A Handbook of Magnetochemical Formulae

982 Pages·2012·34.563 MB·English

by  Boca, Roman (Auth.)

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
suchaquantity.
Tensor product. The tensor product, denoted by (cid:1), may be applied in different
contextstovectors,matrices,tensors,vectorspaces,etc.Ineachcasethesignificance
ofthesymbolisthesame:themostgeneralbilinearoperation.
A representative case is the Kronecker product of any two rectangular arrays,
consideredasmatrices
0 1 0 1
b a b a b a b
1 1 1 2 1 3 1
B C B C
B@bb2CA(cid:1) ða1 a2 a3Þ5B@aa1bb2 aa2bb2 aa3bb2CA ð1:1Þ
3 1 3 2 3 3 3
b a b a b a b
4 1 4 2 4 3 4
Here,resultantrank52,resultantdimension(4,3)5433512.Therankdenotes
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
usedinitstensorsenseandshouldnotbeinterpretedasmatrixrank.
(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
expressedexplicitly.)
Outer product. Given a tensor A with rank a and dimensions (i ,...,i ), and a
1 a
tensorBwith rank b and dimensions (j ,..., j ),their outer product C5A(cid:1)B has
1 b
rank a1b and dimensions (k1,...,ka1b) which are the i dimensions followed by
thejdimensions.
For example, if A has rank 3 and dimensions (3, 5, 7) and B has rank 2 and
dimensions(10,100),theirouterproductChasrank5anddimensions(3,5,7,10,100).
Inotherwords,outerproductontensors5tensorproduct.
AHandbookofMagnetochemicalFormulae.DOI:10.1016/B978-0-12-416014-9.00001-X
©2012ElsevierInc.Allrightsreserved.
4 AHandbookofMagnetochemicalFormulae
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
oftheouterproductisarank-2tensorwithdimension(M,N).
Inner product. The result of an inner product between two tensors of rank-q and
rank-risthegreaterof(q1r22)and0.
(cid:1) Theinnerproductoftwomatriceshasthesamerankastheouterproduct(ortensorproduct)
oftwovectors.
(cid:1) TheinnerproductoftwomatricesAwithdimensions(I,M)andBwithdimensions(M,J)is
XM
C 5 A B whereiAf1;...; Ig and jAf1;...; Jg ð1:2Þ
ij im mj
m
Directproductoftwomatricesis
(cid:2) (cid:3) (cid:2) (cid:3)
a a b b
A3B5 11 12 3 11 12
a a b b
0 21 22 21 22 1
a b a b a b a b
B 11 11 11 12 12 11 12 12C ð1:3Þ
Ba b a b a b a b C
5B 11 21 11 22 12 21 12 22C6¼B3A
@a b a b a b a b A
21 11 21 12 22 11 22 12
a b a b a b a b
21 21 21 22 22 21 22 22
Theresultingrankisnot4butonly2,sothatthisisakindoftheinnerproduct.
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 ). If everything
n
!
in the universe undergoes a rotation (e.g. the displacement vector r is transformed
with the rotation matrix R to!r05R!r), then any vector V! must be transformed in
thesameway(V!05RV!).
The polar vector is a contravariant vector (a tensor of contravariant rank one).
Examples of the polar vectorsare: position (displacement) vector!r; velocity!v and
linearmomentum!p :
Under inversion through the origin (i^5S^ ), the true vector alters its sign: V!
2
goesto 2V!:
1.1.3 Pseudovector (Axial Vector)
!
Thepseudovector(syn.axialvector,P)transformsunderrotationsaccordingtothe
formula
! !
