A handbook of magnetochemical formulae

A Handbook of Magnetochemical Formulae


A Handbook of
Magnetochemical
Formulae

Roman Bocˇa
Faculty of Natural Sciences
University of SS Cyril and Methodius
Trnava
Slovakia
and
Institute of Inorganic Chemistry
Slovak University of Technology
Bratislava
Slovakia

AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD
PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO
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Elsevier
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First edition 2012
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Preface

Magnetochemists (who deal mainly with molecular entities and their assemblies)
and magnetophysicists (who predominantly investigate atomic/ionic solids) are
facing an enormous increase in theoretical knowledge about magnetic properties of
materials. The magnetic materials under investigation:
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differ in the composition from simple units (mononuclear, dinuclear complexes) to very
complex assemblies like polynuclear clusters, oligomers and dendrimers;
cover pure organic polyradicals, pure inorganic phases, transition metal complexes or
organometallic compounds and hybrid materials that combine different functionalities;
exhibit magnetic disorder (paramagnets) or ordered phases in a simple (collinear
ferro/ferri/antiferromagnets) or complex manner (non-collinear and canted magnets); and
in the case of novel materials, cover the single-molecule magnets and single-chain magnets.

The above objects are subjected to a deep investigation by combining several
techniques, such as magnetometry (MM), susceptometry (SM), calorimetry (CM,
DSC), electronic spectroscopy (UV, VIS, MID, FAR), vibrational (infrared À IR
and Raman) spectroscopy, electron spin resonance (ESR, high-frequency/high-field
ESR), frequency-domain magnetic resonance spectroscopy (FDMRS), nuclear
magnetic resonance (NMR), Moăssbauer spectroscopy (MS), inelastic neutron scattering (INS), magnetic circular dichroism (MCD), synchrotron techniques (EXAFS,
XANES, X-MCD) and many others. Some techniques are rather unique, such as
microSQUID and the nanoSQUID or the positron annihilation. The external stimuli
are represented by temperature, pressure, magnetic field, electric field and irradiation
by electromagnetic waves over a wide range of energies. There are three principal
responses that can be recorded:
1. thermodynamic functions at the thermal equilibrium, i.e. magnetisation M, magnetic
susceptibility χ and heat capacity (the isofield CH and the isomagnetisation CM heat
capacities) for MM, SM and CM techniques;
2. absorption (transmission) for a number of spectroscopic methods;
3. scattering of X-ray photons (single crystal and powder diffraction) or neutrons (INS).

Thus the research of magnetic materials becomes more and more complex. The
key factors accompanied with the research of magnetic materials represent:
(a) determination of the structure (positions of atoms, magnetic structure, distribution of the
spin density),
(b) reconstruction of the energy levels at the different degree of resolution and complexity
(electron spinÀorbit coupling, electron-nuclear spin coupling, nuclear spinÀspin coupling)
in zero and in an applied magnetic field.


xvi

Preface

The magnetic phenomena originate at the microscopic level. Thus the quantum
mechanics is the tool that brings answers about the individual energy levels εi and
their evolution in an applied magnetic field. Then the application of the apparatus
of the statistical thermodynamics brings all macroscopic thermodynamic functions
(F, S, G, H, U, Cp, CV, M, χ, CM, CP
H and also K and k) in the form of derivatives
of the partition function ZðB; TÞ 5
expðεi =kTÞ. In other words: by substituting
the energy levels into the partition function and by performing the corresponding
derivatives, we can arrive at the macroscopic observables like magnetisation M 5
(@F/@B) where the Helmholtz energy is F 5 kT ln Z. This is how a magnetochemical
formula can be generated for an individual model system. As an example, the Brillouin
function for the magnetisation and the Curie law for the magnetic susceptibility
can be derived for a Curie paramagnet.
The energy levels result as eigenvalues of the model Hamiltonian whose matrix
elements must be expressed in an appropriate basis set of wave functions. The
actual form of the model Hamiltonian must be postulated for the system under

A
study. It contains an operator part (usually the operators of the spin S^a and orbital
A
A
angular momentum L^a ; sometimes also the nuclear spin I^a ) and a parameter part À the
magnetic constants pfgα ; D; E; F; a; Bqk ; J; D; d; A; Dαβ ; b; . . .g: When the eigenvalues
are closed functions of the parameters and of the magnetic field, εi 5 f ({p},B), then
the partition function is an analytical function (though a rather complex function),
allowing to get its derivatives. In this way magnetochemical formulae can be
obtained in a closed form.
In most of the contemporary problems a numerical diagonalisation of the
Hamiltonian matrix yields a discrete set of eigenvalues εi({p},Bm) for a ‘working
field’ Bm. Three sets of Bm, Bm 1 δ and Bm 12δ allow us to apply a parabolic fit
and then the numerical construction of Z, Z0 and Zv. This facilitates a modelling of
the magnetic functions M(T,B) and χ(T,B). This also allows a fitting of the experimental magnetic data to the chosen model and determination of the magnetic parameters {p}. Necessary tools are a computer (in most cases a PC is enough) and an
appropriate software. And this is the core of this book À a presentation:
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of the derived closed formulae for the magnetisation M(B,T0,{p}) and the magnetic
susceptibility χ(T,B0,{p}) for a number of simple cases,
of the programmable matrix elements for the model Hamiltonian that describes the
magnetic systems of various complexity,
a modelling of the magnetic functions for a number of important cases.

In order to get those matrix elements the reader should be familiar with the fundamentals of quantum mechanics and with the theory of angular momentum and irreducible tensor operators (ITO). Common university courses provide only a brief
insight into the electronic structure of atoms and complex ions surrounded by ligands.

