WEAKLYCONVEX ANDCONVEXDOMINATIONNUMBERS

扰动模型算法
扰动模型算法是一种用于处理数据隐私保护的方法。

它通过对原始数据进行一系列的扰动操作，使得输出的数据在保持一定的数据分布特征的同时，避免了对个体隐私的直接泄露。

扰动模型算法的主要原理是在原始数据中引入一定的噪声，从而对数据进行模糊化处理，使得攻击者无法从数据中直接获取个体的敏感信息。

常见的扰动模型算法包括拉普拉斯机制和指数机制。

拉普拉斯机制是一种基于指数分布的扰动模型算法。

它通过在查询结果中添加服从拉普拉斯分布的噪声，来对数据进行扰动。

具体来说，对于一个查询结果的真实值x，拉普拉斯机制会在
结果中增加一个服从均值为0，尺度参数为1/ε的拉普拉斯分
布的随机噪声。

指数机制是一种基于指数分布的扰动模型算法。

它通过对每个个体的敏感程度进行量化，并根据敏感程度决定对查询结果的扰动程度。

具体来说，指数机制会根据个体的敏感程度和查询结果的近似值，计算每个个体对查询结果的得分，然后按照得分的指数分布进行随机选择，选取一个个体作为查询结果的扰动项。

这些扰动模型算法在数据隐私保护领域得到了广泛的应用。

它们可以在保护数据隐私的同时，提供一定程度的数据可用性，使得数据可以仍然用于一些常见的数据分析任务。

量子力学英语
随着量子力学的发展和应用，许多新的概念和术语相继出现。

掌握量子力学英语不仅有利于学习和研究，还可以更好地沟通和交流。

以下是一些常用的量子力学英语词汇：
1. Quantum mechanics 量子力学
2. Wave function 波函数
3. Schrdinger equation 薛定谔方程
4. Uncertainty principle 不确定性原理
5. Superposition principle 叠加原理
6. Entanglement 纠缠
7. Quantum state 量子态
8. Eigenvalue 特征值
9. Eigenfunction 特征函数
10. Hamiltonian 哈密顿量
11. Operator 算符
12. Commutation relation 对易关系
13. Quantum tunneling 量子隧穿
14. Quantum entanglement 量子纠缠
15. Quantum superposition 量子叠加
以上是一些常用的量子力学英语词汇，学习量子力学英语需要不断积累和运用。

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1、microscopic world 微观世界2、macroscopic world 宏观世界3、quantum theory 量子[理]论4、quantum mechanics 量子力学5、wave mechanics 波动力学6、matrix mechanics 矩阵力学7、Planck constant 普朗克常数8、wave-particle duality 波粒二象性9、state 态10、state function 态函数11、state vector 态矢量12、superposition principle of state 态叠加原理13、orthogonal states 正交态14、antisymmetrical state 正交定理15、stationary state 对称态16、antisymmetrical state 反对称态17、stationary state 定态18、ground state 基态19、excited state 受激态20、binding state 束缚态21、unbound state 非束缚态22、degenerate state 简并态23、degenerate system 简并系24、non-deenerate state 非简并态25、non-degenerate system 非简并系26、de Broglie wave 德布罗意波27、wave function 波函数28、time-dependent wave function 含时波函数29、wave packet 波包30、probability 几率31、probability amplitude 几率幅32、probability density 几率密度33、quantum ensemble 量子系综34、wave equation 波动方程35、Schrodinger equation 薛定谔方程36、Potential well 势阱37、Potential barrien 势垒38、potential barrier penetration 势垒贯穿39、tunnel effect 隧道效应40、linear harmonic oscillator线性谐振子41、zero proint energy 零点能42、central field 辏力场43、Coulomb field 库仑场44、δ-function δ-函数45、operator 算符46、commuting operators 对易算符47、anticommuting operators 反对易算符48、complex conjugate operator 复共轭算符49、Hermitian conjugate operator 厄米共轭算符50、Hermitian operator 厄米算符51、momentum operator 动量算符52、energy operator 能量算符53、Hamiltonian operator 哈密顿算符54、angular momentum operator 角动量算符55、spin operator 自旋算符56、eigen value 本征值57、secular equation 久期方程58、observable 可观察量59、orthogonality 正交性60、completeness 完全性61、closure property 封闭性62、normalization 归一化63、orthonormalized functions 正交归一化函数64、quantum number 量子数65、principal quantum number 主量子数66、radial quantum number 径向量子数67、angular quantum number 角量子数68、magnetic quantum number 磁量子数69、uncertainty relation 测不准关系70、principle of complementarity 并协原理71、quantum Poisson bracket 量子泊松括号72、representation 表象73、coordinate representation 坐标表象74、momentum representation 动量表象75、energy representation 能量表象76、Schrodinger representation 薛定谔表象77、Heisenberg representation 海森伯表象78、interaction representation 相互作用表象79、occupation number representation 粒子数表象80、Dirac symbol 狄拉克符号81、ket vector 右矢量82、bra vector 左矢量83、basis vector 基矢量84、basis ket 基右矢85、basis bra 基左矢86、orthogonal kets 正交右矢87、orthogonal bras 正交左矢88、symmetrical kets 对称右矢89、antisymmetrical kets 反对称右矢90、Hilbert space 希耳伯空间91、perturbation theory 微扰理论92、stationary perturbation theory 定态微扰论93、time-dependent perturbation theory 含时微扰论94、Wentzel-Kramers-Brillouin method W. K. B.近似法95、elastic scattering 弹性散射96、inelastic scattering 非弹性散射97、scattering cross-section 散射截面98、partial wave method 分波法99、Born approximation 玻恩近似法100、centre-of-mass coordinates 质心坐标系101、laboratory coordinates 实验室坐标系102、transition 跃迁103、dipole transition 偶极子跃迁104、selection rule 选择定则105、spin 自旋106、electron spin 电子自旋107、spin quantum number 自旋量子数108、spin wave function 自旋波函数109、coupling 耦合110、vector-coupling coefficient 矢量耦合系数111、many-partic le system 多子体系112、exchange forece 交换力113、exchange energy 交换能114、Heitler-London approximation 海特勒-伦敦近似法115、Hartree-Fock equation 哈特里-福克方程116、self-consistent field 自洽场117、Thomas-Fermi equation 托马斯-费米方程118、second quantization 二次量子化119、identical particles全同粒子120、Pauli matrices 泡利矩阵121、Pauli equation 泡利方程122、Pauli’s exclusion principle泡利不相容原理123、Relativistic wave equation 相对论性波动方程124、Klein-Gordon equation 克莱因-戈登方程125、Dirac equation 狄拉克方程126、Dirac hole theory 狄拉克空穴理论127、negative energy state 负能态128、negative probability 负几率129、microscopic causality 微观因果性本征矢量eigenvector本征态eigenstate本征值eigenvalue本征值方程eigenvalue equation本征子空间eigensubspace (可以理解为本征矢空间)变分法variatinial method标量scalar算符operator表象representation表象变换transformation of representation表象理论theory of representation波函数wave function波恩近似Born approximation玻色子boson费米子fermion不确定关系uncertainty relation狄拉克方程Dirac equation狄拉克记号Dirac symbol定态stationary state定态微扰法time-independent perturbation定态薛定谔方程time-independent Schro(此处上面有两点)dinger equati on 动量表象momentum representation角动量表象angular mommentum representation占有数表象occupation number representation坐标（位置）表象position representation角动量算符angular mommentum operator角动量耦合coupling of angular mommentum对称性symmetry对易关系commutator厄米算符hermitian operator厄米多项式Hermite polynomial分量component光的发射emission of light光的吸收absorption of light受激发射excited emission自发发射spontaneous emission轨道角动量orbital angular momentum自旋角动量spin angular momentum轨道磁矩orbital magnetic moment归一化normalization哈密顿hamiltonion黑体辐射black body radiation康普顿散射Compton scattering基矢basis vector基态ground state基右矢basis ket ‘右矢’ket基左矢basis bra简并度degenerancy精细结构fine structure径向方程radial equation久期方程secular equation量子化quantization矩阵matrix模module模方square of module内积inner product逆算符inverse operator欧拉角Eular angles泡利矩阵Pauli matrix平均值expectation value （期望值）泡利不相容原理Pauli exclusion principle氢原子hydrogen atom球鞋函数spherical harmonics全同粒子identical partic les塞曼效应Zeeman effect上升下降算符raising and lowering operator 消灭算符destruction operator产生算符creation operator矢量空间vector space守恒定律conservation law守恒量conservation quantity投影projection投影算符projection operator微扰法pertubation method希尔伯特空间Hilbert space线性算符linear operator线性无关linear independence谐振子harmonic oscillator选择定则selection rule幺正变换unitary transformation幺正算符unitary operator宇称parity跃迁transition运动方程equation of motion正交归一性orthonormalization正交性orthogonality转动rotation自旋磁矩spin magnetic monent(以上是量子力学中的主要英语词汇，有些未涉及到的可以自由组合。

