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

3.3 Symmetry of Localized Crystalline Orbitals. Wannier Functions

87

functions of energy bands in k-space determines the degree of localization of the corresponding Wannier functions in r-space [44, 45]. A useful concept of band (induced) representations has been introduced in the theory of crystals, according to which the position of symmetry localization and the symmetry properties of Wannier functions for a given energy band define unambiguously the symmetry properties of the corresponding Bloch functions [13]. But the last decade has been marked by an outburst of interest in WFs, owing to the development of new theories related to them, and e ective methods for their generation [5, 46–48].

Due to their high localization, WFs are especially beneficial when applied in the following areas of solid-state theory.

1.Linear scaling (so-called O(N )) algorithms for calculation of crystalline electronic structure are based on the concept of WFs [50].

2.WFs play an important role in the theory of electronic polarization in insulators

[51].The electronic polarization itself and related properties can be expressed simply via the centroids of WFs, connected with the valence-band states.

3.Wannier functions are often used as a convenient basis for describing the local phenomena in solids, such as point defects, excitons, surfaces, etc. [52].

4.They can serve as a useful tool in the solution of the problem of electron correlation in crystals [41].

5.WFs, being spatially localized combinations of Bloch functions, thus form a natural basis for analysis of chemical bonding in crystals [47, 53, 54].

The background of LO s symmetry analysis is the theory of BR s of space groups G. This analysis is equally applicable to localized Wannier orbitals (LWO s) i.e. to an orthonormal set of LO s. We describe the main principles of this theory related to the examined problem.

The LO s Vi(1βp)(r) ≡ Vi(β)(r − q(1p)) are the basis functions of the irreducible representation (IR) β of the site-symmetry group Gq(p) G corresponding to their

centering point (centroid) q(1p) (index p distinguishes the symmetry nonequivalent

points in the space of the crystal):
(Rˆ|vR)Vi(1βp)(r) = i	di(βi)(R)Vi(β1p) (r)	(3.71)

where (Rˆ|vR) Gq(p) , d(β)(R) is the matrix mapping the element (Rˆ|vR) in the IR β. For example, in crystal SrZrO3 index p = 1, 2, 3 numbers symmetry nonequivalent Sr, Zr and one of the three oxygen atoms. Applying symmetry operations from the decomposition of the group G into the left cosets with respect to the site group Gq(p)

		G = (Rj |vj + an) · Gq(p)	(3.72)
		j
to the functions V	(β)	(r)
	i1p
		Vijp(β)(r − an) ≡ (Rj |vj + an)Vi(1βp)(r)	(3.73)

one can obtain the complete basis in the space Ωp(β) of the reducible representation of the group G induced from the IR β of the group Lq(p) . The functions Vijp(β)(r − an) ≡

88 3 Symmetry and Localization of Crystalline Orbitals

Vi(β)(r − q(j,pn) −an) are centered at the points q(j,pn) ≡ (Rj |vj + an)q(1p) = Rj q(1p) + vj +

an. Such a basis consisting of the LO s Vijp(β)(r − an) ≡ Vτ (r − an) ≡ Vt(r) is called the q -basis (Sect. 3.2.2). The index τ replaces i, j, β, p, and the index t replaces τ, n

to simplify a traditional group-theoretical notation. This basis is fully determined by its single representative (for example, V11(βp)(r)). All the other basis functions of the same BR can be obtained from it by the symmetry operations, (3.73). A BR is characterized in q -basis by the site q(p) (the centroid of Vi(1βp)(r)) and the IR β of the site group Gq(p) : (q(p), β) is a symbol of the BR in q -basis. For example, in MgO crystal oxygen 2s -functions transform according to β = a1g IR of the site-symmetry group Gq(1) = Oh and decomposition (3.72) consists of 1 term as the point group

(a1g )	(r − an) are centered at the oxygens in a
of the crystal is Oh. The functions Vi1p

whole direct lattice. Silicon crystal has space group Oh7 , diamond-type lattice with two atoms per unit cell, occupying the Wycko position a with the site-symmetry group Ga = Td. The calculated upper valence and lower conduction bands shown in Fig. 3.5 are composite (for each wavevector value there are 4 + 4 = 8 Bloch states). The corresponding induced representation is engendered by a1, t2 irreps of the point group Ga, corresponding by symmetry to the s, p states of Si atom. The upper valence band in silicon crystal is connected with the one-electron states localized not on the Si atom but on the Si−Si bond middle (Wycko position c with the site-symmetry group D3d). This can be found by inducing the four-sheet upper valence band states from the identity representation a1g of the point group D3d. The lower conduction-band states symmetry is defined by the induction from the irrep a2u of the same point group. Therefore, these two band representations may be labeled by the symbols (c, a1g ),(c, a2u).

