# Comparing first-principles density functionals plus corrections for the
lattice dynamics of YBa2Cu3O6

Jinliang Ning  Department of Physics and Engineering Physics, Tulane
University, New Orleans, Louisiana 70118, United States  Christopher Lane
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA  Bernardo Barbiellini  Department of Physics, School of Engineering
Science, LUT University, FI-53851 Lappeenranta, Finland  Department of
Physics, Northeastern University, Boston, MA 02115, USA  Robert S. Markiewicz
Department of Physics, Northeastern University, Boston, MA 02115, USA  Arun
Bansil  Department of Physics, Northeastern University, Boston, MA 02115, USA
Adrienn Ruzsinszky<EMAIL_ADDRESS>Department of Physics and Engineering
Physics, Tulane University, New Orleans, Louisiana 70118, United States  John
P. Perdew<EMAIL_ADDRESS>Department of Physics and Engineering Physics,
Tulane University, New Orleans, Louisiana 70118, United States  Jianwei Sun
<EMAIL_ADDRESS>Department of Physics and Engineering Physics, Tulane
University, New Orleans, Louisiana 70118, United States

###### Abstract

The enigmatic mechanism underlying unconventional high-temperature
superconductivity, especially the role of lattice dynamics, has remained a
subject of debate. Theoretical insights have long been hindered due to the
lack of an accurate first-principles description of the lattice dynamics of
cuprates. Recently, using the r2SCAN meta-GGA functional, we were able to
achieve accurate phonon spectra of an insulating cuprate YBa2Cu3O6, and
discover significant magnetoelastic coupling in experimentally interesting
Cu-O bond stretching optical modes [Phys. Rev. B 107, 045126 (2023)]. We
extend this work by comparing PBE and r2SCAN performances with corrections
from the on-site Hubbard U and the D4 van der Waals (vdW) methods, aiming at
further understanding on both the materials science side and the density
functional side. We demonstrate the importance of vdW and self-interaction
corrections for accurate first-principles YBa2Cu3O6 lattice dynamics. Since
r2SCAN by itself partially accounts for these effects, the good performance of
r2SCAN is now more fully explained. In addition, the performances of the Tao-
Mo series of meta-GGAs, which are constructed in a different way from
SCAN/r2SCAN, are also compared and discussed.

##  I Introduction

Despite the decades of vigorous efforts devoted to the understanding of
unconventional high-temperature superconductivity in the cuprates, a consensus
on the underlying mechanism has yet to be reached Bednorz and Müller (1986);
Anderson (1987); Tsuei and Kirtley (2000); Damascelli, Hussain, and Shen
(2003); Keimer  _et al._ (2015); Sobota, He, and Shen (2021). Early
theoretical works Bohnen, Heid, and Krauss (2003); Giustino, Cohen, and Louie
(2008); Heid  _et al._ (2009) suggested that the conventional BCS theory
(electron-phonon coupling mechanism) Cooper (1956); Bardeen, Cooper, and
Schrieffer (1957a, b) could not account for such high critical temperatures in
cuprate superconductors. However, a more intricate and intriguing picture has
been suggested by recent experimental findingsMcQueeney  _et al._ (2001);
Reznik  _et al._ (2006); Lanzara  _et al._ (2001); d’Astuto  _et al._ (2002);
McQueeney  _et al._ (1999); Reznik  _et al._ (2006); Gweon  _et al._ (2004);
He  _et al._ (2018). Strong anomalies in Cu-O bond-stretching modes are found
near optimal doping, which is associated with charge inhomogeneity and beyond
previous pictures and understandings Reznik  _et al._ (2006). Optical
spectroscopy results indicate that the electron-phonon coupling contributes at
least 10% the bosonic pairing glue, although antiferromagnetic spin
fluctuations are deemed as the main mediators Dal Conte  _et al._ (2012).
Moreover, the electronic interactions and the electron-phonon coupling are
found to reinforce each other in a positive-feedback loop, which in turn
enhances superconductivity, as suggested by recent ARPES observations He  _et
al._ (2018).

