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LPEAD-tag and IPEAD-tag: Derivative of the orbitals w.r.t. the k-point

IPEAD= 1 | 2 | 3 | 4

Default: LPEAD=.FALSE. and IPEAD=4 (available as of VASP.5.2).

The derivative of the cell-periodic part of the orbitals w.r.t. $ {\bf k}$, $ \vert{\bf\nabla_{k}} u_{n{\bf k}} \rangle$, may be written as:

$\displaystyle \vert{\bf\nabla_{k}} u_{n{\bf k}} \rangle = \sum_{n\neq n'} \frac...
... {\bf k}} \vert u_{n{\bf k}} \rangle}{\epsilon_{n{\bf k}}-\epsilon_{n'{\bf k}}}$ (6.46)

where $ H({\bf k})$ and $ S({\bf 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:

$\displaystyle \left[H({\bf k})-\epsilon_{n{\bf k}}S({\bf k})\right] \vert{\bf\n...
...silon_{n{\bf k}}S({\bf k})\right]} {\partial {\bf k}}\vert u_{n{\bf k}} \rangle$ (6.47)

(See Sec. 6.72.4).

Alternatively one may compute $ \vert{\bf\nabla_{k}} u_{n{\bf k}} \rangle$ from finite differences (see Eqs. (96) and (97) in Ref. [87]):

$\displaystyle \frac{\partial \vert u_{n{\bf k}_j} \rangle}{\partial k}= \frac{i...
...t u_{m{\bf k}_{j-1}} \rangle S^{-1}_{mn}({\bf k}_j,{\bf k}_{j-1})\rangle\right]$ (6.48)

where $ m$ runs over the $ N$ occupied bands of the system, $ \Delta k={\bf k}_{j+1}-{\bf k}_j$, and

$\displaystyle S_{nm}({\bf k}_j,{\bf k}_{j+1})= \langle u_{n{\bf k}_{j}}\vert u_{m{\bf k}_{j+1}}\rangle$ (6.49)

As mentioned in Ref. [87] one may derive analoguous expressions for $ \vert{\bf\nabla_{k}} u_{n{\bf k}} \rangle$ using higher-order finite difference approximations.

When LPEAD=.TRUE., VASP will compute $ \vert{\bf\nabla_{k}} u_{n{\bf k}} \rangle$ using the aforementioned finite difference scheme. The order of the finite difference approximation can be specified by setting the IPEAD-tag.

These tags may be used in combination with LOPTICS=.TRUE. (Sec. 6.72.1) and LEPSILON=.TRUE. (Sec. 6.72.4).

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