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Calculations of chemical shifts of x-ray emission
spectra and effective states of Nb atom in the Niobates
Yuriy V. Lomachuk,∗Daniil A. Maltsev, Yuriy A. Demidov, Nikolay S. Mosyagin, Lev A.
Batalov, Edward Fomin, Roman V. Bogdanov, Andrei V. Zaitsevskii, and Anatoly V. Titov†
National Research Centre “Kurchatov Institute” B.P.Konstantinov Petersburg Nuclear Physics Institute,
Gatchina, Leningrad district 188300, Russia
Department of Chemistry, Saint Petersburg State University,
Saint Petersburg, Petrodvoretz 198504, Russia and
Department of Chemistry, Moscow State University,
Vorob’evy Gory 1-3, Moscow 119991, Russia
(Dated: Thursday 12th January, 2017)
A computational approach to evaluation of chemical shifts of characteristic X-ray emission
lines of dand felement in different chemical compounds is proposed. It is based on modeling
the embedded cluster electronic structure using relativistic pseudopotential approximation
that is followed by restoration of (all-electron) wavefunction in the core region. The proposed
approach is applied to the Nb Kαlines in fersmite. The line positions and their shifts
with respect to those of metallic Nb were evaluated via the restoration of the all-electron
wavefunction in the Nb core region. The results are in satisfactory agreement with the
experiment, overestimating the measured value by ∼20 %.
PACS numbers: 31.15.V-;31.15.A-;32.30.Rj;31.15.es; 33.15.-e; 31.15.ae
Keywords: Electron correlation calculations for atoms ions and molecules, Properties of atoms,
Theory of electronic structure electronic transitions and chemical binding, Properties of molecules
Introduction
According to modern concepts of closed
nuclear fuel cycle, the immobilization and con-
servation of the actinide fraction is provided for
in the form of chemically stable and radiation-
resistant matrices, which is to be followed by
their long-term storage in deep geological for-
mations. As the most promising matrix ma-
terials, some synthetic mineral-like composites
are regarded and some uranium and thorium
containing accessor minerals can serve as the
composites natural analogues, many of which
have been in existence for hundreds of mil-
lions or even for billions of years. Hence,
a study on these natural formations, of their
stability to aggressive factors of the environ-
ment, of post-effects of nuclear-chemical- and
radiation-induced chemical processes, of the
physico-chemical state of the f- and d-elements
in such minerals, could provide important prog-
∗E-mail: jeral2007@gmail.com
†URL: http://www.qchem.pnpi.spb.ru
nostic information on the immobilization prop-
erties of synthetic matrices to be used for high-
level radioactive waste (HLW) disposal. In par-
ticular, the niobium-(titanium-tantalum) based
complex oxides AB2O6are considered to be
some of perspective actinide-containing matri-
ces; their natural analogues being minerals eu-
xenite, aeschynite polycrase, and the general
chemical formula, (Ca,TR,U,Th)(Nb,Ta,Ti)2O6,
where TR stands for “terrae rarae” (lat.), or
“rare earth” elements.
The metamict degradation, as an amor-
phization of the mineral structure, is an im-
portant consequence of the radioactive decay.
Therefore, one of the important issues here is
chemical consequences of the mineral matamic-
tization, i.e. evolution of the physical-chemical
states of atoms in the A- and B-groups and,
in particular, red-ox states of the polyvalent
atoms which are f- and d-elements here. While
studying aeschynite and euxenite, Ewing et al.
have found that amorphization of the mineral
occurs in atomic displacement cascades initi-
ated by alpha-recoil atoms and affecting several
thousands of lattice atoms [1]. As the alpha-
2Yuriy V. Lomachuk, Daniil A. Maltsev et al.
decay dose increases, individual cascades over-
lap, and gradually occupy the mineral volume.
A chemical consequence of amorphization of eu-
xenite and aeschynite [2] consists of breaking of
the dioctahedric chains in (Nb,Ta,Ti)2O10 and
a spatial distortion of the octahedrons formed
by the Nb, Ta and Ti atoms with the oxygen
ligands. At the same time, there are insignifi-
cant changes observed in the first coordination
sphere of the B-position (Nb-O6etc.) atoms: a
small decrease of the coordination number and
of the bond lengths. When there is a transfor-
mation occuring of euxenite and aeshynite into
the metamict state, the main damage is observed
in the second coordination sphere, namely, it is
the inclination angle of the cationic coordina-
tion polyhedron that is changed. The damage of
the second coordination sphere that leads to the
loss of the long-range order and of the crystal
periodicity, was clearly observed in the EXAFS
and XANES [3] spectra, but there was no signif-
icant difference found between the Nb-O bond
lengths of the crystal and metamict samarskite
(AB1.7O5.3). As to the stability of the AB2X6
group of minerals to post-effects of the radioac-
tive decay, there is a positive influence observed
of the high ionicity of the Nb-O bond. The pa-
pers cited here are only few among those devoted
to studies of physical-chemical state of the group
B atoms. However, these are related rather to
structural or geometrical changes in the octahe-
dral polyhedron system. There were no signifi-
cant data obtained on the electronic subsystems
of these atoms.