P5ðdetRÞðRPÞ ð1:4Þ
MolecularSymmetry 5
Ittransformsundertheproperrotationslikeapolarvector,butundertheimproper
rotations(rotationsofmirrorimage,S^ 5σ^ C^ )gainsanadditionalsignflip.
n h n
Examples of the axial vector are: angular momentum!l ; magnetic induction !B;
torqueandvorticity.Forinstance,theangularmomentumisdefinedthroughthecross
product!l 5 !r 3!p :Oninversion:!r !2!r and!p !2p^;while!l !!l :
Propertiesforadditionandmultiplication
!P 6!P 5!P; a!P 5 !P; V! 6 V! 5 !V; aV! 5 !V ð1:5Þ
1 2 1 1 2 1
!P 6V! 2undefined ð1:6Þ
1 2
Propertiesforthecrossproduct
V! 3V! 5 !P; !P 3!P 5 !P; V! 3!P 5 !V; !P 3V! 5 !V ð1:7Þ
1 2 1 2 1 2 1 2
A common way of introducing a pseudovector is by taking the cross product of
polarvectors.Forinstance,themagneticinductionis
! ! !
B 5 r 3 A ð1:8Þ
1.2 Point Groups
1.2.1 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 (cid:3) 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
themolecule) [61284].
A brief summary of the properties of the symmetry point groups is presented in
Table1.2.Someadditionaldefinitionsfollow:
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;theyhavethesamedefiningrelationsandthesamemultiplicationandcharacter
tables;
3. ifthere are two groups G and G , having only their identity in common and possessing
a b
elements R^ (for a51,...,h ) and R^ (for b51,...,h ), then the direct product group
a a b b
G5G 3G isdefinedasthesetofalldistinctelementsR^ R^ 5R^ R^ forallaandb.
a b a b b a
6 AHandbookofMagnetochemicalFormulae
Table1.1 SymmetryOperations
Symmetry Property Inverse
OperationR^k OperatorR^k21
E^orI^ Identity;rotationthroughanangle2πinsinglegroups; E^
rotationthroughanangle4πindoublegroups
C^ Rotationaboutangle2π/n C^n2k forC^k
n n n
σ^ðσ^ ;σ^ ;σ^ Þ Mirrorplane(horizontal,vertical,diagonal) σ^
h v d
^ ^ ^
iorS Inversecentre i
S^ 5σ^2 C^ Rotationfollowedbythemirrorplane S^n2k forS^k
n h n n n
Q^ orR^ Rotationbyanangle2πbutdifferingfromtheidentity Q^
^ ^
operation(Q6¼E);applicabletodoublegroups
C^kQ^ Arotationthroughϕ52π12πk/n;applicabletodouble C^n2k
n n
groups
Table1.2 ElementaryTermsintheSymmetryPointGroupTheory
Term Property Note
(a)PropertiesofthesymmetrygroupGoforderh
Existenceofaproduct R^R^ 5R^ ðR^;R^;R^ ÞAG
i j k i j k
Associativelaw R^ðR^R^ Þ5ðR^R^ÞR^
i j k i j k
Existenceofidentity R^ E^5E^R^ 5R^
Existenceofinversion R^kR^215R^2k1R^ 5k E^ ðR^ ;R^21ÞAG
k k k k k k
(b)ClassTofthegroupG
Asetofoperatorsobeying R^ 5R^21R^R^ ðR^;R^ÞAT,R^ AG
i s j s i j s
The symmetry point groups along with their important characteristics are classi-
fied in Table 1.3 (Crystals have no C , C , C , etc. axes and this fact restricts the
5 7 8
possiblegroupsto32pointgroups.)
1.2.2 Representations
A set of matrices D(R ), transforming coordinates in the same way as the symmetry
k
operator R^ ; forms a representation Γ of the group G (Table 1.4). The irreducible
k
representations (IRs) used to be denoted by two conventions: according to Mulliken
(Table1.5)oraccordingtoBethe(simplyΓ ,Γ ,etc.).
