光学性质

光学计算问题交流讨论

CASTEP中的光学计算是以电子结构计算为基础的，因为传统DFT在能带计算方面的问题，所以光学计算的准确性受到很大影响，但还是可以得到一些有用信息的。

而且对于一些strong Coulomb correlation的问题也可以通过LDA+U，LDA+SIC

等等进行修正。

因此此方面也会得到更多发展，应用。

我抛砖引玉先提出一个问题，希望高手解答，大家讨论。

对于光学各向异性的晶体，我们要考虑方向性，CASTEP中提供了两个选项，分别是polarized和unpolarized，可以提供各向异性的考虑。分别解释如下：

Polarized - optical properties are calculated for plane polarized with the specified polarization direction;

Unpolarized - optical properties are averaged over polarization directions perpendicular to the specified incident direction.

但是这两种情况究竟分别适用与研究什么类型材料呢？

下面以wur结构为例，此种提法：the electric field parallel （E平行c）和perpendicular （E垂直c）to the crystallographic c axis，分别对应于CASTEP中的哪个选项呢？

还有一种提法是分成两个分量：two components, the in-plane component is the average over the x and y directions and the z component which is perpendicular to x-y plane. 这样z分量和x-y plane分量分别可以和CASTEP中的哪种情况对应呢？polarization vectors perpendicular (E垂直c)and parallel（E平行c）to the crystallographic c axis

偏振矢量（or 极化矢量）分别垂直和平行c轴两种情况，这两种情况如果通过MS中对polarized和unpolaried的说明，其实都可以实现的，不知道具体有什么区别？选择两个选项的具体原则该是什么呢？

大家多多讨论

在回答上面问题的之前，我绝对有必要了解一下CASTEP计算光学性质的主要原理，CASTEP计算的光学性质主要电子能带结构中最基本的跃迁方式，其他的考虑不多，如声子（晶格振动吸收），激子，自由电子气光学响应等，在CASTEP里面也有这个说明了，比如：Limitations of the methodLocal field effectsThe level of approximation used here does not take any local field effects into account. These arise from the fact that the electric field experienced at a particular site in the system is screened by the polarizability of the system itself. So, the local field is different from the applied external field (that is, the photon electric field). This can have a significant effect on the spectra calculated (see the example of bulk silicon calculation below) but it is prohibitively expensive to calculate for general systems at present.

Quasiparticles and the DFT bandgapIn order to calculate any spectral properties it is necessary to identify the Kohn-Sham eigenvalues with quasiparticle energies. Although there is no formal connection between the two, the similarities between the Schrödinger-like equation for the quasiparticles and the Kohn-Sham equations allow

the two to be identified. For semiconductors, it has been shown computationally (by comparing GW and DFT band structures) that most of the difference between Kohn-Sham eigenvalues and the true excitation energies can be accounted for by arigid shift of the conduction band upward with respect to the valence band . This is attributed to a discontinuity in the exchange-correlation potential as the system goes from (N)-electrons to (N+1)-electrons during the excitation process. There can, in some systems, be considerable dispersion of this shift across the Brillouin zone, and the scissor operator used here will be insufficient.

Excitonic effectsIn connection with the absence of local field effects, excitonic effects are not treated in the present formalism. This will be of particular importance for ionic crystals (for example NaCl) where such effects are well known.

Other limitations

∙The nonlocal nature of the GGA exchange-correlation functionals is not taken into account when evaluating the matrix elements but it is expected that this

will have a small effect on the calculated spectra.

∙Phonons and their optical effects have been neglected.

∙Finally, there is an intrinsic error in the matrix elements for optical transition due to the fact that pseudowavefunctions have been used (that is they deviate

from the true wavefunctions in the core region). However, the selection rules

will not be changed when going from pseudo- to all-electron wavefunctions ∙比如第一条所说的局域场效应，我们在计算光学跃迁的时候，外界跃迁激发电场在材料内部认为是没有衰减的，实际上由于内场的作用，一部分电场会被Screen了，但我们没有考虑。其次提到了，DFT计算的单粒子激发谱方面存在的低估问题，这个可以通过一个对能带的刚性平移实现，也就是常说的剪刀工具。上面也提到了赝势，Exc等效应对光学性质的影响。

∙中所周知的是Optical Properties （OP）计算主要是从复合介电方程开始的，介电方程中虚部表示了和能带之间跃迁有关的信息，峰值可能和The First Brillouin Zone 的Van－HOff singularity 有关系，现在DFT计算结构比较复杂，要解析这些关系，即具体的解析出BZ结构中不同k点附近的Van－Hoff奇点是很困难的，不过在20世界50年代以后的很多文献对一些简单的半导体结构做了计算，如As，Si，AsP

等，这些物质晶体结构比较简单，因此可以比较详细的了解Van－Hoff奇点到底在BZ区那个位置。

∙

关于Van－Hoff奇点后面在说，首先看看DFT计算OP的原理，这些公示虽然很多文献都在使用，但这里为了说明问题还是要重新写一下；首先看有关的计算公示：同时也给出了在施加剪刀修正的时候只需要在Delta函数项添加修正带隙能量数值即可！（图2）

接下来是两个比较重要的概念，首先我们再给出CASTEP里面光学性质的计算公示，其次解释介电常数虚部和Joint Density of STates之间的关系，Joint Densty of States也和Van－Hoff奇点有关，Van－Hoff奇点总共有四类，他们再JDOS上面曲线形式是不同的，有两类是拐点（Saddle point），其他分别是最大和最小值点。下面分别给出CASTEP中OP 计算和JDOS的定义：图3