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Effective Hamiltonian Crystal Field: Present Status and Applications to Iron Compounds

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We review the basics of the Effective Hamiltonian Crystal Field (EHCF) method originally targeted for calculations of the intra-shell excitations in the d-shells of coordination compounds of the first row transition metal. The formalism employs in the concerted way the McWeeny's group-function approximation and the Lowdin partition technique. It is needed for description of the transition metal complexes with partially filled d-shells where the (static) electronic correlations are manifested. These features are particularly important for electron fillings close to " half shell " ones occurring, for example, in the Fe 21 and Fe 31 ions. Recently we extended this methodology to polynuclear coordination compounds to describe magnetic interactions of the effective spins residing in several open d-shells. This improves the accuracy from about 1000 cm 21 to that of about 100 cm 21 , that is, eventually by an order of magnitude. This approach implemented in the MagAixTic package is applied here to a series of binuclear Fe(III) complexes featuring l-oxygen super-exchange pathways. The results of calculations are in a reasonable agreement with available experimental data and other theoretical studies of protonated bridges. Further we discuss the application of the EHCF to analysis of Mosbauer experiments performed on two organometallic solids: FeNCN and Fe(HNCN) 2 and conjecture a new thermal effect in the latter material. V
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Effective Hamiltonian Crystal Field: Present Status
and Applications to Iron Compounds
Andrei L. Tchougr
eeff,*
[a,b]
Alexander V. Soudackov,
[c]
Jan van Leusen,
[d]
Paul K
ogerler,
[e]
Klaus-Dieter Becker,
[f]
and Richard Dronskowski
[g,h]
We review the basics of the Effective Hamiltonian Crystal
Field (EHCF) method originally targeted for calculations of
the intra-shell excitations in the d-shells of coordination com-
pounds of the first row transition metal. The formalism
employs in the concerted way the McWeeny’s group-function
approximation and the Lowdin partition technique. It is
needed for description of the transition metal complexes
with partially filled d-shells where the (static) electronic corre-
lations are manifested. These features are particularly impor-
tant for electron fillings close to “half shell” ones occurring,
for example, in the Fe
21
and Fe
31
ions. Recently we
extended this methodology to polynuclear coordination com-
pounds to describe magnetic interactions of the effective
spins residing in several open d-shells. This improves the
accuracy from about 1000 cm
21
to that of about 100 cm
21
,
that is, eventually by an order of magnitude. This approach
implemented in the MagAixTic package is applied here to a
series of binuclear Fe(III) complexes featuring l-oxygen super-
exchange pathways. The results of calculations are in a rea-
sonable agreement with available experimental data and
other theoretical studies of protonated bridges. Further we
discuss the application of the EHCF to analysis of Mosbauer
experiments performed on two organometallic solids: FeNCN
and Fe(HNCN)
2
and conjecture a new thermal effect in the
latter material. V
C2015 Wiley Periodicals, Inc.
DOI: 10.1002/qua.25016
Introduction
The currently dominating paradigm in quantum chemistry can
be characterized as “monistic mechanism”. Fundamentally peo-
ple believe that a molecule must be calculated by a single
program in one ultimate setting. Deviations from this para-
digm are admitted only as concessions to temporary technical
complications which will be certainly overcome in the future.
During the times when the career of the older coauthors of
this article evolved this ultimate method was consequently
MO LCAO, ab initio, and currently—the DFT. The real life if con-
sidered from more physical point of view seems to be very
much different from this ideal and much more interesting.
Quantum chemistry largely reduces to searching approximate
solutions of the “Schr
odinger equation (SE)”:
HW5EW(1)
with the molecular electronic Hamiltonian:
H51
2XN
iDi11
2XA
ab
ZaZb
j~
Ra2~
Rbj
11
2XN
ij
1
j~
ri2
~
rjj2XAN
ai
Za
j~
Ra2
~
rij
(2)
The quotation marks applied to the SE Eq. (1) indicate a
rarely recognized fact that Eqs. (1) and (2) refer not to one SE
rather to a wide family of SE’s parameterized by sets of inte-
gers, Zaand three-dimensional-vectors ~
Ra.
The dominating paradigm mentioned above bases on certain
existence theorems, one of which is the L
owdin theorem of
expansion
[1]
stating that provided a complete basis of one-
electron states is given the complete basis of N-electronic func-
tionsisformedbyalltheN-electronic Slater determinants so that
the solutions of Eqs. (1) and (2) can be approximated with
[a] A. L. Tchougr
eeff
Moscow Center for Continuous Mathematical Education, Moscow, 119002,
Russia
E-mail: tch@elch.chem.msu.ru
[b] A. L. Tchougr
eeff
Moscow State University (Lomonosov), Moscow, 119992, Russia
[c] A. V. Soudackov
Department of Chemistry, University of Illinois at Urbana-Champaign,
Urbana, Illinois, 61801
[d] J. van Leusen
Institute of Inorganic Chemistry, RWTH Aachen University, Aachen,
D-52056, Germany
[e] P. K
ogerler
Institute of Inorganic Chemistry, RWTH Aachen University, Aachen,
D-52056, Germany
[f] K.-D. Becker
Institute of Physical and Theoretical Chemistry, Braunschweig Technical
University, Braunschweig, D-38023, Germany
[g] R. Dronskowski
Institute of Inorganic Chemistry, RWTH Aachen University, Aachen,
D-52056, Germany
[h] R. Dronskowski
J
ulich–Aachen Research Alliance (JARA-HPC), RWTH Aachen University,
Aachen, 52056, Germany
Contract grant sponsor: RFBR; contract grant number: 14-03-00867.
V
C2015 Wiley Periodicals, Inc.
Standard notation is used.
282 International Journal of Quantum Chemistry 2016,116, 282–294 WWW.CHEMISTRYVIEWS.ORG
REVIEW WWW.Q-CHEM.ORG
arbitrary precision if enough terms is spent. The physical fact,
however, is that solutions of Eqs. (1) and (2) in different areas of
the fZa;~
Ragparameter space are very different in nature.
[2]
Thus
the convergence by itself does not guarantee the success, since
the required expansion can become too long. In this situation a
legitimate question is how to regroup the series so that the con-
vergence is achieved faster.
Similarly, in case of the DFT the famous Hohenberg-Kohn
and Kohn-Sham theorems
[3]
state the existence of an univer-
sal exact functional producing the ground state energy of an
N-electronic system through its one-electron density only.
The problems which arise in this relation are exemplified by
the following pairs of the two-orbital, two-electronic
functions
Alexander V. Soudackov is a Research Assistant Professor at the University of Illinois at
Urbana-Champaign (UIUC) in USA. He studied Chemistry at the Lomonosov Moscow
State University (MSU) (MSc, 1986). His Ph.D. Thesis (1992) was devoted to the develop-
ment of the Effective Hamiltonian–Crystal Field (EHCF) methodology and its applications
to transition metal complexes. After postdoctorate in Germany (1994–1996) as an
Alexander von Humboldt Fellow and in the USA (1998–2002) he joined the Chemistry
Department of the Pennsylvania State University in 2002 and then moved to UIUC in 2013.
His current research is focused on the theoretical studies of the dynamics of charge trans-
fer reactions in complex environments.
Jan van Leusen was born in Kempen, Germany, in 1973. He studied physics at RWTH
Aachen University, and was awarded the Dipl.-Phys. in 2000 and the Dr. rer. nat. in theo-
retical particle physics in 2004. After a few years in industry, in 2008 he returned to his for-
mer alma mater to study chemistry. After receiving the B.Sc. in 2011 and M.Sc. in 2013,
he is now a Ph.D. student at RWTH Aachen performing research that focuses on the mag-
netochemistry of molecular compounds.
