Wannier function及其应用
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关注：1） &Wannier function的构造及其物理含义2) & Wannier function的应用： GW计算能带结构绘制& Thanks to Xiegang Zhu&认识1：第一性原理计算是为了获得wannier函数，有了wannier函数后就可以不需要第一性原理，直接通过wannier函数得到电子结构，尤其是表面态信息。认识2： num_wann，num_band, Projections参数设置认识3：画能带时高对称性k点位置的确定，可在wannier90_band.gnu文件中找到& vi wannier90_band.gnu set data style dotsset nokeyset xrange [0: 5.73583]set yrange [-49.7533]set arrow from &0.51430, -49.75490 to &0.5533 noheadset arrow from &1.76040, -49.75490 to &1.7533 noheadset arrow from &2.27470, -49.75490 to &2.2533 noheadset arrow from &3.52080, -49.75490 to &3.5533 noheadset arrow from &4.59941, -49.75490 to &4.5533 noheadset arrow from &5.11371, -49.75490 to &5.1533 noheadset xtics (& G & &0.00000,& A & &0.51430,& H & &1.76040,& K & &2.27470,& G & &3.52080,& M & &4.59941,& L & &5.11371,& H & &5.73583) plot &wannier90_band.dat&&&#num_wann = & &64 &! set to NBANDS by VASPnum_wann=8
& &【nm_bands数量必须大于num_wann数量，从计算的64条能带中选取8条(num_bands)，对这8条轨道采用wannier函数投影（wannier函数是实空间的函数）；exclude_bands不是必须的；Nbands必须大于num_wann，否则会存在误投影情形】num_bands=8 # for GW uncomment exclude_bands 9-64 Begin Projections & Si:sp3 End Projections&【 这是初始猜测轨道，必须有；对Si来说，1个硅sp3的共4条，两个硅原子8条；对于ScH3，Sc以spd计算，则一个Sc原子需1+3+5=9条轨道，一个氢原子1条轨道，所以一个分子式的ScH3需要9+3=12条轨道，一个单胞有两个分子式，则需要24条轨道，因此，num_wann=24,num_bands=30 (大于24即可)；如下方式设置投影&Begin ProjectionsSc:l=0;l=1;l=2H: l=-0End Projections】& dis_froz_max=9 dis_num_iter=1000 guiding_centres=true&&&网络摘录1：&0.1 下载wannier90-1.2，并编译安装&在 http://www.wannier.org/download.html 下载wannier90-1.2&修改make.sys如果装了ifort编译器LIBS = -lmkl_intel_lp64 &-lmkl_sequential -lmkl_core然后 make all&0.2 重新编译VASP&之后修改vasp LIB & & = -L../vasp.5.lib -ldmy &\ & & & ../vasp.5.lib/linpack_double.o \ & & & ../wannier90-1.2/libwannier.a $(SCA) $(LAPACK) $(BLAS) CPP & &= $(CPP_) -DMPI &-DHOST=\&LinuxIFC\& -DIFC \ & & &-DCACHE_SIZE=4000 -DPGF90 -Davoidalloc &\ & & &-DMPI_BLOCK=8000 -Duse_collective -DscaLAPACK -DNGZhalf -DVASP2WANNIER90之后重新make分五步计算&&1. step1：静态计算& INCAR ISMEAR = &0 SIGMA &= &0.05&2.得到虚轨道（导带上方的空轨道） INCAR ISMEAR = &0 SIGMA &= &0.05 ALGO = Exact NBANDS &= 64 LOPTICS=.TRUE. NEDOS=2000&3. GW计算 INCAR ALGO = GW0 ; LSPECTRAL = .TRUE. ; NOMEGA = 50 NBANDS=64 LRPA = .FALSE. LWANNIER90=.TRUE.&前三步运行vasp &【第三步运行VASP前，需写好wannier90.win参数（可先不写wannier90.win文件，试运行在第三步运行VASP，自动产生符合POSCAR和KPOINTS的wannier90.win文件，再修改该文件，运行一遍VASP），以使得运行VASP后产生 wannier90.amn, wannier90.mmn, wannier90.eig三个文件，供后续第四步的wannier90.x程序调用】&4. step4：运行 wannier90.x wannier90 wannier90.win &【第三步运行完成之后得到的参数文件】 num_wann=8 & & 【参数含义？】 num_bands=8 # for GW uncomment exclude_bands 9-64 Begin Projections & 【开始轨道投影？】 