关注：

1）强关联相互作用及其参数的设置

格式为：
LDAUU = 0.00  2.00  0.00
LDAUJ =  0.00  0.20  0.00

其他的:
LDAU =.TRUE.
LDAUTYPE = 2

LDAUL =-1  3    -1

Wikipedia

Typically, strongly correlated materials have incompletely filled d- or f-electron shells with narrow energy bands.

   One can no longer consider any electron in the material as being in a "sea" of the averaged motion of the others (also known as mean field theory).

    Each single electron has a complex influence on its neighbors.

    The term strong correlation refers to behavior of electrons in solids that is not well-described (often not even in a qualitatively correct manner) by simple one-electron theories such as the local-density approximation (LDA) of density-functional theory or Hartree–Fock theory.

  For instance, the seemingly simple material NiO has a partially filled 3d-band (the Ni atom has 8 of 10 possible 3d-electrons) and therefore would be expected to be a good conductor.

   However, strong Coulomb repulsion (a correlation effect) between d-electrons makes NiO instead a wide-band gap insulator.

   Thus, strongly correlated materials have electronic structures that are neither simply free-electron-like nor completely ionic, but a mixture of both.

摘录：

紧密相连的半导体和 近藤绝缘体。 物理学中有个Kondo effect 即近藤效应，指的是含有极少量磁性杂质的晶态金属在低温下出现电阻极小的现象。
   日本科学家近藤指出，电阻极小值的出现，是与杂质原子局域磁矩的存在相联系的，是磁性杂质离子与传导电子气交换耦合作用的结果（见交换作用）。交换耦合作用引起传导电子被局域磁性原子散射,使磁性原子自旋反向,传导电子本身也反向；随后，倒向的磁性原子又作用于该传导电子，这一多次散射过程相当于对电子运动的障碍，是使电阻增加的原因。
   自从1930年以来，实验上发现某些掺有磁性杂质原子的非磁性金属（例如，以铜、金、银等为基，掺入杂质铬、锰、铁等的稀固溶体）的电阻－温度曲线在低温下出现一个极小值。 按照通常的电阻理论（见固体的导电性），稀固溶体的电阻应随温度下降而单调下降，最后趋于由杂质散射决定的剩余电阻，因此，难以理解上述现象。1964年，近藤淳对这个现象作了正确的解释，因此人们常把它称作近藤效应。

网络摘录3：  http://blog.sina.com.cn/s/blog_6a8a6ba40100pvfb.html

在平均场近似或者说是一般得LDA计算中，能带的自旋分裂是由一个称为Stoner 参数I来主导得，而且平均场近似中认为这个交换分裂能是小于带宽的，一般而言I的数值在LSDA中大概在1eV左右，这样即使对于某些氧化物如 NiO，FeO，MnO等即使能带发生交换分裂，整个体系依然是金属性的，试验观察到得这些氧化物实际上是绝缘体，主要原因在于在这些氧化物中d轨道的能级位置不是由I来决定的，而是Hubbard参数U决定，U也称为On－site Coulomb作用能，相当于把两个电子放在空间同一个位置需要得能量，U数值一般在10eV左右，如此大的分裂能足以将Fermi面附近连续分布得d能级分开，从而得到正确得基态性质，目前广泛采用得LDA＋U的算法就是针对某些定域轨道，如d或者f，这些轨道占据存在强烈的在位Coulomb排斥。正是U使得带隙分裂，而不是轨道极化参量I。
        U计算目前仍然是一个研究热点，最近由人采用线性响应理论，同时结合轨道束缚的DFT计算自洽的求解了部分体系的U参数，在大部分情况下U数值要根据计算结果和试验参数的符合程度而定。
        LDA＋U计算核心思路是：首先将研究体系的轨道分隔成两个子体系（subsystem），其中一部分是一般的DFT算法（如LSDA，GGA）等可以比较准确描述的体系，另外是定域在原子周围的轨道如d或者f轨道，这些轨道在标准的DFT计算下不能获得正确的能量与占据数之间的关系（如DFT总是认为分数占据是能量最小的，而不是整数占据）【什么是分数占据？】；

       对于d或者f轨道，能带模型采用Hubbard模型【方程是什么样子的？】，而其他轨道仍然是按照Kohn－SHam方程求解；

       d以及f 轨道电子之间的关联能采用一个和轨道占据以及自旋相关的有效U表示；整体计算的时候需要将原来DFT计算过程中已经包含的部分关联能扣除，这部分一般叫 Double Counting part，并且用一个新的U来表示，最终的结果是在DFT计算的基础上新增加一个和d或者f轨道直接相关的分裂势的微扰项，这部分能量可以采用一般微扰理论计算。
        在CASTEP最新的版本中增加的LDA＋U的计算，U参数的设置一般主要是针对过渡金属氧化物（Charge transfer type insulator），包含非满层f轨道的元素等，高温超导体强关联体系。在

