DOI: 10.1038/NCHEM.1754 Caesium in high oxidation states and as a p-block element Mao-sheng Miao Materials Research Laboratory, University of California, Santa Barbara, CA 93106-5050, USA and Beijing Computational Science Research Center, Beijing 10084, P. R. China (Dated: July 12, 2013) NATURE CHEMISTRY www.nature.com/naturechemistry 1
Supplemental Material Section I. CsFn structures Page 3 Section II. Enthalpy of formation Page 10 Section III. Structure and stability of Cs + [F 3 ] Page 11 Section IV. Structure and stability of CsF and F 2 under pressure Page 12 Section V. Phonon spectra and dynamic stability Page 14 Section VI. Molecular orbitals Page 16 Section VII. Validity test 1: functionals Page 16 Section VIII. Validity test 2: PAW Page 18 Section IX. Validity test 3: TB-LMTO DOS Page 21 NATURE CHEMISTRY www.nature.com/naturechemistry 2
I. CSFn STRUCTURES FIG. 1: The selected structures of CsF 2. a. The I4/mmm structure (XeF 2 ). b. The Cmcm structure. Large green balls and small grey balls represent Cs and F atoms. The structure parameters including the lattice constants, angles and Wyckoff positions are shown in Table I. FIG. 2: The selected structures of CsF 3. a. C2/m structure; b. Pnnm structure; c. I4/mmm structure. Large green balls and small grey balls represent Cs and F atoms. The structure parameters including the lattice constants, angles and Wyckoff positions are shown in Table I. NATURE CHEMISTRY www.nature.com/naturechemistry 3
FIG. 3: The selected structures of CsF 4. a. C2/m structure; b. P 1 structure. Large green balls and small grey balls represent Cs and F atoms. The structure parameters including the lattice constants, angles and Wyckoff positions are shown in Table II. FIG. 4: The selected structures of CsF 5. a. Fdd2 structure; b. Pna2 1 structure. Large green balls and small grey balls represent Cs and F atoms. The structure parameters including the lattice constants, angles and Wyckoff positions are shown in Table III. NATURE CHEMISTRY www.nature.com/naturechemistry 4
FIG. 5: The selected structures of CsF 6. a. P 1 structure; b. P2 1 structure. Large green balls and small grey balls represent Cs and F atoms. The structure parameters including the lattice constants, angles and Wyckoff positions are shown in Table IV. NATURE CHEMISTRY www.nature.com/naturechemistry 5
TABLE I: The calculated lattice parameters and atomic positions for the selected CsF 2 and CsF 3 structures. space group pressure lattice parameters atomic positions CsF 2 I4/mmm 10 a =3.8135 Cs 0.50000 0.79492 0.25000 b = 3.8135 F 0.0000 0.0000 0.65241 c = 7.0245 F 0.0000 0.0000 0.34759 CsF 2 Cmcm 20 a =4.7105 Cs 0.50000 0.79492 0.25000 b = 6.8724 Cs 0.50000 0.20508 0.75000 c = 5.4557 F 0.00000 0.00000 0.50000 F 0.00000 0.00000 0.00000 F 0.50000 0.13157 0.25000 F 0.50000 0.86843 0.75000 CsF 3 C2/m 100 a =6.5042 Cs 0.00000 0.50000 0.00000 b = 3.5707 F 0.23443 0.50000 0.75680 c = 3.35266 F 0.76557 0.50000 0.24320 β = 109.88 F 0.00000 0.00000 0.50000 CsF 3 Pnnm 100 a =6.3620 Cs 0.50000 0.00000 0.50000 b = 3.3939 Cs 0.00000 0.50000 0.00000 c = 3.3674 F 0.77256 0.09388 0.00000 F 0.27256 0.40612 0.50000 F 0.22744 0.90612 0.00000 F 0.72744 0.59388 0.50000 F 0.50000 0.50000 0.00000 F 0.00000 0.00000 0.50000 CsF 3 I4/mmm 100 a =3.3165 Cs 0.00000 0.50000 0.00000 b = 3.31652 F 0.00000 0.00000 0.50000 c = 6.59586 F 0.50000 0.00000 0.25000 F 0.00000 0.50000 0.25000 NATURE CHEMISTRY www.nature.com/naturechemistry 6
TABLE II: The calculated lattice parameters and atomic positions for the selected CsF 4 structures. CsF 4 C2/m 100 a =5.6545 Cs 0.00000 0.00000 0.00000 b = 3.6256 F 0.71575 0.50000 0.38106 c = 3.8827 F 0.28425 0.50000 0.61894 β = 59.22 F 0.11847 0.50000 0.22372 F 0.88153 0.50000 0.77628 CsF 4 P 1 100 a =5.4932 Cs 0.00000 0.00000 0.00000 b = 3.9141 Cs 0.50000 0.50000 0.50000 c = 3.7707 F 0.41634 0.21025 0.93515 α = 89.13 F 0.58366 0.78975 0.06485 β = 87.70 F 0.19638 0.62695 0.97315 γ = 96.43 F 0.80362 0.37305 0.02685 F 0.13812 0.36414 0.48448 F 0.86188 0.63586 0.51552 F 0.29172 0.91120 0.45565 F 0.70828 0.08880 0.54435 NATURE CHEMISTRY www.nature.com/naturechemistry 7
