Energetics of point defects in aluminum via orbitalfree density functional theory
Abstract
The formation and migration energies for various point defects, including vacancies and selfinterstitials in aluminum are reinvestigated systematically using the supercell approximation in the framework of orbitalfree density functional theory. In particular, the finitesize effects and the accuracy of various kinetic energy density functionals are examined. The calculated results suggest that the errors due to the finitesize effect decrease exponentially upon enlarging the supercell. It is noteworthy that the formation energies of selfinterstitials converge much slower than that of vacancy. With carefully chosen kinetic energy density functionals, the calculated results agree quite well with the available experimental data and those obtained by KohnSham density functional theory which has exact kinetic term.
keywords:
Formation energy, migration energy, point defects, orbitalfree density functional theory, aluminumsort&compress
1 Introduction
Neutron and other energetic particles produced by nuclear reactions usually induce significant changes in the physical properties of irradiated materials. Since the radiation defects are often very small and hence not readily accessible to an experimental observation, many atomic, mesoscopic, and continuumlevel models have been developed to understand the irradiation effect on materials in the past Gibson1960 (); Bai2010 (); Fu2005 (); Robinson1974 (). Though great achievements have been obtained with these empirical models, they all make assumptions about the physical laws governing the behaviors of the materials Carter2008 (). By contrast, firstprinciples modeling, which is based on the laws of quantum mechanics, only requires input of the atomic numbers of the elements.
One of the widely used methods for firstprinciples modeling is the KohnSham density function theory (dubbed as KSDFT) Hohenberg1964 (); Kohn1965 (). It has been proven that the KSDFT method can provide reliable information about the structure of nanoscale defects produced by irradiation, and the nature of shortrange interaction between radiation defects, defect clusters, and their migration pathways Dudarev2013 (). However, the traditional KSDFT method is not linear scaling, and at most only a few thousands of atoms can be treated with it using modern supercomputer. Obviously, it is far away from the requirement of simulating the large atomic system for radiation effect. The orbitalfree density functional theory (dubbed as OFDFT) method provides another choice for simulating the radiation effect. Unlike KSDFT, which uses singleelectron orbitals to evaluate the noninteracting kinetic energy, OFDFT relies on the electron density as the sole variable in the spirit of the HohenbergKohn theorem Hohenberg1964 () and is significantly less computationally expensive. The accuracy of OFDFT depends upon the quality of the used kinetic energy density functional (KEDF), which is usually based on the linear response of a uniform electron gas. Note that similar to the exchangecorrelation density functional (XCDF), the exact form of KEDF is not known except in certain limits. Currently, the most popular KEDFs are the WangGovindCarter (WGC) Wang1999 (); Wang2001 () and WangTeter (WT) Wang1992 () functionals. Both were designed to reproduce the Lindhard linear response of a freeelectron gas Lindhard1953 ().
In order to apply the OFDFT method to study radiation defects in realistic materials, it is essential to evaluate the accuracy of various KEDFs. The formation and migration energetics of typical point defects in simple metal aluminum are very useful test beds. Actually when a new KEDF or formulation was proposed, the vacancy formation energy in Al was always calculated to make a comparison with the experimental value and the KSDFT results Wang1992 (); Perrot1994 (); Smargiassi1994 (); Foley1996 (); Wang1998 (); Wang1999 (); Gavini2007b (). For example, Foley and Madden Foley1996 () generalized the WTKEDF and calculated the relaxed vacancy formation energy in Al on 32 and 108site cells, and later Jesson et al. Jesson1997 () use the same KEDF to calculate the formation and migration energies of various selfinterstitials using 108 and 256site cell. But the estimated values differ strongly from the experimental values and the KSDFT results using the same supercell. Later Carter’s group Wang1999 () proposed the densitydependent WGCKEDF and also calculated the vacancy formation energy in Al using a 4site and 32site cell. The estimated value (0.6100.628) is in very good agreement with the experimental value (0.67) and the KSDFT result in this work (0.626). Lately the same group Ho2007 () used various supercell with number of sites up to 1372 to calculate the vacancy formation and migration energies. But it was found that the calculated results from OFDFT systematically underestimated the measured values or the KSDFT results by 0.2 eV. Recently Gavini’s group Gavini2007 (); Gavini2007b (); Radhakrishnan2010 (); Das2015 () proposed the nonperiodic realspace formulation for OFDFT and used this formulation to calculate the vacancy formation energy and also the unrelaxed case. The size effect is also considered here. The results are in good agreement with those obtained from the OFDFT calculation using planwave basis Wang1999 (); Ho2007 (). Despite those above abundant researches on the properties of point defect in Al, relatively little is known regarding the OFDFT study of the selfinterstitials and the corresponding size effect. Thus it is necessary to systematically calculate the formation and migration energies of vacancy and various selfinterstitials by employing the OFDFT method with various KEDFs and XCDFs.