It can be briefly summarised that the electron configurations represent only a starting
point in generating the atomic terms which are further modulated by the spinÀorbit
interaction yielding the atomic multiplets. Since magnetic properties originate in the


Preface

xvii

fine-structure energetic spectrum, an advanced knowledge of the atomic multiplets is
an ultimate requirement. However, this book is not a substitute for other, excellent
monographs on this subject developed by pioneers like Condon and Shortley, Slater,
Racah, Griffith, Sugano and Tanabe, Judd, Sobelmann, Jucys and Saukynas, to name
just a few.
In treating the atomic energy levels, a fluent knowledge of the angular momentum (orbital as well as spin) and their addition rules is necessary. The basis set
transformation, called the coupling, is provided by the vector coupling coefficients
and, in atoms, the set of ClebschÀGordan coefficients, 3j-symbols, 6j-symbols and
9j-symbols, and/or Racah V-coefficients and W-coefficients, is met. These enter an
algebra of the irreducible tensor operators which provides a great advantage in a
direct evaluation of the matrix elements between the atomic kets without knowing
the explicit form of the state functions.
In passing to the crystal (ligand) field, the group theory adopts a great advantage. In the subgroups of the rotational group the vector coupling coefficients are
again met: the 3Γ-coefficients and the 6Γ-coefficients. The reference state can be
represented by the atomic terms which, in the weak-field limit, subduce the weakfield terms. Alternatively, new electron configurations arise in the strong-field
limit, which represent a new basis in generating the strong-field terms and finally
the strong-field multiplets. In evaluating the necessary matrix elements of the
model Hamiltonian again the irreducible tensor approach meets its full utilisation.
The monographs about the magnetism involve the theoretical aspects to a different
extent and complexity (from a very limited to a very broad presentation) [1À24].
There has been great progress in the electron spin resonance (ESR) which could be

considered one of the complementary experimental techniques [25À33]. However, a
full description of the situation requires much more effort and we will see later
that there is a need to combine the information from the electronic structure of
atoms [34À39], the crystal/ligand field theory [40À48], fundaments of the angular
momentum [49À52], and finally the irreducible tensor operator approach [53À60] as
a real working tool. Let us note that the contemporary monographs utilise the ITO
approach as a common standard [22,23,32,46]. The group theory cannot be omitted
from consideration and the traditional point groups of symmetry must be extended by
knowledge of the symmetry (permutational) group [61À84].
Notations used are as follows:
1. SI units are used consistently throughout; χmol [SI] 5 4π 3 1026 χmol [cgs and emu].
2. The energy quantities E (like ε, J, D, E, a, F, etc.) are presented either in the form of the
corresponding wave number, i.e. E/hc and given in units of cm21, or in the form of the
corresponding temperature E/k and given in units of K.
3. The angular momentum operators bring the reduced Planck constant ¯h when operating to
a corresponding wavefunction (a ket).
4. The fundamental physical constants (c, ε0, μ0, NA, k 5 kB, R, μB, e, h, ¯h) adopt
their usual meaning; they enter the reduced Curie constant C0 5 NA μ0 μ2B =k
5 4:7141997 3 1026 K m3 mol 21 :


xviii

Preface

5. The CondonÀShortley phase convention is utilised along with the pseudo-standard phase
system for the irreducible tensor operators.
! !
¯ 22 :
6. The isotropic exchange constants are uniformly thought in the form2Jij ðSiUSj Þh

7. The individual interaction terms entering the spin Hamiltonian for a diad are termed as
follows:
!

!

H^ AB 5 2 JAB ðSA U SB Þh
¯ 22 . . . isotropic ðbilinearÞ exchange
!

!

!

¯ 21 . . . Zeeman term
1μB B U ðgA U SA 1 gB U SB Þh

9

! !
! !
A A
B B
1 DA ẵS^z S^z 2SA U SA ị=3h
22 1 DB ẵS^z S^z 2 SB U SB ị=3h
22 . . . axial single-ion anisotropy =

1EA ðS^x S^x 2 S^y S^y Þh
¯ 22 1 EB ðS^x S^x 2 S^y S^y Þh
¯ 22 . . . rhombic single-ion anisotropy

A A

A A

B B

B B

;

. . . zero-field splitting

9
! !
A B
A B
A B 22
1DAB ẵS^z S^z 2 SA U SB ị=3h
22 1EAB ðS^x S^x 2 S^y S^y Þh
¯ . . . asymmetric exchange =
. . . anisotropic exchange
!
!
!
;
¯ 22 . . . antisymmetric exchange
1dAB U ðSA 3 SB Þh
1biquadratic exchange 1 triquadratic exchange 1 double exchange:

Roman Boˇca

1 Molecular Symmetry
1.1

Some Definitions

1.1.1

Tensors

A tensor is a generalised linear ‘quantity’ or ‘geometrical entity’ that can be
expressed as a multi-dimensional array relative to a choice of basis. However, as
an object in and of itself, a tensor is independent of any chosen frame of reference.
The rank of a particular tensor is the number of array indices required to describe
such a quantity.
Tensor product. The tensor product, denoted by  , may be applied in different
contexts to vectors, matrices, tensors, vector spaces, etc. In each case the significance
of the symbol is the same: the most general bilinear operation.
A representative case is the Kronecker product of any two rectangular arrays,
considered as matrices
0

