4 Mg(BH $_4$ ) $_2$ at high pressure

For the high pressure region, the tetragonal $I4_1/acd$ and trigonal $P\bar{3}m1$ phases are found to be the most stable ones in structure searches at 2-5 GPa and 10-20 GPa, respectively. Interestingly, within the whole pressure range (up to 20 GPa), we did not find the $P4_2nm$ structure proposed by Filinchuk et al. for the $\delta$ phase ¹²⁶, but instead found the $I4_1/acd$ phase with four formula units (44 atoms) per cell and $P\bar{4}$ phase with 2 formula units per cell at pressures below 5 GPa (see Fig. 4.2). Given that the $P4_2nm$ structure is dynamically unstable at ambient pressure, and based on our enthalpy calculations, we hypothesized that the $I4_1/acd$ and $P\bar{4}$ structures might correspond to the experimentally observed $\delta$ and $\delta ’$ phases. Further investigation confirmed this suggestion, as we will show below.

$\includegraphics[scale=1.0]{chapter4/pdf/Fig2.png}$

Figure 4.3: Enthalpy curves (relative to the $\gamma$ phase) of various structures of Mg(BH $_4$ ) $_2$ as a function of pressure. Enthalpies are given per formula unit. The inset shows the energy per formula unit of $P\bar{4}$ , $I4_1/amd$ , and $I4_1/acd$ structures including vdW interactions at zero pressure relative to the $P4_2nm$ structure.

Lattice constants of the most interesting candidate structures are listed in Table I. The calculated zero-pressure lattice constant ( $a$ = $b$ = $c$ = 15.79 ) of the $\gamma$ phase is in excellent agreement with the experimental value (15.76 ) ¹²⁶, which gives a benchmark of the typical accuracy to expect of DFT simulations for this system. For the $I4_1/acd$ structure, the Mg atom occupies the crystallographic $8b$ site at (0.5, 0, 0), the B atom occupies the $16e$ site at (0.533, 0.25, 0.375), and the H atoms are at the $32g$ sites with coordinates (0.439, 0.247, 0.295) and (0.627, 0.123, 0.369). For the $P\bar{4}$ structure, the Mg atoms occupies the $1a$ site at (0, 0, 0) and $1d$ site at (0.5, 0.5, 0.5), the B atom occupies at the $4h$ site at (0.25, 0.25, 0.25), and the H atoms are at the $4h$ sites with coordinates (0.459, 0.282, 0.210), (0.139, 0.327, 0.089), (0.223, 0.038, 0.286), and (0.174, 0.354, 0.416). From Table I, one can see that relaxing the experimental $P4_2nm$ structure, one gets unexpectedly large changes in the lattice constants – so large that, in fact, the relaxed lattice constants of the $P\bar{4}$ structure are the closest match to the experimental ones ¹²⁶. One has to keep in mind that what is called the “experimental" cell parameters in many cases is a non-unique result of indexing powder XRD spectra, and this is the case here.

$\includegraphics[scale=1.0]{chapter4/pdf/Fig3.png}$

Figure 4.4: Simulated XRD patterns of the $I4_1/acd$ , $P\bar{4}$ , and $P4_2nm$ structures of Mg(BH $_4$ ) $_2$ with the X-ray wavelength of 0.770518 at ambient pressure (a) and 0.36814 at 10 GPa (b) in comparison with the corresponding experimental results (Ref. [123] and Ref. [125]).

Table 4.1: Lattice constants, density ( $\rho$ ), bulk modulus (B $_0$ ) and its pressure derivative ( $B’$ ), and the total energy of the polymorphs of Mg(BH $_4$ ) $_2$ . The energy difference ( $\Delta E$ ) including vdW interactions ( $\Delta E’$ ) is relative to the $P4_2nm$ structure.

Symmetry	$I4_1/acd$	$I4_1/amd$	$P\bar{4}$	$P4_2nm$	$P4_2nm$
				(Theory)	(Expt)
$a$ ()	7.69	8.32	5.58	5.79	5.44
$b$ ()	7.69	8.32	5.58	5.79	5.44
$c$ ()	12.30	10.52	5.99	5.73	6.15
$\rho$ (g/cm $^{-3}$ )	0.986	0.984	0.963	0.933	0.987
B $_0$ (GPa)	24.0	31.0	31.7	28.5	28.5
$B’$	4.3	3.6	3.6	3.6	5.8
$\Delta E$ (kJ/mol)	-17.0	-15.5	-13.6	0
$\Delta E’$ (kJ/mol)	-21.2	-19.3	-15.4	0

