2008,78,052513.48.Furche,F.Phys Rev B2001,64,195120.49.Furche,F.J Chem Phys2008,129,114105.50.Harl,J.;Kresse,G.Phys Rev Lett2009,103,056401.51.Perdew,J.P.;Schmidt,K.In Density Functional Theoryand Its Applications to Materials;Van Doren,V.E.,Van Alsenoy,K.,Geerlings,P.,Eds.;American Institute of Physics:Melville,NY,2001;p1.52.Perdew,J.P.;Constantin,L.A.;Sagvolden,E.;Burke,K.Phys Rev Lett2006,97,223002.53.Csonka,G.I.;Vydrov,O.A.;Scuseria,G.E.;Ruzsinszky,A.;Perdew,J.P.J Chem Phys2007,126,244107.54.Perdew,J.P.;Ruzsinszky,A.;Csonka,G.I.;Constantin,L.A.;Zhou,X.;Vydrov,O.A.;Scuseria,G.E.;Burke,K.PhysRev Lett2008,100,136406;Erratum:Perdew,J.P.;Ruz-sinszky, A.;Csonka,G.I.;Constantin,L. A.;Zhou,X.;Vydrov,O.A.;Scuseria,G.E.;Burke,K.Phys Rev Lett 2009,102,039902.55.Ruzsinszky,A.;Perdew,J.P.;Csonka,G.I.;Vydrov,O.A.;Scuseria,G.E.J Chem Phys2006,125,194112.56.Ruzsinszky,A.;Perdew,J.P.;Csonka,G.I.;Vydrov,O.A.;Scuseria,A.J Chem Phys2007,126,104102.57.Vydrov,O.A.;Scuseria,G.E.;Perdew,J.P.J Chem Phys2007,126,154109.58.Tsuchimochi,T.;Scuseria,G.E.J Chem Phys2009,131,121102.59.Cohen,A.J.;Mori-Sanchez,P.;Yang,W.Science2008,321,792.60.Yan,Z.;Perdew,J.P.;Kurth,S.Phys Rev B2000,61,16430;Erratum:Yan,Z.;Perdew,J.P.;Kurth,S.Phys Rev B2010,81,169902.61.Ruzsinszky,A.;Perdew,J.P.;Csonka,G.I.J Chem TheoryComput2010,6,127.62.Talman,J.D.;Shadwick,W.F.Phys Rev A1976,14,36.63.Kuemmel,S.;Perdew,J.P.Phys Rev Lett2003,90,043004.64.Zhao,Q.;Morrison,R.C.;Parr,R.G.Phys Rev A1994,50,2138.65.Sahni,V.Quantal Density Functional Theory;Springer:Berlin,2004,and references therein.66.Runge,E.;Gross,E.K.U.Phys Rev Lett1984,52,997.67.Marques,M.A.L.;Ullrich,C.A.;Nogueira,F.;Rubio,A.;Burke,K.;Gross,E.K.U.,Eds.Time-Dependent Density Functional Theory;Springer Lecture Notes in Physics;Springer:Berlin,2006;Vol.706.FOURTEEN EASY LESSONS IN DENSITY FUNCTIONAL THEORY VOL.000,NO.000DOI10.1002/qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY7。
courses3详解
有机电子学米技术研究院1. Born-Oppenheimer ApproximationHΨ于电子与核的坐标米技术研究院1. Born-Oppenheimer Approximation复杂，难以求解！米技术研究院2. Einstein Transition Probability米技术研究院Rotational transitions~10 meV↓↑S1米技术研究院且，米技术研究院且，米技术研究院且，米技术研究院3. Selection rule for optical transitionR米技术研究院米技术研究院|||><ef ei χχ通常，将光能的退降称为“Stoke 位移”米技术研究院米技术研究院米技术研究院金属重原子可产生轨道光Metal complexes is intensively studied for OLED信息材料与纳4. 分子中的激子米技术研究院1米技术研究院米技术研究院米技术研究院影响）给体荧光光谱与受体吸收光谱的重叠程度；的情况下，给体分子的辐射速率；内，受体摩尔消光系数的大小；距离。
