Empirical Energy Corrections

The thermodynamic properties resulting from a density functional calculation most of the time depend dramatically on the exchange-correlation density functional employed. This is the main source of error in the calculation of thermodynamic properties provided a good temperature model is used. In order to correct these discrepancies, empirical energy corrections (EEC) have been proposed. See Phys. Rev. B 84 (2011) 024109 and Phys. Rev. B 84 (2011) 184103.

The EEC are corrections are applied to the static energy curve and are designed to correct the E(V) curve in such a way that one or two experimental data are exactly reproduced. In the current implementation, the data may be the experimental equilibrium volume at ambient conditions alone and, possibly, the experimental bulk modulus at ambient conditions. Several EECs have been implemented:

• PSHIFT: shift the energy with a constant pressure term.

• APBAF: add a constant times 1/V.

• BPSCAL: corrects the energy using the observation that $p/B_0$ versus $V/V_0$ is similar for all functionals and experimental data.

Empirical energy corrections are applied to a phase using the EEC option to the PHASE keyword:

PHASE ... [EEC NOSCAL|PSHIFT vexp.r|BPSCAL vexp.r bexp.r|APBAF vexp.r|USE phase.i]
          [EEC_P pext.r] [EEC_T text.r] 

The EEC option is followed by the type of EEC to be applied. The additional keyword following EEC can be one of:

• NOSCAL: do not use any correction. This is the default.

• PSHIFT: use a correction of the type $E_{\rm corr} = E + \Delta p * V$, with empirical parameter $\Delta p$ determined to make the ambient temperature and pressure volume equal to vexp.r.

• BPSCAL: use a correction of the type

  $$E_{\rm corr} = E + \frac{B_{\rm exp}V_{\rm exp}}{B_0V_0} \times \left[E\left(\frac{V\times V_0}{V_{\rm exp}}\right)-E(V_0)\right]$$

  with parameters $V_{\rm exp}$ and $B_{\rm exp}$ chosen to reproduce the experimental volume vexp.r and bulk modulus bexp.r at ambient conditions.

• APBAF: use a correction of the type:

  $$E_{\rm corr} = E + \frac{\alpha}{V}$$

  with the parameter $\alpha$ chosen to reproduce the volume at ambient conditions (vexp.r).

• USE: if the current phase is not stable at ambient conditions, it is not possible to calculate the equilibrium volume and bulk modulus and apply the EEC. The USE keyword copies the EEC applied to a different phase with index phase.i. This is useful in the context of calculating phase transitions, where an EEC is applied to all phases involved..

Sometimes, the experimental data are available only under pressure and temperature that are different from ambient conditions. In that case, the pressure and temperature of the EEC can be changed to match the experimental data with the EEC_P and EEC_T options to PHASE:

PHASE ... EEC ... EEC_P pext.r

The EEC_P option specifies the external pressure (in GPa) of the experimental data from which the EEC parameters obtained. The default is zero.

PHASE ... EEC ... EEC_T text.r

The EEC_T option specifies the temperature (in K) of the experimental data from which the EEC parameters obtanied. The default is room temperature (298.15 K).