量子力学第一性原理概述
第一性原理
第一性原理第二章第一性原理计算方法与软件介绍19世纪末，科学家们发现经典力学和经典电动力学在描述物质微观系统方面存在明显缺陷，无法对实验中的许多现象做出真实合理的解释。
有鉴于此，20世纪初，物理学家在旧量子理论的基础上建立了量子力学，主要研究原子、分子和凝聚态物质等内部微粒子的结构、运动规律和其他性质。
目前，它已广泛应用于物理、化学、材料等学科领域。
随着量子力学理论的不断完善和计算机技术的日益成熟，量子计算模拟已经成为现代科学中不可或缺的研究手段之一。
第一原理计算，也称为从头计算。
这种计算方法可以根据量子力学的基本原理，基于密度泛函理论，从理论上预测材料微系统的状态和性质。
在计算过程中，它不需要使用任何经验参数，只需要使用一些基本的物理量（电子电荷质量e、电子静质量M0、光速C、普朗克常数h、玻尔兹曼常数KB）。
本文选择的计算程序是Materials Studio软件中的CASTEP量子力学模块，这是一个基于密度泛函理论的从头算量子力学程序。
本章将简要介绍密度泛函理论和CASTEP计算模块。
2.1密度泛函理论概述第一性原理的主要研究对象是多原子体系。
它基于量子力学原理，在没有任何实验参数的情况下，将多原子系统视为由自由电子和原子核组成的多粒子系统。
然而，量子力学中处理多粒子系统的起点是著名的Schr？丁格方程。
施尔？丁格方程是量子力学的基本方程，也是第一原理计算方法的核心。
它是由奥地利物理学家施罗德提出的？1926年的丁格。
这个方程可以用来描述微粒子的运动规律，所以也叫Schr？丁格波动方程。
其稳态方程描述如下：2[?2??2?v(r)]?(r,t)?i?(2-1)?(r,t)?t哪里是约化普朗克常数；μ和V（R）分别代表粒子质量和势场；R和T是系统中所有电子和原子核的位置坐标；ψ（R，t）是系统的波函数，即移动的微观粒子在v(r)势场下的波函数。
但schr?dinger方程在描述真实的复杂系统时求解过程非常困难，只能处理氢原子等简单的电子体系。
新能源材料研究中的第一性原理计算
新能源材料研究中的第一性原理计算近年来，随着节能减排和环保意识的逐步加强，新能源的开发和利用已成为世界各国共同关注的焦点。
而为了更有效地提高新能源的利用效率和降低成本，科学家们开始转向新能源材料的研究和开发。
在这一过程中，第一性原理计算发挥着越来越重要的作用。
第一性原理计算是指基于量子力学理论和数学方法对材料的电子结构和性质进行计算和模拟。
这种计算方法的好处在于既能提供高精度的计算结果，又能对材料的微观结构和电子能带等性质进行深入分析，为新材料的设计和开发提供有力的支持。
在新能源材料研究中，第一性原理计算可以帮助科学家们确定材料的电子结构、晶格结构、热力学性质、光电特性等重要参数。
以太阳能电池材料为例，研究者可以通过第一性原理计算预测材料的光吸收性能、载流子输运特性和光电转换效率等重要指标，从而优化材料的能带结构和界面特性，提高太阳能电池的转化效率。
除了太阳能电池材料之外，第一性原理计算在其他新能源领域的研究中也发挥着重要作用。
比如，在固态氢储存材料的研究中，第一性原理计算可以用来预测材料的结晶形态、氢吸附能力和释放能力等关键性质，为研发更高效、更安全的氢储存材料提供支持。
在燃料电池材料的研究中，第一性原理计算可以预测氧化还原反应的能垒、电子传输特性和催化活性等参数，为提高燃料电池的效率和寿命提供重要帮助。
需要指出的是，尽管第一性原理计算具有高计算精度和深入分析的优点，但该方法也存在一些挑战和限制。
其中，计算复杂度是最主要的问题之一。
由于第一性原理计算需要对大量的原子和电子进行计算，因此计算量非常大，需要使用高性能计算机进行处理。
而由于计算复杂度高，一些材料的性质无法通过第一性原理计算来预测，需要通过实验来验证。
另一方面，第一性原理计算还需要与实验相结合，以验证计算结果的准确性和可靠性。
特别是在新能源材料研究中，第一性原理计算和实验之间的结合非常重要。
通过实验，科学家们可以验证计算结果，并不断优化计算模型，提高计算精度和可靠性。
第一性原理计算
第一性原理计算第一性原理计算是指利用基本的物理学原理和数学方程，通过计算机模拟来预测材料的性质和行为。
它是材料科学和凝聚态物理领域中一种非常重要的研究方法，可以帮助科学家们快速、高效地设计新材料，优化材料结构，预测材料的性能等。
首先，第一性原理计算是建立在量子力学原理之上的。
量子力学是描述微观世界中粒子运动和相互作用的理论，它提供了描述原子和分子行为的数学框架。
基于量子力学的第一性原理计算方法可以准确地描述原子和分子的结构、能量、电子结构等性质，为材料科学和工程领域提供了重要的理论基础。
其次，第一性原理计算的核心是求解薛定谔方程。
薛定谔方程是描述微观粒子运动的基本方程，通过求解薛定谔方程可以得到材料的电子结构和能量。
基于薛定谔方程的第一性原理计算方法可以准确地预测材料的电子能带结构、电子云分布、原子间相互作用等信息，为理解材料的性质和行为提供了重要的手段。
第三，第一性原理计算方法包括密度泛函理论、量子分子动力学、格林函数方法等。
这些方法在计算材料的结构、热力学性质、电子输运性质等方面都有重要应用。
通过这些方法，科学家们可以快速地筛选材料候选者，预测材料的稳定性和反应活性，设计新型的功能材料等。
第一性原理计算在材料科学和工程领域有着广泛的应用。
它可以帮助科学家们理解材料的基本性质，预测材料的性能，加速材料研发过程，降低研发成本。
同时，随着计算机技术的不断发展，第一性原理计算方法的计算速度和精度也在不断提高，为材料科学和工程领域的发展带来了新的机遇和挑战。
综上所述，第一性原理计算是一种基于量子力学原理的计算方法，可以准确地预测材料的性质和行为。
它在材料科学和工程领域有着重要的应用价值，可以帮助科学家们加快材料研发过程，推动材料科学的发展。
随着计算机技术的不断进步，第一性原理计算方法将会发挥越来越重要的作用，成为材料研发的重要工具。
第一性原理计算
第一性原理计算引言第一性原理计算是一种基于量子力学原理的计算方法，用于研究材料的性质和行为。
它通过解析薛定谔方程，从头开始计算材料的性质，而不依赖于经验参数或已知的实验数据。
这使得第一性原理计算成为研究材料性质的重要工具，也为材料设计和开发提供了新的途径。
原理和方法第一性原理计算的核心是薛定谔方程的求解。
薛定谔方程描述了量子力学系统的行为，通过求解薛定谔方程可以得到体系的能量、电子结构、晶体结构、力学性能等信息。
然而，薛定谔方程的精确求解是不可行的，因此需要使用一些近似方法来简化计算过程。
其中最常用的方法是密度泛函理论（DFT）。
密度泛函理论的基本思想是将体系中的电子密度视为基本变量，通过最小化体系的总能量来确定电子密度。
这可以通过Kohn-Sham方程来实现，其中包括了交换-相关能的近似处理。
通过求解Kohn-Sham方程，可以得到体系的电子结构和能量。
此外，还有一些其他的方法被用于提高计算精度，如GW近似、自洽Poisson方程、多体微扰理论等。
这些方法的选择取决于研究问题的特点和需要。
应用领域第一性原理计算在材料科学、物理学和化学等领域有着广泛的应用。
1.材料设计：第一性原理计算可以用于预测新材料的性质，从而加速材料的设计和开发过程。
它可以通过计算和优化材料的能带结构、晶体结构等来寻找具有特定性能的材料。
2.反应动力学：第一性原理计算还可以用于研究化学反应的动力学过程。
通过计算反应的势能面和反应路径，可以预测反应速率和产物选择性。
3.催化剂设计：催化剂是许多化学反应中的关键组分。
第一性原理计算可以帮助设计和优化催化剂的表面结构和活性位点，从而提高催化剂的效率和选择性。
4.电子器件：第一性原理计算在电子器件领域的应用也日益重要。
它可以用于模拟和优化半导体器件的性能，如晶体管、太阳能电池等。
5.生物物理学：第一性原理计算在生物物理学研究中也发挥着重要作用。
它可以用于预测蛋白质的结构和稳定性，研究生物分子的相互作用以及药物分子的设计等。
第一性原理的基本概念
Condensed matter physicsFrom Wikipedia, the free encyclopediaCondensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. In particular, it is concerned with the "condensed" phases that appear whenever the number of constituents in a system is extremely large and the interactions between the constituents are strong. The most familiar examples of condensed phases are solids and liquids, which arise from the bonding and electromagnetic force between atoms. More exotic condensed phases include the superfluid and the Bose-Einstein condensate found in certain atomic systems at very low temperatures, the superconducting phase exhibited by conduction electrons in certain materials, and the ferromagnetic and antiferromagnetic phases of spins on atomic lattices.Condensed matter physics is by far the largest field of contemporary physics. Much progress has also been made in theoretical condensed matter physics. By one estimate, one third of all American physicists identify themselves as condensed matter physicists. Historically, condensed matter physics grew out of solid-state physics, which is now considered one of its main subfields. The term "condensed matter physics" was apparently coined by Philip Anderson and Volker