Revealing the regularities of electron correlation energies associated with valence electrons in atoms in the first three rows of the periodic table
G. -Q. Hai L. Cândido B. G. A. Brito Y. Liu





Corresponding author: hai@ifsc.usp.br




^aInstituto de Física de São Carlos, Universidade de São Paulo, 13560-970, São Carlos, SP, Brazil
^bInstituto de Física,  Universidade Federal de Goiás, 74.001-970, Goiânia, GO, Brazil
^cDepartamento de Física, Universidade Federal do Triângulo Mineiro, 38.025-180, Uberaba, MG, Brazil
^dResearch School of Chemistry, The Australian National University, ACT 2601, Australia




Electronic correlation is a complex many-body effect and the correlation energy depends on the specific
electronic structure and spatial distribution of electrons in each atom and molecule. Although the total
correlation energy in an atom can be decomposed into different components such as inter-orbital and
intra-orbital pair-correlation energies (PCE), it is generally believed that the PCEs in different atoms
cannot be the same. In this work, we investigate the correlation energies of the atoms in the first
three rows of the periodic table (He to Ar). It is found that when the correlation energy
is defined as the difference between the exact ground-state energy and the unrestricted Hartree-Fock (UHF) energy,
the inter- and intra-orbital PECs associated with the valence electrons of the atoms in the same row
of the periodic table have the same values. These PCEs are not entangled and their values depend only on
the electron orbitals. For two specific orbitals, the inter-orbital correlation energy is
the same between two electrons of parallel spins or anti-parallel spins.
We also show that the effects of orbital relaxation on the correlation energy are surprisingly small.


Revealing the regularities of electron correlation energies associated with valence electrons
in atoms in the first three rows of the periodic table
    Y. Liu^d 
    
====================================================================================================================================================



Electron correlation energy (E_ c) in atoms and molecules is defined as the difference between
the ground-state energy (E_ exact) and the Hartree-Fock energy (E_ HF) in the complete basis
set limit<cit.>, i.e., E_ c=E_ exact - E_ HF,
where E_ exact is obtained from the exact solution of the non-relativistic electronic
Schrödinger equation with fixed nuclei. The Hartree-Fock (HF) energy contributes to more
than 99% of the total ground-state energy in most atoms and molecules. Although E_ c
is only a small fraction of the total energy, it is a quantity at the heart of chemistry
playing a crucial role in the determination of the physicochemical properties of atoms and molecules.
In order to obtain the correlation energy of a many-electron atom or a molecule,
one must perform sophisticated calculations to find out the total energy of each many-body system.
Although accurate knowledge of correlation energies is essential for understanding and predicting
the behavior of molecules as well as their thermochemical properties,
many aspects of electron correlation and the properties of the correlation energy
are still poorly understood.<cit.>

In the long history of investigation of the electron correlation, some regularities are
observed by examining trends in the correlation energy of atoms as we move across the periodic
table or within a specific group of elements.<cit.>
The correlation energy is influenced by the specific electron configuration and the effective
nuclear charge experienced by the electrons. It varies significantly depending on the position
of the atom in the periodic table. Elements in the same group tend to have similar valence
electron configurations and exhibit similar correlation energy patterns. Generally, for neutral atoms,
E_ c increases roughly linearly with increasing the atomic number but the
ratio E_ c/E_exact decreases. In dealing with the correlation energy in an atom,
it is possible to separate the correlation energy into different components such as inter-orbital
and intra-orbital pair-correlation energies (PCE). For different atoms, however, these PCEs
are unlikely to be the same for different atoms.
In this work, we attempt to establish some regularities in the electron correlation energies of the atoms.
We show for the first time that, there are simple relations for the PCEs associated with
the valence electrons of different atoms in the same period of the periodic table.
These PCEs are not entangled and their values depend only on the electron orbitals.
We have obtained very accurate values of the PCEs for atoms in the first three rows of the periodic table.
For example, all the PCEs associated with the valence electrons in the 2p orbitals in B, C, N, O, F and Ne atoms
share the same values ε_2p2p=-7.3± 0.2mE_ h (for two electrons
in different 2p orbitals), ε_2p2s=-3.9± 0.2mE_ h
(for one electron in 2p orbital and the other in 2s orbital),
and ε_2p^2=-35.5± 0.8mE_ h (for two electrons in the same 2p orbital).
Our calculations show that the inter-orbital PCEs have the same value between two electrons with parallel spins
or anti-parallel spins. It means that the spin states do not directly affect the PCEs in atoms.
We will also show that the orbital relaxation effects on the correlation energy is very small.
These results provide valuable insights into the properties of electron correlation energy in atoms.

