Abstract
The crystal, electronic and magnetic structures of solid oxygen in the epsilon phase have been investigated using the strongly constrained appropriately normed (SCAN) + rVV10 method and the generalized gradient approximation (GGA) + vdWD + U method. The spinpolarized SCAN + rVV10 method with an 8atom primitive unit cell provides lattice parameters consistent with the experimental results over the entire pressure range, including the epsilonzeta structural phase transition at high pressure, but does not provide accurate values of the intermolecular distances d_{1} and d_{2} at low pressure. The agreement between the intermolecular distances and the experimental values is greatly improved when a 16atom conventional unit cell is used. Therefore, the SCAN + rVV10 method with a 16atom unit cell can be considered the most suitable model for the epsilon phase of solid oxygen. The spinpolarized SCAN + rVV10 model predicts a magnetic phase at low pressure. Since the lattice parameters of the predicted magnetic structure are consistent with the experimental lattice parameters measured at room temperature, our results may suggest that the epsilon phase is magnetic even at room temperature. The GGA + vdWD + U (with an ad hoc value of U_{eff} = 2 eV at low pressure instead of the firstprinciples value of U^{lr}_{eff} ~ 9 eV) and hybrid functional methods provide similar results to the SCAN + rVV10 method; however, they do not provide reasonable values for the intermolecular distances.
Introduction
Oxygen, the third most abundant element in the universe^{1}, makes up approximately 46% of the components of the Earth’s crust and 21% of the atmosphere. Oxygen can exist not only on the Earth’s surface but also deep inside the mantle layers. According to molecular theory, an oxygen molecule O_{2} has two unpaired electrons that make O_{2} an active substance. Oxygen can therefore easily form compounds with many elements in the Earth’s mantle. The depth of the Earth’s mantle ranges from approximately 100 km to 3000 km, and the corresponding pressure ranges from approximately a few GPa up to 150 GPa. Under such high pressure, O_{2} gas must be solidified. To understand O_{2} and its compounds in the mantle, the structural evolution, electronic and magnetic properties of molecular oxygen under high pressure are investigated in this article.
The O_{2} molecule under high pressure has been widely studied over many decades^{2,3,4,5,6,7,8,9,10,11}. It was shown in a neutron diffraction experiment^{12} that solid oxygen transforms from the antiferromagnetic delta phase to the nonmagnetic epsilon phase at 7.6~8 GPa, while recent generalized gradient approximation (GGA) + U calculations also suggested a lowpressure antiferromagnetic epsilon phase of oxygen (from ~10 GPa up to 20 GPa) before it completely transforms into the nonmagnetic epsilon phase^{13,14}. To examine the ability of stateoftheart density functional theory (DFT) methods to predict the crystal and electronic structures of solid oxygen, the results of two quasilocal density functionals combined with van der Waals (and Hubbard U) corrections, i.e., the GGA + vdWD + U method and strongly constrained appropriately normed (SCAN) + rVV10 method^{15}, are compared with experimental results^{9,16}. The GGA + vdWD + U method uses a GGA (PBE^{17}) functional combined with a semiempirical vdW interaction^{18} and a Hubbard U correction for the onsite Coulomb interaction between the p orbitals of the oxygen atom. The SCAN + rVV10 method^{15} uses a metaGGA SCAN functional, which is accurate for short and intermediate ranges, combined with a firstprinciples longrange van der Waals interaction (rVV10). Thus, this approach is expected to accurately reproduce the structures and electronic properties of molecular crystalline solid oxygen. The SCAN + rVV10 method does not require a Hubbard U correction since it describes the onsite Coulomb interaction with sufficient accuracy.
Computational Methods
DFT calculations were performed using the Quantum ESPRESSO package^{19} with normconserving pseudopotentials^{20} and the VASP package^{21} with a projector augmented wave (PAW) pseudopotential. The number of k points in the irreducible Brillouin zone was equal to 88 (5 × 5 × 7 MonkhostPack sampling). The kinetic energy cutoff for the wavefunctions was set at 150 Ry with a 10^{−8} Ry total energy convergence for one selfconsistent field (SCF) cycle. Variablecell optimization was carried out to optimize both the lattice parameters and atomic coordinates. A primitive cell consisting of 8 oxygen atoms was used unless otherwise noted. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm^{22} was used for optimization of both the ion positions and unit cell vectors in compression. The pressure was increased by 10 GPa from 10 GPa to 140 GPa. The convergence threshold of the forces for ionic minimization was set at 5 × 10^{−4} a.u.
