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

9.1 Theoretical Analysis of Chemical Bonding in Crystals

347

Fig. 9.1. Geometric structure of [Ti2O129 −] cluster, modeling Ti2O3 crystal in corundum structure

density matrix o diagonal elements are not taken into account in DFT approximation for exchange).

In Ti2O3 crystal the same di erence in the total energy for Sz = 0 and Sz = 1 cases was found: In HF and HBS calculations, antiferromagnetic ordering gives an energy gain compared with the diamagnetic picture (compare in Table 9.8 the total energy for Sz = 0 and Sz = 1 in UHF and HBS calculations); however, Becke exchange gives the opposite order of the total energies (compare Sz = 0 and Sz = 1 for KSS calculations). As for covalencies of oxygen atoms and WTi−O, one can draw from Tables 9.7 and 9.8 the following conclusions: 1. Only spin correlation of Ti atoms gives small changes for these properties of electronic structure in both crystals (compare in Tables 9.7 and 9.8 RHF-UHF, KS-KSS, and HB-HBS results). 2. Inclusion of Coulomb correlation with Becke exchange gives a more covalent picture of chemical bonding (compare the results for RHF and KS calculations), but only the correlation correction gives small changes of the covalence of chemical bonding (compare the results for RHF and HB calculations).

As was demonstrated in [571], the full valence of the metal atom, calculated for di erent metal oxides according to definition (9.13), correlates with the oxidation state in the semiempirical calculations, implicitly taking into account the correlation e ects (see Chap. 6). In TiO2 and Ti2O3 crystals the Ti atom oxidation state should be +4 and +3, respectively. The calculated total valence VTi is close to oxidation state 4 for all calculations of TiO2 (see Table 9.7) but strongly depends on the spin-correlation description for Ti2O3 (see Table 9.8). The calculated total valence VO appears to be close to a reasonable value, 2.0, in all cases.

348 9 LCAO Calculations of Perfect-crystal Properties

For rutile TiO2 the LCAO results for Mulliken charges and the Ti–O overlap population can be compared with those found in DFT-PW calculations [601] after projecting the eigenstates onto a localized basis set of atomic pseudo-orbitals. In KSLCAO calculations, QTi = 1.88 and overlap populations of Ti–O bonds are 0.11 and 0.10 for nearest and next-nearest oxygens, respectively. In PW calculations [20], QTi = 1.46, and the overlap populations are 0.35 and 0.43, respectively. This comparison shows that the calculated local properties’ numerical values depend on the choice of basis set used in calculations (LCAO or PW sets). Moreover, in PW calculations the procedure of projection is used to receive atomic orbitals (see Sect. 9.1.5), which is not necessary in LCAO calculations.

From the discussion of the results obtained it follows that the spin-correlation effects are described in an essentially di erent manner in HF and DFT calculations for Ti2O3 crystal and as a result give di erent descriptions of Ti–Ti interaction: in the HF method, this interaction is weak (for energetically more favorable antiferromagnetic ordering); in the DFT method, the ferromagnetic ordering appears to be more favorable and Ti–Ti interaction remains essential.

It seems reasonable to use post-HF descriptions of spin-correlation e ects other than that used in HF and DFT methods. For this, one can perform, say, UHF and CASSCF electronic-structure calculations of reasonably selected clusters of the crystal under consideration to get single-determinant UHF and multiconfigurational CASSCF wavefunctions. Then, expanding UHF β-MOs over an α set and inserting these expansions in the UHF determinant, we obtain a certain multiconfigurational wavefunction that is, in general, a mixture of di erent spin states. Annihilation of the highest-spin contaminants leads to the multiconfigurational wavefunction that should be compared with the CASSCF one. If these functions are similar, that is, in both functions the same determinants are dominating, then one can state that the UHF method accounts for part of the Coulomb-correlation e ects. Estimation of relevant energies may give an impression of what part of the correlation energy (in comparison with the CASSCF method) is obtained on the UHF level. Then, all qualitative conclusions concerning a cluster may be directly applied to the corresponding crystal. In particular, if local molecular characteristics (such as atomic charges, bond orders, valences, and covalences) of a cluster are notably a ected by correlation e ects and UHF cluster calculation reproduces this tendency, then the UHF calculation of a crystal should certainly display the same tendency.

