VI. Choice of Initial Wave Functions for the Electronic States
VI.电子态初始波函数的选择
B. Vanderbilt potentials B.范德比尔特潜能
C. Real-space projection; nonuniqueness of spherical harmonic projection
C.实空间投影；球谐投影的非唯一性
D. Potentials for real-space projection
D.实空间投影的潜力
X. Parallel Computers X.并行计算机
XI. Concluding Remarks XI.结束语

Acknowledgments 致谢
References 参考资料
are still debated, it is clear that whether or not a more complete description of the world is possible, those things that modern quantum theory does predict are predicted with incredible accuracy. One outstanding example of this accuracy is the calculation of the gyromagnetic ratio of the electron, which agrees with the experimental result to the limit of the measurement, some 10 significant figures. Quantum theory has also proven correct and provided fundamental understanding for a wide variety of phenomena, including the energy levels of atoms, the covalent bond, and the distinction between metals and insulators. Further, in every instance of its application to date, the equations of quantum mechanics have yet to be shown to fail. There is, therefore, every reason to believe that an understanding of a great number of phenomena can be achieved by continuing to solve these equations.
尽管人们对量子理论的发展仍有争议，但显而易见的是，无论是否有可能对世界进行更完整的描述，现代量子理论都能以惊人的精确度预言那些确实能预言的事情。这种准确性的一个突出例子就是对电子回旋磁比的计算，它与实验结果的吻合程度达到了测量的极限，大约是 10 个有效数字。量子理论也被证明是正确的，并为原子能级、共价键、金属和绝缘体的区别等各种现象提供了基本的理解。此外，迄今为止，在量子力学的每一个应用实例中，量子力学方程都还没有被证明是失效的。因此，我们完全有理由相信，通过继续求解这些方程，可以理解大量现象。

As we shall soon see, the ability of quantum mechanics to predict the total energy of a system of electrons and nuclei, enables one to reap a tremendous benefit from a quantum-mechanical calculation. In fact this entire article is dedicated to just this one type of quantummechanical calculation, the foundation for which is quite strong. The quantum-mechanical rules, or Hamiltonians, for calculating the total energy of simple one-atom systems have provided some of the most precise tests of the theory, and the rules for calculating the energies of more complicated systems are simple, straightforward extensions of these atomic Hamiltonians. It is therefore eminently reasonable to expect quantum mechanics to predict accurately the total energies of aggregates of atoms as well. So far, this expectation has been confirmed time and time again by experiment.
我们很快就会看到，量子力学能够预测电子和原子核系统的总能量，这使我们能够从量子力学计算中获得巨大收益。事实上，这篇文章就是专门讨论这一种量子力学计算的，其基础是相当牢固的。计算简单单原子系统总能量的量子力学规则或称汉密尔顿法则，为理论提供了一些最精确的检验，而计算更复杂系统能量的规则，则是这些原子汉密尔顿法则简单明了的延伸。因此，我们完全有理由期待量子力学也能准确预测原子集合体的总能量。迄今为止，实验一次又一次地证实了这一预期。

A few moments’ thought shows that nearly all physical properties are related to total energies or to differences between total energies. For instance, the equilibrium lattice constant of a crystal is the lattice constant that minimizes the total energy; surfaces and defects of solids adopt the structures that minimize their corresponding total energies. If total energies can be calculated, any physical property that can be related to a total energy or to a difference between total energies can be determined computationally. For example, to predict the equilibrium lattice constant of a crystal, a series of total-energy calculations are performed to determine the total energy as a function of the lattice constant. As shown in Fig. 1, the results are then plotted on a graph of energy versus lattice constant, and a smooth curve is constructed through the points. The theoretical value for the equilibrium lattice constant is the value of the lattice constant at the minimum of this curve. Total-energy techniques also have been successfully used to predict with accuracy equilibrium lattice constants, bulk moduli, phonons, piezoelectric constants, and phase-transition pressures and temperatures (for reviews see Cohen, 1984; Joannopoulos, 1985; Pickett, 1989).
稍加思考就会发现，几乎所有的物理特性都与总能量或总能量之间的差异有关。例如，晶体的平衡晶格常数就是使总能最小化的晶格常数；固体的表面和缺陷采用的结构使其相应的总能最小化。如果可以计算总能量，那么任何与总能量或总能量之差相关的物理特性都可以通过计算确定。例如，为了预测晶体的平衡晶格常数，需要进行一系列总能计算，以确定总能与晶格常数的函数关系。如图 1 所示，然后将计算结果绘制在能量与晶格常数的关系图上，并通过各点绘制出一条平滑的曲线。平衡晶格常数的理论值就是这条曲线最小处的晶格常数值。总能量技术还被成功用于精确预测平衡晶格常数、体模量、声子、压电常数以及相变压力和温度（相关评论见 Cohen, 1984; Joannopoulos, 1985; Pickett, 1989）。

Part of our aim in this article is to introduce the usefulness of quantum total-energy techniques to a broad
本文的部分目的是向广大读者介绍量子总能技术的用途。

FIG. 1. Theoretical determination of an equilibrium lattice constant. Calculations (open circles) at various possible lattice constants are performed and a smooth function is fitted through the points. The predicted lattice constant is determined by the minimum in the curve.
图 1.平衡晶格常数的理论测定。在各种可能的晶格常数下进行计算（开圆圈），并通过各点拟合一个平滑函数。预测的晶格常数由曲线的最小值决定。
range of scientists, including, for example, chemists, biologists, and geophysicists, who can at last benefit from these techniques. It is often suggested that quantum mechanics was primarily developed to describe events on the atomic scale, raising the question, “Of what use are quantum-mechanical calculations in a science not concerned directly with events on the atomic scale?” Since our world is composed of and defined by the interactions of atoms and molecules, a detailed and fundamental understanding of the world must ultimately rest on comprehending these interactions. The good news that this article brings is that the search for this kind of understanding is no longer a mere idle philosophical musing but rather a practical methodology.
量子力学在科学领域中的应用范围越来越广，包括化学家、生物学家和地球物理学家，他们终于可以从这些技术中获益。人们常常认为，量子力学的发展主要是为了描述原子尺度上的事件，这就提出了一个问题："量子力学计算对于一门不直接涉及原子尺度上事件的科学有什么用呢？既然我们的世界是由原子和分子的相互作用构成和定义的，那么对世界的详细和基本理解最终必须建立在对这些相互作用的理解之上。这篇文章带来的好消息是，寻求这种理解不再仅仅是一种空洞的哲学思辨，而是一种实用的方法论。

There are many examples of connections between atomic and macroscopic levels being made every day. The “lock and key” mechanism in biological systems has led to the development of “designer drugs” whose shapes correspond to the “key” in the relevant biological reaction. In turn the shapes of the drugs are known and understood only because of the quantum-mechanical description of covalent bonds. Although no drug has yet been designed by determining the shape of the molecule by solving the Schrödinger equation, this particular application of quantum mechanics is not far away. There are also examples of the application of quantum mechanics beyond the atomic scale in materials science, for instance in the onset of failure in a material. The failure of a material starts on the atomic scale when one bond is stressed beyond its yield-stress and breaks. Thus it is obvious that where and when a material actually starts to fail is determined by quantum mechanics. Though there are many examples in materials science of significant progress having been made without any need for quantummechanical modeling, this progress is often limited as one pushes forward and encounters the atomic world. For instance, the understanding of the properties and behavior of dislocations comes from classical elasticity theory, but even in these cases very little is known about the core of a dislocation, precisely because this part of the disloca-
在原子和宏观层面之间建立联系的例子比比皆是。生物系统中的 "锁与钥匙 "机制导致了 "特制药物 "的开发，这些药物的形状与相关生物反应中的 "钥匙 "相对应。反过来，只有通过对共价键的量子力学描述，才能知道和理解药物的形状。虽然目前还没有通过求解薛定谔方程来确定分子形状的药物，但量子力学的这种特殊应用离我们并不遥远。在材料科学中，量子力学的应用也有超越原子尺度的例子，例如材料失效的开始。当一个键受力超过其屈服应力并断裂时，材料的失效就在原子尺度上开始了。因此，材料在何时何地开始失效显然是由量子力学决定的。尽管在材料科学领域有许多无需量子力学建模就能取得重大进展的例子，但当我们向前推进并进入原子世界时，这种进展往往会受到限制。例如，人们对位错特性和行为的理解来自经典弹性理论，但即使在这种情况下，人们对位错的核心也知之甚少，这正是因为位错的这一部分--"位错的核心"--"位错的核心"--"位错的核心"--"位错的核心"。
tion requires detailed quantum-mechanical modeling. Moreover, even classical elasticity theory, which can only be applied on a macroscopic scale, is directly related to the atomic world through the elastic constants, the parameters of elasticity theory determined by the quantum-mechanical behavior of the material. Though materials constants such as the elastic constants may often be simply measured in the laboratory, the geophysicist modeling a continental drift does not have the luxury of performing experiments to bypass quantummechanical calculations. It is not possible, at present, to generate geophysical pressures in the laboratory. Therefore the relevant high-pressure parts of the phase diagrams of the materials that constitute the earth are unknown. Geophysicists will greatly benefit from quantum-mechanical modeling, which can provide them with the parameters needed to pursue their research.
这需要详细的量子力学建模。此外，即使是只能在宏观尺度上应用的经典弹性理论，也是通过弹性常数（由材料的量子力学行为决定的弹性理论参数）与原子世界直接相关的。虽然弹性常数等材料常数通常可以在实验室中简单测量，但建立大陆漂移模型的地球物理学家却无法绕过量子力学计算进行实验。目前还不可能在实验室中产生地球物理压力。因此，构成地球的材料相图中的相关高压部分是未知的。量子力学建模可以为地球物理学家提供开展研究工作所需的参数，他们将从量子力学建模中获益匪浅。
When should a scientist consider quantum-mechanical modeling? The examples given above suggest that quantum-mechanical modeling be considered in situations where experiment is impossible. This principle is not limited, as in the geophysics example, to situations that are completely inaccessible to experiment, but also includes the performance of “computational experiments,” which afford far greater “experimental” control than their physical counterparts. For instance, one can “reach into” a theoretical chemical calculation and, at will, bend bonds at experimentally unstable and inaccessible angles to gain insight into the processes controlling chemical reactions. Or, one can study the properties of isolated defects in a material in which segregation of impurities towards those defects tends to cloud the experimental results.
科学家何时应该考虑量子力学建模？上述例子表明，在不可能进行实验的情况下应考虑量子力学建模。这一原则并不像地球物理学的例子那样，仅限于完全无法进行实验的情况，还包括 "计算实验"，因为计算实验的 "实验 "控制能力远远超过物理实验。例如，我们可以 "深入 "到理论化学计算中，随意以实验中不稳定和难以接近的角度弯曲化学键，从而深入了解控制化学反应的过程。或者，我们可以研究一种材料中孤立缺陷的特性，在这种材料中，杂质向这些缺陷的偏析往往会使实验结果变得模糊不清。