To investigate the effects of various conventional functionals, we carry out our calculations for the local density approximation (LDA), BeckeLeeYangParr (BLYP), PerdewBurkeErnzerhof (PBE) and metaGGA (M06L) functionals. The metaGGA (M06L) functional predicts the epsilonzeta transitional pressure (P_{T}) to be 30 GPa with inaccurate lattice parameters compared to the experimental parameters. The BLYP and PBE functionals are the best among these functionals, showing the transition at 40 GPa with better lattice parameters. Moreover, we consider both semiempirical^{18} and nonlocal van der Waals functionals^{23,24}. We find that the van der Waals functionals give similar results to the GGA (PBE) without the van der Waals prediction (transitional pressure of P_{T} = 40 GPa), while the semiempirical GGA + vdWD method^{18} (Grimme potential) results in a small improvement (P_{T} = 50 GPa) with lattice parameters very close to those of the conventional GGA (PBE). The combination of the van der Waals functionals and the Hubbard U correction shows no improvement over the GGA + U method only (P_{T} = 70 GPa). While the combination of semiempirical vdWD^{18} and Hubbard U shows a small improvement with P_{T} = 80 GPa, the lattice parameters calculated with vdWD + U and those with Hubbard U are very similar^{25}. The effect of the van der Waals interaction, therefore, is small compared to the effect of the Hubbard U correction. The details of the comparison between different functionals are mentioned in the Supplementary Materials and reference^{25}. In this study, we use the semiempirical vdWD^{18} (Grimme potential) functional only and vary the value of the constant U_{eff} to investigate the effect of the Hubbard correction. For results with various conventional functionals, please refer to the Supplementary Materials. For the Hubbard term, we use two methods: gradually varying the values of U and the firstprinciples rotationally invariant scheme of Cococcioni^{26}, which can uniquely determine the value of U.
Structural models
In Fig. 1(a), we show top and birdseye views of the conventional unit cell of the epsilon phase and its lattice parameters, where the angle β between the a and c axes is larger than 90 degrees and the other angles equal 90 degrees. The intermolecular distances d_{1} and d_{2} are also defined as the distances between the centres of the O_{2} molecule in an (O_{2})_{4} cluster and between (O_{2})_{4} clusters as indicated in Fig. 1(a). The crystal structure has C2/m symmetry in which four O_{2} molecules gather to form an (O_{2})_{4} cluster^{8}. The primitive cell which consists of 8 atoms is shown in Fig. 1(b). There are three symmetrically inequivalent atoms, O1, O2, and O3, in the C2/m structure, as shown in Fig. 1(c), where the initial spin configurations for the spinpolarized calculations are labelled Nonmagnetic, Antiferromagnetic 1 and Antiferromagnetic 2. We classify the initial configurations into group A (Nonmagnetic and Antiferromagnetic 1) and group B (Antiferromagnetic 2). The final structures after structural optimization in the spinpolarized calculations are either ferrimagnetic or antiferromagnetic depending on the initial spin configurations and the value of the Hubbard energy as indicated in Fig. 1(c). Since the total magnetization becomes zero for both the antiferromagnetic and nonmagnetic configurations, the absolute magnetization M is defined to estimate the degree of spin polarization in the ferrimagnetic and antiferromagnetic phases by:
where \({{\boldsymbol{n}}}_{{\boldsymbol{up}}}(\overrightarrow{{\boldsymbol{r}}})\,\,\)and \({{\boldsymbol{n}}}_{{\boldsymbol{down}}}(\overrightarrow{{\boldsymbol{r}}})\,\,\)are spinup and spindown densities, respectively.
