# Phonon self-energy corrections: To screen, or not to screen

Jan Berges  U Bremen Excellence Chair, Bremen Center for Computational
Materials Science, and MAPEX Center for Materials and Processes, University of
Bremen, D-28359 Bremen, Germany  Nina Girotto  Institute of Physics, HR-10000
Zagreb, Croatia  Tim Wehling  I. Institute of Theoretical Physics, University
of Hamburg, D-22607 Hamburg, Germany  The Hamburg Centre for Ultrafast
Imaging, D-22761 Hamburg, Germany  Nicola Marzari  Theory and Simulation of
Materials (THEOS), École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne,
Switzerland  National Centre for Computational Design and Discovery of Novel
Materials (MARVEL), École Polytechnique Fédérale de Lausanne, CH-1015
Lausanne, Switzerland  U Bremen Excellence Chair, Bremen Center for
Computational Materials Science, and MAPEX Center for Materials and Processes,
University of Bremen, D-28359 Bremen, Germany  Samuel Poncé  Institute of
Condensed Matter and Nanosciences, Université catholique de Louvain, BE-1348
Louvain-la-Neuve, Belgium  National Centre for Computational Design and
Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de
Lausanne, CH-1015 Lausanne, Switzerland

###### Abstract

First-principles calculations of phonons are often based on the adiabatic
approximation, and Brillouin-zone samplings that might not always be
sufficient to capture the subtleties of Kohn anomalies. These shortcomings can
be addressed through corrections to the phonon self-energy arising from the
low-energy electrons. A well-founded correction method exists [Phys. Rev. B
82, 165111 (2010)], which only relies on adiabatically _screened_ quantities.
However, many-body theory suggests to use one _bare_ electron-phonon vertex in
the phonon self-energy [Rev. Mod. Phys. 89, 015003 (2017)] to avoid double
counting. We assess the accuracy of both approaches in estimating the low-
temperature phonons of monolayer TaS2 and doped MoS2. We find that the former
yields excellent results at low computational cost due to its designed error
cancellation to first order, while the latter becomes exact in the many-body
limit but is not accurate in approximate contexts. We offer a third strategy
based on downfolding to _partially screened_ phonons and interactions [Phys.
Rev. B 92, 245108 (2015)] to keep both advantages.

##  I Introduction

The interplay between interacting electrons and lattice vibrations in solids
gives rise to diverse phenomena ranging from quantitative effects, such as
changes in the electric [4] or thermal conductivity [5], to qualitative
effects, such as instabilities toward charge [6] and superconducting order
[7]. Accordingly, the simulation of lattice dynamics is an exceptionally well
established branch of computational materials science [8]. Readily accessible
total energies from density-functional theory (DFT) [9, 10] and the
development of density-functional perturbation theory (DFPT) [11, 12, 8] have
significantly advanced this field. Nowadays, phonon frequencies and normal
modes can be obtained routinely and at reasonable computational cost, allowing
for high-throughput calculations [13, 14]. For a wide range of materials,
especially semiconductors [15, 16] but also metals [17], the agreement between
theory and experiment is remarkable.

Nevertheless, DFPT still depends on crucial approximations: First, the
adiabatic or Born-Oppenheimer approximation [18] implicitly assumes that the
dynamics of electrons and ions happens on two well separated energy scales.
However, in some materials the relevant energy scale of the electrons is
similar to or even smaller than that of the phonons, leading to nonadiabatic
effects [19, 20, 21, 1, 22, 23, 24, 25, 26]. Second, calculations are in
practice limited to a sparse sampling of the Brillouin zone (BZ) or,
equivalently, short-range interatomic force constants. This poses a problem
for materials close to a lattice instability, signalled by Kohn anomalies [27]
driven by a strong long-range electronic response. Furthermore, the exchange-
correlation energy is described by an approximate functional. Thus, DFT-based
methods will inevitably fail for strongly correlated materials where the
single-electron picture breaks down [28].

