LPEAD: Difference between revisions

Latest revision as of 07:59, 19 July 2022

LPEAD = .TRUE. | .FALSE
Default: LPEAD = .FALSE.

Description: for LPEAD=.TRUE., the derivative of the cell-periodic part of the orbitals w.r.t. k, |∇_ku_nk⟩, is calculated using finite differences.

The derivative of the cell-periodic part of the orbitals w.r.t. k, k, |∇_ku_nk⟩, may be written as:

{\displaystyle |\mathbf{\nabla_{k}} \tilde{u}_{n\mathbf{k}} \rangle = \sum_{n\neq n'} \frac{| \tilde{u}_{n'\mathbf{k}} \rangle \langle \tilde{u}_{n'\mathbf{k}} | \frac{\partial\left[H(\mathbf{k})-\epsilon_{n\mathbf{k}}S(\mathbf{k})\right]}{\partial \mathbf{k}} | \tilde{u}_{n\mathbf{k}} \rangle}{\epsilon_{n\mathbf{k}}-\epsilon_{n'\mathbf{k}}} }

where H(k) and S(k) are the Hamiltonian and overlap operator for the cell-periodic part of the orbitals, and the sum over n´ must include a sufficiently large number of unoccupied states.

It may also be found as the solution to the following linear Sternheimer equation (see LEPSILON):

{\displaystyle \left[H(\mathbf{k})-\epsilon_{n\mathbf{k}}S(\mathbf{k})\right] |\mathbf{\nabla_{k}} \tilde{u}_{n\mathbf{k}} \rangle =-\frac{\partial\left[H(\mathbf{k})-\epsilon_{n\mathbf{k}}S(\mathbf{k})\right]} {\partial \mathbf{k}}|\tilde{u}_{n\mathbf{k}} \rangle }

Alternatively one may compute $\nabla_{\mathbf{k}} \tilde{u}_{n\mathbf{k}}$ from finite differences:

{\displaystyle \frac{\partial | \tilde{u}_{n\mathbf{k}_j} \rangle}{\partial k}= \frac{ie}{2\Delta k} \sum^N_{m=1} \left[ | \tilde{u}_{m\mathbf{k}_{j+1}} \rangle S^{-1}_{mn}(\mathbf{k}_j,\mathbf{k}_{j+1})\rangle - | \tilde{u}_{m\mathbf{k}_{j-1}} \rangle S^{-1}_{mn}(\mathbf{k}_j,\mathbf{k}_{j-1})\rangle\right] }

where m runs over the N occupied bands of the system, Δk=k_j+1-k_j, and

{\displaystyle S_{nm}(\mathbf{k}_j,\mathbf{k}_{j+1})= \langle \tilde{u}_{n\mathbf{k}_{j}}| \tilde{u}_{m\mathbf{k}_{j+1}}\rangle }

.

As mentioned in the context of the self-consistent response to finite electric fields one may derive analoguous expressions for |∇_ku_nk⟩ using higher-order finite difference approximations.

When LPEAD=.TRUE., VASP will compute |∇_ku_nk⟩ using the aforementioned finite difference scheme. The order of the finite difference approximation can be specified by means of the IPEAD-tag (default: IPEAD=4).

These tags may be used in combination with LOPTICS=.TRUE. and LEPSILON=.TRUE..

N.B. Please note that LPEAD = .TRUE. is not supported for metallic systems.