The aim of the present work is to provide
some theoretical estimates for chemical shifts of
the Kα-line of niobium, which enable, as we be-
lieve, some correct data on the chemical state of
niobium in metamict and crystalline samples of
the AB2O6group of minerals, as well as ther-
mochemical processes in various minerals of this
type to be studied.
I. Cluster modeling technique
Theoretical modeling of localized phenom-
ena in solids comprising d- and f-elements by
density functional theory (DFT) techniques for
extended periodic systems encounters serious
problems arising from the unavailability of ad-
vanced meta-gradient and, particularly, hybrid
functionals and basis set restrictions. An at-
tractive alternative consists in cutting a cluster
which can be studied by non-periodical (“molec-
ular”) techniques (i.e. using so-called embedded
cluster model [4]). Provided that the process (or
property) of interest is localized at a single atom,
one should reproduce some important character-
istics of the surroundings of this atom to preserve
the electronic structure around the atom with
appropriate accuracy.
To determine an appropriate set of em-
bedding data for the cluster modeling we have
performed the calculations of periodic CaNb2O6
structure at experimental geometry [5] using the
OpenMX package [6, 7] and applying there the
LSDA-CA functional [8]. To minimize uncer-
tainties of the Mulliken population analysis (uti-
lized in OpenMX) we used the minimal basis set:
one radial function per each occupied shell.
While there are different approaches to con-
struct appropriate cluster model (see Ref. [9–11]
and references), in this paper we follow a way
based on fitting the cluster characteristics such
as charges or force field distributions to those in
the solid.
FIG. 1. The cluster model of the CaNb2O6. The cen-
tral niobium atom Nb1 is surrounded by the six oxy-
gen atoms O1–O6 in the first coordination sphere and
niobium atoms Nb2–Nb7 and calcium atoms Ca1–
Ca4 in the second coordination sphere. The first co-
ordination sphere displayed as a crystal polyhedron.
In this scheme the central atom (Nb) and
its first coordination sphere (6 oxygen atoms)
Nonlinear Phenomena in Complex Systems Vol. , No. , 2016
Calculations of chemical shifts of x-ray emission spectra and effective states of Nb atom in the
Niobates 3
are included into the cluster as usually (being
all-electron or conventional RECP-modeled ele-
ments). In turn, the atoms of second coordina-
tion sphere (Ca1–Ca4, Nb2–Nb7) are simulated
by boundary “pseudo-atoms” (having a prede-
fined reduced pseudo-charge and/or an opera-
tor describing the embedding) in order to make
whole cluster neutral and with the closed shells
only, while preserving most essential characteris-
tics of the electronic distribution inside the clus-
ter.
The boundary “pseudo-atoms” are cur-
rently described by some embedding potentials
for corresponding atoms with reduced nuclear
pseudo-charges. We assume here that the va-
lence electrons of each boundary atom in the
original solid crystal can be divided by some lo-
calization procedure into two parts: the first one
that is located “inside” the cluster of interest,
and the other one located “outside” the cluster.
Next, we expect that effective core charge of a
corresponding “pseudo-atom” should be equal to
the number of electrons in first part, located in-
side the cluster. Thus, the cluster will be electri-
caly neutral, while each border atom will have a
charge, proportional to it’s electronic contribu-
tion to a cluster area in the crystal.
To estimate electronic distribution of
boundary atoms, we assume that the total va-
lence electronic density of crystal can be divided
into atomic (lone pairs, if exist) and diatomic
parts (bonds), supposing that such bonds exist
only between closest neighbours and that the va-
lence electrons of each atom are distributed only
among its bonds and lone pairs. If we find this
distribution, we will know partial charges of the
atoms.
In the case of simple crystals, like MgO [10]
or ZrO2[9], this distribution can be trivially ob-
tained from the symmetry of atomic surround-
ings. However, for more complex systems, like
CaNb2O6under study, which includes 5 non-
equivalent atoms and 9 non-equivalent bonds,
precise results cannot be easily determined, so
that some estimations should be done.
