# Ab Initio Studies on Interactions in K3C60 under High Pressure

Jianyu Li  School of Science, Harbin Institute of Technology, Shenzhen,
518055, China  Zhangkai Cao  School of Science, Harbin Institute of
Technology, Shenzhen, 518055, China  Jiahao Su  School of Science, Harbin
Institute of Technology, Shenzhen, 518055, China  Ruipeng Wang  School of
Materials Science and Physics, China University of Mining and Technology,
Xuzhou, 221116, China  Haipeng Li  School of Materials Science and Physics,
China University of Mining and Technology, Xuzhou, 221116, China  Yusuke
Nomura  Department of Applied Physics and Physico-Informatics, Keio
University, Yokohama 223-8522, Japan  Xiaosen Yang<EMAIL_ADDRESS>Department of Physics, Jiangsu University, Zhenjiang, 212013, China.  Ho-Kin
Tang<EMAIL_ADDRESS>School of Science, Harbin Institute of
Technology, Shenzhen, 518055, China

###### Abstract

Fullerene solids doped with alkali metals (A3C60, A = K, Rb, Cs) exhibit a
superconducting transition temperature ($T_{c}$) as high as 40 K, and their
unconventional superconducting properties have been a subject of debate. With
application of high pressure on K3C60 and Rb3C60, the experiments demonstrate
the decrease of $T_{c}$. In this paper, we focus on K3C60 and derive the
structure of K3C60 under different pressures based on first-principles
calculations, exploring the trends of Coulomb interactions at various
pressures. By utilizing the Maximally Localized Wannier function approach,
Constrained Density Functional Perturbation Theory (cDFPT), and Constrained
Random Phase Approximation (cRPA), we construct a microscopic low-energy model
near the Fermi level. Our results strongly indicate that, in the K3C60 system,
as pressure increases, the effect of phonons is the key to intraorbital
electron pairing. There is a dominance of the phonon-driven superconducting
mechanism at high pressure.

###### pacs:

Valid PACS appear here

##  I Introduction

Since their discovery three decades ago, fullerenes have commanded substantial
scientific attention Stankevich  _et al._ (1984); Smalley (1997); Wang  _et
al._ (2022a, 2023). Notably, the intriguing hallmark of these molecules
resides in their distinctive geometric structure: C60 molecules exhibit the
symmetry of an icosahedral group, a symmetry class of paramount significance
within the realm of symmetric operations. The recent realization of monolayer
fullerene materials Hou  _et al._ (2022); Yu  _et al._ (2023); Tromer  _et
al._ (2022); Yu  _et al._ (2022); Peng (2022); Ying  _et al._ (2023); Dong
_et al._ (2023); Yuan  _et al._ (2023) further underscores the importance of
investigating fullerene materials’ physical properties, as these insights are
pivotal in unlocking their full technological potential.

After successfully synthesizing solid C60 (fullerene crystals), a series of
extensive solid-state experiments has been conducted, revealing numerous
significant physical properties Xiang  _et al._ (1992); Klein  _et al._
(1992); Hou  _et al._ (1995); Hebard  _et al._ (1993). For instance, fcc K3C60
demonstrates metallic behavior, with the resistivity increasing as the
temperature rises Xiang  _et al._ (1992). A distinct Drude peak is observed in
the optical conductivity, and photoemission studies have identified a finite
density of states near the Fermi level Chen  _et al._ (1991); Benning  _et
al._ (1993); Hesper  _et al._ (2000); Yang  _et al._ (2003). Pressure tuning
of light-induced superconductivity in K3C60 has also made significant progress
Cantaluppi  _et al._ (2018); Rowe  _et al._ (2023). Measurements of electron
spin resonance Tanigaki  _et al._ (1995); Robert  _et al._ (1998) and nuclear
magnetic resonance Tycko  _et al._ (1992); Stenger  _et al._ (1995);
Pennington and Stenger (1996) indicate a Pauli-like susceptibility behavior in
the normal state, which is nearly temperature-independent.

A groundbreaking achievement was made by Hebard et al. as they successfully
identified superconductivity in potassium-doped C60, notably characterizing a
superconducting transition at a critical temperature of $T_{c}$ = 18 K Hebard
_et al._ (1991). Subsequent investigations into the underlying superconducting
mechanisms have revealed resemblances to the Bardeen-Cooper-Schrieffer (BCS)
theory, supported by consistent experiments. These include the confirmation of
a full-gap $s$-wave pairing Zhang  _et al._ (1991a, b); Tycko  _et al._
(1992); Kiefl  _et al._ (1993), reduction in spin susceptibility within the
superconducting phase Stenger  _et al._ (1995), observation of the Hebel-
Slichter peak Hebel and Slichter (1959) in nuclear magnetic resonance (NMR)
Sasaki  _et al._ (1994); Stenger  _et al._ (1995), and the correlation between
lattice constant and $T_{c}$ Fleming  _et al._ (1991).

Moreover, there have been thorough examinations of pressure-induced
alterations in $T_{c}$ Sparn  _et al._ (1991); Schirber  _et al._ (1991); Wang
_et al._ (2022b) and lattice parameter Zhou  _et al._ (1992); Wang  _et al._
(2022b) within the K3C60 system. In non-Cs-doped C60 superconductors, the
observed fact of $T_{c}$ monotonically decreasing under lattice compression
has been attributed to the band widening caused by lattice contraction,
resulting in a reduction of the density of states at the Fermi level
Rosseinsky  _et al._ (1991); Fleming  _et al._ (1991); Sparn  _et al._ (1991);
Zhou  _et al._ (1992); Tanigaki  _et al._ (1992); Yildirim  _et al._ (1995).
In the context of the general reduction in $T_{c}$ under pressure, these
experimental findings provide the conditions for further exploration of the
superconducting mechanism in K3C60.

Given the intricate interplay between electronic correlations and complex
electron-phonon coupling (EPC), the mechanisms underpinning superconductivity
in fullerenes remain contentious. While initial perceptions categorized
fullerene superconductors as conventional BCS systems, the discourse has
expanded to encompass an array of unconventional theories. These include
propositions of polaron-driven superconductivity Takahashi  _et al._ (1992);
Tiwari  _et al._ (2009), local pairing facilitated by Jahn-Teller phonons and
Coulomb repulsion Han  _et al._ (2003a); Nomura  _et al._ (2016), the
intriguing concept of negative Hund’s coupling stabilized by EPC Capone  _et
al._ (2002); Nomura  _et al._ (2015), and even scenarios of pure electronic
pairing Jiang and Kivelson (2016), among other possibilities.