The effect of supercell size on the defect energetics is also a major concern of this work. Due to the periodical boundary condition (PBC) in the routine DFT calculation using the popular planewave basis, there may be crossboundary effects and defectdefect interactions, and therefore a different system other than the intent was studied. For KSDFT, the study on the size effect is limited by the small size of the system studied and the imperfection of the Brillouin zone sampling Hine2009 (); Dabrowski2015 (), while these disadvantages disappear for OFDFT. For example, Ho et al. Ho2007 () found that the formation energy is converged within 3 meV by supercell and the migration energy is converged within 1 meV at supercell. This finitesize errors from the PBC could also been circumvented by using a nonperiodic formulation of DFT Gavini2007 (); Gavini2007b (). The corresponding nonperiodic cellsize effect on the energetics of vacancy and divacancy in aluminum using OFDFT was investigated in References Gavini2007b (); Radhakrishnan2010 (); Das2015 (); Radhakrishnan2016 (). It was revealed that more than 2000 sites are required to obtain a converged value for the divacancy.
In the present work, we employ the OFDFT method with WGCKEDF and WTKEDF to calculate the formation and migration energies of typical point defects in facecentered cubic (fcc) aluminum. The simulation cell ranges from to supercells. By comparison, these energies are also calculated by using the KSDFT method and a supercell. The rest of this paper is organized as follows. The computation methods and details are described in Sec. 2. In this section, the typical KEDFs used in the calculations are introduced. The detailed results are presented and discussed in Sec. 3. Section 4 serves as a conclusion.
2 Computational Methods and Details
According to the HohenbergKohn theorem Hohenberg1964 (), the ground state density of interacting electrons in some external potential determines this potential uniquely, and the ground state energy could be obtained variationally,
(1) 
under the constraint that the electron density is nonnegative and normalized to the number of electrons . The minimum is the ground state electron density for a nondegenerate ground state, which determines all the properties of an electronic ground state. In the framework of density functional theory, the total energy functional in Eq. (1) could be expressed as
(2) 
where , and represent the kinetic energy of the ground state of the noninteracting electrons with density , the Hartree electrostatic energy and the exchangecorrelation energy, respectively.
In the framework of OFDFT method, the kinetic term is approximated using KEDF. Here we adopt the most accurate functionals available, i.e., the WangGovindCarter (WGC) Wang1999 (); Wang2001 () and WangTeter (WT) Wang1992 () KEDFs. They consist of the ThomasFermi (TF) functional Thomas1927 (); Fermi1927 (), the von Weizsäcker (vW) functional Weizsaecker1935 (), and a linear response term. In reduced units, the TFKEDF is given by
(3) 
where is the Fermi wave vector of a uniform electron gas of density and is the mean kinetic energy per electron of such a gas. The vWKEDF reads
(4) 
which could be obtained from a singleorbital occupied system. Response functions of an electronic system is of vital importance to understand its physical properties, and for a noninteracting electron gas, the correct linear response behavior was derived analytically by Lindhard Lindhard1953 (). In Lindhard’s theory, the static electric susceptibility in reciprocal space is given by
(5) 
where is the average electron density. But for the TFKEDF and vWKEDF, the corresponding susceptibility functions are given by and , respectively. In order to remedy this, one must introduce a linear response term and then the resulting KEDF could be given by
(6) 
For the WTKEDF Wang1992 (), the linear response kernel takes the local form,
(7) 
where denotes the Fourier transform. While for the WGCKEDF Wang1999 (); Wang2001 (), the kernel takes the nonlocal form,
(8) 
and the exact functional form could be determined by solving a secondorder differential equation Wang1999 (); Wang2001 (). Actually, the WGCKEDF kernel is evaluated by performing a Taylor series of with being a reference density and usually chosen to be the average density . In this work, the Taylor series expansion are evaluated up to second order. The exponent and could be treated as fitting parameters Wang1992 (); Pearson1993 (); Foley1994 (); Smargiassi1994 (); Smargiassi1995 (); Foley1996 (); Jesson1997 () or determined from an asymptotic analysis Wang1998 (); Wang1999b (); Wang2001b (). In the following, we choose and for WGCKEDF and for WTKEDF. The exponent is a materialspecific adjustable parameter. According to the literatures, is found to be optimal for Al Wang1999 (); Wang2001 ().
