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.

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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 |∇_ku_nk⟩ $\nabla_{k} \tilde{u}$ from finite differences:

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

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

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

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..

Related Tags and Sections

IPEAD, LEPSILON, LOPTICS, LCALCEPS, EFIELD_PEAD, Berry phases and finite electric fields

Example Calculations using this Tag

bandgap of Si in GW, Dielectric properties of Si using BSE, dielectric properties of Si, dielectric properties of SiC, Model BSE calculation on Si

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