Geometries, interaction energies and bonding in [Po(H2O)_n]^4+ and
[PoCl_n]^4-n complexes†

Nadiya Zhutova,a,b Florent Réal,c Valérie Vallet,∗c and Rémi Maurice∗a,b

Polonium ($Z$ = 84) is one of the rarest elements on Earth. More than a
century after its discovery, its chemistry remains poorly known and even basic
questions are not yet satisfactorily addressed. In this work, we perform a
systematic study of the geometries, interactions energies and bonding in basic
polonium(IV) species, namely the hydrated [Po(H2O)_n]^4+ and chlorinated
[PoCl_n]^4-n complexes by means of gas-phase electronic structure
calculations. We show that while up to nine water molecules can fit in the
first coordination sphere of the polonium(IV) ion, its coordination sphere can
already be filled with eight chloride ligands. Capitalising on previous
theoretical studies, a focused methodological study based on interaction
energies and bond distances allows us to validate the MP2/def2-TZVP level of
theory for future ground-state studies. After discussing similarities and
differences between complexes with the same number of ligands, we perform
topological analyses of the MP2 electron densities in the quantum theory of
atoms in molecules (QTAIM) fashion. While the water complexes display typical
signatures of closed-shell interactions, we reveal large Po–Cl delocalisation
indices, especially in the hypothetical [PoCl]^3+ complex. This “enhanced”
covalency opens the way for a significant spin-orbit coupling (SOC) effect on
the corresponding bond distance, which has been studied by two independent
approaches (i.e. one a priori and one a posteriori). We finally conclude by
stressing that while the SOC may not affect much the geometries of high-
coordinated polonium(IV) complexes, it should definitely not be neglected in
the case of low-coordinated ones.

††footnotetext: a Subatech, UMR CNRS 6457, IN2P3/IMT Atlantique/Université de
Nantes, 4 rue A. Kastler, 44307 Nantes Cedex 3, France††footnotetext: b Univ
Rennes, CNRS, ISCR (Institut des Sciences Chimiques de Rennes) – UMR 6226,
F-35000 Rennes, France; E-mail<EMAIL_ADDRESS>c
Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers, Atomes et Molécules,
F-59000 Lille, France; E-mail<EMAIL_ADDRESS>†
Electronic Supplementary Information (ESI) available: Additional interaction
energies, bond distances and atomic charges obtained at various levels of
theory (PDF) and MP2/def2-TZVP molecular geometries for all the studied
complexes (XYZ files compressed into two ZIP files).

##  1 Introduction

Since the discovery of polonium in 1898 by Marie and Pierre Curie, not a lot
of experiments has been carried out to unravel its intriguing chemical
properties. While most of polonium chemistry remains unknown, its extreme
toxicity, being it either chemical or radiological, is notorious 1, 2. Though
its radioactivity may be used to detect it, it is still complicated to perform
analyses to determine polonium traces in, for example, sea water 3, tobacco 4,
5, plants 6, soils 7 and other samples. Meanwhile, former usages of polonium
as a heat source in the space machine industry, as a $\alpha$-emission source,
or as static eliminators have now become obsolete because of alternatives
based on materials that are safer and easier to get. In the case of accidental
or criminal ingestion of polonium, no specific antidote is known, and one can
administrate British anti-Lewisite (BAL), the antidote for heavy metals and in
particular arsenic 8, unfortunately without any guarantee of success.

Polonium has more than 40 known isotopes. While only seven of them are
naturally occurring 9, ^210Po is the usual isotope for performing experiments,
even if other ones may be scarcely used, e.g. ^208Po for atomic spectroscopy
10. Four most oxidation states of polonium are usually considered: $-$II, +II,
+IV and +VI. In solution, polonium(IV) is meant to be the most stable one 11,
so that it is clearly an oxidation state whose chemistry deserves to be
studied. Nonetheless, evidences for polonides (Na2Po, ZnPo, etc.) 12, oxides
(PoO2, PoO3) 13, halides and dihalides (PoCl2, PoCl4, PoBr4, PoI4) 14, 15, 16
and complexes with organic ligands such as the acetylacetonate anion (acac-),
the ethylenediaminetetraacetic acid trianion (EDTA^3-), the oxalate dianion
(C2O4^2-), etc. were reported in the literature 17.

Computational chemistry is, of course, free of any direct toxicological
impact. Its increased accuracy and applicability observed in the last few
decades may open the way for future predictions aiming at designing new
successful experiments with polonium. However, prior to this, methodological
studies have to be performed in order to ensure good results for the good
reasons (or at least identified error cancellations). In fact, only a limited
number of electronic structure theory studies were dedicated to polonium,
among which a few emblematic ones will be discussed hereafter. Note that the
electronic structure of the free Po^4+ ion is [Xe]4f145d106s2, which has two
main consequences for the calculations, (i) single-configurational and single-
reference approaches may correctly describe the ground state wave functions of
polonium(IV) species and (ii) relativistic effects must be accounted for
explicitly or implicitly (e.g. scalar relativistic effects lead to a strong
stabilisation of the 6s orbital 18).

From 2008 to 2010, Ayala et al. published three computational works related to
polonium(IV) hydration and to the formation of hydroxo complexes in solution
19, 20, 21. In 2015, Borschevsky et al. calculated very accurately with a
fully relativistic coupled cluster method the ionisation potentials and
electron affinity of the free Po atom 22. Note that their first ionization
potential value was in good accord with the early value of Charles 10, and
proved to be also in good accord with the yet to come new experimental values
derived from two independent experiments 23, 24. In 2019, Stoïanov et al.
revisited from theory the UV-Vis absorption spectra of polonium chloride
complexes 25. The same year, chalcogen bonding with polonium was examined by
Zierkiewicz et al. 26. In fact, performing relativistic and correlated
calculations is always quite challenging, and validating approximate schemes
from scratch may require tedious methodological studies due to the lack of
experimental reference data (one should then validate more approximate schemes
by comparison with more accurate ones, i.e. based on very demanding
calculations, see for instance the work of Rota et al. on the polonium
hypothetical dimer 27).

