# Effects of Non-local Pseudopotentials on the Electrical and Thermal
Transport Properties of Aluminum: A Density Functional Theory Study

Qianrui Liu (刘千锐)  HEDPS, CAPT, College of Engineering and School of Physics,
Peking University, Beijing 100871, P. R. China  Mohan Chen (陈默涵)
<EMAIL_ADDRESS>(Corresponding author) HEDPS, CAPT, College of
Engineering and School of Physics, Peking University, Beijing 100871, P. R.
China  AI for Science Institute, Beijing 100080, P. R. China

###### Abstract

Accurate prediction of electron transport coefficients is crucial for
understanding warm dense matter. Utilizing the density functional theory (DFT)
with the Kubo-Greenwood formula is widely used to evaluate the electrical and
thermal conductivities of electrons. By adding the non-local potential
correction term that appears in the dynamic Onsager coefficient and using two
different norm-conserving pseudopotentials, we predict the electrical and
thermal conductivities of electrons for liquid Al (1000 K) and warm dense Al
(0.2 to 10 eV). We systematically investigate the effects of non-local terms
in the pseudopotentials and the frozen-core approximation on the
conductivities. We find that taking into account the non-local potential
correction and validating the frozen core approximation is essential for
accurately calculating the electrical and thermal transport properties of
electrons across a wide range of temperatures.

##  I Introduction

Warm dense matter (WDM) is a state of matter characterized by extremely high
temperature, high density, and high pressure that commonly exists and plays a
significant role in the interiors of giant planets [1, 2] and inertial
confinement fusion [3]. Consequently, research on WDM has emerged as one of
the most prominent areas in high energy density physics. Nonetheless,
generating and analyzing WDM in laboratory settings presents a significant
challenge, making computational modeling an indispensable tool for
understanding this unique state of matter. Furthermore, given that WDM
comprises partially ionized electrons and strongly coupled ions, incorporating
quantum mechanics is essential for accurately simulating and investigating its
properties. Nevertheless, even with the use of modern high-performance
computing, it is still challenging to simulate the structural, dynamical, and
transport properties of WDM at extremely high temperatures [4, 5].

In recent years, various first-principles methods, such as the Kohn-Sham
density functional theory (KSDFT) [6, 7, 8], the orbital-free density
functional theory (OFDFT) [9, 10], and the path integral Monte Carlo [11, 12,
13], have been developed to investigate WDM across a wide range of
temperatures and pressures [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24].
Additionally, several KSDFT-based methods have been proposed to overcome the
challenge of simulating WDM, including the extended first-principles molecular
dynamics [25, 26, 27, 28], the stochastic DFT [29, 30], and the mixed
deterministic-stochastic DFT [31, 32, 33], etc. Within the framework of DFT,
the exchange-correlation (XC) functionals considering temperature effects have
been developed [34, 35] to yield more accurate results for the matter under
extreme conditions. In addition, machine-learning-based molecular dynamics
[36, 37], known for their ability to efficiently simulate large systems while
retaining high accuracy, have recently been widely utilized to investigate WDM
with larger systems or longer trajectories [38, 39, 40, 23, 41, 24, 33].

Several methods have been developed to describe the ion-electron interactions
within the framework of DFT. First, adopting norm-conserving pseudopotentials
(NCPPs) [42] in DFT calculations provides an accurate yet efficient way to
describe the behavior of valence electrons while neglecting the fast-varying,
highly localized core electron wavefunctions. Among the NCPPs, the Kleinman-
Bylander (KB) approach [43] employs a single local radial potential and a few
separable $l$-dependent non-local projectors. Recently, Hamann [44, 45] has
developed optimized norm-conserving Vanderbilt (ONCV) pseudopotentials with
two projectors, which enhances the accuracy and lower cutoff energies when
compared to the traditional NCPPs [46, 47]. Second, the ultrasoft
pseudopotentials (USPPs) [48] remove the norm-conserving condition and provide
a smoother potential to describe the electron-ion interaction, allowing for a
lower energy cutoff and thus reducing computational costs. Third, another
notable approach is the projector augmented-wave (PAW) method [49], which
allows recovery of all-electron quantities while still benefiting from reduced
computational costs compared to the all-electron methods. Although the USPPs
and PAW methods can reduce the energy cutoff and improve efficiency, they come
with additional complexity in the implementation of certain calculations, such
as density-functional perturbation theory (DFPT) [50]. On the other hand,
NCPPs remain widely used due to their simplicity and ease of implementation.
In this work, we adopt the NCPPs.

The electrical and thermal transport properties are essential in fields such
as laser heating [51], hydrodynamic instability [52], and metal-nonmetal
transitions [53]. Both electrons and ions contribute to the electrical and
thermal transport, but the former becomes more important in WDM. [24] In this
regard, we focus on calculating the electrical and thermal conductivities
driven by electrons. To do this, KSDFT-based molecular dynamics and the Kubo-
Greenwood (KG) method are employed, which have been successfully applied to
various systems, including liquid metals [54], silica [55], and plastics [56],
etc. In particular, various studies have adopted this method to study
transport properties of WDM and obtained valuable results [20, 21, 22, 24].

Notably, when employing the KG formula, the presence of non-local potentials
in NCPPs leads to a correction term in this formula (vide infra).
Unfortunately, some previous works have overlooked the influences of these
corrections on the computed properties [57, 58, 24]. To the best of our
knowledge, when the NCPPs are employed to study the conductivities of WDM, the
effects of non-local potential corrections that appear in the Kubo-Greenwood
formula are still inconclusive. In this regard, we have implemented the non-
local potential corrections with the plane wave basis sets and under periodic
boundary conditions in the ABACUS package[59, 60, 32]. Next, we select Al at
two densities (2.35 and 2.70 g/cm$3$) and the temperature ranges from 0.086 to
10 eV. We first compute the electrical and thermal conductivities of electrons
for aluminum (Al) with the revised formula. Next, we thoroughly investigate
the effects of using two different pseudopotentials with or without the non-
local potential corrections on the computed conductivities. Finally, We
analyze the density of states, the decomposed electrical conductivity, and the
Lorents number of Al across a wide range of temperatures and elucidate our
findings.

The paper is organized as follows. In Sec. II, we provide a brief introduction
to the Kubo-Greenwood formula and explain the formulas to implement the non-
local potential corrections with the plane wave basis set. Additionally, we
outline the setup for calculating the electrical and thermal conductivities of
Al. In Sec. III, we present our results and analysis for Al. The concluding
remarks are shown in Sec. IV.