值得注意的是，根据第二点，体是发光的；根据第三点，受体分子要有较大的吸收才有利于通常都很小，因此以的作用，通常是可以忽略的。
米技术研究院D米技术研究院激子与周围物种的作用：（(2)米技术研究院Transient analysis of organic electrophosphorescence:I. Transient analysis of triplet energy transferM. A. BaldoLiF/Al (1000 Å)BPhenHost: Triplet emitter (400 Å)NPB(600 Å)ITO glass (30米技术研究院1) Dexter energy Transfer via host triplet:2) Förster energy Transfer via host triplet:米技术研究院Triplet Energy Transfer Probability:米技术研究院信息材料与纳1)To get triplet energy level by phosphorescent spectra米技术研究院信息材料与纳2)To get triplet exciton米技术研究院米技术研究院发光过程：复合荧光或磷光米技术研究院τ0=1 μs Ir(ppy)3Results:1)2)Ir: 2.4米技术研究院米技术研究院功课：下次上课前，提交拟作报告的大纲：By the way, if you have questions or not understanding in the paper you want to present, pls feel free to contact me at anytime.。
纳米尺度理论与计算-第一节理论计算方法之密度泛函理论简介
2022 纳米尺度理论与计算课程理论计算方法之密度泛函理论简介1946年,人类第一台计算机诞生Walter Nernst Fritz London Walter Heitler John Dalton Walter Kohn John Pople19世纪下半叶207020世纪80年代建立简单理论模型说明实验现象建立化学热力学和化学统计力学吸纳量子力学形成量子化学吸纳计算机和计算数学成果，逐步掌握化学变化规律理论与计算化学 化学理论理论化学Theoretical chemistry , from the standpoint of Avogadro’s law and thermodynamics.1893 Walter Nernst20次/100次现在的计算机对于化学家来说，如同试管一样重要。
化学逐步发展为依靠“实验、理论、计算” 三方面协同推动的科学理论与计算化学利用基于密度泛函理论软件发表文章数目l牛顿三大定律和麦克斯韦方程组，经典世界，一切都是确定的l微观世界我们如何描述粒子？l粒子l波l经典世界：牛顿三大定律和麦克斯韦方程组l微观世界：薛定谔（Schrödinger）方程第一原理的基本思想：将多原子构成的体系理解为由电子和原子核组成的多粒子系统，在解体系薛定谔方程的过程中，最大限度地进行“非经验性”处理，即不涉及任何经验参数，所要输入的只是原子的核电荷数和一些模拟环境参量。
计算所求得的结果是体系薛定谔方程的本征值和本征函数（波函数），有了这两项结果，就可研究体系的基本物理性质。
First Principle ，ab initio， DFTl从微观角度看，一块材料是由大量（每立方厘米约1023个）原子核和游离于原子核之间的电子组成。
因此，材料的性质（如硬度、电磁和光学性质）和发生在固体内的物理和化学过程是由它所包含的原子核及其电子的行为决定的。
l理论上，给定一块固体化学成分（即所含原子核的电荷和质量），我们就可以计算这些固体的性质。
中国科学技术大学第一性原理讲义(6)
Levy-Lieb functional applies to degenerate ground states Levy-Lieb functional is defined within a large variational space of electron density, namely, any density that can be obtained from a N-particle anti-symmetric wave function. This is the so-called “N representability”. Within the domain of “V-representable” densities, the two functionals yield the same results.
