# Insights into the Electron-Electron Interaction from Quantum Monte Carlo
Calculations

Carl A. Kukkonen<EMAIL_ADDRESS>Kun Chen<EMAIL_ADDRESS>Center for Computational Quantum Physics, Flatiron Institute, 162 Fifth
Avenue, New York, NY 10010

###### Abstract

The effective electron-electron interaction in the electron gas depends on
both the density and spin local field factors. Variational Diagrammatic
Quantum Monte Carlo calculations of the spin local field factor are reported
and used to quantitatively present the full spin-dependent, electron-electron
interaction. Together with the charge local field factor from previous
Diffusion Quantum Monte Carlo calculations, we obtain the complete form of the
effective electron-electron interaction in the uniform three-dimensional
electron gas. Very simple quadratic formulas are presented for the local field
factors that quantitatively produce all of the response functions of the
electron gas at metallic densities.

Exchange and correlation become increasingly important at low densities. At
the compressibility divergence at rs = 5.25, both the direct (screened
Coulomb) term and the charge-dependent exchange term in the electron-electron
interaction at q=0 are separately divergent. However, due to large
cancellations, their difference is finite, well behaved, and much smaller than
either term separately. As a result, the spin contribution to the electron-
electron interaction becomes an important factor. The static electron-electron
interaction is repulsive as a function of density but is less repulsive for
electrons with parallel spins.

The effect of allowing a deformable, rather than rigid, positive background is
shown to be as quantitatively important as exchange and correlation. As a
simple concrete example, the electron-electron interaction is calculated using
the measured bulk modulus of the alkali metals with a linear phonon
dispersion. The net electron-electron interaction in lithium is attractive for
wave vectors $0-2k_{F}$, which suggests superconductivity, and is mostly
repulsive for the other alkali metals.

††preprint: APS/123-QED

##  I Introduction

The electron-electron interaction is important for pairing and
superconductivity, spin and magnetic phenomena, transport properties, and as
an input for numerical calculations. Many interesting phenomena occur in
exotic materials, some with reduced dimensionality and unusual topologies, and
different theoretical approaches have been employed to explain experiment.

This paper looks back to the well-studied three-dimensional electron gas to
examine the quantitative effects of exchange and correlation on the electron-
electron interaction to see if any insight may be gained for problems of
modern interest.

The equation for the spin dependent effective electron-electron interaction in
the three-dimensional electron gas is well-established in the mean field local
approximation, and is given in terms of the local field factors that define
all of the response functions of the electron gas Kukkonen and Overhauser
(1979). Sum rules specify the local field factors at small and large wave
vector $q$, and the difficult problem is to calculate the intermediate wave
vector dependence. The Quantum Monte Carlo method is considered to produce
accurate results. The density local field factor was calculated many years ago
by numerous methods including Diffusion Quantum Monte Carlo. Results for the
spin local field factor using the Variational Diagrammatic Monte Carlo (VDMC)
method were first published in 2019Chen and Haule (2019) and additional
results are presented here. The spin local field factor completes the
specification of the effective electron-electron interaction and has motivated
this paper. The local field factors are also known as exchange and correlation
kernels in Time-Dependent Density Functional Theory.

The equation for the electron-electron interaction itself is indicative, but
it is difficult to understand the relative importance of the terms until the
actual local field factors are used to show quantitative results.

The numerically calculated values for the density and spin local field factors
are discussed, and it is shown that they can approximated by very simple
formulas that produce all of the response functions of the electron gas.

##  II Electron-electron Interaction

Kukkonen and OverhauserKukkonen and Overhauser (1979) (KO) demonstrated that
the standard self-consistent perturbation theory based on Hartree-Fock theory
and linear response theory could be used to calculate the effective many-body
interaction $V_{ee}$ between two electrons in a simple metal, modeled by the
electron gas, in terms of the density local field factor $G_{+}(q,\omega)$ and
spin local field factor $G_{-}(q,\omega)$.

$\displaystyle V_{e\vec{\sigma}_{1},e\vec{\sigma}_{2}}$ $\displaystyle=$
$\displaystyle\frac{4\pi
e^{2}}{q^{2}}\left(\frac{\left(\omega^{2}-\omega_{0}^{2}\right)/\left(\omega^{2}-\omega_{q}^{2}\right)}{\left(1-G_{+}Q\right)\left[1+\left(1-G_{+}\right)Q\right]}\right.$
(1)
$\displaystyle\left.-\frac{G_{+}^{2}Q}{1-G_{+}Q}-\frac{G_{-}^{2}Q}{1-G_{-}Q}\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}\right)$

This is the interaction to be used for calculating matrix elements between two
electrons with momenta $k_{1}$ and $k_{2}$ and spins $\sigma_{1}$ and
$\sigma_{2}$. For parallel spins, the wave functions must be properly anti-
symmetric.

The electron gas is characterized by the density parameter $r_{s}$,
($n=4\pi(r_{s}a_{0})^{3}\\!/3$, $a_{0}$ is the Bohr radius). $Q=v\Pi^{0}$
where $v=4\pi e^{2}/q^{2}$ is the Coulomb potential and $\Pi^{0}(q,\omega)$ is
the Lindhard function. For convenience, we will not usually explicitly present
the wave vector and frequency dependence. We will also not explicitly use the
word effective in discussing interactions. With the deformable background, the
standard phonon frequencies and background (lattice) screening results were
obtained and are represented by the frequency dependence of the first term in
Eq. (1). The intuitive physics concepts behind the electron-electron
interaction in Eq. (1) are presented in Ref. Kukkonen and Overhauser (1979).
The frequencies $\omega_{q}$ and $\omega_{0}$ refer to the frequency response
of the deformable background which is discussed in Ref. Kukkonen and
Overhauser (1979) and Section VII of this paper.

The first calculations of the screened interaction in the electron gas were
the Thomas Fermi interaction and its quantum mechanical extension by
LindhardGiuliani and Vignale (2005). The Lindhard result is recovered by
setting both $G$’s equal to zero which results in

$\displaystyle V_{\text{Lindhard}}=\frac{4\pi e^{2}}{q^{2}(1+Q)}$ (2)

In this approximation, the electron-electron, electron-test charge and test
charge-test charge interactions are all the same. These latter interactions
are discussed in Appendix B.

The KO electron-electron interaction has been verified by many-body
calculations Vignale and Singwi (1985a), and has been extended to multicarrier
Vignale and Singwi (1985b), spin polarized and two-dimensional systemsGiuliani
and Vignale (2005). The potential for superconductivity without phonons using
the KO interaction was examined by TakadaTakada (1993) and by Richardson and
Ashcroft(Richardson and Ashcroft, 1996). Takada found superconductivity for a
single carrier system and Richardson and Ashcroft did not. Richardson and
Ashcroft stated that an important difference was that each group used
different values of $G_{+}(q)$ and $G_{-}(q)$. Richardson and Ashcroft
calculated their own $G$’s including frequency dependence. Takada used
frequency independent $G$’s. The $G$’s considered here are the static local
field factors.

Connecting with Feynman diagrams, the frequency independent factor in the
first term can also be written as $\Lambda^{2}/\epsilon$ – two vertex
corrections divided by the dielectric function Kukkonen and Wilkins (1979).
The deformable background (lattice) screens the first term, the Coulomb
interaction, but not the exchange and correlation and spin response terms
which arise from summing ladder diagrams.

For a rigid lattice, the frequency dependence in the first term of Eq. (1)
equals one, and the first term is repulsive. The second term is attractive.
The spin dependent term is repulsive for opposite spins (singlet) and
attractive for parallel spins (triplet). We initially consider the rigid
background and discuss the deformable background in section VII below.

