Accurate relativistic density functional calculations for the solid-state of metallic francium

Abstract

Bulk properties of the heaviest known Group 1 element francium are investigated by using density functional theory within the projector augmented wave method including scalar relativistic and spin–orbit corrections as well as zero-point vibrational effects. The accuracy of our method is tested for lithium and sodium where we find that the PBEsol functional describes the properties of these elements extremely well. This gives confidence for the accurate prediction of the cohesive energy differences between the different proposed phases of francium. We find that francium fits nicely within the periodic trend of the Group 1 metals with a predicted low temperature close packed structure (hcp or fcc or Barlow mixtures between those two), and with the bcc phase being slightly less stable.

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Introduction

Francium is a naturally occurring but highly radioactive alkali metal.^1,2 The most stable isotope is ²²³₈₇Fr, a dominant β-emitter (with a minuscule 0.02% α branching ratio) and a half-life of 21.8 m.³ Unlike the lighter element radon,^4,5 where the most stable isotope, ²²²₈₆Rn, has a half-life of 3.8235 d, spectroscopic and chemical investigations into francium are very scarce^1,6,7 and bulk properties of this element have never been observed. Instead, one has to rely on predictions from periodic trends^8–11 or accurate computational simulations.^12–14 For a historical overview on francium and its chemistry see Adloff and Kauffman¹ or Orna.¹⁵ Here we mention that francium has received recent attention for the study of parity-violation in electronic transitions and the search for a permanent electric dipole moment in order to test the standard model of physics.^16–19

Only very few theoretical investigations into bulk properties of francium are available.^20,21 It is however well known that the Group 1 elements up to cesium adopt the less dense packed body-centered cubic (bcc) structure at normal conditions²² with a phase transition into a closed-packed structure with possible stacking faults of the hexagonal sheets at low temperatures,^23–25 albeit there is some recent controversy on the low temperature structure of lithium.^21,26,27 For the alkaline metals up to cesium, computational studies, with few exceptions,^28,29 all involve density functional theory^30–32 with only a few correcting for long-range dispersive type of interactions or zero-point vibrational effects.^21,33 Upadhyaya et al. showed already in 1980 by using simple models that van der Waals dispersive type of interactions cannot be neglected anymore for the cohesive energy differences between the different phases of the heavier alkali metals, because of the increase in the Group 1 dipole polarizability with nuclear charge,¹⁴ and that these interactions slightly favor the bcc arrangement compared to the close packed structures.³⁴ This may seem perhaps counter-intuitive because of the very different coordination numbers, but changes in the nearest neighbor distances play a crucial role as well. Moreover, the transition path from bcc to the more densely packed face-centered cubic (fcc) arrangement is predicted to be rather shallow for the Group 1 metals,^21,33 and the stability of various phases depends critically on the pressure and temperature applied. In addition, for francium one should include spin–orbit effects as the higher lying 7p levels are spin–orbit split by 0.209 eV³⁵ and the core 6p levels by 5.26 eV.³⁶ This may influence the relative stability of the different phases considered here. It is now clear that predicting correctly the small energy differences between the different low-temperature and pressure phases for francium constitutes a major challenge to current computational solid-state theory. In this respect, the accuracy of the density functional approximation for the solid state has been discussed intensively in the literature.^37–39

In this work we try to identify the most stable low-temperature bulk phase of francium by using relativistic density functional theory including spin–orbit effects and dispersive type of interactions through the Grimme energy correction,^40–42 as well as accounting for zero-point vibrational energies.^21,33 The bulk phases studied are the ones generally considered for the bulk Group 1 metals, i.e. bcc, fcc and hcp (hexagonal close packing). The solid-state data provided are not only useful for discussing periodic trends,⁴³ but also for future atom-at-a-time chemistry with Fr or element 119 to estimate adsorption enthalpies on various surfaces such as gold.^44,45

Computational details

To accurately obtain the small energy differences expected for the three low-energy phases, hcp, bcc and fcc, we need to treat the hexagonal and cubical unit cells on an equal basis to consistently sample the k-space in our electronic structure calculations. For this we use the basis vectors as defined in ref. 46 describing the different phases through a common underlying bi-lattice, which will briefly be described here.

