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Note

First-pass extraction of display equations from rmp17.tex. Labels have been preserved for Sphinx math cross-references.

(1)\[\begin{split}\begin{aligned} \hat{H} & = & \sum_{n\mathbf{k}} \varepsilon_{n\mathbf{k}} \hat{c}_{n\mathbf{k}}^\dagger \hat{c}_{n\mathbf{k}} + \sum_{\mathbf{q}\nu} \hbar\omega_{\mathbf{q}\nu} (\hat{a}_{\mathbf{q}\nu}^\dagger \hat{a}_{\mathbf{q}\nu}+1/2) \\ & + & N_p^{-\frac{1}{2}} \sum_{\substack{\mathbf{k},\mathbf{q} \\ m n \nu }} g_{mn\nu}(\mathbf{k},\mathbf{q}) \hat{c}_{m\mathbf{k}+\mathbf{q}}^\dagger \hat{c}_{n\mathbf{k}}(\hat{a}_{\mathbf{q}\nu}+\hat{a}_{-\mathbf{q}\nu}^\dagger) \\ \Bigg[& + & N_p^{-1} \sum_{\substack{\mathbf{k},\mathbf{q},\mathbf{q}'\\m n \nu\nu'}} g^{\mathrm DW}_{mn\nu\nu'}(\mathbf{k},\mathbf{q},\mathbf{q}') \hat{c}_{m\mathbf{k}+\mathbf{q}+\mathbf{q}'}^\dagger \hat{c}_{n\mathbf{k}} \\ &\times& (\hat{a}_{\mathbf{q}\nu}+\hat{a}^\dagger_{-\mathbf{q}\nu}) (\hat{a}_{\mathbf{q}'\nu'}+\hat{a}^\dagger_{-\mathbf{q}'\nu'}) \Bigg]. \end{aligned}\end{split}\]

Ab initio self-consistent field calculations

(2)\[g_{mn\nu}(\mathbf{k},\mathbf{q}) = \langle u_{m\mathbf{k}+\mathbf{q}} | \Delta_{\mathbf{q}\nu} v^{\mathrm KS} | u_{n\mathbf{k}} \rangle_{\mathrm uc},\]

(3)\[\begin{split}U = U_0 + \frac{1}{2} \sum_{\substack{\kappa\alpha p\\\kappa'\alpha' p'}} \frac{\partial^2 U}{\partial \tau_{\kappa\alpha p}\partial \tau_{\kappa'\alpha' p'}}\Delta \tau_{\kappa\alpha p} \Delta \tau_{\kappa'\alpha' p'},\end{split}\]

(4)\[C_{\kappa\alpha p,\kappa'\alpha' p'} = \partial^2 U/\partial \tau_{\kappa\alpha p}\partial \tau_{\kappa'\alpha' p'}.\]

(5)\[D^{\mathrm dm}_{\kappa\alpha,\kappa'\alpha'}(\mathbf{q}) = (M_\kappa M_{\kappa'})^{-\frac{1}{2}} {\sum}_p C_{\kappa\alpha 0,\kappa'\alpha' p} \exp(i\mathbf{q}\cdot\mathbf{R}_p),\]

(6)\[{\sum}_{\kappa'\alpha'} D^{\mathrm dm}_{\kappa\alpha,\kappa'\alpha'}(\mathbf{q}) e_{\kappa'\alpha',\nu}(\mathbf{q}) = \omega_{\mathbf{q}\nu}^2 e_{\kappa\alpha,\nu}(\mathbf{q}).\]

(7)\[\begin{split}\begin{aligned} {\sum}_{\nu} e_{\kappa'\alpha',\nu}^*(\mathbf{q}) e_{\kappa\alpha,\nu}(\mathbf{q}) & = & \delta_{\kappa\kappa'} \delta_{\alpha\alpha'}, \\ {\sum}_{\kappa\alpha} e_{\kappa\alpha,\nu}^*(\mathbf{q}) e_{\kappa\alpha,\nu'}(\mathbf{q}) & = & \delta_{\nu\nu'}. \end{aligned}\end{split}\]

(8)\[\omega_{-\mathbf{q}\nu}^2 = \omega_{\mathbf{q}\nu}^2; \qquad e_{\kappa\alpha,\nu}(-\mathbf{q}) = e_{\kappa\alpha,\nu}^*(\mathbf{q}).\]

(9)\[\begin{split}\hat{H}_{\mathrm p} = \frac{1}{2} \sum_{\substack{\kappa\alpha p\\\kappa'\alpha' p'}} C_{\kappa\alpha p,\kappa'\alpha' p'} \Delta \tau_{\kappa\alpha p} \Delta \tau_{\kappa'\alpha' p'} -\sum_{\kappa\alpha p}\frac{\hbar^2}{2M_\kappa}\frac{\partial^2}{\partial \tau_{\kappa\alpha p}^2},\end{split}\]

(10)\[\Delta\tau_{\kappa\alpha p} = \left(\frac{M_0}{N_p M_\kappa}\right)^{\frac{1}{2}} \sum_{\mathbf{q}\nu} e^{i\mathbf{q}\cdot\mathbf{R}_p} e_{\kappa\alpha,\nu}(\mathbf{q}) l_{\mathbf{q}\nu} (\hat{a}_{\mathbf{q}\nu}+\hat{a}^\dagger_{-\mathbf{q}\nu}),\]

(11)\[l_{\mathbf{q}\nu} = [\hbar/(2M_0\omega_{\mathbf{q}\nu})]^{1/2}.\]

(12)\[\hat{H}_{\mathrm p} = {\sum}_{\mathbf{q}\nu} \hbar\omega_{\mathbf{q}\nu} \left( \hat{a}_{\mathbf{q}\nu}^\dagger \hat{a}_{\mathbf{q}\nu} + 1/2 \right),\]

Kohn-Sham Hamiltonian

(13)\[\hat{H}^{\mathrm KS} = -\frac{\hbar^2}{2m_e}\nabla^2 + V^{\mathrm KS}(\mathbf{r};\{\tau_{\kappa\alpha p}\}).\]

(14)\[V^{\mathrm KS} = V^{\mathrm en} + V^{\mathrm H} + V^{xc}.\]

(15)\[V^{\mathrm en}(\mathbf{r};\{\tau_{\kappa\alpha p}\}) = {\sum}_{\kappa p,\mathbf{T}} V_\kappa(\mathbf{r}-\boldsymbol{\tau}_{\kappa p} - \mathbf{T}),\]

(16)\[V_\kappa(\mathbf{r}) = -\frac{e^2}{4\pi\epsilon_0} \frac{Z_\kappa}{|\mathbf{r}|},\]

(17)\[V^{\mathrm H}(\mathbf{r};\{\tau_{\kappa\alpha p}\}) = \frac{e^2}{4\pi\epsilon_0}{\sum}_{\mathbf{T}}\int_{\mathrm sc} \frac{n(\mathbf{r}';\{\tau_{\kappa\alpha p}\})} {|\mathbf{r}-\mathbf{r}'-\mathbf{T}|} d\mathbf{r}',\]

(18)\[V^{xc}(\mathbf{r};\{\tau_{\kappa\alpha p}\}) = \delta E^{xc}[n] / \delta n \big|_{n(\mathbf{r};\{\tau_{\kappa\alpha p}\})}.\]

(19)\[\psi_{n\mathbf{k}}(\mathbf{r}) = N_p^{-\frac{1}{2}} u_{n\mathbf{k}}(\mathbf{r}) e^{i\mathbf{k}\cdot\mathbf{r}},\]

(20)\[\hat{H}_{\mathrm e} = \sum_{n\mathbf{k},n'\mathbf{k}'} \langle \psi_{n\mathbf{k}} | \hat{H}^{\mathrm KS} | \psi_{n'\mathbf{k}'} \rangle \hat{c}_{n\mathbf{k}}^\dagger \hat{c}_{n'\mathbf{k}'} = \sum_{n\mathbf{k}} \varepsilon_{n\mathbf{k}} \hat{c}_{n\mathbf{k}}^\dagger \hat{c}_{n\mathbf{k}}.\]

Electron-phonon coupling Hamiltonian to first- and second-order in the atomic displacements ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

(21)\[V^{\mathrm KS}(\{ \boldsymbol{\tau}_{\kappa p} \}) = V^{\mathrm KS}(\{ \boldsymbol{\tau}^0_{\kappa p} \}) + {\sum}_{\kappa\alpha p} \frac{\partial V^{\mathrm KS}}{\partial \tau_{\kappa\alpha p}} \Delta \tau_{\kappa\alpha p}.\]

(22)\[V^{\mathrm KS} = V^{\mathrm KS}(\{ \boldsymbol{\tau}^0_{\kappa p} \}) + N_p^{-\frac{1}{2}}\sum_{\mathbf{q}\nu} \Delta_{\mathbf{q}\nu} V^{\mathrm KS} (\hat{a}_{\mathbf{q}\nu}+\hat{a}^\dagger_{-\mathbf{q}\nu}),\]

(23)\[\begin{split}\begin{aligned} \Delta_{\mathbf{q}\nu} V^{\mathrm KS} & = & e^{i\mathbf{q}\cdot\mathbf{r}} \Delta_{\mathbf{q}\nu} v^{\mathrm KS}, \\ \Delta_{\mathbf{q}\nu} v^{\mathrm KS} & = & l_{\mathbf{q}\nu} {\sum}_{\kappa\alpha} (M_0/M_\kappa)^\frac{1}{2} e_{\kappa\alpha,\nu}(\mathbf{q}) \partial_{\kappa\alpha,\mathbf{q}}v^{\mathrm KS}, \hspace{0.3cm} \\ \partial_{\kappa\alpha,\mathbf{q}}v^{\mathrm KS} & = & {\sum}_{p}e^{-i\mathbf{q}\cdot(\mathbf{r}-\mathbf{R}_p)} \left.\frac{\partial V^{\mathrm KS}} {\partial \tau_{\kappa\alpha}}\right|_{\mathbf{r}-\mathbf{R}_p}. \end{aligned}\end{split}\]

\[\hat{H}_{\mathrm ep} = \sum_{n\mathbf{k},n'\mathbf{k}'} \langle \psi_{n\mathbf{k}}| V^{\mathrm KS}(\{ \boldsymbol{\tau}_{\kappa p} \})-V^{\mathrm KS}(\{ \boldsymbol{\tau}^0_{\kappa p} \}) |\psi_{n'\mathbf{k}'}\rangle \hat{c}_{n\mathbf{k}}^\dagger \hat{c}_{n'\mathbf{k}'},\]

(24)\[\begin{split}\hat{H}_{\mathrm ep} = N_p^{-\frac{1}{2}} \sum_{\substack{\mathbf{k},\mathbf{q} \\ m n \nu }} g_{mn\nu}(\mathbf{k},\mathbf{q}) \hat{c}_{m\mathbf{k}+\mathbf{q}}^\dagger \hat{c}_{n\mathbf{k}}(\hat{a}_{\mathbf{q}\nu}+\hat{a}_{-\mathbf{q}\nu}^\dagger),\end{split}\]