@@ Line 6: / Line 6: @@
 :<math>
-|\mathbf{\nabla_{k}} u_{n\mathbf{k}} \rangle =
+|\mathbf{\nabla_{k}} \tilde{u}_{n\mathbf{k}} \rangle =
 \sum_{n\neq n'}
-\frac{| u_{n'\mathbf{k}} \rangle \langle u_{n'\mathbf{k}} |
+\frac{| \tilde{u}_{n'\mathbf{k}} \rangle \langle \tilde{u}_{n'\mathbf{k}} |
 \frac{\partial\left[H(\mathbf{k})-\epsilon_{n\mathbf{k}}S(\mathbf{k})\right]}{\partial \mathbf{k}}
-| u_{n\mathbf{k}} \rangle}{\epsilon_{n\mathbf{k}}-\epsilon_{n'\mathbf{k}}}
+| \tilde{u}_{n\mathbf{k}} \rangle}{\epsilon_{n\mathbf{k}}-\epsilon_{n'\mathbf{k}}}
 </math>
@@ Line 19: / Line 19: @@
 :<math>
 \left[H(\mathbf{k})-\epsilon_{n\mathbf{k}}S(\mathbf{k})\right]
-|\mathbf{\nabla_{k}} u_{n\mathbf{k}} \rangle
+|\mathbf{\nabla_{k}} \tilde{u}_{n\mathbf{k}} \rangle
 =-\frac{\partial\left[H(\mathbf{k})-\epsilon_{n\mathbf{k}}S(\mathbf{k})\right]}
-{\partial \mathbf{k}}|u_{n\mathbf{k}} \rangle
+{\partial \mathbf{k}}|\tilde{u}_{n\mathbf{k}} \rangle
 </math>
-Alternatively one may compute |&nabla;<sub>'''k'''</sub>u<sub>n'''k'''</sub>&rang; from finite differences:
+Alternatively one may compute <math>\nabla_{\mathbf{k}} \tilde{u}_{n\mathbf{k}}</math> from finite differences:
 :<math>
-\frac{\partial | u_{n\mathbf{k}_j} \rangle}{\partial k}=
+\frac{\partial | \tilde{u}_{n\mathbf{k}_j} \rangle}{\partial k}=
 \frac{ie}{2\Delta k} \sum^N_{m=1}
-\left[ | u_{m\mathbf{k}_{j+1}} \rangle
+\left[ | \tilde{u}_{m\mathbf{k}_{j+1}} \rangle
 S^{-1}_{mn}(\mathbf{k}_j,\mathbf{k}_{j+1})\rangle -
-| u_{m\mathbf{k}_{j-1}} \rangle
+| \tilde{u}_{m\mathbf{k}_{j-1}} \rangle
 S^{-1}_{mn}(\mathbf{k}_j,\mathbf{k}_{j-1})\rangle\right]
 </math>
-where ''m'' runs over the ''N'' occupied bands of the system, <math>\Delta k=\mathbf{k}_{j+1}-\mathbf{k}_j</math>, and
+where ''m'' runs over the ''N'' occupied bands of the system, &Delta;''k''='''k'''<sub>j+1</sub>-'''k'''<sub>j</sub>, and
 :<math>
 S_{nm}(\mathbf{k}_j,\mathbf{k}_{j+1})=
-\langle u_{n\mathbf{k}_{j}}| u_{m\mathbf{k}_{j+1}}\rangle
+\langle \tilde{u}_{n\mathbf{k}_{j}}| \tilde{u}_{m\mathbf{k}_{j+1}}\rangle
 </math>.
+As mentioned in the context of [[Berry_phases_and_finite_electric_fields#Self-consistent_response_to_finite_electric_fields|the self-consistent response to finite electric fields]] one may derive analoguous expressions for |&nabla;<sub>'''k'''</sub>u<sub>n'''k'''</sub>&rang; using higher-order finite difference approximations.
 When {{TAG|LPEAD}}=.TRUE., VASP will compute |&nabla;<sub>'''k'''</sub>u<sub>n'''k'''</sub>&rang; using the aforementioned finite difference scheme. The order of the finite difference approximation can be specified by means of the {{TAG|IPEAD}}-tag (default: {{TAG|IPEAD}}=4).
 These tags may be used in combination with {{TAG|LOPTICS}}=.TRUE. and {{TAG|LEPSILON}}=.TRUE..
+----
+*N.B. Please note that {{TAG|LPEAD}} = .TRUE. '''is not supported for metallic systems'''.
-== Related Tags and Sections ==
+== Related tags and articles ==
-{{TAG|LPEAD}},
 {{TAG|IPEAD}},
 {{TAG|LEPSILON}},
 {{TAG|LOPTICS}},
+{{TAG|LCALCEPS}},
+{{TAG|EFIELD_PEAD}},
 [[Berry_phases_and_finite_electric_fields|Berry phases and finite electric fields]]
+{{sc|LPEAD|Examples|Examples that use this tag}}
 ----
-[[The_VASP_Manual|Contents]]
-[[Category:INCAR]][[Category:Berry phases]]
+[[Category:INCAR tag]][[Category:Linear response]][[Category:Dielectric properties]][[Category:Berry phases]]