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ISSN 00360244, 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

everexpanding 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 lowlying

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 UrbanaChampaign, Urbana, IL, 61801 USA

email: tch@elch.chem.msu.ru

Received March 5, 2014

Abstract

—The fundamentals of the Effective Hamiltonian Crystal Field (EHCF) method, used originally to

calculate intrashell 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 superexchange 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 DFTbased 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 oneelectron

density in practical calculations. Despite the enor

mous success claimed for DFTbased 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 semiempirical 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 semiquantita

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 lowspin 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 lowenergy excitations in

some iron(II) complexes and ionic crystals [26], and

for quadrupole splittings in the Mössbauer spectra of

spinactive 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 lowestenergy 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

Ⰶ

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 lowlying 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 antisymmetric.)

Wave function (1) cannot be exact: the oneelectron

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 ligandtometal and metalto

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

Ψ=Φ ∧Φ

()

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 energydependent 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 oneelectron 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 lowlying

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) oneelectron operator

describing interactions between the electrons in the

d

shell and the atomic core of the TMI and its sur

roundings, and is the twoelectron 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

Ᏼ

0

dlcr r

HH HVV H V

=+++=+

d

H

l

H

()

−

=+ −

=+

eff 1

00

0

()

().

rr

R

EPHPPVQEQQHQQVP

PH P V E

Ᏼ

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

oneelectron 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

Ᏼ

H

λ

eff

()

E

λ

Ᏼ

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

=+ −

+−

+− −

Ᏼ

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 oneelectron 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 oneelectron 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

×κ−−κ−δ

+ββ κ λ+δ

+ββ − − 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, oneelectron 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 oneelectron transfer. Wave functions

,

of

the individual TMIs in (10) are obtained via EHCF,

and the oneelectron hopping in the second order of

the perturbation theory does not alter them. In

PTMCs, secondorder treatment of oneelectron

hopping produces additional matrix elements with

states in two subspaces in the outer space that emerge

only now: a metaltometal charge transfer (MMCT)

subspace with characteristic energies and a

double ligandtometal or metaltoligand 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 oneelectron

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 oneelectron 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

−

→

Ᏸ

(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

Ᏸ

(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 metaltometal charge transfer (MMCT)

configurations and double ligandtometal or metalto

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

Hund’s rule) as was shown in [44–46]. 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

Ᏸ

Ᏸ

Ᏸ

Ᏸ

(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 EHCFbased 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 numberofelectrons 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 superexchange 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

DDᏰ

(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 superexchange

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 +

−

→

=κD

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 dAO 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)

Oneelectron 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 bridgecentered 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 superexchange

,(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 semiempirism 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, twocenter 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 quantumchemical 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. 1003

00155 and 140300867.

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