Si:sp3 End Projections dis_froz_max=9 dis_num_iter=1000 guiding_centres=true begin unit_cell_cart & & &2.7150000 & & 2.7150000 & & 0.0000000 & & &0.0000000 & & 2.7150000 & & 2.7150000 & & &2.7150000 & & 0.0000000 & & 2.7150000 end unit_cell_cart begin atoms_cart Si & & & 0.0000000 & & 0.0000000 & & 0.0000000 Si & & & 1.3575000 & & 1.3575000 & & 1.3575000 end atoms_cart mp_grid = & & 4 & & 4 & & 4 begin kpoints & & &0.0000000 & & 0.0000000 & & 0.0000000 & & &0.2500000 & & 0.0000000 & & 0.0000000 & & &0.5000000 & & 0.0000000 & & 0.0000000 & &........ & & &0.2500000 & &-0.5000000 & &-0.2500000 & & -0.2500000 & & 0.2500000 & &-0.5000000 end kpoints运行 wannier90.x wannier90&5.---step5:修改wannier90.win参数，再次运行 wannier90.x wannier90；能带即为wannier90_band.dat num_wann=8 num_bands=8 # for GW uncomment exclude_bands 9-64 Begin Projections Si:sp3 End Projections dis_froz_max=9 dis_num_iter=1000 guiding_centres=true # Bandstructure plot restart & & & & = &plot bands_plot & & &= &true begin kpoint_path L 0.500 0.5000 G 0.000 0.0000 G 0.000 0.0000 X 0.500 0.5000 X 0.500 0.5000 K 0.300 0.0000 K 0.300 0.0000 G 0.000 0.0000 end kpoint_path bands_num_points 40 bands_plot_format gnuplot xmgrace begin unit_cell_cart & & &2.7150000 & & 2.7150000 & & 0.0000000 & & &0.0000000 & & 2.7150000 & & 2.7150000 & & &2.7150000 & & 0.0000000 & & 2.7150000 end unit_cell_cart begin atoms_cart Si & & & 0.0000000 & & 0.0000000 & & 0.0000000 Si & & & 1.3575000 & & 1.3575000 & & 1.3575000 end atoms_cart mp_grid = & & 4 & & 4 & & 4 begin kpoints & & &0.0000000 & & 0.0000000 & & 0.0000000 & & &0.2500000 & & 0.0000000 & & 0.0000000 & & &0.5000000 & & 0.0000000 & & 0.0000000 & & &0.2500000 & & 0.2500000 & & 0.0000000 & & &0.5000000 & & 0.2500000 & & 0.0000000 & &.......... & & &0.2500000 & &-0.5000000 & &-0.2500000 & & -0.2500000 & & 0.2500000 & &-0.5000000 end kpoints再次运行 wannier90.x wannier90；能带即为wannier90_band.dat如果你有gnuplot，可以直接执行wannier90_band.gnu。&NBANDS的确定原则 & & Furthermore the method selected using &LOPTICS=.TRUE. requires an appreciable number of empty conduction band states. Reasonable results are usually only obtained, if the parameter NBANDS is roughly doubled or tripled in &the INCAR file with respect to the VASP default.【Default:&&NBANDS=NELECT/2 + NIONS/2 (non-spinpolarized)&=0.6*NELECT + NMAG &(spin-polarized)】
& & Furthermore it is emphasized that the routine works properly even for Hartree-Fock and screened exchange &type calculations and hybrid functionals. In this case, finite differences are used to determine the derivatives of the Hamiltonian with respect to .
