关注：

1) LDA、GGA设置上的区别：赝势上

2) LDA与GGA哪一个赝势更准确，看计算值与实验值 的比较

3) LDA、GGA背后的物理含义/图像

学习摘录：

1) LDA与GGA设置上的区别：赝势上

2) LDA与GGA哪一个赝势更准确，看计算值与实验值 的比较，如晶格参数、体模量、电子结构、光学性质(与能找到的实验值对比)

引子:

Besides, we canevaluate the spin and orbital contribution to the magnetic moment (μ_s and μ_l).

In our calculations,we constrain the direction of the magnetic moments along z-axis in LDA and LDA +U.

In LDA +U +SOC,instead, the magnetic moments are automatically set to be noncollinear.

However, in the finalself-consistent solution, they result again all aligned along the z-axis.

For the sake of the computation efficiencyof so many calculation models,we decide to leave the 3-k structure outside of scope for our present calculations. Therefore, only the results ofthe collinear 1-k AFM configurations related to the (100) lattice directions are discussed.

参考handsonIV手册、Wiki算例、百度12算例

手册参考：

MAGMOM-tag：      http://cms.mpi.univie.ac.at/vasp/vasp/MAGMOM_tag.html

LNONCOLLINEAR:    http://cms.mpi.univie.ac.at/vasp/vasp/LNONCOLLINEAR_tag.html

LSORBIT-tag：          http://cms.mpi.univie.ac.at/vasp/vasp/LSORBIT_tag.html  

问题：

考虑3-k磁性优化后，面心立方晶格常数三边不等！？

only DFT-U：加入SOC、FM后，会影响晶格常数，进而影响能量值及动力学稳定性；

后续，或可考虑加入SOC、FM后，保持晶格参数不变，仅仅做一次自洽计算？

LNONCOLLINEAR = .TRUE.  【非共线磁性计算？需设置三个方向的磁矩？】

赝势的类型LDA  vs PBE

pot_91  

pot_LDA_UP

pot_paw

pot_paw_GGA

pot_paw_PBE

POTCAR里的信息

赝势的类型从LEXCH参数（如LEXCH  = 91）中可以看出。

....

一些INCAR例子：

（1） From Dr. Ruizhi for pure X :   LDA+U

POSCAR中 4个X原子

X-delta-fm3m                          
  1.00000000000000    
    4.6124086025297766   -0.0000000000000000    0.0000000000000000
    0.0000000000000000    4.6124086025297766   -0.0000000000000000
   -0.0000000000000000    0.0000000000000000    4.6124086025297766
 X
    4
Direct
 0.0000000000000000 -0.0000000000000000  0.0000000000000000
-0.0000000000000000  0.5000000000000000  0.5000000000000000
 0.5000000000000000 -0.0000000000000000  0.5000000000000000
 0.5000000000000000  0.5000000000000000 -0.0000000000000000

PREC = Accurate
LSORBIT = .TRUE.
MAGMOM = 0 0 4 0 0 -4 0 0 4 0 0 -4
ENCUT = 700
EDIFF = 1E-6
ISMEAR = 1
SIGMA = 0.1
VOSKOWN = 1
LDAU=.TRUE.
LDAUTYPE=1
LDAUL=3
LDAUU=4.0
LDAUJ=0.7
LDAUPRINT=2
EDIFFG = -1E-3
NSW = 100
IBRION = 1
POTIM = 0.1
NELMIN = 6
ISIF = 7
LWAVE = .FALSE.
LCHARG = .FALSE.
LREAL = .FALSE.
LMAXMIX = 6

