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

		9.1	Theoretical Analysis of Chemical Bonding in Crystals									337
Table 9.3. Atomic charges and full atomic valencies in crystalline metal oxides*
Crystal			Atomic charges				Full atomic valencies
	Cu, Pb, Ni O1				O2	Me	Cu, Pb, Ni		O1	O2	Me
Cu2O	0.97			–1.94			1.03		2.00
							(1.29)		(2.58)
YCuO2	0.99			–1.59		2.18	1.08		2.03		3.06
LaCuO2	0.97			–1.73		2.49	1.02		2.01		3.03
							(1.25)		(2.16)		(3.07)
CuO	1.60			–1.60			2.18		2.01
							(2.04)		(2.04)
NiO	1.87			–1.87			2.00		2.00
Cu4O3	0.96	1.57		–1.72	–1.56		1.02	2.11	2.00	2.01
							(1.25)	(1.99)	(2.15)	(2.17)
Li2CuO2	1.43			–1.71		0.99	1.92		1.99		1.00
							(1.84)		(1.99)
SrCuO2	1.43			–1.60	–1.74	1.90	1.98		1.99	2.00	1.99
							(2.01)		(2.08)	(1.80)	(1.87)
Sr2CuO3	1.38			–1.84		1.99	1.81		1.99	1.97	2.00
							(1.90)		(1.82)	(2.16)	(1.96)
MgCu2O3	1.31			–1.49		1.98	1.72		2.01	1.98	2.00
Y2Cu2O5	1.59	1.67		–1.20	–1.32	1.68	2.25	2.29	2.09	2.06	3.13
				–1.58	–1.32	1.67			2.02	2.06	3.14
				–1.18					2.10
NaCuO2	1.89			–1.44		0.99	2.56		2.01		1.00
							(2.52)		(1.91)		(1.29)
KCuO2	2.08			–1.54		0.99	2.73		2.00		1.00
							(2.50)		(1.73)		(0.95)
LaCuO3	2.19			–1.54		2.45	2.82		2.02		3.02
							(2.96)		(1.90)		(2.75)
La2O3				–1.68, –1.60		2.43			2.01	2.03	3.03
La2CuO4 orth	1.56			–1.52,–1.54		2.28	2.18		2.00	1.98	3.01
							(2.38)		(2.19)	(1.67)	(2.67)
La2NiO4 orth	1.86			–1.76,–1.65		2.45	2.01		2.00		3.00
La2CuO4 tetr	1.70			–1.61,–1.63		2.39	2.38		2.01	1.98	3.00
							(2.45)		(2.28)	(1.46)	(2.52)
La2NiO4 tetr	1.88			–1.76,–1.67		2.48	2.01		2.00		3.00
YBa2Cu3O6
Cu–O chains	0.97			–1.97		Y:2.44	1.00		2.00		3.03
							(1.42)		(1.94)		(2.89)
Cu–O planes	1.57			–1.65			2.12		2.01
							(2.12)		(2.13)
YBa2Cu3O7
Cu–O chains	2.04			–1.78,–1.64		Y:2.44	2.66		1.99	1.97	3.03
							(2.21)		(1.87)	(2.00)	(2.80)
Cu–O planes	1.52			–1.74,–1.51			2.06		2.01	1.99
							(2.13)		(2.03)	(2.03)
				to be continued

338		9 LCAO Calculations of Perfect-crystal Properties
		Crystal			Atomic charges					Full atomic valencies
			Cu, Pb, Ni			O1	O2	Me	Cu, Pb, Ni			O1	O2	Me
		α -PbO	0.91			–0.91			2.38			2.21
		β -PbO	0.88			–0.88			2.41			2.21
		α -PbO2	2.78			–1.39			3.95			2.04
		β -PbO2	2.80			–1.40			3.95			2.05
		Pb2O3	0.94	2.80		–1.11,–1.24			2.32		3.94	2.11	2.07
						–1.38						2.06
		Pb3O4	0.74	3.09		–1.05,–1.24			2.34		3.97	2.10	2.09
		Cu, Pb, Ni –	atoms	with		variable valence; oxygen			and Me (I,II,III) – atoms with
		constant valence; crystallographic atomic valencies are given in brackets.

nonstoichiometric composition, which introduces some uncertainty into their crystalstructure interpretation.

