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The fundamentals of the Effective Hamiltonian Crystal Field (EHCF) method, used originally to calculate intra-shell excitations in the d-shells of coordination compounds of the first row transition metals, are reviewed. The formalism of effective operators is applied to derive an explicit form of the effective operator for a dipole moment in d-shell electronic subspace, allowing us to calculate the oscillator strengths of optical d-d transitions, which are otherwise forbidden when treated in the standard EHCF approach. EHCF methodology is also extended to describing magnetic interactions of the effective spin in several open d-shells of polynuclear coordination compounds. The challenging task of improving a precision of ∼1000 cm−1 (describing the excitation energies of single d-shells by EHCF) to one of ∼100 cm−1 for the energies required to reorient spins by an order of magnitude is considered within the same paradigm as EHCF: the targeted use of McWeeny’s group function approximation and the Löwdin partition technique. These are applied to develop an effective description of a d-system. This approach is tested on a series of binuclear complexes [{(NH3)5M}2O]4+ of trivalent cations featuring oxygen super-exchange paths in order to confirm the reproducibility of the trends in the series of exchange constants values for compounds that differ in the nature of their metal ions. The results from calculations are in reasonable agreement with the available experimental data and other theoretical methods.
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ISSN 00360244, Russian Journal of Physical Chemistry A, 2014, Vol. 88, No. 11, pp. 1904–1913. © Pleiades Publishing, Ltd., 2014.
Published in Russian in Zhurnal Fizicheskoi Khimii, 2014, Vol. 88, No. 11, pp. 1721–1731.
1904
1
INTRODUCTION
Every more or less important contribution to our
everexpanding knowledge of modern theoretical
physical chemistry starts from an idea or recognition
of a previously ignored or overlooked fact. Further
developments resulting in new approaches or meth
odologies often overshadow the initial idea but in no
way diminish the importance of its origin. It is diffi
cult now to say exactly what was historically or meth
odologically the most important contribution or
effect produced by I.A. Misurkin. So far as we can
tell, however, it began with his most important work,
in which I.A. Misurkin and A.A. Ovchinnikov real
ized that a physically valid result can be reproduced
by employing a specialized form of electronic wave
function [1], specifically the Hartree–Fock function
with broken spin and translation symmetry, known
nowadays as the unrestricted spin Hartree–Fock
1
The article was translated by the authors.
function
2
that guarantees correct reproduction of the
gap in the spectrum of the long polyenes responsible
for the familiar colors of autumn leaves, tomatoes, and
carrots. Without this optical gap, the world would
appear gray or black. It is likely because of the impor
tance of finding a physically correct approach that
Misurkin formulated the maxim “Choose the correct
wave function!” for his students. It is difficult to say if
his advice was followed (it was a bit too vague), but it
definitely played the role of a catalyst after it was real
ized in [2, 3] that the catalytic activity of transition
metal complexes (TMCs) was directly related to the
spectrum of their lowlying
d
d
excitations. This
shifted our efforts toward seeking methods capable of
delivering the information we needed.
At that time, it was obvious to us that ab initio meth
ods, though they are potentially capable of solving prob
2
They referred to their approach as “the generalized Hartree–Fock
method,” abbreviated in Russian as OMHF, which some immedi
ately interpreted as “Ovchinnikov–Misurkin–Hartree–Fock.”
Effective Hamiltonian Crystal Fields:
Present Status and Applicability to Magnetic Interactions
in Polynuclear Transition Metal Complexes
1
A. L. Tchougréeff
a, b, c
and A. V. Soudackov
d
a
Institute of Inorganic Chemistry, RWTH Aachen University, D–52056 Aachen, Germany
b
Moscow Center for Continuous Mathematical Education, Moscow, 119002 Russia
c
Faculty of Chemistry, Moscow State University, Moscow, 119991 Russia
d
Department of Chemistry, University of Illinois at UrbanaChampaign, Urbana, IL, 61801 USA
email: tch@elch.chem.msu.ru
Received March 5, 2014
Abstract
—The fundamentals of the Effective Hamiltonian Crystal Field (EHCF) method, used originally to
calculate intrashell excitations in the
d
shells of coordination compounds of the first row transition metals,
are reviewed. The formalism of effective operators is applied to derive an explicit form of the effective operator
for a dipole moment in
d
shell electronic subspace, allowing us to calculate the oscillator strengths of optical
d–d
transitions, which are otherwise forbidden when treated in the standard EHCF approach. EHCF meth
odology is also extended to describing magnetic interactions of the effective spin in several open
d
shells of
polynuclear coordination compounds. The challenging task of improving a precision of ~1000 cm
–1
(describ
ing the excitation energies of single
d
shells by EHCF) to one of ~100 cm
–1
for the energies required to reori
ent spins by an order of magnitude is considered within the same paradigm as EHCF: the targeted use of
McWeeny’s group function approximation and the Löwdin partition technique. These are applied to develop
an effective description of a
d
system. This approach is tested on a series of binuclear complexes
[{(NH
3
)
5
M}
2
O]
4+
of trivalent cations featuring oxygen superexchange paths in order to confirm the repro
ducibility of the trends in the series of exchange constants values for compounds that differ in the nature of
their metal ions. The results from calculations are in reasonable agreement with the available experimental
data and other theoretical methods.
Keywords
: crystal field, effective Hamiltonian, binuclear complexes, exchange interactions.
DOI:
10.1134/S0036024414110053
THEORY OF ATOMIC
MOLECULAR PROCESSES
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 88 No. 11 2014
EFFECTIVE HAMILTONIAN CRYSTAL FIELDS 1905
lems, cannot be systematically applied to systems of real
interest in this area. Even today, they can hardly be used
for chemical problems, i.e., to establish or reproduce
trends in a number of similar compounds, rather than
obtaining a unique number for a single molecule.