As mentioned before, the +IV oxidation state of polonium dominates in
solution. Following the pioneer works of Ayala et al. 19 and Stoïanov et al.
25, we will focus on two families of Po(IV)-complexes, namely the hydrated
[Po(H2O)_n]^4+ and the chlorinated [PoCl_n]^4-n complexes. The first series is
of fundamental interest since it relates to the general context of the
hydration of heavy-element cations in solution 28. The second series is of
practical interest since polonium chloride complexes can form in acidic
chloride solutions 12, 28. We will focus on ground-state properties
(geometries, interaction energies and bonding descriptors) and assess the
dependence of the theoretical results with respect to specific degrees of
freedom such as the choice of the correlation method, of the atomic basis sets
or for instance of correcting for the basis set superposition error (BSSE)
when computing interaction energies, which was not fully done in those earlier
works 19, 25. To limit the discussion to pure electronic-structure theory
aspects, only gas-phase results will be considered (future work may deal with
solvation). Similarities and differences between the hydrated and chlorinated
complexes will be discussed, in particular concerning the first coordination
sphere of the polonium(IV) cation.

##  2 Computational details

Our most extended methodological investigation was done on the [Po(H2O)_n]^4+
complexes (where in fact $n$ = 1–9), even though some tests were repeated on
the [PoCl_n]^4-n ones ($n$ = 1–8, see Results and discussion section and
Supporting information). MP2 29, 30, 31, 32, 33 and B3LYP 34, 35 calculations
were performed to determine the molecular geometries and the corresponding
total interaction energies of all the complexes. Because of the close
agreement of the MP2 and B3LYP results and for the sake of simplicity, we will
only report MP2 results in the remainder of the text. For the [Po(H2O)_n]^4+
($n$ = 7–9) complexes, initial structures were built from previous thorium
structures 36. Note that additional reference calculations with the coupled
cluster with single, double and perturbative triple excitations 37, 38,
CCSD(T), method were also performed on specific complexes, namely the
[Po(H2O)n]^4+ with $n$ = 1–3. All the MP2, B3LYP and CCSD(T) calculations were
carried out with the Gaussian 16 39 software. Note that every molecular
geometry is optimised unless mentioned otherwise.

The Dunning 40 aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ, and redefined
Ahlrichs 41 def2-TZVP and def2-QZVP basis sets together with the relativistic
60-electron small core ECP60MDF pseudopotential 40 for Po and its associated
aug-cc-pV$N$Z-PP or def2-$N$ZVP-PP basis set were used ($N$ = D, T, Q). All
the basis sets and pseudopotential parameters were taken from the Basis Set
Exchange software 42.

Fig. 1: Hartree-Fock/def2-TZVP orbital energies (in a.u.) of the PoCl4
complex. The frozen (blue) and unfrozen (red) orbitals are displayed, by
default (left) or with our recommended scheme (right).

By default, in post-Hartree-Fock calculations, Gaussian 16 correlates all
outer-core (5s, 5p, 5d) orbitals and the occupied 6s valence orbital of
Po(IV), while freezing the 5s and 5p orbitals is wiser to ensure a significant
energy gap between the frozen and the correlated orbitals in a molecular
system, as illustrated by Figure 1 related to the PoCl4 molecule, where the Po
5s molecular orbital (MO 21) is energetically close to the frozen combinations
of the chlorine 2s (MOs 9–12) and 2p orbitals (MOs 13–20). To generally avoid
artefacts in the correlated calculations (when discussing trends, one should
only compare results for which the same shells are frozen), the default frozen
core choice was overwritten with the $window$ keyword of Gaussian, with which
the user may define the first unfrozen orbital, i.e. the number of frozen
cores –NFC– plus one). In fact, in the current study, it was manually set to
freeze 4 orbitals for Po (5s and 5p pseudo-orbitals), 1 for each water (the 1s
orbital of the O atom), and 5 for Cl (the 1s, 2s and 2p shells). Moreover,
though it is a bit technical, we would like to warn the reader that attention
should be paid in computing the frequencies in a second step to ensure that
Gaussian correctly applies the desired freezing scheme.

The interaction energies were calculated using the following formula:

$E_{int}=E_{complex}-E_{Po^{4+}}-nE_{ligand}+\Delta E_{\text{BSSE}}$ (1)

The basis set superposition error correction ($\Delta E_{\text{BSSE}}$) was
calculated by the means of the counterpoise correction at the (fixed) geometry
of the complex:

$\Delta
E_{\text{BSSE}}=E_{\text{Po}^{4+}}-E_{\text{Po}^{4+}}^{\text{ghosts}}-E_{(\text{ligands})_{n}}+E_{(\text{ligands})_{n}}^{\text{ghost}}$
(2)

where the _ghost(s)_ superscript refers to calculations performed with ghost
ligand atoms ($E_{\text{Po}^{4+}}^{\text{ghosts}}$) or with a ghost Po atom
($E_{(\text{ligands})_{n}}^{\text{ghost}}$), meaning that the basis functions
of the corresponding atoms are applied but their electrons are not (neither
their nuclear charges). Note that unlike Equation 1, which dissociates the
complex in Po^4+ + $n$ ligands, Equation 2 considers clusters of $n$
interacting ligands, denoted $(\text{ligands})_{n}$.

To describe the nature of the chemical bonds in the systems under study,
quantum theory of atoms in molecules (QTAIM) 43 analyses were done with the
AIMALL 44 software using the MP2/def2-TZVP electron densities. Note that more
details concerning the meaning of the derived descriptors will be directly
given in the Results and discussion section.

