# Measuring correlated electron motion in atoms with the momentum-balance
density

Lucy G. Todd  Department of Chemistry, University of Winnipeg, Winnipeg,
Manitoba, R3B 2E9, Canada  Joshua W. Hollett<EMAIL_ADDRESS>Department of Chemistry, University of Winnipeg, Winnipeg, Manitoba, R3B 2E9,
Canada  Department of Chemistry, University of Manitoba, Winnipeg, Manitoba,
R3T 2N2, Canada

###### Abstract

Three new measures of relative electron motion are introduced: equimomentum,
antimomentum, and momentum-balance. The equimomentum is the probability that
two electrons have the exact same momentum, whereas the antimomentum is the
probability their momenta are the exact opposite. Momentum-balance (MB) is the
difference between the equimomentum and antimomentum, and therefore indicates
if equal or opposite momentum is more probably in a system of electrons. The
equimomentum, antimomentum and MB densities are also introduced, which are the
local contribution to each quantity. The MB and MB density of the
extrapolated-Full Configuration Interaction wave functions of atoms of the
first three rows of the periodic table are analyzed, with a particular focus
on contrasting the correlated motion of electrons with opposite and parallel
spin. Coulomb correlation between opposite-spin electrons leads to a higher
probability of equimomentum, whereas Fermi correlation between parallel-spin
electrons leads to a higher probability of antimomentum. The local
contribution to MB, given an electron is present, is a minimum at the nucleus
and generally increases as the distance from the nucleus increases. There are
also interesting similarities between the effects of Fermi correlation and
Coulomb correlation (of opposite-spin electrons) on MB.

††preprint: MBDatoms v1

##  I Introduction

Most of the difficulties encountered in the study of electronic structure,
both experimentally and computationally, arise from the complexity of the
many-electron wave function. Therefore, both stand to benefit substantially
from the unravelling this complexity. The unravelling can begin by
understanding the relative motion of electrons. Not only does this help
demystify the mesmerizing many-electron wave function, but it also lends
itself to the development of more suitable models of electronic structure and
a more complete picture of quantum mechanics.

Much has been learned about the relative position and momentum of electrons,
and most is due to the analysis of intracules.Coulson and Neilson (1961);
Lester Jr and Krauss (1966); Coleman (1967); Banyard and Ellis (1972); Thakkar
and Smith Jr. (1977); Banyard and Reed (1978); Sharma and Thakkar (1984);
Thakkar, Tripathi, and Smith Jr (1984); Boyd, Sarasola, and Ugalde (1988);
Wang, Tripathi, and Smith Jr (1992); Dominguez  _et al._ (1992); Aguado  _et
al._ (1992); Sarasola  _et al._ (1992); Banyard and Sanders (1993); Cann,
Boyd, and Thakkar (1993a, b, c); Cioslowski and Liu (1996); Fradera, Duran,
and Mestres (1997); Sarsa, Gálvez, and Buendia (1998, 1999); Fradera, Duran,
and Mestres (2000); Koga (2001, 2002); Koga and Matsuyama (2003); Gill,
O’Neill, and Besley (2003); Toulouse, Assaraf, and Umrigar (2007); Dumont,
Crittenden, and Gill (2007); Crittenden, Dumont, and Gill (2007); Crittenden
and Gill (2007); Bernard, Crittenden, and Gill (2008); Piris, Lopez, and
Ugalde (2008); Pearson  _et al._ (2009); Per, Russo, and Snook (2009); Hollett
and Gill (2011); Loos and Gill (2010); Hollett, McKemmish, and Gill (2011);
Hollett and Gill (2012); Bernard, Loos, and Gill (2013); Mercero  _et al._
(2016); Hollett and Li (2017); Rodríguez-Mayorga  _et al._ (2018, 2019); Via-
Nadal  _et al._ (2019) Intracules are two-electron probability distribution
functions.Gill  _et al._ (2006) Constructed from the $N$-electron wave
function (approximate or exact), these reduced probability densities reveal
information regarding the relative position,Coulson and Neilson (1961);
Banyard and Ellis (1972); Sharma and Thakkar (1984); Koga (2002); Hollett and
Gill (2012) momentum,Banyard and Ellis (1972); Banyard and Reed (1978); Koga
(2002); Hollett and Gill (2012) or position and momentum of electrons.Gill and
Chien (2003); Bernard, Loos, and Gill (2013) The analyses often involve the
calculation of intracule holes,Coulson and Neilson (1961); Cioslowski and Liu
(1998) which are the difference between the exact and the Hartree-Fock
intracule. An intracule hole explicitly describes the effects of Coulomb
correlationTew, Klopper, and Helgaker (2007) on that particular intracule, and
subsequent inferences are made regarding the correlated motion of the
electrons. Such studies have led to a better understanding of the relative
motion of electrons due to radial and angular correlation in atoms,Banyard and
Ellis (1972); Hollett, McKemmish, and Gill (2011); Hollett and Gill (2012) the
correlated motion of electrons during bond dissociation,Hollett, McKemmish,
and Gill (2011) and their correlated motion in the presence of a static
electric field.Hollett and Li (2017) However, for most intracules the spatial
information (i.e. absolute location of the electrons) is lost, which impedes
the completion of a picture of relative electron motion in atoms and
molecules.

By using a more complex object, such as the two-electron density, correlated
electron movement can be analyzed in real space through the construction of
Coulomb,Sanders and Banyard (1992); Banyard and Sanders (1993) exchange, or
exchange-correlationWang, Kim, and Baerends (2010) holes. In this context, the
hole is the two-electron density as a function of the position of the second
electron when a test position is chosen for the first.Cooper and Pounder
(1978); Tew, Klopper, and Helgaker (2007) Such an approach can be used to
demonstrate Fermi correlation and the Fermi hole, which occur between
electrons of parallel-spin and is due to the antisymmetry of the electronic
wave function.Tew, Klopper, and Helgaker (2007) This approach is also used to
design and test models for density functional approximations.Baerends (2001)
By choosing a position in the molecule for the test electron, the complexity
of the two-electron density is reduced but a complete picture of the relative
motion of the electrons must be synthesized from the combination of densities
produced from a collection of test electron positions.

Alternatively, there are intracule-like probability densities that are not
reduced to a single (1D or 3D) variable. Such probability densities, like the
intex distribution,Proud and Pearson (2010, 2012); Valderrama, Mercero, and
Ugalde (2001) retain absolute position information, that is often lost with
simpler intracules, and can provide information regarding the relative
positions of electrons at different locations in an atom or molecule. Similar
distributions also exist for momentum space.Reed and Banyard (1980)

In an effort to unravel some of the complexity of the correlated motion of
electrons this article presents multiple new quantum mechanical properties.
The properties are the equi- and antimomentum and momentum-balance, along with
the corresponding equi- and antimomentum density and momentum-balance density.
The momentum-balance is a measure of the correlated motion in an electronic
system, and the momentum-balance density describes the local contribution of
the electronic wave function to the correlated electron motion. In Section II,
expressions for the equimomentum, antimomentum, and momentum balance are
derived, along with the corresponding densities. This is followed by a
description of the basic algorithm for the calculation of these properties in
atoms and molecules using Gaussian basis sets. Section IV presents an analysis
and discussion of the momentum-balance and momentum-balance density in atoms
of the first 17 many-electron elements of the periodic table. Then finally,
the results are summarized and future applications of these properties are
discussed in Section V. Atomic units are used throughout.

