CMBJ: Difference between revisions

Latest revision as of 15:15, 27 February 2025

CMBJ = [real (array)]
Default: CMBJ = calculated self-consistently

Description: defines the [math]\displaystyle{ c }[/math] parameter in the MBJ potential.

The CMBJ tag can be set in the following ways:

Specify a constant that is used at every point of space [math]\displaystyle{ \mathbf{r} }[/math]
```
CMBJ = c
```

Specify one entry per atomic type
```
CMBJ = c_1 c_2 .. c_n
```
where the order and number [math]\displaystyle{ n }[/math] is in accordance with atomic types in your POSCAR file. The MBJ exchange potential at a point [math]\displaystyle{ \mathbf{r} }[/math] will then be calculated using the parameter [math]\displaystyle{ c_{i} }[/math] belonging to the atomic species of the atomic site nearest to [math]\displaystyle{ \mathbf{r} }[/math].

If CMBJ is not set, [math]\displaystyle{ c }[/math] is calculated at each electronic step as the average of [math]\displaystyle{ \left\vert\nabla n\right\vert/n }[/math] in the unit cell, as explained in the description of the METAGGA tag.

References

@@ Line 1: / Line 1: @@
-{{TAGDEF|CMBJ|[real (array)]|calculated selfconsistently}}
+{{TAGDEF|CMBJ|[real (array)]|calculated self-consistently}}
-Description: defines the <math>c</math> parameter in the modified Becke-Johnson meta-GGA potential.
+Description: defines the <math>c</math> parameter in the MBJ potential.
 ----
-The modified Becke-Johnson exchange potential in combination with LDA-correlation{{cite|becke:jcp:06}}{{cite|tran:prl:09}} ({{TAG|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:
-:<math>
- \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})}}.
-</math>
-where &rho;<sub>&sigma;</sub> denotes the electron density, &tau;<sub>&sigma;</sub> the kinetic energy density, and V<sup>BR</sup>('''r''') the Becke-Roussel potential:
-:<math>
- \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})}].
-</math>
-The Becke-Roussel potential was introduced to mimic the Coulomb potential created by the exchange hole. It is local and completely determined by &rho;<sub>&sigma;</sub>, &nabla;&rho;<sub>&sigma;</sub>, &nabla;<sup>2</sup>&rho;<sub>&sigma;</sub>, and &tau;<sub>&sigma;</sub>.
-The function b<sub>&sigma;</sub> is given by:
-:<math>
- b_{\sigma} = [x^3_{\sigma}e^{-x_{\sigma}}/(8\pi\rho_{\sigma})]^{\frac{1}{3}},
-</math>
-and
-:<math>
-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}
-</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 &alpha;=&minus;0.012 (dimensionless) and &beta;=1.023 bohr<sup>1/2</sup> 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}}-tag.
-The MBJ functional is a ''potential-only'' functional, ''i.e.'', there is no corresponding MBJ exchange-correlation energy.
 The {{TAG|CMBJ}} tag can be set in the following ways:
-*One may specify one entry per atomic type<pre>CMBJ = c_1 c_2 .. c_n</pre> where the order and number ''n'' is in accordance with atomic types in your {{FILE|POSCAR}} file. The MBJ exchange potential at a point '''r''' will then be calculated using the parameter ''c''<sub>i</sub> belonging to the atomic species of the atomic site nearest to '''r'''.
+*Specify a constant that is used at every point of space <math>\mathbf{r}</math><pre>CMBJ = c</pre>
-*Specify a constant<pre>CMBJ = c</pre>
+*Specify one entry per atomic type<pre>CMBJ = c_1 c_2 .. c_n</pre> where the order and number <math>n</math> is in accordance with atomic types in your {{FILE|POSCAR}} file. The MBJ exchange potential at a point <math>\mathbf{r}</math> will then be calculated using the parameter <math>c_{i}</math> belonging to the atomic species of the atomic site nearest to <math>\mathbf{r}</math>.
-If {{TAG|CMBJ}} is not set, it will be calculated from the density at each electronic step, in accordance with {{TAG|CMBJA}} and {{TAG|CMBJB}}, from the formula given above.
+If {{TAG|CMBJ}} is not set, <math>c</math> is calculated at each electronic step as the average of <math>\left\vert\nabla n\right\vert/n</math> in the unit cell, as explained in the description of the {{TAG|METAGGA}} tag.
-== Related Tags and Sections ==
+== Related tags and articles ==
 {{TAG|METAGGA}},
 {{TAG|CMBJA}},
 {{TAG|CMBJB}},
+{{TAG|CMBJE}},
+{{TAG|SMBJ}},
+{{TAG|RSMBJ}},
 {{TAG|LASPH}},
 {{TAG|LMAXTAU}},
@@ Line 49: / Line 28: @@
 ----
-[[The_VASP_Manual|Contents]]
+[[Category:INCAR tag]][[Category:Exchange-correlation functionals]]
-[[Category:INCAR]][[Category:Exchange-correlation functionals]][[Category:metaGGA]]