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100种思维模型——第一性原理

100种思维模型——第一性原理讲个笑话，问如何把一个大象装进冰箱里。

答：共三步。

1、打开冰箱；2、把大象放进去；3、关上冰箱。

哈哈哈哈哈哈哈哈哈，确实是个笑话。

今天，和大家分享的是美国著名企业家埃隆·马斯克曾多次提到的思维模型，第一性原理。

他曾提出一个问题：我们如何才能确认自己不是在根据有瑕疵或者不完整的信息来解决问题？第一性原理就是让我们拨开问题的层层表象，回溯事物的本质！也只有这样，才能真正解决问题。

大家应该都听说过，马斯克创建的SpaceX公司吧，没错，就是那家民营造火箭的公司。

因为他有一个宏伟的计划，帮助人类移民火星！！目的是避免因为地球气候变迁导致的人类灭绝。

不管这个想法是否过于超前，我觉得只要人类有梦想，就应该不断去探索广袤宇宙的奥秘。

当然，马斯克不仅仅停留在了想的层面，他切切实实在不断落实这个想法。

第一性原理就是他的最好帮手之一。

首先他成立私人宇航公司Space，商业是技术最好的助推器。

有人愿意为此买单，那么梦想就有了实际的物质基础。

但是他很快发现，这项业务开展不下去了，因为火箭造价高昂，根本发不起。

第三方公司报价6500万美元一个。

好家伙，这么贵，有钱人也经不起折腾啊。

于是，马斯克运用第一性原理，将一个火箭的回归到最本质的组成上面：一个弹头需要什么材料，一个弹体需要什么材料，发动机需要什么材料，镍基合金，钴基合金,铝材质....分别需要多少，市场价格是多少，经过这么一推算，他很快就发现，火箭造价虚高了。

马斯克就决定自己来造火箭。

内包了整个流程，购买原材料，自己制造火箭。

短短几年内，SpaceX就大大削减了发射火箭的成本——有报告称，减到了他早前估计的10%。

起初，火箭上天后就无法返回进行重新使用，在此之后，马斯克和团队又不断开发，实现了对火箭的精准回收，对资源的消耗进一步降低，成本也大幅度缩减。

可见，第一性原理的使用，深入问题的本质，竟然能促进人类技术快速进步。

马斯克的例子就很好的诠释了这一点。

第一性原理简介

2第一性原理的作用以密度泛函理论(DFT)为基础以及在此基础上发展起来的简单而具有一定精度的局域密度近似(LDA)和广义梯度近似(GGA)的第一性原理电子结构计算方法,与传统的解析方法一样,不但能够给出描述体系微观电子特性的物理量如波函数、态密度、费米面、电子间互作用势等,以及在此基础上所得到的体现体系宏观物理特性的参量如结合能、电离能、比热、电导、光电子谱、穆斯堡尔谱等等,而且它还可以帮助人们预言许多新的物理现象和物理规律。

密度泛函计算的一些结果能够与实验直接进行比较,一些应用程序的发展乃至商业软件的发布,导致了基于密度泛函理论的第一原理计算方法的广泛应用。

密度泛函理论(DFT)为第一性原理中的一类，在物理系、化学、材料科学以及其他工程领域中，密度泛函理论（DFT）及其计算已经快速发展成为材料建模模拟的一种“标准工具”。

密度泛函理论可以计算预测固体的晶体结构、晶格参数、能带结构、态密度（DOS）、光学性能、磁性能以及原子集合的总能等等。

3第一性原理怎么用？目前我所学到的利用第一性原理的软件为Material Studio、V ASP软件。

其中Materials Studio（简称MS）是专门为材料科学领域研究者开发的一款可运行在PC上的模拟软件。

使化学及材料科学的研究者们能更方便地建立三维结构模型，并对各种晶体、无定型以及高分子材料的性质及相关过程进行深入的研究。

模拟的内容包括了催化剂、聚合物、固体及表面、晶体与衍射、化学反应等材料和化学研究领域的主要课题。

模块简介Materials Studio采用了大家非常熟悉的Microsoft标准用户界面，允许用户通过各种控制面板直接对计算参数和计算结果进行设置和分析。

目前，Materials Studio软件包括如下功能模块：Materials Visualizer:提供了搭建分子、晶体及高分子材料结构模型所需要的所有工具，可以操作、观察及分析结构模型，处理图表、表格或文本等形式的数据，并提供软件的基本环境和分析工具以及支持Materials Studio的其他产品。

第一性原理

第一性原理在混沌大学，到底要学什么？怎么学？•思维模型•刻意练习混沌大学三大思维模型1.非连续性2.第一原理3.第二曲线“用第一性原理，跨越非连续性，发现第二曲线”《世界觀: 現代年輕人必懂的科學哲學和科學史》第1讲『第一性原理』Part 1常人的逻辑思维，有且仅有两种：归纳法、演绎法。

一、归纳法归纳法是人类最基础、最常见的用智方式，借助感觉和经验来积累知识，从特殊到一般。

几千年来，我们一直使用这个简单的归纳推理，99%人类知识建立在基于经验的归纳法上。

无师自通，也理所当然。

培根《新工具》提倡科学的归纳法，迄今，归纳法仍是常规科学的主要工具。

创业者最擅长归纳法，比如商业计划书中的上升曲线。

休谟是经验论者，他相信“知识源于感觉经验”。

但他第一个发现了归纳法的谬误：即使所有前提都正确，结论依然可能错误。

归纳法是对经验事实的简约处理，仅能收集部分信息，却得出普遍判断。

归纳法有一特征：“只能证伪，不能证明”。

换句话说，归纳法得出的知识一定是错误的，就像我们的眼睛看到外界的事实，一定会进行简约化和扭曲一样。

归纳法，我们的逻辑对经验事实进行处理的结果，也一定是简约和扭曲的。

既然每个人的经验都来自归纳法，你必须承认自己的认知有可能是错的。

——这就是“可证伪性” 的态度。

为什么我们要用归纳法呢？我们是要求存，而不是求真。

二、演绎法常人的思维逻辑只有两种，归纳法有根本问题，演绎法是不是好一点呢？•归纳法：从具象到抽象。

•演绎法：从抽象到抽象。

可以从已知思想推出未知思想。

演绎法的一大特征/好处：你可以从已知的知识里面推演出未知的知识来。

1%的人类知识来自演绎法，但这却是最重要的那一小部分知识。

哲科思维的重要特征“假设与检验” 或者“假设与证明”所有大科学家都是演绎法。

——张守晟抽象思维演绎法的坏处是速度慢；好处是可迁移性。

可迁移性：抽象东西，它生下来就是抽象的，抽象的东西可以从不同领域里边来迁移。

一旦在逻辑上导通一个共同的抽象概念，与此相关的所有具象问题，立即全部化解。

First-Principles-Based Thermodynamic Description of Solid Copper Using the Tight-Binding Ap