The experimental Cu–O distance in the orthorhombic modification of La2CuO4, see Fig. 2.13, is larger than in the tetragonal one. The corresponding CV of copper is less in the orthorhombic modification. This tendency is reproduced by the calculated full valence values. However, in both La2CuO4 modifications the copper CV values are larger than two (stoichiometric value) while the full valences are closer to this value. The crystal valence of copper atoms also changes when the oxygen vacancies concentrate on the 01 site (oxygen in chains) in YBa2Cu3O7, (see Fig. 2.15), and change the composition to YBa2Cu3O6, and symmetry from orthorhombic to tetragonal. In YBa2Cu3O7 both Cul copper atoms in chains and Cu2 copper atoms in planes have CV values larger than that in CuO crystal (see Table 9.3) and the CV of the Cul atom is somewhat larger. The absence of oxygens on the Ol site destroys the chain structure of Cul atoms that become linearly coordinated by two oxygen atoms in YBa2Cu3O6, and have a CV value close to that of Cu I copper. The five-coordinated Cu2 atom has a crystal valence close to that of Cu II (see Table 9.3). The calculation of not only the full valencies but also other local properties of electronic structure gives more detailed information about the chemical bonding in crystals of metal oxides.

As is seen from Table 9.3, the copper atomic charge QCu is close to +1 in the Cu I oxide compounds. This means that the Cu I–O bonding is essentially ionic. The calculated 4s population of copper atoms appears to be near zero so that the copper atomic configuration in Cu I oxide crystals is close to 3d10. The results for Cu IIcontaining copper oxide compounds indicate that Cu II–O bonding is more covalent compared to Cu I–O bonding; the copper atomic charge in Cu II oxide compounds lies between 1.38 and 1.70 so that the copper atomic configuration appears to be between 3d9.6 and 3d9.3. The Cu III atomic charge is close to +2 (see Table 9.3); calculated atomic populations correspond to a 3d9 configuration. Thus, there are no Cu3+ ions in cuprates, although these compounds do contain Cu III atoms, which lose two electrons to form an ionic bond with oxygen and supply one electron to form the covalent bonds. Calculated atomic charges on alkali and alkali-earth metal atoms (Me) are close to the number of their valence electrons, so that these metal atoms are present as Me+ and Me2+ ions, respectively. As shown in Table 9.3 the calculated charge of the rare-earth atoms Y and La are close to +3.

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In Table 9.3 the results suggest that the absolute value of the atomic charge in plumbic oxides PbO2 are larger than those in plumbous oxides PbO. However, all the lead–oxygen crystals are very far from purely ionic compounds.

The nonequivalent metal atoms in the miscellaneous oxides Cu4O3, Pb2O3, Pb3O4 have di erent atomic charges, and the charge values are close to those in either higher (CuO, PbO2) or lower (Cu2O, PbO) oxides.

We see that the calculated atomic-charge values agree qualitatively with those expected from the chemical point of view. On the other hand, the electronic-charge distribution calculated by other approaches (more traditional for solid-state physics) appears to be unrealistic: for example, it was found in an LAPW calculation of the Cu2O crystal [589] that the atomic charges (estimated by the density integration inside the atomic spheres) are positive for both atoms: QCu = 2.05 and QO = 0.40. LCAO calculations of the electron density and direct integration over a spherical region near atomic cores [590] also do not allow one to distinguish the atomic charges on atoms with di erent oxidation states (the numerical values for the copper atomic charge in Cu2O and CuO crystals are practically the same :+0.58 and +0.48, respectively). These examples demonstrate the advantage of quantum-chemical LCAO calculations of the local electronic-structure properties to describe reasonably both the valence and charge states of metal atoms in crystals. This conclusion is important for the study of new complicated crystalline structures, synthesized experimentally. As an example, we refer to the high-Tc superconductors. It is well known that a small variation of the composition or the way in which the material is prepared may essentially change the superconducting properties. Evidently this is connected with a change of the electronic structure of the superconductor.

