关注：

1)  成键作用的理论分析方法：

     Charge density、Charge transfer、Electron localization function、Density of states、Crystal orbital Hamiltonian population、Mulliken population

     Bader 、Phonon、Phonon-electron

2） ELF图的分析

3) ELF的计算似乎没有走通：ScH4-i4mmm、ScH6-cmcm

(一)Charge density

(二) Charge transfer

(三) Electron localization function

(四)Density of states

(五)Crystal orbital Hamiltonian population

(六) Mulliken population

(七) Bader analysis

(八) Vibration frequency

(一)Charge density

(二) Charge transfer

(三) Electron localization function

“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；反之，ELF越小，找到自旋平行的电子对概率就大”

离子键：similar ELFs for LiH  as NaCl

   ELF=1  局域在某一个原子之上，离子键？

共价键：and for B􀀀2H􀀀6 like C.

        局域在原子之间，共价键？

金属键： In the metallic ScH
2 hydrides, we obtain low ELFs compare to Na.

     分布在间隙区域：金属键？电子跑到晶格当中去了

Thanks to Xinxin Zhang

vasp xcrysden/vaspview

一．ELF简介

1. “Electron localization 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。

“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；反之，ELF越小，找到自旋平行的电子对概率就大”

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。

这个函数是由Axel D. 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。

一个专门的ELF介绍网站：

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

Concerning the interpretation of the absolute values of ELF in the original paper [BECKE1990] the following hint was given: "... the upper limit ELF = 1 corresponding to perfect localization and the value ELF=½ corresponding to electron-gas-like pair probability".

So ELF can be said to represent the organization of chemical bonding in direct space. Although it has been termed "electron localization function" its relation with the physical concept of localized and itinerant (delocalized) electrons (orbital picture) seems to be more subtle (see above).

      The absolute value of η at critical points does not (yet) play a general role. Instead, the topology is analyzed (see section Topological Analysis). Chemical information can be obtained from ELF attractors taking the other topological elements into account as well.

A suitable way to characterize the η(r) representation of chemical bonding for a compound is the construction of a "bifurcation diagram" or "basin interconnection diagram". The attractors can be attributed to

I) bonds,
II) lone pairs,
III) atomic shells and
IV) other elements of chemical bonding.
The attribution is done in an empirical way, as there is no direct proof that relates ELF with these conceptual aspects of chemistry.

The physical meaning of an ELF basin is unknown, as there is no quantum mechanical motivation yet for the definition of a surface of zero flux in the gradient vectors of ELF. The integrated electron density in an ELF basin (electronic basin population) does not correlate in an obvious manner with the energetical aspects of the bonding. However, the electronic basin population characterizes the spatial organization of the bonding in terms of ELF and the electron density. It may not be expected to resemble the bonding analysis in Hilbert space ("population analysis"). An investigation about the relative basin population with respect to a suitable standard bond [CHESNUT2001A] revealed a correlation with the chemical bond order for a selected set of test examples.

以TiB2为例，画其ELF，并简单分析结果

1. 将vasp计算得的ELFCAR转化成ELFCAR.xsf后导入xcrysden, 在tools工具栏里找到Data Grid，点击ok即可，如图所示，在右边的“Isosuface/Property Contonls”里就可以控制画图了。三维的空间分布，第一个Isosurface控制，其中Isovalue的框是填写等值面的ELF值(无单位？)，根据情况在0～1之间选取，一般在最小和最大的“grid value”（Isovalue 上面的两个值）中间选取。

这个值是等值面ELF值大小，可以根据需要修改

每个Ｂ周围都有三个局域最大值（Isovalue=0.8），这种局域最大值一般出现在atom，bonding,anti-bonding处，上图可见电子主要局域在Ｂ－Ｂ成健路径上，说明Ｂ－Ｂ之间成共价健。

From  P. Vajeeston

-2A规则 Chemical-bonding and high-pressure studies on hydrogen-storage materials

According to Eqn. 5.2, the ELF takes the value one either for a single-electron

wave function or for a two-electron-singlet wave function.

In a many-electron system ,

 ELF is close to one(ELF=1) in regions where electrons are paired

such as covalent bonds, or for unpaired lone electrons of dangling bonds,

while the ELF is small in low-density regions.

In a homogeneous electron gas ELF equals 0.5 (ELF=0.5)at any electron density,

and ELF values of this order in homogeneous systems indicates

regions where the bonding has a metallic character.