$I4_1/acd$ Mg(BH $_4$ ) $_2$ becomes more stable than the $\gamma$ phase at pressures above 0.7 GPa (Fig. 4.3). In the room-temperature experiment, a pressure-induced structural transformation is observed for the porous $\gamma$ phase, and occurs in two steps: The $\gamma$ phase turns into a diffraction-amorphous phase at 0.4-0.9 GPa, and then at approximately 2.1 GPa into the $\delta$ phase ¹²⁶. The calculated phase transition pressure from the $\gamma$ phase to the proposed $\delta$ phase with $P4_2nm$ symmetry is 1.2 GPa (the corresponding phase transition pressure for $P\bar{4}$ phase is 0.8 GPa), which are in good agreement with the experimental values (0.4-0.9 GPa). We note a tiny enthalpy difference between $I4_1/acd$ and $P\bar{4}$ structures at pressures around 1 GPa. As pressure increases to 9.8 GPa, the $P\bar{3}m1$ structure becomes the most stable one, in agreement with earlier predictions ^{128; 136}. Bil et al. ¹³¹ indicated that it is important to treat long-range dispersion interactions to get the ground state structures of magnesium borohydrides correctly. We have examined the energetic stability of the considered structures through a semi-empirical Grimme correction to DFT energies, stresses and forces ⁶⁷ (see the inset of Fig. 4.3). When this correction is included, the $I4_1/acd$ and $P\bar{4}$ structures once again come out as more stable than the $P4_2nm$ structure, by 21.2 kJ/mol and 15.4 kJ/mol, respectively. Energetic stability seems to correlate with the degree of disparity of bond lengths and atomic Bader charges. The $P4_2nm$ structure has two inequivalent Mg-H distances, 2.26 and 2.07 , compared to 2.11 and 2.07 in the $I4_1/acd$ structure, and 2.12 and 2.06 in the $P\bar{4}$ structure. As we can see, the more homogeneous bond lengths, the greater stability. Bader charges show the same picture: for H atoms, we find them to be -0.63 and -0.59 $e$ in the $P4_2nm$ structure, -0.63 and 0.62 $e$ in the $P\bar{4}$ structure, and -0.63 and -0.61 $e$ in the $I4_1/acd$ structure ¹³⁷. More homogeneous Bader charges and bond lengths in the $I4_1/acd$ and $P\bar{4}$ structures correlate with their greater thermodynamic stability at ambient pressure, in agreement with proposed correlations between local bonding configurations and energetic stability ¹³⁵.

Our calculations suggest that the $P4_2nm$ structure, proposed by experiment for the $\delta$ phase, is unstable. This implies that either density functional theory calculations are inaccurate for this system, or experimental structure determination was incorrect. To assess these possibilities, we simulated the XRD patterns of the $I4_1/acd$ and $P\bar{4}$ structures, and compared them with the experimental XRD pattern of the $\delta$ phase at ambient pressure (see Fig. 4.4a). One observes excellent agreement, both for the positions and the intensities of the peaks (including both strong and weak peaks), of the $I4_1/acd$ structure with experiment ¹²⁶. The situation is very peculiar: two structures, $I4_1/acd$ and $P4_2nm$ , have nearly identical XRD patterns, both compatible with the experiment – but one, $I4_1/acd$ , is the true thermodynamic ground state (global minimum of the enthalpy), whereas the other, $P4_2nm$ , is not even a local minimum of the enthalpy (dynamically unstable structure, incapable of sustaining its own phonons). In this situation, the true structure is clearly $I4_1/acd$ . This example gives a clear real-life example of the fact that very different structures can have very similar powder XRD patterns, making structure determination from powder data dangerous, and in such cases input from theory is invaluable. The $P\bar{4}$ structure also has a rather similar XRD pattern, but the peak positions are slightly shifted. Comparison with an independent experimental XRD pattern collected at 10 GPa (Fig. 4.4b) shows that the peak positions and intensities of the $I4_1/acd$ structure are once again in excellent agreement with the experimental data ¹²⁸, while the strong peaks of the $P\bar{4}$ structure at $9.9^\circ$ , $11.6^\circ$ , and $11.8^\circ$ obviously deviate from the observed ones. This reinforces our conclusion that the $I4_1/acd$ structure is the best candidate for the high pressure $\delta$ phase. At pressures below 10 GPa a mixture of $I4_1/acd$ and $P\bar{4}$ phases is possible, as the XRD peaks of these two structures are quite similar. We remind that in the experiment, the $\delta$ and $\delta ’$ phases are nearly indistinguishable ¹²⁶.

$\includegraphics[scale=1.0]{chapter4/pdf/Fig4.png}$

Figure 4.5: Phonon density of states of (a) the $P\bar{4}$ phase and (b) the $I4_1/acd$ phase at ambient pressure.

Filinchuk et al. demonstrated the bulk modulus of the $\delta$ phase (28.5 GPa) is almost three times higher than that (10.2 GPa) reported by George et al. by fitting the Murnaghan equation of state ^{126; 128}. Our third-order Birch-Murnaghan ¹³⁸ fits of the equation of state yielded bulk moduli of the $I4_1/acd$ and $P\bar{4}$ structures equal to 24 GPa and 31.7 GPa, respectively, consistent with the measured value (28.5 GPa) ¹²⁶. The observed large density difference with respect to the $\gamma$ phase at ambient conditions (44%) is equally consistent with 45% (43%) for $I4_1/acd$ ( $P\bar{4}$ ) structures ¹²⁶. Therefore, it is difficult to discriminate between the $I4_1/acd$ and $P\bar{4}$ structures by their compression behavior, density or bulk modulus. Our calculations show that the $I4_1/acd$ structure does not only match all experimental observations for the $\delta$ phase and has the lowest enthalpy among all sampled structures at the relevant pressure range, but is also dynamically stable – phonons were computed at 0, 5 and 10 GPa. The phonon densities of states (PDOS) of $P\bar{4}$ and $I4_1/acd$ phases at ambient pressure are shown in Fig. 4.5, and once again we see a great degree of similarity. The similarity of all characteristics of these two phases parallels the observed similarity of characteristics of the $\delta$ and $\delta ’$ phases and invites one to propose that while the I41/acd structure corresponds to the $\delta$ phase, the $\delta ’$ phase may have the $P\bar{4}$ structure.

4 Mg(BH) at high pressure

4 Mg(BH $_4$ ) $_2$ at high pressure