(25)\[g_{mn\nu}(\mathbf{k},\mathbf{q}) = \langle u_{m\mathbf{k}+\mathbf{q}} | \Delta_{\mathbf{q}\nu} v^{\mathrm KS} | u_{n\mathbf{k}} \rangle_{\mathrm uc}.\]

(26)\[\begin{split}\begin{aligned} \hat{H}_{\mathrm ep}^{(2)} & = & N_p^{-1} \sum_{\substack{\mathbf{k},\mathbf{q},\mathbf{q}'\\m n \nu\nu'}} g^{\mathrm DW}_{mn\nu\nu'}(\mathbf{k},\mathbf{q},\mathbf{q}') \hat{c}_{m\mathbf{k}+\mathbf{q}+\mathbf{q}'}^\dagger \hat{c}_{n\mathbf{k}} \\ &\times& (\hat{a}_{\mathbf{q}\nu}+\hat{a}^\dagger_{-\mathbf{q}\nu}) (\hat{a}_{\mathbf{q}'\nu'}+\hat{a}^\dagger_{-\mathbf{q}'\nu'}), \end{aligned}\end{split}\]

(27)\[g^{\mathrm DW}_{mn\nu\nu'}(\mathbf{k},\mathbf{q},\mathbf{q}') = \frac{1}{2} \langle u_{m\mathbf{k}+\mathbf{q}+\mathbf{q}'}| \Delta_{\mathbf{q}\nu} \Delta_{\mathbf{q}'\nu'} v^{\mathrm KS} |u_{n\mathbf{k}}\rangle_{\mathrm uc}.\]

Calculation of electron-phonon matrix elements using density-functional perturbation theory ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

\[\left.\frac{\partial V^{\mathrm KS}} {\partial \tau_{\kappa\alpha p}}\right|_{\boldsymbol{\tau}_{\kappa p}^0} \simeq \left[ V^{\mathrm KS}(\mathbf{r};\tau_{\kappa\alpha p}^0+b) - V^{\mathrm KS}(\mathbf{r};\tau_{\kappa\alpha p}^0) \right]/b.\]

(28)\[\begin{aligned} \partial_{\kappa\alpha,\mathbf{q}}v^{\mathrm KS} & = & \partial_{\kappa\alpha,\mathbf{q}}v^{\mathrm en} + \partial_{\kappa\alpha,\mathbf{q}}v^{\mathrm H} + \partial_{\kappa\alpha,\mathbf{q}}v^{xc}. \end{aligned}\]

(29)\[\partial_{\kappa\alpha,\mathbf{q}}v^{\mathrm en}(\mathbf{G}) = -i (\mathbf{q}+\mathbf{G})_\alpha V_\kappa(\mathbf{q}+\mathbf{G}) e^{-i(\mathbf{q}+\mathbf{G})\cdot\boldsymbol{\tau}_\kappa},\]

(30)\[\partial_{\kappa\alpha,\mathbf{q}}v^{\mathrm H}(\mathbf{G}) = \Omega v^{\mathrm C}(\mathbf{q}+\mathbf{G}) \partial_{\kappa\alpha,\mathbf{q}} n(\mathbf{G}),\]

(31)\[\partial_{\kappa\alpha,\mathbf{q}}v^{xc}(\mathbf{G}) = \Omega {\sum}_{\mathbf{G}'} f^{xc}(\mathbf{q}+\mathbf{G},\mathbf{q}+\mathbf{G}') \partial_{\kappa\alpha,\mathbf{q}}n(\mathbf{G}'),\]

(32)\[f^{xc}(\mathbf{r},\mathbf{r}') = \left.\frac{\delta^2 E^{xc}[n]}{\delta n(\mathbf{r})\delta n(\mathbf{r}')} \right|_{n(\mathbf{r};\{\boldsymbol{\tau}_{\kappa p}^0\})}.\]

(33)\[\left(\hat{H}^{\mathrm KS}_{\mathbf{k}+\mathbf{q}} - \varepsilon_{v\mathbf{k}}\right)\partial u_{v\mathbf{k},\mathbf{q}} = -\partial_{\kappa\alpha,\mathbf{q}}v^{\mathrm KS} u_{v\mathbf{k}},\]

(34)\[\left(\hat{H}^{\mathrm KS}_{\mathbf{k}+\mathbf{q}} - \varepsilon_{v\mathbf{k}}\right)\partial \tilde{u}_{v\mathbf{k},\mathbf{q}} = -(1-\hat{P}^{ \mathrm occ}_{\mathbf{k}+\mathbf{q}}) \partial_{\kappa\alpha,\mathbf{q}}v^{\mathrm KS} u_{v\mathbf{k}}.\]

(35)\[\partial n_{\kappa\alpha,\mathbf{q}}(\mathbf{r}) = 2 N_p^{-1}{\sum}_{v\mathbf{k}} u^*_{v\mathbf{k}} \partial \tilde{u}_{v\mathbf{k},\mathbf{q}}.\]

Operators and distinguishability

(36)\[\hat{H} = \hat{T}_{\mathrm e} + \hat{T}_{\mathrm n} + \hat{U}_{\mathrm ee} + \hat{U}_{\mathrm nn} + \hat{U}_{\mathrm en},\]

(37)\[\hat{T}_{\mathrm e} = -\frac{\hbar^2}{2m_{\mathrm e}}\intd\mathbf{x} \hat{\psi}^\dagger(\mathbf{x}) \nabla^2 \hat{\psi}(\mathbf{x}),\]

(38)\[\hat{U}_{\mathrm ee} = \frac{1}{2}\intd\mathbf{r}\int d\mathbf{r}' \hat{n}_{\mathrm e}(\mathbf{r})\left[\hat{n}_{\mathrm e}(\mathbf{r}') - \delta(\mathbf{r}-\mathbf{r}')\right]v(\mathbf{r},\mathbf{r}'),\]

(39)\[\hat{U}_{\mathrm nn} = \frac{1}{2}\sum_{ \kappa' p' \ne \kappa p} Z_\kappa Z_{\kappa'} v( \boldsymbol{\tau}^0_{\kappa p}+\Delta\hat{\bm\tau}_{\kappa p}, \boldsymbol{\tau}^0_{\kappa' p'}+\Delta\hat{\bm\tau}_{\kappa' p'}).\]

(40)\[\hat{U}_{\mathrm en} = \intd\mathbf{r}\int d\mathbf{r}' \hat{n}_{\mathrm e}(\mathbf{r}) \hat{n}_{\mathrm n}(\mathbf{r}') v(\mathbf{r},\mathbf{r}'),\]

(41)\[\hat{n}_{\mathrm n}(\mathbf{r}) = - {\sum}_{\kappa p} Z_\kappa \delta(\mathbf{r}-\boldsymbol{\tau}^0_{\kappa p}-\Delta\hat{\bm\tau}_{\kappa p}).\]

Equation of motion and self-energy

(42)\[\begin{split}\begin{aligned} \hat{H} & = & \hat{T}_{\mathrm n} + \hat{U}_{\mathrm nn} +\int d\mathbf{x} \hat{\psi}^\dagger(\mathbf{x}) \left[-\frac{\hbar^2}{2m_{\mathrm e}}\nabla^2 + \hat{V}_{\mathrm n}(\mathbf{r})\right] \hat{\psi}(\mathbf{x}) \\ & + & \frac{1}{2}\intd\mathbf{x} d\mathbf{x}' v(\mathbf{r},\mathbf{r}')\hat{\psi}^\dagger(\mathbf{x})\hat{\psi}^\dagger(\mathbf{x}')\hat{\psi}(\mathbf{x}')\hat{\psi}(\mathbf{x}), \end{aligned}\end{split}\]

(43)\[\hat{V}_{\mathrm n}(\mathbf{r}) = \int d\mathbf{r}' v(\mathbf{r},\mathbf{r}') \hat{n}_{\mathrm n}(\mathbf{r}').\]

(44)\[G(\mathbf{x} t, \mathbf{x}' t') = -\frac{i}{\hbar}\langle 0 | \hat{T} \psi(\mathbf{x} t) \psi^\dagger(\mathbf{x}' t') | 0 \rangle,\]

(45)\[\hat{\psi}(\mathbf{x} t) = e^{i t\hat{H}/\hbar} \hat{\psi}(\mathbf{x}) e^{-i t\hat{H}/\hbar},\]

(46)\[i\hbar\frac{\partial}{\partial t}\hat{\psi}(\mathbf{x} t) = [\hat{\psi}(\mathbf{x} t),\hat{H}].\]

(47)\[i\hbar\frac{\partial}{\partial t}\hat{\psi}(\mathbf{x} t) = \left[-\frac{\hbar^2}{2m_{\mathrm e}}\nabla^2 + \intd\mathbf{r}' v(\mathbf{r},\mathbf{r}') \hat{n}(\mathbf{r}'t) \right] \hat{\psi}(\mathbf{x} t),\]

(48)\[\begin{split}\begin{aligned} && \left[ i\hbar \frac{\partial}{\partial t} + \frac{\hbar^2}{2m_{\mathrm e}}\nabla^2 -\varphi(\mathbf{r} t) \right] G(\mathbf{x} t, \mathbf{x}' t') = \delta(\mathbf{x} t,\mathbf{x}' t') \\ && -\frac{i}{\hbar} \intd\mathbf{r}''dt'' v(\mathbf{r} t,\mathbf{r}'' t'') \langle \hat{T} \hat{n}(\mathbf{r}''t'') \psi(\mathbf{x} t) \psi^\dagger(\mathbf{x}' t') \rangle.\hspace{0.2cm} \end{aligned}\end{split}\]

(49)\[\frac{\delta \langle \hat{T} \hat{a}(t_1)\hat{b}(t_2)\rangle }{\delta \varphi(\mathbf{r}'' t'')} = -\frac{i}{\hbar} \langle \hat{T}\left[ \hat{n}(\mathbf{r}'' t'')-\langle \hat{n}(\mathbf{r}'' t'') \rangle \right] \hat{a}(t_1)\hat{b}(t_2)\rangle.\]

(50)\[\begin{split}\begin{aligned} && \left[ i\hbar \frac{\partial}{\partial t} + \frac{\hbar^2}{2m_{\mathrm e}}\nabla^2 -V_{\mathrm tot}(\mathbf{r} t) -i\hbar\int d\mathbf{r}''dt'' v(\mathbf{r} t+\eta,\mathbf{r}'' t'') \right. \\ &&\qquad\qquad \left. \times \frac{\delta }{\delta \varphi(\mathbf{r}'' t'')} \right] G(\mathbf{x} t, \mathbf{x}' t') = \delta(\mathbf{x} t,\mathbf{x}' t'), \end{aligned}\end{split}\]

\[V_{\mathrm tot}(\mathbf{r} t) = \intd\mathbf{r}' v(\mathbf{r},\mathbf{r}') \langle \hat{n}(\mathbf{r}' t) \rangle + \varphi(\mathbf{r} t).\]