1
b1
B b2 C
B C  ða1
@ b3 A
b4

0

a2

a1 b1
B a1 b2
a3 Þ 5 B
@ a1 b3
a1 b4

a 2 b1
a 2 b2
a 2 b3
a 2 b4

1
a3 b1
a3 b2 C
C
a3 b3 A
a3 b4

ð1:1Þ

Here, resultant rank 5 2, resultant dimension (4,3) 5 4 3 3 5 12. The rank denotes
the number of requisite indices, while dimension counts the number of degrees of
freedom in the resulting array. It should be emphasised that the term rank is being
used in its tensor sense and should not be interpreted as matrix rank.
(You can arbitrarily add many leading or trailing one dimensions to a tensor
without fundamentally altering its structure. These one dimensions would alter the

character of operations on these tensors, so any resulting equivalences should be
expressed explicitly.)
Outer product. Given a tensor A with rank a and dimensions (i1, . . . , ia), and a
tensor B with rank b and dimensions (j1, . . . , jb), their outer product C 5 A  B has
rank a 1 b and dimensions (k1, . . . , ka 1 b) which are the i dimensions followed by
the j dimensions.
For example, if A has rank 3 and dimensions (3, 5, 7) and B has rank 2 and
dimensions (10, 100), their outer product C has rank 5 and dimensions (3, 5, 7, 10, 100).
In other words, outer product on tensors 5 tensor product.
A Handbook of Magnetochemical Formulae. DOI: 10.1016/B978-0-12-416014-9.00001-X
© 2012 Elsevier Inc. All rights reserved.


4

A Handbook of Magnetochemical Formulae

To understand the matrix definition of an outer product in terms of the tensor
definition of outer product, you can interpret the vector v as a rank-1 tensor with
dimension (M), and the vector u as a rank-1 tensor with dimension (N). The result
of the outer product is a rank-2 tensor with dimension (M,N).
Inner product. The result of an inner product between two tensors of rank-q and
rank-r is the greater of (q 1 r 22) and 0.
●

●

The inner product of two matrices has the same rank as the outer product (or tensor product)
of two vectors.
The inner product of two matrices A with dimensions (I,M) and B with dimensions (M,J) is

Cij 5

M
X

where iAf1; . . . ; Ig and jAf1; . . . ; Jg

Aim Bmj

ð1:2Þ

m

Direct product of two matrices is


a11

a12





b11

b12



3
a21 a22
b21 b22
0
1
a11 b11 a11 b12 a12 b11 a12 b12
Ba b
C
B 11 21 a11 b22 a12 b21 a12 b22 C
5B
C 6¼ B 3 A
@ a21 b11 a21 b12 a22 b11 a22 b12 A

A3B5

a21 b21

a21 b22

a22 b21

ð1:3Þ

a22 b22

The resulting rank is not 4 but only 2, so that this is a kind of the inner product.

1.1.2

Physical Vector (Polar Vector)
!

A true vector (syn. polar vector, V ) is required to have components that transform
in a certain way under a proper rotation (rotation about an axis, C^ n ). If everything
!
in the universe undergoes a rotation (e.g. the displacement vector r is transformed
!
!0
!
with the rotation matrix R to r 5 R r ), then any vector V must be transformed in
!
!
the same way (V 0 5 R V ).
The polar vector is a contravariant vector (a tensor of contravariant rank one).
!
!
Examples of the polar vectors are: position (displacement) vector r ; velocity v and
!
linear momentum p :
!
Under inversion through the origin (i^5 S^2 ), the true vector alters its sign: V
!
goes to 2V :

1.1.3

Pseudovector (Axial Vector)
!

The pseudovector (syn. axial vector, P ) transforms under rotations according to the
formula
!

!

P 5 ðdet RÞðR P Þ

ð1:4Þ


Molecular Symmetry

5

It transforms under the proper rotations like a polar vector, but under the improper
sign flip.
rotations (rotations of mirror image, S^n 5 σ^ h C^ n ) gains an additional
!
!
Examples of the axial vector are: angular momentum l ; magnetic induction B ;
torque and
vorticity. For instance, the angular momentum is defined
through
the cross
!
!
!
!
!

!
!
!
product l 5 r 3 p : On inversion: r ! 2r and p ! 2^p; while l ! l :
Properties for addition and multiplication
!

!
P1

6 P2 5 P ;

!

!

!
P1

6 V 2 2 undefined

!

!

aP1 5 P ;

!

!

!

!

V1 6 V2 5 V ;

aV 1 5 V

ð1:5Þ

!

ð1:6Þ

Properties for the cross product
!

!

!

!
P1

V1 3 V2 5 P;

!

!

3 P2 5 P ;

!

!

!

V 1 3 P2 5 V ;

!
P1

!

!

3 V2 5 V

ð1:7Þ

A common way of introducing a pseudovector is by taking the cross product of
polar vectors. For instance, the magnetic induction is
!

!

!

B 5r3 A

1.2
1.2.1

ð1:8Þ

Point Groups
Elementary Terms

The molecular symmetry originates in the fact that there exist symmetry operations
(transformations of the nuclear coordinates) that transform the molecule into a
nuclear configuration identical with an initial one. The symmetry elements (axis,
plane and inversion centre À Table 1.1) remain unchanged. Molecules belong to
the point groups of symmetry as all the symmetry operations have at least one
point in common (this point does not necessarily be identified with any atom of
the molecule) [61 284].
A brief summary of the properties of the symmetry point groups is presented in
Table 1.2. Some additional definitions follow:
1. a subgroup G0 is a set of elements within a group G which, on their own constitute a
group;
2. two groups are isomorphous when there exists a one-to-one correspondence between their
operations; they have the same defining relations and the same multiplication and character
tables;
3. if there are two groups Ga and Gb, having only their identity in common and possessing
elements R^a (for a 5 1, . . . , ha) and R^b (for b 5 1, . . . , hb), then the direct product group
G 5 Ga 3 Gb is defined as the set of all distinct elements R^a R^b 5 R^b R^a for all a and b.