There exists one simple but important case where more detailed analysis is possible. Indeed, let us consider a molecular system (cluster) with a nondegenerate singlet ground state that can be described by a MCSCF wavefunction with dominating contribution of only two configurations:

ΨMCSCF = C1 det | . . . ϕhαϕhβ| + C2 det | . . . ϕlαϕlβ|

(9.28)

With the aid of a nondegenerate (nonorthogonal) transformation (for the standard case C1 > 0 and C2 < 0) generalized valence bond orbitals may be constructed:

ϕa = -	1
				4C1		ϕh +	4|C2		|ϕl		(9.29)
		2
ϕb = -	1
	2				C1	ϕh −		|C2	|ϕl		(9.30)
			4				4

9.1 Theoretical Analysis of Chemical Bonding in Crystals

349

Simple algebraic manipulations lead to the conclusion that

ΨMCSCF Φab + Φba	(9.31)
where
Φab = det | . . . ϕaαϕbβ|	(9.32)
and
Φba = det | . . . ϕbαϕaβ|	(9.33)

are typical single-determinant UHF configurations. These determinants correspond to zero projection of the total spin but are not, in general, S2 eigenfunctions. Their symmetrical combination gives the initial singlet wavefunction (9.28) whereas the antisymmetrical combination gives the related triplet state. In general, the nondiagonal

| ˆ |

matrix element Φab H Φab di ers from zero and its absolute value may serve as a certain measure of the UHF method’s capability to account for Coulombcorrelation e ects. Indeed, if this matrix element is small (in absolute value) then singlet and triplet states are energetically close and the UHF method for a singlet state accounts for about the same part of the correlation e ects as the MCSCF one with wavefunction (9.28).

Calculations of the lowest singlet and triplet states of cluster [Ti2O129 −], see Fig. 9.1, were performed by UHF and CASSCF methods using the GAMESS program set for molecular calculations [35]. The cluster includes two nearest Ti atoms and their nearest oxygen neighbors. The initial MOs for this cluster were obtained by the RHF method with C3h as the point-symmetry group. The active space of the CASSCF method involved five highest occupied and three lowest virtual MOs (active space [2a 2a 2e ]10). From Table 9.9 it is seen that the cluster chosen reasonably reproduces the Ti atom’s local properties compared with periodic calculations for Sz = 0 (see Table 9.8). In UHF calculations, QTi = 2.22 (cluster model), QTi = 2.32 (periodic model), CTi = 1.37 (cluster model), and CTi = 1.23 (periodic model). It turned out that the singlet state of the cluster under consideration is described by

two configurational wavefunctions.
0.8[ϕh2 ϕl0] − 0.6[ϕh0 ϕl2]	(9.34)

whereas the related triplet state with SZ = 1 is reasonably described by a single determinant.

The comparison of the total energies for the cluster shows (see Table 9.9) that the UHF calculation reproduces well the small di erence of the CASSCF energies for the total spin values S = 0 and S = 1. Moreover, as seen from Table 9.9, the UHF calculation describes correctly the influence of the correlation e ects on the Ti– Ti bond-order value. Indeed, in RHF calculations this bond order constitutes 0.88, whereas in UHF calculations the bond-order values are 0.03 for Sz = 0 and 0.01 for Sz = 1, in full accordance with CASSCF calculations (0.09 for S = 0 and 0.01 for S = 1). At the same time, the Ti atomic charges are practically not a ected by the method used.