In [582] the comparative theoretical analysis of chemical bonding was made in two pairs of copper oxide compounds : (a) orthorhombic and tetragonal modifications of La2CuO4 (the latter modification has a structure similar to that of the superconductor La2−x, SrxCuO4) and (b) nonsuperconducting tetragonal YBa2Cu3O6 and superconducting orthorhombic YBa2Cu3O7 crystals. In accordance with the results of previous band-structure calculation, it was found that La2CuO4, and YBa2Cu3O7 are metallic with small values of the density of states at the Fermi level, but YBa2Cu3O6 is a semiconductor with a forbidden energy gap of 2.5 eV. It was also found that the density of states for the YBa2Cu3O7 crystal contains 3d Cu I (copper in chains) peak around 13 eV below Fermi level, but in the case of YBa2Cu3O6 this peak was absent. An analysis of chemical bonding in high-Tc superconductors and related nonsuperconductor crystals shows changes that take place in the local electronic structure of these substances. As seen in Table 9.3 the charge distribution around the copper atom in the orthorhombic modification of the La2CuO4 crystal is practically the same as in the CuO crystal. Contrary to this, in the tetragonal modification of La2CuO4 the copper atomic valence exceeds the ordinary Cu II valence in copper oxide compounds. The calculated atomic charge on the copper atoms in La2CuO4 crystals correlates well with that from it ab-initio Hartree–Fock cluster calculation (1.79e) [591]. If we consider the following defect-formation process, the substitution of La III by Sr II atoms, or the addition of oxygen atoms (for example, La2CuO4.13 compound exists) it still further increases the copper valence in La2CuO4. A comparison of the atomic charges and bond orders in YBa2Cu3O6 and YBa2Cu3O7 crystals shows that electron-density distributions are practically identical in Cu–O planes, see Table 9.3.

340 9 LCAO Calculations of Perfect-crystal Properties

The main changes in the chemical bonding take place in Cu–O chains. The results in Table 9.3 show that the charge state of Cu I atoms (in Cu–O chains) in YBa2Cu3O6 crystal is close to the state of copper atoms in Cu I oxides, but that in YBa2Cu3O7 they are close to copper atoms in Cu III oxides.

The analysis of calculated bond orders shows that there is no peroxide bonding in copper oxygen superconducting materials. It was also shown by atom–atom potential modeling that even additional oxygen atoms in La2CuO4+x do not cause the formation of peroxide bonds.

One of the questions to be considered is the role of the tetragonal–orthorhombic transition in superconductor materials. To analyze the changes in the electronic structure that are connected with the tetragonal–orthorhombic transition the YBa2Cu3O7 crystal with an oxygen vacancy was considered. The composition of this phase is YBa2Cu3O6, but the lattice parameters and space symmetry are the same as those in the YBa2Cu3O7 crystal. It was found [571] that di erences in atomic charges between perfect YBa2Cu3O6 and orthorhombic YBa2Cu3O6 do not exceed 0.05. These results show the strong correlation between Cu III (though not Cu3+) atoms presence and high-Tc superconductivity in copper oxides.

The correlating bonding and structure in Cu II and Ni II mixed oxides is a matter of high interest. The comparative study of the chemical bonding in NiO–CuO and La2NiO4–La2CuO4 crystals was made in [582, 583]. NiO crystals were calculated in rhombohedral structure, corresponding to nonmetallic antiferromagenetic phase of this crystal. Analysis of local properties of electronic structure of crystals containing a Ni II–O bond with those containing Cu II–O bond, allows us to conclude that the former is more ionic than the latter (see Table 9.3). The increase of ionicity degree in the Ni II–O bonding with respect to Cu II–O bonding is understood in terms of the mutual position of the oxygen 2p and metal 3d subbands that lie nearly in the same energy interval for Cu, whereas the 3d Ni subband lies approximatively 2 eV above the center of gravity of the oxygen 2p level.

The ionic part of the Ni II–O bonding is the same for NiO and La2NiO4 (see Table 9.3). As regards the Cu II–O bonding, the data obtained for La2CuO4 as compared to CuO and other Cu II mixed oxides such as Sr2CuO3, Sr2CuO2, Li2CuO2, atomic charges that are found in the range 1.4 to 1.6, point to a rather high ionic part of the Cu II–O bonding. Comparing the full atomic valence for Ni II and Cu II (Table 9.3) evidences a striking feature: the quantum-chemical value of the valence fits perfectly the formal oxidation state of Ni, namely (II), whereas for Cu, the quantumchemical valence is larger than the oxidation state, with an enhancement for La2CuO4, (VCu = 2.38). Such a data is likely to depend on an increase of the covalence of the Cu II–O bonding, as compared to the corresponding Ni II–O bonding.

As regards the problem of the true meaning of the (II) valence in tetragonal La2NiO4, quantum-chemical data cannot be considered as contradictory to the occurrence of some Ni II–Ni III oxidation coupled with the presence of interstitial oxygen, which is rather usual for air-prepared polycrystalline La2NiO4.