Paul K
ogerler graduated with a Dr. rer. nat. degree with Prof. Achim M
uller at the Univer-
sity of Bielefeld (Germany) in 2000, followed by a postdoctoral research stay at the Depart-
ment of Physics and Astronomy at Iowa State University (USA). In 2003, he was
appointed as a tenured Associate Scientist at the U.S. DOE Ames Laboratory, before
returning to Germany in 2006 as a Professor of Chemistry at the Institute of Inorganic
Chemistry at RWTH Aachen University and Group Leader for Molecular Magnetism at the
Peter Gr
unberg Institute (PGI-6) at Research Centre J
ulich.
Klaus-Dieter Becker studied physics in G
ottingen and received his doctorate in 1972. He
achieved his habilitation in Physical Chemistry in Bochum in 1979. In 1992, he joined Uni-
versity Hannover. In 1995, he became Professor of Physical Chemistry at Technische Uni-
versit
at Braunschweig (TUBS). 2010–2015 he was appointed Niedersachsen Professor at
TUBS. His research interest focuses on the application of spectroscopic techniques to the
elucidation of atomic defects/disorder, diffusion, and reactivity of solids.
Richard Dronskowski, born 1961, studied chemistry and physics in M
unster and received
his doctorate in Stuttgart in 1990. After a one-year stay as a scientific visitor with Roald
Hoffmann, he achieved his habilitation in Dortmund in 1995. In 1996 he went to RWTH
Aachen University where he is currently holding the Chair of Solid-State and Quantum
Chemistry. His interests lie in synthetic solid-state chemistry, in neutron diffraction, and in
the quantum chemistry of the solid state (electronic structure, magnetism, linear methods,
phase prediction, thermochemistry).
REVIEWWWW.Q-CHEM.ORG
International Journal of Quantum Chemistry 2016,116, 282–294 283
1
ffiffi
2
pjr#l"j6jl#r"j

;1
ffiffi
2
pjr#r"j6jl#l"j

;(3)
where in the first case the upper sign corresponds to the sin-
glet and the lower one to the triplet state of two electrons. As
one can easily check, the one-electron densities corresponding
to these states are equal, whereas the energies are not (the fact
known since the seminal work Ref. [4a]; if not earlier, the analy-
sis of its negative consequences for the DFT treatment of TMC’s
electronic structure is given in Ref. [4b]).
[4]
The universal exact
functional had it been known would definitely produce the cor-
rect energy, but in any case it would not be known to what
spin state it belongs. It had been shown many times that the
one-electron density is insensitive to the total spin (Ref. [5] and
references therein). That means that the available (pragmatic)
density functionals cannot distinguish the key element impor-
tant for description of many-electron states stemming from
(weakly split) “d-shell” AOs—the states being in relations shown
by the examples Eq. (3). This is, however, important since walk-
ing through the above mentioned fZa;~
Ragparameter space
may lead from the area with the singlet ground state to that
with the triplet ground state and vice versa (not talking about
further possibilities) not having any effect on the one-electron
density. These manifestations known as static correlations (or
under a more technical nickname of “multi-reference states”) are
fairly localized: they pertain to electrons in the d-shells and do
not manifest in the “ligands” (also including the valence s- and
p-shells of the transition metal ions—TMIs) for which the
dynamic correlations are more characteristic. Incidentally this
problematic area (or, more precisely, the problematic subspace
of one-electron states) is also responsible for the physics of this
class of objects: the low-energy electronic excitations are as well
fairly localized in the d-shells. This is reflected in the way the
spectral information is classified: namely by the (integer) number
of d-electrons.
[6]
The precise orbital composition of the d-shells
within (different) complexes is, of course, unknown, but it does
not affect the possibility of the classification which relies upon
the structure of the space of n-electronic functions in a five-
dimensional orbital space specific for each n.
This above mentioned defect of the pragmatic DFT setting
leads to known problems in describing truly correlation depend-
ent effects in quantum chemistry of transition metal complexes
(TMCs)—the relative energies of spin and orbital multiplets. These
features are particularly ubiquitous in the d-shells close to half fill-
ing, for example, in Fe
21/31
featuring the largest possible dimen-
sionalities of the configuration subspaces of each accessible total
spin. Namely this prevents available DFT methods, for example,
from correct description of the singlet-quintet separation in iron(II)
complexes (even the sign can be wrong)
[7]
which was one of the
reasons to undertake studies described in this review.
Crystal Field Theory and Effective
Hamiltonian of Crystal Field
Basics of EHCF and mononuclear setting
The key moment was to realize that the difficulties faced by
quantum chemistry methods when addressing iron complexes
are in a shear contradiction with the physically transparent
and successful phenomenological picture provided by the
crystal field theory (CFT ).
[8]
The CFT is physically justified by the observation that the
lowest-energy electronic excitations of the mononuclear TMCs
are those of their d-shells. These energies are controlled by
the effective crystal field induced by the ligands. The CFT for-
malizes this by reducing the consideration to the states of the
d-shells only. This comprises the correct half of the original
Bethe’s conjecture:
[8a]
the ground state spin and symmetry of
TMCs are those of electrons in the d-shell. Implicitly it refers to
the wave function (WF) of all electrons having the form of a
product of those in the d-shell and of the further unspecified
function of the remaining electrons:
Wn5UnðÞ
dÙUl(4)
Here UnðÞ
dis the nth full configuration interaction function of nd
electrons in the d-shell of the TMI and Ulis the function of all
other (nl) electrons of the system; the sign Ùindicates that the
resulting function is anti-symmetric with respect to permutations
of all electronic coordinates. Equation (4) represents a regrouping
of the terms of a potentially infinite series for the exact WF
adequate for the considered class of problems/molecules.
Originally, the effect of Uland the ligand nuclei was mod-
eled by the Coulomb field of the surrounding effective charges
induced on the d-shell. This is known as the ionic model of
the CFT. Although it perfectly reproduces the symmetry, at
best 20% of the observed splitting even if unrealistically large
effective charges are ascribed to the ligands can be repro-
duced. The irrelevance of the ionic model to experiment is
clearly seen from the integral results of the spectroscopic
measurements: the spectrochemical series
[6,8b]
ranging differ-
ent ligands according to the strengths of the crystal fields (the
10Dq parameter) they induce:
I2<Br2<S22<N3
2<F2<OH2<Cl2<1=2Ox22
<O22<H2O<SCN2<NH3;py <1=2en <SO322
<NO2
2<CN2<CO
The crystal fields are systematically weaker for charged ligands
than for the uncharged ones with the utter example of CO
inducing the strongest crystal field, although bearing neither
charge nor even significant dipole moment. Thus the strengths
of the crystal fields observed in the experiment must have
some other origin.
The WF Eq. (4) cannot be exact: the one-electron hopping
terms in the Hamiltonian of a TMC induce electron transfers
between the d-shell and the rest of the complex and mix the
states of the form Eq. (4) spanning the model subspace with
those in the outer subspace (following the terminology
[9]
)
spanned by the ligand-to-metal and metal-to-ligand charge
transfer (LMCT and MLCT) states. Including the surroundings of
the d-shell into consideration explicitly, opens the way for evalu-
ating the amount of the crystal field felt by d-electrons which
was not accessible in the phenomenological version of the CFT.