Ψ →n ( r )
= F LL [ n ( r )]+∫ d r V ext ( r ) n ( r )
3
Second , minimize E LL [ n ( r )] with respect to n(r)
min E LL [ n ( r )]= E LL [ n0 ( r )]= E 0
the same or different
j
j 0
If more than 1 degenerate ground-state wave functions | Ψ 0 > yield the same density n ̃ 0 ( r ), then
j ̂ j ∣Ψ 0 E LL [ n 〉 | Ψ 0j > is any of the degenerate ̃ 0 ( r )]=〈 Ψ 0∣H
N
台湾清华大学 DSGE模型讲义 lecture5
Bayesian Estimation of Linearized DSGE ModelsDr.Tai-kuang HoAssociate Professor.Department of Quantitative Finance,National Tsing Hua University,No. 101,Section2,Kuang-Fu Road,Hsinchu,Taiwan30013,Tel:+886-3-571-5131,ext.62136,Fax: +886-3-562-1823,E-mail:tkho@.tw.1Posterior distributionChapter12of Tsay(2005)provides an elegant introduction to Markov Chain Monte Carlo Methods with applications.Greenberg(2008)provides a very good introduction to fundamentals of Bayesian inference and simulation.Geweke(2005)provides a more advanced treatment of Bayesian econometrics.Bayesian inference combines prior belief(knowledge)with empirical data to form posterior distribution,which is the basis for statistical inference.:the parameters of a DSGE modelY:the empirical dataP( ):prior distribution for the parametersThe prior distribution P( )incorporates the prior belief and knowledge of the parameters.f(Y j ):the likelihood function of the data for given parametersBy the de…nition of conditional probability:f( j Y)=f( ;Y)f(Y)=f(Y j )P( )f(Y)(1)The marginal distribution f(Y)is de…ned as:f(Y)=Z f(Y; )d =Z f(Y j )P( )d f( j Y)is called the posterior distribution of .It is the probability density function(PDF)of given the observed empirical data Y.Omit the scale factor and equation(1)can be expressed as:f( j Y)/f(Y j )P( )Bayes theoremposterior P DF/(likelihood function) (prior P DF)Expressed in logarithm:log(posterior P DF)/log(likelihood function)+log(prior P DF)2Markov Chain Monte Carlo(MCMC)methodsThis section draws from Chapter7of Greenberg(2008).An advanced textbook is Carlin and Louis(2009),Bayesian Methods for Data Analysis,CRC Press.The basis of an MCMC algorithm is the construction of a transition kernel, denoted by p(x;y),that has an invariant density equal to the target density.Given such a kernel,we can start the process at x0to yield a draw x1from p(x0;x1),x2from p(x1;x2),x3from p(x2;x3),...,and x g from p x g 1;x g .The distribution of x g is approximately equal to the target distribution after a transient period.Therefore,MCMC algorithms provide an approximation to the exact posterior distribution of a parameter.How to…nd a kernel that has the target density as its invariant distribution? Metropolis-Hasting algorithm provides a general principle to…nd such kernels. Gibbs sampler is a special case of the Metropolis-Hasting algorithm.2.1Gibbs algorithmThe Gibbs algorithm is applicable when it is possible to sample from each con-ditional distribution.Suppose we want to sample from the joint distribution f(x1;x2).Further suppose that we are able to sample from the two conditional distributions f(x1j x2)and f(x2j x1).Gibbs algorithm1.Choose x(0)2(you can also start from x(0)1)2.The…rst iterationdraw x(1)1from f x1j x(0)2draw x(1)2from f x2j x(1)1 3.The g-th iterationdraw x(g)from f x1j x(g 1)21draw x(g)from f x2j x(g)124.Draw until the desired number of iterations is obtained. We discard some portion of the initial sample.This portion is the transient or burn-in sample.Let n be the number of total iterations and m be the number of burn-in sample. The point estimate of x1(similarly for x2)and its variance are:^x1=1n mnX j=m+1x(j)1^ 21=1n m 1nX j=m+1 x(j)1 ^x1 2The invariant