From the data of Table 9.3, one can get an insight into the anisotropy of the M II–O bonding in La2MO4, i.e. the electronic anisotropy in the MO6 octahedra (see Fig. 2.13), in terms of the two following features:

(i) The Ni II–O bonding in La2NiO4 does not exhibit any significant anisotropic trend: the covalence is rather low and equally distributed in the equatorial plane and

9.1 Theoretical Analysis of Chemical Bonding in Crystals

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the apical direction. Consequently, the nature of the Ni II–O bonding in the N iO6 octahedra is nearly unchanged when going from the binary oxide NiO to the composite oxide La2NiO4.

(ii) The increase of the covalent bond order of the Cu II–O bonding, as compared to the Ni II–O one, concerns the Cu II–O inplane bonds: such Cu–O equatorial bonds are very similar to the Cu II–O bonding in the binary oxide CuO. Conversely, the Cu–O apical bonds are slightly covalent and look like the corresponding Ni–O apical bonds in the NiO6 octahedra of La2NiO4 and NiO, as well. The strong ionic character of the Cu–O apical bonds in La2CuO4 results in an unusual overall increase of the atomic charge of Cu (Table 9.3), when compared to CuO and other Cu II mixed oxides. The absence of any significant anisotropic covalent e ect of the Ni–O bonding in La2NiO4 found in calculations, fully agrees with the conclusions of magnetization density studies, which point to an equal population of dx2−y2 and dz2 orbitals and no significant covalence in the M–O plane [592]. Moreover, as also stated in [592], the covalent e ect to be found in the apical direction of the octahedra is likely to be due not to Ni II but to La III, in terms of a “hybridization between the La 5d and O 2p bands”. This is consistent with the lowering of the atomic charge of the apical oxygen, as compared to the equatorial one, modeled in La2NiO4 (Table 9.3); such a lowering cannot depend on an increase of the covalent e ect of the Ni II–O bonding in the apical direction, but more likely it can be ascribed to the covalent character of the La III–O bonding (Table 9.3).

From the main di erences in the Ni II–O and Cu II–O bonding emphasized above, it is possible to clear up the problem of the actual nature of the anisotropic e ects in the Ni II and Cu II octahedral oxygen coordination in La2MO4 oxides (M = Cu, Ni), namely to appreciate the mutual electronic and geometric or steric contributions. In this respect, Ni II and Cu II behave in a completely di erent way:

(i)The electronic contribution to the overall anisotropy of the NiO6 octahedra in La2NiO4 is nearly absent (4Ni − Oeq: 1.934 ˚A, 2Ni − Oap: 2.243 ˚A) [593]). This means that the tetragonal distortion of the NiO6 octahedra largely depends on steric e ects originating in the impossibility to match properly the usual La–O and Ni–O bond lengths: 2.57 and 2.06 ˚A, respectively, when building the structure of La2NiO4, as proposed in [594].

(ii)The electronic contribution of Cu II in terms of a significant covalent e ect of the Cu II–O inplane bonding, results in a cooperative shortening/lengthening of the

Cu − Oeq and Cu − Oap, bonds, respectively: 4 Cu − Oeq : 1.904 ˚A, 2 Cu − Oap: 2.397 ˚A. The enhancement of the tetragonal distortion of the CuO6 octahedra results from adding the electronic contribution of Cu II to the typical steric e ect of the La2CuO4.

Finally, the quantum-chemical data, as used to compare the anisotropy of the M II–O bonding in the M II octahedral oxygen coordination of La2MO4 oxides (M = Ni, Cu), allow the following two features to be emphasized:

(i) The so–called Jahn–Teller e ect of the d9 Cu II must be understood in terms of an anisotropy of bonding, namely a strong covalent character of the Cu II–O inplane bonding that no longer exists in the Cu II–O apical bonding. Such an anisotropic covalent bonding character cannot be found in the high-spin d8 Ni II: as a consequence, the occurrence of a distortion of the NiO6 octahedra is related to the steric e ects of the geometric structure in question: the rocksalt structure of NiO permits only a very

342 9 LCAO Calculations of Perfect-crystal Properties

slight distortion of the octahedra, as the structure of La2NiO4 results in significant steric constraints on the octahedra.

(ii) The overall anisotropy of the MO6 octahedra in the La2MO4 oxides (M = Ni, Cu) cannot be fully understood in terms of the typical M II–O bonding properties. The role of La III–O bonding can be held as crucial in enhancing the covalent character of the apical oxygen and consequently increasing the tetragonal distortion of the MO6 octahedra. This is clear from the data obtained for La2NiO4.