##  II Theory

The connection between electron position and momentum is through the Fourier
transform. The $N$-electron momentum-space wave function, $\Phi$, in terms of
the position-space wave function, $\Psi$, is given as

$\Phi(\mathbf{p}_{1},\mathbf{p}_{2},\dots,\mathbf{p}_{N})=\frac{1}{(2\pi)^{3N}}\int
e^{-i\sum_{k=1}^{N}\mathbf{p}_{k}\cdot\mathbf{r}_{k}}\\\
\times\Psi(\mathbf{r}_{1},\mathbf{r}_{2},\dots,\mathbf{r}_{N})d\mathbf{r}_{1}d\mathbf{r}_{2}\dots
d\mathbf{r}_{N},$ (1)

where $\mathbf{p}_{k}$ and $\mathbf{r}_{k}$ are the electron momenta and
positions, respectively. The wave functions are spinless, that is, they have
been previously integrated with respect to the electron spin-coordinates. When
addressing the relative motion of electrons, it is simpler to focus on
electron pairs rather than all $N$-electrons at once. Therefore, the two-
electron reduced density matrix (2-RDM),
$\Gamma(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime})$,
is formed via integration over the $N$-electron wave function,

$\Gamma(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime})=\int\Psi^{*}(\mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime},\dots,\mathbf{r}_{N})\\\
\times\Psi(\mathbf{r}_{1},\mathbf{r}_{2},\dots,\mathbf{r}_{N})d\mathbf{r}_{3}\dots
d\mathbf{r}_{N},$ (2)

where the diagonal of the 2-RDM is the two-electron density,
$\Gamma(\mathbf{r}_{1},\mathbf{r}_{2})=\Gamma(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1},\mathbf{r}_{2})$.
Following Equation (1), the two-electron momentum density,
$\Pi(\mathbf{p}_{1},\mathbf{p}_{2})$, can be formed from the position-space
2-RDM,

$\Pi(\mathbf{p}_{1},\mathbf{p}_{2})=\frac{1}{(2\pi)^{6}}\int
e^{i\left[\mathbf{p}_{1}\cdot(\mathbf{r}_{1}^{\prime}-\mathbf{r}_{1})+\mathbf{p}_{2}\cdot(\mathbf{r}_{2}^{\prime}-\mathbf{r}_{2})\right]}\\\
\times\Gamma(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime})d\mathbf{r}_{1}d\mathbf{r}_{2}d\mathbf{r}_{1}^{\prime}d\mathbf{r}_{2}^{\prime}.$
(3)

The two-electron momentum density gives the simultaneous probability that one
electron has momentum $\mathbf{p}_{1}$ and another has $\mathbf{p}_{2}$. By
inserting a Dirac delta function, $\delta(x)$, and using the identity,
$\delta(x)=\frac{1}{2\pi}\int e^{-ikx}dk$, the probability that two electrons
have the exact same momentum can be calculated,

$\displaystyle\lambda^{+}=$
$\displaystyle\int\Pi(\mathbf{p}_{1},\mathbf{p}_{2})\delta(\mathbf{p}_{1}-\mathbf{p}_{2})d\mathbf{p}_{1}d\mathbf{p}_{2}$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\int\Gamma(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1}+\mathbf{q},\mathbf{r}_{2}-\mathbf{q})d\mathbf{q}d\mathbf{r}_{1}d\mathbf{r}_{2},$
(4)

which will be referred to as the equimomentum. Similarly, the probability that
two electrons have the exact opposite momentum is given by,

$\displaystyle\lambda^{-}=$
$\displaystyle\int\Pi(\mathbf{p}_{1},\mathbf{p}_{2})\delta(\mathbf{p}_{1}+\mathbf{p}_{2})d\mathbf{p}_{1}d\mathbf{p}_{2}$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\int\Gamma(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1}+\mathbf{q},\mathbf{r}_{2}+\mathbf{q})d\mathbf{q}d\mathbf{r}_{1}d\mathbf{r}_{2},$
(5)

which will be referred to as the antimomentum. The difference between the two,
will be referred to as the momentum-balance (MB),

$\mu=\lambda^{+}-\lambda^{-}$ (6)

The MB is positive for a system in which it is more probable that the
electrons have the same momentum, negative for a system in which opposite
momenta are more probable, and zero when both scenarios are equally likely.

If the integrand for the equi- or antimomentum [Equation (II) or (II)] is
integrated over only one of the electron coordinates the result is the local
contribution to each quantity,

$\lambda^{\pm}(\mathbf{r})=\frac{1}{(2\pi)^{3}}\int\Gamma(\mathbf{r},\mathbf{r}_{2},\mathbf{r}+\mathbf{q},\mathbf{r}_{2}\mp\mathbf{q})d\mathbf{r}_{2}d\mathbf{q},$
(7)

the equimomentum density, $\lambda^{+}(\mathbf{r})$, and the antimomentum
density, $\lambda^{-}(\mathbf{r})$. The difference between the two is the MB
density,

$\mu(\mathbf{r})=\lambda^{+}(\mathbf{r})-\lambda^{-}(\mathbf{r})$ (8)

Although $\lambda^{+}(\mathbf{r})$ and $\lambda^{-}(\mathbf{r})$ are not
probability densities, the MB density can still be analyzed to reveal the
regions of a wave function that contribute to electrons having equal or
opposite momenta. Such analysis is analogous to that seen with various types
of energy densities.Ayers, Parr, and Nagy (2002); Anderson, Ayers, and
Hernandez (2010); Bacskay and Nordholm (2017) The sign of the MB density
reveals whether there is a local contribution to equi- or antimomentum,

$\displaystyle\mu(\mathbf{r})>0\>$ $\displaystyle\implies$ contributes to
equal momenta $\displaystyle\mu(\mathbf{r})<0\>$ $\displaystyle\implies$
contributes to opposite momenta

Inspection of the definitions of $\lambda^{+}(\mathbf{r})$ and
$\lambda^{-}(\mathbf{r})$ [Equation (S11)] in terms of the 2-RDM reveals a
strong dependence of those densities and the MB density on the electron
density at $\mathbf{r}$. Therefore, a reduced MB density, $m(\mathbf{r})$, is
defined (similar to an energy potential) to eliminate that dependence,

$m(\mathbf{r})=\frac{\mu(\mathbf{r})}{\rho(\mathbf{r})},$ (9)

where $\rho(\mathbf{r})$ is the one-electron density. It is expected that
$m(\mathbf{r})$ will reveal more fine details of the relative momenta of
electrons in a system in comparison to $\mu(\mathbf{r})$.

For wave functions constructed from an orbital basis, the equi- and
antimomentum can be calculated by contracting the 2-RDM in the given basis,
$\Gamma_{abcd}$, with the appropriate two-electron integrals,

$\lambda^{\pm}=\sum_{abcd}\Gamma_{abcd}\left[abcd\right]_{\lambda^{\pm}}.$
(10)

The two-electron integrals are given by

$\left[abcd\right]_{\lambda^{\pm}}=\frac{1}{(2\pi)^{3}}\int\phi_{a}(\mathbf{r}_{1})\phi_{b}(\mathbf{r}_{1}+\mathbf{q})\\\
\times\phi_{c}(\mathbf{r}_{2})\phi_{d}(\mathbf{r}_{2}\mp\mathbf{q})d\mathbf{q}d\mathbf{r}_{1}d\mathbf{r}_{2},$
(11)

where the orbitals, $\phi_{a}$, are assumed to be real. The integrals exhibit
the very practical identity,

$\left[abcd\right]_{\lambda^{-}}=\left[abdc\right]_{\lambda^{+}}.$ (12)

The equi- and antimomentum densities can be calculated in the same manner,
except the four-orbital integral is replaced by the product of an orbital and
a three-orbital integral,

$\lambda^{\pm}(\mathbf{r})=\sum_{abcd}\Gamma_{abcd}\phi_{a}(\mathbf{r})\left[bcd\right]_{\lambda^{\pm}}.$
(13)

The three-orbital integral is given by,

$\left[bcd\right]_{\lambda^{\pm}}=\frac{1}{(2\pi)^{3}}\int\phi_{b}(\mathbf{r}+\mathbf{q})\\\
\times\phi_{c}(\mathbf{r}_{2})\phi_{d}(\mathbf{r}_{2}\mp\mathbf{q})d\mathbf{q}d\mathbf{r}_{2}$
(14)

which also has a useful identity,

$\left[bcd\right]_{\lambda^{-}}=\left[bdc\right]_{\lambda^{+}}.$ (15)

Recurrence relations for the practical calculation of the molecular integrals,
$\left[abcd\right]_{\lambda^{\pm}}$ and $\left[bcd\right]_{\lambda^{\pm}}$,
over Gaussian-type basis functions of arbitrary angular momentum are provided
in the supplementary material.