##  II Methods

###  II.1 Electrical conductivity and thermal conductivity

To calculate the electrical and thermal conductivities, we first need to
compute the dynamic Onsager coefficients $L_{mn}(\omega)$ ($m,n=1,2$) using
the Kubo-Greenwood formula. The equation is as follows:

$\displaystyle L_{mn}(\omega)=(-1)^{m+n}\frac{2\pi e^{2}}{3\omega\Omega}$ (1)
$\displaystyle\times\sum_{ij\alpha\mathbf{k}}W(\mathbf{k})\left(\frac{\epsilon_{i\mathbf{k}}+\epsilon_{j\mathbf{k}}}{2}-\mu\right)^{m+n-2}|\langle\Psi_{i\mathbf{k}}|\hat{v}_{\alpha}|\Psi_{j\mathbf{k}}\rangle|^{2}$
$\displaystyle\times[f(\epsilon_{i\mathbf{k}})-f(\epsilon_{j\mathbf{k}})]\delta(\epsilon_{j\mathbf{k}}-\epsilon_{i\mathbf{k}}-\hbar\omega),$

where $\omega$ represents the frequency, $e$ is the elementary charge,
$\Omega$ denotes the volume of the cell, $\mu$ refers to the chemical
potential. $\Psi_{i\mathbf{k}}$ represents the wave function of the $i$-th
band with $\mathbf{k}$ being a point in the first Brillouin zone, and the
corresponding eigenvalue is $\epsilon_{i\mathbf{k}}$. $\hat{v}_{\alpha}$
denotes the $\alpha$-th component of the velocity operator $\hat{\mathbf{v}}$.
The Fermi-Dirac distribution function of electrons is $f$, and $\delta$ is the
delta function. In practice, the delta function is approximated by a Gaussian
function:

$\delta(E)\approx\frac{1}{\sqrt{2\pi}\Delta E}\exp\left(-E^{2}/(2\Delta
E^{2})\right),$ (2)

where $\Delta E$ determines the width of the Gaussian function. In this work,
we set the full width at half maximum ($\mathrm{FWHM}=2.3548\Delta E$) to 0.1
eV.

With the above formulas, the electrical conductivity $\sigma$ and thermal
conductivity $\kappa$ of electrons respectively take the form of

$\sigma=\lim_{\omega\to 0}L_{11}(\omega),$ (3) $\kappa=\lim_{\omega\to
0}\frac{1}{e^{2}T}\left(L_{22}(\omega)-\frac{L_{12}^{2}(\omega)}{L_{11}(\omega)}\right).$
(4)

In addition, to allow for extrapolation to zero frequency, we employ specific
functions to fit the frequency-dependent lines, which can be found in
Supplementary Material. It is worth noting that the choices of different
fitting functions do not substantially affect the results when the fitting
range is close to the zero frequency $\omega$=0.

###  II.2 Non-local potential correction

The velocity operator [61] in quantum mechanics, denoted as
$\hat{\mathbf{v}}$, is defined as

$\hat{\mathbf{v}}=\frac{i}{\hbar}\left[\hat{H},\hat{\mathbf{r}}\right],$ (5)

Here, $\hat{H}$ represents the Hamiltonian operator, and $\hat{\mathbf{r}}$
represents the position operator. For Hamiltonian that contains the non-local
pseudopotential, an additional commutator enters the formalism as [61, 62]

$\hat{\mathbf{v}}=\frac{\hat{\mathbf{p}}}{m_{e}}+\frac{i}{\hbar}\left[\hat{V}_{\mathrm{NL}},\hat{\mathbf{r}}\right],$
(6)

where $\hat{\mathbf{p}}$ is the momentum operator, $m_{e}$ is the mass of
electrons, and $\hat{V}_{\mathrm{NL}}$ depicts the non-local pseudopotential
operator. When dealing with the Hamiltonian that only contains the local
potential terms, the velocity operator can be represented by the momentum
operator alone. However, for systems with non-local potential terms, the
correction term in the second part of Eq. (6) cannot be ignored.

In this work, we adopt the plane-wave basis set with the periodic boundary
conditions and the $k$-point sampling method [63], with which the electronic
wave function can be written as

$|\Psi_{i\mathbf{k}}\rangle=\sum_{\mathbf{G}}c_{i\mathbf{k}}(\mathbf{G})|\mathbf{k+G}\rangle,$
(7)

where $\mathbf{G}$ depicts the plane wave basis and $\\{c_{i\mathbf{k}}\\}$
are the coefficients of plane wave basis sets for band $i$. We denote the wave
vector as $\mathbf{q}=\mathbf{k+G}$, which satisfies
$\sum_{\mathbf{q}}|\mathbf{q}\rangle\langle\mathbf{q}|=I$ with $I$ being an
identity matrix. $c_{i\mathbf{k}}(\mathbf{G})$ takes the formula of

$c_{i\mathbf{k}}(\mathbf{G})=\langle\mathbf{q}|\Psi_{i\mathbf{k}}\rangle=\Psi_{i\mathbf{k}}(\mathbf{q}).$
(8)

Thus, the non-local correction term in the velocity matrix has the form of

$\displaystyle\langle\Psi_{i\mathbf{k}}|[\hat{V}_{\mathrm{NL}},\hat{\mathbf{r}}]|\Psi_{j\mathbf{k}}\rangle$
(9)
$\displaystyle=\sum_{\mathbf{qq^{\prime}}}\langle\Psi_{i\mathbf{k}}|\mathbf{q}\rangle\langle\mathbf{q}|[\hat{V}_{\mathrm{NL}},\hat{\mathbf{r}}]|\mathbf{q^{\prime}}\rangle\langle\mathbf{q^{\prime}}|\Psi_{j\mathbf{k}}\rangle,$
$\displaystyle=\sum_{\mathbf{qq^{\prime}}}\Psi_{i\mathbf{k}}^{*}(\mathbf{q})\langle\mathbf{q}|[\hat{V}_{\mathrm{NL}},\hat{\mathbf{r}}]|\mathbf{q^{\prime}}\rangle\Psi_{j\mathbf{k}}(\mathbf{q^{\prime}}).$

Since the position operator $\hat{\mathbf{r}}$ is ill-defined in a periodic
system, the $[\hat{V}_{\mathrm{NL}},\hat{\mathbf{r}}]$ operator can be
calculated in the reciprocal space [64] as

$\langle\mathbf{q}|[\hat{V}_{\mathrm{NL}},\hat{\mathbf{r}}]|\mathbf{q^{\prime}}\rangle=(\nabla_{\mathbf{q}}+\nabla_{\mathbf{q^{\prime}}})V_{\mathrm{NL}}(\mathbf{q},\mathbf{q^{\prime}})$
(10)

