How to calculate the enthalpy of formation?

Question

I am following this paper to calculate the formation enthalpy of MgCu$_2$, in which the formation enthalpy is defined as follows:

where $E$(Mg$_x$X$_y$) is the total energy of the compound, and $E$(Mg) and $E$(X) are the total energies of pure elements in their stable structures. Since the influence of pressure on the condensed phases is ignored and the energies are calculated at $0$ K without any entropic contributions, the energy of formation is taken to be the enthalpy of formation.

Based on this, I calculated the formation enthalpy of MgCu$_2$. The calculation flow is shown below:

• The ground state energy for the conventional cell of bulk MgCu$_2$ with Fd$\bar{3}$m space group:

  $E$(MgCu2)=-.75231981E+02 (eV)

• The ground state energy for the conventional cell of bulk Mg with P6$_3$/mmc space group:

  $E$(Mg)=-.30124735E+01 (eV)

• The ground state energy for the conventional cell of bulk Cu with Fm$_3$m space group:

  $E$(Cu)=-.14921761E+02 (eV)

Then the enthalpy of formation is estimated as:

$\Delta E_f$=$E$(MgCu2) - $\dfrac{1}{3}$ $E$(Mg) - $\dfrac{2}{3}$ $E$ (Cu) = -64.2799825 (eV)

In the reference paper, the unit kJ/mol is utilized. Due to 1eV=96 kJ/mol, then the enthalpy of formation in kJ/mol is:

$\Delta E_f$ = -64.2799825 (eV) = -6170.87832 (kJ/mol)

which is very different from the result of the reference paper:

What I am missing?

PS: the lattice constants for all structures are kept the same and all results are also calculated by VASP package with consistent parameters.

Answer 1

I haven't been able to resolve everything, but here are some thoughts.

First, the form of Equation (1) assumes the energies are given on a per atom basis.

Second, try comparing the Materials Project energies against your energies. For example, with 24-, 2-, and 4-atom cells for MgCu2, Mg, and Cu, respectively:

• MgCu2:  -3.3975 vs. -3.1347 [= -75.231981 / 24] eV/atom;
• Mg:        -1.5908 vs. -1.5062 [= -3.0124735 / 2] eV/atom;
• Cu:        -4.0992 vs. -3.7304 [= -14.921761 / 4] eV/atom.

All three agree reasonably well.

Finally, note that inserting the Materials Project energies into Equation (1) yields -0.134 eV/atom, matching the Materials Project formation energy given for MgCu2. Doing the same for your energies yields -0.146 eV/atom. However, these values convert to -12.9 kJ/mol-atoms and -14.0 kJ/mol-atoms, respectively, which makes me a little skeptical of the -4.75 kJ/mol-atoms given in the paper.