Results and Discussion
Nonspinpolarized calculations (with constant U _{eff})
Figure 2(a–c) shows the calculated lattice parameters β, a, b, and c, and the intermolecular distances d_{1} and d_{2} calculated using the GGA + vdWD + U method with U_{eff} = 0, 2, 4, 6, 9.6, 12, and 14 eV in comparison with those for the SCAN + rVV10 method and the experimental data adopted from the Xray diffraction measurement by Weck et al.^{9} (a, b) and Fujihisha et al.^{16} (c). In our calculations, we manually increase the value of U_{eff} from 0 eV to 14 eV. We call this the constant U_{eff} method. In addition, we calculate U^{lr}_{eff} from the firstprinciples linear response method for each inequivalent atom at 10 GPa. The values of U^{lr}_{eff} for each inequivalent atom are very similar, and the average value of U^{lr}_{eff} is approximately 9.6 eV. We consider this value as the result of the firstprinciples linear response method and apply this value at various pressures in increments of 10 GPa in order to compare the results with the constant U_{eff} method with U_{eff} = 0, 2, 4, 6, 12, and 14 eV. As U_{eff} increases from 0 eV to 9.6 eV and 14 eV, the epsilonzeta transition pressure increases from 50 GPa to 90 GPa and 110 GPa. Therefore, U_{eff} should be between 9.6 eV and 14 eV to be consistent with the experimental transition pressure of 96 GPa. This fact demonstrates that the linear response method, which gives U^{lr}_{eff} ~ 9.6 eV, works well in the highpressure regime.
In general, both the GGA + vdWD (U_{eff} = 0 eV) and GGA + vdWD + U calculations underestimate the lattice parameters of the epsilon phase at low pressure. As a result, the calculated volume of the unit cell for the epsilon phase is underestimated at low pressure, as indicated in Fig. 2(d). The results from the SCAN + rVV10 calculation are closer to the experimental data than those from the GGA + vdWD and GGA + vdWD + U calculations. In particular, the SCAN + rVV10 and GGA + vdWD + U calculations with U_{eff} ≤ 2 eV give the most consistent volumes at low pressure. This suggests that U_{eff} should be smaller than 2 eV at low pressure (≤20 GPa) and approximately 9.6 eV at higher pressure.
Nonspinpolarized calculations (with updated firstprinciples U ^{lr} _{eff})
To determine the correct values of the Hubbard U parameter at each pressure, we performed a firstprinciples linear response calculation of the Hubbard U parameter for the structure optimized at each pressure with the GGA + vdWD method. In Fig. 3(a), the firstprinciples U^{lr}_{eff} values for the inequivalent atoms are plotted as a function of pressure. The values exhibit a very small variation with respect to the inequivalent atoms and exhibit a small pressure dependence ranging from 9.5 eV (P ≤ 50 GPa) to 10.2 eV (P > 50 GPa) with a jump due to the epsilonzeta structural transition at 50 GPa calculated with the GGA + vdWD method. Since this jump is still much smaller than the absolute value of U, the pressuredependent firstprinciples U may be approximated by the constant U^{lr}_{eff} = 9.6 eV for all inequivalent atoms at all pressures. The convergence with the cell size was confirmed by a calculation with a larger cell. Interestingly, this firstprinciples result turns out to be inconsistent with the empirical estimation in the previous paragraph where U_{eff} should be smaller than 2 eV at low pressure (≤20 GPa) and approximately 9.6 eV at higher pressure. This problem will be further discussed in the paragraphs related to the spinpolarized calculation.
As shown in Fig. 3(b), the lattice parameters calculated with the empirical constant U_{eff} = 9.6 eV and those calculated with the firstprinciples U^{lr}_{eff} values are very similar. The firstprinciples U^{lr}_{eff} predicts a transition pressure of 90 GPa, which is closer to the experimental pressure than the transition pressure of 80 GPa predicted with a constant U_{eff} = 9.6 eV. In the following, the results with the firstprinciples U^{lr}_{eff} are used for the GGA + vdWD + U calculation unless otherwise stated.
In Fig. 4(a–d), we show the lattice parameters a, b, and c, the angle β, the intermolecular distances d_{1} and d_{2}, and the unit cell volume calculated using the SCAN + rVV10 and GGA + vdWD + U methods in comparison with the experimental measurements^{9,16}. The parameters from the GGA + vdwD + U method are close to those from the SCAN + rVV10 method, although the SCAN + rVV10 method gives parameters closer to the experimental values overall. In general, the nonspinpolarized GGA + vdWD + U (U^{lr}_{eff} ~ 9.6 eV) calculation can predict reasonable lattice parameters of the epsilon phase of solid oxygen at pressures above 20 GPa and can even predict the epsilonzeta transition pressure at 90 GPa, which is very close to the experimental value of 96 GPa. Interestingly, below 20 GPa, both the GGA + vdWD + U and SCAN + rVV10 calculations significantly underestimate d_{1}. This problem will be discussed in a later part of the paper.
