Quantum Chemistry of Solids / 13-Symmetry and Localization of Crystalline Orbitals

3.1 Translation and Space Symmetry of Crystalline Orbitals. Bloch Functions

57

Those elements of the point-symmetry group F0 of the reciprocal lattice that transform a point k into itself or into some equivalent k point form the wavevector point-symmetry group Fk F0. For example, for the direct face-centered cubic lattice (F0 = Oh) BZ points Γ (0,0,0), X (1/2,1/2,0), L(1/2,1/2,1/2),W(1/4,1/2,3/4) (see Fig. 3.2) have point symmetry groups Oh, D4h, D3d, D2d, respectively. The point group Fk of the k vectors F, Z, M, A (Fig. 3.7) for a simple tetragonal lattice coincides with the D4h point group of the tetragonal lattice itself; the point group D2h of the k vectors X and R is a subgroup of D4h.

Fig. 3.7. Primitive unit cell and Brillouin zone for simple tetragonal lattice

All the mentioned points of the BZ are called points of symmetry. By definition, k is a point of symmetry if there exists a neighborhood of k in which no point k has the same symmetry group Fk and Fk Fk. The Γ (k = 0) point of the Brillouin zone is usually a symmetry point; exceptions here are the space groups of the crystallographic classes Cs, Cnv, Cn. All the other symmetry points are situated on the surface of the Brillouin zone and are usually denoted in a more or less unique way by capital Roman letters as in Fig. 3.7 for a simple tetragonal lattice. Kovalev [31], however, used ordinal numbers to denote k vectors.

If, in any su ciently small neighborhood of k, there is a line (plane) of points passing through k and having the same point group Fk then k is said to be on a line (plane) of symmetry. The lines of symmetry are denoted both by Roman (on the surface of the Brillouin zone) and Greek (inside the Brillouin zone) capital letters. A symmetry line may be denoted by two symmetry points at the ends of this line: Λ − Γ L(C3v ), Σ − Γ M (C2v ), ∆ − Γ X(C2v ) (Fig. 3.2a); the corresponding Fk groups are in parentheses. It is evident that the point groups C2v of di erent wavevectors, being isomorphic to each other, do not coincide for all symmetry lines. For example, the second-order symmetry axis C2 is along the X-coordinate axis for the Γ X line and along the XY symmetry axis for the Γ M line.

Tables 3.1, 3.2 and 3.3 of k vector types for BZ symmetry points and symmetry lines (space groups 221, 225 and 136) are taken from the site [16]. In fact, for all the space groups referring to the same crystal class and the same lattice type the k vector

58 3 Symmetry and Localization of Crystalline Orbitals

types are the same (groups 221 – 224 with simple cubic lattice, groups 225 – 228 with face-centered cubic lattice and groups 123 – 138 for a simple tetragonal lattice). This follows from the fact that in reciprocal space only the point-symmetry operations R transform the k vectors. In the space-group sets mentioned the di erence in symmetry operations appears only in the direct lattice where the improper translations are di erent. The columns labeled CDML and ITA in Tables 3.1–3.3 mean the k points notations used in [17] (accepted in solid-state theory) and in [16] as Wycko positions of the reciprocal lattice space group. The latter notations are practically not used as the k points labels. In the next section we discuss the generation of the space-group irreps from those of translation group.

Table 3.1. The k-vector types of group 221 [Pm3m] (Table for arithmetic crystal class m3mP, Pm3m-Oh1 (221) to Pn3m-Oh4 (224)). Reciprocal-space group (Pm3m)-Oh1 (221)

k-vector label		Wycko position					Parameters
	CDML			ITA			ITA
Γ	0, 0, 0	1	a				0, 0, 0
				m3m
R	1/2, 1/2, 1/2	1	b				1/2, 1/2, 1/2
				m3m
M	1/2, 1/2, 0	3	c	4/mm.m			1/2, 1/2, 0
X	0, 1/2, 0	3	d	4/mm.m			0, 1/2, 0
∆	0,u,0	6	e	4m.m			0, y, 0 : 0 < y < 1/2
T	1/2,1/2,u	6	f	4m.m			1/2, 1/2, z : 0 < z < 1/2
Λ	u,u,u	8	g	.3m			x, x, x : 0 < x < 1/2
Z	u,1/2,0	12	h	mm2..			x, 1/2, 0 : 0 < x < 1/2
Σ	u,u,0	12	i	m.m2			x, x, 0 : 0 < x < 1/2
S	u,1/2,u	12	j	m.m2			x, 1/2, x : 0 < x < 1/2