A popular approach in cases where the abovementioned approximations cannot be
applied is to complement the results with suitable corrections to the self-
energy of electrons [29, 30] and phonons [31, 32, 1, 2]. Here, we usually face
a difficulty referred to as “double counting” or “overscreening” [33, 34, 35]:
An effect that is accounted for in DFT must be removed before a more elaborate
description of the same effect can be applied. This is closely related to the
concept of “downfolding” [36, 37, 38]: The full problem is mapped to an _ab
initio_ low-energy effective system with a significantly reduced number of
degrees of freedom [39]. Established downfolding methods for the electron-
electron and electron-phonon interaction [40] are the constrained random-phase
approximation (cRPA) [36, 41] and the constrained density-functional
perturbation theory (cDFPT) [3], respectively. While for the former several
implementations in popular simulation software exist [42, 43, 44, 45, 46], a
general workflow for the latter is at an earlier development stage [3, 47,
48].

In this work, we revisit the problem of obtaining nonadiabatic phonons that
are converged with respect to the sampling of the BZ, originally addressed in
Ref. 1. In doing so, we also react to the recently revived controversy about
the correct screening of the electron-phonon vertices in the phonon self-
energy: There are strong arguments [1] that the traditional choice [49, 20,
21, 50, 51, 52] of two screened vertices can not only be ascribed to the
absence of a better alternative, but is indeed a good option as long as
screening effects are approximated at the DFT level. This is however not
universally acknowledged, since both perturbative [33, 34, 35] and
nonperturbative treatments of the problem based on the Hedin-Baym equations
[2] yield a phonon self-energy with one bare and one screened vertex – a fact
that is by no means disputed in Ref. 1.

Here, by comparing the two approaches and supporting them with numerical
results from DFPT and cDFPT, we will detail the following options: (i) The
established approach with two screened vertices [1] yields excellent results
at low cost for a wide parameter range. Its robustness can be explained by a
designed cancellation of errors to first order. (ii) Working with one bare
vertex is exact, avoids overscreening, and thus allows for systematic
improvements. However, the bare vertex is not routinely obtained in a
pseudopotential framework, and the quality of the result depends on the
achievable precision of the screened vertex. Also, the computational cost is
increased by the necessity to sum over many electronic bands, but a possible
way around this problem has recently been published [53]. In practice, some
authors have approximated the bare vertex by _unscreening_ the DFPT one with
model dielectric functions [23, 54], but their accuracy is limited by the
validity of these models.

We also illustrate a third option (iii) where downfolding [3] is used and can
be shown to be equivalent to (ii) with a computational cost similar to (i). By
splitting the electronic transitions into an “active subspace” and its
complement, the “rest subspace,” we can gradually switch between the bare and
the screened vertex and settle for the optimum. Importantly, not only the bare
but also the partially screened quantities are, in good approximation,
adiabatic – in an Engelsberg-Schrieffer sense [55, 56] – and independent of
the electronic temperature since the rest system is gapped. As a consequence,
they are also smooth in large parts of reciprocal space, which facilitates
interpolation and ensures convergence already at coarse BZ sampling. This is
the main advantage of our proposed downfolding approach. Nevertheless, there
is also a drawback since the absence of these screening effects is accompanied
by the emergence of long-range Fröhlich terms. Since the associated
discontinuities in the derivatives of phonon dispersion and electron-phonon
coupling render a straightforward Fourier interpolation impossible, an
accurate description of these long-range terms, which allows to subtract and
add them before and after interpolation, is needed.

We implemented approaches (i), (ii), and (iii), in Quantum ESPRESSO [57, 58,
59] and the EPW code [60, 61, 62], see Supplemental Material. On this basis,
we can efficiently study fine features such as nonadiabatic Kohn anomalies and
Peierls instabilities [63], which all arise from the active subspace, at the
required (and otherwise prohibitive) ultradense BZ sampling, low electronic
temperature, and with frequency dependence.

We showcase the simplicity and importance of methods (i) and (iii) by
investigating the electron-phonon interaction and resulting peculiarities of
the phonon dispersions of monolayer TaS2 and n-doped MoS2. In the former, we
can access the sharp features of the doping-dependent Kohn anomalies,
indicating instability toward charge order, and significant nonadiabatic
phonon renormalization, which only emerge at low electronic temperatures. In
the latter, we find that the observed doping-induced phonon softening is due
to a combination of high intervalley susceptibility and electron-phonon
coupling, while the maximum of the coupling is still _dormant_ and could be
activated by doping or band engineering.

