原载 http://blog.sina.com.cn/s/blog_729a92140102wc7i.html 本短文用“实变换时”表明：限于Tpwp中，特别地假定被处理信号和滤波器冲激响应，都是实数序列。不管Tpwp之外的变换，因为使用DFT的条件和方法不一样。这里，不涉及人们从文献中可能见到的奇偶时间点序列分组、FIR以及时间延迟之类的问题，给定的一个被处理序列x和滤波器序列F（高通或低通），必须靠“移位方向参量为空值”另外指明它们都是频域区间 的结果仍等于v。 以博文《减少小波包变换的DFT算法模式中的DFT的次数和点数》（2015-10-26）为基础，减少了部分复数乘法，但是，新增了复共轭、向量倒序的处理内容，改变了那里的“频域数据重复”与频域的乘积运算的顺序。 用这两个函数子程序，覆盖掉以前用的旧子程序，重新运行博文《冗余小波包合成以及降噪试验和能量守恒定标》（2015-12-27）的最后三个降噪实验。可见在Scilab-5.3.3的命令窗中显示了相同的数字（format(‘v’,10) 格式）结果。但是，有些遗憾，没有见到实验的总耗时量明显减少，甚至反而可见高达百分之十几的增加，这可能与引入的其它处理以及系统工作平台中整体的解释、管理等有关。据此类经验，比较算法的性能时，居士希望尽量交代清楚方案、条件、环境。 新浪赛特居士SciteJushi-2016-05-13。 图片 1. 减少了紧凑DFT算法模式中实变换时的频域乘法的次数的程序段 附 . 最近，搜索小波的最新研究进展，在《百度百科》中，注意到了中国已有研究成果“比小波分析创始人Mallat提出的著名Mallat算法快一个数量级以上，而且效果和质量远优于Mallat算法”。看到这很振奋又好奇，然而还未搜到演示资料或方法细述，故特将其复制为留念于下。原来，还有了国际小波分析应用研究中心。虽然，姓名，是居士在博客中多次提到（因个人认为，翻译得已很不错）的小波名著“十讲”的译者，也是前面（2016-04-20）的实变换的滤波器构造中曾用过那篇中文参考文献的作者。但是，居士真不记得当年离开小波的研究时，对这个姓名已有印象，实际上，除去博文已明确提到的部分，至今仍几乎未知其具体研究（注意，在此无褒贬之意）。 ： 李建平（电子科技大学教授） 简介?编辑 1986年获重庆大学应用数学理学学士，1989年获西安交通大学软件工程、计算数学双硕士学位，1998年获重庆大学计算机科学理论工学博士学位，1999年至2001年到香港浸会大学做博士后研究，2002年至2003年到法国普罗旺斯大学做访问学者，2005年至2006年到加拿大圭尔夫大学、多伦多大学做访问学者. 教学情况 现为电子科技大学计算机科学与工程学院、示范性软件学院副院长，国际小波分析应用研究中心主任，国际学术进展 International Progress on Wavelet Active Media Technology and Information Processing主编，国际学术期刊International Journal of Wavelet Multiresolution and Information Processing副主编，兼任国家科学技术奖励评审委员、国家自然科学基金项目评审委员、中华人民共和国公安部技术顾问等十几个学术、社会职务，是2004年国际计算机学术大会、第三届国际小波分析及其应用学术大会（ICWAA）、第二届智能体媒介技术国际学术大会（ICAMT）主席，是国际上小波分析与信息处理研究领域十分活跃的知名专家。 科研情况 他在国际上独立提出并系统建立了“小波变换的加速方法”、“矢量积小波变换理论”、“基于小波分析的电子签名系统”等系列理论与方法，该理论比小波分析创始人 Mallat提出的著名Mallat算法快一个数量级以上，而且效果和质量远优于Mallat算法，这为小波理论在信号处理、信息分析等许多方面的应用提供了先进的算法，为小波分析的实时处理和产品化提供了理论基础，为小波分析开辟了广阔的应用前景。他在国际上首次提出“基于‘三大特征’（机器特征、文档特征、人体特征）的信息最安全传输的模型与方法”，为当今研究热点的网络与信息安全做出了重要贡献。他先后主持国家863高技术项目、国家自然科学基金项目等30项。他在国内外著名学术期刊上发表重要论文150篇，被国际三大检索机构SCI、EI、ISTP等检索收录论文48篇，出版学术专著15部，其中2部被多次修订重印。他主持研制的“小波指纹加密系统”、“分布式网络监控系统”等高技术产品产生了广泛的经济效益和社会影响
7th International School and Workshop on Time-Dependent Density-Functional Theory Dear Colleagues, Apologies for possible cross-posting. This is to announce the CECAM / Psi-k funded 7th International School and Workshop on Time-Dependent Density-Functional Theory: Prospects and Applications which will be held September 12th - September 23rd, 2016, at the beautiful Centro de Ciencias de Benasque Pedro Pascual (Benasque, Spain; http://benasque.org/ ) Please see http://www.benasque.org/2016tddft/ for details, and read more below! Summary: This School Workshop is the seventh of a very successful series that started in 2004. The positive response to this first event, also held in the “Centro de Ciencias de Benasque Pedro Pascual”, from August 28th to September 12th 2004, encouraged the organization of the sequels in the same place with an approximate periodicity of two years. The purpose has in all occasions been to (1) make a very intense introduction to both the theory, the practice, and the numerical implementation of time-dependent density-functional theory (TDDFT), mainly (but not exclusively) oriented to young scientists willing to initiate or strengthen their knowledge and skills on TDDFT, followed by (2) a workshop on the subject in which all the main aspects are to be covered by the leading experts. All the students of the school are expected to participate in the workshop, in order to learn about the state-of-the-art of the subject, after being exposed to the fundamentals. Since TDDFT is a rapidly evolving field of Science, the precise content of both school and workshop have changed over the years –although the format of the events has been largely unaltered. In all occasions there has been a very large number of applicants for the school, that has increased every edition to become more than 150. This is not only a testimony of the strong pulse of the scientific field itself, but also of the good quality of the school. Since we want to maximize the learning experience of the students via a close interaction with the teachers (and also due to the logistic limitations imposed by the hands-on tutorial), the participants of the school will not exceed 40. The total number of participants in the full School Workshop has been close to 100 in all occasions. It is worth mentioning that participants came from all over the world, making this series of schools and workshops a truly global event. Location/Timing: This event will take place at the “Centro de Ciencias de Benasque Pedro Pascual”, Benasque, Spain ( http://benasque.org/ ), from September 12th -- 23rd, 2016 (travel to Benasque on the 11th). Benasque is a beautiful town in the heart of the Pyrenees. The school will take place from September 12th to September 19th, while the workshop will be from September 20th to September 23rd. Participants The call for participation will be mainly directed