Part of the reason why the role of phonons was dismissed by the theoretical
community was that, previous density functional theory (DFT) calculations at
the local density approximation (LDA) and generalized gradient approximation
(GGA) levels failed to find strong electron-phonon coupling in related
cupratesGiustino, Cohen, and Louie (2008). This issue is related to and
compounded by the fact that these density functional approximations (DFAs)
cannot stabilize the correct electronic and magnetic ground state in the
parent phase, let alone its evolution with dopingZhang  _et al._ (2020a);
Furness  _et al._ (2018). While corrections such as the Hubbard U Anisimov,
Zaanen, and Andersen (1991); Cococcioni and de Gironcoli (2005); Kasinathan
_et al._ (2006); Dudarev  _et al._ (1998) method can stabilize the
antiferromagnetic (AFM) ground state Sterling and Reznik (2021), their
structural predictions can be unexpected and uncontrollable Jarlborg  _et al._
(2014). Obviously, an ab initio treatment is required to capture
simultaneously the electronic, magnetic and lattice degrees of freedom.

Recently, utilizing the r2SCAN meta-GGA functional Furness  _et al._ (2020a),
some of usNing  _et al._ (2023) were able to stabilize the AFM state of the
pristine oxide YBa2Cu3O6, and faithfully reproduce the experimental phonon
dispersions. We further found significant magnetoelastic coupling in numerous
high-energy Cu-O bond stretching optical branches, where the AFM results
improve over the soft nonmagnetic phonon bandsNing  _et al._ (2023). Moreover,
these phonons correspond to breathing modes within the CuO2 plane, suggesting
a sensitive dependence on magnetoelastic coupling, which may facilitate a
positive-feedback loop between electronic, magnetic, and lattice degrees of
freedom. The r2SCAN functional is a modified and improved version of the
strongly-constrained and appropriately-normed (SCAN) meta-GGA functional Sun,
Ruzsinszky, and Perdew (2015); Sun  _et al._ (2016), which satisfies 17 exact
constraints, has demonstrated excellent performance across a diverse range of
bonding environments. For cuprates, SCAN accurately predicts the correct half-
filled AFM ground state and the observed insulator-metal transition upon
doping Zhang  _et al._ (2020a); Furness  _et al._ (2018). Moreover, SCAN
provides improved estimates of lattice constants, across correlated and
transition metal compounds Sun, Ruzsinszky, and Perdew (2015); Sun  _et al._
(2016); Zhang  _et al._ (2020a); Furness  _et al._ (2018); Zhang  _et al._
(2020b, 2019); Kitchaev  _et al._ (2016); Peng and Perdew (2017a); Ning  _et
al._ (2020); Lane  _et al._ (2020). Thus, SCAN is promising in accurate
descriptions of lattice dynamics of cuprates and associated electron-phonon
couplings, by virtue of its ability to capture the electronic and magnetic
ground states. Unfortunately, efficiently obtaining reliable phonon spectra
from SCAN calculations can be challenging due to numerical instability
problems. By design, r2SCAN Furness  _et al._ (2020a) solves the numerical
instability problem and delivers accurate, transferable, and reliable lattice
dynamics for various systems with different bonding characteristicsNing,
Furness, and Sun (2022). We thus chose r2SCAN instead of SCAN for the study of
lattice dynamics of YBa2Cu3O6 and achieved remarkable success.

