关注：

1） ELF的定义；

2） 计算原理及流程。

题记：赠人玫瑰，手留余香。

     相逢是一种缘。旅途中，有一种人，当你不经意间想起，总是满怀感激......XinXin Zhang 和 Meiguang Zhang即属于这种人。

新问题：正确设置INCAR，VASP533版本千万不要加IBRION=2，否则会陷入死循环

EXEC=/db/home/yexq/software/vasp533/vasp.5.4.1/vasp.5.4.1/bin/vasp_std  可快速成功计算ELF的VASP版本

一般设置0.5 为起点

isosurface value of 0.5 and 0.7,

SYSTEM=Static- self-consistent

PREC = Accurate

ENCUT = 600

EDIFF = 1e-6

ISTART = 0; IBRION = 2

POTIM = 0.1

ISMEAR =1; SIGMA = 0.2

LELF=T

正确的INCAR设置-from  Shichang Li

SYSTEM=Static- self-consistent

PREC = Accurate

ENCUT = 400

EDIFF = 1e-6

ISTART = 0

POTIM = 0.1

ISMEAR =1; SIGMA = 0.2

KSPACING=0.2

KGAMMA=.TRUE.

LELF=.TRUE.

关于IBRION参数

IBRION=2 A conjugate-gradient algorithm (a simple discussion of this algorithm can be found for instance in [28]) is used to relax the ions into their instantaneous groundstate

关于KSPACING参数

KSPACING-tag and KGAMMA-tag KSPACING = [real]
KGAMMA = [logical]

The tag KSPACING determines the number of k-points if the KPOINTS file is not present (see Sec. 5.5). KSPACING is the smallest allowed spacing between k-points in units of Å. The number of k-points increases when the spacing is decreased. The number of k-points in the direction of the first, second and third reciprocal lattice vector is determined by the equations

These values are rounded to the next integer. The generated grid is either centred at the point (e.g. includes the point) (KGAMMA=.TRUE.) or is shifted away from the point, as usually done for Monkhorst Pack grids (KGAMMA=.FALSE.) (compare Sec. 5.5.3). Per default, the grids include the point.

哪一种方法设置的K点更密：KSPACING or KPOINTS？  可通过查看IBZKPT中k点个数，来看哪一种更密

The three qualities of k-point separation for CASTEP (1/Angstrom) >>>

          coarse.le..08 & medium.le..05 & fine.le..04

Please input the quality of Monkhorst-Pack grid (Default is .035) >>>

0.03

Reciprocal lattice parameters >>>

      0.84446874   2.56760472   2.41391726

Mesh parameters of Monkhorst-Pack grid >>>

1. “Electronlocalization function”，电子局域化函数。用来描述以某个位置处的电子为参考，在其附近找到与他同自旋的电子的概率，可以表征这个作为参考的电子的局域化程度，也是一种描述在多电子体系中的电子对概率的方法。

2. 一些公式：

 :电子局域化函数

    0≤ELF≤1 , ELF=1 对应完全局域化，ELF=1/2，对应类电子气型的成对概率。

　　　：均匀电子气的，自旋密度等于(r)的局域值

　　：反映电子局域化的信息

　　：在r位置，s半径内找到两个同自旋的电子的几率，做Taylor 展开，中括号中的项就是上边的D ,  这个值越小，说明在上述区域找到相同自旋的电子的几率越小，那么这个电子的局域性越高。

3. 这个函数是由AxelD. Becke和K. E.Edgecombe在1990年定义的[1].详细的定义和公式请看参考文献和下面给出的链接。

    这个函数能很有效的分析电子局域化程度，比如分析重元素的电子壳层排布结构，在分子中，通过ELF可以清晰的分出核态和价态，也能显示出共价键和未共用的电子对。这在我们分析成键中可能会有很大的帮助。