REVIEW WWW.Q-CHEM.ORG
284 International Journal of Quantum Chemistry 2016,116, 282–294 WWW.CHEMISTRYVIEWS.ORG
The required moves are performed in two steps. First, the
L
owdin partition
[10]
of the complete electronic Hamiltonian is
performed and the energy-dependent effective total Hamilto-
nian Heff EðÞacting in the model subspace is obtained. This
process is illustrated by Figure 1. Second, the variables describ-
ing the ligand electrons are integrated out by averaging the
interaction parts of Heff EðÞwith the WF Ul, which yields the
effective Hamiltonian for the electrons in the d-shell only, pre-
cisely as conjectured by Bethe:
[8a]
Heff
d5hUljHeff E0
ðÞjUli5Hcf 1Hee (5)
(here E0is the ground state energy of the Hamiltonian in the
outer subspace). In the last formula, Hcf stands for the (effective)
one-electron operator describing interactions of the electrons in
the d-shell with the atomic core of the TMI and its entire sur-
rounding, and Hee is the two-electron operator describing the
Coulomb interactions within the d-shell. The symmetry proper-
ties of the CFT Hamiltonian Hcf and Heff
dcoincide. However, the
matrix elements of Heff
dare not taken as parameters, but are cal-
culated within the EHCF procedure. For example, for the pair of
d-AOs mmthe effective crystal field matrix element is:
[11]
X
j
bljbmj njD11ðÞ
jðÞ212nj
ðÞD12ðÞ
jðÞ
hi
(6)
where njare occupancies (0 for an empty, 1 for a doubly
filled) of the MO’s in Ul(see caption of Fig. 1 for further nota-
tion). It ultimately comes from the mixing of the states in the
model configuration subspace—that with the fixed number of
electrons in the d-shell—with those in the outer subspace—
one spanned by the MLCT and LMCT states as depicted in Fig-
ure 1. This comprises original form of the effective Hamiltonian
crystal field (EHCF) theory.
[12]
It inherits the form of the WF
describing the ground and low-lying excited states of a TMC
which the CFT uses implicitly. This move turned out to be very
much successful numerically as we described previously many
times. It was the first example of using explicitly the group
product in quantum chemistry at least in the semi-empirical
context. It helped us that time to overcome the inherent
defects of the Hartree-Fock-based methods when applied to
open d-shell TMC’s. Two summits had been reached by this
technique: first, a long lasting story
[13]
of the ground state of
iron(II) porphyrin had been resolved and the correct one (3EÞ
had been reproduced.
[14]
Second, the ground states of spin-
active complexes of iron (II) had been correctly described at the
respective experimental geometries: they come out high-spin at
the high-spin geometry and low-spin at the low-spin one.
[15]
Moreover, the QM/MM extension of the EHCF approach
[16]
turned out to be very successful as shown in Figure 2. There
we depicted numerous iron(II) complexes with rather involved
organic ligands the ground state spins of which we were capa-
ble to reproduce. This has been reached by taking into account
the energies of the respective d-shells calculated by the EHCF
method. Since the intrashell static correlations were of crucial
importance here, the hybrid QM/MM-like incarnation of the
EHCF contained its local version which can be briefly character-
ized as a method of sequential derivation and independent esti-
mation
[12]
of parameters of the Angular Overlap Model
(AOM)
[5,17]
—the successful empirical systematics of the spectro-
chemical data combined with the correlated calculation of the
d-shell energy. It represents the crystal filed felt by the d-shells
as a superposition of ligand-specific increments elknown as
AOM parameters determined from experiment. The local EHCF
transforms the crystal field matrix elements Eq. (4) expressed
through ligand MOs jis into basis of local orbitals Lyielding
el’s explicitly:
el5b2
lLX
j
D16ðÞ
LL jðÞ (7)
where D16ðÞ
LL jðÞare elements of the Green’s functions in the
local basis. More details can be found elsewhere.
[12]
There are
Implications for extending this approach to the solid state.
[18]
Nephelauxetic effect
The success of the EHCF method based ultimately on using
the physical conditions for electrons in TMCs. Specifically, we
employed the information about the relative strength of the
crystal fields induced by different ligands (donor atoms) to
identify the (most probable) dominating contribution to the
effective crystal field and concentrated on reproducing it.
Another physical effect known from spectroscopic studies of
TMCs is the nephelauxetic effect manifesting in a reduction of
the amount of electron-electron interaction in the d-shells of
TMCs as compared to the corresponding energies
Figure 1. Pictorial representation of the partition of the mononuclear TMC
Hamiltonian matrix relevant for the EHCF method. The quantities in the
yellow blocks couple the model subspace with the LMCT/MLCT subspaces:
b
lj
are the one-electron hopping integrals between the lth d-AO and the
jth ligand MO; [D
(16)
(j)]
21
are the inverse energies of the LMCT/MLCT
excited states so that D
(16)
’s are the energy denominators with the signs in
the superscript referring to the ionization state of the ligands. The matrix
elements of the effective crystal field induced by the ligands appear in the
cyan triangle as sums over the index jof the ligand MOs of the products
of the multipliers shown in the rectangles/square. Reproduced with permis-
sion from A. L. Tchougr
eeff, R. Dronskowski, J. Phys. Chem. A, 2013, 117
(33), pp 7980–7988, Copyright (2013) American Chemical Society.
REVIEWWWW.Q-CHEM.ORG
International Journal of Quantum Chemistry 2016,116, 282–294 285
characteristic for free TMIs.
[19]
The experimental data about it
are organized in so called nephelauxetic series:
F2<H2O<urea O
ðÞ
<NH3<En <1=2Ox22<NCS2
<Cl2<CN2<Br2<N3
2<I2<S22<Se22<Te22;
which in many cases demonstrates not only deviations from the
spectrochemical series, but even inversions: F
2
versus I
2
;Cl
2
versus Br
2
, and so forth. Thus the physical nature of the nephe-
lauxetic effect cannot be the same as of the crystal field: the vir-
tual charge transfers between the d-shells and ligands which
were crucial (not less than 80% of the observed splitting) for
reproducing the effective crystal field acting on d-electrons are
not that important for explaining the reduction of the intrashell
electron-electron repulsion. By contrast, if the nephelauxetic
series is rewritten in terms of donor atoms,
[5]
it becomes:
F<O<N<Cl <Br <I<Se <Te
which fairly well corresponds to the order of the atomic polar-
izabilities.
[19]
This suggests the extension of the outer configu-
ration subspace on account of excitations of the ligands. The
formal derivation and necessary semi-quantitative estimates
have been performed recently,
[20]
so we do not overload the
present text by the formalism, but give a pictorial presentation
of the relevant segmentation of the outer space in Figure 3. In
variance with that presented in Figure 1, the outer space nec-
essary to reproduce the nephelauxetic effect (not present in
the original EHCF) is extended by the configurations contain-
ing singly excited Slater determinants of the ligands as multi-
pliers at the functions describing the d-shells. They represent
yet another way of regrouping terms in the expansion of the
exact many-electron WF of a TMC. This accounts for the polar-
ization of the ligands and through the Coulomb matrix ele-
ments of the form lmjllðÞtaken into account in the second
order of perturbation theory reduces the electron–electron
interaction matrix element lmjqrðÞwithin the d-shell by the
quantity:
lmjllðÞqrjl0l0
ðÞPll0
where Pll0are the elements of the polarization propagator of
the ligands. Unfortunately, the published formulae
[20]
have not
been implemented yet although the preliminary estimates are
Figure 2. The examples of the high- and low-spin iron(II) complexes with bulky organic ligands as described by the hybrid QM/MM method combining the
EHCF procedure for emulating the effective crystal field acting on the d-shells Color coding: the spin values indicated by green are correctly reproduced
by the EHCF-based procedure, those in red are wrong. In case of spin crossover compounds the lower of two possible ground state is always low spin,
which changes to the high-spin at the corresponding geometry.