      参数设置方面主要是需要注意d和f轨道，至于s以及p轨道一般不需要设置，当然由文献也报道p轨道的这种关联性。过渡金属氧化物的有效U如下：
Species       U              J                    U－J（Ueff）
NiO              8.0           0.95                7.1
CoO          7.8          0.92                 6.9
FeO          6.8          0.89                 5.9
MnO          6.9          0.86                 10.3
VO             6.7          0.81                 5.9
TiO              6.6          0.78                5.8
Reference: Band theory and Mott insulators: Hubbard U insteat of Stoner I, PRB Vol44 No 3 (1991);
       3d轨道U和J计算如下所示：主要原理是改变d轨道的占据，在自旋极化的前提下计算不同自旋轨道能量的差值提取U和J，U微Coulomb排斥能，J是交换能，U在所有电子中都存在，不管自旋是否相同，J只存在于自旋相同的电子上。
下面给出过渡金属（不包括稀土元素）U和J参数的选取(uint in Ry,1Ry=13.6eV)：
3d series:
Elements       U                J
V                0.25             0.05
Cr                0.26             0.053
Mn                0.28             0.055
Fe                0.3                 0.058
Co                0.31             0.059
Ni                    0.31             0.06
4d series:
Elements       U                    J
Nb                    0.19             0.04
Mo                    0.2                 0.04
Tc                    0.21             0.042
Ru                    0.22             0.042
Rh                    0.25             0.044
Pd                    0.29             0.044
5d series:
Elements           U                   J
Ta                    0.19             0.039
W                   0.20             0.038
Re                    0.205              0.039
Os                    0.2                0.039
Ir                       0.21             0.038
Pt                   0.215              0.038
        对于其他过渡金属化合物U一般在5－10eV之间。如在PRB73，134418（2006）这个文献中作者在计算Co掺杂的ZnO时采用的U是6和 8eV。过渡金属的U数值和d电子排列以及价态有关系，因此上面给出的数值只是一个大概的估算数值，具体文献见Physical Review B Vol50,No23,1994.
      LDA+U 算法主要原创作者是俄罗斯金属研究所的V.I. Anisimov，重要文献有：
Corrected atomic limit in the local density approximation and the electronic structure of d impurities in Rb, Phys.Rev.B 50,23 (1994）；
Band theory and Mott insulators: Hubbard U instead of Stoner I, Phys.Rev.B. 44 No.3 (1991);
Materials Studio 4.3版本中也给出了一些元素默认的U数值（实际是Ueff＝U－J）：

网络摘录2：http://blog.sina.com.cn/s/blog_6a8a6ba40100pvev.html

问题：
在VASP的instruction中有这么一行文字：
NB: LDAUL, LDAUU, and LDAUJ must be specified for all atomic species!
也就是说U和J的值要对每种原子设。但是并没有给出例子。
有没有高手用过的，给个例子告诉我一下在有多种原子的体系中关于LDA+U的部分怎么对每种原子设定其U-J值。
谢谢。

回答：
LDAUU LDAUJ 两个参数是根据体系的POTCAR 中原子的种类来确定其值的个数，也就是说如果POTCAR 有三类原子的话，你想对其中的一类原子如V 加U 修正（假设加的U 为6.0V）的话，并且V 在POSCAR 中的顺序是第2类原子，而其他元素的原子不想加U 的话，其相应值设为0 即可。

格式为：
LDAUU = 0.00  2.00  0.00
LDAUJ =  0.00  0.20  0.00

其他的:
LDAU =.TRUE.
LDAUTYPE = 2
LDAUL =-1  3    -1

LDAUPRING =2  ,此参数一般可以不设。

总之注意加 U 要和各种元素原子对应起来，这样才行。
LDAUJ 的值一般是LDAUU 值的1/10 或稍大一些，总之前者和后者差一个数量级。
LDAU = .TRUE. Switches on the L(S)DA+U.
• LDAUTYPE = 1|2|4 Type of L(S)DA+U (Default: LDAUTYPE = 2)
1 Rotationally invariant LSDA+U according to Liechtenstein et al.
4 Idem 1., but LDA+U instead of LSDA+U (i.e. no LSDA exchange splitting)
2 Dudarev’s approach to LSDA+U (Default)
• LDAUL = L .. l-quantum number for which the on site interaction is added
(-1: no on site terms added, 1: p, 2: d, 3: f, Default: LDAUL = 2)
• LDAUU = U .. Effective on site Coulomb interaction parameter
• LDAUJ = J .. Effective on site Exchange interaction parameter
• LDAUPRINT = 0|1|2 Controls verbosity of the L(S)DA+U module

网络摘录1：http://blog.sciencenet.cn/blog-588243-488216.html

1.手册介绍：

On site Coulomb interaction: L(S)DA+U (Supported as of VASP.4.6.)

