12. Calculation of grand potential using thermodynamic integration¶

12.1. Theoretical backgrounds¶

Thermodynamic integration

[TI]

A van de Walle and M Asta, Modell. Simul. Mater. Sci. Eng. 10, 521 (2002).

$\beta_1\phi(\beta_1,\mu_1) = \beta_0\phi(\beta_0,\mu_0) + \int_{(\beta_0,\mu_0)}^{(\beta_1,\mu_1)} (E+\mu x, \beta x )d(\beta,\mu)$

Low-temperature expansion for grand potential

See [LTE].

[LTE]

A F Kohan, P D Tepesch, G Ceder and C Wolverton, Comput. Mater. Sci. 9, 389 (1998).

12.2. ti¶

Calculating grand potential using semi-grand canonical MC and thermodynamic integration.

Input files

TI.in
path_data

Output

Free energy
Grand potential

12.3. lte¶

Estimating grand potential at a temperature and a chemical potential using the low temperature expansion.
Available only for binary systems.

Input files

LTE.in
UPOSCAR
UPOTCAR
POSCAR
POTCAR
ECI
of.out

Output

Grand potential

12.4. TI.in¶

12.4.1. INITPOT tag¶

Value of grand-potential at the initial point of the path. When the temperature of the initial point of the path is 0 K, the grand potential at a finite temperature evaluated using the low-temperature expansion should be used. The initial point of the path should be set to the finite temperature.

Default : 0
Example : INITPOT = -0.016338475

12.5. LTE.in¶

Calculating the grand potential at T=100K and mu=0.0 using the low temperature expansion around T=0. The size of the stable structure at T=0 and mu=0.0 is the same as the size of the 2x2x2 expansion of UPOSCAR. The stable structure at T=0 and mu=0.0 is specified by POSCAR.

TEMP = 100
MU = 0.00
ISUB = 1
SPIN = 1 -1
NAMEPOT = Mg Zn
NUCELL = 4 4 4
NUCELLPOSCAR = 2 2 2

12.5.1. TEMP tag¶

Temperature for the low temperature expansion.

Default : none
Example : TEMP = 10

12.5.2. MU tag¶

Chemical potential for the low temperature expansion.

Default : none
Example : MU = 0.20

12.5.3. NUCELL tag¶

Low temperature expansion is performed using a supercell constructed by the NUCELL expansion of the unit cell. The grand potential calculated using the low temperature expansion converges as the supercell size increases.

Default : none
Example : NUCELL = 4 4 4

12.5.4. ISUB tag¶

Sublattice indexes for performing the CE. The CE is performed on ISUBth lattice sites of UPOSCAR (line 6).

Default : 1
Example : ISUB = 1 3

12.5.5. NUCELLPOSCAR tag¶

Number of unit cells included in POSCAR (initial structure).

Default : none
Example : NUCELLPOSCAR = 2 1 1

12.5.6. NAMEPOT tag, SPIN tag¶

Atom names and spin values for the atom names. Do not set spin values for atoms which are not used for the CE.

Default : none
Example : NAMEPOT = Mg Zn O
Example : SPIN = 1 -1

In the above example, spin values for Mg and Zn are 1 and -1, respectively.