To investigate the evolution of the chemical bonds inside the (O_{2})_{4} structure and between (O_{2})_{4} structures in compression, we consider the charge density difference (CDD) as the charge density minus the superposition of atomic densities. A positive CDD indicates that the electron density increases, and a negative CDD indicates that the electron density decreases with respect to the charge density of isolated individual atoms. Therefore, the CDD can indicate how the chemical bonds form and evolve. Figure 5 shows a crosssection of the CDD in the ab plane (the purple plane) in three cases: (a) GGA + vdWD (U_{eff} = 0 eV), (b) SCAN + rVV10, and (c) GGA + vdWD + U (with firstprinciples U^{lr}_{eff}). Compared to the GGA + vdWD calculation, both the SCAN + rVV10 and GGA + vdWD + U calculations result in an increase in the electron density inside the (O_{2})_{4} cluster and a decrease in the electron density between the (O_{2})_{4} clusters, which means that the electron is more localized within the (O_{2})_{4} region. This means that the enhancement of the localization of p orbitals using the Hubbard U correction is necessary for describing the electronic structure of solid oxygen. Interestingly, in the zeta phase, the electron charge around O2 and O3 connects to those of the neighbouring (O_{2})_{4,} while O1 is still isolated from the neighbouring (O_{2})_{4}. These results suggest that the metallization of epsilon phase may start by the connection of electron charge density between either O2 or O3 of an (O_{2})_{4} cluster to O2 or O3 of neighboring cluster and completely finish when the electron charge density spreads to all atoms O1, O2, and O3. That indicates the metallization should occur gradually. Our results also confirm the experimental results^{8,16} that epsilon phase has (O_{2})_{4} structure and suggest the C2/m zeta phase^{11}.
In summary, the nonspinpolarized GGA + vdWD + U method can predict the direct transition from the epsilon phase to the zeta phase for the first time due to the enhancement of the electron density localization at 90 GPa with the values of U^{lr}_{eff} (~ 9.6 eV) obtained from the firstprinciples linear response method^{26}. The result from the firstprinciples SCAN + rVV10 functional is 70 GPa. The GW calculations^{27,28} predicted an insulatormetal transition at ~100 GPa (but a structural transition at 50 GPa). This value of U^{lr}_{eff} (~9.6 eV) is not too high for the GGA + vdWD + U method if we compare other studies in which DFT + U in combination with van der Waals density functionals reported that U_{eff} = 5 eV and U_{eff} = 12 eV are needed for the prediction of the lattice parameters of the alpha phase of solid oxygen using the revPBE and optB86b exchanges, respectively^{29}. At low pressure (below 20 GPa), all the DFT calculations underestimate all of the lattice parameters, especially the intermolecular distance d_{1}.
Spinpolarized calculations
For the spinpolarized calculations, the structural optimization is started with three different initial atomic spin configurations as indicated in Fig. 1(b), i.e., nonmagnetic and antiferromagnetic 1 and 2 with C2/m symmetry. The structures are then relaxed without a symmetry restriction using a spinpolarized electronic calculation. We use the experimental data measured at 17.5 GPa^{8} as the initial structure for the optimization at 10 GPa. The pressure step size is 10 GPa. The optimized structures are divided into ferrimagnetic and antiferromagnetic structures. The ferrimagnetic structures are similar to the antiferromagnetic arrangement but have unequal magnetic moments. The ferrimagnetic configuration agrees well with the C2/m symmetry of the unit cell, which has three inequivalent atoms: four O1 atoms, two O2 atoms and two O3 atoms. Four O1 atoms are located on the diagonal of the (O_{2})_{4} structure. The spins of these four O1 atoms are parallel and equal, while the spins of the O2 and O3 atoms are antiparallel to those of the O1 atom and not equal to those of the O1 atoms. The antiferromagnetic spin arrangement agrees with Fmmm symmetry in which all atoms are equivalent.