6

A Handbook of Magnetochemical Formulae

Table 1.1 Symmetry Operations
Symmetry
^k
Operation R

Property

Inverse
^ 21
Operator R
k

E^ or I^

Identity; rotation through an angle 2π in single groups;
rotation through an angle 4π in double groups
Rotation about angle 2π/n
Mirror plane (horizontal, vertical, diagonal)
Inverse centre
Rotation followed by the mirror plane

E^

C^ n
σð
^ σ^ h ; σ^ v ; σ^ d Þ
i^ or S^2

S^n 5 σ^ h C^ n
Q^ or R^
k
C^ n Q^

Rotation by an angle 2π but differing from the identity
^ applicable to double groups
operation (Q^ 6¼ E);
A rotation through ϕ 5 2π 1 2πk/n; applicable to double
groups

n2k
k
C^ n for C^ n
σ^
i^
n2k
k
S^
for S^
n

n

Q^

n2k
C^ n

Table 1.2 Elementary Terms in the Symmetry Point Group Theory

Term

Property

(a) Properties of the symmetry group G of order h
Existence of a product
R^i R^j 5 R^k
Associative law
R^i ðR^j R^k Þ 5 ðR^i R^j ÞR^k
Existence of identity
R^k E^ 5 E^ R^k 5 R^k
21
21
Existence of inversion
R^k R^k 5 R^k R^k 5 E^
(b) Class T of the group G
21
A set of operators obeying
R^i 5 R^s R^j R^s

Note
ðR^i ; R^j ; R^k ÞAG
21

ðR^k ; R^k ÞAG
ðR^i ; R^j ÞAT, R^s AG

The symmetry point groups along with their important characteristics are classified in Table 1.3 (Crystals have no C5, C7, C8, etc. axes and this fact restricts the
possible groups to 32 point groups.)

1.2.2

Representations

A set of matrices D(Rk), transforming coordinates in the same way as the symmetry
operator R^k ; forms a representation Γ of the group G (Table 1.4). The irreducible
representations (IRs) used to be denoted by two conventions: according to Mulliken
(Table 1.5) or according to Bethe (simply Γ1, Γ2, etc.).
The representation is reducible, Γr, when by the same similarity transformation
U a block-diagonal form of matrices D(Rk) is obtained
U−1D(Rk)U = Dbd(Rk) =

⎛ D1
⎜
⎝ 0

⎛
⎜
D2 ⎝
0

(1.9)


Table 1.3 Symmetry Point Groups of Molecules and Atoms
Symbol

Symmetry Operations

Order h

Number of IR

Note

Generators

Non-axial
C1
Cs
Ci

E^
^ σ^ h
E;
^ i^
E;

1
2
2

1
2
2

C1 5 C
Cs 5 C1h 5 C1v 5 S1
Ci 5 S2

E^
σ^
i^

Axial, cyclic
Cn
S2n

^ C^ n
E;
^ C^ n ; S^2n
E;

n
2n

n
2n

n 5 2,3,. . .
S6 5 C3i

C^ n
S^2n

Axial, non-cyclic
^ C^ n ; σ^ h ; S^n
Cnh
E;
^ C^ n ; nσ^ v

Cnv
E;

2n
2n

2n
(n 1 3)/2
(n 1 6)/2

For odd n
For even n

Axial, dihedral
^ C^ n ; nC^ 02
E;
Dn

2n

For odd n
For even n; D2 5 V
For odd n
For even n; D2h 5 Vh
Dnd 5 S2nv; D2d 5 Vd

C^ n ; σ^ h
C^ n ; σ^ v
C^ n ; C^ 02

Dnh

^ C^ n ; nC^ 02 ; S^n ; σ^ h ; nσ^ v
E;

4n

Dnd

^ C^ n ; nC^ 02 ; S^2n ; nσ^ d
E;

4n

(n 1 3)/2
(n 1 6)/2
n13
n16
n13

Axial, linear
CNv
DNh

^ C^ N ; Nσ^ v
E;
^ C^ N ; Nσ^ v ; S^N ; NC^ 02
E;

N

N

N
N

12

4

Rotations of the tetrahedron

24

8

Th 5 Ci 3 T

C^ 3 ; C^ 2 ðzÞ
C^ 3 ; C^ 2 ðzÞ; i^

24

5

Regular tetrahedron

3
C^ 3 ; S^4 ðzÞ

Cubica

T
Th
Td

^ 4C^ 3 ; 4C^ 23 ; 3C^ 2
E;
^ 4C^ 3 ; 4C^ 23 ; 3C^ 2 ; i;^ 4S^56 ; 4S^6 ; 3σ^ v
E;
^ 8C^ 3 ; 3C^ 2 ; 6S^4 ; 6σ^ d
E;

C^ n ; C^ 02 ; σ^ h
C^ n ; C^ 02 ; σ^ d
C^ N ; σ^ v
C^ N ; σ^ h ; C^ 0

2

(Continued)


Table 1.3 (Continued)
Symbol
O
Oh

Symmetry Operations
^ 8C^ 3 ; 6C^ 02 ; 6C^ 4 ; 3C^ 2
E;
^ 8C^ 3 ; 6C^ 2 ; 6C^ 4 ; 3C^ 2 ; i;^ 6S^4 ; 8S^6 ; 3σ^ h ; 6σ^ d

E;

