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| Description: defines the <math>c</math> parameter in the MBJ potential. | | Description: defines the <math>c</math> parameter in the MBJ potential. |
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| The modified Becke-Johnson potential {{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 is computationally less expensive. The computational time of an iteration is comparable to the case of a standard GGA calculation, however more iterations to schieve self-consistent field convergence are usually required with the MBJ potential.
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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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| 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})}}.
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| </math>
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| where <math>n_{\sigma}</math> denotes the electron density, <math>\tau_{\sigma}</math> the kinetic-energy density, and <math>v^{\rm BR}</math> the Becke-Roussel potential:
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| :<math>
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| 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 <math>n_{\sigma}</math>, <math>\nabla n_{\sigma}</math>, <math>\nabla^{2}n_{\sigma}</math> and <math>\tau_{\sigma}</math>. The function <math>b_{\sigma}</math> 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 <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.
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| Alternatively one may also set the <math>c</math> parameter directly, by means of the tag {{TAG|CMBJ}}.
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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: |
Revision as of 19:10, 7 April 2022
CMBJ = [real (array)]
Default: CMBJ = calculated selfconsistently
Description: defines the 
parameter in the MBJ potential.
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 
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 .
- 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