The effective nuclear potential experienced by the electrons in an atomic system
may strongly affect the correlation energy. For the same electron number and
configuration, the correlation energy in the ground state of an atomic ion is dependent
on the nuclear charge Z.<cit.> For example, the correlation energy E_ c
of the four-electron Be-like ion increases almost linearly with increasing Z. Therefore,
in order to facilitate our investigation to better understand the nature of
the electron correlation, we will focus on the correlation energies associated with the valence
electrons in the neutral state of the atoms. The correlation energies of the valence
electrons in the outermost shell will be analyzed with the neutral atomic state as a
reference. We will investigate the correlation-energy gain
Δ E_c(Z) = E^(0)_ c(Z) - E^(+)_ c(Z)
in the formation of the neutral atom (with atomic number Z) by adding one valence
electron to its cation, where E^(0)_ c(Z) and E^(+)_ c(Z) are the correlation
energies of the neutral atom and the corresponding cation, respectively.
The inverse of the above process “ion + electron → neutral atom” is atomic
ionization and the ionization energy is experimentally measurable.
Analyzing the correlation energies associated to the valence electrons, we may minimize
the influences of the nuclear potential as well as the so-called exclusion effects.<cit.>
In fact, the correlation energies of these valence electrons are of great
importance tending to be responsible for the element’s chemical properties.

The correlation energy E_ c depends upon whether a restricted (E_ RHF) or unrestricted
Hartree-Fock energy (E_ UHF) is used in its definition.<cit.>.
Thus, there are different correlation energies  E_ c(R) = E_ exact - E_ RHF
defined by Löwdin<cit.> and E_ c(U) = E_ exact - E_ UHF defined by Pople and Binkley<cit.>.
The restricted Hartree-Fock (RHF) theory uses a single atomic or molecular orbital twice,
one multiplied by the α spin function and the other multiplied by the β spin
function in the Slater determinant, then the resulting approximate wavefunction is
an eigenfunction of the spin-squared operator S^2 and the spin component operator S_z,
simulating the corresponding properties of the exact wavefunction at the non-relativistic
limit.<cit.>
A disadvantage of the RHF theory is manifested in its incorrect limit for molecule dissociation.
On the other hand, in the unrestricted Hartree–Fock (UHF) theory, the α and β
spin functions are assigned to two completely independent sets of spatial orbitals.
It has the advantage of dissociation to correct fragments for most molecular ground states.<cit.>
The additional flexibility in UHF permits a lower limiting energy to be achieved but its total
wavefunction is not an eigenfunction of the total spin-squared operator S^2.
It was pointed out by Pople and Binkley that, according to the variational theorem
E_ UHF is lower than E_ RHF because there are no symmetry constraints are placed
on the orbitals in the UHF theory.<cit.> Therefore, E_ UHF obtained within
the complete basis set limit is the limiting energy which can be reached within the mean-field theory.
Consequently, the relations E_ RHF≥ E_ UHF > E_ exact
and |E_ c(R)| ≥ |E_ c(U)| hold.

In the literature, most studies of the electron correlation energies in atoms
are about the correlation energy E_ c(R) with the RHF energy as the reference.
Different correlation energies E_ c(U) and E_ c(R) were calculated and compared
for many atoms and molecules<cit.> by O'Neill and Gill. It was pointed out from the point
of view of static and dynamical electron correlation<cit.> that the UHF energy can capture
part of the static correlation energy and, consequently, the correlation energy E_ c(U)
is more robust than E_ c(R). In order to better understand the electron correlation,
we will analyze both correlation energy gains
Δ E_ c(R)(Z) = E^(0)_ c(R)(Z) - E^(+)_ c(R)(Z)
and Δ E_ c(U)(Z) = E^(0)_ c(U)(Z) - E^(+)_ c(U)(Z)
in the process “ion + electron → neutral atom”.