The ELF distribution of typical

examples for the bonds in the three test cases are given in Fig. 5.1g􀀀i.

For the

NaCl case ELF is around 0.9 at the Cl site(局域在某一个原子之上，离子键？) and only a non significant ELF value

is found at the Na site, thus clearly reflecting ionic bonding.

In the C case, a ELF  value of around one is

found where the shared electrons are present (in between

the C atoms)(局域在原子之间，共价键？).

In the third case, maximum values of ELF is present around the

interstitial region where the delocalized electrons are found(分布在间隙区域：金属键？电子跑到晶格当中去了).

From the above examples,one can clearly visualize different bonding nature in solids. We extended  the corresponding analyzes to hydrides and found almost

离子键：similar ELFs for LiH  as NaCl

共价键：and for B􀀀2H􀀀6 like C.

金属键： In the metallic ScH
2 hydrides, we obtain low ELFs compare to Na.

We like to note that our test calculations for the metals Ni, Co,

and Cu also show such low ELF values and this is the general characteristic for

transition metals ( electrons).

We have used ELF to interpret the short H􀀀H separation

in the InH
 ( = La, Ce, Pr, Nd;  = Ni, Pd, Pt) series (publications

II, III, and IV). We have also conducted such type of analysis for the complex hydrides

in order to identify the reason for their high decomposition temperature.

!!!P. Vajeeston-Violation of the Minimum H-H Separation “Rule” for Metal Hydrides

The ELF is  an informative tool to distinguish different bonding interactions

in solids [15] and ELF for LaNiInH1.333 in

(100) is given in Fig. 3c.

The large value of ELF at  the H site indicates strongly paired electrons with local

bosonic(玻色子) character.

Another manifestation of covalent   bonding between Ni and H should have been paired electron

distribution between these atoms.

The negligibly small ELF between Ni and H indicates that the probability

of finding parallel spin electrons close together is

rather high (correspondingly small for antiparallel spin

pairs) in this region confirm metallic bonding consistent   (Ni-H之间可以忽略的ELF值，表示找到自旋平行的电子对的概率非常高-相对于自旋相反的电子对)

“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；反之，ELF越小，找到自旋平行的电子对概率就大”

with the result obtained from charge transfer analysis

and the detailed analysis show that delocalized metallic

Ni(2c)-d electrons are distributed in this region.

Even  though the charge distribution between Ni and H looks

like a typical covalent bonding

the charge transfer and  ELF analyzes clearly show that the electron distribution

between Ni and H are having parallel spin alignment and

purely from Ni site. (Ni-H之间自旋平行电子对，且完全来自与Ni)

Hence, chemical bonding between Ni and H is dominated by metallic components with considerable

ionic weft.  (具有一定离子特征的金属键)

The partial density of state analysis also show that the H-s states are well separated from

the Ni-d states in the whole valence band indicates the

presence of ionic bonding between Ni and H.（没有重叠的DOS,表明离子键成分？）

Due to the repulsive interaction between the negatively charged H

electrons, the ELF contours are not spherically shaped

but polarized towards La and In.

The localized nature  of the electrons at the H site and their polarization towards

La and In reduce significantly the H−H repulsive interaction and this can explain the unusually short H−H

separation in this compound.【局域与极化特征-费球形局域-降低了H-H之间的负电荷排斥作用？】

The ELF between the H atom takes a significant value of 0.35. Considering the

small charge density, this indicates a weak metallic type of interaction between the hydrogen atoms.【氢原子之间(ELF=0.35)呈现若的金属键特征？】

About “the 2-°A rule”

it is found that the metal hydrides RTInH1.333 (R = La, Ce, Pr, or Nd;

T = Ni, Pd, or Pt) possess unusually short H−H separations. The most extreme value (1.454 °A)

ever obtained for metal hydrides occurs for LaPtInH1.333. This finding violates the empirical rule for  metal hydrides, which states that the minimum H  H separation is 2°A. Electronic structure, charge

density, charge transfer, and electron localization function analyses on RTInH1.333 show dominant

metallic bonding with a non-negligible ionic component between T and H, the H−H interaction

being weakly metallic.

  The paired, localized, and bosonic nature of the electron distribution at the

H site are polarized towards La and In which reduces the repulsive interaction between negativelycharged H atoms.

This could explain the unusually short H−H separation in these materials. Also,R−R interactions contribute to shielding of the repulsive interactions between the H atoms.