To compute the above term, we need to first calculate the non-local
pseudopotential term in the plane wave basis, which is

$V_{\mathrm{NL}}(\mathbf{q},\mathbf{q^{\prime}})=\sum_{\kappa=1}^{N_{\mathrm{t}}}S^{\kappa}(\mathbf{q^{\prime}-q})V_{\mathrm{NL}}^{\kappa}(\mathbf{q,q^{\prime}}),$
(11)

where $N_{\mathrm{t}}$ is the number of atom types,
$V_{\mathrm{NL}}^{\kappa}(\mathbf{q,q^{\prime}})$ is the non-local
pseudopotential of the $\kappa$-th element, and the first part $S^{\kappa}$ is
the structure factor of the $\kappa$-th element, given by

$S^{\kappa}(\mathbf{G})=\sum_{j=1}^{N_{\kappa}}\exp(i\mathbf{G}\cdot\mathbf{\tau}_{\kappa
j}),$ (12)

where $N_{\kappa}$ denotes the number of $\kappa$-type atoms and
$\mathbf{\tau}$ represents the position of the atom. In this case, we find

$(\nabla_{\mathbf{q}}+\nabla_{\mathbf{q^{\prime}}})S^{\kappa}(\mathbf{q^{\prime}-q})=0.$
(13)

Furthermore, the second part is the non-local potential for the $\kappa$-th
element and can be calculated using the formula of

$V_{\mathrm{NL}}^{\kappa}(\mathbf{q,q^{\prime}})=\sum_{ll^{\prime}mm^{\prime}}D_{lm,l^{\prime}m^{\prime}}^{\kappa}\beta^{\kappa}_{lm}(\mathbf{q})\beta^{\kappa*}_{l^{\prime}m^{\prime}}(\mathbf{q^{\prime}}),$
(14)

where $\beta_{lm}^{\kappa}(\mathbf{q})$ is the non-local projector of NCPPs
expanded with the plane wave basis. Additionally, the atom type is $\kappa$
and the coefficient matrix of projectors is $D_{lm,l^{\prime}m^{\prime}}$. In
detail, $\beta_{lm}^{\kappa}(\mathbf{q})$ is written as

$\beta_{lm}^{\kappa}(\mathbf{q})=\frac{4\pi(-i)^{l}}{\sqrt{\Omega}}f_{l}^{\kappa}(q)Y_{lm}(\hat{\mathbf{q}}).$
(15)

Here, the radial part is obtained by

$f_{l}^{\kappa}(q)=\int_{0}^{+\infty}F_{l}^{\kappa}(r)j_{l}(qr)r^{2}\mathrm{d}r.$
(16)

In this equation, $rF_{l}^{\kappa}(r)$ is a one-dimensional non-local
projector read from the pseudopotential file, $j_{l}$ depicts the spherical
Bessel function and $Y_{lm}(\hat{\mathbf{q}})$ is the spherical harmonic
function.

Based on the above formulas, the key part of the non-local potential
correction term can be expressed as

$\displaystyle(\nabla_{\mathbf{q}}+\nabla_{\mathbf{q^{\prime}}})V_{\mathrm{NL}}(\mathbf{q,q^{\prime}})$
(17)
$\displaystyle=\sum_{\kappa=1}^{N_{\mathrm{t}}}S^{\kappa}(\mathbf{q^{\prime}-q})(\nabla_{\mathbf{q}}+\nabla_{\mathbf{q^{\prime}}})V_{\mathrm{NL}}^{\kappa}(\mathbf{q,q^{\prime}})$
$\displaystyle=\sum_{\kappa=1}^{N_{\mathrm{t}}}S^{\kappa}(\mathbf{q^{\prime}-q})\sum_{ll^{\prime}mm^{\prime}}D_{lm,l^{\prime}m^{\prime}}^{\kappa}$
$\displaystyle\times\Big{[}\mathbf{g}^{\kappa}_{lm}(\mathbf{q})\beta^{\kappa*}_{l^{\prime}m^{\prime}}(\mathbf{q^{\prime}})+\beta^{\kappa}_{lm}(\mathbf{q})\mathbf{g}^{\kappa*}_{l^{\prime}m^{\prime}}(\mathbf{q^{\prime}})\Big{]},$

where $\mathbf{g}_{lm}^{\kappa}(\mathbf{q})$ represents the gradient of
$\beta_{lm}^{\kappa}(\mathbf{q})$ with respect to $\mathbf{q}$ and has the
following form

$\displaystyle\mathbf{g}_{lm}^{\kappa}(\mathbf{q})$
$\displaystyle=\frac{4\pi(-i)^{l}}{\sqrt{\Omega}}$ (18)
$\displaystyle\times\left[q^{l}Y_{lm}(\hat{\mathbf{q}})\nabla_{\mathbf{q}}\left(\frac{f^{\kappa}_{l}(q)}{q^{l}}\right)+\frac{f^{\kappa}_{l}(q)}{q^{l}}\nabla_{\mathbf{q}}\left(q^{l}Y_{lm}(\hat{\mathbf{q}})\right)\right].$

It should be noted that the gradient of $q^{l}Y_{lm}(\hat{\mathbf{q}})$ is
used instead of the gradient of $Y_{lm}(\hat{\mathbf{q}})$ to avoid the
singularity at $\mathbf{q=0}$.

In total, the velocity matrix
$\langle\Psi_{i\mathbf{k}}|\hat{v}_{\alpha}|\Psi_{j\mathbf{k}}\rangle$ in Eq.
(1), can be evaluated in the reciprocal space with plane wave basis as

$\displaystyle\sum_{\mathbf{qq^{\prime}}}\Psi_{i\mathbf{k}}^{*}(\mathbf{q})\left(\frac{\hbar\mathbf{q}}{m_{e}}\delta(\mathbf{q-q^{\prime}})+\frac{i}{\hbar}\langle\mathbf{q}|[\hat{V}_{\mathrm{NL}},\mathbf{r}]|\mathbf{q^{\prime}}\rangle\right)\Psi_{j\mathbf{k}}(\mathbf{q^{\prime}}),$
(19)

where the second term is calculated through Eqs. (10) and (17).