a r X i v :c o n d -m a t /0201141v 1 [c o n d -m a t .m t r l -s c i ] 9 J a n 2002First-Principles-Based Thermodynamic Description of SolidCopper Using the Tight-Binding ApproachSven P.Rudin,1M.D.Jones,2C.W.Greeﬀ,1and R.C.Albers 11Los Alamos National Laboratory,Los Alamos,NM 875452Department of Physics and Center for Computational Research,University at Buﬀalo,The State University of New York,Buﬀalo,NY 14260(February 1,2008)Abstract A tight-binding model is ﬁt to ﬁrst-principles calculations for copper that include structures distorted according to elastic constants and high-symmetry phonon modes.With the resulting model the ﬁrst-principles-based phonon dispersion and the free energy are calculated in the quasi-harmonic approximation.The resulting thermal expansion,the temperature-and volume-dependence of the elastic constants,the Debye temperature,and the Gr¨u neisen parameter are compared with available experimental data.PACS numbers:63.20.Dj,64.30.+t,65.40.De,65.40.GrTypeset using REVT E XI.INTRODUCTIONDensity-functional theory(DFT)ﬁrst-principles electronic-structure methods describe anomaly-free solids such as elemental copper successfully.They achieve high accuracy for quantities such as bulk properties,1surface relaxation and lattice dynamics of the surface,2 as well as the epitaxial Bain path and elastic constants.3DFT methods are routinely used to compute the zero-temperature internal energy,Φ0(V),but also can be used to calculate the free energy contributions from the ions,F I(V,T),and the electrons,F E(V,T),resulting in a complete equation of state,F(V,T)=Φ0(V)+F I(V,T)+F E(V,T).(1)However,the required computational eﬀort is expensive,and an alternative eﬃcient evalu-ation at all volumes and temperatures would be desirable.In this paper we use the computationally less demanding tight-binding(TB)total energy model in conjunction with well chosenﬁrst-principles calculations.In particular,we use the functionalﬁtting forms developed at the U.S.Naval Research Laboratory(NRL)for com-puting the total energy within the TB formalism,i.e.,without an external potential.4The model isﬁt to and accurately reproduces a set ofﬁrst-principles calculations with a speed-up of many orders of magnitude.In addition,transferability(i.e.,a TB parameterization that is accurate for a wide variety of crystal structures and atomic arrangements)has been successfully demonstrated for semiconductors as well as for simple and transition metals.4 We believe that the TB method can be used as a highly accurate,but computationally more eﬃcient,surrogate for a fullﬁrst-principles-based approach to calculate the equation of state for solids.Copper is frequently used as a test material for theoretical methods.5In this paper we have(1)developed an improvedﬁt for copper that is accurate for phonons,and(2)used this model to calculate a wide range of temperature-and volume-dependent thermodynamic quantities.Copper is furthermore widely employed as a pressure standard in high-pressure research.7 This use is based on correcting P(V)data taken along the shock Hugoniot8to room tem-perature.Such corrections employ model assumptions about the volume dependence of the Gr¨u neisen parameterγ(V),which is diﬃcult to measure independently.Shock heating increases with pressure,making the corrections more signiﬁcant at high pressure.It is there-fore important to develop theoretical techniques for accurate prediction ofγfor copper at high pressure.Phonons play a major role in the calculations of thermodynamic quantities,and the TBﬁts are adjusted to more accurately calculate them.Structures corresponding to high-symmetry phonon modes are shown here to aid in reﬁning the model;the resulting phonon density of states can then be used to determine the free energy and hence all thermodynamic quantities of interest.The precision required to calculate phonon frequencies is an order of magnitude higher than that for the lattice constant or bulk modulus,9making this a stringent test for the validity of the tight-binding approach in general and the copper model in particular.The ion–ion free energy of Eq.1is often separated into harmonic and anharmonic parts,F I(V,T)=F H(V,T)+F A(V,T).(2)Normally,the harmonic component is not a function of volume,but is calculated from the eﬀect of small displacements about the zero-temperature equilibrium lattice.In our calcula-tions,we use the quasi-harmonic approximation,which considers small displacements at any ﬁxed volume(lattice constant)within the harmonic approximation,and hence our phonon frequencies become volume dependent.However,our phonon frequencies are calculated at zero temperature for any given volume,and are not temperature dependent.The anharmonic part of the free energy involves terms that arise from the potential energy of the lattice when it is expanded beyond the harmonic part to higher than second order. Such terms are needed at high temperatures,when the phonon amplitudes are large,and ultimately lead to melting.They are also needed to explain thermal expansion eﬀects whenthe harmonic part is based on the equilibrium volume.The quasi-harmonic approximation can handle thermal expansion and the Gr¨u neisen parameter accurately through the volume dependence of the phonons at low temperature.At suﬃciently high temperatures,the quasi-harmonic approximation breaks down when the phonon amplitudes become large,and additional anharmonic phonon-phonon corrections are necessary(as indicated in Eq.2).We have not included these anharmonic types of eﬀects in our calculations.Hence we always set F A(V,T)=0,and our calculations will become less reliable at very high temperatures (near melting).In the following section we introduce the basic ideas of the tight-binding method and the ﬁrst-principles method used to generate theﬁtting database,and then describe our TBﬁtting procedures.In the subsequent section we present calculated results for the thermodynamic properties and compare them with experiment.II.FITTING THE MODELA.Tight-binding electronic structureThe tight-binding approach is essentially a parameterized version of theﬁrst-principles calculations and hence is orders of magnitude more computationally eﬃcient.In DFT meth-ods the secular equation,Hψi,v=ǫi,v Sψi,v,(3)is constructed directly from approximate solutions to the full many-body Hamiltonian,and involves a self-consistent potential that is solved iteratively;whereas in the TB approach the elements of the Hamiltonian(and the overlap matrix)themselves have been parameter-ized.Only two-center terms are considered.10For the non-orthogonal tight-binding model described here this requires73ﬁtted parameters.Of those parameters,thirty each are used to describe the inter-site matrix elements of the Hamiltonian and of the overlap matrix.For each combination of symmetries(ll′m)theform11ish ll′m(r)=(a ll′m+b ll′m r)e−c2ll′m r f c(r),(4)s ll′m(r)= ¯a ll′m+¯b ll′m r e−¯c2ll′m r f c(r),(5) where f c=1/(1+e2(r−r0))is a multiplicative factor included to ensure a smooth cutoﬀwithincreasing distance.In our calculations we have set r0=16.0Bohr radii.The remaining13parameters determine the on-site terms,which allows the parameter-ization to be applied to structures not included in theﬁtting database.A measure of thevalence electron density,ρ= i=j e−λ2r ij f c(r ij),(6) where r ij is the interatomic distance,serves to describe the on-site energy,eα=e0α+e1αρ2/3+e2αρ4/3+e3αρ2,(7)for the three orbital typesα,i.e.,s,p,and d.These terms are somewhat similar to an embedded-atom-like form in that the energy changes depending on the nearby arrangements of atoms,and may approximately account for self-consistency eﬀects as the atoms move around.B.Full potential LAPW methodTheﬁrst-principles quality of the tight-binding model results fromﬁtting to full potential linear augmented plane wave(LAPW)calculations using the reliable WIEN97program suite.12The parameters for theﬁrst-principles calculations are listed in Table I.The LAPW method divides space into spherical regions centered on the atoms and the remaining interstitial region.The radius of the spheres,the muﬃn-tin radius R m,must be chosen such that the spheres do not overlap.The basis functions used to represent the wave function are adapted to the regions:radial solutions to the Schr¨o dinger equation in thespheres,plane waves in the interstitial region.The wave functions then are found iteratively within density-functional theory,constrained to match at the boundaries of the diﬀerent regions.C.Initial Fitting ProcedureWeﬁrstﬁt the TB method to predict energy diﬀerences between the ground-state and non-equilibrium structures.Theﬁtting database includedﬁrst-principles energies calculated for the cubic structures.In addition to the total energies of these structures,it proved to be crucial toﬁt the energy bands at high-symmetry points in reciprocal space.15,16By decom-posing the electronic wave function in terms of the symmetry character of the eigenvalues17 the bands are guided to the correct ordering.The total energies and the band energies can be calculated by starting with a very crude initial tight-binding model that ignores intersite terms;15the errors are then minimized utilizing standard nonlinear least squares algorithms.18Figure1shows the T=0phonon dispersion for fcc copper calculated with the initial model.15The long-wavelength modes nearΓare well described,the short-wavelength modes near the zone boundary display somewhat high frequencies,in particular the longitudinal modes.The reasonable agreement for phonons near the zone centerΓcan be understood by considering the elements of theﬁtting database.The bulk modulus,i.e.,a linear combination of the elastic constants,is implicitly included in theﬁt.While this does not guarantee accurate elastic constants,i.e.,good agreement for the slopes of the dispersion nearΓ,it does set the right scale.Furthermore,theﬁt includes the bcc structure,which is related to the fcc crystal by a tetragonal strain corresponding to the long-wave-vector limit of the longitudinal mode in the[00ξ]direction.The database lacks any information related to the short-wave-vector modes.D.Fitting procedure with distorted structuresIn order to construct a model with an improved phonon dispersion the database was expanded to include additional information on the phonons,in particular,structures that are snapshots of the crystal deformed by particular phonon modes,i.e.,frozen phonons. The undistorted and distorted crystal structures are treated on the same footing in the ﬁrst-principles calculations and theﬁtting procedure,implicitly including the diﬀerences in energy and hence the frequencies of the phonon modes.The longitudinal and the transverse mode at the high-symmetry point X(q=(0,0,1)) were chosen because of the large discrepancy in frequency(see Fig.1)and because the distorted structures require only a doubling of the unit cell.These distorted structures are considered as additional,distinct structures in the database,to beﬁt to over a range of volumes.The initialﬁt for copper already contains some of the character of distortions related to the elastic constants:the bulk modulus is explicitly included in the energy as a function of volume,and the tetragonal distortion of the fcc crystal is somewhat reﬂected byﬁtting to the bcc structure.For completeness,tetragonally-and trigonally-distorted fcc crystals were added to theﬁt as distinct structures.These additional structures barely inﬂuence the model resulting from theﬁt;however,theﬁtting process converges much more quickly when they are included.The cubic structures that were included in the initialﬁt diﬀer from each other by an energy scale of fractions of electron volts.Phonons require a model tuned to discern energies on a scale that is approximately an order of magnitude smaller.This could be a problem since the minimization procedure tends to ignore small energy diﬀerences.For frozen phonons at the zone boundary,where neighboring atoms move against each other,it turns out that amplitudes which are still within the harmonic regime can produce energies that diﬀer from the undistorted structure by fractions of electron volts.The distortions corresponding to elastic constants,however,need to be exaggerated for them to give large enough energydiﬀerences.The trigonal distortion used here compresses the base angle from90◦to75◦, while the tetragonal distortion changes the c/a ratio from unity to1.9.Figure2shows the energy values in theﬁtting database alongside those of the initial and improved tight-binding models.The volumes of theﬁrst-principles calculations are limited to structures where the muﬃn-tin radius R m is smaller than the nearest-neighbor distance, particularly for the strongly-distorted fcc structures the choice of R m=2.0a.u.prohibits strong compression.No such limitations exist for the tight-binding approach;the volumes for which the model is appropriate will become clear in the next section.Figure3shows the errors in the improved model’sﬁpared to the initialﬁt,errors for the simple,cubic structures remain about the same.The errors for the tetragonally-distorted structures are small around the equilibrium volume(11.93˚A3),but show a ten-dency to increase as the crystal is compressed.The form of the matrix elements(Eq.4) cannot be expected to allow a high-qualityﬁt at all volumes;indeed when only a subset of data points are included in theﬁt the errors show no radical change.Including the distorted structures in theﬁt improves the transferability of the model. Figure4shows the improved agreement between tight-binding andﬁrst-principles energies for the diamond structure,which is not included