Despite this success, there still exists a notable residual softening trend in
the Cu-O bond-stretching optical phonon branches from r2SCAN, especially in
the full-breathing modes, for which we achieved further improvements when a
Hubbard U correction is applied to r2SCAN Ning  _et al._ (2023). Furthermore,
it is not fully understood why r2SCAN/SCAN perform so well on cuprates.
Although in general we can attribute it to the power of satisfying exact
constraints by design in SCAN/r2SCANSun, Ruzsinszky, and Perdew (2015);
Furness  _et al._ (2020a, 2022); Kaplan, Levy, and Perdew (2023), more
specific and physical knowledge will be helpful and highly required. Previous
studies suggest that vdW corrections are important for first-principles
prediction of lattice constants and cohesive energy of ionic solids and heavy
metals Tao  _et al._ (2017). In addition, combining vdW correction and self-
interaction correction (SIC) is of critical importance for ground state
electronic, structural and energetic properties of transition metal monoxides
Peng and Perdew (2017a). Therefore, it is expected that the vdW correction and
its combination with SIC are crucial for obtaining accurate phonon dispersions
of cuprates based on DFT.

To confirm this, in this work we extend our lattice dynamics study of
YBa2Cu3O6 by comparing the PBE and r2SCAN performances with corrections from
the Hubbard U (applied to the $d$ orbitals of Cu) and the D4 van der Waals
(vdW) correction methods Caldeweyher, Bannwarth, and Grimme (2017);
Caldeweyher  _et al._ (2020, 2019); Ehlert  _et al._ (2021), aiming at further
understanding both the physics of cuprate lattice dynamics and density
functionals. We demonstrate the importance of vdW interactions and SIC for
YBa2Cu3O6 lattice dynamics. Since r2SCAN by itself provides a partial account
of these effects to a greater degree than PBE, the better performance of
r2SCAN is more fully explained. In addition, the performances of the Tao-Mo
family of meta-GGAs (TMs) Tao and Mo (2016); Jana, Sharma, and Samal (2019);
Jana  _et al._ (2021) are also compared and discussed. The original Tao-Mo
meta-GGA (TM) Tao and Mo (2016) is constructed based on a density matrix
expansion of the exchange hole model, while revTM Jana, Sharma, and Samal
(2019) and rregTM Jana  _et al._ (2021) are two successors with modifications.
The revTM includes a correlation correction obtained from the full high-
density second-order gradient expansion, while rregTM includes a regulation to
the order-of-limit problemFurness  _et al._ (2020b), paired with a one-
electron self-interaction-free correlation energy functional. In comparison,
SCAN and r2SCAN are constructed by satisfying exact-constraints on the
exchange-correlation energySun, Ruzsinszky, and Perdew (2015); Furness  _et
al._ (2020a, 2022); Kaplan, Levy, and Perdew (2023). Due to the inherently
different way the Tao-Mo meta-GGAs are constructed compared to SCAN/r2SCAN, a
comparison of their performances will be interesting and is expected to shed
light on both the materials science side in cuprate lattice dynamics and the
DFT side. Due to the absence of D4 parametrizations matched to TM functionals,
the effects of vdW corrections to TMs are not considered in this work.

The synergy of long-range vdW corrections and +U SIC that we find here for
cuprate lattice dynamics has also been foundPeng and Perdew (2017a) for
structural properties and structural phase transitions in MnO, FeO, CoO, and
NiO.

##  II Methods

First-princiles calculations were performed using the pseudopotential
projector-augmented wave methodBlöchl (1994); Kresse and Joubert (1999) with
the Vienna ab initio simulation package (VASP) Kresse and Furthmüller (1996);
Kresse and Hafner (1993) with an energy cutoff of 600 eV for the plane-wave
basis set. Several exchange-correlation functionals including PBE at the GGA
level, and r2SCAN Furness  _et al._ (2020a); Ning, Furness, and Sun (2022),
TMTao and Mo (2016), revTMJana, Sharma, and Samal (2019), and rregTMJana  _et
al._ (2021) at the meta-GGA level were used. For the D4 vdW correction, we use
the literature parametrizations fitted separately for PBE
($s\mathrm{{}_{6}}$=1.0000, $s\mathrm{{}_{8}}$=0.9595,
$a\mathrm{{}_{1}}$=0.3857, $a\mathrm{{}_{2}}$=4.8069) Caldeweyher  _et al._
(2019) and for r2SCAN ($s\mathrm{{}_{6}}$=1.0000, $s\mathrm{{}_{8}}$=0.6019,
$a\mathrm{{}_{1}}$=0.5156, $a\mathrm{{}_{2}}$=5.7734)Ehlert  _et al._ (2021).
A Gamma-centered 8$\times$8$\times$4 mesh for the k-space sampling is used for
the relaxation of the unit cell with a G-type AFM structure, while a
2$\times$2$\times$2 mesh is used for the 2$\times$2$\times$1 supercells for
force constant calculations of the phonon dispersion. All atomic sites in the
unit cell along with the cell dimensions were relaxed using a conjugate
gradient algorithm to minimize the energy with an atomic force tolerance of
0.001 eV/Å and a total energy tolerance of $10^{-7}$ eV. The harmonic force
constants were extracted from VASP using the finite displacement method
(0.015Å) as implemented in the Phonopy code Togo and Tanaka (2015).