Very accurate values of the correlation-energy gains Δ E_ c(R) and Δ E_ c(U) are
needed in our investigation. The most accurate non-relativistic ground-state energies
E^(0)_ GS(Z) and E^(+)_ GS(Z) (for Z ≤ 18) of the atoms and cations,
respectively, were obtained by Davidson et al.<cit.>.
They took an empirical/theoretical
approach and combined experimental ionization energies with corrections to remove
the effect of reduced mass, nuclear shape, and relativistic corrections.
The ground-state energies obtained by Davidson et al. are widely accepted as the reference
values for atoms and ions with Z ≤ 18. In their work, they also calculated
the RHF energies of the atoms and ions and gave the corresponding correlation energies
E^(0)_ c(R)(Z) and E^(+)_ c(R)(Z). However, there are no such accurate
results available for larger atoms with more than 18 electrons. For example,
the correlation energies calculated by McCarthy and Thakkar<cit.> for atoms
from K to Kr (Z=19 to 36) have errors from 22 to 55 mE_ h.
Very recent calculations by Annaberdiyev et al.<cit.> based on the configuration
interaction, coupled cluster, and quantum Monte Carlo methods covering elements up to Kr
have achieved an accuracy of about 1 to 10 mE_ h for K to Zn atoms (Z= 19 to 30).
The difficulty in obtaining accurate correlation energies in many-electron atoms and
molecules is a major obstacle in the study of correlation effects.



Using the results given by Chakravorty and Davidson<cit.> of the non-relativistic ground-state
energies E_ GS(Z) and the correlation energies E_ c(R)(Z) of the atoms and cations,
we may obtain the correlation-energy gains for Z ≤ 18. To obtain the correlation
energy E_ c(U)(Z), we also need the UHF energies of the atoms and cations.
We performed calculations using the cc-pVQZ, cc-pV5Z, and cc-pV6Z basis sets<cit.>
to extrapolate to the complete basis set limit<cit.>. Because the ground-state energies obtained
in Ref. Davidson96 for most atoms and ions are with four decimal digits (i.e., they are accurate
to within 0.1 mE_ h), we can determine the correlation energies to an accuracy
in the order of 0.1 mE_ h. The obtained E^(0)_ UHF(Z) and E^(+)_ UHF(Z)
are given in Table I together with the ground-state energies E^(0)_ GS(Z) and E^(+)_ GS(Z)
from Ref. Davidson96. Their differences are the correlation energies E^(0)_ c(U)(Z)
for the atoms and E^(+)_ c(U)(Z) for the ions. The above calculations yield
the correlation-energy gain Δ E_ c(U)(Z) = E^(0)_ c(U)(Z)-E^(+)_ c(U)(Z).
The correlation-energy gain Δ E_ c(R)(Z) = E^(0)_ c(R)(Z)-E^(+)_ c(R)(Z)
can be estimated using the correlation energies E^(0)_ c(R)(Z) and E^(+)_ c(R)(Z)
given in Ref. Davidson96. The obtained values of Δ E_ c(U)(Z) and Δ E_ c(R)(Z)
are shown in the last two columns in Table I and they are plotted as a function of Z in Fig. 1.



Fig. 1 shows that both the correlation-energy gains Δ E_ c(U) and Δ E_ c(R)
are large when the added electron occupies an orbital in which there is already another electron
in it. This happens at Z = 2, 4, 8, 9, 10, 12, 16, 17, and 18, and a new electron pair forms
in the same orbital.
Otherwise, the correlation-energy gains are small. The largest difference between the values of
Δ E_ c(U) and Δ E_ c(R) is not more than 10 mE_ h.
But the most significant difference between them is manifested in their dependence
on the atomic number Z. The energy Δ E_ c(U) versus Z for Z = 5 to 7, as well
as for Z = 8 to 10 (which are associated with the 2p valence electrons) shows very good linear relations.
The same is happening for the correlation-energy gain Δ E_ c(U) associated with
the 3p valence electrons for Z = 13 to 18. But Δ E_ c(R)(Z) does not show
such a linear behavior for p electrons. The piecewise linear dependence of the energy Δ E_ c(U)(Z)
on Z reveals that the inter-orbital and intra-orbital correlation energies
associated with the valence electrons are not interdependent.
But it occurs only for the correlation energy
E_ c(U) beyond the UHF energy defined by Pople and Binkley<cit.> due mostly to
the dynamical correlation. Therefore, we will focus on the correlation-energy gains
Δ E_ c(U) for the valence electrons to show the regularities of
the electron correlation energy associated with the valence electrons in atoms.

