Quantum Chemistry of Solids / 22-LCAO Calculations of Perfect-crystal Properties

9.1 Theoretical Analysis of Chemical Bonding in Crystals

367

e

of the site group (D

4h

) of the oxygen atom, the linear combination W (e,a1) =

u

+

1

W (d,a1g )

+ W

(d,a2u)

of two others engenders the LWO sited on the bond line

√2

!

Zr–O and transforming according to IR a

of the symmetry group C4v of the bond.

1

!

!

(e,a1)

1

(d,a1g )

(d,a2u )

This LWO induces another one W−

=

√

2

W

− W

centered on the

Zr–O bonding line but at the other side of the same oxygen atom. On the whole, the
!	!	!

number of LWOs per unit cell remains the same. These functions describe the covalent bonding of the oxygen atom with its two neighboring Zr atoms. The numerical value of covalency can be evaluated from the centroid positions of these LWOs counted o the oxygen atom.

!

To find the centroid positions of the LWOs Wi(r) it is su cient to symmetrize the

!

LWOs Wi(r) calculated without symmetry by the usual group-theoretical methods. Instead of extracting the LWO s themselves from the results of computer code [23] application it was done in [612] by using the relation (9.58) with the normality condition and the orthogonality of the matrix α in (9.58) connecting the symmetrized LWOs and those found in the direct calculations. This matrix was found in numerical form and analyzed in [612]. It was found that LWOs Wi(r), i = 1, 2 almost coin-

cide with W

(d,eu )(r) although they have a noticeable admixture of LWO W (d,a1g )(r).

LWOs W

i

(r), i = 3, 4 are formed essentially by W (d,a2u )(r) and W (d,a1g )(r) although

i

!

W

(d,eu )

!

they have a noticeable admixture of LWO

(r). Using (9.59), one obtains

(d,ag )

(d,a2u )

!

!

W

(r)|x3|W

(r) = 0.912. The

centroid positions relative to oxygen atoms

!

for the LWOs W (r)(i = 1, 2, 3, 4), r

−

r

= 0 and the corresponding polarization frac-

!

!i

0

(d,eu )

(r) and W

(d,eu )

(r), P = 1.

tion Pi = 1

for all four LWOs. In particular, for W

1

2

!

They

describe lone pairs on the oxygen atom in the crystal. But for the linear com-

(e,a1)

!

!

binations W±

(r) (the IR a1 of the site group C4v of the Wycko position e on

the bond line Zr–O), r± − r0 = ±0.912e3

(in a.u.) and the polarization fractions

P

±

= 0.

54. These LWOs describe the partly covalent binding of atoms O and Zr in

!

the crystal SrZrO3. The other numerical characteristics (9.53) and (9.54) of this bond are: Q(O) = −1.54, Q(Zr) = −0.46, IO−Zr = 0.54, KO−Zr = 0.46. Taking into account the choice of valence and semicore electrons, atomic cores have the charges: +6(O), +8(Sr), +4(Zr). According to the q-basis index of the BR (9.60) the electron structure of the crystal can be represented by 15 LWOs per primitive cell, every orbital being occupied by two electrons:

charge +2=8–6		!
with WTAOs.
	2. Six orbitals W (d,eu )(r) of symmetry eu (px, py-type) are localized on O atoms
(site group D		, three atoms per primitive cell) and bring to every atom the charge
–4.	4h	!

1. Three orbitals W (b,t1u)(r) of symmetry t1u (p-type) are localized on Sr (site

group Oh) and bring to it the charge –6. Finally, atom Sr has in the crystal the rigorously. This value is close to that found by population analysis

!(e,a1)

3.Six orbitals W (r) of symmetry a1 (spz -type) are localized on the bonding line O–Zr (site group C4v , six bonding lines per primitive cell). The charge –2 associated with every orbital is partitioned between oxygen (–1.54) and zirconium (–0.46) atoms. As there are two bonds for the oxygen atom, its charge in the crystal is –1.08 (=+6-4-1.54 × 2). Equally for the zirconium atom, one has +1.24(=+4–0.46 × 6; 6 bonds for Zr). The traditional Mulliken population analysis gives values –1.29 and

368 9 LCAO Calculations of Perfect-crystal Properties

+2.02, the projection techniques –1.14 and +1.63, WTAO population analysis gives more ionic charges –1.67 +3.00, [611].