to students and scientists specialized on computational physics, quantum chemistry and biophysics. We call for students willing to participate at the School (and attend the workshop), and for scientists willing to present a contributed talk or poster at the Workshop. We will limit the number of students to the school to 40 and participants to the workshop to less than 100, in order to ensure a maximum interaction between all the scientists participating. Attendance of graduate students and post-docs will be strongly encouraged through the inclusion of short contributed talks and a poster session. Furthermore, we will award to PhD students who present an outstanding poster short oral presentations. Applications/Support: All persons who wish to participate should fill out the application form at: http://benasque.org/2016tddft/nbsp ; In the comments section, please indicate if you wish to participate in the Summer Summer School or in the Workshop (or in both). For participants coming from the USA, please check the following address for support: http://www.mcc.uiuc.edu/ School only: As we have a very limited number of places for the school, students will be selected from among an open pool of applicants who have demonstrated a strong interest in computational sciences, applied to chemistry, physics, materials science and biology.
原载 http://blog.sina.com.cn/s/blog_729a92140101oxbx.html 在Matlab-R2011a中，用 =wavefun('bior2.2',10); subplot(2,1,1); plot(x,w); axis tight; subplot(2,1,2); plot(x,w1); axis tight; 看到国际标准 JPEG2000的小波的“连续时间函数”，像感觉到分形曲线的味道。至于得到的bior3.1、bior3.3、bior3.5的函数波形，如果是居士自己搞的，那么很可能自认为它们是错误的。不过，尖叫“Matlab牛”后，居士还是偏爱“基向量”，并以为，本短文对简化、理清文献中的连续时间小波包基函数、其自然排序、其频率排序等问题的长期缠绕，有重要意义。 无先验知识以明确被分析数据与尺度函数的关系，用Tpwp的“小波包基向量”，无需工具箱生成函数（Building Wavelet Packets, Wavelet Packet Atoms）,即可方便地确定小波包变换树上任意位置的频率中心(功率或能量谱中心，区别正频率、负频率和直流)。 常用三个索引指标，尺度、频率、位置，标示一个小波包。但是，频率指标值越大，不同于1兆赫兹大于1千赫兹这样的事情，也不必意味着相应的小包中的高频能量（功率）就更大、或在低频时更小。实际上，尺度指标，常与水平（level）、深度（depth）混用，也并不直接就是小波函数概念中的“尺度值”。频率指标是，把“基本 小波和尺度函数 (或序列) ”以适当方式“运算、组合”而 形成某个“波的包裹” 的过程的一种编码。在离散变换中，可以简明地看到“高通、低通”滤波器的组合过程，但是注意，“下抽样”使频谱搬移了、使高低频率的位置转换了！ 普通离散小波分解中，第一级分解后，得到的“细节系数序列”中的“直流部分”，对应原信号中的“高频成分”。 在某个尺度上，如果它的节点数（频率指标的个数）不超过被处理序列的长度N的一半，那么简单用功率谱的中心、波形过零率等物理直观真实的正频率概念，区别小波包，是自然的。 然而，在最大的尺度上，节点数（频率指标的个数）等于N，那么这些普通的“频率分辨观念”就可能显得不方便。例如，实数序列的离散傅里叶变换DFT中正频率部分（0，fs/2）中,只有N/2-1点，难道，小波包分解能分辨出约N个“正”频率点吗？ 这时，小波包基向量，存在配对关系：幅度谱相同，则其定义的频率中心就相同，然而，相谱不同、有不相等的频率索引指标。这类似于DFT中的正负频率、共轭对称、幅相谱等的状况和问题。单个节点内，不再存在，可以用于分辨相位的“位置指标”。 如果遵从那些小波工具箱的处理，用滤波器和信号序列的长度，来限制分解深度，那么就遇不到这种状况。 在《随机设置小波包变换及其优选基的随机降秩矩阵》（2014-03-19）的基础上，居士做试验程序PwpRandNegf.sce，附于本文末。运行 clear ; LessLevel = 0 ; rand ( 'seed' , 1e9 ) ; exec ( 'PwpRandNegf.sce' ) ; 可得图片 1。 假设信号序列长度，随机地取为16、32、64、128，相应的最大分解深度分别为4、5、6、7。在最大分解尺度上，频率分辨率达到极限。抽出最大尺度上的所有小波包基向量，做两两比较，检验它们的FFT幅度谱的关系。 两基向量，其规范化幅度谱之间的误差，RMSE，er0，小于千之一，则视为是配对的。对于正交变换，附带地确认：两基向量为等腰三角形的直角边，即不可能是因为两向量本身相同所以幅度谱才相同，如若不然，程序报告错误。 从数据序列长度的二分之一中，减去(倒记数)相配的对数，得图片1.中的第一幅曲线图。结果始终为1，与滤波器、小波包变换的随机设置、序列长度都无关。 余下一对（故结果恒为1）：涉及直流成分，和高频向量。980次实验的直流基向量偏离理想值的误差(RMSE)的累加总和，TestDC = 0.000038，很小。高频向量，被归入某对的计数结果，TestHF = 0。 配对向量的幅度谱之间的实际误差er0的总和，示于图片1.中的第二幅曲线图。“总和”用的数值的数目，与序列长度有关，但结果仍都很小。 在一个稳定的变换中，尺度参数、频率参数都相同而只是位置参数不同的小波包基向量，它们也具有相同的幅度谱，但与图片1.反应的情形不同。例如，没有直流基向量；配对向量，经历的数字运算过程更相近，所以它们的幅度谱之间的数字误差更小。 运行 clear ; LessLevel = 1 ; rand ( 'seed' , 1e9 ) ; exec ( 'PwpRandNegf.sce' ) ; 可得图片 2，反应“次最大”尺度上的基向量的情况。无单纯的直流基向量，所以TestDC = 125.14948,很大；TestHF = 980，即，等于随机实验次数，所以高频向量也总是成对的。 由于，必须覆盖相同的物理正频率带，所以， LessLevel =0 时的向量对，与 LessLevel =1 时的向量对，不可能有实质上不同的正频率带宽度，然而频率中心须错位。这种错位，改变了正、负频率中心的距离，提供了一种频率信息“分辨能力”。在 LessLevel =0 时，配对向量，可来自前面的尺度上的不同频率指标块。为此转换问题，居士杜撰“负频率幽灵”一词，供有兴趣者参考。 程序的Matlab版的处理，约更快1倍。 附程序: PwpRandNegf-2014-03-27.zip // PwpRandNegf.sce // for the concept: wavelet packet basis vector // test the random settings of wavelet packet vectors // and the ghost/compare of negative frequency of DFT basis // in order to run with fixed seed, use one of following lines: // clear; LessLevel=0; rand('seed',1e9); exec('PwpRandNegf.sce'); // clear; LessLevel=1; rand('seed',1e9); exec('PwpRandNegf.sce'); // in order to run with variable seed, use one of following lines: // clear; LessLevel=0; exec('PwpRandNegf.sce'); // clear; LessLevel=1; exec('PwpRandNegf.sce'); // note: TpwpSubs.bin, TpwpSubs_E01.bin, MatOrtWlts.bin are // in Scilab current directory // reference: PwpRandPsbm.sce, uploaded, 2014-03-19 // in Scilab-5.3.3,Baiyu Tang( tang.baiyu@gmail.com ) // last revised,2014-Mar xdel(winsid()); // kill all figures mode(0); ieee(1); // clear; // LessLevel=0; // rand('seed',1e9); // ---start timing date1=getdate(); date1=date1( ); disp( ); tic(); load('TpwpSubs.bin'); // load function subroutines load('TpwpSubs_E01.bin'); // extended set of subroutines for PreSet Basis load('MatOrtWlts.bin'); // load the filter cases TestDC=0; // DC vector error, all, total TestHF=0; // count high frequency vectors in pairs TestCount=zeros(98,10); // count matrix TestError=zeros(98,10); // error matrix // ---compute for ii=1:98; // filter case index. 