##  III Results

Figure 1: (a) $\sqrt{2}\times\sqrt{2}\times 1$ crystal structure of YBa2Cu3O6
where the related G-type AFM structure is highlighted by coloring the corner-
sharing Cu-O pyramids blue(orange) for spin up(down). (b) A schematic of the
nonmagnetic (black dashed line) and G-type AFM (blue dashed line) Brillouin
zones and high-symmetry k-points. (c) Schematic of the typical $ac$-plane and
3D full-breathing modes. Table 1: Calculated lattice constants, volume, Cu
magnetic moment, Cu-O plane buckling angle $\angle$O-Cu-O, and Cu-O bond
lengths for YBa2Cu3O6 in the G-type AFM phase, along with the available
experimental data. The Cu-O bond length for the two adjacent Cu-O planes in
the $ab$ plane, where Cu shows local magnetic moment $m$, is denoted by
$d_{\mathrm{Cu-O}}$, while $z_{\mathrm{Cu-O_{ap}}}$ (nonmagnetic Cu-apical O
bond) and $z^{\prime}_{\mathrm{Cu-O_{ap}}}$ (magnetic Cu-apical O bond) denote
the Cu-O bond lengths along the $c$ direction. For PBE and r2SCAN, the Hubbard
U and vdW (D4 and rVV10) corrections are considered. The choice of Hubbard U
values is guided by both experimental lattice constants and Cu magnetic
moment.

DFA | U | vdW |  $a$ (Å) |  $c$ (Å) |  $V$ (Å3) |  $m$ ($\mu_{B}$) |  $d_{\mathrm{Cu-O}}$ (Å) |  $\angle$O-Cu-O (∘) |  $z_{\mathrm{Cu-O_{ap}}}$ (Å) |  $z^{\prime}_{\mathrm{Cu-O_{ap}}}$ (Å)
---|---|---|---|---|---|---|---|---|---|---
PBE | 0 | – | 3.8819 | 12.1905 | 183.70 | 0.00 | 1.948 | 170.32 | 1.816 | 2.672
| 6 | – | 3.8750 | 12.0379 | 180.76 | 0.59 | 1.950 | 167.06 | 1.811 | 2.551
| 6 | D4 | 3.8525 | 11.8702 | 176.17 | 0.59 | 1.938 | 167.25 | 1.803 | 2.492
r2SCAN | 0 | – | 3.8570 | 11.9417 | 177.65 | 0.45 | 1.937 | 169.28 | 1.805 | 2.554
| 5 | – | 3.8562 | 11.8321 | 175.95 | 0.66 | 1.941 | 167.00 | 1.795 | 2.472
| 4 | D4 | 3.8485 | 11.8032 | 174.82 | 0.62 | 1.936 | 167.41 | 1.794 | 2.469
| 4 | rVV10 | 3.8482 | 11.7638 | 174.21 | 0.62 | 1.936 | 167.41 | 1.793 | 2.451
TM | 0 | – | 3.8651 | 11.8224 | 176.61 | 0.15 (0.3) | 1.939 | 170.31 | 1.812 | 2.502
| 5 | – | 3.8626 | 11.7186 | 174.84 | 0.62 | 1.943 | 167.56 | 1.803 | 2.421
revTM | 0 | – | 3.8649 | 11.8952 | 177.68 | 0.09 (0.3) | 1.940 | 170.09 | 1.810 | 2.535
| 5 | – | 3.8621 | 11.7919 | 175.89 | 0.61 | 1.943 | 167.52 | 1.802 | 2.451
rregTM | 0 | – | 3.8982 | 11.9688 | 181.88 | 0.39 | 1.956 | 170.17 | 1.832 | 2.548
| 5 | – | 3.8941 | 11.8648 | 179.92 | 0.63 | 1.960 | 166.74 | 1.824 | 2.445
Expt. |  |  | 3.8544a | 11.8175a | 175.57a | 0.55b | 1.940 | 166.78 | 1.786 | 2.471