3.1.3 Stars of Wavevectors. Little Groups. Full Representations of Space Groups

The representation theory of space groups uses a theorem that the translation group T is an invariant subgroup of G (T G see Sect. 2.1.2). Therefore, the little-group method [13] may be used for the generation of irreps of space group G from irreps of translation subgroup T. As was shown in Sect. 3.1.2, the one-dimensional irreps of translation group T transform under point-symmetry operations R of space group G according to relation (3.25) i.e. the Bloch function with wave vector k transforms to a Bloch function with wavevector Rk.

Let us suppose that D(g) is an irrep of G acting in a space L of dimension n. The operators D(g) for g = ta T form a rep of T that is in general reducible. Let it contain irreps of T characterized by the vectors k = k1, k2, . . . , kn. Therefore a basis can be found in space L that consists of Bloch functions ϕ(k1, r), . . . , ϕ(kn, r). With respect to this basis, the elements of T are represented in the irrep D(g) by diagonal matrices with elements exp(−ikpa), p = 1, 2, . . . , n. The fact that D(g) is a rep of G implies that if we start with ϕ(k, r) and generate the Bloch functions

3.1 Translation and Space Symmetry of Crystalline Orbitals. Bloch Functions

59

Table 3.2. The k-vector types of group 225 [Fm3m] (Table for arithmetic crystal class m3mF, Fm3m−Oh5 (225) to Fd3c−Oh8 (228)). Reciprocal-space group (Im3m)−Oh9 (229)

k-vector label

Wycko position

Parameters

CDML

ITA

ITA

Γ

0, 0, 0

2

a

0, 0, 0

m3m

X

1/2, 0, 1/2

6

b

4/mm.m

0, 1/2, 0

L

1/2, 1/2, 1/2

8

c

1/4, 1/4, 1/4

.3m

W

1/2, 1/4, 3/4

12

d

1/4, 1/2, 0

4m.2

∆

u, 0, u

12

e

4m.m

x, 0, 0 : 0 < x < 1/2

Λ

u, u, u

16

f

.3m

x, x, x : 0 < x < 1/4

V

1/2, u, 1/2 + u

24

g

mm2..

x, 1/2, 0 : 0 < x < 1/4

Σ

u, u, 2u

24

h

m.m2

x, x, 0 : 0 < x < 3/8

S

1/2 + u, 2u, 1/2 + u

24

h

m.m2

x, 1/2, x : 0 < x < 1/8

S S1[KM]

24

h

m.m2

x, x, 0 : 3/8 < x < 1/2

SM + S1[GMM]

24

h

m.m2

x, x, 0 : 0 < x < 1/2

Q

1/2, 1/4 + u, 3/4 − u

48

i

..2

1/4, 1/2 − y, y : 0 < y < 1/4

Table 3.3. The k-vector types

of

group 136 [P42/mnm] (Table for arithmetic crys-

tal class 4/mmmP, P4/mmm–D41h (123) to P42/ncm–D416h (138)). Reciprocal-space group (P4/mmm)-D41h (123)

