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# CMBJ

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:

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

and

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

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.