As the third example of the band representations generated we consider SrZrO3

crystal. The configurations of valence electrons in free atoms are:
O – 2s22p4,	Sr – 4p65s2,	Zr – 4d25s2	(3.74)

The semicore 4p -states of Sr atom are included in (3.74) as they take part in the valence-band structure. The cubic phase of SrZrO3 with the space group

P m3m

(Oh1 , a simple cubic lattice) consists of one formula unit (5 atoms) in the primitive unit cell. Atoms Sr, Zr and O occupy Wycko positions (Cartesian coordinates are given in units of lattice constant of the crystal) b (1/2,1/2,1/2), a (0,0,0) (site group Oh for both), and d (0,0,1/2) (site group D4h), respectively. Fig. 3.3 shows the valence energy bands of this crystal calculated by HF LCAO method with the help of CRYSTAL03 code [23]. 30 electrons per primitive unit cell of SrZrO3 crystal occupy 15 one-electron levels for each k-point in the BZ. The 15-sheeted valence band consists of two subbands (6 and 9 sheets). The analysis of the projected densities of states shows that the lower 6-sheeted subband is formed by Sr 4p - and O 2s -states, the upper 9-sheeted band is connected mainly with O 2p -states. These free-atom electron states are transformed by the interatomic interaction in crystalline LO s. The latter basis (just as the basis of canonical Bloch one-electron states) may be used to describe the electronic structure of the crystal and calculate its properties. In the SrZrO3 crystal, two induced BR s

3.3 Symmetry of Localized Crystalline Orbitals. Wannier Functions

89

correspond to sixand nine-sheeted bands. As is seen from Table 3.16, the states of six-sheeted band are induced by LO of the symmetry a1g of oxygen site-symmetry group D4h (Wycko position d) and by LO s of the symmetry t1u of Sr-atom sitesymmetry group Oh (Wycko position b), i.e. by O 2s- and Sr 4p-atomic-like states. The states of the nine-sheeted band are induced by LO s of the symmetry a2u and eu of oxygen site-symmetry group D4h, i.e. by O 2p-atomic-like states. In SrZrO3 crystal oxygen 2s -functions transform according to β = a1g IR of the site-symmetry group Gq = D4h and decomposition consists of 3 terms as the point group of the crystal is

Oh. The functions V (a1g )(r − an) are centered on the oxygens in a whole direct lattice

ijp

(j = 1, 2, 3 numbers oxygens in one primitive cell). This is an example of the q-basis corresponding to a 3-sheeted BR, induced by O 2s -type atomic states. This basis is fully determined by a 2s-function of one of the three oxygens in the primitive cell and may be labeled by the symbol (d, a1g ) as oxygen atoms occupy Wycko position d in space group Oh1 .

Resolving this BR into IR s of the space group G, one gets the indices of the BR in k -basis (Bloch basis). The short symbol of the BR in k -basis contains only the indices of the small IR s for the most symmetrical points of the BZ, because the indices for all other IR s contained in the BR are determined with the help of compatibility relations. For example, in Table 3.16 the BR (d, a1g) is given in k-basis (Γ, R, M, X are the symmetry points of the BZ).

In our example, the BR corresponding to the 6-sheeted valence band is a composite one as it is formed by two simple band representations (d, a1g ) and (b, t1u) induced by O 2s- and Sr 4p-states, respectively.

From the theory of the band representations of space groups, it follows that the generation of LO s corresponding to a given simple or composite energy band is possible only if the canonical (Bloch) orbitals of this band form the basis of some simple or composite BR. This analysis permits not only to establish the principal possibility to construct LO s, but also to define the possible positions of their centroids q(p) and their symmetry with respect to the site-symmetry group Gq(p) . The latter is not always unambiguous due to the fact that there are the BR s that have di erent symbols in q -basis, but the same index in k -basis or there are the composite BR s that can be decomposed into simple ones by several ways (see Sect. 3.2.4).

Thus, the symmetry analysis consists of a procedure of identifying the localizedorbitals symmetry from the symmetry of the canonical orbitals of the energy band under consideration, or of establishing the fact that the construction of LO s from the canonical orbitals chosen is impossible for the reasons of symmetry.