98 filter cases for jj=1:10; // random test index. 10 tests per filter-case // ---random length and decomposition depth of test signal Wp.DecL=4+round(3*rand(1,1,'uniform'));// maximal level, 4,5,6 or 7 Wp.DatL=2^Wp.DecL; // data length, 16,32,64 or 128 Wp.DecL=Wp.DecL-LessLevel; // less-level decomposition CIndx0= *(Wp.DecL+1); // coefficient index, Last-Level // ---random settings Set0=rand(1,9,'normal')*1e6; Wp.RightShift=Set0(1); Wp.ExchangeR_D=Set0(2); Wp.FlipFirst=Set0(3); Wp.MatchFilter=Set0(4); Wp.LowT_zero=fix(Set0(5)); // integer Wp.HighT_shift=fix(Set0(6)); // integer Wp.FlipTime=Set0(7); Wp.PeriodFirst=Set0(8); Wp.AlterL4fSign=Set0(9); // ---get filters if Wp.ExchangeR_D0 then =WltFilters(ii); else =WltFilters(ii); end if length(hcoef(:))2 then if length(rhcoef(:))1 then hcoef=rhcoef; rhcoef= =TpwpAllGH(hcoef,rhcoef,Wp.DatL,Wp.DecL,... Wp.MatchFilter,Wp.PeriodFirst,Wp.LowT_zero,Wp.HighT_shift,Wp.FlipTime); if Wp.AlterL4fSign0 then // forced sign. usually not used hlen=max(round(length(hcoef)/2),round(length(rhcoef)/2))/2; if fix(hlen)==hlen then pgcoef=-pgcoef; // rpgcoef=-rpgcoef; end end =PsbMatrix(CIndx0,Wp.DecL,Wp.DatL,phcoef,pgcoef,Wp.RightShift); // ---match, compare x1=tM0(1,:)-1/sqrt(Wp.DatL); // DC error, realization and idea TestDC=TestDC+sqrt(mean(x1.*x1)); // DC RMSE accumulation TestCount(ii,jj)=Wp.DatL/2; // half the length of vectors and signals tM0f=zeros(Wp.DatL,Wp.DatL); for k1=1:Wp.DatL; x0=fft(tM0(k1,:)); // fast Fourier Transform, DFT x1=abs(x0); // magnitude spectrum tM0f(k1,:)=x1/sqrt(sum(x1.*x1)); // normalized with power end for k1=1:(Wp.DatL-1); // check the pairs of base vectors k12=0; // count vectors in one group for k2=(k1+1):Wp.DatL; x1=tM0f(k1,:)-tM0f(k2,:); er0=sqrt(mean(x1.*x1)); // magnitude RMSE if er00.001 then // equal if ii81 then // orthogonal cases x1=tM0(k1,:)-tM0(k2,:); er1=sqrt(sum(x1.*x1)); // vector error, l_2 norm if er11.4 then // confirm difference, two disp( ); error('Strange Vector Pair.'); end end if (k1==(Wp.DatL/2+1))|(k2==(Wp.DatL/2+1)) then TestHF=TestHF+1; // count, high frequency end k12=k12+1; TestCount(ii,jj)=TestCount(ii,jj)-1; // count, pairs TestError(ii,jj)=TestError(ii,jj)+er0; // RMSE accumulation,total end end if k121 then disp( ); error('The group has too many vectors for PwpRandNegf.sce .'); end end end end // ---display TestDC, TestHF, subplot(2,1,1); plot2d('nn',TestCount, rect= ); set(gca(),'tight_limits','on'); set(gca(),'box','on'); set(gca(),'grid', ); xlabel('Filters: sym1-sym35,coif1-coif5,dmey,db4-db42,18 biorthogonal cases'); ylabel('Length by 2 minus Magnitude-Match Count'); AuLabel1= ; xstring(1,-4+%eps,AuLabel1); Note1= ; if LessLevel0 then title('PwpRandNegf.sce: One Level Less than Maximal Decomposition'); xstring(1,2,Note1( )); else title('PwpRandNegf.sce: Tpwp Transform Trees Hold Negative Frequency of DFT ?'); xstring(1,1.5,Note1( )); end subplot(2,1,2); plot2d('nl', TestError+%eps, rect= ); set(gca(),'tight_limits','on'); set(gca(),'box','on'); set(gca(),'grid', ); xlabel('Filters: sym1-sym35,coif1-coif5,dmey,db4-db42,18 biorthogonal cases'); ylabel('Error of Magnitude of Matched Pairs'); xstring(1,2e-16,AuLabel1); if LessLevel0 then title('PwpRandNegf.sce: One Level Less than Maximal Decomposition'); xstring(1,1e-5,Note1( )); else title('PwpRandNegf.sce: Tpwp Transform Trees Hold Negative Frequency of DFT ?'); xstring(1,1e-5,Note1( )); end // ---timing Time_In_second=toc(), // --clear; LessLevel=0; rand('seed',1e9); exec('PwpRandNegf.sce'); // !Date and Time: 2014 3 25 8 41 23 ...... ! // TestDC = // 0.0000380 // TestHF = // 0. // Time_In_second = // 209.516 // --clear; LessLevel=1; rand('seed',1e9); exec('PwpRandNegf.sce'); // !Date and Time: 2014 3 25 8 46 55 ...... ! // TestDC = // 125.14948 // TestHF = // 980. // Time_In_second = // 194.25 // ***** The Version in Matlab ***** // clear; LessLevel=0; rng(1e9); PwpRandNegf; // Date and Time: 2014 3 25 8 57 ...... // TestDC = // 4.6382e-005 // TestHF = // 0 // Time_In_second = // 84.647 // clear; LessLevel=1; rng(1e9); PwpRandNegf; // Date and Time: 2014 3 25 8 59 ...... // TestDC = // 124.64 // TestHF = // 980 // Time_In_second = // 81.178 新浪赛特居士SciteJushi-2014-03-27。 图片 1。用小波包基向量从变换树上捕捉DFT中负频率的影子 图片 2。用同样的处理检验在次最大尺度上小波包基向量