in theﬁt.The transferability to a structure of such a diﬀerent coordination is not guaranteed,and our initial model did not reproduce the diamond energies well,nor did the model of the NRL group.5Figure5shows the phonon dispersion calculated with the improved model.Including the distorted fcc structures clearly reﬁnes the agreement with the measured values,though the curves do not overlap perfectly:the dispersion of the low-lying transverse modes in the[0ξ1] direction shows a diﬀerent character,and the high-frequency longitudinal modes remain somewhat large.The discrepancy of the longitudinal frequency at L suggests including this data point in theﬁt.However,aﬁrst-principles,frozen-phonon calculation of this mode shows better agreement with the tight-binding model than with experiment and was therefore not added to the database.Figure6shows the phonon density of states calculated with the improved model.Thegeneral shape agrees with the data calculated from the Born-von K´a rm´a n force constants ﬁtted to the experimental phonon dispersion along high-symmetry directions.19,20The diﬀer-ence in maximum frequencies and the peak near7THz can be attributed to the discrepancy in the dispersion of the longitudinal mode near L in the[ξξξ]direction.The tight-binding density of states displays more structure around4THz,which may be due to modes in low-symmetry directions that are not part of the experimental force-constant model.The distorted structures added to theﬁt indeed make for a model that is better suited for phonon calculations.However,while the additional constraints improve the total energies described by the model,the electronic band structure deteriorates.Figure7shows the electronic band structure along two sample high-symmetry directions of fcc copper at the experimental volume.While the initial model agrees well with theﬁrst-principles band structure,the model improved for thermodynamic quantities loses the good agreement. The resulting electronic density of states,shown in Fig.8,shows the same discrepancy; however,the density of states at the Fermi energy is quite similar,which is important for the temperature-dependent inﬂuence of the electrons(see below).It is possible that a better or moreﬂexible functional form for the distance dependence of the intersite Hamiltonian and overlap matrices are necessary to keep the good transferability and the good agreement with the individual energy bands.III.CALCULATIONS WITH THE TB MODELA.Force ConstantsThe force constants are calculated from the tight-binding model by the direct-force method,21–24which relies on evaluating the forces on all atoms in a simulation cell in which a reference atom(0,i)has been displaced.The large simulation cell consists of primitive cells transposed by vectorsℓ.Due to periodic boundary conditions on the simulation cell, the force on an atom(ℓ,j)is in response to the displaced reference atom(0,i)as well as itsimages transposed by vectors L,F(ℓ,j)=− LφC(ℓ,j;0,i)= Lφ∂uα(0,i)≈−Fβ(ℓ,j)(q),which in turn is the Fourier transform of the system’s force constants,Dαβ(q)=1that break the inversion symmetry with respect to the reference atom have to be duplicated (with adjusted weight)and transposed with a basis vector of the simulation cell to reinstate the symmetry.The cubic symmetry of the fcc crystal allows the calculation of the force constants at a particular volume with a single displacement of the basis atom.Distorted fcc structures no longer have the cubic symmetry,the calculation of the force constants therefore requires the forces to be evaluated for the basis atom displaced in all three Cartesian directions separately. For all calculations the simulation cell contained108atoms and a mesh of4×4×4k-points was used.B.ThermodynamicsAs indicated by Eq.1,the free energy is the internal energy from the tight-binding calculation with entropic terms added from the electrons and the ions.In both terms the relevant physical quantity is the density of states(DOS).The electronic DOS,n(E),the occupation of which is given by the Fermi distribution f(E,T)=[e(E−E f)/(k B T)+1]−1, determines the electrons’contribution to the entropy,S el(T)=−k B [f ln f+(1−f)ln(1−f)]n(E)dE.(12) The phonon DOS,g(ω),contributes through the zero-point energy,1U zero=the phonons,although at low temperatures(where both contributions are very small)and small volumes the percentage rises to about10%.Figure9shows the resulting free-energy as a function of volume for temperatures between 0K to1400K(at ambient pressure copper melts at1356K;melting is an anharmonic eﬀect that lies outside the scope of the quasi-harmonic treatment)in100K increments.A comparison with the free energy for the bcc phase shows the fcc structure at lower free energy for all temperatures and volumes,indicating that the model agrees with experiment in that respect.The free energy as a function of volume and temperature determines the thermal ex-pansion.The temperature-dependent lattice constant derived from the tight-binding model is shown in the inset of Fig.9along with the experimental values.As is typical for GGA-calculations,the tight-binding model overestimates the equilibrium volume by1.4%.The calculated linear expansion coeﬃcient is compared to experimental data in Fig.10and shows good agreement,in particular the characteristic temperature,which is determined by the phonon characteristic temperatures(see below).The shape of the free energy as a function of volume and temperature directly provides the temperature-dependence of the bulk modulus,B(T),which is calculated byﬁtting a second order Birch equation of state.26The bulk modulus is related to two of the elastic constants by B(T)=1accounted for byﬁnding the equilibrium volume for each temperature and then calculating the eﬀect of the strain on the free energy of that volume.Figure12shows the calculated T=0elastic constants as a function of volume.The phonon characteristic temperatures,which are deﬁned as moments of the phonon density of states,28ln(k Bθ0)= ln(¯hω) BZ,(15)k Bθ1=43(¯hω)2 BZ 1/2,(17) are shown in Fig.13.The approximate rule of thumbθ2≈θ1≈e1/3θ0holds nicely for the calculated values(inset).At temperatures below the phonon characteristic temperatures individual phonon modes must be considered separately,because they contribute to the crystal’s thermal properties with weights depending on their frequency relative to the temperature.The weight of a mode of branch s with wave vector q is determined by the heat capacity for that mode,c s(q)=∂eβ¯hωs(q)−1.(18)The sum of these individual heat capacities as a function of temperature agrees well with calorimetric data;the comparison is plotted in Fig.14in terms of the Debye temperature θD,which is found such that the Debye model’s heat capacityc V=9k B T(e x−1)2dx(19) is the same as the heat capacity calculated for the tight-binding model at the same temper-ature.The shape of the Debye temperature plotted against temperature remains very similar with compression;the curve itself is shifted upwards with the same volume-dependence as the characteristic phonon temperatures.The heat capacity of each individual phonon mode,combined with the Gr¨u neisen Pa-rameter of that mode,d lnωs(q)γq,s=−q,s c v,s(q).(21) At high temperatures(T>θ2),where all phonon modes contribute equally,γ≈γ0= d lnθ0/d lnρ.At low temperatures only the acoustic phonon modes contribute.Figure15compares the tight-binding results for the Gr¨u neisen parameter with available data.For densities up to near13g/cm3the results roughly agree with the rule of thumb thatγ·ρ=constant.Our values are slightly below the experimental values,indicating that the phonon frequencies do not increase with compression as rapidly as they should.Figure15also shows the calculated temperature-dependence of the Gr¨u neisen parameter. At low temperatures(T<40K)the plot shows a fair amount of structure relative to the high-∼temperature curve.This can be understood from the phonon dispersion shown in Fig.5, where the lowest branch is in the[ξξξ]direction and becomesﬂat around3THz,frequencies that become relevant in their contribution to the speciﬁc heat at temperatures around a third of their energy,i.e.,around50K.This branch is the lowest and hence appearsﬁrst with increasing temperature,furthermore it appears with a lot of weight as there are eight spatial directions corresponding to these modes.At low temperatures the phonon contribution to the heat capacity is proportional to T3 and vanishes more rapidly than the electronic contribution,which is linear in temperature. Figure16shows the calculated coeﬃcient of the electronic contribution to the heat capacity,π2γel=which is proportional to the density of states at the Fermi energy,n(E F).Compression of the crystal reduces n(E F),i.e.,γel decreases monotonically.IV.SUMMARYThe work presented here is aimed at(1)improving the tight-bindingﬁt of copper specif-ically for the calculation of thermodynamic properties,and(2)investigating the transfer-ability and range of applicability of the improved model.For the model to be reliable in calculating thermodynamic properties,it must produce a phonon dispersion in good agreement with experiment.The initial model wasﬁt to ﬁrst-principles calculations of the total energy at a series of diﬀerent volumes for the cubic crystal structures.The database ofﬁrst-principles calculations was extended here to include fcc structures distorted to reﬂect high-symmetry phonon modes and the elastic constants;ﬁtting to the extended database yields the improved model which indeed delivers phonon frequencies signiﬁcantly closer to the experimental values.From the phonon density of states the free energy was calculated,in the quasi-harmonic approximation,as a function of volume and temperature.The temperature-dependence of the minimum of the free energy directly yields the thermal expansion and the linear expansion coeﬃcient,both in good agreement with experiment.The elastic constants are somewhat improved over the initial model,though discrepancies with experiment remain evident.The quantities in the previous paragraph depend on volumes only in the vicinity of the T=0equilibrium volume.The volumes used for the cubic and the distorted fcc structures in theﬁt extend over a wide range;the equilibrium volume is not treated any diﬀerently than other values(down to9.7˚A3,the smallest volume for which distorted structures were ﬁt).This gives some conﬁdence that the model applies to a range beyond the equilibrium volume and its immediate vicinity.Within the quasi-harmonic approximation the volume dependence of the phonon fre-quencies gives a non-zero Gr¨u neisen parameter;the results calculated from the TB model roughly agrees with the empiricalγ·ρ=constant.The magnitude is somewhat low,i.e., the compression-induced stiﬀening of the crystal remains somewhat weaker than is experi-mentally measured.The compression at which the model clearly fails can be seen from the Gr¨u neisen param-eter as well as the volume dependences of the elastic constants,the electronic contribution to the heat capacity,and the characteristic phonon temperatures.All of these entities vary monotonically with compression until the volume reaches approximately8˚A3,i.e.,a density of roughly13g/cm3,at which point unphysical behavior appears.The unphysical behavior points to the limitations of the model.The Hamiltonian and overlap matrix elements are described by a functional form which can at best approximate the actual behavior within a limited range.For an extended range either the functional form must be modiﬁed,e.g.by including higher-order terms in Eq.4,as has been done in a more recent NRL TB copper potential used in Ref.5.The need for modiﬁcation can also be seen in the electronic band structure,which is degraded by theﬁtting to distorted fcc structures.V.ACKNOWLEDGMENTWe thank Jon Boettger,Matthias Graf,David Schiferl,and Duane Wallace for helpful and encouraging discussions.This research is supported by the Department of Energy un-der contract W-7405-ENG-36.All FLAPW calculations were performed using the Wien97 package.12Some of the calculations were performed at the National Energy Research Scien-tiﬁc Computing Center(NERSC),which is supported by the Oﬃce of Science of the U.S. Department of Energy under Contract No.DE-AC03-76SF00098REFERENCES1N.Troullier,J.L.Martins,Phys.Rev.B43,1993(1991).2C.Y.Wei,S.P.Lewis,E.J.Mele,and A.M.Rappe,Phys.Rev.B57,10062(1998).3F.Jona and P.M.Marcus,Phys.Rev.B63,094113/1(2001).4R.E.Cohen,M.J.Mehl,and D.A.Papaconstantopoulos,Phys.Rev.B50,14694(1994); M.J.Mehl and D.A.Papaconstantopoulos,Phys.Rev.B54,4519(1996);S.H.Yang, M.J.Mehl,and D.A.Papaconstantopoulos,Phys.Rev.B57,R2013(1998).5Y.Mishin,M.J.Mehl,D.A.Papaconstantopoulos,A.F.Voter,and J.D.Kress,Phys. Rev.B63,224106(2001).6W.J.Nellis,J.A.Moriarty,A.C.Mitchell,M.Ross,R.G.Dandrea,N.W.Ashcroft,N.C.Holmes,and G.R.Gathers,Phys.Rev.Lett.60,1414(1988).7H.K.Mao,P.M.Bell,J.W.Shaner,and D.J.Steinberg,J.Appl.Phys.49,3276(1978). 8R.G.McQueen,S.P.Marsh,J.W.Taylor,J.N.Fritz,and W.J.Carter,in High Velocity Impact Phenomena,R.Kinslow Ed.,(Academic,New York,1970).9S.G.Louie in Electronic Structure,Dynamics,and Quantum Structured Properties of Condensed Matter,edited by D.T.Devreese and P.van Camp(Plenum,New York, 1985),p.335.10J.C.Slater and G.F.Koster,Phys.Rev.94,1498(1954).11The(ll′m)symmetries are(ssσ),(spσ),(sdσ),(ppσ),(pdσ),(ddσ),(ppπ),(pdπ),(ddπ), and(ddδ),asﬁrst suggested by J.C.Slater and G.F.Koster,Phys.Rev.94,1498(1954). 12P.Blaha,K.Schwarz,and J.Luitz,WIEN97,A Full Potential Linearized Augmented Plane Wave Package for Calculating Crystal Properties,(Techn.Universit¨a t Wien,Aus-tria,1999).。