This paper is organized as follows: In Sec. II, we present the relevant
theoretical concepts and formulas. Subsequently, in Sec. III, we describe how
these are implemented in practice. On this basis, in Sec. IV, we discuss our
numerical results. We finish by summarizing our work and discussing possible
generalization of the scheme in Sec. V.

##  II Theory

In this section, we review the theoretical background, including the phonon
Green’s function [Sec. II.1], downfolding [Sec. II.2], the diagrams describing
the screening of phonons and interactions [Sec. II.3], approximate phonon
self-energies [Sec. II.4], long-range effects [Sec. II.5], and the basics of
Wannier-Fourier interpolation [Sec. II.6]. We employ Rydberg atomic units,
where in particular $\hbar=e=1$.

###  II.1 Phonon Green’s function and dynamical matrix

The lattice dynamics of a material can be described by the phonon Green’s
function or, more precisely, the retarded displacement-displacement
correlation function 111Strictly speaking, the phonon Green’s function is the
correlation function of phonon ladder operators instead of displacement
operators [2] and thus differs from the latter by a factor of $2\omega$, which
thus appears in Eq. (3) for the phonon spectral function.,

$G_{\bm{R}-\bm{R}^{\prime}\kappa\alpha\kappa^{\prime}\beta}(T,t-t^{\prime})=-\mathrm{i}\Theta(t-t^{\prime})\langle[\hat{u}_{\bm{R}\kappa\alpha}^{\vphantom{\dagger}}(t),\hat{u}_{\bm{R}^{\prime}\kappa^{\prime}\beta}^{\dagger}(t^{\prime})]\rangle_{T},$
(1)

where $\langle\ldots\rangle_{T}$ denotes an ensemble average, which depends on
the electronic temperature $T$, and $[{\cdots},{\cdots}]$ the commutator. The
Heaviside function $\Theta$ ensures the time ordering $t>t^{\prime}$ of the
Green’s function. $u_{\bm{R}\kappa\alpha}$ describes the displacement in the
$\alpha$th Cartesian direction of the $\kappa$th basis atom in the unit cell
of lattice vector $\bm{R}$. Knowledge of the correlations between any two
ionic displacements at different times and positions as in Eq. (1) completely
characterizes the lattice dynamics.

Due to translational invariance, Eq. (1) depends on _differences_ of times
$t-t^{\prime}$ and lattice vectors $\bm{R}-\bm{R}\mathrlap{{}^{\prime}}$,
which allows for a Fourier transform to frequency $\omega$ and phonon momentum
$\bm{q}$,

$G_{\bm{q}}(T,\omega)=\frac{\mathds{1}}{\omega^{2}\mathds{1}-D_{\bm{q}}(T,\omega)}$
(2)

with the screened dynamical matrix $D$. Here, $\omega$ is defined on the whole
complex plane and the _retarded_ phonon Green’s function is obtained at
$\omega+\mathrm{i}0^{+}$ with $\omega$ real and $0^{+}$ a positive
infinitesimal. The right-hand side of Eq. (2) is a matrix inversion in the
basis of displacements $\kappa\alpha,\kappa^{\prime}\beta$ and $\mathds{1}$ is
the corresponding identity matrix.

The link to experiments such as inelastic neutron- or x-ray-scattering
spectroscopy [23] is the phonon spectral function. It assigns an intensity to
each combination of energies $\omega$ and crystal momenta $\bm{q}$ and thus
provides the quasiparticle bandstructure, where applicable, including many-
body effects such as broadening and satellites. We compute the phonon spectral
function as [65]

$A_{\bm{q}}(T,\omega)=-\frac{2\omega}{\pi}\operatorname{Tr}\operatorname{Im}G_{\bm{q}}(T,\omega+\mathrm{i}0^{+}).$
(3)

Instead, the static ($\omega=0$) adiabatic phonon frequencies
$\omega_{\bm{q}\nu}$ as obtained from DFPT, usually at high electronic
smearing $\sigma$, follow from the eigenvalue equation for the dynamical
matrix,

$D_{\bm{q}}(\sigma,0)\bm{e}_{\bm{q}\nu}^{\vphantom{0}}=\omega^{2}_{\bm{q}\nu}\bm{e}_{\bm{q}\nu}^{\vphantom{0}},$
(4)

where $\bm{e}_{\bm{q}\nu}$ are the phonon eigenvectors with mode index $\nu$.

