verypsp(verypsp网站去哪了)

前面介绍了线性响应情况下的介电常数和波恩有效电荷，下面将介绍使用有限差分技术计算这些物理性质，基于Berry相理论基础，相关理论的参考文章总结和教程可见Abinit：https://docs.abinit.org/tutorial/ffield/。

# Finite difference calculation of the Born effective charges of AlP# The calculation proceeds by computing the change in cell polarization# by a Berrys phase formula, due to moving one of the atoms in the cell# Berry phase calculation of the polarization# berryopt -1 triggers the implementation of M. Viethenberryopt -1rfdir 1 1 1#Definition of the unit cell# these cell parameters were derived from a relaxation run done # Note that for a cubic space group as found here, the primitive unit cell is often represented as (0,a/2,a/2); (a/2,0,a/2); (a/2,a/2,0). # Here abinit uses 1/sqrt(2) as the normalizer and adjusts acell (the cell length scale) accordingly. acell 7.2728565836E+00 7.2728565836E+00 7.2728565836E+00 Bohrrprim 0.0000000000E+00 7.0710678119E-01 7.0710678119E-01 7.0710678119E-01 0.0000000000E+00 7.0710678119E-01 7.0710678119E-01 7.0710678119E-01 0.0000000000E+00#Definition of the atom types and pseudopotentials ntypat 2 # two types of atoms znucl 15 13 # the atom types are Phosphorous and Aluminum# the following norm-conserving pseudopotentials are stored in the abinit distribution, but are freely available through www.pseudo-dojo.org# this set uses the Perdew-Wang parameterization of LDA for the xc model pp_dirpath "$ABI_PSPDIR" pseudos "Pseudodojo_nc_sr_04_pw_standard_psp8/P.psp8, Pseudodojo_nc_sr_04_pw_standard_psp8/Al.psp8"#Definition of the atoms natom 2 # two atoms in the cell typat 1 2 # type 1 is Phosphorous, type 2 is Aluminum (order defined by znucl above and pseudos list) nband 4 # nband is restricted here to the number of filled bands only, no empty bands. The theory of # the Berrys phase polarization formula assumes filled bands only. Our pseudopotential choice # includes 5 valence electrons on P, 3 on Al, for 8 total in the primitive unit cell, hence 4 filled bands.#atomic positions. Here we do three separate computations, where the Al atom is moved (manually)#from its equilibrium position at 0,0,0 in calculation 1, to (+/-0.01,0,0) offsets in calculations #2 and 3. The P position is again taken from the value in the relaxation calculation done previously.ndtset 3jdtset 1 2 3 xcart1 2.5713431044E+00 2.5713431044E+00 2.5713431044E+00 0.0000000000E+00 0.0000000000E+00 0.0000000000E+00 xcart2 2.5713431044E+00 2.5713431044E+00 2.5713431044E+00 1.0000000000E-02 0.0000000000E+00 0.0000000000E+00 xcart3 2.5713431044E+00 2.5713431044E+00 2.5713431044E+00 -1.0000000000E-02 0.0000000000E+00 0.0000000000E+00#Numerical parameters of the calculation : planewave basis set and k point gridecut 5 # this value is very low but is used here to achieve very low calculation times. # in a production environment this should be checked carefully for convergence and a more reasonable value is probably around 20ecutsm 0.5dilatmx 1.05ngkpt 6 6 6nshiftk 4 # this Monkhorst-Pack shift pattern is used so that the symmetry of the shifted grid # is correct. A gamma-centered grid would also have the correct symmetry but would be less efficient.shiftk 0.5 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.5#Parameters for the SCF proceduretoldfe 1.0d-15nstep 8# suppress printing of density, wavefunctions, etc for this tutorialprtwf 0prtden 0prteig 0

在原子坐标部分使用了3个计算，第2和3个是Al原子位移了+/-0.01，（较典型的位移矢量），使用了笛卡尔坐标而不是xred分数坐标是为了在输出结果中更好的理解极化。使用了berryopt参数，设为1或-1都可，这里采用-1，使用-1时可输出笛卡尔坐标的结果，而且kptopt可采用1或2，而使用berryopt=1时，kptopt只能使用2.