When a helium cation binds one more electron becoming a neutral atom,
the correlation-energy gain Δ E_c(2) is determined by the intra-orbital PCE
between two electrons occupying the same 1s orbital (ε_1s^2) of the He atom. From Table I
we obtain Δ E_ c(U)(2) = -42.00 mE_ h = ε_1s^2.
For a lithium cation, there are two electrons in the core 1s orbital with correlation energy
E^(+)_ c(3)= ε_ core(1s^2). When it catches one more electron becoming a neutral Li atom,
the third electron occupies the 2s orbital and the total correlation energy of the Li atom is given by
E^(0)_ c(3)= ε_ core(1s^2) + ε_2s1s^2 with the correlation-energy gain
Δ E_ c(3)= ε_2s1s^2.
In this case, the influence of the third electron on the core-electron correlation in the 1s orbital is very small
and can be ignored. The correlation-energy gain Δ E_ c(3) is due to the inter-shell correlation energy
ε_2s1s^2≡ε_2s^ I between one valence electron in the 2s orbital
and two core-electrons in the 1s orbital in the K shell. Using the value of Δ E_ c(U)(3) we obtain
ε_2s^ I = Δ E_ c(U)(3) = -1.76 mE_ h.

Similarly, adding one electron to a beryllium cation the correlation-energy gain from the 3-electron cation
to 4-electron neutral atom is given by Δ E_ c(4)
=ε_2s^21s^2-ε_2s^ I +ε_2s^2, where ε_2s^21s^2
is the inter-shell correlation energy between the two 2s valence electrons and two 1s core electrons
and ε_2s^2 is the intra-orbital PCE between the two electrons in the same 2s orbital.
The energy ε_2s^21s^2 should be different from 2ε_2s^ I
because the one electron density distribution in a singly occupied 2s orbital is different
from that in the doubly occupied. We may introduce a constant η_2s and denote
ε_2s^21s^2= 2η_2sε_2s^ I.
Then the correlation-energy gain is given by
Δ E_ c(4) = ε_2s^'^ I+ε_2s^2 with
ε_2s^'^ I= (2η_2s-1)ε_2s^ I.

For Z=2 to 7, we can obtain the following expressions for the correlation-energy gain,

    Δ E_c(2)= ε_1s^2, 
          Δ E_c(3)= ε_2s1s^2≡ε_2s^ I, 
          Δ E_c(4)=  ε_2s^'^ I + ε_2s^2, 
          Δ E_c(5)=  ε_2p^ II,
          Δ E_c(6)= ε_2p^ II + ε_2p2p, 
          Δ E_c(7)= ε_2p^ II + 2ε_2p2p.

Eq. (<ref>) shows that Δ E_c(5) for boron atom is due to the (inter-shell-inter-subshell) correlation energy
ε_2p^ II≡ε_2p(2s^2 1s^2) between one 2p valence electron and the rest four
in the 2s subshell and 1s core. In carbon atom (Z=6), two 2p electrons occupy different orbitals by Hund's rule,
being with the inter-orbital PCE ε_2p2p between them.
Thus Δ E_c(6) is given by ε_2p^ II for the added electron and the inter-orbital
PCE ε_2p2p.
Eqs. (<ref>-<ref>) shows that Δ E_c(Z) has a linear dependence on Z for
Z=5 to 7 with the slope ε^(2p)_ slope=ε_2p2p.
It means that the inter-orbital correlation energies ε_2p2p and
ε_2p^ II for 2p valence electrons in B, C, and N atoms should have
the same values. The best linear fitting to the values of Δ E_c(Z) versus Z for Z=5 to 7
in Fig. 1 yields the slope ε^(2p)_ slope=ε_2p2p=-7.3± 0.2
mE_ h and the intercept ε_2p^ II=-9.6± 0.3 mE_ h,
with the standard errors of a few tenths millihartree. Therefore, our analysis
is confirmed by the best numerical results of the correlation energies in atoms and,
within the accuracy of a few tenths mE_ h, the inter-orbital correlation energies
ε_2p2p and ε_2p^ II are not
dependent on each other.