The Hydra of DFTbased methods had just started
to raise its heads, but Misurkin was again able to see
the main weakness of this family of approaches: the
fundamentally noncorrelated character of the elec
tronic wave function used to construct oneelectron
density in practical calculations. Despite the enor
mous success claimed for DFTbased techniques
when applied to TMCs, it could be demonstrated only
for
d
0
or
d
10
complexes, complexes of the second and
third transition row, or carbonyls or the other organo
metallic compounds cited in abundance in [4], where
the effects of the static correlation crucial to TMCs
with open
d
shells are relatively unimportant [5, 6].
TMCs also pose almost insurmountable difficulties
for traditional semiempirical quantum chemistry,
which could be one reason for its visible decline in
recent decades. Although considerable efforts [7–11]
were made in this direction, the success achieved is
still far from being satisfactory, particularly with regard
to detailed descriptions of open
d
shells, an area basic
to describing TMCs. We realized that the situation in
(semiempirical) quantum chemistry of TMCs was in
astonishing contrast to the general theoretical under
standing of the physics of TMCs based on Bethe’s
crystal field theory (CFT) [12] and its semiquantita
tive descendants like the angular overlap model
(AOM) [13–15]. This striking contradiction had to be
explained somehow, and Misurkin’s “wave function
maxim” was crucial to our understanding the problem
and dealing with it further. The solution [16] was to
realize that the CFT’s success was based primarily on
the type of electronic wave function for a TMC (or a
crystal with an impurity transition metal ion) that was
implicitly used in constructing the theory.
The proposed approach, dubbed the Effective
Hamiltonian Crystal Field (EHCF), turned out to be
enormously successful when applied to mononuclear
TMCs. EHCF was parameterized for calculations of
various complexes of the first series of transition met
als with mono– and polyatomic ligands. The parame
ters for ligands with donor atoms N, C, O, F, Cl and
doubly and triply charged transition metal ions V, Cr,
Mn, Fe, Co, Ni were fitted to reproduce the experi
mental
d–d
spectra of these complexes [16–19]. These
parameters are characteristic for each pair metal
donor atom. The dependence of the effective field on
details of the geometry and chemical composition of
the ligands are reproduced using the standard semiem
pirical CNDO procedure based on Hartree–Fock. In
[20, 21] EHCF was extended for calculations of
ligands using the INDO and MINDO/3 parameter
izations. In all cases, the experimental multiplicity
(spin) and spatial symmetry of the corresponding
ground states were reproduced correctly. The peak of
this approach was reached in [22] with calculations on
cis
[Fe(NCS)
2
(bipy)
2
]
. Its molecular geometry is
known for both high and lowspin isomers. The cal
culations reproduced the respective ground state spins
and the spectra of low lying
d–d
excitations in
remarkable agreement with experiments.
Another semiempirical implementation of EHCF
is based on the SINDO1 scheme [23–25], which has
certain features that seem to be important in light of
EHCF. Details of the EHCF/SINDO1 implementa
tion were described in detail in [26]. The
EHCF/SINDO1 method has proved to be useful for
calculations of the spectra of lowenergy excitations in
some iron(II) complexes and ionic crystals [26], and
for quadrupole splittings in the Mössbauer spectra of
spinactive iron complexes [5]. In all cases, the
method reproduces not only the experimentally
observed spin and symmetry of the electronic ground
state but also provides excitation energies with accu
racy sufficient for modeling the Mössbauer spectra.
In this review, we describe the basics of EHCF and
its most recent developments: its use in describing the
effective exchange interactions in polynuclear TMCs
(PTMCs) and ways of including calculations of optical
oscillator strengths. Some conclusions and prospects
are discussed in the last section.
THEORY
EHCF Model of d–d Spectra of Mononuclear TMCs
The physical foundation of the CFT is the observa
tion that the lowestenergy electronic excitations of the
mononuclear TMCs are those of
d
shells. Their ener
gies are controlled by the effective crystal field induced
by ligands. This is the correct half of Bethe’s original
conjecture [12]: the (optical) spectrum of TMCs is that
of electrons in the
d
shell. It was formalized in the CFT
by considering the states of
d
shells only, thereby
implicitly taking the wave function of all electrons as a
product of the one in a
d
shell and of a further unspec
ified function of the ones that remain. The incorrect
half was an ionic model of the CFT that assumed the
field felt by
d
electrons was purely electrostatic.
Although symmetry is perfectly reproduced by the ionic
model, it accounts at best for 20% of the observed split
ting even if unrealistically high effective charges are
ascribed to ligands. The contradictions between the
ionic model and experiments can be clearly seen from
the integral results of spectroscopic measurements: a
spectrochemical series [27, 28] in which different
ligands are ranged according to the strengths of the
crystal fields they induce (the
10
Dq
parameter):
−−− −
−−− −
−−−
<<<<<
<< << <
<<<<
IBrS NFOH
Cl Ox O H O < SCN NH
py en SO NO CN CO.
2
3
22
23
2
32
1
,
2
1
2
1906
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 88 No. 11 2014
TCHOUGRÉEFF, SOUDACKOV
The crystal fields are systematically weaker for charged
ligands than for uncharged ones, with the example of
CO inducing the strongest crystal field, though it has
neither charge nor even a noticeable dipole moment.
The strengths of the crystal fields observed in the
experiment thus cannot be explained by the ionic
model of the environment, and electrostatic effects
can be only of minor significance.