REVIEW WWW.Q-CHEM.ORG
286 International Journal of Quantum Chemistry 2016,116, 282–294 WWW.CHEMISTRYVIEWS.ORG
promising and correct within the order of the described effect.
This will be done in the future releases of the EHCF
package.
[21]
Magnetic exchange in polynuclear TMCs
The success of the EHCF approach tested on the examples of
mononuclear TMCs were quite encouraging. Even very subtle
effects like the geometry dependence of the effective crystal
field leading to the corresponding dependence of the ground
state spin of iron (II) complexes could be satisfactorily repro-
duced. It is remarkable that this physics appears as interplay
of the virtual charge transfer between the d-shells and the
ligands and the electronic correlations in the d-shells. Further
exploitation of these interactions logically leads to the known
Van Vleck–Anderson superexchange model
[22–25]
of magnetic
interactions between the effective electronic spins residing in
more than one d-shells of polynuclear TMC (PTMC).
The theory itself is known for decades, but it had not been
implemented in the semi-empirical context. Suitable partition-
ing of the configuration space into model and outer subspaces
is formalized by setting the WFs in the model subspace
as:
[26,27]
Wn5UAnA;SA
ðÞÙUBnB;SB
ðÞÙ... ÙUl(8)
where UA;B;... nA;B;...;SA;B;...

are the ground states of nA;B;... elec-
trons in the d-shells of TMI’s A,B, ... derived from the one-
center EHCF calculations. The model space in Eq. (8) assumes
that unique fixed numbers of electrons can be ascribed to
each of the d-shells of a PTMC. This is formalized applying a
second L
owdin partitioning in addition to that of Eq. (5) as
depicted in Figure 4. Applying the original partition procedure
to a PTMC induces interactions with two additional compo-
nents of the outer space, namely, (i) with those with electron
transfers between d-shells located on different TMIs of a
PTMC—the configuration subspace dubbed as metal to metal
charge-transfer states (MMCT) and (ii) the double ligand-to-
metal or metal-to-ligand charge-transfer ((LM)
2
CT/(ML)
2
CT)
subspace, where the ligands get doubly ionized in either
sense. The L
owdin projection fixing the distribution of elec-
trons among the multiple d-shells of a PTMC generates the
terms of formally fourth order with respect to the one-
electron hopping operator (i.e., of the fourth power in its
parameters b’s), and ultimately contribute to the effective
exchange interactions.
The wave functions UA;B;... nA;B;...;SA;B;...

in Eq. (8) result
from EHCF calculations for individual d-shells. This takes into
account the one-electron hopping up to second-order of per-
turbation theory. The energies of the one-electron states in
the d-shells get renormalized by the second-order procedure
described by Eq. (6) that is, shifted and split by the effective
crystal field. Incidentally, definite values of electronic spin for
the respective numbers of electrons SA;B;... can be attributed to
them. The corrections to the energy stemming from either the
MMCT or ((LM)
2
CT/(ML)
2
CT) subspaces
[28–30]
are sensitive to
the way how the effective spins SA;B;...arrange in the state of
the PTMC with the total spin S, causing the sought splitting of
the states of the different total spin in the model subspace of
the PTMC. The target effective exchange constants are rather
tedious sums described elsewhere.
[26]
The calculations of the
effective inter-d-shell exchange constants JAB have been imple-
mented as a program suite MAGAI
ˆXTIC
that accepts standard
quantum-chemical input (molecular composition and geome-
try).
[31]
The suite fully implements the effective Hamiltonian
scheme for the d-shells of a PTMC. Current implementation
[31]
Figure 3. Pictorial representation of the partition of the TMC Hamiltonian
matrix relevant for estimating nephelauxetic effect in the d-shell of an TMI.
The model subspace is spanned by the functions of the form of Eq. (5).
The original outer LMCT/MLCT subspace is complemented by the L!L*
one—that formed by the one-electron excitations in the ligands.
Figure 4. Pictorial representation of the partition of the PTMC Hamiltonian
matrix relevant for estimating magnetic interactions between electrons in
the d-shells of the Ath and Bth TMIs. The model subspace is spanned by
the functions of the form of Eq. (5). The outer subspace further decom-
poses in three subspaces: the old LMCT/MLCT and two more subspaces -
that of the metal-to-metal charge transfer (MMCT) configurations and the
double ligand-to-metal or metal-to-ligand charge transfer states ((LM)
2
CT/
(ML)
2
CT) when the ligands get doubly ionized in either sense. Reproduced
with permission from A. L. Tchougr
eeff, R. Dronskowski, J. Phys. Chem. A,
2013, 117 (33), pp 7980–7988, Copyright (2013) American Chemical Society.
Pronounces as “majestic” and stands for magnetic exchange from Aix-la-
Chapelle (old-fashioned French for Aachen, pronounces as [ekslaÐapel]) by
means of Theoretical Chemistry.
REVIEWWWW.Q-CHEM.ORG
International Journal of Quantum Chemistry 2016,116, 282–294 287
of the program available through the NetLaboratory system is
configured for 500 atoms, of which up to five can be transition
metal ones, and 2000 atomic orbitals. Recently, it has been
tested on a series of binuclear l-oxo Cr(III) dimers
[27]
and on a
series of binuclear complexes [{(NH
3
)
5
M}
2
O]
41
of trivalent cati-
ons featuring l-oxygen super-exchange paths.
[32]
These studies
confirmed the reproducibility of the trends in the series of val-
ues of exchange constants for the compounds differing by the
nature of TMI.
Although significant progress had been achieved in the field
of calculation of exchange parameters by the DFT methods,
[33]
our very simplistic method, nevertheless, keeps some advan-
tages allowing to lift two types of concerns. First, the DFT
methods are not particularly successful when applied to iso-
lated d-shells. Specifically, even the splitting between the
states of individual shells having different total spin (see
above), which in principle requires much lower precision than
the estimates of the magnetic parameters are problematic.
Thus the required estimates may be possible after fine tuning
of the applied procedures (including functionals), but no uni-
fied picture, covering simultaneously the description of individ-
ual d-shells and their magnetic interactions is possible.
Second, the calculations of the magnetic interactions in the
DFT context strongly rely upon the possibility to obtain
the broken-symmetry solutions. Although trivial for binuclear
complexes, it may become very problematic when odd-
numbered cycles of spins coupled through antiferromagnetic
exchange become dominant and induce magnetic frustration,
so that there is no chance of obtaining the exchange con-
stants for it. The EHCF approach either for the mononuclear or
the PTMCs does not require any converged solution of the
electronic problem involving the d-shells; only one for the
ligands is needed which is much easier to obtain and which
provides necessary information for calculating the matrix ele-
ments of the effective Hamiltonian for the d-system (com-
posed either of one or several d-shells) thus yielding the
targeted entities: either crystal field or Heisenberg-van Vleck
Hamiltonians.
Latest Applications
Antiferromagnetic Exchange in Iron (III) Dimers
In this work, we extend the EHCF method to assess exchange
parameters for a selection of the l-oxo Fe(III) binuclear com-
pounds. The phenomenological theory of exchange interac-
tions in such dimers has been developed,
[34]
extending the
model
[28–30]
expressing effective exchange through a pair of
energy parameters e0
rand e0
pand the /5/MOM angle pro-
posed for other MOM dimers. The model
[34]
turned out to be
more sophisticated and required many more assumptions and
estimates of the parameters than the original one.