When calculating the electronic structure of a crystal its cyclic model is used i.e. the model of a finite crystal with periodic boundary conditions – a supercell, consisting of N unit cells. Generally, when a numerical integration over the BZ is carried out as a summation over a set of special points of the BZ, it means that a cyclic model of a certain size is introduced for the crystal. The relation between the symmetry group G of the model of an infinite crystal and the symmetry group G(N ) of a corresponding cyclic model and their IR s and BR s has been studied in detail in [55]. The localized orbitals of the model of an infinite crystal are well approximated by the localized orbitals of a cyclic models with a size slightly exceeding the region of their localization.

90 3 Symmetry and Localization of Crystalline Orbitals

We assume that the canonical delocalized orbitals ϕ(mµγ) (k, r) ≡ ϕσ (k, r) ≡ ϕs(r) (the index m numbers the basis vectors of IR γ with wavevector k, µ discriminates between the independent bases of equivalent IR s; the index σ replaces γ, m, µ, and s replaces σ, k) of an energy band under consideration form a basis in the space Ω of some simple BR of the group G(N) of a crystal (index p is omitted for simplicity). The same space Ω is spanned also by the set of LO s Vij(β)(r − an) ≡ Vt(r). The delocalized and localized bases are bounded by a linear transformation in the space Ω:

Vτ (r − an) = exp(−ikan)		Uσ,τ (k)ϕσ(k, r),	or	(3.75)
	k	σ
		Ust = Uσ,τ (k)· exp(−ikan)
Vt(r) = Ust · ϕs(r) with				(3.76)

s

If the matrix U is unitary and the functions ϕs(r) are orthonormalized, the functions Vt(r) form an orthonormal system of LWO s Vt(r) = Wt(r).

Their localization extent depends on the choice of the matrix Uσ,τ (k) in (3.76). The existence of Wannier functions decreasing exponentially at infinity (for the model of an infinite crystal) is established in many cases. The di erent localization criteria used for the generation of localized orbitals in crystals are considered in the next subsection.

3.3.2 Localization Criteria in Wannier-function Generation

The general method of the most localized Wannier-function (WF) generation exists only for the one-dimensional case and was o ered by Kohn [57].

It is a variational method based on the first-principles approach, i.e. without preliminary knowledge of Bloch-type delocalized functions. The localized functions with the symmetry of Wannier functions and depending on some number of parameters are used.

If spin variables are ignored the one determinant wavefunction for the system of

M electrons occupying M Bloch-type states can be written as											(3.77)
Ψ0	= (M !)−1/2 det		Φ(ϕ)	τ k,i		= (M !)−1/2 det		ΦW	sn,i

where Φ(ϕ) and ΦW are Bloch and Wannier functions, respectively

	Φ(ϕ)	= ϕτ k,i(ri)			(3.78)
		τ k,i
	ΦW sn,i = Ws(ri − an)				(3.79)
and ri – coordinate of the ith electron (i = 1, 2, . . . , M ).
The total energy E0 per cell, is
E0 = M −1 Ψ0, M	HiΨ0 = M −1		Eτ,k =	(Ws(r), HWs(r))	(3.80)
i=1		τ,k		s

The better the initial modeling localized functions approximate WF Ws(r), the closer to the minimal value E0 is the value of the total energy calculated on these

3.3 Symmetry of Localized Crystalline Orbitals. Wannier Functions

91

modeling functions. Thus, the variational principle assuming a variation of parameters of modeling functions is used to minimize the value of the total energy calculated on such functions.

Before calculation of the total energy on the basis of the trial localized functions fs(r, α, β, . . .) (where α, β, . . . are variational parameters), their orthonormalization is made. Functions fs(r − an) are orthogonalized by means of a procedure suggested by Des Cloizeaux [56]. This procedure consists of construction of the orthonormalized trial Bloch functions ψs(r, k):

			fs (r − an) S(ψ)−1/2 (k) s s
ψs(r, k) = M −1/2	n	exp(ikan) s		(3.81)

where S(ψ)(k) is an overlap matrix of trial Bloch functions, directly constructed of trial localized functions fs(r − an):

S(ψ)(k) s s =		(fs (r), fs(r − an)) exp(ikan)	(3.82)
	n

Then the modeling localized functions Ws(m)(r)=Ws(m)(r, α, β, . . .) are defined. These functions depend on the same parameters as function fs(r, α, β, . . .) does:

Ws(m)(r) = M −1/2 ψs(r, k)	(3.83)

k

and are substituted in (3.83) with the purpose of minimization of the total energy E0. The advantages of the method are the following: WF generation does not require preliminary Bloch-function calculation, the WF symmetry is taken into account at a stage of choice of trial modeling functions. Disadvantages of the method: the procedure is laborconsuming in realization; the WF obtained are approximate and their di erences from the exact WF depend on the choice of the trial modeling functions; there is no universal procedure of a choice of basis of trial functions for di erent crystals; use of the one-determinant approximation for vacant states WF is not quite

correct.