First, for simplicity, we discuss the GGA + vdWD + U calculation with a constant U_{eff}. In Fig. 6(a–d), we show the optimized lattice parameters β, a, b, and c for the initial spin configuration of groups A and B calculated with U_{eff} = 1, 2, 4, and 9.6 eV, respectively. The initial spin configurations are either C2/m nonmagnetic or C2/m antiferromagnetic, as shown in Fig. 1(b). With U_{eff} = 1 eV, there is no difference between groups A and B in the spinpolarized calculation and in the nonspinpolarized calculations. All calculations predict nonmagnetic properties for the epsilon phase. When U_{eff} increases to 2 eV, the lowpressure magnetic epsilon phase appears in the pressure range from 10 GPa to 20 GPa in both groups A and B. The absolute magnetization collapses when the pressure becomes higher than 20 GPa. Then, the epsilon phase changes to a highpressure nonmagnetic phase, and at 50 GPa, it transforms into the zeta phase. Moreover, the range of the lowpressure magnetic epsilon phase increases as U_{eff} increases: from 10–20 GPa with U_{eff} = 2 eV to 10–50 GPa with U_{eff} = 4 eV in group A or 10–70 GPa with U_{eff} = 4 eV in group B (Fig. 6(b,c)). In a neutron diffraction experiment^{12}, the magnetization was found to collapse at ~7–8 GPa at 1.4 ~ 4 K.
When U_{eff} increases to 4 and 9.6 eV, the optimized structures retain the C2/m symmetry in group A or change to Fmmm symmetry in group B. With U_{eff} = 4 eV, the epsilon phase in group B is predicted as an Fmmm antiferromagnetic structure from 10 GPa to 70 GPa. After 70 GPa, it transforms to a nonmagnetic Cm phase, which also has semiconductor characteristics. The symmetry in group A is C2/m for both the magnetic and nonmagnetic phases. As U_{eff} increases to 9.6 eV, the epsilon phase is predicted to have either C2/m or P1 symmetry in group A and Fmmm antiferromagnetic order in group B. In groups A and B, the enthalpies of the magnetic structures are always lower than those of the nonmagnetic structures before the epsilonzeta transition occurs. Details of the enthalpy and magnetization of groups A and B are described in the Supplementary Materials.
Finally, we perform a spinpolarized SCAN + rVV10 calculation for a comparison with the GGA + vdWD + U calculation. Figure 7(a) shows the difference in enthalpies between the spinpolarized and nonspinpolarized calculations for the SCAN + rVV10 method and the GGA + vdWD + U method at U_{eff} = 2 eV. With both methods, the magnetic phase is predicted to be more stable than the nonmagnetic phase up to 20 GPa. The spinpolarized SCAN + rVV10 calculation also predicts ferrimagnetic order. The magnetization collapses at 20 GPa (Fig. 7(b)). The result from the SCAN + rVV10 calculation is consistent with the that of the GGA + vdWD + U calculation at U_{eff} = 2 eV, where the ferrimagnetic phase is also predicted up to 20 GPa. This suggests that U_{eff} = 2 eV is a suitable value for epsilonoxygen at low pressures (below 20 GPa).
In Fig. 7(c,d), we investigate the dependence of U^{lr}_{eff} calculated from the firstprinciples linear response method for the nonmagnetic and magnetic structures with respect to compression (Fig. 7(c)) and the unit cell size at 10 GPa (Fig. 7(d)). We use the structures from either the nonspinpolarized optimization or spinpolarized optimization with fixed U_{eff} = 2 eV to calculate the value of U^{lr}_{eff} in compression. The difference in the value of U^{lr}_{eff} obtained from the linear response method in nonspinpolarized and spinpolarized calculations shows how the conventional linear response method predict for the nonmagnetic and magnetic structures. As we can see from Fig. 7(c) that the firstprinciples values of U^{lr}_{eff} for the magnetic structures are still high (~8 eV) but, interestingly, smaller than those for the nonmagnetic structures (~9.6–10.2 eV) at pressures below 20 GPa. The difference becomes almost zero at pressures above 20 GPa, where the enthalpy comparison shows that the structures are nonmagnetic. We also check the size dependence of U^{lr}_{eff} at 10 GPa, as shown in Fig. 7(d). The values of U^{lr}_{eff} virtually do not depend on the size of the unit cell in both cases with and without spin polarization. The limitation of the linear response method in accounting for the screening effect of the opposite spin channel of the same site was demonstrated in a recent paper^{30}. This may be a reason why the linear response method applied in this study predicts U^{lr}_{eff} ~ 9.6 eV at pressures below 20 GPa. Another posibility is that we calculated the selfconsistent U based on the GGA functionals which is less accurate than SCAN. Recent study of transition metal monoxides^{31} shows that the SCAN + vdW + U with the selfconsistent U calculated based on SCAN predicts good ground state of FeO.