distribution of the Gibbs kernel is the target distribution. Proof:x=(x1;x2)y=(y1;y2)p(x;y):x!y,x1!y1;x2!y2The Gibbs kernel is:p(x;y)=f(y1j x2) f(y2j y1)f(y1j x2):draw x(g)1from f x1j x(g 1)2f(y2j y1):draw x(g)2from f x2j x(g)1 It follows that:Z p(x;y)f(x)dx=Z f(y1j x2) f(y2j y1)f(x1;x2)dx1dx2=f(y2j y1)Z f(y1j x2)f(x1;x2)dx1dx2=f(y2j y1)Z f(y1j x2)f(x2)dx2=f(y2j y1)f(y1)=f(y1;y2)=f(y)This proves that f(y)is the invariant distribution for the Gibbs kernel p(x;y).The invariant distribution of the Gibbs kernel is the target distribution is a nec-essary,but not a su¢cient condition for the kernel to converge to the target distribution.Please refer to Tierney(1994)for a further discussion of such conditions.The Gibbs sampler can be easily extended to more than two blocks.In practice,convergence of Gibbs sampler is an important issue.I will use Brooks and Gelman(1998)’s method for convergence check.2.2Metropolis-Hasting algorithmMetropolis-Hasting algorithm is more general than the Gibbs sampler because it does not require the availability of the full set of conditional distribution forsampling.Suppose that we want to draw a random sample from the distribution f(X). The distribution f(X)contains a complicated normalization constant so that a direct draw is either too time-consuming or infeasible.However,there exists an approximate distribution(jumping distribution,proposal distribution)for which random draws are easy to obtain.The Metropolis-Hasting algorithm generates a sequence of random draws from the approximate distribution whose distributions converge to f(X).MH algorithm1.Given x,draw Y from q(x;y).2.Draw U from U(0;1).3.Return Y if:U (x;Y)=min(f(Y)q(Y;x);1)f(x)q(x;Y)4.Otherwise,return x and go to step1.5.Draw until the desired number of iterations is obtained.q(x;y)is the proposal distribution.The normalization constant of f(X)is not needed because only a ratio is used in the computation.How to choose the proposal density q(x;y)?The proposal density should generate proposals that have a reasonably good probability of acceptance.The sampling should be able to explore a large part of the support.Two well-known proposal kernels are the random walk kernel and the independent kernel.A.Random walk kernel:y=x+uh(u)=h( u)!q(x;y)=q(y;x)! (x;y)=f(y)f(x) B.Independent kernel:q(x;y)=q(y)q(x;y)=q(y)! (x;y)=f(y)=q(y)f(x)=q(x)2.2.1Metropolis algorithmThe algorithm uses a symmetric proposal function,namely q(Y;x)=q(x;Y). Metropolis algorithm1.Given x,draw Y from q(x;y).2.Draw U from U(0;1).3.Return Y if:U (x;Y)=min(f(Y)f(x);1)4.Otherwise,return x and go to step1.5.Draw until the desired number of iterations is obtained.2.2.2Properties of MH algorithmThis part draws from Chib and Greenberg(1995),“Understanding the Metropolis-Hasting Algorithm,”The American Statistician,49(4),pp.327-335.A kernel q(x;y)is reversible if:f(x)q(x;y)=f(y)q(y;x)It can be shown that f is the invariant distribution for the reversible kernel q de…ned above.Now we begin with a kernel that is not reversible:f(x)q(x;y)>f(y)q(y;x)We make the irreversible kernel into a reversible kernel by multiplying both sides by a function .f(x) (x;y)q(x;y)|{z}|{z}=f(y) (y;x)q(y;x)p(x;y) (x;y)q(x;y)f(x)p(x;y)=f(y)p(y;x)This turns the irreversible kernel q(x;y)into the reversible kernel p(x;y). Now set (y;x)=1.f(x) (x;y)q(x;y)=f(y)q(y;x)(x;y)=f(y)q(y;x)f(x)q(x;y)<1By letting (x;y)< (y;x),we equalize the probability that the kernel goes from x to y with the probability that the kernel goes from y to x.Similar consideration for the general case implies that:;1)(x;y)=min(f(y)q(y;x)f(x)q(x;y)2.2.3Metropolis-Hasting algorithm with two blocksSuppose we are at the(g 1)-th iteration x=(x1;x2)and want to move to the g-th iteration y=(y1;y2).MH algorithm1.Draw Z1from q1(x1;Z j x2).2.Draw U1from U(0;1).3.Return y1=Z1if:U1 (x1;Z1j x2)=f(Z1;x2)q1(Z1;x1j x2)f(x1;x2)q1(x1;Z1j x2) 4.Otherwise,return y1=x1.5.Draw Z2from q2(x2;Z j y1).6.Draw U2from U(0;1).7.Return y2=Z2if:U2 (x2;Z2j y1)=f(y1;Z2)q(Z2;x2j y1)f(y1;x2)q(x2;Z2j y1) 8.Otherwise,return y2=x2.3Estimation algorithmKaragedikli et al.