To assess the role of the spin-orbit coupling (SOC) on molecular geometries,
two types of calculations were performed. First, zeroth-order regular
approximation (ZORA) 45 relativistic Hartree-Fock calculations were performed
using the AMS driver of the Amsterdam Modeling Suite 46 software and Slater-
type TZ2P 47 basis sets, without freezing core orbitals. In this case, the SOC
is included a priori, but the interplay between SOC and electron correlation
is completely missing. Alternatively, we have performed two-step SOC
calculations with ORCA 48 (v. 5.0.0), for which the SOC is introduced a
posteriori, as a perturbation of the correlated spin-orbit free (SOF) level,
obtained with the Douglas-Kroll-Hess Hamiltonian 49, 50, 51. In this case,
specific scalar relativistic recontracted (DKH-def2-TZVP) or originally
generated (SARC-DKH-TZVP) all-electron basis sets 52, 53 were used for all the
atoms. There, the role of the SOC may this time be underestimated, and only a
part of the interplay between electron correlation and the SOC is by
construction treated (we have only used the contracted spin-orbit
configuration interaction, SOCI, scheme). More details and explanation will be
directly given in the Results and discussion section. In short, these two
complementary approaches do not suffer from the same pitfalls, and if a
physical effect is observed with both methodologies, one may thus be quite
confident in its actual occurrence.

##  3 Results and discussion

###  3.1 Electronic structure theory calculations

As mentioned earlier, we will start the discussion with the interactions
energies that were obtained for the [Po(H2O)_n]^4+ and [PoCl_n]^4-n complexes,
with special emphasis on the water complexes, previously studied by Ayala et
al. 19. We performed a partial benchmark to find the best accuracy/cost
methodology for ground-state studies of polonium(IV) complexes. We recall here
some of their important conclusions, confirmed by our independent
calculations:

  * •

up to nine water molecules can fit in the polonium(IV) coordination sphere,
meaning that $n$ = 1–9 for the hydrated [Po(H2O)_n]^4+ complexes under study;

  * •

the interaction energies evolve monotonously with $n$;

  * •

MP2 and to a lesser extend B3LYP calculations provide interactions energies in
very good agreement with the CCSD(T) ones when the aug-cc-pVDZ basis set are
used.

We have thus retained the MP2 method, and focused on two main degrees of
freedom: the nature and size of the basis set and the use of a BSSE correction
(see Eqs. 1 and 2). Furthermore, since we have proposed to change the number
of polonium frozen orbitals (see Computational details section), we have also
reported new CCSD(T) and MP2 calculations with the aug-cc-pVDZ basis set, as
well as original ones with triple zeta and quadruple zeta aug-cc or def2 basis
sets (see Table 1, as well as Tables S1 and S2 for an investigation of the
impact of the BSSE correction).

Table 1:  Interaction energies with BSSE correction ($\mathrm{kcal}\text{\,}{\mathrm{mol}}^{-1}$) obtained at various levels of theory for polonium(IV) complexes with water and chlorides, computed according to Eqs. 1 and 2. |  H2O |  Cl-
---|---|---
$n$(ligands) | CCSD(T) / | MP2 / | MP2 / | MP2 / | MP2 / | MP2 /
| aug-cc-pVDZ | aug-cc-pVDZ | aug-cc-pVTZ | def2-TZVP | def2-QZVP | def2-TZVP
1 | $-236.2$ | $-249.2$ | $-257.9$ | $-262.0$ | $-263.8$ | $-883.7$
2 | $-409.5$ | $-439.0$ | $-454.7$ | $-461.4$ | $-464.9$ | $-1415.0$
3 | $-541.1$ | $-586.4$ | $-607.7$ | $-616.1$ | $-621.4$ | $-1749.2$
4 |  | $-694.9$ | $-718.8$ | $-728.9$ |  | $-1911.4$
5 |  | $-791.8$ | $-817.6$ | $-829.4$ |  | $-1974.3$
6 |  | $-875.8$ | $-903.1$ | $-918.2$ |  | $-1947.8$
7 |  | $-941.2$ | $-971.1$ | $-987.8$ |  | $-1861.4$
8 |  | $-1004.4$ | $-1035.9$ | $-1054.0$ |  | $-1650.3$
9 |  | $-1052.7$ | $-1085.5$ | $-1104.4$ |  |

One can quickly see that results for both the MP2 and CCSD(T) methods, in
combination with the aug-cc-pVDZ basis set, are in fair agreement, though the
accord is less impressive than in the paper of Ayala et al. 19. Since the BSSE
corrections are similar with both methods (see Tables S1 and S2), the observed
difference essentially comes from the number of frozen orbitals. However, it
is known that wave-function theory calculations, especially correlated ones of
the coupled-cluster type, slowly converge with the basis-set size. Because of
the computational cost, we have not systematically performed CCSD(T)
calculations with larger basis sets and have essentially investigated the
effect of the basis-set nature and size only at the MP2 level. However, the
interested reader may find a couple of CCSD(T) optimised structures obtained
with the aug-cc-pVTZ and def2-TZVP basis sets in Table S2. Interestingly, the
accord between MP2 and CCSD(T) improves with basis set enlargement,
independent from its nature (i.e. aug-cc vs. def2). Indeed, the difference
between the two methods is divided by two with respect to the basis-set size.
Anyway, due to the single reference nature of the ground state of polonium(IV)
complexes, we indeed expect to obtain accurate values with MP2 provided that
convergence of the results with respect to the basis-set size is reached. This
is why we chose to pursue of analysis of the results dependence with respect
to the basis set with solely the MP2 method.

Nowadays, computational facilities allow us to proceed with larger basis sets
than fifteen years ago. For instance, we have reported calculations with the
aug-cc-pVTZ, def2-TZVP and def2-QZVP basis sets (see Table 1). By comparing
the aug-cc-pVDZ and aug-cc-pVTZ results, it is clear that the MP2 interaction
energies are not yet converged with aug-cc-pVDZ basis set, which somehow
justifies a posteriori the present benchmark. Thus, at least a triple-zeta
basis set should be used. As an alternative to the aug-cc basis sets, we have
employed the def2-TZVP and def2-QZVP ones (see Table 1). Notably, the
def2-TZVP basis set show very similar results to the ones obtained with the
aug-cc-pVTZ basis set, despite being built with fewer functions. The largest
difference between the corresponding sets of results is below
$20\text{\,}\mathrm{kcal}\text{\,}{\mathrm{mol}}^{-1}$ for the $n$ = 9
complex, which translates into
$\sim$$2\text{\,}\mathrm{kcal}\text{\,}{\mathrm{mol}}^{-1}$ per ligand. Since
the calculations with aug-cc-pVTZ basis set take roughly 5 to 20 times more
time than the def2-TZVP one, it is clear that the best accuracy/cost ratio is
expected for the def2 basis sets, in particular the def2-TZVP one.