k-vector label		Wycko position			Parameters
	CDML			ITA	ITA
Γ	0, 0, 0	1	a	4/mmm	0, 0, 0
Z	0, 0, 1/2	1	b	4/mmm	0, 0, 1/2
M	1/2, 1/2, 0	1	c	4/mmm	1/2, 1/2, 0
A	1/2, 1/2, 1/2	1	d	4/mmm	1/2, 1/2, 1/2
R	0, 1/2, 1/2	2	e	mmm.	0, 1/2, 1/2
X	0, 1/2, 0	2	f	mmm.	0, 1/2, 0
Λ	0, 0, u	2	g	4mm	0, 0, z : 0 < z < 1/2
V	1/2, 1/2, u	2	h	4mm	1/2, 1/2, z : 0 < z < 1/2
W	0, 1/2, u	4	i	2mm.	0, 1/2, z : 0 < z < 1/2
Σ	u, u, 0	4	j	m.2m	x, x, 0 : 0 < x < 1/2
S	u, u, 1/2	4	k	m.2m	x, x, 1/2 : 0 < x < 1/2
∆	0, u, 0	4	l	m2m.	0, y, 0 : 0 < y < 1/2
U	0, u, 1/2	4	m	m2m.	0, y, 1/2 : 0 < y < 1/2
Y	u, 1/2, 0	4	n	m2m.	x, 1/2, 0 : 0 < x < 1/2
T	u, 1/2, 1/2	4	o	m2m.	x, 1/2, 1/2 : 0 < x < 1/2

60 3 Symmetry and Localization of Crystalline Orbitals

tvRϕ(k, r) = ϕ˜(Rk, r), where tvR G then we obtain some linear combination of n Bloch functions of the initial basis. This means that Rk is one of the vectors k, k2, . . . , kn.

In n-dimensional space L there are no nonzero subspaces invariant with respect to D(g) for all g G.

Therefore, as we run over all elements of G operating on ϕ(k, r) by D(g), we generate the entire space L, i.e. each of the vectors k1, . . . , kn appears as the transform of k under some element of the point group F (point group of the space group G). It may be that the vectors k1, . . . , kn are not all di erent. A set of ns distinct (nonequivalent) k vectors chosen from the set k1, . . . , kn is called the star of wavevector k and is denoted as k. A star can be generated from one of its members by operating on it by elements of point group F. The Γ point of BZ forms one-ray star for any space group as it is a coordinate system origin for all the pointsymmetry group F transformations in reciprocal space. The stars X, L and W (see Fig. 3.2 of the Brillouin zone for the fcc direct lattice) consist of 3, 4 and 6 rays, respectively.

The point group of the wavevector k (little cogroup of k) Fk F , by definition, consists of all the rotations or reflections Rk(i = 1, 2, ..., nk) that rotate k into itself or an equivalent vector Rk = k + Bm.

In Sect. 3.1.2 points and lines of symmetry in the Brillouin zone were defined for the case when F = F0 (holosymmetric space groups, in particular Oh5 and D414h). In the same manner the points and lines of symmetry may be defined for the point group F F0. Then, instead of the basic domain of the Brillouin zone the representation domain is introduced.

We can write F as a sum of left cosets with respect to the subgroup Fk:

ns
i	RiFk	(3.26)
F =
=1

where ns = nF /nk is the number of k-vectors in the star k. If Rik = ki then all elements of the left coset RiFk transform k into ki.

By definition, the little group Gk of wavevector k consists of all elements gjk = tv(jk)+aRj(k), j = 1, 2, . . . , nk, where Rj(k) Fk. The group Gk G is a space group of order N · njk so that we can write G as a sum of left cosets with respect to the subgroup Gk:

ns
j
G = gj Gk, gj = tvj Rj / Gk	(3.27)
=1

Any element of coset gj Gk transforms a Bloch function ϕ(k, r) into a Bloch function ϕ(kj , r). Thus, if the vectors k, k2, . . . , kn characterizing the rep D(g) of dimension n = nαns are not all unique, then the star k, k2, . . . , kns is repeated exactly nα times in the set k1, k2, . . . , kn.

Let the basis set of space L(1) be formed by Bloch functions ϕ(iα)(k, r), (i = 1, 2, . . . , nα).

These functions form a basis of the so-called small (allowed) irrep of the space group Gk:

	nα
	i	,α)(g)ϕi(α)(k, r), g Gk
D(g)ϕi(α)	(k, r) = Dii(k		(3.28)
	=1

3.1 Translation and Space Symmetry of Crystalline Orbitals. Bloch Functions

61

The wavevector k is called a canonical wavevector.