Using the symmetry analysis in MgO crystal [612] for the Bloch states of the four upper valence bands one obtains the q-basis of symmetry

(a, a1g + t1u)

(9.62)

with the oxygen atom in the Wycko position a. Electronic states of these energy bands are originated certainly from 2s, 2p-O atomic states. The points on the bond line of the atoms Mg and O (Wycko position e) have the symmetry C4v . Even the one-dimensional IR a1 of this group induces in the group Oh the six-dimensional representation a1(C4v ) ↑ Oh = a1g + eg + t1u(Oh). This means that there exists no LWO centered on this bond that could generate a four-sheeted upper valence energy band in the MgO crystal. One can say the same about the LWOs centered on the Wycko position f (x,x,x) with site group C3v : they could engender an eight-sheeted energy band.

Therefore, there is no other alternative but (9.62) for the BR corresponding to valence bands in the MgO crystal. For a reasonable interpretation of bonding in the

!	!	!	i	! i	−
crystal only the functions W1	(r) = W	(a,a1g )(r), Wi(r) = W1(a,t1u )(r), i = 2, 3, 4 are
suited. They are centered on an oxygen atom and r				= rO, Q(O) =		2, Ii = Pi =

tal corresponds to the

1, Ki = 0(i = 1, 2, 3, 4). The centroid positions and the values of other numerical characteristics of chemical bonding are not changed evidently for any linear combination of the functions Wi(r), i = 1, 2, 3, 4 in the space of IR t1u. The charges on atoms are: +2(Mg), –2(O) strongly. Such a picture of chemical bonding in MgO crysaccepted one for this ionic system. Nevertheless, it is necessary

!

to introduce such numerical characteristics of LWOs that could describe the small covalent part of Mg-O bonding and relative change of this part in the row MgO– CaO–SrO–BaO.

This discussion demonstrates the importance of the symmetry analysis in the procedure of the generation of LWOs in crystals. The theory of space group BR gives a powerful tool to determine the centroid positions of LWOs.

(1) If this centroid is at the symmetry point in the direct lattice, its position is defined exactly by the BR theory. If this position is occupied by an atom, the corresponding LWO describes an ionic bond or a lone pair. This is the case for the

!(d,eu ) !(d,eu )

MgO crystal and the LWOs W1 and W2 in the SrZrO3 crystal. If it is a middle point between two atoms, the corresponding LWO describes a pure covalent bond. This is the well-known example of the Si crystal.

(2) If the centroid is located on a symmetry line (possible site groups are

C6v , C4v , C3v , C2v , C6, C4, C3, C2), its exact position cannot be defined by the BR theory, and numerical calculations are necessary. The corresponding LWO describes a

!(e,a1)

two-atom partly covalent bond. This is the case of the LWO W (site group C4v ) in the SrZrO3 crystal with the following numerical characteristics of the bond:

IO−Zr = PO−Zr = 0.54, KO−Zr = 0.46, Q(O) = −1.54, Q(Zr) = −0.46

(3) If the centroid is located at a symmetry plane (possible site group is Cs), its exact position cannot be defined by the BR theory, and numerical calculations are also necessary. The corresponding LWO describes a three-(and more)-atom bond.

9.1 Theoretical Analysis of Chemical Bonding in Crystals

369

As we saw in the example of SrZrO3 crystal the absolute atomic-charge values di er in the di erent approaches to their calculation. Do di erent approaches (including those based on PW calculations) reproduce correctly at least the relative changes in the chemical bonding in crystals with analogous structure? This point is discussed in the next section.

9.1.6 Projection Technique for Population Analysis of Atomic Orbitals. Comparison of Di erent Methods of the Chemical-bonding Description in Crystals

At present, the electronic structure of crystals, for the most part, has been calculated using the density-functional theory in a plane-wave (PW) basis set. The one-electron Bloch functions (crystal orbitals) calculated in the PW basis set are delocalized over the crystal and do not allow one to calculate the local characteristics of the electronic structure. As a consequence, the functions of the minimal valence basis set for atoms in the crystal should be constructed from the aforementioned Bloch functions. There exist several approaches to this problem. The most consistent approach was considered above and is associated with the variational method for constructing the Wanniertype atomic orbitals (WTAO) localized at atoms with the use of the calculated Bloch functions. Another two approaches use the so-called projection technique to connect the calculated in PW basis Bloch states with the atomic-like orbitals of the minimal basis set.