计算结果输出：

Computing the polarization (Berry phase) for recIProcal vector: 0.16667 0.00000 0.00000 (in reduced coordinates) -0.01620 0.01620 0.01620 (in cartesian coordinates - atomic units) Number of strings: 144 Number of k points in string: 6 Summary of the results Electronic Berry phase 2.206976733E-03 Ionic phase -7.500000000E-01 Total phase -7.477930233E-01 Remapping in [-1,1] -7.477930233E-01 Polarization -1.632453164E-02 (a.u. of charge)/bohr^2 Polarization -9.340041842E-01 C/m^2

Remapping in [-1,1]是为了避免量子极化，其物理意义参考：http://dx.doi.org/10.1103/physrevb.86.054107。

进一步的极化结果：3个不同坐标，0，0.01，-0.01

Polarization in cartesian coordinates (a.u.): Total: -0.282749182E-01 -0.282749182E-01 -0.282749182E-01 Polarization in cartesian coordinates (C/m^2): Total: -0.161774270E+01 -0.161774270E+01 -0.161774270E+01

Polarization in cartesian coordinates (a.u.): Total: -0.281920467E-01 -0.282749119E-01 -0.282749119E-01 Polarization in cartesian coordinates (C/m^2): Total: -0.161300123E+01 -0.161774234E+01 -0.161774234E+01

Polarization in cartesian coordinates (a.u.): Total: -0.283577762E-01 -0.282749119E-01 -0.282749119E-01 Polarization in cartesian coordinates (C/m^2): Total: -0.162248340E+01 -0.161774234E+01 -0.161774234E+01

使用以下公式获得波恩有效电荷值：

作为对比，使用上节中的DFPT方法计算：

# Linear response calculation for AlP# Perturbation: atomic displacements, electric fields, & strains# Finite difference calculation of the ddk# DFPT calculations require a series of computations to ndtset 3#DATASET1 : scf calculation: GS WF in the BZ#******************************************** prtden1 1 prtwf1 1 kptopt1 1 tolvrs1 1.0D-18 nstep1 8#DATASET2 : non scf calculation: GS WF in the whole BZ#***************************************************** getden2 1 kptopt2 2 iscf2 -2 getwfk2 1 tolwfr2 1.0d-22 berryopt2 -2 # berryopt -2 provides the DDK perturbation through a finite # difference formula, roughly |du/dk> = (|u_k+dk> - |u_k-dk>)/(2*dk) # It would also be possible to use rfddk 1 here, and compute the DDK # wavefunctions from within the DFPT formalism rfdir2 1 1 1 prtwf2 1#DATASET3 : linear response to atomic displacements#************************************************** getwfk3 2 rfdir3 1 1 1 rfphon3 1 rfatpol3 1 2 rfstrs3 3 rfelfd3 3 getddk3 2 tolvrs3 1.0d-12 kptopt3 2 nstep3 8#Definition of the unit cell# these cell parameters were derived from a relaxation run done acell 7.2728565836E+00 7.2728565836E+00 7.2728565836E+00 Bohr rprim 0.0000000000E+00 7.0710678119E-01 7.0710678119E-01 7.0710678119E-01 0.0000000000E+00 7.0710678119E-01 7.0710678119E-01 7.0710678119E-01 0.0000000000E+00#Definition of the atom types and pseudopotentials ntypat 2 # two types of atoms znucl 15 13 # the atom types are Phosphorous and Aluminum#Definition of the atoms natom 2 # two atoms in the cell typat 1 2 # type 1 is Phosphorous, type 2 is Aluminum (order defined by znucl above and pseudos list) nband 4 # nband is restricted here to the number of filled bands only, no empty bands. #atomic positions. xred # atomic positions are given in units of the primitive cell vectors, 1/4 1/4 1/4 # P position0 0 0 # Al position#Numerical parameters of the calculation : planewave basis set and k point gridecut 5 # this value is very low but is used here to achieve very low calculation times. # in a production environment this should be checked carefully for convergence and # a more reasonable value is probably around 20ecutsm 0.5dilatmx 1.05ngkpt 6 6 6nshiftk 4 # this Monkhorst-Pack shift pattern is used so that the symmetry of the shifted grid # is correct. A gamma-centered grid would also have the correct symmetry but would be # less efficient.shiftk 0.5 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.5# by default, prtwf 0prtden 0prteig 0