Calculations of the electronic structure of crystals TiO2 and Ti2O3 and corresponding cluster show that numerical results for local properties of the electronic structure of crystals (atomic charges, covalences, free and total valences, and bond

350 9 LCAO Calculations of Perfect-crystal Properties

Table 9.9. Energies and local characteristics of electronic structure of molecular cluster [Ti2O9]12− in UHF and MCSCF calculations

	Characteristics of	UHF		MCSCF
	electronic structure
	Sz	Sz =0 Sz =1		Sz =0 Sz =1
	E (a.u.)	–137.874 –137.867		–137.913 –137.904
	QTi	2.22	2.28	2.11	2.12
	QO1	–1.68	–1.68	–1.63	–1.64
	QO2	–1.90	–1.90	–1.89	–1.89
	WTi−Ti	0.03	0.01	0.09	0.01
	WTi−O1	0.17	0.17	0.19	0.20
	WTi−O2	0.27	0.27	0.31	0.32
	CTi	1.37	1.34	1.62	1.56
	CO1	0.63	0.62	0.70	0.71
	CO2	0.21	0.21	0.23	0.24
	VTi	3.01	3.02	3.07	3.04
	VO1	2.02	2.02	2.02	2.03
	VO2	2.01	2.01	2.01	2.01

orders) depend on (a) Choice of basis for calculation (LCAO, PW); (b) Choice of Hamiltonian (RHF, UHF, DFT); (c) Choice of population analysis (POPAN) scheme (L¨owdin, Mulliken). The e ects of electron correlation on local properties may be estimated using (a) periodic and cluster DFT and hybrid HF–DFT calculations in the periodic model; (b) post-HF calculations in the molecular cluster model; (c) the results of (a) and (b) agree when the appropriate cluster choice is made and the same basis and population analysis are used.

To minimize the essential dependence of results on the basis-set choice and POPAN, it is reasonable to generate Wannier-type atomic functions that are orthogonal and localized on atomic sites. These functions can be generated from Bloch-type functions (after relevant symmetry analysis), see Chap. 3. In the next section it is shown that the results of POPAN with use of Wannier-type atomic functions weakly depend on the basis choice in Bloch function LCAO calculations.

9.1.4 Wannier-type Atomic Functions and Chemical Bonding in Crystals

As we have seen above chemical bonding in crystals (as well as in molecules) is analyzed in terms of the local properties of the electronic structure, obtained from the one-electron density matrix, written in a localized basis. Since local properties of electronic structure are essential ingredients of a number of theories and models (for example, the numerical values of atomic charges are used in the atom–atom potentials of the shell model), their estimation is of great importance. Traditionally, the same AO basis is used both in LCAO SCF calculations and in the local properties definition, as was noted above. However, this approach is not always reliable, since the results of the population analyses are often strongly dependent on an inclusion of di use orbitals into the basis (useful for the electronic-structure calculations) and on the scheme chosen for the population analysis.

9.1 Theoretical Analysis of Chemical Bonding in Crystals

351

As discussed in Chap. 3 the variational method of localized functions generation allows development of an approach of the chemical-bonding analysis, which is much less basis-set dependent than the conventional approaches. The population analysis in this case is done using the density matrix given in the minimal valence basis of Wanniertype atomic orbitals (WTAOs) – Wannier functions centered on atoms and having the behavior of atomic-valence states in the cores of atoms [47, 603, 604]. WTAOs are analogous to atomic functions in form and by symmetry but, in contrast to the initial LCAO basis, they are directly connected with the calculated electronic band structure and the Bloch states involved. WTAOs are constructed from the occupied and some vacant Bloch states, chosen so that they assure the maximal localization for the corresponding WTAOs (see Chap. 3). The summary of the results of WTAO application for the calculations of local properties of the electronic structure of periodic systems is given in [605].