(51)\[\begin{split}\begin{aligned} && \hspace{-0.7cm}\left[ i\hbar \frac{\partial}{\partial t_1} + \frac{\hbar^2}{2m_{\mathrm e}}\nabla^2(1) -V_{\mathrm tot}(1) \right. \\ && \hspace{0.5cm} \left. -i\hbar\int d3 v(1^+ 3) \frac{\delta }{\delta \varphi(3)} \right] G(1 2) = \delta(12), \end{aligned}\end{split}\]

(52)\[V_{\mathrm tot}(1) = \intd2 v(12) \langle \hat{n}(2) \rangle + \varphi(1).\]

(53)\[\epsilon^{-1}(12) = \delta V_{\mathrm tot}(1) / \delta \varphi(2).\]

(54)\[\frac{\delta G(12)}{\delta \varphi(3)} = -\int d(45) G(14) \frac{\delta G^{-1}(45)}{\delta \varphi(3)} G(52).\]

(55)\[\frac{\delta G^{-1}(45)}{\delta \varphi(3)} = \int d6 \frac{\delta G^{-1}(45)}{\delta V_{\mathrm tot}(6)} \frac{\delta V_{\mathrm tot}(6)}{\delta \varphi(3)}.\]

(56)\[\Gamma(123) = - \delta G^{-1}(12) / \delta V_{\mathrm tot}(3).\]

(57)\[\begin{split}\begin{aligned} && \hspace{-1cm}\left[ i\hbar \frac{\partial}{\partial t_1} + \frac{\hbar^2}{2m_{\mathrm e}}\nabla^2(1) -V_{\mathrm tot}(1) \right]G(12) \\ &&\hspace{2cm} - \intd3 \Sigma(13) G(32) = \delta(12), \end{aligned}\end{split}\]

(58)\[\Sigma(12) = i\hbar\int d(34) G(13) \Gamma(324) W(41^+),\]

(59)\[W(12) = \int d3 \epsilon^{-1}(13) v(32) = \int d(3) v(13) \epsilon^{-1}(23).\]

(60)\[\Gamma(123) = \delta(12)\delta(13) +\int d(4567) \frac{\delta\Sigma(12)}{\delta G(45)} G(46) G(75) \Gamma(673).\]

The screened Coulomb interaction

(61)\[W(12) = v(12) + \int d(34) v(13) \frac{\delta \langle \hat{n}(3) \rangle }{\delta V_{\mathrm tot}(4)} W(42).\]

(62)\[P(12) = \frac{\delta \langle \hat{n}(1) \rangle }{\delta V_{\mathrm tot}(2)},\]

(63)\[W(12) = v(12) + \int d(34) v(13) P(34) W(42).\]

(64)\[\epsilon(12) = \delta(12) - \int d(3) v(13) P(32).\]

(65)\[P_{\mathrm e}(12) = \frac{\delta \langle \hat{n}_{\mathrm e}(1) \rangle }{\delta V_{\mathrm tot}(2)} = -i\hbar \sum_{\sigma_1} \int d(34) G(13) G(41^+) \Gamma(342).\]

(66)\[\langle \hat{n}_{\mathrm e}(1) \rangle = -i\hbar {\sum}_{\sigma_1} G(11^+).\]

(67)\[W_{\mathrm e}(12) = v(12) + \int d(34) v(13) P_{\mathrm e}(34) W_{\mathrm e}(42),\]

(68)\[\epsilon_{\mathrm e}(12) = \delta(12) - \int d3 v(13) P_{\mathrm e}(32).\]

(69)\[W(12)= W_{\mathrm e}(12)+\int d(34) W_{\mathrm e}(13) \frac{\delta \langle \hat{n}_{\mathrm n}(3) \rangle }{\delta \varphi(4)} v(42).\]

(70)\[\frac{\delta \langle \hat{n}_{\mathrm n}(1)\rangle }{\delta \varphi(2)} = -\frac{i}{\hbar} \langle \hat{T}\left[ \hat{n}(2)-\langle \hat{n}(2) \rangle \right] [\hat{n}_{\mathrm n}(1)-\langle\hat{n}_{\mathrm n}(1)\rangle]\rangle.\]

(71)\[\frac{\delta \langle \hat{n}(1)\rangle }{\delta J(2)} = -\frac{i}{\hbar} \langle \hat{T}\left[ \hat{n}_{\mathrm n}(2)-\langle \hat{n}_{\mathrm n}(2) \rangle \right] [\hat{n}(1)-\langle\hat{n}(1)\rangle]\rangle.\]

(72)\[\frac{\delta \langle \hat{n}_{\mathrm n}(1)\rangle }{\delta \varphi(2)} = \frac{\delta \langle \hat{n}(2)\rangle }{\delta J(1)}.\]

(73)\[\frac{\delta \langle \hat{n}(1)\rangle }{\delta J(2)} = \int d3 \epsilon_{\mathrm e}^{-1}(13) \frac{\delta \langle \hat{n}_{\mathrm n}(3)\rangle }{\delta J(2)}.\]

(74)\[\frac{\delta \langle \hat{n}_{\mathrm n}(1)\rangle }{\delta J(2)} = -\frac{i}{\hbar} \langle \hat{T}\left[ \hat{n}_{\mathrm n}(2)-\langle \hat{n}_{\mathrm n}(2) \rangle \right] \hat{n}_{\mathrm n}(1)\rangle,\]

\[\delta \langle \hat{n}_{\mathrm n}(1)\rangle / \delta J(2) = D(21),\]

(75)\[D(12) = -\frac{i}{\hbar} \langle \hat{T}\left[ \hat{n}_{\mathrm n}(1)-\langle \hat{n}_{\mathrm n}(1) \rangle \right] \left[ \hat{n}_{\mathrm n}(2)-\langle \hat{n}_{\mathrm n}(2) \rangle \right]\rangle.\]

(76)\[W(12)= W_{\mathrm e}(12)+ W_{\mathrm ph}(12),\]

(77)\[W_{\mathrm ph}(12) = \int d(34) W_{\mathrm e}(13) D(34) W_{\mathrm e}(24).\]

Nuclear contribution to the screened Coulomb interaction

(78)\[\delta(\mathbf{r}-\mathbf{u}) = \delta(\mathbf{r}) - \mathbf{u}\cdot \nabla \delta(\mathbf{r}) + \frac{1}{2}\mathbf{u} \cdot \nabla \nabla \delta(\mathbf{r}) \cdot \mathbf{u},\]

(79)\[\begin{split}\begin{aligned} \hat{n}_{\mathrm n}(\mathbf{r}) & = & n_{\mathrm n}^0(\mathbf{r}) + {\sum}_{\kappa p} Z_\kappa \Delta\hat{\bm\tau}_{\kappa p}\cdot \nabla \delta(\mathbf{r}-\bm\tau_{\kappa p}^0) \\ & - &\frac{1}{2}{\sum}_{\kappa p} Z_\kappa \Delta\hat{\bm\tau}_{\kappa p} \cdot \nabla \nabla \delta(\mathbf{r}-\bm\tau_{\kappa p}^0) \cdot \Delta\hat{\bm\tau}_{\kappa p}, \end{aligned}\end{split}\]

\[\begin{split}D(12) = \sum_{\substack{\kappa \alpha p \\ \kappa' \alpha' p'}} Z_\kappa \nabla_{1,\alpha} \delta(\mathbf{r}_1-\bm\tau_{\kappa p}^0) D_{\kappa \alpha p,\kappa' \alpha' p'}(t_1 t_2)\end{split}\]

(80)\[\hspace{0.5cm}\times Z_{\kappa'} \nabla_{2,\alpha'} \delta(\mathbf{r}_2-\bm\tau_{\kappa' p' }^0).\]

(81)\[D_{\kappa \alpha p,\kappa' \alpha' p'}(t t') = -\frac{i}{\hbar} \langle \hat{T} \Delta \hat{\tau}_{\kappa \alpha p}(t) \Delta\hat{\tau}_{\kappa' \alpha' p'}(t') \rangle.\]

(82)\[\begin{split}\begin{aligned} && W_{\mathrm ph}(12)= \sum_{\substack{\kappa \alpha p \\ \kappa' \alpha' p'}} \int d(34) \epsilon_{\mathrm e}^{-1}(13) \nabla_{3,\alpha} V_\kappa(\mathbf{r}_3-\bm\tau_{\kappa p}^0) \\ &&\times D_{\kappa \alpha p,\kappa' \alpha' p'}(t_3 t_4) \epsilon_{\mathrm e}^{-1}(24) \nabla_{4,\alpha'} V_{\kappa'}(\mathbf{r}_4-\bm\tau_{\kappa'p'}^0). \end{aligned}\end{split}\]

Phonon Green’s function

(83)\[M_\kappa \frac{\partial^2}{\partial t^2} \Delta\hat{\bm\tau}_{\kappa p} = -\frac{M_\kappa}{\hbar^2}[[\Delta\hat{\bm\tau}_{\kappa p},\hat{H}],\hat{H}].\]

(84)\[\begin{split}\begin{aligned} &&M_\kappa \frac{\partial^2}{\partial t^2} \Delta\hat{\bm\tau}_{\kappa p}(t) = Z_\kappa \intd\mathbf{r} d\mathbf{r}' \hat{n}^{(\kappa p)}(\mathbf{r} t) v( \mathbf{r}, \mathbf{r}') \\ && \times \left\{ -\nabla^\prime \delta(\mathbf{r}'-\boldsymbol{\tau}_{\kappa p}^0) + \nabla^\prime \left[\nabla^\prime \delta(\mathbf{r}'-\boldsymbol{\tau}_{\kappa p}^0) \cdot \Delta\hat{\bm\tau}_{\kappa p}(t)\right] \right\}. \qquad \end{aligned}\end{split}\]

\[\frac{\delta \langle \Delta\hat{\tau}_{\kappa \alpha p}(t) \rangle}{\delta F_{\kappa' \alpha' p'}(t')} = D_{\kappa \alpha p, \kappa' \alpha' p'}(t t').\]

(85)\[\begin{split}\begin{aligned} &&\hspace{-0.7cm}M_\kappa \frac{\partial^2}{\partial t^2} D_{\kappa \alpha p, \kappa' \alpha' p'} (t t')= -\delta_{\kappa \alpha p,\kappa' \alpha' p'} \delta(t t') \\ && \hspace{-0.7cm}+Z_\kappa \intd\mathbf{r} d\mathbf{r}' \left[ -\frac{\delta \langle \hat{n}^{(\kappa p)}(\mathbf{r} t) \rangle }{\delta F_{\kappa' \alpha' p'}(t')} v( \mathbf{r}, \mathbf{r}') \nabla^\prime_{\alpha} \delta(\mathbf{r}'-\boldsymbol{\tau}_{\kappa p}^0) \right. \\ && \hspace{-0.7cm}\left. + \langle \hat{n}^{(\kappa p)}(\mathbf{r} t)\rangle v( \mathbf{r}, \mathbf{r}') \nabla^\prime_{\alpha} \nabla^\prime_{\gamma} \delta(\mathbf{r}'-\boldsymbol{\tau}_{\kappa p}^0) D_{\kappa \gamma p, \kappa' \alpha' p'}(t t') \vphantom{\int}\right], \end{aligned}\end{split}\]