Icosahedralb
2
I
E; 12C^ 5 ; 12C^ 5 ; 20C^ 3 ; 15C^ 2
Ih

fE; 12C^ 5 ; 12C^ 5 ; 20C^ 3 ; 15C^ 2 ;
2

Order h

Number of IR

Note

24

5

Rotations of the octahedron

48

10

Oh 5 Ci 3 O regular octahedron

C^ 3 ; C^ 4 ðzÞ

C^ 3 ; C^ 4 ðzÞ; i^

60

5

Rotations of the icosahedron

C^ 3 ; C^ 5 ðzÞ

120

10

Ih 5 Ci 3 I regular icosahedron

C^ 3 ; C^ 5 ðzÞ; i^

N
N

R3 5 SO(3)
O(3) 5 Ci 3 R3

3
^
i;^ 12S^10 ; 12S^10 ; 20S^6 ; 15σg

Rotational
R3

O(3)
SU(2)

^ C^ 2 ; C^ 3 ; . . .
E;

N
N
A group of unitary matrices of order 2 having determinant 5 1

C3 axis inclined at an angle 54.74 to the C2(z) axis.
Angle C3 2 C5(z) 5 37.38 .

a

b

Generators


Molecular Symmetry

9

Table 1.4 Representation Γ of the Group G
Term

Property

Conditions

Existence of transformation
matrices
Matrix elements

R^k .DðRk Þ

For kAh1,hi
Frequently real

Action on the basis (f1,. . .,fl)

Complex [D(Rk)]nm
P
R^k fm 5 n51 ẵDRk ịnm fn

Properties of matrices

D(Ri)D(Rj) 5 D(Rk)

Dimension of the
representation
Equivalent representations

l 5 dimension of D
B(Rk) 5 U21D(Rk)U

For kAh1,hi and mAh1,l i
When R^i R^j 5 R^k ; for kAh1,hi

U2unitary matrix, kAh1,hi

Table 1.5 Mulliken Classification of IRs
Representation

Name

A
B
E
T (or F)
G (or U)
H (or V)
A0
Av
Ag
Au
A1 (or Σ1 ), E1 (or Π)
A2 (or Σ2 ), E2 (or Δ)
A3, E3 (or Φ)

One-dimensional
One-dimensional
Two-dimensional
Three-dimensional
Four-dimensional
Five-dimensional
Symmetric
Antisymmetric
Even (gerade)

Odd (ungerade)
Symmetric
Antisymmetric

Property χα(Rk) 5 m
R^k
C^ n
C^ n

m
11
21

σ^ h
σ^ h

11
21
11
21
11
21

σ^ v
σ^ v
Specific properties

The reducible representation consists of IRs. The decomposition of a reducible
representation into its irreducible components may be written as follows
Γr 5

X
α

nα Γ α

ð1:10Þ

where nα is a multiplicity of inequivalent IRs (an integer). Their orders obey the
relationship
lr 5

X
α

nα lα

ð1:11Þ

where lα is the dimension of the α-th block Dα in the reducible representation
matrix Dbd.


10

A Handbook of Magnetochemical Formulae

The matrix elements of IRs satisfy the orthogonality relation (The Great
Orthogonality Theorem)
h

X

ẵA Rk ị ẵB Rk ފλσ 5 hðlα lβ Þ21=2 δαβ δμλ δνσ

ð1:12Þ

k51

The IRs are fully characterised by their characters; these are formed by the
traces of the transformation matrices
χα ðRk Þ 5 TrfDα ðRk Þg 5

l
X

ẵD Rk ịii ;

ẵfor k51; 2; . . . ; hŠ

ð1:13Þ

i

The characters of the irreducible representations possess these properties
1. The number of IRs of a group is equal to the number of classes in the group, Nirep5 Nclass.
2. In a given representation the characters of all matrices belonging to operations in the same
class are equal.
3. The orthogonality relationship
h
X

χα ðRk ÞU χβ ðRk Þ 5 hUδαβ

ð1:14Þ

k51

or, when the summation runs over classes of operation, then
N
class
X

gðRi ÞU χα ðRi ÞUχβ ðRi Þ 5 hUδαβ

ð1:15Þ

i51

where g(Ri)2 number of symmetry operations in the i-th class.
4. The sum of the squares of the characters under the identity of the IRs equals to the order
of the group
Nirep
X

ðχi ðEÞÞ2 5 h

ð1:16Þ

i

5. The multiplicity nα of the IR Γa in the reducible representation Γr is given by the formula
n 5

h
class
1X
1 NX
ẵ Rk ị U r Rk ị 5
gRi ÞU ½χα ðRi ފà Uχr ðRi Þ
h k51
h i51

ð1:17Þ

Of numerous applications of the group theory the following theorem is of a great
importance: the matrix element
^ ji
Pij 5 hΨi jPjΨ

ð1:18Þ


Molecular Symmetry

11

is non-zero only when the triple direct product of the involved IRs
^ 3 ΓðΨj Þ 5 Γr 5 Γ1 1 ?
ΓðΨi Þ 3 ΓðPÞ

ð1:19Þ

(which is a reducible representation Γr) contains the totally symmetric representation
Γ1 of the relevant point group. Alternatively, the double direct product of the IRs of
state vectors should contain a representation of the operator P^
X
^ 1?
nα Γα 5 ΓðPÞ
ð1:20Þ
ΓðΨi Þ 3 ΓðΨj Þ 5 Γr 5
α

The characters of the representations Γi3j, spanned by a direct product Γi 3 Γj, are
obtained by multiplying corresponding characters of the contributing representations
χi 3 j ðRk Þ 5 χi ðRk ÞU χj ðRk Þ

ð1:21Þ

The reducible representation is then decomposed by using the formula
h
1X
Rk ịU ẵ Rk ị
n 5
h k51 i3j

1:22ị

The direct product of IRs follows the rules compiled in Table 1.6. For degenerate representations the rules are more complex and specific for the given group
(Tables 1.7 and 1.8).
The direct product of a k-fold degenerate IR Γk with itself may be resolved into a

symmetric component, ½Γ2k Š; and an antisymmetric component ðΓ2k Þ
Γk 3 Γk 5 ½Γ2k Š 1 ðΓ2k Þ