Analysis of the relative motion of electrons via MB and the MB density, is
aided by decomposing the 2-RDM into its various spin-components,

$\displaystyle\Gamma(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime})=$
$\displaystyle\Gamma^{\alpha\alpha}(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime})+\Gamma^{\beta\beta}(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime})$
$\displaystyle\Gamma^{\alpha\beta}(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime})+\Gamma^{\beta\alpha}(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime}),$
(16)

where $\alpha$ denotes “spin-up” and $\beta$ denotes ”spin-down”. To
demonstrate the relationship between $\mu$ and correlated electron motion,
consider the Hartree-Fock wave function. Inserting the spin-resolved HF 2-RDM
into Equation (S12) yields an expression for the HF equi- and antimomentum,

$\lambda^{\pm}_{\text{HF}}=\frac{1}{2}\sum_{ij}n^{\alpha}_{i}n^{\alpha}_{j}\left(\left[iijj\right]^{\alpha\alpha}_{\lambda^{\pm}}-\left[ijji\right]^{\alpha\alpha}_{\lambda^{\pm}}\right)\\\
+n^{\beta}_{i}n^{\beta}_{j}\left(\left[iijj\right]^{\beta\beta}_{\lambda^{\pm}}-\left[ijji\right]^{\beta\beta}_{\lambda^{\pm}}\right)\\\
+n^{\alpha}_{i}n^{\beta}_{j}\left(\left[iijj\right]^{\alpha\beta}_{\lambda^{\pm}}+\left[iijj\right]^{\beta\alpha}_{\lambda^{\pm}}\right),$
(17)

where $n_{i}^{\sigma}$ is the occupancy of $\sigma$-spin orbital $i$, with
$\sigma=\alpha$ or $\beta$. The integral identity relating equimomentum and
antimomentum [Equation (S9)] implies that the Coulomb-type integral for both
quantities are equivalent,

$\left[iijj\right]_{\lambda^{-}}=\left[iijj\right]_{\lambda^{+}}.$ (18)

Considering this, and the decomposition of $\mu$ into spin-components,

$\mu_{\text{HF}}=\mu^{\alpha\alpha}_{\text{HF}}+\mu^{\beta\beta}_{\text{HF}}+\mu^{\alpha\beta}_{\text{HF}}+\mu^{\beta\alpha}_{\text{HF}},$
(19)

the following simplification can be made,

$\mu^{\sigma\sigma}_{\text{HF}}=\frac{1}{2}\sum_{ij}n^{\sigma}_{i}n^{\sigma}_{j}\left(\left[ijji\right]^{\sigma\sigma}_{\lambda^{-}}-\left[ijji\right]^{\sigma\sigma}_{\lambda^{+}}\right).$
(20)

The HF MB between electrons of parallel-spin is the difference between the
equi- and antimomentum contributions that result from the exchange interaction
(i.e. antisymmetry). Given that there is no exchange interaction between
electrons of opposite-spin, the HF MB of opposite-spin electrons is exactly
zero,

$\mu^{\sigma\sigma^{\prime}}_{\text{HF}}=0.$ (21)

An analogous result is found for the MB density,

$\mu^{\sigma\sigma}_{\text{HF}}(\mathbf{r})=\frac{1}{2}\sum_{ij}n^{\sigma}_{i}n^{\sigma}_{j}\phi_{i}^{\sigma}(\mathbf{r})\left(\left[jji\right]^{\sigma\sigma}_{\lambda^{-}}-\left[jji\right]^{\sigma\sigma}_{\lambda^{+}}\right),$
(22)

and

$\mu^{\sigma\sigma^{\prime}}_{\text{HF}}(\mathbf{r})=0.$ (23)

It is clear from this result that non-zero MB and MB density only exists when
electron motion is correlated. In other words, if electron motion is
uncorrelated then equimomentum and antimomentum are equally probable,
$\lambda^{+}=\lambda^{-}$ and
$\lambda^{+}(\mathbf{r})=\lambda^{-}(\mathbf{r})$. In the case of the HF wave
function, correlation occurs only between parallel-spin electrons and is due
to the antisymmetry of the wave function, often referred to as Fermi
correlation.Tew, Klopper, and Helgaker (2007) Therefore, the MB and MB density
of correlated (e.g. post-HF) wave functions reveal the presence of correlation
between opposite-spin electrons and the change in correlation between
parallel-spin electrons.

##  III Method

All wave functions (including 1 and 2-RDMs) were obtained using a determinant-
driven selected configuration interaction (sCI) method known as CIPSI
(Configuration Interaction using a Perturbative Selection made
Iteratively)Huron, Malrieu, and Rancurel (1973); Giner, Scemama, and Caffarel
(2013, 2015) in which the energies are extrapolated to the full configuration
interaction (FCI) result using multireference perturbation theory.Garniron
_et al._ (2017); Loos  _et al._ (2018); Garniron  _et al._ (2019) The all-
electron extrapolated-FCI (exFCI) calculations were performed using Quantum
Package 2.0.Garniron  _et al._ (2019) The wave functions and RDMs for He were
obtained using the cc-pVTZ, aug-cc-pVTZ and aug-cc-pVQZ/f basis sets,Dunning
Jr. (1989); Woon and Dunning Jr. (1994) and all calculations beyond He were
performed using the aug-cc-pCVTZ/f basis set.Prascher  _et al._ (2011); Feller
(1996); Schuchardt  _et al._ (2007); Pritchard  _et al._ (2019)

Using the RDMs provided by Quantum Package 2.0, the equimomentum,
antimomentum, and MB, along with their densities were calculated using
MUNgauss.Poirier, Hollett, and Warburton (2015) The densities were calculated
on a two-dimensional mesh, with a grid-spacing of 0.05 bohr, for
visualization. The densities were also numerically integrated on the SG-1
gridGill, Johnson, and Pople (1993) for comparison to analytically calculated
equimomentum, antimomentum and MB values.

##  IV Results

###  IV.1 Opposite-spin correlation

As an initial assessment of basis set dependence, the MB density of the He
atom obtained using three different Dunning basis sets is shown in Figure 1.

Figure 1:  Comparison of momentum-balance density of He atom for multiple
Dunning correlation-consistent basis sets.

It is seen that there is significant deviation between the MB density
calculated using the cc-pVTZ basis set and the density calculated using the
aug-cc-pVTZ basis set. The deviation is most pronounced at the nucleus and
decays as the distance from the nucleus increases. Therefore, it appears the
addition of diffuse functions, or “augmenting” the basis set, (cc-pVTZ to aug-
cc-pVTZ) is more important to an accurate description of the MB density than
adding an extra set of valence basis functions (aug-cc-pVTZ to aug-cc-pVQZ).
Furthermore, the MB density is far more sensitive to basis set than the one-
electron density (Figure 2). This may not be surprising considering it is a
comparison between a one-electron property and a two-electron property. All
subsequent properties of He that follow are obtained using the aug-cc-pVQZ
basis set.

