LPEAD

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:

|\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):

\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 |∇_k\tilde{u}_nk⟩ from finite differences:

{\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=k_j+1-k_j, and

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..