The effective Hamiltonian crystal field (EHCF)
theory uses the CFT form of the wave function
describing the ground and lowlying excited states of
aTMC:
,(1)
which the CFT uses implicitly. (In Eq. (1) is the
n
th full configuration interaction function of
n
d
elec
trons in the
d
shell of the TMI and is the function
of all other (
n
l
) electrons of the system; the sign
indi
cates that the resulting function is antisymmetric.)
Wave function (1) cannot be exact: the oneelectron
hopping terms in the Hamiltonian of a TMC effect
electron transfers between the
d
shell and the rest of a
complex molecule and thus mix the states as in (1),
creating a model subspace with states from the outer
subspace formed by ligandtometal and metalto
ligand charge transfer (LMCT and MLCT). With this
physically based classification of the electronic states,
the Hamiltonian matrix acquires the form presented
()n
nd l
()
n
d
Φ
l
Φ
schematically in Fig. 1. Explicitly considering the sur
roundings of the
d
shell opens the way to assessing the
amount of the crystal field felt by
d
electrons.
Formally, this reduction can be performed using the
Löwdin partition technique [29]. In this formalism, we
first obtain energydependent effective total Hamilto
nian active in the model subspace but still pro
viding eigenvalues that coincide with the eigenvalues of
total Hamiltonian
H
for the entire system:
.(2)
Here, and are the Hamiltonians for
d
electrons
and the remaining electrons (
l
electrons) in the sys
tem, respectively;
V
c
is the operator of the Coulomb
interaction between metal
d
electrons and
l
electrons;
and
V
r
is the oneelectron resonance (electron hop
ping) operator describing the electron transfer between
the
d
shell and the ligands. By introducing the compli
mentary projector operators
P
and
Q
= 1 –
P
for the
model and outer subspaces, respectively, the effective
Hamiltonian can be written in the form
(3)
The above equation serves as a basis for deriving differ
ent forms of operator perturbation theory [30]. In the
lowest order, the state of the electrons outside the
d
shell
does not change: it is described by wave function
Φ
l
. The
variables of
n
l
electrons can thus be integrated out by
averaging the interaction parts of the effective Hamil
tonian with the wave function
Φ
l
, and the lowlying
excitations in TMC are described by the effective
Hamiltonian for the electrons in the
d
shell only, just
as proposed by Bethe [12]:
(4)
(where
E
0
is the ground state energy of Hamiltonian
H
0
in the outer subspace). In the last formula,
stands for the (effective) oneelectron operator
describing interactions between the electrons in the
d
shell and the atomic core of the TMI and its sur
roundings, and is the twoelectron operator
describing the (renormalized) Coulomb interactions
within the
d
shell. The symmetry properties of
are those of the CFT Hamiltonian. However, the
matrix elements of are not taken as parameters,
but are calculated using the EHCF procedure. For the
pair of
d
АО
and , the effective crystal field matrix
element is [16]:
.(5)
Quantities are the inverses of the energy
denominators and denotes the occupation
number of the ligand MOs. The individual contribu
tions in (5) ultimately comes from the interaction
between the states in the model configuration sub
space and those in the outer subspace.
eff
()
E
0
dlcr r
HH HVV H V
=+++=+
d
H
l
H
()
=+ −
=+
eff 1
00
0
()
().
rr
R
EPHPPVQEQQHQQVP
PH P V E
eff
cf ee
0
() l
ddlcR l
n
HH VVE HH
=+Φ+ Φ= +
cf
H
ee
H
eff
d
H
cf
H
µ
ν
()
(1 ) (1 )
[()1 ()]
nD n D
+−
μκ νκ κ κ
κ
ββ κ− − κ
(1 )
()
D±
κ
κ
=
0, 1
n
βμ
k
Model
LMCT &
MLCT
D
(1±)
(
k
)
βν
k
βν
k
βμ
k
×
D
(1±)
(
k
)
Fig. 1.
Pictorial representation of the partition of the
mononuclear TMC Hamiltonian matrix in EHCF [47].
The quantities in the square blocks associate the model
subspace with the LMCT/MLCT subspaces: are the
oneelectron hopping integrals between the
μ
th,
d
AO,
and
κ
th ligand MO; are the inverse energies of
the LMCT/MLCT excited states (
D
denotes the corre
sponding Green functions). The matrix elements of the
effective crystal field induced by the ligands appear in the
lower triangle as sums over index of the ligand MOs of the
products of the multipliers shown in the square blocks.
μκ
β
(1 ) 1
[()]
D
±−
κ
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 88 No. 11 2014
EFFECTIVE HAMILTONIAN CRYSTAL FIELDS 1907
Observables in the Model Subspace:
Effective Operator Formalism
Despite its qualitatively correct form, the CFT
wavefunction given in (1) for the model subspace has
an important drawback: it generally cannot be used
directly to calculate the expected values of operators
corresponding to observables other than energies. As
the total system Hamiltonian, the operators of observ
ables can include terms that mix the states in the
model subspace with those from the outer subspace.
An effective operator must be defined for such an
observable [31, 32]. As with an effective Hamiltonian,
an effective operator will provide exact expected values
despite operating in a model subspace (in our case, a
model subspace with a fixed number of
d
electrons
created by the wave functions defined in (1)). Explicit
expressions for effective operators can be derived using
the technique of double perturbation theory [30].
Total Hamiltonian (2) for the entire system is modified
by adding a perturbation:
,(6)
where
A
is an operator of interest and
λ
is a parameter
corresponding to the external field interacting with a
first order observable described by operator
A
. Effec
tive Hamiltonian corresponding to the modi
fied (perturbed) Hamiltonian is then obtained by
means of Löwdin partitioning, and effective operator
A
eff
is extracted as the derivative of with respect
to the parameter :
3
(7)
Finally, the effective operator for the
d
shell is found
by averaging over wave function
Φ
l
,
. (8)
Note that the above expressions for the effective oper
ator are formally exact and can be represented as an
infinite perturbation series with respect to the opera
tors mixing the states from the model and outer sub
spaces.