1 2
The representation is reducible, Γ, when by the same similarity transformation
r
Uablock-diagonalformofmatricesD(R )isobtained
k
⎛ ⎛
U−1D(R)U = D (R) = ⎜D1 0 ⎜ (1.9)
k bd k ⎝ ⎝
0 D2
Table1.3 SymmetryPointGroupsofMoleculesandAtoms
Symbol SymmetryOperations Orderh NumberofIR Note Generators
Non-axial
C E^ 1 1 C 5C E^
1 1
C E^;σ^ 2 2 C 5C 5C 5S σ^
s h s 1h 1v 1
C E^;i^ 2 2 C 5S i^
i i 2
Axial,cyclic
C E^;C^ n n n52,3,... C^
n n n
S E^;C^ ;S^ 2n 2n S 5C S^
2n n 2n 6 3i 2n
Axial,non-cyclic
C E^;C^ ;σ^ ;S^ 2n 2n C^ ;σ^
nh n h n n h
C E^;C^ ;nσ^ 2n (n13)/2 Foroddn C^ ;σ^
nv n v n v
(n16)/2 Forevenn
Axial,dihedral
D E^;C^ ;nC^0 2n (n13)/2 Foroddn C^ ;C^0
n n 2 n 2
(n16)/2 Forevenn;D 5V
2
D E^;C^ ;nC^0;S^ ;σ^ ;nσ^ 4n n13 Foroddn C^ ;C^0;σ^
nh n 2 n h v n 2 h
n16 Forevenn;D 5V
2h h
D E^;C^ ;nC^0;S^ ;nσ^ 4n n13 D 5S ;D 5V C^ ;C^0;σ^
nd n 2 2n d nd 2nv 2d d n 2 d
Axial,linear
CNv E^;C^N;Nσ^v N N C^N;σ^v
DNh E^;C^N;Nσ^v;S^N;NC^02 N N C^N;σ^h;C^02
Cubica
T E^;4C^ ;4C^2;3C^ 12 4 Rotationsofthetetrahedron C^ ;C^ ðzÞ
3 3 2 3 2
T E^;4C^ ;4C^2;3C^ ;i^;4S^5;4S^ ;3σ^ 24 8 T 5C 3T C^ ;C^ ðzÞ;i^
h 3 3 2 6 6 v h i 3 2
T E^;8C^ ;3C^ ;6S^ ;6σ^ 24 5 Regulartetrahedron C^ ;S^3ðzÞ
d 3 2 4 d 3 4
(Continued)
Table1.3 (Continued)
Symbol SymmetryOperations Orderh NumberofIR Note Generators
O E^;8C^ ;6C^0;6C^ ;3C^ 24 5 Rotationsoftheoctahedron C^ ;C^ ðzÞ
3 2 4 2 3 4
O E^;8C^ ;6C^ ;6C^ ;3C^ ;i^;6S^ ;8S^ ;3σ^ ;6σ^ 48 10 O 5C 3Oregularoctahedron C^ ;C^ ðzÞ;i^
h 3 2 4 2 4 6 h d h i 3 4
Icosahedralb
I E;12C^ ;12C^2;20C^ ;15C^ 60 5 Rotationsoftheicosahedron C^ ;C^ ðzÞ
5 5 3 2 3 5
I fE;12C^ ;12C^2;20C^ ;15C^ ; 120 10 I 5C 3Iregularicosahedron C^ ;C^ ðzÞ;i^
h 5 5 3 2 h i 3 5
i^;12S^ ;12S^3 ;20S^ ;15σ^g
10 10 6
Rotational
R E^;C^ ;C^ ;... N N R 5SO(3)
3 2 3 3
O(3) N N O(3)5C 3R
i 3
SU(2) Agroupofunitarymatricesoforder2havingdeterminant51
aC3axisinclinedatanangle54.74(cid:4)totheC2(z)axis.
bAngleC 2C(z)537.38(cid:4).