Quantitative evaluation of the electron-electron interaction is made in
section V using the values of $G_{+}(q)$ discussed in Appendix A and the new
accurate values of $G_{-}(q)$ reported below.

##  III Spin Local Field Factor $\mathbf{G_{-}(q)}$

A Quantum Monte Carlo method using renormalized Feynman diagrams was developed
by Chen and Haule(Chen and Haule, 2019). The resulting Variational
Diagrammatic Monte Carlo (VDMC) method is a generic many-body solver that was
tested on the electron gas. The VDMC method was used to calculate the spin and
density responses in the electron gas. It is well suited for finite
temperatures. The description of the VDMC method, including grouping of
Feynman diagrams, numerical approach and high precision results are given in
Ref. Chen and Haule (2019).

The data reported here are new VDMC calculations of the static spin local
field factor with higher accuracy for densities $r_{s}=1-4$ from
$q=0-2.34\,k_{F}$, and additional results for $r_{s}=5$. The calculation
temperature is $T=0.025\,T_{F}$ which is equivalent to $T=0$. The method
provides the highest accuracy calculations of the $q=0$ susceptibility, and
our calculations at finite $q$ also have high accuracy, but less than at
$q=0$. Typical error bars are shown with the data.

Figure 1: Spin local field factor $G_{-}(q)$ versus $q/k_{F}$ for $r_{s}=1-5$
for the three-dimensional electron gas calculated with the Variational
Diagrammatic Monte Carlo method. Error bars are shown by shading.

Figure 1 shows the wave vector dependence of $G_{-}(q)$ which demonstrates
that it initially follows the quadratic behavior required by the
susceptibility sum rule. The behavior changes dramatically near $q=2\,k_{F}$.

The susceptibility enhancement at $q=0$ was calculated more accurately than
the values at finite $q$. The results from Ref.Chen and Haule (2019) and the
new result at $r_{s}=5$ are reported in Table 1.

$\mathbf{r_{s}}$ | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
$\mathbf{\chi}/\mathbf{\chi_{0}}$ | 1.152(2) | 1.296(6) | 1.432(9) | 1.576(9) | 1.683(15)
Table 1: Susceptibility enhancement at $q=0$ for the three-dimensional
electron gas calculated by Variational Diagrammatic Monte Carlo method.
Uncertainty is indicated by the number in parentheses.

In order to clearly see the small $q$ behavior and the effect of the
susceptibility sum rule, the quantity $G_{-}(q)/(q/q_{TF})^{2}$ is plotted in
Fig. 2. The Thomas Fermi screening wave vector is defined by
$q_{TF}^{2}=4k_{F}/\pi a_{0}$. The spin exchange and correlation kernel for
Time Dependent Density Function Theory is $f_{xc}^{\text{spin}}=-4\pi
G_{-}(q)/(q/q_{TF})^{2}$.

Figure 2: Spin local field factor divided by the wave vector divided by Thomas
Fermi wave vector squared, $G_{-}(q)/(q/q_{TF})^{2}$ versus $q/k_{F}$ for
$r_{s}=1-5$. Error bars are shown by shading.

Figure 2 shows that the spin local field factor $G_{-}(q)$ follows the
quadratic well and does not fall below the quadratic behavior until near
$2\,k_{F}$. In fact, for high density $r_{s}=1$, $G_{-}(q)$ rises
significantly above the quadratic before it falls below. The close adherence
to the quadratic behavior, suggests that in the metallic region $r_{s}=2-5$,
that a simple quadratic that satisfies the susceptibility sum rule is adequate
to calculate the spin response function.

The wave vector dependent susceptibility enhancement is given by

$\frac{\chi(q)}{\chi_{0}(q)}=\frac{1}{1-G_{-}Q}\;,$ (3)

and shown in Figure 3.

Figure 3: Susceptibility enhancement $\chi(q)/\chi_{0}$ plotted versus
$q/k_{F}$ for $r_{s}=1-5$. The data points use the actual values of $G_{-}(q)$
calculated here and reported in Figure 1. The solid lines are the simple
quadratic function Eq. (5) set by the susceptibility sum rule at $q=0$. Error
bars are shown by shading. Note that the $Y$ axis starts at 1.0.

The $q=0$ value of the susceptibility enhancement is entirely set by the
susceptibility sum rule. The enhancement is modest because there is no
divergence in the susceptibility near the metallic region. The simple
quadratic, which is the horizontal line in Fig. 2, fits the data quite well
and is adequate for the discussions in this paper and for comparison with
experiment. If higher accuracy is needed, the actual data shown in Figure 2
can be used.

The application of the VDMC method to the density local field factor
$G_{+}(q)$ is briefly discussed in Appendix A.

##  IV Simple Expressions for local field factors $\mathbf{G_{+}(q)}$ and
$\mathbf{G_{-}(q)}$

The recommended simple quadratic forms for $G_{+}(q)$ and $G_{-}(q)$ are:

$G_{+}(q)=\left(1-\frac{\kappa_{0}}{\kappa}\right)\left(\frac{q}{q_{TF}}\right)^{2}$
(4)
$G_{-}(q)=\left(1-\frac{\chi_{0}}{\chi}\right)\left(\frac{q}{q_{TF}}\right)^{2}\;.$
(5)

These expressions are exact at small $q$ and accurately represent the QMC data
up to almost $q=2\,k_{F}$ for the metallic region $r_{s}=2-5$. $G_{+}(q)$ is
discussed in Appendix A. Although these simple quadratic approximations are
not accurate beyond $2\,k_{F}$, they are suitable for the electron gas
response functions which are cut off by the Lindhard function above
$q=2\,k_{F}$. For any application that requires values of $G$ at larger $q$,
we recommend the interpolation formula discussed in Appendix A for $G_{+}$ or
the actual data above for $G_{-}$.

The simple quadratic approximation to $G_{-}(q)$ given in Eq. (5) and used to
calculate the susceptibility enhancement in Fig. 3 fits the VDMC data quite
well. The fit is exact at $q=0$, falls below by $2\%$ at $q=1.5\,k_{F}$ and
slightly above at $2\,k_{F}$. The average values are within $1\%$. The
susceptibility is the product of the enhancement times the Lindhard function
and the Bohr magneton. The falloff of the Lindhard function above $2\,k_{F}$
makes this region unimportant for most applications.

The recommended values of the compressibility are taken from Perdew and
WangPerdew and Wang (2018) and susceptibility ratios are given in Table 1.
Both are plotted in Figure 4 and are accurately fitted by quadratic
interpolation formulas.

Figure 4: Compressibility ratio $\kappa_{0}/\kappa$ and susceptibility ratio
$\chi_{0}/\chi$ for the three-dimensional electron gas with a rigid uniform
positive background.

The curves in Fig 4 are fits to the data in the metallic region and fit the
data to less than $0.5\%$. Note that the compressibility and susceptibility
ratios must equal 1 at $r_{s}=0$. Since we are only interested in the metallic
region $r_{s}=1-5$, the fitting curves were not required to have an intercept
of 1 at $q=0$, which results in a simpler and more accurate equations in the
metallic region.

The compressibility and susceptibility ratios at $q=0$ are well fitted from
$r_{s}=1-5$ by the following equations.

$\frac{\chi_{0}}{\chi}=0.9821-0.1232\,r_{s}+0.0091\,r_{s}^{2}$ (6)
$\frac{\kappa_{0}}{\kappa}=1.0025-0.1721\,r_{s}-0.0036\,r_{s}^{2}$ (7)

With these $G$’s and the compressibility and susceptibility ratios, all of the
response functions for the three-dimensional electron gas with a rigid
background can be quantitatively calculated. The same approach can be used for
a spin polarized or two component electron gas.