As an additional step of analysis, we compared the results obtained with the
def2-TZVP and def2-QZVP basis sets. As can be seen in Supporting Information
(Tables S1 and S2), interaction energies differ by less than
$2\text{\,}\mathrm{kcal}\text{\,}{\mathrm{mol}}^{-1}$ per ligand when the BSSE
correction is included (the correction effectively reducing this difference).
Considering the higher computational cost of calculations with the def2-QZVP
basis set, we conclude that the MP2/def2-TZVP level of theory reaches the best
accuracy to cost ratio out of all the tested methods for computing interaction
energies between the Po^4+ ion and the coordinated water ligands. Furthermore,
we do not expect a notable improvement of the interactions energies by
performing a complete basis set extrapolation 54, 55, 56 based on the
def2-TZVP and def2-QZVP calculations.

We have repeated similar calculations on the chlorinated complexes. First, we
have found that eight chloride ligands are sufficient to saturate the
coordination sphere of polonium(IV) ions. Second, we observe a change in the
evolution of the interaction energies (see Table 1), which is a priori more
visible in Figure 2. While in the case of water, the evolution is monotonous
with respect to ligand addition, the curve for chloride ligands displays a
minimum for $n$ = 5 (note that all the structures are stable though). We
connect this phenomenon with the negative charges of chloride ligands: for
each ligand addition, we earn a coordination bond but we also enhance the
inter-ligand repulsion in a much stronger way than with neutral ligands. We
have also repeated a similar benchmark as what was done for the step-wise
water addition (see Tables S3 and S4), leading to the exact same conclusion:
the MP2/def2-TZVP level of theory is a good compromise to compute accurate
interaction energies.

Fig. 2: MP2/def2-TZVP interaction energies
($\mathrm{kcal}\text{\,}{\mathrm{mol}}^{-1}$) as functions of the number of
ligands ($n$) for polonium(IV) complexes with water and chlorides.

As a part of our benchmark study, we have been looking at molecular
geometries, with emphasis on the mean Po-O and Po-Cl bond lengths. Prior to
fully digging into this, we start by an overall description of the structures.
Perspective views are given in Figure 3, together with the symmetry point
groups (SPGs). For the polonium(IV) complexes with water, we obtain similar
structures as the ones reported by Ayala et al. 19, if we exclude the
[Po(H2O)7]^3- complex from the discussion (a different SPG may be found
depending on the applied level of theory). Concerning this specific complex,
note that we report a $C_{2}$ SPG, obtained at the MP2/def2-TZVP level of
theory.

When comparing the [Po(H2O)_n]^4+ and [PoCl_n]^4-n structures two-by-two for
$n=$ 1–8, several situations occurs:

  * •

For $n$ = 2, the same SPG is observed;

  * •

For $n$ = 1, 3, 6 and 8, a direct subgroup of the SPG of the [PoCl_n]^4-n
complex is found for the corresponding [Po(H2O)_n]^4+ complex;

  * •

For $n$ = 4, 5 and 7, the [Po(H2O)_n]^4+ complexes display, if any, much less
symmetry elements than the corresponding [PoCl_n]^4-n ones. Those structures
are thus considered as qualitatively different.

Though it is obvious that the presence of the hydrogen atoms of the water
ligands may lower the symmetry, it is actually interesting to note that in
some of the cases ($n$ = 4, 5 and 7) quite different structures were obtained.

Concerning more specifically the PoCl4 complex, Stoïanov et al. reported in
their article 25 a $C_{2v}$ SPG for the gas-phase structure, with one set of
bonds being slightly longer than the other. In fact, we found with our level
of theory that the gas phase $C_{2v}$ structure is only
$1.6\text{\,}\mathrm{kcal}\text{\,}{\mathrm{mol}}^{-1}$ higher in energy than
the $T_{d}$ one, meaning that both are technically possible. We have thus
performed several test calculations with various levels of theory, and
concluded that though the potential energy hypersurface may be quite flat, the
$T_{d}$ structure must be the correct gas-phase one (we recall here that a
structure similar to the one of the corresponding water complex is expected
when a solvation model is applied 25). Finally, we would like to mention that
the structure of the [Po(H2O)9]^4+ complex, which has no match in the
chlorinated complex series, is highly symmetric ($D_{3}$ SPG), with 6 and 3 by
symmetry related bonds, respectively.

The dependence of the mean Po-O and Po-Cl bond lengths with respect to the
basis-set sizes may be seen from Tables 2, S5 and S6, and is also displayed in
Figure 4 for the polonium(IV) complexes with water. In line with the
interaction energies, we note significant shortening of the bonds by going
from the aug-cc-pVDZ basis set to the aug-cc-pVTZ one, a good agreement
between the aug-cc-pVTZ and def2-TZVP results, and little change by going from
def2-TZVP to def2-QZVP, meaning that the def2-TZVP basis set is again a good
accuracy/cost ratio performer.

Table 2:  Mean MP2 Po-O and PoCl bond lengths ($\mathrm{\text{\r{A}}}$), obtained for polonium(IV) complexes with water and chlorides, respectively |  H2O | Cl-
---|---|---
n(ligands) | aug-cc-pVDZ | aug-cc-pVTZ | def2-TZVP | def2-TZVP
1 | $2.08$ | $2.04$ | $2.03$ | $2.25$
2 | $2.11$ | $2.08$ | $2.07$ | $2.28$
3 | $2.15$ | $2.11$ | $2.11$ | $2.34$
4 | $2.20$ | $2.17$ | $2.17$ | $2.47$
5 | $2.25$ | $2.22$ | $2.22$ | $2.54$
6 | $2.30$ | $2.27$ | $2.27$ | $2.60$
7 | $2.34$ | $2.31$ | $2.31$ | $2.72$
8 | $2.37$ | $2.34$ | $2.34$ | $2.83$
9 | $2.41$ | $2.38$ | $2.39$ |
Fig. 3: Ball-and-stick representations and symmetry point groups of the
studied polonium(IV) complexes with water and chlorides. Colour code: polonium
(gray), chlorine (yellow), oxygen (red) and hydrogen (white). Fig. 4: Mean MP2
Po-O bond lengths
($\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{m}$),
obtained for polonium(IV) complexes with water for various basis sets, as
functions of the number of ligands ($n$).