The connection between the cubical lattices including both bcc and fcc along a so-called Bain path of deformation^21,33,47 can be further generalized to include the hexagonal close-packed structure.⁴⁶ This can be achieved by introducing a bi-lattice with corresponding lattice vectors and parameters which smoothly connect the hexagonal with the cuboidal lattices, resulting in two atoms for the unit cell. This implies that in addition to the (a,c) (variable) lattice parameters as shown in Fig. 1 we have an angle between the two base vectors of the hexagonal prism, plus the position of the second atom inside the unit cell. For the entire transformation between the different phases we thus have an underlying monoclinic unit cell with two atoms.

Fig. 1 Monoclinic unit cell (shown as the hexagonal cell here) with a two-atom basis. Here we set |b⃑₁| = |b⃑₂| = a and |b⃑₃| = c. See ref. 27 for details.

Our lattice-parameter space for the bi-lattice is defined by (α,a,γ,β), where α indicates the type of structure (cuboidal for α = 1, hexagonal for α = 0, or monoclinic for other α values) used previously for describing the Burgers deformation path, which correlates directly with the angle between the vectors at the base of the unit cell, both with magnitude a (see Fig. 1), γ = c/a as the height of the unit cell with respect to a, and β as the second component of the position vector of the atom in the center of the unit cell, that is, . Using these parameters, the monoclinic skeleton for the different lattices is defined by

and the positions of the atoms within the unit cell are

The volume of the unit cell can be used to restrict the range of values for α and it is written as

The definition of a common unit cell for hcp, fcc and bcc results in volumes and densities that are very similar. Since V > 0, a physically meaningful α parameter lies in the interval (−1,3). For α = 0, the structure is of hexagonal type with ∠(b⃑₁,b⃑₂) = 60°, whereas for α = 1 the same angle becomes 90° with a unit cell being a rectangular prism. Hence, the ideal hcp structure is obtained when . On the other hand, for the ideal fcc structure we have and the ideal bcc lattice is reached at (α,γ,β)_bcc = (1,1,1). It is worth pointing out that the Bain transformation path defined in previous works^21,33 by a single parameter A is a special case of the transformation above for fixed α = 1 and β = 1 and by setting .

Our definition of the bi-lattice is applied to the Group 1 elemental metals Li, Na, and Fr in order to determine their solid-state properties and possible ground state structures. The two metallic elements Li and Na were used to test the reliability of the density functional approximations against experimental data. For a fixed value of α, an optimization of the three lattice parameters (a,γ,β) using an hybrid method of Bayesian optimization^48,49 (BO) and the Broyden–Fletcher–Goldfarb–Shanno⁵⁰ (BFGS) algorithm is performed using density functional theory (DFT) at different levels of approximations, i.e. the local spin-density approximation (LSDA, with Slater exchange⁵¹ and Volko–Wilk–Nusair correlation functional⁵²) and the generalized gradient approximation (GGA, with the Perdew–Burke–Ernzerhof (PBE) exchange–correlation,⁵³ along with PBEsol⁵⁴). In addition, dispersion corrections were incorporated to the PBE functional through the DFT-D3 approach including the Becke–Johnson damping function.^40,41 Spin–orbit (SO) interactions are also considered when using the PBEsol functional. All electronic structure calculations were carried out with the VASP 6.4.0 version^55–58 with the atomic cores as described by Blöchl's Projector Augmented Wave (PAW) method^59,60 with the recommended scalar relativistic pseudopotentials. As the focus was on the total energy of the system, the electronic minimization was done using the tetrahedron smearing method⁶¹ with Blöchl corrections with an energy width of 0.1 eV. Convergence tests regarding k-spacing and energy cut-off were performed guaranteeing a variation of less than 1 meV in the total energy with respect to more demanding accuracy settings. The energy cut-off was set to 500 eV for Li, Na, and Fr with a k-point grid set by using KSPACING = 0.07 and centered at the Γ point. The calculation of the cohesive energy requires the energy of the isolated atom, which was obtained by building a lattice of one atom in a large and slightly orthorhombic unit cell of size 14 × 14.001 × 14.002 Å.