Figure 4 shows the well-known divergence and sign change of the
compressibility at $r_{s}=5.25$. This causes the vertex function and thus the
dielectric function to diverge and become negative. The rigid uniform positive
background prevents the overall model electron gas from becoming unstable.

##  V Electron-Electron Interaction: Quantitative Results

Using $G_{+}(q)$ and $G_{-}(q)$, we plot the electron-electron interaction. We
plot each term in Eq. (1) separately to show their relative importance. These
are denoted $V_{ee}1$, $V_{ee}2$ and $V_{ee}3$. The first term $V_{ee}1$ is
the coefficient of the frequency dependent factor (which is equal to $1$ for a
rigid background). $V_{ee}1$ is intrinsically positive. The second term
$V_{ee}2$ subtracts from the first term. $V_{ee}3$, the spin dependent term is
positive (repulsive) for opposite spins (singlet) and negative (attractive)
for parallel spins (triplet).

These three terms are plotted together in Fig. 5 for $r_{s}=2$ and $5$.

Figure 5: The three terms in the equation for $V_{ee}(q)$ for $r_{s}=2$ (a)
and $r_{s}=5$ (b). The first term $V_{ee}1$ is the screened Coulomb
interaction. The second two terms are additional effects of exchange and
correlation. The second term $V_{ee}2$ is subtracted from the first term and
the third (spin dependent) term $V_{ee}3$ is subtracted for parallel spins and
added for antiparallel (opposite) spins. The potential is measured in units of
$4\pi e^{2}$.

The magnitude of the first term $V_{ee}1$ at $q=0$ is
$(\kappa/\kappa_{0})/q_{TF}^{2}$ which is divergent at the compressibility
divergence. The results look “normal” at $r_{s}=2$ where the second and third
terms are small corrections to $V_{ee}1$ (the screened Coulomb interaction).
$V_{ee}1$ is screened by the deformable background (lattice) in the usual
fashion. This seems consistent with the idea of perturbation theory where the
corrections are small (except for the effect of the compressibility sum rule
which is large even at $r_{s}=2$).

At $r_{s}=5$, $V_{ee}1$ is very large. $V_{ee}1$ diverges as a function of
$r_{s}$ at the compressibility divergence approximately as $1/(1-r_{s}/5.25)$.
When calculating with Feynman diagrams, this term arises as the direct
screened interaction with two vertex corrections
$V_{ee}1=(\Lambda^{2}/\epsilon)V_{ext}=\Lambda V_{et}$. This apparent
divergence is one concern.

The second term $V_{ee}2$ is also large and apparently diverging, and is not
screened by the lattice. $V_{ee}2$ is subtracted from $V_{ee}1$. The fact that
this term is large brings into question the use of perturbation theory and
linear response. However, upon closer examination, $V_{ee}2$ completely tracks
$V_{ee}1$ and for a rigid background, they formally and exactly partially
cancel each other to yield a finite value. This is due to a massive
cancellation of Feynman diagrams using that approach. For a rigid background,
the difference $V_{ee}1-V_{ee}2$ at $q=0$ is given as

$V_{ee}1(0)-V_{ee}2(0)=\frac{\kappa}{\kappa_{0}}\left(\frac{1-(\kappa/\kappa_{0})^{2}}{q_{TF}^{2}}\right)\;=\;\frac{2-\kappa_{0}/\kappa}{q_{TF}^{2}}\;.$
(8)

For a rigid background, the net spin independent terms of the electron-
electron interaction are completely well behaved and have no divergence at the
compressibility divergence. This was also observed by KOKukkonen and
Overhauser (1979). The deformable lattice will be discussed in Section VII
below.

Figure 6: The net electron-electron interactions $V_{ee}(q)$ for opposite and
parallel spins compared to the Lindhard potential at $r_{s}=2$ (a) and $5$
(b). A rigid background is assumed for the electron gas.

Figure 6 shows the net electron-electron interaction for electrons with
parallel and opposite spins. This is the interaction to be used for
calculating matrix elements and superconductivity.

$V_{ee}^{\uparrow\downarrow}=V_{ee}1-V_{ee}2+V_{ee}3$ (9)

is the interaction for opposite spins, and

$V_{ee}^{\uparrow\uparrow}=V_{ee}1-V_{ee}2-V_{ee}3$ (10)

is the interaction for parallel spins.

As mentioned above $V_{ee}1-V_{ee}2$ is a smooth function and lies midway
between parallel and opposite spins. The overall electron-electron interaction
is smooth and has no remaining evidence of the large effects in $V_{ee}1$ and
$V_{ee}2$ individually. At $r_{s}=2$, $V_{ee}2$ and $V_{ee}3$ have effects at
the $10\%$ level. However at $r_{s}=5$, the first two terms nearly cancel, and
$V_{ee}3$, the spin dependent term, is relatively important. The overall
electron-electron interaction is considerably less repulsive for parallel
spins.

The values of the electron-electron interaction at $q=0$ are completely
determined by the compressibility and susceptibility sum rules. At large $q$
(short distances), the electron-electron interaction follows the Lindhard
function to the bare Coulomb interaction. The Lindhard interaction is shown
for comparison.

This quantitative evaluation of the electron-electron interaction shows that
with a rigid background, the static electron-electron interaction is well
behaved and repulsive throughout the metallic region. The electron-electron
interaction (and the other interactions in the electron gas discussed in
Appendix B) are completely specified with simple equations provided in this
paper. However, care must be taken with quantitative comparison with
experiment. The effective mass, renormalization factor $z$, core polarization
and a deformable lattice can have significant effects. Some of these were
discussed by Kukkonen and WilkinsKukkonen and Wilkins (1979). The deformable
lattice will be discussed in section VII.

##  VI Effect of the Compressibility Sum Rule

We have emphasized that the compressibility and susceptibility sum rules,
which are derived from changes in the total electron gas energy, completely
determine the various interactions at $q=0$. This is illustrated Fig. 7 where
we plot the ratio of the different interactions to the Lindhard interaction at
$q=0$ versus $r_{s}$.

Figure 7: The ratio of the electron gas interactions at $q=0$ to the Lindhard
(Thomas Fermi) interaction as a function of $r_{s}$ for a rigid background.
The top curve is the electron-electron interaction for electrons with opposite
spins. The next curve down is the electron- electron interaction for parallel
spins. The electron-test charge interaction $V_{et}$ at $q=0$ is equal to the
Thomas Fermi interaction at all $r_{s}$. The bottom curve is the test charge-
test charge interaction $V_{tt}$ which always falls below Thomas Fermi
interaction and becomes negative at the compressibility instability.

Figure 7 shows that all of the interactions are the same and equal to the
Lindhard and Thomas Fermi interactions at high density near $r_{s}=0$. At
lower density (larger $r_{s}$), the effects of exchange and correlation
manifest themselves through the sum rules. Only the electron-electron
interaction depends on spin and that is shown by the two curves for parallel
and opposite spins.

With a rigid uniform positive background, the electron-electron and electron-
test charge interactions show no unusual behavior as $r_{s}$ approaches the
compressibility divergence at $r_{s}=5.25$. However, the test charge-test
charge interaction at $q=0$ becomes negative (attractive) above the
instability. This is due to the fact that the dielectric function has become
negative. This interaction is discussed in Appendix B.