In the KSDFT method Kohn1965 (), the KEDF could be expressed in terms of the KohnSham orbitals ,
(9) 
where the sum is over the lowestenergy orbitals and represents the corresponding occupancy in the orbital . The KohnSham orbitals satisfies . In principles, the expression in Eq. (9) is exact, while those in Eq. (3), (4), and (6) are not.
For KSDFT, the commonly used pseudopotential schemes, which represent in Eq. (2), are usually nonlocal and could be expressed by using KohnSham orbitals. However, since there is no orbitals in OFDFT, the socalled bulkderived local pseudopotential (BLPS) Zhou2004 (); Huang2008 () is used here for describing the potential . Both in KSDFT and OFDFT, the exchangecorrelation energy functionals in Eq. (2) are described by using the local density approximation (LDA) Perdew1981 () and the generalized gradient approximation (GGA) of Perdew, Burke and Ernzerhof (PBE) Perdew1996 ().
In the present work, we use the PROFESS code to do the OFDFT calculations Ho2008 (); Hung2010 (); Chen2015 (). Multiple simulation box sizes, ranging from to supercells, were tested here. Let us denote the supercell size as , then the corresponding number of atoms in the perfect supercell of fccAl is given by which ranges from 108 to 10976. For electron density optimization, the kinetic energy cutoff is set to 1200 eV, and the square root truncated Newton minimization method is used with energy convergence threshold being 2.7210 eV. For ion relaxation, dynamical boundary conditions are employed, allowing cell volume and cell shape to change, along with the relaxation of atom positions. In addition, both the conjugate gradient and quickmin algorithm are used with convergence threshold for the maximum force component on any atom being 2.5710 eVÅ.
For comparison, we also calculated the point defect energies in a supercell using the VASP code which implemented the KSDFT method Kresse1996 (). The projector augmented wave (PAW) Blochl1994 (); Kresse1999 () pseudopotential for Al with the valence electronic configuration is chosen. The kinetic energy cutoff is taken as 400 eV and the point meshes based on MonkhorstPack scheme Monkhorst1976 () are adopted. Methfessel and Paxtonâs smearing method Methfessel1989 () of the first order is used with a width of 0.1 eV to determine the partial occupancies for each KohnSham orbitals. Relaxations are performed by employing the dynamical boundary conditions and using the conjugate gradient and quasiNewton algorithm with a convergence criterion of 1 meV with regards to the total free energy of the system.
3 Results and Discussion
3.1 Equilibrium lattice constant
Method  KEDF  XCDF  (Å) 

KSDFT    LDA  3.9830 
KSDFT    GGA  4.0387 
OFDFT  WGC  LDA  3.9725 
OFDFT  WGC  GGA  4.0579 
OFDFT  WT  LDA  3.9849 
OFDFT  WT  GGA  4.0676 
KSDFT Narasimhan1995 (); Huang2008 (); Das2015 ()    LDA  3.9453.968 
KSDFT Zhuang2016 ()    GGA  4.063 
OFDFT Carling2003b ()  Perrot Perrot1994 ()  LDA  4.06 
OFDFT Carling2003b ()  SM Smargiassi1994 ()  LDA  3.96 
OFDFT Carling2003b ()  WT Wang1992 ()  LDA  4.04 
OFDFT Carling2003b ()  WGC Wang1999 (); Wang2001 ()  LDA  4.034.04 
OFDFT Radhakrishnan2010 ()  WGC Wang1999 (); Wang2001 ()  LDA  4.022 
OFDFT Das2015 ()  WGC  LDA  3.973 
OFDFT Zhuang2016 ()  WGC  GGA  4.039 
OFDFT Jesson1997 ()  FM Foley1996 ()  LDA  4.0270 
Experiment Bandyopadhyay1978 ()      4.0315 
For the bulk properties and the energies of several competing phases of bulk Al, Shin et al. Shin2009 () has already made detailed comparison between the OFDFT and KSDFT methods with various exchangecorrelation density functionals (XCDFs) and KEDFs. It was found that the OFDFT results accurately reproduced those by the KSDFT method. This manifests that the OFDFT method with reliable BLPS and KEDF is an accurate simulating tool for perfect crystal. Before introducing the point defect, here we preformed a structural relaxation on the perfect supercell for fccAl to figure out the equilibrium lattice constant . In Table 1 we compare the lattice constants of fccAl obtained from various OFDFT and KSDFT calculations with the experimental value and the previous theoretcial results. The experimental value shown in Table 1 is obtained by extrapolating to 0 K using the polynomial proposed in Ref. Bandyopadhyay1978 (). Clearly, all the lattice constants obtained from OFDFT are accurate enough to perform the further investigation of defect energetics.