For the [Po(H2O)_n]^4+ ($n$ = 1–3) complexes, we obtain similar MP2/aug-cc-
pVDZ bond distances than the ones of Ayala et al. 19, meaning that our
proposed change in the NFCs as virtually no effect on the molecular
geometries. Consequently, the good agreement of the MP2 and CCSD(T) structures
that they observed is preserved in the present study. Concerning the
[PoCl_n]^4-n ($n$ = 4–6) complexes, the mean B3LYP/aug-cc-pVTZ bond distances
reported by Stoïanov et al. 25 ($2.522.592.67\text{\,}\mathrm{\text{\r{A}}}$,
respectively), are significantly longer than the MP2/def2-TZVP ones (by
$\sim$$0.05\text{\,}\mathrm{\text{\r{A}}}$), which is in line with the
$\sim$$0.03\text{\,}\mathrm{\text{\r{A}}}$ observed by Ayala et al. 19 for the
[Po(H2O)_n]^4+ ($n$ = 1–9) series. Note that the application of a dispersion
correction 57 slightly diminishes those bond distances (by
$\sim$$0.01\text{\,}\mathrm{\text{\r{A}}}$ in the [PoCl_n]^4-n, $n$ = 4–6,
complexes), which thus only explains part of the difference between B3LYP and
MP2. Since the bond lengths of the [Po(H2O)_n]^4+ (n = 1–3) complexes tend to
be overestimated by $\sim$$0.02\text{\,}\mathrm{\text{\r{A}}}$ 19, we conclude
that our new MP2/def2-TZVP structures for the [PoCl_n]^4-n ($n$ = 4–6)
complexes are actually more accurate than the previously reported ones 25.

###  3.2 QTAIM analyses

To better characterise the nature of the bonds in the studied polonium
complexes, QTAIM 43 analyses were performed. Several indicators may be
defined, in particular at the bond critical points (BCPs). Such indicators may
be used to classify the nature of the bonds, as proposed by Nakanishi and
Hayashi 58 and validated by Pilmé et al. 59 for astatine species (a neighbour
of polonium in the periodic table). We report for convenience the bond
lengths, and more specifically the electron density values at the BCPs
($\rho$), the Laplacian of the density at the BCP ($\nabla^{2}\rho$), the
kinetic energy density (G), the potential energy density (V), the ratio
between the absolute value of the potential energy density to the kinetic
energy one (|V|/G), the delocalisation indices ($\delta$), and the Po and O or
Cl atomic charges (q) in Tables 3 and 4.