1.2.3

ð1:23Þ

Rotation Group R3

A free atom belongs to the continuous rotation group R3. The IRs of the group R3
are labelled with the quantum number l. The spherical harmonic functions Yl,m  jl,mi
form the basis of the IR of R3 with the dimension 2l 1 1.
Table 1.6 Rules for the Direct Product of One-Dimensional IRs
1. For the representation A
A3A5A
A3B5B
A 3 Ek 5 Ek
A3T5T

2. For the representation B
B 3 B 5 Aa
B3E5E

3. For the lower indices
xg 3 xg 5 xg
xu 3 xg 5 xu
xu 3 xu 5 xg

4. For the upper indices
x0 3 x0 5 x 0

xv 3 x0 5 xv
xv 3 xv 5 x0

a

For all groups except D2 and D2h.

5. For the numerical indicesa
x1 3 x1 5 x1
x1 3 x2 5 x2
x2 3 x2 5 x1


12

A Handbook of Magnetochemical Formulae

Table 1.7 Irreducible Components of the Direct Product of Multi-dimensional IRs for Axial
Groups Gn 5 Cn, Cnh, Cnv, Dn, Dnh, Dnd, Sna
G3 1 S6
E

E
A1,(A2),E

G4 1 D2d 2 D4d
B
E

B

A
E

E
.
A1,(A2),B1,B2

G5 1 S10
E1
E2

E1
A1,(A2),E2
E1,E2

E2
.
A1,(A2),E1

G6 2 S6 2 D6d
B
E1
E2

B
A
E2
E1

E1

.
A1,(A2),E2
B1,B2,E1

E2
.
.
A1,(A2),E2

G7
E1
E2
E3

E1
A1,(A2),E2
E1,E3
E2,E3

E2
.
A1,(A2),E3
E1,E2

E3
.
.
A1,(A2),E1

G8 1 D4d 2 D8d

B
E1
E2
E3

B
A
E3
E2
E1

E1
.
A1,(A2),E2
E1,E3
B1,B2,E2

E2
.
.
A1,(A2),B1,B2
E1,E3

E3
.
.
.
A1,(A2),E2

The antisymmetric component of the direct product Γi 3 Γi is placed in parentheses; the rest is the symmetric

component. Points show symmetry equivalent result by means of the commutation property for the direct product,
Γ i 3 Γ j 5 Γ j 3 Γ i.

a

The operation of rotation through an angle α about the z-axis yields
R^α jl; mi 5 expðiαh
¯ 21 L^z Þjl; mi 5 expðimαÞjl; mi

ð1:24Þ

Such a rotation has a representation expressed through the 2l11 dimensional matrix
0

expẵil
B
0
Dl R ị 5 B
@ ...
0

1
0
...
0
C
expẵil 2 1ị . . .
0
C
A

...
...
...
0
0 expẵ2il

1:25ị

Therefore the character of this operation (a trace of the transformation matrix) is
a sum of the geometric series, i.e.
l R ị 5

sinẵ2l 1 1ị=2ị sinẵl 1 1=2ị
5
sin=2ị
sin=2ị

1:26ị


Molecular Symmetry

13

Table 1.8 Irreducible Components of the Direct Product of IRs for Special Groups
(a) Groups D2 and D2h
B1
B2
B3
E

B1
A
B3
B2
A

B2
.
A
B1
E

B3
.
.
A
E

E
.
.
.
A,B1,B2,B3

E2
.
.
A1,E4,(A2)
E1,E5

B1,B2,E2
E3,E5

E3
.
.
.
A1,(A2),B1,B2
E1,E5
E2,E4

(b) Groups D6d and S12
B
E1
E2
E3
E4
E5

B
A
E5
E4
E3
E2
E1

E1
.
A1,E2,(A2)

E1,E3
E2,E4
E3,E5
B1,B2,E4

E4
.
.
.
.
A1,E4,(A2)
E1,E3

E5
.
.
.
.
.
A1,E2,(A2)

(c) Linear groups (CNv,DNh)
Σ
Π
Δ

Σ
Σ
Π
Δ

Π
.
Σ1,(Σ2),Δ
Π,Φ

Δ
.
.
Σ1,(Σ2),Γ

(d) Cubic groups (T,Th,Td,O,Oh)
E
T1
T2

E
A1,(A2),E
T1,T2
T1,T2

T1
.
A1,E,(T1),T2
A2,E,T1,T2

T2
.
.
A1,E,(T1),T2

T2
.
A,H,(T2)
T1,G,H
T1,T2,G,H

G
.
.
A,G,H,(T11 T2)
T1,T2,G,H

(e) Icosahedral groups (I,Ih)
T1
T2
G
H

T1
A,H,(T1)
G,H
T2,G,H
T1,T2,G,H

H
.
.
.
A,G,2H,(T1 1 T2 1 G)

Then one can arrive at the progression
α
0
2π/2
2π/3

R^α
E^
C^ 2
C^ 3

4π/2

C^ 4

...

...

χl ðRα Þ
2l 1 1
(21)l
1, l 5 3k
0, l 5 3k 1 1
21, l 5 3k 1 2
(21)l/2, even 1
(21)(l21)/2, odd 1
...