The sum rules dictate the $q=0$ behavior. At large $q$, all the interactions
fall off as $1/q^{2}$ which reflects the bare Coulomb interaction at short
distances, and the wave vector dependence of the local field factors and the
cut off of the Lindhard function at $2\,k_{F}$ set the intermediate $q$
behavior which has been discussed above.

##  VII Deformable Background

KO considered a smooth but elastically deformable background which led to the
frequency dependence in Eq.(1). The result was verified in Ref.Giuliani and
Vignale (2005). Introducing the deformable background naturally results in the
phonon frequenciesKukkonen and Overhauser (1979).

$\displaystyle\omega_{q}^{2}$ $\displaystyle=$
$\displaystyle\frac{Nq^{2}}{M}\left(V_{ii}^{\text{bare
}}-\frac{\left(V_{ei}^{\text{bare
}}\right)^{2}}{v}+\frac{\left(V_{ei}^{\text{bare
}}\right)^{2}}{v\epsilon}\right)$ (11) $\displaystyle\equiv$
$\displaystyle\omega_{0}^{2}+\frac{Nq^{2}\left(V_{ei}^{\text{bare
}}\right)^{2}}{Mv\epsilon}$

Where $N$ is the density and $M$ is the mass of the background (ions),
$V_{ii}^{\text{ bare}}$ is the bare ion-ion interaction,
$V_{ei}^{\text{bare}}$ is the bare electron-ion interaction, and $\epsilon$ is
the dielectric function.

It is instructive to rewrite the frequency dependence in equation (1) as

$\frac{\omega^{2}-\omega_{0}^{2}}{\omega^{2}-\omega_{q}^{2}}=1+\frac{\omega_{q}^{2}-\omega_{0}^{2}}{\omega^{2}-\omega_{q}^{2}}\;.$
(12)

This now multiplies the first term. The factor 1 allows the frequency
independent part of the first term and second term to be formally combined
which cancels the compressibility divergence in the static interaction, and
Eq.(1) can be rewritten as

$\displaystyle V_{e\vec{\sigma}_{1},e\vec{\sigma}_{2}}$
$\displaystyle=\frac{4\pi
e^{2}}{q^{2}}\left(\frac{\left(\omega^{2}_{q}-\omega_{0}^{2}\right)/\left(\omega^{2}-\omega_{q}^{2}\right)}{\left(1-G_{+}Q\right)\left[1+\left(1-G_{+}\right)Q\right]}\right.+$
(13)
$\displaystyle\left.\frac{1+(1-G_{+})G_{+}Q}{1+(1-G_{+})Q}-\frac{G_{-}^{2}Q}{1-G_{-}Q}\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}\right)$

The new first term $V_{ee-{\text{phonon}}}$ is divergent at $q=0$ at the
compressibility divergence, but the second two terms have no divergence.

$V_{ee-\text{phonon}}$ represents the additional screening of the Coulomb
interaction by the background (lattice). At $\omega=0$, the numerator of the
first term is negative which is the expected result for static screening by
the positive background.

$-\frac{\omega_{q}^{2}-\omega_{0}^{2}}{\omega_{q}^{2}}=\displaystyle{\frac{-Nq^{2}\left(V_{ei}^{\text{bare}}\right)^{2}\Big{/}(Mve)}{(Nq^{2})/M\displaystyle{\left(V_{ii}^{\text{bare}}-\frac{\left(V_{ei}^{\text{bare
}}\right)}{v}+\frac{\left(V_{ei}^{\text{bare
}}\right)^{2}}{\epsilon\,v}\right)}}}$ (14)

The screening depends on the stiffness of the background represented by
$V_{ii}^{\text{bare}}$ and the properties of the electron gas. The rigid
background is obtained when $V_{ii}^{\text{bare}}$ (and thus $\omega_{q}^{2}$)
goes to infinity and this term goes zero.

Another interesting limit occurs if all of the interactions including
$V_{ii}^{\text{b}are}$ are Coulomb interactions. In this case

$-\frac{\omega_{q}^{2}-\omega_{0}^{2}}{\omega_{q}^{2}}=-1$ (15)

and $V_{ee-\text{phonon}}$ is large and negative (attractive) At $q=0$, the
first term has the value set by the compressibility sum rule as

$\displaystyle V_{ee-\text{phonon}}(q=0)$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{\kappa_{0}}V_{et}(q=0)$ (16) $\displaystyle=$
$\displaystyle-\frac{\kappa}{\kappa_{0}}V_{TF}(q=0)$

With all Coulomb interactions, this negative first term is larger than the
other two terms combined and the overall electron-electron interaction at
$q=0$ is attractive. Note this term diverges at the compressibility
divergence.

An instructive intermediate case is to consider that $V_{ei}^{\text{bare}}$ in
the numerator of Eq.(14) is equal to the Coulomb interaction $v=4\pi
e^{2}/q^{2}$, and to take $\omega_{q}$ from experiment. The background
(lattice) screening term can be rewritten as

$\displaystyle V_{ee-\text{phonon}}(q)$ $\displaystyle=$
$\displaystyle-\frac{1}{\omega_{q}^{2}}\left(\frac{v}{(\epsilon(1-G_{+}Q))^{2}}\right)\frac{Nq^{2}}{M}$
(17) $\displaystyle=$
$\displaystyle-\frac{1}{\omega_{q}^{2}}V_{et}^{2}(q)\frac{Nq^{2}}{M}$

All of the effects of the deformable background are in $\omega_{q}^{2}$, the
phonon frequencies which depend on the electron gas parameters as well as the
bare ion-ion interaction. The electron test charge interaction appears because
the lattice appears as a test charge to the electron gas. The electron test
charge interaction is equal to the Thomas Fermi and Lindhard interactions at
$q=0$, but differs at larger q as shown in Appendix B.

As the simplest example, we model the phonons by the relationship between
$\omega_{q}$ and the bulk modulus B which is the inverse of the
compressibility $\kappa$.

$\omega_{q}^{2}=\frac{Bq^{2}}{NM}$ (18)

where $NM$ is the mass density of the background, and $\sqrt{B/NM}$ is the
speed of sound. Using this relationship and the bulk modulus of the non-
interacting electron gas $B_{0}=1/(n^{2}V_{TF}(q=0))$, and the fact that the
ion density and the electron density are the same for the monovalent alkali
metals, one obtains the equation for the background screening contribution to
the electron-electron interaction

$V_{ee-\text{phonon}}(q)=-\frac{V_{et}(q))^{2}}{V_{et}(0)}\frac{B_{0}}{B_{\text{experiment}}}$
(19)

The measured bulk moduli, free electron valuesAshcroft and Mermin (1976), and
their ratios for the alkali metals are given in Table 2.

| $r_{s}$ | $B_{0}$ | $B_{\text{experiment}}$ | $B_{0}/B_{\text{experiment}}$
---|---|---|---|---
Li | 3.25 | 23.9 | 11 | 2.173
Na | 3.93 | 9.23 | 6.3 | 1.465
K | 4.86 | 3.19 | 3.1 | 1.029
Rb | 5.2 | 2.28 | 2.5 | 0.912
Cs | 5.62 | 1.54 | 1.6 | 0.963
Table 2: Free electron and experimentally measured bulk moduli for the alkali
metals. Figure 8: Static $(\omega=0)$ electron-electron interaction at $q=0$
for the electron gas at the density of alkali metals (in units of the Lindhard
(Thomas Fermi) interaction). The top two curves are the electron-electron
interaction for a rigid background with opposite spins, and parallel spins.
The data at the lower right are the electron-electron interaction including
the deformable background screening for the alkali metals using the measured
bulk modulus for opposite and parallel spins.