Figure 2:  Comparison of total one-electron density of He atom for multiple
Dunning correlation-consistent basis sets.

In Figure 3, the He atom MB density, $\mu(\mathbf{r})$, reduced MB density,
$m(\mathbf{r})$, and one-electron density, $\rho(\mathbf{r})$ are plotted
together.

Figure 3:  Total momentum-balance density and reduced momentum-balance density
of He, with the one-electron density.

Evident from Figures 1 and 3, $\mu(\mathbf{r})$ for He is positive everywhere,
and results in a total MB of 0.00373. The MB is the difference between an
equimomentum $\lambda^{+}=0.02481$ and an antimomentum $\lambda^{-}=0.02107$.
In other words, if one could measure the simultaneous momenta of the two
electrons of ground state He 100 000 times, they would find the electrons have
the exact same momentum 2 481 times and the exact opposite momentum 2 107
times. The difference, which is the MB, is due to Coulomb correlation between
the $\alpha$ and $\beta$ electron. Their correlated motion favours
equimomentum over antimomentum, which is consistent with earlier studies of
angular correlation in the He atom.Banyard and Reed (1978); Koga and Matsuyama
(2003); Hollett and Gill (2012)

From Figures 1 and 2 it is clear that $\mu(\mathbf{r})$ closely follows the
shape of the $\rho(\mathbf{r})$, albeit with a much smaller magnitude and
slower decay. As mentioned previously, this is due to the dependence of the
two-electron density on the one-electron density (i.e. an electron must be
present to interact). This dependency is removed in the reduced MB density,
$m(\mathbf{r})$, which can be viewed as a MB potential. It is seen that
$m(\mathbf{r})$ is actually a minimum at the nucleus and the potential
contribution to MB increases as the distance from the nucleus increases. Of
course, while $m(\mathbf{r})$ continues to grow as $r$ increases,
$\rho(\mathbf{r})$ decays and eventually extinguishes the contribution.

Figure 4 presents the $\alpha\beta$ MB density of He, Li and Be divided by the
number of $\alpha\beta$ electron pairs.

Figure 4:  Renormalized $\mu^{\alpha\beta}(\mathbf{r})$ of He, Li, Be and
Be(fc) [frozen core]. The difference between the all-electron and frozen core
$\mu^{\alpha\beta}(\mathbf{r})$ for Be is included.

It also includes $\mu^{\alpha\beta}(\mathbf{r})$ of Be calculated using a
frozen-core approximation to ascertain the role of the valence electrons in
$\mu^{\alpha\beta}(\mathbf{r})$. The frozen-core
$\mu^{\alpha\beta}(\mathbf{r})$ closely resembles the fully correlated
$\mu^{\alpha\beta}(\mathbf{r})$ which, unlike He and Li, has a node at small
$r$ and a negative peak at the nucleus. It is evident that the valence
electrons are mainly responsible for the structure of
$\mu^{\alpha\beta}(\mathbf{r})$ for Be. The difference between the frozen-core
and fully correlated $\mu^{\alpha\beta}(\mathbf{r})$ reveals the contribution
from the core, and core-valence, correlation and it is very similar to the
$\mu^{\alpha\beta}(\mathbf{r})$ of He and Li.

The reduced MB densities, divided by the number of $\beta$ electrons, of Be,
Li and He are presented in Figure 5. Note, that for the spin-decomposed
reduced MB density,
$m^{\sigma\sigma^{\prime}}(\mathbf{r})=\frac{m^{\sigma\sigma^{\prime}}(\mathbf{r})}{\rho^{\sigma}(\mathbf{r})}$.

Figure 5:  Renormalized $m^{\alpha\beta}(\mathbf{r})$ of He, Li, and Be.

The $m^{\alpha\beta}(\mathbf{r})$ are similar, in that they have a minimum at
the nucleus and are everywhere positive. However, the
$m^{\alpha\beta}(\mathbf{r})$ of Li reaches a maximum at $\sim$1.4 bohr and
then decays due to the fact that $\rho^{\alpha}(\mathbf{r})$ decays much
slower than $\rho^{\beta}(\mathbf{r})$, unlike He and Be. The amplitude of the
Be $m^{\alpha\beta}(\mathbf{r})$, which is dominated by the valence electrons,
is significantly larger than that of He and Li. Near the nucleus
$m^{\alpha\beta}(\mathbf{r})$ of Be is negative and relatively flat. It
reaches a minimum at 0.374 bohr and has node at 0.549 bohr. After crossing the
node, $m^{\alpha\beta}(\mathbf{r})$ increases sharply with a shoulder just
beyond $r=1$ bohr. The $m^{\alpha\beta}(\mathbf{r})$ of He also exhibits a
shoulder, although it is less pronounced and further from the nucleus ($\sim$
2.4 bohr).

The $\mu^{\alpha\beta}(\mathbf{r})$ for Be to Ne are presented in Figure 6.

Figure 6:  Renormalized $\mu^{\alpha\beta}(\mathbf{r})$ of the atoms Be to Ne
(above) and N to Ne (below).

With the exception of N, all the atoms have a $\mu^{\alpha\beta}(\mathbf{r})$
with the the same shape as that of Be. Proceeding from Be to C the amplitude
per $\alpha\beta$ electron pair decreases and the node contracts toward the
nucleus, in the same manner as the node of a 2$s$ atomic orbital with
increasing nuclear charge. Interestingly, $\mu^{\alpha\beta}(\mathbf{r})$ for
the N atom has a positive peak at the nucleus, a minimum near $r=0.3$ bohr,
and is positive everywhere. This indicates a reduction in the type of
correlated motion between the valence electrons, exhibited by the electrons of
Be to C, to such an extent that other correlated motion dominates
$\mu^{\alpha\beta}(\mathbf{r})$, such as the relative motion of core, and
core-valence electron pairs. With the addition of $\beta$-spin $p$-electrons,
the $\mu^{\alpha\beta}(\mathbf{r})$ of O to Ne have a shape similar to the
$\mu^{\alpha\beta}(\mathbf{r})$ of those atoms before N. From O to Ne, the
amplitude of the maximum per electron pair decreases, while the amplitude of
the minimum at the nucleus increases, and the location of the node remains
relatively stationary.

For a description of the correlated momemnta that is unbiased by
$\rho^{\alpha}(\mathbf{r})$, the reduced MB densities, divided by the number
of $\beta$ electrons, of Be to Ne are presented in Figure 7.

Figure 7:  Renormalized $m^{\alpha\beta}(\mathbf{r})$ of the atoms Be to Ne
(above) and N to Ne (below).

As expected from the analysis of $\mu^{\alpha\beta}(\mathbf{r})$, all
$m^{\alpha\beta}(\mathbf{r})$ have the same shape with the exception of that
for N, which is positive everywhere. The amplitude of
$m^{\alpha\beta}(\mathbf{r})/N^{\beta}$ decreases substantially moving from Be
to C. Far from the nucleus ($r>1.5$ bohr), the magnitude of
$m^{\alpha\beta}(\mathbf{r})/N^{\beta}$ also decreases from O to Ne. However,
the amplitude of $m^{\alpha\beta}(\mathbf{r})/N^{\beta}$ near the nucleus is
less ordered.

The $\mu^{\alpha\beta}(\mathbf{r})$, per $\alpha\beta$ electron pair, for the
third period atoms are presented in Figure 8.

Figure 8:  Renormalized $\mu^{\alpha\beta}(\mathbf{r})$ of the atoms Na to Ar.