One of the most interesting applications of this for
malism for TMC is calculating the oscillator strengths
(intensities) of the electronic
d–d
transitions using
EHCF. In the CFT model subspace, these transitions
are forbidden because of the parity (Laporte) selection
rules, and the matrix elements of dipole moment oper
ator
M
between wave functions thus vanish. The prob
lem can be resolved by using an effective operator of
3
Higher order observables like polarizabilities etc. would require
higher derivatives.
HHA
λ
=+λ
eff
()
E
λ
H
λ
eff
()
E
λ
λ
()
()
()()
eff
eff
0
1
1
11
()
()
.
E
AE
PAP PHQ EQ QHQ QAP
PAQ EQ QHQ QHP
PHQEQ QHQ QAQEQ QHQ QHP
λ
λ=
−−
=
∂λ
=+ −
+−
+− −
eff eff
dl l
AA
=Φ Φ
the dipole moment obtained using a third order
perturbation theory expansion of the general expres
sion given in (7). Omitting the algebraic details of the
derivation, we write the final expression for matrix ele
ments of oneelectron effective dipole moment opera
tor in the basis of
d
АО
s:
(9)
where denotes the electron hopping integrals
between the
d
shell and ligand MOs, and are
the matrix elements of the oneelectron dipole
moment operator, represents the occupation num
bers of the ligand MOs, and stands for the
inverse excitation energies corresponding to the
MLCT and LMCT excited states.
In the expression above, the first sum represents the
diagonal constant contribution from ligand MOs, and
the remaining sums combine the second and third
order corrections arising from the electron hopping
(resonance) operator mixing the electronic states in
the
d
shell with the states in the outer subspace
(ligands). All of the quantities in (9) can be calculated
using the standard EHCF approach, so the oscillator
strengths for
d–d
transitions can readily be found by
simply counting the matrix elements of between
the ground and excited state
d
electron wavefunc
tions.
Extending EHCF to PTMCs
The success of EHCF as documented in [17, 18,
22, 33–37] (for the most recent successful application
to the object as complex as
3
d
decorated polyoxomo
lybdates see [38]) prompted the search for other possi
ble applications, of which describing the exchange
interactions in polynuclear TMCs (PTMCs) is the one
most logical but at the same time quite challenging. It
would involve extending the applicability of EHCF
from estimating the spectroscopic
10
Dq
parameters
that lay in the range of 5000–20000 cm
–1
and their
lower symmetry analogues to estimating the (effec
tive) exchange constants
J
AB
laying in the range of
200–500 cm
–1
with a corresponding improvement in
precision.
The magnetic interactions in PTMCs stem basi
cally from electrons transferring between
d
shells
mediated by bridging ligand states. Anderson [39, 40]
eff
d
M
eff
d
M
()
()
()
()()
κκ κ μν
κ
μκ νκ νκ μκ
κ
+−
κκμν
++
μκ νλ κλ κ λ μν
κλ
−−
μκ νλ κλ κ λ
κλ
μν= δ
+β +β
×κκδ
β κ λ+δ
β κ λ
eff
2(1) 2 (1)
2(1) (1)
(1 ) (1 )
[()(1)()]1
() ()1
11 ()(),
d
MMn
MM
nD n D
MnnD D
MnnDD
μκ
β
M
κκ
M
μκ
n
κ
(1 )
()
D±
κ
eff
d
M
1908
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 88 No. 11 2014
TCHOUGRÉEFF, SOUDACKOV
proposed using something very similar to EHCF in
order to calculate these: “One tries to explicitly sepa
rate two very different aspects of the problem: (a) The
first is … obtaining the wave function of a magnetic ion
that is surrounded by various diamagnetic groups …
while excluding the exchange effects of other magnetic
ions. (b) The second … centers around the question of
the way in which two magnetic ions defined in the
above way interact when they approach one another.”
Remarkably, the methods currently used in quantum
chemistry to estimate effective exchange parameters
do not follow Anderson’s procedure, although they
claim to. The procedures currently in use recommend
we first obtain different broken symmetry solutions for
a PTMC and then combine them to derive exchange
constants from the systems of linear equations for their
energy differences [41, 42]. This is not Anderson’s rec
ommendation, nor is it always feasible, particularly in
cases where the local electronic spins in
d
shells form
frustrated (e.g., triangular) clusters that prevent the
formation of broken symmetry solutions.
Suitably dividing the configuration space into
model and outer subspaces formalizing the above pro
cedure poses no conceptual problem. It is done in two
steps: First, the total number of electrons in all
d
shells
is fixed; second, unique numbers are established for
electrons in each of the
d
shells of a PTMC. The sec
ond step requires physical substantiation: the numbers
of electrons in the individual
d
shells of a PTMC must
indeed be good quantum numbers. This may differ for
mixed valence complexes where significant fluctua
tions of the numbers of particles in
d
shells can occur.
The division of the configuration space needed for cal
culating the effective magnetic exchange in PTMCs is
shown in Fig. 2. The wave functions in the model sub
space that corresponds to EHCF treatment of a
PTMC are thus written as
(10)
where functions are those of the respec
tive ground states
n
A
,
B
of electrons in the
d
shells of
A
and
B
obtained from EHCF calculations so that defi
nite values of total spin
S
A
,
B
can be attributed to them
(for the sake of simplicity, we restrict ourselves to the
case of a binuclear complex; the details of derivation,
which are quite cumbersome, can be found elsewhere
[43]).
