ELPH_DECOMPOSE

ELPH_DECOMPOSE = [string]
Default: ELPH_DECOMPOSE = VDPR

Description: Chooses which contributions to include in the computation of the electron-phonon matrix elements.

The electron-phonon matrix element can be formulated in the projector-augmented-wave (PAW) method in terms of individual contributions^[1]. Each contribution can be included by specifying the associated letter in ELPH_DECOMPOSE. We suggest two different combinations to define matrix elements:

ELPH_DECOMPOSE = VDPR: "All-electron" matrix element^[1]^[2]
ELPH_DECOMPOSE = VDQ: "Pseudo" matrix element^[1]^[3]

Available contributions

V - Derivative of pseudopotential, $\tilde{v}$: ${\displaystyle g^{(\text{V})}_{m \mathbf{k}', n \mathbf{k}, a} \equiv \langle \tilde{\psi}_{m \mathbf{k}'} | \frac{\partial \tilde{v}}{\partial u_{a}} | \tilde{\psi}_{n \mathbf{k}} \rangle }$
D - Derivative of PAW strength parameters, $D_{a, ij}$: ${\displaystyle g^{(\text{D})}_{m \mathbf{k}', n \mathbf{k}, a} \equiv \sum_{bij} \langle \tilde{\psi}_{m \mathbf{k}'} | \tilde{p}_{b i} \rangle \frac{\partial D_{b, ij}}{\partial u_{a}} \langle \tilde{p}_{b j} | \tilde{\psi}_{n \mathbf{k}} \rangle }$
P - Derivative of PAW projectors, $|\tilde{p}_{ai}\rangle$: ${\displaystyle \begin{split} g^{(\text{P})}_{m \mathbf{k}', n \mathbf{k}, a} & \equiv \sum_{ij} \langle \tilde{\psi}_{m \mathbf{k}'} | \frac{\partial \tilde{p}_{a i}}{\partial u_{a}} \rangle ( D_{a, ij} - \varepsilon_{n \mathbf{k}} Q_{a, ij} ) \langle \tilde{p}_{a j} | \tilde{\psi}_{n \mathbf{k}} \rangle \\ & + \sum_{ij} \langle \tilde{\psi}_{m \mathbf{k}'} | \tilde{p}_{a i} \rangle ( D_{a, ij} - \varepsilon_{m \mathbf{k}'} Q_{a, ij} ) \langle \frac{\partial \tilde{p}_{a j}}{\partial u_{a}} | \tilde{\psi}_{n \mathbf{k}} \rangle \end{split} }$
R - Derivative of PAW partial waves, $|\phi_{ai}\rangle$ and $|\tilde{\phi}_{ai}\rangle$: ${\displaystyle g^{(\text{R})}_{m \mathbf{k}', n \mathbf{k}, a} \equiv (\varepsilon_{n \mathbf{k}} - \varepsilon_{m \mathbf{k}'}) \sum_{ij} \langle \tilde{\psi}_{m \mathbf{k}'} | \tilde{p}_{a i} \rangle R_{a, ij} \langle \tilde{p}_{a j} | \tilde{\psi}_{n \mathbf{k}} \rangle }$; with ${\displaystyle R_{a, ij} \equiv \langle \phi_{a i} | \frac{\partial \phi_{a j}}{\partial u_{a}} \rangle - \langle \tilde{\phi}_{a i} | \frac{\partial \tilde{\phi}_{a j}}{\partial u_{a}} \rangle }$
Q - Derivative of PAW projectors, $|\tilde{p}_{ai}\rangle$ (different eigenvalues): ${\displaystyle \begin{split} g^{(\text{Q})}_{m \mathbf{k}', n \mathbf{k}, a} & \equiv \sum_{ij} \langle \tilde{\psi}_{m \mathbf{k}'} | \frac{\partial \tilde{p}_{a i}}{\partial u_{a}} \rangle ( D_{a, ij} - \varepsilon_{n \mathbf{k}} Q_{a, ij} ) \langle \tilde{p}_{a j} | \tilde{\psi}_{n \mathbf{k}} \rangle \\ & + \sum_{ij} \langle \tilde{\psi}_{m \mathbf{k}'} | \tilde{p}_{a i} \rangle ( D_{a, ij} - \varepsilon_{n \mathbf{k}} Q_{a, ij} ) \langle \frac{\partial \tilde{p}_{a j}}{\partial u_{a}} | \tilde{\psi}_{n \mathbf{k}} \rangle \end{split} }$