###  II.3 Computational Details

With the above formulas implemented, we performed a systematic study on the
electrical and thermal conductivities of Al at temperatures ranging from 0.2
to 10 eV. The densities were chosen to be 2.35 and 2.70 $\mathrm{g/cm^{3}}$.
First, to obtain atomic configurations of Al at various temperatures, we
utilized Born-Oppenheimer molecular dynamics (BOMD) simulations based on
OFDFT. The Wang-Teter (WT) kinetic energy density functional [65] was employed
and has been validated in previous works [23, 24]. During BOMD simulations,
the Nosé-Hoover thermostat [66, 67] was adopted in the NVT ensemble. We
utilized 256 atoms in a cell for temperatures up to 5 eV. For higher
temperatures, we used 108 atoms. The energy cutoff for the electron density
was set to 80 Ry. The simulations were performed for 4000 steps with time
steps being $\frac{r_{s}}{60\bar{v}}$, where $r_{s}$ was the Weigner-Setz
radius and $\bar{v}=\sqrt{T/m}$ was the average velocity of atoms. Here, $T$
is the temperature and $m$ is the mass of an Al atom. From the BOMD
trajectories, we selected 5 atomic configurations to calculate the electrical
and thermal conductivities of electrons in the Al system, which has been
tested to be converged.

Second, we employed the Kohn-Sham density functional theory (KSDFT) and the
Kubo-Greenwood formula, as shown in Eqs. (1), (3), and (4), to calculate the
Onsager coefficients and subsequently the electrical and thermal
conductivities of Al. We used an energy cutoff of 50 Ry to describe the wave
functions in KSDFT calculations. In particular, we utilized two NCPPs, which
are named NC11 (11 valence electrons) and NC3 (3 valence electrons). On the
one hand, the NC11 pseudopotential, generated using the optimized norm-
conserving Vanderbilt pseudopotential method via the ONCVPSP package [44, 45],
had 11 valence electrons and a cutoff radius of 0.50 Å. On the other hand, the
NC3 pseudopotential was generated through the PSlibrary package [68] using the
Troullier-Martins method [69]. In addition, the NC3 pseudopotential contained
3 valence electrons, and the cutoff radius was set to 1.38 Å. To ensure
convergence, we selected a larger number of KS bands at higher temperatures,
ensuring that the occupation of the last band was smaller than 1e-5. For
temperatures of 1000 K and 0.2 eV, we used a $5\times 5\times 5$ $k$-point and
$3\times 3\times 3$ $k$-point mesh [63], respectively. For temperatures
ranging from 0.5 to 1.0 eV, we employed a $2\times 2\times 2$ $k$-point mesh,
and we only considered the $\Gamma$ point at higher temperatures.

Finally, both OFDFT and KSDFT simulations employed the Perdew-Burke-Ernzerhof
(PBE) [70] XC functional. A previous work [24] had demonstrated that using
either PBE or LDA [7] XC did not significantly affect the results.
Additionally, we also tested a temperature-dependent XC functional called KSDT
[34, 35] and found that it did not substantially influence the results within
the temperature and density range studied in this work (see Supplementary
Material). All of the above simulations were performed using the ABACUS v3.2
package [71, 59].

##  III Results and Discussion

###  III.1 Liquid Al

Figure 1: (a) Frequency-dependent electrical ($\sigma$ in
$10^{6}~{}\rm{Sm^{-1}}$) and (b) thermal conductivities ($\kappa$ in
$\rm{10^{2}~{}Wm^{-1}K^{-1}}$) of liquid Al at $T$ = 1000 K. The NC11 label
refers to a norm-conserving pseudopotential with 11 valence electrons. The
corr. label depicts the line obtained by considering the non-local
pseudopotential correction, while the fitting label depicts the line obtained
by a fitting method. Here, the electrical conductivities are fitted with a
Drude model, while the thermal conductivities are linearly fitted.
Computational results from Recoules et al. [54] and Knyazev et al. [72], as
well as available experimental data at $\omega=0$ eV are shown for comparison
[73, 74, 75, 76].

Figs. 1(a) and (b) respectively illustrate the calculated frequency-dependent
electrical and thermal conductivities of liquid Al with and without the non-
local pseudopotential corrections. In addition, first-principles data from
previous works [54, 72] and experimental results are displayed for comparison.
All of the above data are obtained for liquid Al at 1000 K with a density of
2.35 $\mathrm{g/cm^{3}}$. Our calculations utilize a simulation cell
containing 256 atoms and a 5$\times$5$\times$5 shifted $k$-point mesh to
ensure convergence. We use the NC11 pseudopotential with 11 valence electrons
for liquid Al. In particular, to extrapolate the frequency-dependent
electrical and thermal conductivities to zero frequency and obtain the
transport properties, we employ the Drude formula to fit the frequency-
dependent (dynamic) electrical conductivity and use a linear function to fit
the thermal conductivity. The fitting range is from 0.15 to 1.00 eV. Note that
the experimental data of Mills et al. [73] and Rhim et al. [74] are obtained
by substituting 1000 K into the corresponding fitting equation, while the
results of Abdellah et al. [75] and Brandt et al. [76] are obtained by
linearly interpolating their results from the table to 1000 K.

Figure 2: (a) Electrical ($\sigma$ in $\rm{Sm^{-1}}$) and (c) thermal
conductivities ($\kappa$ in $\rm{10^{3}~{}Wm^{-1}K^{-1}}$) of liquid and warm
dense Al at a density of 2.35 $\mathrm{g/cm^{3}}$ and temperatures ranging
from 0.086 to 10 eV. The simulation cell contains 256 atoms for temperatures
no more than 5 eV and 108 atoms for higher temperatures. The NC3 and NC11
labels refer to two norm-conserving pseudopotentials with valence electrons
being 3 and 11, respectively. The core. label depicts the data obtained by
considering the non-local pseudopotential correction. (b) and (d) plot the
relative differences of $\sigma$ and $\kappa$, respectively. The label “NCx
diff.“ represents the relative difference computed by the formula of
$|V_{\mathrm{NCx}}-V_{\mathrm{NCx\ corr.}}|/V_{\mathrm{NCx\ corr.}}$, where
$x$ is 3 or 11 and $V$ is $\sigma$ or $\kappa$. Additionally, the term “NC3&11
diff.” depicts $|V_{\mathrm{NC3\ corr.}}-V_{\mathrm{NC11\
corr.}}|/V_{\mathrm{NC11\ corr.}}$.