（2） From Dr. Baotian for pure XH2  : LDA+U

SYSTEM = XH2
EDIFF=0.000001
ENCUT = 550
PREC = Accurate
ISTART = 0
ICHARG = 2
#####relax
IALGO = 48
#NSW = 60
#EDIFFG=-0.02
ISMEAR = -5
#SIGMA = 0.01
#IBRION = -1
#ISIF = 2
############
NPAR = 6
#NEDOS = 3000
#RWIGS = 1.588 0.3700
#NGX = 27
#NGY = 27
#NGZ = 27
#NELMDL = -6
#LORBIT = 10
#EMAX = 20
#EMIN = -
################
ISYM = 0
LSORBIT = .TURE.
LORBMOM = .TURE.
LMAXMIX = 6
SAXIS = 0 0 1      
################## the switch of L(S)DA + U calculation
ISPIN = 2        
MAGMOM = 0 0 4 0 0 -4 0 0 -4 0 0 4 24*0  #貌似3-k设定方式
LDAU = .TRUE.
LDAUTYPE = 2
LDAUL = 3 -1
LDAUU =4.51 0
LDAUJ = 0.51 0

POSCAR

XH2
  5.2
  1.0000000000000     0.000000000000000   0.000000000000000
  0.000000000000000   1.0000000000000     0.0000000000000
  0.000000000000000   0.0000000000000     1.0000000000000
  4   8
Direct
  0.000000000000000   0.000000000000000   0.000000000000000
  0.000000000000000   0.500000000000000   0.500000000000000
  0.500000000000000   0.000000000000000   0.500000000000000
  0.500000000000000   0.500000000000000   0.000000000000000
  0.250000000000000   0.250000000000000   0.250000000000000
  0.750000000000000   0.750000000000000   0.250000000000000
  0.750000000000000   0.250000000000000   0.750000000000000
  0.250000000000000   0.750000000000000   0.750000000000000
  0.250000000000000   0.250000000000000   0.750000000000000
  0.750000000000000   0.750000000000000   0.750000000000000
  0.750000000000000   0.250000000000000   0.250000000000000
  0.250000000000000   0.750000000000000   0.250000000000000

（3） From  Prof. GaoTao

System=XH3
PREC=Accurate
GGA=PE
ISMEAR=0
SIGMA=0.05
ENCUT=550
EDIFF=0.1E-5
EDIFFG=-0.01
LREAL=.FALSE.
IBRION=8
IALGO=38
LORBIT=11
LDAU=.TRUE.
LMAXMIX=6
LDAUTYPE=2
LDAUL=  -1  3
LDAUJ= 0.0  0.51
LDAUU= 0.0  4.5
LASPH=.TRUE.
ISPIN=2
MAGMOM= 0 0 0 0 0 0 2 2   #1个原子设一个磁矩？1-k方式？
KSPACING=.251327412287
KGAMMA=.TRUE.

POSCAR

H3 X1                                  
  1.00000000000000    
    3.8205048939691340    0.0000000000000000    0.0000000000000000
   -1.9102524469845670    3.3086542934220629    0.0000000000000000
    0.0000000000000000    0.0000000000000000    6.8513561488925525
  H    X
    6     2
Direct
 0.0000000000000000  0.0000000000000000  0.2500000000000000
 0.0000000000000000  0.0000000000000000  0.7500000000000000
 0.3333333429999996  0.6666666870000029  0.5945038509985161
 0.6666666269999979  0.3333333129999971  0.4054961490014841
 0.6666666269999979  0.3333333129999971  0.0945038509985160
 0.3333333429999996  0.6666666870000029  0.9054961490014840
 0.3333333429999996  0.6666666870000029  0.2500000000000000
 0.6666666269999979  0.3333333129999971  0.7500000000000000

 0.00000000E+00  0.00000000E+00  0.00000000E+00
 0.00000000E+00  0.00000000E+00  0.00000000E+00
 0.00000000E+00  0.00000000E+00  0.00000000E+00
 0.00000000E+00  0.00000000E+00  0.00000000E+00
 0.00000000E+00  0.00000000E+00  0.00000000E+00
 0.00000000E+00  0.00000000E+00  0.00000000E+00
 0.00000000E+00  0.00000000E+00  0.00000000E+00
 0.00000000E+00  0.00000000E+00  0.00000000E+00