Source: http://web.mit.edu/newsoffice/2009/explained-fourier.html Larry Hardesty, MIT News Office Science and technology journalists pride themselves on the ability to explain complicated ideas in accessible ways, but there are some technical principles that we encounter so often in our reporting that paraphrasing them or writing around them begins to feel like missing a big part of the story. So in a new series of articles called Explained, MIT News Office staff will explain some of the core ideas in the areas they cover, as reference points for future reporting on MIT research. In 1811, Joseph Fourier, the 43-year-old prefect of the French district of Isère, entered a competition in heat research sponsored by the French Academy of Sciences. The paper he submitted described a novel analytical technique that we today call the Fourier transform, and it won the competition; but the prize jury declined to publish it, criticizing the sloppiness of Fourier’s reasoning. According to Jean-Pierre Kahane, a French mathematician and current member of the academy, as late as the early 1970s, Fourier’s name still didn’t turn up in the major French encyclopedia the Encyclopædia Universalis. Now, however, his name is everywhere. The Fourier transform is a way to decompose a signal into its constituent frequencies, and versions of it are used to generate and filter cell-phone and Wi-Fi transmissions, to compress audio, image, and video files so that they take up less bandwidth, and to solve differential equations, among other things. It’s so ubiquitous that “you don’t really study the Fourier transform for what it is,” says Laurent Demanet, an assistant professor of applied mathematics at MIT. “You take a class in signal processing, and there it is. You don’t have any choice.” The Fourier transform comes in three varieties: the plain old Fourier transform, the Fourier series, and the discrete Fourier transform. But it’s the discrete Fourier transform, or DFT, that accounts for the Fourier revival. In 1965, the computer scientists James Cooley and John Tukey described an algorithm called the fast Fourier transform, which made it much easier to calculate DFTs on a computer. All of a sudden, the DFT became a practical way to process digital signals. To get a sense of what the DFT does, consider an MP3 player plugged into a loudspeaker. The MP3 player sends the speaker audio information as fluctuations in the voltage of an electrical signal. Those fluctuations cause the speaker drum to vibrate, which in turn causes air particles to move, producing sound. An audio signal’s fluctuations over time can be depicted as a graph: the x-axis is time, and the y-axis is the voltage of the electrical signal, or perhaps the movement of the speaker drum or air particles. Either way, the signal ends up looking like an erratic wavelike squiggle. But when you listen to the sound produced from that squiggle, you can clearly distinguish all the instruments in a symphony orchestra, playing discrete notes at the same time. That’s because the erratic squiggle is, effectively, the sum of a number of much more regular squiggles, which represent different frequencies of sound. “Frequency” just means the rate at which air molecules go back and forth, or a voltage fluctuates, and it can be represented as the rate at which a regular squiggle goes up and down. When you add two frequencies together, the resulting squiggle goes up where both the component frequencies go up, goes down where they both go down, and does something in between where they’re going in different directions. The DFT does mathematically what the human ear does physically: decompose a signal into its component frequencies. Unlike the analog signal from, say, a record player, the digital signal from an MP3 player is just a series of numbers, each representing a point on a squiggle. Collect enough such points, and you produce a reasonable facsimile of a continuous signal: CD-quality digital audio recording, for instance, collects 44,100 samples a second. If you extract some number of consecutive values from a digital signal — 8, or 128, or 1,000 — the DFT represents them as the weighted sum of an equivalent number of frequencies. (“Weighted” just means that some of the frequencies count more than others toward the total.) The application of the DFT to wireless technologies is fairly straightforward: the ability to break a signal into its constituent frequencies lets cell-phone towers, for instance, disentangle transmissions from different users, allowing more of them to share the air. The application to data compression is less intuitive. But if you extract an eight-by-eight block of pixels from an image, each row or column is simply a sequence of eight numbers — like a digital signal with eight samples. The whole block can thus be represented as the weighted sum of 64 frequencies. If there’s little variation in color across the block, the weights of most of those frequencies will be zero or near zero. Throwing out the frequencies with low weights allows the block to be represented with fewer bits but little loss of fidelity. Demanet points out that the DFT has plenty of other applications, in areas like spectroscopy, magnetic resonance imaging, and quantum computing. But ultimately, he says, “It’s hard to explain what sort of impact Fourier’s had,” because the Fourier transform is such a fundamental concept that by now, “it’s part of the language.”