In Figure 8, the repulsive electron-electron interactions at $q=0$ for
opposite and parallel spins in a rigid background is plotted versus $r_{s}$ as
in Figure 7. Also plotted is the net electron-electron interaction including
screening by the deformable background using the experimentally measured bulk
modulus for the alkali metals.

The electron-electron interactions for a rigid background are simply the
second two terms of Eq. (13) evaluated at $q=0$. These are completely
specified by the compressibility and susceptibility sum rules, and the values
are given in Fig. 4.

$V_{ee}^{\uparrow\downarrow}(0)=\left[\left(2-\frac{\kappa_{0}}{\kappa}\right)+\frac{\chi}{\chi_{0}}\left(1-\frac{\chi_{0}}{\chi}\right)^{2}\right]V_{\text{Lindhard}}(0)$
(20)
$V_{ee}^{\uparrow\uparrow}(0)=\left[\left(2-\frac{\kappa_{0}}{\kappa}\right)-\frac{\chi}{\chi_{0}}\left(1-\frac{\chi_{0}}{\chi}\right)^{2}\right]V_{\text{Lindhard}}(0)$
(21)

The attractive background screening contribution to the electron-electron
interaction is given by Eq. (19) evaluated at $q=0$. The electron-test charge
interaction $V_{et}(q)$ is given in Eq. (26) in Appendix B, and is equal to
the Lindhard and Thomas Fermi potentials at $q=0$.

$V_{et-\text{phonon}}(0)=\left[\frac{B_{0}}{B_{\text{experiment}}}\right]V_{\text{Lindhard}}(0)$
(22)

This term is spin independent and is subtracted from both the opposite and
parallel spin electron-electron interactions for a rigid background and net
electron-electron interaction is plotted at the $r_{s}$ values of the alkali
metals.

Figure 9: Electron-electron interaction at densities of lithium (a) and sodium
(b) comparing the repulsive interaction calculated for a rigid background to
the net interaction when the deformable background is included using the
measured bulk modulus. The top two curves are for the rigid lattice for
opposite and parallel spins. The bottom two curves subtract the deformable
background contribution from the top two curves. They represent the net
electron-electron interaction.

Note that all of the electron-electron interactions at $q=0$ are proportional
to $V_{\text{Lindhard}}(q=0)=1/q_{TF}^{2}=r_{s}\,a_{0}/(2.95)^{2}$ in units of
$4\pi e^{2}$. The electron-electron interactions divided by
$V_{\text{Lindhard}}(q=0)$ at each $r_{s}$ are plotted in Fig. 8 which shows
the relative importance of exchange and correlation and the deformable
background.

Figure 8 shows that the effects of screening by the deformable background are
as large as the effects of exchange and correlation. The $q=0$ value of the
net electron-electron for lithium is attractive for both parallel and opposite
spins, while all of the other alkali metals are repulsive. Lithium is the only
alkali metal that exhibits superconductivity at ambient pressure. Although the
net electron-electron for lithium is more attractive for parallel spins than
opposite spins, this does not necessarily imply triplet pairing because the
spatial part of the overall wave function must be anti-symmetric. This is
discussed briefly after Fig. 10.

Figure 10: $V_{ee}(r)$, the Fourier transform of $V_{ee}(q)$, the electron-
electron interaction shown in Fig. 9, at the densities of lithium (a) and
sodium (b). This figure compares the repulsive interaction calculated for a
rigid background to the net electron-electron interaction when a deformable
background is included. The top two curves are for the rigid background for
opposite and parallel spins. The bottom two curves include the deformable
background contribution.

The wave vector dependent electron-electron interaction for this simple model
of lattice screening in the alkali metals can be calculated using Eq. (13),
Eq. (19) and Eq. (26). This assumes the linear phonon dispersion relation of
Eq. (18). A linear phonon dispersion should be the correct behavior at small
$q$, but overestimates $\omega_{q}$ as $q$ approaches $2\,k_{F}$ (and a
reciprocal lattice vector). Since $\omega_{q}^{2}$ is in the denominator, a
smaller value near $2\,k_{F}$ would imply an even more attractive potential.
The results for lithium and sodium are shown in Fig. 9.

The electron-electron interaction in a rigid background is repulsive. The
repulsion is less for parallel spins than for opposite spins. The deformable
background at low frequencies contributes a negative term due the additional
screening by the deformable background/lattice. The background screening is
due to the net Coulomb interaction and is independent of spin for a non-
magnetized electron gas.

The most interesting feature is that the net electron-electron interaction in
lithium is attractive from $q=0$ to above $2\,k_{F}$, for both parallel and
opposite spins. A more careful treatment of the background/phonons is needed,
but this result is likely to be qualitatively correct. The strong attractive
region in lithium may explain the source of the observed superconductivity.

The other alkali metals with lower densities have more repulsive electron-
electron interactions and smaller effects of the deformable lattice as
measured by the bulk modulus. The lattice screening is still a significant
effect. The repulsive interaction in the rigid lattice falls off more quickly
with wave vector than the attractive contribution from the deformable
background. The resulting net electron-electron interaction has a minimum near
$1.5-2\,k_{F}$. This minimum is slightly repulsive for opposite spins, and
slightly attractive for parallel spins for all the alkali metals. The small
attraction for parallel spins near $2\,k_{F}$ may lead to interesting physics.
Sodium is shown in Fig. 9. The curves for the other alkali metals are similar.

Figure 10 shows $V_{ee}(r)$, the Fourier transform of the electron-electron
interactions in Fig 9. The energy scale is $eV$, and the Fermi energy for
lithium is $4.74\,eV$ and for sodium it is $3.23\,eV$. With a rigid lattice,
the electron-electron interaction is repulsive as shown in the top two curves
of Figure 10. The classical turning points for scattering, where the repulsive
potential is equal to the Fermi energy, are approximately $0.75\,r_{s}$ for
lithium and $0.9r_{s}$ for sodium. Electron-electron scattering does not
contribute to the electrical resistivity because momentum and charge are
conserved in the scattering event (except for a small effect due to umklapp
scattering in a real-lattice as opposed to a uniform background). Electron-
electron scattering does however contribute to the thermal resistivity.

When the deformable background is included, the overall interaction potential
for lithium is attractive for both parallel and opposite spins in the region
$0.25-1.0\,r_{s}$ with a depth comparable to the Fermi energy, with a
repulsive core at shorter distances. At larger distances, not shown in Fig 10,
much smaller oscillations similar to Friedel oscillations are seen. The
interaction or scattering of two electrons in the electron gas with an
attractive potential due to the deformable background could include
resonances, virtual bound states or even bound states. We note that although
the electron-electron attraction is stronger for parallel spins, the overall
wave function which must be anti-symmetric would imply that the spatial part
is anti-symmetric (p-wave) and the probability that the electron is found at
small distances is lower. Thus it is likely that the opposite spin electrons
with a symmetric (s-wave) wave function would sample more of the region of the
attractive potential.

For sodium, only the electron-electron interaction for parallel spins becomes
significantly attractive.

We present this Fourier transform real space calculation to stimulate thinking
within the single particle picture of the electron gas– which has limitations.
The usual thinking is in momentum and frequency space. Equation (13) gives the
explicit frequency dependence. The local field factors used here are
independent of frequency, but the frequency dependence of the Lindhard
function is known.

The electron-electron interaction in this paper can be used in calculations of
superconductivity. The simple inclusion of the deformable lattice shows that
lithium is substantially different from the other alkali metals, and lithium
is the only alkali that shows superconductivity at ambient pressure. Cesium
exhibits superconductivity at high pressure. The net electron-electron
interaction is sensitive to exchange and correlation and to the deformable
lattice. Richardson and AshcroftRichardson and Ashcroft (1997) considered
superconductivity with the KO electron-electron interaction and phonons
treated on an equal basis, and applied the theory to several metals including
lithium. The considerable difficulties in comparison with experiment are
discussed in that paper.