In the ground state of oxygen atom (Z=8), there is a doubly occupied 2p orbital where
two electrons of opposite spins occupying the same orbital. The corresponding correlation-energy gain is given by

    Δ E_c(8)= ε_2p^2(2s^2 1s^2) -ε_2p^ II + 2 ε_2p^22p
    - 2 ε_2p2p + ε_2p^2.

Because the charge distribution for each electron in doubly occupied 2p orbital is different from
that of the singly occupied one, in principle it leads to ε_2p^22p 2 ε_2p2p.
Thus, we introduce a dimensionless parameter η_2p such that
ε_2p^22p= 2 ε^'_2p2p with
ε^'_2p2p = η_2pε_2p2p and also
ε_2p^2(2s^2 1s^2)=2ε^'_2p (2s^2 1s^2) with
ε^'_2p (2s^2 1s^2) =  η_2pε_2p^ II
to quantify this alteration with η_2p a constant of value close to but different from 1.
Then, the expression for Δ E_c(8) becomes Δ E_c(8)= (2η_2p-1)(ε_2p^ II
+2ε_2p2p) + ε_2p^2≡ε^ III_2p.

For fluorine (Z=9) and neon (Z=10) atoms with two and three doubly occupied 2p orbitals,
respectively, the inter-orbital correlation energy ε_2p^22p^2 between
two 2p electrons in one orbital and the other two in another 2p orbital appears.
Using the parameter η_2p, we have ε_2p^22p^2
= 2 η_2pε_2p^22p = 4 η^2_2pε_2p2p.
The expressions for Δ E_c(Z) can be written as, for Z=8, 9 and 10,

    Δ E_c(8)= ε^ III_2p, 
          Δ E_c(9)= ε^ III_2p+ (2η_2p-1)^2 ε_2p2p,
          Δ E_c(10)= ε^ III_2p+ 2 (2η_2p-1)^2ε_2p2p.

We see that Δ E_c(Z) versus Z for Z=8 to 10 also has a linear relation but with
the slope ε^(2p)_ slope^'= (2η_2p-1)^2 ε_2p2p.
Therefore, the valence electrons in doubly occupied 2p orbitals in O, F and Ne atoms also share
the same values of the intra- and inter-orbital pair-correlation energies.
The linear regression of Δ E_ c(U)(Z) for Z=8 to 10 in Fig. 1 yields the slope
ε^(2p)_ slope^'=(2η_2p-1)^2 ε_2p2p=-6.8± 0.3
mE_ h and the intercept ε^ III_2p=-58.8± 0.4 mE_ h.
Again, we observe that the standard errors for both the slope and intercept obtained from
the fitting are a few tenths millihartree. From the above results, we obtain η_2p=0.982± 0.013 and
ε_2p^2=-35.5± 0.8  mE_ h. Thus, the inter-orbital PCE ε^'_2p2p
is given by ε^'_2p2p=η_2pε_2p2p = -7.2± 0.2 mE_ h
which is only about 2% smaller than ε_2p2p due to double occupation of a 2p orbital.
It demonstrates that the strong direct Coulomb repulsion between two electrons in the same orbital
affects very slightly the related inter-orbital PCEs.

It is known that there is orbital relaxation when we take one electron out from the atom or
add one electron into the cation. As a matter of fact, the most significant orbital
relation occurs when we remove one electron from a doubly occupied orbital or
when we add one electron into an orbital where there is already one electron in it.
It is generally believed that strong Coulomb repulsion between two electrons occupying the same orbital
alters the electron density distribution and induces strong orbital relaxation effects on
the correlation energy. Here, we show the difference
ε_2p2p-ε^'_2p2p=(1- η_2p) ε_2p2p≃ 0.1± 0.2 mE_ h is quantitatively responsible for the change of inter-orbital PCE result from
orbital relaxation. This is the most important ingredient for the correlation energy change
due to orbital relaxation. We find that the effects of the orbital relaxation on the correlation
energy are surprisingly small, contributing less that 2%  change of the correlation energy in this case.