LDAU= .TRUE. | .FALSE.

LDAUTYPE= 1 | 2 | 4

LDAUL= [0 | 1 | 2 | 3  array] LDAUU= [real array] LDAUJ= [real array]

LDAUPRINT= 0 | 1 | 2

The L(S)DA often fails to describe systems with localized (strongly correlated) and  electrons (this manifests itself primarily in the form of  unrealistic one-electron energies).

     In some cases this can be remedied by introducing a strong intra-atomic interaction in a (screened) Hartree-Fock like manner, as an on site replacement of the L(S)DA. This approach is commonly known as the L(S)DA+U method.
 Setting LDAU=.TRUE. in the INCAR file switches on the L(S)DA+U.

By means of the LDAUTYPE-tag on specifies which type of L(S)DA+U approach will be used:

• LDAUTYPE=1: The rotationally invariant LSDA+U introduced by Liechtenstein et al. [90], which is of the form

   

  

   

  and is determined by the PAW on site occupancies

   

  

   

  and the (unscreened) on site electron-electron interaction

   

  

   

  ( are the spherical harmonics)

  The unscreened e-e interaction can be written in terms of Slater's integrals , , , and (f-electrons). Using values for the Slater integrals calculated from atomic orbitals, however, would lead to a large overestimation of the true e-e interaction, since in solids  the Coulomb interaction is screened (especially ).

  In practice these integrals are therefore often treated as parameters,  i.e., adjusted to reach agreement with experiment in some sense: equilibrium volume, magnetic moment, band gap, structure. They are normally specified in terms of the effective on site Coulomb- and exchange parameters, and . ( and are sometimes extracted from constrained-LSDA calculations.)

  These translate into values for the Slater integrals in the following way  (as implemented in VASP at the moment):

  • -electrons: ,

  • -electrons: , , and

  • -electrons: , , , and

  The essence of the L(S)DA+U method consists of the assumption that one may now write the total energy as:

   

  

   

  where the Hartree-Fock like interaction replaces the L(S)DA on site due to the fact that one subtracts a double counting energy () which supposedly equals the on site L(S)DA contribution to the total energy,

   

  (6.57)

  
  

   

   

   

   

   

   

   

   

   

   

   

   

   

   

   

   

   

• LDAUTYPE=2 (Default): The simplified (rotationally invariant) approach to the LSDA+U, introduced by Dudarev et al. [91]. This flavour of LSDA+U is of the following form:  

   

  

   

  This can be understood as adding a penalty functional to the LSDA total energy expression that forces the on site occupancy matrix in the direction of idempotency, i.e., . (Real matrices are only idempotent when their eigenvalues are either 1 or 0, which for an occupancy matrix translates to either fully occupied or fully unoccupied levels.)

  Note: in Dudarev's approach the parameters and do not enter seperately, only the difference is  meaningfull.  

   

   

   

   

   

   

   

   

   

   

• LDAUTYPE=4: Same as LDAUTYPE=1, but LDA+U instead of LSDA+U (i.e. no LSDA exchange splitting). In the LDA+U case the double counting energy is given by,

   

  (6.58)

  
  

   

   

   

   

   

LDAUL= ... specifies the -quantum number (one number for each species) for which the on-site interaction is added.
(-1=no on-site terms added, 1= p, 2= d, 3= f, Default: LDAUL=2)

LDAUU= ... specifies the effective on-site Coulomb interaction parameters.

LDAUJ= ... specifies the effective on-site Exchange interaction parameters.

NB: LDAUL, LDAUU, and LDAUJ must be specified for all atomic species!

LDAUPRINT= 0 | 1 | 2  Controls the verbosity of the L(S)DA+U module.
(0: silent, 1: Write occupancy matrix to OUTCAR, 2: idem 1., plus potential matrix dumped to stdout, Default: LDAUPRINT=0)

It is important to be aware of the fact that when using the L(S)DA+U, in general the total energy will depend on the parameters and . It is therefore not meaningful to compare the total energies resulting from calculations with different and/or (c.q. in case of Dudarev's approach).  

Furthermore, since LDA+U usually results in aspherical charge densities at and atoms we recommend to set LASPH = .TRUE. in the INCAR file for gradient corrected functionals (see Sec. 6.44). For CeO for instance, identical results to the FLAPW methods can be only obtained setting  LASPH = .TRUE.

Note on bandstructure calculation:  The CHGCAR file also contains only information up to LMAXMIX (defaulted to  2) for the on-site PAW occupancy matrices. When the CHGCAR file is read and kept fixed in the course of the calculations (ICHARG=11), the results will be necessarily not identical to a selfconsistent run. The deviations can be (or actually are) large for L(S)DA+U calculations.  For the calculation of band structures within the L(S)DA+U approach, it is hence strictly required to increase LMAXMIX to 4 (d elements) and 6 (f elements).  (see Sec. 6.63).