The Bayesian optimization was performed through a custom algorithm where we used a Gaussian Process surrogate with a kernel of the form

k(x,x′) = σ²_fk_Matern(x,x′;ν) + σ²_nδ(x,x′) (4)

where k_Matern(x,x′;ν) is the Matérn kernel⁶² with a smoothness parameter ν = 5/2, σ²_f is the prior signal variance (learned during training), and a fixed noise variance σ²_n = 10⁻⁵via the WhiteKernel class from the Python package scikit-learn.⁶³ The acquisition function employed was Expected Improvement^64–66 with a jitter value of 0.01 and its optimization was done using the limited-memory BFGS (L-BFGS) method with 10 random restarts. The search started with 10 initial random samples and 40 Bayesian steps, then a BFGS optimization of the electronic energy was performed starting with the best point after BO convergence.

The zero-point energy (ZPE) of both sodium and francium is obtained in an approximate way using Badger's rule⁶⁷ together with density functional perturbation theory (DFPT) phonon calculations for the lithium metal as implemented in VASP 6.4.0 with the aid of Phonopy 2.14.0.⁶⁸ For this we used a 6 × 6 × 6 supercell with a Γ -centered 3 × 3 × 3 k-point grid. This approximation uses a reference element (RE), whose phonon calculations are less computationally demanding and with a similar phonon density of states to that of the heavier atom (X). Then, the ratio between the zero-point energies for a specific phase becomes

where R is the nearest neighbor distance of the solid. This simple model has proven to be in good agreement with the vibrational ZPE obtained from the Debye temperature for the alkali and coinage metals.²¹ However, care must be taken in choosing the right reference element with a similar phonon dispersion, such as lithium for the Group 1 elements.

To calculate the bulk modulus at the minimum energy structure, we first fix the value of γ = γ(a) to the optimized value (shown in Table 1 for each metal) and then proceed to vary the value of a around the minimum. The energy variation with respect to the volume, V, is then fitted into the fourth-order Birch-Murnaghan equation of state (BM-EOS),^69,70

where η = (V₀/V)^2/3, E₀, and V₀ are the total energy and equilibrium volume, respectively, B₀ is the bulk modulus at zero pressure and its first and second pressure derivative, respectively. We notice that a fit to a third-order BM-EOS⁷¹ is not sufficient in some cases leading to larger deviations from the fourth-order result. We also include the bulk moduli obtained from a fourth-order polynomial fit in B(V) leading to results close to our BM-EOS values.

Table 1 Properties of optimized structures for Li, Na, and Fr. Lattice parameter a for the common bi-lattice is given in Å, and cohesive and zero-point energies are in kJ mol⁻¹. The deviation in γ is with respect to the ideal hard-sphere value with γ^bcc_ideal = 1, . The nearest neighbor distance for the chosen unit cell is R_NN = a for the closed-packed structures (assuming for hcp), while for bcc we have