Notably, we find both zero-frequency electrical and thermal conductivities
($\omega=0$) of Al obtained with the non-local potential correction (labeled
as NC11 corr. fitting) yield substantially smaller values than the ones
without the correction (labeled as NC11 fitting). In detail, the calculated
electrical conductivity at $\omega=0$ with the non-local potential correction
shown in Fig. 1(a) is $3.89\times 10^{6}$ $\mathrm{Sm^{-1}}$, which is 40.1%
lower than the value of $5.45\times 10^{6}$ $\mathrm{Sm^{-1}}$ without
correction. For comparison, the DC electrical conductivity ($\omega=0$)
measured by Abdellah et al. [75], Rhim et al. [74], and Brandt et al. [76]
from experiments are $4.07\times 10^{6}$, $3.99\times 10^{6}$, and $4.05\times
10^{6}$ $\mathrm{Sm^{-1}}$, respectively. We conclude that the value with the
non-local potential correction matches better with the experimental data. In
Fig. 1(b), the thermal conductivity at $\omega=0$ calculated with the non-
local potential correction (labeled as NC11 corr. fitting) is 96
$\mathrm{Wm^{-1}K^{-1}}$, which is 40.6% lower than the value of 135
$\mathrm{Wm^{-1}K^{-1}}$ without correction (labeled as NC11 fitting). On the
other hand, experimental measurements by Mill et al. [73], Rhim et al. [74],
and Brandt et al. [76] yield values of 103, 98, and 96
$\mathrm{Wm^{-1}K^{-1}}$, respectively. Again, we find the calculated data
with the non-local potential correction agrees well with the experimental
results.

We also compare the calculated results with previous computational works by
Recoules et al. [54], and Knyazev et al. [72]. In detail, Knyazev et al. [72]
used 3 valence electrons in a USPP to describe Al, and the non-local potential
corrections are not included in the results. More discussions are in the later
session. On the other hand, Recoules et al. [54] utilized the density
functional perturbation theory (DFPT) to account for the effects of non-local
potential in NCPPs and obtained better results compared to experimental
results. The above results demonstrate the importance of considering the non-
local potential correction in computed DC electrical and thermal
conductivities for liquid Al.

###  III.2 Warm dense Al with a density of 2.35 $g/cm^{3}$

Besides the non-local potential correction, the use of frozen core
approximation in pseudopotentials may substantially affect the computed
electrical and thermal conductivities of Al. Figs. 2(a) and (c) show the
calculated electrical and thermal conductivities of Al in terms of using two
different pseudopotentials, respectively. The temperatures range from 0.086 to
10 eV. The two used NCPPs are named as the NC3 and NC11 pseudopotentials.
Here, the NC3 and NC11 labels refer to two norm-conserving pseudopotentials
with valence electrons being 3 and 11, respectively. In addition, we compute
the electrical and thermal conductivities with and without the non-local
pseudopotential corrections. We have the two following findings in Figs. 2(a)
and (c).

Firstly, in the temperature range studied in Fig. 2(a), the computed
electrical conductivity decreases as temperature increases, which can be
explained by an increase in the collision frequency of electrons with elevated
temperatures. Conversely, Fig. 2(c) illustrates the computed thermal
conductivity becomes larger with rising temperatures. Secondly, comparing the
data with and without non-local potential corrections reveals that the
corrected values are systematically lower than those without corrections for
both electrical and thermal conductivities.

Figure 3:  (a-b) Electrical ($\sigma$ in $10^{6}~{}\rm{Sm^{-1}}$) and (c-d)
thermal conductivities ($\kappa$ in $\rm{10^{3}~{}Wm^{-1}K^{-1}}$) of liquid
and warm dense Al at a density of 2.70 $\mathrm{g/cm^{3}}$ and temperatures
ranging from 0.086 to 10 eV. The simulation cell contains 256 Al atoms for
temperatures no more than 5 eV and 108 atoms for higher temperatures. The NC11
labels refer to the norm-conserving pseudopotentials with valence electrons
being 11. The “corr.” term depicts the calculated data obtained by considering
the non-local pseudopotential correction. Calculation results from Knyazev et
al. [72, 77], Sjostrom et al. [21], and Witte et al. [22], as well as the
experimental results from Milchberg et al. [78], Sperling et al. [79], and
McKelvey et al. [80] are shown for comparison. PBE and HSE refer to two
different exchange-correlation functionals used in DFT calculations.

Figs. 2 (b) and (d) further illustrate the relative differences for the
calculated electrical and thermal conductivities, respectively. Interestingly,
we find the non-local potential corrections for the NC3 pseudopotential
exhibit a trend of becoming substantially larger at around 2 eV and higher
temperatures. This can be explained by the fact that the NC3 pseudopotential
has 8 fewer electrons than the NC11 pseudopotential in the frozen core
approximation; the missing 8 electrons in the NC3 pseudopotential affect the
computed conductivities at high temperatures, so the NC3 pseudopotential
approximates the ion-electron interactions worse than the NC11 pseudopotential
at high temperatures. Consequently, the non-local pseudopotential loses
transferability and suffers more errors at higher temperatures.

On the contrary, when the NC11 pseudopotential is used, we observe that the
non-local potential corrections tend to exhibit another trend of becoming
substantially smaller at around 2 eV and higher temperatures. Since the errors
introduced by an insufficient number of electrons are minimized, we attribute
this trend to the velocity operator shown in Eq. 6 comprising both the kinetic
energy term and non-local potential term of electrons, with the former one
becoming increasingly larger than the latter one as the temperature rises.

We also compare the relative differences of electrical and thermal
conductivities obtained by the two different pseudopotentials, i.e., NC3 and
NC11. At low temperatures below 2 eV, the two pseudopotentials yield similar
results, but as the temperature rises over 2 eV, the differences become
increasingly larger, indicating again that the invalidity of the frozen-core
approximation for the NC3 pseudopotential at higher temperatures.

###  III.3 Warm dense Al with a density of 2.7 $g/cm^{3}$

Figs. 3 (a) and (c) illustrate the computed temperature-dependent electrical
and thermal conductivities as obtained from the NC11 pseudopotential,
respectively. The temperature ranges from 0.2 to 10 eV while the density of Al
is chosen to be $\rho$=2.7 $\mathrm{g/cm^{3}}$. We also compute the electrical
and thermal conductivities by using the NC3 pseudopotential. In addition,
available experimental results are added for comparison. Although the trends
exhibited by the conductivities at the density of $\rho$=2.7 g/cm3 are similar
to those of $\rho$=2.35 g/cm3 shown in Fig. 2, there are some points worth
noting.