(2010)provide an overview of the Bayesian estimation of a simple RBC model.This paper provides internet linkages of several sources of useful computation code.The program appendix includes a whole set of DYNARE programs to estimate, simulate the simple RBC model by using the U.S.output data,as well as to diagnose the convergence of MCMC.The references include a rich and most updated literature on estimation of DSGE models.However,the paper is con…ned to Bayesian estimation,and it is not useful for researchers who want to understand the computational details and to build their own programs.An and Schorfheide(2007)review Bayesian estimation and evaluation techniques that have been developed in recent years for empirical work with DSGE models.Why using Bayesian method to estimate DSGE models?Bayesian estimation of DSGE models has3characteristics(An and Schorfheide, 2007).First,compared to GMM estimation,Bayesian estimation is system-based.(This is also true for maximum likelihood estimation)Second,the estimation is based on likelihood function generated by the DSGE model,rather than the discrepancy between model-implied impulse responses and VAR impulse responses.Third,prior distributions can be used to incorporate additional information into the parameter estimation.Counter-argument:Fukaµc and Pagan(2010)show that DSGE models should be not estimated and evaluated only with full information methods.If the assumption that the complete system of equations is speci…ed properly seems dubious,limited information estimation,which focuses on speci…c equa-tions,can provide useful complementary information about the adequacy of the model equations in matching the data.3.1Draw from the posterior by Random Walk Metropolis algo-rithmRemember that posterior is proportional to likelihood function times prior.f( j Y)/f(Y j )P( )How to compute posterior moments?Random Walk Metropolis(RWM)algorithm allows us to draw from the posterior f( j Y).RWM algorithm belongs to the more general class of Metropolis-Hasting algo-rithm.RWM algorithm1.Initialize the algorithm with an arbitrary value 0and set j=1.2.Draw j from j= j 1+" N j 1; " .3.Draw u from U(0;1).4.Return j= j ifu j 1; j 1 =min 8<:fY j j P j f Y j j 1 P j 1 ;19=;5.Otherwise,return j= j 1.6.If j N then j!j+1and go to step2.Kalman…lter is used to evaluate the above likelihood values f Y j j and f Y j j 1 .3.2Computational algorithmSchorfheide(2000)and An and Schorfheide(2007).e a numerical optimization routine to maximize log f(Y j )+log P( ).2.Denote the posterior model by~ .3.Denote by~ the inverse of the Hessian computed at the posterior mode~ .4.Specify an initial value for 0,or draw 0from N ~ ;c20 ~ .5.Set j=1and set the number of MCMC N.6.Evaluate f(Y j 0)and P( 0)A evaluate P( 0)for given 0B use Paul Klein’s method to solve the model for given 0C use Kalman…lter to evaluate f(Y j 0)7.Draw j from j= j 1+" N j 1;c2 ~ .8.Draw u from U(0;1).9.Evaluate f Y j j and P jA evaluate P j for given jB use Paul Klein’s method to solve the model for given jC use Kalman…lter to evaluate f Y j j10.Return j= j ifu j 1; j 1 =min 8<:fY j j P j