Another relevant consideration is what can be calculated quantum mechanically and at what cost. The boundary of feasible quantum-mechanical calculations has shifted significantly, to the extent that it may now be more cost effective to employ quantum-mechanical modeling even when experiments do offer an alternative. Moreover, many fields of science, not just physics and basic materials science, may benefit. It is true that the original modeling of the covalent bond was not quantitatively accurate, and it did not give the correct value for the energy associated with the bond. However, chemists solving the Schrödinger equation nowadays accurately calculate the energy of covalent bonds as well as their equilibrium lengths, force constants, and polarizabilities. Physicists have developed many methods that can be used to calculate a wide range of physical properties of materials, such as lattice constants and elastic constants. These methods, which require only a specification of the ions present (by their atomic number), are usually referred to as $ab$$ab$aba b initio methods.
另一个相关的考虑因素是什么可以用量子力学计算，成本又是多少。可行的量子力学计算的边界已经发生了重大变化，以至于现在即使在实验确实提供了替代方案的情况下，采用量子力学建模可能更具成本效益。此外，许多科学领域，而不仅仅是物理学和基础材料科学，都可能从中受益。诚然，最初的共价键建模在定量上并不准确，也没有给出与共价键相关的能量的正确值。然而，如今化学家在求解薛定谔方程时，可以准确计算出共价键的能量及其平衡长度、力常数和极化性。物理学家已经开发出许多方法，可用于计算材料的各种物理性质，如晶格常数和弹性常数。这些方法只需指定存在的离子（按原子序数），通常被称为 $ab$$ab$aba b 初始方法。

Many of the $ab$$ab$aba b initio methods that physicists and chemists use have existed for more than a decade, and it is not the purpose of this article to describe or compare the wide range of methods that exist. All of the ab initio methods have been continuously refined over recent years
物理学家和化学家使用的许多 $ab$$ab$aba b 初始方法已经存在了十多年，本文的目的不是描述或比较现有的各种方法。近年来，所有的原子弹初始化方法都在不断改进
and all have benefited from the availability of increasingly powerful computers. A decade ago most ab initio methods were capable only of modeling systems of a few atoms, and hence applicability to real-world systems at that point was extremely limited. Most methods can now model systems that contain some tens of atoms and are used to study a small but significant range of interesting problems. Of all the methods, one, the total-energy pseudopotential method, stands alone. A decade ago this method was also capable of modeling only few-atom systems. Now, however, this method can model thousandatom systems, and it is already clear that this number will increase by at least a factor of 10 within the next five years. Pushing back the limits of quantum-mechanical modeling to this extent, the total-energy pseudopotential method opens up a wide range of interesting problems to quantum-mechanical calculation, and the future should bring the application of quantum mechanics to many new fields of science. The increase in the number of atoms that can be handled is directly due to an increase in computational efficiency of the ab initio pseudopotential method, which also means that there is an increasing class of problems for which it is more cost effective to use quantum-mechanical modeling than experiment to determine the physical parameters of systems. One purpose of this article is to heighten awareness amongst scientists in a range of scientific disciplines that the world is quantum mechanical and that there now exists an ab initio method that allows the quantum mechanics to be solved and incorporated into everyday science.
所有这些都得益于日益强大的计算机。十年前，大多数自创方法只能模拟由几个原子组成的系统，因此对现实世界系统的适用性极为有限。现在，大多数方法都能模拟包含几十个原子的系统，并被用于研究少量但范围广泛的有趣问题。在所有方法中，有一种方法，即总能假势法，独树一帜。十年前，这种方法还只能模拟少原子系统。但现在，这种方法可以模拟千原子系统，而且很明显，在未来五年内，这个数字将至少增加 10 倍。将量子力学建模的极限推后到这一程度，总能伪势法为量子力学计算开辟了广泛的有趣问题，未来量子力学将应用于许多新的科学领域。可处理原子数量的增加直接归因于ab initio 伪势方法计算效率的提高，这也意味着有越来越多的问题，使用量子力学建模比实验确定系统的物理参数更具成本效益。这篇文章的目的之一是提高各学科科学家的认识，让他们知道世界是量子力学的，现在有一种自证方法可以解决量子力学问题并将其纳入日常科学。

There is an economy of scale to ab initio total-energy calculations because so many physical properties are related to total energies. While just one piece of theoretical “apparatus” is needed to calculate all the physical properties that are related to total energies, completely different sets of experimental apparatus are required to measure each class of physical property of a material. This represents an enormous advantage of quantummechanical modeling over experimental measurements. Comparing the decreasing cost of computers with the cost of a large number of different pieces of experimental apparatus needed to carry out the same functions, one sees that the cost effectiveness of quantum-mechanical modeling methods over physical experimentation will continue to increase with time. This, then, is the time for researchers in a wide range of scientific disciplines to consider very seriously whether quantum-mechanical modeling may be applied in a cost-effective way to their own field of research, even if the field of research is far removed from what is usually assumed to be the quantum-mechanical world.
由于许多物理性质都与总能量有关，因此从头开始的总能量计算具有规模经济性。计算与总能量相关的所有物理性质只需要一套理论 "仪器"，而测量材料的每一类物理性质则需要完全不同的实验仪器。这是量子力学建模相对于实验测量的巨大优势。将计算机成本的不断降低与执行相同功能所需的大量不同实验仪器的成本进行比较，我们可以发现，量子力学建模方法相对于物理实验的成本效益将随着时间的推移而不断提高。因此，现在正是各种科学学科的研究人员认真考虑是否可以将量子力学建模以经济有效的方式应用于自己的研究领域的时候，即使研究领域与通常认为的量子力学世界相去甚远。

Total-energy pseudopotential calculations do require significant amounts of computer time, even for systems containing a few atoms in the unit cell, and the computational time required to perform the calculations always increases with the number of atoms in the unit cell. Thus, for large systems containing hundreds of atoms in the unit cell, it is essential to use the most efficient nu-
总能伪势计算确实需要大量的计算机时间，即使是对于单胞中含有少量原子的系统也是如此，而且进行计算所需的计算时间总是随着单胞中原子数的增加而增加。因此，对于单元格中含有数百个原子的大系统，必须使用最高效的 Nu-
merical algorithms. In the following pages we shall discuss in detail the latest methods for doing this. Among the methods we shall discuss, two have revolutionized the field of $ab$$ab$aba b initio total-energy calculation, each increasing the number of atoms in a calculable system by more than an order of magnitude over previously existing techniques. As we shall discuss in detail, the moleculardynamics method developed by Car and Parrinello (1985) transformed the way in which we viewed quantummechanical calculations and hence total-energy pseudopotential calculations; instead of finding a coupled selfconsistent solution to a descretized partial differential equation through matrix techniques, Car and Parrinello minimized a single function through simulated annealing. The Car-Parrinello method can be used to perform calculations for systems containing on the order of one hundred atoms in the unit cell. However, severe difficulties are encountered in certain cases when one attempts to use this method to perform calculations on much larger systems. Recently, conjugate-gradients methods have been developed (Teter et al. 1989; Gillan, 1989) that overcome the difficulties encountered with the molecular-dynamics technique. Conjugate-gradients methods have again transformed total-energy pseudopotential calculations by replacing simulated annealing minimization with a direct, completely self-consistent second-order search for the minimum. Using these methods, one can perform calculations for systems containing many hundreds, and soon thousands, of atoms.
我们将在下文中详细讨论实现这一目标的最新方法。下面我们将详细讨论实现这一目标的最新方法。在我们将要讨论的方法中，有两种方法彻底改变了 $ab$$ab$aba b 起始总能计算领域，每种方法都将可计算系统中的原子数量增加了一个数量级以上，超过了之前已有技术的数量级。正如我们将详细讨论的那样，Car 和 Parrinello（1985 年）开发的分子动力学方法改变了我们对量子力学计算的看法，进而改变了我们对总能伪势计算的看法；Car 和 Parrinello 不再通过矩阵技术寻找解具体化偏微分方程的耦合自洽解，而是通过模拟退火最小化单个函数。Car-Parrinello 方法可用于计算单元格中含有约 100 个原子的系统。然而，在某些情况下，当人们试图使用这种方法对更大的系统进行计算时，会遇到严重的困难。最近开发的共轭梯度方法（Teter 等，1989 年；Gillan，1989 年）克服了分子动力学技术所遇到的困难。共轭梯度方法以直接、完全自洽的二阶最小值搜索取代了模拟退火最小化，从而再次改变了总能伪势计算。利用这些方法，我们可以对包含数百个原子甚至数千个原子的系统进行计算。

The molecular-dynamics and conjugate-gradients methods allow pseudopotential calculations to be performed for much larger systems than was possible using conventional matrix diagonalization methods. They also allow, for the first time, tractable ab initio quantummechanical simulations to be performed for systems at nonzero temperatures. While these capabilities offer the obvious advantage of permitting more complex systems to be studied, there is yet another benefit to be gained by using these new computational methods. By increasing the efficiency of the total-energy pseudopotential technique, they have greatly extended the range of application of this technique by allowing, for the first time, the inclusion of noble and transition-metal atoms and firstrow elements such as oxygen in large pseudopotential calculations. Until recently it was widely believed that computations including such elements would be completely intractable with pseudopotentials in a plane-wave representation. Recent work (Allan and Teter, 1987; Bar-Yam et al., 1989; Rappe et al., 1990; Vanderbilt, 1990; Trouillier and Martins, 1991) has shown that pseudopotential calculations can be performed for systems containing these atoms by employing a substantial but manageable number of plane waves in the basis set. With their increased efficiency, molecular-dynamics and conjugate-gradients methods can handle very large plane-wave basis sets and therefore permit large totalenergy pseudopotential calculations to be performed with these new pseudopotentials, thus opening the way for
通过分子动力学和共轭梯度方法，可以对更大的系统进行伪势计算，这比使用传统的矩阵对角化方法要容易得多。它们还首次允许在非零温度下对系统进行可控的原子序数量子力学模拟。虽然这些功能具有允许研究更复杂系统的明显优势，但使用这些新计算方法还有另一个好处。通过提高总能伪势技术的效率，它们首次允许在大型伪势计算中加入惰性原子、过渡金属原子和第一排元素（如氧），从而大大扩展了该技术的应用范围。直到最近，人们还普遍认为包括这些元素在内的计算将完全无法用平面波表示的伪势进行。最近的研究（Allan 和 Teter，1987 年；Bar-Yam 等人，1989 年；Rappe 等人，1990 年；Vanderbilt，1990 年；Trouillier 和 Martins，1991 年）表明，通过在基集中使用大量但易于管理的平面波，可以对含有这些原子的系统进行伪势计算。随着效率的提高，分子动力学和共轭梯度方法可以处理非常大的平面波基集，因此可以用这些新的伪势进行大的总能伪势计算，从而为以下方面开辟了道路
study of a larger variety of chemical systems than was previously possible.
对更多化学体系的研究，这在以前是不可能实现的。

