ELPH DECOMPOSE: Difference between revisions

Revision as of 10:56, 27 February 2025

ELPH_DECOMPOSE = [string]
Default: ELPH_DECOMPOSE = VDPR

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

Mind: Available as of VASP 6.5.0

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, $\tilde{v}$: ${\displaystyle g^{(\text{V})}_{m \mathbf{k}', n \mathbf{k}, a} \equiv \langle \tilde{\psi}_{m \mathbf{k}'} | \frac{\partial \tilde{v}}{\partial u_{a}} | \tilde{\psi}_{n \mathbf{k}} \rangle }$; This term is the pure plane-wave contribution to the total PAW matrix element. If the PAW augmentation region were vanishingly small, this would be the sole contribution.
D - Derivative of PAW strength parameters, $D_{a, ij}$: ${\displaystyle g^{(\text{D})}_{m \mathbf{k}', n \mathbf{k}, a} \equiv \sum_{bij} \langle \tilde{\psi}_{m \mathbf{k}'} | \tilde{p}_{b i} \rangle \frac{\partial D_{b, ij}}{\partial u_{a}} \langle \tilde{p}_{b j} | \tilde{\psi}_{n \mathbf{k}} \rangle }$; This contribution stems from the PAW treatment of the electronic Hamiltonian. It is of the same nature as $g^{(\text{V})}$ but is treated in the local basis inside the augmentation region. For a detailed discussion of the PAW strength parameters, we refer to Ref. ^[4].
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): ${\displaystyle \begin{split} g^{(\text{Q})}_{m \mathbf{k}', n \mathbf{k}, a} & \equiv \sum_{ij} \langle \tilde{\psi}_{m \mathbf{k}'} | \frac{\partial \tilde{p}_{a i}}{\partial u_{a}} \rangle ( D_{a, ij} - \varepsilon_{n \mathbf{k}} Q_{a, ij} ) \langle \tilde{p}_{a j} | \tilde{\psi}_{n \mathbf{k}} \rangle \\ & + \sum_{ij} \langle \tilde{\psi}_{m \mathbf{k}'} | \tilde{p}_{a i} \rangle ( D_{a, ij} - \varepsilon_{n \mathbf{k}} Q_{a, ij} ) \langle \frac{\partial \tilde{p}_{a j}}{\partial u_{a}} | \tilde{\psi}_{n \mathbf{k}} \rangle \end{split} }$; This contribution is very similar to $g^{(\text{P})}$ . The only difference is in the Kohn-Sham eigenvalues. While $g^{(\text{P})}$ uses the eigenvalues of both the initial and final state (so $\varepsilon_{n \mathbf{k}}$ and $\varepsilon_{m \mathbf{k}'}$ ), $g^{(\text{Q})}$ only uses the eigenvalues of the initial state ( $\varepsilon_{n \mathbf{k}}$ ).

References

[engel:prb:2022-1] M. Engel, H. Miranda, L. Chaput, A. Togo, C. Verdi, M. Marsman, and G. Kresse, Zero-point renormalization of the band gap of semiconductors and insulators using the projector augmented wave method, Phys. Rev. B 106, 094316 (2022).

[chaput:prb:2019-2] L. Chaput, A. Togo, and I. Tanaka, Finite-displacement computation of the electron-phonon interaction within the projector augmented-wave method, Phys. Rev. B 100, 174304 (2019).

[engel:prb:2020-3] M. Engel, M. Marsman, C. Franchini, and G. Kresse, Electron-phonon interactions using the projector augmented-wave method and Wannier functions, Phys. Rev. B 101, 184302 (2020).

[kresse:prb:99-4] I. G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).

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@@ Line 27: / Line 27: @@
      \rangle
 </math>
+:This term is the pure plane-wave contribution to the total PAW matrix element. If the PAW augmentation region were vanishingly small, this would be the sole contribution.
 ;D - Derivative of PAW strength parameters, <math>D_{a, ij}</math>
 :<math>
@@ Line 42: / Line 43: @@
      \rangle
 </math>
+:This contribution stems from the PAW treatment of the electronic Hamiltonian. It is of the same nature as <math>g^{(\text{V})}</math> but is treated in the local basis inside the augmentation region. For a detailed discussion of the PAW strength parameters, we refer to Ref. {{Cite|kresse:prb:99}}.
 ;P - Derivative of PAW projectors, <math>|\tilde{p}_{ai}\rangle</math>
 :<math>
@@ Line 134: / Line 136: @@
      \end{split}
 </math>
+:This contribution is very similar to <math>g^{(\text{P})}</math>. The only difference is in the Kohn-Sham eigenvalues. While <math>g^{(\text{P})}</math> uses the eigenvalues of both the initial and final state (so <math>\varepsilon_{n \mathbf{k}}</math> and <math>\varepsilon_{m \mathbf{k}'}</math>), <math>g^{(\text{Q})}</math> only uses the eigenvalues of the initial state (<math>\varepsilon_{n \mathbf{k}}</math>).
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