In this context, our present study initiates an $ab$ $initio$ exploration of
interactions within the K3C60 system under the influence of elevated
pressures. Utilizing experimentally acquired data on lattice constants under
various pressures Wang  _et al._ (2022b), we aim to employ a combination of
Density Functional Theory (DFT) and the model-calculation approach Kotliar
_et al._ (2006); Held (2007); Imada and Miyake (2010) to compute essential
interaction parameters. We will determine the effective Coulomb interaction
within the low-energy subspace using the constrained random phase
approximation (cRPA) Aryasetiawan  _et al._ (2004). Additionally, we will
calculate partially renormalized phonon frequencies and electron–phonon
couplings through Constrained Density Functional Perturbation Theory (cDFPT)
Nomura  _et al._ (2014); Nomura and Arita (2015). Our primary objective is to
unveil the alterations in physical properties, ranging from electronic
characteristics to phonon behavior, caused by pressure fluctuations. By
closely examining the pressure-induced modifications in various interaction
parameters, we strive to discern the most plausible superconducting mechanism
amid the changing pressure conditions.

The rest of this paper is outlined as follows. In Sec. II, we choose the
target band and show the low-energy models from $ab$ $initio$ calculations. In
Sec. III, we calculate the global band structure using DFT and construct
MLWO’s within the low-energy band subspace. Detailed computations are
presented in this section. We obtained parameter results for the Coulomb
interaction and electron-phonon interaction and analyzed their trends with
varying pressure. In Sec. IV, we discuss changes in electronic and phononic
properties resulting from pressure fluctuations and compare various
explanations for the unconventional superconducting mechanism in K3C60.
Finally, we present the conclusion in Sec. V.

##  II Method and Model

Figure 1: (Color online) (a) illustrates the atomic structure of K3C60, which
is set at a size of $2\times 2\times 2$, and the real-space Wannier function
displayed using VESTA software Momma and Izumi (2008). K atoms are depicted as
purple spheres, while C atoms are represented by brown spheres. Positive
isosurfaces are highlighted in yellow, while negative isosurfaces are shown in
blue. (b) Brillouin zone of the face-centered cubic (fcc) lattice, path:
$\Gamma-X-U-K-\Gamma-L-K-W-X$ as described in Ref. Setyawan and Curtarolo
(2010).

Close to the Fermi energy, $t_{1u}$ band is the focus of our investigation
which we call target band. We use a lattice Hamiltonian encompassing the
electron and phonon degrees of freedom specifically associated with the
$t_{1u}$ orbitals that has been formulated in Refs. Nomura  _et al._ (2012,
2015). Nearly all the excitation processes occur in $t$-subspace, the subspace
which the target bands span (For later use, we define $r$-subspace as the rest
of the Hilbert space). The Hamiltonian assumes the following general form:

$\displaystyle\mathcal{H}$
$\displaystyle=\sum_{ij\mathbf{k}\sigma}{\mathcal{H}_{ij}^{(0)}}(\mathbf{k})c_{i\mathbf{k}}^{\sigma\dagger}c_{j\mathbf{k}}^{\sigma}$
$\displaystyle\quad+\sum_{\mathbf{qkk}^{\prime}iji^{\prime}j^{\prime}}{\sum_{\sigma\sigma^{\prime}}{U_{ij,i^{\prime}j^{\prime}}}}(\mathbf{q})c_{i\mathbf{k}+\mathbf{q}}^{\sigma\dagger}c_{j^{\prime}\mathbf{k}^{\prime}}^{\sigma^{\prime}\dagger}c_{i^{\prime}\mathbf{k}^{\prime}+\mathbf{q}}^{\sigma^{\prime}}c_{j\mathbf{k}}^{\sigma}$
$\displaystyle\quad+\sum_{ij\mathbf{k}\sigma\mathbf{q}\nu}{\sum_{ij}{g_{ij}^{\nu}(}}\mathbf{k},\mathbf{q})c_{i\mathbf{k}+\mathbf{q}}^{\sigma\dagger}c_{j\mathbf{k}}^{\sigma}(b_{\mathbf{q}\nu}+b_{-\mathbf{q}\nu}^{\dagger})$
$\displaystyle\quad+\sum_{\mathbf{q}\nu}{\omega_{\mathbf{q}\nu}}b_{\mathbf{q}\nu}^{\dagger}b_{\mathbf{q}\nu},$
(1)

This Hamiltonian operates on the fcc lattice, where each lattice site
corresponds to a C60 molecule. The terms $\mathcal{H}^{(0)}$, $U$, $g$, and
$\omega$ respectively denote the one-body electron Hamiltonian, Coulomb
interaction, electron-phonon coupling, and phonon frequency. All these
quantities are indexed by Wannier orbitals $i$, $j$, $i^{\prime}$,
$j^{\prime}$, spin $\sigma$, $\sigma^{\prime}$, momentum $\mathbf{k}$,
$\mathbf{q}$, and the phonons are further distinguished by a branch index
$\nu$.

To ensure the short-ranged nature of hopping, we have constructed maximally
localized Wannier orbitals Marzari and Vanderbilt (1997) as the basis (Fig.
1(a)). These parameters are determined using state-of-the-art $ab$ $initio$
techniques, with particular emphasis on the utilization of cRPA Aryasetiawan
_et al._ (2004) for the calculation of electron-related terms
$U_{ij,i^{\prime}j^{\prime}}$ and the cDFPT Nomura  _et al._ (2014) for the
calculation of phonon-related terms $g_{ij}^{\nu}$ and
$\omega_{\mathbf{q}\nu}$, wherein the influence of high-energy bands is
incorporated into the parameter values.

The parameters $\mathcal{H}_{ij}^{(0)}(\mathbf{k})$ represent an on-site
energy $\left(\bf{k}=\mathbf{0}\right)$ and hopping integrals
$\left(\bf{k}\neq\mathbf{0}\right)$, which are described with the
translational symmetry as

$\displaystyle\mathcal{H}_{ij}^{(0)}(\mathbf{k})=\left<\phi_{i\mathbf{k}}\right|\mathcal{H}_{\mathrm{KS}}\left|\phi_{j\mathbf{0}}\right>,$
(2)

where $|\phi_{i\mathbf{k}}\rangle=c_{i\mathbf{k}}^{\dagger}|0\rangle$ and
$\mathcal{H}_{\mathrm{KS}}$ is the Kohn-Sham Hamilton.