f Y j j 1 P j 1 ;19=;11.Otherwise,return j= j 1.12.If j N then j!j+1and go to step7.13.Approximate the posterior expected value of a function h( )by:E[h( )j Y]=1N simN simX j=1h jN sim=N N burn inIt is recommended to adjust the scale factor c so that the acceptance rate is roughly25percent in WRM algorithm.4An example:business cycle accountingThis example illustrates Bayesian estimation of the wedges process in Chari, Kehoe and McGrattan(2007)’s business cycle accounting.The wedges process is s t= ^A t;^ lt;^ xt;^g t .s t+1=P s t+Q"t+1P=26664p11p12p13p14 p21p22p23p24p31p32p33p34p41p42p43p4437775Q=26664q11000 q21q2200q31q32q330q41q42q43q4437775"t+1 N(04 1;I4 4)We estimate the lower triangular matrix Q to ensure that the estimate of V= QQ0is positive semide…nite.The matrix Q has no structural interpretation.Given the wedges,which are functionally similar to shocks,the next step is to solve the log-linearized model.We use Paul Klein’s MATLAB code to solve the log-linearized model.The state variables of the model are: ^k t;s t = ^k t;^A t;^ lt;^ xt;^g tThe control variables of the model are: ^c t;^x t;^y t;^l t The observed variables of the model are: ^y t;^x t;^l t;^g t Here again the log-linearized model:~c ~y ^c t+~x~y^x t+~g~y^g t=^y t(1.c)^y t=^A t+ ^k t+(1 )^l t(2.c)^c t=^A t+ ^k t " +l(1 l)#^l t 1(1 l)^ lt(3.c)^ xt (1+ )+(1+ x)(1+ )E t^c t+1 (1+ x)(1+ )^c t(4.c)=E t ~y~k ^y t+1 ^k t+1 +(1 )^ xt+1(1+ n)(1+ )^k t+1=(1 )^k t+~x~k^x t(5.c)In Lecture4,I have shown the maximum likelihood estimation of the wedges process.Going from MLE to Bayesian estimation is straightforward.The…rst step is to set the priors.The choice of the priors follows Saijo,Hikaru(2008),"The Japanese Depres-sion in the Interwar Period:A General Equilibrium Analysis,"B.E.Journal of Macroeconomics.The prior for diagonal terms of matrix P is assumed to follow a beta distribution with mean0:7and standard deviation0:2.The prior for non-diagonal terms of matrix P is assumed to follow a normal distribution with mean0and standard deviation0:3.The prior for diagonal terms of matrix Q is assumed to follow an uniform distri-bution between0and0:5.The prior for non-diagonal terms of matrix Q is assumed to follow an uniform distribution between 0:5and0:5.The table below summarizes the priors.PriorName Domain Density Parameter1Parameter2 p11;p22;p33;p44[0;1)Beta0:0350:015p12;p13;p14R Normal00:3p21;p23;p24R Normal00:3p31;p32;p34R Normal00:3p41;p42;p43R Normal00:3q11;q22;q33;q44[0;0:5]Uniform00:5q21;q31;q32;q41;q42;q43[ 0:5;0:5]Uniform 0:50:5If the sequence obtained from MCMC were i.i.d.,we could use a central limit theorem to derive the standard error(known as the numerical standard error) associate with the estimate.Koop,Gary,Dale J.Poirier,and Justin L.Tobias(2007),Bayesian Econometric Methods,page119.E[ j y]=^ =1RR X r=1 (r)V ar( j y)=^ 2=1RRX r=1 (r) ^ 2p R ^ N 0;^ 2。
A Birdeye View of DFT
Preface
This text is the outgrowth of lectures the author gave at the Chemistry Institute of the University of S˜ ao Paulo at S˜ ao Carlos, Brazil, and at the VIII’th Summer School on Electronic Structure of the Brazilian Physical Society [1]. It is an attempt to introduce density-functional theory (DFT) in a language accessible for students entering the field or researchers from other fields. It is not meant to be a scholarly review of DFT, but rather an informal guide to its conceptual basis and some recent developments and advances. The Hohenberg-Kohn theorem and the Kohn-Sham equations are discussed in some detail, including comparisons with the equations of Thomas-Fermi, Hartree-Fock, and