Figure 7(e–h) shows a comparison between the spinpolarized SCAN + rVV10 calculation, the GGA + vdWD + U calculation and the experiments. The lattice parameters a, b, and c predicted from the SCAN + rVV10 calculation are more accurate than those from the GGA + vdWD + U calculation, but the β value is less accurate, especially at pressures below 20 GPa. Overall, at low pressure (10 ~ 20 GPa), our calculations suggest the existence of a magnetic epsilon phase at 0 K. Our results also suggest that the value of the Hubbard U parameter at low pressure should be approximately 2 eV, which is much smaller than 9.6 eV predicted by the firstprinciples linear response method.
It should be mentioned that the optimized structures in spinpolarized SCAN + rVV10 can be either P1 or C2/m symmetric for 8atom primitive unit cell. But the lattice parameters of the two structures are very close including d_{1} and d_{2}. Therefore in Fig. 7(e–h), we only show the data of C2/m structures. The structures for the nonmagnetic phase are always C2/m symmetric.
As for the intermolecular distances d_{1} and d_{2}, the spinpolarized SCAN + rVV10 method shows an improvement in the calculation of the d_{1} closer to the experimental value, but the d_{2} is underestimated. The behaviour is similar to that of the spinpolarized GGA + vdWD + U method as shown in Fig. 7(f). In the next section, we use a larger unit cell in geometry optimization to solve the problem of the underestimations of the d_{1}, d_{2}.
SCAN + rVV10 geometry optimization with a 16atom supercell (conventional unit cell)
In the previous sections, we consider a primitive unit cell consisting of 8 atoms. In this section, we investigate a larger unit cell that includes 16 atoms. The structures of the two unit cells are shown in Fig. 1(a,b). A 16atom conventional unit cell provides more degrees of freedom for the oxygen atoms to relax. Therefore, we expect that the optimization can reach a more stable state. In this section, we only discuss the results from the SCAN + rVV10 calculation. As shown in Fig. 8(a), the magnetic phase is still predicted at low pressure (with 16atom unit cell, the magnetization collapses at 30 GPa). Above 30 GPa, both the spinpolarized and nonspinpolarized SCAN + rVV10 calculations predict nonmagnetic structures.
For the optimizations of primitive 8atom unit cell, the d_{1} is closed to the measurement while the d_{2} is underestimated. The optimized structures of the 16atom unit cell can be either P1 symmetry or C2/m symmetry. The enthalpy of the P1 structure is about 0.06 eV/atom lower than that of the C2/m structure. And the lattice parameters of P1 structure are similar to those of primitive 8atom unit cell, which means the d_{1} is closed to the measurement while the d_{2} is underestimated. Therefore we do not report the lattice parameters of the P1 structure. The lattice parameters of the C2/m structure of the 16atom unit cell are shown in Fig. 8(b–e). The magnetic structures at 10 and 20 GPa have more accurate lattice parameters than the nonmagnetic structures. Especially, the intermolecular distances d_{1} and d_{2} are improved much. In Fig. 8(f), we compare the enthalpy of the 16atom structure and the 8atom structure with magnetic configuration (spinpolarized calculation) and with nonmagnetic configuration (nonspinpolarized calculation). The 16atom unit cell with the magnetic order is the most stable and the lattice parameters are also the most consistent with the measurements.
Summary and Conclusion
The crystal, electronic and magnetic structures of solid oxygen in the epsilon phase have been investigated using the SCAN + rVV10 method and the GGA + vdWD + U method. The spinpolarized SCAN + rVV10 method with an 8atom model provides lattice parameters consistent with the experimental results over the entire pressure range except for the intermolecular distances d_{1} and d_{2}. When the size of the unit cell is extended to 16 atoms, the agreement between the intermolecular distances and the experimental values is greatly improved. Therefore, the SCAN + rVV10 method with a conventional 16atom unit cell is the most suitable model for the epsilon phase of solid oxygen. The spinpolarized SCAN + rVV10 models predict a magnetic phase at low pressure. Since the lattice parameters of the predicted magnetic structure are consistent with the experimental lattice parameters measured at room temperature, our results may suggest that the epsilon phase is magnetic even at room temperature.