Table 3:  MP2/def2-TZVP Po-O bond distances ($\mathrm{\text{\r{A}}}$) and QTAIM bonding descriptors for polonium(IV) complexes with water $n$(H2O) |  $r$(Po-O), $\mathrm{\text{\r{A}}}$ |  $\rho$, a.u. | $\nabla^{2}\rho$ | G, a.u. | V, a.u. | |V|/G |  $\delta$(Po-O) | q(Po), a.u. | q(O) , a.u. | q(H) , a.u.
---|---|---|---|---|---|---|---|---|---|---
1 | $2.03$ |  | $0.20$ | $0.42$ | $0.15$ | $-0.20$ | $1.30$ | $1.1$ | $3.36$ | $-1.04$ | $0.84$
2 | $2.07$ |  | $0.12$ | $0.36$ | $0.13$ | $-0.17$ | $1.31$ | $0.8$ | $3.03$ | $-1.13$ | $0.80$
3 | $2.11$ |  | $0.11$ | $0.31$ | $0.11$ | $-0.15$ | $1.31$ | $0.6$ | $2.91$ | $-1.20$ | $0.78$
4 | $2.13$ |  | $0.10$ | $0.29$ | $0.10$ | $-0.14$ | $1.31$ | $0.5$ | $2.87$ | $-1.23$ | $0.77$
| $2.22$ | (x2) | $0.08$ | $0.24$ | $0.08$ | $-0.11$ | $1.28$ | $0.6$ |  | $-1.25$ |
| $2.12$ |  | $0.10$ | $0.30$ | $0.11$ | $-0.14$ | $1.31$ | $0.6$ |  | $-1.21$ |
5 | $2.13$ |  | $0.10$ | $0.30$ | $0.10$ | $-0.13$ | $1.29$ | $0.4$ | $2.85$ | $-1.23$ | $0.75$
| $2.24$ | (x2) | $0.08$ | $0.22$ | $0.08$ | $-0.10$ | $1.27$ | $0.5$ |  | $-1.26$ | $0.74$
| $2.24$ | (x2) | $0.08$ | $0.22$ | $0.08$ | $-0.10$ | $1.27$ | $0.5$ |  | $-1.26$ | $0.74$
6 | $2.27$ |  | $0.07$ | $0.21$ | $0.07$ | $-0.09$ | $1.26$ | $0.4$ | $2.84$ | $-1.26$ | $0.73$
7 | $2.32$ | (x2) | $0.07$ | $0.18$ | $0.06$ | $-0.07$ | $1.23$ | $0.3$ | $2.88$ | $-1.27$ | $0.71$
| $2.34$ | (x2) | $0.06$ | $0.18$ | $0.06$ | $-0.07$ | $1.23$ | $0.3$ |  | $-1.27$ | $0.72$
| $2.29$ |  | $0.07$ | $0.20$ | $0.06$ | $-0.08$ | $1.24$ | $0.3$ |  | $-1.26$ |
| $2.28$ | (x2) | $0.07$ | $0.20$ | $0.07$ | $-0.08$ | $1.24$ | $0.4$ |  | $-1.26$ |
8 | $2.34$ |  | $0.06$ | $0.18$ | $0.06$ | $-0.07$ | $1.21$ | $0.3$ | $2.90$ | $-1.27$ | $0.70$
9 | $2.43$ | (x3) | $0.05$ | $0.15$ | $0.04$ | $-0.05$ | $1.17$ | $0.2$ | $2.92$ | $-1.26$ | $0.69$
Table 4:  MP2/def2-TZVP Po-Cl bond distances ($\mathrm{\text{\r{A}}}$) and QTAIM bonding descriptors for polonium(IV) complexes with chlorides $n$(Cl-) |  $r$(Po-Cl), $\mathrm{\text{\r{A}}}$ |  $\rho$, a.u. | $\nabla^{2}\rho$ | G, a.u. | V, a.u. | |V|/G |  $\delta$(Po-Cl) | q(Po) , a.u. | q(Cl) , a.u.
---|---|---|---|---|---|---|---|---|---
1 | $2.25$ |  | $0.12$ | $0.26$ | $0.12$ | $-0.17$ | $1.46$ | $1.8$ | $2.52$ | $0.48$
2 | $2.28$ |  | $0.12$ | $0.18$ | $0.10$ | $-0.15$ | $1.53$ | $1.2$ | $2.06$ | $-0.03$
3 | $2.34$ |  | $0.11$ | $0.13$ | $0.08$ | $-0.12$ | $1.57$ | $0.9$ | $1.94$ | $-0.31$
4 | $2.47$ |  | $0.08$ | $0.13$ | $0.06$ | $-0.09$ | $1.46$ | $0.7$ | $1.98$ | $-0.50$
5 | $2.56$ | (x2) | $0.07$ | $0.11$ | $0.05$ | $-0.07$ | $1.43$ | $0.6$ | $2.06$ | $-0.64$
| $2.53$ | (x3) | $0.07$ | $0.11$ | $0.05$ | $-0.07$ | $1.44$ | $0.6$ |  | $-0.59$
6 | $2.60$ |  | $0.06$ | $0.10$ | $0.04$ | $-0.06$ | $1.41$ | $0.5$ | $2.13$ | $-0.69$
7 | $2.75$ | (x5) | $0.05$ | $0.07$ | $0.03$ | $-0.04$ | $1.34$ | $0.4$ | $2.13$ | $-0.74$
| $2.63$ | (x2) | $0.06$ | $0.09$ | $0.04$ | $-0.05$ | $1.39$ | $0.5$ |  | $-0.71$
8 | $2.83$ |  | $0.04$ | $0.07$ | $0.02$ | $-0.03$ | $1.28$ | $0.4$ | $2.12$ | $-0.77$

With the value of the electron density at the BCP of interest, one can
determine the nature of the bond, in particular, determine if the interaction
is of the _shared-shell_ ($\rho\geq 0.2$ a.u.) or _closed-shell_ type. This
classification may also be done based on the following two relations for the
Laplacian ($\nabla^{2}\rho(\boldsymbol{r})$) and the total electron energy
densities at BCPs ($H(\boldsymbol{r}$)) 58:

$\displaystyle\frac{1}{4}\nabla^{2}\rho(\boldsymbol{r})$
$\displaystyle=2G(\boldsymbol{r})\ +\ V(\boldsymbol{r})$ (3) $\displaystyle
H(\boldsymbol{r})$ $\displaystyle=G(\boldsymbol{r})\ +\ V(\boldsymbol{r})$ (4)

where $G(\boldsymbol{r})$ is positive and $V(\boldsymbol{r})$ is negative.
Actually, if the Laplacian of the density at the BCP is positive, as in all
the studied complexes, then the kinetic energy density dominates, meaning that
we are dealing with a _closed-shell_ interaction 58.

Another criterion that can help us with the description of chemical bonds is
the ratio between the absolute value of the potential energy density to the
kinetic energy one. If this ratio is smaller than 1, we expect a _pure closed-
shell_ interaction (thus a pure ionic bond or a typical non-covalent
interaction). If this ratio is larger than 1, which is the case in all the
studied complexes, it means that the potential energy is large (electrons are
thus stabilized at the BCP 58), we face a _regular closed-shell_ interaction.

For both the types of complexes we can see the same trend: the larger the
complex, the longer the bond, the smaller the values of electron density at
the BCP, and so do the values of the Lagrangian, potential and kinetic energy
densities. This is somehow logical since the addition of ligand will lead to
smaller charge transfers from each ligand to the polonium(IV) ion, i.e. to
reduced “covalency”, and since inter-ligand repulsion is also enhanced (which
obviously is much more pronounced for chloride ligands, as already mentioned).

The delocalisation index ($\delta$) 60, 61, is also another important
criterion, meant to be related to the bond multiplicity. A plot of the
comparison between the mean $\delta$(Po-O) and $\delta$(Po-Cl) values is
depicted in Figure 5, allowing a quicker view at the general trends than by
inspecting Tables 3 and 4. First, we can see that almost all the values are
lower or equal to 1, that represents an ideal single bond (the two atoms
exactly sharing one electron pair). In fact, half of the complexes presented
here have $\delta$’s that are smaller than 0.5, meaning that the corresponding
bonds are far from being covalent. Since a $\delta$ value around 1 may be
indicative of several bonding patterns (i.e. covalent, charge-shift 62 or even
iono-covalent), these are not incompatible with our previous conclusion
(_regular closed-shell_ interactions). In fact, the more striking value is the
one observed for the [PoCl]^3+ complex: 1.8. This value somehow resembles what
is expected for a double bond, meaning that this bond must bear a strong
covalent character.