Heine when they renamed their research group at Cavendish Laboratory - previously "solid-state theory" - in 1967. In 1978, the Division of Solid State Physics at the American Physical Society was renamed as the Division of Condensed Matter Physics. Condensed matter physics has a large overlap with chemistry, materials science, nanotechnology and engineering.One of the reasons for calling the field "condensed matter physics" is that many of the concepts and techniques developed for studying solids actually apply to fluid systems. For instance, the conduction electrons in an electrical conductor form a type of quantum fluid with essentially the same properties as fluids made up of atoms. In fact, the phenomenon of superconductivity, in which the electrons condense into a new fluid phase in which they can flow without dissipation, is very closely analogous to the superfluid phase found in helium 3 at low temperatures.Fermi energyThe Fermi energy is a concept in quantum mechanics usually referring to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature. This article requires a basic knowledge of quantum mechanics.IntroductionIn quantum mechanics, a group of particles known as fermions (for example, electrons, protons and neutrons are fermions) obey the Pauli exclusion principle. This principle states that two identical fermions can not be in the same quantum state. The states are labeled by a set of quantum numbers. In a system containing many fermions (like electrons in a metal) each fermion will have a different set of quantum numbers. To determine the lowest energy a system of fermions can have, we first group the states in sets with equal energy and order these sets by increasing energy. Starting with an empty system, we then add particles one at a time, consecutively filling up the unoccupied quantum states with lowest-energy. When all the particles have been put in, the Fermi energy is the energy of the highest occupied state. What this means is that even if we have extracted all possible energy from a metal by cooling it down to near absolute zero temperature (0 kelvins), the electrons in the metal are still moving around, the fastest ones would be moving at a velocity that corresponds to a kinetic energy equal to the Fermi energy. This is the Fermi velocity. The Fermi energy is one of the important concepts of condensed matter physics. It is used, for example, to describe metals, insulators, and semiconductors. It is a very important quantity in the physics of superconductors, in the physics of quantum liquids like low temperature helium (both normal 3He and superfluid 4He), and it is quite important to nuclear physics and to understand the stability of white dwarf stars against gravitational collapse.The Fermi energy (E F) of a system of non-interacting fermions is the increase in the ground state energy when exactly one particle is added to the system. It can also be interpreted as the maximum energy of an individual fermion in this ground state. The chemical potential at zero temperature is equal to the Fermi energy.Illustration of the concept for a one dimensional square wellThe one dimensional infinite square well is a model for a one dimensional box. It is a standard model-system in quantum mechanics for which the solution for a single particle is well known. The levels are labeled by a single quantum number n and the energies are given by.Suppose now that instead of one particle in this box we have N particles in the box and that these particles are fermions with spin 1/2. Then only two particles can havethe same energy i.e. two particles can have the energy of , or two particles can have energy E2 = 4E1 and so forth. The reason that two particles can have the same energy is that a spin-1/2 particle can have a spin of 1/2 (spin up) or a spin of -1/2 (spin down), leading to two states for each energy level. When we look at the total energy of this system, the configuration for which the total energy is lowest (the ground state), is the configuration where all the energy levels up to n=N/2 are occupied and all the higher levels are empty. The Fermi energy is therefore.The three-dimensional caseThe three-dimensional isotropic case is known as the fermi sphere.Let us now consider a three-dimensional cubical box that has a side length L (see infinite square well). This turns out to be a very good approximation for describing electrons in a metal. The states are now labeled by three quantum numbers n x, n y, and n z. The single particle energies aren x, n y, n z are positive integers.There are multiple states with the same energy, for example