Metal		a ^hcp	a ^fcc	a ^bcc	Δγ^hcp	Δγ^fcc	Δγ^bcc	E ^hcpcoh	E ^fcccoh	E ^bcccoh	E ^hcp0	E ^fcc0	E ^bcc0
Li	LDA	2.9943	2.9873	3.3653	0.0027	0.0026	0.0000	173.7099	173.7508	173.5187	—	—	—
	PBE	3.0632	3.0621	3.4443	−0.0012	−0.0005	−0.0020	155.1413	155.1418	154.9859	3.658	3.971	3.842
	PBE-D3	2.9549	2.9599	3.3197	0.0024	−0.0002	0.0012	172.2305	172.2587	172.0417	—	—	—
	PBEsol	3.0627	3.0615	3.4433	−0.0012	−0.0003	−0.0018	162.0904	162.0913	161.9339	—	—	—
	PBEsol + SO	3.0618	3.0619	3.4355	−0.0031	−0.0003	0.0034	162.0775	162.0786	161.9217	—	—	—
Na	LDA	3.6118	3.6112	4.0886	0.0014	0.0013	−0.0151	123.2987	123.3046	123.2073	—	—	—
	PBE	3.7419	3.7436	4.1966	0.0006	−0.0023	0.0004	109.1666	109.1563	109.0883	[1.488]	[1.617]	[1.565]
	PBE-D3	3.6368	3.6364	4.0891	0.0017	0.0010	−0.0027	124.5986	124.6103	124.6097	—	—	—
	PBEsol	3.7186	3.7201	4.1713	0.0007	0.0000	0.0001	111.2742	111.2743	111.2035	—	—	—
	PBEsol + SO	3.7187	3.7185	4.1709	0.0007	−0.0014	0.0002	111.2772	111.2770	111.2065	—	—	—
Fr	LDA	5.0920	5.1199	5.7048	−0.0009	−0.0006	0.0009	78.5290	78.3809	78.2760	—	—	—
	PBE	5.4485	5.4855	6.1478	0.0150	0.0014	0.0003	61.0826	61.0616	60.8333	[0.268]	[0.292]	[0.283]
	PBE-D3	5.3706	5.3660	6.0073	−0.0043	0.0007	−0.0001	70.6475	70.6072	70.4832	—	—	—
	PBEsol	5.3178	5.3124	5.9629	−0.0052	0.0027	0.0035	70.4832	68.9475	68.7412	—	—	—
	PBEsol + SO	5.2804	5.2707	5.9923	−0.0056	0.0005	−0.0227	70.0579	70.0329	69.8153	—	—	—

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The Debye temperature used in this work is obtained from the equation⁷²

where r_s is the Wigner–Seitz radius in a.u., B₀ the bulk modulus in GPa, and M is the atomic mass in amu. This property can be then used to calculate the vibrational ZPE through

Results and discussions

Table 1 shows the lattice parameters for lithium, sodium and francium where the values of γ are presented as the deviation from the ideal values for each lattice, that is, Δ^X_γ = γ^X_opt − γ^X_ideal, with X = {bcc,fcc,hcp}. As the largest deviation found for β is Δβ^X = β^X_opt − β^X_ideal = −0.0049 due numerical noise, the difference with respect to the ideal values is omitted for this lattice parameter in our discussion.

The quality of our DFT approximation applied can be evaluated from the available experimental data for both lithium and sodium. For bcc lithium we have an experimental nearest neighbor distance of R_NN = 3.023 Å,²⁴ a cohesive energy of E_coh = 158 kJ mol⁻¹,⁷³ and a bulk modulus of B = 12.95 GPa (for 7-Li) from Felice et al.⁷⁴ or 12.65 GPa from Anderson and Swenson⁷⁵ (extrapolated to 0 K). These are in very good agreement with our PBEsol + SO results of 2.972 Å, 158.1 kJ mol⁻¹ (including ZPE) and 13.4 GPa respectively (see Tables 1 and 2), albeit the change of the bulk modulus due to spin–orbit coupling is most likely overestimated. Here we took the bcc phase for comparison, but note that the experimental phase for lithium has also been postulated to be a Barlow 9R structure with an (ABACABAC) sequence of hexagonal layers (often referred to as stacking faults) as deducted from neutron diffraction studies at low temperatures.^25,76 We can interpret this sequence as a mixture of of hcp with of fcc phase.⁷⁷ We have recently shown that different Barlow packings⁷⁸ lie close in energy and vary little in their basic properties with varying sequence of hexagonal layers, and the main properties can accurately be determined from the two limiting cases hcp and fcc alone together with their packing fraction.⁷⁹ Indeed, our data show the fcc and hcp cohesive energies are very close. Furthermore, the low temperature phase of lithium is still being debated as the possible phases are quasi-degenerate in energy.²⁶ We also mention that the PBE + D3 results including long-range dispersion also give very good results, albeit the cohesive energy is overestimated by 6.5%. They do bring the bcc phase closer to the close-packed structures, albeit this effect is quite small. Dispersion effects are important as they contribute 9.9% to the cohesive energy of lithium.