To evaluate effective interaction parameters
$U_{ij,i^{\prime}j^{\prime}}(\mathbf{q})$, we convert
$U_{ij,i^{\prime}j^{\prime}}(\mathbf{q})$ into the onsite Coulomb repulsion
$U_{ij}$ and the exchange interaction $J_{ij}$ and calculate the screened
Coulomb interaction $W$ at the low-frequency limit. We first calculate the
noninteracting-polarization function $\chi$, excluding polarization processes
within the target bands. Note that screening by the target electrons is
considered when we solve the effective models so that we have to avoid double
counting of it when we derive the effective models. With the resulting $\chi$,
the $W$ interaction is calculated as $W=(1-v\chi)^{-1}v$, where $v$ is the
bare Coulomb interaction potential
$v\left(\mathbf{r},\mathbf{r}^{\prime}\right)=\frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}$.
We consider a decomposition of the total irreducible polarization $\chi^{0}$
into the one involving only the $t$-subspace electrons $\chi_{t}^{0}$ and the
rest $\chi_{r}^{0}$ Aryasetiawan  _et al._ (2004) :

$\chi^{0}=\chi_{t}^{0}+\chi_{r}^{0}.$

Note that $\chi_{r}^{0}$ contains not only the processes involving only the
$r$-subspace degrees of freedom but also the ones involving both the
$t$-subspace electrons and the $r$-subspace electrons. We define the partially
screened Coulomb interaction $W^{(p)}$ as

$\displaystyle W^{(p)}=\left(1-v\chi_{r}^{0}\right)^{-1}v=\epsilon_{r}^{-1}v$
(3)

The onsite Coulomb repulsion $U_{ij}$ and the exchange interaction $J_{ij}$
are evaluated as

$\displaystyle U_{ij}$ $\displaystyle=\iint
d\mathbf{r}d\mathbf{r}^{\prime}\left|\phi_{i\mathbf{0}}(\mathbf{r})\right|^{2}W^{(p)}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\left|\phi_{j\mathbf{0}}\left(\mathbf{r}^{\prime}\right)\right|^{2}$
$\displaystyle=\frac{4\pi
e^{2}}{N\Omega}\sum_{\mathbf{q}}\sum_{\mathbf{G}\mathbf{G}^{\prime}}\rho_{ii}(\mathbf{q}+\mathbf{G})W_{\mathbf{G},\mathbf{G}^{\prime}}^{(p)}(\mathbf{q})\rho_{jj}^{*}\left(\mathbf{q}+\mathbf{G}^{\prime}\right)$
(4)

and

$\displaystyle J_{ij}$ $\displaystyle=\iint
d\mathbf{r}d\mathbf{r}^{\prime}\phi_{i\mathbf{0}}^{*}(\mathbf{r})\phi_{j\mathbf{0}}(\mathbf{r})W^{(p)}\left(\mathbf{r},\mathbf{r}^{\prime}\right)\phi_{j\mathbf{0}}^{*}\left(\mathbf{r}^{\prime}\right)\phi_{i\mathbf{0}}\left(\mathbf{r}^{\prime}\right)$
$\displaystyle=\frac{4\pi
e^{2}}{N\Omega}\sum_{\mathbf{q}}\sum_{\mathbf{GG}^{\prime}}\rho_{ij}(\mathbf{q}+\mathbf{G})W_{\mathbf{G},\mathbf{G}^{\prime}}^{(p)}(\mathbf{q})\rho_{ij}^{*}\left(\mathbf{q}+\mathbf{G}^{\prime}\right),$
(5)

respectively, where $\Omega$ is the volume of the unit cell, and
$\rho_{ij}(\mathbf{q}+\mathbf{G})$ is given, with the Wannier-gauge Bloch
functions $\psi_{i\mathrm{k}}^{(w)}$, by

$\displaystyle\rho_{ij}(\mathbf{q}+\mathbf{G})=\frac{1}{N}\sum_{\mathbf{k}}\left\langle\psi_{i\mathbf{k}+\mathbf{q}}^{(w)}\left|e^{i(\mathbf{q}+\mathbf{G})\cdot\mathbf{r}}\right|\psi_{j\mathbf{k}}^{(w)}\right\rangle.$
(6)

To facilitate a comparative analysis with the cRPA results, we compute
interaction parameters using the unscreened case, representing the bare
Coulomb interaction. In order to differentiate it from the cRPA results, we
refer to this as the “bare” interaction.

In solids, the individual ion can be identified as the $\kappa$-th ion in the
$p$-th unit cell and the direction of the displacement $\alpha$ = $x$, $y$,
$z$. To evaluate effective electron-phonon-coupling interaction parameters, we
use

$\displaystyle
g_{ij}^{\nu}(\mathbf{k},\mathbf{q})=\sum_{\kappa\alpha}{\sqrt{\frac{\hbar}{2M_{\kappa}\omega_{\mathbf{q}\nu}}}}e_{\kappa}^{\left(p\right)\alpha}(\mathbf{q}\nu)\left.\langle\phi_{i\mathbf{k}+\mathbf{q}}\left|\frac{\partial
V^{\left(p\right)}(\mathbf{r})}{\partial
u_{\kappa}^{\alpha}(\mathbf{q})}\right|\phi_{j\mathbf{k}}\right.\rangle,$ (7)

where we employ the Wannier-gauge for the electrons, and the superscript $(p)$
indicates the partially renormalized quantities.

Here, we consider the real-phonon contributions the intraorbital density-
density-type interaction $U_{\rm ph}$, interorbital density-density-type
interaction $U_{\rm ph}^{\prime}$ and exchange-type interaction $J_{\rm ph}$ :

$\displaystyle U_{\rm ph}$
$\displaystyle=-\frac{1}{N_{\mathbf{q}}}\sum_{\mathbf{q}\nu}{\frac{2\left|\tilde{g}_{ii}^{(p)}(\mathbf{q},\nu)\right|^{2}}{\omega_{\mathbf{q}\nu}^{(p)}},}$
(8) $\displaystyle U_{\rm ph}^{\prime}$
$\displaystyle=-\frac{1}{N_{\mathbf{q}}}\sum_{\mathbf{q}\nu}{\frac{2\tilde{g}_{ii}^{(p)}(\mathbf{q},\nu)\tilde{g}_{jj}^{(p)*}(\mathbf{q},\nu)}{\omega_{\mathbf{q}\nu}^{(p)}},}$
(9) $\displaystyle J_{\rm ph}$
$\displaystyle=-\frac{1}{N_{\mathbf{q}}}\sum_{\mathbf{q}\nu}{\frac{2\tilde{g}_{ij}^{(p)}(\mathbf{q},\nu)\tilde{g}_{ij}^{(p)*}(\mathbf{q},\nu)}{\omega_{\mathbf{q}\nu}^{(p)}}}$
$\displaystyle=-\frac{1}{N_{\mathbf{q}}}\sum_{\mathbf{q}\nu}{\frac{2\tilde{g}_{ij}^{(p)}(\mathbf{q},\nu)\tilde{g}_{ji}^{(p)*}(\mathbf{q},\nu)}{\omega_{\mathbf{q}\nu}^{(p)}},}$
(10)

where the orbital dependences of $U_{\rm ph}$, $U_{\rm ph}^{\prime}$ and
$J_{\rm ph}$ do not exist by symmetry, and
$\tilde{g}_{ij}^{(p)}(\mathbf{q},\nu)$ is calculated as