14

A Handbook of Magnetochemical Formulae

The full orthogonal group is O(3) 5 Ci 3 R3 and has the full symmetry of the
sphere.
For some applications, two modes of the mathematical description of rotations
are met [66]. A vector r is written through components {ri} and basis elements {ei}
are organised into the row-matrix (ej and the column-matrix jr), respectively; the
row-column product is
0 1
r1
X
ð1:27Þ
ei ri 5 ðe1 e2 e3 Þ@ r2 A 5 ðejjrÞ
r5
i
r3
a. The passive interpretation means a passage from one basis (ej to another ðej (a rotation
of the reference frame alone; the observer is sitting at the object). The bar denotes the
new basis

ðej 5 ðejR

ð1:28Þ

When the rotation through an angle ϑ about the z-axis is concerned, then
(Figure 1.1)
0

ðej  ðe 1

e2

e 3 Þ 5 ðe1

e2

cos ϑ
e3 Þ@ sin ϑ
0

2 sin ϑ
cos ϑ
0

1
0
0A
1

ð1:29Þ

New components of a fixed vector are
jrÞ 5 R 21 jrÞ

ð1:30Þ

The new column jrÞ contains the components of a fixed vector relative to a new

(rotated) basis ðej: The basis vectors and the components of a fixed vector are said
to transform contragrediently
X
X
ej Rji ; ri ! r i 5
ðR21 Þij rj
ð1:31Þ
ei ! e i 5
j

j

The vector r is regarded as an invariant (remains fixed)
r 5 ðejjrÞ 5 ðejRR 21 jrÞ 5 ðejjrÞ

e2

e2

r
e1

θ

e1

Figure 1.1 Change of basis (passive interpretation).

ð1:32Þ

Molecular Symmetry

15

The last equation can be rewritten as a transformation of components of a fixed
r accompanying an inverse rotation of the basis
r 5 ðejjrÞ 5 ðejR21 RjrÞ 5 ðe0 jjr0 Þ

ð1:33Þ

ðej ! ðe0 j 5 ðejR21

ð1:34Þ

with

b. The active interpretation means an actual rotation of a vector r in a fixed basis; such a
mapping associates an image (new vector r0 ) to the old vector r: r!r0 . The components jr)
(column matrices) are transformed with the use of a rectangular matrix R as

jr0 Þ 5 RjrÞ

ð1:35Þ

This matrix equation shows how all vectors are sent into their images under a
common rotation. The new column jr0 ) contains the components of a rotated vector
relative to the fixed basis (ej. Then
r ! r0 5 ðejjr0 Þ 5 ðejRjrÞ 5 ðe0 jjrÞ

ð1:36Þ

The basis vectors can be regarded as images of the original set
ðej ! ðe0 j 5 ðejR

ð1:37Þ

The above transformations can be rewritten to the form
X

ri ! r 0i 5

ð1:38Þ

Rij rj

j

ei ! e0i 5

X

ej ðR21 Þji 5

j

X

Qij ej

ð1:39Þ

j

The latter transcription allows us to put the basis set to be transformed on the
extreme right with the matrix
Q 5 ðRT Þ21

ð1:40Þ

A tensorial set is any set of quantities which, when the basis is changed, must
be replaced by a new set:
a. contravariant set transforms cogrediently to the coordinates but contragrediently to the
basis vectors
Ti ! T 0i 5

X
j

Rij Tj

ð1:41Þ


16

A Handbook of Magnetochemical Formulae

b. covariant set transforms cogrediently to the basis vectors
Ti ! T 0i 5

X

ð1:42Þ

Qij Tj

j

(In the case of real-orthogonal transformations the distinction between the covariance
and contravariance disappears, since Q 5 (RT)21 5 R.)
Let us consider two vectors jX) and jY) related through a tensor T
jYÞ 5 TjXÞ

ð1:43Þ

After the active transformation with the matrix R
jY 0 Þ 5 RjYÞ;

jX 0 Þ 5 RjXÞ

ð1:44Þ

we get
jY 0 Þ 5 ðRTR21 ÞjX 0 Þ 5 T 0 jX 0 Þ

ð1:45Þ

T 0 5 RTR21

ð1:46Þ

However, with the passive transformation we get
jY Þ 5 R21 jYÞ;

jX Þ 5 R21 jXÞ;

jXÞ 5 RjX Þ

jY Þ 5 ðR21 TRÞjX Þ 5 T jX Þ

ð1:47Þ
ð1:48Þ

and then
T 5 R21 TR

ð1:49Þ

In the active interpretation, the nine elements of T transform according to
Tij ! T 0ij 5

X

Rik Tkl ðR21 Þlj 5

k;l

X

Rik Qjl Tkl

ð1:50Þ

k;l

This means that the second-rank tensor has one degree of contravariance and
one of covariance. In general, the transformation law of the form
Ti...j... ! T 0i...j... 5

X

Rik . . . Qjl . . . Tk...l...

ð1:51Þ

k;l...

implies that the tensorial set of rank (r 1 q) has r-degrees of contravariance (matrices R
occur r-times) and q-degrees of covariance (matrices Q occurring q-times).


Molecular Symmetry

17

More complex operations R^ of the point group G are expressed through a set of
Euler angles (α,β,γ); the sequence of the Euler rotations is:
a. rotation about the z-axis through an angle 0 # α , 2π, giving rise to {x0 ;y0 ;z0 5 z};
b. rotation about the y0 -axis through an angle 0 # β , π, yielding {xv;yv5 y0 ;zv};

c. rotation about the zv-axis through an angle 0 # γ, 2π, yielding {xw;yw;zw5 zv}.