转自： http://cms.mpi.univie.ac.at/vasp/vasp/vdW_DF_functional_Langreth_Lundqvist_et_al.html vdW-DF functional of Langreth and Lundqvist et al.The vdW-DF proposed by Dion et al. is a non-local correlation functional that approximately accounts for dispersion interactions . In VASP the method is implemented using the algorithm of Roman-Perez and Soler which transforms the double real space integral to reciprocal space and reduces the computational effort. Several propsed versions of the method can be used : the original vdW-DF , vdW-DF with exchange functionals optimised for the correlation part , and the vdW-DF2 of Langreth and Lundqvist groups . N.B. : This feature has been implemented by J. Klimeš. If you make use of the vdW-DF functionals presented in this section, we ask that you cite the following paper: J. Klimeš, D. R. Bowler, and A. Michaelides, Phys. Rev. B 83 , 195131 (2011). Correlation functionals The method is invoked by setting LUSE_VDW = .TRUE. Moreover, the PBE correlation correction needs to be removed since only LDA correlation is used in the functionals. This is done by setting AGGAC = 0.0000 The two tags above need to be used for all the following functionals. Exchange functionals To use the different exchange functionals, the GGA tag needs to be set appropriately. The original version of Dion et al uses revPBE which can be set by GGA = RE More accurate exchange functionals for the vdW correlation functional have been proposed in and . They can be used by setting GGA = OR for optPBE, GGA = BO PARAM1 = 0.1833333333 PARAM2 = 0.2200000000 for the optB88 functional , or GGA = MK PARAM1 = 0.1234 PARAM2 = 1.0000 for the optB86b functional . For the vdW-DF2 functional the rPW86 exchange functional is used: GGA = ML moreover, the vdW functional needs to be changed to the vdW2 correlation which requires only a change of a parameter: Zab_vdW = -1.8867 An overview of the performance of the different approaches can be found for example in for gas phase clusters and in for solids. Important remarks : The method needs a precalculated kernel which is distributed via the VASP download portal ( VASP - src - vdw_kernel.bindat ) and on the ftp server ( vasp5/src/vdw_kernel.bindat ). If VASP does not find this file, the kernel will be calculated. This, however, is rather demanding calculation. The kernel needs to be either copied to the VASP run directory for each calculation or can be stored in a central location and read from there. The location needs to be set in routine PHI_GENERATE. This does not work on some clusters and the kernel needs to be copied into the run directory in such cases. Currently the evaluation of the vdW energy term is not done fully within the PAW method but the sum of the pseudo-valence density and partial core density is used. This approximation works rather well, as is discussed in , and the accuracy generally increases when the number of valence electrons is increased or when harder PAW datasets are used . For example, for adsorption it is recommended to compare the adsorption energy obtained with standard PAW datasets and more-electron POTCARs for both PBE calculation and vdW-DF calculation to asses the quality of the results. The optimisation of the cell (ISIF=3 and higher) is currently not possible. The spin polarised calculations are possible, but strictly speaking the non-local vdW correlation is not defined for spin-polarised systems. For spin-polarised calculation the non-local vdW correlation energy is evaluated on the sum of the spin-up and spin-down densities. The evaluation of the vdW energy requires some additional time. Most of it is spent on performing FFTs to evaluate the energy and potential. Thus the additional time is determined by the number of FFT grid points in the calculation, basically size of the simulation cell. It is almost independent on the number of the atoms in the cell. Thus the relative cost of the vdW-DF method depends on the ``filling" of the cell and increases with the amount of vacuum in the cell. The relative increase is high for isolated molecules in large cells, but small for solids in smaller cells with many k-points. This feature has been implemented by J. Klimeš. If you make use of the vdW-DF functionals presented in this section, we ask that you cite the following paper: J. Klimeš, D. R. Bowler, and A. Michaelides, Phys. Rev. B 83 , 195131 (2011). http://cms.mpi.univie.ac.at/vasp/vasp/DFT_D2_method_Grimme.html#tab:grimme DFT-D2 method of Grimme LVDW= .TRUE. | .FALSE. (Available as of VASP.5.2.11) Default: LVDW=.FALSE. Popular