$\displaystyle\tilde{g}_{ij}^{(p)}(\mathbf{q},\nu)=\frac{1}{N_{\mathbf{k}}}\sum_{\mathbf{k}}g_{ij}^{(p)\nu}(\mathbf{k},\mathbf{q})$
(11)

##  III RESULT

###  III.1 Calculation details

We construct maximally localized Wannier orbitals (MLWOs) (Fig. 1) from the
DFT $t_{1u}$ band, and calculate the model parameters in the lattice
Hamiltonian [Eq. (1)] as described in Methods. We employed the Local Density
Approximation (LDA) exchange-correlation functional with the parameterization
by Perdew-Burke-Ernzerhof and employed the Troullier-Martins conservative
pseudopotentials within the Kleinman-Bylander representation. The
configurations used to generate the pseudopotentials for C and K, are
$(2s)^{2}(2p)^{2}$ and $(3p)^{6}(4s)^{0}(3d)^{0}$, respectively. A partial
core correction was applied to the pseudopotential for the alkali metal K. The
unit cell volume data for K3C60 at different pressures were obtained from
experimental sources Wang  _et al._ (2022b). Calculations were performed using
the open-source software Quantum Espresso Giannozzi  _et al._ (2020, 2017,
2009) and RESPACK Fujiwara  _et al._ (2003); Nohara  _et al._ (2009); Nakamura
_et al._ (2008, 2009, 2016, 2021). We perform the structure optimization with
fixing the lattice constants to the values employed Wang  _et al._ (2022b) and
ignoring the orientational disorder of C60 molecules.

In DFT calculations, crystal lattice is defined as a type 2 Bravais lattice,
with the lattice parameter $a$ determined from Fig. 4(a). The calculation
includes 350 electronic bands. Electron occupation numbers are determined
using the ”smoothing” method with a Gaussian smoothing of 0.025 Ry. In our
Wannier orbital calculations, we have specified the following parameters: a
crystal cell with a multiplicity of 2, a computation for 3 Wannier orbitals,
and an energy window ranging from $E_{\mathrm{F}}-$0.5 to $E_{\mathrm{F}}+$0.5
eV ($E_{\mathrm{F}}$ is Fermi energy). We have employed a set of 6 initial
Gaussian functions as a foundational basis. Moreover, we have defined Gaussian
function coefficient matrices for distinct orbital types, such as $p_{x}$,
$p_{y}$, and $p_{z}$. For example, in the case of the $p_{x}$ orbital, the
coefficient matrix comprises four values: 1.0, $-$0.243, 0.243, and 0.243,
which represent the weights assigned to each Gaussian function utilized in
constructing the $p_{x}$ orbital. Analogously, $p_{y}$ and $p_{z}$ orbitals
possess analogous coefficient matrices for specifying their Gaussian function
weights. Additionally, we have established a path for symmetric
$\mathbf{k}$-point interpolation, encompassing specific high-symmetry points:
$\Gamma$ (0, 0, 0), $X$ (0.5, 0, 0.5), $U$ (0.625, 0.25, 0.625), $K$ (0.375,
0.375, 0.75), $L$ (0.5, 0.5, 0.5), $W$ (0.5, 0.25, 0.75) (Fig. 1(b)). Then, we
calculate the band structure for the optimized structure.

In order to maximize the utilization of computational resources, different
cutoff energies for wave functions and charge densities, as well as
$\mathbf{k}$-point sampling grids, were set because the calculation
requirements vary when computing Coulomb interactions using the cRPA method
and electron-phonon interactions using the cDFPT method. In the calculation of
effective parameters for Coulomb interaction, the wave function and charge
density were truncated with energy cutoffs of 36 Ry and 144 Ry, respectively.
A $\mathbf{k}$-point sampling grid of $5\times 5\times 5$ $\mathbf{k}$-points
was utilized. But in the calculation of effective parameters for electron-
phonon interactions, the cutoff energy for the wave functions is set to be 50
Ry, and we employ $4\times 4\times 4$ $\mathbf{k}$-points. The cDFPT
calculations are performed with $2\times 2\times 2$ $\mathbf{q}$-mesh.

###  III.2 Band structure

Figure 2: (Color online) Calculated $ab$ $initio$ electronic band structure of
fcc-K3C60 at different pressures. The pressure parameters are set as 0, 0.53,
1.42, 2.05, 2.81, 3.34, 3.74, 5.03 GPa. The horizontal axis is labeled with
the special points in the Brillouin zone: $\Gamma$ (0, 0, 0), $X$ (0.5, 0,
0.5), $U$ (0.625, 0.25, 0.625), $K$ (0.375, 0.375, 0.75), $L$ (0.5, 0.5, 0.5),
$W$ (0.5, 0.25, 0.75). Fermi energy is depicted as blue dashed lines. Figure
3: (Color online) Our calculated density of states (DOS) for the $t_{1u}$ band
of fcc-K3C60 at different pressures. The pressure parameters are set as 0,
0.53, 1.42, 2.05, 2.81, 3.34, 3.74, 5.03 GPa. Figure 4: (Color online) The
pressure dependence of (a) lattice constant $a$, (b) bandwidth $W$ and (c)
Fermi velocity $v_{\mathrm{F}}$. We fix the lattice constants to the
experimental values Wang  _et al._ (2022b) and calculate the bandwidth of the
$t_{1u}$ band by Quantum Espresso Giannozzi  _et al._ (2020, 2017, 2009). We
combine Eq. (14) to obtain the Fermi velocity from the energy band results.

The C60 solid possesses a three-dimensional crystal structure Heiney  _et al._
(1991); Fischer and Heiney (1993) owing to the relatively isotropic shape of
the C60 molecule. The centers of these molecules form an fcc lattice.
Considering that undoped C60 constitutes a molecular solid, an insightful
initial approach to comprehend its electronic structure is to examine the
molecular limit. In this limit, numerous molecular orbitals degenerate due to
the high symmetry inherent to the molecule itself: specifically, the C60
molecule boasts the fivefold-degenerate HOMO, threefold-degenerate LUMO, and
threefold degenerate LUMO+1 orbitals, denoted as $h_{u}$, $t_{1u}$ and
$t_{1g}$ orbitals, respectively, based on their symmetry properties. In the
context of a solid, these orbitals acquire a dispersion as a result of
electron transfers taking place between the molecules. However, owing to the
relatively small magnitude of the transfer integral, the bandwidth of each
orbital remains narrow (typically around $\sim 0.5$ $\rm{eV}$). Consequently,
there is minimal overlap observed between the bands associated with different
molecular orbitals. Meanwhile, the C60 solid manifests as a band insulator in
which the LUMO $t_{1u}$ band remains unoccupied, while the HOMO $h_{u}$ band
is fully populated.