& Note that the number of frequency grid points is determined by the parameter NEDOS (see Sec. ). In many cases it is desirable to increase this parameter significantly from its default value. Values around 2000 are strongly recommended. & & & Wannier90.x 函数关键参数介绍&&&integer :: num_wannNumber of WF to be found.No default.integer :: num_bandsTotal number of bands passed to the code in the seedname.mmn file.Default num_bands=num_wann&Projection & [轨道投影需事先在xx.win文件中写好，何为轨道投影，投影后的能带与不投影的能带有何不同？]The projections block defines a set of localised functions used to generate an initial guess for the unitary transformations. This data will be written in the seedname.nnkp file to be used by a first-principlescode.begin projections..end projectionsIf guiding_centres=true, then the projection centres are used as the guiding centres in the Wannierisation routine.For details see Section 3.1.&integer :: exclude_bands(:)A k-point independent list of states to excluded from the calculation of
for example to select only valence states, or ignore semi-core states. This keyword is passed to the first-principles code via the seedname.nnkp file. For example, to exclude bands 2, 6, 7, 8 and 12:exclude_bands : 2, 6-8, 12&logical :: guiding_centresUse guiding centres during the minimisation, in order to avoid local minima.The default value is false.&character(len=20) :: restart & & &If restart is present the code will attempt to restart the calculation from the seedname.chk file.The value of the parameter determines the position of the restartThe valid options for this parameter are:– default. Restart from the point at which the check file seedname.chk was written– wannierise. Restart from the beginning of the wannierise routine– plot. Go directly to the plotting phase– transport. Go directly to the transport routines&real(kind=dp) :: dis_froz_minThe lower bound of the inner energy window for the disentanglement procedure. Units are eV.If dis_froz_max is given, then the default for dis_froz_min is dis_win_min.&real(kind=dp) :: dis_froz_maxThe upper bound of the inner (frozen) energy window for the disentanglement procedure. If dis_froz_maxis not specified, then there are no frozen states. Units are eV.No default.&integer :: dis_num_iterIn the disentanglement procedure, the number of iterations used to extract the most connected subspace.The default value is 200.&Specification of projections in seedname.win[轨道投影需事先在xx.win文件中写好，何为轨道投影，投影后的能带与不投影的能带有何不同？] &【即是所谓的特征能带？】Here we describe the projection functions used to construct the initial guess 【A(k)mn 】for the unitary transformations.Each projection is associated with a site and an angular momentum state defining the projectionfunction. Optionally, one may define, for each projection, the spatial orientation, the radial part, thediffusivity, and the volume over which real-space overlaps Amn are calculated.The code is able to1. project onto s,p,d and f angular momentum states, plus the hybrids sp, sp2, sp3, sp3d, sp3d2.2. control the radial part of the projection functions to allow higher angular momentum states, e.g.,both 3s and 4s in silicon.We use the following format to specify projections in &seedname&.win:Begin Projections[units]site:ang_mtm:zaxis:xaxis:radial:zona...End Projections&Notes:轨道投影参数指定：&& Wannier函数自带算例To run the examples, follow the instructions in theWannier90 Tutorial, which may be found in the 'doc'directory of the Wannier90 distribution.The examples are as follows:Examples with A and M matrices provided---------------------------------------example01: Gallium Arsenide, valence bandsexample02: Lead, 4 Fermi surfaceexample03: Silicon, 4 valence bands + 4 & & & & &interpolated bandstructureexample04: Copper, states around the F Fermi surface&Examples that use the pw2wannier90 interface with PWSCF-------------------------------------------------------example05: Diamond, valence statesexample06: Copper, states around the F Fermi surfaceexample07: Silane, valence statesexample08: Iron, states around the Fermi levelexample09: Barium Titanateexample10: Graphiteexample11: Siliconexample12: Benzene (gamma-only branch of algorithms)example13: (5,5) Carbon Nanotube (transport)example14: Linear chain of Na atoms (LCR transport)example15: (5,0) Carbon Nanotube (LCR transport)&&& wannier函数方法论： & & wannier90 computes maximally-localised Wannier functions (MLWF) following the method of Marzari and Vanderbilt (MV) [1].
& & & For entangled energy bands, the method of Souza, Marzari and Vanderbilt(SMV) [2] is used.