It is important to note that no ad hoc parameters are required in the SCAN + rVV10 calculation. The GGA + vdWD + U (with an ad hoc value of U_{eff} = 2 eV at low pressure instead of the firstprinciples value U^{lr}_{eff} ~ 9 eV)^{13} and hybrid functional methods^{14} provide similar results to the SCAN + rVV10 method; however, they do not provide reasonable values for the intermolecular distances. Recent study of transition metal monoxides^{31} shows that the SCAN + vdW + U with the selfconsistent U calculated based on SCAN predicts good ground state of FeO. The SCAN + vdW + U may be a good choice for the calculation of solid oxygen’s ground state.The possibilities of further improving the result of the SCAN + rVV10 calculation are discussed in the Supplementary Materials.
References
 1.
Thiemens, M. K. Oxygen origin. Nature Chemistry 4, 66 (2012).
 2.
Schiferl, D., Cromer, D. T. & Mills, R. L. Structure of O_{2} at 5.5 GPa and 299 K. Acta Cryst. B37, 1329–1332 (1981).
 3.
Agnew, S. F., Swanson, B. I. & Jones, L. H. Extended interactions in the ε phase of oxygen. J. Chem. Phys. 86, 5239–5245 (1987).
 4.
Desgreniers, S., Vohra, Y. K. & Ruoff, A. L. Optical response of very high density solid oxygen to 132 GPa. J. Phys. Chem. 94, 1117–1122 (1990).
 5.
Akahama, Y. et al. New highpressure structural transition of oxygen at 96 GPa associated with metallization in a molecular solid. Phys. Rev. Lett. 74(23), 4690–4693 (1995).
 6.
Weck, G., Loubeyre, P. & LeToullec, R. Observation of structural transformations in metal oxygen. Phys. Rev. Lett. 88, 035504 (2002).
 7.
Freiman, Y. A. Magnetic properties of solid oxygen under pressure (Review Article). Low Temp. Phys. 41, 847 (2015).
 8.
Lundegaard, L. F. et al. Observation of an O8 molecular lattice in the ɛ phase of solid oxygen. Nature 443, 7108 (2006).
 9.
Weck, G., Desgreniers, S., Loubeyre, P. & Mezouar, M. SingleCrystal structural characterization of the metallic phase of oxygen. Phys. Rev. Lett. 102, 25 (2009).
 10.
OchoaCalle, A. J., ZicovichWilson, C. M. & RamírezSolís, A. Solid oxygen ζ phase and its transition from ɛ phase at extremely high pressure: A firstprinciples analysis. Phys. Rev. B 92, 8 (2015).
 11.
Ma, Y., Oganov, A. R. & Glass, C. W. Structure of the metallic ζphase of oxygen and isosymmetric nature of the ε−ζ phase transition: Ab initio simulations. Phys. Rev. B 76, 064101 (2007).
 12.
Igor, N. Goncharenko, Evidence for a magnetic collapse in the epsilon phase of solid oxygen. Phys. Rev. Lett. 94, 205701 (2005).
 13.
Crespo, Y. et al. Collective spin 1 singlet phase in highpressure oxygen. Proc. Natl. Acad. Sci. USA 111(29), 10427–10432 (2014).
 14.
RamirezSolis, A. et al. Antiferromagnetic vs. nonmagnetic ε phase of solid oxygen. Periodic density functional theory studies using a localized atomic basis set and the role of exact exchange. Phys.Chem.Chem.Phys. 19, 2826 (2017).
 15.
Sun, J., Ruzsinszky, A. & Perdew, J. P. Strongly Constrained and Appropriately Normed Semilocal Density Functional. Phys. Rev. Lett. 115, 036402 (2015).
 16.
Fujihisa, H. et al. O8 Cluster structure of the epsilon phase of solid oxygen. Phys. Rev. Lett. 97, 085503 (2006).
 17.
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
 18.
Grimme, S. Semiempirical GGAtype density functional constructed with a longrange dispersion correction. J. Comp. Chem. 27(15), 1787–1799 (2006).
 19.