density functionals are unable to describe correctly van der Waals interactions resulting from dynamical correlations between fluctuating charge distributions. A pragmatic method to work around this problem has been given by the DFT-D approach , which consists in adding a semi-empirical dispersion potential to the conventional Kohn-Sham DFT energy: (90) In the DFT-D2 method of Grimme , the van der Waals interactions are described via a simple pair-wise force field, which is optimized for several popular DFT functionals. The dispersion energy for periodic systems is defined as (91) where the summations are over all atoms and all translations of the unit cell , the prime indicates that for , is a global scaling factor, denotes the dispersion coefficient for the atom pair , is a position vector of atom after performing translations of the unit cell along lattice vectors. In practice, terms corresponding to interactions over distances longer than a certain suitably chosen cutoff radius contribute only negligibly to and can be ignored. The term is a damping function (92) whose role is to scale the force field such as to minimize contributions from interactions within typical bonding distances. Combination rules for dispersion coefficients and vdW radii are (93) and (94) respectively. The global scaling parameter has been optimized for several different DFT functionals such as PBE ( ), BLYP ( ), and B3LYP ( ). The DFT-D2 method can be activated by setting LVDW=.TRUE. Optionally, the forcefield parameters can be controlled using the following flags (the default values are listed): VDW_RADIUS = 30.0 cutoff radius () for pair interactions VDW_SCALING = 0.75 global scaling factor VDW_D = 20.0 damping parameter VDW_C6 = ,... parameters ( ) for each species defined in POSCAR VDW_R0 = ,... parameters () for each species defined in POSCAR The default values for VDW_C6 and VDW_R0 are compiled in Tab. 2 . As the potential energy, interatomic forces as well as stress tensor are corrected by adding contribution from the forcefield, simulations such as the atomic and lattice relaxations, molecular dynamics, and vibrational analysis can be performed. The number of atomic pairs contributing to and the estimated vdW energy are written in OUTCAR (check lines following the expression 'Grimme's potential'). The forces and stresses written in OUTCAR contain the vdW correction but the corrected energy should be read from OSZICAR (energies in OUTCAR do not contain the vdW term). IMPORTANT NOTE: The defaults for VDW_C6 and VDW_R0 are defined only for the first five rows of periodic table of elements (see Tab. 2 ) - if the system contains other elements the user must provide the corresponding parameters. Table 2: Parameters used in the empirical force-field of Grimme . Element C R Element C R Jnm mol Jnm mol H 0.14 1.001 K 10.80 1.485 He 0.08 1.012 Ca 10.80 1.474 Li 1.61 0.825 Sc-Zn 10.80 1.562 Be 1.61 1.408 Ga 16.99 1.650 B 3.13 1.485 Ge 17.10 1.727 C 1.75 1.452 As 16.37 1.760 N 1.23 1.397 Se 12.64 1.771 O 0.70 1.342 Br 12.47 1.749 F 0.75 1.287 Kr 12.01 1.727 Ne 0.63 1.243 Rb 24.67 1.628 Na 5.71 1.144 Sr 24.67 1.606 Mg 5.71 1.364 Y-Cd 24.67 1.639 Al 10.79 1.716 In 37.32 1.672 Si 9.23 1.716 Sn 38.71 1.804 P 7.84 1.705 Sb 38.44 1.881 S 5.57 1.683 Te 31.74 1.892 Cl 5.07 1.639 I 31.50 1.892 Ar 4.61 1.595 Xe 29.99 1.881
APS的观点评论最近刊出了一篇关于 DFT 计算的viewpoint，作者是 Neepa T. Maitra ， Department of Physics and Astronomy, Hunter College and the City University of New York, 695 Park Avenue, New York, NY 10065, USA。 DFT （density-functional theory）方法是计算材料学和量子化学中计算电子结构的一种重要方法。它最初于1964年被提出，已经被应用了许多年，相关的文献和研究成果恐怕已经无法计数 。而 TD-DFT (time dependentdensity-functional theory)是DFT的扩展，主要用于处理非平衡态的电子与受外场束缚的电子 ，但是什么才是正确的方法通过“弯曲”有吸引力的原子核位势去描述不相互作用的电子（或者用文中生动的方法，叫做编排电子的舞蹈），这些电子恰恰又正好包含了真实的时间相关系统的准确信息。这个问题困扰了很久。然而限制看来，除了少数双电子系统由于计算水平限制之外，其他大部分系统的问题恐怕都可以得到解决。最近的 Phys. Rev. Lett 上，NYU的 James Ramsden 等人提出了一种新的算法来解决这个问题，其实也不是很陌生，它就是含时的Kohn-Sham potential，他们已经将此种方法应用到含有运动电子的半导体材料中 Exact Density-Functional Potentials for Time-Dependent Quasiparticles.pdf ，之前TDDFT忽略了一些 Kohn-Sham potential中重要的细节。 原文链接： http://physics.aps.org/articles/v5/79 References: P. Hohenberg and W. Kohn, “Inhomogeneous Electron Gas,” Phys. Rev.136, B864 (1964) . W. Kohn and L. J. Sham, “Self-Consistent Equations Including Exchange and Correlation Effects,” Phys. Rev.140, A1133 (1965) . E. Runge and E. K. U. Gross, “Density-Functional Theory for Time-Dependent Systems,” Phys. Rev. Lett.52, 997 (1984) . Fundamentals of Time-Dependent Density Functional Theory, edited by M. A. L. Marques, N. T. Maitra, F. Nogueira, E. K. U. Gross, and A. Rubio, Lecture Notes in Physics Vol. 837 (Springer, Berlin, 2012) . J. D. Ramsden and R. W. Godby, ”Exact Density-Functional Potentials for Time-Dependent Quasiparticles,” Phys. Rev. Lett.109, 036402 (2012) .