The first approach (A) consists in using the technique for projecting the Bloch functions calculated in the PW basis set onto atomic orbitals of the minimal valence basis set for free atoms [617, 618]. Within this procedure the atomic-like functions are generated using the pseudopotentials chosen for the PW band structure calculations. Such functions are neither orthonormal nor complete in the sense of spanning the space of the occupied states. To measure how completely the localized atomic orbitals represent the eigenstates a so-called spilling parameter is introduced, which varies between one and zero. If the spilling parameter is nonzero a special projection procedure is needed to correctly define the density-operator for the incomplete basis of atomic-like functions. Since this constructed localized basis is not orthogonal, as in the case of LCAO calculations both Mulliken and L¨owdin schemes may be used.

The second approach (B), proposed for constructing quasiatomic minimal basis orbitals (QUAMBO) in [609] is also closely related to the projection technique (being in fact a projection reverse to the first one): the projection of a given minimal atomic basis is made on the Bloch states obtained in the PW calculations. This method can be regarded as a sort of localization procedure, and it describes the electronic structure of periodic solids in terms of localized quasiatomic minimal basis orbitals (QUAMBO).

Both variants of the projection technique were used for calculations of the local characteristics in the bulk of crystals [601, 609, 618] and, in a number of cases, led to results that are rather di cult to explain from the chemical standpoint. For example, the calculated charge of Mg atom in the ionic crystal MgO equals 0.76 [601]. Note that the construction of the WTAOs from the Bloch functions obtained in the PW basis set permitted almost ionic charges at atoms in the MgO crystal to be derived, see Table 9.14. However, it should also be noted that comparison of the data available

370 9 LCAO Calculations of Perfect-crystal Properties

in the literature on di erent characteristics obtained by the projection technique is complicated because the crystal orbitals are calculated with di erent variants of the DFT method, di erent pseudopotentials for excluding the core electrons from consideration, di erent atomic basis sets, etc. In order to interpret the results obtained by the projection technique, it is necessary to perform a comparative analysis of the density matrices (in the atomic orbital basis set) determined by the two aforementioned projection variants. In [619] such an analysis is carried out for the first time, and a simplified version of technique B was proposed to avoid cumbersome calculations of a large number of vacant crystal orbitals. Thus, in both variants A and B of the projection technique when the atomic orbital set for free atoms or free ions is assumed to be known, the population analysis can be carried out without invoking vacant crystal orbitals. We refer the reader for the mathematical details of this analysis to the publications [611, 619]. Here, we discuss only the numerical results, obtained in these studies for the local properties of electronic structure of crystals in both projection techniques and compare them with those obtained in traditional AO and WTAO Mulliken population analysis, including Bloch functions of vacant states. The comparison is also made with the results obtained with localized Wannier functions, generated only for the valence bands by method [63].

In Table 9.18 we give the local properties of electronic structure (atomic charges QA, covalencies CA, bond orders WAB and overlap populations RAB ) calculated by projection techniques A and B for the crystals with di erent nature of chemical bonding: Si, SiC, GaAs, MgO, cubic BN, and TiO2 with a rutile structure, R0 is the nearest-neighbor distance, given in ˚A.

Table 9.18. Local properties of electronic structure in projection technique, [619]

Crystal	R0		QA		CA	spl (10−3)		WAB		RAB
		A	B	A	B	A	B	A	B	A	B
MgO	2.107	1.609 1.607		0.630 0.632		1.4	1.1	0.112 0.113		0.095 0.096
BN	1.565	0.681 0.715		3.292 3.263		3.7	3.0	0.804 0.800		0.709 0.700
TiO2	1.951	1.730 1.739		3.474 3.459		2.2	1.4	0.552 0.550		0.305 0.302
Si	2.364	0.000 0.000		3.823 3.801		9.6	6.5	0.894 0.889		0.756 0.744
SiC	1.901	1.260 1.284		3.497 3.472		8.9	5.3	0.831 0.827		0.760 0.751
GaAs	2.451	0.361 0.380		3.231 3.202		1.8	1.8	0.768 0.765		0.636 0.629