WTAOs are defined as the Wannier functions that are constructed from a set of specially chosen occupied and vacant bands and have a definite symmetry (they are centered on atoms and transform via irreducible representations (irreps) of the corresponding site groups). Thus, the index t in (3.114) may be substituted by several

indices – i, j, a, β, µ – that reflect the symmetry properties of the WTAOs – Wijaβµ(r − an): the index a marks the symmetrically nonequivalent atoms, the index j runs

through the set of symmetrically equivalent atoms of the type a; the Wycko positions of these atoms qa are characterized by their site groups, whose irreps are labeled by the index β, the index i numbers the basis functions of the irrep β and the index µ discriminates between the independent sets of the functions transforming according to the same irrep β. The requirement for WTAOs to have fixed symmetry properties implies additional restrictions on the matrices U (k) in (3.112). These restrictions can be completely taken into account by using, instead of canonical Bloch orbitals, their linear combinations, symmetrized according to the desired irreps of the site-symmetry groups of atoms [47].

WTAOs were generated by the variational method [604], which allows both orthogonal Wijaβµ(r − an) and nonorthogonal Vijaβµ(r − an) sets of symmetry adapted

	WTAOs to be obtained. In this method the functional
	Iβα =							2
		ωβα		V11(β,a1)(r)				dr	(9.35)
	with the weight function:				−
	ωβα = πr2	−3/2 exp				(r − qa)2			(9.36)
	βα						2rβα2

is maximized (rβα2 is a parameter, for which a value of 1 ˚A was used). The orthogonal

WTAOs Wijaβµ(r − an) are obtained from the nonorthogonal ones Vijaβµ(r − an) via L¨owdin’s procedure:

−1/2

W = V S(V )

(9.37)

where S(V ) is the overlap matrix of nonorthogonal WFs Vijaβµ(r − an), and W and V are row vectors constituted by the sets of the orthogonal and nonorthogonal WFs, cor-

respondingly. The orthogonalization procedure can be performed either in the direct or reciprocal space. The latter is computationally faster.

352 9 LCAO Calculations of Perfect-crystal Properties

Before constructing the minimal basis of WTAOs one should choose the energy bands whose states are to be used in the process of the WTAOs generation. For each of the WTAOs a band or a group of bands is chosen according to the following criteria: the band states should be compatible with the corresponding WTAOs by symmetry, in the context of the induced representations (indreps) theory, see Chap. 3, and provide the maximal localization for them. Hereafter, we mark such bands by the type of the WTAOs corresponding to them (e.g. s-band of the oxygen atom: the s-WTAOs centered on oxygen and constructed from the states of this band appear to be the most localized comparing to those constructed from any other states). After the desired bands for all the WTAOs are chosen the final WTAOs are constructed using the states from this group of bands. When the WTAOs (orthogonal or nonorthogonal) are found, the density matrix is calculated in the localized basis of these functions. The local properties are expressed via the matrix elements of this matrix.

WTAOs’ analog of Mulliken population analysis is made using nononorthogonal WTAOs.

Let us now consider WTAOs VνA(r −an) (ν-th WTAO of atom A in the nth cell)

as the linear combinations of Bloch functions:
VνA(r − an) = bτ k,νAnϕrk(r), or V = ϕb, ϕ = Vb−1	(9.38)

τ k

where Bloch functions ϕrk(r) are given in LCAO or PW basis. The density matrix in the corresponding basis ρ(V ) can be written in the following way:

ρ(V ) = 2¯b−1¯b

(9.39)

¯
where b is the truncated matrix b with rows corresponding only to the occupied states.
For the orthogonalized WTAOs:
WνA(r − an) =
	dτ k,νAnϕrk(r), or W = ϕd, ϕ = Vdf †	(9.40)
	τ k
the density matrix ρ(V ) is given by
	ρ(V ) = 2d¯†d¯	(9.41)

The electronic population NAn of an atom A in the nth primitive cell and an order of the bond WA0,Bn between an atom A of the reference cell and an atom A in the nth cell can be obtained from the density matrix (9.41):

NAn = NA0 = ρννA0,A0	(9.42)
ν
WA0,Bn = ρµνA0,BnρνµBn,A0	(9.43)

µν

Using these quantities one can obtain the atomic charge QA (9.10) of an atom A, the covalence CA (9.12) and total valence VA (9.13) of this atom.