Switendick

[3] observed from a compilation of experimental  structure data that the minimum H−H separation in ordered  metal hydrides is >2°A (“the 2-°A rule”). This empirical

pattern is later [4] supported by band-structure

calculations which ascribe the effect to repulsive interaction

generated by the partially charged hydrogen atoms.

A practical consequence of this repulsive H−H interaction

in metal hydrides is that it puts a limit to the amount

of hydrogen which can be accommodated within a given

structural framework. So, if H−H separations less than

2°A would be possible this could open for new efforts to

identify potential intermetallics for higher hydrogen storing

capacity. However, there are indeed metal hydrides

which do violate “the 2-°A rule” and we have here identified

the origin for such behavior.

As the total energy curves increase steadily on reduction  of the H−H separations, the possibility of stabilization  of hydrogen in the form of molecular H2-like units  seems completely ruled out.

   【氢化物中出现氢分子概率还是很小】

Another important observation is that the bonding interaction between the hydrogens

is small, which further confirms that the short H−H separation in these materials are not rooted in hydrogen pairing or formation of H2-like molecular units.   【H-H 相互作用很弱，没有形成氢原子对或类氢分子单元】

Chemical bondings:High pressure partially ionic phase of water ice

   The unexpected formation of OH and H3O units invites us to perform a thorough analysis on the chemical bondings of P21 structure.

  We first calculated the electron localization

functions (ELF), known to be an informative tool to distinguish

different bonding interactions in solid17.

    The isosurface plots at ELF = 0.75 (a typical good number for characterization of covalent

bondings)17 clearly illustrate the covalent bonding nature of the I-42d structure (Fig. 1c) and confirm the formation of OH and H3O units in P21 structure (Fig. 1d; Supplementary Fig. S2).

   However, we also noticed a small charge distribution localized in between OH and H3O, an indicative of their covalent interaction.

20122JCP: Quai-molecular and atomic phases of dense solid hydrogen【准分子和原子相氢】

Analysis of the electron localization function (Figure 3b) revealed the remarkable feature that the localized charges were spread out over the interstitial areas within the H3 clusters, and there is no charge localization between H3 clusters.

Notably, the electron localization function within H₃ clusters is small in the range 0.57−0.64.

This indicated that some electrons of H₃ clusters might have been delocalized into the lattice, forming H₃ ⁺-like triangles.【不解，如何判断是H3+】

This is not unexpected since the rational stability of the isolated H₃ ⁺ ion has previously been documented in the literature.

PRB2012 Ab initio study revealing a layered structure in hydrogen-rich KH_{6} under high pressure

Figure 5 (right and panels) plots the (ELF)29,30 of the C2/m and C2/c structures along the planes where the potassium and hydrogen atoms lie at selected pressures.

The 1D insulated chains can be formed by the potassium atoms in both structures.

For the layered metallic phase C2/c, the conductivity comes from the contributions of the electrons around the hydrogen atoms, where the value of ELF is about 0.5 to ～0.6.

It is worthwhile to note that only hydrogen atoms take part in conducting electricity and potassium ones do not.【为什么会有这样的结论，从价带看出来的吗】 So, the C2/c structure becomes a layered and 1D conductor, which conducts electricity along the direction of  the 1D network formed by the hydrogen atoms

(四)Density of states

(五)Crystal orbital Hamiltonian population

(六) Mulliken population

(七) Bader analysis

We subsequently

performed a topological analysis of the static electron density through Bader’s quantum theory of atoms-in-molecules18, which has been successfully applied to the determination of bonding

interactions through the values of the density and its Laplacian

at bond critical points.

The calculated data are summarized in

Table 1.

Again, the analysis gives strong evidence on the formation

of OH and H3O units as indicted by the very negative values of  

the Laplacian of charge density at the critical points. (为什么为负数，即显示有相互作用，大到一定程度，则显示为形成了团簇units?）

In agreement with above ELF results, our calculation also supports the covalent interaction (though relatively weak) between OH and H3O layers as seen from the noticeable negative Laplacian value (Table 1) between O1 and H2/H3/H4 (d2, d3 and d4 in Fig. 1b). 团簇之间的相互作用，弱的共价作用

This interaction is understandable because P21 structure is a highly packed structure at such an extreme pressure of 14 Mbar (at higher pressure 20 Mbar, this interaction becomes even stronger as shown in Table 1).