E100 = E010 = E001. Now let's put N non-interacting fermions of spin 1/2 into this box. To calculate the Fermi energy, we look at the case for N is large.If we introduce a vector then each quantum state corresponds to a point in 'n-space' with EnergyThe number of states with energy less than E f is equal to the number of states that liewithin a sphere of radius in the region of n-space where n x, n y, n z are positive. In the ground state this number equals the number of fermions in the system.The free fermions that occupy the lowest energy states form a sphere in momentum space. The surface of this sphere is the Fermi surface.the factor of two is once again because there are two spin states, the factor of 1/8 is because only 1/8 of the sphere lies in the region where all n are positive. We findso the Fermi energy is given byWhich results in a relationship between the fermi energy and the number of particles per volume (when we replace L2 with V2/3):The total energy of a fermi sphere of N0 fermions is given byTypical fermi energies White dwarfsStars known as White dwarfs have mass comparable to our Sun, but have a radius about 100 times smaller. The high densities means that the electrons are no longer bound to single nuclei and instead form a degenerate electron gas. The numberdensity of electrons in a White dwarf are on the order of 1036 electrons/m3. This means their fermi energy is:Another typical example is that of the particles in a nucleus of an atom. The radius of the nucleus is roughly:where A is the number of nucleons.The number density of nucleons in a nucleus is therefore:Now since the fermi energy only applies to fermions of the same type, one must divide this density in two. This is because the presence of neutrons does not affect the fermi energy of the protons in the nucleus, and vice versa.So the fermi energy of a nucleus is about:The radius of the nucleus admits deviations around the value mentioned above, so a typical value for the fermi energy usually given is 38 MeV.Fermi levelThe Fermi level is the highest occupied energy level at absolute zero, that is, all energy levels up to the Fermi level are occupied by electrons. Since fermions cannot exist in identical energy states (see the exclusion principle), at absolute zero, electrons pack into the lowest available energy states and build up a "Fermi sea" of electron energy states. [1] In this state (at 0 K), the average energy of an electron is given by:where E f is the Fermi energy.The Fermi momentum is the momentum of fermions at the Fermi surface. The Fermi momentum is given by:where m e is the mass of the electron.This concept is usually applied in the case of dispersion relations between the energy and momentum that do not depend on the direction. In more general cases, one must consider the Fermi energy.The Fermi velocity is the velocity of fermions at the Fermi surface. It is defined by:where m e is the mass of the electron.Below the Fermi temperature, a substance gradually expresses more and more quantum effects of cooling. The Fermi temperature is defined by:where k is the Boltzmann constant.Quantum mechanicsAccording to quantum mechanics, fermions -- particles with a half-integer spin, usually 1/2, such as electrons -- follow the Pauli exclusion principle, which states that multiple particles may not occupy the same quantum state. Consequently, fermions obey Fermi-Dirac statistics. The ground state of a non-interacting fermion system is constructed by starting with an empty system and adding particles one at a time, consecutively filling up the lowest-energy unoccupied quantum states. When the desired number of particles has been reached, the Fermi energy is the energy of the highest occupied molecular orbital (HOMO). Within conductive materials, this is equivalent to the lowest unoccupied molecular orbital (LUMO); however, within other materials there will be a significant gap between the HOMO and LUMO on the order of 2-3 eV.Pinning of Fermi levelWhen the energy density of surface states is very high (>1013/cm2), the position of the Fermi level is determined by the neutral level of the Surface states [2] and becomes independent of Work Function [3] variations.Free electron gasIn the free electron gas, the quantum mechanical version of an ideal gas of fermions, the quantum states can be labeled according to their momentum. Something similar can be done for periodic systems, such as electrons moving in the atomic lattice of a metal, using something called the "quasi-momentum" or "crystal momentum" (see Bloch wave). In either case, the Fermi energy states reside on a surface