We compare our calculated data in Fig. 3 (a) with the experimental data
reported by Milchberg et al. [78] and Sperling et al. [79]. We observe that
the two experiments exhibit discrepancies in electrical conductivity. In fact,
accurately measuring the conductivity of WDM from experiments has always been
challenging. For example, the experiment of Milchberg et al. did not directly
measure the temperature [78] and it was re-estimated by Dharma-wardana et al.
[81] based on the degree of ionization calculated using KSDFT. In another
first-principles work by Mo et al. [17], the measured temperature from the
experiment of Sperling et al. is argued to be lower. In general, our
calculated electrical conductivities (NC11 corr.) are larger than those from
Sperling et al. [79], and we suspect the discrepancies may not be caused by
the insufficient number of electrons or the lack of non-local potential
corrections in the NC11 pseudopotential. The exchange-correlation functional
may cause certain errors [82, 22]. Meanwhile, our calculated electrical
conductivities align with the data from Milchberg et al. [78] when the
temperature is larger than 2 eV.  However, it is worth noting that Sperling et
al. and Milchberg et al. are ultrafast experiments, where electrons are heated
significantly faster than ions, and some studies suggested that they should be
analyzed with a two-temperature model [81, 83, 84].

Fig. 3(c) illustrates the thermal conductivities of Al. We compare our
calculated results with the experimental data from proton-heated warm dense Al
by McKelvey et al. [80]. Our results fall within the error bars of the
experimental data for Al densities ranging from 1.7 $\mathrm{g/cm^{3}}$ to 2.7
$\mathrm{g/cm^{3}}$.

Figs. 3 (b) and (d) compare the calculated results with previous first-
principles computational works by Knyazev et al. [72], Sjostrom et al. [21],
and Witte et al. [22]. Our results, incorporating the non-local potential
corrections and utilizing NC11 pseudopotential, are systematically lower than
the results obtained from Knyazev et al. To explain the differences, we notice
that a USPP with 3 valence electrons without non-local potential correction
was used for Al in the work of Knyazev et al. For comparison, when using the
NC3 and NC11 pseudopotential, the electrical and thermal conductivities
calculated without non-local potential corrections are systematically higher
than those with corrections, as respectively shown in Figs. 2(a) and (c) for
Al density being 2.35 g/cm3; the conclusion holds when the density increases
to 2.7 g/cm3. Thus, we suspect the discrepancies between our results and those
from Knyazev et al. may come from the correction term.

Furthermore, both works of Sjostrom [21] and Witte et al. [22] adopted the PAW
method, but their results differ, especially at temperatures higher than 2 eV.
On the one hand, Sjostrom et al. employed the PAW method with 3 valence
electrons for Al, and only the $\Gamma$ point was used with a cell containing
64 Al atoms. In addition, the electron densities used in KSDFT calculations to
yield Kohn-Sham orbitals were obtained from OFDFT calculations without the
self-consistent loop, suggesting that the electron densities were
approximated. Moreover, to compute the velocity operator, the transversal
expression was adopted. Here, the transversal expression means directly
substituting $\hat{\mathbf{v}}$ with $-i\hbar\nabla/m_{e}$ in Eq. 1,
considering the PAW method’s ability to recover all-electron wavefunctions.
Nonetheless, Dajdoš et al. [85] pointed out that PAW typically truncates the
one-center expansion, leading to inaccurate transversal expression.
Consequently, they proposed incorporating a term that describes the dipole
moments within the one-center sphere, known as the longitude form [85]. On the
other hand, Witte et al. employed the PAW method with 11 valence electrons.
The cell size and $k$-point sampling were checked to yield converged results.
Importantly, the longitude form [85, 86] with corrections was adopted, as the
different results of transversal and longitude forms were also shown by
Demyanov and Knyazev’s recent work et al. [87] These findings suggest that
although the PAW method is generally considered an all-electron method, the
truncation of partial waves introduces additional corrections that should not
be overlooked. [85]

Recently, Witte et al. [22] employed both PBE and HSE (hybrid functional)
exchange-correlation functionals. In general, the HSE functional yields
smaller electrical conductivity than PBE, but the results from the two
functionals converge when the temperature exceeds 4 eV. Comparing with these
PBE and HSE data, our results with the non-local potential correction and NC11
agree well with the PBE data at 0.2 eV and with the HSE data at higher
temperatures. As illustrated in Fig. 3(d), unlike electrical conductivities,
both PBE and HSE data from Witte et al. [22] lead to similar thermal
conductivities, and our results with the non-local potential correction and
the NC11 pseudopotential match well with these data. The above results
highlight the importance of incorporating the non-local potential corrections
when calculating the electrical and thermal conductivities of electrons.

###  III.4 Density of states

Fig. 4 shows the calculated density of states (DOS) of Al (2.35
$\mathrm{g/cm^{3}}$) at temperatures of 0.5, 5.0, and 10.0 eV, where $\mu$ is
the chemical potential. The DOS contributed by 2s and 2p electrons of Al are
labeled. In particular, we plot the DOS of Al by using two NCPPs (NC3 and
NC11). In this regard, the 2s and 2p electrons are absent in the NC3
pseudopotential with 3 valence electrons but are included in the NC11
pseudopotential with 11 valence electrons. Interestingly, we observe that the
2s and 2p electrons become more dispersed and shift towards the Fermi energy
as temperature increases, and this physical phenomenon can only be captured by
DFT calculations with the NC11 pseudopotential.

Additionally, we plot the Fermi-Dirac function for each temperature. We can
see that as temperatures increase, the slope of the Fermi-Dirac function
becomes smoother. By checking the occupations of these energy states, we find
at temperatures of 0.5 and 5 eV, the 2s and 2p orbitals of Al are fully
occupied, and the electrons of the 2p orbitals start to ionize at 10 eV. The
ionization degree of the 2p orbitals is 0.36% at the temperature of 10 eV, as
evaluated from the occupation number. This is further confirmed by seeing that
the Fermi-Dirac distribution function at 10 eV begins to deviate from 1 at the
peak position of the 2p orbitals. Meanwhile, the DOS of the scattering states
(3s3p and other higher-energy orbitals of Al) around the Fermi level shifts to
higher energies as temperature arises, which is captured by both NC3 and NC11
pseudopotentials. In conclusion, even though core electrons are almost non-
ionized at the temperature of 5 eV, the energy levels of 2s and 2p electrons
move towards higher energies and affect the computed conductivities,
suggesting that a pseudopotential with 11 valence electrons is a more
appropriate choice than the NC3 pseudopotential for warm dense Al.