(86)\[\begin{split}\begin{aligned} \hspace{-0.5cm}\frac{\delta \langle \hat{n}^{(\kappa p)}(\mathbf{r} t) \rangle }{\delta F_{\kappa' \alpha' p'}(t')} & = & \int d\mathbf{r}'' dt'' \epsilon_{\mathrm e}^{-1}(\mathbf{r} t, \mathbf{r}'' t'') \frac{\delta \langle \hat{n}_{\mathrm n}(\mathbf{r}'' t'') \rangle }{\delta F_{\kappa' \alpha' p'}(t')} \\ & - & {\sum}_\gamma Z_\kappa D_{\kappa \gamma p, \kappa' \alpha' p'}(t t') \nabla_{\gamma} \delta(\mathbf{r}-\bm\tau_{\kappa p}^0). \end{aligned}\end{split}\]

(87)\[\begin{split}\begin{aligned} && \hspace{-0.8cm}M_\kappa\frac{\partial^2}{\partial t^2} D_{\kappa \alpha p, \kappa' \alpha' p'} (t t')= -\delta_{\kappa \alpha p,\kappa' \alpha' p'}\delta(tt') \\ &&\hspace{-0.7cm} -\sum_{\kappa'' \alpha'' p''}\int dt'' \Pi_{\kappa \alpha p, \kappa'' \alpha'' p''}(t t'') D_{\kappa'' \alpha'' p'',\kappa' \alpha' p'}(t''t'). \end{aligned}\end{split}\]

(88)\[\begin{split}\begin{aligned} &&\Pi_{\kappa \alpha p, \kappa' \alpha' p'}(t t') = \\ && \intd\mathbf{r} d\mathbf{r}' \left[ Z_\kappa \nabla_{\alpha} \delta(\mathbf{r}-\boldsymbol{\tau}_{\kappa p}^0) W_{\mathrm e}(\mathbf{r} t, \mathbf{r}' t') Z_{\kappa'}\nabla^{\prime}_{\alpha'} \delta(\mathbf{r}'-\bm\tau_{\kappa' p'}^0) \right. \\ && \left. \hspace{0.73cm}+ \delta_{\kappa p, \kappa' p'}\delta(t t') \nabla_\alpha \langle \hat{n} (\mathbf{r})\rangle v(\mathbf{r},\mathbf{r}') Z_{\kappa'} \nabla^\prime_{\alpha'} \delta(\mathbf{r}'-\boldsymbol{\tau}_{\kappa' p'}^0) \right]. \\ \end{aligned}\end{split}\]

(89)\[\begin{split}\begin{aligned} && \hspace{-0.8cm}{\sum}_{\kappa'' \alpha'' p''} \left[M_\kappa\omega^2 \delta_{\kappa \alpha p,\kappa'' \alpha'' p''} - \Pi_{\kappa \alpha p, \kappa'' \alpha'' p''}(\omega) \right] \\ &&\hspace{0.8cm} \times D_{\kappa'' \alpha'' p'', \kappa' \alpha' p'} (\omega) = \delta_{\kappa \alpha p,\kappa' \alpha' p'}, \end{aligned}\end{split}\]

(90)\[\begin{split}\begin{aligned} && \Pi_{\kappa \alpha p, \kappa' \alpha' p'}(\omega) = \intd\mathbf{r} d\mathbf{r}' \left[ Z_\kappa \nabla_{\alpha} \delta(\mathbf{r}-\boldsymbol{\tau}_{\kappa p}^0) W_{\mathrm e}(\mathbf{r}, \mathbf{r}',\omega) \right. \\ &&\hspace{0.3cm}\left. + \delta_{\kappa p, \kappa' p'} \nabla_\alpha \langle \hat{n} (\mathbf{r})\rangle v(\mathbf{r},\mathbf{r}') \right] Z_{\kappa'} \nabla^\prime_{\alpha'} \delta(\mathbf{r}'-\boldsymbol{\tau}_{\kappa' p'}^0). \end{aligned}\end{split}\]

(91)\[{\sum}_{\kappa'p'}\Pi_{\kappa \alpha p, \kappa' \alpha' p'}(\omega=0) = 0 \mbox{ for any } \alpha,\alpha'.\]

(92)\[\begin{split}\begin{aligned} && \hspace{-0.7cm}\Pi_{\kappa \alpha p, \kappa' \alpha' p'}(\omega) = \sum_{\kappa''p''} Z_\kappa Z_{\kappa''} \left. \frac{\partial^2}{\partial r_\alpha \partial r'_{\alpha'}} \right|_{{\mathbf{r} = \boldsymbol{\tau}_{\kappa p}^0, \mathbf{r}' = \boldsymbol{\tau}_{\kappa'' p''}^0}} \\ && \hspace{0.0cm} \Big[ \delta_{\kappa' p',\kappa'' p''} W_{\mathrm e}(\mathbf{r}, \mathbf{r}',\omega) -\delta_{\kappa p, \kappa' p'} W_{\mathrm e}(\mathbf{r}, \mathbf{r}',0) \Big], \end{aligned}\end{split}\]

(93)\[\langle\hat{n}_{\mathrm n}(\mathbf{r} t)\rangle = n_{\mathrm n}^0(\mathbf{r}) -\frac{i\hbar}{2}\sum_{\kappa p, \alpha \alpha'} Z_\kappa \frac{\partial^2 \delta(\mathbf{r}-\bm\tau_{\kappa p}^0)}{\partial r_\alpha \partial r_{\alpha'}} D_{\kappa \alpha p,\kappa \alpha' p}(t^+ t).\]

Phonons in the Born-Oppenheimer adiabatic approximation

(94)\[{\mathbf D}(\omega) = \left[ {\mathbf M} \omega^2 - {\mathbf \Pi}(\omega) \right]^{-1},\]

(95)\[\Omega_\nu(\omega) - \omega =0, \mbox {with } \nu=1,\dots 3M,\]

(96)\[{\mathbf \Pi}(\omega) = {\mathbf \Pi}^{\mathrm A} + {\mathbf \Pi}^{\mathrm NA}(\omega),\]

(97)\[\begin{split}\begin{aligned} &&\hspace{-0.7cm}\Pi_{\kappa \alpha p,\kappa' \alpha' p'}^{\mathrm A} = \sum_{\kappa'' p''} (\delta_{\kappa' p', \kappa'' p''}-\delta_{\kappa p, \kappa' p'}) \\ &&\hspace{0.0cm} \times \left[ \int d\mathbf{r} \frac{\partial \langle\hat{n}_{\mathrm e}(\mathbf{r})\rangle}{\partial \tau_{\kappa'' \alpha' p''}} \frac{\partial V^{\mathrm en}(\mathbf{r})}{\partial \tau_{\kappa \alpha p}} + \frac{\partial^2 U_{\mathrm nn}}{\partial \tau_{\kappa \alpha p}\partial \tau_{\kappa'' \alpha' p''}} \right]. \end{aligned}\end{split}\]

(98)\[\frac{\partial \langle\hat{n}_{\mathrm e}(\mathbf{r})\rangle}{\partial \tau_{\kappa \alpha p}^0} = -{Z_\kappa}\int d\mathbf{r}' [\epsilon_{\mathrm e}^{-1}(\mathbf{r},\mathbf{r}';0) - \delta(\mathbf{r},\mathbf{r}')] \nabla'_{\alpha} \delta(\mathbf{r}'-\boldsymbol{\tau}^0_{\kappa p}).\]

(99)\[\begin{split}\begin{aligned} && \hspace{-0.8cm}\Pi_{\kappa \alpha p, \kappa' \alpha' p'}^{\mathrm A}= \int d\mathbf{r} \frac{\partial \langle\hat{n}_{\mathrm e}(\mathbf{r})\rangle}{\partial \tau_{\kappa' \alpha' p'}} \frac{\partial V^{\mathrm en}(\mathbf{r})}{\partial \tau_{\kappa \alpha p}} \\ &&\hspace{-0.1cm}+ \int d\mathbf{r} \langle \hat{n}_{\mathrm e}(\mathbf{r})\rangle \frac{\partial^2 V^{\mathrm en}(\mathbf{r})}{\partial \tau_{\kappa \alpha p}\partial \tau_{\kappa' \alpha' p'}} + \frac{\partial^2 U_{\mathrm nn}}{\partial \tau_{\kappa \alpha p}\partial \tau_{\kappa' \alpha' p'}}. \end{aligned}\end{split}\]

(100)\[{\mathbf D}^{\mathrm A}(\omega) = \left[ {\mathbf M} \omega^2 - {\mathbf \Pi}^{\mathrm A} \right]^{-1}.\]

(101)\[D^{\mathrm A}_{\kappa \alpha p,\kappa' \alpha' p'}(\omega) ={\sum}_\nu \int \frac{d\mathbf{q}}{\Omega_{\mathrm BZ}} S^*_{\mathbf{q}\nu,\kappa\alpha p} S_{\mathbf{q}\nu,\kappa'\alpha' p'} \frac{2\omega_{\mathbf{q}\nu}}{\omega^2-\omega_{\mathbf{q}\nu}^2},\]

(102)\[S_{\mathbf{q}\nu,\kappa\alpha p} = e^{i\mathbf{q}\cdot\mathbf{R}_{p}} (2 M_{\kappa} \omega_{\mathbf{q}\nu})^{-1/2} e_{\kappa\alpha,\nu}(\mathbf{q}).\]

(103)\[\begin{aligned} D^{\mathrm A}_{\mathbf{q}\nu\nu'}(\omega) = 2\omega_{\mathbf{q}\nu}/ (\omega^2-\omega_{\mathbf{q}\nu}^2) \delta_{\nu\nu'}. \end{aligned}\]

(104)\[D^{\mathrm A}_{\mathbf{q}\nu\nu'}(tt') = -i\langle \hat{T} [\hat{a}^\dagger_{\mathbf{q}\nu}(t) \hat{a}_{\mathbf{q}\nu}(t')+ \hat{a}_{-\mathbf{q}\nu}(t)\hat{a}^\dagger_{-\mathbf{q}\nu}(t') ]\rangle \delta_{\nu\nu'}.\]

(105)\[D^{\mathrm A}_{\mathbf{q}\nu\nu}(\omega) = \frac{1}{\omega-\omega_{\mathbf{q}\nu}+i\eta} - \frac{1}{\omega+\omega_{\mathbf{q}\nu}-i\eta},\]