As a mononuclear TMC, oneelectron hopping
when applied to states in a model subspace results in
mixing with the LMCT/MLCT states. The only for
mal difference at this stage is that diagonal energies
in this subspace are additionally indexed
with the TMI label of the
d
shell affected by
the oneelectron transfer. Wave functions
,
of
the individual TMIs in (10) are obtained via EHCF,
and the oneelectron hopping in the second order of
the perturbation theory does not alter them. In
PTMCs, secondorder treatment of oneelectron
hopping produces additional matrix elements with
states in two subspaces in the outer space that emerge
only now: a metaltometal charge transfer (MMCT)
subspace with characteristic energies and a
double ligandtometal or metaltoligand charge
transfer (
(LM)
2
CT/(ML)
2
CT
) subspace where the
ligands are doubly ionized with any sign. (The diago
nal energies in this subspace are .) As was
shown in [44–46], the amounts of both admixtures
depend on how local spins
S
A
and
S
B
of the
d
shells are
arranged in the overall state of total spin
S
, resulting in
the required splitting of states with different total spin
in the model subspace of the PTMC.
It is thus clear that the effective exchange constants
describing the matrix elements in the model subspace
(the triangles in Fig. 2) do not require any additional
quantities for their calculation except those that are
already available in EHCF. These are oneelectron
hopping integrals and the orbital energies of the
ligand MOs. The amount of splitting between states of
different total spin is given [43] by expressions that are
the sums of contributions containing products of four
integrals of oneelectron hopping between the
d
shells
and the ligands:
(11)
(; ) (; ) ,
AA A BB B l
nS nS
Ψ=Φ ×Φ ×Φ
,, ,
(; )
AB AB AB
nS
Φ
(1 ) 1
[()]
Dj
±−
κ
,
jAB
=
A
Φ
B
Φ
(0) 1
[]
ij
(2 ) 1
[]
ij
D
±−
κλ→
()
j
νκ
β
() () () (),
ABAB
μκ νκ μλ νλ
ββββ
βν
k
(
B
)
βμ
k
(
A
)
×
D
(1±)
(
Bk
)
βν
k
(
B
)
βμλ
(
A
)
D
(2±)
kk
AB
βμ
k
(
A
)
βν
k
(
B
)
×
D
(1±)
(
Bk
)
D
(1±)
(
Bk
)
Model
LMCT &
MLCT
MMCT
(LM)
2
CT &
(ML)
2
CT
(0)
A
B
Fig. 2.
Pictorial representation of the partition of the
PTMC Hamiltonian matrix for estimating magnetic inter
actions between electrons in the
d
shells of the
i
th and
j
th
TMIs [47]. The model subspace is created by functions in
the form of Eq. (5). The outer subspace further decom
poses in three subspaces: the old LMCT/MLCT and two
more: those of metaltometal charge transfer (MMCT)
configurations and 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.
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 88 No. 11 2014
EFFECTIVE HAMILTONIAN CRYSTAL FIELDS 1909
further supplied by the products of the energy denom
inators:
(12)
The energy denominators in the products of (12)
allow for combinations of the local spins in the
MMCT and
(LM)
2
CT/(ML)
2
CT
states. Increasing
and decreasing local spins in any of the
d
shells differ
by exchange energy of the respective
d
shell
( and
K
i
is the parameter of intrashell exchange
interaction (the one responsible for conforming with
Hunds rule) as was shown in [4446]. Each of the
terms (a)–(f) can be used once or several times for
each combination of ligand MOs and
d
–AOs
;
with numerical nonpairity factors
being combinations of Rach’s
6
j
symbols and
genealogical coefficients [46] assembled in Table 1.
Each such case predetermines the scale of nonpairity
factor . The products of the expressions in
(11), (12) and are further multiplied by aspin
factor dependent on the combination of local spins
;
characteristic of the
MMCT or
(LM)
2
CT/(ML)
2
CT
state, and then
summed over the allowed combinations.
DESCRIBING MAGNETIC
INTERACTIONS IN PTMCS
Our calculations of the products in (11), (12) and
their summation to effective exchange constants
J
AB
describing interaction between different
d
shells were
performed using the
M
AG
A
Î
X
T
IC
software package
(0)
(0)
(0)
(0)
(2 )
++
κλ →
+−
κλ →
−+
κλ →
−−
κλ →
++ + +
κλ κλ
κλ
−− κ λ
−− κ λ
−− κ λ
κκ+λ
a
b
c
d
e
f
(1 ) (1 )
(1 ) (1 )
(1 ) (1 )
(1 ) (1 )
(1 ) (1 ) (1 )
() ( ) ( ) ,
() 1 ) ( ) ( ) ,
() (1 ) ( ) ( ) ,
()(1 )(1 ) ( ) ( ) ,
() ( )[ ( ) ( )] ,
()
AB
AB
AB
AB
AB
nnD A D B
nnDBDA
nnD A D B
nnDADB
nnD B D A D B D
(2 )
κλ
−−
→κλ
−− λ
×κ+λ
(1 )
(1 ) (1 )
(1 ) (1 ) ( )
[() ()] .
AB
nnDA
DA DBD
(1)
ii
nK
+
,
iAB
=
κλ
A
μ∈
B
ν∈
(, )
AB
Un n
(, )
AB
Un n
(, )
AB
Un n
12
AA
SS
→±
12
BB
SS
→±
[43] with standard quantum chemical input data. The
package was tested using compounds of the
Cr(III)OCr(III) [47] family as examples. In this work,
we tested the applicability of EHCFbased procedures
to calculating the effective exchange constants for a
series of model symmetric linear dimers of the compo
sition
[(NH
3
)
5
MOM(NH
3
)
5
]
4+
(M = Ti, V, Cr, Mn,
Fe). The sole known experimental geometry of Cr(III)
complex was substituted for their geometries [48].