This article provides a detailed description of the total-energy pseudopotential method, the moleculardynamics method, and conjugate-gradient minimization. The article also discusses related techniques and developments in a number of areas that have played a role in increasing the computational efficiency of these methods. It is hoped that the information presented here is sufficiently detailed and at the leading edge of the work being done to be useful to scientists working both in and outside the field of ab initio quantum-mechanical calculations.
本文详细介绍了总能伪势法、分子动力学法和共轭梯度最小化法。文章还讨论了一些领域的相关技术和发展，这些技术和发展在提高这些方法的计算效率方面发挥了作用。希望本文所提供的信息足够详细，并处于当前工作的前沿，从而对从事原子量子力学计算领域内外工作的科学家有所帮助。

This section describes the total-energy pseudopotential method. An extremely useful review of the pseudopotential method can be found in articles by Ihm et al. (1979) and Denteneer and van Haeringen (1985). These articles are essential reading for anyone intending to implement codes for total-energy pseudopotential calculations. Total-energy calculations can only be performed if a large number of simplifications and approximations are used. These simplifications and approximations are described in the following sections.
本节介绍总能假势法。Ihm 等人（1979 年）以及 Denteneer 和 van Haeringen（1985 年）的文章对伪势方法进行了非常有用的综述。这些文章是任何打算实施总能伪势计算代码的人的必读文章。只有使用大量的简化和近似方法，才能进行总能计算。下文将介绍这些简化和近似方法。

Prediction of the electronic and geometric structure of a solid requires calculation of the quantum-mechanical total energy of the system and subsequent minimization of that energy with respect to the electronic and nuclear coordinates. Because of the large difference in mass between the electrons and nuclei and the fact that the forces on the particles are the same, the electrons respond essentially instantaneously to the motion of the nuclei. Thus the nuclei can be treated adiabatically, leading to a separation of electronic and nuclear coordinates in the many-body wave function-the so-called BornOppenheimer approximation. This “adiabatic principle” reduces the many-body problem to the solution of the dynamics of the electrons in some frozen-in configuration of the nuclei.
要预测固体的电子和几何结构，就必须计算系统的量子力学总能量，然后根据电子和原子核坐标将该能量最小化。由于电子和原子核的质量相差很大，而且粒子所受的力是相同的，因此电子对原子核运动的反应基本上是瞬时的。因此，可以对原子核进行绝热处理，从而在多体波函数中分离电子坐标和核坐标--即所谓的玻恩-奥本海默近似。这种 "绝热原理 "将多体问题简化为电子在核的某种冻结构型中的动力学求解。

Even with this simplification, the many-body problem remains formidable. Further simplifications, however, can be introduced that allow total-energy calculations to be performed accurately and efficiently. These include density-functional theory to model the electron-electron interactions, pseudopotential theory to model the electron-ion interactions, supercells to model systems with aperiodic geometries, and iterative minimization techniques to relax the electronic coordinates.
即使进行了这样的简化，多体问题仍然十分棘手。然而，进一步的简化可以使总能计算更精确、更高效。这些简化方法包括：建立电子-电子相互作用模型的密度泛函理论、建立电子-离子相互作用模型的伪势理论、建立非周期性几何体系模型的超胞，以及放松电子坐标的迭代最小化技术。

Very briefly, the essential concepts are the following:
简而言之，基本概念如下：
(i) Density-functional theory (Hohenberg and Kohn,
(i) 密度功能理论（Hohenberg 和 Kohn、

1964; Kohn and Sham, 1965) allows one, in principle, to map exactly the problem of a strongly interacting electron gas (in the presence of nuclei) onto that of a single particle moving in an effective nonlocal potential. Although this potential is not known precisely, local approximations to it work remarkably well. At present, we have no a priori arguments to explain why these approximations work. Density-functional theory was revitalized in recent years only because theorists performed totalenergy calculations using these potentials and showed that they reproduced a variety of ground-state properties within a few percent of experiment. Thus the acceptance of local approximations to density-functional theory has only emerged, a posteriori, after many successful investigations of many types of materials and systems. Generally, total-energy differences between related structures can be believed to within a few percent and structural parameters to at least within a tenth of an $\text{Å}$$\text{Å}$"Å"\AA. Cohesive energies, however, can be in error by more than $10\mathrm{\%}$$10\mathrm{\%}$10%10 \%.
1964；Kohn 和 Sham，1965）原则上允许人们将强相互作用电子气（在有原子核存在的情况下）的问题精确地映射到在有效非局部势中运动的单个粒子的问题上。尽管我们对这个势还不十分了解，但对它的局部近似却非常有效。目前，我们还没有先验的论据来解释这些近似为什么有效。密度泛函理论之所以能在近几年重新焕发生机，完全是因为理论家们使用这些势进行了总能计算，结果表明它们在实验的百分之几的范围内再现了各种基态性质。因此，对密度函数理论的局部近似的接受是在对多种类型的材料和系统进行了许多成功的研究之后才出现的。一般来说，相关结构之间的总能差异可以被认为在百分之几以内，结构参数至少在十分之一 $\text{Å}$$\text{Å}$"Å"\AA 以内。然而，内聚能的误差可能超过 $10\mathrm{\%}$$10\mathrm{\%}$10%10 \% 。
(ii) Pseudopotential theory (Phillips, 1958; Heine and Cohen, 1970) allows one to replace the strong electron ion potential with a much weaker potential-a pseudopotential - that describes all the salient features of a valence electron moving through the solid, including relativistic effects. Thus the original solid is now replaced by pseudo valence electrons and pseudo-ion cores. These pseudoelectrons experience exactly the same potential outside the core region as the original electrons but have a much weaker potential inside the core region. The fact that the potential is weaker is crucial, however, because it makes the solution of the Schrödinger equation much simpler by allowing expansion of the wave functions in a relatively small set of plane waves. Use of plane waves as basis functions makes the accurate and systematic study of complex, low-symmetry configurations of atoms much more tractable.
(ii) 伪电势理论（菲利普斯，1958 年；海因和科恩，1970 年）允许我们用一个弱得多的电势--伪电势--来取代强电子离子电势，该电势描述了价电子在固体中移动的所有显著特征，包括相对论效应。因此，原来的固体现在被伪价电子和伪离子核心所取代。这些伪电子在核心区域外的电势与原始电子完全相同，但在核心区域内的电势要弱得多。然而，电势较弱这一事实是至关重要的，因为它允许在一组相对较小的平面波中展开波函数，从而使薛定谔方程的求解简单得多。使用平面波作为基函数，使得对复杂的低对称性原子构型进行精确而系统的研究变得更加容易。
(iii) The supercell approximation allows one to deal with aperiodic configurations of atoms within the framework of Bloch’s theorem (see Ashcroft and Mermin, 1976). One simply constructs a large unit cell containing the configuration in question and repeats it periodically throughout space. By studying the properties of the system for larger and larger unit cells, one can gauge the importance of the induced periodicity and systematically filter it out. This approach has been successfully tested against “exact” Koster-Slater Green’s-function methods (see Baraff and Schluter, 1979), which are only tractable for very-high-symmetry configurations.
(iii) 超单元近似允许我们在布洛赫定理的框架内处理原子的非周期性构型（见 Ashcroft 和 Mermin，1976 年）。我们只需构建一个包含相关构型的大单元，并在整个空间周期性地重复该构型。通过研究越来越大的单元格的系统特性，我们可以衡量诱导周期性的重要性，并系统地将其过滤掉。这种方法已经成功地与 "精确 "的 Koster-Slater 格林函数方法（见 Baraff 和 Schluter，1979 年）进行了比较，后者只适用于对称性非常高的构型。
(iv) Finally, new iterative diagonalization approaches (Car and Parrinello, 1985; Payne et al., 1986; Williams and Soler, 1987; Gillan, 1989; Stich et al., 1989; Teter et al., 1989) can be used to minimize the total-energy functional. These are much more efficient than the traditional diagonalization methods. These new methods allow expedient calculation of ionic forces and total energies and significantly raise the level of modern totalenergy calculations. These methods are the subject of Secs. III, IV, and V.
(iv) 最后，新的迭代对角化方法（Car 和 Parrinello，1985 年；Payne 等人，1986 年；Williams 和 Soler，1987 年；Gillan，1989 年；Stich 等人，1989 年；Teter 等人，1989 年）可用于最小化总能量函数。这些方法比传统的对角化方法更有效。这些新方法可以方便地计算离子力和总能，大大提高了现代总能计算的水平。这些方法是第 III、IV 和 V 章的主题。

The most difficult problem in any electronic structure calculation is posed by the need to take account of the effects of the electron-electron interaction. Electrons repel each other due to the Coulomb interaction between their charges. The Coulomb energy of a system of electrons can be reduced by keeping the electrons spatially separated, but this has to balanced against the kineticenergy cost of deforming the electronic wave functions in order to separate the electrons. The effects of the electron-electron interaction are briefly described below.
电子结构计算中最困难的问题是需要考虑电子-电子相互作用的影响。由于电子间电荷的库仑相互作用，电子会相互排斥。电子系统的库仑能可以通过保持电子在空间上的分离而降低，但这必须与为分离电子而使电子波函数变形所付出的动能代价相平衡。下面简要介绍电子-电子相互作用的影响。

The wave function of a many-electron system must be antisymmetric under exchange of any two electrons because the electrons are fermions. The antisymmetry of the wave function produces a spatial separation between electrons that have the same spin and thus reduces the Coulomb energy of the electronic system. The reduction in the energy of the electronic system due to the antisymmetry of the wave function is called the exchange energy. It is straightforward to include exchange in a totalenergy calculation, and this is generally referred to as the Hartree-Fock approximation.
多电子系统的波函数在交换任意两个电子时必须是不对称的，因为电子是费米子。波函数的不对称性会在具有相同自旋的电子之间产生空间分隔，从而降低电子系统的库仑能。由于波函数的不对称性而导致的电子系统能量的减少称为交换能。将交换能直接纳入总能计算，一般称为哈特里-福克近似。

The Coulomb energy of the electronic system can be reduced below its Hartree-Fock value if electrons that have opposite spins are also spatially separated. In this case the Coulomb energy of the electronic system is reduced at the cost of increasing the kinetic energy of the electrons. The difference between the many-body energy of an electronic system and the energy of the system calculated in the Hartree-Fock approximation is called the correlation energy (see Fetter and Walecka, 1971). It is
如果具有相反自旋的电子也在空间上分离开来，电子系统的库仑能就会降低到哈特里-福克值以下。在这种情况下，电子系统库仑能的降低是以电子动能的增加为代价的。电子系统的多体能与哈特里-福克近似计算的系统能之间的差值称为相关能（见 Fetter 和 Walecka，1971 年）。它是
extremely difficult to calculate the correlation energy of a complex system, although some promising steps are being taken in this direction using quantum Monte Carlo simulations of the electron-gas dynamics (Fahy et al., 1988; Li et al., 1991). At present these methods are not tractable in total-energy calculations of systems with any degree of complexity, and alternative methods are required to describe the effects of the electron-electron interaction.
要计算一个复杂系统的相关能是极其困难的，尽管人们正在利用电子-气体动力学的蒙特卡罗量子模拟（Fahy 等人，1988 年；Li 等人，1991 年）朝这个方向迈出一些有希望的步伐。目前，在计算任何复杂程度的系统的总能量时，这些方法都是不可行的，因此需要采用其他方法来描述电子-电子相互作用的影响。