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(147)iStdDev标准偏差指标 (148)iStdDevOnArray (148)iStochastic随机震荡指标 (148)iWPR威廉指标 (149)Timeseries access时间序列图表数据 (150)iBars柱的数量 (150)iClose (150)iHigh (151)iHighest (151)iLow (152)iLowest (152)iOpen (152)iTime (153)iVolume (153)Trading functions交易函数 (155)Execution errors (155)OrderClose (157)OrderCloseBy (158)OrderClosePrice (158)OrderCloseTime (158)OrderComment (159)OrderCommission (159)OrderDelete (159)OrderExpiration (160)OrderLots (160)OrderMagicNumber (160)OrderModify (160)OrderOpenPrice (161)OrderOpenTime (161)OrderPrint (162)OrderProfit (162)OrderSelect (162)OrderSend (163)OrdersHistoryTotal (164)OrderStopLoss (164)OrdersTotal (164)OrderSwap (165)OrderSymbol (165)OrderTakeProfit (165)OrderTicket (166)OrderType (166)Window functions窗口函数 (167)HideTestIndicators隐藏指标 (167)Period使用周期 (167)RefreshRates刷新预定义变量和系列数组的数据 (167)Symbol当前货币对 (168)WindowBarsPerChart可见柱总数 (168)WindowExpertName智能交易系统名称 (169)WindowFind返回名称 (169)WindowFirstVisibleBar第一个可见柱 (169)WindowHandle (169)WindowIsVisible图表在子窗口中可见 (170)WindowOnDropped (170)WindowPriceMax (170)WindowPriceMin (171)WindowPriceOnDropped (171)WindowRedraw (172)WindowScreenShot (172)WindowTimeOnDropped (173)WindowsTotal指标窗口数 (173)WindowXOnDropped (173)WindowYOnDropped (174)Obsolete functions过时的函数 (175)MetaQuotes Language 4 (MQL4) 是一种新的内置型程序用来编写交易策略。