[28–30]
The EHCF procedure, when applied to FeOFe type com-
plexes, immediately yields the crystal-field states of the individ-
ual iron ions. The splitting pattern has an expected form
characteristic for the tetragonally distorted octahedron (see
below Fig. 9). All iron (III) species studied in this work feature
high-spin states so that the levels depicted in Figure 9 are sin-
gly occupied. In order to have a reference data set to compare
results of our approach we address the phenomenological
model for the FeOFe dimer.
[34]
In it the parameter J
eff
describ-
ing effective exchange between the 5/2 spins located in the d-
shells of the Fe
31
ions is contributed nontrivially by the ele-
mentary exchanges J
nn
,J
gg
,J
gh
,J
hg
, and J
hh
:
Jeff 54
25 Jnn1Jgg 1Jhh1Jgh 1Jhg

;
where the orbital indices are: f5b2;n;g5e;e5b1;h5a1with
the symmetry labels assigned according to the C4vlocal sym-
metry group and the parameters
Jlm5h2
lm=Ueff (9)
describing coupling between the individual electronic spins
residing in the respective d-orbitals of two centers through
the effective hopping between the d-orbitals centered on the
TMIs Aand Bmediated by the jth ligand MO
hlm5X
j
blj AðÞbmj BðÞD16ðÞ
BjðÞ (10)
and the effective energy of the charge-transfer states acquire
the form:
[23,30]
U21
eff 5D11ðÞ
ðOpÞ1Dð0Þ
M!M:
The factor 4/25 is the combination of the unpairity and the spin
factors relevant for a pair of high2spin d
5
2shells. Comparing
Eqs. (4), (8), and (9) allowed authors
[28–30]
to express the
elementary exchanges through the AOM parameters and to
produce unified picture of the one-center excitations and two-
center exchanges. The model
[35]
formally involves much more
(up to eight) elementary exchanges, but these as in the
model
[34]
are all expressed through the same set of the AOM
parameters e
pp
,e
pr
,ande
sr
. Nevertheless, neither of fitting pro-
cedures
[34,35]
allows to uniquely determine these parameters.
As in the case of the l-oxo bridged Cr(III) dimers, we deter-
mine the AOM parameters independently from the calculation
of the hypothetical model [(NH
3
)
5
FeOFe(NH
3
)
5
]
41
, from now
on referred to as FeOFe, taken at the geometry of the dichro-
mium “basic rhodo” compound (CrOCr).
[36]
The AOM parame-
ter e
pr
(strength of the p
r
interaction) can be evaluated by a
single term in Eq. (7), namely by the dimer HOMO formed by
75% of the Op
r
AO. The value e
p
of the l-O is estimated to be
0.316 eV (2550 cm
21
) whereas e
pr
and e
sr
are 9255 and
3710 cm
21
, respectively. These values are derived, using the
crystal field renormalized energies of the charge-transfer
states.
§
§
This in a way implements gradual/iterative renormalization prescribed by the
Density Matrix Renormalization Group (DMRG) approach (see below) which in
its turn is one of innumerable ways to construct/regroup the perturbation
series terms existing in the intermediary between the Rayleigh-Schrodinger
and Brillouin-Wigner versions of the theory.
REVIEW WWW.Q-CHEM.ORG
288 International Journal of Quantum Chemistry 2016,116, 282–294 WWW.CHEMISTRYVIEWS.ORG
The effective MMCT/(LM)
2
CT energy U
eff
is estimated to be
4.1 eV from the intermediate EHCF quantities. That small value
as compared to the deceptive estimate derived from one-
center repulsion is due to mentioned crystal field induced
renormalization and to incorporation of the (LM)
2
CT energies
into this value (first term in the definition for U21
eff ). The larger
crystal field effect observed for the FeOFe dimer as compared
to the CrOCr dimer and other compounds of the dichromium
series
[27]
appears due to involvement of the horbitals stronger
affected by the crystal field than the f,n, and gones which
produces larger effects on the exchange parameters.
Combining so obtained elementary exchange parameters
according to Eq. (8) we obtain a very good estimate of the
effective exchange in the FeOFe dimers: 212 cm
21
. Applying
the MAGAI
ˆXTIC with the geometry of the experimentally known
basic rhodo CrOCr dimer
[37]
for the model FeOFe dimer results
in the effective exchange of 244 cm
21
. The latter result of the
complete summation yields a much better estimate in terms
of its closeness to the upper boundary of the experimental
values of other FeOFe dimers which all fit in the range 160–
265 cm
21
with /values of about 1808as well accumulated
close to the upper boundary.
[34]
The reason is that due to
strong overlap (hopping) of the horbitals, many terms gener-
ated from pairs of orbitals of different mirror symmetry appear
in the total summation. Applying the phenomenological
model
[34]
to the complexes and using the crystal field renor-
malized values of the AOM parameters e
k
results in the values
of the exchange parameters given in the right (dubbed as
“WG interpolation,” where WG stands for the “Weihe-G
udel”)
of two columns “J
phen
” in Table 1. The phenomenological for-
mulae
[35]
are not universal; they apply to specific types of
compositions and numbers of the bridges (superexchange
paths). We applied the MAGAI
ˆXTIC package to a selection of
compounds listed in Table 1. The complete summation by the
package (the numerical values given in the J
calc
column of
Table 1) generally underestimates the experimental values by
20–40 cm
21
. This is acceptable in comparison to, for example,
DFT calculations that usually overestimate J
exp
by a factor of
about two.
[33]
The level of systematic error of the MAGAI
ˆXTIC
package is, with adequate precaution, acceptable for the prac-
tical needs when considering compounds featuring the
exchange values at the upper boundary of the interval of the
observed ones (hundreds of wave numbers). It is, however,
too high as compared to the values of the effective exchange
parameters characteristic for the protonated or alkylated
bridges: that is, on the scale of several tens of wave numbers.
Nevertheless, the qualitative effect of the oxo-bridge protona-
tion is not only reproduced, but eventually exaggerated.
Characteristic examples are provided by the pairs DIBXAN/
DIBWUG and CACZIP/COCJIN which allow to illustrate the
effects of the bridge protonation and geometry variation. For
the first pair, the respective rows of Table 1 show that the pro-
tonation together with the variation of the geometry of the
bridge results in a significant reduction (more than 20 times)
of the calculated effective exchange (more precisely, of the
kinetic superexchange contribution to it). In the second pair the
Table 1. The exchange constants in a series of l-oxo bridged Fe(III) dimers: experimental values and those calculated with use of the MAGAI
ˆXTIC package.