(4) My own XH3-fm3m：fm-U

#ISYM=0
#LREAL = .FALSE.
LCHARG = FALSE
LWAVE = FALSE

优化后POSCAR

xh3-0g-225-opt0G                      
  1.00000000000000    
    5.3376219386840846    0.0000000000000000    0.0000000000000000
    0.0000000000000000    5.3376219386840846    0.0000000000000000
    0.0000000000000000    0.0000000000000000    5.3376219386840846
  H    Pu
   12     4
Direct
 0.5000000000000000  0.5000000000000000  0.5000000000000000
 0.5000000000000000  0.0000000000000000  0.0000000000000000
 0.0000000000000000  0.5000000000000000  0.0000000000000000
 0.0000000000000000  0.0000000000000000  0.5000000000000000
 0.7500000000000000  0.2500000000000000  0.2500000000000000
 0.2500000000000000  0.7500000000000000  0.7500000000000000
 0.2500000000000000  0.7500000000000000  0.2500000000000000
 0.7500000000000000  0.2500000000000000  0.7500000000000000
 0.2500000000000000  0.2500000000000000  0.7500000000000000
 0.7500000000000000  0.7500000000000000  0.2500000000000000
 0.7500000000000000  0.7500000000000000  0.7500000000000000
 0.2500000000000000  0.2500000000000000  0.2500000000000000
 0.0000000000000000  0.0000000000000000  0.0000000000000000
 0.0000000000000000  0.5000000000000000  0.5000000000000000
 0.5000000000000000  0.0000000000000000  0.5000000000000000
 0.5000000000000000  0.5000000000000000  0.0000000000000000

(5) my own   SOC-fm-U:XH3-fm3m

优化后POSCAR:  fm3m-三边长不一样哈

Xh3-0g-225-opt0G                      
  1.00000000000000    
    5.3489016055428795    0.0000003074117130    0.0000000000000000
   -0.0000003074117130    5.3489016055428795    0.0000000000000000
    0.0000000000000000    0.0000000000000000    5.3659230080051943
  H    X
   12     4
Direct
 0.5000000000000000  0.5000000000000000  0.5000000000000000
 0.5000000000000000  0.0000000000000000  0.0000000000000000
 0.0000000000000000  0.5000000000000000  0.0000000000000000
 0.0000000000000000  0.0000000000000000  0.5000000000000000
 0.7500000000000000  0.2500000000000000  0.2500000000000000
 0.2500000000000000  0.7500000000000000  0.7500000000000000
 0.2500000000000000  0.7500000000000000  0.2500000000000000
 0.7500000000000000  0.2500000000000000  0.7500000000000000
 0.2500000000000000  0.2500000000000000  0.7500000000000000
 0.7500000000000000  0.7500000000000000  0.2500000000000000
 0.7500000000000000  0.7500000000000000  0.7500000000000000
 0.2500000000000000  0.2500000000000000  0.2500000000000000
 0.0000000000000000  0.0000000000000000  0.0000000000000000
 0.0000000000000000  0.5000000000000000  0.5000000000000000
 0.5000000000000000  0.0000000000000000  0.5000000000000000
 0.5000000000000000  0.5000000000000000  0.0000000000000000

(6)  AFM-U

优化后POSCAR

xh3-0g-225-opt0G                      
  1.00000000000000    
    5.2331968739349097    0.0000000000000000    0.0000000000000000
    0.0000000000000000    5.5911222151299125    0.0000000000000000
    0.0000000000000000   -0.0000000000000000    5.2331968739349097
  H    x
   12     4
Direct
 0.5000000000000000  0.5000000000000000  0.5000000000000000
 0.5000000000000000  0.0000000000000000 -0.0000000000000000
 0.0000000000000000  0.5000000000000000 -0.0000000000000000
 0.0000000000000000  0.0000000000000000  0.5000000000000000
 0.7500000000000000  0.2500000000000000  0.2500000000000000
 0.2500000000000000  0.7500000000000000  0.7500000000000000
 0.2500000000000000  0.7500000000000000  0.2500000000000000
 0.7500000000000000  0.2500000000000000  0.7500000000000000
 0.2500000000000000  0.2500000000000000  0.7500000000000000
 0.7500000000000000  0.7500000000000000  0.2500000000000000
 0.7500000000000000  0.7500000000000000  0.7500000000000000
 0.2500000000000000  0.2500000000000000  0.2500000000000000
 0.0000000000000000  0.0000000000000000  0.0000000000000000
-0.0000000000000000  0.5000000000000000  0.5000000000000000
 0.5000000000000000 -0.0000000000000000  0.5000000000000000
 0.5000000000000000  0.5000000000000000  0.0000000000000000