请先看： 胡新根：你相信金属锌（Zn）有+3价氧化态吗？ 卤素，F, Cl, Br, I具有强烈的吸电子能力。其吸电子能力可以通过计算它们的电子亲和能得到，即X吸收一个电子形成X-所需的能量。Cl具有卤素中最高的电子亲和能，3.6eV。但是，你能不能设计一个具有更高电子亲和能的物质？这种物质对工业中所需要的强氧化过程非常有用。 可以。BO2和AuF6就是这样的具有更高电子亲和能的物质，称为superhalogen，“高级卤素”，其形成BO2(-)和AuF6(-)阴离子的电子亲和能分别为4.5和8.4。 还能不能找到更高电子亲和能的物质？能，通过把F, BO2， 和AuF6这样的superhalogen和过渡金属阳离子结合形成配合物，它们的电子亲和能会更高，称为hyperhalogen，“超级卤素”。这就是弗吉尼亚理工的Puru Jena用电子亲和能玩出来的概念。他和他的研究生Devleena Samanta不断设计新的具有高电子亲和能的各种配体。 设计能够稳定存在的“超级卤素”的技巧，关键是要让“高级卤素”和极度缺电子的过渡金属阳离子结合，而且金属阳离子氧化态越高越好。过渡金属由于d轨道不满，容易形成多种高配位数的配合物。 他们最近发表在JACS上的一篇文章，“Zn的+3价氧化态”，不仅设计出了电子亲和能高达9.4的超级卤素Zn(AuF6)3，而且理论上预测了一种稳定存在的非常不寻常的Zn的氧化态，+3（作者强调为氧化态，或者配位态，而非阳离子价态）。我们知道Zn的电子组态时3d10,4s2，因此，一般只有+2价；但是，既然同族的Hg被证实可以有+4价，为什么Zn不可以有+3价？（Hg的+4价也是理论预测存在，并在长达20年后被实验证实存在的。）他们在这篇文章中，通过纯的DFT理论计算，预测了超级卤素Zn(AuF6)3的稳定存在。具体是不是真的存在。。。who knows? 纯理论计算发JACS还是比较难的，尤其是纯的B3LYP/6-311+g*，除非这个计算背后的理论或者概念足够重要。 不过，我认为这篇文章的对Zn的+3价的确定并非无懈可击，甚至可以说有重大缺陷，令人很难信服。这篇预测一种新型Zn氧化态配合物的文章背后涉及两个基本的计算理论问题：第一，怎么证明你设计的，或者通俗说你用GaussView搭出来的一堆原子组成的化合物真的能够稳定存在，尤其是一个从来没有人观测到过的Zn(III)价态？第二，你怎么确定你的那堆配合物中Zn真的是Zn(III)氧化态？因为对于非离子型化合物，确定原子化合价是比较困难的。 对于第一个问题，即怎么验证一个假想中的化合物能在实际中稳定存在，作者的回答还是比较好的。确定假想化合物热力学能够稳定存在，需要分析这个假想化合物各种不同的可能组成形式(构象)，和各种不同的可能的分解形式，然后说明哪种状态下能量最低：是处于化合态时的能量低，还是处于解离态能量低。比如， H3明显是热力学不稳定的，因为分别优化计算H3, H2和H得到其能量，可知H3--H2 + H反应是放热的。因此H3只是假象中的化合物，实际中即使短时间碰到一起也会立即分解。 （注：有些化合物是动力学稳定的，即它虽然是热力学不稳定的高能态，但解离途径存在能垒，导致化合物可以短暂呆在这个状态；此文说的稳定是指热力学稳定，即只考虑产物态势能，而不考虑过渡态的势能） 因此，在这篇文章中，作者对中性ZnX3的各种解离产物进行结构优化，并比较各种解离路线是放热还是吸热的。如果所有解离路线都是吸热的，那么OK，说明ZnX3是能够稳定存在的。这对于ZnF3不难，但是对于Zn(AuF6)3就不容易了，要考虑Zn(AuF6)2 + AuF4 + F2这样的路线，和ZnF2 + Au2F10 + AuF4 + F2 这样的路线，等等。文章中称这种解离能为片段化能Fragmentation energies. 文章附件对各种体系罗列了总共数十种可能的解离方式，十分细致。他们发现，ZnF3不稳定，分解成ZnF2和(1/2)F2单质要放热；而中性Zn(BO2)3虽然能够稳定存在，但其中Zn是+2价的。只有Zn(AuF6)3这个东东可以稳定存在且其中Zn为3配位的。 对于第二个问题，即他们怎么确定配合物Zn(AuF6)中Zn的氧化价态，作者给出的理由并不充分，关键处含混其词。基本上来说，他们对优化好的结构做了NBO布局分析何轨道占据分析，然而却又指出NBO分析并不能确定无误地给出氧化价态的结果，最后给出了一个自己的判别条件，认为只要3个配体在优化中没有塌缩到一起的倾向，就能称之为3价氧化态存在。这实际上还是从几何结构稳定性上分析，认为3配位等于3价氧化态。不过，这3个配体却并非C3对称，而是C2对称，即有两个配体更接近一些。对于这一点，作者甚至回避了解释。并总结说明，预测了一个“配体不会聚集”的化合物，以强调其3配位态的稳定性。但是，3配位态等于3价氧化态吗？这个化合物中是真的存在3根配位键，还是2根配位键加一个很弱的分子间作用？我认为这是本文暗含的一个逻辑缺陷。 结论是，这是一篇值得看看的文章。如果有兴趣，可以接着讨论如何确定其中Zn的氧化态到底是否Zn(III)；或者，沿着作者的思路，看看能不能和其它过渡态金属结合，预测新的，有更高电子亲和能的配合物。 参考文献： （1）JACS原文: Zn in the +3 Oxidation State DOI: 10.1021/ja3029119 （2）Phys.org 报道1: Enhancing the chemistry of zinc （3）Phys.org 报道2: Researchers discover a new class of highly electronegative chemical species （3）Angewandte通信: Hyperhalogens: Discovery of a New Class of Highly Electronegative Species 后续：Zn真的有Zn(III)价的氧化态吗？--JACS刊登反驳文章 http://bbs.sciencenet.cn/home.php?mod=spaceuid=38740do=blogquickforward=1id=590937
我们大家知道，傅立叶变换是我们很多科学领域的重要的数学工具。有人说没有傅立叶变换就没有我们现代科学一些新分支或者很多学科就可能不太完善，我觉得这句话一点没有错。我们在地球物理的信号，无线电信号，现代的通讯技术、光学、声学甚至音乐中都大量应用傅立叶变换，尤其是离散的傅立叶变换。美国麻省理工学院（MIT）最近在她的网站主页上用非数学的语言给普通民众大谈离散傅立叶变换（DFT），这篇文章很通俗，推荐给大家共享，希望大家多多讨论。 网站链接： http://web.mit.edu/newsoffice/2009/explained-fourier.html Explained: The Discrete Fourier Transform The theories of an early-19th-century French mathematician have emerged from obscurity to become part of the basic language of engineering. Larry Hardesty, MIT News Office November 25, 2009 Science and technology journalists pride themselves on the ability to explain complicated ideas in accessible ways, but there are some technical principles that we encounter so often in our reporting that paraphrasing them or writing around them begins to feel like missing a big part of the story. So in a new series of articles called Explained, MIT News Office staff will explain some of the core ideas in the areas they cover, as reference points for future reporting on MIT research. In 1811, Joseph Fourier, the 43-year-old prefect of the French district of Isre, entered a competition in heat research sponsored by the French Academy of Sciences. The paper he submitted described a novel analytical technique that we today call the Fourier transform, and it won the competition; but the prize jury declined to publish it, criticizing the sloppiness of Fouriers reasoning. According to Jean-Pierre Kahane, a French mathematician and current member of the academy, as late as the early 1970s, Fouriers name still didnt turn up in the major French encyclopedia the Encyclopdia Universalis. Now, however, his name is everywhere. The Fourier transform is a way