CMBJ: Difference between revisions

Revision as of 15:33, 7 April 2022

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 LDA-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:

{\displaystyle v_{x,\sigma}^{\rm MBJ}(\mathbf{r}) = cv_{x,\sigma}^{\rm BR}(\mathbf{r}) + (3c-2)\frac{1}{\pi}\sqrt{\frac{5}{12}}\sqrt{\frac{2\tau_{\sigma}(\mathbf{r})}{n_{\sigma}(\mathbf{r})}}. }

where $n_{\sigma}$ denotes the electron density, $\tau_{\sigma}$ the kinetic-energy density, and $v^{\rm BR}$ } the Becke-Roussel potential:

{\displaystyle \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 $n_{\sigma}$ , $\nabla n_{\sigma}$ , $\nabla^{2}n_{\sigma}$ and $\tau_{\sigma}$ . The function $b_{\sigma}$ is given by:

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

and

{\displaystyle 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 $\alpha=-0.012$ (dimensionless) and $\beta=1.023$ 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 tag CMBJ.

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 $n$ $n$ is in accordance with atomic types in your POSCAR file. The MBJ exchange potential at a point $\mathbf{r}$ $\mathbf{r}$ will then be calculated using the parameter $c_{i}$ $c_{i}$ belonging to the atomic species of the atomic site nearest to $\mathbf{r}$ $\mathbf{r}$ .

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

Examples that use this tag

References

Contents

[becke:jcp:06-1] A. D. Becke and E. R. Johnson, J. Chem. Phys. 124, 221101 (2006).

[tran:prl:09-2] F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401 (2009).

[1]

[2]

@@ Line 22: / Line 22: @@
 \int_{\mathrm{cell}}\frac{|\nabla \rho(\mathbf{r}')|}{\rho(\mathbf{r}')}d{\mathbf{r}'}\right)^{1/2}
 </math>
-where &alpha; and &beta; are two free parameters, that may be set by means of the {{TAG|CMBJA}} and {{TAG|CMBJB}} tags, respectively. The defaults of<math>\alpha=-0.012</math> (dimensionless) and <math>\beta=1.023</math> bohr<math>^{1/2}</math> were chosen such that for a constant electron density roughly the LDA exchange is recovered.
+where &alpha; and &beta; are two free parameters, that may be set by means of the {{TAG|CMBJA}} and {{TAG|CMBJB}} tags, respectively. The defaults of <math>\alpha=-0.012</math> (dimensionless) and <math>\beta=1.023</math> bohr<math>^{1/2}</math> were chosen such that for a constant electron density roughly the LDA exchange is recovered.
 Alternatively one may also set the <math>c</math> parameter directly, by means of the tag {{TAG|CMBJ}}.