The crystal orbitals were calculated by the DFT method in the plane-wave basis set with the CASTEP code [377] in the GGA density functional. A set of special points k in the Brillouin zone for all the crystals was generated by the supercell method (see Chap. 3) with a 5 × 5 × 5 diagonal symmetric extension, which corresponds to 125 points. In all cases, the pseudopotentials were represented by the normconserving optimized atomic pseudopotentials [621], which were also used to calculate the atomic potentials of free atoms. In the framework of both techniques (A, B), the population analysis was performed in the minimal atomic basis set; i.e. the basis set involved only occupied or partially occupied atomic orbitals of free atoms. It is well known that the inclusion of di use vacant atomic orbitals in the basis set can substantially change the results of the population analysis. For example, if the Mg 2p vacant atomic orbitals are included in the basis set, the charge calculated by technique A for the Mg

9.1 Theoretical Analysis of Chemical Bonding in Crystals

371

atom in the MgO crystal decreases from 1.61 to 1.06. Consequently, the inclusion of the Mg 2p functions leads to a considerable decrease in the ionic component of the bonding and, correspondingly, to an increase in the covalence of atoms. Chemically, it is di cult to explain this high covalency of the bonding in the MgO crystal.

The spilling parameter spl values are also given in Table 9.18, which characterize the accuracy of the projection of occupied crystal orbitals onto the space of the atomic orbitals (technique A) and the accuracy of the projection of atomic orbitals onto the Bloch functions (technique B).

The atomic charges obtained by technique A can be compared with those calculated in [601] with the use of a similar method. The atomic charges and the spilling parameters presented in [601] are considerably smaller than those in the calculation discussed. In our opinion, the main factor responsible for this disagreement is the di erence between the atomic basis sets that are used for the projection. This assumption is confirmed by the following fact noted in [601]. The removal of the Si d vacant orbital from the atomic basis set leads to an increase in the charge at the Si atom in the SiC crystal from 0.66 to 1.25 and an increase in the spilling parameter from 2 ×10−3 to 9 ×10−3. It follows from Table 9.18 that, in calculations in the minimal basis set (without the Si d function), these quantities are equal to 1.26 ×10−3 and 8.9 × 10−3, respectively, which agrees well with the results obtained in [601] (without the Si d function). It is evident that an increase in the size of the basis sets should result in a decrease in the spilling factor in method A, because this is accompanied by an increase in the space of the atomic orbitals onto which the occupied crystal orbitals are projected. However, we believe that it is incorrect to decrease the spilling parameter in the population analysis of the atomic orbitals by increasing the atomic basis sets, as was actually done in [601]. For example, if the basis set of the atomic orbitals is increased to a complete set, the spilling coe cient can be reduced to zero; however, the population analysis of the atomic orbitals in this basis set loses all physical meaning.

For the purely covalent cubic Si crystal, the results obtained by technique B can be compared with the data reported in [609], for Si–Si bond order WAB = 0.885 and the Mulliken overlap population is RAB = 0.756. As can be seen from Table 9.18, the results obtained by technique B (WAB = 0.889, RAB = 0.744) are very close to those given in [609]. The insignificant di erences can be associated with the use of di erent variants of the DFT method in the plane-wave basis set. Note that technique B used in [619] is a simpler modified variant of the method proposed in [609] and does not deal with the vacant Bloch states.

It can be seen from Table 9.18 that the data calculated by the techniques A and B are in good agreement. It should only be noted that the ionic component of the bonding (QA) in technique B is somewhat larger than the analogous component in technique A. Correspondingly, the covalent component (CA, WAB , RAB ) is somewhat smaller in technique B. Moreover, the spilling parameters in technique B are regularly smaller in magnitude.

In Table 9.19 we compare the local characteristics of the electronic structure determined for the TiO2 crystal with the use of the projection technique and those obtained in an earlier study [623], in which the Wannier-type atomic functions of the minimal valence basis set of titanium and oxygen atoms were constructed by the variational method. In [623], the Bloch functions were calculated by the DFT method

372 9 LCAO Calculations of Perfect-crystal Properties

Table 9.19. Local characteristics of the electronic structure of the TiO2 (rutile) crystal*

Method	QTi	CTi	WTi−O	VTi
A	1.79	3.36	0.51	4.13
B	1.80	3.34	0.51	4.12
LCAOM	1.78	3.30	0.52	4.08
LCAOL	1.04	4.29	0.66	4.52
WTAOM	1.98	3.16	0.51	4.11
WTAOL	1.92	3.23	0.52	4.12