Phonons beyond the adiabatic approximation

(106)\[{\mathbf D}(\omega) = {\mathbf D}^{\mathrm A}(\omega) + {\mathbf D}^{\mathrm A}(\omega){\mathbf \Pi}^{\mathrm NA}(\omega){\mathbf D}(\omega).\]

(107)\[D^{-1}_{\mathbf{q}\nu\nu'}(\omega) = \frac{1}{2\omega_{\mathbf{q}\nu}} \left[ \delta_{\nu\nu'}(\omega^2-\omega_{\mathbf{q}\nu}^2) -2\omega_{\mathbf{q}\nu}\Pi_{\mathbf{q}\nu\nu'}^{\mathrm NA}(\omega)\right],\]

(108)\[\tilde{\Omega}_{\mathbf{q}\nu}^2=\omega_{\mathbf{q}\nu}^2 +2\omega_{\mathbf{q}\nu} \Pi_{\mathbf{q}\nu\nu}^{\mathrm NA}(\tilde{\Omega}_{\mathbf{q}\nu}),\]

(109)\[\begin{split}\begin{aligned} \gamma_{\mathbf{q}\nu} & = & -\frac{\omega_{\mathbf{q}\nu}}{\Omega_{\mathbf{q}\nu}} {\mathrm Im} \Pi_{\mathbf{q}\nu\nu}^{\mathrm NA}(\Omega_{\mathbf{q}\nu}-i\gamma_{\mathbf{q}\nu}), \\ \Omega_{\mathbf{q}\nu}^2 & = & \omega_{\mathbf{q}\nu}^2+\gamma_{\mathbf{q}\nu}^2 + 2\omega_{\mathbf{q}\nu} {\mathrm Re} \Pi_{\mathbf{q}\nu\nu}^{\mathrm NA}(\Omega_{\mathbf{q}\nu}-i\gamma_{\mathbf{q}\nu}). \end{aligned}\end{split}\]

Expressions for the phonon self-energy used in it ab~initio

(110)\[\begin{split}\begin{aligned} && \hspace{-0.5cm}\Pi^{\mathrm NA}_{\kappa \alpha p, \kappa' \alpha' p'}(\omega) = \intd\mathbf{r} d\mathbf{r}' Z_\kappa \nabla_{\alpha} \delta(\mathbf{r}-\boldsymbol{\tau}_{\kappa p}^0) \\ && \hspace{-0.3cm} \times \left[W_{\mathrm e}(\mathbf{r}, \mathbf{r}',\omega) -W_{\mathrm e}(\mathbf{r}, \mathbf{r}',0) \right] Z_{\kappa'} \nabla^\prime_{\alpha'} \delta(\mathbf{r}'-\boldsymbol{\tau}_{\kappa' p'}^0). \end{aligned}\end{split}\]

(111)\[\begin{split}\begin{aligned} \hspace{-.6cm}\hbar \Pi^{\mathrm NA}_{\mathbf{q}\nu,\mathbf{q}'\nu'}(\omega) & = & \int d\mathbf{r} d\mathbf{r}' g^{\mathrm b}_{\mathbf{q}\nu}(\mathbf{r}) P_{\mathrm e}(\mathbf{r},\mathbf{r}',\omega) g^{\mathrm cc}_{\mathbf{q}'\nu'}(\mathbf{r}',\omega) \\ & - & \int d\mathbf{r} d\mathbf{r}' g^{\mathrm b}_{\mathbf{q}\nu}(\mathbf{r}) P_{\mathrm e}(\mathbf{r},\mathbf{r}', 0) g^*_{\mathbf{q}'\nu'}(\mathbf{r}',\hspace{0.9pt}0), \end{aligned}\end{split}\]

(112)\[g^{\mathrm b}_{\mathbf{q}\nu}(\mathbf{r}) = \Delta_{\mathbf{q}\nu} V^{\mathrm en}(\mathbf{r}),\]

(113)\[\begin{split}\begin{aligned} g_{\mathbf{q}\nu}(\mathbf{r},\omega) & = & \intd\mathbf{r}' \epsilon_{\mathrm e}^{-1}(\mathbf{r},\mathbf{r}',\omega) g^{\mathrm b}_{\mathbf{q}\nu}(\mathbf{r}'), \\ g^{\mathrm cc}_{\mathbf{q}\nu}(\mathbf{r},\omega) & = & \intd\mathbf{r}' \epsilon_{\mathrm e}^{-1}(\mathbf{r},\mathbf{r}',\omega) g^{\mathrm b,*}_{\mathbf{q}\nu}(\mathbf{r}'). \end{aligned}\end{split}\]

(114)\[\begin{split}\begin{aligned} &&\hspace{-0.7cm}\hbar \Pi^{\mathrm NA}_{\mathbf{q}\nu\nu'}(\omega) = 2 \sum_{mn} \int \frac{d\mathbf{k}}{\Omega_{\mathrm BZ}} g^{\mathrm b}_{mn\nu}(\mathbf{k},\mathbf{q}) g^*_{mn\nu'}(\mathbf{k},\mathbf{q}) \\ &&\hspace{-0.3cm}\times \left[ \frac{f_{m\mathbf{k}+\mathbf{q}}-f_{n\mathbf{k}}}{\varepsilon_{m\mathbf{k}+\mathbf{q}}-\varepsilon_{n\mathbf{k}}-\hbar(\omega+i\eta)} - \frac{f_{m\mathbf{k}+\mathbf{q}}-f_{n\mathbf{k}}}{\varepsilon_{m\mathbf{k}+\mathbf{q}}-\varepsilon_{n\mathbf{k}}} \right]. \end{aligned}\end{split}\]

(115)\[\begin{split}\begin{aligned} \frac{1}{\tau^{\mathrm ph}_{\mathbf{q}\nu}} & = & \frac{2\pi}{\hbar} 2 \sum_{mn} \int \frac{d\mathbf{k}}{\Omega_{\mathrm BZ}} |g_{mn\nu}(\mathbf{k},\mathbf{q})|^2 (f_{n\mathbf{k}}-f_{m\mathbf{k}+\mathbf{q}}) \\ &\times& \delta(\varepsilon_{m\mathbf{k}+\mathbf{q}}-\varepsilon_{n\mathbf{k}}-\hbar\omega_{\mathbf{q}\nu}). \end{aligned}\end{split}\]

Electron self-energy: Fan-Migdal and Debye-Waller terms

(116)\[\begin{split}\begin{aligned} && \hspace{-0.7cm}\left[ i\hbar \frac{\partial}{\partial t_1} + \frac{\hbar^2}{2m_{\mathrm e}}\nabla^2(1) -V_{\mathrm tot}^{\mathrm cn}(1) \right]G^{\mathrm cn}(12) \\ &&\hspace{1cm} - \intd3 \Sigma_{\mathrm e}^{\mathrm cn}(13) G^{\mathrm cn}(32) = \delta(12). \end{aligned}\end{split}\]

\[V_{\mathrm tot}^{\mathrm cn}(1) = \intd 2 v(1,2) \langle \hat{n}^{\mathrm cn}(2)\rangle,\]

(117)\[\langle \hat{n}^{\mathrm cn}(1)\rangle = -i\hbar {\sum}_{\sigma_1} G^{\mathrm cn}(11^+) + n_{\mathrm n}^0(\mathbf{r}_1).\]

(118)\[\Sigma_{\mathrm e}^{\mathrm cn}(12) = i\hbar \int d(34) G^{\mathrm cn}(13) \Gamma^{\mathrm cn}(324) W_{\mathrm e}^{\mathrm cn}(41^+).\]

(119)\[G(12) = G^{\mathrm cn}(12) + \intd(34) G^{\mathrm cn}(13) \Sigma^{\mathrm ep}(34) G(42),\]

(120)\[\Sigma^{\mathrm ep} = \Sigma^{\mathrm FM}+\Sigma^{\mathrm DW}+\Sigma^{\mathrm dGW},\]

(121)\[\begin{split}\begin{aligned} \hspace{-0.5cm} \Sigma^{\mathrm FM}(12) & = & i\hbar\int d(34) G(13) \Gamma(324) W_{\mathrm ph}(41^+), \\ \hspace{-0.5cm} \Sigma^{\mathrm DW}(12) & = & \intd3 v(13) \left[\langle \hat{n}(3) \rangle -\langle \hat{n}^{\mathrm cn}(3)\rangle \right]\delta(12), \\ \hspace{-0.5cm} \Sigma^{\mathrm dGW}(12) & = & \Sigma_{\mathrm e}(12) - \Sigma_{\mathrm e}^{\mathrm cn}(12). \end{aligned}\end{split}\]

Expressions for the electron self-energy used in it ab~initio

(122)\[\begin{split}\begin{aligned} && \hspace{-0.7cm} \Sigma^{\mathrm FM}(12) = i\sum_{\nu\nu'} \int \frac{d\omega}{2\pi}\frac{d\mathbf{q}}{\Omega_{\mathrm BZ}}d(34) e^{-i\omega(t_4-t_1^+)} \\ &&\hspace{-0.2cm} \times G(13) \Gamma(324) g_{\mathbf{q}\nu}^{\mathrm cc}(\mathbf{r}_4,\omega) D_{\mathbf{q}\nu\nu'}(\omega) g_{\mathbf{q}\nu'}(\mathbf{r}_1,\omega). \end{aligned}\end{split}\]

(123)\[\begin{split}\begin{aligned} && \Sigma^{\mathrm FM}_{n n'\mathbf{k}}(\omega) = \frac{1}{\hbar} \sum_{m\nu} \int \frac{d\mathbf{q}}{\Omega_{\mathrm BZ}} g_{mn\nu}^*(\mathbf{k},\mathbf{q}) g_{mn'\nu}(\mathbf{k},\mathbf{q}) \\ && \times \left[ \frac{1-f_{m\mathbf{k}+\mathbf{q}}}{\omega-\varepsilon_{m\mathbf{k}+\mathbf{q}}/\hbar-\omega_{\mathbf{q}\nu}+i\eta} + \frac{f_{m\mathbf{k}+\mathbf{q}}}{\omega-\varepsilon_{m\mathbf{k}+\mathbf{q}}/\hbar+\omega_{\mathbf{q}\nu}-i\eta} \right]. \\ \end{aligned}\end{split}\]

(124)\[\hspace{-0.05cm}\left[ \frac{1-f_{m\mathbf{k}+\mathbf{q}}}{\cdots + i\eta} + \frac{f_{m\mathbf{k}+\mathbf{q}}}{\cdots -i\eta} \right] \rightarrow \left[ \frac{1-f_{m\mathbf{k}+\mathbf{q}}+n_{\mathbf{q}\nu}}{\cdots+i\eta} + \frac{f_{m\mathbf{k}+\mathbf{q}}+n_{\mathbf{q}\nu}} {\cdots + i\eta} \right],\]

