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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

ﬃﬃﬃ

2

pjr#l"j6jl#r"j

;1

ﬃﬃﬃ

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

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

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

ﬃﬃﬃ

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

163ﬃﬃﬃ

3

p0

63ﬃﬃﬃ

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

3ﬃﬃﬃ

3

p0

12b

11b

3ﬃﬃﬃ

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

13ﬃﬃﬃ

3

pD

2kBT0

3ﬃﬃﬃ

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€

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Received: 8 July 2015

Revised: 26 August 2015

Accepted: 27 August 2015

Published online 7 October 2015

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294 International Journal of Quantum Chemistry 2016,116, 282–294 WWW.CHEMISTRYVIEWS.ORG