Dyson. Approximate density functionals, selected aspects of applications of DFT, and a variety of extensions of standard DFT are also discussed, albeit in less detail. Throughout it is attempted to provide a balanced treatment of aspects that are relevant for chemistry and aspects relevant for physics, but with a strong bias towards conceptual foundations. The text is intended to be read before (or in parallel with) one of the many excellent more technical reviews available in the literature. The author apologizes to all researchers whose work has not received proper consideration. The limits of the author’s knowledge, as well as the limits of the available space and the nature of the intended audience, have from the outset prohibited any attempt at comprehensiveness.1
Conceptual Density Functional Theory
1793
Conceptual Density Functional Theory
P. Geerlings,*,† F. De Proft,† and W. Langenaeker‡
Eenheid Algemene Chemie, Faculteit Wetenschappen, Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussels, Belgium, and Department of Molecular Design and Chemoinformatics, Janssen Pharmaceutica NV, Turnhoutseweg 30, B-2340 Beerse, Belgium Received April 2, 2002
* Corresponding author (telephone +32.2.629.33.14; fax +32.2.629. 33.17; E-mail pgeerlin@vub.ac.be). † Vrije Universiteit Brussel. ‡ Janssen Pharmaceutica NV.
Contents
I. Introduction: Conceptual vs Fundamental and Computational Aspects of DFT II. Fundamental and Computational Aspects of DFT A. The Basics of DFT: The Hohenberg−Kohn Theorems B. DFT as a Tool for Calculating Atomic and Molecular Properties: The Kohn−Sham Equations C. Electronic Chemical Potential and Electronegativity: Bridging Computational and Conceptual DFT III. DFT-Based Concepts and Principles A. General Scheme: Nalewajski’s Charge Sensitivity Analysis B. Concepts and Their Calculation 1. Electronegativity and the Electronic Chemical Potential 2. Global Hardness and Softness 3. The Electronic Fukui Function, Local Softness, and Softness Kerndness Kernel 5. The Molecular Shape FunctionsSimilarity 6. The Nuclear Fukui Function and Its Derivatives 7. Spin-Polarized Generalizations 8. Solvent Effects 9. Time Evolution of Reactivity Indices C. Principles 1. Sanderson’s Electronegativity Equalization Principle 2. Pearson’s Hard and Soft Acids and Bases Principle 3. The Maximum Hardness Principle IV. Applications A. Atoms and Functional Groups B. Molecular Properties 1. Dipole Moment, Hardness, Softness, and Related Properties 2. Conformation 3. Aromaticity C. Reactivity 1. Introduction 2. Comparison of Intramolecular Reactivity Sequences 1793 1795 1795 1796 1797 1798 1798 1800 1800 1802 1807 1813 1814 1816 1819 1820 1821 1822 1822 1825 1829 1833 1833 1838 1838 1840 1840 1842 1842 1844 V. VI. VII. VIII. IX.
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1 N − ∑ ϕi ∇2 ϕi 2 i=1
2 2 1 1 N N + ∑∑ ∫∫ ϕi (r ϕ j (r 1) 2 ) dr 1dr 2 2 i=1 j=1 r 12
+ Exc [ρ]
Exc[ρ] includes everything which is unknown:
- exchange energy - correlation energy - correction of kinetic energy (T-TKS)
Correlated methods are not feasible for medium and large sized molecules!
Why DFT
Alternative: DFT
The electron density
- it is the central quantity in DFT - is defined as:
Basic Theory:
The electron density is the essential
Properties of the electron density
Function: y=f(x) Functional: y=F[f(x)]
ρ= ρ(x,y,z) E=F[ρ(x,y,z)]
Question: How can we uniquely determine the orbitals in our non-interacting reference system? How can we define a potential such that it provides us with a slater determinant which is characterized by the same density as our real system?