Compound
[a]
d
Fe–O
(A
˚)/5/FeOFe ()
J
calc
(cm
21
)
this work
[m]
J
phen
(cm
21
)
J
exp
(cm
21
)
[34,35]
fit according
to [34,35]
WG
interpolation
[p]
DIBXAN
[b]
1.800 119.7 187 224
[n]
215 238
VABMUG
[c]
1.785 123.9 212 243
[n]
235 264
WIHSAH
[d]
1.785 174.7 210 203
[n]
238 232
JIGNUI
[e]
1.802 122.8 194 214
[n]
207 240
CACZIP
[f]
1.784 123.6 205 246
[n]
237 243
deprot COCJIN
[g]
1.956 123.1 60 63
[n]
61 –
COCJIN
[h]
1.956 123.1 8 33
[o]
61 34
DIBWUG
[i]
1.987 113.2 8 23
[o]
52 –
deprot DIBWUG 1.987 113.2 40 54
[n]
52 –
prot DIBXAN 1.800 119.7 41 82
[o]
215 –
ZOVWUC
[j]
1.997 104.3 8 19
[o]
54 57
YUBLEM
[k]
2.089 117.5 1 14
[o]
22 12
PIMTEK
[l]
1.972 129.1 6 30
[o]
52 30
[a] CCSD Code. [b] DIBXAN 5[(Me
3
tacn)
2
Fe
2
(O)(acO)
2
]
21
,Me
3
tacn 51,4,7-trimethyl-1,4,7-triazacyclononane, acO here and below stands for acetate
anion—for geometry see Ref. [38]. Protonated DIBXAN is obtained by adding a proton to the bridging oxygen atom at a distance of 1 A
˚and all other
geometry parameters kept as in original DIBXAN. [c] VABMUG5[(bpy)
2
Cl
2
Fe
2
(O)(acO)
2
], bpy stands for 2,20-bipyridine—for geometry see Ref. [39]. [d]
WIHSAH 5[(tpa)
2
Cl
2
Fe
2
(O)]
21
, tpa stands for tris(2-pyridylmethyl)amine—for geometry see Ref. [40]. [e] JIGNUI 5[Fe
2
O(acO)
2
(TMIP)
2
]
21
, TMIP stands for
tris(N-methylimidazol-2-y1)phosphine—for geometry see Ref. [41]. [f] CACZIP 5real l-deprotonated COCJIN. [g] Model l-deprotonated COCJIN on the
geometry of COCJIN. [h] COCJIN 5[(HB(pz)
3
)
2
Fe
2
(OH)(acO)
2
]
1
, HB(pz)
3
here and below stands for hydridotris(pyrazolyl)borate—for geometry see Refs.
[42 and 43]. [i] DIBWUG stands here not for the real compound DIBWUG which is a binuclear complex of Fe(II), but for the l-protonated DIBXAN – the
model Fe(III) complex taken on the experimental geometry of DIBWUG; its key elements are given in the Table. Thus no experimental value of the
exchange is available in this case. Deprotonated DIBWUG is obtained by removing a proton from the bridging oxygen atom keeping all other geome-
tries intact. [j] ZOVWUC 5[Fe
2
(chp)
4
(MeO)
2
(phen)
2
], chp stands for 6-chloro-2-pyridone anion, phen stands for 1,10-phenanthroline—for geometry see
Ref. [44]. [k] YUBLEM 5[Fe
2
(biomp)(acO)
2
]
1
, biomp stands for 2,6-bis[(2-hydroxybenzyl)((1-methylimidazol-2-yl)-methyl)-aminomethyl]-4-methylphenol
anion—geometry see Ref. [45]. [l] PIMTEK 5[Fe
2
(OH)(O
2
P(Ph)
2
)
2
(HB(pz)
3
)
2
]
1
—for geometry see Ref. 43. [m] Calculated with use of the MAGAI
ˆXTIC pack-
age. [n] Calculated with use of the interpolation formulae of [34]. [o] Calculated with use of the interpolation formulae of [35]. [p] Calculated in the
present work using the AOM parameters extracted from the MAGAI
ˆXTIC, with the value of btaken from Ref. 34.
REVIEWWWW.Q-CHEM.ORG
International Journal of Quantum Chemistry 2016,116, 282–294 289
difference of the exchange constants must be attributed to two
effects: the protonation itself and significant adjustment of the
geometry in two forms. Comparing CACZIP with the deproto-
nated form of COCJIN indicates that the geometry relaxation
(predominantly the increase of the Fe-O separations) decreases
the kinetic superexchange by a factor of about three to four.
Comparing the pairs DIBXAN/protonated DIBXAN and DIBWUG/
deprotonated DIBWUG show a reasonable stability of the factor
describing the transition between the protonated and nonpro-
tonated forms taken at fixed geometries of the molecule.
Remarkably enough the estimates according to the formulae
[34]
fed with the above EHCF estimates of the AOM parameters as
well produce a similar ratio of the exchange constants. The pro-
tonation further reduces the exchange by an additional factor
of about eight. The parameterization
[35]
results in an exchange
value that is very close to the experimental value. Thus, apply-
ing the parameter sets and formulae
[34,35]
within the range of
those intervals (and therefore presumably to a specific bridging
type) usually results in a smaller deviation of model from experi-
ment. Although the experimental sets
[34,35]
contain a wide vari-
ety of compounds, there is a strong correlation of the geometry
parameters (distance/angle) to the type of bridging. Thus, all
formulas given in both works
[34,35]
are fairly valid for the respec-
tive ranges of geometry parameters, but the parameters fitted
in these formulae are indirectly loaded by their composition.
Further dimers examplify the alkoxo, phenoxo, and phosphi-
nato superexchange paths accompanied by the carboxylate
bridges as well (three last rows of Table 1). The phenomeno-
logical model
[34]
fed by the above EHCF estimates of the AOM
parameters gives surprisingly good results, which should be
considered as a coincidence for compounds containing Fe(III)
centers since the setting of the model
[34]
is different: the com-
pounds contain several potentially active bridges, neither of
which is a single l-oxo bridge for which the model
[34]
applies
sensibly. A somewhat similar observation was made with
respect to asymmetrically bridged complexes:
[35]
the shortest,
single substituted oxo-bridges (alkoxo, phenoxo, phosphinato,
etc.) determine the most efficient superexchange path way.
Shorter mean Fe-O distances which may possibly appear due
to three-atomic brigdes like, for example, carboxylates should
be neglected. For the compounds with substituted oxo-
bridges the MAGAI
ˆXTIC package provides a correct order of
magnitude of the kinetic superexchange. The crude effect of
reducing the (effective) number/efficiency of the superex-
change paths under introducing substituents to the l-oxo
bridge is fairly reproduced. However, the obtained numerical
values are too small in comparison with the experiment. One
has to admit that the MAGAI
ˆXTIC package provides only the
kinetic superexchange contribution to the total exchange con-
stant whereas other contributions (spin-polarization, etc.) are
left aside. They will be included in the future releases of MAG-
AI
ˆXTIC. Nevertheless, this study in a way negatively confirms
the unimportance of the carboxylate bridges/superexchange
paths: they are explicitly included in the summation performed
by the MAGAI
ˆXTIC package, but it does not affect the final result
and the values of the exchange parameters for the dimers
with carboxylate bridges are almost the same as for those
without the bridges, having similar geometry. This cannot be a
coincidence. The hypothesis which we are going to check in
the future is that both symmetric and occupied antisymmetric
p-MO’s of the carboxylate group (symmetry notation is given
with respect to the local symmetry of the ACO
2
fragment)
mediate the superexchange. The resonance (hopping) matrix
elements (b’s) between the p-MO’s and the d-AOs fon either
TMI become nonvanishing. However, as it was shown
[32]
if
quasi-degenerate symmetric and asymmetric MOs contribute
to the effective exchange, the transfers mediated by these
orbitals almost compensate each other so that the ACO
2
bridges are not real superexchange paths.
To conclude this Section we mention that, indeed, the theo-
retical evaluations not only rest on DFT methods, but also on
alternative methods such as CASSCF/CASPT2, extended CI calcu-
lations or Density Matrix Renormalization Group (DMRG)
approach. They recently had been applied to assessment of
exchange integrals in diiron complexes.
[46]
Obviously, such
approaches are in principle capable to reproduce whatever
characteristics of molecules “in the infinite limit.”
[2]
For example,
only the infinite DMRG summation yields the adequate result
for diiron complex. However, this achievement must be consid-
ered as an accidental one since analogous treatment applied to
dichromium complex results in not more than 60% of the
experimental amount. In this situation the efficiency issues can-
not be ignored:
[27]
for complex of comparable number of atoms
the computer time required by DFT-based methods ranges
from 1 to 5 h as compared to the minute scale required by our
method on comparable processors, not talking about the
resource requirements of the DMRG methods.