Qualitative Description
General expressions (11), (12) are considerably
simplified if we assume that only fully occupied orbit
als of monoatomic bridge
O
2–
contribute to the pro
cesses of virtual transfer and the molecules are sym
metric (the M–O distances and the composition of the
ligand spheres of each TMI are the same). Under these
assumptions, the products of (12) obey the rules of
selection, which guarantee that superexchange is pos
sible only between
d
–AOs of one symmetry
with respect to local group
(
ζ ≡
b
2
;
ξ
,
η
e
g
;
ε ≡
b
1
;
θ ≡
a
1
)
of each
M(NH
3
)
5
unit. The ligand
MOs can be further classified as symmetric or anti
symmetric (
s
or
a
) with respect to the mirror plane
running through the oxygen atom perpendicular to the
MOM axis. Due to phases
s
and
a
MO, the reso
nance integrals between them and TMIs
A
and
B
sat
isfy the relations
The effective exchange constant in a symmetric lin
ear MOM dimer is reduced to the sum of elementary
exchange constants
J
µµ
and summation over extends
to all singly occupied orbitals in each of the
d
shells:
. (13)
,,,,
μ=ζξηεθ
4
v
C
() () ,
() () .
sss
aaa
AB
AB
µµµ
µµµ
β=β=β
β=β=β
µ
AB
JJ
µµ
µ
=
Table 1.
Spin and numberofelectrons dependent factors scaling the contributions—the products in Eqs. (11), (12)
Occupancies of the
d
AOs
μ
,
ν
involved
in the
A
µ
B
ν
MMCT process Unpairity factors
U
(
n
A
,
n
B
)Spin factors
[1/2]
[1/2] +1;
S
A
S
A
– 1/2;
S
B
S
B
– 1/2
[1/2]
[0] +1;
S
A
S
A
– 1/2;
S
B
S
B
– 1/2
–1;
S
A
S
A
– 1/2;
S
B
S
B
+ 1/2
[1]
[1/2] +1;
S
A
S
A
– 1/2;
S
B
S
B
– 1/2
–1;
S
A
S
A
+ 1/2;
S
B
S
B
– 1/2
[1]
[0] +1;
S
A
S
A
±
1/2;
S
B
S
B
±
1/2
–1;
S
A
S
A
±
1/2;
2
AB
nn
()
2
1
AB
nn
+
()
2
1
AB
nn
+
()()
2
11
AB
nn
++
/
12
AA
SS
1910
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 88 No. 11 2014
TCHOUGRÉEFF, SOUDACKOV
Elementary exchange constants
J
μμ
are in turn sums
over superexchange paths that include ligand MOs
and of suitable symmetry:
(14)
where factor is determined by symmetry indi
ces
s
and
a
, which are respectively even and odd. The
TMI labels in energy denominators can be omit
ted since they are identical for both TMI. After intro
ducing individual contributions from the ligand MOs
into the crystal field according to
,(15)
we notice that they coincide with the EHCF expres
sions for AOM parameters and [49].
Results for Individual Metal Dimers
Ti.
For Ti compound, the only occupied
d
AO in
either of the ions is the lowest crystal field state of the
ζ
type. In the phenomenological model, elementary
exchange
J
ζζ
is assumed to be vanishingly small, since
ζ
AOs do not overlap with the oxygen AOs. This, how
ever, does not prevent them from overlapping with
ligand MOs of suitable symmetry (e.g., those pro
duced by ammonia lone pairs). Our direct calculations
of the hopping integrals between
ζ
AOs and ligand
MOs in combination with the formulas for an effective
crystal field yield an estimate of 2360 cm
–1
for the
crystal field component, which is by no means negligi
ble. This quantity is the sum of contributions from two
almost degenerate occupied ligand MOs with close
values of the hopping integrals with
ζ
AOs. The con
tributions to the crystal filed can thus be denoted as
and , and the above quantity is their sum: .
Elementary exchange, however, is the sum of contri
butions from MMCT processes with two different
occupied MOs in the intermediate LMCT states. Four
terms responsible for the transfer through ligand MOs,
κ
λ
()
κ+λ +
μμ μκ
κλ
++
μλ → κλ→
=−βκ
×β λ +
2(1)
2(1) (0) (2)
(, ) 1 [ ()]
[()][ 2],
AB
AB AB
JUnn D
DD
(1)
κ+λ
(1 )
D
+
2 (1) 2 (1)
(), ()
ss aa
Ds Da
++
µµ µµ
Δ=β μ Δ =β μ
e
σ
e
π
s
Δ
a
Δ
sa
Δ+Δ
, and
; ;
are
included with opposite signs due to phase factor
in (9). We thus see that in fourth order pertur
bation theory, terms almost equal in absolute value
but having opposite signs are included in the
summation so that
ζ
type elementary exchange is
,(16)
where the expression
(17)
is identified with the inverse effective MMCT energy.
We should therefore not be surprised by the results
presented in Table 2 from the complete summation of
all terms using our program suite.
V.
The vanadium (III) dimer is an exception among
the other members of the series. Due to the degener
acy of the singly filled
ξ
,
the
η
subset in this case is not
covered by the summation procedure programmed in
our package. Nevertheless, a (semi)quantitative anal
ysis is possible if we take as a rule of thumb that the
exchange parameter supplied by the program suite
must be used with the opposite sign. With this in mind,
we performed estimates along the same lines as in the
case of the Ti dimer. Effective exchange in the case of
the V dimer must appear as the sum of elementary
exchanges corresponding to two superexchange
paths: through the
π
orbital for
d
AO
ξ
and through
the
δ
orbital for the
d
AOs
ζ
. The symmetry reflec
tions used for the Ti dimer apply to all members of the
series and elementary exchange
J
ζζ
is thus negligibly
small, although the contribution from the ligand
δ
MOs
to splitting is in this case far from negligible.