1 N TKS = − ∑ Ψi ∇2 Ψi 2 i=1
– kinetic energy of a fictitious non-interacting system of the same density ρ(r) - are the orbitals for the non-interacting system Ψi (KS orbitals) Tks is not equal to the true kinetic energy of the system, but contains the major fraction of it T = Tks + (T-Tks)
- variational principle
For any trial density ρ(r), which satisfies the necessary boundary conditions such as: ρ(r)≥0 and and which is associated with some external potential Vext, the energy obtained from the functional of FHK represents an upper bound to the true ground state energy E0.
The orbitals satisfying the Kohn-Sham orbitals have no physical significance. Their only connection to the real world is that the sum of their squares add up to the exact density. However, many authors recommend the KS orbitals as legitimate tools in qualitative MO considerations and this is due to the fact that the KS orbitals are not only associated with a one electron potential which includes all non-classical effects, but they are also consistent with the exact ground state density. Thus, in a sense, these orbitals are in a sense much closer to the real systems than the HF orbitals that neither reflect correlation effects nor do they yield the exact density. On the other hand, the Slater determinant generated from the KS orbitals will not be confused with the true many-electron wave function! The exact wave function of the target system is simply not available in density functional theory! Accordingly, the eigenvalues εi connected to the KS orbitals do not have a strict physical meaning. In Kohn-Sham theory there is no equivalent of Koopmans’ theorem, which could relate orbital energies to ionization energies. There is one exception though: as a direct consequence of the long range behavior of the charge density (its asymptotic exponential decay for large distances from all nuclei)
First attempt: Thomas-Fermi model (1927)
L.H. Thomas, Proc. Camb. Phil. Soc., 23, 542-548 (1927) 3 2 2/3 5/3 TTF [ρ(r )] = (3π ) ∫ ρ (r )dr E. Fermi, Rend. Acad., Lincei, 6, 602-607 (1927) 10 ρ ( r ) ρ ( r 3 ρ(r ) 1 1 2) 2 2/3 5/3 ETF [ρ(r )] = (3π ) ∫ ρ (r )dr − Z ∫ r dr + 2 ∫∫ r 12 dr1dr2 10
Problem in TF approximation: ignore correlation, and use local density approximation
Kohn and Sham (1965)
T[ρ] – kinetic energy of the system
Kohn and Sham proposed to calculate the exact kinetic energy of a non-interacting system with the same density as for the real interacting system.
TKS
Hohenberg-Kohn (1964) and Kohn-Sham (1965) ---Modern DFT
F ] = TKS[ρ] + J[ρ] + Enon-cl[ρ] HK [ρ
E[ρ] = ENe [ρ] + TKS[ρ] + J[ρ] + Exc [ρ] = -∑∫∑
i=1 N 2 ZA ϕi (r ) dr1 1 A=1 r 1A M
with:
Kohn-Sham Formalism
Kohn-Sham equations
1 2 − ∇ + v(r) + ρ(rห้องสมุดไป่ตู้) dr' − ∑ Ki (r) ϕ j = εjϕ j ∫ r − r' 2 i Hartree-Fock equations
Kohn-Sham Orbitals
Hohenberg–Kohn Theorems
ρ(r)
∫ ρ(r)dr = N
First HK Theorem (HK1)
The external potential Vext(r) is (to within a constant) a unique functional of ρ(r).
Since, in turn Vext(r) fixes H, the full many particle ground state is a uniquefunctional of ρ(r). Thus, the electron density uniquely determines the Hamiltonian operator and thus all the properties of the system.
N
ν(r)
H
ˆ Ψ = EΨ H
E
Proof:
Ψ’ as a test function for H:
Ψ as a testfunction for H’:
Summing up the last two inequalities: Contradiction!
Variational Principle in DFT Second HK Theorem