The region of largest static attraction between two electrons in lithium is at
very short distances. The role of this short range attraction in the dynamical
theory of superconductivity is not known to us.

A strong word of caution is needed when comparing electron gas calculations
with experiment, particularly at low density close to the compressibility
divergence at $r_{s}=5.25$. Factors such as effective mass, core polarization
and renormalization factor $z$ have to be carefully considered when comparing
with experiment. These effects were discussed in Kukkonen and WilkinsKukkonen
and Wilkins (1979) and in RefRichardson and Ashcroft (1997). A simple example
is Cesium with $r_{s}=5.62$ which is beyond the compressibility instability.
Cesium and the other alkali metals have polarizable cores with a core
polarization dielectric function $\epsilon_{B}$ that is frequency independent
in the regions of interest. The correct theoryKukkonen and Wilkins (1979) is
an electron gas with electrons of effective mass $m^{\ast}$ in a neutralizing
uniform positive background with dielectric constant $\epsilon_{B}$. The
result is obtained by scaling the known solutions for the electron gas with
mass $m$ and a non-polarizable background, but at a different density
$r_{s}^{\ast}=r_{s}(m^{\ast}/m\epsilon_{B})$ and evaluated at a scaled wave
vector. For cesium, $\epsilon_{B}=1.27$ and this re-normalizes $r_{s}$ from
5.62 to 4.44 which is below the compressibility divergence. The core
polarization corrections for lithium and sodium are quite small.

##  VIII Summary

Variational Diagrammatic Monte Carlo (VDMC) calculations of the wave vector
dependent spin local field factor (exchange and correlation kernel) have been
presented and utilized for all calculations. Using the density and spin local
field factors and explicit equations presented in this paper, all of the
response functions of the three-dimensional electron gas can be easily and
quantitatively calculated. Exchange and correlation are fully included within
the self-consistent local field approximation and the compressibility and
susceptibility sum rules at $q=0$ are satisfied.

The full spin dependent electron-electron interaction is calculated using
these field factors. For a rigid background, the electron-electron interaction
shows no divergence at the compressibility divergence, is repulsive for all
$r_{s}$ in the metallic region.

Considering a deformable background, the $\omega=0$ (static) screening by the
background is shown to be very important with an effect that is as large as
exchange and correlation. A simple calculation shows that with a deformable
background modelled by using the measured bulk modulus, the net electron-
electron interaction including exchange and correlation is attractive
(negative) in a large range of momentum and real space for lithium which does
exhibit superconductivity at ambient pressure, and is mostly positive
(repulsive) for all the other alkali metals.

The compressibility divergence does not appear in the electron-electron and
electron-test charge interactions, but still shows up in the test charge-test
charge interaction as a divergence in the dielectric function and in the
screening by the deformable background.

The quantitative spin dependent electron-electron interaction can be used in
other calculations such as superconductivity, and can be also used as a
starting point for improved numerical calculations. The self-consistent local
field calculation of the KO interaction using Feynman diagrams may lead to new
techniques for identifying massive cancellations of divergent diagrams and
allow new perturbation techniques around the self-consistent solution.

More detailed numerical calculations of the density local field factor
$G_{+}(q)$ from $q=0$ to $3\,k_{F}$ would be welcomed, as well as a physical
explanation for the behavior.

## Acknowledgements

The authors are grateful to William Halperin, Jan Herbst, Giovanni Vignale and
David Ceperley for discussions, and to Giovanni Onida and Massimiliano
Corradini for providing the exact coefficients for their analytic function for
$G_{+}(q)$ that fits the QMC data, and to Tori Hagiya for comparing his
experimental data with our theory.

C.A.K. acknowledges support from the US Social Security Administration. K.C.
appreciates support from the Simons Foundation.

##  Appendix A DENSITY LOCAL FIELD FACTOR $\mathbf{G_{+}(q)}$

The density local field factor has been a subject of research for more than 60
years. $G_{+}(q)$ is needed to calculate the dielectric function and vertex
correction. These quantities are sufficient to calculate all of the
interactions and response functions that do not depend on spin. Many proposals
have been made for the dielectric function and $G_{+}(q)$ particularly by
HubbardHubbard (1958), GeldartGeldart and Vosko (1966) and collaborators, and
SingwiVashishta and Singwi (1972) and collaborators. When it was realized that
the compressibility sum rule was of paramount importance at $q=0$, the
dielectric functions and $G$’s were manually adjusted to satisfy the sum rule
even though the calculations themselves did not satisfy the sum rule. The
calculations were used to interpolate the wave vector dependence above $q=0$.
The behavior of $G_{+}(q)$ at $2\,k_{F}$ and above has been the subject of
considerable research and debate, but is not important for the calculations of
the static response properties of the electron gas because the Lindhard
function cuts off the response quickly above $2\,k_{F}$ .

The Quantum Monte Carlo method was used by Moroni, Ceperley and SenatoreMoroni
et al. (1995) to calculate $G_{+}(q)$ from $q=k_{F}$ to $q=4\,k_{F}$ for
$r_{s}=2,5$ and $10$. Corradini, Del Sole, Onida and PalummoCorradini et al.
(1998) fitted the QMC data with an analytical function that reflected the
appropriate small and large limits. This expression is given in terms of
derivatives of the electron gas energy. The Quantum Monte Carlo calculations
are considered the most accurate, but there are no data below $q=k_{F}$.
Richardson and AshcroftRichardson and Ashcroft (1994) calculated the local
field corrections at finite frequencies, and their static results are similar,
but differ somewhat in detail from the QMC results. They emphasized the
importance of the sum rules. Richardson and Ashcroft also provided an
interpolation formula. Retrospective discussions of the local field factors
were given by Simion and GiulianiSimion and Giuliani (2008) and by Hellal,
Gasser and IssolahHellal et al. (2003).

We plot the Quantum Monte Carlo results of Ref. Hubbard (1958) for $G_{+}(q)$
together with the analytic interpolation function Ref.Geldart and Vosko
(1966), and the even simpler quadratic formula that quite accurately
reproduces the response functions of the electron gas in Fig. 11. Although
there are no data below $q=k_{F}$ , the QMC results above $k_{F}$ show that
$G_{+}(q)$ follows the quadratic required by the compressibility sum rule up
until nearly $2\,k_{F}$ and then falls significantly below the initial
quadratic. Theory predicts that the large $q$ behavior will also be a
quadratic but with a different coefficient. Other earlier versions of
$G_{+}(q)$ Hubbard (1958) had a much smaller value at $2\,k_{F}$.

(a)

(b)

Figure 11: Density local field function $G_{+}(q)$ plotted versus $q/k_{F}$
for $r_{s}=2$ (a) and $r_{s}=5$ (b). Data points are the Quantum Monte Carlo
calculations from Ref. Hubbard (1958). The solid curves that fit the data are
the analytic function from Ref. Geldart and Vosko (1966). The quadratic Eq.
(4) is the proposed simple approximation to $G_{+}(q)$ for calculating the
response functions. Error bars are shown for all data points. If they are not
evident, the error is smaller than the data point.

Figure 11 emphasizes the large $q$ behavior of $G_{+}(q)$. However the static
response functions of the electron gas depend mostly on the low $q$ behavior,
because the Lindhard functions cuts off the effect of $G_{+}(q)$ quickly above
$q=2\,k_{F}$ .