M
oßbauer spectra of materials with N-ligated iron(II)
Iron takes an exceptional position among elements for many
reasons. The presence of rather easily accessible M
oßbauer
active isotope
57
Fe allows an access to details of the electronic
distribution in the d-shell of the iron ions. Here we discuss in
more details the results of M
oßbauer experiments and corre-
sponding calculations of two recently synthetized iron(II) con-
taining materials FeNCN and Fe(NCNH)
2
.
[47,48]
The EHCF
method in its semiempirical implementation based on the
SINDO1 parameterization
[49]
has proven to be successful in cal-
culations of the quadrupole splitting (QS, DEQ) and their tem-
perature dependence observed in M
oßbauer spectra of
divalent iron complexes.
[50]
The QS arises due to the interac-
tion of the quadrupole moment Qof the
57
Fe nucleus in its
excited state and the electric field gradient (EFG) at the posi-
tion of the Fe nucleus. The EFG is the traceless tensor Vab;a;
b5x;y;zand its components are expectation values of the
corresponding quantum mechanical operator to be calculated
for the ground and the excited electronic states of the TMC
represented by the WF Eq. (4). In a line with the partition of
the total electronic WF adopted in EHCF method the compo-
nents of the total EFG tensor for the n-th electronic state
express as a sum of contributions from 3d-electrons V3dðÞ
ab nðÞ,
metal valence 4p-electrons V4pðÞ
ab and a lattice contribution VL
ab
REVIEW WWW.Q-CHEM.ORG
290 International Journal of Quantum Chemistry 2016,116, 282–294 WWW.CHEMISTRYVIEWS.ORG
induced by the effective charges residing on the atoms of the
ligand environment:
Vab nðÞ512RðÞV3dðÞ
ab nðÞ1V4pðÞ
ab
hi
112c1
ðÞVLðÞ
ab
where 12Rand 12c1are the Sternheimer factors accounting
for the shielding and antishielding effects from the inner shell
electrons and the lattice effective charges, respectively. The
total EFG tensor is obtained by the thermal averaging its com-
ponents for the individual states:
Vab5Z21X
n
Vab nðÞexp 2n
kBT

Z5X
n
exp 2n
kBT

This applies provided the interstate fluctuations (inverse life-
time of each individual electronic state) are faster than a
characteristic value
[51,52]
of about 10
7
s
21
. If the interstate fluc-
tuations are slower than the above characteristic value, differ-
ent electronic states manifest separately as individual species
with corresponding characteristic QS values.
The QS is given by:
DEQ51
2eQsgn VZZ
ðÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
V2
ZZ 11
3VXX 2VYY
ðÞ
2
r
where, eis the unit charge (positive) Q51:87 barn (10
228
m
2
)
is the quadrupole moment of
57
Fe nucleus in its excited state,
Vbb are the eigenvalues of the appropriately averaged EFG ten-
sor Vwith jVZZ j5maxbjVbbj

. Further details as well as the
values of parameters used in the calculations described below
are given in the review.
[53]
FeNCN. The FeNCN material having unique magnetic prop-
erties
[49]
is an extended three-dimensional framework
formed by Fe(II) ions, each coordinated to six linear [NCN]
22
bridges (Fig. 5). Each Fe(II) ion is located at the center of a
trigonally distorted octahedron formed by six [NCN]
22
groups. First, we have performed
[54]
EHCF calculations on a
cluster model depicted in Figure 5. In the cluster calculation,
the 3d-electrons were explicitly included only for the central
Fe(II) ion, the remaining Fe(II) ions were modeled by Be
atoms (no d-electrons) with semiempirical parameters for
valence s-andp-orbitals corresponding to 4s-and4p-atomic
orbitals of Fe(II) and predicted the ground state to be high-
spin
5
A
1
state in agreement with the observed magnetic
moment. More detailed consideration
[55]
confirms this find-
ing which is agreement with the negative sign of the QS in
the observed M
oßbauer spectrum.
[55]
In the surrounding of
nitrogen donor atoms the Fe
21
ion at the experimental
geometry of FeNCN is in the high-spin state. The
5
A
1
state
originates from the
5
T
2g
term in octahedral symmetry which
splits into
5
A
1
and
5
Eterms as a result of a trigonal distortion
(Fig. 6). That latter derives from the spherically symmetric S5
5=2 determinant of five electrons supplemented by one
more occupied d-orbital so that the symmetry of the elec-
tronic distribution around the nucleus coincides with that of
the only doubly occupied orbital. The electrons arranged in
the spherically symmetric state do not contribute to the EFG
and the latter derives from the nth function of the additional
electron:
V3dðÞ
ab nðÞ512RðÞ
1
7ehr23i3dhnjfab
lm jni
where fab
lm are the angular factors;
[53]
the diagonal ones (a5b
and l5mwith land mreferring to the cubic d-harmonics) are
tabulated,
[51]
and hr23i3dis the dimensional parameter charac-
terizing the spatial extension of the iron d-shell. In the case of
the trigonally distorted octahedron this setting suffice for
describing QS in the ground state since the latter is nonde-
generate and the diagonal components of the EFG tensor suf-
fice for this purpose. The diagonal angular factor with the
maximal absolute value for the ground state equals—4. For
the first excited
5
Estate the off-diagonal (Vxy ) components of
the EFG tensor appear for the individual components of the
spatially degenerate state. They exactly compensate each
other so that the EFG (and the QS) in the
5
Estate is also given
by the corresponding diagonal components, maximal by
Figure 5. The structure of the unit cell of FeNCN (left) and a cluster model
[Fe(NCN)
6
(Fe*)
6
]
21
used in EHCF calculations. The pseudo-iron atoms Fe*
for which the Be atoms with iron parameters for s- and p-orbitals were
used in calculation are shown in pink.
Figure 6. The splitting diagram of the effective d-levels for a cluster model
of FeNCN obtained in EHCF calculation. Notation refers to the local
symmetry.
REVIEWWWW.Q-CHEM.ORG
International Journal of Quantum Chemistry 2016,116, 282–294 291
absolute value of which, equals to 2. Thus the d-shell contribu-
tion to the QS in the units of 1
14 12R
ðÞ
e2Qhr23i3damounts
241232exp 2
kBT

112exp 2
kBT
 (11)
which is nothing but the Ingalls formula
[56]
adjusted for our
situation. Although the qualitative picture could be guessed
without calculation on purely symmetry grounds the relative
positions of the a1and elevels in the d-shell is not obvious.
The calculated QS for the ground state is (2)3.535 mm/s
which overestimates the experimental value of (2)2.6 mm/s
measured at low temperature. The possible sources of this dis-
crepancy may be related to underestimation of the lattice con-
tribution by the cluster of a finite size. The temperature
dependence of the quadrupole splitting observed in the
experiment arises from the thermal population of the first
excited doubly degenerate
5
Estate. In the EHCF calculation
the
5
A
1
!
5
Eexcitation energy is 223 cm
21
or 320 K which
leads to the temperature dependence which is too strong
compared to the experiment (the calculated ratio of the split-
ting at 5 K to that at 300 K is about 2.32 instead of the experi-
mental value of 1.3). That indicates that the splitting between
5
A
1
and
5
Estates should be most probably larger than 320 K.
In fact, adjusting it to the value of 700–770 K results in a very
reasonable agreement between the calculation and the experi-
ment as shown in Figure 7.