The shape of $\mu^{\alpha\beta}(\mathbf{r})$ for the Na atom is the same as
that for Ne. This is due to the fact that the $\beta$-electron configuration
remains the same. Once a $\beta$ electron is added (Mg atom and beyond), the
magnitude of $\mu^{\alpha\beta}(\mathbf{r})$ per $\alpha\beta$ electron pair
increases substantially, similar to that seen for Be in the second period.
Unlike atoms of the second period, the $\mu^{\alpha\beta}(\mathbf{r})$ of Mg
and beyond have two nodes (like the $3s$ atomic orbital) and are positive at
the nucleus. Yet again, like the second period atoms,
$\mu^{\alpha\beta}(\mathbf{r})>0$ in the valence region and beyond.

The $\alpha\beta$ reduced MB densities, per $\beta$ electron, for the third
period atoms are presented in Figure 9.

Figure 9:  Renormalized $m^{\alpha\beta}(\mathbf{r})$ of the atoms Na to Ar.

As suggested by their $\mu^{\alpha\beta}(\mathbf{r})$, the
$m^{\alpha\beta}(\mathbf{r})$ of Na closely resembles that of Ne. The
$m^{\alpha\beta}(\mathbf{r})$ of Mg, like Be of the second period, has the the
largest amplitude per $\beta$ electron. Close to the nucleus, the features of
$m^{\alpha\beta}(\mathbf{r})$ are the same as those for the second period
atoms, except the sign is reversed. These features are nested inside the same
features (a minimum and a node) repeated, but with the same sign as those of
the second period atoms. It is also interesting to note that, unlike N, the
$m^{\alpha\beta}(\mathbf{r})$ for the $\ce{P}$ atom has the same number of
nodes as the other atoms of the period. However, like N, the amplitude of
$m^{\alpha\beta}(\mathbf{r})$ for P is relatively minuscule.

The trend in $\mu^{\alpha\beta}$ for the second and third period atoms is
illustrated in Figure 10.

Figure 10:  Opposite-spin momentum-balance of 2nd and 3rd row atoms.

For all atoms $\mu^{\alpha\beta}>0$, which corresponds to opposite-spin
electrons preferring to have the same momentum rather than opposite, and
periodicity is followed quite closely. There is a sharp increase in
$\mu^{\alpha\beta}$ moving from group 1 to 2 (i.e. one valence electron to
two) which illustrates the importance of valence electron motion to
$\mu^{\alpha\beta}$. As atomic number increases, and more electrons are added
to the valence shell, the value of $\mu^{\alpha\beta}$ decreases, which
suggests less correlated motion.

For open-shell systems
$\mu^{\alpha\beta}(\mathbf{r})\neq\mu^{\beta\alpha}(\mathbf{r})$, and
subsequently $m^{\alpha\beta}(\mathbf{r})\neq m^{\beta\alpha}(\mathbf{r})$.
However, the features observed in $\alpha\beta$ densities are very similar to
those observed in the $\beta\alpha$ densities. Therefore, plots of
$\mu^{\beta\alpha}(\mathbf{r})$ and $m^{\beta\alpha}(\mathbf{r})$ can be found
in the supplementary material.

###  IV.2 Parallel-spin correlation

As mentioned previously, Fermi correlation contributes to the MB of parallel-
spin electrons. Not surprisingly, the contribution of Fermi correlation is
orders of magnitude larger than that of Coulomb correlation. Consider the Ne
atom, where $\mu_{\text{HF}}^{\sigma\sigma}(\mathbf{r})=-0.08644$ and
$\mu^{\sigma\sigma}(\mathbf{r})=-0.08663$, which means Coulomb correlation
contributes $\sim$0.2% to the overall parallel-spin MB. Furthermore, Fermi
correlation favours $\mu^{\sigma\sigma}<0$, which is parallel-spin electrons
having the exact opposite momentum.

The $\alpha\alpha$ MB density per electron pair,
$\frac{\mu^{\alpha\alpha}(\mathbf{r})}{N^{\alpha}(N^{\alpha}-1)/2}$, for atoms
of the second period are presented in Figure 11.

Figure 11:  Renormalized $\mu^{\alpha\alpha}(\mathbf{r})$ of the atoms Li to
Ne.

It is seen that $\mu^{\alpha\alpha}(\mathbf{r})$ is negative everywhere,
meaning all regions contribute to electrons having the exact opposite
momentum. Similar to the opposite-spin MB density, the amplitude of
$\mu^{\alpha\alpha}(\mathbf{r})$ is relatively small for Li. For the Be atom,
which has only one valence $\alpha$-electron, the amplitude of
$\mu^{\alpha\alpha}(\mathbf{r})$ is much more appreciable. But, it is still
significantly smaller than that for atoms with multiple valence
$\alpha$-electrons. The addition of $p$-electrons for B, and the proceeding
atoms, creates a shoulder in $\mu^{\alpha\alpha}(\mathbf{r})$ further from the
nucleus. Moving from B to Ne the shoulder contracts to the nucleus and the
amplitude of $\mu^{\alpha\alpha}(\mathbf{r})$ per electron pair decreases.

Figure 12 presents the $\alpha\alpha$ reduced MB density, divided by the
number of $\alpha$ electrons, for the second period atoms.

Figure 12:  Renormalized $m^{\alpha\alpha}(\mathbf{r})$ of the atoms Li to Ne.

The amplitude of $m^{\alpha\alpha}(\mathbf{r})/N^{\alpha}$ increases slightly
from Li to Be, and then substantially from Be to B. This is accompanied by a
change in the shape of $m^{\alpha\alpha}(\mathbf{r})$ due to the addition of a
$p$-electron. The amplitude for B is the largest amongst the atoms of the
second period, and the amplitude gradually decreases from B to Ne. As with
opposite-spin MB density, the magnitude of the reduced MB is a minimum at the
nucleus, and the $m^{\alpha\alpha}(\mathbf{r})/N^{\alpha}$ converge to a small
range of values at the origin.

The $\mu^{\alpha\alpha}(\mathbf{r})$ per electron pair for the third period
atoms are plotted in Figure 13.

Figure 13:  Renormalized $\mu^{\alpha\alpha}(\mathbf{r})$ of the atoms Na to
Ar.

Similar to the $\mu^{\alpha\alpha}(\mathbf{r})$ of the second period atoms,
the $\mu^{\alpha\alpha}(\mathbf{r})$ of the third period are negative
everywhere. The $\mu^{\alpha\alpha}(\mathbf{r})$ of the third period atoms
have extra features due to the presence of additional core electrons. The
$\mu^{\alpha\alpha}(\mathbf{r})$ of Na and Mg resemble that of Ne. The
secondary minimum near the nucleus decreases in magnitude and almost
disappears as $\alpha$ $p$-electrons are added from Al to P. For those atoms
with $\alpha$ $p$-electrons, there is a less pronounced shoulder at larger
$r$.

The $\alpha\alpha$ reduced MB density per $\alpha$ electron is presented for
the third period atoms in Figure 14.

Figure 14:  Renormalized $m^{\alpha\alpha}(\mathbf{r})$ of the atoms Na to Ar.

The transition from atoms without valence $p$-electrons to those with them is
obvious. The $m^{\alpha\alpha}(\mathbf{r})$ of both Na and Mg have minima in
their valence region (similar to Li and Be), whereas the atoms beyond Mg
exhibit a shoulder and continue to increase in amplitude. Also similar to the
second period, the maximum amplitude of
$m^{\alpha\alpha}(\mathbf{r})/N^{\alpha}$ is reached by the atom with only one
$p$-electron and the amplitude decreases across the period. Furthermore, the
values of $m^{\alpha\alpha}(\mathbf{r})/N^{\alpha}$ converge to a short range
of near-zero values close to the nucleus.