The method of WF generation using the Slater–Koster interpolation procedure was suggested in [58]. In this method a group of one-electron energy bands is chosen and described by model Hamiltonian matrices H(k):

Hνµ(k)eµτ (k) = Eτ keντ (k)

(3.84)

µ

where e(k) and Eτ k are eigenvectors and eigenvalues of a matrix model Hamiltonian H(k) that should coincide with the eigenvalues of the exact Hamiltonian matrix in the given point k. For the model Hamiltonian Fourier decomposition is used

H(k) = ε(an) exp(−ikan)	(3.85)

n

The WF is defined as

92	3 Symmetry and Localization of Crystalline Orbitals
	Ws(ran) =	Va		dk exp(−ikan)		eτ s(k)ϕτ (r, k)	(3.86)
		(2π)3			τ

so that Fourier coe cients εs s(an) (Slater–Koster parameters) are elements of a

Hamiltonian matrix in the basis of WF:

εs s(an) = Ws (r − an)HWs(r)dr (3.87)

For well-localized WF values εs s(an) quickly decrease with increase of |an|. Therefore, in this case in decomposition (3.90) it is enough to consider a small amount of the matrices ε(an) corresponding to small values of translation vectors. Additional reduction of the number of independent Slater–Koster parameters is possible when the symmetry is taken into account. The remaining independent elements of matrices ε(an) can be found by means of a least squares method, adjusting eigenvalues of a matrix H(k) to the calculated band structure. Matrices of eigenvectors e(k) for well-adjusted modeling Hamiltonian are matrices of the unitary transformation U (k) connecting Bloch functions with localized WF.

There is still an uncertainty in the choice of relative phase multipliers for matrices U (k) (or) e(k) at di erent k (this problem can be solved only in crystals with inversion symmetry [59]). Let us(r, k) be periodic parts of Bloch functions ψs(r, k):

us(r, k) = exp(−ik · r)ψs(r, k)

(3.88)

Phase multipliers at matrices U (k) need to be chosen so that decomposition coe - cients Vj (s, k) of functions us(r, k) on plane waves:

us(r, k) =	Vj (s, k) exp(iKj · r)	(3.89)
	j

remained real (this is possible if the space group of symmetry of a crystal contains an inversion).

Advantages of the method: for WF generation it is possible to choose any groups of energy band – occupied, vacant or both; the use of expected WF symmetry reduces the number of independent Slater–Koster parameters.

Disadvantages of the method: an e cient procedure for the case when the lattice has no the center of inversion it is not developed; the WF maximal localization criteria is not formulated; the method is complex enough in realization.

The recently developed WF generation method of Marzari and Vanderbilt [46, 60] extends to crystalline solids the Boys [38] criteria of localized MO generation (see Sect. 3.3.1). The localized WF (let their number be N for the primitive unit cell) are found by the minimization of the functional

N
I = s=1		r2	s − r s2	(3.90)
where
r2 s =		r2 |Ws(r)|2 dr		(3.91)
r s =		r |Ws(r)|2 dr		(3.92)

3.3 Symmetry of Localized Crystalline Orbitals. Wannier Functions

93

Minimization of (3.90) is made in reciprocal space, for which expressions (3.91)

and (3.92) can be rewritten in the form:
r s = i	Va			dk us(r, k) |k			| us(r, k)	(3.93)
	(2π)3
r2 s = −	Va			dk	us(r, k)	k2	us(r, k)	(3.94)
	(2π)3

where us(r, k) are periodic parts of the Bloch functions defined by (3.89).