We used several approaches to determine
distribution of “atomic” electrons among bonds.
In first one, we assume that each bond can be as-
sociated with one or more doubly occupied bond
orbitals [9]. Requiring that total contribution of
each atom to all its bonds is equal to its initial
number of valence electrons, and sum of contri-
butions to each bond from two involved atoms
is twice the number of occupied orbitals, corre-
sponding to this bond. We get a system of linear
equations, which is undetermined, but we can
choose a least-norm solution. The main draw-
back of this method is that it is “more mathe-
matical than chemical”, so that the results are
fully determined by the number of valence elec-
trons of each atom and bond topology.
In the second method we associate each
bond with the part of valence electronic den-
sity between two atoms, which is is not obliged
to be related to bond orbitals in any way. We
calculate a ”characteristic order” of each bond,
in the present case, Mulliken overlap population
and Mayer index, and assume that the valence
electrons of each atom are distributed among
its bonds proportionally to these orders. This
method is much more ”chemical” in its nature.
However, unlike the previous one, it does not au-
tomatically provide integer number of electrons
in the cluster, so, the results generally should be
corrected.
Table I. Additional charges on atoms of the second
coordination sphere of Nb in CaNb2O6cluster model.
I II III IV
Ca1 1.08 0.42 0.50 0.35
Ca2 1.08 0.42 0.17 0.30
Ca3 1.08 0.17 0.33 0.35
Nb2 0.45 1.59 1.31 1.22
Ca4 1.08 0.17 0.33 0.30
Nb3 0.45 0.67 0.71 0.60
Nb4 0.45 0.67 0.33 0.35
Nb5 0.45 1.59 1.32 1.30
Nb6 0.45 0.67 0.99 0.91
Nb7 0.45 0.67 0.99 0.91
Additional charges obtained by (I) fitting the Nb1
Bader charge to that from periodic structure calculations
(Bader atomic charges are given in the Table II); (II) asso-
ciating the bonds with double-occupied orbitals; (III) dis-
tributing valence electrons according to Mulliken overlap
population; (IV) distributing valence electrons according
to Mayer index.
Additional positive charges located at the
atoms of the second coordination sphere are
Nonlinear Phenomena in Complex Systems Vol. , No. , 2016
4Yuriy V. Lomachuk, Daniil A. Maltsev et al.
listed in Table I. We also provide the cluster
model with the additional charges chosen to fit
central Nb atom Bader [12, 13] charge to that
from periodic structure calculations.
II. Computations of the X-ray
emission spectra chemical shifts of Nio-
bium in CaNb2O6
Chemical shifts of Niobium Kαlines in solid
CaNb2O6with respect to metallic Nb are calcu-
lated using the cluster model described above.
We employed the method described in our pre-
vious papers [14–16]. This method is based on
the relativistic pseudopotential model and one-
center restoration procedure [17, 18] to recover a
proper all-electron structure in heavy-atom cores
after the pseudopotential simulation of chemi-
cal compounds. In [14] the effective one-electron
operator is constructed to obtain the chemical
shifts values as difference between its mean val-
ues for considered compounds. In [15, 16] further
improvement and generalization of this method
are provided.
The results of these calculations are com-
pared to the experimental datum on the niobium
X-ray emission spectra (XES) chemical shift in
the wiikite mineral with respect to the metallic
niobium and chemical shifts computed as differ-
ences between corresponding transition energies
obtained from the all-electron calculations of the
periodic structures CaNb2O6and metallic nio-
bium.
Electronic structure calculations of the em-
bedded cluster are performed in the framework
of two-component DFT and relativistic pseu-
dopotential model [19–21] using the code de-
scribed in Ref. [22].
The exchange-correlation functional pbe0
[23] was used. For central niobium atom we em-
ployed the pseudopotential leaving 13 electrons
for explicit treatment, while the atoms of the
second coordination sphere were modeled with
zero electron pseudopotentials.
In the equilibrium state of the system, the
forces acting on each atom of the system com-
pensate each other. We can express this con-
dition in the following way. Let us denote the
coordinates of the central atom and the atoms
in the first coordination sphere as {~
Ri}N
i=1, the
additional charges of atoms of 2nd coordination
sphere as {Qj}M
j=1 and consider the total energy
of the system Uas function of {~
Ri}and {Qj}.
Then we define the quantity
Φ = X|~
Φi|2=X
∂U
∂~
Ri
2
(1)
depending on the additional charges values and
measuring the deviation of the force field in the
cluster from that in the solid.