To demonstrate the e ciency of WTAO use for chemical-bonding analysis, we consider at first the cubic crystals with the perovskite structure [603]. The LCAO

9.1 Theoretical Analysis of Chemical Bonding in Crystals

353

approximation in the HF and DFT methods was used to calculate the band structure and the Bloch functions of bulk SrTiO3 (STO), BaTiO3 (BTO), PbTiO3 (PTO) and LaMnO3 (LMO) crystals. The Hay–Wadt pseudopotentials [483] were adopted: smallcore for Ti, Sr, Ba, La and Mn atoms, large-core for Pb atoms. The oxygen atoms were included with the all-electron basis set. The optimized Gaussian-type atomic basis was taken from [606], where the same pseudopotentials were used. The basis optimization made it possible to obtain the lattice constant, bulk modulus and elastic constants of the considered crystals that were in good agreement with the experimental data. For the exchange-correlation functional the hybrid HF-DFT form (B3PW) was used, as it reproduces, for optimized basis sets, optical band gaps close to the experimental values [606].

The Monkhorst–Pack [14] mesh of 4×4×4 = 64 k-points, used in the calculations, corresponds to a cyclic model with 64 primitive unit cells in the direct lattice, see Chap. 3. An increase of this number in the bandstructure calculations up to 8×8×8 = 512 does not lead to any significant changes in the values of the one-electron energies.

The cubic structure of perovskite-like ABO3 compounds is characterized by the primitive cubic lattice with the Oh1 (P m3m) space group, see Sect. 2.3.2. The A-atoms and B-atoms occupy the Wycko positions a (0,0,0) and b (0.5,0.5,0.5), correspondingly, with the site group Oh. The oxygen atoms occupy c (0.5,0.5,0) positions, site group D4h. The minimal atomic-like basis for the crystals under consideration consists of s- and p-type functions of the oxygen atoms, s- and d-type functions of the Ti atoms and s- and p-type functions of Sr, Ba and Pb in STO, BTO, PTO, or s- and d-type functions of La and Mn in LMO. So these are the types of WTAOs to be used in the population analysis for the compounds under consideration. Table 9.10 lists the symmetries of these functions.

Table 9.10. The symmetry properties of the WTAOs of the minimal valence basis for STO, BTO, PTO and LMO crystals

Atom	Function’s type	Symmetry
Sr, Ba, Pb	s	(a, a1g )
	p	(a, t1u)
La	s	(a, a1g )
	d	(a, t2g ) + (a, eg )
Ti, Mn	s	(b, a1g )
	d	(b, t2g ) + (b, eg )
O	s	(c, a1g )
	p	(c, a2u) + (c, eu)

Once the types of WTAOs are determined one should find the energy-bands states corresponding to these WTAOs. First, the bands to be chosen should be symmetry compatible with the corresponding WTAOs (see [604] for details). Secondly, the WTAOs constructed from the Bloch states of the chosen bands should be principally the most localized WTAOs of the corresponding type. The analysis of the electronic structure of STO and BTO has shown (excluding the semicore states from consideration) that the oxygen s- and p-bands and the Sr (Ba) atom p-bands form 15 upper

354 9 LCAO Calculations of Perfect-crystal Properties

valence bands states, see also next section. The Ti atom d-states form the five lowest conduction bands, while the Sr (Ba) and the Ti atom s-states are located among vacant conduction-band states very high in energy. In the case of PTO the di erence in the location of the bands is that the Pb atom s-band is one of the valence bands and the Pb atom p-bands are vacant and lie high in the conduction band. This agrees with the results of DFT LDA calculations [607].

Unlike STO, BTO and PTO, the electronic structure of cubic LMO is metallic both in HF and DFT calculations (a nonzero bandgap in LMO appears only if an orthorhombic structure with four formula units in the primitive cell is considered). The s- and p-bands of the oxygen atoms are occupied. As in STO and BTO the s- bands of the metal atoms are located at high energies. But the d-bands of both metal atoms are located in the region of the Fermi energy. These bands are partly occupied and partly vacant, which is caused by the metallic nature of the calculated electronic structure.