TABLE III: The calculated lattice parameters and atomic positions for the selected CsF 5 structures. space group pressure lattice parameters atomic positions CsF 5 Fdd2 150 a = 13.18831 Cs 0.25000 0.25000 0.05639 b = 5.08634 Cs 0.50000 0.50000 0.80639 c = 4.93811 F 0.57813 0.34579 0.51392 F 0.42187 0.65421 0.51392 F 0.67187 0.59579 0.76392 F 0.82813 0.90421 0.76392 F 0.25000 0.25000 0.43896 F 0.50000 0.50000 0.18896 F 0.58813 0.74479 0.44452 F 0.41187 0.25521 0.44452 F 0.66187 0.49479 0.19452 F 0.33813 0.50521 0.19452 CsF 5 I4/mmm 10 a =3.8135 Cs 0.50000 0.79492 0.25000 b = 3.8135 F 0.0000 0.0000 0.65241 c = 7.0245 F 0.0000 0.0000 0.34759 NATURE CHEMISTRY www.nature.com/naturechemistry 8
TABLE IV: The calculated lattice parameters and atomic positions for the selected CsF 6 structures. space group pressure lattice parameters atomic positions CsF 6 P 1 200 a =3.5822 Cs 0.50000 0.50000 0.00000 b = 3.6452 F 0.79496 0.64890 0.57386 c = 3.5975 F 0.20504 0.35110 0.42614 α = 96.11 F 0.63003 0.08171 0.28901 β = 102.35 F 0.36997 0.91829 0.71099 γ = 102.63 F 0.95263 0.21008 0.85119 F 0.04737 0.78992 0.14881 CsF 6 P2 1 200 a =3.5557 Cs 0.70715 0.69217 0.27102 b = 3.4458 Cs 0.29285 0.19217 0.72898 c = 7.3855 F 0.05954 0.26561 0.27702 β = 81.03 F 0.94046 0.76561 0.72298 F 0.71199 0.27961 0.85801 F 0.28801 0.77961 0.14199 F 0.15724 0.27214 0.00022 F 0.84276 0.77214 0.99978 F 0.60564 0.25443 0.13053 F 0.39436 0.75443 0.86947 F 0.07157 0.75555 0.43419 F 0.92843 0.25555 0.56581 F 0.49787 0.14820 0.42308 F 0.50213 0.64820 0.57692 NATURE CHEMISTRY www.nature.com/naturechemistry 9
II. ENTHALPY OF FORMATION The enthalpy of formation for each CsF n compound are calculated as the enthalpy difference of the following reaction: CsF + n 1 F 2 CsF n (1) 2 The enthalpy per atom (divided by total number of atoms in CsF n unit) for selected structures of CsF n compounds are calculated by the following formula: h f (CsF n )=(H(CsF n ) H(CsF) (n 1)H(F 2 )/2)/(n + 1), (2) in which H(CsF n ), H(CsF) and H(F 2 ) are the enthalpies of CsF n, CsF and F 2 under desired pressure. The results are shown in Fig. 6. FIG. 6: Enthalpy of formation per atom for CsF n calculated in this work as function of pressure. For each n, the 0 (dashed line) is the corresponding enthalpy of CsF plus n 1 2 F 2. The convex hull as shown in Fig. 1 in the paper is a compact way of comparing the enthalpies of formation of all the compounds. If the value of one compound is above the connecting line of a compound left to it and another compound right to it, the compound is not stable and will decompose into the two compounds. For example, as shown in Fig. 1 in the paper, CsF 2 is not stable at a pressure of 50 GPa and will decompose into CsF and NATURE CHEMISTRY www.nature.com/naturechemistry 10
CsF 3 : 2CsF 2 CsF + CsF 3 (3) III. STRUCTURE AND STABILITY OF Cs + [F 3 ] The relaxation of the high pressure CsF 3 structure at 0 GPa will lead to a structure in the same space group (C2/m) featuring F 3 polyfluoride ions. Lattice parameters are: a=7.094 Å, b=5.478 Å, c=4.502 Å, β=108.03. Cs atoms locate at 2b (0.0 0.5 0.0) and F atoms locate at 4i (0.1826 0.0 0.3161) and 2c (0.0 0.0 0.5). The F-F distance is 1.741 Å, which is significantly larger than the F-F bond length in F-F molecules, however is much shorter than the F-F distance of 2.227 Å in CsF 3 at 100 GPa. The nearest Cs-F distance is 3.012 Å, which is much larger than the Cs-F bond length of 2.015 Åin CsF 3 at 100 GPa. Thus the geometry clearly shows that the Cs-F compound is actually Cs + [F 3 ]. FIG. 7: The enthalpy difference between the CsF 3 and the Cs + [F 3 ] structures under pressure from 0 to 50 GPa. The formation of Cs + [F 3 ] indicate that Cs is not in high oxidation state at lower pressure. In order to examine the pressure that Cs become stable in high oxidation state, we calculate and compare the enthalpy of CsF 3 and Cs + [F 3 ] under a series of pressures from 0 GPa to 50 GPa. We started from the two strutcures at 50 GPa and 0 GPa respectively and increase NATURE CHEMISTRY www.nature.com/naturechemistry 11