方法解析：

From Dr Baotian

Our first-principlestotal energy calculations are performed by means of the

Vienna ab initio simulation package (VASP) [14],

based onthe frozen-core projected augmented wave (PAW) method of Blöchl [15].

The exchange andcorrelation effects are described with the local density approximation (LDA) orgeneralized gradient approximation (GGA) [16,17], and a cutoff energy of 550 eV is used for the set of plane waves.

The MonkhorstePack(MP) [18] 9 × 9 × 9 mesh is used in the Brillouin zone(BZ) for the fcc unit cell of both PuH₂ and PuH₃, which turns out to be sufficient to get results converged to less than 1.0 × 10^-6 eV【怎样测试】. The Pu 6s²7s²6p⁶6d²5f⁴and the H1s¹ orbitals are treated as valence electrons 【看不出采用的是哪个赝势哈？！】.

The strong on-siteCoulomb repulsion among the localized Pu 5f electrons is described by using the LDA/GGA +Uapproach formulated by Dudarev et al. [19-21], where the doublecounting correction【双重抵消】 has already beenincluded as in the fully localized limit (FLL)[22].

In this paper, we study several values of the Hubbard parameter U, while we keep the Hund's exchange parameter fixed to J =0.51 eV.

One can notice that only the difference between U and J is significant, and we will henceforth refer to it as a single parameter, named U for sake of simplicity.

Due to the strong SOC effects on the electronicstructure of these plutonium hydrides, as already reported in a earlier DFTstudy [13], in our present work we also discuss the energy,electronic structure, and phonon spectrum with and without inclusion ofthe SOC.

From Dr.Ao

Both XO₂ and XO₂crystallize in the face-centered cubic (fcc, space group: 225/Fmm) fluorite structures as shown inFigure 1, with the lattice parameters of 5.470 Å and 5.396 Å, respectively.

 Inorder to reasonably reflect the experimental hydrogen concentration in the twodioxides, here we use their 2 × 2 × 2 supercells containing 32 metal atoms and64 O atoms (An₃₂O₆₄;An = U and Pu.) to build thecomputational configurations. For the sake of markingthe pathway of hydrogen, only a conventional cell unit (An₄O₈) is shown in Figure 1. 【这个计算量该多大 哈】

       For their  magnetic orders of UO₂ and PuO₂, we use the widely accepted collinear 1-k a ntiferromagnetic(AFM) states along (100) lattice direction.⁴

    In fact,our calculations and some similar calculations by other researchers show that the 1-k AFM states are energetically more favorable than nonmagnetic (NM) and ferromagnetic (FM) states.

Twenty-one incorporationsites for atomic hydrogen, i.e., along the pathway from its nearest neighboringlattice oxygen to the octahedral interstitial site, are considered, asindicated by the red arrow in Figure 1. The potential existence states ofhydrogen in perfect UO₂ and PuO₂ crystals are supposed tobe approximately determined by the formation energy of hydrogen in the above 21incorporation sites. More complicated incorporation sites, such as variousdefects which might accommodate hydrogen and require the larger supercells tobuild the defect configurations, are currently not considered.