Density-functional theory, developed by Hohenberg and Kohn (1964) and Kohn and Sham (1965), provided some hope of a simple method for describing the effects of exchange and correlation in an electron gas. Hohenberg and Kohn proved that the total energy, including exchange and correlation, of an electron gas (even in the presence of a static external potential) is a unique functional of the electron density. The minimum value of the total-energy functional is the ground-state energy of the system, and the density that yields this minimum value is the exact single-particle ground-state density. Kohn and Sham then showed how it is possible, formally, to replace the many-electron problem by an exactly equivalent set of self-consistent one-electron equations. For more details about density-functional theory see von Barth (1984), Dreizler and da Providencia (1985), Jones and Gunnarson (1989), and Kryachko and Ludena (1990).
霍恩伯格（Hohenberg）和科恩（Kohn）（1964 年）以及科恩和沙姆（Sham）（1965 年）提出的密度函数理论为描述电子气中的交换和相关效应提供了一种简单的方法。霍恩伯格和科恩证明，电子气（即使存在静态外部电势）的总能量（包括交换和相关）是电子密度的唯一函数。总能量函数的最小值就是系统的基态能量，而产生这个最小值的密度就是精确的单粒子基态密度。随后，科恩和沙姆展示了如何从形式上用一组完全等效的自洽单电子方程组取代多电子问题。有关密度函数理论的更多详情，请参阅 von Barth (1984)、Dreizler 和 da Providencia (1985)、Jones 和 Gunnarson (1989) 以及 Kryachko 和 Ludena (1990)。

The Kohn-Sham total-energy functional for a set of doubly occupied electronic states ${\psi }_i$${\psi }_i$psi_(i)\psi_{i} can be written
一组双占电子态 ${\psi }_i$${\psi }_i$psi_(i)\psi_{i} 的 Kohn-Sham 总能量函数可写成

where $E_{\text{ion}}$$E_{\text{ion }}$E_("ion ")E_{\text {ion }} is the Coulomb energy associated with interactions among the nuclei (or ions) at positions $\left\{{\mathbf{R}}_I\right\}$$\left\{{\mathbf{R}}_I\right\}${R_(I)}\left\{\mathbf{R}_{I}\right\}, $V_{\text{ion}}$$V_{\text{ion }}$V_("ion ")V_{\text {ion }} is the static total electron-ion potential, $n(\mathbf{r})$$n(\mathbf{r})$n(r)n(\mathbf{r}) is the electronic density given by
其中， $E_{\text{ion}}$$E_{\text{ion }}$E_("ion ")E_{\text {ion }} 是与位置 $\left\{{\mathbf{R}}_I\right\}$$\left\{{\mathbf{R}}_I\right\}${R_(I)}\left\{\mathbf{R}_{I}\right\} 处的原子核（或离子）之间的相互作用相关的库仑能量， $V_{\text{ion}}$$V_{\text{ion }}$V_("ion ")V_{\text {ion }} 是静态电子-离子总电势， $n(\mathbf{r})$$n(\mathbf{r})$n(r)n(\mathbf{r}) 是电子密度，其计算公式为

and $E_{XC}[n(\mathbf{r})]$$E_{XC}[n(\mathbf{r})]$E_(XC)[n(r)]E_{X C}[n(\mathbf{r})] is the exchange-correlation functional.
和 $E_{XC}[n(\mathbf{r})]$$E_{XC}[n(\mathbf{r})]$E_(XC)[n(r)]E_{X C}[n(\mathbf{r})] 是交换相关函数。
Only the minimum value of the Kohn-Sham energy functional has physical meaning. At the minimum, the Kohn-Sham energy functional is equal to the groundstate energy of the system of electrons with the ions in positions $\left\{{\mathbf{R}}_I\right\}$$\left\{{\mathbf{R}}_I\right\}${R_(I)}\left\{\mathbf{R}_{I}\right\}.
只有 Kohn-Sham 能量函数的最小值才具有物理意义。在最小值处，Kohn-Sham 能量函数等于位置 $\left\{{\mathbf{R}}_I\right\}$$\left\{{\mathbf{R}}_I\right\}${R_(I)}\left\{\mathbf{R}_{I}\right\} 的离子电子系统的基态能量。

It is necessary to determine the set of wave functions ${\psi }_i$${\psi }_i$psi_(i)\psi_{i} that minimize the Kohn-Sham energy functional. These are given by the self-consistent solutions to the Kohn-Sham equations (Kohn and Sham, 1965):
有必要确定一组使 Kohn-Sham 能量函数最小化的波函数 ${\psi }_i$${\psi }_i$psi_(i)\psi_{i} 。这些由 Kohn-Sham 方程的自洽解给出（Kohn 和 Sham，1965 年）：
$[\frac{-{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V_{\text{ion}}(\mathbf{r})+V_H(\mathbf{r})+V_{XC}(\mathbf{r})]{\psi }_i(\mathbf{r})={\epsilon }_i{\psi }_i(\mathbf{r})$$\left[\frac{-{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V_{\text{ion }}(\mathbf{r})+V_H(\mathbf{r})+V_{XC}(\mathbf{r})\right]{\psi }_i(\mathbf{r})={\epsilon }_i{\psi }_i(\mathbf{r})$[(-ℏ^(2))/(2m)grad^(2)+V_("ion ")(r)+V_(H)(r)+V_(XC)(r)]psi_(i)(r)=epsi_(i)psi_(i)(r)\left[\frac{-\hbar^{2}}{2 m} \nabla^{2}+V_{\text {ion }}(\mathbf{r})+V_{H}(\mathbf{r})+V_{X C}(\mathbf{r})\right] \psi_{i}(\mathbf{r})=\varepsilon_{i} \psi_{i}(\mathbf{r}),
where ${\psi }_i$${\psi }_i$psi_(i)\psi_{i} is the wave function of electronic state $i,{\epsilon }_i$$i,{\epsilon }_i$i,epsi_(i)i, \varepsilon_{i} is the Kohn-Sham eigenvalue, and $V_H$$V_H$V_(H)V_{H} is the Hartree potential of the electrons given by
其中， ${\psi }_i$${\psi }_i$psi_(i)\psi_{i} 是电子状态的波函数， $i,{\epsilon }_i$$i,{\epsilon }_i$i,epsi_(i)i, \varepsilon_{i} 是科恩-沙姆特征值， $V_H$$V_H$V_(H)V_{H} 是电子的哈特里电势，其值为

The exchange-correlation potential, $V_{XC}$$V_{XC}$V_(XC)V_{X C}, is given formally by the functional derivative
交换相关势 $V_{XC}$$V_{XC}$V_(XC)V_{X C} 由函数导数正式给出

The Kohn-Sham equations represent a mapping of the interacting many-electron system onto a system of noninteracting electrons moving in an effective potential due to all the other electrons. If the exchange-correlation energy functional were known exactly, then taking the functional derivative with respect to the density would produce an exchange-correlation potential that included the effects of exchange and correlation exactly.
Kohn-Sham 方程是将相互作用的多电子系统映射到在所有其他电子的有效电势中运动的非相互作用电子系统上。如果交换相关能函数是已知的，那么对该能函数进行与密度相关的导数运算，就可以得到一个交换相关势，其中精确地包含了交换和相关的效应。

The Kohn-Sham equations must be solved selfconsistently so that the occupied electronic states generate a charge density that produces the electronic potential that was used to construct the equations. The sum of the single-particle Kohn-Sham eigenvalues does not give the total electronic energy because this overcounts the effects of the electron-electron interaction in the Hartree energy and in the exchange-correlation energy. The Kohn-Sham eigenvalues are not, strictly speaking, the energies of the single-particle electron states, but rather the derivatives of the total energy with respect to the occupation numbers of these states (Janak, 1978). Nevertheless, the highest occupied eigenvalue in an atomic or molecular calculation is nearly the unrelaxed ionization energy for that system (Perdew et al., 1982).
Kohn-Sham 方程必须自洽地求解，这样被占据的电子态才会产生电荷密度，从而产生用于构建方程的电子势。单粒子 Kohn-Sham 特征值之和并不能得出总的电子能量，因为它过多地计算了哈特里能量和交换相关能量中电子-电子相互作用的影响。严格来说，Kohn-Sham 特征值并不是单粒子电子态的能量，而是总能量相对于这些态的占据数的导数（Janak，1978 年）。不过，原子或分子计算中的最高占据特征值几乎就是该系统的非松弛电离能（Perdew 等人，1982 年）。

The Kohn-Sham equations are a set of eigenequations, and the terms within the brackets in Eq. (2.3) can be regarded as a Hamiltonian. The bulk of the work involved in a total-energy pseudopotential calculation is the solution of this eigenvalue problem once an approximate expression for the exchange-correlation energy is given.
Kohn-Sham 方程是一组特征方程，式 (2.3) 中括号内的项可视为哈密顿方程。一旦给出交换相关能的近似表达式，总能伪势计算的大部分工作就是求解这个特征值问题。

The Hohenberg-Kohn theorem provides some motivation for using approximate methods to describe the exchange-correlation energy as a function of the electron density. The simplest method of describing the exchange-correlation energy of an electronic system is to use the local-density approximation (LDA; Kohn and Sham, 1965), and this approximation is almost universally used in total-energy pseudopotential calculations. In the local-density approximation the exchange-correlation energy of an electronic system is constructed by assuming that the exchange-correlation energy per electron at a point $r$$r$rr in the electron gas, ${\epsilon }_{XC}(\mathbf{r})$${\epsilon }_{XC}(\mathbf{r})$epsi_(XC)(r)\varepsilon_{X C}(\mathbf{r}), is equal to the exchange-correlation energy per electron in a homogeneous electron gas that has the same density as the electron gas at point $\mathbf{r}$$\mathbf{r}$r\mathbf{r}. Thus
霍恩伯格-科恩（Hohenberg-Kohn）定理为使用近似方法描述作为电子密度函数的交换相关能提供了一些动力。描述电子系统交换相关能的最简单方法是使用局部密度近似（LDA；Kohn 和 Sham，1965 年），这种近似方法几乎普遍用于总能伪势计算。在局域密度近似中，电子系统的交换相关能是通过假设电子气中 $r$$r$rr 点 ${\epsilon }_{XC}(\mathbf{r})$${\epsilon }_{XC}(\mathbf{r})$epsi_(XC)(r)\varepsilon_{X C}(\mathbf{r}) 的每个电子的交换相关能等于与 $\mathbf{r}$$\mathbf{r}$r\mathbf{r} 点电子气密度相同的均相电子气中每个电子的交换相关能来构建的。因此

and 和

with 与

The local-density approximation assumes that the exchange-correlation energy functional is purely local. Several parametrizations exist for the exchangecorrelation energy of a homogeneous electron gas (Wigner, 1938; Kohn and Sham, 1965; Hedin and Lundqvist, 1971; Vosko et al., 1980; Perdew and Zunger, 1981), all of which lead to total-energy results that are very similar. These parametrizations use interpolation formulas to link exact results for the exchangecorrelation energy of high-density electron gases and calculations of the exchange-correlation energy of intermediate and low-density electron gases.
局部密度近似假设交换相关能函数是纯局部的。对于均质电子气的交换相关能，存在几种参数化方法（Wigner，1938；Kohn 和 Sham，1965；Hedin 和 Lundqvist，1971；Vosko 等人，1980；Perdew 和 Zunger，1981），所有这些方法得出的总能结果都非常相似。这些参数使用插值公式将高密度电子气体交换相关能的精确结果与中低密度电子气体交换相关能的计算结果联系起来。