& &We introduce briefly the methods and key definitions here, but full details can be found in the original papers and in Ref. [3]. & &First-principles codes typically solve the electronic structure of periodic materials in terms of Bloch states, .nk. These extended states are characterised by a band index n and crystal momentum k.【not bond index】An alternative representation can be given in terms of spatially localised functions known as Wannierfunctions (WF). The WF centred on a lattice site R, Wn*R(r), is written in terms of the set of Blochstates as&wannier90 requires two ingredients from an initial electronic structure calculation.&&&&&&轨道投影需事先在xx.win文件中写好，何为轨道投影，投影后的能带与不投影的能带有何不同？&&&&&&VASPwiki介绍&LWANNIER90 = .TRUE. | .FALSE. Default: LWANNIER90 = .FALSE. &Description: LWANNIER90=.TRUE. switches on the interface between VASP and . N.B.: At present the VASP2WANNIER90 interface works with WANNIER90 v1.2, not with WANNIER90 v2.0 (yet). & For LWANNIER90=.TRUE., VASP will run wannier_setup in library mode (see Chapter 6 of the ). The VASP2WANNIER90 interface will write the following files that WANNIER90 needs as input: 1. If this file does not exist, VASP will create it and write the following information onto it: num_wann = begin unit_cell_cart &... ... ... &... ... ... &... ... ...end unit_cell_cart&begin atoms_cart & ... ... ... & ... ... ... & ... ... ... & ... ... ... &end atoms_cart&mp_grid = .. .. ..&begin kpoints & ... ... ... & ... ... ... & ... ... ... & ... ... ...end kpoints&Where the unit_cell_cart, atoms_cart, and kpoints blocks, and mp_grid array, will be set in accordance with the setup of the VASP calculation (i.e., basically the information from the
files).&&&For the meaning of these tags and blocks please refer to the . &&2. If the
file already exists, VASP will only add the the aforementioned information in so far it is not already present. This means that VASP will check, for instance, whether or not the
file contains a kpoints-block, and add one if not. If it finds a kpoints-block, however, it will not check whether this block tallies [符合]with the present set of k-points used in the VASP calculation! &&3. The user may create a
file prior to running VASP and executing the VASP2WANNIER90 interface, and specify any tag and/or block that is understood by wannier_setup and/or wannier_run.
& The most common example of this is probably the projections-block that specifies the initial guess for the maximally localized Wannier functions (see Chapter 3 of the ). If a projections-block has been specified in the
file, VASP writes the projections of the Bloch functions onto the relevant projectors to the
file (see Chapter 3 of the ). If =.TRUE. is set in the
file, VASP writes the cell-periodic part of the wave functions in spin-channel s at k-point p to the file . &&– lead.win The master input file– lead.mmn The overlap matrices M(k;b)– lead.amn Projection A(k) of the Bloch states onto a set of trial localised orbitals– lead.eig The Bloch eigenvalues at each k-point. For interpolation only&Mind: VASP needs to be compiled with the following additional precompiler flag: -DVASP2WANNIER90and the variable LIB in the