###  II.2 Screened, partially screened, and bare quantities

In a typical _ab initio_ calculation, all electronic states are treated
equally. However, the relevant physics often takes place in a small subset of
these states, namely the low-energy states close to the Fermi level, which we
will refer to as active states. In particular, if an _ab initio_ calculation
for chosen parameters and approximations fails to converge or capture the
effects of interest, this is likely due to processes within this subset of
active states alone, while the rest might already be properly described.
Before resorting to more elaborate treatments, it is thus instructive to
consider the different subsets of states and corresponding energy scales
separately.

While pure _ab initio_ approaches always address the full system, the
_downfolding_ approach [36, 41, 40, 3] uses _ab initio_ calculations to
construct tractable low-energy systems with active states only, thus reducing
the number of degrees of freedom significantly. This approach is exact,
provided the complexity reduction is properly compensated by the partial
screening of the system parameters. While the full system consists of simple
_bare_ elementary particles and interactions, the parameters of the downfolded
system usually acquire dependences on the involved quantum numbers and are in
general also frequency dependent [36]. The chosen active states must thus span
an energy window large enough that low-energy effects and related dependences
can be safely neglected and at the same time small enough to keep the
computational cost affordable.

Figure 1: Visualization of active ($A$) and rest ($R$) subspaces of electronic
transitions based on a generic band structure. $A$ only includes transitions
between a chosen set of low-energy states highlighted in orange, $R$ all
remaining transitions. Note that there is in general an infinite number of
empty states beyond the shown energy range.

A typical choice of active and rest states is sketched in Fig. 1. Note that
here, for convenience but departing from what is usually done, we define the
active _subspace_ not as the set of active _states_ but as the set of
_transitions_ between them.

The full system with bare parameters would be recovered if _all_ states were
counted among the active states. Hence, special care has to be taken in the
context of pseudopotentials. Since quasiparticle energies and interactions
decrease with screening, the _bare_ quantities in an all-electron calculation
are larger than those in a pseudopotential framework with less core states.

###  II.3 Random-phase approximation

Table 1: List of symbols used in this manuscript. Subscript momenta and band indices have been omitted for brevity. Phonon Green’s function | Electron-phonon coupling | Bare electron susceptibility | Miscellaneous
---|---|---|---
$G^{\text{b}}$ | bare | $g^{\text{b}}$ | bare | $\chi^{\text{b}}(T,\omega)$ | full | $\omega$ | frequency argument
$G^{\text{p}}(T,\omega)$ | partially screened | $g^{\text{p}}(T,\omega)$ | partially screened | $\chi^{\text{b,R}}(T,\omega)$ | rest subspace | $T$ | electronic temperature
$G(T,\omega)$ | screened | $g(T,\omega)$ | screened | $\chi^{\text{b,A}}(T,\omega)$ | active subspace | $\sigma$ |  _ab initio_ smearing
Dynamical matrix | Electron-electron interaction | Phonon self-energy | $\bm{R}$ | Bravais lattice
$D^{\text{b}}$ | bare | $v$ | bare | $\varPi(T,\omega)$ | $=$ | $g^{\text{b}}$ | $\chi^{\text{b}}(T,\omega)$ | $g(T,\omega)$ | $\bm{G}$ | reciprocal lattice
$D^{\text{p}}(T,\omega)$ | partially screened | $U(T,\omega)$ | partially screened | $\varPi^{\text{R}}(T,\omega)$ | $=$ | $g^{\text{b}}$ | $\chi^{\text{b,R}}(T,\omega)$ | $g^{\text{p}}(T,\omega)$ | $\bm{k}$ | fermion momentum
$D(T,\omega)$ | screened | $W(T,\omega)$ | screened | $\varPi^{\text{A}}(T,\omega)$ | $=$ | $g^{\text{p}}(T,\omega)$ | $\chi^{\text{b,A}}(T,\omega)$ | $g(T,\omega)$ | $\bm{q}$ | boson momentum
Basis indices | Long-range electrostatics | $\varPi^{\text{00}}(T,\omega)$ | $=$ | $g(\sigma,0)$ | $\chi^{\text{b,A}}(T,\omega)$ | $g(\sigma,0)$ | $\varepsilon$ | electron energy
$s,p$ | electron orbitals | $\epsilon$ | dielectric constant | $\varPi^{\text{p0}}(T,\omega)$ | $=$ | $g^{\text{p}}(\sigma,0)$ | $\chi^{\text{b,A}}(T,\omega)$ | $g(\sigma,0)$ | $f$ | electron occupation
$m,n$ | electron bands | $\bm{Z}^{*}$ | Born effective charge | $\varPi^{\text{p$T$}}(T,\omega)$ | $=$ | $g^{\text{p}}(\sigma,0)$ | $\chi^{\text{b,A}}(T,\omega)$ | $g(T,0)$ | $M$ | atomic mass
$\kappa,\kappa^{\prime}$ | basis atoms | $Q$ | quadrupole tensor | $\varPi^{\text{b0}}(T,\omega)$ | $=$ | $g^{\text{b}}$ | $\chi^{\text{b,A}}(T,\omega)$ | $g(\sigma,0)$ | $\bm{\tau}$ | atomic position
$\alpha,\beta$ | Cartesian directions | $L$ | separation parameter | $\varPi^{\text{b$T$}}(T,\omega)$ | $=$ | $g^{\text{b}}$ | $\chi^{\text{b,A}}(T,\omega)$ | $g(T,0)$ | $\bm{u}$ | atomic displacement