(decrease) the pressure incrementally. If the step is too large, the structure will jump to CsF 3 (Cs + [F 3 ] ) at high (low) pressure. The results are shown in Fig. 7. As shown in the figure, CsF 3 is higher in energy than Cs + [F 3 ] structure at lower pressures, however it become more stable than the latter while the pressure is higher than 30 GPa. These results undoubtedly show the preference of forming Cs 3+ under higher pressure. It also reveals that the F 3 polyfluoride ions might be stabilized in the proper pressure range. IV. STRUCTURE AND STABILITY OF CsF AND F 2 UNDER PRESSURE In order to correctly predict the stability of CsF n under pressure, it is important to know the structures and the enthalpies of CsF and F solids under pressure. We conducted a thorough structure search for CsF and F in a pressure range from 0 to 200 GPa. We then calculated and compared selected structures from the structure search as well as the structures reported in the previous experimental and theoretical studies. The results are shown in Fig. 8 and Fig. 9. FIG. 8: Enthalpies per CsF unit for CsF structures calculated as function of pressure. The results show that besides a structure change from rocksalt to CsCl at low pressure, CsF does not undergo other phase transition and remains in CsCl structure below 200 GPa. Fluorine changes from a structure with C12/m1 symmetry[1] to another with C12/c1 symmetry[2] at about 15 GPa, and remains in that structure till 200 GPa. Both structures NATURE CHEMISTRY www.nature.com/naturechemistry 12
FIG. 9: Enthalpy per F atom for CsF calculated as function of pressure. FIG. 10: Two most stable structures of F 2 under pressures up to 200 GPa. are molecular crystals and have been observed by experiments. The structures are shown in Fig. 10. We also calculated the formation enthalpy of CsF against the decomposition into Cs and F. The results are shown in Fig. 11. NATURE CHEMISTRY www.nature.com/naturechemistry 13
FIG. 11: Enthalpy of formation for CsF as function of pressure. V. PHONON SPECTRA AND DYNAMIC STABILITY In order to check the dynamic stability of the structures, we calculated the phonon spectra of all the structures in pressure range from 0 GPa to 200 GPa. We found that there is no imaginary modes for the structures in their stable pressure range. The following shows the phonon spectra of three most important structures, including the XeF 2 structure for CsF 2, the C2/m structure for CsF 3 and the Fdd2 structure for CsF 5 at 10 GPa, 100 GPa and 100 GPa, respectively. The vibration modes are calculated by use of the density functional perturbation (DFPT) method(ibrion=7,8 in VASP), and the spectra are constructed by use of the Phonopy program.[3] NATURE CHEMISTRY www.nature.com/naturechemistry 14
FIG. 12: From top to bottom: the phonon spectra of CsF 2 in XeF 2 structure at 10 GPa, of CsF 3 in C2/m structure at 100 GPa, and of CsF 5 in Fdd2 structure at 100 GPa. NATURE CHEMISTRY www.nature.com/naturechemistry 15
VI. MOLECULAR ORBITALS FIG. 13: Molecular orbitals of CsF 2 molecules calculated by Gaussian03 program.[4] The Cs-F bond length of 2.358 Å as in CsF 2 at 20 GPa is used. B3LYP hybrid functional and the LANL2DZ Gaussian basis are used. HOMO and LUMO denotes the highest occupied and lowest unoccupied molecular orbitals for CsF + 2 or XeF 2 molecules. In CsF 2 molecule, there is one extra electron occupying the LUMO. VII. VALIDITY TEST 1: FUNCTIONALS One very important issue for DFT calculations is the choice of the exchange-correlation functionals. Currently, the most popularly used functional in condensed matter physics and computational materials is the generalized gradient approximated (GGA) functional, including for example the PBE functional used in our work. Although these semi-local functionals are known to have problem in calculating the excited state properties, their capability for ground state properties have been demonstrated in many systems. They are reliable in producing properties such as total energies and the ground state geometries. They have been successfully used in predicting numerous novel structures and compounds under high pressure. It is important and interesting to test how the choice of functionals may change the results of our calculations. We thus choose the stability of CsF 3 as an example and calculate its enthalpy of formation, H f, from following reaction: CsF + F 2 CsF 3. (4) NATURE CHEMISTRY www.nature.com/naturechemistry 16