Total energy calculations are performed withVASP code, the projector augmented wave (PAW) method, and relativisticeffective core potentials (ECPs).^30-32

X 6s²7s²6p⁶6d²5f²,

Y 6s²7s²6p⁶6d²5f⁴, O 2s²2p⁴and H 1s¹ are treated as valence electrons, respectively.

The exchange and correlation interactions are described by the spin-polarized generalized gradient approximation (GGA) in the Perdew-Wang 91 (PW91) functional.

Other functionalssuch as Perdew-Burke-Ernzerhof (PBE) and local density approximation (LDA) havebeen demonstrated to have a slight influence on the energetics of impurities incorporation into UO₂ and PuO₂ in our previouscalculations.

The Hubbard model is used to treat strong on-site Coulombinteraction within the DFT + U methodin the Dudarev formalism.³³ An effective U (U_eff= U – J; i.e., thedifference between the Coulomb U and exchange J parameters,hereafter referred to as U) value of 4 eV is selected for both U and Pu5f electrons.

This value has been demonstrated by our previous calculations to be reasonable in reproducing the experimental lattice parameter, bulk modulus, bandgap and reaction energy of UO₂ and PuO₂.

Complete relaxationwithout symmetry constraints is employed for the perfect UO₂ and PuO₂.This means that the positions of the atoms as well as the lattice parameters ofthe unit cells are fully relaxed. We find that the total energies of the configurations that are relaxed without symmetry constraints are always smaller than those relaxed with symmetry constraints. With the incorporation ofhydrogen, the hydrogen atom and one of its nearest oxygen atoms are fixed foreach configuration, and the remaining part of each configuration is stillcompletely relaxed. Convergence is reached when the total energies convergewithin 1×10⁻⁵ eV andthe Hellmann-Feynman forces on each ion are less than 0.02 eV/Å. The use of aplane-wave kinetic energy cutoff of 500 eV and 3 × 3 × 3 Monkhorst-Pack k-point sampling are shown to giveaccurate energy convergence. For the total energy and density of state (DOS)calculations, the tetrahedron method with Blöchl correction is used for theBrillouin-zone integration.³⁴ Pure spin-polarized DFT calculationsare performed to determine the total energy of a hydrogen atom in molecularstate. We use half the total energy of an H₂ molecule as the totalenergy of a hydrogen atom. Owing to the well-known disadvantages of pure DFT indescribing molecule, the scheme selected to calculate the total energy of an H₂molecule is similar to the one proposed by Korzhavyi et al.³⁵ The total energy of an H₂ molecule isobtained by the sum of the energy of free hydrogen atom and the well-establisheddimerization energy (i.e., the reaction energy of 2H = H₂).³⁶The total energy of a free hydrogen atom is calculated by using a periodiccubic cell with a lattice constant of 15 Å and only one k point Γ in theframework of spin-polarized DFT.

From Ao'2

As can be found in the literature, therelative stabilities of impurity atoms in nuclear fuels and other materialsderived by different authors are in discrepancies.^1,23,53,54 Apartfrom the theoretical schemes, the definition on the energetics of defects maybe attributed to the discrepancies. Generally, the terminology “formation energy of defect”, denoted as E_f,is widelyused, whereas “incorporation energy”, denoted as E_i, is used not so widely.

E_f is defined as the total energy difference of theconfigurations in the perfect atom arrangement and defective atom arrangement.

E_i is the total energy difference of the configurationsbefore and after the formation of the investigated defect.

Therefore, E_f is the sum of E_iof all of the defects if the configuration contains more than one defect.

Among all of the definitions on energetics,it is worth keeping in mind that the more positive the incorporation energy andthe formation energy are, the less stable the defective models are, whereas thecase of the binding energy is inverse. In addition, vacancy formation energy isthe basic parameter for the study of defect behavior in solid state materials.The vacancy formation energies of V_O and V_Pu in Pu₄O₈are calculated by the following equations:

Among all of thedefinitions on energetics, it is worth keeping inmind that the more positive the incorporation energy and the formation energyare, the less stable the defective models are, whereas the case ofthe binding energy is inverse.