Not all the irreps of space group Gk are small irreps of this group since the small irrep basis functions ϕ(iα)(k, r), (i = 1, 2, . . . , nα) have an additional property: they are Bloch functions with the same vector k, i.e. the matrices of the small irrep are diagonal for the pure translations ta and have nα equal diagonal elements exp(−ika). The Bloch functions ϕi(kj , r) = tvj Rj ϕi(k, r) form the basis set of irrep kj = Rj k of the translation group T. The space group elements g(kj ) = gj g(k)gj−1 form the group

Gk of the wavevector kj . The point group Fkj consists of elements R(kj ) = Rj R(k)Rj−1.

The basis functions ϕi(α)(kj , r), (i = 1, 2, . . . , nα) of a space L(j) transform over the
irrep D(k,α)(gj−1g(kj )gj )						i	Dii(k,α) g(k) ϕi (kj , r)
gˆ(kj )ϕi(kj , r) = gˆj gˆ(k)ϕi(k, r) =
	=	i		Dii(k,α) g−1g(k)gj ϕi (kj , r)				(3.29)
The space L =	L(j) of n	n		functions ϕ	(k		, r), (i = 1, . . . , nα, j = 1, . . . , ns) is
j	s		α	i		j		( k,α)	(g)
irreducible under G. These functions form a basis of the so-called full irrep D

of the space group. The full irrep D( k,α)(g) of G is induced by the small irrep D(k,α)(g) of the little group Gk G: D( k,α)(g) = D(k,α)(g) ↑ G,

( k,α)	(kα)	gj−1ggj δj j	(3.30)
Di j ,ij	(g) = Dii

The little group Gk is itself a space group. The small irreps of Gk can be found if projective irreps of the point group Fk are known.

3.1.4 Small Representations of a Little Group. Projective Representations of Point Groups

The small reps of little groups are su cient for many purposes in solid-state theory, such as classifying states within electron energy bands and vibration frequencies within phonon band spectra, and for generating the symmetry-adapted wavefunctions.

The little group Gk is itself a space group. Whereas in the reps of G the translation ta is represented by a diagonal matrix, in the small irreps D(k,α) of Gk that we are looking for ta is represented by a scalar matrix

D(k,α)(ta) = exp(−ik · a)Enkα

(3.31)

where nkα is the dimension of the small irrep D(k,α) and Enkα is the unit matrix of order nkα.

We decompose Gk into left cosets with respect to the translation group T :

Gk = gj T, gj = tvj Rj	(3.32)
j

The coset representatives in (3.32) obey the multiplication rule

62

3 Symmetry and Localization of Crystalline Orbitals

tvi Ritvj Rj = RiRj tvi +Rivj = taij +vs Rs

(3.33)

where

and aij = vi + Rivj − vi

(3.34)

taij T

Because aij is a translation it follows that

D(k,α)(gi)D(k,α)(gj ) = exp(−ik · aij )D(k,α)(gs)

(3.35)

If we now set for all g Gk

˜

D

(k,α)

(k,α)

(R)

(3.36)

(g) = exp(−ik · v)D

then from (3.35) we obtain

˜

(k,α)

˜ (k,α)

(Rj ) = exp(−iBi

˜

(k,α)

(Rs)

(3.37)

D

(Ri)D

· vj )D

where the reciprocal lattice vector Bi is defined by the relation

Ri−1k = k + Bi

(3.38)

˜

Equations (3.36) together imply that the matrices D(k,α)(R) are the same for all members of any fixed coset in (3.32), i.e. these matrices are in correspondence with the elements of the factor group Gk/T . The factor group Gk/T is isomorphic with

˜

the little cogroup Fk so that D(k,α) is a matrix-valued function on the elements of the point group of wavevector k with the multiplication law

˜ (k,α)	˜ (k,α)	˜	(k,α)	(RiRj )	(3.39)
D	(Ri)D	(Rj ) = exp(−iBi · vj )D

˜

Matrices D(k,α) form a so-called projective rep of the point group Fk with the factor system

ω(Ri, Rj ) = exp(−iBi · vj )

(3.40)

where Bi is defined by (3.38).