in momentum space known as the Fermi surface. For the free electron gas, the Fermi surface is the surface of a sphere; for periodic systems, it generally has a contorted shape (see Brillouin zones). The volume enclosed by the Fermi surface defines the number of electrons in the system, and the topology is directly related to the transport properties of metals, such as electrical conductivity. The study of the Fermi surface is sometimes called Fermiology. The Fermi surfaces of most metals are well studied both theoretically and experimentally.The Fermi energy of the free electron gas is related to the chemical potential by the equationwhere E F is the Fermi energy, k is the Boltzmann constant and T is temperature. Hence, the chemical potential is approximately equal to the Fermi energy at temperatures of much less than the characteristic Fermi temperature E F/k. The characteristic temperature is on the order of 105 K for a metal, hence at room temperature (300 K), the Fermi energy and chemical potential are essentially equivalent. This is significant since it is the chemical potential, not the Fermi energy, which appears in Fermi-Dirac statistics.ReferencesKroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. ISBN 0-7167-1088-9.Table of fermi energies, velocities, and temperatures for various elements.a discussion of fermi gases and fermi temperatures.Fermi Surface:In condensed matter physics, the Fermi surface is an abstract boundary useful for predicting the thermal, electrical, magnetic, and optical properties of metals, semimetals, and doped semiconductors. The shape of the Fermi surface is derived from the periodicity and symmetry ofthe crystalline lattice and from the occupation of electronic energy bands. The existence of a Fermi surface is a direct consequence of the Pauli exclusion principle, which allows a maximum of one electron per quantum state.TheoryFormally speaking, the Fermi surface is a surface of constant energy in -spacewhere is the wavevector of the electron. At absolute zero temperature the Fermi surface separates the unfilled electronic orbitals from the filled ones. The energy of the highest occupied orbitals is known as the Fermi energy E F which, in the zero temperature case, resides on the Fermi level. The linear response of a metal to an electric, magnetic or thermal gradient is determined by the shape of the Fermi surface, because currents are due to changes in the occupancy of states near the Fermi energy.Free-electron Fermi surfaces are spheres of radius determinedby the valence electron concentration where is the reduced Planck's constant. A material whose Fermi level falls in a gap between bands is an insulator or semiconductor depending on the size of the bandgap. When a material's Fermi level falls in a bandgap, there is no Fermi surface.A view of the graphite Fermi surface at the corner H points of the Brillouin zone showing the trigonal symmetry of the electron and hole pockets.Materials with complex crystal structures can have quite intricate Fermi surfaces. The figure illustrates the anisotropic Fermi surface of graphite, which has both electron and hole pockets in its Fermi surface due to multiple bands crossing the Fermi energyalong the direction. Often in a metal the Fermi surface radius k F is larger than the size of the first Brillouin zone which results in a portion of the Fermi surface lying in the second (or higher) zones. As with the band structure itself, the Fermi surface canbe displayed in an extended-zone scheme where is allowed to have arbitrarilylarge values or a reduced-zone scheme where wavevectors are shown modulowhere a is the lattice constant. Solids with a large density of states at the Fermi level become unstable at low temperatures and tend to form ground states where the condensation energy comes from opening a gap at the Fermi surface. Examples of such ground states are superconductors, ferromagnets, Jahn-Teller distortions and spin density waves.The state occupancy of fermions like electrons is governed by Fermi-Dirac statistics so at finite temperatures the Fermi surface is accordingly broadened. In principle all fermion energy level populations are bound by a Fermi surface although the term is not generally used outside of condensed-matter physics.Experimental determinationde Haas-van Alphen effect. Electronic Fermi surfaces have been measured through observation of the oscillation of transport properties in magnetic fields H, for example the de Haas-van Alphen effect (dHvA) and the Shubnikov-De Haas effect (SdH). The former is an oscillation in magnetic susceptibility and the latter