The WTAOs that are used in the population analysis are built from the space of all the chosen bands taken together. The density matrix is then calculated using the coe cients connecting the orthogonal or nonorthogonal WTAOs and the occupied Bloch states.

In Table 9.11 the atomic charges for STO, BTO, PTO, correspondingly, calculated by the WTAOs technique, and using the traditional Mulliken population-analysis scheme are compared.

Table 9.11. Atomic charges in cubic ATiO3 (A=Sr,Ba,Pb)

Method of electronic-	Population-analysis	Atomic charges, |e|
structure calculation	scheme
		Sr	Ti	O
	nonorthogonal WTAO	2.00	3.20	–1.73
HF	Orthogonal WTAO	2.00	3.13	–1.71
	Mulliken	1.92	2.76	–1.56
	nonorthogonal WTAO	2.00	2.68	–1.56
DFT	Orthogonal WTAO	2.00	2.63	–1.54
	Mulliken	1.87	2.35	–1.40
		Ba	Ti	O
	nonorthogonal WTAO	2.00	3.19	–1.73
HF	Orthogonal WTAO	2.00	3.16	–1.72
	Mulliken	1.86	2.82	–1.56
	nonorthogonal WTAO	2.00	2.65	–1.55
DFT	Orthogonal WTAO	2.01	2.60	–1.54
	Mulliken	1.80	2.37	–1.39
		Pb	Ti	O
	nonorthogonal WTAO	1.99	3.27	–1.75
HF	Orthogonal WTAO	1.89	3.14	–1.68
	Mulliken	1.59	2.79	–1.46
	nonorthogonal WTAO	1.99	2.72	–1.57
DFT	Orthogonal WTAO	1.93	2.64	–1.52
	Mulliken	1.34	2.34	–1.23

9.1 Theoretical Analysis of Chemical Bonding in Crystals

355

The Mulliken atomic charges coincide with those given in [606]. The results of the WTAO population analysis in these three crystals show the following: 1. The results of population analysis performed on orthogonal and nonorthogonal WTAOs are close.

2.The fully ionic charge +2 for the Sr, Ba or Pb atoms remains practically the same for both methods of the electronic-structure calculation – Hartree–Fock and DFT. The only exception is the orthogonal WTAOs analysis in PTO, but even in this case the charge is very close to +2. The reason for such a high ionicity is related to the high-energy location of the s-bands of the Sr and Ba atoms and the p-bands of Pb atom, which leads to a negligible contribution of the corresponding WTAOs to the covalence of these atoms. The switch from the HF calculations to DFT practically does not a ect the location of high-lying vacant bands of the Sr, Ba and Pb atoms and, therefore, no covalence for these atoms appears. On the contrary, the traditional Mulliken analysis displays some covalence for these atoms, which increases in the case of DFT calculations. This is especially noticeable in a PTO crystal. 3. The Ti and O atom charges di er from the purely ionic ones (indicating a partial covalence of the Ti and O atoms). For these atoms the covalence is higher in the DFT calculations than in the HF ones. This correlates with the decrease of the bandgaps in DFT calculations compared to the HF ones (in DFT calculations the d-bands of the Ti atom shift down and become closer to the valence oxygen bands). As a result, mixing of the vacant states of the Ti atom d-bands with the oxygen WTAOs increases, generating the larger values of the covalence for these atoms. 4. The results of the WTAO analysis demonstrate a mixed (ionic-covalent) character of chemical bonding in ABO3 crystals. However, the ionicity, calculated by the WTAOs method, appears to be larger than that obtained in the traditional Mulliken population analysis. 5. The values of atomic charges in all the three considered perovskite-type crystals are close.

The values of some other local characteristics of the electronic structure of STO, BTO, PTO – bond orders, covalences and full valences of atoms – are listed in Table

9.12.These values correspond to nonorthogonal WTAOs analysis. The results given in Table 9.12 allow the following to be concluded. 1. The numerical values of the local characteristics of the electronic structure of the cubic STO, BTO, PTO are very close. 2. The covalence in these crystals appears only in the nearest-neighbor Ti–O bonds (all the other bonds manifest almost no covalence both in HF and DFT calculations). Each Ti atom has six bonds of this type, and each oxygen atom two bonds. In DFT calculations the value of the bond order is larger than in HF ones.

