关注：

1) 在满足要求的前提下，降低参数的收敛标准，提高计算速度

声子谱计算

PREC = Accurate
ENCUT = 300.0    【ENCUT为500 eV时，计算非常耗时，远远超过300 eV时的十倍，甚至数十倍！】
LREAL = .False
ISMEAR = 1
EDIFF = 1E-7

#LSDA-plus-U
ISPIN=2
LDAU = .TRUE.
LDAUTYPE = 2
LDAUL = -1  3
LDAUU = 0.00 4.10
LDAUJ = 0.0   0.1
LDAUPRINT = 0
LMAXMIX = 6

#PSTRESS = 2500
IBRION = 8
LWAVE = .FALSE
LCHARG = .FALSE
ADDGRID = .TRUE

Thanks to Huayun:

All total energy calculations were carried out within the framework of DFT^25,26based on the frozen core all-electron projector-augmented wave (PAW) method,^27,28as implemented in the VASP code.^29,30

The spin-polarized GGA was employed, andthe exchange-correlation energy of electrons was parameterized by the Perdew-Burke-Ernzerhof(PBE) functional.³¹ Akinetic energy cutoff of 500 eV was used for theplane wave basis set, and the electronic self-consistency convergence was setto 10^-6 eV for all calculations. 【电子自洽收敛10^-7,则可能收敛太慢；只有竞争相能量相差非常小时，或可设置更高精度】

Integrations over reciprocal space were performed on an automatically generated Monkhorst-Pack grid,³²which includes at least 18 nonequivalent k pointsin the irreducible Brillouin zone for cubic phases, and more than 32-48 nonequivalent kpoints for cotunnite and other noncubicphases.

Lattice parameters and atomic positions thatcannot be fixed by symmetry were fully optimized at fixed volumes, witha Hellmann-Feynman force tolerance less than 2 × 10^-3 eV/Å for all structures of Z = 1-4,and 2 ×10^-2 eV/ Å for Z = 8structures (Pbca and P112₁/a in the Supplementary Material³³).【不同大小的胞，力的收敛可不一样，或可提高速度？】

To improve the convergence of integration over reciprocal space we adoptedthe smearing scheme of tetrahedron method with Blöchl corrections,³⁴which is fairly accurate. 【smear能改善优化收敛】

For all calculations, electrons of 6s²6p⁶5f³6d¹7s² for uranium, 6s²6p⁶5f⁶7s² for plutonium, and 2s²2p⁴ for oxygen weretreated as valence configurations.

The on-site Coulomb interaction among localized 5felectrons of uranium and plutoniumatoms was described by a Hubbard U term,after the DFT+U scheme introduced by Anisimov et al.^35–38This method has proved successful in capturing the band structuralfeatures of strongly correlated systems. In all DFT+U calculations,we adopted the Dudarev version,38 and set the semi-empiricalparameters U= 4.5 eV, J = 0.51 eV forUO₂ and U= 4 eV, J= 0.7 eV for PuO₂, which have beenextensively checked by many authors.^7,39–41

Thanks to ABY:

Here, we employ the non-local vdW densityfunctional (vdW-DF) of Dion et al.^49,50 (i.e., revPBE-vdW) to evaluate the influence of vdWcorrections on the incorporation energies of impurity atoms.

DFT methods

  Total energy calculations are performedwith VASP code,⁴³ projector augmented-wave (PAW) method and  relativistic effective core potentials (ECPs) potentials【在哪个参数上体现？】.