In order to estimate the elementary exchange con
stant for the
π
super exchange path, we note first of all
that in the considered model, only one orbital of the
bridge has the required symmetry. According to our
calculations, the highest occupied
π
MO of the
ligands lies at the 90% level contributed by the
p
π
AO
of the bridging
O
2–
ions. It is symmetric with respect to
the mirror plane and has no asymmetric counterpart.
,
sa
λ=κ=ζ ζ
,
sa
λ=ζ ζ
,
as
κ=ζ ζ
λ≠κ
(1)
κ+λ
sa
Δ≈Δ
()
eff
/
2
sa
JU
ζζ ζ ζ
≈Δ −Δ
()
()
eff
1
1(0)
+
AB
U D +
D
Table 2.
Characteristic quantities for series [(NH
3
)
5
MOM(NH
3
)
5
]
4+
of linear
µ
oxo bridged M(III) model analogs of the
basic rhodo compound
M
Parameter 10
Dq
in [M(H
2
O)
6
]
3+
,
cm
–1
[28]
AOM parameter, cm
–1
J
WG
, cm
–1
J
calcd
, cm
–1
EHCF renorm. [46] phenomenological model
EHCF renorm.
Ti 20200 2360 2360 0 88 88 0
V 19950 4340 3818 –480 –490 –343 –332
Cr 17400 4920 3850 427 733 389 408
Mn 21100 16820 7615 240 6400 786 900
Fe 13700 16915
5
9040 210 1756 356 220
AOM parameter
:
e
δ
,
e
π
,
e
π
,
e
π
, and
e
p
σ
for singly occupied dAO in Ti
, V, Cr, Mn, and Fe
compound, respectively
.
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 88 No. 11 2014
EFFECTIVE HAMILTONIAN CRYSTAL FIELDS 1911
In accordance with [44, 45, 50] we immediately obtain
elementary
π
exchange
.(18)
Oneelectron hopping integral is 2.17 eV
and the corresponding energies are 8.76
(bare) and 9.97 eV (renormalized by the crystal field).
The phenomenological value of the effective exchange
is satisfactorily reproduced using bare estimates of
AOM parameters
e
π
. In contrast, we obtain a number
quite close to the one determined by our package if we
use the renormalized values.
Cr.
The Cr dimer (rhodo complex) holds a special
place among other members of the considered series.
It is the only truly existing compound that has been
studied experimentally. The highest occupied
π
MO
of the ligands in the Cr compound lies at the 90% level
contributed by the
p
π
AO of the bridging O atom. One
electron hopping integral for the relevant pairs
of
d
and bridgecentered MOs is 1.86 eV. Due to the
localized character of the highest occupied
π
MO
ligand, only one LMCT state contributes to AOM
parameter
e
π
. Corresponding energy is
5.67 eV for the rhodo compound, which satisfactorily
fits into the 5–7 eV range suggested in [44, 45, 50].
The
e
π
value thus derived is somewhat higher than the
empirical value of 4000 cm
–1
(0.5 eV) proposed in [44,
45, 50]. Using the same energies of the LMCT states,
we obtain a strong overestimate of the exchange con
stant in this compound. In contrast, our package
yields a much more realistic estimate for the rhodo
compound, since it uses the LMCT energies renor
malized by the crystal field, reducing the value of
parameter
e
π
. The exchange constant obtained by the
package is thus comparable to the known experimen
tal value of 450 cm
–1
.
Mn.
Another important feature of the Cr dimer is
that it is the last one in the considered series where
only the
d
AOs stemming from the weakly overlapping
octahedral
t
2
g
subset are occupied. Starting with the
Mn dimer, strongly overlapping (
ε
and
θ
)
d
AOs
stemming from the doubly degenerate
e
g
subset
strongly destabilized by the effective crystal field are
occupied and thus enter into play.
The arguments applied above to elementary
exchange
J
ζζ
are universal. As before, the two pairs of
almost degenerate ligand
ε
MOs lying in the energy
ranges of –34.49 to –33.59 eV and –11.25 to –11.21 eV
overlap strongly with
ε
d
AOs. Within each of these
pairs, however, one of the orbitals is symmetric and
another asymmetric with respect to the mirror plane
crossing the MnOMn axis. As in the case of
ζ
exchange,
the full value of elementary exchange
J
εε
is the sum of
differences of almost equal quantities and is negligibly
small on our scale. Only elementary exchanges
contribute in the case of a linear MnOMn
dimer. The magnitude of these obtained using the bare
eff
/
2
JeU
ξξ π
V
()
ξπ
β
O
(1 ) 1
[()]
D
+−
π
Cr
()
ξπ
β
O
(1 ) 1
[()]
D
+−
π
JJ
ξξ ηη
=
energy denominators is completely unrealistic. The
values obtained using the renormalized energy
denominators with the phenomenological formulas
and with our software package are much better, but are
still well above either the fitted or the experimental val
ues characteristic of linear MnOMn dimers [51]. The
question thus remains: Where does the difference
between the theoretical and experimental values come
from? For the time being, we are inclined to believe
that more elaborate parameterization is needed for Mn.
Fe.
Finally, we address the FeOFe model dimer.
Elementary exchange
J
θθ
contains its effective
exchange parameter. With respect to symmetry, it is
contributed through two paths of superexchange
,(19)
one running through the
O
p
z
σ
orbital, and one run
ning through the
O
s
σ
orbital. These are correspond
ingly antisymmetric and symmetric with respect to the
mirror plane, but have no degenerate counterparts of
other symmetry. Due to the phase multiplier, elemen
tary exchange
J
θθ
is expressed as the squared difference
between the respective AOM parameters. AOM
parameter
e
p
σ
characterizing the strength of
p
σ
inter
action is easily estimated and is the only contribution
from the HOMO of the dimer, which is 75%
O
p
σ
AO.