The above results indicate clearly that the inter-orbital
PCEs ε_2p2p and ε^'_2p2p do not intertwine
with the intra-orbital PCE ε_2p^2. The inter-orbital PCEs are also independent of the spins
of the electrons because two electrons with the same or opposite spins have the same value of inter-orbital PCE.
Notice that the intra-orbital PCE in atoms is always for two electrons of opposite spins.




We now come back to exam the correlation energy gain Δ E_ c(U)(4) given by Eq. (<ref>)
and its components ε_2s^'^ I= (2η_2s-1)ε_2s^ I and ε_2s^2.
Because the present analysis cannot provide exact value for η_2s, considering the result obtained
above for η_2p we may assume that the double occupation of the 2s orbital affects very little
the related inter-orbital correlation energy and approximate η_2s≃ 1, i.e.,
ε_2s^'^ I≃ε_2s^ I= -1.76 mE_ h.
Using the value Δ E_ c(U)(4) = -47.10 mE_ h we obtain
ε_2s^2≃ -45.3 mE_ h.

For the atoms in the third row of the periodic table, following the similar discussions and notations
used above for the atoms in the second row, the correlation-energy gains Δ E_c(Z) for Z=11 to 18 can be written as,

    Δ E_c(11)= ε_3s (core)≡ε_3s^ I, 
          Δ E_c(12)= ε_3s^'^ I + ε_3s^2, 
          Δ E_c(13)= ε_3p(3s^2 core)≡ε_3p^ II,
          Δ E_c(14)= ε_3p^ II + ε_3p3p, 
          Δ E_c(15)= ε_3p^ II + 2 ε_3p3p, 
          Δ E_c(16)=  ε^ III_3p, 
          Δ E_c(17)=  ε^ III_3p+ (2η_3p-1)^2 ε_3p3p , 
          Δ E_c(18)= ε^ III_3p+ 2 (2η_3p-1)^2ε_3p3p.

In Eq. (<ref>), ε_3s^ I is the inter-shell correlation energy between a 3s
valence electron and all the core electrons in the K and L shells.
From Table I we obtain ε_3s^ I =Δ E_ c(U)(11)= -6.8 mE_ h. Being similar to
Eq. (<ref>), Δ E_c(12) in Eq. (<ref>) is a sum of ε_3s^'^ I
and the intra-orbital PCE ε_3s^2.
Assuming ε_3s^'^ I≃ε_3s^ I
and using the value Δ E_ c(U)(12)=-38.0 mE_ h, we obtain ε_3s^2≃ 31.2 mE_ h.
Based on the relation indicated by Eqs. (<ref>-<ref>), a linear regression to the values of Δ E_ c(U)(Z)
vs. Z (for Z=13 to 15) in Fig. 1 yields the slope ε^( 3p)_ slope
=ε_3p3p=-5.1± 0.2 mE_ h
and the intercept ε_3p^ II=-14.0± 0.3 mE_ h, where ε_3p^ II is
the correlation energy between a 3p valence electron and the rest in the 3s subshell and in the core (K and L shells).
To characterize the difference in the inter-orbital PCE for singly and doubly occupied 3p orbitals,
we follow the technique used for 2p electrons and introduce the parameter η_3p such that
ε_3p^23p= 2 ε^'_3p3p with
ε^'_3p3p = η_3pε_3p3p and
ε_3p^2(3s^2 core)=2 η_3pε_3p^ II.
In this way, the quantity ε^ III_3p appeared in Eqs. (<ref>-<ref>) is given by
ε^ III_3p≡ (2η_3p-1)(ε_3p^ II +2ε_3p3p) + ε_3p^2.
The best linear fitting to the values of Δ E_ c(U)(Z) vs. Z (for Z=16 to 18) in Fig. 1 yields the slope
ε^(3p)_ slope^' =(2η_3p-1)^2 ε_3p3p =-4.0± 0.1 mE_ h
and the intercept ε^ III_3p=-40.6± 0.2 mE_ h.
Using the above results from the linear fitting, we obtain η_3p=0.943± 0.010,
ε^'_3p3p=4.8± 0.2 mE_ h, and ε_3p^2=-19.2± 0.7 mE_ h.
We see that the difference between ε_3p3p and ε^'_3p3p
due to orbital relation
is larger than that between ε_2p2p and ε^'_2p2p, but it
is still quite small being close to the standard error from the fitting. The above results show that for
the valence electrons in the third period, the inter-orbital and intra-orbital correlation energies
exhibit similar behaviors to those of the second period.