The right-handed coordinate system is used and the sense of the rotation is
that positive rotations carry a right-handed screw forward along the rotation axis
(a corkscrew advancing along the positive direction of the rotation axis) À Figure 1.2.
(Some other conventions can be met in the literature.)
The operator for the active rotation is
D^ Rz ðϑÞ 5 expð1iϑh
¯ 21 L^z Þ

ð1:52Þ

whereas the inverse operator is
B^Rz ðϑÞ 5 expð2iϑh
¯ 21 L^z Þ

ð1:53Þ

When several rotations are applied, the components of a fixed vector r transform
contragrediently as follows
jrÞ ! jr0 Þ 5 R2 R1 jrÞ

ð1:54Þ

ðe0 j 5 ðejR121 R221 5 ðejðR2 R1 Þ21

ð1:55Þ

A sequence of rotations, each defined with respect to the floating (temporary)
basis, is equivalent to a sequence applied in a reverse order with the rotations defined

in the fixed basis.
The rotation operator R^ corresponding to an active rotation through the Euler
angles in the temporary coordinates above (primed) is
^
¯ 21 L^y0 Þexpð1iαh
¯ 21 L^z Þ
RðγβαÞ
5 expð1iγh
¯ 21 L^zv Þexpð1iβh

y'

y
y

e1

θ

ð1:56Þ

y

Image r'

e1

θ

x'

x Object r

x

x

Figure 1.2 Sense of the corkscrew rotation. The z-axis is perpendicular to the x-y plane and
pointing to the reader. Left À active interpretation, the observer
is sitting at the frame;
right À passive interpretation, the observer residues at the object.


18

A Handbook of Magnetochemical Formulae

Alternatively the fixed rotation axes {x;y;z} are used instead of the temporary
ones and then the elementary rotation operators act in a reversed order [49]
^
RðαβγÞ
5 expð1iαh
¯ 21 L^z Þexpð1iβh
¯ 21 L^y Þexpð1iγh
¯ 21 L^z Þ

ð1:57Þ

The convention used to generate representation matrices is the active-fixed
^
interpretation RðαβγÞ;

i.e. step 12γ, step 2 2 β and step 3 2 α. This matches to
those in Tinkham [64] and has been used by Pyykkoă and Toivonen [81].
^
The results of the action of the operations RðαβγÞ
on the set of spherical harmonic
functions are obtained as follows
^
Rịjl;
mi 5

1l
X
m0 52l

ẵDl ịm0 ;m jl; m0 i

1:58ị

Here the 2l11 dimensional transformation (Wigner rotation) matrix has the
elements [49]
ẵDl ịm0 ;m

é Ã
0 ^
^
5 Yl;m
0 RYl;m dV 5 hl; m jRðαβγÞjl; mi
21 ^
0
¯ Lz Þexpð2iβh

¯ 21 L^y Þexpð2iγh
¯ 21 L^z Þjl; mi
5 hl; m jexp2ih
0
5 exp2im ịexp2imị
max
X
0
ẵl 1 mị!l 2 mị!l 1 m0 Þ!ðl 2 m0 Þ!Š1=2
3
ð21Þm 2m1κ
ðl 2 m0 2 κÞ!ðl 1 m 2 ị!!m0 2 m 1 ị!
5min
0

0

3 ẵcos=2ị2l1m2m 22 ½sinðβ=2ފ2κ1m 2m
ð1:59Þ
with the limiting values
κmin 5 maxf0; l2m0 g

ð1:60Þ

κmax 5 minfl2m0 ; l1mg

ð1:61Þ

that secure non-negative arguments of the factorials. For some frequent cases
there is

ẵDl ; 5 0; ịm0 ;m 5 m0 ;m exp2im0 2imị

1:62ị

ẵDl ; 5 ; ịm0 ;m 5 ð21Þ2l1m δm0 ; 2m expð2im0 α 2 imγÞ

ð1:63Þ

For the half-integral angular momentum, the functions jl,mi need to be replaced
by jj,mi thus losing the meaning of the spherical harmonic functions; however, it
holds true that
^
Rịjj;
mi 5

1j
X
m0 52j

ẵDj ịm0 ;m jj; m0 i

ð1:64Þ


Molecular Symmetry

19

Rotation by an angle 2π yields
^

Rð2πÞjj;
mi 5 expð2i2πmÞjj; mi

ð1:65Þ

so that for half integral m 5 6 n/2 5 6 1/2, 6 3/2,. . . we arrive at
^
R2ịjj;
6n=2i 5 expẵ2i26 n=2ịjj; mi 5 ẵcosnị 7 i sinnịjj; mi
5 2jj; mi

1:66ị

But
^
R4ịjj;
6 n=2i 5 expẵ2i46n=2ịjj; mi
5 ẵcos2nị 7 i sin2nịjj; mi 5 1jj; mi

ð1:67Þ

Therefore the wave function is not a single-valued function of position within
the ordinary group; this dilemma is solved by the concept of the double groups.

1.2.4

Representation Matrices for a Group

The matrices that transform the Cartesian coordinates under the symmetry operations
are compiled in Table 1.9.

Every element of the group G can be expressed as a product containing only
integer powers (positive or negative) of the group generators, g^i . These are listed
in Table 1.10. Irreducible matrix representations may be found by multiplying the
generator matrices Dðg^i Þ according to the relationships given for the symmetry
operations of the point group. Non-degenerate representations are contained in the
character tables. Other groups can be obtained either through the isomorphism or
through the direct product groups.
For cyclic groups (Cn or S2n) there is
Operation

C^ n

2
C^ n

...

k
C^ n

...

n
C^ n 5 E^

Representation

ω1

ω2

...

ωk

...

ωn 5 1

where ω is an n-th root of unity, i.e.
ωk 5 expði2πk=nÞ 5 ðωà Þ2k ;

k 5 1; 2; . . . ; n

ð1:68Þ

The doubly degenerate representations occur in conjugate pairs:
DEk ðC^ n Þ 5
k



ωk
0

0
ðωk ÞÃ


ð1:69Þ