第一性原理

Ab initio study of the phase diagram of epitaxial BaTiO3Oswaldo Diéguez,*Silvia Tinte,A.Antons,Claudia Bungaro,J.B.Neaton,†Karin M.Rabe,and David Vanderbilt Department of Physics and Astronomy,Rutgers University,Piscataway,New Jersey08854-8019,USA(Received1April2004;published8June2004;corrected5August2004)Using a combination ofﬁrst-principles and effective-Hamiltonian approaches,we map out the structure ofBaTiO3under epitaxial constraints applicable to growth on perovskite substrates.We obtain a phase diagramin temperature and misﬁt strain that is qualitatively different from that reported by Pertsev,Zembilgotov,andTagantsev[Phys.Rev.Lett.80,1988(1998)],who based their results on an empirical thermodynamic potentialwith parametersﬁtted at temperatures in the vicinity of the bulk phase transitions.In particular,weﬁnd aregion of“r phase”at low temperature where Pertsev et al.have reported an“ac phase.”We expect our resultsto be relevant to thin epitaxialﬁlms of BaTiO3at low temperatures and experimentally achievable strains.DOI:10.1103/PhysRevB.69.212101PACS number(s):77.55.ϩf,77.80.Bh,77.84.Dy,81.05.ZxThe perovskite oxide barium titanate͑BaTiO3͒is a proto-typical ferroelectric,an insulating solid whose macroscopic polarization can be reoriented by the application of an elec-tricﬁeld.1In the perovskite ferroelectrics,it is well known both experimentally and theoretically that the polarization is also strongly coupled to strain,2and thus that properties such as the ferroelectric transition temperature and polarization magnitude are quite sensitive to external stress.Experimentally,the properties of ferroelectrics in thinﬁlm form generally differ signiﬁcantly from those in the bulk.3 While many factors are expected to contribute to these dif-ferences,it has been shown that the properties of perovskite thinﬁlms are strongly inﬂuenced by the magnitude of the epitaxial strain resulting from lattice matching theﬁlm to the substrate.For example,Yoneda et al.4used molecular-beam epitaxy to grow BaTiO3(lattice constant of4.00Å)on (001)-oriented SrTiO3(lattice constant of 3.91Å);they found that the ferroelectric transition temperature exceeds 600°C,to be compared to the bulk Curie temperature of T C=130°C.Other studies have shown that the amount of strain in BaTiO3/SrTiO3superlattices on SrTiO3substrates strongly inﬂuences properties including the observed polar-ization,phase transition temperature,and dielectric constant.5–8In a seminal paper,Pertsev,Zembilgotov and Tagantsev9 introduced the concept of mapping the equilibrium structure of a ferroelectric perovskite material versus temperature and misﬁt strain,thus producing a“Pertsev phase diagram”(or Pertsev diagram)of the observable epitaxial phases.The ef-fect of epitaxial strain is isolated from other aspects of thin-ﬁlm geometry by computing the structure of the bulk mate-rial with homogeneous strain tensor constrained to match a given substrate with square surface symmetry.10In addition, short-circuit electrical boundary conditions are imposed, equivalent to ideal electrodes above and beneath theﬁlm.9 Given the recognized importance of strain in determining the properties of thin-ﬁlm ferroelectrics,Pertsev diagrams have proven to be of enormous interest to experimentalists seek-ing to interpret the results of experiments on epitaxial thin ﬁlms and heterostructures.In Ref.9the mapping was carried out with a phenomeno-logical Landau–Devonshire model taken from the literature.This should give excellent results in the temperature/strain regime in which the model parameters wereﬁtted,but will generally be less accurate when extrapolated to other re-gimes.In Fig.1,we compare two Pertsev diagrams for BaTiO3computed using two different sets of Landau–Devonshire parameters,used by Pertsev and co-workers in Refs.9and11,respectively.While both give the same be-havior near the bulk T C and small misﬁt strains,they predict completely different low-temperature phase behavior.Withﬁrst-principles methods,it is possible not only to resolve such discrepancies arising in phenomenogical theo-ries,but also to generate a wealth of microscopic information about the structure and properties of epitaxial phases at vari-ous temperatures and substrate lattice constants.In this pa-per,using parameter-free total-energy methods based on den-sity functional theory(DFT),we map out the equilibrium structure of BaTiO3as a function of epitaxial constraints at zero temperature,and then extend the results toﬁnite tem-perature via an effective-Hamiltonian approach.The Pertsev diagram obtained in this way has a similar global topology as that of Ref.11(but not to the one in Ref.9).This allows us to predict the impact of misﬁt strain on the magnitudeand FIG.1.Phase diagrams of epitaxial BaTiO3as predicted by the theory of Pertsev et al.(see Ref.9).(a)Using the parameters quoted in Ref.9.(b)Using the parameters quoted in Ref.11.The second-andﬁrst-order phase transitions are represented by thin and thick lines,respectively.PHYSICAL REVIEW B69,212101(2004)orientation of the polarization and Curie temperature ofBaTiO 3.Our results should thus be of considerable impor-tance for understanding experimental growth of high-quality,coherent epitaxial thin ﬁlms of BaTiO 3on perovskite sub-strates,as well as more generally illustrating the utility of ﬁrst-principles Pertsev diagrams.The ﬁrst-principles DFT calculations are carried out in the Kohn–Sham framework 12using the V ASP software package.13The electron-ion interaction is described by the projector augmented wave method;14semicore electrons are included in the case of Ba ͑5s 25p 66s 2͒and Ti ͑3s 23p 64s 23d 2͒.The calculations employ the Ceperley–Alder 15form of the local-density approximation (LDA )exchange-correlationfunctional,16a 700eV plane-wave cutoff,and a 6ϫ6ϫ6Monkhorst–Pack sampling of the Brillouin zone.17We begin by systematically performing optimizations of the ﬁve-atom unit cell of the cubic perovskite structure (space group Pm 3¯m )in the six possible phases considered by Pertsev and co-workers in Ref.9.A description of these phases is given in Table I.Starting from a structure in which the symmetry is established by displacing the Ti and O at-oms,we relax the atomic positions and the out-of-plane cell vector until the value of the Hellmann–Feynman forces and zz ,yz and zx stress tensor components fall below some given thresholds (0.001eV/Åand 0.005eV,respectively ).In Fig.2we present the computed energy for each phase as a function of the misﬁt strain s =a /a 0−1,where a 0is our DFT lattice constant for free cubic BaTiO 3͑3.955Å͒.For large compressive strains,the lowest energy corresponds to the c phase;for large tensile strains,the aa phase is favored.At a misﬁt strain of s max ͑c ͒=−6.4ϫ10−3͑a =3.930Å͒,there is a second-order transition from the c phase to the r phase,with the polarization in the r phase continuously rotating away from the z direction as the misﬁt strain increases.At misﬁt strain s min ͑aa ͒=6.5ϫ10−3͑a =3.981Å͒,the r phase polarization completes its rotation into the xy plane,resulting in a continuous transition to the aa phase.The minimum energy r phase is at misﬁt strain of 2.2ϫ10−3͑a =3.964Å͒;lattice matching to the substrate would be optimal at this point.At the misﬁt strain of the c →r transition,the polar-ization could also begin a continuous rotation into the (010)plane,corresponding to the ac phase.However,it is clear from the ﬁgure that the energy of the ac phase is alwayshigher than that of the r phase,which makes sense given that the r phase is an epitaxial disortion of the ground-state rhom-bohedral phase of bulk BaTiO 3,while the ac phase is related to the higher-energy bulk orthorhombic phase.We conclude that the phase sequence at low temperatures is not c →ac →aa as given in Ref.9,but c →r →aa .Figure 3shows the computed behavior of the atomic dis-placements for the lowest-energy phase with increasing mis-ﬁt strain.For large compressive strains,the pattern of dis-placements corresponds to the c phase,and atoms relax only along the [001]direction.As the in-plane strain increases,we observe a second-order phase transition ͑c →r ͒,and while the magnitude of the atomic displacements continues to di-minish along [001],the displacements in the xy plane begin to grow.With increasing tensile strain,the displacements along [001]vanish at the r →aa transition,while the dis-TABLE I.Summary of possible epitaxial BaTiO 3phases.In-plane cell vectors are ﬁxed at a 1=axˆ,a 2=ay ˆ.Columns list,respec-tively:phase;space group;out-of-plane lattice vector;number of free internal displacement coordinates;and form of the polarization vector.Phase SG a 3N p Polarization p P 4mmm cz ˆ00c P 4mm cz ˆ3P z z ˆaa Amm 2cz ˆ4P ͑x ˆ+y ˆ͒aPmm 2cz ˆ4Px ˆac Pm c ␣x ˆ+cz ˆ8Px ˆ+P z z ˆrCmc ␣͑xˆ+y ˆ͒+cz ˆ7P ͑xˆ+y ˆ͒+P z z ˆFIG.2.(Color online )Energies of the possible epitaxial BaTiO 3phases for different misﬁt strains,as obtained from the full ab initio calculations.The vertical lines denote the phase transition points given by the stability analysis.FIG.3.(Color online ).Displacements of the atoms from the cubic perovskite cell positions,for the most energetically favorable conﬁguration at a given misﬁt strain.The vertical lines denote the phase transition points obtained from the stability analysis.