In the fcc K3C60, each alkali atom donates about one electron into the
$t_{1u}$ band, so the $t_{1u}$ band becomes half-filled. As a result, K3C60 is
no longer an insulator; instead, it exhibits partial metallic characteristics.
Fig. 2 illustrates the computed band structures of fcc
$\mathrm{K}_{3}\mathrm{C}_{60}$ at various pressure levels within the
framework of LDA. These compounds exhibit shared characteristics in their band
structures, particularly notable are the presence of narrow bands in proximity
to the Fermi level, which are well-isolated from other bands. Within the
K3C60, there exists a set of threefold-degenerate states, known as the
$t_{1u}$ band, which occupies a prominent position in the vicinity of the
Fermi level. This distinctive feature renders them particularly suitable for
selection as target bands in the construction of an effective model.

We show in Fig. 3 the calculated density of states (DOS) of the $t_{1u}$ band
for fcc-K3C60. DOS configuration profile shows only relatively small change,
in which the profile generally obeys the DOS behavior of the Fermi gas. This
suggests we can use the Fermi gas model to quantify Fermi parameters. For the
Drude model, we define the density of states $N(E)$, Fermi energy
$E_{\mathrm{F}}$ and Fermi velocity $v_{\mathrm{F}}$, respectively:

$\displaystyle N(E)$
$\displaystyle=\frac{2V}{(2\pi)^{2}}(\frac{2m^{*}}{\hbar^{2}})^{3/2}E^{1/2},$
(12) $\displaystyle E_{\mathrm{F}}$
$\displaystyle=\hbar^{2}\frac{k_{\mathrm{F}}^{2}}{2m^{*}},$ (13)
$\displaystyle v_{\mathrm{F}}$
$\displaystyle=\hbar\frac{k_{\mathrm{F}}}{m^{*}}.$ (14)

We find that $N(E)\propto v_{\mathrm{F}}$ and $E_{F}\propto
v^{2}_{\mathrm{F}}$, demonstrating a consistent alignment between theoretical
and computational trends. Increasing pressure causes a reduction in the
lattice constant of K3C60 (Fig. 4(a)), leading to an upward shift of the Fermi
level and an increase in the Fermi velocity. Additionally, alterations in
lattice constant impact the electronic band structure, resulting in a wider
bandwidth and influencing the density of states. Both bandwidth $W$ and Fermi
velocity $v_{\mathrm{F}}$ exhibit monotonic increase as the lattice constant
decreases , $W$ rises from 0.51 eV to 0.76 eV (Fig. 4(b)), representing an
approximate 50% increase, and $v_{\mathrm{F}}$ rises from 1.42 $\times
10^{6}$m/s to 1.58 $\times 10^{6}$m/s (Fig. 4(c)), representing an approximate
11% increase.

###  III.3 Coulomb interaction parameters of K3C60

Figure 5: (Color online) (a)-(c) $U$, $U^{\prime}$ and $J$ with different
screening levels [unscreened (bare) and constrained RPA (cRPA)]. For bare and
cRPA $U$, $U^{\prime}$ and $J$ values, the units are given in eV. (d)
$\epsilon_{M}$ is the cRPA-macroscopic-dielectric constant in Eq. (15). Figure
6: (Color online) Diagonal terms of of the inverse dielectric matrix for the
all $\bf q$ points $\epsilon_{\mathbf{GG}}^{-1}\left(\mathbf{q},\omega\right)$
at the first frequency $\omega$ = 0. The horizontal axis is limited within the
range of 0 to 0.5 Bohr-1, where it represents $|\mathbf{q}+\mathbf{G}|$.
(Inset): The horizontal axis represents the entire range of
$|\mathbf{q}+\mathbf{G}|$.  Figure 7: (Color online) Pressure dependence of
(a) the average of the on-site effective Coulomb repulsion $\bar{U}$, (b) the
on-site effective exchange interaction $J$, (c) the average of the off-site
effective Coulomb repulsion between neighboring sites $\bar{V}$, and (d) the
correlation strength $(\bar{U}-\bar{V})/W$, which are derived within the cRPA
method.

In this section, we analyze the effective Coulomb interaction parameters for
K3C60. Specifically, we focus on the parameters $U=U_{ii}(\mathbf{0})$,
$U^{\prime}=U_{ij}(\mathbf{0})$, and $J=J_{ij}(\mathbf{0})$, where $i\neq j$,
$U_{ij}(\mathbf{R})$ in Eq. (4) and $J_{ij}(\mathbf{R})$ in Eq. (5). $U$ is
the intraorbital Coulomb interaction, $U^{\prime}$ is the interorbital
interaction and $J$ is the exchange interactions. Fig. 5 shows our calculated
interaction parameters $U$, $U^{\prime}$ and $J$ with two screening levels
(bare and cRPA). It can be observed that the values of effective interaction
parameters decrease due to the screening process. For instance, at 0 GPa, the
bare Coulomb repulsion for K3C60 is approximately 3.32 eV, but considering the
cRPA screening effects, this value decreases to around 0.83 eV.

Fig. 6 shows the inverse dielectric matrix $\left\\{\epsilon_{\mathrm{{\bf
GG}}}^{-1}(\mathbf{q},\omega)\right\\}$ and the macroscopic dielectric
constant was defined as

$\displaystyle\epsilon_{M}^{\mathrm{cRPA}}=\lim_{\mathbf{Q}\rightarrow
0,\omega\rightarrow
0}\frac{1}{\epsilon_{\mathbf{GG}}^{\mathrm{cRPA}^{-1}}(\mathbf{q},\omega)},$
(15)

where $\omega$ is the frequency, $\mathbf{Q}=\mathbf{q}+\mathbf{G}$, with
$\mathbf{q}$ being the wave vector in the first Brillouin zone, and
$\mathbf{G}$ being the reciprocal lattice vector. Fig. 5(d) present our
calculated cRPA-macroscopic-dielectric constant $\epsilon_{M}^{\mathrm{cRPA}}$
in Eq. (15). The diagonal elements of the inverse dielectric matrix
demonstrate lower values with increasing pressure (Fig. 6), eventually
converging to 1 with increasing $|\mathbf{q}+\mathbf{G}|$ (Fig. 6(Inset)).