通信中的数学原理英文Mathematical principles in communication1. Encoding and decoding: Communication systems often involve encoding information into a format suitable for transmission and decoding it back into its original form at the receiving end. This process relies on mathematical principles to ensure the accurate transmission and recovery of information.2. Modulation: Modulation refers to the process of encoding information onto a carrier signal for efficient transmission. It involves mathematical operations such as amplitude modulation, frequency modulation, or phase modulation, which are used to represent the information in a wave form.3. Signal processing: Signal processing is a fundamental component of communication systems, and it involves mathematical operations such as convolution, Fourier transforms, and filtering. These operations are used to manipulate and analyze signals to optimize their transmission or extraction of information.4. Error detection and correction: In communication systems, errors canoccur during transmission due to noise or other impairments. Mathematical principles such as error detection codes (e.g., parity check) and error correction codes (e.g., Reed-Solomon codes) are used to detect and correct these errors, ensuring the accuracy of the transmitted data.5. Channel capacity: The channel capacity of a communication system refers to the maximum rate at which information can be reliably transmitted through a given channel. Shannon's theorem, a mathematical result derived by Claude Shannon, provides a fundamental limit on the channel capacity and allows for the optimization of communication systems.6. Data compression: Data compression techniques, such as Huffman coding or arithmetic coding, rely on mathematical principles to reduce the size of information for efficient transmission or storage. These techniques exploit patterns in the data and use mathematical algorithms to encode the information more efficiently.7. Cryptography: Cryptographic algorithms, which are used to secure the confidentiality and integrity of communication, rely on mathematical principles such as modular arithmetic, prime numbers, and discretelogarithms. These algorithms ensure the protection of sensitive information from unauthorized access or tampering.8. Information theory: Information theory is a branch of mathematics that studies the quantification, storage, and communication of information. It provides mathematical models and principles to analyze and optimize communication systems, including coding theory, data compression, and channel capacity.Overall, mathematics plays a crucial role in various aspects of communication, from encoding and modulation to error detection and correction, channel capacity, data compression, cryptography, and information theory. These mathematical principles enable efficient and reliable communication in various domains.。