in resistivity. The oscillations are periodic versus 1 / H and occur because of the quantization of energy levels in the plane perpendicular to a magnetic field, a phenomenon first predicted by Lev Landau. The new states are called Landau levels and are separated by an energywhere ωc = eH / m*c is called the cyclotron frequency, e is the electronic charge, m* is the electron effective mass and c is the speed of light. In a famous result, Lars Onsager proved that the period of oscillation ΔH is related to the cross-section of theFermi surface (typically given in ) perpendicular to the magnetic field directionby the equation . Thus the determination of the periods ofoscillation for various applied field directions allows mapping of the Fermi surface.Observation of the dHvA and SdH oscillations requires magnetic fields large enough that the circumference of the cyclotron orbit is smaller than a mean free path. Therefore dHvA and SdH experiments are usually performed at high-field facilities like the High Field Magnet Laboratory in Netherlands, Grenoble High Magnetic Field Laboratory in France, the Tsukuba Magnet Laboratory in Japan or the National High Magnetic Field Laboratory in the United States.Fermi surface of BSCCO measured by ARPES. The experimental data shown as an intensity plot in yellow-red-black scale. Green dashed rectagle represents the Brillouin zone of the CuO2 plane of BSCCO.Angle resolved photoemission. The most direct experimental technique to resolve the electronic structure of crystals in the momentum-energy space (see reciprocal lattice), and, consequently, the Fermi surface, is the angle resolved photoemission spectroscopy (ARPES). An example of the Fermi surface of superconducting cuprates measured by ARPES is shown in figure.Two photon positron annihilation.With positron annihilation the two photons carry the momentum of the electron away; as the momentum of a thermalized positron is negligible, in this way also information about the momentum distribution can be obtained. Because the positron can be polarized, also the momentum distribution for the two spin states in magnetized materials can be obtained. Another advantage with De Haas-Van Alphen-effect is that the technique can be applied to non-dilute alloys. In this way the first determination of a smeared Fermi surface in a 30% alloy has been obtained in 1978.ReferencesN. Ashcroft and N.D. Mermin, Solid-State Physics, ISBN 0-03-083993-9.W.A. Harrison, Electronic Structure and the Properties of Solids, ISBN0-486-66021-4.VRML Fermi Surface DatabaseBrillouin zoneFrom Wikipedia, the free encyclopediaJump to: navigation, searchBrillouin zoneIn mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell of the reciprocal lattice in the frequency domain. It is found by the same method as for the Wigner-Seitz cell in the Bravais lattice. The importance of the Brillouin zone stems from the Bloch wave description of waves in a periodic medium, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.Taking the surfaces at the same distance from one element of the lattice and its neighbours, the volume included is the first Brillouin zone. Another definition is as the set of points in k-space that can be reached from the origin without crossing any Bragg plane.There are also second, third, etc., Brillouin zones, corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin, but these are used more rarely. As a result, the first Brillouin zone is often called simply the Brillouin zone. (In general, the n-th Brillouin zone consist of the set of points that can be reached from the origin by crossing n− 1 Bragg planes.)A related concept is that of the irreducible Brillouin zone, which is the first Brillouin zone reduced by all of the symmetries in the point group of the lattice.The concept of a Brillouin zone was developed by Leon Brillouin (1889-1969), a French physicist.Critical pointsFirst Brillouin zone of FCC lattice showing symmetry labels for high symmetry lines and points Several points of high symmetry are of special interest – these are called critical points.[1]Symbol DescriptionΓ Center of the Brillouin zoneSimple cubeM Center of an edgeR Corner pointX Center of a faceFace-centered cubicK Middle of an edge joining two hexagonal facesL Center of a hexagonal faceU Middle of an edge joining a hexagonal and a square faceW Corner pointX Center of a square faceBody-centered cubicH Corner point joining four edgesN Center of a faceP Corner point joining three edgesHexagonalA Center of a hexagonal faceH Corner pointK Middle of an edge joining two rectangular facesL Middle of an edge joining a hexagonal and a rectangular faceM Center of a rectangular faceReferences1.