In this section, we review the formulas that describe the electronic screening
of phonons and interactions in the framework of the random-phase approximation
(RPA) [66, 67, 68, 38]. In this approximation, the electronic response is
given by the polarizability of the system. In terms of Feynman diagrams, we
consider all possible diagrams consisting of bare phonons, the bare electron-
phonon interaction, the bare electron-electron interaction, and electron-hole
“bubbles.” In the static case ($\omega=0$), this is equivalent to the
screening in DFPT as long as the electrons are given by the adiabatically
screened Kohn-Sham states and the exchange-correlation kernel is included in
the bare electron-electron interaction [3, 2]. A summary of the most relevant
symbols used in the following is provided in Table 1.

####  II.3.1 Electron-electron interaction

We start with the screening of the electron-electron interaction, for which
the RPA has been originally derived [66]. In the basis of electronic
eigenstates, used throughout the manuscript, the bare Coulomb interaction can
be written as

$v_{\bm{q}\bm{k}mn\bm{k}^{\prime}m^{\prime}n^{\prime}}=\langle\bm{k}n;\bm{k}^{\prime}{+}\bm{q}m^{\prime}|\hat{v}|\bm{k}{+}\bm{q}m;\bm{k}^{\prime}n^{\prime}\rangle$
(5)

with electron momentum $\bm{k}$, band indices $m,n$, and the virtual photon
momentum transfer $\bm{q}$. We purposely used the symbol $\bm{q}$ as in Eq.
(2) to highlight the fact that in RPA the phonon and photon momentum transfers
have to be the same – this is no longer true beyond RPA. More precisely, Eq.
(5) quantifies the scattering of two electrons from single-particle states
$|\bm{k}{+}\bm{q}m\rangle$ and $|\bm{k}^{\prime}n^{\prime}\rangle$ into states
$|\bm{k}n\rangle$ and $|\bm{k}^{\prime}{+}\bm{q}m^{\prime}\rangle$,
respectively. In the basis of electronic positions, it has the well-known
diagonal representation
$\langle\bm{r};\bm{r}^{\prime}|\hat{v}|\bm{r};\bm{r}^{\prime}\rangle=1/|\bm{r}-\bm{r}^{\prime}|$.

The interaction between two electrons is however screened by the
polarizability of all other electrons. Taking the formation of any number of
electron-hole pairs into account, the screened electron-electron interaction
$W(T,\omega)$ is related to the bare Coulomb interaction $v$ via

