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CMBJ = [real (array)]
Default: CMBJ = calculated selfconsistently 

Description: defined the c parameter in the modified Becke-Johnson meta-GGA potential.

The modified Becke-Johnson exchange potential in combination with L(S)DA-correlation[1][2] (METAGGA=MBJ), yields band gaps with an accuracy similar to hybrid functional or GW methods, but computationally less expensive (comparable to standard DFT calculations). The modified Becke-Johnson potential is a local approximation to an atomic exact-exchange potential plus a screening term and is given by:

 \text{V}_{x,\sigma}^{\rm MBJ}(\mathbf{r}) = c\text{V}_{x,\sigma}^{\rm BR}(\mathbf{r}) + (3c-2)\frac{1}{\pi}\sqrt{\frac{5}{12}}\sqrt{\frac{2\tau_{\sigma}(\mathbf{r})}{\rho_{\sigma}(\mathbf{r})}}.

where ρσ denotes the electron density, τσ the kinetic energy density, and VBR(r) the Becke-Roussel potential:

 \text{V}_{x,\sigma}^{\rm BR}(\mathbf{r}) = -\frac{1}{b_{\sigma}(\mathbf{r})} [1-e^{-x_{\sigma}(\mathbf{r})}-\frac{1}{2}x_{\sigma}(\mathbf{r})e^{-x_{\sigma}(\mathbf{r})}].

The Becke-Roussel potential was introduced to mimic the Coulomb potential created by the exchange hole. It is local and completely determined by ρσ, ∇ρσ, ∇2ρσ, and τσ. The function bσ is given by:

 b_{\sigma} = [x^3_{\sigma}e^{-x_{\sigma}}/(8\pi\rho_{\sigma})]^{\frac{1}{3}},


c=\alpha+\beta \left(\frac{1}{V_{\mathrm{cell}}}
\int_{\mathrm{cell}}\frac{|\nabla \rho(\mathbf{r}')|}{\rho(\mathbf{r}')}d{\mathbf{r}'}\right)^{1/2}

where α and β are two free parameters, that may be set by means of the CMBJA and CMBJB tags, respectively. The defaults of α=−0.012 (dimensionless) and β=1.023 bohr1/2 were chosen such that for a constant electron density roughly the LDA exchange is recovered. Alternatively one may also set the c parameter directly, by means of the CMBJ-tag.

The MBJ functional is a potential-only functional, i.e., there is no corresponding MBJ exchange-correlation energy.

The CMBJ tag can be set in the following ways:

  • One may specify one entry per atomic type
    CMBJ = c_1 c_2 .. c_n
    where the order and number n is in accordance with atomic types in your POSCAR file. The MBJ exchange potential at a point r will then be calculated using the parameter ci belonging to the atomic species of the atomic site nearest to r.
  • Specify a constant
    CMBJ = c

If CMBJ is not set, it will be calculated from the density at each electronic step, in accordance with CMBJA and CMBJB, from the formula given above.

Related Tags and Sections



  1. A. D. Becke and E. R. Johnson, J. Chem. Phys. 124, 221101 (2006).
  2. F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401 (2009).


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