Fig. 7 summarizes the results of the cRPA calculations: the on-site Coulomb
repulsion $\bar{U}$ averaged over the MLWOs derived from the target band, the
on-site exchange interaction $J$, the off-site interaction $\bar{V}$ averaged
over the nearest-neighbor sites, and the ratio $(\bar{U}-\bar{V})/W$ which
measures the correlation strength in the system. Note that the net interaction
is estimated as $\bar{U}-\bar{V}$, based on the analysis of the extended
Hubbard model. Throughout the entire process, the relationship $U^{\prime}\sim
U-2J$ remains valid. As pressure increases, all parameters exhibit varying
degrees of reduction, with the on-site Coulomb repulsion being notably
affected by pressure. Specifically, $\bar{U}$ decreases from 0.80 eV to 0.50
eV, representing a decrease of approximately 40%. The value of $J$ undergoes a
relatively minor change, decreasing from 0.034 eV to 0.029 eV, corresponding
to a decrease of approximately 15%. Additionally, $\bar{V}$ decreases from
0.24 eV to 0.16 eV, approximately declining by 31%. $\bar{U}$ is approximately
three times $\bar{V}$ and twenty times $J$. $\bar{U}$ and $\bar{V}$ decrease
almost synchronously with increasing pressure, while the rate of decrease in
$J$ with pressure is slower than that of $\bar{U}$ and $\bar{V}$. In terms of
ratios, $\bar{U}$/$J$ decreases from 23.4 to 17.0, while $\bar{V}$/$J$ only
decreases from 7.0 to 5.7.

In general, a small value of $J$ tends to favor low-spin states, consistent
with experimental observations. It is worth mentioning that there is a
proposal suggesting that the Jahn-Teller coupling may exert greater influence
than the Hund’s rule coupling $J$, and when coupled with a sufficiently large
$U$, lead to the emergence of superconductivity. We will discuss this
pertinent topic in Sec. IV. The ratio $(\bar{U}-\bar{V})/W$, which measures
the correlation strength in the system, shows simple monotonic decrease with
applying pressure. It is noteworthy that at 0 GPa, the value of
$(\bar{U}-\bar{V})/W$ is approximately 1, suggesting the potential for C60
superconductors to be strongly correlated electronic systems. The results at 0
GPa are consistent with the $(\bar{U}-\bar{V})/W\sim 1$ Nomura  _et al._
(2012) exhibited by other C60 superconductors. However, as pressure increases,
this value gradually decreases, indicating a departure from a strongly
correlated electronic system.

###  III.4 Electron-Phonon Interactions and Phonon Frequencies

The phonon modes associated with K3C60 that have the potential to interact
with the $t_{1u}$ electrons can be categorized into distinct classes,
including libration modes, intermolecular modes, optical modes involving
alkali cations and C${}_{60}^{3-}$ anions, and intramolecular modes. It has
been demonstrated that, among these modes, the interactions of libration,
intermolecular, and alkali-ion modes with the $t_{1u}$ electrons are
relatively weak in comparison to those of the intramolecular modes Gunnarsson
(1997); Christides  _et al._ (1992); Antropov  _et al._ (1993); Ebbesen  _et
al._ (1992); Burk  _et al._ (1994a, b). Therefore, our primary focus lies on
investigating the characteristics of intramolecular phonons.

Given the high symmetry of the C60 molecule, only specific types of
intramolecular phonon modes can couple with the $t_{1u}$ electrons due to
symmetry considerations. If we consider the ideal icosahedral symmetry of the
isolated C60 molecule, a total of $60\times 3-6=174$ intramolecular
vibrational modes exist (where the subtraction of six modes pertains to the
molecule’s translational and rotational degrees of freedom). These modes are
further categorized into two $A_{g}$ modes, one $A_{u}$ mode, three $T_{1g}$
modes, four $T_{1u}$ modes, four $T_{3g}$ modes, five $T_{3u}$ modes, six
$G_{g}$ modes, six $G_{u}$ modes, eight $H_{g}$ modes, and seven $H_{u}$
modes.

The $A_{g(u)}$ modes exhibit no degeneracy, whereas the $T_{1g(u)}$ and
$T_{3g(u)}$ modes, the $G_{g(u)}$ modes, and the $H_{g(u)}$ modes possess
three-fold, four-fold, and five-fold degeneracies, respectively. Among the
intramolecular modes, only the phonon modes with $A_{g}$ and $H_{g}$
symmetries exhibit finite electron-phonon couplings to the $t_{1u}$ electrons
Varma  _et al._ (1991); Lannoo  _et al._ (1991).

Figure 8: (Color online) Partially renormalized electron-phonon coupling
parameter of different phonon modes at $\Gamma$ point calculated by cDFPT
method. The phonon modes are as follows: (a) $H_{g}$(1)$\sim$(8), (b)
$A_{g}$(1) and $A_{g}$(2). Figure 9: (Color online) Pressure dependence of
effective phonon-mediated interaction. $U_{\rm ph}$ is the intraorbital
density-density-type interaction, $U^{\prime}_{\rm ph}$ is the interorbital
density-density-type interaction, $J_{\rm ph}$ is the exchange-type
interaction, respectively. Table 1: Partially renormalized phonon frequencies
of different modes at $\Gamma$ point calculated by cDFPT method. The unit is
in $\mathrm{cm}^{-1}\left(1\ \mathrm{eV}=8065.54\mathrm{~{}cm}^{-1}\right)$.

Pressure (GPa) |  $H_{g}$(1) |  $H_{g}$(2) |  $H_{g}$(3) |  $H_{g}$(4) |  $H_{g}$(5) |  $H_{g}$(6) |  $H_{g}$(7) |  $H_{g}$(8) |  $A_{g}$(1) |  $A_{g}$(2)
---|---|---|---|---|---|---|---|---|---|---
0 | 259.41 | 427.28 | 685.73 | 779.26 | 1114.59 | 1273.56 | 1402.96 | 1531.60 | 504.82 | 1503.33
0.53 | 260.77 | 429.27 | 685.52 | 781.82 | 1117.53 | 1277.27 | 1407.12 | 1534.45 | 509.91 | 1504.94
1.42 | 262.92 | 432.06 | 685.25 | 785.49 | 1121.59 | 1282.41 | 1413.03 | 1538.64 | 514.54 | 1507.33
2.05 | 264.71 | 434.33 | 684.98 | 788.66 | 1124.99 | 1287.02 | 1418.17 | 1542.50 | 518.17 | 1509.58
2.81 | 266.85 | 437.14 | 684.52 | 792.61 | 1129.16 | 1292.71 | 1424.41 | 1547.37 | 522.46 | 1512.30
3.34 | 267.89 | 438.62 | 684.22 | 794.29 | 1130.64 | 1294.10 | 1427.68 | 1549.32 | 525.02 | 1514.18
3.74 | 268.91 | 439.71 | 684.05 | 796.28 | 1132.94 | 1297.92 | 1430.22 | 1552.10 | 527.33 | 1515.02
5.03 | 271.29 | 450.40 | 686.52 | 777.20 | 1104.26 | 1256.63 | 1374.34 | 1506.26 | 520.53 | 1453.92

Table 1 summarizes our calculated partially renormalized phonon frequencies of
the all modes at $\Gamma$ point under different pressures. The high phonon
frequencies up to $\sim 1500\mathrm{~{}cm}^{-1}(\sim 0.2$ $\rm{eV})$ can be
ascribed to the stiff C$-\mathrm{C}$ bonds and the lightness of the carbon
atoms.