At small $q$, the compressibility sum rule specifies that

$\displaystyle G_{+}(q\rightarrow
0)=\left(1-\frac{\kappa_{0}}{\kappa}\right)\left(\frac{q}{q_{TF}}\right)^{2}$
(23)

To emphasize the low q behavior, $G_{+}(q)/(q/q_{TF})^{2}$ is plotted in Fig.
12 below, where the intercept at $q=0$ is $(1-\kappa_{0}/\kappa)$. Note that
the density exchange and correlation kernel needed for Time-Dependent Density
Functional Theory is given by $f_{xc}=-4\pi G_{+}(q)/(q/q_{TF})^{2}$.

(a)

(b)

Figure 12: $G_{+}(q)/(q/q_{TF})^{2}$, the density local field function divided
by $(q/q_{TF})^{2}$ is plotted versus $q/k_{F}$ for $r_{s}=2$ (a) and
$r_{s}=5$ (b). Data points are the Quantum Monte Carlo calculations from Ref.
Moroni et al. (1995). The solid curves that fit the data are the analytic
function from Ref. Corradini et al. (1998). The straight line represents the
quadratic Eq. (4) that is the proposed simple approximation to $G_{+}(q)$ for
calculating the response functions. Error bars are shown for all data points.

Note again that there are no Quantum Monte Carlo data below $q=k_{F}$. The
$q=0$ values of the quadratic and the analytic function are set by the
compressibility sum rule. The compressibility is also calculated by Monte
Carlo methods, and the $q=0$ value is much more accurate than the q dependent
data. The compressibility sum rule dictates $G_{+}(0)$ and that the initial
behavior will be quadratic, Eq. (23), as represented by the constant
horizontal line. The analytic interpolation formula (14) apparently fits the
data above $1.5k_{F}$ quite well, but misses substantially the data at $k_{F}$
for $r_{s}=2$. This illustrates the problem of global curve fitting with
analytical functions with a limited number of parameters. The simple quadratic
does at least as good a job below $2k_{F}$ and is substantially different at
larger $q$, but the effect at large $q$ is not very important for the response
functions as will be shown below. Looking carefully at the QMC data between
$k_{F}$ and $2\,k_{F}$, there is an intriguing hint of structure in
$G_{+}(q)$, with data both below and above the quadratic. Further Quantum
Monte Carlo calculations from $q=0-3\,k_{F}$ would be informative. The vertex
function which embodies the effect of exchange and correlation in the response
functions is plotted in Fig 13, where the simple quadratic is compared to the
QMC data and the fitting function for this data.

(a)

(b)

Figure 13: Vertex function $\Lambda=1/(1-G_{+}Q)$ plotted versus $q/k_{F}$ for
$r_{s}=2$ (a) and $r_{s}=5$ (b). Data points are the Quantum Monte Carlo
calculations from Ref. Moroni et al. (1995). The dotted curve that fits the
data is the analytic function from Ref. Corradini et al. (1998). The solid
curve uses Eq. (4), the proposed simple quadratic approximation for
calculating the response functions. Error bars are shown for all data points.
If they are not evident, the error is smaller than the data point. Note that
the $y-\text{axis}$ starts at 1.0 for $r_{s}=2$, in order to emphasize small
differences.

The vertex function $\Lambda=1/(1-G_{+}Q)$ is required to calculate the
dielectric function, and the other interactions in the electron gas. Figure 13
shows the vertex function using the actual QMC data and the analytic function
as well as the simple quadratic. The first point to note is that the $q=0$
value of the vertex function is exactly given by the compressibility sum rule
and is equal to $\Lambda(0)=\kappa/\kappa_{0}$ which diverges at the
compressibility divergence as approximately $1/(1-r_{s}/5.25)$. At $r_{s}=2$,
the vertex enhancement is substantial, but it is not near the compressibility
divergence. At $r_{s}=5$, the electron gas is very near the divergence and the
vertex enhancement is very large. The error bar in the value of $G_{+}(q)$ at
$q=k_{F}$ reaches nearly a point of instability and its effect is magnified
because it appears in the denominator. The vertex function is extremely
sensitive to small changes at $r_{s}=5$.

The simple quadratic function for $G_{+}(q)$ yields a vertex function that
satisfies the compressibility sum rule and fits the vertex function derived
from the QMC data as well as the fitting formula of Ref. Geldart and Vosko
(1966). This is despite the large differences at large $q$, because the
contributions at large $q$ are cut off by the Lindhard function. The fitting
function should be used for any calculations that depend on $q$ substantially
above $2k_{F}$.

The fact that the vertex function resulting from the analytic function is
larger than the quadratic for $q$ greater than zero and less than $k_{F}$ is
entirely due to the curve fitting, and there are no QMC data in this region.

The simple quadratic approximation for $G_{+}(q)$ was suggested by Taylor
Taylor (1978) 40 years ago and we concur.

Figure 14: Density local field factor $G_{+}(q)$ from VDMC calculations for
$r_{s}=1$ plotted versus $q/k_{F}$ (a), and $G_{+}(q)/(q/q_{TF})^{2}$ (b).
Error bars are shown.

The current VDMC method works well when the vertex correction or
susceptibility enhancement is modest. This corresponds to $\chi_{0}/\chi$ or
$\kappa_{0}/\kappa<0.6$ in Figure 4. For the spin susceptibility, this is near
$r_{s}=5$. For the compressibility, this corresponds to $r_{s}=2$. As a test,
we have calculated the density local field factor $G_{+}(q)$ for $r_{s}=1$ and
$2$, and the data are shown in Figures 14 and 15. At $r_{s}=2$, we compare the
new VDMC calculations to QMC results of Ref. Moroni et al. (1995) and the
corresponding interpolation formula of Ref. Corradini et al. (1998).

At $r_{s}=1$, Fig. 14 shows that the density local field factor $G_{+}(q)$ has
the same qualitative behavior as the spin local field factor $G_{-}(q)$ which
are shown in Fig. 1 and Fig. 2. The error bars are acceptable. Both show that
$G$ rises above the quadratic at approximately $1.2\,kF$ and then falls below
at $2\,kF$.

Figure 15: Density local field factor $G_{+}(q)$ from VDMC calculations for
$r_{s}=1$ plotted versus $q/k_{F}$ (a), and $G_{+}(q)/(q/q_{TF})^{2}$ (b).
Error bars are shown.

The new data for $G_{+}(q)$ in Fig. 15 demonstrates the limitations of the
current version of the VDMC approach. Ordinarily, we would not show data with
such large error bars. However, we want to compare with the QMC data Ref.
Moroni et al. (1995) and to provide new data below $k_{F}$. The VDMC data with
error bars overlap the QMC data with its error bars except for two points near
$2\,k_{F}$, and even there, the agreement is quite close. The simple quadratic
approximation for $G_{+}(q)$ given in Eq. (4) represents the VDMC data quite
well up to $2\,k_{F}$. The interpolation curve of Ref. Corradini et al. (1998)
fits the data well above $2\,k_{F}$. The data for $G_{+}(q)$ at $r_{s}=2$ are
qualitatively similar to the data for $G_{-}(q)$.

The VDMC data for $G_{-}$ and the limited data for $G_{+}$ show that both of
these local field factors are smooth functions of wave vector. Although the
data rise slightly above the quadratic between 1.5 and $2\,k_{F}$, there is no
evidence of a large peak. We have not developed a physical intuition for this
behavior.