不同的是加入了berryopt参数，使用了有限差分方法，作为对比，获得波恩有效电荷为2.254，和上面的计算结果相吻合。使用后处理程序Anaddb获得压电常数结果，输入以下input和files文件：

# General information#******************** rfmeth 1 chneut 2# Flags#******* elaflag 3 piezoflag 3 instrflag 1

tffield_3.abitffield_3.abotffield_2o_DS3_DDBtffield_3_band2epstffield_dummy1tffield_dummy2tffield_dummy3

得出压电常数结果：

Proper piezoelectric constants (clamped ion) (unit:c/m^2) 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 -0.00000000 -0.00000000 -0.00000000 -0.64263948 0.00000000 0.00000000 0.00000000 -0.64263948 0.00000000 0.00000000 0.00000000 -0.64263948 .... Proper piezoelectric constants (relaxed ion) (unit:c/m^2) 0.00000000 0.00000000 -0.00000000 0.00000000 -0.00000000 -0.00000000 0.00000000 -0.00000000 -0.00000000 0.13114427 0.00000000 -0.00000000 0.00000000 0.13114427 -0.00000000 -0.00000000 -0.00000000 0.13114427

利用贝利相方法同样可以计算压电常数：

禁锢离子：

# The calculation proceeds by computing the change in cell polarization by a Berrys phase formula# due to changes in the strain of the unit cell. #"Clamped-ion" refers to the fact that the ions move with the strains but are not # allowed to relax to their new equilibrium positions# Berry phase calculation of the polarization# berryopt -1 triggers the implementation of M. Viethenberryopt -1rfdir 1 1 1# three cell geometries will be computedndtset 3#Definition of the unit cellacell 3*7.2728565836E+00# rprim = strain tensor x rprim0, where rprim0 is the unstrained rprim# strain: e4 = 0.00rprim1 0.0000000000E+00 7.0710678119E-01 7.0710678119E-017.0710678119E-01 0.0000000000E+00 7.0710678119E-017.0710678119E-01 7.0710678119E-01 0.0000000000E+00 #strain: e4 = 0.01rprim2 0.0000000000E+00 7.1064231509E-01 7.1064231509E-017.0710678119E-01 3.535533906E-03 7.0710678119E-017.0710678119E-01 7.0710678119E-01 3.535533906E-03# strain: e4 = -0.01rprim30.0000000000E+00 7.0357124728E-01 7.0357124728E-017.0710678119E-01 -3.535533906E-03 7.0710678119E-017.0710678119E-01 7.0710678119E-01 -3.535533906E-03tolsym 1.0e-8 # Since we make rather small lattice vector changes, with the default value of tolsym # ABINIT will recognize that we are close to a more symmetric # configuration, and will resymmetrize the primitive vectors, something we want to avoid ...#Definition of the atom types ntypat 2 # two types of atoms znucl 15 13 # the atom types are Phosphorous and Aluminum#Definition of the atoms natom 2 # two atoms in the cell typat 1 2 # type 1 is Phosphorous, type 2 is Aluminum (order defined by znucl above and pseudos list) nband 4 # nband is restricted here to the number of filled bands only, no empty bands. The theory of # the Berrys phase polarization formula assumes filled bands only. xred 1/4 1/4 1/40 0 0#Numerical parameters of the calculation : planewave basis set and k point gridecut 5 # this value is very low but is used here to achieve very low calculation times. # in a production environment this should be checked carefully for convergence and # a more reasonable value is probably around 20ecutsm 0.5dilatmx 1.05ngkpt 6 6 6nshiftk 4 # this Monkhorst-Pack shift pattern is used so that the symmetry of the shifted grid # is correct. A gamma-centered grid would also have the correct symmetry but would be # less efficient.shiftk 0.5 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.5#Parameters for the SCF proceduretoldfe 1.0d-15nstep 8# suppress printing of density, wavefunctions, etc for this tutorialprtwf 0prtden 0prteig 0