*LCAOM(L) – traditional Mulliken (L¨owdin) population analysis with basis set used in LCAO calculations; WTAOM(L) – Mulliken (L¨owdin) population analysis with WTAO basis set

in the LCAO approximation with the CRYSTAL code [23] and the atomic pseudopotentials taken from [484]. It follows from Table 9.19 that the local characteristics of the electronic structure of the TiO2 crystal are close for two projection variants (techniques A, B) and di er only slightly from the results of the Mulliken (not L¨owdin!) population analysis performed in the basis set of the atomic orbitals at the LCAO level. It is seen that the traditional L¨owdin population analysis with the initial LCAO basis set containing di use atomic functions leads to an overestimated covalence of chemical bonding.

When analyzing the populations in the WTAO basis set, the L¨owdin orthogonalization of the basis set leads to insignificant di erences as compared to the results of the Mulliken population analysis (due to the localized character of the Wannier-type atomic functions). The L¨owdin atomic charges and the atomic covalences correspond to a somewhat lower degree of ionicity. An important conclusion can be drawn from comparison of WTAO results with those obtained in the projection technique: despite the use of substantially di erent basis sets for calculating the crystal orbitals (plane waves in the projection technique, LCAO in the WTAO method), the local characteristics of the rutile electronic structure obtained within the two approaches are in good agreement. Meanwhile, the results of traditional population analysis di er from those obtained by the projection technique (see Table 9.19). This di erence is demonstrated also in Table 9.20, where for SrTiO3 and SrZrO3 crystals the results of traditional Mulliken population analysis are compared with those obtained by the projection technique (both crystals were taken in cubic perovskite structure with the space group P m3m.

The results, given in Table 9.20, were obtained by the DFT PBE method in LCAO and PW calculations (the details of the LCAO basis set and other parameters choice can be found in [611]). In the projection technique the atomic orbitals were numerically calculated for occupied states of free atoms using the same pseudopotentials as in the PW band-structure computations. As can be seen in Table 9.20, the results of projection-techniques application, unlike the traditional population analysis, exhibit the correct trends in the chemical-bonding change when the Ti atom is replaced by the Zr atom in the same structure (the experimental lattice constants were used for both cases). The population analysis based on the projection technique shows that ionicity of chemical bonds M–O (M= Ti, Zr) in SrZrO3 is larger than in SrTiO3. This is in agreement with the experimental data on the bandgaps in SrTiO3 (3.2

9.1 Theoretical Analysis of Chemical Bonding in Crystals

373

Table 9.20. Local properties of electronic structure of SrMO3 crystals (M=Ti, Zr) in Mulliken population analysis, projection techniques and WTAO population analysis [611]

Quantity	Mulliken		Projection A		Projection B		WTAO
	LCAO		PW		PW		LCAO
	SrTiO3	SrZrO3	SrTiO3	SrZrO3	SrTiO3	SrZrO3	SrTiO3	SrZrO3
Q(Sr)	1.85	1.85	1.83	1.80	1.83	1.80	2.00	2.00
Q(M)	2.24	2.02	1.46	1.63	1.46	1.63	2.54	3.00
Q(O)	–1.36	–1.29	–1.09	–1.14	–1.09	–1.14	–1.51	–1.67
WSr−O	0.036	0.033	0.025	0.026	0.026	0.027	0.00	0.00
WM−O	0.461	0.496	0.626	0.607	0.663	0.606	0.389	0.269

eV) and SrZrO3 (5.9 eV) and the relative Pauling electronegativities of Ti (1.4) and Zr (1.2) atoms. Both variants of the projection approach give very close results for the local properties of electronic structure, possibly because the projection method A produces almost orthogonal crystalline orbitals. From Table 9.20 one can conclude that the traditional Mulliken population analysis within the framework of LCAO calculations does not correctly reproduce the relative ionicity of chemical bonding in SrTiO3 and SrZrO3 crystals, while the WTAO-based populations and projectiontechniques application allows one to reflect properly the more ionic nature of SrZrO3 in comparison with SrTiO3.

Table 9.21 gives the localization indices of Wannier functions in SrTiO3 and SrZrO3 crystals calculated for the valence bands by the method used in [63].