The local-density approximation, in principle, ignores corrections to the exchange-correlation energy at a point $\mathbf{r}$$\mathbf{r}$r\mathbf{r} due to nearby inhomogeneities in the electron density. Considering the inexact nature of the approximation, it is remarkable that calculations performed using the LDA have been so successful. Recent work has shown that this success can be partially attributed to the fact that the local-density approximation gives the correct sum rule for the exchange-correlation hole (Harris and Jones, 1974; Gunnarsson and Lundqvist, 1976; Langreth and Perdew, 1977). A number of attempts to improve the LDA, for instance by using gradient expansions, have not shown any improvement over results obtained using the simple LDA. One of the reasons why these “improvements” to the LDA do so poorly is that they do not obey the sum rule for the exchange-correlation hole. Methods that do enforce the sum rule appear to offer a consistent improvement over the LDA (Langreth and Mehl, 1981, 1983).
局部密度近似原则上忽略了由于附近电子密度的不均匀性而对 $\mathbf{r}$$\mathbf{r}$r\mathbf{r} 点的交换相关能的修正。考虑到该近似方法的不精确性，使用本地相关性近似方法进行的计算能够取得如此大的成功实属不易。最近的研究表明，这种成功部分归功于局域密度近似给出了交换相关空穴的正确总和规则（Harris 和 Jones，1974 年；Gunnarsson 和 Lundqvist，1976 年；Langreth 和 Perdew，1977 年）。许多改进 LDA 的尝试，例如使用梯度展开，与使用简单 LDA 得到的结果相比，并没有显示出任何改进。这些对 LDA 的 "改进 "之所以效果不佳，原因之一是它们不遵守交换相关孔的和规则。执行总和规则的方法似乎比 LDA 有持续的改进（Langreth 和 Mehl，1981 年，1983 年）。

The LDA appears to give a single well-defined global minimum for the energy of a non-spin-polarized system of electrons in a fixed ionic potential. Therefore any energy minimization scheme will locate the global energy minimum of the electronic system. For magnetic materials, however, one would expect to have more than one local minimum in the electronic energy. If the energy functional for the electronic system had many local minima, it would be extremely costly to perform total-energy calculations because the global energy minimum could only be located by sampling the energy functional over a large region of phase space.
LDA 似乎为固定离子势中的非自旋极化电子系统的能量提供了一个定义明确的全局最小值。因此，任何能量最小化方案都能找到电子系统的全局能量最小值。然而，对于磁性材料来说，我们会发现电子能量的局部最小值不止一个。如果电子系统的能量函数有许多局部最小值，那么进行总能量计算的成本就会非常高，因为只有在相空间的较大区域内对能量函数进行取样，才能找到全局能量最小值。

In the preceding section it was demonstrated that certain observables of the many-body problem can be mapped into equivalent observables in an effective single-particle problem. However, there still remains the formidable task of handling an infinite number of noninteracting electrons moving in the static potential of an
上一节已经证明，多体问题中的某些观测值可以映射到有效单粒子问题中的等效观测值。然而，要处理在静态势能中运动的无穷多个非相互作用电子，仍然是一项艰巨的任务。
infinite number of nuclei or ions. Two difficulties must be overcome: a wave function must be calculated for each of the infinite number of electrons in the system, and, since each electronic wave function extends over the entire solid, the basis set required to expand each wave function is infinite. Both problems can be surmounted by performing calculations on periodic systems and applying Bloch’s theorem to the electronic wave functions.
无限多个原子核或离子。必须克服两个困难：必须为系统中无限多个电子中的每个电子计算一个波函数；由于每个电子波函数扩展到整个固体，因此扩展每个波函数所需的基集是无限的。这两个问题都可以通过对周期系统进行计算并对电子波函数应用布洛赫定理来解决。

Bloch’s theorem states that in a periodic solid each electronic wave function can be written as the product of a cell-periodic part and a wavelike part (see Ashcroft and Mermin, 1976),
布洛赫定理指出，在周期性固体中，每个电子波函数都可以写成单元周期部分和波状部分的乘积（见 Ashcroft 和 Mermin，1976 年）、

The cell-periodic part of the wave function can be expanded using a basis set consisting of a discrete set of plane waves whose wave vectors are reciprocal lattice vectors of the crystal,
波函数的晶胞周期部分可以用一个基集展开，该基集由离散的平面波组成，其波向量是晶体的倒易晶格向量、

where the reciprocal lattice vectors $\mathbf{G}$$\mathbf{G}$G\mathbf{G} are defined by $\mathbf{G}\cdot \mathit{l}=2\pi m$$\mathbf{G}\cdot \mathit{l}=2\pi m$G*l=2pi m\mathbf{G} \cdot \boldsymbol{l}=2 \pi m for all $\mathit{l}$$\mathit{l}$l\boldsymbol{l} where $\mathit{l}$$\mathit{l}$l\boldsymbol{l} is a lattice vector of the crystal and $m$$m$mm is an integer. Therefore each electronic wave function can be written as a sum of plane waves,
其中倒易晶格矢量 $\mathbf{G}$$\mathbf{G}$G\mathbf{G} 由 $\mathbf{G}\cdot \mathit{l}=2\pi m$$\mathbf{G}\cdot \mathit{l}=2\pi m$G*l=2pi m\mathbf{G} \cdot \boldsymbol{l}=2 \pi m 对所有 $\mathit{l}$$\mathit{l}$l\boldsymbol{l} 进行定义，而 $\mathit{l}$$\mathit{l}$l\boldsymbol{l} 是晶体的晶格矢量， $m$$m$mm 是整数。因此，每个电子波函数都可以写成平面波之和、

Electronic states are allowed only at a set of $\mathbf{k}$$\mathbf{k}$k\mathbf{k} points determined by the boundary conditions that apply to the bulk solid. The density of allowed $\mathbf{k}$$\mathbf{k}$k\mathbf{k} points is proportional to the volume of the solid. The infinite number of electrons in the solid are accounted for by an infinite number of $k$$k$kk points, and only a finite number of electronic states are occupied at each $\mathbf{k}$$\mathbf{k}$k\mathbf{k} point. The Bloch theorem changes the problem of calculating an infinite number of electronic wave functions to one of calculating a finite number of electronic wave functions at an infinite number of $\mathbf{k}$$\mathbf{k}$k\mathbf{k} points. The occupied states at each $\mathbf{k}$$\mathbf{k}$k\mathbf{k} point contribute to the electronic potential in the bulk solid so that, in principle, an infinite number of calculations are needed to compute this potential. However, the electronic wave functions at $\mathbf{k}$$\mathbf{k}$k\mathbf{k} points that are very close together will be almost identical. Hence it is possible to represent the electronic wave functions over a region of $\mathbf{k}$$\mathbf{k}$k\mathbf{k} space by the wave functions at a single $\mathbf{k}$$\mathbf{k}$k\mathbf{k} point. In this case the electronic states at only a finite number of $\mathbf{k}$$\mathbf{k}$k\mathbf{k} points are required to calculate the electronic potential and hence determine the total energy of the solid.
电子态仅允许存在于一组 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点，这些点由适用于块状固体的边界条件决定。允许的 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点的密度与固体的体积成正比。固体中的无限多个电子由无限多个 $k$$k$kk 点组成，而每个 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点只占据有限个电子状态。布洛赫定理将计算无限多个电子波函数的问题转变为计算无限多个 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点上的有限多个电子波函数的问题。每个 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点的占位态都会对体固体中的电子势产生影响，因此原则上需要进行无限次计算才能计算出电子势。然而，相距很近的 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点的电子波函数几乎是相同的。因此，可以用单个 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点的波函数来表示 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 空间区域的电子波函数。在这种情况下，只需要有限数量的 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点上的电子状态来计算电子势，从而确定固体的总能量。

Methods have been devised for obtaining very accurate approximations to the electronic potential and the contri-
为了获得非常精确的电子电势近似值，我们设计了一些方法。
bution to the total energy from a filled electronic band by calculating the electronic states at special sets of $\mathbf{k}$$\mathbf{k}$k\mathbf{k} points in the Brillouin zone (Chadi and Cohen, 1973; Joannopoulos and Cohen, 1973; Monkhorst and Pack, 1976; Evarestov and Smirnov, 1983). Using these methods, one can obtain an accurate approximation for the electronic potential and the total energy of an insulator or a semiconductor by calculating the electronic states at a very small number of $\mathbf{k}$$\mathbf{k}$k\mathbf{k} points. The electronic potential and total energy are more difficult to calculate if the system is metallic because a dense set of $\mathbf{k}$$\mathbf{k}$k\mathbf{k} points is required to define the Fermi surface precisely.
通过计算布里渊区中特殊的 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点处的电子状态，可以从填充电子带得出总能量（Chadi 和 Cohen，1973 年；Joannopoulos 和 Cohen，1973 年；Monkhorst 和 Pack，1976 年；Evarestov 和 Smirnov，1983 年）。利用这些方法，我们可以通过计算极少数 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点的电子状态，获得绝缘体或半导体的电子势能和总能量的精确近似值。如果系统是金属，则电子势和总能量的计算就比较困难，因为需要一组密集的 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点来精确定义费米面。

The magnitude of any error in the total energy due to inadequacy of the $\mathbf{k}$$\mathbf{k}$k\mathbf{k}-point sampling can always be reduced by using a denser set of $\mathbf{k}$$\mathbf{k}$k\mathbf{k} points. The computed total energy will converge as the density of $\mathbf{k}$$\mathbf{k}$k\mathbf{k} points increases, and the error due to the $\mathbf{k}$$\mathbf{k}$k\mathbf{k}-point sampling then approaches zero. In principle, a converged electronic potential and total energy can always be obtained provided that the computational time is available to calculate the electronic wave functions at a sufficiently dense set of $\mathbf{k}$$\mathbf{k}$k\mathbf{k} points. The computational cost of performing a very dense sampling of $k$$k$kk space can be significantly reduced by using the $\mathbf{k}\cdot \mathbf{p}$$\mathbf{k}\cdot \mathbf{p}$k*p\mathbf{k} \cdot \mathbf{p} total-energy method (Robertson and Payne, 1990, 1991). In this technique solutions on the dense set of $\mathbf{k}$$\mathbf{k}$k\mathbf{k} points are generated from the solutions on a much coarser grid of $\mathbf{k}$$\mathbf{k}$k\mathbf{k} points using $\mathbf{k}\cdot \mathbf{p}$$\mathbf{k}\cdot \mathbf{p}$k*p\mathbf{k} \cdot \mathbf{p} perturbation theory.
由于 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点取样不足而导致的总能量误差，可以通过使用更密集的 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点来减小。随着 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点密度的增加，计算出的总能量将趋于收敛，而 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点取样造成的误差将趋近于零。原则上，只要有足够的计算时间在足够密集的 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点上计算电子波函数，就总能得到收敛的电子势和总能量。通过使用 $\mathbf{k}\cdot \mathbf{p}$$\mathbf{k}\cdot \mathbf{p}$k*p\mathbf{k} \cdot \mathbf{p} 总能量方法（Robertson 和 Payne，1990 年，1991 年），可以大大降低对 $k$$k$kk 空间进行高密度采样的计算成本。在这种技术中， $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点密集集上的解是通过 $\mathbf{k}\cdot \mathbf{p}$$\mathbf{k}\cdot \mathbf{p}$k*p\mathbf{k} \cdot \mathbf{p} 干涉理论从 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点更粗网格上的解生成的。