弛豫离子：

# The calculation proceeds by computing the change in cell# polarization by a Berrys phase formula, due to changes in the strain of the unit cell. In this case, the ion positions# will be allowed to relax to to their new equilibrium positions within the strain cell.# Berry phase calculation of the polarization# berryopt -1 triggers the implementation of M. Viethenberryopt -1rfdir 1 1 1# three cell geometries will be computedndtset 3#Definition of the unit cellacell 3*7.2728565836E+00# rprim = strain tensor x rprim0, where# rprim0 is the unstrained rprim# strain: e4 = 0.00rprim1 0.0000000000E+00 7.0710678119E-01 7.0710678119E-017.0710678119E-01 0.0000000000E+00 7.0710678119E-017.0710678119E-01 7.0710678119E-01 0.0000000000E+00 #strain: e4 = 0.01rprim2 0.0000000000E+00 7.1064231509E-01 7.1064231509E-017.0710678119E-01 3.535533906E-03 7.0710678119E-017.0710678119E-01 7.0710678119E-01 3.535533906E-03# strain: e4 = -0.01rprim30.0000000000E+00 7.0357124728E-01 7.0357124728E-017.0710678119E-01 -3.535533906E-03 7.0710678119E-017.0710678119E-01 7.0710678119E-01 -3.535533906E-03tolsym 1.0e-8 # Since we make rather small lattice vector changes, with the default value of tolsym # ABINIT will recognize that we are close to a more symmetric # configuration, and will resymmetrize the primitive vectors, something we want to avoid ...#Definition of the atom types and pseudopotentials ntypat 2 # two types of atoms znucl 15 13 # the atom types are Phosphorous and Aluminum#Definition of the atoms natom 2 # two atoms in the cell typat 1 2 # type 1 is Phosphorous, type 2 is Aluminum (order defined by znucl above and pseudos list) nband 4 # nband is restricted here to the number of filled bands only, no empty bands. The theory of # the Berrys phase polarization formula assumes filled bands only. #atomic positions, given in units of the cell vectors. Thus as the cell vectors#change due to strain the atoms will move as well.xred 1/4 1/4 1/40 0 0# Relaxation of atomic positionsoptcell 0 # the cell geometry is not allowed to changedionmov 2 # the ion positions will be relaxedtolmxf 1.0d-5ntime 13#Numerical parameters of the calculation : planewave basis set and k point gridecut 5 ecutsm 0.5dilatmx 1.05ngkpt 6 6 6nshiftk 4 # this Monkhorst-Pack shift pattern is used so that the symmetry of the shifted grid # is correct. A gamma-centered grid would also have the correct symmetry but would be # less efficient.shiftk 0.5 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.5#Parameters for the SCF proceduretolvrs 1.0D-19nstep 8# suppress printing of density, wavefunctions, etc for this tutorialprtwf 0prtden 0prteig 0

应变的施加：e4应变张力乘以晶格矢量：

获得压电常数：未弛豫情况：-0.6427；弛豫情况：0.1310；

在输出结果中：

== DATASET 2 ==========================================================.... Polarization in cartesian coordinates (C/m^2): Total: -0.162420887E+01 -0.162587046E+01 -0.162587046E+01.... == DATASET 3 ========================================================== ... Polarization in cartesian coordinates (C/m^2): Total: -0.161135421E+01 -0.160969264E+01 -0.160969264E+01

+0.01的结果-0.162420887E+01，-0.01的结果-0.161135421E+01，两者之差除以0.02，即可得-0.6427。

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