to decompose a signal into its constituent frequencies, and versions of it are used to generate and filter cell-phone and Wi-Fi transmissions, to compress audio, image, and video files so that they take up less bandwidth, and to solve differential equations, among other things. Its so ubiquitous that you dont really study the Fourier transform for what it is, says Laurent Demanet, an assistant professor of applied mathematics at MIT. You take a class in signal processing, and there it is. You dont have any choice. The Fourier transform comes in three varieties: the plain old Fourier transform, the Fourier series, and the discrete Fourier transform. But its the discrete Fourier transform, or DFT, that accounts for the Fourier revival. In 1965, the computer scientists James Cooley and John Tukey described an algorithm called the fast Fourier transform, which made it much easier to calculate DFTs on a computer. All of a sudden, the DFT became a practical way to process digital signals. Summing together only three discrete frequencies can produce a much more erratic composite. The Fourier transform provides a way to decompose signals into their constituent frequencies. To get a sense of what the DFT does, consider an MP3 player plugged into a loudspeaker. The MP3 player sends the speaker audio information as fluctuations in the voltage of an electrical signal. Those fluctuations cause the speaker drum to vibrate, which in turn causes air particles to move, producing sound. An audio signals fluctuations over time can be depicted as a graph: the x-axis is time, and the y-axis is the voltage of the electrical signal, or perhaps the movement of the speaker drum or air particles. Either way, the signal ends up looking like an erratic wavelike squiggle. But when you listen to the sound produced from that squiggle, you can clearly distinguish all the instruments in a symphony orchestra, playing discrete notes at the same time. Thats because the erratic squiggle is, effectively, the sum of a number of much more regular squiggles, which represent different frequencies of sound. Frequency just means the rate at which air molecules go back and forth, or a voltage fluctuates, and it can be represented as the rate at which a regular squiggle goes up and down. When you add two frequencies together, the resulting squiggle goes up where both the component frequencies go up, goes down where they both go down, and does something in between where theyre going in different directions. The DFT does mathematically what the human ear does physically: decompose a signal into its component frequencies. Unlike the analog signal from, say, a record player, the digital signal from an MP3 player is just a series of numbers, representing very short samples of a real-world sound: CD-quality digital audio recording, for instance, collects 44,100 samples a second. If you extract some number of consecutive values from a digital signal 8, or 128, or 1,000 the DFT represents them as the weighted sum of an equivalent number of frequencies. (Weighted just means that some of the frequencies count more than others toward the total.) The application of the DFT to wireless technologies is fairly straightforward: the ability to break a signal into its constituent frequencies lets cell-phone towers, for instance, disentangle transmissions from different users, allowing more of them to share the air. The application to data compression is less intuitive. But if you extract an eight-by-eight block of pixels from an image, each row or column is simply a sequence of eight numbers like a digital signal with eight samples. The whole block can thus be represented as the weighted sum of 64 frequencies. If theres little variation in color across the block, the weights of most of those frequencies will be zero or near zero. Throwing out the frequencies with low weights allows the block to be represented with fewer bits but little loss of fidelity. Demanet points out that the DFT has plenty of other applications, in areas like spectroscopy, magnetic resonance imaging, and quantum computing. But ultimately, he says, Its hard to explain what sort of impact Fouriers had, because the Fourier transform is such a fundamental concept that by now, its part of the language.
标签: DFT