We compare the PBE results with rpbe, the revised PBE functional,[5] and with HSE, a hybrid functional in the framework of Heyd-Scuseria-Ernzerhof.[6] As shown in Fig. 14, the choice of the different functionals will not change the general results of the work, i.e. Cs will be oxidized into higher charge state under pressure, although the prediction of the transition pressures may vary. The hybrid functional with screen exchange term can usually give more reliable results. It is interesting to see that at lower pressure range, it behaves more like rpbe, while at high pressure range the calculated enthalpies of formation of CsF 3 are closer to those of PBE. Although more accurate, HSE functionals usually take 10-20 time more computing time comparing with local and semi-local functionals such as PBE etc. Thus they are rarely used in predicting high pressure structures, which usually involves the geometry optimization and total energy calculations of many structures. FIG. 14: Comparing the enthalpy of formation of CsF 3 calculated by use of PBE, revised PBE and HSE functionals. Furthermore, the geometries obtained by different functionals are also very close. The HSE lattice parameters of CsF 3 at 50 GPa are: a=6.593 Å, b=3.788 Å, c=3.512 Å, and β=110.20. Cs atoms locate at 2b (0.0 0.5 0.0) and F atoms locate at 4i (0.728 0.0 0.767) and 2c (0.0 0.0 0.5) sites. The differences from PBE are in general less than 4% for both lattice parameters and inter-atomic distances. For instance, The Cs-F bond lengths in CsF 3 NATURE CHEMISTRY www.nature.com/naturechemistry 17
at 50 GPa are 2.014 Å and 1.937 Åfor PBE and HSE respectively. VIII. VALIDITY TEST 2: THE PAW POTENTIALS We used the Projector Augmented Wave (PAW) potentials to describe the ionic potentials of Cs and F. The potentials are taken from the VASP PAW potential library. Because of the short bond length in F-F molecule and the short distances between Cs and F in CsF n compounds under high pressure, we used the hard PAW potential of F. For Cs, the Cs sv potential is used in which both 5s and 5p electrons are kept as valence together with 6s electrons. The core radii for Cs and F are 2.5 and 1.1 a.u. respectively. The summation of the core radii is 3.6 a.u. or 1.905 Å. This value is smaller than most of the Cs-F bond lengths in CsF n under the pressure range, except for some Cs-F bonds in CsF 5 and CsF 6 under 200 GPa. However, the core-core overlaps are well corrected in PAW scheme. As a matter of fact, more severe core-core overlap occurs in many other high pressure systems. FIG. 15: Comparing the total energies obtained by PAW and LAPW methods for a. CsF and b. F 2. The straight line shows where the PAW and LAPW energies are identical. In order to test the validity of the PAW potentials for Cs and F provided by the VASP PAW library, we calculated the equation of states (EOS) for CsF in CsCl structure, F 2 in the Cmca structure and the CsF n in selected structures. The calculations are performed by use of PAW method as implemented in VASP code as well as a full-potential linearized augmented plane wave (FP-LAPW) method as implemented in the ELK code.[7] The calculation of the stresses are not implemented in the FP-LAPW code. It is not our intention to study the NATURE CHEMISTRY www.nature.com/naturechemistry 18
