ELPH DECOMPOSE

ELPH_DECOMPOSE = [string]
Default: ELPH_DECOMPOSE = VDPR 

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

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The electron-phonon matrix element can be formulated in the projector-augmented-wave (PAW) method in terms of individual contributions. 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

ELPH_DECOMPOSE = VDQ

"Pseudo" matrix element

Available contributions

V - Derivative of pseudopotential, $\tilde{v}$

$g_{m{\mathbf{𝐤}}^{\prime },n\mathbf{𝐤},a}^{(\text{V})}\equiv ⟨{\tilde{\psi }}_{m{\mathbf{𝐤}}^{\prime }}|\frac{\partial \tilde{v}}{\partial u_a}|{\tilde{\psi }}_{n\mathbf{𝐤}}⟩$

D - Derivative of PAW strength parameters, $D_{a,ij}$

$g_{m{\mathbf{𝐤}}^{\prime },n\mathbf{𝐤},a}^{(\text{D})}\equiv \sum _{bij}⟨{\tilde{\psi }}_{m{\mathbf{𝐤}}^{\prime }}|{\tilde{p}}_{bi}⟩\frac{\partial D_{b,ij}}{\partial u_a}⟨{\tilde{p}}_{bj}|{\tilde{\psi }}_{n\mathbf{𝐤}}⟩$

P - Derivative of PAW projectors, $|{\tilde{p}}_{ai}⟩$

Failed to parse (unknown function "\begin{split}"): {\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, $|{\varphi }_{ai}⟩$ and $|{\tilde{\varphi }}_{ai}⟩$

$g_{m{\mathbf{𝐤}}^{\prime },n\mathbf{𝐤},a}^{(\text{R})}\equiv ({\epsilon }_{n\mathbf{𝐤}}-{\epsilon }_{m{\mathbf{𝐤}}^{\prime }})\sum _{ij}⟨{\tilde{\psi }}_{m{\mathbf{𝐤}}^{\prime }}|{\tilde{p}}_{ai}⟩R_{a,ij}⟨{\tilde{p}}_{aj}|{\tilde{\psi }}_{n\mathbf{𝐤}}⟩$

with $R_{a,ij}\equiv ⟨{\varphi }_{ai}|\frac{\partial {\varphi }_{aj}}{\partial u_a}⟩-⟨{\tilde{\varphi }}_{ai}|\frac{\partial {\tilde{\varphi }}_{aj}}{\partial u_a}⟩$

Q - Derivative of PAW projectors, $|{\tilde{p}}_{ai}⟩$ (different eigenvalues)

Failed to parse (unknown function "\begin{split}"): {\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} }

For more details, please refer to Ref., and consult our documentation on the projector-augmented-wave_formalism.