Computational Details
In this study, two models have been used to understand the arrangement
of the environment of Bk(IV) in carbonate and carbonate-hydroxide
solutions. All calculations herein, except ligand-field DFT (LFDFT),
were performed using the ORCA 4.1 package.29 For DFT
optimizations of the molecular structures, we utilized the scalar
relativistic all-electron Douglas-Kroll-Hess Hamiltonian (DKH), the PBE0
hybrid functional, and a SARC-DKH-TZVP basis set for the Bk(IV) atom and
def2-TZVP for all the other atoms. No symmetry constraints were used.
TD-DFT was used to model the UV-vis spectrum of the two optimized
models. Kohn-Sham density functional theory with approximate functionals
has proven to be remarkably successful given its relative simplicity and
efficiency. However, it is well known that DFT produces an important
delocalization error responsible for overestimation of covalency, which
has an impact in the correct prediction of the ground state and
spectroscopic properties. These problems can be solved in part using
detailed parameterizations principally in the interelectronic terms. A
good example of these functionals is the Coulomb attenuating method,
CAM-B3LYP functional,30 which was developed to
minimize deviations in charge-transfer excitation energies and used in
this work to reproduce the absorption spectra of both carbonate
compounds. It is important to note the recent study of Yang et
al.31, which is the closer DFT study to heavy
actinides using this class of functionals. In this context, this work
represents one of the first studies where the performance of CAM-B3LYP
functional is applied to heavy actinides.
The Complete Active Space Self-Consistent Field (CASSCF) approach
combined with NEVPT2 for the inclusion of both, static and dynamic
correlation, allows the construction of the configuration of all
possible multiplicities within a determined active orbital space.
Additionally, state mixing due to spin-orbit (SO) coupling can be
accounted for via quasi-degenerate perturbation theory approach (QDPT).
The effect of the dynamic correlation is included in the QDPT step as a
diagonal correction of the nonrelativistic state energies. This
combination of tools has been shown to be an effective methodology in
the description of actinide compounds. The CASSCF/NEVPT2 approach
provides a reliable treatment of the multiconfigurational character
present in open-shell actinide systems and can be used with all-electron
relativistic Hamiltonians. The CASSCF calculations were carried out
using, initially, a minimal active space, which corresponds to the seven
5f electrons of the Bk(IV) ion in seven 5f orbitals. This active space
was then augmented to a CAS(11,9) and finally to a CAS(13,10), including
two and three symmetry adapted ligand orbitals respectively, which were
selected based on its Bk–O bonding or antibonding character and 5f
contribution. The 6d shell of orbitals was also considered. However, the
occupation numbers of these orbitals were less than 0.02, so they were
omitted in the results and discussion section.
Ligand-field theory was also considered for this study from two
perspectives, ab initio (AILFT) and DFT (LFDFT)32. The
former was calculated from the CASSCF matrix within ORCA, whereas the
latter was performed in ADF201933,34. LFDFT was
performed based on Kohn-Sham orbitals calculated at ZORA/PBE0/TZP level
of theory.
To obtain a better understanding of the nature of the chemical bond,
Natural Bond Orbital (NBO) analyses were performed on correlated
wavefunctions to localize orbitals. The NBO analysis is helpful to
understand the Lewis picture of bonding by performing a transformation
of the wavefunction into a localized form. NBOs are conceived as a set
of localized bonds and lone pairs as basic units, where delocalization
effects are missing from the molecular orbital (MO) perspective.
However, there is an intermediate approach that recovers the MO
perspective in the natural localized framework called the natural
localized molecular orbitals (NLMOs).35 These
calculations were carried out using the Weinhold’s package
NBO636 attached to ORCA.
Another tool especially developed to gain a deeper comprehension on the
nature of bonding in molecular systems is the Bader’s quantum theory of
atoms in molecules (QTAIM).37 QTAIM parameters at the
bond critical points (BCPs) such as the electron density
ρBCP(r), Laplacian of the electron density
∇ρBCP(r), delocalization δBCP(r) and
localization l indices, and total energy densities
HBCP(r) are commonly used to characterize the nature of
bonds. It is well-known that covalent bonds or open-shell interactions,
in terms of QTAIM metrics, present ρBCP(r)
> 0.2 a.u. with ∇ρBCP(r) < 0 and
HBCP(r) < 0, while ρBCP (r)
< 0.1 a.u. with ∇ρ (r) > 0 and
HBCP(r) > 0 represent ionic, van der Waals
or hydrogen bonds (closed-shell interactions). However, as Kerridge
pointed out, there is no a priori reason to assume this
convention for f-element chemistry.13 On the other
hand, delocalization and localization indices are useful to quantify the
bond order and calculate the partial oxidation states,
respectively.38
Additionally, the Interacting Quantum Atom (IQA)27energy partition analysis have been carried out. This complementary tool
to understand the chemical bond under the framework of nonoverlapping
quantum atoms decomposes the total energy of interaction between two
topological atoms into Coulomb and exchange-correlation contributions.
These calculations are obtained by 6-dimensional integration over the
whole electron density between the two topological atoms, which uses the
2-electron density matrix (2EDM).
QTAIM parameters and IQA energies were calculated using the AIMAll
software developed by Todd Keith.39 All parameters
reported herein were based on multiconfigurational wavefunctions. For
post-HF wavefunctions, AIMAll uses the Muller
approximation40 to obtain the 2EDM due to the high
computational costs involved in obtaining the exact matrix.