^ Harald Ibach & Hans Lüth, Solid-State Physics, An Introduction to Principles of MaterialsScience, corrected second printing of the second edition, 1996, Springer-Verlag, ISBN3-540-58573-72.Charles Kittel, Introduction to Solid State Physics (Wiley: New York, 1996).3.Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).4.Léon Brillouin Les électrons dans les métaux et le classement des ondes de de Brogliecorrespondantes Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, 191, 292 (1930). (original article)Bloch waveFrom Wikipedia, the free encyclopediaA Bloch wave or Bloch state, named after Felix Bloch, is the wavefunction of a particle (usually, an electron) placed in a periodic potential. It consists of the product of a plane wave envelope function and a periodic function (periodic Bloch function)which has the same periodicity as the potential:Bloch wave in siliconThe result that the eigenfunctions can be written in this form for a periodic system is called Bloch's theorem. The corresponding energy eigenvalue is Єn(k)= Єn(k + K), periodic with periodicity K of a reciprocal lattice vector. Because the energies associated with the index n vary continuously with wavevector k we speak of an energy band with band index n. Because the eigenvalues for given n are periodic in k,all distinct values of Єn(k) occur for k-values within the first Brillouin zone of the reciprocal lattice.More generally, a Bloch-wave description applies to any wave-like phenomenon in a periodic medium. For example, a periodic dielectric in electromagnetism leads to photonic crystals, and a periodic acoustic medium leads to phononic crystals. It is generally treated in the different forms of the dynamical theory of diffraction.The plane wave wavevector (or Bloch wavevector) k (multiplied by the reduced Planck's constant, this is the particle's crystal momentum) is unique only up to a reciprocal lattice vector, so one only needs to consider the wavevectors inside the first Brillouin zone. For a given wavevector and potential, there are a number of solutions, indexed by n, to Schrödinger's equation for a Bloch electron. These solutions, called bands, are separated in energy by a finite spacing at each k; if there is a separation that extends over all wavevectors, it is called a (complete) band gap. The band structure is the collection of energy eigenstates within the first Brillouin zone. All the properties of electrons in a periodic potential can be calculated from this band structure and the associated wavefunctions, at least within the independent electron approximation.A corollary of this result is that the Bloch wavevector k is a conserved quantity in a crystalline system (modulo addition of reciprocal lattice vectors), and hence the group velocity of the wave is conserved. This means that electrons can propagate without scattering through a crystalline material, almost like free particles, and that electrical resistance in a crystalline conductor only results from things like imperfections that break the periodicity.It can be shown that the eigenfunctions of a particle in a periodic potential can always be chosen in this form by proving that translation operators (by lattice vectors) commute with the Hamiltonian. More generally, the consequences of symmetry on the eigenfunctions are described by representation theory.The concept of the Bloch state was developed by Felix Bloch in 1928, to describe the conduction of electrons in crystalline solids. The same underlying mathematics, however, was also discovered independently several times: by George William Hill (1877), Gaston Floquet (1883), and Alexander Lyapunov (1892). As a result, a variety of nomenclatures are common: applied to ordinary differential equations, it is called Floquet theory (or occasionally the Lyapunov-Floquet theorem). Variousone-dimensional periodic potential equations have special names, for example, Hill's equation:[1],where the θ's are constants. Hill's equation is very general, as the θ-related terms may viewed as a Fourier series expansion of a periodic potential. Other much studied periodic one-dimensional equations are the Kronig-Penney model and Mathieu's equation.References1.^ W Magnus and S Winkler (2004). Hill's Equation. Courier Dover, p. 11. ISBN0-0486495655.Further reading∙Charles Kittel, Introduction to Solid State Physics (Wiley: New York, 1996).∙Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).∙Felix Bloch, "Über die Quantenmechanik der Elektronen in Kristallgittern," Z. Physik52, 555-600 (1928).