Numerical atomic orbitals for linear-scaling calculationsJavier Junquera,1O´scar Paz,1Daniel Sa´nchez-Portal,2,3and Emilio Artacho41Departamento de Fı´sica de la Materia Condensada,C-III,Universidad Auto´noma,28049Madrid,Spain2Department of Physics and Materials Research Laboratory,University of Illinois,Urbana Illinois618013Departamento de Fı´sica de Materiales and DIPC,Facultad de Quı´mica,UPV/EHU,Apdo.1072E-20080San Sebastia´n,Spain 4Department of Earth Sciences,University of Cambridge,Downing Street,Cambridge CB23EQ,United Kingdom͑Received6April2001;published28November2001͒The performance of basis sets made of numerical atomic orbitals is explored in density-functional calcula-tions of solids and molecules.With the aim of optimizing basis quality while maintaining strict localization ofthe orbitals,as needed for linear-scaling calculations,several schemes have been tried.The best performanceis obtained for the basis sets generated according to a new scheme presented here,aflexibilization of previousproposals.Strict localization is maintained while ensuring the continuity of the basis-function derivative at thecutoff radius.The basis sets are tested versus converged plane-wave calculations on a significant variety ofsystems,including covalent,ionic,and metallic.Satisfactory convergence is obtained for reasonably smallbasis sizes,with a clear improvement over previous schemes.The transferability of the obtained basis sets istested in several cases and it is found to be satisfactory as well.DOI:10.1103/PhysRevB.64.235111PACS number͑s͒:71.15.Ap,71.15.MbI.INTRODUCTIONIn order to make intelligent use of the increasing power of computers for thefirst-principles simulation of ever larger and more complex systems,it is important to develop and tune linear-scaling methods,where the computational load scales only linearly with the number of atoms in the simula-tion cell.The present status of these methods and their ap-plications can be found in several reviews.1–3Essential for linear scaling is locality,and basis sets made of localized wave functions represent a very sensible basis choice.It is not only the scaling that matters,however,the prefactor be-ing also important for practical calculations.The prefactor depends significantly on two aspects of the basis:͑i͒the number of basis functions per atom,and͑ii͒the size of the localization regions of these functions.Atomic orbitals offer efficient basis sets since,even though their localization ranges are larger than those of some other methods,4the number of basis functions needed is usu-ally quite small.The price to pay for this efficiency is the lack of systematics for convergence.Unlike with plane-wave5or real-space-grid6related methods,there is no unique way of increasing the size of the basis,and the rate of convergence depends on the way the basis is enlarged.This fact poses no fundamental difficulties,it just means that some effort is needed in the preparation of unbiased basis sets,in analogy to the extra work required to prepare pseudo-potentials to describe the effect of core electrons.Maximum efficiency is achieved by choosing atomic or-bitals that allow convergence with small localization ranges and few orbitals.It is a challenge again comparable to the one faced by the pseudopotential community,where transfer-ability and softness are sought.7For atomic wave functions the optimization freedom is in the radial shape.Gaussian-type orbitals have been proposed for linear scaling,8–10con-necting with the tradition of quantum chemistry.11,12These bases are,however,quite rigid for the mentioned optimiza-tion,imposing either many Gaussians or large localization ranges.Numerical atomic orbitals͑NAO’s͒are moreflexible in this respect.Different ideas have been proposed in the litera-ture,originally within tight-binding contexts concentrating on minimal͑single␨)bases.They are obtained byfinding the eigenfunctions of the isolated atoms confined within spherical potential wells of different shapes,13–15or directly modifying the eigenfunctions of the atoms.16These schemes give strictly localized orbitals,i.e.,orbitals that are strictly zero beyond given cutoff radii r c.Afirst extension towards more complete basis sets was proposed using the excited states of the confined atoms,17but the quite delocalized char-acter of many excited states made this approach inefficient unless very stringent confinement potentials were used.18 For multiple␨,a better scheme was proposed based on the split-valence idea of quantum chemistry,11,12but adapted to strictly localized NAO’s.19In the same work,a systematic way was proposed to generate polarization orbitals suited for these basis sets.The scheme of Ref.19has proven to be quite efficient,systematic,and reasonable for a large variety of systems͑for short reviews,see Refs.19and20͒.In this work we go beyond previous methodologies be-cause of two main reasons:͑i͒It is always desirable to obtain the highest possible accuracy given the computational re-sources available,and͑ii͒it is important to know and show what is the degree of convergence attainable by NAO basis sets of reasonable sizes.We explore these issues by variationally optimizing basis sets for a variety of condensed systems.The parameters de-fining the orbitals are allowed to vary freely to minimize the total energy of these systems.This energy is then compared with that of converged plane-wave calculations for exactly the same systems,including same density functional and pseudopotentials.The optimal basis sets are then tested monitoring structural,and elastic properties of the systems.The transferability of the basis sets optimized for particu-lar systems is then checked by transferring them to other systems and testing the same energetical,structural,and elas-tic parameters.Finally,the effect of localizing the orbitalsPHYSICAL REVIEW B,VOLUME64,235111tighter than what they variationally choose is explored on a demanding system.II.METHODThe calculations presented below were all done using density-functional theory21,22͑DFT͒in its local-density23ap-proximation͑LDA͒.Core electrons were replaced by norm-conserving pseudopotentials7in their fully separable form.24 The nonlocal partial-core exchange-correlation correction25 was included for Cu to improve the description of the core-valence interactions.Periodic boundary conditions were used for all systems. Molecules were treated in a supercell scheme allowing enough empty space between molecules to make intermo-lecular interactions negligible.For solid systems,integra-tions over the Brillouin zone were replaced by converged sums over selected kជsets.26Thus far the approximations are exactly the same for the two different sets of calculations performed in this work: based on NAO’s and on plane-waves͑PW’s͒.The calcula-tions using NAO’s were performed with the SIESTA method, described elsewhere.18,27Besides the basis itself,the only additional approximation with respect to PW’s is the replace-ment of some integrals in real space by sums in afinite three-dimensional͑3D͒real-space grid,controlled by one single parameter,the energy cutoff for the grid.27This cutoff, which refers to thefineness of the grid,was converged for all systems studied here͑200Ry for all except for Si and H2,for which80and100Ry respectively,were used͒.Similarly,the PW calculations were done for converged PW cutoffs.28 Cohesive curves for the solids were obtained byfitting calculated energy values for different unit-cell volumes to cubic,quartic,and Murnaghan-like29curves,a