$W(T,\omega)$$=$$v$$+$$v$$W(T,\omega)$$\chi^{\text{b}}(T,\omega)$ (6)

or, translated to a formula,

$W_{\bm{q}\bm{k}mn\bm{k}^{\prime}m^{\prime}n^{\prime}}(T,\omega)=v_{\bm{q}\bm{k}mn\bm{k}^{\prime}m^{\prime}n^{\prime}}+\frac{2}{N}\sum_{\bm{k}^{\prime\prime}m^{\prime\prime}n^{\prime\prime}}v_{\bm{q}\bm{k}mn\bm{k}^{\prime\prime}m^{\prime\prime}n^{\prime\prime}}\\\
\chi_{\bm{q}\bm{k}^{\prime\prime}m^{\prime\prime}n^{\prime\prime}}^{\text{b}}(T,\omega)W_{\bm{q}\bm{k}^{\prime\prime}m^{\prime\prime}n^{\prime\prime}\bm{k}^{\prime}m^{\prime}n^{\prime}}^{\vphantom{0}}(T,\omega),$
(7)

where the $2$ accounts for the spin and the summation is over $N$ momenta and
_all_ pairs of band indices. $W(T,\omega)$ as defined here is also used in the
$GW$ approximation [69, 70, 71]. The summation over an infinite number of
bands can be circumvented via the Sternheimer approach [72, 73, 74, 53].

The bare electronic susceptibility or “polarizability,” from which any $T$ or
$\omega$ dependence originates, reads

$\chi_{\bm{q}\bm{k}mn}^{\text{b}}(T,\omega)=\frac{f(\varepsilon_{\bm{k}n}/T)-f(\varepsilon_{\bm{k}+\bm{q}m}/T)}{\varepsilon_{\bm{k}n}-\varepsilon_{\bm{k}+\bm{q}m}+\omega},$
(8)

where $\varepsilon$ and $f$ are the electronic energies and occupations.

The most important contributions in Eq. (8) come from transitions between
occupied and empty states with similar energies, i.e., from the low-energy
electrons near the Fermi level. This suggests to split the screening in Eq.
(6) into two steps, namely the _downfolding_ to a low-energy system and the
_renormalization_ to recover physical results. We label the subset of
transitions between properly chosen low-energy active states as $A$ and the
remaining transitions as the rest $R$ [cf. Fig. 1]. The bare susceptibility
can then be decomposed as

$\chi_{\bm{q}\bm{k}mn}^{\text{b}}(T,\omega)=\chi_{\bm{q}\bm{k}mn}^{\text{b,A}}(T,\omega)+\chi_{\bm{q}\bm{k}mn}^{\text{b,R}}(T,\omega),$
(9)

where the first term is nonzero only if the transition from $|\bm{k}n\rangle$
to $|\bm{k}{+}\bm{q}m\rangle$ is part of $A$ and the second term vice versa.

First, in the downfolding step, the partially screened electron-electron
interaction $U(T,\omega)$ is calculated from the bare Coulomb interaction $v$,

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}\pgfsys<EMAIL_ADDRESS>(10)

This is known as the cRPA [36, 41]. The corresponding formula is equivalent to
Eq. (7), except that the summation is constrained to the transitions in $R$.
In general, the partially screened $U(T,\omega)$ depends on $T$ and $\omega$.
However, excluding low-energy transitions makes the system effectively gapped
and the dependence on $T$ and $\omega$ can be controlled via the size of the
active subspace [75].

Second, in the renormalization step, the screened $W(T,\omega)$ can be
recovered from the partially screened $U(T,\omega)$,

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}\pgfsys<EMAIL_ADDRESS>(11)

The corresponding formula is again equivalent to Eq. (7), but now the
summation is constrained to the finite number of active bands. This allows to
evaluate the low-energy response, which requires a dense BZ sampling at low
electronic temperature, at an affordable computational cost.

####  II.3.2 Electron-phonon interaction

The change of the electronic energies upon ionic displacements is also
screened by the surrounding electrons. Without this screening, the bare
electron-phonon coupling in the electronic eigenbasis reads

$\displaystyle
g_{\bm{q}\kappa\alpha\bm{k}mn}^{\text{b}}=\frac{1}{\sqrt{M_{\kappa}}}\langle\bm{k}{+}\bm{q}m|\frac{\partial\hat{V}^{\text{b}}}{\partial
u_{\bm{q}\kappa\alpha}}|\bm{k}n\rangle$ (12)

with the atomic mass $M$, and the bare external potential $V^{\text{b}}$
acting on an electron amid the ensemble of ions. In the position
representation, we have
$\langle\bm{r}|\hat{V}^{\text{b}}|\bm{r}\rangle=-\sum_{\bm{R}\kappa}Z_{\kappa}/|\bm{R}+\bm{\tau}_{\kappa}+\bm{u}_{\bm{R}\kappa}-\bm{r}|$,
where $Z_{\kappa}$ and $\bm{\tau}$ are ionic charges and equilibrium positions
within the unit cell.