∙George William Hill, "On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon," Acta. Math.8, 1-36 (1886). (This work wasinitially published and distributed privately in 1877.)∙Gaston Floquet, "Sur les équations différentielles linéaires à coefficients périodiques,"Ann. École Norm. Sup.12, 47-88 (1883).∙Alexander Mihailovich Lyapunov, The General Problem of the Stability of Motion (London: Taylor and Francis, 1992). Translated by A. T. Fuller from Edouard Davaux'sFrench translation (1907) of the original Russian dissertation (1892).Retrieved from "/wiki/Bloch_wave"Density of statesFrom Wikipedia, the free encyclopediaIn statistical and condensed matter physics, the density of states (DOS) of a system describes the number of states at each energy level that are available to be occupied.A high DOS at a specific energy level means that there are many states available for occupation. A DOS of zero means that no states can be occupied at that energy level.ExplanationWaves, or wave-like particles, can only exist within quantum mechanical (QM) systems if the properties of the system allow the wave to exist. In some systems, the interatomic spacing and the atomic charge of the material allows only electrons of certain wavelengths to exist. In other systems, the crystalline structure of the material allows waves to propagate in one direction, while suppressing wave propagation in another direction. Waves in a QM system have specific wavelengths and can propagate in specific directions, and each wave occupies a different mode, or state. Because many of these states have the same wavelength, and therefore share the same energy, there may be many states available at certain energy levels, while no states are available at other energy levels. For example, the density of states for electrons in a semiconductor is shown in red in Fig. 2. For electrons at the conduction band edge, very few states are available for the electron to occupy. As the electron increases in energy, the electron density of states increases and more states become available for occupation. However, because there are no states available for electrons to occupy within the bandgap, electrons at the conduction band edge must lose at least E g of energy in order to transition to another available mode. The density of states can be calculated for electron, photon, or phonon in QM systems. The DOS is usually represented by one of the symbols g, ρ, D, n, or N, and can be given as a function of either energy or wavevector k. To convert between energy and wavevector, the specific relation between E and k must be known. For example, the formula forelectrons isAnd for photons, the formula isDerivationThe density of states is dependent upon the dimensional limits of the object itself. The role dimensions play is evident from the units of DOS (Energy-1Volume-1). In the limit that the system is 2 dimensional a volume becomes an area and in the limit of 1 dimension it becomes a length. It is important to note that the volume being referenced is the volume of k-space,the space enclosed by the constant energy surface of the system derived through a dispersion relation that relates E to k. An example of a specific k-space is given in Fig. 1. It can be seen that the dimensionality of the system itself will confine the momentum of particles inside the system.。
第六讲第一原理计算方法简介及Castep使用
www.tcm.ph / castep/ www.pwscf.o rg/
Pseudo, PAW Pseudo Pseudo Pseudo, PAW all-electron
Windows Linux
Linux
Linux
cms.mpi.un ivie.ac.at/v asp
CASTEP模块 Cambridge Serial Total Energy Package)
CASTEP是特别为固体材料学而设计的一个现代的量子力学基本程序，其 使用了密度泛函(DFT）平面波赝势方法，进行第一原理量子力学计算，以 探索如半导体，陶瓷，金属，矿物和沸石等材料的晶体和表面性质。 典型的应用包括表面化学，键结构，态密度和光学性质等研究， CASTEP 也可用于研究体系的电荷密度和波函数的3D形式。此外， CASTEP可用于 有效研究点缺陷（空位，间隙和置换杂质）和扩展缺陷（如晶界和位错） 的性质。 Material Studio使用组件对话框中的CASTEP选项来准备，启动，分析和 监测CASTEP计算工作。
密度泛函理论
赝势(pseudo potential) 赝势就是把离子实的 内部势能用假想的势能 取代真实的势能，但在 求解波动方程时，不改 变能量本征值和离子实 之间区域的波函数。模 守恒赝势NCP (Norm Conserving Pseudopotential) 和 超软赝势 USPP(Ultrasoft Pseudoptential)
模拟方法（LDA, GGA）。
5. 可模拟的内容：催化剂、聚合物、固体化学、
结晶学、晶粉衍射以及材料特性等。
主要模块：
建模模块 计算和分析模块 Amorphous Cell Blends GULP MesoDyn Morphology
第一性原理计算方法在材料科学中的应用
第一性原理计算方法在材料科学中的应用1.引言第一性原理计算方法（First Principles Calculation）是近年来发展的新型计算方法，用于准确计算分子和固体物质的能量、结构和物理性质。
它的优势在于不依赖于实验数据，可以直接从基本原理推导出体系的特性。
在材料科学领域，第一性原理计算方法已经成为研究材料的重要工具，可以为合成新材料和设计功能材料提供理论依据，并指导实验研究。
2.第一性原理计算方法的基本原理第一性原理计算方法的基本原理是量子力学中的密度泛函理论，它的基本假设是所有粒子的运动都可以描述为波函数的运动。
根据波函数理论，一个由N个电子和原子核组成的体系的波函数可以用N个单电子波函数表示。
通过求解薛定谔方程，可以确定体系的基态能量和电子的密度，从而得到体系的性质。
3.第一性原理计算方法在材料科学中的应用（1）材料合成第一性原理计算方法可以模拟材料的结构、动力学和化学反应，为材料合成提供理论指导。
例如，使用第一性原理计算方法可以预测材料的稳定性、生长机制和晶体缺陷，从而为材料的设计和制备提供指导。
（2）材料性能第一性原理计算方法可以计算材料的电子结构、热力学性质、光电性质和磁学性质等，从而为材料的性能研究提供理论基础。