 The exchange and correlation interactions are comparatively described bygeneralized gradient  approximation (GGA)   in the Perdew–Burke–Ernzerhof (PBE)and Perdew–Wang 91  (PW91) functional.^44,45 

The Hubbard model is usedto treat strong correlations within DFT + Umethod in Dudarev formalism.^46-48 An effective U(U_eff= U−J,i.e., the difference between the Coulomb U and exchange J parameters, herein after  referred to as U)  value of 4 eV is selected for the localized 5f electrons of X.  This value has been  proven to be reasonable in reproducing experimental lattice parameter, bulk modulus and band gap of XO₂,  and the reaction energies  related to XO₂ by our previous calculations.³⁶ 

In order to describe the weak interactions of the systems containing RG atoms, van der Waals (vdW)  dispersion  correctionshad better be considered. Here, we employ the non-local vdW density functional (vdW-DF)of Dion et al.^49,50(i.e., revPBE-vdW) to evaluate the influence of vdW corrections on the incorporation energies of impurity atoms.

 Our previous calculations showed that antiferromagnetic (AFM) states are energetically more  favorable than  nonmagnetic (NM) and ferromagnetic(FM) states, and the magnetic orders such as collinear 1-k  and noncollinear 2-k configurations have insignificant influence on the energetics of the  incorporated O  atom. The more  complicated non-collinear 3-k AFM configurations have been focused only on UO₂.^16-18 For the sake of the computation efficiency of so many calculation models, we decide to leave the 3-ks tructure outside of scope for our present calculations.

Therefore, only theresults of the collinear 1-k AFM configurations related to the (100)  lattice directions  are discussed. 

Fully relaxations without symmetry constraints are adopted. We find that the total energies of the configurations relaxed without symmetry constraints are always smaller than those relaxed with symmetryconstraints. Convergence is reached when the total energies converge within 1×10⁻⁵ eV  and Hellmann-Feynman forces on each ion are less than 0.02 eV/Å.

【总能和力的精度不高，可显著提高速度，又可满足精度需求】

The use of a plane-wave kinetic energycutoff of 500 eV and 9 × 9 × 9 Monkhorst-Pack k-point sampling are shown to  give accurate convergence of total energies【如何确保总能收敛？】.

For the total energy and density of state (DOS) calculations, the tetrahedron method with Blöchl correction  is used for  the Brillouin-zoneintegration.⁵¹

Spin-orbit coupling (SOC) interactions have also been proven to be negligible for the calculations of energetics  of  impurity atoms, despite that SOC may play an important role in reproducing ground-state of actinide materials.^27,28 This is mainly due to the fact that the possible calculated energy errors in XO₂with and without impurity atoms using the same theoretical methods areapproximately in the very close level.

Pure spin-polarized DFT calculations are performed in the calculations of the total energies of impurity  atoms  in their molecular states. This means that we use a half of the total energy of a diatomic molecule  (e.g., H₂, O₂,  N₂,F₂ and Cl₂) as the total energy of an impurity atom.

Forthe well-known disadvantages of pure DFT in describing molecules, the calculation  scheme of total energy of a diatomic molecule similar to the one proposed by Korzhavyiet al.³³ is adopted.

For example, the total energy of H₂ is obtained by the sum of the energy of free atom H and the well-established  dimerization  energy,⁵²i.e., the reaction energy of 2H = H₂. The total energy of a freeatom  is calculated  by using  a periodic cubic cell with a lattice constants of 15 Å and only one k point Γ. Certainly,  the total  energies of  R G atoms are calculated by the method. Both B and C are in the states of molecule crystals  under standard conditions; therefore, we calculate the total energies of their molecule crystals, i.e., α-B and graphite, respectively, to obtain  the total  energies of single B atom andsingle C atom.

Interestingly,one can find that the volume of X₄O₇ is larger than thatof X₄O₈. In fact, the phenomenon can be viewed as the results of the opposite effects of size effects and charge-transfer effects, in other words,  the opposite  effects  of expansion and contraction, just as the case that the volume of X₄O₉ is slightly smaller  than that  of  X₄O₈. When an O atom is removed from X₄O₈, f electrons of X become more localized, facilitating the  formation   of lower valence X ion with larger atomic radius. If the charge-transfer effects prevail  the size  effects, the models  contract and viceversa.³²