The screened electron-phonon interaction can be written as
$g=g^{\text{b}}\epsilon^{-1}$ with the inverse dielectric function
$\epsilon^{-1}=1+\chi^{\text{b}}W$, similarly to the screened electron-
electron interaction $W=v\epsilon^{-1}$ of Eq. (6), or, using diagrams, as
[cf. Appendix I of Ref. 76]

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{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
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Translated to a formula, it reads

$g_{\bm{q}\kappa\alpha\bm{k}mn}^{\vphantom{0}}(T,\omega)=g_{\bm{q}\kappa\alpha\bm{k}mn}^{\text{b}}+\frac{2}{N}\sum_{\bm{k}^{\prime}m^{\prime}n^{\prime}}g_{\bm{q}\kappa\alpha\bm{k}^{\prime}m^{\prime}n^{\prime}}^{\text{b}}\\\
\chi_{\bm{q}\bm{k}^{\prime}m^{\prime}n^{\prime}}^{\text{b}}(T,\omega)W_{\bm{q}\bm{k}^{\prime}m^{\prime}n^{\prime}\bm{k}mn}^{\vphantom{0}}(T,\omega).$
(14)

Also the renormalization of the electron-phonon coupling can be split into two
steps: First, the partially screened $g^{\text{p}}(T,\omega)$ is obtained from
the bare $g^{\text{b}}$,

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}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@moveto{58.90552pt}{0.0pt}\pgfsys@curveto{58.90552pt}{1.10458pt}{58.0101pt}{2.0pt}{56.90552pt}{2.0pt}\pgfsys@curveto{55.80093pt}{2.0pt}{54.90552pt}{1.10458pt}{54.90552pt}{0.0pt}\pgfsys@curveto{54.90552pt}{-1.10458pt}{55.80093pt}{-2.0pt}{56.90552pt}{-2.0pt}\pgfsys@curveto{58.0101pt}{-2.0pt}{58.90552pt}{-1.10458pt}{58.90552pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{54.3413pt}{7.47745pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$g\mathrlap{{}^{\text{b}}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$ (15)

which can be accomplished using cDFPT [3]. The screened $g(T,\omega)$ follows
from the partially screened $g^{\text{p}}(T,\omega)$,