⌬3͑Ti ͒labels the displacement of the Ti atom in units the third lattice vector,etc.placements in the xy plane continue to grow smoothly.Simi-lar results are found when we analyze the c→ac→a se-quence(not shown),where what was said for the xy plane applies now to the[100]direction.The clear change in char-acter of the displacement pattern within the r phase wit-nessed here illustrates the quantitative limitations of using a single misﬁt-strain-independent local mode to model the phase diagram.A stability analysis provides the precise limits of phase stability shown in Figs.2and3.At each value of misﬁt in the c phase,for example,we carry outﬁnite-difference calcula-tions of x forces and xz stress as the atomic x coordinates and xz strain are varied.The zero crossing of the lowest eigen-value of the resulting6ϫ6Hessian matrix identiﬁes the critical misﬁt.A similar analysis is used to consider z dis-placements and shear strains in the a and aa phases.By properly considering zone-center phonons,elastic shear,and linear cross coupling between them,this analysis allows us to locate the second-order phase boundaries much more pre-cisely than is possible through direct comparison of total energies.18Having established theﬁrst principles zero-temperature phase diagram,we now extend our study of epitaxial BaTiO3 toﬁnite temperatures using the effective Hamiltonian ap-proach of Zhong,Vanderbilt,and Rabe.19In this method,the full Hamiltonian is mapped onto a statistical mechanical model by a subspace projection,and parametrized through ab initio calculations of small distortions of bulk BaTiO3in the cubic perovskite structure.The reduced subspace is com-posed of a set of relevant degrees of freedom identiﬁed for ferroelectric perovskites as the unit cell distortions corre-sponding to local polarization,expressed in the form of local modes.This subspace is augmented by the inclusion of the homogeneous strain.It is straightforward to impose the constraint ofﬁxed in-plane strain byﬁxing three of the six tensor strain compo-nents during the Monte Carlo(MC)simulations.For each value of in-plane strain,MC thermal averages are obtained for the unconstrained components of the homogeneous strain and the average polarization,20and phase transitions are identiﬁed by monitoring the symmetry of these quantities. Following Ref.19,all the simulations were performed at the same negative external pressure of P=−4.8GPa.Misﬁt is deﬁned relative to a0=3.998Å,the lattice constant at the bulk cubic-to-tetragonal transition as computed with this approach.19The resulting phase diagram appears in Fig.4, where all phase lines represent second-order transitions.The Pertsev diagrams of Figs.1(a),1(b),and4share the same topology above and just below T C:p at high tempera-ture,c at large compressive misﬁt,aa at large tensile misﬁt, and a four-phase point connecting these phases with the r phase at T C.At lower temperature,there is a drastic differ-ence between Figs.1(a)and1(b),with our theory supporting the latter.While our theory underestimates the temperature of the four-phase crossing point in Fig.4by about100°C, this is the price we pay for insisting on aﬁrst-principles approach;indeed,this effective Hamiltonian underestimates the temperature of the bulk cubic-to-tetragonal transition by about the same amount.At low temperature,our Pertsev diagram shows the se-quence of second-order phase transitions c→r→aa.The r phase is predicted to exist in a range that is more than twice as broad as that shown in Fig.2.We have found that this range is reduced to about1.5times that of Fig.2when the negative-pressure correction is not included.The remaining discrepancy is related to technical differences between the DFT calculations used in Ref.21to obtain the parameters for the effective Hamiltonian method and the DFT calculations we report here.22We should also mention that the effectiveHamiltonian used does not include the physics related to thezero-point motion of the ions.This quantum effect shouldalter the shape of the lines of the diagram at very low tem-peratures,and it would result in those lines approaching themisﬁt-strain axis with inﬁnite derivative(see,for example,Ref.23).In any case,at zero temperature,the phase se-quence is quite unambiguously established by theﬁrst-principles results.This clearly indicates that the low-temperature extrapolation of the Landau–Devonshireparametersﬁtted near T C can give rise to spurious results,such as the stability of the ac phase obtained in Ref.9.Finally,we comment on the effect of the assumptionsmade in the construction of thisﬁrst-principles Pertsev dia-gram.In principle,we should consider the possibility ofequilibrium structures with larger unit cells,particularlythose with cell-doubling octahedral rotations,which havebeen shown to be important in SrTiO3,and could condensein BaTiO3under sufﬁciently large misﬁt strains.As an ex-ample,we have checked that the paraelectric phase of theﬁlm is stable with respect to octahedral rotations about the [001]direction(with M3symmetry)up to an epitaxial com-pressive strain of−70.9ϫ10−3͑a=3.675Å͒,far larger than those likely to be experimentally relevant.In addition,whilewe have studied only the effects of epitaxial strain,otherphysical effects may also be relevant to the structure andproperties of thinﬁlms,such as atomic rearrangements at theﬁlm-substrate interface and free surface,and the instability toformation of multiple domain structures.24To summarize,we have performed density functionaltheory calculations in order to obtain the Pertsev diagram ofepitaxial BaTiO3at zero temperature.The results we obtaindiffer from those computed previously9using a Landau–Devonshire theory where the parameters needed wereob-FIG.4.Phase diagram of epitaxial BaTiO3obtained using the effective Hamiltonian of Zhong,Vanderbilt and Rabe(see Ref.19).tained from experimental information about bulk BaTiO3at the phase transitions temperatures.Alternatively,the use of a similar theory where the constants of the model are com-puted using an ab initio method is consistent with both the ﬁrst principles results at zero temperature,and with the work of Pertsev and co-workers9at high temperature.It is a pleasure to thank JorgeÍñiguez,Javier Junquera, Marcelo Stachiotti,and JoséJuan Blanco-Pillado for useful comments and discussions.This work was supported by ONR Grant Nos.N0014-97-1-0048,N00014-00-1-0261,and N00014-01-1-0365,and DOE Grant No.DE-FG02-01ER45937.*Electronic address:dieguez@†Present address:The Molecular Foundry,Materials Sciences Divi-sion,Lawrence Berkeley National Laboratory,Berkeley,CA 94720.1M.E.Lines and A.M.Glass,Principles and Applications of Ferroelectrics and Related Materials(Clarendon,Oxford, 1977).2R.E.Cohen,Nature(London)358,136(1992).3C.H.Ahn,K.M.Rabe,and J.-M.Triscone,Science303,488 (2004).4Y.Yoneda,T.Okabe,K.Sakaue,H.Terauchi,H.Kasatani,and K.Deguchi,J.Appl.Phys.83,2458(1998).5W.Chang,C.M.Gilmore,W.-J.Kim,J.M.Pond,S.W.Kirch-oefer,S.B.Qadri,D.B.Chrisey,and J.S.Horwitz,J.Appl.Phys.87,3044(2000).6H.Li,A.L.Roytburd,S.P.Alpay,T.D.Tran,L.Salamanca-Riba,and R.Ramesh,Appl.Phys.Lett.78,2354(2001).7T.Shimuta,O.Nakagawara,T.Makino,S.Arai,H.Tabata,and T.Kawai,J.Appl.Phys.91,2290(2002).8J.B.Neaton and K.M.Rabe,Appl.Phys.Lett.82,1586(2003). 9N.A.Pertsev,A.G.Zembilgotov,and A.K.Tagantsev,Phys.Rev.Lett.80,1988(1998).10The applicability of our study is thus not limited to cubic sub-strates;it also applies,e.g.,to tetragonal perovskite substrates. 11N.A.Pertsev,A.G.Zembilgotov,and A.K.Tagantsev,Ferro-electrics223,79(1999).12P.Hohenberg and W.Kohn,Phys.Rev.136,B864(1964);W.Kohn and L.J.Sham,ibid.140,A1133(1965).13G.Kresse and J.Hafner,Phys.Rev.B47,558(1993);G.Kresse and J.Furthmüller,ibid.54,11169(1996).14P.E.Blöchl,Phys.Rev.B50,17953(1994);G.Kresse and D.Joubert,ibid.59,1758(1999).15D.M.Ceperley and B.J.Alder,Phys.Rev.Lett.45,566(1980). 16Using the generalized gradient approximation instead of the LDA does not lead to substantial improvements in the case of BaTiO3;see D.J.Singh,Ferroelectrics164,143(1995).17H.J.Monkhorst and J.D.Pack,Phys.Rev.B13,5188(1976). 18Similarly,the a phase is stable against the ac phase down to s min͑a͒=10.3ϫ10−3͑a=3.993Å͒.Omitting the strain and working only with the5ϫ5Hessian matrix results in very littleerror:s max͑c͒=−6.2ϫ10−3͑a=3.931Å͒,s min͑aa͒=6.1ϫ10−3͑a=3.980Å͒,and s min͑a͒=9.3ϫ10−3͑a=3.992Å͒.19W.Zhong,D.Vanderbilt,and K.M.Rabe,Phys.Rev.Lett.73, 1861(1994);Phys.Rev.B52,6301(1995).20MC simulations were performed using a12ϫ12ϫ12supercell.Typically30000MC sweeps were used to equilibrate the sys-tem,and an additional50000to obtain averages of local-modesvariables with a statistical error below10%.The temperaturewas increased in steps of5K.21R.D.King-Smith and D.Vanderbilt,Phys.Rev.B49,5828 (1994).22In particular,the soft-mode eigenvalue has been found to be very sensitive to theﬁneness of the grid used to evaluate the Fourier transforms.This discrepancy vanishes almost completely if in-stead of working with the soft-mode eigenvalue calculated in Ref.21͑0.0350a.u.͒we use the one given by our new DFT calculations(0.0223a.u.),that was calculated using aﬁner grid. 23J.Íñiguez and D.Vanderbilt,Phys.Rev.Lett.89,115503(2002). 24Y.L.Li,S.Choudhury,Z.K.Liu,and L.Q.Chen,Appl.Phys.Lett.83,1608(2003).。