In line with the delocalisation index, the QTAIM atomic q(Cl) charge in
[PoCl]^3+ is positive (strong Cl- to Po^4+ charge transfer). Knowing that
atomic charges are strongly dependent on the underlying charge model, we have
also computed the Mulliken 63 and Hirschfeld 64 atomic charges (see Table S7).
Since all the applied charge models lead to a positive q(Cl) in [PoCl]^3+, we
conclude for a strong charge transfer from Cl- to Po^4+ in this diatomic. The
trends observed by increasing the number of ligand is identical for the three
charge models and is in fact not surprising: q(Cl) always decrease with
increasing $n$.

Fig. 5: Mean MP2/def2-TZVP Po-O and Po-Cl delocalisation indices as functions
of the number of ligands ($n$) for polonium(IV) complexes with water and
chlorides, respectively.

###  3.3 Spin-orbit coupling calculations

As already mentioned, the free Po^4+ ion has a [Xe]4f145d106s2 electronic
structure. We recall here that within the non-relativistic or scalar-
relativistic picture (i.e. when one considers orbitals and not spinors), the
SOC is only active with the occurrence of partially filled p, d, f, etc.
shells (note that this is a necessary but not sufficient condition).
Therefore, electron donation to the Po^4+ ion is required to activate the SOC,
and in fact we have just seen that such a strong charge transfer occurs for
the [PoCl]^3+ complex, which encouraged us to analyse the role of the SOC on
this Po-Cl bond, in particular concerning the bond distance.

At the non-relativistic or scalar-relativistic Hartree-Fock level, three
bonding valence orbitals are doubly occupied and have no populated antibonding
counterparts (see Figure 6). These are the doubly degenerate $\pi$-bonding
HOMO and HOMO-1 orbitals and the HOMO-2 $\sigma$-bonding orbital. This
$\sigma$-bonding orbital is lower in energy than the $\pi$-bonding ones,
meaning that the molecular orbital diagram is uncorrelated. Below these, the
two anti bonding and bonding $\sigma$ orbitals are essentially formed by
linear combinations of the 6s orbital of Po and of the 3s orbital of Cl. Those
two extra valence orbitals are doubly occupied, giving a net zero contribution
to the formal bond order. Overall, the formal bond order in this system is 3,
and the populated bonding orbitals are (i) delocalised on the two atoms (see
Figure 6) and (ii) essentially formed by linear combinations of the 6p
orbitals of Po and of the 3p orbitals of Cl.

Fig. 6: Representations of the HOMO, HOMO$-$1 and HOMO$-$2 $\pi$ and $\sigma$
bonding orbitals of the [PoCl]^3+ complex. Note that HOMO and HOMO$-$1 are
degenerate by symmetry (those are the $\pi$ orbitals).

Without explicitly computing effective bond orders (EBOs) 65, 66, 67, 68, 69,
we would like to stress that two effects will tend to weaken the bond and thus
enlarge the bond distance:

  * •

electron correlation may promote electron pairs from a bonding to an
antibonding orbital, here especially from the $\sigma$ orbital to the
$\sigma$* one (correlation induced reduction of the $\sigma$ contribution to
the EBO);

  * •

the SOC may promote single electrons to antibonding orbitals, here especially
the $\pi$ orbitals to the $\sigma$* one (SOC induced reduction of both the
$\pi$ and $\sigma$ contributions to the EBO).

As a consequence, the effective bond order will significantly differ from the
formal one, meaning that one should not see the Po-Cl bond in [PoCl]^3+ as
triple (and we have already seen with the delocalisation index, in the absence
of the SOC, is more likely to be a double bond).

In previous works 19, 25, the role of the SOC on the molecular geometries of
polonium(IV) complexes was neglected, based on reference to the electronic
structure of the free Po^4+ ion. However, since we have here evidenced that a
strong effect can potentially occur in the hypothetical [PoCl]^3+ complex, we
have computed molecular geometries in the absence and in the presence of the
SOC, with two different frameworks, which are in fact complementary (see
Computational details section):

  * •

ZORA scalar relativistic and two-component Hartree-Fock calculations (no
electron correlation, a priori inclusion of the SOC);

  * •

two-step SOCI calculations, with a first scalar relativistic step and then
diagonalisation of the H${}_{\text{tot}}$ = E${}_{\text{el}}$ \+
H${}_{\text{SOC}}$ matrix in the basis of the $M_{S}$ components of the states
of the previous step (spin-orbit free electron correlation, a posteriori
inclusion of the SOC).

We just recall here that the first approach may overestimate the impact of the
SOC (by lacking correlation) and that the second one (contracted SOCI) may
underestimate it (by neglecting the SOC polarisation 70).

First, we tried out the _a priori_ inclusion of the SOC, using the AMS
software. For the [PoCl]^3+ complex, we observed an increase in the bond
length from
$2.17\text{\,}\mathrm{\text{\r{A}}}2.29\text{\,}\mathrm{\text{\r{A}}}$, i.e. a
$0.12\text{\,}\mathrm{\text{\r{A}}}$ SOC-induced bond lengthening. This is a
significant effect, clearly larger than the expected error of the underlying
quantum mechanical method (meaning that this effect is significant and thus
not negligible). As we have seen that “covalency”, a key ingredient to active
the SOC in polonium(IV) complexes, quickly diminishes in the [PoCl_n]^4-n
series, we have performed calculations for the $n$ = 2, 3, 4 and 8 complexes.
For the [PoCl2]^2- complex, the SOC triggers a
$0.01\text{\,}\mathrm{\text{\r{A}}}$ bond lengthening, which becomes even more
insignificant for larger $n$ values. From this first SOC study, we can already
conclude that the SOC is negligible in high-coordinated polonium(IV)
complexes, implying that its inclusion should not be key to perform future
calculations of polonium(IV) complexes in solution (solvent molecules will
always surround the heavy cation). On the contrary, a significant SOC effect
can potentially occur on low-coordinated polonium(IV) complexes, meaning that
SOC should somehow generally be probed in diatomics or systems with only a
coupled of coordinated ligands (those systems are mostly of theoretical
interest since gas phase experiments with polonium are not that safety
friendly).