(125)\[\begin{split}\Sigma^{\mathrm DW}(12) = \delta(12) \frac{i\hbar}{2}\sum_{\substack{\kappa \alpha p\\\kappa' \alpha' p'}} \frac{\partial^2 V_{\mathrm tot}(1)} {\partial\tau_{\kappa\alpha p}^0\partial\tau_{\kappa'\alpha' p'}^0} D_{\kappa \alpha p,\kappa' \alpha' p'}(t_1^+,t_1).\end{split}\]

(126)\[\Sigma^{\mathrm DW}_{nn'\mathbf{k}} = {\sum}_\nu \int \frac{d\mathbf{q}}{\Omega_{\mathrm BZ}} g^{\mathrm DW}_{nn'\nu\nu}(\mathbf{k},\mathbf{q},-\mathbf{q}),\]

(127)\[g^{\mathrm DW}_{nn'\nu\nu}(\mathbf{k},\mathbf{q},-\mathbf{q}) \rightarrow g^{\mathrm DW}_{nn'\nu\nu}(\mathbf{k},\mathbf{q},-\mathbf{q}) (2n_{\mathbf{q}\nu}+1),\]

Temperature-dependence of electronic band structures

(128)\[G^{-1}_{nn'\mathbf{k}}(\omega) = G^{{\mathrm cn},-1}_{nn'\mathbf{k}}(\omega) - \Sigma^{\mathrm ep}_{nn'\mathbf{k}}(\omega).\]

(129)\[G^{-1}_{nn'\mathbf{k}}(\omega) = (\hbar\omega - \tilde{\varepsilon}_{n\mathbf{k}})\delta_{nn'} - \Sigma^{\mathrm ep}_{nn'\mathbf{k}}(\omega),\]

(130)\[\begin{split}\begin{aligned} E_{n\mathbf{k}} & = & \varepsilon_{n\mathbf{k}} + {\mathrm Re} \Sigma^{\mathrm ep}_{nn\mathbf{k}}(\tilde{E}_{n\mathbf{k}}/\hbar), \\ \Gamma_{n\mathbf{k}} & = & {\mathrm Im} \Sigma^{\mathrm ep}_{nn\mathbf{k}}(\tilde{E}_{n\mathbf{k}}/\hbar). \end{aligned}\end{split}\]

(131)\[\begin{split}\begin{aligned} E_{n\mathbf{k}} & = & \varepsilon_{n\mathbf{k}}+{\sum}_\nu\int \frac{d\mathbf{q}}{\Omega_{\mathrm BZ}} \sum_{m} |g_{mn\nu}(\mathbf{k},\mathbf{q})|^2 \\ & \times & \left. {\mathrm Re} \left[ \frac{1-f_{m\mathbf{k}+\mathbf{q}}+n_{\mathbf{q}\nu}}{E_{n\mathbf{k}}-\varepsilon_{m\mathbf{k}+\mathbf{q}}-\hbar\omega_{\mathbf{q}\nu} +i\Gamma_{n\mathbf{k}}} \right. \right. \\ && \left. + \frac{\qquad f_{m\mathbf{k}+\mathbf{q}}+n_{\mathbf{q}\nu}}{E_{n\mathbf{k}}-\varepsilon_{m\mathbf{k}+\mathbf{q}}+\hbar\omega_{\mathbf{q}\nu}+i\Gamma_{n\mathbf{k}}} \right] \\ & +& {\sum}_\nu\int \frac{d\mathbf{q}}{\Omega_{\mathrm BZ}} g^{\mathrm DW}_{nn\nu\nu}(\mathbf{k},\mathbf{q},-\mathbf{q}) (2n_{\mathbf{q}\nu}+1). \end{aligned}\end{split}\]

(132)\[\begin{split}\begin{aligned} E_{n\mathbf{k}} = \varepsilon_{n\mathbf{k}}& + &\sum_\nu\int \frac{d\mathbf{q}}{\Omega_{\mathrm BZ}} \left[ \sum_{m} \frac{|g_{mn\nu}(\mathbf{k},\mathbf{q})|^2}{\varepsilon_{n\mathbf{k}}-\varepsilon_{m\mathbf{k}+\mathbf{q}}} \right. \\ & + & \left. g^{\mathrm DW}_{nn\nu\nu}(\mathbf{k},\mathbf{q},-\mathbf{q}) \vphantom{\sum_{m}} \right] (2n_{\mathbf{q}\nu}+1), \end{aligned}\end{split}\]

(133)\[E_{\mathrm g}(T) = E_{\mathrm g}^{\mathrm cn} - \left|\Delta E_{\mathrm g}^{\mathrm ZP}\right| [1+ 2 n(\hbar\omega_0/ k_{\mathrm B}T)],\]

Carrier lifetimes

(134)\[\begin{split}\begin{aligned} && \hspace{-0.65cm} \frac{1}{\tau_{n\mathbf{k}}} = \frac{2\pi}{\hbar} \sum_{m\nu} \int \frac{d\mathbf{q}}{\Omega_{\mathrm BZ}} |g_{nm\nu}(\mathbf{k},\mathbf{q})|^2 \\ && \times \left| (1-f_{m\mathbf{k}+\mathbf{q}})\delta(\varepsilon_{n\mathbf{k}} -\hbar\omega_{\mathbf{q}\nu}-\varepsilon_{m\mathbf{k}+\mathbf{q}}) \right. \\ && \left. \hspace{0.9cm} -f_{m\mathbf{k}+\mathbf{q}} \delta(\varepsilon_{n\mathbf{k}}+\hbar\omega_{\mathbf{q}\nu} -\varepsilon_{m\mathbf{k}+\mathbf{q}})\right|. \end{aligned}\end{split}\]

(135)\[\begin{split}\begin{aligned} && \hspace{-0.5cm} \frac{1}{\tau_{n\mathbf{k}}} = \frac{2\pi}{\hbar} \sum_{m\nu} \int \frac{d\mathbf{q}}{\Omega_{\mathrm BZ}} |g_{nm\nu}(\mathbf{k},\mathbf{q})|^2 \\ && \times \left[ (1-f_{m\mathbf{k}+\mathbf{q}}+n_{\mathbf{q}\nu})\delta(\varepsilon_{n\mathbf{k}} -\hbar\omega_{\mathbf{q}\nu}-\varepsilon_{m\mathbf{k}+\mathbf{q}}) \right.+ \\ && \left. \hspace{1cm} (f_{m\mathbf{k}+\mathbf{q}}+n_{\mathbf{q}\nu})\delta(\varepsilon_{n\mathbf{k}}+\hbar\omega_{\mathbf{q}\nu} -\varepsilon_{m\mathbf{k}+\mathbf{q}})\right]. \end{aligned}\end{split}\]

Kinks and satellites

\[A(\mathbf{k},\omega) = -\frac{1}{\pi} {\sum}_n{\mathrm Im} G^{ \mathrm ret}_{nn\mathbf{k}}(\omega),\]

(136)\[A(\mathbf{k},\omega) = \sum_n \frac{-(1/\pi) {\mathrm Im } \Sigma^{\mathrm ep}_{nn\mathbf{k}}(\omega)}{\left[\hbar\omega-\varepsilon_{n\mathbf{k}}- {\mathrm Re} \Sigma^{\mathrm ep}_{nn\mathbf{k}}(\omega)\right]^2+\left[{\mathrm Im } \Sigma^{\mathrm ep}_{nn\mathbf{k}}(\omega)\right]^2}.\]

(137)\[A(\mathbf{k},\omega) = \sum_n Z_{n\mathbf{k}} \frac{-(1/\pi) Z_{n\mathbf{k}} {\mathrm Im } \Sigma^{\mathrm ep}_{nn\mathbf{k}}(E_{n\mathbf{k}}/\hbar)} {\left[\hbar\omega-E_{n\mathbf{k}}\right]^2+\left[Z_{n\mathbf{k}} {\mathrm Im } \Sigma^{\mathrm ep}_{nn\mathbf{k}}(E_{n\mathbf{k}}/\hbar)\right]^2}.\]

(138)\[Z_{n\mathbf{k}}= \left[1 - \hbar^{-1}\partial{\mathrm Re}\Sigma^{\mathrm ep}_{nn\mathbf{k}}(\omega)/\partial\omega\big|_{\omega=E_{n\mathbf{k}}/\hbar}\right]^{-1}.\]

Maximally-localized Wannier functions

(139)\[\mathrm{w}_{m p} (\mathbf{r}) = N_p^{-1} {\sum}_{n\mathbf{k}} e^{i\mathbf{k}\cdot (\mathbf{r}-\mathbf{R}_p)} U_{nm\mathbf{k}} u_{n\mathbf{k}}(\mathbf{r}),\]

(140)\[u_{n\mathbf{k}}(\mathbf{r}) = {\sum}_{mp} e^{-i\mathbf{k}\cdot (\mathbf{r}- \mathbf{R}_{p})} U^\dagger_{mn\mathbf{k}} \mathrm{w}_{m p} (\mathbf{r}).\]

Interpolation of electron-phonon matrix elements

(141)\[g_{mn\kappa\alpha}(\mathbf{R}_p,\mathbf{R}_{p'}) = \langle \mathrm{w}_{m0} (\mathbf{r}) | \frac{\partial V^{\mathrm KS}} {\partial \tau_{\kappa\alpha}} (\mathbf{r}-\mathbf{R}_{p'})| \mathrm{w}_{n 0}(\mathbf{r}-\mathbf{R}_p)\rangle_{\mathrm sc},\]

(142)\[\begin{split}\begin{aligned} && g_{mn\nu}(\mathbf{k},\mathbf{q}) = {\sum}_{pp'} e^{i(\mathbf{k}\cdot \mathbf{R}_p+\mathbf{q}\cdot \mathbf{R}_{p'})} \\ && \hspace{0.2cm} \times\sum_{ m'n'\kappa\alpha} U_{m m'\mathbf{k}+\mathbf{q}} g_{m'n'\kappa\alpha}(\mathbf{R}_p,\mathbf{R}_{p'}) U^\dagger_{n'n\mathbf{k}} {\mathrm u}_{\kappa\alpha,\mathbf{q}\nu}, \hspace{0.6cm} \end{aligned}\end{split}\]

(143)\[\begin{split}\begin{aligned} &&g_{mn\kappa\alpha}(\mathbf{R}_p,\mathbf{R}_{p'}) = \frac{1}{N_p N_{p'}} \sum_{\mathbf{k},\mathbf{q}} e^{-i(\mathbf{k}\cdot \mathbf{R}_p+\mathbf{q}\cdot\mathbf{R}_{p'})} \\ && \hspace{0.4cm}\times \sum_{m'n'\nu} {\mathrm u}_{\kappa\alpha,\mathbf{q}\nu}^{-1} U^\dagger_{m m'\mathbf{k}+\mathbf{q}} g_{m'n'\nu}(\mathbf{k},\mathbf{q}) U_{n'n\mathbf{k}}, \hspace{0.6cm} \end{aligned}\end{split}\]