procedure giving values to the lattice parameter,the bulk modulus and the cohesive energy of each system.The bulk moduli given by the Murnaghan and quarticfits deviate from each other by around3%,the Murnaghan values being the lowest and the ones shown in the tables.The deviations between Mur-naghan and cubicfits are of the order of7%.The other cohesive parameters do not change appreciably with thefits.The atomic-energy reference for the cohesive energy was taken from the atomic calculations within the same DFT and pseudopotentials,always converged in the basis set.They are hence the same reference for NAO’s and for PW’s,the dif-ference in cohesive energies between the two accounting for the difference in the total energy of the solid.The isolated-atom calculations included spin polarization.III.BASIS OF NUMERICAL ATOMIC ORBITALSThe starting point of the atomic orbitals that conform the basis sets used here is the solution of Kohn-Sham’s Hamil-tonian for the isolated pseudoatoms,solved in a radial grid, with the same approximations as for the solid or molecule ͑the same exchange-correlation functional and pseudopoten-tial͒.A strict localization of the basis functions is ensured either by imposing a boundary condition,by adding a con-fining͑divergent͒potential,or by multiplying the free-atom orbital by a cutting function.We describe in the following three main features of a basis set of atomic orbitals:size, range,and radial shape.A.Size:Number of orbitals per atomFollowing the nomenclature of quantum chemistry,we es-tablish a hierarchy of basis sets,from single␨to multiple␨with polarization and diffuse orbitals,covering from quick calculations of low quality to highly converged ones,as con-verged as thefinest calculations in quantum chemistry.A single␨͑also called minimal͒basis set͑SZ in the following͒has one single radial function per angular momentum chan-nel,and only for those angular momenta with substantial electronic population in the valence of the free atom.Radialflexibilization is obtained by adding a second func-tion per channel:double␨͑DZ͒.Several schemes have been proposed to generate this second function.In quantum chem-istry,the split valence11,30scheme is widely used:starting from the expansion in Gaussians of one atomic orbital,the most contracted Gaussians are used to define thefirst orbital of the double␨and the most extended ones for the second. Another proposal defines the second␨as the derivative of thefirst one with respect to occupation.31For strictly local-ized functions there was afirst proposal17of using the ex-cited states of the confined atoms,but it would work only for tight confinement.An extension of the split valence idea of quantum chemistry to strictly localized NAO’s was proposed in Ref.19and has been used quite successfully in a variety of systems.It consists of suplementing each basis orbital with a new basis function that reproduces exactly the tail of the original orbital from a given matching radius r m out-wards.The inner part goes smoothly towards the origin as r l(aϪbr2),where a and b are chosen to ensure continuity of the function and its derivative at r m.We follow this scheme in this work,which generalizes to multiple␨trivially by adding more functions generated with the same procedure.Angularflexibility is obtained by adding shells of higher angular momentum.Ways to generate these so-called polar-ization orbitals have been described in the literature,both for Gaussians11,12and for NAO’s.19In this work,however,they will be obtained variationally,as the rest,within theflexibili-ties described below.B.Range:Cutoff radii of orbitalsStrictly localized orbitals͑zero beyond a cutoff radius͒are used in order to obtain sparse Hamiltonian and overlap ma-trices for linear scaling.The traditional alternative to this is based on neglecting interactions when they fall below a tol-erance or when the atoms are beyond some scope of neigh-bors.For long ranges or low tolerances both schemes are essentially equivalent.They differ in their behavior at shorter ranges,where the strict-localization approach has the advan-tage of remaining in the Hilbert space spanned by the basis, remaining variational,and avoiding numerical instabilities no matter how short the range becomes.For the bases made of strictly localized orbitals,the prob-lem isfinding a balanced and systematic way of defining all the different cutoff radii,since both the accuracy and theJUNQUERA,PAZ,SA´NCHEZ-PORTAL,AND ARTACHO PHYSICAL REVIEW B64235111computational efficiency in the calculations depend on them.A scheme was proposed19in which all radii were defined by one single parameter,the energy shift,i.e.,the energy raise suffered by the orbital when confined.In this work,however, we step back from that systematic approach and allow the cutoff radii to vary freely in the optimization procedure͑up to a maximum value of8a.u.͒.C.ShapeWithin the pseudopotential framework it is important to keep the consistency between the pseudopotential and the form of the pseudoatomic orbitals in the core region.This is done by using as basis orbitals the solutions of the same pseudopotential in the free atom.The shape of the orbitals at larger radii depends on the cutoff radius͑see above͒and on the way the localization is enforced.Thefirst proposal13used an infinite square-well potential͑see Fig.1͒.It has been widely and successfully used for minimal bases within the ab initio tight-binding scheme of Sankey and collaborators13us-ing the FIREBALL program,but also for moreflexible bases using the methodology of SIESTA.This scheme has the disadvantage,however,of generating orbitals with a discontinuous derivative at r c as seen in Fig.1.This discontinuity is more pronounced for smaller r c’s and tends to disappear for long enough values of this cutoff.It does remain,however,appreciable for sensible values of r c for those orbitals that would be very wide in the free atom.It is surprising how small an effect such a kink produces in the total energy of condensed systems͑see below͒.It is,never-theless,a problem for forces and stresses,especially if they are calculated using a͑coarse͒finite three-dimensional grid.Another problem of this scheme is related to its defining the basis considering the free atoms.Free atoms can present extremely extended orbitals,their extension being,besides problematic,of no practical use for the calculation in con-densed systems:the electrons far away from the atom can be described by the basis functions of other atoms.Both problems can be addressed simultaneously by add-ing a soft confinement potential to the atomic Hamiltonian used to generate the basis orbitals:it smooths the kink and contracts the orbital as variationally suited.Two soft confine-ment potentials have been proposed in the literature͑Fig.1͒, both of the form V(r)ϭV o r n,one for nϭ2͑Ref.14͒and the other for nϭ6.15They present their own inconveniences,however.First,there is no radius at which the orbitals be-come strictly zero,they have to be neglected at some point. Second,these confinement potentials affect the core region spoiling its adaptation to the pseudopotential.This last problem affects a more traditional scheme as well,namely,the one based on the radial scaling of the or-bitals by suitable scale factors.In addition to very basic bonding arguments,32it is soundly based on restoring virial’s theorem forfinite bases,in the case of Coulombic potentials ͑all-electron calculations͒.33The pseudopotentials limit its applicability,allowing