Alternatively, we have also performed SOCI calculations on the [PoCl]^3+
complex, which proved to be interesting in terms of the SOC. We recall here
that the two-step SOCI implementations do not yet propose any numerical or
analytical gradients, meaning that the user has to explore the potential
energy surfaces, here the potential energy curve (obviously with respect to
the bond distance), by manual scans 71. Several degrees of freedom are
inherent to the SOCI approach: the size of the active space in the state-
average complete active space (SA-CASSCF) calculation, the number of states
that are included in the SA-CASSCF calculation and the way post-CASSCF
electron correlation is introduced. In our calculations, we have considered 6
active electrons within 6 active orbitals (the $\sigma$, $\pi$, $\pi$* and
$\sigma$* orbitals that are essentially formed by linear combinations of the
6p orbitals of Po and of the 3p orbitals of Cl). Note that test calculations
showed that enlargement of the active space hardly affected our conclusions.
With a ground state spin-singlet state, only spin-triplet spin components are
directly coupled to it. After some tests, we have set up the SA-CASSCF state
space to 11 spin-triplet and 1 spin-singlet states. Post-CASSCF electron
correlation has been introduced by replacing the E${}_{\text{el}}$ diagonal
elements of the SOCI matrix 72, 73 by $N$-electron valence space perturbation
theory at second order (NEVPT2) ones 74, 75, 76. With this setup, we observed
a SOC-induced bond lengthening of $0.08\text{\,}\mathrm{\text{\r{A}}}$ (from
$2.27\text{\,}\mathrm{\text{\r{A}}}2.35\text{\,}\mathrm{\text{\r{A}}}$).

Before concluding on the role of the SOC on the bond distance, it is important
to comment on the difference between both the scalar relativistic bond
distances reported above ($2.17\text{\,}\mathrm{\text{\r{A}}}$ vs.
$2.27\text{\,}\mathrm{\text{\r{A}}}$). For the sake of pedagogy and
completeness, we would like to mention that three main factors may affect this
difference: the basis set (TZ2P vs. a combination of DKH-def2-TZVP and SARC-
DKH-TZVP, depending on the atom), the scalar relativistic Hamiltonian (ZORA
vs. Douglas-Kroll-Hess) and the reference electronic structure method
(Hartree-Fock vs. NEVPT2). As mentioned before, electron correlation is meant
to weaken the bond order and in fact, also enlarge the bond distance. To probe
this, we have performed an additional Hartree-Fock scan with ORCA with the
same setup as in the second set of calculations, reproducing an equilibrium
bond distance of $2.17\text{\,}\mathrm{\text{\r{A}}}$, as in the first set of
calculations. We thus essentially attribute the observed difference in scalar
relativistic bond distances to electron correlation.

Since the second approach is expected to underestimate the effect of SOC on
the bond distance while the first one should overestimate it (see above), we
finally conclude for a $\sim$$0.1\text{\,}\mathrm{\text{\r{A}}}$ SOC-induced
bond lengthening in [PoCl]^3+. This lengthening is of the same order of
magnitude as the one observed for the AtX (X = F–At) diatomic molecule for
instance 69, for which the importance of the SOC was more obvious (the free At
atom displaying a strong SOC splitting of $\sim$$2.9\text{\,}\mathrm{eV}$ 67
in its ground 2P state). Also, it was shown in these systems that the SOC
reduces the covalent character of the bond in terms of EBO 69 and of
topological indicators 77, 59, 78 such as the delocalisation index or the
population of the bonding basin of the electron localization function 79, 80,
81.

##  4 Conclusions

In this work, two series of polonium(IV) complexes were investigated,
respectively with water and chloride ligands. A new benchmark study performed
on the water complexes showed that the MP2/def2-TZVP level of theory is both
accurate and efficient.

We have seen that the coordination sphere of the polonium(IV) ion may fit up
to 9 water molecules or 8 chloride ligands, respectively. The interaction
energy curves built by increasing $n$ do not follow the exact same trend, a
fact which we have attributed to the neutral vs. charged nature of the
ligands.

We have also analysed the chemical bonds in the studied systems, generally
qualified as _regular closed-shell_ interactions. Because of signs of a strong
“covalency” in the [PoCl]^3+ complex, we have investigated the influence of
the SOC on the Po-Cl bond distance, quite unexpectedly revealing a SOC-induced
bond lengthening of $\sim$$0.1\text{\,}\mathrm{\text{\r{A}}}$ in this system,
which vanishes for larger coordinations (in fact already starting from $n$ =
2). This allows us to formulate a more general conclusion concerning heavy-
element chemistry:

  * Even if the reference free ion of a given heavy-element cation do not display any SOC, electron donation from the ligands to the cation may be strong enough to active it and trigger sizeable effects. As a consequence, computational chemists should definitely not neglect the role of the SOC a priori (i.e. without any testing), especially in the case of low-coordinated complexes.

Two main perspectives arise from this work, (i) exploring more deeply and
largely the role of the SOC in low-coordinated polonium(IV) complexes as well
as with valence iso-electronic complexes such as bismuth(III) ones and (ii)
studying thoroughly the role of solvation on the molecular geometries,
interaction energies and bonding in those complexes, which was out of the
scope of the present work.

## Author Contributions

All the calculations were performed by N.Z. All the authors contributed to the
scientific development of this work and jointly discussed the interpretation
of the results. N.Z. wrote the first draft of the manuscript that has been
then amended by all the co-authors.

## Conflicts of interest

There are no conflicts of interest to declare.

## Acknowledgements

This work has been supported by the MITI of CNRS (80|Prime project MSM4Po).
HPC resources from the CCIPL (“Centre de calcul intensif des Pays de la
Loire”) have been used. FR and VV acknowledge support from PIA ANR project
CaPPA (ANR-11-LABX-0005-01), I-SITE ULNE projects OVERSEE and MESONM
International Associated Laboratory (LAI) (ANR-16-IDEX-0004), and the French
Ministry of Higher Education and Research, region Hauts de France council and
European Regional Development Fund (ERDF) project CPER CLIMIBIO.

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