例如，通过计算材料的电子结构，可以预测材料的导电性、热导率和热电性能等，为相关应用提供指导。
（3）材料改性第一性原理计算方法可以模拟材料的界面和表面结构，研究材料的改性效果。
例如，可以通过计算材料与其他材料的界面能量来评估材料的附着性和界面稳定性，从而指导材料的改性设计。
（4）功能材料设计借助第一性原理计算方法，可以针对具体的应用需求，设计出具有特定功能的材料。
例如，通过计算材料的光电性质、催化活性和磁学性质等，可以指导材料的功能设计，为实现特定的应用提供理论指导。
4.发展趋势随着材料科学和计算科学的发展，第一性原理计算方法的应用前景越来越广阔。
未来，第一性原理计算方法将会与机器学习和高通量计算等技术结合，为材料科学的研究提供更多的可能性。
第一性原理计算公式
第一性原理计算公式引言第一性原理计算是一种基于量子力学原理的理论和计算方法，可以用于研究和预测材料的物理和化学性质。
它是一种从头开始的计算方法，不依赖于任何经验参数和实验数据，因此被广泛应用于材料科学、化学、物理等领域的研究和设计。
在第一性原理计算中，通过求解薛定谔方程来得到体系的电子结构和能量。
这些计算需要使用一系列的公式和算法，本文将重点介绍一些常见的第一性原理计算公式，帮助读者理解这一领域的基本原理和方法。
基本概念在介绍具体的计算公式之前，我们先来回顾一些基本概念。
哈密顿算符哈密顿算符是量子力学中描述体系总能量和动力学演化的算符。
对于单电子体系，哈密顿算符可以写为：H = T + V其中T表示动能算符，V表示势能算符。
对于多电子体系，哈密顿算符则需要加入电子之间的相互作用算符，形式更加复杂。
波函数和薛定谔方程波函数是描述量子力学体系的状态的函数。
在薛定谔方程中，波函数满足以下的时间无关薛定谔方程：Hψ = Eψ其中H是哈密顿算符，ψ是波函数，E是能量。
求解薛定谔方程可以得到体系的能级结构和波函数。
密度泛函理论密度泛函理论是一种处理多电子体系的方法。
其核心思想是将多电子体系的性质建立在电子密度上。
密度泛函理论的基本方程是：E = T[n] + V[n] + E_{ee}[n]其中E是总能量，T[n]是电子动能的泛函，V[n]是外势能的泛函，E_{ee}[n]是电子之间相互作用的泛函。
第一性原理计算公式赝势方法赝势方法是一种快速计算材料电子结构的方法。
在赝势方法中，原子核和一部分芯层电子对价层电子的作用通过赝势进行描述。
赝势方法的基本方程是：H_{KS}ψ = Eψ其中H_{KS}是Kohn-Sham方程中的赝势哈密顿算符，ψ是波函数，E是能量。
平面波基组展开法平面波基组展开法是一种基于平面波基函数的展开方法。
平面波基组展开法的基本方程是：ψ(r) = ∑ c_k exp(ik·r)其中ψ(r)是波函数，c_k是展开系数，k是波矢。
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量子力学第一性原理：仅需五个物理基本常数——电子质量、电子电量、普郎克常数、光速和玻耳兹曼常数，通过求薛定谔方程得到材料的电子结构，而不依赖于任何经验常数即可以预测微观体系的状态和性质，预测材料的组分、结构、性能之间的关系，进一步设计具有特定性能的新材料。
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)。
如果某些原理或数据来源于第一性原理，但推演过程中加入了一些假设(这些假设当然是很有说服力的)，那么这些原理或数据就称为“半经验的”。
量子化学的第一性原理是指多电子体系的Schrödinger方程，但是光有这个方程是无法解决任何问题的，量子力学能够准确的解决的问题很少很少，绝大多数都是有各种各样的近似，为此计算量子力学提出一个称为“从头计算”的原理作为第一性原理，除了Schrödinger 方程外还允许使用下列参数和原理：(1) 物理常数，包括光速c、Planck常数h、电子电量e、电子质量me以及原子的各种同位素的质量，尽管这些常数也是通过实验获得的。
(在国际单位值中，光速是定义值，Planck 常数是测量值，在原子单位制中则相反。
)(2) 各种数学和物理的近似，最基本的近似是“非相对论近似”(Schrödinger方程本来就是非相对论的原理)、“绝热近似”(由于原子核质量比电子大得多，而把原子核当成静止的点处理)和“轨道近似”(用一个独立函数来描述一个独立电子的运动)。
量子化学的从头计算方法就是在各种近似上作的研究。
如果只考虑一个电子，而把其他电子对它的作用近似的处理成某种形式的势场，这样就可以把多电子问题简化成单电子问题，这种近似称为单电子近似，也称为平均场近似，例如最基本的从头计算方法哈特里－富克（Hartree-Fock）方法，是平均场近似的一种，它把所有讨论的电子视为在离子势场和其他电子的平均势场中的运动。
但是哈特里－富克近似程度过大，忽略了电子之间的交换和相关效应，使得计算的精度受到一定的限制，为了解决这一问题，P Hohenberg和W Kohn于1964年提出密度泛函理论（density functional theory, DFT），这一理论将电子之间的交换相关势表示为密度泛函，然后使薛定谔方程在考虑了电子之间的复杂相互作用后利用建立在自洽场近似的方法求解，DFT认为：粒子的哈密顿量取决于电子密度的局域值，由此可以得出局域密度近似（local density approximation）方法。
由于诸多近似方法的使用，“从头计算”方法并不是真正意义上的第一性原理，但是其近似方法的运用使得量子计算得以实现。
从头计算的结果具有相当的可靠程度，某些精确的从头计算产生的误差甚至比实验误差还小。
话说第一性原理的基本感念是指不采用经验参数……（此后省略若干千字）不采用经验参数，但也得有近似才能计算。
在处理原子的时候就采用了波恩-奥本海默近似（既绝热近似）这个近似的主要内容就是电子运动速度远远大于原子核，于是近似原子核不动，只考虑电子运动。
于是，这个近似带来的效果就是体系在绝对零度时候的性质。
@@@什么是第一性原理呢？其实就是指从最基本的原理出发，不掺杂任何经验参数，而得出所有的现象。
其实这个是很难的。
但是能验证规律的正确性。
而本文所提到的第一性原理，主要是指从量子力学的基本假设出发，而推导各种物理，化学等现象。
也指从头算，即ab initio。
最早期的从头算主要是一些量子化学的专家在搞。
其实这还要从头说起，薛定谔搞出了薛定谔方程。
方程写起来虽然简单，而且直观。
（动能算符+势能算符）*波函数=能量的本征值*波函数。
这是定态波函数，我就不写含时的方程了。
这个方程主要是用来求基态。
方程虽然容易写，但是求解起来实在是太难了。
有点像经典力学里的混沌现象。
虽然每个粒子都服从牛顿三大定律，但是求解是不可能的。
首先，求解的难题是电子和核的相互作用项难以分开。
还好奥本海默提出了绝热近似，即核的质量和电子的质量不是一个数量级的，所以速度也不是一个数量级的。
认为核缓慢的能跟上电子的运动。
这样，我们即可将核的方程和电子的方程分开。
而在真实的材料中，电子的作用是很大。
例如成键主要是指电子和电子的相互作用。
因此我们只需要仔细求解电子的薛定谔方程即可。
对于N个电子的系统，方程是3N维的。
其求解仍然很难。
所以，科学家们就想办法。
最后的办法是，利用自洽的方法求解。
自洽其实是一个很有意思的办法，说白了，就是指，A和B有数学的依赖关系，但A和B的方程是解不出来的。
给出一个A我可以求B，有B又反求出一个A，这样循环下去直到|A_new - A_old|<无穷小，或者|B_new -B_old|<无穷小。
方程自洽，运算结束。
这只是简单的举例。
首先，这个由hartree和fock提出了hartree-fock近似，他们将多电子的波函数写成了单个电子波函数的积的形式，考虑了全同粒子的交换性，而哈密顿量写成了密度的泛函，进而求得基态能量，可以说这是一个很大的进步。
但确定是没有考虑电子间的关联作用。
接着，由kohn等人进一步改善了薛定谔方程，首先他们认为基态可以写成电子密度的泛函，真实的材料的所有性质都可以有电子密度求出。
这就是密度泛函理论，详细的推导还是很繁琐的。
k-s方程中，动能是用的是无关联的自由电子动能，势能用的是无交换的电子相互作用能和离子对电子的势能，剩下的就是交换关联能。
虽然方程是相当完美的，也很准确，但很可惜。
交换关联能的具体形式是无法求解的。
没办法，总有一些人是乐于奉献的，有些人花一辈子在找一个比较好的交换关联能的形式。
LDA是一个很好的近似，认为电子可以分成各个部分均匀的电子气，进而得出了交换关联能的形式，本来这是一个近似。
但是，在总能计算过程中，是相当的准确，这主要是由于交换能的低估和关联能的高估得到了很好的补偿。
最后再讲一下，KS方程的基本流程。
首先，给出一个试探的电荷密度，可以写出哈密顿量，从而求得本证值和本证波函数。
而本证波函数的模的平方就是电荷密度，这样看电荷密度是否自洽。
不自洽就继续，直到自洽。
当然我这里也只是笼统的说。
在具体实现中，还有很多细节。
需要仔细推导公式。
如波函数是用平面波展开还是用轨道展开。
轨道主要指的是TB的方法，即紧束缚方法。
还有介于他们之间，如LMTO。
还有，为了减少平面波的数量，计算加快是采用赝势的方法。
很多很多细节问题需要主要，还是需要找一本书仔细推导。
但基本思想是，电子的作用相当关键，如果我们能解决好电子的问题，我们就能解决很多难题，如高温超导等问题。
第一性原理之所以能得到如此发展，很大的程度依赖于计算机的发展，现在的集群越来越强大。
要求越来越多的人，既懂理论，又会编程。
这样才能由理论到程序，到计算，最后得到结果，解释很多现象，探索基本的规律。
想对规律研究，紧紧靠推导理论是难以实现的，还是需要在新现象或反常现象中去摸索。
这样，就需要我们做一些具体的计算。
也正因为如此，第一性原理还有很大的发展空间，是很有潜力。
但需要你去认真的推导理论，热爱编程。
@@@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（ab initio）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”。
爱因斯坦晚年致力与“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。
目前主要的计算软件：PWscf，Material Studio，V ASP，Wein2k等.个人推荐PW.需要精通的编程语言：fortran，C.@@@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（ab initio）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”。