only for extremely small deviationsfrom unity(ϳ1%)in the scale factors obtained variationally ͑with the exception of hydrogen that can contract up to 25%͒.34An alternative scheme to avoid the kink has also beenproposed:16Instead of modifying the potential,it directlymodifies the orbitals of the atom.Following ideas of previ-ous mixed-basis schemes37the atomic orbital is multiplied by1Ϫexp͓Ϫ␣(rϪr c)2͔for rϽr c and zero otherwise.16In Ref.16it is the hard confined wave function which is thenmodified,while in Ref.37it is the free atom wave function.We follow Ref.37.This method is tested in the next section. This scheme does provide strict localization beyond r c,but introduces a different problem:for large␣and small r c a bump appears in the orbital close to r c,which becomes a discontinuity in the wave function in the limit of infinite␣͑Ref.37͒͑this is not the case in Ref.16͒.FIG.1.Shape of the3s orbital of Mg in MgO for the different confinement schemes͑a͒and corresponding potentials͑b͒.NUMERICAL ATOMIC ORBITALS FOR LINEAR-...PHYSICAL REVIEW B64235111In this work we propose a new soft confinement potential avoiding the mentioned deficiencies.It is shown in Fig.1.It isflat͑zero͒in the core region,starts off at some internal radius r i with all derivatives continuous,and diverges at r c ensuring the strict localization there.It isV͑r͒ϭV o eϪ(r cϪr i)/(rϪr i)r cϪr.͑1͒In the following the different schemes are compared,theirdefining parameters being allowed to change variationally.Finally,the shape of an orbital is also changed by theionic character of the atom.Orbitals in cations tend to shrink, and they swell in anions.Introducing a␦Q in the basis-generating free-atom calculations gives orbitals betteradapted to ionic situations in the condensed systems.IV.OPTIMIZATION PROCEDUREGiven a system and a basis size,the range and shape ofthe orbitals are defined by a set of parameters as describedabove.The parameters are described in the following.Per atomic species there is a global␦Q,an extra charge͑positive or negative͒added to the atom at the time of solving theatomic DFT problem for obtaining the basis orbitals͑see below͒.Confinement is imposed separately for each angular mo-mentum shell,with its corresponding parameters that depend on the scheme used.Hard confinement implies one param-eter per shell(r c),and our soft confinement implies three (r c,r i,and V o).One parameter(V o)is needed only in the r n-confinement schemes,14,15and two parameters in the scheme of Elsaesser et al.37(r c and the width of the cutting function͒.Finally,for each␨beyond thefirst,there is a matching radius as mentioned above.19The values of these parameters are defined variationally, according to the following procedure:͑i͒Given a set of parameters,the Kohn-Sham Hamil-tonian͑including the pseudopotential͒is solved for the iso-lated atom,in the presence of the confining potential and the extra charge␦Q.͑In the case of the scheme of Elsaesser et al.,37there is no confining potential,but an a posteriori modification of the solution wave functions.͒This is done for all the relevant l shells of all the different atomic species. The multiple zetas are built from the former using the match-ing procedure described above,19according to the r m’s within the set of parameters.This procedure gives a basis set for each set of parameter values.͑ii͒Given the basis set,a full DFT calculation is per-formed of the system for which the basis is to be optimized, normally a condensed system,solid or molecule.The Kohn-Sham total energy of this system becomes then a function of that set of parameters.Note that neither the extra charge nor the confinement potentials are added to the Kohn-Sham Hamiltonian of the system,they were just used to define the basis.The total-energy calculations are performed for given structural parameters of the studied system.We have chosen to work with experimental structures.This choice is,how-ever,of no great importance since the basis sets are supposed to be transferable enough to render any bias negligible.This is certainly the case at the DZP level,not so much for mini-mal bases.See the section on transferability below.͑iii͒The previous two steps are built in as a function into a minimization algorithm.As a robust and simple minimiza-tion method not requiring the evaluation of derivatives,we have chosen the downhill simplex method.38We have not dedicated special efforts to maximizing the efficiency of the minimization procedure since the systems used for basis op-timization typically involve a small number of atoms and the total-energy calculations are quick.The possible improve-ment in the minimization efficiency is therefore of no rel-evance to the present study.We have no argument to discard the existence of several local minima in the energy function.For the systems studied here there may be sets of parameters giving better bases than the ones we obtain.We systematically tested their robustness by restarting new simplex optimizations from the already optimized sets.More systematic searches for absolute minima,however,would require much more expensive tech-niques,which would not be justified at this point.We have thus satisfied ourselves with the ones obtained,that show good and consistent convergence characteristics.The values obtained for the parameters in the optimizations described below can be obtained from the authors.39V.RESULTSparison of different confinement schemesTable I shows the performance for MgO of the different schemes described above for constructing localized atomic orbitals.The basis sets of both magnesium and oxygen were variationally optimized for all the schemes.Mg was chosen because the3s orbital is very extended in the atom and both the kink and the confinement effects due to other orbitals are very pronounced.Results are shown for a SZ͑single s and p TABLE parison of different confinement schemes on the cohesive properties of MgO,for SZ and DZP basis sets.The gen-eralization of the different schemes to DZP is done as explained in the text.Unconfined refers to using the unconfined pseudoatomic orbitals as basis.a,B,and E c stand for lattice parameter,bulk modulus,and cohesive energy,respectively.The PW calculations were performed with identical approximations as the NAO ones except for the basis.Experimental values were taken from Ref.40.SZ DZPBasis a B E c a B E c scheme͑Å͒͑GPa͒͑eV͒͑Å͒͑GPa͒͑eV͒Unconfined 4.25119 6.49Sankey 4.1722210.89 4.1216511.82 Elsaesser 4.1622811.12 4.1216311.84 Porezag 4.1819611.17 4.0918311.83 Horsfield 4.1522111.26 4.1116811.86 This work 4.1522611.32 4.1016711.87PW 4.1016811.90 Expt. 4.2115210.3JUNQUERA,PAZ,SA´NCHEZ-PORTAL,AND ARTACHO PHYSICAL REVIEW B64235111channels for both species͒and a DZP basis͑double s and p channels plus a single d channel͒.Figure1shows the shape of the optimal3s orbital for the different schemes,and the shape of the confining potentials.The following conclusions can be drawn from the results:͑i͒Within the variational freedom offered here,the3s orbital of Mg wants to be confined to a radius of around6.5bohr, irrespective of scheme,which is extremely short for the free atom.This confinement produces a pronounced kink in the hard scheme.͑ii͒The total energy is relatively insensitive to the scheme used to generate the basis orbitals,as long as there is effective confinement.͑iii͒The basis made of uncon-fined atomic orbitals is substantially worse than any of the others.