must contain an entry that points to libwannier.a. For instance: LIB & & = -L../vasp.5.lib -ldmy &\ & & ../vasp.5.lib/linpack_double.o $(SCA) $(LAPACK) $(BLAS)might be changed to LIB & & = -L../vasp.5.lib -ldmy &\ & & ../vasp.5.lib/linpack_double.o ../wannier90-1.2/libwannier.a $(SCA) $(LAPACK) $(BLAS)depending on where you have installed WANNIER90, obviously. , , , &&网络摘录：瓦尼尔函数（Wannier function）：　　瓦尼尔函数是比较定域化的一种函数。能带电子的Bloch函数可以写成瓦尼尔函数之和的形式[即瓦尼尔函数的Bloch和]。在紧束缚近似成立的情况下，瓦尼尔函数就是各个格点上孤立原子的波函数，它在讨论电子空间局域性起重要作用的问题时比较有用 & &电子结构理论里的Wannier函数：简单说来，Wannier函数是一套完整的正交函数，在理论计算里，可以以它为基础计算固体物理的电子结构、键等性质。要理解它，需要理论知识。& & & 何力新，教授。 1994年毕业于中国科技大学物理系， 并于1997年获中国科技大学硕士学位。1998 - 2003年在美国Rutgers 物理系跟随David Vanderbilt教授学习第一性原理的计算方法，并获博士学位。6.1在美国国家再生能源实验室Dr. Alex Zunger领导的固体理论小组从事半导体量子点的理论研究工作。2006年1月作为国外杰出人才（百人计划）引进至中国科技大学中科院量子信息重点实验室工作。 截至2006年3月止， 共发表论文13篇，其中以第一作者发表PRL 3 篇，PRB 7 篇 （其一为 Rapid Comm.）。共被他人引用160余次。 & & &主要工作有：1. 研究了中局域化的性质。首次发现并证明了在一维晶格中， Wannier函数和密度矩阵的一般渐进形式为幂衰减与指数衰减之积；⒉预言了InAs/InSb 量子点中存在自发形成的电子-空穴对（即强关联的激子基态）。此外的工作还包括介电，铁电材料性质的研究；耦合量子点电子结构和纠缠态的研究；等等。
& & 主要研究兴趣：主要从事计算研究,发展和应用第一性原理方法研究物质材料的基本物理性质；研究量子点的电、性质，及其在计算和量子信息中的应用&固体中Wannier函数的计算及其应用该篇博士论文的主要工作分作两部分,一是理论算法的推广和数值代码的具体实现,二是将前一部分的数值成果具体应用于拓扑绝缘体相关的、新奇物理性质的研究。　　 首先,我们在BSTATE(Beijing Simulation Tool for Atom TEchnology)第一原理程序包中具体实现了Marzari及其合作者提出的从布洛赫波函数变换得到最局域Wannier函数(Maximally Localized Wannier Function,MLWF)的算法；并且,我们把算法从处理简单的非自旋极化情况推广到自旋极化情况情况,特别是考虑自旋轨道耦合相互作用情况。基于局域Wannier函数跃迁的紧束缚哈密顿量,我们可以方便...作者 ： 张海军 &导师姓名 ： 方忠 &&&摘录2wannier函数在实际计算中有那些应用呢？ 从wannier90计算得到的系统某些电子态的最大局域化wannier函数中可以得到系统的哪些性质呢？ Wannier基函数是一套正交而且空间局域的基函数，不仅能够精确重构赝势平面波基函数计算得到的本征态，而且基组规模较一般基函数小得多，从而方便针对特定分子轨道研究其在电子输运过程中的贡献。那如果我要比较两种结构中过渡金属原子的d电子态的话，是不是得到的d电子wannier函数局域化程度越强，就说明这种噢乖结构下d带的关联作用更强呢？很多的应用的。你看看psi-k 上找找看，有一期的highllight介绍了wannier function。楼上提到的，还有帮组分析化学键、构造模型哈密顿量中的参数、计算电声耦合参数、在wannier function的基础上做GW的计算、插值的方法得到能带结构图、以及电极化的计算等。Wannier函数就是晶体波函数之间的关系就是一个傅立叶变换。H哈密顿矩阵元在k空间和实空间也存在一个简单的变换关系。下面的一篇是一个很好的例子，通过Wannier函数来构造模型哈密顿量的参数：Roman Kováčik and Claude Ederer PHYSICAL REVIEW B 81, 10) Calculation of model Hamiltonian parameters for LaMnO3 using maximally localized Wannier functionshttp://dx.doi.org/10.1103/PhysRevB.81.245108有兴趣的可以重复该文的工作。Wannier函数的角标是能带数的索引(n)，另一变量是实空间变量r,即 w_n(r-R_i)，n是能带数的索引，r是位置坐标, R_i晶格位置（与平移对称性有关)。如果考虑单带模型，即 n=1的话，Wannier函数同哈密顿量左右作用后得到的就是哈密顿量的矩阵（类似紧束缚近似)。 & Wannier函数的角标是能带数的索引(n)，另一变量是实空间变量r,即 w_n(r-R_i)，n是能带数的索引，r是位置坐标, R_i晶格位置（与平移对称性有关)。如果考虑单带模型，即 n=1的话，Wannier函数同哈密顿量左右作用后得到的就是哈密顿量的矩阵（类似紧束缚近似)。&摘自：&最近在做vasp接口wannier90计算，虽然最新版的wannier90-2.0已经发布，但是我的vasp5.2.12却不能很好地和其做好接口，可能支持vasp5.3吧，最后还是用的wannier90-1.2，然后中科大一位师兄的帮助下，能开始计算了，现在贴出来希望会用的大神能指导一下： 第一步：先在INCAR里加入LWANNIER90 = T ，这样可以让vasp计算完后自动写wannier90.win参数，不过奇怪的是，明明LWANNIER90_RUN的默认是FALSE，但是我写完这个后都会自动调用wannier90 进行计算，会得到WOUT文件和mmn以及eig文件，不知道为什么？ 第二步：根据上一步写的win参数，你可以进行修改，和加入你需要的参数，不过目前好像我觉得都不需要修改，在INCAR里加入你需要的wannier参数，比如计算UNK和画图的，然后重新跑一步vasp； 第三步，在win中加入plot或者fermi_surface参数，可以画图或者计算fermi surface等等； 目前我就知道这么多，不过我想知道的是，如果我想用wannier90画出我的d轨道的话，是不是就是计算MLWFs for the valence bands，然后同UNK文件画图，得到很多xcrysden读的xsf文件?我也计算了fermi_surface，虽然需要很大很大的K点，不过计算的结果还是很令我满意的，就是K点太大了需要。 希望大神们能指导指导，有太多的不懂，同时也贴出来给需要的同学互相学习，交流。vasp接口wannier90计算完了，能设置费米能级嘛？&可以，直接fermi_energy = 5.2676。我算fermi surface的时候，需要用到这个，直接写到win里面，&&fermi surface也是学习中，不会分析，都是算好了，给老师分析的，关键是结果我还觉得怪怪的，可能还没有完全弄明白。我这里有一篇，wannier90论坛里的一篇文章，里面有feimi surface，但是我还没有看过，你先看看：PRL 107, 11)&你第一步的计算，仅仅是为了得到win的输入文件，这样你可以修改后继续再计算，这个时候的vasp和wannier90是连接在一起的，也就是计算完wannier90也就计算完了，当然你也可以只用vasp计算一次，然后再剩下的用wannier90计算的，但是投影的问题可能你就不好处理了。至于fermi surface我也是初学者，不是很明白，虽然算了一些，你有什么结果可以一起讨论下。&&附摘自：Wannier function&The Wannier functions are a complete set of
& They were introduced by . & &The Wannier functions for different lattice sites in a
are orthogonal, allowing a convenient basis for the expansion of
states in certain regimes. Wannier functions have found widespread use, for example, in the analysis of binding forces they have proven to be in general localized, at least for insulators, in 2006. Specifically, these functions are also used in the analysis of