$\leavevmode\hbox to167.81pt{\vbox
to37.73pt{\pgfpicture\makeatletter\hbox{\hskip
18.45915pt\lower-18.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}
\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{ } {{}}
\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.0pt}{0.0pt}\pgfsys@curveto{3.0pt}{1.65688pt}{1.65688pt}{3.0pt}{0.0pt}{3.0pt}\pgfsys@curveto{-1.65688pt}{3.0pt}{-3.0pt}{1.65688pt}{-3.0pt}{0.0pt}\pgfsys@curveto{-3.0pt}{-1.65688pt}{-1.65688pt}{-3.0pt}{0.0pt}{-3.0pt}\pgfsys@curveto{1.65688pt}{-3.0pt}{3.0pt}{-1.65688pt}{3.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-15.12614pt}{8.033pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$g(T,\omega)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{10.33748pt}{-1.83437pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$=$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}
\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{ } {{}}
\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@moveto{30.45276pt}{0.0pt}\pgfsys@curveto{30.45276pt}{1.10458pt}{29.55734pt}{2.0pt}{28.45276pt}{2.0pt}\pgfsys@curveto{27.34818pt}{2.0pt}{26.45276pt}{1.10458pt}{26.45276pt}{0.0pt}\pgfsys@curveto{26.45276pt}{-1.10458pt}{27.34818pt}{-2.0pt}{28.45276pt}{-2.0pt}\pgfsys@curveto{29.55734pt}{-2.0pt}{30.45276pt}{-1.10458pt}{30.45276pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.77106pt}{-13.033pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$g^{\text{p}}(T,\omega)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{38.79024pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$+$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}
\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}} }{{{{}{}{{}}
}}{{}}}{{{{}{}{{}} }}{{}}{{}}} {{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}}
{}{}{} }{{{{}{}{{}} }}{{}} {{}} } {{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@rectcap\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{113.81104pt}{0.0pt}\pgfsys@lineto{114.66461pt}{0.0pt}\pgfsys@curveto{115.91461pt}{0.0pt}{116.53961pt}{2.5pt}{117.78961pt}{2.5pt}\pgfsys@curveto{119.03961pt}{2.5pt}{119.66461pt}{0.0pt}{120.91461pt}{0.0pt}\pgfsys@lineto{142.2638pt}{0.0pt}\pgfsys@lineto{142.2638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{110.05882pt}{-13.033pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$W(T,\omega)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}
\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{ } {{}}
\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}{}{}}{{}{}}{{}}
{{{}}{{}}}{{}}{{}{}{}{}}{{{}}{{}}}{{}{}{}}{}{{}}{}{}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@curveto{72.59729pt}{-15.69177pt}{98.11926pt}{-15.69177pt}{113.81104pt}{0.0pt}\pgfsys@curveto{98.11926pt}{15.69177pt}{72.59729pt}{15.69177pt}{56.90552pt}{0.0pt}\pgfsys@closepath\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }
{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}}{{}{}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {\pgfsys@beginscope\pgfsys@invoke{ }{{}} {{}}
{{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.00005}{-0.00005}{1.0}{86.1501pt}{-11.76875pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}
{{{}}} }{{}{}}{{}{}}{{}{}{}{}{{}}{}{{}} {{{}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@roundcap\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@curveto{72.59729pt}{-15.69177pt}{98.11926pt}{-15.69177pt}{113.81104pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }
{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}}{{}{}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {\pgfsys@beginscope\pgfsys@invoke{ }{{}} {{}}
{{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{-0.00005}{0.00005}{-1.0}{84.56636pt}{11.76875pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}
{{{}}} }{{}{}}{{}{}}{{}{}{}{}{{}}{}{{}} {{{}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@roundcap\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{113.81104pt}{0.0pt}\pgfsys@curveto{98.11926pt}{15.69177pt}{72.59729pt}{15.69177pt}{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{65.23457pt}{-1.07729pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\chi^{\text{b,A}}(T,\omega)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}
\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{ } {{}}
\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@moveto{58.90552pt}{0.0pt}\pgfsys@curveto{58.90552pt}{1.10458pt}{58.0101pt}{2.0pt}{56.90552pt}{2.0pt}\pgfsys@curveto{55.80093pt}{2.0pt}{54.90552pt}{1.10458pt}{54.90552pt}{0.0pt}\pgfsys@curveto{54.90552pt}{-1.10458pt}{55.80093pt}{-2.0pt}{56.90552pt}{-2.0pt}\pgfsys@curveto{58.0101pt}{-2.0pt}{58.90552pt}{-1.10458pt}{58.90552pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{53.79326pt}{8.033pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\mathllap{g^{\text{p}}(T,{}}\omega\mathrlap{)}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(16)

The corresponding formulas are equivalent to Eq. (14).

Using the alternative expression for the dielectric function
$\epsilon=1-\chi^{\text{b}}v$ and the corresponding formulas for the active
and rest subspace, we can also write Eqs. (13), (15), and (16) as

$\displaystyle\leavevmode\hbox to164.99pt{\vbox
to38.62pt{\pgfpicture\makeatletter\hbox{\hskip 18.45915pt\lower-19.7549pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}
\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{ } {{}}
\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.0pt}{0.0pt}\pgfsys@curveto{3.0pt}{1.65688pt}{1.65688pt}{3.0pt}{0.0pt}{3.0pt}\pgfsys@curveto{-1.65688pt}{3.0pt}{-3.0pt}{1.65688pt}{-3.0pt}{0.0pt}\pgfsys@curveto{-3.0pt}{-1.65688pt}{-1.65688pt}{-3.0pt}{0.0pt}{-3.0pt}\pgfsys@curveto{1.65688pt}{-3.0pt}{3.0pt}{-1.65688pt}{3.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-15.12614pt}{8.033pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$g(T,\omega)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{10.33748pt}{-1.83437pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$=$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}
\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{ } {{}}
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