Table 9.21. Localization indices of localized Wannier functions in SrTiO3 and SrZrO3 crystals [611]

Crystal	LWF Wi(r)	λi	pi	qO	qOs	qOpx	qOpy	qOpz
SrTiO3	a (i=1)	1.145	0.978	0.934	0.191	0.731	0.011	0.002
	b (i=2)	1.145	0.978	0.934	0.191	0.011	0.731	0.002
	c (i=3)	1.252	0.627	0.890	0.311	0.064	0.064	0.453
	d (i=4)	1.230	0.670	0.890	0.296	0.117	0.117	0.373
SrZrO3	e (i=1)	1.183	0.999	0.918	0.136	0.005	0.778	0.000
	f (i=2)	1.183	0.999	0.918	0.136	0.778	0.005	0.000
	g (i=3)	1.302	0.610	0.870	0.343	0.064	0.064	0.403
	h (i=4)	1.300	0.610	0.870	0.341	0.066	0.066	0.400

Localized Wannier functions (LWFs) have been calculated for three upper valence bands in SrTiO3 and SrZrO3, represented mainly by O 2p, Sr 4p, and O 2s atomic states (in the case of SrZrO3 the last two bands overlap considerably). A total of 15 crystalline orbitals have been used to generate, correspondingly, 15 LWFs per primitive unit cell: in both crystals under consideration, three oxygen atoms occupy the same Wycko positions, and four LWFs can be attributed to each oxygen atom. It was found by calculations with CRYSTAL03 code [23] that the centroids of four functions are positioned near the center of one oxygen (at distances of about 0.3 ˚A).

374 9 LCAO Calculations of Perfect-crystal Properties

Table 9.21 presents the following localization indices: the Mulliken population of LWF (9.46) qO due to the oxygen atom and corresponding contributions from 2s and 2p- oxygen functions. The atomic delocalization index of LWF

λi = " (qAig )2	#	−1
		(9.63)
A g

is a measure of the extent of the LWF in terms of the number of “contributing” atoms. This index is close to 1.0 for much localized LWF, and it is > 1.0 for more di use LWFs. The polarization fraction, (9.49) for all LWFs is also given.

Figure 9.3 presents the fragment of crystal containing four primitive unit cells to show the axes orientation: the oxygen atom is positioned in the center and the two nearest Ti or Zr atoms lie on the z-axis.

The counter density maps of the obtained LWFs are shown in Fig. 9.4.

As is well seen from Fig. 9.4 and Table 9.21 (functions a, b and e, f) two of the four LWFs for each crystal are perpendicularly oriented toward the M–O–M line and lie along x- (Fig. 9.4, a and e) and y- ( Fig. 9.4, b and f) axes, respectively.

Fig. 9.3. Local environment of oxygen atom in cubic perovskites SrMO3 (M= Ti, Zr).

The polarization fractions of LWFs a, b for SrTiO3 and e, f for SrZrO3 are very close to 1.0; i.e., these LWFs practically represent the electron lone pairs of oxygen ions O2− with a predominant contribution of oxygen 2px- and 2py -orbitals. Two other LWFs (see Fig. 9.4 c, d, g and h) are directed along the z-axis. As can be seen from the populations (Table 9.21), they closely correspond to hybrid spz orbitals of oxygens. The polarization fractions are substantially less than 1.0 for these two orbitals, revealing the rather strong covalent character of Ti (Zr)–O bonds in both crystals (these covalency e ects are well known for transition-metal–oxygen compounds). Thus, localized Wannier functions, generated with the inclusion of only the valenceband states, appear to be a useful tool in analyzing the electronic distribution and

9.2 Electron Properties of Crystals in LCAO Methods

375

Sr	Sr Sr	SrO	Ti	O O	Ti	O
	9	9	9		9
a	b	c			d
Sr	Sr Sr	Sr O	Ti	O O	Ti	O
Sr	Sr Sr	Sr O	Zr	O O	Zr	O
	9	9	9		9
e	f	g			h
Sr	Sr Sr	Sr O	Zr	O O	Zr	O

Fig. 9.4. Localized orbitals for oxygen atom in cubic SrTiO3 (a,b,c,d) and SrZrO3 (e,f,g,h). Functions a,b,e,f in Sr-plane; functions c,d,g,h in MO plane (M= Ti, Zr)

chemical bonds in solids. It should be noted that the LWFs obtained are not entirely symmetry-agreed: they do not transform via the representations of the oxygen atom site-symmetry group under the point-symmetry operations because there is no explicit symmetry constraints in the localization procedure [63]. Further symmetrization of WTAOs is desirable to correlate the results with those qualitatively known in quantum chemistry. This symmetrization procedure was discussed in Sect. 9.1.5 and allows the LWFs centered on oxygen atoms and transforming over eu and a1u irreducible representations of the oxygen site-symmetry group D4h to be obtained. The symmetrization of LWF is considered in general form in [109].