͑iv͒The pronounced kink obtained in Sankey’s hard confinement scheme is not substantially affecting the total energy as compared with the other schemes.It does perturb, however,by introducing inconvenient noise in the energy variation with volume and other external parameters,and especially in the derivatives of the energy.͑v͒The scheme proposed in this work is variationally slightly better than the other ones,but not significantly.Its main advantage is the avoidance of known problems.In the remainder of the paper, the confinement proposed in this work will be used unless otherwise specified.B.Basis convergenceTable II shows how NAO bases converge for bulk silicon.This is done by comparing different basis sizes,each of them optimized.The results are compared to converged͑50Ry͒PW results͑converged basis limit͒keeping the rest of the calculation identical.Figure2͑a͒shows the cohesion curves for this system.Even though the main point of this work is testing the convergence of NAO basis sets independently of other is-sues,we consider it interesting to gauge the relevance of the errors introduced by the basis by comparing them with other typical errors that appear in these calculations.The NAO and PW results are thus compared to all-electron LDA results49to compare basis errors with the ones produced by the pseudo-potentials.Experiment gives then reference to the error co-mitted by the underlying LDA.The comparisons above are made with respect to the converged-basis limit,for which we used PW’s up to very high cutoffs.It is important to distinguish this limit from the PW calculations at lower cutoffs,as used in many computa-tions.To illustrate this point,Fig.2͑b͒compares the energy convergence for PW’s and for NAO’s.Even though the con-vergence of NAO results is a priori not systematic with the way the basis is enlarged,the sequence of bases presented in thefigure shows a nice convergence of total energy with respect to basis size͑the number of basis functions per atom are shown in parentheses in thefigure͒:the convergence rate is similar to the one of PW’s͑DZP has three times more orbitals than SZ,and a similar factor is found for their equivalents in PW’s͒.For the particular case of Si,Fig.2 shows that the polarization orbitals(3d shell͒are very im-portant for convergence,more than the doubling of the basis. This fact is observed from the stabilization of SZP with re-spect to SZ,which is much larger than for DZ.Figure2shows that an atomic basis at the DZP level requires ten times less functions than its͑energetically͒equivalent PW basis,being Si the easiest system forPW’s.FIG.2.Convergence of NAO basis sets for bulk Si.͑a͒Cohe-sive curve for different basis sets.The lowest curve shows the PW results,filled symbols the NAO bases of this work(opt),and open symbols the NAO bases following Ref.19.Basis labels are like in Table II.͑b͒Comparison of NAO convergence with PW conver-gence.In parentheses is the number of basis functions per atom.TABLE II.Basis comparisons for bulk Si.a,B,and E c stand forlattice parameter͑inÅ͒,bulk modulus͑in GPa͒,and cohesive en-ergy͑in eV͒,respectively.SZ,DZ,and TZ stand for single␨,double␨,and triple␨.P stands for polarized,DP for doubly polar-PW results were taken from Ref.41,and the experimentalvalues from Ref.42.SZ DZ TZ SZP DZP TZP TZDP PW LAPW Expt.a 5.52 5.49 5.48 5.43 5.40 5.39 5.39 5.38 5.41 5.43B85878597979797969698.8E c 4.70 4.83 4.85 5.21 5.31 5.32 5.34 5.37 5.28 4.63NUMERICAL ATOMIC ORBITALS FOR LINEAR-...PHYSICAL REVIEW B64235111For other systems the ratio is much larger,as shown in Table III.It is important to stress that deviations smaller than the ones due to the pseudopotential or the DFT used are obtained with a relatively modest basis size as DZP.This fact is clearin Table II for Si,and in Table IV for other systems.Table IV summarizes the cohesion results for a variety of solids of different chemical kind.They are obtained with optimal DZP basis sets.It can be observed that DZP offers results in good agreement with converged-basis numbers,showing the con-vergence of properties other than the total energy.The devia-tions are similar or smaller than those introduced by LDA or by the pseudopotential.47VI.TRANSFERABILITYTo what extent do optimal bases keep their performance when transferred to different systems than the ones they were optimized for?This is an important question,since if the performance does not suffer significantly,one can hope to tabulate basis sets per species,to be used for whatever sys-tem.If the transferability is not satisfactory,a new basis set should then be obtained variationally for each system to be studied.Of course the transferability increases with basis size,since the basis has more flexibility to adapt to different environments.In this work we limit ourselves to try it on DZP bases for a few representative systems.Satisfactory transferability has been obtained when check-ing in MgO the basis set optimized for Mg bulk and O in a water molecule.Similarly,the basis for O has been tested in H 2O and O 2,and the basis for C in graphite and diamond.TABLE III.Equivalent PW cutoff (E cut )to optimal DZP basesfor different parison of number of basis functions per atom for both bases.For the molecules,a cubic unit cell of 10Åof side was used.System No.funct.DZPNo.funct.PWE cut ͑Ry ͒H 251129634O 2134544286Si1322722Diamond 1328459␣-quartz1392376TABLE IV .Basis comparisons for different solids.a ,B ,and E c stand for lattice parameter ͑in Å͒,bulk modulus ͑in GPa ͒,and co-hesive energy ͑in eV ͒,respectively.ExpLAPW Other PW PW DZP Aua 4.08a 4.05b 4.07c 4.05 4.07B 173a 198b 190c 191188E c 3.81a -- 4.19 4.03MgOa 4.21d 4.26e - 4.10 4.11B 152d 147e -168167E c 10.30d 10.40e -11.9011.87Ca 3.57a3.54f 3.54g 3.53 3.54B 442a 470f 436g 466453E c 7.37a 10.13f 8.96g 8.908.81Sia 5.43a 5.41h 5.38g 5.38 5.40B 99a 96h 94g 9697E c 4.63a 5.28h 5.34g 5.37 5.31Naa 4.23a 4.05i 3.98g 3.95 3.98B 6.9a 9.2i 8.7g 8.89.2E c 1.11a 1.44j 1.28g 1.22 1.22Cua 3.60a 3.52b 3.56g - 3.57B 138a 192b 172g -165Ec 3.50a 4.29k 4.24g - 4.37Pba 4.95a - 4.88- 4.88B 43a -54-64E c2.04a- 3.77- 3.51aC.Kittel,Ref.42.bA.Khein,D.J.Singh,and C.J.Umrigar,Ref.43.cB.D.Yu and M.Scheffler,Ref.44.dF.Finocchi,J.Goniakowski,and C.Noguera,Ref.40.eJ.Goniakowski and C.Noguera,Ref.45.fN.A.W.Holzwarth et al.,Ref.46.gM.Fuchs,M.Bockstedte,E.Pehlke,and M.Scheffler,Ref.47.hC.Filippi,D.J.Singh,and C.J.Umrigar,Ref.41.iJ.P.Perdew et al.,Ref.48.jM.Sigalas et al.,Ref.49.kP.H.T.Philipsen and E.J.Baerends,Ref.50.TABLE V .Transferability of basis sets.‘‘Transf.’’stands for the DZP basis transferred from other systems,while ‘‘Opt.’’refers to the DZP basis optimized for the particular system.For MgO the basis was transferred from bulk Mg and an H 2O molecule,for graphite the basis was transferred from diamond,and for H 2O it was taken from H 2and O 2.a ,B ,and E c stand for lattice parameter,bulk modulus,and cohesive energy,respectively.⌬E stands for the energy difference per atom between graphite and a graphene plane.E b is the binding energy of the molecule.System BasisProperties MgOa ͑Å͒B ͑GPa ͒E c ͑eV ͒Transf. 4.1315711.81Opt. 4.1016711.87PW 4.1016811.90Expt.4.2115210.30Graphitea ͑Å͒c ͑Å͒⌬E ͑meV ͒Transf. 2.456 6.5038PW a 2.457 6.7224Expt.b2.456 6.67423c H 2Od O-H ͑Å͒␪H-O-H ͑deg ͒E b ͑eV ͒Transf.0.975105.012.73Opt.0.972104.512.94PW 0.967105.113.10LAPW d 0.968103.911.05Expt.e0.958104.510.08a M.C.Schabel and J.L.Martins,Ref.51.bY .Baskin and L.Mayer,Ref.52.cL.A.Girifalco and dd,Ref.53.dP.Serena,A.Baratoff,and J.M.Soler,Ref.54.eG.Herzberg,Ref.55.JUNQUERA,PAZ,SA´NCHEZ-PORTAL,AND ARTACHO PHYSICAL REVIEW B 64235111。