The gradient in k-space, used in (3.93) and (3.94) is correctly defined only in the model of an infinite crystal when a variable k is continuous. In a cyclic model of a crystal the gradient of the function is replaced with the approximate finitedi erences expressions, and integration on the Brillouin zone is made by summation

(k,b)	be defined as the scalar product:
over a discrete set of wavevectors k. Let Mmn
Ms(ks,b) = us (r, k) | us(r, k + b)		(3.95)

where b is a vector connecting each k-point from a discrete set, corresponding to the

considered cyclic model, with one of its nearest neighbors. Then

r2

and

r

s

can

be approximately calculated using (3.93) and (3.94):

s

r s = −

1

wbb Im ln Mss(k,b)

(3.96)

L

k,b

r2

s =

Mss(k,b)

+

Im ln Mss(k,b)

(3.97)

L k,b wb 1 −

1

2

2

The multiplier wb depends on the type of Bravais lattice.

Let δUrs be the changes of functional (3.90) value at infinitesimal transformation

δU (k) of matrices U (k):
δUrs(k) = δrs + d∆rs(k)	(3.98)

Here, d∆(k) is the antihermitian matrix of infinitesimal transformation. It is possible to show that the gradient of functional (3.91) in the space of matrices U is expressed

by the following formula:

dI

= 4

b

wb

A R(k,b) − S T (k,b)

(3.99)

G(k) = d∆(k)

where

R(k,b) = M (k,b)

M (k,b)

(3.100)

s

s

ss

(k,b)s

s

(k,b)

Ms s

(k,b)

Ts s

=

Im ln Mss

+ b r

(3.101)

Mss(k,b)

and A and S are superoperators:

A

B

} ≡

B − B†

2

{

S{B} ≡

B + B†

(3.102)

2i

94 3 Symmetry and Localization of Crystalline Orbitals

The algorithm of functional (3.90) minimization is based on expression (3.99) and consists in an iterative procedure of the steepest descent. For this purpose, on each iteration the matrix d∆(k) is chosen in the form of:

d∆(k) = εG(k)

(3.103)

where ε is a positive small constant. As is seen from (3.99), a matrix G(k) is anti-

hermitian (G† =	−	G), and for this choice the functional change:
		dI = −ε |Gss (k)|2
			(3.104)
		k ss

is less than zero. Hence, on each step of the iterative procedure the value of the sum of WF dispersions (3.90) decreases.

Thus, the method consists in a choice of initial approximation to WF and the subsequent iterative procedure of construction as much as possible localized WF (in sense of a minimality of the sum of their dispersions). Subsequently, this method has been advanced and generalized to the case of entangled energy bands [61].

Advantages of the method: the method is universal and can practically be applied for any system; the method is e cient and can be automated to apply in computer code [62]. Disadvantages of the method: in the method the criterion of localization is fixed as the Boys localization criterion, therefore there is no opportunity to receive WF as much as possible localized concerning any another criterion; the symmetry use is not included, in some cases the number of iterations strongly depends on the initial approximation; the use of a su ciently dense grid of wavevectors is necessary to obtain a good accuracy in the gradient calculation.

This method of WF generation is now widely used in PW DFT calculations of Bloch functions. In the next two sections we consider the LCAO approach to maximally localized WF generation.

3.3.3 Localized Orbitals for Valence Bands: LCAO approximation

The method of the localized WF generation for occupied energy bands was suggested in [53, 63, 64] and implemented with the CRYSTAL code for LCAO calculations of periodic systems. In this method, WFs are sought in the form of a linear combination of atomic orbitals (AO):

Ws(r) = cnµsφµ(r − sµ − an) (3.105)

µn

where φµ(r−sµ −an) is the µth AO, centered in a point sµ +an , the index s numbers the localized functions in one primitive cell, and cnµs are the coe cients connecting WF and AO. Construction of the localized functions is made by means of an iterative procedure where each step consists of two basic stages.

Let us designate WF after the (n − 1)th step through Ws(n−1)(r). At the first

stage, named by authors “Wannierization”, the localized functions W (sn−1)(r) that maximize the functional are sought

	3.3 Symmetry of Localized Crystalline Orbitals. Wannier Functions								95
		N
	IW(n)	= s=1	W s(n)(r)					(r)	(3.106)
						Vs(n−1)	Ws(n−1)

where the operator Vs(n−1) cuts o the tails of functions Ws(n−1)(r):

V	W	(r) = p	A,s	cGA,s	(3.107)
s	s			µs

Aµ A

Here, index A runs over atoms in a cell, multipliers pA,s are equal to unity only for those atoms on which the Mulliken populations of WF Ws(r) are more than the, fixed in advance, chosen threshold (for other atoms they are supposed to be equal to zero), and the vector GA,s defines a cell in which this population is maximal. Thus, the application of the operator Vs(n−1) leaves in the sum (3.107) only the members corresponding to atomic functions of one cell and not necessarily over all the atoms

of this cell. The renormalization of functions Vs(n−1)		(r) makes them, and,
	Ws(n−1)

hence, the functions W (sn)(r), more localized. To carry out the Wannierization step at the first cycle of the iterative procedure an initial guess built up from Γ point coe cients in Bloch functions is used in (3.106)