3.2 Formation energy of vacancy
Method  KEDF  XCDF  (eV) 

KSDFT    LDA  0.7291 
KSDFT    GGA  0.6646 
OFDFT  WGC  LDA  0.7947 
OFDFT  WGC  GGA  0.7213 
OFDFT  WT  LDA  1.3456 
OFDFT  WT  GGA  1.3001 
KSDFT Chetty1995 (); Turner1997 (); Wang1999 (); Carling2000 (); Baskes2001 (); Carling2003 (); Uesugi2003 (); Lu2005 (); Ho2007 (); Iyer2014 ()    LDA  0.660.73 
KSDFT Carling2000 (); Carling2003 (); Lu2005 ()    GGA  0.540.55 
OFDFT Foley1996 ()  FM Foley1996 ()  LDA  0.29 
OFDFT Wang1999 (); Ho2007 (); Gavini2007b (); Radhakrishnan2010 ()  WGC  LDA  0.480.72 
OFDFT Ho2007 ()  WGC  GGA  0.387 
Experiment Ullmaier1991 ()      0.670.03 
Typical defects were then introduced into the fully relaxed supercells, and the structural relaxation without any symmetry constraints was performed again for the given supercell to calculate formation energies. For a single vacancy defect, it is created by eliminating one central atom in the supercell and the corresponding formation energy is defined as Jesson1997 ()
(10) 
where is the total energy for the supercell containing an Al atoms and one vacancy, and is the total energy of a perfect aluminum supercell containing Al atoms. For an Al interstitial defect, it is generated by adding a single Al atom into the supercell in different interstitial positions around the center. The corresponding formation energy is given by
(11) 
where is the total energy for the supercell containing Al atoms at lattice sites and one interstitial aluminum atom. Since the full relaxation is performed on the prefect and defectcontaining system, and are the formation energy at constant zero pressure Gillan1989 (); Ho2007 (); Iyer2014 ().
We at first try to test the size dependence of vacancy formation energy . The results are shown in Figure 1. The axis is the size of supercell . For a specified defect, the distance between it and its image defect located in the neighboring cell is given by . Surely, the increment of leads to larger , and thus reduces the defectdefect interaction. As a consequence, the formation energy approaches the converged value asymptotically upon increasing . This tendency is clearly seen in Figure 1 for different computational methods. In particular, the WGCKEDF converges more rapidly than the WTKEDF, and the LDAXCDF performed better than the GGAXCDF. Note that the WGCKEDF plus LDAXCDF performed extremely well and was chosen in the previous literatures Ho2007 (); Radhakrishnan2010 (); Das2015 ().
In order to model the convergent tendency, we used a simple exponential function
(12) 
to fit the vacancy formation energy. The fitting curves are also shown in Figure 1 and apparently, the fit is very well. Since the fit always give a positive value of in equation (12), the value of may be regarded as the formation energy in the thermodynamic limit (). Here we realized that finitesize errors decrease exponentially as is increased, and finitesize scaling could give a reliable value.
The convergence of with respect to was already found in the previous literatures Ho2007 (); Radhakrishnan2010 (); Iyer2014 (); Das2015 (). For WGCKEDF, Ho et al. Ho2007 () found that the formation energy is converged within 3 meV by supercell, which is also confirmed by Gavini’s group Radhakrishnan2010 (); Iyer2014 (); Das2015 () and our calculation (see Figure 1). But for WTKEDF, the convergence of with respect to is not as quick as that for WGCKEDF. For a convergence criterion of 1 meV, a supercell size of is required for the vacancy formation energy.
