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:

{\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..