aPowder neutron diffraction at temperature of 5 KCava  _et al._ (1990).
bSingle crystal neutron scatteringCasalta  _et al._ (1994).

Figure 2: Comparison of the $ac$-plane full-breathing branch (highest branch
of $\Delta 1$) and the 3D full-breathing branch (highest branch of $\Sigma 1$)
for YBa2Cu3O6, calculated from PBE, PBE+U (6 eV), PBE+D4+U (6 eV), r2SCAN,
r2SCAN+U (5 eV) and r2SCAN+D4+U (4 eV) methods, with the experimental data
(open circles). The Brillouin zone and high-symmetry k-points are shown in
Fig. 1. The phonon results are obtained with supercell interatomic forces
calculated with the same DFA (+U+D4) methods as for geometry relaxations, as
detailed in Table 1. Figure 3: Same as Fig. 2, but calculated results are from
TM, TM+U (5 eV), revTM, revTM+U (5 eV), rregTM, rregTM+U (5 eV), and
r2SCAN+D4+U (4 eV) methods. Due to the complicated band crossing, the bare TM
results are not shown for the $ac$-plane full-breathing branch.

Table 1 compares various properties calculated from various methods to
available experimental values for YBa2Cu3O6. The bare DFA results are compared
with those with Hubbard U or/and vdW corrections. For the bare DFAs considered
here, they all overestimate the lattice constants and underestimate the
magnetic moments, in different degrees. In particular, PBE overestimates the
lattice constants the most and at the same time is not able to stabilize the
correct G-type AFM ground state, while r2SCAN and the TMs at the meta-GGA
level can stabilize the G-type AFM ground state, and improve over PBE for
structural properties. In addition, notable difference can be observed in the
performances of these different meta-GGAs. TM gives the closest structural
properties but notably underestimates the magnetic moments. The revTM
functional performs similarly with TM in structural properties but worse in
magnetic moments. The rregTM functional improves in magnetic moments but
worsens in structural properties. The r2SCAN functional in general predicts a
good combination of structural and magnetic properties. What’s more, TM and
revTM predict different magnetic moments for the G-type AFM unit cell and the
2$\times$2$\times$1 supercells used for interatomic force constants
calculations. This suggests that they could stabilize some spin-density-wave
states rather than the simple G-type AFM ground state.