Note that these frequencies are inputs for the low-energy solvers and thus can
not be directly compared with the experimentally observed frequencies. In
general, the electron-phonon coupling of the individual mode is not large (see
Fig. 8), while the accumulation of the contributions leads to the total
electron-phonon coupling of $\lambda\sim 0.5-1.0$ Gunnarsson (2004).

In Fig. 8, we can visually observe the trends of various phonon modes as they
change with pressure. With increasing pressure, the majority of phonon modes
exhibit a decreasing trend in their intensities, with the exception of the
$H_{g}$(1) and $H_{g}$(8) modes. This observed trend aligns with the overall
decrease in phonon coupling strengths as the pressure increases.

Fig. 9 summarizes the values of the static parts. We also find that the
relation $U_{\rm ph}^{\prime}\sim U_{\rm ph}-2J_{\rm ph}$ holds well. The
negative values of $U_{\rm ph}$, $U_{\rm ph}^{\prime}$, and $J_{\rm ph}$
indicate that the interactions are attractive. Therefore, they will compete
with the repulsive onsite Coulomb interactions. It is evident that an increase
in pressure leads to a reduction in the absolute magnitudes of $U_{\rm ph}$
and $J_{\rm ph}$. The weakening trend is consistent for intra-orbital density-
density-type interactions and exchange-type interactions. Additionally, the
magnitude of $U_{\rm ph}^{\prime}$ decreases to approximately 3 GPa before
undergoing a sign reversal, transitioning from negative to positive. This sign
change indicates a gradual transition from attraction to repulsion in the
inter-orbital density-density-type interactions. As for the density-density
channel, since the Coulomb interaction $U$ for the $t_{1u}$ electrons (see
Fig. 7(a)) is approximately five times larger in magnitude than electron-
phonon coupling $U_{\rm ph}$, the repulsive Coulomb interaction dominates over
the phonon mediated attraction. However, remarkably, the situation changes for
the exchange type interaction: at 0 GPa the absolute values of $\left|J_{\rm
ph}\right|\sim 0.05$ $\mathrm{eV}$ are larger than those of the Hund’s
coupling $J\sim 0.034$ $\mathrm{eV}$ (see Fig. 7(b)). It indicates that,
effectively, negative exchange interactions are realized. Thus the systems
will obey the inverted Hund’s rule, which favors the low-spin state rather
than the high-spin state. This fact naturally explains the origin of the low-
spin state observed in the Mott-insulating phase Ganin  _et al._ (2010).

On the other hand, the enhancement of the negative $J_{\rm ph}$ due to the
strong coupling between the Jahn-Teller modes and the $t_{1u}$ electrons. As
is already mentioned in the discussion above, the Jahn-Teller $H_{g}$ modes
are the non-density-type electron-phonon coupling and $A_{g}$ modes are
density-type electron-phonon coupling. It shows the $H_{g}$ modes contributes
to $J_{\rm ph}$, the $A_{g}$ modes do not contribute. We focus our attention
on the behavior of the $H_{g}$ and $A_{g}$ modes, as they play key roles in
these interactions. Notably, $A_{g}$ modes predominantly contribute to
$|U_{\rm ph}|$, and their trends exhibit substantial similarity. Conversely,
$H_{g}$ modes contribute primarily to $|J_{\rm ph}|$, and it is evident that
the trends of $H_{g}$(2)$\sim$$H_{g}$(7) modes are largely consistent with
$|J_{\rm ph}|$, although the intensities of $H_{g}$(1) and $H_{g}$(8) modes
increase as pressure decreases. Given that increasing pressure does not alter
the relationship between $U_{\rm ph}^{\prime}$ and $U_{\rm ph}-2J_{\rm ph}$,
it can be inferred that the contributions of $H_{g}$(1) and $H_{g}$(8) modes
become more prominent as pressure increases compared to other modes. This
leads to a relatively smaller reduction in $J_{\rm ph}$ compared to $U_{\rm
ph}$, resulting in the reversal of the inter-orbital density-density-type
interaction $|U_{\rm ph}^{\prime}|$ from attraction to repulsion.

###  III.5 Effective interactions

Figure 10: (Color online) Pressure dependence of effective interaction $U_{\rm
eff}(U+U_{\rm ph})$, $U^{\prime}_{\rm eff}(U^{\prime}+U^{\prime}_{\rm ph})$
and $J_{\rm eff}(J+J_{\rm ph})$. $U[U_{\rm ph}]$,
$U^{\prime}\left[U^{\prime}_{\rm ph}\right]$, and
$J\left[J_{\mathrm{ph}}\right]$ are the intraorbital, interorbital, and
exchange components of the cRPA Coulomb [cDFPT phonon-mediated] interactions,
respectively.  Figure 11: (Color online) Pressure dependence of the comparison
of effective interaction. $\bar{U}[\bar{U}_{\rm ph}]$, and
$J\left[J_{\mathrm{ph}}\right]$ are the on-site effective Coulomb repulsion
and exchange components of the cRPA Coulomb [cDFPT phonon-mediated]
interactions, respectively.

Fig. 10 shows the pressure dependence of the effective interactions between
the $t_{1u}$ electrons ($U_{\rm eff}=U+U_{\rm ph}$, $U^{\prime}_{\rm
eff}=U^{\prime}+U^{\prime}_{\rm ph}$, $J_{\rm eff}=J+J_{\rm ph}$), which are
given by the sum of the Coulomb $U$, $U^{\prime}$, $J$ and the retarded
phonon-mediated interactions $U_{\rm ph}$, $U^{\prime}_{\rm ph}$ and $J_{\rm
ph}$. In the context of K3C60 subjected to variable external pressure
conditions, several notable observations have emerged. Firstly, it has been
observed that both $U_{\rm eff}$ and $U^{\prime}_{\rm eff}$ remain positive
across the entire range of pressures studied, exhibiting substantial
variations as their values decrease from 0.7 eV to 0.45 eV. Notably, these two
parameters exhibit a highly synchronized trend, demonstrating a strong degree
of synchronicity. Secondly, the effective exchange interaction parameter,
$J_{\rm eff}$, consistently takes negative values under all pressure
conditions, with its variation confined to a narrow range, specifically within
the limits of $-$0.016 eV to $-$0.011 eV. It is pertinent to note that $U_{\rm
eff}$ and $U^{\prime}_{\rm eff}$ exhibit pronounced repulsive tendencies
throughout the studied pressure range, which are greater than the attractive
tendencies exhibited by $J_{\rm eff}$.