and condensed .&Contents&&&&&&&&&&&&&&&&&&[] An example of WF in Barium Titanate.&&&&Although Wannier functions can be chosen in many different ways, the original, simplest, and most common definition in solid-state physics is as follows. Choose a single
in a perfect crystal, and denote its
bywhere uk(r) has the same periodicity as the crystal. Then the Wannier functions are defined by,whereR is any lattice vector (i.e., there is one Wannier function for each );N is the numbeThe sum on k includes all the values of k in the
(or any other
of the ) that are consistent with
on the crystal. This includes N different values of k, spread out uniformly through the Brillouin zone. Since N is usually very large, the sum can be written as an integral according to the replacement rule:where &BZ& denotes the , which has volume Ω.[]On the basis of this definition, the following properties can be proven to hold:For any lattice vector R' ,In other words, a Wannier function only depends on the quantity (r − R). As a result, these functions are often written in the alternative notationThe Bloch functions can be written in terms of Wannier functions as follows:,where the sum is over each lattice vector R in the crystal.The set of wavefunctions
for the band in question.Wannier functions have been extended to nearly periodic potentials as well.[]The Bloch states ψk(r) are defined as the eigenfunctions of a particular Hamiltonian, and are therefore defined only up to an overall phase. By applying a phase transformation eiθ(k) to the functions ψk(r), for any (real) function θ(k), one arrives at an equally valid choice. While the change has no consequences for the properties of the Bloch states, the corresponding Wannier functions are significantly changed by this transformation.One therefore uses the freedom to choose the phases of the Bloch states in order to give the most convenient set of Wannier functions. In practice, this is usually the maximally-localized set, in which the Wannier function ϕR is localized around the point R and rapidly goes to zero away from R. For the one-dimensional case, it has been proved by Kohn that there is always a unique choice that gives these properties (subject to certain symmetries). This consequently applies to any
the general conditions are not established, and are the subject of ongoing research.[]Wannier functions have recently found application in describing the
in crystals, for example, . The modern theory of polarization is pioneered by Raffaele Resta and David Vanderbilt. See for example, Berghold, and Nakhmanson, and a power-point introduction by Vanderbilt. The polarization per unit cell in a solid can be defined as the dipole moment of the Wannier charge density:where the summation is over the occupied bands, and Wn is the Wannier function localized in the cell for band n. The change in polarization during a continuous physical process is the time derivative of the polarization and also can be formulated in terms of the
of the occupied Bloch states.[][]^ ^ ^ A Bohm, A Mostafazadeh, H Koizumi, Q Niu and J Zqanziger (2003). . Springer. pp. §12.5, p. 292 ff. &.Theory of generalized Wannier functions for nearly periodic potentials Physical Review B 48, 1993, Analytic Properties of Bloch Waves and Wannier Functions, Phys. Rev. 115, 809 (1959)General and efficient algorithms for obtaining maximally localized Wannier functions Spontaneous polarization and piezoelectricity in boron nitride nanotubes, 2008Berry phases and Curvatures in Electronic Structure Theory.C. Pisani (1994).
(Proceedings of the IV School of Computational Chemistry of the Italian Chemical Society ed.). Springer. p. 282. &.&
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