Electronic Contributions to the Free Energy

Metals possess electronic degrees of freedom that contribute to the free energy and the heat capacity, in addition to the vibrational contributions described above. These electronic contributions arise from their band structure: the electrons are relatively free to move through the metal because there are empty states at small energies above the Fermi level. Usually, the electronic contribution to the free energy is negligible compared to the vibrational one. However, due to its simplicity, gibbs2 provides a means for including the electronic free energy using two different models: the Sommerfeld model of free and independent electrons (SOMMERFELD) and a model that reads the coefficients of a polynomial fitted to results of finite temperature DFT calculations (see Mermin, 1965). The option for including the electronic contribution in a given PHASE is ELEC:

PHASE ... [ELEC {SOMMERFELD [FREE|icol.i|]| POL4 [icol1.i]}] [NELEC nelec.i]

The ELEC is followed by the type of electronic contribution:

• SOMMERFELD: the Sommerfeld model of free and independent electrons is used. If the additional FREE keyword is used, the free electron model is used. If SOMMERFELD is followed by an integer icol.i, then the occupation at the Fermi level is read from an additional column in the data file. Note that SOMMERFELD requires giving the number of conduction electrons using the NELEC keyword.

• POL4: reads the result of a finite-temperature DFT calculation. When POL4 is used, gibbs2 rads eight columns starting at icol.i (first column is icol.i, second column is icol.i+1, etc.). The first four columns are the coefficients of a fourth-degree polynomial fit to the electronic free energy ($F_{\rm el}$) with respect to temperature:

  $$F_{\rm el} = i_4 \times T^4 + i_3 \times T^3 + i_2 \times T^2 + i_1 \times T$$

  Columns 5 to 8 represent a fourth-degree polynomial fit to $-T*S_{\rm el}$ as a function of temperature:

  $$-TS_{\rm el} = i_8 \times T^4 + i_7 \times T^3 + i_6 \times T^2 + i_5 \times T$$

  Both $F_{\rm el}$ and $-T*S_{\rm el}$ can be calculated using an electronic structure program. The fitted free energy and entropy contributions must correspond to NAT times Z atoms.

By default, no electronic contribution is added. The SOMMERFELD keyword requires:

NELEC nelec.i

The NELEC keyword gives the number of conduction electrons in the solid. This value is appropriate only in combination with SOMMERFELD. Note this keyword is not related to NELECTRONS, used for the AP2 fits. Default: zero.