Bloch’s theorem states that the electronic wave functions at each $\mathbf{k}$$\mathbf{k}$k\mathbf{k} point can be expanded in terms of a discrete plane-wave basis set. In principle, an infinite plane-wave basis set is required to expand the electronic wave functions. However, the coefficients $c_{i,\mathbf{k}+\mathbf{G}}$$c_{i,\mathbf{k}+\mathbf{G}}$c_(i,k+G)c_{i, \mathbf{k}+\mathbf{G}} for the plane waves with small kinetic energy ( ${\hslash }^2/2m$${\hslash }^2/2m$ℏ^(2)//2m\hbar^{2} / 2 m ) $|\mathbf{k}+\mathbf{G}{|}^2$$|\mathbf{k}+\mathbf{G}{|}^2$|k+G|^(2)|\mathbf{k}+\mathbf{G}|^{2} are typically more important than those with large kinetic energy. Thus the plane-wave basis set can be truncated to include only plane waves that have kinetic energies less than some particular cutoff energy. If a continuum of plane-wave basis states were required to expand each electronic wave function, the basis set would be infinitely large no matter how small the cutoff energy. Application of the Bloch theorem allows the electronic wave functions to be expanded in terms of a discrete set of plane waves. Introduction of an energy cutoff to the discrete plane-wave basis set produces a finite basis set.
布洛赫定理指出，每个 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点的电子波函数都可以用离散的平面波基集展开。原则上，展开电子波函数需要一个无限平面波基集。然而，动能较小的平面波的系数 $c_{i,\mathbf{k}+\mathbf{G}}$$c_{i,\mathbf{k}+\mathbf{G}}$c_(i,k+G)c_{i, \mathbf{k}+\mathbf{G}} ( ${\hslash }^2/2m$${\hslash }^2/2m$ℏ^(2)//2m\hbar^{2} / 2 m ) $|\mathbf{k}+\mathbf{G}{|}^2$$|\mathbf{k}+\mathbf{G}{|}^2$|k+G|^(2)|\mathbf{k}+\mathbf{G}|^{2} 通常比动能较大的系数更为重要。因此，平面波基集可以截断，只包含动能小于某个特定截止能量的平面波。如果需要连续的平面波基态来扩展每个电子波函数，那么无论截断能量有多小，基集都将无限大。布洛赫定理的应用使得电子波函数可以用一组离散的平面波来展开。在离散的平面波基集上引入能量截止，就会产生一个有限的基集。

The truncation of the plane-wave basis set at a finite cutoff energy will lead to an error in the computed total energy. However, it is possible to reduce the magnitude of the error by increasing the value of the cutoff energy. In principle, the cutoff energy should be increased until the calculated total energy has converged, but it will be shown later that it is possible to perform calculations at lower cutoff energies.
在有限截止能量处截断平面波基集会导致计算出的总能量出现误差。不过，可以通过增加截止能量的值来减小误差的幅度。原则上，截止能量应该一直增加，直到计算出的总能量收敛为止，但稍后将证明可以在较低的截止能量下进行计算。

One of the difficulties associated with the use of planewave basis sets is that the number of basis states changes discontinuously with cutoff energy. In general these
使用平面波基集的困难之一在于，基态的数量会随着截止能量的变化而发生不连续的变化。一般来说，这些
discontinuities will occur at different cutoffs for different $\mathbf{k}$$\mathbf{k}$k\mathbf{k} points in the $\mathbf{k}$$\mathbf{k}$k\mathbf{k}-point set. (In addition, at a fixed-energy cutoff, a change in the size or shape of the unit cell will cause discontinuation in the plane-wave basis set.) This problem can be reduced by using denser k-point sets, so that the weight attached to any particular plane-wave basis state is reduced. However, the problem is still present even with quite dense $\mathbf{k}$$\mathbf{k}$k\mathbf{k}-point samplings. It can be handled by applying a correction factor which accounts approximately for the difference between the number of states in a basis set with infinitely large number of $\mathbf{k}$$\mathbf{k}$k\mathbf{k} points and the number of basis states actually used in the calculation (Francis and Payne, 1990).
在 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点集中，不同的 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点会在不同的截止点出现不连续性。(此外，在固定能量截止点上，单元尺寸或形状的改变也会导致平面波基集的不连续性）。这个问题可以通过使用更密集的 k 点集来减少，从而降低任何特定平面波基态的权重。不过，即使使用密度相当高的 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点采样，这个问题依然存在。这个问题可以通过应用校正因子来解决，校正因子大致考虑了具有无限多 $\mathbf{k}$$\mathbf{k}$k\mathbf{k} 点的基集中的状态数与计算中实际使用的基态数之间的差异（Francis 和 Payne，1990 年）。
4. Plane-wave representation of Kohn-Sham equations
4.Kohn-Sham 方程的平面波表示法

When plane waves are used as a basis set for the electronic wave functions, the Kohn-Sham equations assume a particularly simple form. Substitution of Eq. (2.9) into (2.3) and integration over $\mathbf{r}$$\mathbf{r}$r\mathbf{r} gives the secular equation
当使用平面波作为电子波函数的基集时，Kohn-Sham 方程的形式特别简单。将式 (2.9) 代入 (2.3)，并对 $\mathbf{r}$$\mathbf{r}$r\mathbf{r} 进行积分，即可得到世俗方程

In this form, the kinetic energy is diagonal, and the various potentials are described in terms of their Fourier transforms. Solution of Eq. (2.10) proceeds by diagonalization of a Hamiltonian matrix whose matrix elements $H_{\mathrm{k}+\mathrm{G},\mathrm{k}+{\mathrm{G}}^{\prime }}$$H_{\mathrm{k}+\mathrm{G},\mathrm{k}+{\mathrm{G}}^{\prime }}$H_(k+G,k+G^('))H_{\mathrm{k}+\mathrm{G}, \mathrm{k}+\mathrm{G}^{\prime}} are given by the terms in the brackets above. The size of the matrix is determined by the choice of cutoff energy ( ${\hslash }^2/2m$${\hslash }^2/2m$ℏ^(2)//2m\hbar^{2} / 2 m ) ${|\mathbf{k}+{\mathbf{G}}_c|}^2$${\left|\mathbf{k}+{\mathbf{G}}_c\right|}^2$|k+G_(c)|^(2)\left|\mathbf{k}+\mathbf{G}_{c}\right|^{2}, and will be intractably large for systems that contain both valence and core electrons. This is a severe problem, but it can be overcome by use of the pseudopotential approximation, as discussed in Sec. II.D.
在这种形式中，动能是对角的，而各种电势则用它们的傅里叶变换来描述。公式 (2.10) 的求解是通过对哈密顿矩阵进行对角化来实现的，其矩阵元素 $H_{\mathrm{k}+\mathrm{G},\mathrm{k}+{\mathrm{G}}^{\prime }}$$H_{\mathrm{k}+\mathrm{G},\mathrm{k}+{\mathrm{G}}^{\prime }}$H_(k+G,k+G^('))H_{\mathrm{k}+\mathrm{G}, \mathrm{k}+\mathrm{G}^{\prime}} 由上面括号中的项给出。矩阵的大小由截止能量 ( ${\hslash }^2/2m$${\hslash }^2/2m$ℏ^(2)//2m\hbar^{2} / 2 m ) ${|\mathbf{k}+{\mathbf{G}}_c|}^2$${\left|\mathbf{k}+{\mathbf{G}}_c\right|}^2$|k+G_(c)|^(2)\left|\mathbf{k}+\mathbf{G}_{c}\right|^{2} 的选择决定，对于同时包含价电子和核电子的系统来说，矩阵的大小会大得难以处理。这是一个严重的问题，但可以通过使用伪势近似来克服，详见第 II.D 节。

The Bloch theorem can be applied neither to a system that contains a single defect nor in the direction perpendicular to a crystal surface. A continuous plane-wave basis set would be required for the defect calculation, and, although the plane-wave basis set for the surface calculation would be discrete in the plane of the surface, it would be continuous in the direction perpendicular to the surface. Hence an infinite number of plane-wave basis states would be required for both of these calculations, no matter how small the cutoff energy chosen for the basis set. Calculations using plane-wave basis sets can only be performed on these systems if a periodic supercell is used. The supercell for a point-defect calculation is illustrated schematically in Fig. 2. The supercell contains the defect
布洛赫定理既不适用于包含单一缺陷的系统，也不适用于垂直于晶体表面的方向。缺陷计算需要连续的平面波基态集，虽然表面计算的平面波基态集在表面平面上是离散的，但在垂直于表面的方向上却是连续的。因此，无论为基集选择的截止能有多小，这两种计算都需要无限多的平面波基态。只有在使用周期性超级簇的情况下，才能在这些系统上使用平面波基集进行计算。点缺陷计算的超级囚笼示意图如图 2 所示。超级囚笼包含缺陷

FIG. 2. Schematic illustration of a supercell geometry for a point defect (i.e., vacancy) in a bulk solid. The supercell is the area enclosed by the dashed lines.
图 2.块状固体中点缺陷（即空位）的超级胞几何示意图。超级电池是虚线所包围的区域。
surrounded by a region of bulk crystal. Periodic boundary conditions are applied to the supercell so that the supercell is reproduced throughout space, as implied in the figure. Therefore the energy per unit cell of a crystal containing an array of defects is calculated, rather than the energy of a crystal containing a single defect. It is essential to include enough bulk solid in the supercell to prevent the defects in neighboring cells from interacting appreciably with each other. The independence of defects in neighboring cells can be checked by increasing the volume of the supercell until the computed defect energy has converged. It can then be assumed that defects in neighboring unit cells no longer interact.
周围是块状晶体区域。如图所示，周期性边界条件被应用于超级晶胞，从而使超级晶胞在整个空间再现。因此，计算的是含有缺陷阵列的晶体的单位晶胞能量，而不是含有单一缺陷的晶体的能量。在超晶胞中必须包含足够的块状固体，以防止相邻晶胞中的缺陷发生明显的相互作用。相邻晶胞中缺陷的独立性可以通过增大超级晶胞的体积来检验，直到计算出的缺陷能量趋同为止。这时可以假定相邻单元格中的缺陷不再相互作用。