Concluding this section we note that the projection technique with PW calculations gives reasonable results for local properties of electronic structure, close to those obtained in LCAO calculations after using WTAO population analysis. In general, WTAO results correspond to the more ionic bonding. The useful information about the atomic nature of the band states is obtained in the PW calculations of solids from the orbital and site decompositions of the densities of states (DOS). As an example, we refer to the DFT PW calculations [620] of transparent conductive oxides In4Sn3O12 and In5SnSbO12 – complicated rhombohedral structures with 19 atoms in the primitive unit cell. It was found by DOS analysis that the tops of the valence states of both materials are formed by oxygen 2p states, whereas the bottom of the conduction bands are due primarily to the Sn 5s electrons. For such crystals LCAO calculations with a posteriori WTAO generation or the projection technique application would be very cumbersome.

The analysis of DOS and other one-electron properties of crystals in the LCAO calculations is discussed in the next section.

9.2 Electron Properties of Crystals in LCAO Methods

9.2.1 One-electron Properties: Band Structure, Density of States, Electron Momentum Density

The local properties of chemical bonding in crystals considered above are defined by the electron-density distribution in real space described by a one-electron density

376 9 LCAO Calculations of Perfect-crystal Properties

matrix (DM), see (4.125)-(4.127). The latter is calculated self-consistently for the finite set of L discrete k-points in BZ and corresponds to the cyclic cluster of L-primitive unit cells modeling the infinite crystal:

1			M	L
			i
ρµν (Rn) = L				exp(−ikRn)Ciµ(kj )Ciν (kj )ni(kj )	(9.64)

=1 j=1

The total number of bands M in (9.64) equals the number of AOs associated with the primitive unit cell. The one-electron energy levels form energy bands consisting of L levels in each band. Therefore, each energy band can allocate 2 × L electrons; if in the unit cell there are n electrons, and the bands do not cross, the lowest n/2 bands are occupied and are separated from the empty bands. In this case the occupation numbers in (9.64) are ni = 2, 0 for the occupied and empty bands, respectively (insulators and semiconductors). However, if n is odd, or if the valence and conduction bands cross, more than n/2 bands are partially occupied (metal).

The eigenvalue spectrum of an infinite periodic system does not consist of discrete energy levels as the wavevector k changes continuously along the chosen direction of the BZ. The DM of an infinite crystal is written in the form where the summation over the discrete k-vectors of the BZ is replaced by the integration over the BZ. In the LCAO approximation written for the cyclic-cluster (4.126) for DM in the coordinate space is replaced by

				M
				i
				ρµν (Rn) = ρµν(i) (Rn)
				=1
		M
	1	BZ
=	VBZ i=1			exp(−ikRn)Ciµ(k)Ciν (k) (εF − εi(k)) dk	(9.65)

where ρ(µνi) (Rn) is the contribution of the ith energy band.

For the infinite crystal at each cycle of the SCF process, an energy εF (the Fermi energy) must be determined, such that the number of one-electron levels with energy below εF is equal to the number of electrons (or, in other words, the number of filled bands below εF is equal to half the number of electrons in the unit cell). The Fermi surface is the surface in reciprocal space that satisfies the condition εi(k) = εF . By limiting the integration over the BZ to states with energy below εF , a Heaviside step function θ(εF − ε) excludes the empty states from the summation over the energy bands. In fact, the band structure of the infinite crystal is obtained after the cycliccluster self-consistent calculation by the interpolation of the one-electron energy levels considered as continuous functions ε(k) of the wavevector. The band structure of solids is an important feature, defining their optical, electrostatic and thermal properties. Both the electron charge distribution and the band structure are defined by the selfconsistent DM of a crystal.

As we have seen, two binary oxides – MgO in sodium chloride structure and TiO2 in rutile structure (see Chap. 2 for descriptions of these structures) – di er significantly in the character of chemical bonding, which is due to the electron-density distribution being much di erent in them. We discuss now the di erences in the band structures of these crystals.