Table 2 Bulk moduli B₀ in GPa and corresponding unit cell volumes V₀ in ų at the optimized stable structures. BM: derived from a fourth-order Birch–Murnaghan fit. P: from a fourth-order polynomial fit in B(V)

Metal		B ^hcp0 (BM)	B ^fcc0 (BM)	B ^bcc0 (BM)	B ^hcp0 (P)	B ^fcc0 (P)	B ^bcc0 (P)	V ^hcp0	V ^fcc0	V ^bcc0
Li	LDA	14.04	16.14	14.15	14.11	16.11	14.20	38.04	37.86	38.12
	PBE	13.99	12.84	13.38	14.01	12.80	13.51	40.65	40.58	40.77
	PBE-D3	16.17	12.64	15.84	16.11	13.07	15.81	36.61	36.66	36.74
	PBEsol	13.75	12.49	12.79	13.77	12.45	12.84	40.63	40.55	40.77
	PBEsol + SO	15.21	15.40	13.29	15.21	15.48	13.41	40.64	40.50	40.79
Na	LDA	8.70	10.63	9.82	8.68	10.63	9.87	66.68	66.83	66.86
	PBE	7.65	7.61	7.88	7.66	7.60	7.65	74.13	74.02	73.96
	PBE-D3	8.40	9.81	11.74	8.37	9.73	10.70	68.19	68.03	68.38
	PBEsol	7.65	8.08	8.46	7.66	8.06	8.40	72.71	72.66	72.57
	PBEsol + SO	7.64	8.22	7.91	7.65	8.24	7.91	72.71	72.65	72.57
Fr	LDA	2.42	2.64	2.95	2.43	2.64	2.92	186.64	189.75	185.65
	PBE	2.62	1.95	1.89	2.58	1.95	1.89	231.30	233.73	232.45
	PBE-D3	1.84	2.46	2.91	1.77	2.44	3.01	218.79	218.64	218.10
	PBEsol	1.68	1.75	2.03	1.86	1.78	2.08	211.97	211.18	213.31
	PBEsol + SO	2.08	2.57	2.67	2.08	2.58	2.71	207.50	207.36	209.36

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Concerning sodium we have the experimental values for the nearest neighbor distance of R_NN = 3.766 Å (at 20 K),⁸⁰ cohesive energy of E_coh = 107 kJ mol⁻¹,⁷³ and bulk moduli of B = 7.7 GPa (extrapolated to 0 K),⁸¹ so we can compare to our bcc results which are 3.612 Å, 109.6 kJ mol⁻¹ (corrected for ZPE) and 7.9 GPa respectively for the PBEsol + SO functional. This is again in very good agreement with experiment. However, as for lithium, the low-temperature phase could be a Barlow 9R structure rather than bcc. Again, PBE-D3 overestimates slightly the cohesive energy by 2.6%. Except for the bulk moduli, SO effects are rather small for both lithium and sodium as one expects. LDA overestimates cohesive energies and are included only for comparison. We note that the performance of various DFT approximations for metals including the Group 1 elements has been discussed before, and the good overall performance of PBEsol for the alkali metals up to Cs has also been noted.^82,83 This is clearly seen in the Fig. 2–4.

Fig. 2 Trend in cohesive energies (in kJ mol⁻¹) for the Group 1 metals for the fcc and bcc structures using various density functionals. Data for K, Rb and Cs are taken from ref. 21 and 83. Experimental data are taken from ref. 83. Exp. + ZPVE is the cohesive energy with added ZPVE for better comparison with the density functional results which are not corrected for ZPVE. The dissociation energies for the Group 1 dimers are also shown for comparison (experimental data for Li₂ to Cs₂ from ref. 84 and 85, calculated data for Fr₂ from ref. 14).