Giannozzi, P. et al. QUANTUM ESPRESSO: a modular and opensource software project for quantum simulations of materials, J. Phys.: Condens. Matter 21, 395502; The pseudopotentials were downloaded from, http://www/quantumespresso.org/pseudopotentials/ (2009).
 20.
Troullier, N. & Martine, J. L. Efficient pseudopotentials for planewave calculations. Phys. Rev. B 43 (1993).
 21.
Kresse, G. & Furthmüller, J. Efficiency of abinitio total energy calculations for metals and semiconductors using a planewave basis set. Comput. Mat. Sci. 6(1), 15–50 (1996).
 22.
Shanno, D. F. Conditioning of quasiNewton methods for function minimization. Math. Comp. 24, 647 (1970).
 23.
Dion, M. et al. Van der Waals Density Functional for General Geometries. Phys. Rev. Lett. 92, 246401 (2004).
 24.
Klimes, J., Bowler, D. R. & Michaelides, A. Van der Waals density functionals applied to solids. Phys. Rev. B 83, 195131 (2011).
 25.
Anh, L. T., Wada, M., Fukui, H. & Iitaka, T. The impact of Hubbard and van der Waalscorrections on the DFT calculation of epsilonzeta transition pressure in solid oxygen. Preprint at, https://arxiv.org/abs/1803.06619 (2018).
 26.
Cococcioni, M. & Gironcoli, S. Linear response approach to the calculation of the effective interaction parameters in the LDA + U method. Phys. Rev. B 71, 035105 (2005).
 27.
Tse, J. S., Klug, D. D., Yao, Y. & Desgreniers, S. Electronic structure of εoxygen at high pressure: GW calculations. Phys. Rev. B 78, 132101 (2007).
 28.
Kim, D. Y. et al. Structurally induced insulatormetal transition in solid oxygen: A quasiparticle investigation. Phys. Rev. B 77, 092104 (2008).
 29.
Kasamatsu, S., Kato, T. & Sugino, O. Firstprinciples description of van der Waals bonded spinpolarized systems using the vdWDF + U method: Application to solid oxygen at low pressure. Phys. Rev. B 95, 235120 (2017).
 30.
Linscott, E. B., Cole, D. J., Payne, M. C. & O’Regan, D. D. Role of spin in the calculation of Hubbard U and Hund’s J parameters from first principles. Phys. Rev. B 98, 235157 (2018).
 31.
Peng, H. & Perdew, J. P. Synergy of van der Waals and selfinteraction corrections in transition metal monoxides. Phys. Rev. B 96, 100101 (R) (2017).
Acknowledgements
This research was supported by MEXT through the “Exploratory Challenge on PostK computer” (Frontiers of Basic Science: Challenging the Limits) and the RIKEN iTHES Project. This research used the computational resources of the K computer provided by the RIKEN Center for Computational Science through the HPCI System Research project (Project ID: hp160251/hp170220/hp180175) and the Hokusai supercomputer system provided by the RIKEN Advanced Center for Computing and Communication.
Author information
Affiliations
Contributions
L.T.A. performed all the firstprinciples calculations. L.T.A., M.W., H.F., T.K. and T.I. contributed to the analysis of the results and commented on the results.
Corresponding author
Ethics declarations
Competing Interests
The authors declare no competing interests.
Additional information
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Anh, L.T., Wada, M., Fukui, H. et al. Firstprinciples calculations of the epsilon phase of solid oxygen. Sci Rep 9, 8731 (2019). https://doi.org/10.1038/s41598019453149
Received:
Accepted:
Published:
Further reading

A2AgCrCl6 (A = Li, Na, K, Rb, Cs) halide double perovskites: a transition metalbased semiconducting material series with appreciable optical characteristics
Physical Chemistry Chemical Physics (2020)

Physical and optoelectronic features of leadfree A2AgRhBr6 (A = Cs, Rb, K, Na, Li) with halide double perovskite composition
Journal of Materials Chemistry C (2020)

The Cs2AgRhCl6 Halide Double Perovskite: A Dynamically Stable LeadFree TransitionMetal Driven Semiconducting Material for Optoelectronics
Frontiers in Chemistry (2020)

Stability and metallization of solid oxygen at high pressure
Physical Chemistry Chemical Physics (2020)

Electronic structure of dense solid oxygen from insulator to metal investigated with Xray Raman scattering
Proceedings of the National Academy of Sciences (2019)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.