The vacancy formation energies of fccAl obtained with the OFDFT method using various XCDFs and KEDFs are collected and summarized in Table 2. For comparison, the related KSDFT values for a supercell, the previous theoretical results from KSDFT and OFDFT, and the experimental value are also shown. Clearly, the calculated formation energies from the OFDFT method with WGCKEDF are close to that of KSDFT. However, since the kernel in WTKEDF is densityindependent, it isn’t suitable for systems where the electron density varies rapidly, such as vacancies or surfaces Wang1999 (); Wang2001 (). It is not surprised that the results obtained with the WTKEDF deviate from the others apparently. In addition, both the values from KSDFT and OFDFT with WGCKEDF are in good agreement with the experimental value.
3.3 Formation energies of selfinterstitials
Now let’s turn to the selfinterstitials in fccAl. Five types of selfinterstitials, including dumbbell, dumbbell, dumbbell, octahedral and tetrahedral interstitials, are considered in the present work. The corresponding formation energies are denoted as , , , , and , respectively. The relaxed geometry configurations from KSDFT are shown in Figure 2. Those configurations obtained by the OFDFT method are very similar and wouldn’t be shown here.
Being analogous to the vacancy formation energies as discussed before, the interstitial formation energies also show a convergent behavior with respect to the supercell size. As an example, the formation energies of dumbbell interstitial as a function of and the fitting curves are shown in Figure 3. Note that the values obtained from small supercell sizes deviate from the fitting curves and were not used in the fitting process. The convergence speed of is much slower than that of . For a rough convergence criterion of 5 meV, a supercell size of is required for the formation energy of the dumbbell interstitial. If we only focus on for the convergent behavior, the WGCKEDF does not perform better than the WTKEDF, which differs from the case of vacancy formation energy. Here the values in Eq. (12) from the exponential fit are also taken as the formation energies in the thermodynamic limit.
Table 3.3 shows the interstitial formation energies of fccAl obtained from the OFDFT and KSDFT methods with various density functionals. The pervious theoretical results from KSDFT Iyer2014 () and OFDFT Jesson1997 () using supercell, and the experimental value were also shown for comparison. Note that the experimental value was estimated from the experimental value of Frenkel pair formation energy, 3.7 eV Ullmaier1991 () and the vacancy formation energy, 0.670.03 eV Ullmaier1991 (). We met convergence problems when we performed OFDFT calculations with WGCKEDF except for the dumbbell configuration. In order to cure this trouble, we repeated the OFDFT calculations by setting in the WGCKEDF (denoted as WGC* in the following). The calculated results show that the modified WGCKEDF leads to slightly larger formation energies.
[th]
Method KEDF XCDF KSDFT  LDA 2.6073 2.9809 3.1821 2.9485 3.2941 KSDFT  GGA 2.4597 2.7508 3.0183 2.7945 3.1072 OFDFT WGC LDA 2.5862 OFDFT WGC^{*} LDA 2.6031 2.8854 3.0986 2.8728 3.1466 OFDFT WGC GGA 2.3925 OFDFT WGC^{*} GGA 2.4200 2.6856 2.8784 2.6724 2.9217 OFDFT WT LDA 2.4257 2.7000 2.8507 2.6672 2.8749 OFDFT WT GGA 2.2943 2.5422 2.6789 2.5302 2.6998 KSDFT Iyer2014 ()  LDA 2.9   3.1 3.5 OFDFT Jesson1997 () FM Foley1996 () LDA 1.579 1.869 1.959 1.790 1.978 Exp. Ullmaier1991 ()   3.0

* Here in WGCKEDF is set to be zero.

The average value of formation energy, not the value of .
It is surprising to found that the OFDFT results with WGCKEDF reproduce extremely well the KSDFT values with errors being less than 0.15 eV. In addition, we find that for LDAXCDF, both OFDFT and KSDFT calculations yield,
It is consistent with the experimental finding that the dumbbell is the favorite interstitial configuration Ehrhart1973 (); Ehrhart1974 (); Robrock1976 (). In addition, this order was the same as that of OFDFT calculation with the FMKEDF Jesson1997 () and that of KSDFT calculation Iyer2014 (). For GGAXCDF, the only exception is for the KSDFT case.
The results from KSDFT and OFDFT with WGCKEDF are both consistent with the experimental value 3.0 eV Ullmaier1991 (). But the results from OFDFT plus WTKEDF and FMKEDF deviate from the experimental value since all the calculated values for different geometries are smaller than 3.0 eV. The results Jesson1997 () from FMKEDF differ strongly from our values, and also the KSDFT and experimental data. Those discrepancies may stem from the local pseudopotentials. For KSDFT, the estimated values in our work is in agreement with the previous theoretical results and the difference may result from the different supercells and computational parameters.
