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# LPEAD

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

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

The derivative of the cell-periodic part of the orbitals w.r.t. k, k, |∇kunk〉, 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 ${\displaystyle \nabla _{{{\mathbf {k}}}}{\tilde {u}}_{{n{\mathbf {k}}}}}$ from finite differences:

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

where m runs over the N occupied bands of the system, Δk=kj+1-kj, 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 |∇kunk〉 using higher-order finite difference approximations.

When LPEAD=.TRUE., VASP will compute |∇kunk〉 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.