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# CMBJ

**CMBJ** = [real (array)]

Default: **CMBJ** = calculated selfconsistently

Description: defines 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:

where ρ_{σ} denotes the electron density, τ_{σ} the kinetic energy density, and V^{BR}(**r**) the Becke-Roussel potential:

The Becke-Roussel potential was introduced to mimic the Coulomb potential created by the exchange hole. It is local and completely determined by ρ_{σ}, ∇ρ_{σ}, ∇^{2}ρ_{σ}, and τ_{σ}.
The function b_{σ} is given by:

and

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 bohr^{1/2} 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*c*_{i}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

METAGGA, CMBJA, CMBJB, LASPH, LMAXTAU, LMIXTAU

## References

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