Fe(HNCN)
2
.Protonated analogue of FeNCN, the crystalline
Fe(NCNH)
2
(Fig. 8), shows as well antiferromagnetic cou-
pling
[48]
and as well calls for a M
oßbauer study. The local sym-
metry differs from that of the FeNCN by the pattern of the
local symmetry perturbation. Although the low-energy d-states
originate as well from the
5
T
2g
term in octahedral classification,
this symmetry, first, lowers to the tetragonal one so that the
splitting is one shown in Figure 9. The variance with the trigo-
nal case (Fig. 6) is that the ground state is the
5
B
2
one. It pro-
duces the QS of 4 in the units of 1
14 12RðÞe2Qhr23i3dwhich in
the case of FeNCN converted into the observed value of
2.6 mm/s (observe the opposite sign of QS as compared to
FeNCN). This perfectly corresponds to the larger of the two
values observed in the experiment (Fig. 10) and cannot be
reproduced without explicit EHCF calculation yielding the rela-
tive order of the b
2
and elevels.
However, in Figure 10 one can see the second (smaller) value
of the QS. This is enigmatic enough since no sign of two types
of iron atoms can be traced structurally.
[48]
This is reminiscent
to the situation occurring in spin-active compounds where the
coexisting components of different total spin manifest as two
doublet signals (see Ref. 56 for a recent example of such a
behavior). This means that the interstate fluctuation in the
spin-active compounds takes place with a smaller rate than
the characteristic one of 10
7
s
21
. Our calculation performed
in the cluster approximation shows that the splitting follows
the tetragonal pattern as shown in Figure 9 and places the e
d-AO manifold at about 450 cm
21
above the b2one. How-
ever, the
5
Estate is further slightly—by about 100 cm
21
Figure 7. Quadrupole splitting in FeNCN versus temperature. Red crosses:
experimental data.56 Green curve is obtained by adjusting the Sternheimer
factor for 3d-electrons by a factor of 0.745 and setting the first excitation
energy to 700 K. Magenta curve is alternatively obtained by setting the lat-
tice contribution 0.56 mm/s and the first excitation energy to 770 K.
Figure 8. The crystal structure of the Fe(HNCN)
2
. Reprinted with permission
from X. Liu, L. Stork, M. Speldrich, H. Lueken, R. Dronskowski, Chem. Eur. J.,
2009, 15, 1558–1561. Copyright (2009) American Chemical Society.
Figure 9. The splitting diagram of the effective d-levels for a cluster model
of Fe(HNCN)
2
obtained in EHCF calculation. Approximate local symmetry in
meant.
REVIEW WWW.Q-CHEM.ORG
292 International Journal of Quantum Chemistry 2016,116, 282–294 WWW.CHEMISTRYVIEWS.ORG
split due to digonal perturbation (the equatorial /NFeN
angles—those between the atoms bearing hydrogens—are
988and 828and form a rectangle with the sides 3.335 and
2.873 A
˚, rather than a square). Despite the small amount Dof
the splitting the WFs in the originally degenerate manifold
adjust to the diagonal perturbation and acquire the form:
1
ffiffi
2
pex6ey

;
prescribed by the degenerate perturbation theory. Although,
in Fe(HNCN)
2
there is no indication of the variation of spin
states of individual Fe
21
ions, we assume that the observed
larger and smaller QS components can be identified with
those form the
5
B
2
ground state and the split
5
Estate, respec-
tively occurring in the slow interstate fluctuation regime with
the constant smaller than 10
7
s
21
.
[51]
The EFG tensors corre-
sponding to the perturbed states stemming from the tetrago-
nal
5
Estate in units of 1
712RðÞehr23i3dare
163ffiffi
3
p0
63ffiffi
3
p10
002
0
B
B
@1
C
C
A
:
The EFG tensor for the excited states derives from the above
expression by averaging the components differing by the signs
of the off-diagonal matrix elements with the Boltzmann factors
b5exp 2D=kBTðÞso that the EFG tensor becomes:
1
7
112b
11b

3ffiffi
3
p0
12b
11b

3ffiffi
3
p10
002
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
:
The above calculated splitting Dof the
5
Estate is small com-
pared to k
B
Twhich results in the following EFG tensor:
1
7
13ffiffi
3
pD
2kBT0
3ffiffi
3
pD
2kBT10
002
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
:
The QS in the units of 1
14 12RðÞe2Qhr23i3dis approximately:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
419D
kBT

2
s
which at the temperature of about 300 K yields a value of the
QS of about 0.8 of that for the ground state: 2.16 mm/s in a
fair agreement with the measured smaller QS component (Fig.
10), provided already mentioned overestimation of the QS
derived from the d-contribution only. The spectral weights of
the larger and smaller QS components, which we identify with
the
5
B
2
ground state and the split
5
Estate, yield the splitting
of about 300 cm
21
in a fair agreement with our EHCF calcula-
tion. Apparently, the situation requires further experimental
clarification which will be undertaken in the future.
Conclusions
The effective Hamiltonian crystal field approach building a
bridge between semi-empirical quantum chemistry and the
phenomenological crystal field theory successfully described
iron molecular complexes with a wide variety of ligands and
related solid materials. Both the ground and lowest energy
excited states are successfully reproduced as well as many
subtle effects including spin transitions depending on molecu-
lar geometry, effective exchange constants, and the M
oßbauer
quadrupole splitting and its temperature dependence.
Acknowledgments
Dr. M. Herlitschke is acknowledged for kindly providing the results
of M
oßbauer measurements
[55]
in a numeric form. The authors are
thankful to the Referee for the valuable comments which for sure
helped to improve the manuscript.
Keywords: effective Hamiltonian for crystal field iron com-
pounds effective exchange parameters M
oßbauer quadru-
pole splitting
How to cite this article: A. L. Tchougr
eeff, A. V. Soudackov, J.
van Leusen, P. K
ogerler, K.-D. Becker, R. Dronskowski. Int. J.
Quantum Chem. 2016,116, 282–294. DOI: 10.1002/qua.25016
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294 International Journal of Quantum Chemistry 2016,116, 282–294 WWW.CHEMISTRYVIEWS.ORG
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This chapter discusses that the subject of exchange in magnetic materials is divided into two parts, referring to insulators and to metals. This distinction is useful from the magnetic point of view because in insulators the spins and magnetic moments whose alignments lead to magnetic effects are certainly localizable so that phenomenologically is described by a spin Hamiltonian that contains spin operators and exchange terms of Heisenberg type. It discusses that there is relationship among the mechanisms important in metals, such as conduction electron polarization, and those in insulators. The chapter provides historical discussion of the subjects of antiferromagnetism and of exchange in insulators. It focuses on the Heisenberg Hamiltonian, with a brief derivation and a discussion of some of the statistical theories of magnetism based upon it, primarily molecular field theory, which is by far the most generally useful in the experimental measurement of exchange. The chapter also describes older theories and presents ideas about super-exchange, and gives a discussion and a diagrammatic classification of all the possible higher-order processes.
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The crystal structures of Fe(NCNH)2 and Co(NCNH)2, isotypical with Ni(NCNH)2, have been refined by means of combined X-ray and neutron powder diffraction data (SPODI, FRM II). The lattice parameters are a = 6.6655(7), b = 8.7923(8), c = 3.3304(3) Å for Fe(NCNH)2 and a = 6.5696(2), b = 8.8058(2), c = 3.2622(1) Å for Co(NCNH)2 in the orthorhombic system Pnmm (no. 58). The positions of the hydrogen atoms have been clearly resolved.
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