Table II. Bader atomic charges of central Nb atom
and its 1st coordination sphere and deviations of the
force field from that in solid.
IaII III IV cryst.b
Nb 2.60 2.74 2.73 2.73 2.59
O -1.38 -1.19 -1.17 -1.18 -1.06
O -1.46 -1.29 -1.28 -1.28 -1.16
O -1.38 -1.40 -1.36 -1.33 -1.16
O -1.34 -1.06 -1.99 -1.01 -1.16
O -1.33 -1.40 -1.51 -1.45 -1.16
O -1.28 -1.09 -1.14 -1.08 -1.16
Φc0.040 0.021 0.026 0.029
aI, II, III and IV: see the legend in Table I.
bData from CaNb2O6periodic structure calculations.
cTotal values of the squared force functional acting on
the central atom and its first coordination sphere (defined
in (1)).
Another criterion of the quality of the cho-
sen cluster model is provided by the Bader
charge analysis. The results of the Bader charge
analysis and Φ values for all discussed cluster
models are presented in Table II.
It follows from data listed in Table II that
these two criteria can hardly be satisfied simul-
taneously.
The chemical shifts values for niobium in
CaNb2O6with respect to the metallic Nb are
presented in Table III. Metallic Nb was mod-
elled in two ways: as Nb2molecule and the
single atom. The difference between chemical
shifts values for these two references is about 10
meV by the order of magnitude. The different
additional charges distributions give the results
which differ from each other up to 100 meV. This
demonstrates the importance of proper choice of
Nonlinear Phenomena in Complex Systems Vol. , No. , 2016
Calculations of chemical shifts of x-ray emission spectra and effective states of Nb atom in the
Niobates 5
Table III. XES chemical shifts of the Nb Kαlines in
the CaNb2O6with respect to the neutral atom and
Nb2, meV.
2p1/2→1s1/22p3/2→1s1/2
CaNb2O6, Ia– Nb2-331 -349
CaNb2O6, II – Nb2-393 -412
CaNb2O6, III – Nb -375 -406
CaNb2O6, III – Nb2-397 -416
CaNb2O6, IV – Nb2-391 -410
CaNb2O6
b– Nb (met.) -220 -215
expt.c−264 ±7
aI, II, III and IV: see the legend in Table I.
bResult of all-electron calculations of the CaNb2O6
periodic structure and metallic niobium performed by
wien2k code [24].
cExperimental datum on niobium chemical shifts in wi-
ikite mineral with respect to metallic Nb, provided by
one of the authors of the paper (E. F.) obtained by the
method described in Ref. [25]. Although Fersmite and
Wiikite are two different minerals, they share common
NbO6structure modelled in our study, so one can expect
the qualitative agreement between computational and ex-
perimental results.
distribution of additional charges. The exper-
imental value 264 meV, is in qualitative agree-
ment with the results obtained for the first model
and the results obtained from all-electron DFT
linearized augmented plane wave calculations of
the periodic CaNb2O6structure and metallic
Niobium with using wien2k code. These calcu-
lations give results close to the experimental da-
tum, although their quality cannot be considered
as high, so, the close agreement can be rather in-
terpreted as fortunate.
Conclusions
A computational approach to evaluation
of chemical shifts of characteristic X-ray emis-
sion lines of dand felement in different chem-
ical compounds is proposed. It is based on
modeling the embedded cluster electronic struc-
ture using relativistic pseudopotential approxi-
mation that is followed by restoration of (all-
electron) wavefunction in the core region. The
proposed approach is applied to the Nb Kαlines
in fersmite. The embedded cluster comprises the
central atom (Nb) and those of its first coordi-
nation sphere as well as a set of pseudoatoms
simulating the second coordination sphere. The
pseudoatoms are simulated in such a way that
the electrical neutrality of the cluster is main-
tained and the distribution of their electrons
among chemical bonds is reflected. To deter-
mine the pseudoatoms uniquely, additional re-
quirements were imposed (minimizing the force
field in the inner part of the cluster or repro-
ducing the infinite-crystal atomic charges for the
central atom an its first coordination sphere).
The emission line positions and their shifts with
respect to those of metallic Nb were evaluated
via the restoration of the all-electron wavefunc-
tion in the Nb core region. The results are in sat-
isfactory agreement with the experiment, over-
estimating the measured value by ∼20 %.
Acknowledgements
This work is supported by the Russian Sci-
ence Foundation grant No. 14-31-00022.
We are grateful to Professor
I. V. Abarenkov for many fruitful remarks
and discussions.
Thanks are due to Christoph van V¨ullen for
supplying us with code [22].
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