A surface may have periodicity in the plane of the surface, but it cannot have periodicity perpendicular to the surface. The supercell for a surface calculation is illustrated schematically in Fig. 3. The supercell contains a
曲面可以在曲面平面内具有周期性，但不能在垂直于曲面的方向上具有周期性。图 3 是曲面计算超级小区的示意图。超级小区包含一个

FIG. 3. Schematic illustration of a supercell geometry for a surface of a bulk solid. Same convention as in Fig. 2.
图 3.块状固体表面的超级椭球几何示意图。与图 2 相同。

FIG. 4. Schematic illustration of a supercell geometry for a molecule. Same convention as in Fig. 2.
图 4.分子超级囚室几何示意图。与图 2 相同。
crystal slab and a vacuum region. The supercell is repeated over all space, so the total energy of an array of crystal slabs is calculated. To ensure that the results of the calculation accurately represent an isolated surface, the vacuum regions must be wide enough so that faces of adjacent crystal slabs do not interact across the vacuum region, and the crystal slab must be thick enough so that the two surfaces of each crystal slab do not interact through the bulk crystal.
晶体板和真空区域。超级电池在所有空间重复进行，因此可以计算出晶体板阵列的总能量。为了确保计算结果准确地代表一个孤立的表面，真空区域必须足够宽，以便相邻晶体板的表面不会穿过真空区域发生相互作用，晶体板必须足够厚，以便每个晶体板的两个表面不会穿过块状晶体发生相互作用。

Finally, even molecules can be studied in this fashion (Joannopoulos et al., 1991), as illustrated in Fig. 4. Again, the supercell needs to be large enough so that the interactions between the molecules are negligible.
最后，如图 4 所示，甚至分子也可以用这种方法进行研究（Joannopoulos 等人，1991 年）。同样，超级囚笼需要足够大，这样分子之间的相互作用才可以忽略不计。

Although Bloch’s theorem states that the electronic wave functions can be expanded using a discrete set of plane waves, a plane-wave basis set is usually very poorly suited to expanding electronic wave functions because a very large number of plane waves are needed to expand the tightly bound core orbitals and to follow the rapid oscillations of the wave functions of the valence electrons in the core region. An extremely large plane-wave basis set would be required to perform an all-electron calculation, and a vast amount of computational time would be required to calculate the electronic wave functions. The pseudopotential approximation (Phillips, 1958; Heine and Cohen, 1970; Yin and Cohen, 1982a) allows the electronic wave functions to be expanded using a much smaller number of plane-wave basis states.
虽然布洛赫定理指出电子波函数可以用一组离散的平面波来展开，但平面波基集通常非常不适合展开电子波函数，因为需要大量的平面波来展开紧密结合的核心轨道，并跟踪核心区域价电子波函数的快速振荡。要进行全电子计算，需要极大的平面波基集，而计算电子波函数则需要大量的计算时间。伪势近似（Phillips，1958 年；Heine 和 Cohen，1970 年；Yin 和 Cohen，1982a）允许使用数量少得多的平面波基态来扩展电子波函数。

It is well known that most physical properties of solids are dependent on the valence electrons to a much greater extent than on the core electrons. The pseudopotential approximation exploits this by removing the core electrons and by replacing them and the strong ionic potential by a weaker pseudopotential that acts on a set of
众所周知，固体的大多数物理性质对价电子的依赖程度远远大于对核心电子的依赖程度。伪电势近似利用了这一点，去掉了核心电子，用一个较弱的伪电势代替了核心电子和强离子电势，这个伪电势作用于一组
pseudo wave functions rather than the true valence wave functions. An ionic potential, valence wave function and the corresponding pseudopotential and pseudo wave function are illustrated schematically in Fig. 5. The valence wave functions oscillate rapidly in the region occupied by the core electrons due to the strong ionic potential in this region. These oscillations maintain the orthogonality between the core wave functions and the valence wave functions, which is required by the exclusion principle. The pseudopotential is constructed, ideally, so that its scattering properties or phase shifts for the pseudo wave functions are identical to the scattering properties of the ion and the core electrons for the valence wave functions, but in such a way that the pseudo wave functions have no radial nodes in the core region. In the core region, the total phase shift produced by the ion and the core electrons will be greater by $\pi$$\pi$pi\pi, for each node that the valence functions had in the core region, than the phase shift produced by the ion and the valence electrons. Outside the core region the two potentials are identical, and the scattering from the two potentials is indistinguishable. The phase shift produced by the ion core is different for each angular momentum component of the valence wave function, and so the scattering from the pseudopotential must be angular momentum dependent. The most general form for a pseudopotential is
而不是真正的价波函数。离子电势、价波函数以及相应的伪电势和伪波函数示意图见图 5。由于核心电子占据的区域具有很强的离子势，该区域的价波函数会快速振荡。这些振荡保持了核波函数和价波函数之间的正交性，这是排除原理所要求的。理想情况下，伪势的构造应使其对伪波函数的散射特性或相移与离子和价电子对价波函数的散射特性相同，但要使伪波函数在核心区域没有径向节点。在核心区域，对于价函数在核心区域的每个节点，离子和核心电子产生的总相移将比离子和价电子产生的相移大 $\pi$$\pi$pi\pi 。在核心区之外，两种电势完全相同，两种电势的散射也没有区别。离子核心产生的相移对于价电子波函数的每个角动量分量都是不同的，因此来自伪电势的散射必须与角动量有关。伪电势的最一般形式是

where $|lm⟩$$|lm⟩$|lm:)|l m\rangle are the spherical harmonics and $V_l$$V_l$V_(l)V_{l} is the pseudopotential for angular momentum $l$$l$ll. Acting on the electronic wave function with this operator decomposes the wave function into spherical harmonics, each of which is then multiplied by the relevant pseudopotential $V_l$$V_l$V_(l)V_{l}.
其中 $|lm⟩$$|lm⟩$|lm:)|l m\rangle 是球面谐波， $V_l$$V_l$V_(l)V_{l} 是角动量 $l$$l$ll 的伪势。用这个算子作用于电子波函数，可将波函数分解为球面谐波，然后将每个球面谐波乘以相关的伪势 $V_l$$V_l$V_(l)V_{l} 。

FIG. 5. Schematic illustration of all-electron (solid lines) and pseudoelectron (dashed lines) potentials and their corresponding wave functions. The radius at which all-electron and pseudoelectron values match is designated $r_c$$r_c$r_(c)r_{c}.
图 5.全电子（实线）和伪电子（虚线）电势及其相应波函数的示意图。全电子和伪电子值相匹配的半径为 $r_c$$r_c$r_(c)r_{c} 。

A pseudopotential that uses the same potential for all the angular momentum components of the wave function is called a local pseudopotential. A local pseudopotential is a function only of the distance from the nucleus. It is possible to produce arbitrary, predetermined phase shifts for each angular momentum state with a local potential, but there are limits to the amount that the phase shifts can be adjusted for the different angular momentum states, while maintaining the crucial smoothness and weakness of the pseudopotential. Without a smooth, weak pseudopotential it becomes difficult to expand the wave functions using a reasonable number of plane-wave basis states.
对波函数的所有角动量分量使用相同电势的伪势称为局部伪势。局部伪势仅是原子核距离的函数。利用局部位势可以为每个角动量态产生任意的、预定的相移，但在保持伪势的关键平滑性和弱性的同时，不同角动量态的相移调整量是有限的。如果没有平滑、微弱的伪势，就很难使用合理数量的平面波基态来展开波函数。

In total-energy calculations, the exchange-correlation energy of the electronic system is a function of the electron density. If the exchange-correlation energy is to be desired accurately, it is necessary that outside the core regions the pseudo wave functions and real wave functions be identical, not just in their spatial dependences but also in their absolute magnitudes, so that the two wave functions generate identical charge densities. Adjustment of the pseudopotential to ensure that the integrals of the squared amplitudes of the real and the pseudo wave functions inside the core regions are identical guarantees the equality of the wave function and pseudo wave function outside the core region. One of the first attempts to construct pseudopotentials of this type was by Starkloff and Joannopoulos (Joannopoulos et al. 1977, Starkloff and Joannopoulos 1977). They introduced a class of local pseudopotentials that described the valence energies and wave functions of many heavy atoms accurately.
在总能计算中，电子系统的交换相关能是电子密度的函数。如果要精确地求得交换相关能，那么在核心区域之外，伪波函数和实波函数必须是完全相同的，不仅在空间依赖性上，而且在绝对幅度上，这样两个波函数才能产生完全相同的电荷密度。调整伪势以确保核心区域内的实波函数和伪波函数的平方振幅积分相同，就能保证核心区域外的波函数和伪波函数相等。斯塔克洛夫和琼诺普洛斯（Joannopoulos et al.）他们引入了一类局部伪势，准确地描述了许多重金属原子的价能和波函数。

Of course, in general, the scattering from the ion core is best described by a nonlocal pseudopotential that uses a different potential for each angular momentum component of the wave function. Various groups (Redondo et al., 1977; Hamann et al., 1979; Zunger and Cohen, 1979; Kerker, 1980; Bachelet et al., 1982; Shirley et al., 1989) have now introduced nonlocal pseudopotentials of this type that work extremely well. Moreover, as pointed out by Hamann, Schluter, and Chiang (1979), a match of the pseudo and real wave functions outside the core region also assures that the first-order energy dependence of the scattering from the ion core is correct, so that the scattering is accurately described over a wide range of energy. A method for the construction of pseudopotentials that corrects even the higher-order energy dependence of the scattering has recently been introduced by Shirley et al. (1989). Local and nonlocal pseudopotentials of these types are currently termed ab initio or norm conserving and are capable of describing the scattering due to the ion in a variety of atomic environments, a property referred to as transferability.
当然，一般来说，离子核心的散射最好用非局部伪势来描述，即对波函数的每个角动量分量使用不同的势。目前已有多个研究小组（Redondo 等人，1977 年；Hamann 等人，1979 年；Zunger 和 Cohen，1979 年；Kerker，1980 年；Bachelet 等人，1982 年；Shirley 等人，1989 年）引入了这类效果极佳的非局部伪势。此外，正如 Hamann、Schluter 和 Chiang（1979 年）所指出的那样，核心区域外的伪波函数和实波函数的匹配也保证了离子核心散射的一阶能量依赖性是正确的，因此可以在很宽的能量范围内准确地描述散射。Shirley 等人（1989 年）最近提出了一种构建伪势的方法，它甚至可以修正散射的高阶能量依赖性。这些类型的局部和非局部伪势目前被称为ab initio 或 norm conserving，能够描述各种原子环境中的离子散射，这一特性被称为可转移性。