3.Both in the DFT and HF calculations the values of the total calculated valence of the metal atoms practically coincide with the values of the oxidation state used by chemists for these compounds.

The population analysis is not entirely correct for metallic crystals, because they are characterized by a special metallic type of bonding. But since the nonconducting state in LMO arises already within small distortions from the cubic structure, a formal population analysis performed on the cubic structure can show some tendencies in the local properties. Their values for the cubic LMO obtained with the help of the orthogonal WTAOs in HF calculations are given in Table 9.13.

Analyzing these values one can conclude the following. 1. Some of the results are principally di erent from those of STO, BTO and PTO. These are a nonzero covalence of the A (La) atom, a nonzero La–O bond order, the calculated valence of the B (Mn) atom is di erent from the expected one. Probable reasons for these results

356 9 LCAO Calculations of Perfect-crystal Properties

Table 9.12. Atomic charges in cubic ATiO3 (A=Sr,Ba,Pb)

	Method of electronic-	Order of	Covalence			Full valence
	structure calculation	Ti–O bond
			Sr	Ti	O	Sr	Ti	O
	HF	0.24	0.00	1.47	0.51	2.00	4.01	2.01
	DFT	0.36	0.00	2.28	0.82	2.00	4.06	2.02
			Ba	Ti	O	Ba	Ti	O
	HF	0.24	0.00	1.46	0.51	2.00	4.01	2.01
	DFT	0.37	0.01	2.33	0.83	2.01	4.06	2.02
			Pb	Ti	O	Pb	Ti	O
	HF	0.22	0.02	1.35	0.47	2.00	4.01	2.01
	DFT	0.35	0.04	2.23	0.80	2.02	4.06	2.02

Table 9.13. Local properties of electronic structure of cubic LaMnO3

Atomic charges			Bond orders		Covalence			Full valence
La	Mn	O	La–O	Mn–O	La	Mn	O	La	Mn	O
2.48	2.09	–1.53	0.08	0.16	1.00	2.48	0.80	3.04	3.68	1.98

are connected with the metallic nature of the considered crystal in cubic structure and with the di erence of the LMO crystal from STO, BTO or PTO. 2. On the other hand, the value of the La atomic charge is relatively close to the fully ionic one +3. Besides, the calculated valences of the La and oxygen atoms are similar to those obtained in STO, BTO and PTO crystals. This indicates the similarity between these crystals and LMO, manifested in spite of the metallic nature of the latter. 3. The largest value of a bond order in LMO corresponds to the nearest-neighbor Mn–O bond. This correlates with the interatomic charge distribution in STO, BTO and PTO. The di erence is in the value of the bond order – in LMO this bond order is smaller, while the covalence (which is, actually, the sum of all the bond orders) of the Mn atoms practically coincides with that of the Ti atoms in the other three considered crystals. This is, apparently, a consequence of the metallic nature of LMO, which leads to smearing of the nearest-neighbor Mn–O bond charge over the other bonds. Finally, we note that the local properties for STO, BTO and PTO not that close to each other might be obtained if noncubic low-temperature phases are studied. It is especially important to consider the orthorhombic phase of LMO since in this phase LMO is no longer metallic.

WTAO population analysis can be made also using the Bloch functions found in the calculations with plane-wave basis (as was noted, the majority of modern computer codes for periodic systems calculations use this basis and the DFT Hamiltonian).

The results of chemical-bonding analysis for crystal MgO, performed via the conventional and WF schemes of population analysis, were compared in [608]. The MgO crystal is a good testing system for studying the accuracy of such methods, since on the one hand, the nature of chemical bonding in it is well known to be ionic, and on the other hand, some of the methods give contradictory results of chemical-bonding analysis in this crystal (see below and [601, 609]).