The two identities are too close to not be covalently interacted. However,

this covalent coupling is, in fact, not intrinsic as we find the interaction decreases significantly with lowering pressure to 7 Mbar (Table 1), whereas the extraordinarily strong O–H bondings within OH and H3O units remain nearly unaltered, and at an extreme case by extrapolation to about 2 Mbar, no interaction was found at all.

To support our argument, we have theoretically compressed the perfect

ionic NH4 + NH2 − solid derived from NH3 (ref. 19) and found a similar covalent interaction of NH4 + and NH2 − at a pressure 7 Mbar. On the basis of Bader theory18, the approximate charge values of oxygen and hydrogen ions of P21 structure are calculated at 14 Mbar as listed in Table 2. We found different charges for inequivalent oxygen and hydrogen ions as determined by their distinct chemical environments.

It is seen that the charge transfer from H3O to OH is about 0.62e at 14 Mbar, illustrating the ionic nature of P21 structure with a notation of (OH)δ − (H3O)δ + (δ = 0.62).

(八) Vibration frequency

Remarkably, the branches

of the highest frequencymodes are mainly due to the minimum

H-H bond stretching in the H3 or H2 units.

The highest phonon  frequencies of C2/m KH6 at 100 GPa and C2/c KH6 at

166 GPa are about 108 and 97 THz, respectively, which are reduced

from the vibrations of about 134 and 128.9 THz in solid

hydrogen at corresponding pressure, respectively.

A similar

reduction of bond length and vibration frequency is also found

in LiH6, which is believed to be the electrons’ transfer from

alkali metal to hydrogen.

Pressure-Induced Hydrogen Uptake and Destabilization of H3 −: Formation of BaH8, BaH12, and BaH10

In order to further visualize the interactions between the atoms in the polymeric hydrogenic sublattice, the electron localization function (ELF) of BaH10 was plotted in Figure 8d.

The contour plot is taken in the ac plane, and the 0.6 ELF value located between the hydrides and the elongated dihydrogens is significantly larger than the ELF value between either one of these species and the nearest short hydrogen molecules (ELF <0.5), suggesting that the former two comprise a twodimensional hydrogenic network.

A Bader charge analysis66 of Cmmm-BaH10 assigned charges of +1.13/−0.27/−0.13/−0.02e to the Ba/H−/H2(long)/H2(short) atoms, which is qualitatively consistent with an overall net charge being transferred from the alkaline earth metal atom to the hydrogenic lattice shown in the ELF plot.

It should be mentioned that, even though throughout this manuscript our notation implies that both valence electrons are ionized from the barium atoms into the hydrogenic sublattice (for simplicity), in actuality, the results of our computations are indicative of an incomplete electron transfer from the alkaline earth metal atom.

PNAS2012: Superconductive sodalite-like clathrate calcium

hydride at high pressures

The three-dimentional sodalite cage in CaH₆ is the result of

interlink of other H₄ units via each H atom at the corner of

one H4 unit.

So, what is the electronic factor promoting the formation

of these H4 units?

To answer this question, the electron

localization functions (ELF) of a hypothetical bare bcc Ca lattice

with the H atoms removed and CaH6 hydride (Fig. 3A and 3B)

were examined.

In bare body-centered cubic (bcc) Ca, regions

with ELF values of 0.58 were found to localize at the H atom

sites in the H4 units on the faces of the cube.

The ELF of CaH6

hydride suggested that no bonds were present between the Ca

and H.【Ca-H之间没有成键】

A weak “pairing” covalent interaction with an ELF of

0.61, however, was found between the H atoms that formed a

square H4 lattice. 0.61

Their formation resulted from the accommodation

by H2 of excess electrons from the Ca.

An electron topological

analysis also showed the presence of a bond-critical

point (21) along the path connecting neighboring H atoms.

The integrated charge within the Hatomic basin was 1.17 e, which

corresponded to a charge transfer of 1.02 e from each Ca.

A partially

ionized Ca was also clearly supported by the band structure

and the density of states as reported in Fig. 3C and Fig. S15.【Ca部分离子化了】

At

150 GPa, Ca underwent an s–d hybridization with an electron

transferred from the 4 s to the 3 d orbital.

In CaH6, the Ca site

symmetry was mˉ3 m (Oh) and the Ca 3 d manifold was clearly

split into the eg and t2g bands, with the lower energy eg band

partially occupied.