We see from (3.36) that the small irreps D(k,α) of a little group are found if the

˜

projective irreps D(k,α)(R) of the point group Fk with the factor system (3.40) are known.

A factor system is specified by n2Fk coe cients ω(Ri, Rj ), where nFk is the order of the point group of wave vector Fk. These coe cients must satisfy the following

identities for any Ri, Rj , Rk:
(Ri, Rj Rk)(Rj , Rk) = (RiRj , Rk)(Ri, Rj )	(3.41)

implied by the associative law of group multiplication Ri(Rj Rk) = (RiRj )Rk. However, conditions (3.41) do not define a factor system uniquely [27]. If D(R) is a projective representation of the group belonging to the factor system ω(Ri, Rj ) then any other representation D (R) = D(R)/u(R) where u(R) is an arbitrary singlevalued function on the group, |u(R)| = 1, also defines a projective representation of the group, but with factor system ω (Ri, Rj )= ω(Ri, Rj )u(Ri, Rj )/u(Ri)u(Rj ). The factor systems ω(Ri, Rj ) and ω (Ri, Rj ) representations satisfying (3.41) are said to be projectively equivalent or p equivalent. The set of all p equivalent factor systems is called a class of factor systems. Note that two di erent p equivalent

3.1 Translation and Space Symmetry of Crystalline Orbitals. Bloch Functions

63

representations may belong to the same factor system. For this to occur it is necessary that u(Ri)u(Rj ) = u(RiRj ), i.e. function u(R) defines some ordinary (with all ω(Ri, Rj ) = 1) one-dimensional representation of the group. The group may have several classes of factor systems. It can be shown [27] that if ω(Ri, Rj ) = ω(Rj , Ri) for any pair of commuting elements (RiRj = Rj Ri) the factor system ω(Ri, Rj ) is p equivalent to identity factor system ω(Ri, Rj ) = 1 and the corresponding representation is p equivalent to ordinary representation. Although the total number of possible factor systems is infinite, it can be shown that for a finite group the number of classes K of factor systems is finite [27]. For every group there is class K0 with all ω(Ri, Rj ) = 1, which is called vector or ordinary representation. The other representations of the class K0 with ω(Ri, Rj ) = 1 are projectively equivalent to vector representations. The dimensions of all p equivalent representations are the same, the factor systems may di er. If a group has several classes of factor systems then only the class K0 may correspond to one-dimensional representations and there will be no one-dimensional representations for classes K0 =Kp.

Projective representations were first introduced by Shur [32], who developed a general theory of projective representations and worked out methods for constructing projective representations of finite groups. The connection between projective representations of point groups and representations of space groups was demonstrated by Lyubarskii, Kovalev, Bir [27, 31, 33]. To find D(k,α) with the factor system (3.40) Herring’s approach [34] may be useful. In this approach the problem is reduced to finding ordinary irreps of abstract groups with order greater than that of Fk but not very large.

In [30] it is shown that these groups for di erent wavevectors in the Brillouin zone of one space group or for the same wavevector in the Brillouin zone of di erent space groups may be isomorphic. In [30] all the irreps of all the abstract groups that occur among reps of space groups are completely identified. In [17] the small irreps of little groups are found from those of the translation group by successive augmentations, each augmenting operator being chosen so that the augmented group contains the unaugmented one as an invariant subgroup.

The classes of all factor systems and the corresponding projective representations characters can be found in [27] for all 32 crystallographic point groups. Ten groups (C1, S2, Cs, C2, S4, C4, C3, S6, C6, C3h) are cyclic, only one factor system K0 belongs to these groups, i.e. for these groups all the projective representations are p equivalent to ordinary representations.

