ab initio Phase Diagram for LiH and Li

Introduction

In this project the phase diagram for LiH and Li will constructed. To do this three single point energy calculations must be preformed, one for the LiH crystal, the Li crystal and $H_2$ molecule. Before the single point energies could be calculated, the geometry of the crystals and molecule were optimized.

The calculations were performed with CASTEP [2] and used the GGA XC-functional PBE [1], OTFG ultrasoft pseudopotentials, the Koelling-Hamon relativistic treatment, a k-point grid of 11x11x11 for the crystals and 2x2x2 for $H_2$ for the and a energy cutoff of 600eV. The SCF tolerance was set to $10^{-7}$ eV/atom with a convergence window of three steps, For the geometry optimizations the energy convergence was set to $10^{-6}$ eV/atom, the max force was set to 0.01 eV/A, the max stress was set to 0.02 GPa, and the max displacement was set to 0.0005A. For the two crystals a full cell optimization was used with hard comprehensibility.

The k-point grid used converged the energy of the Li and LiH crystals to within 0.005eV and energy cutoff used converged the energy of the Li and LiH crystals to within 0.02eV.

Pseudo atomic configuration Li: 1s2 2s1

Pseudo atomic configuration H: 1s1

Optimization

The geometries were optimized using the settings above and the results are shown in Figure 1, 2 and 3

Figure 1: Primitive bcc lattice of Li, a = 2.983Å

Figure 2: Primitive NaCl type lattice of LiH, a = 2.835Å

Figure 3: hydrogen molecule with bond length 0.752Å in a 15Å sided cube of vacuum.

The Grand Partition Function

To find the equilibrium point between LiH and Li the grand partition functions, $\Omega$ [3] of the two were set to be equal and the chemical potential of atomic hydrogen, $\mu_H$ , was solved for

$\displaystyle \Omega_{Li} = E_{Li} - \Omega^M_S$ (eq 1)

$\displaystyle \Omega_{LiH} = E_{LiH} - \mu _{H} N_{H,LiH} - \Omega^{M}_{S}$ (eq 2)

where $E_{Li}$ and $E_{LIH}$ are the internal energies of Li and LiH crystals respectively, $N_{H,LIH}$ is the number of hydrogen atoms in the LiH crystal and $\Omega^{M}_{S}$ is an additive constant that is the same for all materials. When those two equations are combined the equilibrium chemical potential of atomic hydrogen can be expressed as,

$\displaystyle \mu_H = \frac{ E_{LiH} - E_{Li} } { N_{H,LiH} }$ (eq 3)

In addition the, chemical potential of atomic hydrogen can be expressed in terms of the chemical potential of molecular hydrogen $\mu_{H_2}$ ,

$\mu_H = \frac{1} {2} \mu_{H_2}$ (eq 4)

Assuming that the hydrogen behaves like an ideal gas one can write the chemical potential of molecular hydrogen as,

$\mu_{H_2} = E^{Total}_{H_2} + \hat{\mu}_{H_2}(T, p^o) +kTln(p/p^o)$ (eq 5)

where $E^{Total}_{H_2}$ is the total energy of an isolated $H_2$ molecule at T=0K, $\hat{\mu}_{H_2}(T, p^o)$ is the difference in the chemical potential of $H_2$ between T=0K and the temperature of interest at the reference pressure, $p^o$ is the reference pressure, p is the pressure of the system, and T is the temperature of the system.

Chemical Potential Difference

The values for the difference in the chemical potential of $H_2$ between T=0K and the temperature of interest at the reference pressure can be evaluated using data tabulated in the NIST-JANAF Thermochemical Tables and the following equation

$\hat{\mu}_{H_2}(T, p^o) = [H^o (T) - H^o (T_r) ] - TS (T) - [H^o (0) - H^o (T_r)]$ (eq 6)

Where $T_r = 298.15K$ ; $[H^o (T) - H^o (T_r) ]$ , the difference in the enthalpy at temperature T and the enthalpy at the reference temperature; and S(T), the entropy at temperature T, are values given in the reference table. Selected values are listed in Table 1, along with the calculated value of $\hat{\mu}_{H_2}(T, p^o)$ .

Table 1: Values from NIST-JANAF Thermochemical Tables and the calculated difference in chemical potential.

Results

Single point calculations were performed on the optimized geometries for the Li crystal, the LiH crystal and the molecular hydrogen, the results of which are listed in Table 2. Using equations 3 and 4 the value of the chemical potential of atomic hydrogen and molecular hydrogen were found, the resulting values can be found in Table 2.

Table 2: Results of single point energy calculations for the three systems.

As we know the value of $\mu_{H_2}$ , $E^{Total}_{H_2}$ and $\hat{\mu}_ {H_2}(T, p^o)$ equation 5 can be rewritten to solve for p.

$\displaystyle p(T) = p^0 e^{ \bigg( \frac{\mu_{H_2} - E^{Total}_{H_2} - \hat{\mu}_{H_2}(T, p^o) } { kT } \bigg) }$ (eq 7)

Since the relationship of temperature to pressure is known at the equilibrium point between LiH and Li, the phase diagram can be constructed and is presented in Figure 4.

Figure 4: Phase diagram of Li and LiH

References

1. Perdew, J. P.; Burke, K.; Ernzerhof, M. Physical Review Letters 1996, 77 (18), 3865–3868.

2. Clark, S. J.; Segall, M. D.; Pickard, C. J.; Hasnip, P. J.; Probert, M. I. J.; Refson, K.; Payne, M. C. Zeitschrift für Kristallographie – Crystalline Materials 2005, 220 (5/6).

3. Sholl, D.; Steckel, J. A.; Sholl. Density Functional Theory: a Practical Introduction; Wiley: Somerset, 2011.

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