二．计算和画图

1.vasp进行自恰计算

   使用一般自恰计算的四个输入文件，在INCAR中一定要加入开关LELF=T,其他设置不变。例如：

SYSTEM=Static- self-consistent

PREC = Accurate

ENCUT =800

EDIFF = 1e-6

ISTART = 0; IBRION = 2

POTIM = 0.1

ISMEAR =1; SIGMA = 0.2

LELF=T

  这样是为了让包含“电子局域密度函数”信息的ELFCAR文件保存下来。这个文件的格式和CHGCAR文件的类似，详细可以参见vasp手册，这里不做介绍。

2.绘图的时候可以使用xcrysden和vaspview两种软件。

在使用xcrysden查看ELFCAR文件的时候需要先要将这个文件用v2xsf程序将其转化为*.xsf格式的xcrysden可读文件。

xcrysden软件的使用大家一般比较熟悉，它可以画xy, xz, yz二维平面图，也可以画三维空间图。Vaspview同样可以画二维和三维两种图。

一个专门的ELF介绍网站：

http://www.cpfs.mpg.de/ELF/index.php?content=01quant/01def.txt

附-网言摘录

希望大家先帮看看我的计算流程原则上是不是正确的，然后说说计算精度的问题。
我的计算流程： geo-opt---->scf---->elf and bader
scf 后, mkdir elf bader
cp scf/*  elf/.
INCAR做如下修改：
ICHARG =11
#MAGMOM = *  * *
LELF = T
LWAVE = F
其它不变。k-mesh不变
cp scf/* bader/.
INCAR做如下修改：
ICHARG = 1
#MAGMOM=* * *
LAECHG = T
LWAVE = F
其他不变。k-mesh不变。
以上两个性质计算中，elf是做非自洽计算的，我的理解是，elf仅仅作为初始电荷密度的数值处理，不需要做自洽计算。而bader需要原子价层电子和内层电子，一般vasp的PAW方法不提供内层电子，所以需要以上一步scf电荷密度做输入，来补全电荷密度。原则上这两个任务设置有问题没？
精度方面，elf计算为了提高精度（图的饱满平滑），需要调高NGX(Y,Z)，然后重新做静态自洽计算。这个不必提高k采样密度，保持不变就好。
bader计算说要适当调高NGX(Y,Z)，使得总电子数等于实际电子数。这个总电子数是不是为bader分析结果文件ACF.dat的最后一行给出的电荷数？

附-from 上述网站的介绍

Original definition of ELF

The electron localization function (ELF) was introduced by Becke and Edgecombe as a "simple measure of electron localization in atomic and molecular systems" [BECKE1990]. The original formula is based on the Taylor expansion of the spherically averaged conditional same-spin pair probability density to find an electron close to a same-spin reference electron. The main aspect of this formulation is that thus defined ELF is a property of the same-spin pair density.

The same-spin pair probability density  P₂(r, r') is the probability density to simulaneously find two like-spin electrons at positions r and r'. In Hartree-Fock (HF) approximation:

P₂(r, r')  =   &rho(r) ρ(r')  −    |ρ₁(r, r')|2

The conditional same-spin pair probability density  P_cond(r, r') is the probability density to find an electron at some position r' if a like-spin reference electron is located with certainty at position r. In Hartree-Fock (HF) approximation:

P_cond(r, r')  =   ρ(r')  −    |ρ₁(r, r')|2 ⁄ ρ(r)

with the electron densities  ρ(r) and  ρ(r'), and the σ-spin one-particle density matrix  ρ₁(r, r') of the HF determinant:

ρ₁(r, r')  =   ∑_i^σ ψ_i^*(r')ψ_i(r)

where the summation runs over all occupied  σ-spin (i.e. either up or down spin) orbitals  ψ_i(r).

The probability density to find a like-spin electron at a distance s from the reference point  r can be found by a Taylor expansion of the spherically averaged conditional same-spin probability density  P_cond(r, s) (the spherical average is on a shell of radius s around the reference point  r). The first (s independent) term of the Taylor expansion vanishes, because the conditional probability density to find two like-spin electrons at the same position  r is, as a direct consequence of the Pauli principle, equal to zero. The linear term is dependent on the gradient of the HF Fermi hole at  r - thus it vanishes as well. The leading (quadratical) term of the Taylor expansion of the spherically averaged conditional same-spin probability density is [BECKE1990]:

P_cond(r, s)  =   1⁄3 [∑_i^σ |∇ψ_i(r)|2   −   ¼ |∇ρ(r)|2 ⁄ ρ(r)] s2   +   ...

The expression in the brackets is besides a ρ factor proportional to the Fermi hole mobility function of Luken and Culberson [LUKEN1982] and is related to the curvature of the HF Fermi hole at  r as shown by Dobson[DOBSON1991].

Becke and Edgecombe associated the localization of an electron with the probability density to find a second like-spin electron near the reference point. The smaller this probability density, i.e. the smaller the expression

D(r)   =   ∑_i^σ |∇ψ_i(r)|2   −   ¼ |∇ρ(r)|2 ⁄ ρ(r)

of the quadratic term, the higher localized an electron is. Thus, the Pauli repulsion between two like-spin electrons, described by the smallness of  D(r), is taken as a measure of the electron localization. Using the corresponding factor found for uniform electron gas  D_h(r) Becke and Edgecombe defined ELF as follows:

η(r)  =   1 ⁄ [1 + χ2_BE(r)]

with

χ_BE(r)  =   D(r) ⁄ D_h(r)

where

D_h(r)  =   3/5 (6π2)^2/3 ρ(r)^5/3

Given by the definition, ELF values are bound between 0 and 1.