Notice that ε_2s^ I (ε_3s^ I) is the correlation energy between 1 valence electron
in 2s (3s) orbital and 2 (10) core electrons. And ε_2p^ II (ε_3p^ II) is
that between 1 valence electron in 2p (3p) orbital and the rest ones in the 2s (3s) subshell and the ionic core.
It is reasonable to assume that the 2s and 2p (3s and 3p) valence electrons have the same inter-shell
correlation energy with the core electrons, i.e.,
ε_2p1s^2 =ε_2s1s^2≡ε_2s^ I
(ε_3p core =ε_3s core≡ε_3s^ I ).
Within such approximations, we obtain the inter-subshell PCEs
ε_2p2s = -3.9± 0.2 mE_ h and ε_3p3s≃ -3.6± 0.2 mE_ h.
Consequently, the average inter-shell pair-correlation energy between a 2s or a 2p valence electron
and a core electron is given by ε̅_2^ I= ε_2s^ I/2 = -0.88 mE_ h.
And that between a 3s or a 3p valence electron and a core electron is given by
ε̅_3^ I= ε_3s^ I/10 = -0.68 mE_ h.

In summary, we have investigated and compared the different correlation energies
E_ c(R) and E_ c(U) of atoms based on the most accurate ground-state energies
for Z ≤ 18. We show that the correlation-energy gain Δ E_ c(U)(Z) associated
with the valence electrons in atoms is given by a linear combination of the intra- and inter-orbital PCEs.
We reveal that within the chemical accuracy, the PECs associated with the valence
electrons of the atoms in the same row of the periodic table have the same values.
These PCEs are not entangled and their values depend only on the electron orbitals.
For two specific orbitals, the inter-orbital correlation energy is
the same for two electrons of parallel spins or anti-parallel spins.
Furthermore, We find that the effects of orbital relaxation on the correlation energy are very small.



The obtained values of the intra-orbital, inter-orbital, and average inter-shell PCEs for
the atoms in the first three rows of the periodic table are given in Table II.
Using the obtained PCEs in this table, the correlation energy gains Δ E_ c(U)(Z)
for the valence electrons shown in Table I can be recovered according to Eqs. (1-17).
The PCEs associated with the valence electrons in 2p orbitals in B, C, N, O, F and Ne atoms
in the second row have the same value ε_2p^2, ε_2p2p,
ε^'_2p2p and ε_2p2s given in Table II,
and those associated with the valence electrons in 3p orbitals in Al, Si, P, S, Cl and Ar atoms
in the third row have the same PCEs ε_3p^2, ε_3p3p,
ε^'_3p3p and ε_3p3s, etc.
Quantitatively, the intra-orbital PCEs are much larger than the inter-orbital
and inter-shell ones. The inter-orbital PCE for electrons in the same shell is about one order
of magnitude smaller than the intra-orbital one. The inter-shell PCE between a valence
and a core electron is even smaller. The distinction between the intra-orbital and inter-orbital
correlation may have important implications because they are not entangled and
the intra-orbital PCE is much larger. Two electrons occupying the same orbital
may lose their single-particle identity within the correlation effects acting as a new
entity of a bound electron pair, which should manifest itself
in the properties of some molecules and materials.<cit.>



§ ACKNOWLEDGMENTS

This research was supported by FAPESP (São Paulo Research Foundation, under the grant No. 2024/00484-2),
FAPEMIG, FAPMG, and CNPq (Brazil). The Authors appreciate helpful
discussions with Tomaz Catunda.




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