The typical method for generating an ionic pseudopotential for an atom of species $\alpha ,v_{\alpha }$$\alpha ,v_{\alpha }$alpha,v_(alpha)\alpha, v_{\alpha} is illustrated in Fig. 6 and proceeds as follows. All-electron calculations are performed for an isolated atom in its ground state and some excited states, using a given form for the exchangecorrelation density functional. This provides valence electron eigenvalues and valence electron wave functions for the atom. A parametrized form for the ionic pseudopotential is chosen. The parameters are then adjusted, so that a pseudoatom calculation using the same form for exchange-correlation as in the all-electron atom gives both pseudowave functions that match the valence wave functions outside some cutoff radius $r_c$$r_c$r_(c)r_{c} and pseudoeigenvalues that are equal to the valence eigenvalues. The ionic pseudopotential obtained in this fashion is then used, without further modification, for any environment of the atom. The electronic density in any new environment of the atom is then determined using both the ionic pseudopotential obtained in this way and the same form of exchange-correlation functional as employed in the construction of the ionic pseudopotential. A generalization of this pseudopotential construction procedure for solutions of the atom that are not normalizable has recently been introduced by Hamann (1989).
图 6 展示了生成 $\alpha ,v_{\alpha }$$\alpha ,v_{\alpha }$alpha,v_(alpha)\alpha, v_{\alpha} 原子离子假势的典型方法，其过程如下。使用给定形式的交换相关密度函数，对处于基态和某些激发态的孤立原子进行全电子计算。这提供了原子的价电子特征值和价电子波函数。选择离子伪势的参数化形式。然后对参数进行调整，以便在使用与全电子原子相同的交换相关形式进行伪原子计算时，既能得到与某个截止半径 $r_c$$r_c$r_(c)r_{c} 以外的价电子波函数相匹配的伪波函数，又能得到与价电子特征值相等的伪特征值。以这种方式获得的离子假势无需进一步修改，即可用于原子的任何环境。原子在任何新环境中的电子密度都可以使用以这种方式得到的离子赝势和在构建离子赝势时使用的相同形式的交换相关函数来确定。最近，Hamann（1989 年）提出了针对不可归一化原子解的伪势构建程序的推广方法。
Finally, it should be noted that ionic pseudopotentials are constructed with $r_c$$r_c$r_(c)r_{c} ranging typically from one to two times the value of the core radius. It should also be noted that, in general, the smaller the value of $r_c$$r_c$r_(c)r_{c}, the more “transferable” the potential. (The entire procedure for solving the problem of a solid, given an ionic pseudopotential, is outlined in Sec. II.F.)
最后，需要指出的是，离子赝势的 $r_c$$r_c$r_(c)r_{c} 通常是核心半径值的一到两倍。还应注意的是，一般来说， $r_c$$r_c$r_(c)r_{c} 的值越小，势的 "可转移性 "就越强。(给定离子态势后，求解固体问题的整个过程将在第 II.F 节中概述）。

FIG. 6. Flow chart describing the construction of an ionic pseudopotential for an atom.
图 6.描述构建原子离子假势的流程图。

The replacement of the true ionic potential by a weaker pseudopotential allows the electronic wave functions to be expanded using far fewer plane-wave basis states than would be needed to expand the wave functions in a full ionic potential. The rapid oscillations of the valence wave functions in the cores of the atoms have been removed, and the small core electron states are no longer present. The pseudopotential approximation has a number of other advantages in addition to reducing the number of plane-wave basis states needed to expand the electronic wave functions. The removal of the core electrons means that fewer electronic wave functions have to be calculated. More importantly, the total energy of the valence electron system is typically a thousand times smaller than the energy of the all-electron system. The difference between the electronic energies of different ionic configurations appears almost totally in the energy of the valence electrons, so that the accuracy required to determine energy differences between ionic configurations in a pseudopotential calculation is much smaller than the accuracy required in an all-electron calculation. The energy differences are just as large when only the valence electrons are included in the calculation, but the total energies are typically a thousand times smaller in the pseudopotential calculation than in the all-electron calculation. But, of course, the total energy is no longer meaningful. Only differences have meaning.
用较弱的伪电势代替真正的离子电势，可以使用比在全离子电势中扩展波函数所需的平面波基态少得多的平面波基态来扩展电子波函数。原子核内价波函数的快速振荡已被消除，小的核电子态也不复存在。除了减少扩展电子波函数所需的平面波基态数量外，伪势近似还有其他一些优点。去除核心电子意味着需要计算的电子波函数数量减少。更重要的是，价电子系统的总能量通常比全电子系统的能量小一千倍。不同离子构型的电子能量差异几乎完全体现在价电子的能量上，因此在假势计算中确定离子构型间能量差异所需的精度要比全电子计算所需的精度小得多。当计算中只包括价电子时，能量差异同样很大，但在伪势计算中，总能量通常比全电子计算小一千倍。当然，总能量已不再有意义。只有差异才有意义。

One property of a pseudopotential that is not incorporated into the standard generation procedure is the value of the cutoff energy required for the plane-wave basis sets. Obviously, the smaller this value, the smaller the basis set required for any particular calculation, and the faster the calculation. A number of approaches to this problem have been adopted. Some authors add additional constraints in the process of generating the pseudopotential which are intended to produce a more rapidly convergent potential (Trouillier and Martins, 1991). Alternatively, a separate optimization procedure can be applied after generating a pseudopotential using one of the standard techniques (Rappe et al., 1990; Rappe and Joannopoulos, 1991). A rather more radical approach has been suggested by Vanderbilt (Vanderbilt, 1990; Laasonen et al., 1991), which involves relaxing the norm conservation of the pseudopotential. This approach will be described more fully in Sec. IX.B.
伪势的一个未纳入标准生成程序的属性是平面波基集所需的截止能量值。显然，这个值越小，任何特定计算所需的基集就越小，计算速度就越快。在这个问题上，人们采用了多种方法。一些学者在生成伪势的过程中增加了额外的约束条件，目的是产生一个收敛更快的势 （Trouillier 和 Martins，1991 年）。另外，在使用标准技术之一生成伪势后，还可以使用单独的优化程序（Rappe 等人，1990 年；Rappe 和 Joannopoulos，1991 年）。范德比尔特提出了一种更为激进的方法（Vanderbilt，1990；Laasonen 等人，1991），即放宽伪势的规范守恒。这种方法将在第 IX.B 节中详细介绍。

The total ionic potential in a solid is obtained by placing an ionic pseudopotential at the position of every ion in the solid. The information about the positions of the ions is contained in the structure factor. The value of the
固体中的总离子电势是通过在固体中每个离子的位置上放置一个离子伪电势得到的。有关离子位置的信息包含在结构因子中。结构因子的值为
structure factor at wave vector $\mathbf{G}$$\mathbf{G}$G\mathbf{G} for ions of species $\alpha$$\alpha$alpha\alpha is given by
种类为 $\alpha$$\alpha$alpha\alpha 的离子在波矢 $\mathbf{G}$$\mathbf{G}$G\mathbf{G} 处的结构因数为

where the sum is over the positions of all the ions of species $\alpha$$\alpha$alpha\alpha in a single unit cell.
其中，总和是一个单胞中 $\alpha$$\alpha$alpha\alpha 物种的所有离子的位置。

The periodicity of the system restricts the nonzero components of the ionic potential to reciprocal-lattice vectors. Hence it is only necessary to calculate the structure factor at the set of reciprocal-lattice vectors.
系统的周期性将离子势的非零分量限制在倒易点阵向量上。因此，只需计算倒易点阵向量集上的结构因子即可。

The total ionic potential $V_{\text{ion}}$$V_{\text{ion }}$V_("ion ")V_{\text {ion }} is obtained by summing the product of the structure factor and the pseudopotential over all the species of ions. For example, for a local potential $V_{\text{ion}}$$V_{\text{ion }}$V_("ion ")V_{\text {ion }} is given by
离子总电势 $V_{\text{ion}}$$V_{\text{ion }}$V_("ion ")V_{\text {ion }} 是由所有离子种类的结构因子与伪电势的乘积相加而得。例如，局部电势 $V_{\text{ion}}$$V_{\text{ion }}$V_("ion ")V_{\text {ion }} 的计算公式为

At large distances the pseudopotential is a pure Coulomb potential of the form $Z/r$$Z/r$Z//rZ / r, where $Z$$Z$ZZ is the valence of the atom. On taking the Fourier transform, one finds that the pseudopotential diverges as $Z/G^2$$Z/G^2$Z//G^(2)Z / G^{2} at small wave vectors. Therefore the total ionic potential at $\mathbf{G}=\mathbf{0}$$\mathbf{G}=\mathbf{0}$G=0\mathbf{G}=\mathbf{0} is infinite, so the electron-ion energy is infinite. However, there are similar divergences in the Coulomb energies due to the electron-electron interactions and the ion-ion interactions. The Coulomb $\mathbf{G}=\mathbf{0}$$\mathbf{G}=\mathbf{0}$G=0\mathbf{G}=\mathbf{0} contributions to the total energy from the three interactions cancel exactly. This is not surprising because there is no Coulomb potential at $\mathbf{G}=\mathbf{0}$$\mathbf{G}=\mathbf{0}$G=0\mathbf{G}=\mathbf{0} in a charge-neutral system, and so there cannot be a contribution to the total energy from the $\mathbf{G}=\mathbf{0}$$\mathbf{G}=\mathbf{0}$G=0\mathbf{G}=\mathbf{0} component of the Coulomb potential.
在大距离上，伪势是形式为 $Z/r$$Z/r$Z//rZ / r 的纯库仑势，其中 $Z$$Z$ZZ 是原子的价。在进行傅立叶变换时，我们会发现在小波矢量处，伪势发散为 $Z/G^2$$Z/G^2$Z//G^(2)Z / G^{2} 。因此， $\mathbf{G}=\mathbf{0}$$\mathbf{G}=\mathbf{0}$G=0\mathbf{G}=\mathbf{0} 处的总离子势是无限的，所以电子-离子能量也是无限的。然而，由于电子-电子相互作用和离子-离子相互作用，库仑能量也存在类似的发散。三种相互作用对总能量的库仑 $\mathbf{G}=\mathbf{0}$$\mathbf{G}=\mathbf{0}$G=0\mathbf{G}=\mathbf{0} 贡献完全抵消。这并不奇怪，因为在电荷中性系统中， $\mathbf{G}=\mathbf{0}$$\mathbf{G}=\mathbf{0}$G=0\mathbf{G}=\mathbf{0} 处不存在库仑势，因此库仑势的 $\mathbf{G}=\mathbf{0}$$\mathbf{G}=\mathbf{0}$G=0\mathbf{G}=\mathbf{0} 分量不可能对总能量有贡献。

The pseudopotential is, however, not a pure Coulomb potential and therefore not exactly proportional to $Z/G^2$$Z/G^2$Z//G^(2)Z / G^{2} for small $G$$G$GG. There is a constant contribution to the pseudopotential at small $G$$G$GG, equal to the integral of the difference between the pure Coulomb $Z/r$$Z/r$Z//rZ / r potential and the pseudopotential. This constant for species $\alpha$$\alpha$alpha\alpha is
然而，伪电势并非纯库仑电势，因此在小 $G$$G$GG 时，伪电势与 $Z/G^2$$Z/G^2$Z//G^(2)Z / G^{2} 并不完全成正比。在小 $G$$G$GG 时，伪电势有一个常数，等于纯库仑电势 $Z/r$$Z/r$Z//rZ / r 与伪电势之差的积分。对于物种 $\alpha$$\alpha$alpha\alpha 来说，这个常数是