RWIGS and LORBIT (1)

RWIGS and LORBIT 是两个与局域态密度或投影态密度相关的参数；RWIGS指的是Wigner–Seitz radius，LORBIT前面的LO指的就是Local。

这两个参数对于局域轨道的分析具有决定性的作用，下面摘取的是1999年Nature的摘要：

P. Strange, A. Svane, W. M. Temmerman, Z. Szotek & H. Winter，Understanding the valency of rare earths from first-principles theory，Nature 399, 756-758 (24 June 1999) | doi:10.1038/21595。
The rare-earth metals have high magnetic moments and a diverse range of magnetic structures1. Their magnetic properties are determined by the occupancy of the strongly localized 4f electronic shells, while the outer s–d electrons determine the bonding and other electronic properties2. Most of the rare-earth atoms are divalent, but generally become trivalent in the metallic state. In some materials, the energy difference between these valence states is small and, by changing some external parameter (such as pressure), a transition from one to the other occurs. But the mechanism underlying this transition and the reason for the differing valence states are not well understood. Here we report first-principles electronic-structure calculations that enable us to determine both the valency and the lattice size as a function of atomic number, and hence understand the valence transitions. We find that there are two types of f electrons: localized core-like f electrons that determine the valency, and delocalized band-like f electrons that are formed through hybridization with the s–d bands and which participate in bonding. The latter are found only in the trivalent systems; if their number exceeds a certain threshold, it becomes energetically favourable for these electrons to localize, causing a transition to a divalent ground state.

RWIGS and LORBIT 分3个部分分享平僧的一点心得（各位同行若有异议可留言将问题叙述清楚，共同商榷）。

RWIGS and LORBIT (1)是关于RWIGS的设置。 RWIGS的定义见维基百科[1,2] ：The Wigner–Seitz radius, named after Eugene Wigner and Frederick Seitz, is the radius of a sphere whose volume is equal to the mean volume per atom in a solid.This parameter is used frequently in condensed matter physics to describe the density of a system.

RWIGS可以在VASP中的POTCAR文件里面找到：

RWIGS = 2.330; RWIGS = 1.233 wigner-seitz radius (au A)

下面分三点展开：

（1）单位：2.33是以bohr作为单位的，而1.233是以A作为单位的，这个换算只要几秒钟，请各位动手转换一下。INCAR里面是以A作为单位的，所以应取1.233。

（2）取法：在计算块体(bulk)时，直接取POTCAR以及根据OUTCAR中的体积比来取值都是可行的，即vasp手册明确提到的[3]：

For mono-atomic system RWIGS can be defined unambiguously. The sum of the volume of the spheres around each atom should be the same as the total volume of the cell (assuming that you do not have a vacuum region within your cell). This is in the spirit of atomic sphere calculations. VASP writes a line
                                    Volume of Typ 1: 98.5 %
to the OUTCAR file. You should use a RWIGS value which yields a volume of approximately 100%。

网上有个比较好的转帖讲述了RWIGS的取法，原帖出自于ChinaUnix的作者 beyondstar[4] ：

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当用VASP 计算态密度时，采用LORBIT=1是，需要手动设置RWIGS，在INCAR里面的RWIGS是单位为A，添写适当的RWIGS半径，使得用RWIGS计算得到的体积和整个体积近似相等，这个时候RWIGS半径设置最为正确。

V = 4/3 * pi *(sum [N_i * RWIG_i^3])

根据上面的公式，可以计算出得到的采用球近似时体系的体积，这个体积和体系真实体积之比可以从OUTCAR里面找到，采用命令 grep % OURCAR,得可以得到这个百分比；

所以我们知道体积V，原子数N，根据上面的公式就可以直接求出RWIGS，填入INCAR。

若是你要算PDOS，或是需要各个原子的电荷量和磁矩等，可以设置
1. PAW情况下，LORBIT >10，此时无需RWIGS，这里NPAR不等于1，应该也能得到PDOS等。
2. 若LORBIT小于10，要得到PDOS，就计算时把 NPAR 设为 1。
LORBIT和RWIGS的设置，详见VASP说明书（如转载请注明出处）。

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在 beyondstar 的bolg中，有人提到：“不同种类的原子RWIGS半径不一样，如果一个原胞中有好几种原子，怎么一一确定各种原子的RWIGS半径？上述公式是一个条件，但条件不够啊？”

实际上，这个问题在vasp手册中也明确地给出了：

For binary systems there is no unambiguous way to define RWIGS and several choices are possible. In all cases, the sum of the volume of the spheres should be close to the total volume of the cell (i.e the sum of the values given by VASP should be around 100%。
① One possible choice is to set RWIGS such that the overlap between the spheres is minimized.
② However in most cases, it is simpler to choose the radius of each sphere such that they are close to the covalent radius as tabulated in most periodic tables. This simple criterion can be used in most cases, and it relies at least on some “physical intuition”.

（3）适用范围：RWIGS的取法仅仅对块体时使用的，而对于当前DFT领域活跃于二维晶体和一维纳米管，是不能用“体积法”来确定参数取值的。理由：① 对于二维或一维体系含有真空层；② 二维或一维体系与块体的物理本质不一样。做法：① 调试法：通过调整RWIGS来同文献的态密度（需要与PR系列的DFT文献进行比较，可行度高一些）进行比较进而确定；② 经验法：根据该二维或一维的分子或原子轨道形态来经验地判断RWIGS是否合理。

【1】Wigner–Seitz radius：http://en.wikipedia.org/wiki/Wigner%E2%80%93Seitz_radius

【2】Wigner–Seitz cell：http://en.wikipedia.org/wiki/Wigner%E2%80%93Seitz_cell

【3】RWIGS(vasp manual)：http://cms.mpi.univie.ac.at/vasp/vasp/RWIGS.html

【4】VASP计算态密度时RWIGS的取法（原创）：http://blog.chinaunix.net/uid-7726704-id-2045192.html

【5】http://simulation.haotui.com/viewthread.php?tid=11924&extra=&page=1