Finally, the trend in the total MB between parallel-spin electrons for atoms
of the second and third period is illustrated in Figure 15. The actual values
can be found in Table I of the supplementary material.

Figure 15:  Parallel-spin momentum balance of 2nd and 3rd row atoms.

Fermi correlation, or exchange, is mainly responsible for the shape of
$\mu^{\sigma\sigma}(\mathbf{r})$ and the subsequent value of
$\mu^{\sigma\sigma}$. In agreement with the findings of Koga,Koga (2002)
concerning the the angular separation of electron momenta in atoms, the
magnitude of MB is strongly dependent on the $p$-electrons. This is seen with
the dramatic decrease in the value of $\mu^{\alpha\alpha}$ from group 2 to 13,
and the decrease in $\mu^{\beta\beta}$ seen from group 15 to 16.
Interestingly, the minimum value of $\mu^{\alpha\alpha}$ occurs at group 14,
rather than group 15 which has the maximum number of unpaired $p$-electrons.
Also, it is seen that $\mu^{\beta\beta}$ slightly decreases when $\alpha$
$p$-electrons are added (group 2 to 15), and $\mu^{\alpha\alpha}$ decreases
significantly as $\beta$ $p$-electrons are added (group 15 to 17).

The decrease in correlated electron motion, as indicated by the decrease in
the magnitude of $\mu^{\alpha\alpha}$, and the actual overall parallel-spin MB
($\mu^{\alpha\alpha}+\mu^{\beta\beta}$), from group 14 to 18, is similar to
the decrease seen in opposite-spin MB as the number of valence electrons is
increased (10). Due to the calculation of results short of the complete basis
set limit, it is somewhat precarious to draw firm conclusions regarding the
nature of this decrease for opposite-spin electrons alone. However, $>$99% of
parallel-spin MB is due to Fermi correlation which, when considering
magnitude, has been modelled precisely. Therefore, the most reasonable
explanation for the reduction in correlated motion amongst parallel-spin
electrons is the decrease in space available as the electrons are crowded into
the valence region. It is expected that the same phenomenon affects the
Coulomb correlation of opposite-spin electrons.

As with the opposite-spin MB density analysis, the parallel MB densities and
reduced MB densities for $\beta$ electrons are given in the supplementary
material due to the similarities in features and trends compared to the
$\alpha$ electron densities.

###  IV.3 Comparing correlation

As previously mentioned, the effect of Fermi correlation on relative momenta
is orders of magnitude larger than that of Coulomb correlation between
parallel-spin electrons. The effect of Coulomb correlation on opposite-spin
electrons is not as small as that on parallel-spin electrons, but there is a
difference in magnitude compared to Fermi correlation. There is also a
difference in the relative direction of the correlated momenta; Coulomb
correlation of opposite-spin electrons favours equimomentum, whereas Fermi
correlation favours antimomentum. Figures 16 and 17 compare the reduced MB
density for Ne and Ar by scaling the opposite-spin density so that it is
similar in magnitude to that of the parallel-spin density, and by reversing
the sign.

Figure 16:  Comparison between $m^{\alpha\alpha}(\mathbf{r})$ and scaled
$m^{\alpha\beta}(\mathbf{r})$ of Ne. Figure 17:  Comparison between
$m^{\alpha\alpha}(\mathbf{r})$ and scaled $m^{\alpha\beta}(\mathbf{r})$ of Ar.

It is seen for both Ne and Ar, that while $m^{\alpha\alpha}(\mathbf{r})$ and
$m^{\alpha\beta}(\mathbf{r})$ differ in sign, and by an order of magnitude,
their shapes are quite similar. Both $m^{\alpha\alpha}(\mathbf{r})$ and
$m^{\alpha\beta}(\mathbf{r})$ have minimum amplitude at the nucleus and are
relatively flat in the core region. Both increase in magnitude beyond the core
and have a slight shoulder as $r$ increases from the core region to the
valence region. In the valence region, both increase in amplitude towards the
extent of appreciable electron density. At large $r$, the $m(\mathbf{r})$
begin to differ more. For instance, the $m^{\alpha\beta}(\mathbf{r})$ of Ne
approaches a maximum near $r=3$, while $m^{\alpha\alpha}(\mathbf{r})$
continues to increase in amplitude. However, the striking similarities of
these densities, which are intimately connected to how opposite-spin and
parallel-spin electrons avoid each other, cannot be overlooked.

##  V Conclusions

A more thorough understanding of the correlated motion of electrons (i.e. how
electrons avoid each other) is key to unravelling the complexity of the
$N$-electron wave function. Such understanding not only allows for the
conception of new, more effective, models of electronic structure, but it also
increases our basic knowledge of quantum mechanics. The equimomentum,
antimomentum and momentum-balance, along with their corresponding densities,
provide a means for acheiving such understanding. In addition to being a
completely new descriptor of the motion of electrons, MB also serves as a
correlated motion detector.

The MB density, $\mu(\mathbf{r})$, povides a measure of the local contribution
to momentum-balance in a system of electrons. Due to the dependence of
$\mu(\mathbf{r})$ on the probability an electron is found at $\mathbf{r}$, it
resembles the one-electron density. The reduced MB density, $m(\mathbf{r})$,
removes this dependence and amplifies the more subtle aspects of the local
contributions to MB.

The MB, MB density, and reduced MB density can be decomposed into spin
contributions. The analysis of the opposite-spin and parallel-spin MB, MB
density, and reduced MB density of the atoms He to Ar has revealed significant
differences between the correlated motion of opposite-spin and parallel-spin
electrons. Fermi correlation, or exchange, causes parallel-spin electrons to
move with opposite momentum more than equal momentum. For electrons of
opposite spin, Coulomb correlation causes them to move with equal momentum
more than opposite momentum. The amount of correlated motion between parallel-
spin electrons is strongly dependent on the angular momentum of the electrons
present. Whereas, for opposite-spin electrons, the amount of correlated motion
is dependent on simply the number of valence electrons present. Also, the
effect of Fermi correlation on relative electron momenta is an order of
magnitude larger than that of Coulomb correlation.

These new tools, particularly the reduced MB density, have also revealed some
subtle similarities between the correlated motion of parallel-spin and
opposite-spin electrons. For both, correlated motion, given an electron is
present, is a minimum near the nucleus and grows as the distance from the
nucleus is increased.

The quantities presented, their derivations, and practical algorithms for
their calculation are generalized to abitrary systems of multiple nuclei.
Therefore, a natural next step is the investigation of correlated electron
motion in molecules. In addition, because MB and the MB densities reveal the
presence of electron correlation, they are useful tools for understanding the
role of electron correlation in different chemical phenomena (e.g. bond
formation/breakage, excited states, dispersion). They also provide a useful
tool for assessing the quality of various models of electronic structure.
These are just a few of the potential applications of these unique quantum
mechanical properties.

## Supplementary Material

See supplementary material for recurrence relations for integrals over
Gaussian-type basis functions, and additional equations related to the
algorithm for the computation of equimomentum, antimomentum, momentum-balance
and their corresponding densities. Also, see supplementary material for plots
of $\beta\alpha$ and $\beta\beta$ MB densities, as well as numerical values
for total atomic MB.

###### Acknowledgements.

JWH thanks the Natural Sciences and Engineering Research Council of Canada
(NSERC) for a Discovery Grant, Compute/Calcul Canada for computing resources
and the Discovery Institute for Computation and Synthesis for useful
consultations.