Figure 4: Density of States (DOS) of warm dense Al at a density of 2.35
$\mathrm{g/cm^{3}}$ and temperatures of (a) 0.5, (b) 5.0, and (c) 10.0 eV.
NC11 and NC3 refer to two norm-conserving pseudopotentials with 11 and 3
valence electrons, respectively. The Fermi-Dirac function is plotted for each
temperature. Figure 5: Decomposed electrical conductivity
$\Sigma(\omega,\varepsilon)$ (as defined in Eq. 20) of warm dense Al with
respect to different energies $\varepsilon$ at temperatures ($T$) of (a) 0.5,
(c) 5.0, and (d) 10.0 eV. Here $\mu$ is the chemical potential. The inset (f)
shows the decomposed electrical conductivity contributed by the 2s and 2p
orbitals of Al. Besides, we plot the relative difference of
$\Sigma(\omega,\varepsilon)$ between results with and without the non-local
potential correction for temperatures of (b) 0.5, (d) 5.0, and (g) 10.0 eV.
The density of Al is 2.35 $\mathrm{g/cm^{3}}$ and the frequency $\omega$ is
chosen to be 0.2 eV. The area under each curve represents the zero-frequency
electrical conductivity. NC11 and NC3 refer to two norm-conserving
pseudopotentials with 11 and 3 valence electrons, respectively. The term
“corr.” depicts the non-local potential correction. “NCx diff.“ represents the
relative difference computed by the formula of
$|\Sigma_{\mathrm{NCx}}-\Sigma_{\mathrm{NCx\ corr.}}|/\Sigma_{\mathrm{NCx\
corr.}}$, where x is 3 or 11.

###  III.5 Decomposed electrical conductivity

We define a new quantity named decomposed electrical conductivity
$\Sigma(\omega,\varepsilon)$ to analyze the contributions of different
orbitals to the electrical conductivity. The formula of $\Sigma$ is

$\displaystyle\Sigma(\omega,\varepsilon)=\frac{2\pi
e^{2}}{3\omega\Omega}\sum_{ij\alpha\mathbf{k}}W(\mathbf{k})|\langle\Psi_{i\mathbf{k}}|\hat{v}_{\alpha}|\Psi_{j\mathbf{k}}\rangle|^{2}$
(20)
$\displaystyle\times[f(\epsilon_{i\mathbf{k}})-f(\epsilon_{j\mathbf{k}})]\delta(\epsilon_{j\mathbf{k}}-\epsilon_{i\mathbf{k}}-\hbar\omega)\delta(\epsilon_{i\mathbf{k}}-\varepsilon),$

which satisfies the relationship of

$\sigma(\omega)=\int\Sigma(\omega,\varepsilon)\mathrm{d}\varepsilon.$ (21)

Here $\sigma(\omega)$ is the frequency-dependent electrical conductivity. The
results of $\Sigma(\omega,\varepsilon)$ as computed by two different
pseudopotentials (NC3 and NC11) with and without the non-local potential
corrections are displayed in Fig. 5, where $\mu$ is the chemical potential.
Since conductivities are calculated by extrapolating dynamic conductivities at
small frequencies, we choose $\omega$=0.2 eV as a representative frequency to
investigate the effects of pseudopotentials and non-local potential
corrections. According to the above formulas, the area of each line in Fig. 5
represents the value of $\sigma(\omega)$.  As shown in Fig. 2, the
consideration of the non-local corrections and pseudopotentials with different
numbers of electrons results in different behaviors at low and high
temperatures. Here, we calculate $\Sigma(\omega,\varepsilon)$ at a temperature
of 0.5 eV to represent low temperatures, and at temperatures of 5.0 and 10.0
eV to represent high temperatures. As depicted in Fig. 4, the 2p electrons are
not ionized at 5.0 eV but begin to ionize at 10.0 eV. We have the following
findings.

On the one hand, at the low temperature of 0.5 eV, as shown in Fig. 5(a), we
observe that the decomposed electrical conductivity has peaks around the Fermi
surface, indicating that only electrons around the Fermi surface
($\varepsilon-\mu$=0) contribute to the electrical conductivity. This can be
understood by the term of
$f(\epsilon_{i\mathbf{k}})-f(\epsilon_{j\mathbf{k}})$ included in the formula
of $\Sigma(\omega,\varepsilon)$, and $\partial
f(\varepsilon)/\partial\varepsilon$ has the largest value at the peak. We also
find the distribution of $\Sigma(\omega,\varepsilon)$ using the NC11 and NC3
pseudopotentials is similar, indicating that the 2s and 2p electrons in
pseudopotentials barely affect the electrical conductivity of Al at 0.5 eV.
Meanwhile, the inclusion of non-local potential correction significantly
reduces the conductivity, highlighting the importance of considering this
correction.

Furthermore, Fig. 5(b) illustrates the relative differences between $\Sigma$
with and without the non-local potential correction at 0.5 eV. We observe the
relative difference of the NC3 pseudopotential increases with the rise of
eigenenergies of electrons, while that of NC11 decreases. This implies the
correction becomes more prominent for NC3 than NC11 as shown in Fig. 2(b)
because more electrons are excited to high energies with the rise of the
temperature.

On the other hand, as the temperature respectively rises to 5 eV and 10 eV in
Fig. 5(b) and (e), the peaks of $\Sigma$ shift towards higher energies because
the velocity matrix term becomes substantially larger at high-energy bands.

However, the electrical conductivities obtained from the NC3 and NC11
pseudopotentials exhibit substantial discrepancies when the energy
$\varepsilon-\mu$ is around the peak of distributions. While the density of
states (DOS) of Al calculated by NC3 and NC11 are similar in Fig. 4, the
variations in the velocity matrix values calculated with corresponding states,
whether with or without non-local potential corrections, contribute to these
discrepancies. This suggests that the scattering states of the two systems are
inherently different. Consequently, even though the 2p electrons of Al are
either still occupied at 5 eV or barely ionized at 10 eV, the total
conductivities show significant differences at these higher temperatures.
Panel (f) further confirms that the contributions from 2s and 2p electrons are
much smaller compared to the scattering states. These indicate that including
the 2s and 2p orbitals in the NCPP is critical in accurately calculating
correct scattering states and predicting the conductivities.

Besides, the relative difference of the NC11 pseudopotential continues to
decrease to a small value due to the predominance of the kinetic energy,
indicating that the correction becomes less significant with more excited
electrons and a shift of peak to higher energies as the temperature rises. In
contrast, for the NC3 pseudopotential, the relative differences are in general
larger than those from the NC11 pseudopotential, except when $\varepsilon-\mu$
is small.

###  III.6 The Lorenz number

Figure 6: The Lorenz numbers of warm dense aluminum at temperatures ranging
from 0.086 to 10 eV and densities of (a) 2.35 $\mathrm{g/cm^{3}}$ and (b) 2.70
$\mathrm{g/cm^{3}}$. NC11 and NC3 are two norm-conserving pseudopotentials.
The label corr. means the non-local potential correction is included. The cell
contains 256 Al atoms for temperatures no more than 5 eV and 108 atoms for
higher temperatures. Besides, first-principles data from Knyazev et al. [72],
Sjostrom et al. [21], and Witte et al. [22] are also presented for comparison.