Fig. 3 Trend in nearest neighbor distances (in Å) for the Group 1 metals for the fcc and bcc structures using various density functionals. Data for K, Rb and Cs are from ref. 21 and 82. Experimental data are taken from from ref. 82. The equilibrium distances for the Group 1 dimers are also shown for comparison (experimental data for Li₂ to Rb₂ from ref. 84, calculated data for Cs₂ and Fr₂ from ref. 14).

Fig. 4 Trend in bulk moduli (in GPa) for the Group 1 metals (for the bcc structure only) using various density functionals. Data for K, Rb and Cs are from ref. 82. Experimental data from ref. 82.

Turning now to the relative stability of the three different phases we notice that for PBEsol + SO including ZPE corrections we have the sequences in cohesive energies (in kJ mol⁻¹, ZPE included) 158.420 (hcp) > 158.108 (fcc) > 158.080 (bcc) for lithium, and 109.790 (hcp) > 109.660 (fcc) > 109.642 (bcc) for sodium. Thus these three phases (and the infinitely many close-packed Barlow structures) can be seen as being quasi-degenerate with a slight preference for the hcp structure at this level of theory. In our previous paper⁷⁹ we iterated the fact that quasi-degeneracy between fcc and hcp is most likely a condition for the (rather rare) Barlow sequences to be observed experimentally. This quasi-degeneracy is maintained at the PBE-D3 level of theory. It is well known that at higher temperatures the bcc phase becomes dominant in accordance with Landau theory,⁸⁶ which can be traced back to vibrational entropy contributions to the free energy. This gives a consistent picture of what is observed experimentally for the Group 1 elements.^23,24,80 PBE-D3 overestimates the cohesive energies of the Group 1 metals, which could well be due to an overestimation of electron correlation effects in the combination of the PBE functional⁵³ with Grimme's dispersion correction.⁴² Nevertheless, we see that dispersion corrections lead to a substantial stabilization of the Group 1 elements, i.e. by approx. 16% for the cohesive energy of the three phases of francium.

For francium, we see again a quasi-degeneracy of the two closed packed structures with the same sequence compared to lithium and sodium, i.e. (in kJ mol⁻¹) 69.790 (hcp) > 69.741 (fcc) > 69.532 (bcc) at the PBEsol + SO level of theory including vibrational corrections. Hence we predict that in a hypothetical experiment for bulk francium we would observe a low temperature Barlow structure (fcc, hcp or 9R for example) followed by a phase transition to bcc at higher temperatures. In this case francium is not special compared to all the other group 1 elements (beside its nuclear instability), albeit SO effects become a bit more sizable stabilizing both fcc and bcc in contrast to hcp.

Concerning periodic trends we see that francium sits below the cohesive energy of cesium with 77.6 kJ mol⁻¹, thus continuing the trend of decreasing stability of the solid state down the (currently known) Group 1 elements similar to the trend in dissociation energies for the diatomic compounds, see Fig. 2. In this respect, cesium has a nearest neighbor distance of R_NN = 5.235 Å⁷³ close to the one we calculate for hcp francium of 5.280 Å, mainly due to scalar relativistic effects (note that R_NN < a because ), see Fig. 3. Thus we observe a trend of increasing bond distances down the Group 1 elements up to Cs, with Fr being the exception. This is in line with the increasing bond distances and dissociation energies of the Group 1 diatomics.⁸⁴ For the bulk moduli shown in Fig. 4 we see a smooth decreasing trend from Li to Fr, with LDA deviating substantially from the experimental values. Here we note that the (not yet discovered) superheavy element 119 is the first element breaking this trend due to strong scalar relativistic effects.^87–91 We should also mention a previous study on the Group 1 metals by Koufos and Papaconstantopoulos.²⁰ They used a linearized augmented plane wave (LAPW) method together with LDA as well as GGA to discuss trends within the Group 1 element properties obtaining results similar to ours. However, they report energy differences between the different phases rather than directly the cohesive energies and did not include SO effects in their optimizations or ZPE corrections required for the discussion of the small energy differences between the different lattices.