## Data Availability

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

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Supplementary Material: Measuring correlated electron motion in atoms with the
momentum-balance density

## S.I Recurrence relations for $\lambda^{+}$

The equi- and antimomentum, in terms of the spinless 2-RDM, $\Gamma$, are
given by,

$\lambda^{\pm}=\frac{1}{(2\pi)^{3}}\int\Gamma(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1}+\mathbf{q},\mathbf{r}_{2}\mp\mathbf{q})d\mathbf{q}d\mathbf{r}_{1}d\mathbf{r}_{2}.$
(S1)

Expanding the 2-RDM over a general set of basis functions, $\\{\phi_{r}\\}$,
leads to the alternative expression for the equi- and antimomentum,

$\lambda^{\pm}=\sum_{rstu}\Gamma_{rstu}\left[rstu\right]_{\lambda^{\pm}},$
(S2)

where the two-electron integral over four basis functions is given by,

$\left[rstu\right]_{\lambda^{\pm}}=\frac{1}{(2\pi)^{3}}\int\phi_{r}(\mathbf{r}_{1})\phi_{s}(\mathbf{r}_{1}+\mathbf{q})\\\
\times\phi_{t}(\mathbf{r}_{2})\phi_{u}(\mathbf{r}_{2}\mp\mathbf{q})d\mathbf{q}d\mathbf{r}_{1}d\mathbf{r}_{2}.$
(S3)

In the case of a Gaussian basis set, the fundamental $\lambda^{+}$
integralGill, Johnson, and Pople (1991); Gill (1994) over $s$-type functions
with exponents $\alpha$, $\beta$, $\gamma$, $\delta$ and centers $\mathbf{A}$,
$\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$ is given by,

$\left[\mathbf{0}\mathbf{0}\mathbf{0}\mathbf{0}\right]_{\lambda^{+}}=\frac{\pi^{3/2}e^{-\sigma
R^{2}}}{8\left(\alpha\beta\gamma+\beta\gamma\delta+\alpha\beta\delta+\alpha\gamma\delta\right)^{3/2}},$
(S4)

where

$\sigma=\frac{\alpha\beta\gamma\delta}{\alpha\beta\gamma+\beta\gamma\delta+\alpha\beta\delta+\alpha\gamma\delta}$
(S5)

and

$\mathbf{R}=\mathbf{A}+\mathbf{C}-\mathbf{B}-\mathbf{D}.$ (S6)

Integrals of higher angular momentum can be constructed using a five-term
recurrence relation (RR).Hollett and Gill (2011) The RRs for augmenting the
angular momentum of the Gaussians on each centre, $\mathbf{A}$, $\mathbf{B}$,
$\mathbf{C}$ and $\mathbf{D}$, in the $i$-direction (where $i=x,y$ or $z$) are
given by

$\displaystyle[(\mathbf{a}+\bm{1}_{i})\mathbf{b}\mathbf{c}\mathbf{d}]_{\lambda^{+}}=$
$\displaystyle-\frac{\sigma}{\alpha}R_{i}[\mathbf{a}\mathbf{b}\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+a_{i}\frac{\alpha-\sigma}{2\alpha^{2}}[(\mathbf{a}-\bm{1}_{i})\mathbf{b}\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+b_{i}\frac{\sigma}{2\alpha\beta}[\mathbf{a}(\mathbf{b}-\bm{1}_{i})\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle-
c_{i}\frac{\sigma}{2\alpha\gamma}[\mathbf{a}\mathbf{b}(\mathbf{c}-\bm{1}_{i})\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+d_{i}\frac{\sigma}{2\alpha\delta}[\mathbf{a}\mathbf{b}\mathbf{c}(\mathbf{d}-\bm{1}_{i})]_{\lambda^{+}},$
(S7a)
$\displaystyle[\mathbf{a}(\mathbf{b}+\bm{1}_{i})\mathbf{c}\mathbf{d}]_{\lambda^{+}}=$
$\displaystyle\frac{\sigma}{\beta}R_{i}[\mathbf{a}\mathbf{b}\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+a_{i}\frac{\sigma}{2\alpha\beta}[(\mathbf{a}-\bm{1}_{i})\mathbf{b}\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+b_{i}\frac{\beta-\sigma}{2\beta^{2}}[\mathbf{a}(\mathbf{b}-\bm{1}_{i})\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+c_{i}\frac{\sigma}{2\beta\gamma}[\mathbf{a}\mathbf{b}(\mathbf{c}-\bm{1}_{i})\mathbf{d}]_{\lambda^{+}}$
$\displaystyle-
d_{i}\frac{\sigma}{2\beta\delta}[\mathbf{a}\mathbf{b}\mathbf{c}(\mathbf{d}-\bm{1}_{i})]_{\lambda^{+}},$
(S7b)
$\displaystyle[\mathbf{a}\mathbf{b}(\mathbf{c}+\bm{1}_{i})\mathbf{d}]_{\lambda^{+}}=$
$\displaystyle-\frac{\sigma}{\gamma}R_{i}[\mathbf{a}\mathbf{b}\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle-
a_{i}\frac{\sigma}{2\alpha\gamma}[(\mathbf{a}-\bm{1}_{i})\mathbf{b}\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+b_{i}\frac{\sigma}{2\beta\gamma}[\mathbf{a}(\mathbf{b}-\bm{1}_{i})\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+c_{i}\frac{\gamma-\sigma}{2\gamma^{2}}[\mathbf{a}\mathbf{b}(\mathbf{c}-\bm{1}_{i})\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+d_{i}\frac{\sigma}{2\gamma\delta}[\mathbf{a}\mathbf{b}\mathbf{c}(\mathbf{d}-\bm{1}_{i})]_{\lambda^{+}},$
(S7c)
$\displaystyle[\mathbf{a}\mathbf{b}\mathbf{c}(\mathbf{d}+\bm{1}_{i})]_{\lambda^{+}}=$
$\displaystyle\frac{\sigma}{\delta}R_{i}[\mathbf{a}\mathbf{b}\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+a_{i}\frac{\sigma}{2\alpha\delta}[(\mathbf{a}-\bm{1}_{i})\mathbf{b}\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle-
b_{i}\frac{\sigma}{2\beta\delta}[\mathbf{a}(\mathbf{b}-\bm{1}_{i})\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+c_{i}\frac{\sigma}{2\gamma\delta}[\mathbf{a}\mathbf{b}(\mathbf{c}-\bm{1}_{i})\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+d_{i}\frac{\delta-\sigma}{2\delta^{2}}[\mathbf{a}\mathbf{b}\mathbf{c}(\mathbf{d}-\bm{1}_{i})]_{\lambda^{+}},$
(S7d)

where $\mathbf{a}=(a_{x},a_{y},a_{z})$ is a vector of angular momentum quantum
numbers, and $\bm{1}_{i}$ is a unit vector with value 1 in the $i$-direction
[e.g. $\bm{1}_{x}=(1,0,0)$].