We es t im a te d
e
p
σ
to be 16 900 cm
–1
(2.09 eV) using the
bare energies in the LMCT subspace. With the
s
σ
interaction, the situation is less clear, since the spectral
density of the
O
s
orbital is distributed over several
ligand MOs. Using formulas [49], we can reliably esti
mate
e
s
σ
as the sum of contributions coming from two
such MOs supplying 58 and 10% of the
O
s
spectral
density, respectively. The remaining density lies at the
deep ligand levels. These MOs respectively contribute
1200 (the higher one) and 3160 cm
–1
(the deeper one)
to the total value of
e
s
σ
, which is thus 4360 cm
–1
(0.54 eV). The value of the already familiar parameter
e
π
is 3150 cm
–1
when estimated with the bare LMCT
energy. The exchange constant as obtained with the
use of bare energies is, as we might expect, completely
unrealistic. Switching to the renormalized quantities
significantly improves it, although the estimate
remains too high. It is at first glance surprising that
complete summation yields a much better estimate in
terms of its closeness to the experimental values of
FeOFe dimmers, which all fit into the range of 160–
265 cm
–1
[50]. This is because many terms stemming
from pairs of orbitals with different mirror symmetries
appear in the total summation due to the strong over
lap (hopping) of the
θ
orbitals, thereby reducing the
effective exchange constant for the model iron mole
cule.
CONCLUSIONS
We have shown it is possible to extend the Effective
Hamiltonian Crystal Field (EHCF) method originally
intended for crystal fields to their polynuclear analogs
()
eff
/
2
sp
JeeU
θθ σ σ
≈−
1912
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 88 No. 11 2014
TCHOUGRÉEFF, SOUDACKOV
and enable the modeling of magnetic interactions of
the effective spins residing in several open
d
shells.
This challenging problem was solved and a precision of
~1000 cm
–1
(that of describing the excitation energies
of single
d
shells by EHCF) was improved to the
~100 cm
–1
characteristic of the energies required to
reorient spins (i.e., by an order of magnitude). We also
outlined how to employ EHCF for calculating the
oscillator strength for optical
d–d
transitions.
More important for us, however, was stressing the
real roots of this successful development and the entire
concept of semiempirism in quantum chemistry.
Selecting the form of the wave function on the basis of
the observable electronic groups (e.g.,
d
shells,
π
sys
tems, twocenter bonds etc.), i.e., chromophores
characteristic for the target class of molecules, was a
formal expression of a concept of physics that must be
duplicated by quantumchemical means. This type of
reasoning was alien not only to the quantum chemistry
of the 1960s when I.A. Misurkin started his career.
Even today, however, it is best described by the words
Mechanitis
is an occ upati ona l dise ase of one who is so
impressed with modern computing machinery that he
believes a mathematical problem which he can neither
solve or even formulate can be readily answered, once
he has access to a sufficiently expensive machine”
[52]. Misurkin taught us otherwise, and we shall always
remember his lessons—the true source of our success.
ACKNOWLEDGMENTS
This work was partially supported by the Russian
Foundation for Basic Research, grant nos. 1003
00155 and 140300867.
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... By this the problem of describing problematic transition metal compounds separates into two simpler ones: (i) to calculate the effective field induced by the environment upon the d -shell and (ii) to calculate the manyelectronic (multi-reference) states of this shell with the fixed number of electrons. This approach fairly follows the main Cartesian idea: "Diviser chacune des difficultés afin de mieux les examiner et les résoudre" [31] and turns out extremely efficient: dozens of complexes has been calculated (see Fig. 6); none of them has manifested an incorrect ground state spin and symmetry, even highly problematic Fe(II) spin-active complexes could be successfully reproduced [34,35], including the relative energies of the spin-isomers with an unprecedented precision. Our approach allowed us to reduce as much as possible the sizes of the parts of the system (electronic groups) requiring application of numerically demanding methods. ...
... It implements the results of the analysis of the structure of the hybrid methods of molecular/material modeling performed in Refs. [10,[34][35][36]. Presence of several tree like structures calls for a generic tool for representing them. ...
Chapter
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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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Quantum mechanics provides the possibility for the complete description of the electronic properties of molecular systems, their structure, reactivities, etc. However, the computational difficulties encountered in the general case, as well as the magnitude of extraneous information generated by many-electron wave functions, necessitate the development of entire conceptual frameworks in order to apply the quantum theory to chemical systems in a chemically or physically meaningful manner. Thus, far from being a sterile exeicise in applied mathematics, the development of quantum theories of molecular electronic structure has required a great deal of chemical insight and imagination(1–6)
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Article
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Article
The effective Hamiltonian-crystal field (EHCF) method was implemented on the INDO level of approximation. The SINDO1 method was used for the description of the ligand subsystem of a transition-metal complex. The effects of nonorthogonality between the ligand orbitals and d orbitals of a transition-metal atom were taken into account. The effective Hamiltonian of d electrons of a transition-metal atom was constructed with the wave function obtained on the stage of the calculation of the ligand subsystem taking into account Coulomb and exchange interactions between ligand electrons and d electrons of a transition-metal atom. For the whole complex, a non-Hartree-Fock wave function with an explicit inclusion of local d electron correlation was used. The method EHCF/SINDO1 was applied to calculations of the d-d spectra and electronic structure of the ligand sphere of various iron(II) complexes. The results of calculations were compared with available experimental and theoretical data. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 403–418, 1997
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Article
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