This short description of projective representations of point groups allows us to understand information given in di erent tables and on the site [16]. When using di erent existing tables for small representations of little groups one has to remember that the projective representations of point groups can be ordered in di erent ways

and may appear to be p equivalent to each other.
When the space group G is symmorphic all the fractional translations vi may be
˜ k,α)	(Ri)
chosen to be zero (Sect. 3.3) so that all the factors ω(Ri, Rj ) = 1 and D

coincide with the αth irrep of the point group Fk. When k is an interior point of

the Brillouin zone (R−1	k − k) = 0 for all Ri	in the	group F					k	. It can also be that
i				1	k
at some points of the Brillouin zone on its surface (R−						−	k)vj = 0 for all Ri and
			i							k,α	(R) also coincide
vj in the group Gk. In both these cases the projective irreps D

with the usual irreps of the point group of a wavevector. However, for some points

64 3 Symmetry and Localization of Crystalline Orbitals

on the surface of the Brillouin zone in the case of nonsymmorphic space groups the projective irreps Dk,α(R) are not the usual irreps of the point group Fk.

As an example, we consider nonsymmorphic space group Oh7 – the symmetry group of diamond structure. The Brillouin zone for the face-centered-cubic lattice and the symmetry points and the symmetry directions are shown in Fig. 3.2. The small representations of little groups GΓ and GL are p equivalent to the ordinary irreducible representations of the corresponding wavevector point groups Oh and D3d. For the Γ point it is evident as this point is inside (at the center) of the Brillouin zone. For the L point it can be easily shown. Indeed the point group of L(1/2, 1/2, 1/2) consists of rotations C31 , C31−1 through one of four third-order rotation axes, three rotations through second-order axes Uxy− , Uyz− and Uxz− (these operations and identity operation form point group D3). All the remaining operations of the D3d point group are obtained from relation D3d = D3 ×Ci (direct product of D3 and inversion group Ci). As can be seen from [16] these point-symmetry operations are included in space group Oh7 (origin choice 2) with proper translations, so that for any two elements Ri and Rj of point group D3d the multipliers ω(Ri, Rj ) = 1. For X and W points the small representations are not p equivalent to ordinary representations of wavevector point groups D4h and D2d, respectively. To show this it is enough to find in the corresponding little cogroups at least one pair of commuting elements for which ω(Ri, Rj ) =ω(Rj , Ri). The point group D4h = D4 ×Ci of X(1/2, 1/2, 0) includes rotations C4z , C2z and C4−z1 through fourth-order z axis, rotations C2x and C2y through x- and y-coordinate axes and rotations Ux−y, Uxy through second-order axes. The space group Oh7 includes commuting elements C2y and IC2y = σy with improper translations (1/4, 1/2, /3/4) and (1/2, 3/4, 1/4), respectively, given in basic translation vectors a1, a2, a3 (see [16, 19]). For the transformation C2y and k = X we have C2y k −k = b1 + b2 (sum of two basic reciprocal lattice vectors), for the transformation IC2y = σy we have σy k − k = 0. Therefore ω(C2y , σy ) = exp(iπ/2) = i and ω(σy, C2y ) = 1. Considering the commuting elements C2y and σx = IC2x from point group D2d of wavevector W it is easy to show that ω(C2y , σx) = −i and ω(σx, C2y ) = i. In Table 3.4 are given for space group Oh7 labels of the corresponding projective irreducible representations of point groups FX and FW and their dimensions. The point-symmetry group FZ of the symmetry direction XW (12 rays in the star) on the surface of the Brillouin zone (see Fig. 3.2) consists of elements being common for both point groups FX and FW : E, C2y , σx, σz . It was seen that ω(C2y , σx) =ω(σx, C2y ) so that the only wavevector point group C2v two-dimensional irreducible representation is not p equivalent to any of the four ordinary one-dimensional representations of point group C2v (see Table 3.4).

The numbering of some small representations of space groups is di erent in di erent tables. For example, four small twodimensional representations at the X point are ordered in di erent ways on the site [16] and in tables [17]. The ray of stars L, X and W chosen for the small-representation generation can also be di erent, as can the origin choice for space-group description. All these di erences do not change the full representations of space groups.