The momentum-balance (MB) is the difference between the equi- and
antimomentum,

$\mu=\lambda^{+}-\lambda^{-}.$ (S8)

The effort required to calculate MB is halved by making use of the following
identity,

$[rstu]_{\lambda^{-}}=[rsut]_{\lambda^{+}}.$ (S9)

Insertion of the 2-RDM expansion (Equation S12) into Equation S8 and use of
the identity above (Equation S9) leads to a practical equation for the
calculation of MB,

$\mu=\sum_{rstu}\Gamma_{rstu}\left([rstu]_{\lambda^{+}}-[rsut]_{\lambda^{+}}\right).$
(S10)

The equi- and antimomentum densities are given by the following equation,

$\lambda^{\pm}(\mathbf{r})=\frac{1}{(2\pi)^{3}}\int\Gamma(\mathbf{r},\mathbf{r}_{2},\mathbf{r}+\mathbf{q},\mathbf{r}_{2}\mp\mathbf{q})d\mathbf{q}d\mathbf{r}_{2}.$
(S11)

Similar to equi- and antimomentum, their corresponding densities can also be
expanded over a basis set,

$\lambda^{\pm}(\mathbf{r})=\sum_{rstu}\Gamma_{rstu}\phi_{r}(\mathbf{r})\left[stu\right]_{\lambda^{\pm}},$
(S12)

where the three-function integral is given by,

$\left[stu\right]_{\lambda^{\pm}}=\frac{1}{(2\pi)^{3}}\int\phi_{s}(\mathbf{r}+\mathbf{q})\\\
\times\phi_{t}(\mathbf{r}_{2})\phi_{u}(\mathbf{r}_{2}\mp\mathbf{q})d\mathbf{q}d\mathbf{r}_{2}.$
(S13)

If the basis set is comprised of Gaussian-type functions, then the fundamental
$\lambda^{+}(\mathbf{r})$ integral is given by,

$\left[\mathbf{0}\mathbf{0}\mathbf{0}\right]_{\lambda^{+}}=\frac{e^{-\tau
P^{2}}}{8\left(\beta\gamma+\beta\delta+\gamma\delta\right)^{3/2}},$ (S14)

where

$\tau=\frac{\beta\gamma\delta}{\beta\gamma+\beta\delta+\gamma\delta},$ (S15)

and

$\mathbf{P}=\mathbf{r}-\mathbf{B}+\mathbf{C}-\mathbf{D}.$ (S16)

The integrals over functions of higher angular momentum can be calculated
using the following 4-term recurrence relations,

$\displaystyle[(\mathbf{b}+\bm{1}_{i})\mathbf{c}\mathbf{d}]_{\lambda^{+}}=$
$\displaystyle\frac{\tau}{\beta}P_{i}[\mathbf{b}\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+b_{i}\frac{\beta-\tau}{2\beta^{2}}[(\mathbf{b}-\bm{1}_{i})\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+c_{i}\frac{\tau}{2\beta\gamma}[\mathbf{b}(\mathbf{c}-\bm{1}_{i})\mathbf{d}]_{\lambda^{+}}$
$\displaystyle-
d_{i}\frac{\tau}{2\beta\delta}[\mathbf{b}\mathbf{c}(\mathbf{d}-\bm{1}_{i})]_{\lambda^{+}},$
(S17a)
$\displaystyle[\mathbf{b}(\mathbf{c}+\bm{1}_{i})\mathbf{d}]_{\lambda^{+}}=$
$\displaystyle-\frac{\tau}{\gamma}P_{i}[\mathbf{b}\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+b_{i}\frac{\tau}{2\beta\gamma}[(\mathbf{b}-\bm{1}_{i})\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+c_{i}\frac{\gamma-\tau}{2\gamma^{2}}[\mathbf{b}(\mathbf{c}-\bm{1}_{i})\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+d_{i}\frac{\tau}{2\gamma\delta}[\mathbf{b}\mathbf{c}(\mathbf{d}-\bm{1}_{i})]_{\lambda^{+}},$
(S17b)
$\displaystyle[\mathbf{b}\mathbf{c}(\mathbf{d}+\bm{1}_{i})]_{\lambda^{+}}=$
$\displaystyle\frac{\tau}{\delta}P_{i}[\mathbf{b}\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle-
b_{i}\frac{\tau}{2\beta\delta}[(\mathbf{b}-\bm{1}_{i})\mathbf{c}\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+c_{i}\frac{\tau}{2\gamma\delta}[\mathbf{b}(\mathbf{c}-\bm{1}_{i})\mathbf{d}]_{\lambda^{+}}$
$\displaystyle+d_{i}\frac{\delta-\tau}{2\delta^{2}}[\mathbf{b}\mathbf{c}(\mathbf{d}-\bm{1}_{i})]_{\lambda^{+}}.$
(S17c)

The MB density is calculated as the difference between the equi- and
antimomentum densities,

$\mu(\mathbf{r})=\lambda^{+}(\mathbf{r})-\lambda^{-}(\mathbf{r}).$ (S18)

To efficiently calculate $\mu(\mathbf{r})$, the following identity is used,

$[stu]_{\lambda^{-}}=[sut]_{\lambda^{+}}.$ (S19)

If calculating $\mu(\mathbf{r})$ for numerous values of $\mathbf{r}$, it is
desirable to reduce the number of integral evaluations, which can be done with
the following expression,

$\mu(\mathbf{r})=\sum_{rstu}\left(\Gamma_{rstu}-\Gamma_{rsut}\right)\phi_{i}(\mathbf{r})[stu]_{\lambda^{+}}.$
(S20)

## S.II Atomic momentum-balance densities

The $\mu^{\beta\alpha}(\mathbf{r})$, $m^{\beta\alpha}(\mathbf{r})$ and
$\mu^{\beta\beta}(\mathbf{r})$, $m^{\beta\beta}(\mathbf{r})$ of the atoms of
the first three rows of the periodic table are presented in Figures S1 to S8.

Figure S1:  Renormalized $\mu^{\beta\alpha}(\mathbf{r})$ of the atoms Be to
Ne.

Figure S2:  Renormalized $m^{\beta\alpha}(\mathbf{r})$ of the atoms Be to Ne.
Figure S3:  Renormalized $\mu^{\beta\alpha}(\mathbf{r})$ of the atoms Na to
Ar.

Figure S4:  Renormalized $m^{\beta\alpha}(\mathbf{r})$ of the atoms Na to Ar.
Figure S5:  Renormalized $\mu^{\beta\beta}(\mathbf{r})$ of the atoms Be to Ne.

Figure S6:  Renormalized $m^{\beta\beta}(\mathbf{r})$ of the atoms Be to Ne.
Figure S7:  Renormalized $\mu^{\beta\beta}(\mathbf{r})$ of the atoms Na to Ar.

Figure S8:  Renormalized $m^{\beta\beta}(\mathbf{r})$ of the atoms Na to Ar.

## S.III Atomic momentum-balance

The total opposite-spin and parallel-spin momentum-balance for the atoms He to
Ar are reported in Table 1.

Table 1:  Spin-resolved and total momentum-balance of the atoms helium to argon Atom | $\mu^{\alpha\beta}$ | $\mu^{\alpha\alpha}$ | $\mu^{\beta\beta}$ | $\mu$
---|---|---|---|---
He | 0.00373 | 0 | 0 | 0.00373
Li | 0.00107 | -0.00001 | 0 | 0.00212
Be | 0.19320 | -0.00240 | -0.00240 | 0.38160
B | 0.06497 | -0.25768 | -0.00077 | -0.12851
C | 0.02555 | -0.27478 | -0.00028 | -0.22396
N | 0.01029 | -0.24801 | -0.00010 | -0.22752
O | 0.00770 | -0.16412 | -0.06507 | -0.21380
F | 0.00564 | -0.11649 | -0.08387 | -0.18908
Ne | 0.00419 | -0.08663 | -0.08663 | -0.16487
Na | 0.00095 | -0.07077 | -0.04902 | -0.11788
Mg | 0.29053 | -0.04855 | -0.04855 | 0.48398
Al | 0.13933 | -0.76686 | -0.03289 | -0.52109
Si | 0.07348 | -0.89145 | -0.02315 | -0.76764
P | 0.03908 | -0.87708 | -0.01680 | -0.81572
S | 0.03920 | -0.60966 | -0.21962 | -0.75088
Cl | 0.03394 | -0.44856 | -0.30727 | -0.68796
Ar | 0.02828 | -0.34316 | -0.34316 | -0.62976