Electron-phonon matrix elements in polar materials

(144)\[\begin{split}\begin{aligned} && g_{mn\nu}^{\mathcal L}({\mathbf k},{\mathbf q}) = i\frac{4\pi}{\Omega} \frac{e^2 }{4\pi\varepsilon_0} \sum_{\kappa} \left(\frac{\hbar}{2 N_p M_\kappa \omega_{{\mathbf q}\nu}}\right)^{\frac{1}{2}} \sum_{{\mathbf G}\ne -{\mathbf q}} \\ && \frac{ ({\mathbf q}+{\mathbf G})\cdot{\mathbf Z}^*_\kappa \cdot {\mathbf e}_{\kappa\nu}({\mathbf q}) } { ({\mathbf q}+{\mathbf G})\cdot\bm\epsilon^\infty\cdot({\mathbf q}+{\mathbf G})} \langle \psi_{m{\mathbf k+q}} |e^{i({\mathbf q}+{\mathbf G})\cdot({\mathbf r}-\bm\tau_{\kappa})}| \psi_{n{\mathbf k}} \rangle_{\mathrm sc}. \\ \end{aligned}\end{split}\]

(145)\[\frac{\gamma_{\mathbf{q}\nu}}{\pi \omega_{\mathbf{q}\nu}} \simeq 2 \sum_{mn} \int \frac{d\mathbf{k}}{\Omega_{\mathrm BZ}} |g_{mn\nu}(\mathbf{k},\mathbf{q})|^2 \delta(\varepsilon_{n\mathbf{k}}-\varepsilon_{\mathrm F}) \delta(\varepsilon_{m\mathbf{k}+\mathbf{q}}-\varepsilon_{\mathrm F}),\]

(146)\[\begin{split}\begin{aligned} && \Sigma^{\mathrm FM}_{n n\mathbf{k}}(\omega,T) = \int_{-\infty}^{+\infty} d\varepsilon \int_0^\infty d\varepsilon' \alpha^2F_{n\mathbf{k}}(\varepsilon,\varepsilon') \\ && \times \left[ \frac{1-f(\varepsilon/k_{\mathrm B}T)+n(\varepsilon'/k_{\mathrm B}T)}{\hbar\omega-\varepsilon-\varepsilon'+i\hbar \eta} + \frac{f(\varepsilon/k_{\mathrm B}T)+n(\varepsilon'/k_{\mathrm B}T)}{\hbar\omega-\varepsilon+\varepsilon'+i\hbar \eta} \right]. \\ \end{aligned}\end{split}\]

(147)\[\begin{split}\begin{aligned} \alpha^{2}F_{n\mathbf{k}}(\varepsilon,\varepsilon') & = &\sum_{m\nu} \int \frac{d\mathbf{q}}{\Omega_{\mathrm BZ}} |g_{nm\nu}(\mathbf{k},\mathbf{q})|^{^2} \\ & \times &\delta(\varepsilon-\varepsilon_{m\mathbf{k}+\mathbf{q}}) \delta(\varepsilon'-\hbar\omega_{\mathbf{q}\nu}). \end{aligned}\end{split}\]

(148)\[\begin{split}\begin{aligned} && |{\mathrm Im} \Sigma^{\mathrm FM}_{n}(\omega,T)| = \pi \int_0^\infty d\varepsilon' \alpha^2F_n(\hbar\omega,\varepsilon') \{ 1 +2 n(\varepsilon'/k_{\mathrm B}T) \\ && \hspace{1.3cm} +f[(\hbar\omega+\varepsilon')/k_{\mathrm B}T]- f[(\hbar\omega-\varepsilon')/k_{\mathrm B}T] \},\hspace{0.5cm} \end{aligned}\end{split}\]

(149)\[\lambda_{n\mathbf{k}} = Z_{n\mathbf{k}}^{-1}-1 = - \hbar^{-1}\partial {\mathrm Re}\Sigma_{nn\mathbf{k}}(\omega)/\partial\omega\big|_{\omega=E_{n\mathbf{k}}/\hbar}.\]

(150)\[\Sigma^{\mathrm DW}_{nn\mathbf{k}} = \langle u_{n\mathbf{k}}| V_{\mathrm DW}|u_{n\mathbf{k}}\rangle_{\mathrm uc},\]

Electron mass enhancement in metals

(151)\[S_{\mathrm e} = \frac{N_{\mathrm F} k_{\mathrm B} \hbar^3}{(k_{\mathrm B} T)^2}\int_0^\infty \frac{\omega \left[ \omega -\hbar^{-1}{\mathrm Re}\Sigma^{\mathrm FM}_{nn\mathbf{k}}(\omega,T) \right] }{{\mathrm cosh}^2(\hbar\omega/2k_{\mathrm B} T)} d\omega.\]

(152)\[C_{\mathrm e} = T \frac{\partial S_{\mathrm e}}{\partial T} = \frac{2}{3}\pi^2 k_{\mathrm B}^2 N_{\mathrm F} (1+\lambda_{n\mathbf{k}}) T.\]

Perturbative calculations based on the Allen-Heine theory

(153)\[\begin{split}\begin{aligned} &&{\sum}_{\kappa p} \langle \psi_{n\mathbf{k}} |\partial_{\kappa\alpha p} V^{\mathrm KS}| \psi_{n\mathbf{k}} \rangle_{\mathrm sc} = 0, \\ && {\sum}_{\kappa' p'} \langle \psi_{n\mathbf{k}} | \partial^2_{\kappa\alpha p,\kappa'\alpha' p'} V^{\mathrm KS}| \psi_{n\mathbf{k}} \rangle_{\mathrm sc} = - 2 {\mathrm Re} {\sum}_{\kappa' p'}{\sum}_{m\mathbf{q}}{\vphantom{\sum}}^{\prime} \\ && \frac{ \langle \psi_{n\mathbf{k}} | \partial_{\kappa\alpha p} V^{\mathrm KS}| \psi_{m\mathbf{k}+\mathbf{q}} \rangle_{\mathrm sc} \langle \psi_{m\mathbf{k}+\mathbf{q}} | \partial_{\kappa'\alpha' p'} V^{\mathrm KS} | \psi_{n\mathbf{k}} \rangle_{\mathrm sc} }{\varepsilon_{n\mathbf{k}}-\varepsilon_{m\mathbf{k}+\mathbf{q}}}. \qquad \end{aligned}\end{split}\]

(154)\[\Sigma^{\mathrm DW}_{nn\mathbf{k}} = -{\sum}_{\nu m}{\vphantom{\sum}}^{\prime} \int\frac{d\mathbf{q}}{\Omega_{\mathrm BZ}} \frac{g_{mn\nu}^{2,{\mathrm DW}}(\mathbf{k},\mathbf{q})} {\varepsilon_{n\mathbf{k}}-\varepsilon_{m\mathbf{k}}} (2n_{\mathbf{q}\nu}+1).\]

(155)\[\begin{split}\begin{aligned} \hspace{-.5cm} g_{mn\nu}^{2,{\mathrm DW}}(\mathbf{k},\mathbf{q})& =&\sum_{\substack{\kappa\alpha\\ \kappa'\alpha'}} \frac{t_{\kappa\alpha,\kappa'\alpha'}^\nu(\mathbf{q})}{2\omega_{\mathbf{q}\nu}} h_{mn,\kappa\alpha}^*(\mathbf{k}) h_{mn,\kappa'\alpha'}(\mathbf{k}), \\ t_{\kappa\alpha,\kappa'\alpha'}^\nu(\mathbf{q}) & = & \frac{e_{\kappa\alpha\nu}(\mathbf{q})e^*_{\kappa\alpha'\nu}(\mathbf{q})}{ M_\kappa}+ \frac{e_{\kappa'\alpha\nu}(\mathbf{q}) e^*_{\kappa'\alpha'\nu}(\mathbf{q})}{ M_{\kappa'}}, \\ h_{mn,\kappa\alpha}(\mathbf{k}) & = & {\sum}_\nu (M_\kappa \omega_{0\nu})^\frac{1}{2} e_{\kappa\alpha\nu}(0) g_{mn\nu}(\mathbf{k},0). \end{aligned}\end{split}\]

Non-perturbative adiabatic calculations

(156)\[\langle\varepsilon_{n\mathbf{k}}\rangle_{\{n_{\mathbf{q}\nu}\}} = \varepsilon_{n\mathbf{k}} + {\sum}_\nu\int\frac{d\mathbf{q}}{\Omega_{\mathrm BZ}} (n_{\mathbf{q}\nu}+1/2) \frac{\partial \varepsilon_{n\mathbf{k}}}{\partial n_{\mathbf{q}\nu}},\]

Phonon-assisted optical absorption

(157)\[\begin{split}\begin{aligned} &&\alpha(\omega) = \frac{\pi e^2}{\epsilon_0 c \Omega}\frac{1}{\omega n_{\mathrm r}(\omega)} \int \frac{d \mathbf{k} d \mathbf{q}}{\Omega_{\mathrm BZ}^2} \sum_{mn\nu}\sum_{s=\pm 1} (f_{n{\mathbf k}}-f_{m{\mathbf k+q}}) \\ &&\times \Bigg| {\mathbf e}\cdot {\sum}_p\left[ \frac{{\mathbf v}_{np}({\mathbf k})g_{pm\nu}({\mathbf k},{\mathbf q})}{\varepsilon_{p{\mathbf k}} -\varepsilon_{n{\mathbf k}}-\hbar\omega} + \frac{g_{np\nu}({\mathbf k},{\mathbf q}){\mathbf v}_{pm} ({\mathbf k}+{\mathbf q})}{\varepsilon_{p{\mathbf k+q}}-\varepsilon_{n{\mathbf k}} +s\hbar\omega_{{\mathbf q}\nu}} \right]\Bigg|^{2} \\ &&\times \left( n_{{\mathbf q}\nu}+1/2+s/2\right) \delta(\varepsilon_{m{\mathbf k+q}}-\varepsilon_{n{\mathbf k}} -\hbar\omega +s\hbar\omega_{{\mathbf q}\nu}). \end{aligned}\end{split}\]

(158)\[\epsilon_2(\omega;T) =\frac{1}{Z}{\sum}_{\{n_{\mathbf{q}\nu}\}} e^{-E_{\{n_{\mathbf{q}\nu}\}} /k_{\mathrm B}T} \langle \epsilon_2(\omega) \rangle_{\{n_{\mathbf{q}\nu}\}},\]