##  Appendix B TEST CHARGE-TEST CHARGE $\mathbf{V_{tt}}$ AND ELECTRON-TEST
CHARGE $\mathbf{V_{et}}$ INTERACTIONS AND DENSITY RESPONSE FUNCTION

We use the same quadratic function for $G_{+}(q)$ to plot the test charge
test-charge and electron-test charge interactions at $r_{s}=2$ and $5$, and
compare them to the Lindhard interaction. For the general reader, a simple
physically motivated derivation of these interactions and the electron-
electron interaction are in Ref. Kukkonen and Overhauser (1979).

The test charge-test charge interaction $V_{tt}$ is the Coulomb potential
generated by a test charge plus the induced screening cloud, and felt by
another test charge.

Figure 16: $V_{tt}(q)$, the test charge-test charge or Coulomb interaction at
$r_{s}=2$ and $5$. The potential is measured in units of $4\pi e^{2}$. The
dashed line is the Lindhard potential which is equal to the Thomas Fermi
potential at $q=0$ with the value $V_{\text{Lindhard}}(0)=1/q_{TF}^{2}$.

The dielectric function $\epsilon(q,\omega)$ is defined by

$V_{tt}=\frac{V_{ext}}{\epsilon}$ (24)

and is written as

$\epsilon=1+\frac{v\Pi^{0}}{1-G_{+}v\Pi^{0}}=1+\Lambda Q,$ (25)

where $\Pi^{0}$ is the Lindhard free electron response function and $v=4\pi
e^{2}/q^{2}$. Note that others may define the response function with a minus
sign. For convenience, we define $Q=v\Pi_{0}$ and the vertex correction
$\Lambda=1/(1-G_{+}Q)$. The potentials are measured in units of $4\pi e^{2}$
so that the Thomas Fermi and Lindhard potentials at $q=0$ are simply
$1/q_{TF}^{2}$. Without exchange and correlation, $G_{+}=0$ and therefore
$\Lambda=1$ and the Lindhard result is obtained.

The electron-test charge interaction $V_{et}$ is simply the test charge - test
charge interaction multiplied by the vertex function.

$V_{et}=\Lambda\,V_{tt}=\frac{V_{ext}}{1+(1-G_{+})Q}$ (26)

For these interactions and response functions of the electron gas in the
metallic region, the large $q$ behavior of the local field factor is not of
significant importance in most applications because the Lindhard function cuts
off quickly above $q=2\,k_{F}$.

Figure 17: The test charge-test charge interactions for $r_{s}=2$ (a) and 5(b)
have been numerically Fourier transformed and are plotted versus distance. The
Fourier transform of the Lindhard function is shown for comparison. The data
are shown from $r/(r_{s}\,a_{0})=0.5-2.2$.

The $q=0$ value of the interaction is set by the compressibility sum rule.
$V_{tt}(0)=(\kappa_{0}/\kappa)/q_{TF}^{2}$ is always less than the Lindhard or
Thomas Fermi value. Both are virtually identical above $2.5\,k_{F}$. The $q$
dependence of $G_{+}(q)$ is only important between $0$ and $2\,k_{F}$.

$V_{tt}(q)$ in Fig 16 looks qualitatively like the Lindhard potential at
$r_{s}=2$, but is dramatically different at $r_{s}\\!=\\!5$. This is due to
the compressibility sum rule which fixes the value at $q=0$, and $G_{+}(q)$
interpolates between $q=0$ and $2\,k_{F}$. According to the compressibility
sum rule, $V_{tt}(q=0)=0$ at the compressibility divergence at $r_{s}=5.25$,
and becomes negative at larger $r_{s}$. It must turn positive again and match
$1/q^{2}$ at $q$ beyond $2\,kF$. We don’t have a physical intuition for
this“overscreening” behavior resulting from a negative dielectric function.

Figure 18: The test charge-test charge interactions for $r_{s}=2$(a) and 5(b)
have been numerically Fourier transformed and are plotted versus distance. The
Fourier transform of the Lindhard function is shown for comparison. The data
are shown from $r/(r_{s}\,a_{0})=2.2-8$ with a factor of 100 magnification
compared to Fig 17

Near the compressibility divergence, the vertex correction and thus the
dielectric function and $V_{tt}$ are extremely sensitive to small changes, and
pressure may be an interesting variable. When applying this formula to real
metals, the effective mass and core polarization will re-normalize the
equations to make the effective $r_{s}^{\ast}$ lower than the actual physical
$r_{s}$. Another Ward identity specifying the renormalization factor $z$ must
also be considered.

The Fourier transforms of the test charge-test charge potentials are shown in
Fig. 17 and Fig. 18 compared to the Lindhard potential.

Although it is not shown in Fig. 17, $V_{tt}$ and $V_{\text{Lindhard}}$
converge at small distances $r/(r_{s}a_{0})<0.5$ (derived from $q>2\,k_{F}$)
and become equal to the bare interaction at even smaller distances. At
intermediate distances (derived from intermediate $q$), $V_{tt}$ is less
repulsive than $V_{\text{Lindhard}}$. At large distances the oscillations in
$V_{tt}$ are larger than the Friedel oscillations in $V_{\text{Lindhard}}$.
This is dramatically different for $r_{s}=5$ where there is a broad attractive
region around the test charge from $r/(r_{s}a_{0})=0.6$ to $1.4$. The
oscillations at larger distances also have a larger amplitude.

Figure 19: The electron-test charge interaction $V_{et}(q)$ at $r_{s}=2$ and
$5$. The dashed curve is the Lindhard potential.

The lattice spacing of alkali metals is approximately 1.1 times $r_{s}$, and
the diameter of the core electrons is about 0.5 times $r_{s}$. The attractive
minimum for $r_{s}=5$ located at $r/(r_{s}a_{0})=0.8$ has a depth of $0.7\,eV$
compared to a cohesive energy of rubidium of $0.85\,eV$. This attraction can
be part of the explanation for the contraction of the inter-atomic spacing in
liquid rubidium that is observed in x-ray scattering experiments as a function
of pressure and temperatureMatsuda et al. (2007).

The electron-test charge interaction $V_{et}$ is shown in Fig. 19. The
electron-test charge interaction at $q=0$ is equal to $1/q_{TF}^{2}$ which is
the same as the Lindhard and Thomas- Fermi interactions (obtained by setting
$G_{+}=0$). The effect of the vertex correction in the numerator is canceled
by the vertex correction in the dielectric function at $q=0$. The effect of
exchange and correlation only occurs between zero and about $2.5\,k_{F}$. For
$r_{s}=2$, the effects of exchange and correlation are small. The effects are
larger for $r_{s}=5$.

A recent paperHagiya et al. (2020) calculated the static density response
function of lithium from a Kramers-Kronig transformation of the dynamic
structure factor measured by inelastic electron scattering. Figure 20 was
prepared by T. Hagiya, the lead experimental author, using the formula for
$X(q)$ including exchange and correlation provided by us compared to the
Random Phase Approximation.

The density response function is given by:

$\displaystyle\chi(q)=\frac{\Pi^{0}}{(1-G_{+}Q)(1+Q/(1-G_{+}Q))}=\frac{\lambda\Pi^{0}}{\varepsilon}.$
(27)

The RPA response function is obtained by setting $G_{+}\\!=\\!1$.

Figure 20: Measured static density response function for lithium from Ref.
Hagiya et al. (2020) compared to theoretical value with no adjustable
parameters. The top curve includes exchange and correlation using $G_{+}(q)$
and the bottom curve uses the Random Phase Approximation which is obtained by
setting $G_{+}(q)=1$.

The theory has no adjustable parameters. The experimentalists point out that
their data is not accurate enough to definitively distinguish between the
response functions using exchange and correlation and the RPA. Nevertheless,
the experimental results are very impressive as is the data analysis.

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