Applying a Hubbard U correction to these DFAs will reduce the delocalization
error and increase the predicted magnetic moments. At the same time it also
improves the structural properties, which will not be true for LSDA+U as we
found previously since LSDA already underestimates the lattice constantsNing
_et al._ (2023). In general, due to self-interaction reduction, the meta-GGAs
require smaller U corrections than PBEWexler, Gautam, and Carter (2020). The
structural properties can be further improved with additional vdW corrections,
as demonstrated by the PBE+U+D4 and r2SCAN+U+D4 results. Similarly to SCAN,
r2SCAN captures intermediate range vdW interactionsSun  _et al._ (2016) while
PBE captures littlePeng and Perdew (2017b). Therefore, more vdW correction is
needed for PBE than r2SCAN, and correspondingly the structural improvements
from the vdW correction are greater when applied to PBE than to r2SCAN. As a
comparison and cross-check, almost the same structural and magnetic results
are achieved with r2SCAN+rVV10 Ning  _et al._ (2022) and r2SCAN+D4 Ehlert  _et
al._ (2021). This finding is consistent with previous reports that vdW
corrections are important for structural and energetic properties of ionic
solids Tao  _et al._ (2017), and that the combination of vdW and Hubbard U
corrections are important for the ground state electronic, structural and
energetic properties of transition metal monoxides Peng and Perdew (2017a). It
is also consistent with the underestimation of lattice constants in LSDA,
since LSDA tends to overbind weak bonds. Generally speaking, the improvements
of meta-GGAs over GGAs can be attributed to the power of satisfying more exact
constraintsFurness  _et al._ (2022); Kaplan, Levy, and Perdew (2023), but
additionally the self-interaction reduction and better capture of vdW
interactions at least contribute to a major part of their improvements. In
addition, for PBE and r2SCAN, although applying larger Hubbard U alone can
yield structural properties closer to experiment, it could over-localize d
electrons and predict too large magnetic moments, highlighting the different
physics of the two corrections and the necessity of appropriately applying
both together. Since the phonon dispersion is determined by the inter-atomic
forces, which depend sensitively on the ground state electronic structure and
equilibrium atomic positions, the improvements for structural and
electronic/magnetic properties from vdW and Hubbard U corrections bode well
for more accurate predictions of the lattice dynamics.

The phonon dispersion results for the most challenging and also experimentally
most interesting $ac$-plane and three-dimensional (3D) full-breathing
branches, as shown in Fig. 2 and Fig. 3, confirm our expectations. Both the
$ac$-plane and 3D full-breathing modes involve the Cu-O bond-stretching
vibrations within the Cu-O plane, and simultaneously the vibration of the
apical oxygen in the c direction. The difference is, the $ab$ Cu-O plane bond-
stretching vibrations happen along both $a$ and $b$ directions for the 3D
full-breathing modes, while only along either $a$ or $b$ direction for the
$ac$-plane (or $bc$-plane) full-breathing modes, as shown in Fig. 1c. Figure 2
compares the bare PBE and r2SCAN results and those from Hubbard U and D4 vdW
corrections. Both bare PBE and r2SCAN results are too soft for the two
challenging branches, and the PBE results are even softer, similar to the
previous nonmagnetic results from r2SCANNing  _et al._ (2023). This is
consistent with the fact that bare PBE cannot stabilize the AFM ground state
and overestimates lattice constants. With U and vdW corrections, notable
improvements are achieved for both PBE and r2SCAN, and the improvement is more
significant for PBE than for r2SCAN, although the r2SCAN+U+D4 results remain
closest to experiment. This is consistent with the observations and reasons
discussed above for the structural and magnetic properties. To summarize, due
to self-interaction reduction and capture of more intermediate range vdW
interactions, r2SCAN performs much better than PBE in magnetic, structural and
lattice dynamics properties of YBa2Cu3O6, and for that reason the improvements
from vdW and Hubbard U corrections are more significant for PBE than r2SCAN.

Figure 3 includes the bare TMs and Hubbard U corrected results, compared with
r2SCAN+U+D4 and experimental results. All the bare TMs results are too soft
compared to those from r2SCAN, consistent with the fact that r2SCAN gives the
best bare DFA predictions in basic magnetic and structural properties which is
critical for accurate lattice dynamics predictions. With Hubbard U correction,
the improvement is significant and the results are close to but still softer
than the r2SCAN+U+D4 results. Therefore r2SCAN generally has better
performance than TMs for YBa2Cu3O6 in structural, magnetic and lattice
dynamics properties. This could be attributed to the more nonlocal nature of
r2SCAN compared to TMs, since nonlocality is important for descriptions of
semiconductors and insulators. Although both are at the same meta-GGA level,
r2SCAN is more $\alpha$-dependent and thus displays more nonlocality, while
the TMs are more density gradient-dependent and less nonlocal.