VASP第一性原理计算与案例详解

V ASP第一性原理计算与案例详解目录第一章 LINUX命令 (3)1.1 常用命令 (3)1.1.1 浏览目录 (3)1.1.2 浏览文件 (3)1.1.3 目录操作 (3)1.1.4 文件操作 (3)1.1.5 系统信息 (3)第二章 SSH软件使用 (4)2.1 软件界面 (4)2.2 SSH transfer的应用 (5)2.2.1 文件传输 (5)2.2.2 简单应用 (5)第三章 VASP的四个输入文件 (5)3.1 INCAR (5)3.2 KPOINTS (6)3.3 POSCAR (6)3.4 POTCAR (7)第四章实例 (8)4.1 模型的构建 (8)4.2 VASP计算 (11)4.2.1 参数测试（VASP）参数设置 (11)4.2.2 晶胞优化(Cu) (18)4.2.3 Cu(100)表面的能量 (20)4.2.4 吸附分子CO、H、CHO的结构优化 (22)4.2.5 CO吸附于Cu100表面H位 (24)4.2.6 H吸附于Cu100表面H位 (25)4.2.7 CHO吸附于Cu100表面B位 (26)4.2.8 CO和H共吸附于Cu100表面 (28)4.2.9 过渡态计算 (29)第一章 Linux命令1.1 常用命令1.1.1 浏览目录cd: 进入某个目录。

如：cd /home/songluzhi/vasp/CH4cd .. 上一层目录；cd / 根目录；ls: 显示目录下的文件。

注：输入目录名时，可只输入前3个字母，按Tab键补全。

1.1.2 浏览文件cat：显示文件内容。

如：cat INCAR如果文件较大，可用：cat INCAR | more (可以按上下键查看) 合并文件：cat A B > C (A和B的内容合并，A在前，B在后) 1.1.3 目录操作mkdir：建立目录；rmdir：删除目录。

如：mkdir T-CH3-Rh1111.1.4 文件操作rm：删除文件；vi：编辑文件；cp：拷贝文件mv：移动文件；pwd：显示当前路径。

第一性原理——精选推荐

第⼀性原理初步定于１２⽉２⽇下午三点在我的办公室，要求你们作开题报告，内容包括：研究意义，⽂献综述，实验⽅案，所需实验仪器／设备，预期成果等。

第⼀性原理：根据原⼦核和电⼦互相作⽤的原理及其基本运动规律，运⽤量⼦⼒学原理，从具体要求出发，经过⼀些近似处理后直接求解薛定谔⽅程的算法，习惯上称为第⼀原理。

第⼀性原理就是从头计算，不需要任何参数，只需要⼀些基本的物理常量，就可以得到体系基态的基本性质的原理。

第⼀性原理通常是跟计算联系在⼀起的，是指在进⾏计算的时候除了告诉程序你所使⽤的原⼦和他们的位置外，没有其他的实验的，经验的或者半经验的参量，且具有很好的移植性。

作为评价事物的依据，第⼀性原理和经验参数是两个极端。

第⼀性原理是某些硬性规定或推演得出的结论，⽽经验参数则是通过⼤量实例得出的规律性的数据，这些数据可以来⾃第⼀性原理(称为理论统计数据)，也可以来⾃实验(称为实验统计数据)。

背景：21世纪，环境和能源问题已然成为阻碍可持续发展过程的两个重要难题。

⽽半导体光催化剂由于其在净化环境、节约能源⽅⾯的诸多优点⽽越来越受到各国众多学者的研究关注。

在众多半导体光催化剂中TiO2以其超亲⽔性、⽆毒、化学稳定性好、氧化能⼒强和廉价等诸多优点⽽被认为是⽬前最为理想的光催化剂，具有⼴阔的应⽤前景和实⽤价值。

锐钛矿型TIO2的禁带宽度略⼤于⾦红⽯型，但是由于⾦红⽯型⽐表⾯积较⼩，光催化时对O2的吸附能⼒较差，光催化活性会受到⼀定影响，所以对锐钛矿型TIO2的光催化效应的研究较多。

该反应原理是利⽤⼀定能量的光⼦激发某些半导体光催化剂使其产⽣光⽣电⼦-空⽳载流⼦。

当受到能量⼤于或等于其能隙的光照射时，价带电⼦受光激发跃迁到导带成为具有强还原能⼒的⾼活性电⼦e-，同时在价带产⽣具有强氧化能⼒的空⽳h+，此时电⼦和空⽳存在两种可能：⼀是电⼦和空⽳在体内直接复合，即在迁移到表⾯的过程中及迁移到表⾯⽽未发⽣催化氧化反应前复合。

⼆是在⼀定条件下电⼦与空⽳分离并迁移到催化剂表⾯与吸附在那⾥的OH-、H2O 和有机物等发⽣能量和电荷交换，产⽣具有强氧化能⼒的·OH、H2O2、O2-等物种，这些是直接参与化学反应的主要活性物质。

第一性原理

第一性原理第一性原理，英文First Principle，是一个计算物理或计算化学专业名词，广义的第一性原理计算指的是一切基于量子力学原理的计算。

我们知道物质由分子组成，分子由原子组成，原子由原子核和电子组成。

量子力学计算就是根据原子核和电子的相互作用原理去计算分子结构和分子能量（或离子），然后就能计算物质的各种性质。

从头算（abinitio）是狭义的第一性原理计算，它是指不使用经验参数，只用电子质量，光速，质子中子质量等少数实验数据去做量子计算。

但是这个计算很慢，所以就加入一些经验参数，可以大大加快计算速度，当然也会不可避免的牺牲计算结果精度。

那为什么使用“第一性原理”这个字眼呢？据说这是，来源于“第一推动力”这个宗教词汇。

第一推动力是牛顿创立的，因为牛顿第一定律说明了物质在不受外力的作用下保持静止或匀速直线运动。

如果宇宙诞生之初万事万物应该是静止的，后来却都在运动，是怎么动起来的呢？牛顿相信这是由于上帝推了一把，并且牛顿晚年致力于神学研究。

现代科学认为宇宙起源于大爆炸，那么大爆炸也是有原因的吧。

所有这些说不清的东西，都归结为宇宙“第一推动力”问题。

科学不相信上帝，我们不清楚“第一推动力”问题只是因为我们科学知识不完善。

第一推动一定由某种原理决定。

这个可以成为“第一原理”。

爱因斯坦晚年致力于“大统一场理论”研究，也是希望找到统概一切物理定律的“第一原理”，可惜，这是当时科学水平所不能及的。

现在也远没有答案。

但是为什么称量子力学计算为第一性原理计算？大概是因为这种计算能够从根本上计算出来分子结构和物质的性质，这样的理论很接近于反映宇宙本质的原理，就称为第一性原理了。

广义的第一原理包括两大类，以Hartree-Fork自洽场计算为基础的ab initio从头算，和密度泛函理论（DFT）计算。

也有人主张，ab initio专指从头算，而第一性原理和所谓量子化学计算特指密度泛函理论计算。

第一性原理计算方法讲义

第一性原理计算方法引言前面讲述的有限元和有限差分等数值计算方法中，求解的过程中需要知道一些物理参量，如温度场方程中的热传导系数和浓度场方程中的扩散系数等，这些参量随着材料的不同而改变，需要通过实验或经验来确定，所以这些方法也叫做经验或者半经验方法。