3 5
MolecularSymmetry 9
Table1.4 RepresentationΓoftheGroupG
Term Property Conditions
Existenceoftransformation R^ .DðR Þ ForkAh1,hi
k k
matrices
Matrixelements Complex[D(R )] Frequentlyreal
P k nm
Actiononthebasis(f ,...,f) R^ f 5 ½DðR Þ(cid:5) f ForkAh1,hiandmAh1,li
1 l k m n51 k nm n
Propertiesofmatrices D(R)D(R)5D(R ) WhenR^R^ 5R^ ;forkAh1,hi
i j k i j k
Dimensionofthe l5dimensionofD
representation
Equivalentrepresentations B(R )5U21D(R )U U2unitarymatrix,kAh1,hi
k k
Table1.5 MullikenClassificationofIRs
Representation Name Propertyχα(Rk)5m
^
R m
k
A One-dimensional C^ 11
n
B One-dimensional C^ 21
n
E Two-dimensional
T(orF) Three-dimensional
G(orU) Four-dimensional
H(orV) Five-dimensional
A0 Symmetric σ^ 11
h
Av Antisymmetric σ^ 21
h
A Even(gerade) 11
g
A Odd(ungerade) 21
u
A (orΣ1),E (orΠ) Symmetric σ^ 11
1 1 v
A (orΣ2),E (orΔ) Antisymmetric σ^ 21
2 2 v
A ,E (orΦ) Specificproperties
3 3
The reducible representation consists of IRs. The decomposition of a reducible
representationintoitsirreduciblecomponentsmaybewrittenasfollows
X
Γr5 nαΓα ð1:10Þ
α
where nα is a multiplicity of inequivalent IRs (an integer). Their orders obey the
relationship
X
lr5 nαlα ð1:11Þ
α
where lα is the dimension of the α-th block Dα in the reducible representation
matrixD .
bd
10 AHandbookofMagnetochemicalFormulae
The matrix elements of IRs satisfy the orthogonality relation (The Great
OrthogonalityTheorem)
Xh
½AαðRkÞ(cid:5)μν½BβðRkÞ(cid:5)λσ5hðlαlβÞ21=2δαβδμλδνσ ð1:12Þ
k51
The IRs are fully characterised by their characters; these are formed by the
tracesofthetransformationmatrices
Xlα
χαðRkÞ5TrfDαðRkÞg5 ½DαðRkÞ(cid:5)ii; ½for k51;2;...; h(cid:5) ð1:13Þ
i
Thecharactersoftheirreduciblerepresentationspossesstheseproperties
1. ThenumberofIRsofagroupisequaltothenumberofclassesinthegroup,N 5N .
irep class
2. Inagivenrepresentationthecharactersofallmatricesbelongingtooperationsinthesame
classareequal.
3. Theorthogonalityrelationship
Xh
χαðRkÞUχβðRkÞ5hUδαβ ð1:14Þ
k51
or,whenthesummationrunsoverclassesofoperation,then
NXclass
gðRiÞUχαðRiÞUχβðRiÞ5hUδαβ ð1:15Þ
i51
whereg(R)2numberofsymmetryoperationsinthei-thclass.
i
4. ThesumofthesquaresofthecharactersundertheidentityoftheIRsequalstotheorder
ofthegroup
XNirep
ðχðEÞÞ25h ð1:16Þ
i
i
5. ThemultiplicitynαoftheIRΓainthereduciblerepresentationΓrisgivenbytheformula
1Xh 1NXclass
nα5h ½χαðRkÞ(cid:5)(cid:6)UχrðRkÞ5 h gðRiÞU½χαðRiÞ(cid:5)(cid:6)UχrðRiÞ ð1:17Þ
k51 i51
Ofnumerousapplicationsofthegrouptheorythefollowingtheoremisofagreat
importance:thematrixelement
P 5hΨjP^jΨi ð1:18Þ
ij i j
MolecularSymmetry 11
isnon-zeroonlywhenthetripledirectproductoftheinvolvedIRs
ΓðΨÞ3ΓðP^Þ3ΓðΨÞ5Γ 5Γ 1? ð1:19Þ
i j r 1
(whichisa reducible representation Γ) contains the totallysymmetric representation
r
Γ of the relevant point group. Alternatively, the double direct product of the IRs of
1
^
statevectorsshouldcontainarepresentationoftheoperatorP
X
ΓðΨiÞ3ΓðΨjÞ5Γr5 nαΓα5ΓðP^Þ1? ð1:20Þ
α
The characters of the representations Γi3j, spanned by a direct product Γi3Γj, are
obtainedbymultiplyingcorrespondingcharactersofthecontributingrepresentations
χ ðR Þ5χðR ÞUχðR Þ ð1:21Þ
i3j k i k j k
Thereduciblerepresentationisthendecomposedbyusingtheformula
1Xh
nα5 h χi3jðRkÞU½χαðRkÞ(cid:5)(cid:6) ð1:22Þ
k51
The direct product of IRs follows the rules compiled in Table 1.6. For degen-
erate representations the rules are more complex and specific for the given group
(Tables 1.7 and 1.8).