##  IV Conclusions

In summary, we have extended our previous work to a first-principles
comparative study of the most challenging and experimentally important full-
breathing modes of YBa2Cu3O6. We achieve further understanding of both the
lattice dynamics side and the density functional side. By applying both
Hubbard U and the D4 vdW corrections to PBE and r2SCAN, notable improvements
are obtained for structural, electronic, magnetic, and phonon dispersion
predictions of YBa2Cu3O6. The improvements from the combined corrections are
more significant for PBE than for r2SCAN. With the improvements, PBE+U+D4
gives much better full-breathing phonon frequencies, closer but still softer
compared to those from r2SCAN+U+D4 and experimental observations. Considering
the general self-interaction reduction and capturing more intermediate range
vdW inherent in r2SCAN over PBE, we demonstrate the importance of vdW
interactions and SIC in accurate YBa2Cu3O6 lattice dynamics from first-
principles, which in turn contributes to the major reason for the superior
overall performance of r2SCAN over PBE.

In addition, for the family of Tao-Mo meta-GGAs, all the bare DFA results are
too soft compared to those from r2SCAN. With similar Hubbard U corrections,
improvements are notable but still not enough to be as good as r2SCAN with
corrections. Therefore r2SCAN generally has better performance than TMs for
YBa2Cu3O6 in structural, magnetic and lattice dynamics properties, which we
attribute to the more nonlocal nature of r2SCAN compared to TMs. Nevertheless,
the TMs could perform betterKaplan and Perdew (2022) for doped or gapless
systems such as YBa2Cu3O7Pintschovius  _et al._ (2004); Zhang  _et al._
(2020a); Heid  _et al._ (2009), as implied by their good performances in
surface, vacancy, and magnetic properties for metalsJana, Patra, and Samal
(2018); Jana, Sharma, and Samal (2019), which could be further studied in the
future.

Note that, even with the best results we can achieve from r2SCAN+U+D4, there
still exists noticeable softening for the tested full-breathing branches,
especially for the peak at $\textbf{k}\sim(0.2,0,0)$ along the experimental
$ac$-plane full-breathing branch. These residual discrepancies could imply
extra physics or effects we have not included yet. Recent research efforts
have renewed interest in the role of electron-phonon coupling in the mechanism
of high-temperature superconductivity in cuprates Ahmadova  _et al._ (2020).
The findings in the current work provide insights for future first-principles
investigations on cuprates, including phonon anomaliesBohnen, Heid, and Krauss
(2003); Giustino, Cohen, and Louie (2008); Heid  _et al._ (2009), charge
inhomogeneity, cavity-phonon-magnon quasiparticle interactions Curtis  _et
al._ (2022), and phase competition, which in turn contribute to a better
understanding of cuprate high temperature superconducting materials.

###### Acknowledgements.

J.N. and J.S. acknowledge the support of the U.S. Office of Naval Research
(ONR) Grant No. N00014-22-1-2673, with which they designed the project. J.N.,
A.R. and J.P.P acknowledge support from Tulane University’s startup funds
(computations). J.P.P. acknowledges support from the National Science
Foundation under Grant. No. DMR-1939528. Computational work done at Tulane
University was supported by the Cypress Computational Cluster at Tulane and
the National Energy Research Scientific Computing Center. The work at
Northeastern University was supported by the US Department of Energy (DOE),
Office of Science, Basic Energy Sciences Grant No. DE-SC0022216 (modeling
complex states in materials). The work at Los Alamos National Laboratory was
supported by the U.S. DOE NNSA under Contract No. 89233218CNA000001 and by the
Center for Integrated Nanotechnologies, a DOE BES user facility, in
partnership with the LANL Institutional Computing Program for computational
resources. Additional support was provided by DOE Office of Basic Energy
Sciences Program E3B5. B.B. was supported by the Ministry of Education and
Culture (Finland) and by the LUT University INERCOM platform.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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