From the volume per atom of the hcp unit cell of 207.50/2 = 103.75 ų and the molar mass of 223 g mol⁻¹, we can estimate a density at ρ = 3.57 g cm⁻³ at 0 K much larger than the estimated value⁹² of 2.458 g cm⁻³ (at room temperature). However, a simple estimate from the density and molar mass of cesium (ρ = 1.886 g cm⁻³, M = 132.91 g mol⁻¹) we can estimate an upper bound for cesium at room temperature if the volumes of the two metals are close to each other, i.e. ρ_Fr ≈ ρ_CsV_Fr/V_Cs = 3.16 g cm⁻³, closer to our value. Finally, Table 3 contains additional useful data such as Wigner–Seitz radii, Debye temperatures and ZPEs obtained from eqn (8). These estimated ZPEs are in reasonable agreement to what we believe the more accurate data listed in Table 1.

Table 3 Wigner–Seitz radii in atomic units, Debye temperatures, calculated from eqn (7) in K, and zero point energies, calculated from eqn (8) in kJ mol⁻¹, are obtained from the bulk modulus in the Birch–Murnagham equation of state

Metal		r ^hcps	r ^fccs	r ^bccs	Θ ^hcp	Θ ^fcc	Θ ^bcc	E ^hcp0	E ^fcc0	E ^bcc0
Li	LDA	3.129	3.124	3.131	329.38	352.97	330.82	3.081	3.302	3.094
	PBE	3.199	3.198	3.203	332.54	318.44	325.43	3.111	2.979	3.044
	PBE-D3	3.089	3.094	3.093	351.27	310.83	347.91	3.286	2.907	3.254
	PBEsol	3.199	3.197	3.202	329.67	314.13	318.06	3.084	2.938	2.975
	PBEsol + SO	3.199	3.195	3.203	346.77	348.68	324.27	3.244	3.261	3.033
Na	LDA	3.773	3.775	3.775	157.24	173.90	167.10	1.471	1.627	1.563
	PBE	3.909	3.907	3.906	150.06	149.67	150.17	1.404	1.400	1.405
	PBE-D3	3.801	3.799	3.804	155.09	167.56	183.39	1.451	1.567	1.715
	PBEsol	3.883	3.882	3.880	149.59	153.69	157.29	1.399	1.438	1.471
	PBEsol + SO	3.883	3.883	3.881	149.47	155.09	152.09	1.398	1.451	1.423
Fr	LDA	5.317	5.347	5.310	31.63	33.13	34.85	0.296	0.310	0.326
	PBE	5.713	5.732	5.721	34.08	29.45	28.99	0.319	0.276	0.271
	PBE-D3	5.609	5.607	5.606	28.31	32.70	35.59	0.265	0.306	0.333
	PBEsol	5.512	5.545	5.560	26.84	27.45	29.61	0.251	0.257	0.277
	PBEsol + SO	5.508	5.507	5.523	29.81	33.17	33.84	0.279	0.310	0.317

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Conclusions

We obtained of what we believe the most accurate data yet for the solid-state properties of francium at 0 K by using DFT including SO corrections. We calibrated our results on calculations for lithium and sodium where experimental data are available. We found that the PBEsol functional including SO coupling gives excellent results in comparison with experimental data for both lithium and sodium, adding confidence to our results for francium. We predict that bulk francium is very similar to the rest of the Group 1 metals adopting a low temperature closed packed structure (fcc, hcp or Barlow structures) with bcc occurring at higher temperatures in agreement with Landau theory.⁸⁶ We recommend the following solid-state values for francium: a nearest neighbor distance of 5.28 Å, cohesive energy of 70 kJ mol⁻¹, a bulk modulus between 2.1 and 2.6 GPa, and a density at 3.57 g cm⁻³ (at 0 K).

Conflicts of interest

There are no conflicts to declare.

Data availability

Supplementary information (SI): POSCAR files of all optimised structures (18 files in total for Li, Na and Fr with PBE and PBEsol + SO). See DOI: https://doi.org/10.1039/d5cp04548g.

All other data are included in the tables of this article.

Acknowledgements

We like to thank Profs. Heinz Gäggeler and Patrick Steinegger (PSI Villigen) for stimulating discussions.

References

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