(0,2) 插值||(0,2) interpolation0#||zero-sharp; 读作零井或零开。

0+||zero-dagger; 读作零正。

1-因子||1-factor3-流形||3-manifold; 又称“三维流形”。

AIC准则||AIC criterion, Akaike information criterionAp 权||Ap-weightA稳定性||A-stability, absolute stabilityA最优设计||A-optimal designBCH 码||BCH code, Bose-Chaudhuri-Hocquenghem codeBIC准则||BIC criterion, Bayesian modification of the AICBMOA函数||analytic function of bounded mean oscillation; 全称“有界平均振动解析函数”。

BMO鞅||BMO martingaleBSD猜想||Birch and Swinnerton-Dyer conjecture; 全称“伯奇与斯温纳顿－戴尔猜想”。

B样条||B-splineC*代数||C*-algebra; 读作“C星代数”。

C0 类函数||function of class C0; 又称“连续函数类”。

CA T准则||CAT criterion, criterion for autoregressiveCM域||CM fieldCN 群||CN-groupCW 复形的同调||homology of CW complexCW复形||CW complexCW复形的同伦群||homotopy group of CW complexesCW剖分||CW decompositionCn 类函数||function of class Cn; 又称“n次连续可微函数类”。

Cp统计量||Cp-statisticC。

弱无穷小算子算法简介弱无穷小算子算法（Weak Infinitesimal Operator Algorithm）是一种用于解决非线性优化问题的数值计算方法。

它通过将非线性问题转化为一系列线性问题的求解，从而有效地降低了计算的复杂度。

背景在实际问题中，我们经常需要求解非线性优化问题，即最小化或最大化一个非线性目标函数的值。

这类问题通常无法直接应用传统的数值优化方法进行求解，因为非线性函数具有复杂的数学形式，难以找到全局最优解。

弱无穷小算子算法应运而生，为我们提供了一种有效且高效的求解非线性优化问题的方法。

基本思想弱无穷小算子算法基于泰勒展开和牛顿迭代方法，并结合了弱收敛理论。

它将原始的非线性优化问题转化为一系列线性子问题来求解。

具体来说，该算法通过引入一个弱无穷小量（infinitesimal）来逐步逼近原始目标函数，并使用牛顿迭代方法更新当前点的估计值，直到满足收敛准则为止。

算法步骤1.初始化：选择初始点和收敛准则的阈值，设定迭代次数上限。

2.迭代更新：根据泰勒展开，将原始目标函数在当前点进行二阶近似，并引入弱无穷小量。

3.线性子问题求解：将二阶近似后的目标函数转化为一个线性子问题，通过求解线性子问题得到下一步的迭代点。

4.收敛判断：计算当前点与上一步迭代点之间的差异，并与收敛准则进行比较。

如果满足收敛准则，则停止迭代；否则返回第2步继续迭代。

5.输出结果：返回最终收敛的点作为最优解。

算法特点•高效性：弱无穷小算子算法通过将非线性优化问题转化为一系列线性子问题来求解，大大降低了计算复杂度，提高了计算效率。

•全局收敛性：该算法基于牛顿迭代方法，具有全局收敛性。

在合理的初始点选择和适当的参数设定下，可以得到全局最优解。

•鲁棒性：弱无穷小算子算法对于非线性函数形式的要求相对较低，适用于各种类型的非线性优化问题。

•可扩展性：该算法可以与其他优化算法相结合，例如遗传算法、模拟退火等，形成一种混合优化方法，以解决更加复杂的问题。

三维稀疏卷积原理
三维稀疏卷积的原理主要是建立在哈希表的基础上，用于保存特定位置的计算结果。

在输入数据中，只有少量的点（即非零元素或激活输入点）具有实际的值，而大部分点都是零值。

这种稀疏性使得稀疏卷积成为一种有效的计算方式。

在稀疏卷积中，卷积核的定义与传统卷积相同，但输出定义有所不同。

稀疏卷积有两种主要的输出定义方式：regular output definition 和submanifold output definition。

在regular output definition 中，只要卷积核覆盖到一个输入点，就会计算输出点。

而在submanifold output definition中，输出点的计算则更加严格，只有满足特定条件的输入点才会被用于计算输出。

三维稀疏卷积在处理大规模三维数据时具有显著的优势。

由于输入数据的稀疏性，稀疏卷积能够大大减少不必要的计算，从而显著提高计算效率。

此外，稀疏卷积还能够保留输入数据中的关键信息，使得在处理大规模数据时能够保持较高的准确性。

总的来说，三维稀疏卷积是一种针对稀疏数据的高效计算方法，通过利用输入数据的稀疏性来减少计算量，提高计算效率，同时保持较高的准确性。