high pressure structure and stability of CsF n using FP-LAPW method. Rather, we intend to test the accuracy of the PAW potentials in a certain range of hydrostatic compression. FIG. 16: Comparing the total energies obtained by PAW and LAPW methods for a. CsF 2, b. CsF 3, c. CsF 4 and d. CsF 5. The straight line shows where the PAW and LAPW energies are identical. In a., the blue triangles and the orange circles show the values of LAPW with and without geometry relaxation. We calculated the total energies of CsF in CsCl structure under a series of strains by both PAW and LAPW methods. For F 2 and CsF n, there is no single strain representing the hydrostatic compression. We therefore take the PAW optimized structures of these compounds under selected pressures and calculate their total energies by LAPW method. The results are shown and compared in Figs. 15 and 16. The total energies are relative to those of 0 GPa. The core radii of Cs and F in FP-LAPW calculations are 1.8 and 1.2 a.u. The k-mesh, the plane wave cutoff energies and other major accuracy control parameters are well tested for convergency. 10 empty bands are used in calculations for all systems. As shown in Figs. 15 and 16, the EOS curves match remarkably well, despite they NATURE CHEMISTRY www.nature.com/naturechemistry 19
are obtained by two totally different approaches. For all the compounds, the FP-LAPW energies are slightly lower than PAW energies under compressive strains, indicating that the FP-LAPW bulk moduli are slightly smaller than those of PAW. However, the differences are very small and therefore will not change our conclusion of the CsF n stability. For CsF 2, we also relaxed the atomic positions using the forces calculated analytically by the LAPW method. Because the LAPW forces and the PAW forces are very similar, the geometry relaxation in LAPW gives almost identical results as obtained by PAW relaxation, and the LAPW energies are almost identical with and without further relaxation. We also calculated and compared the band structures of CsF 2 (in XeF 2 structure at 10 GPa) and CsF 3 (in C2/m structure at 100GPa) using both PAW and FP-LAPW methods. The results are shown in Fig. 17. PAW and FP-LAPW obtain almost identical band structures, showing again the validity of the PAW potentials used in our study. FIG. 17: The band structures for CsF 2 in XeF 2 structure and XeF 3 in C2/m structure calculated by PAW potential and the FP-LAPW method (ELK). The solid black lines show the results of PAW and the dashed red lines show the results of LAPW. The straight black dashed lines show the fermi levels. NATURE CHEMISTRY www.nature.com/naturechemistry 20
IX. VALIDITY TEST 3: TB-LMTO DOS FIG. 18: Projected density of states (PDOS) of CsF 3 under 100 GPa calculated by TB-LMTO method. The lattice parameters and atomic positions are taken from the structure optimized by VASP method using PBE functional. In the paper, we compared the PDOS and the COHP calculated by VASP and Stuttgart TB-LMTO programs respectively. Here we show the PDOS of CsF 3 at 100 GPa calculated by TB-LMTO. The VASP and the COHP PDOS show similar major features, such as the overlap of the Cs 5p states and F 2p states. The differences between the two are caused by the use of different size of the spheres that the PDOS are projected. The TB-LMTO implements atomic spheres approximation (ASA) and uses significantly larger spheres for each element. [1] L. Meyer, C. S. Barrett, & S. C. Greer, J. Chem. Phys. 49, 1902 (1968). [2] L. Pauling, I. Keaveny & A. B. Robinson, J. Solid State Chem. 2, 225-227 (1970). [3] Phonopy available at http://phonopy.sourceforge.net/. [4] M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Rob, J. R. Cheeseman, J. A. Montgomery Jr., T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, NATURE CHEMISTRY www.nature.com/naturechemistry 21
Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez, and J. A. Pople, Gaussian 03 (Gaussian, Inc., Wallingford, CT, 2003). [5] R. Hammer, L. B. Hansen and J. K. Norskov, Phys. Rev. B 59, 7413 (1999). [6] J. Heyd, G. E. Scuseria, and M. Ernzerhof Hybrid functionals based on a screened Coulomb potential, J. Chem. Phys. 118, 8207,(2003). [7] ELK available at http://elk.sourceforge.net/. NATURE CHEMISTRY www.nature.com/naturechemistry 22