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| {{TAGDEF|CMBJ|[real (array)]|calculated selfconsistently}} | | {{TAGDEF|CMBJ|[real (array)]|calculated self-consistently}} |
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| Description: defines the ''c'' parameter in the modified Becke-Johnson meta-GGA potential. | | Description: defines the <math>c</math> parameter in the MBJ potential. |
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| The modified Becke-Johnson exchange potential in combination with L(S)DA-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).
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| The modified Becke-Johnson potential is a local approximation to an atomic exact-exchange potential plus a screening term and is given by:
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| :<math>
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| \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})}}.
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| </math>
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| where ρ<sub>σ</sub> denotes the electron density, τ<sub>σ</sub> the kinetic energy density, and V<sup>BR</sup>('''r''') the Becke-Roussel potential:
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| :<math>
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| \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})}].
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| </math>
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| The Becke-Roussel potential was introduced to mimic the Coulomb potential created by the exchange hole. It is local and completely determined by ρ<sub>σ</sub>, ∇ρ<sub>σ</sub>, ∇<sup>2</sup>ρ<sub>σ</sub>, and τ<sub>σ</sub>.
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| The function b<sub>σ</sub> is given by:
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| :<math>
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| b_{\sigma} = [x^3_{\sigma}e^{-x_{\sigma}}/(8\pi\rho_{\sigma})]^{\frac{1}{3}},
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| </math>
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| and
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| :<math>
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| c=\alpha+\beta \left(\frac{1}{V_{\mathrm{cell}}}
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| \int_{\mathrm{cell}}\frac{|\nabla \rho(\mathbf{r}')|}{\rho(\mathbf{r}')}d{\mathbf{r}'}\right)^{1/2}
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| </math>
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| where α and β are two free parameters, that may be set by means of the {{TAG|CMBJA}} and {{TAG|CMBJB}} tags, respectively. The defaults of α=−0.012 (dimensionless) and β=1.023 bohr<sup>1/2</sup> were chosen such that for a constant electron density roughly the LDA exchange is recovered.
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| Alternatively one may also set the ''c'' parameter directly, by means of the {{TAG|CMBJ}}-tag.
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| The MBJ functional is a ''potential-only'' functional, ''i.e.'', there is no corresponding MBJ exchange-correlation energy.
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| The {{TAG|CMBJ}} tag can be set in the following ways: | | 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> |
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| *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>. |
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| 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. |
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| == Related Tags and Sections == | | == Related tags and articles == |
| {{TAG|METAGGA}}, | | {{TAG|METAGGA}}, |
| {{TAG|CMBJA}}, | | {{TAG|CMBJA}}, |
| {{TAG|CMBJB}}, | | {{TAG|CMBJB}}, |
| | {{TAG|CMBJE}}, |
| | {{TAG|SMBJ}}, |
| | {{TAG|RSMBJ}}, |
| {{TAG|LASPH}}, | | {{TAG|LASPH}}, |
| {{TAG|LMAXTAU}}, | | {{TAG|LMAXTAU}}, |
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| ---- | | ---- |
| [[The_VASP_Manual|Contents]]
| | [[Category:INCAR tag]][[Category:Exchange-correlation functionals]][[Category:meta-GGA]] |
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| [[Category:INCAR]][[Category:Exchange-correlation functionals]][[Category:metaGGA]] | |
CMBJ = [real (array)]
Default: CMBJ = calculated self-consistently
Description: defines the 
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
CMBJ = c
- Specify one entry per atomic type
CMBJ = c_1 c_2 .. c_n
where the order and number 
is in accordance with atomic types in your POSCAR file. The MBJ exchange potential at a point will then be calculated using the parameter 
belonging to the atomic species of the atomic site nearest to .
If CMBJ is not set, is calculated at each electronic step as the average of 
in the unit cell, as explained in the description of the METAGGA tag.
Related tags and articles
METAGGA,
CMBJA,
CMBJB,
CMBJE,
SMBJ,
RSMBJ,
LASPH,
LMAXTAU,
LMIXTAU
Examples that use this tag
References