Furthermore, the transition of $J_{\rm eff}$ to negative values is attributed
to the contributions from phonon. Within the parameter regime where $J_{\rm
eff}$ becomes negative, as $U^{\prime}_{\rm eff}$ is observed to marginally
surpass $U_{\rm eff}$, indicating that the interorbital repulsion is slightly
greater than the intraorbital repulsion across all pressure conditions. This
scenario favors electron pairing within the orbitals, consistent with the
conclusions drawn regarding intraorbital pairs Naren  _et al._ (2008); Capone
_et al._ (2009, 2001, 2004); Nomura  _et al._ (2015).

Fig. 11 illustrates the pressure-induced variations in the strengths of the
Coulomb interaction and the electron-phonon interaction parameters. In
particular, it compares the behavior of on-site effective interaction
($\bar{U}$) with that of the phonon-mediated interaction ($U_{\rm ph}$).
Notably, $\bar{U}$ exhibits a slower rate of reduction compared to $U_{\rm
ph}$. This implies that as pressure increases, the intraorbital electron
repulsion decreases at a slower rate than the intraorbital electron-phonon
attraction. On the other hand, the exchange interaction parameter ($J$)
exhibits a lower sensitivity to pressure variations. Unlike $\bar{U}$, it does
not display as pronounced changes with increasing pressure.

##  IV discussion

We have carried out a fully $ab$ $initio$ study of all the interactions of a
multiorbital Hubbard model coupled with phonons are computed from first
principles with only the information on the atomic positions. In our study, we
unveil changes in electronic and phononic properties induced by pressure
fluctuations and compare several explanations for the unconventional
superconducting mechanism in K3C60. Referring to the experimental trend of
decreasing $T_{c}$ in K3C60 as pressure increases Wang  _et al._ (2022a);
Sparn  _et al._ (1991), we make the following observations:

1) As pressure increases, the strength of the Coulomb interaction consistently
surpasses that of the electron-phonon interaction, albeit both exhibit
decreasing trends. Furthermore, phonon mediated intaraction decrease in
strength more rapidly than Coulomb interaction. The concurrent weakening of
the superconducting phase and phonon interactions aligns well with the
proposition that local pairing with Jahn-Teller phonons assisted by Coulomb
repulsion predominates as the superconducting mechanism Nomura  _et al._
(2016); Gunnarsson  _et al._ (1995); Han  _et al._ (2003b) in K3C60. This
highlights the impact of phonon interactions on the superconducting mechanism,
even though the phonon interaction strength in K3C60 is relatively modest, due
to the dominant role of strong Coulomb interactions in suppressing kinetic
energy.

2) Through band structure calculations, we assert that the primary source of
electrons within K3C60 relies on the $t_{1u}$ band. However, the density of
states plot in Fig. 3 reveals that the $t_{1u}$ band is entirely contributed
by electrons from the carbon atoms, independent of potassium atoms. This
observation indicates that the hypothesis suggesting that potassium ion phonon
modes exert a strong attractive force on electrons on the C60, sufficient to
overcome Coulomb repulsion and induce superconductivity Zhang  _et al._
(1991c), is incorrect. The role of doping potassium ions merely transitions
C60 from an insulator to a metal but has minimal influence on the
superconducting mechanism.

3) In K3C60, when considering only the Coulomb interaction in isolation, we
observe that $U$ is greater than $U^{\prime}$. However, upon incorporating
electron-phonon interactions, denoted as $U_{\rm ph}$ and $U^{\prime}_{\rm
ph}$, we discern that $U_{\rm eff}$ becomes smaller than $U^{\prime}_{\rm
eff}$. This observation forms the foundational explanation for electron
pairing. Despite the relatively modest strength of phonon interactions within
K3C60, the robust Coulomb interactions effectively curtail the kinetic energy
of electrons. This, in turn, facilitates the interorbital tunneling of
electron pairs through pair-hopping interactions, which is an indispensable
mechanism for the emergence of superconductivity. Throughout the entire
process, both $U_{\rm eff}$ and $U^{\prime}_{\rm eff}$ decrease with
increasing pressure, with $U_{\rm eff}$ consistently being less than
$U^{\prime}_{\rm eff}$. This indicates that interorbital Coulomb interactions
are greater than intraorbital interactions, implying that electron pairing
within the orbitals persists even as Coulomb interactions decrease.
Furthermore, the $A_{g}$ mode give $U_{\rm ph}$ = $U^{\prime}_{\rm ph}$, and
$H_{g}$ mode give negative $J_{\rm ph}$ and negative $J_{\rm eff}$. Hence
$H_{g}$ modes assume a pivotal role in determining the relationship where
$U^{\prime}_{\rm eff}$ = $U_{\rm eff}-2J_{\rm eff}>U_{\rm eff}$, both modes
are indispensable in shaping the dynamics of the superconducting mechanism.

##  V Conlusion

In conclusion, our comprehensive $ab$ $initio$ study of a multiorbital Hubbard
model coupled with phonons under varying pressure conditions sheds light on
the intriguing behavior of the superconducting mechanism in K3C60. Under
increasing pressure, we observe significant changes in the system’s parameters
and theoretical concepts. As pressure increases, we observe that despite the
relatively smaller strength of phonon interactions, their impact on the
superconducting mechanism in K3C60 is significant. This is due to the local
pairing with Jahn-Teller phonons Nomura  _et al._ (2016); Gunnarsson  _et al._
(1995); Han  _et al._ (2003b), which is further assisted by the suppressive
effect of strong Coulomb interactions on kinetic energy.

Furthermore, our study reveals that interorbital Coulomb interactions remain
greater than intraorbital interactions throughout the pressure increase,
indicating that electron pairing within the orbitals persists despite
decreasing Coulomb interactions. Pressure-induced changes in the electronic
and phononic properties of the system, along with variations in key parameters
like $U_{\rm eff}$, $U^{\prime}_{\rm eff}$ and $J_{\rm eff}$, highlight the
intricate interplay between different factors in the superconducting mechanism
of K3C60. This study advances our understanding of unconventional
superconductivity in complex materials and underscores the importance of
considering both electronic and phononic contributions in such systems.

## Acknowledgement

This work is supported by the National Natural Science Foundation of China
(Grant No. 12204130), Shenzhen Start-Up Research Funds (Grant No. HA11409065),
the HITSZ Start-Up Funds (Grant No. X2022000). Yusuke Nomura acknowledges the
support by JSPS KAKENHI (No. JP23H04869 and JP21H01041). Haipeng Li
acknowledges the support by the Fundamental Research Funds for the Central
Universities of China (Grant No. 2019ZDPY16) and Basic Research Project of
Xuzhou City (Grant No. KC22043).

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