At the second stage termed by “localization”, the linear combination of WF, belonging to one cell is searched:

Ws(n)(r) =			s(n)(r)	(3.108)
	Os s	W
	s s
to maximize the functional (similar to that o ered by Boys for molecules)
IB(n) = s<t r s(n) − r t(n) 2				(3.109)

Here r (sn) is a point of Ws(n)(r) centering (centroid position). In other words, this procedure as much as possible “moves apart” WF centroids positions that promotes greater e ciency on the following step of the iterative procedure.

Advantages of the method: the algorithm is e cient and universal, it allows one to construct quickly well localized WF; the procedure is completely automated and included in the computer code CRYSTAL. The applications of this method to actual crystals are discussed in Chap. 9. Disadvantages of the method: in this method it is possible to use only Bloch functions calculated in LCAO approximation (PW-based Bloch functions can not be used); there is no possibility to apply di erent localization criteria in WF generation; the Mulliken atomic populations used in the iteration procedure are essentially dependent on the AO basis chosen and for di use AO are sometimes unrealistic.

The analysis of the basic methods of WF generation shows that though these methods are e ective enough, all of them possess those or other disadvantages. It is possible to specify three basic types of these disadvantages:

i) WF symmetry is not considered at all, or if it is considered, it is not formulated in the mechanism of definition of this symmetry. It leads, first, to an increase in time of calculations, and, secondly, to loss of the control over properties of WF symmetry during their construction.

963 Symmetry and Localization of Crystalline Orbitals

ii)The criterion of WF localization ensuring the generation of maximally localized WF is fixed or not defined. In the framework of the same method it is impossible to compare WF generated for di erent localization criteria.

An attempt to overcome the disadvantages mentioned was made in [42] where a variational method of Wannier-type function generation was suggested. This method is applicable with the di erent localization criteria, the Bloch functions can be calculated both in LCAO and in PW basis, the full symmetry is taken into account. In the next section we consider it in more detail.

3.3.4 Variational Method of Localized Wannier-function Generation on the Base of Bloch Functions

The background of Wannier-function symmetry analysis is the theory of representations (reps) of a space group G induced from the irreducible representations (irreps) of its site subgroup Gq G called, for brevity, induced representations (indreps). From the theory of indreps of space groups it follows that the construction of localized functions corresponding to a given simple or composite energy band is possible only if canonical orbitals of this band form the basis of some simple or composite indrep. This analysis permits us not only to establish the principle possibility to construct localized orbitals but also to define the possible positions of their symmetry localization center q and their symmetry with respect to site-symmetry group Gq.

Thus, the symmetry analysis consists of a procedure of identifying localized functions symmetry from the symmetry of the canonical orbitals of the considered energy band, or of establishing the fact that the construction of localized functions is impossible for the reasons of symmetry.

We assume that the canonical orbitals ϕ(mµγ) (k, r) ≡ ϕσ(k, r) ≡ ϕs(r) (the index m numbers the basis vectors of irrep γ with wavevector k, µ discriminates between the independent bases of equivalent irreps; the index σ replaces γ, m, µ and s replaces σ, k) of an energy band under consideration form a basis in the space Q(q,β) of some indrep of the group G(N ) of a crystal. The localized functions are defined by a unitary transformation in the space of indrep:

Wij(β)(r − an) = N − 21
	exp(−ikan)ψij(β)(k, r)	(3.110)
	k
or
Wτ(β)(r − an) = N − 21
	exp(−ikan)ψτ(β)(k, r)	(3.110a)
	k

In (3.110, 3.110a) the summation is over the set of N special points in the BZ, which corresponds to the cyclic model considered [55], and the quasi-Bloch functions

ψij(β)(k, r) ≡ ψτ(β)(k, r) are linear combinations of the canonical orbitals ϕ(mµγ) (k, r) ≡ ϕσ (k, r) with the same k belonging to the considered energy band:

ψij(β)(k, r) =	Uγmµ , ij (k)ϕmµ(γ) (k, r)	(3.111)
γmµ
or	Uσ , τ (k)ϕσ (k, r)
ψτ(β)(k, r) =		(3.111a)

σ