The Lorenz number [88, 89] characterizes the relationship between the
electrical and thermal conductivities and is still not well understood in the
warm dense matter region. It is defined as

$L=\frac{\kappa}{\sigma T},$ (22)

where $T$ is the temperature. Here $\kappa$ and $\sigma$ refer to the thermal
conductivity and the DC electrical conductivity, respectively. According to
the Wiedemann-Franz law [88, 89, 90], $L$ is a constant for highly degenerate
electrons, approximately equal to $2.445\times 10^{-8}~{}\mathrm{W\Omega
K^{-2}}$. According to our calculation results shown in Fig. 6, we have the
following findings.

First, at temperatures below 3 eV, the Lorenz number calculated with the NC3
and NC11 pseudopotentials exhibit the same trend. However, as the temperature
further increases, the results differ due to the invalidity of the frozen-core
approximation used in the NC3 pseudopotential.

Second, the Lorents number obtained by the NC11 pseudopotential with and
without the non-local potential corrections are close, indicating that the
corrections influence the electrical and thermal conductivities in the same
proportion. In contrast, by utilizing the NC3 pseudopotential at the
temperature of 8 eV and higher, the non-local potential corrections become
pronounced.

Third, at temperatures up to 1 eV, the Lorenz number agrees well with the
Wiedemann-Franz (WF) law, indicating the inherent relationship between
electrical conductivity and thermal conductivity is valid. Meanwhile, the
degeneracy parameter, defined as the ratio of temperature to the Fermi
temperature $T/T_{F}$, is 0.086 at the density of 2.7 $\mathrm{g/cm^{3}}$ and
0.094 at 2.35 $\mathrm{g/cm^{3}}$, indicating a high degree of degeneracy in
these Al systems.

Fourth, importantly, as the temperature exceeds 1 eV, which falls into the
intermediate degenerate range, the Lorenz number starts to decrease and
reaches a minimum at the temperature of approximately 8 eV. Specifically, at
the density of 2.7g/cm3, the Lorenz number is 36.7% lower than that predicted
by the WF law, while at the density of 2.35 g/cm3, it is 35.1% lower. The
observed trend of the Lorentz number demonstrates that in this region, we
cannot calculate thermal conductivity directly from the electrical
conductivity using the Wiedemann-Franz law.

Fifth, we compared our results with those of previous works [54, 72, 77, 21,
22]. The results obtained by Recoules et al. [54] followed the Wiedemann-Franz
law in the temperature range of 70 to 10000 K (0.006 to 0.862 eV) at a density
of 2.35 g/cm3 (not shown in Fig. 6), agreeing well with our results. The PBE
results of Witte et al. [22] also match well with our results when using the
NC11 pseudopotential with non-local potential corrections. However, the HSE
results predict a higher Lorenz number. It should be noted that all of these
results have taken into account the velocity matrix corrections. Next, Knyazev
et al. [72] reported slightly higher values of the Lorents number at low
temperatures. In addition, we also calculate the Lorenz number using $\sigma$
and $\kappa$ values from Sjostrom et al. [21]. Their results agree well with
the Wiedemann-Franz law for all temperatures below 10 eV, yielding unfavorable
results as it fails to exhibit a decrease in the Lorenz number at higher
temperatures as expected. This may be attributed to the approximations used in
their work, such as the non-self-consistent electronic density obtained by
OFDFT.

##  IV Conclusions

Accurate prediction of electron transport coefficients is vital for
comprehending warm dense matter. The Kubo-Greenwood formula is extensively
employed to calculate the electrical and thermal conductivities of electrons.
Nonetheless, the existence of non-local terms in the pseudopotentials and the
use of the frozen-core approximation in first-principles calculations lead to
inaccuracy when the Kubo-Greenwood formula is adopted.

In this work, we implemented the non-local potential corrections with the use
of norm-conserving pseudopotentials and plane wave basis under periodic
boundary conditions within the ABACUS package. We calculated both electrical
and thermal conductivities of liquid Al at 1000 K and warm dense Al with
temperatures ranging from 0.2 to 10 eV. To ensure converged results, we used
converged numbers of atoms and $k$-points in our calculation.

We examined the electrical and thermal conductivities of liquid Al at a
temperature of 1000 K. By incorporating the non-local potential corrections,
we obtained results that agreed well with the experimental data. Next, we
simulated warm dense Al with temperatures ranging from 0.2 eV to 10.0 eV at a
density of 2.35 $\mathrm{g/cm^{3}}$. Compared to the results without using the
non-local potential corrections, we observed lower conductivities when the
corrections were applied. Notably, two different norm-conserving
pseudopotentials, NC3 and NC11, yielded similar electrical and thermal
conductivities at low temperatures. However, as temperatures exceeded 2 eV,
discrepancies emerged due to the inadequate frozen core approximation of the
2s and 2p electrons of Al. We also simulated Al at a density of 2.70
$\mathrm{g/cm^{3}}$. The results exhibited similar behaviors to those at 2.35
$\mathrm{g/cm^{3}}$ and showed reasonable agreement with available
experimental data. We also compared our calculated results to those obtained
by previous works.

To gain further insights into the computed conductivities, we analyzed the DOS
and the decomposed electrical conductivity at temperatures of 0.5, 5, and 10
eV. We found the core electrons are almost non-ionized at the temperature of 5
eV. However, the core electrons affect the high-energy electronic states,
resulting in substantially different conductivities as compared to the N3
pseudopotential. In conclusion, the NC11 pseudopotential provides a more
accurate prediction than the NC3 pseudopotential for warm dense Al.

In summary, our findings emphasize the importance of incorporating non-local
potential corrections when calculating the conductivities of electrons using
pseudopotentials. Additionally, the choice of pseudopotential is crucial in
accurately determining conductivities, and the frozen-core approximation
becomes invalid for calculating conductivities at temperatures before the
ionization of corresponding core electrons. However, limited by high
computational costs for electrons at extremely high temperatures, traditional
KSDFT only allows conductivity calculations for warm dense Al up to a few tens
of eV. To overcome this limitation, recent advancements in stochastic DFT [29,
30] and mixed stochastic-deterministic DFT [31, 32, 91] present promising
approaches for addressing this challenge.

###### Acknowledgements.

This work is supported by the National Science Foundation of China under Grant
No.12122401 and No.12074007. We thank the Electronic Structure Team at AI for
Science Institute in Beijing for contributing to the ABACUS package. We thank
Tao Chen from Peking University for reviewing this manuscript. Part of the
numerical simulations was performed on the High-Performance Computing Platform
of CAPT.

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