而第一性原理计算方法只需要知道几个基本的物理参量如电子质量、电子的电量、原子的质量、原子的核电荷数、布朗克常数、波尔半径等，而不需要知道那些经验或半经验的参数。

第一性原理计算方法的理论基础是量子力学，即对体系薛定额方程的求解。

量子力学是反映微观粒子运动规律的理论。

量子力学的出现，使得人们对于物质微观结构的认识日益深入。

原则上，量子力学完全可以解释原子之间是如何相互作用从而构成固体的。

量子力学在物理、化学、材料、生物以及许多现代技术中得到了广泛的应用。

以量子力学为基础而发展起来的固体物理学，使人们搞清了“为什么物质有半导体、导体、绝缘体的区别”等一系列基本问题，引发了通讯技术和计算机技术的重大变革。

目前，结合高速发展的计算机技术建立起来的计算材料科学已经在材料设计、物性研究方面发挥着越来越重要的作用。

但是固体是具有～1023数量级粒子的多粒子系统，具体应用量子理论时会导致物理方程过于复杂以至于无法求解，所以将量子理论应用于固体系统必须采用一些近似和简化。

绝热近似（Born-Oppenheimei近似）将电子的运动和原子核的运动分开，从而将多粒子系统简化为多电子系统。

Hartree-Fock近似将多电子问题简化为仅与以单电子波函数（分子轨道）为基本变量的单粒子问题。

但是其中波函数的行列式表示使得求解需要非常大的计算量；对于研究分子体系，他可以作为一个很好的出发点，但是不适于研究固态体系。

1964年，Hohenberg和Kohn 提出了严格的密度泛函理论（Density Functional Theory, DFT）。

它建立在非均匀作为基本变量。

1965年，Kohn和Sham 电子气理论基础之上，以粒子数密度()r提出Kohn-Sham方程将复杂的多电子问题及其对应的薛定谔方程转化为相对简单的单电子问题及单电子Kohn-Sham方程。

H2在Al12X X=Al, Ni, Cu团簇表面解离吸附的第一性原理研究

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计算材料-第一性原理共125页

计算材料-第一性原理
36、如果我们国家的法律中只有某种神灵，而不是殚精竭虑将神灵揉进宪法，总体上来说，法律就会更好。—— 马克·吐温 37、纲纪废弃之日，便是暴政兴起之时。— —威·皮物特
38、若是没有公众舆论的支持，法律是丝毫没有力量的。 ——菲力普斯 39、一个判例造出另一个判例，它们迅速累聚，进而变成法律。 ——朱尼厄斯
40、人类法律，事物有规律，这是不容忽视的。— —爱献生
Hale Waihona Puke 1、最灵繁的人也看不见自己的背脊。——非洲 2、最困难的事情就是认识自己。——希腊 3、有勇气承担命运这才是英雄好汉。——黑塞 4、与肝胆人共事，无字句处读书。——周恩来 5、阅读使人充实，会谈使人敏捷，写作使人精确。——培根

现代材料物理研究方法结课作业

现代材料物理研究方法结课作业1.研究方向在美国，由于基础设施的金属腐蚀其每年导致的经济损失达成了三千亿美元，2023年，在欧洲由于材料腐蚀所导致的经济损失达成两千亿欧元。

所以，我们对于超薄的、可以长期有效的保护金属并且对受保护金属的固有性能没有较大影响的防腐蚀材料有急切的研发需求。

石墨烯是原子级的薄层，已经被证实其可以保护涉及Ni，Cu，Ag，Au和Pt等金属，并且它可以作为金属和半导体内部连接的扩散屏障。

高温退火和电化学测试显示，石墨烯纳米片在短期内具有十分优秀的屏障性能。

然而，其他研究显示，当铜长期暴露在空气中时（几个月），石墨烯纳米片覆层会促进铜在空气中的腐蚀。

这种由石墨烯覆层导致的腐蚀促进的作用我们可以理解为石墨烯和铜原电池腐蚀对的结果。

其电化学途径被拟定为石墨烯/氧气/铜的堆叠作用，并且其对腐蚀的速率有加速作用，所以覆层之后的腐蚀速率已经完全超过了洁净的裸露的纯铜。

因此，如何去防止或是克制覆层和被保护金属之间的电化学腐蚀途径的形成就变成了一个非常重要的问题。

为了阻止电化学腐蚀作用，研究者们将石墨烯和氧化石墨烯使用绝缘的聚合物基体复合起来，以提高其屏障的性能并且阻隔其电化学反映途径。

然而，这样做会使制造工艺变得非常复杂，并且它屏障的厚度也会提高，对其本体性能的影响会变大。

六方氮化硼（BN）有着类似于石墨烯的晶格结构、超薄的厚度、很高的电阻率、杰出的热传导性能、大气环境下的稳定性和较高的阻隔性能。

有研究显示，BN有着杰出的阻隔性能，它可以在短期内，在1100℃的温度下，有效的保护Ni 不被氧化。

有研究者证实，BN在短期的高温环境下，有着较为优越的氧化阻隔性能。

研究者们也研究了BN纳米片附着在Cu箔上面在200个小时以内的腐蚀阻隔性能，研究结果显示，BN涂层提高了Cu箔的开路电势。

实验显示，BN比起石墨烯来说，有更好的长期防止腐蚀的性能，这是由于它有很强的绝缘性能，所以它可以较好地阻隔电化学反映。

第一性原理资料

MaterialStudio是一款功能强大的软件，其功能当然不仅仅限于构造模型。

在这里，我们就牛刀小试，用Material Studio来构建我们所感兴趣的吸附表面。

最简单的(111)或者(100)面比较容易，但是稍微复杂点的，比如(1 10)面的构造如果是手工就要稍花力气。

而Material Studio这套软件可以轻易的构造大多数晶体的任意切面。

我们以构造Pt(110)表面为例，主要过程如下：第一, import 一个晶格体系,在structure目录里可以选择metal/pure-metal,找到Pt的晶格文件.倒入后，我们可以看到Pt是fcc密堆结构。

第二, 选择build->cleave surface . 然后在cleave plane对话框里设定需要解理的晶面指数, 这里我们选择1 1 0。

第三, 选择layer 的厚度, 就在depth里选择fraction,也就是重复的晶面数,注意这里不是周期数。

如果是三层的在depth里设定6就好了。

没有把握的话,可以随便设定一定depth值,cleave之后还可以根据厚度来re cleave.第四, 同样在build->crystal选项里选择bulid->vacuum slab 设定真空层厚度。

在这里，我们设定真空层厚度为10A。

到这里就差不多已经搞定Pt(110)表面的构筑了。

构造完的Pt(110)表面如下图所示最后，我们可以用File$\rightarrow$Export菜单把它导出成cif或者pdb文件。

以pdb文件为例，以下是导出的pdb文件REMARK Materials Studio PDB fileREMARK Created: Sat Oct 21 06:15:00 Pacific Standard Time 2006CRYST1 3.924 2.775 25.260 90.00 90.00 90.00 P1ATOM 1 PT1 MOL 2 -7.848 -11.772 0.000 1.00 0.00 PtATOM 2 PT1 MOL 2 -7.848 -7.848 0.000 1.00 0.00 PtATOM 3 PT1 MOL 2 -3.924 -7.848 0.000 1.00 0.00 PtATOM 4 PT1 MOL 2 -3.924 -3.924 0.000 1.00 0.00 PtATOM 5 PT1 MOL 2 0.000 -3.924 0.000 1.00 0.00 PtATOM 6 PT1 MOL 2 0.000 0.000 0.000 1.00 0.00 PtATOM 7 PT1 MOL 2 -7.848 -9.810 1.962 1.00 0.00 PtATOM 8 PT1 MOL 2 -3.924 -5.886 1.962 1.00 0.00 PtATOM 9 PT1 MOL 2 0.000 -1.962 1.962 1.00 0.00 PtATOM 10 PT1 MOL 2 -9.810 -11.772 1.962 1.00 0.00 PtATOM 11 PT1 MOL 2 -5.886 -7.848 1.962 1.00 0.00 PtATOM 12 PT1 MOL 2 -1.962 -3.924 1.962 1.00 0.00 PtTER可见，文件中从第6列到第8列的坐标即为相应的Pt的直角坐标，而第三行中的内容分别是基矢的长度和相对的夹角。

如何分析第一性原理的计算结果

作者: wzxzr标题: [zz]如何分析第一性原理的计算结果时间: Sat May 19 09:48:56 2007点击: 80用第一原理计算软件开展的工作，分析结果主要是从以下三个方面进行定性/定量的讨论：1、电荷密度图（charge density）；2、能带结构（Energy Band Structure）；3、态密度（Density of States，简称DOS）。

电荷密度图是以图的形式出现在文章中，非常直观，因此对于一般的入门级研究人员来讲不会有任何的疑问。

唯一需要注意的就是这种分析的种种衍生形式，比如差分电荷密图（def-ormation charge density）和二次差分图difference charge density）等等，加自旋极化的工作还可能有自旋极化电荷密度图（spin-polarized charge density）。

所谓"差分"是指原子组成体系（团簇）之后电荷的重新分布，"二次"是指同一个体系化学成分或者几何构型改变之后电荷的重新分布，因此通过这种差分图可以很直观地看出体系中个原子的成键情况。

通过电荷聚集（accumulation）/损失（depletion）的具体空间分布，看成键的极性强弱；通过某格点附近的电荷分布形状判断成键的轨道（这个主要是对d轨道的分析，对于s或者p轨道的形状分析我还没有见过）。