The directproduct ofa k-fold degenerate IR Γ withitselfmay beresolvedintoa
k
symmetriccomponent,½Γ2(cid:5);andanantisymmetriccomponentðΓ2Þ
k k
Γ 3Γ 5½Γ2(cid:5)1ðΓ2Þ ð1:23Þ
k k k k
1.2.3 Rotation Group R
3
A free atom belongs to the continuous rotation group R . The IRs of the group R
3 3
arelabelledwiththequantumnumberl.ThesphericalharmonicfunctionsY (cid:7)jl,mi
l,m
formthebasisoftheIRofR withthedimension2l11.
3
Table1.6 RulesfortheDirectProductofOne-DimensionalIRs
1.FortherepresentationA 2.FortherepresentationB
A3A5A B3B5Aa
A3B5B B3E5E
A3E 5E
k k
A3T5T
3.Forthelowerindices 4.Fortheupperindices 5.Forthenumericalindicesa
x 3x 5x x03x05x0 x 3x 5x
g g g 1 1 1
x 3x 5x xv3x05xv x 3x 5x
u g u 1 2 2
x 3x 5x xv3xv5x0 x 3x 5x
u u g 2 2 1
aForallgroupsexceptD2andD2h.
12 AHandbookofMagnetochemicalFormulae
Table1.7 IrreducibleComponentsoftheDirectProductofMulti-dimensionalIRsforAxial
GroupsG 5C ,C ,C ,D ,D ,D ,S a
n n nh nv n nh nd n
G 1S E
3 6
E A ,(A ),E
1 2
G 1D 2D B E
4 2d 4d
B A .
E E A ,(A ),B ,B
1 2 1 2
G 1S E E
5 10 1 2
E A ,(A ),E .
1 1 2 2
E E E A ,(A ),E
2 1, 2 1 2 1
G 2S 2D B E E
6 6 6d 1 2
B A . .
E E A ,(A ),E .
1 2 1 2 2
E E B ,B ,E A ,(A ),E
2 1 1 2 1 1 2 2
G E E E
7 1 2 3
E A ,(A ),E . .
1 1 2 2
E E ,E A ,(A ),E .
2 1 3 1 2 3
E E ,E E ,E A ,(A ),E
3 2 3 1 2 1 2 1
G 1D 2D B E E E
8 4d 8d 1 2 3
B A . . .
E E A ,(A ),E . .
1 3 1 2 2
E E E ,E A ,(A ),B ,B .
2 2 1 3 1 2 1 2
E E B ,B ,E E ,E A ,(A ),E
3 1 1 2 2 1 3 1 2 2
aTheantisymmetriccomponentofthedirectproductΓi3Γiisplacedinparentheses;therestisthesymmetric
component.Pointsshowsymmetryequivalentresultbymeansofthecommutationpropertyforthedirectproduct,
Γi3Γj5Γj3Γi.
Theoperationofrotationthroughanangleαaboutthez-axisyields
R^αjl;mi5expðiα¯h21L^zÞjl;mi5expðimαÞjl;mi ð1:24Þ
Sucharotationhasarepresentationexpressedthroughthe2l11dimensionalmatrix
0 1
exp½ilα(cid:5) 0 ... 0
DlðRαÞ5BB@ .0.. exp½ið.l.2. 1Þα(cid:5) ...... .0.. CCA ð1:25Þ
0 0 0 exp½2ilα(cid:5)
Thereforethe characterofthisoperation(atrace ofthetransformationmatrix) is
asumofthegeometricseries,i.e.
sin½ð2l11Þðα=2Þ(cid:5) sin½ðl11=2Þα(cid:5)
χlðRαÞ5 sinðα=2Þ 5 sinðα=2Þ ð1:26Þ