The dimension of a space-group full representation (degeneracy of energy levels in a crystal) for a given k is equal to the product of the number of rays in the star k and the dimension of the point group Fk irreducible representation (ordinary or projective). In particular, for the space group under consideration at the X point the dimensions of full representations are 6 and at the W point – 12. As to each of the

3.1 Translation and Space Symmetry of Crystalline Orbitals. Bloch Functions

65

) 7 h

(spacepoints Ogroup

Wand

Xfor

d2

Dand

h4

pointof Dgroups

Table 3.4. Characters of projective representations

xy	0
U
x−y	0
U
4z	0
C
−1 4z	0
C
z	0
σ
y	0
σ
x	0
σ
I	0
xy	2
σ
x−y	2
σ
−1 4z	0
S
4z	0
S
2z	2
C
2y	0
C
2x	0
C
E	2
	1
	X

0	-2
0	2
0	0
0	0
0	0
0	0
0	0
0	0
-2	0
-2	0
0	0
0	0
2	-2
0	0
0	0
2	2
2	3
X	X

2
-2
0
0
0
0
0
0
0	xz
	U
0	x−z
	U
0	z
	σ
0	x
	σ
-2	−1 4y
	S
0	4y
	S
0	2y
	C
2	E
4
X

0	0
0	0
0	0	0
0	0	0
1+i	−1−i
1−i	−1+i
0	0	0
2	2	2
W	W	1
		XW
1	2

66 3 Symmetry and Localization of Crystalline Orbitals

degenerated states corresponds the same one-electron energy it is enough to identify energy levels only for one ray of the wavevector star as it is made in the figures showing the electronic band structure. The degeneracy of levels at the symmetry points of the BZ is defined by the dimensions of wavevector point-group representations (ordinary or projective). To identify the one-electron energies at the symmetry lines the compatibility relations are used. In Sect. 3.2.6 we discuss the band structure of some crystals using the considered information.

When the space group is realized in a crystalline structure the atomic states included in the LCAO basis define the symmetry of crystalline orbitals appearing in the electronic-structure calculations. The symmetry connection of atomic and crystalline orbitals is given by induced representations of space groups considered in the next subsection.

3.2 Site Symmetry and Induced Representations of Space

Groups

3.2.1 Induced Representations of Point Groups.

Localized Molecular Orbitals

In the previous section we examined the use of space-group irreducible representations for the classification of the delocalized (Bloch-type) crystalline states. In this traditional approach the crystal is considered as a whole system and the symmetry properties of the environment of constituent atoms are ignored. This results in a loss of information about the connection between the atomic and crystalline states. This information is widely used in the quantum chemistry of solids as it allows the crystalline properties to be explained from the knowledge of the chemical nature of the constituent atoms and their interactions. In the plane-waves methods of electronicstructure calculations the Bloch-type delocalized states are not directly connected with the states of the separate atoms. However, in the LCAO methods the Bloch-type delocalized functions are represented as the linear combination of the functions of separate atoms. Therefore, the symmetry connection between the delocalized Bloch and localized atomic states appears to be important. If we use not only the space symmetry of a crystal as a whole but also the site symmetry of di erent groups of constituent atoms we can considerably extend the possibilities of the group-theory applications. To study this in more detail the reader is referred to our previous book [13] where we examined the theory and the applications of the site-symmetry approach to the electron, phonon, magnetic properties of crystals and in the theory of phase transitions. In this section, we examine only those theoretical aspects of the site-symmetry approach that concern the electron states and allow analysis of the symmetry connection between the delocalized Bloch-type and localized Wannier-type electron states in crystals. We begin from the short description of the site (local) symmetry approach in molecular quantum chemistry.

In the molecular systems with the point symmetry group G the site-symmetry subgroup Hq includes those symmetry operations that keep the point q fixed: hq = q. As an example, we consider a tetrahedral molecular ion [MnO4]− (see Fig. 3.8). The Mn atom site-symmetry group coincides with the whole symmetry group Td. The site-symmetry group of any of the four oxygens is C3v Td.