分析总电荷密度图的方法类似，不过相对而言，这种图所携带的信息量较小。

能带结构分析现在在各个领域的第一原理计算工作中用得非常普遍了。

但是因为能带这个概念本身的抽象性，对于能带的分析是让初学者最感头痛的地方。

关于能带理论本身，我在这篇文章中不想涉及，这里只考虑已得到的能带，如何能从里面看出有用的信息。

首先当然可以看出这个体系是金属、半导体还是绝缘体。

判断的标准是看费米能级和导带（也即在高对称点附近近似成开口向上的抛物线形状的能带）是否相交，若相交，则为金属，否则为半导体或者绝缘体。

第一性原理思维方式案例

第一性原理思维方式案例在解决问题和创造新思路的过程中，第一性原理思维方式是一种十分有效的思维方法。

通过剖析问题的实质，抽丝剥茧，最终找到问题的根源和核心，这种思维方式能够帮助我们找到创新和解决问题的新途径。

下面将通过一个案例来说明第一性原理思维方式的应用。

案例背景假设我们是一家制造飞机的公司，我们希望找到一种新材料，可以提高飞机的燃油效率，从而帮助我们节省成本并减少对环境的影响。

我们的目标是保持飞机的性能不变或者提升，同时降低燃油消耗。

传统思维方式传统思维方式可能会首先考虑对现有的材料进行改进，或者寻找其他已知的材料用于制造飞机。

这种思维方式往往局限于已知的解决方案，并不能真正快速创新或者解决问题。

接下来我们将应用第一性原理思维方式来寻找更好的解决方案。

第一性原理思维方式第一性原理思维方式的核心是将问题一分为二，找到问题的根源和基本原理。

在这个案例中，我们要考虑飞机燃油效率的根本因素。

首先，我们需要明确飞机燃油效率的影响因素。

飞机的燃料消耗主要由两部分组成：飞机的重量和飞机的阻力。

飞机的重量包括飞机本身的重量以及载客和货物的重量。

飞机的阻力则与飞机的外形和空气动力学特性有关。

接下来，我们分析现有的材料是否可以满足我们的需求。

我们发现，虽然现有的材料已经相对轻巧和强度较高，但是在满足飞机的使用要求的同时，仍然存在一定的重量。

因此，我们需要寻找新型材料以更好地满足我们的需求。

针对飞机的阻力问题，我们可以通过改变飞机的外形来减小阻力。

在现代，高度流线型的外形已经成为飞机设计的主流趋势。

然而，我们可以考虑采取更激进的方法，重新设计飞机的外形，以有效降低阻力。

新思路和解决方案基于分析和第一性原理思维方式，我们得出了两个新的思路和解决方案：开发新型材料和重新设计飞机的外形。

对于新型材料的开发，我们需要寻找具有更高强度和更轻质的材料。

这可能需要进行大量的研究和开发，并且可能需要跨行业合作，以利用最新的科学技术。

第一性原理思维模型及应用思考

第一性原理思维模型及应用思考在我们的日常生活中，我们经常会遇到各种问题和挑战，而解决这些问题和应对这些挑战需要我们进行思考和决策。

然而，很多时候我们会受到传统思维模式的限制，无法找到创新和有效的解决方案。

而第一性原理思维模型则为我们提供了一种全新的思考方式，能够帮助我们突破传统思维的束缚，找到问题的本质，并提供创新的解决方案。

第一性原理思维模型源自于哲学和科学的研究方法，它的核心理念是将问题拆解到最基本的原理和事实，然后基于这些原理和事实进行推理和分析。

与传统思维模式不同的是，第一性原理思维模型不依赖于经验、观念和传统的约束，而是从最基本的原理出发，重新审视问题，找到新的解决方案。

一个典型的例子是特斯拉汽车的创始人埃隆·马斯克。

在设计特斯拉电动汽车时，他采用了第一性原理思维模型。

传统汽车行业一直以来都是基于内燃机的设计，而马斯克认为这种设计存在很多限制和问题。

他通过分析和研究汽车的基本原理，发现电动汽车是一种更加高效和环保的选择。

基于这个发现，他开始设计和制造电动汽车，并最终成功地推出了特斯拉汽车。

这个例子表明，通过应用第一性原理思维模型，我们能够突破传统思维的限制，找到创新和有效的解决方案。

除了在科技创新领域，第一性原理思维模型在日常生活中的应用也是非常广泛的。

例如，在做决策时，我们经常会受到他人的影响和意见的干扰，而第一性原理思维模型可以帮助我们独立思考，找到最适合自己的决策。

又如，在解决问题时，我们往往只是针对表面现象进行处理，而第一性原理思维模型可以帮助我们找到问题的根本原因，从而解决问题的根本。

然而，要应用第一性原理思维模型并不容易。

首先，我们需要具备批判性思维和分析能力，能够深入思考问题，并从中找到最基本的原理和事实。

其次，我们需要打破传统思维的束缚，不被经验、观念和传统的限制所困扰。

最后，我们需要有勇气和决心去挑战现状，追求创新和突破。

总的来说，第一性原理思维模型是一种非常有价值的思考方式，它能够帮助我们突破传统思维的束缚，找到问题的本质，并提供创新的解决方案。

第一性计算经验总结2

有限平面波基组的校正采用平面波基组的一个问题是截止能量与基组数量的变化是间断的，一般而言在ｋ点基组（k-point set）中不同ｋ点对应不同能量截止（cutoffs）时就会产生这种不连续性。

此外，在截止能量不变时，晶胞形状和尺寸的变化都会引起平面波基组的间断。

通过采用更加致密的ｋ点基组就可以解决这个问题，与特定平面波基组相关的加权性也会消除。

然而即使在ｋ基组取样很致密的情况下，这个问题依然存在，对其近似的解决方法就是引入一个校正因子（correction factor），利用某个状态基组计算使用了无限数量的ｋ点与实际采用的数量之间的差别来确定。

晶体结构在进行几何优化时如果基组不能真正的达到绝对的收敛，有限基组的纠正就很重要。

比如硅的规范－保守赝势很“软”，在平面波基组截止能量是２００ｅＶ时就已经可以得到准确的计算结果了。

但如果计算状态方程时使用上述截止能量（比如体积与总能量和压强都有关系），能量最小时对应的体积与体系内压为零时对应体积是不同的。

在提高截止能量和增加ｋ点取样基础上重复对状态方程的计算，这两个体积之间的差别会越来越小。

此外截止能量低时计算得到的E-V曲线呈现锯齿状，提高截止能量计算的曲线连续而平滑。

E-V 曲线中出现锯齿状的原因在于平面波基组在相同的截止能量时由于晶体点阵常数不同引起的平面波基组数量的间断。

对总能量进行有限基组的校正，使得我们可以在一个恒定数量基组状态下进行计算，即使采用了恒定的截止能量这个更强制条件也可以纠正计算结果。

Milman等详细的讨论了这种计算方法的细节。

进行这种校正所需要的唯一的参数就是dEtot /d lnEcut，Etot是体系总能量，Ecut是截止能量。

dEtot /d lnEcut的值很好的表示了能量截止和ｋ点取样计算收敛性质。

当它的数值（每个原子）小于0.01 eV/atom时，计算就达到了良好的收敛精度，对于大多数计算0.1 eV/atom就足够了。

冬吴同学会--第一性原理（下）[带语音]

冬吴同学会--第一性原理（下）[带语音]为什么说企业经营的核心是客户价值创造，而非战胜竞争对手？加拿大的太阳马戏团是怎样运用第一性原理，对传统的马戏表演进行颠覆式创新的？坐着打通经济生活任督二脉，冬吴同学会一期一会，本期话题：第一性原理(下)。

01—第一性原理：思维方式之前讲到了伊隆·马斯克所推崇的第一性原理，第一性原理在哲学上、在物理学上、在数学上都有很多类似的应用，其本质上是一种思维方式，就是回到事情最开始的假设条件和回到最开始的初心，然后从这个地方再来看后来的种种演化，会发现往往因为演化时间过长了之后，我们的焦点就会去到那些演化出来的种种事物上，而忘记了最开始的初心和忘记了它最开始那个假设。

就像那个例子一样，很多人在做商业的时候总是去看竞争对手在干什么，忘了你做这个事情的本质焦点应该在消费者要什么，而且要的那个东西的本质是什么，本来要迎合消费者满足消费者的时候。

八仙过海，各显神通问题，这是要过海，结果搞到最后开始斗法，就斗那个神通了，跟那过海都没什么关系了。

包括日常生活当中争论或者争吵，有些时候吵着吵着，就忘了你到底最初是为什么争吵。

02—第一性原理：商业战略竞争、竞争，竞争来竞争去，后来发现你都忘了你到底为什么在竞争了。

所以做商业要关注的是消费者而不是竞争对手，你跟竞争对手的竞争会把你拖向地狱。

这个观念我们以前讲过就是蓝海战略，其蓝海战略的核心就叫客户价值创造，而不是跟竞争对手去比拼。

蓝海战略的经典的案例就是太阳马戏团。

最近几年还时不时到中国来，马戏团一年五百多场，500多场是什么概念，这还不算路费。

普通的马戏团都处在一个非常痛苦拼命的努力，但努力有没有什么效果就越努力越焦虑的那种状态，根本就怎么找到足够多的白老虎聪明的猴子听话的狮子。

因为马戏团这三个字限制了大家，就是这样的有时候会被名字、名相所惑。

就是玩马戏的，不一定要有马，就像马路上也不一定跑马。

后来太阳马戏团对这个行业的习俗假设进行了分析和还原思考，当初我们玩马戏是为了干什么？是为了让观众高兴的！只不过是最初选择了通过动物来讨人的欢心，这本来是个手段，但这个手段一旦成为手段，它就被固化变成目的了。