In the seminal paper of Becke and Edgecombe the ratio  χ_BE(r) was attributed to a dimensionless localization index calibrated with respect to the uniform electron gas as a reference. Nevertheless, it should be mentioned that this reference was chosen arbitrarily (originally, Luken and Culberson had defined a function similar to χ_BE(r), but instead of a division they preferred a subtraction, again arbitrarily choosing the uniform electron gas as a reference). The only measure of the electron localization, as described by the two authors, is the expression  D(r). However, ELF cannot yield the value of  D(r) - i.e. the actual measure of the electron localization - because it depends, through  D_h(r), on the electron density as well. In this sense, ELF is a relative measure of the electron localization.

High ELF values show that at the examined position the electrons are more localized than in a uniform electron gas of the same density.  η(r) = 1⁄2 indicates that the effect of the Pauli repulsion is the same as in the uniform electron gas of the same density. Of course, it cannot be compared with the uniform electron gas with respect to other properties (it is obvious that the electron density gradient in an atom, molecule or solid differs from zero almost everywhere). See also section How to interpret.

ELF for density functionals

In density functional theory the pair density is not explicitly defined. Thus, the original formulation of ELF derived from the pair density is not applicable. Searching for a possibility to use ELF in density functional calculations, Savin et al. [SAVIN1992A] utilized the observation that the Kohn-Sham orbital representation of the Pauli kinetic energy density has the same formal structure as the expression  D(r) of Becke and Edgecombe. The main aspect of Savin's formulation is that thus defined ELF is a property based on the diagonal elements of the one-particle density matrix, i.e. the electron density.

In the Kohn-Sham method the kinetic energy of  N noninteracting electrons is:

T_s   =   ½ ∫ ∑_i^N |∇ψ_i(r)|2 dv

with the Kohn-Sham orbitals  ψ_i(r). The positive definite kinetic energy density  t(r) = ½ ∑_i^N |∇ψ_i(r)|2  is bounded by a minimum value:

t(r)  ≥   1⁄8 |∇ρ(r)|2 ⁄ ρ(r)

when all orbitals are proportional to  √ρ (i.e. like in a bosonic system) . The Pauli kinetic energy is the energy due to the redistribution of the electrons in accordance with the Pauli principle. It is the integral of the Pauli kinetic energy density:

t_P(r)   =   t(r)   −   1⁄8 |∇ρ(r)|2 ⁄ ρ(r)

The Pauli kinetic energy density itself does not resolve the bonding situation. It is the more or less arbitrary division of  t_P(r) by the kinetic energy density of a uniform electron gas of the same electron density (with the Fermi constant  c_F = 3⁄10 (3π2)^2/3)

t_h(r)  =   c_F ρ(r)^5/3

that yields all the information. For a closed shell system the ratio

χ_S(r)  =   t_P(r) ⁄ t_h(r)

is formally identical with the ratio χ_BE(r) in the HF approximation. This identity holds also for an open shell system, when the kinetic energy densities are computed for the corresponding spin part only. Then also the ELF formulas based on χ_BE(r) and χ_S(r) respectively, are identical. In the interpretation of Savin et al. ELF is a measure of the influence of Pauli principle as given by the Pauli kinetic energy density, relative to a uniform electron gas of the same density. Similarly to the original definition, ELF does not mirror  t_P(r).

An expression equivalent to χ_S(r) of Savin et al. was found already 1983 by Deb and Ghosh [DEB1983]. Deb and Ghosh were searching for a "proper local description" of the kinetic energy density. They proposed the following formulation of the kinetic energy density:

t(r)  =   −¼ ∇2ρ(r)   +   1⁄8 |∇ρ(r)|2 ⁄ ρ(r)   +   c_F f(r) ρ(r)^5/3

The right hand side of the above equation consists, besides the density Laplacian that vanishes by an integration over the whole space, of the full Weizsäcker term 1⁄8 |∇ρ(r)|2 ⁄ ρ(r) and a modified Thomas-Fermi term with a correction factor f(r). Substituting for the left hand side the Hartree-Fock expression for the kinetic energy density:

t(r)  =   ½ ∑_i |∇ψ_i(r)|2   −   ¼ ∇2ρ(r)

unveils the correction factor f(r)  of Deb and Ghosh as the ratio  χ_S(r) of Savin. Besides calculating  f(r)  for noble gas atoms (revealing the atomic shell structure) Deb and Ghosh did not further exploit this function.