Phonon-limited mobility

(159)\[\begin{split}\begin{aligned} && \frac{\partial f_{n\mathbf{k}}^0}{\partial\varepsilon_{n\mathbf{k}}} \mathbf{v}_{n\mathbf{k}} \cdot (-e) \mathbf{E} = -{\sum}_\nu\int\frac{d\mathbf{q}}{\Omega_{\mathrm BZ}} \Gamma_{mn\nu}(\mathbf{k},\mathbf{q}) \\ && \qquad\qquad \times \left[ (f_{n\mathbf{k}}-f_{n\mathbf{k}}^0)-(f_{m\mathbf{k}+\mathbf{q}}-f_{m\mathbf{k}+\mathbf{q}}^0)\right], \end{aligned}\end{split}\]

(160)\[\begin{split}\begin{aligned} &&\Gamma_{mn\nu}(\mathbf{k},\mathbf{q}) = {\sum}_{s=\pm 1} \frac{2\pi}{\hbar}|g_{mn\nu}(\mathbf{k},\mathbf{q})|^2 f^0_{n\mathbf{k}}(1-f^0_{m\mathbf{k}+\mathbf{q}}) \\ &&\qquad\times( n_{\mathbf{q}\nu}+1/2 -s/2) \delta(\varepsilon_{n\mathbf{k}}+s\hbar\omega_{\mathbf{q}\nu}-\varepsilon_{m\mathbf{k}+\mathbf{q}}). \qquad \end{aligned}\end{split}\]

McMillan/Allen-Dynes formula

(161)\[k_{\mathrm B}T_{\mathrm c} = \frac{\hbar\omega_{\mathrm log}}{1.2}\exp\left[-\frac{1.04(1+\lambda)}{\lambda - \mu^* (1+0.62 \lambda)} \right].\]

(162)\[\begin{split}\begin{aligned} \alpha^2F(\omega) & = & \frac{1}{N_{\mathrm F}} \int \frac{d\mathbf{k} d\mathbf{q}}{\Omega_{\mathrm BZ}^2} \sum_{mn\nu} |g_{mn\nu}(\mathbf{k},\mathbf{q})|^2 \\ &\times&\delta(\varepsilon_{n\mathbf{k}}-\varepsilon_{\mathrm F})\delta(\varepsilon_{m\mathbf{k}+\mathbf{q}}-\varepsilon_{\mathrm F}) \delta(\hbar\omega-\hbar\omega_{\mathbf{q}\nu}), \qquad \\ \lambda & = & 2\int_0^\infty \frac{\alpha^2F(\omega)}{\omega}d\omega, \\ \omega_{\mathrm log} & = & \exp\left[\frac{2}{\lambda}\int_0^\infty d\omega \frac{\alpha^2F(\omega)}{\omega}\log \omega\right], \end{aligned}\end{split}\]

Anisotropic Migdal-Eliashberg theory

(163)\[\begin{split}\begin{aligned} &&Z_{n\mathbf{k}}(i\omega_j) =1+ \frac{\pi k_{\mathrm B}T}{N_{\mathrm F}} \sum_{n'{\mathbf k}' j'} \frac{ \omega_{j'}/\omega_j }{\sqrt{\hbar^2\omega_{j'}^2+\Delta_{n'{\mathbf k}'}^2(i\omega_{j'})} } \\ &&\hspace{0.3cm}\times \lambda_{n\mathbf{k},n'\mathbf{k}'}(i\omega_{j}-i\omega_{j'}) \delta(\varepsilon_{n'{\mathbf k}'}-\varepsilon_{\mathrm F}), \\ &&Z_{n\mathbf{k}}(i\omega_j) \Delta_{n\mathbf{k}}(i\omega_j) = \frac{\pi k_{\mathrm B}T}{N_{\mathrm F}} \sum_{n'{\mathbf k}' j'} \frac{ \Delta_{n'\mathbf{k}'}(i\omega_{j'}) }{ \sqrt{\hbar^2\omega_{j'}^2+\Delta_{n'\mathbf{k}'}^2(i\omega_{j'})} } \\ &&\hspace{0.3cm}\times\left[ \lambda_{n\mathbf{k},n'\mathbf{k}'}(i\omega_{j}-i\omega_{j'})-N_{\mathrm F} V_{n\mathbf{k},n'\mathbf{k}'}\right]\delta(\varepsilon_{n'{\mathbf k}'}-\varepsilon_{\mathrm F}), \end{aligned}\end{split}\]

\[\lambda_{n\mathbf{k},n'\mathbf{k}'}(i\omega) = \frac{N_{\mathrm F}}{\hbar} {\sum}_{\nu} \frac{2\omega_{\mathbf{q}\nu} }{\omega_{\mathbf{q}\nu}^2+\omega^2} | g_{nn'\nu}(\mathbf{k},\mathbf{q})|^2,\]

(164)\[\begin{split}\begin{aligned} {\sum}_\mathbf{q} \exp(i\mathbf{q}\cdot\mathbf{R}_p) = N_p \delta_{p 0}, {\sum}_p \exp(i\mathbf{q}\cdot\mathbf{R}_p) = N_p \delta_{\mathbf{q} 0}. \\ \end{aligned}\end{split}\]

(165)\[z_{\mathbf{q}\nu} = N_p^{-\frac{1}{2}}\sum_{\kappa\alpha p} e^{-i\mathbf{q}\cdot\mathbf{R}_p} (M_\kappa/M_0)^{\frac{1}{2}} e^*_{\kappa\alpha,\nu}(\mathbf{q}) \Delta\tau_{\kappa\alpha p}.\]

(166)\[z_{-\mathbf{q}\nu} = z^*_{\mathbf{q}\nu}.\]

(167)\[\Delta\tau_{\kappa\alpha p} = N_p^{-\frac{1}{2}} (M_0/M_\kappa)^\frac{1}{2} {\sum}_{\mathbf{q}\nu} e^{i\mathbf{q}\cdot\mathbf{R}_p} e_{\kappa\alpha,\nu}(\mathbf{q}) z_{\mathbf{q}\nu}.\]

(168)\[\begin{split}\begin{aligned} \Delta\tau_{\kappa\alpha p} & = & N_p^{-\frac{1}{2}}(M_0/M_\kappa)^\frac{1}{2} \left[ {\sum}_{\mathbf{q} \in \mathcal{A},\nu} e_{\kappa\alpha,\nu}(\mathbf{q}) x_{\mathbf{q}\nu} \right. \\ & +& \left. 2 {\mathrm Re} {\sum}_{\mathbf{q} \in \mathcal{B},\nu} e^{i\mathbf{q}\cdot\mathbf{R}_p} e_{\kappa\alpha,\nu}(\mathbf{q}) (x_{\mathbf{q}\nu}+iy_{\mathbf{q}\nu})\right]. \hspace{0.5cm} \end{aligned}\end{split}\]

(169)\[\begin{split}\begin{aligned} \hat{H}_{\mathrm p} &= & \frac{1}{2}{\sum}_{\mathbf{q}\in \mathcal{B},\nu} \hbar\omega_{\mathbf{q}\nu}( -\partial^2/\partial \tilde{x}_{\mathbf{q}\nu}^2 -\partial^2/\partial \tilde{y}_{\mathbf{q}\nu}^2 + \tilde{x}_{\mathbf{q}\nu}^2 + \tilde{y}_{\mathbf{q}\nu}^2 ) \\ & + & \frac{1}{2}{\sum}_{\mathbf{q}\in \mathcal{A},\nu} \hbar\omega_{\mathbf{q}\nu} (-\partial^2/\partial \tilde{x}_{\mathbf{q}\nu}^2 + \tilde{x}_{\mathbf{q}\nu}^2 ), \end{aligned}\end{split}\]

(170)\[\begin{split}\begin{aligned} & \tilde{x}_{\mathbf{q}\nu} = x_{\mathbf{q}\nu}/2 l_{\mathbf{q}\nu} \hspace{2.4cm} & \mbox{ for $\mathbf{q}$ in }\mathcal{A}, \\ & \tilde{x}_{\mathbf{q}\nu} = x_{\mathbf{q}\nu}/l_{\mathbf{q}\nu}, \tilde{y}_{\mathbf{q}\nu} = y_{\mathbf{q}\nu}/l_{\mathbf{q}\nu} & \mbox{ for $\mathbf{q}$ in }\mathcal{B}, \end{aligned}\end{split}\]

(171)\[\hat{a}_{\mathbf{q}\nu,x} = (\tilde{x}_{\mathbf{q}\nu}+ \partial/\partial \tilde{x}_{\mathbf{q}\nu})/\sqrt{2},\]

\[\begin{split}\begin{aligned} \hat{H}_{\mathrm p} &= & {\sum}_{\mathbf{q}\in \mathcal{B},\nu} \hbar\omega_{\mathbf{q}\nu} \left( \hat{a}^\dagger_{\mathbf{q}\nu,x} \hat{a}_{\mathbf{q}\nu,x} + \hat{a}^\dagger_{\mathbf{q}\nu,y} \hat{a}_{\mathbf{q}\nu,y} + 1 \right) \\ & + &{\sum}_{\mathbf{q}\in \mathcal{A},\nu} \hbar\omega_{\mathbf{q}\nu} \left( \hat{a}^\dagger_{\mathbf{q}\nu,x} \hat{a}_{\mathbf{q}\nu,x} + 1/2 \right). \end{aligned}\end{split}\]

(172)\[\chi_0(\{\boldsymbol{\tau}_{\kappa p} \}) = A e^{-\frac{1}{2}\left({\sum}_{\mathbf{q}\in\mathcal{A},\nu} \tilde{x}_{\mathbf{q}\nu}^2 +{\sum}_{\mathbf{q}\in\mathcal{B},\nu} \tilde{x}_{\mathbf{q}\nu}^2 + \tilde{y}_{\mathbf{q}\nu}^2 \right)},\]

(173)\[\begin{split}\begin{aligned} \hat{a}^+_{\mathbf{q}\nu} & = & (\hat{a}_{\mathbf{q}\nu,x}+i \hat{a}_{\mathbf{q}\nu,y})/\sqrt{2}, \\ \hat{a}^-_{\mathbf{q}\nu} & = & (\hat{a}_{\mathbf{q}\nu,x}-i \hat{a}_{\mathbf{q}\nu,y})/\sqrt{2}. \end{aligned}\end{split}\]

(174)\[\begin{split}\begin{aligned} \hat{a}_{\mathbf{q}\nu} =& \hat{a}_{\mathbf{q}\nu,x} \hspace{2cm}&\qquad\mbox{ for $\mathbf{q}$ in }\mathcal{A}, \\ \hat{a}_{\mathbf{q}\nu} =& (\hat{a}_{\mathbf{q}\nu,x}+i \hat{a}_{\mathbf{q}\nu,y})/\sqrt{2} &\qquad\mbox{ for $\mathbf{q}$ in }\mathcal{B}, \mathcal{C}. \end{aligned}\end{split}\]

(175)\[z_{\mathbf{q}\nu} = l_{\mathbf{q}\nu} (\hat{a}_{\mathbf{q}\nu}+\hat{a}^\dagger_{-\mathbf{q}\nu}).\]