3.4 Migration energies of vacancy and dumbbell
The general migration pathway of a vacancy is that one atom moves towards the adjacent vacancy, eliminating the vacancy and forming a new vacancy. For the dumbbell, the easiest migration pathway was already discussed by Jesson et al. Jesson1997 (). In this case, one dumbbell atom moves towards an adjacent interstitial site and the other dumbbell atom returns to the lattice site, forming a new or dumbbell. The intermediate configurations for migration of vacancy and dumbbell are depicted in Figure 4. In order to calculate the migration energy, i.e., the potential barrier in the migration pathway, all of the possible intermediate configurations are relaxed at first, and then their energies are calculated. Then the migration energy is given by the difference between the maximum total energy of intermediate configuration and the total energy of initial configuration. For KSDFT with LDAXCDF, the intermediate atomic configurations for migration of vacancy and dumbbell with maximum total energy are shown in Figure 4.
The migration energies also show a convergent behavior with respect to the supercell size . In Figure 5, the supercell size dependence of vacancy migration energy are shown. The convergence speed of is not as good as that of . Especially, it is difficult to obtain converged with the OFDFT method using WTKEDF, which confirms again that the WTKEDF isn’t suitable for describing systems with vacancy defects. In addition, the convergence of is not as good as that of . Thus in the following OFDFT calculations, we just use the results from the supercell.
Table 3.4 shows the migration energies of vacancy and dumbbell obtained from the OFDFT and KSDFT methods. We also show the experimental values and the available theoretical results for comparison. The vacancy migration energy obtained by the OFDFT method with WGCKEDF reproduced well that from the KSDFT method and also the experimental value. As illustrated above, due to the densityindependent kernel, the values from OFDFT with WTKEDF can’t reproduce that from KSDFT and also in bad agreement with the experimental values.
For the dumbbell migration energy , the OFDFT results differ from the results from KSDFT. However, the values from the OFDFT method are in close agreement with the experimental results while the KSDFT results are not. In addition, the deviations of the OFDFT results from the previous theoretical results are acceptable.
[th] Method KEDF XCDF KSDFT  LDA 0.5234 0.2130 KSDFT  GGA 0.4904 0.2179 OFDFT WGC LDA 0.5846 ^{‡} OFDFT WGC^{*} LDA 0.6251 0.1168 OFDFT WGC GGA 0.5569 ^{‡} OFDFT WGC^{*} GGA 0.5964 0.1101 OFDFT WT LDA 0.3135 0.1119 OFDFT WT GGA 0.3041 0.1041 OFDFT Jesson1997 () FM Foley1996 () LDA  0.084 OFDFT Ho2007 () WGC LDA 0.42  Exp. Ullmaier1991 ()   0.610.03 0.1150.025

* Here a modified WGCKEDF is used with .

In this scheme, the electron density does not converge.
4 Conclusion
In summary, the OFDFT method combined with the WGCKEDF and WTKEDF was employed to calculate the formation energies and migration energies of typical point defects in fccAl supercell up to 10976 atoms. The finitesize errors arising from the supercell approximation are examined and could be corrected for using finitesize scaling methods. The convergence of interstitials formation energies is much slower than that of vacancy formation energy. And our cellsize study of dumbbell interstitial has shown that it is converged within 5 meV by a supercell. We compared the accuracy of the commonly used KEDFs. We found that with the WGCKEDF, the calculated results agree quite well with the more accurate data obtained by KSDFT calculations. Sometimes it is not easy to obtain converged results with WGCKEDF. Usually we can apply a slightly modified WGCKEDF in which to overcome this obstacle. On the other hand, the WTKEDF tends to give a wrong estimation especially for vacancy, and is not suitable for the simulating of defectcontaining systems.
Our results suggest that with carefully chosen KEDF, the accuracy of OFDFT calculation is comparable with that of the KSDFT calculation which is more demanding in computer resources. So it is promising to apply the OFDFT method with WGCKEDF to study largescale systems with defects, such as the collision cascade process in irradiated materials.
Acknowledgement
This work was supported by the NSFC (Grant Nos. 11404299, 11305147, and 21471137), the ITER project (No. 2014GB111006), the National 863 Program (No. SQ2015AA0100069), the Foundation of President of CAEP (Nos. 2014158 and 2015208), and the Foundation for Development of Science and Technology of CAEP (Grant No. 9090707).
References
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