Thus, anisotropy of the NiO6 octahedra can be assumed to originate from geometric contributions. Conversely, the anisotropy of the Cu II–O bonding, in terms of a significant covalency of the Cu II–O inplane bonding, is unambiguously settled. More generally, each increase of the covalency e ect of inplane bonding in MO6 octahedra, results in a corresponding lengthening of the apical M–O bonding. Such an e ect, which was previously well evidenced for the anisotropic electronic configurations of some so-called exotic cations such as low-spin Ni3+ or low-spin Cu3+ is presently being investigated in Ni I–O and Cu I –O bonding of the structures of the reduced phases LaNiO2 and LaSrNiO3 [595].

The use of local electronic-structure properties conception in the chemical-bonding analysis was demonstrated also in ab-initio calculations of crystalline metal oxides, discussed in the next section using the example of titanium oxides.

9.1.3 Chemical Bonding in Titanium Oxides: Periodic

and Molecular-crystalline Approaches

As demonstrated above by the results of semiempirical calculations of metal oxides, the full valence of the metal atom, defined according to (9.13), correlates with the metal oxidation state. Such a conclusion was confirmed later in ab-initio HF calculations [574] of titanium oxides. Although the quadrivalent state (oxidation state IV) of the titanium atom is the most stable, the existence of oxygen compounds of titanium in formal oxidation states of III and II, as well as of a series of nonstoichiometric compounds was established. Table 9.4 presents the space-group symbol, the number of formula units in the cell, and the shortest Ti–Ti and Ti–O experimental distances for TiO in hexagonal structure, Ti2O3 in corundum structure and TiO2 (in the rutile (r), anatase (a), and brookite (b) modifications).

Table 9.4. Crystal structure of titanium oxides: space group, number of formula units per cell Z, and nearest interatomic distances (in ˚A)

Structure	TiO(hex)			Ti2O3			TiO2(r)	TiO2(a)	TiO2(b)
data
Group	P 6m2			R3c			P 42/mnm	I41/amd	P bca
Z	2			2			2	2	8
RTi−Ti	3.03			2.58			3.00	3.10	2.95
	3.24			2.99			3.55	3.76	3.06
RTi−O	2.38			2.02			1.95	1.94	1.86
	3.86			2.07			1.97	1.99	1.92

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343

The titanium oxides are of considerable technological interest, so di erent theoretical studies of electronic and atomic structure and properties have been performed both for TiO2 [100,323,596,597] and Ti2O3 [598,599] crystals. We are not aware of the existence of the electronic-structure calculations of TiO. The available publications focus attention primarily on description of the band structure and phase stability of titanium oxides and restrict the discussion of the nature of chemical bonding in these compounds to an analysis of Mulliken atomic charges and overlap populations.

The chemical bonding in titanium oxides was studied in more detail in [574, 575, 577, 600]. The calculations are made by HF-LCAO and DFT-LCAO methods, incorporated in the computer code CRYSTAL [23]. The core electrons of Ti and O atoms are described by Durand–Barthelat [484] pseudopotentials and the atomic basis sets were taken from [323], where the outer exponents of the Gaussians were fitted to reproduce the parameters of rutile structure. For the structural parameters those calculated in [323, 598] were taken. The su ciently good accuracy of the calculations was ensured by the following choice of input computational parameters: 1. The goodquality set of the threshold parameters controlling the accuracy of the bielectronic series in RHF and UHF calculations (10−6, 10−6, 10−6, 10−6, 10−12). 2. The reasonably accurate tolerances controlling the DFT calculations of density, potential, and grid weight (10−9, 10−9, 10−14). 3. The values 10−6 and 10−5 a.u. for the convergence thresholds of eigenvalues and total energy, respectively. 4. For the sampling k-point net of the integration in the reciprocal space Monkhorst–Pack set with shrinking factors s1 = s2 = s3 = 6 was used.

Table 9.5 presents local electronic-structure characteristics of the titanium oxides obtained by the RHF method, namely, the Ti d-orbital populations Pd and atom charges QTi, as well as the atomic valences VA. Also given are the local characteristics calculated in a nonorthogonal atomic basis and a basis orthogonalized according to L¨owdin. As follows from a comparison of the results obtained from population analyses by Mulliken and L¨owdin, the L¨owdin analysis shows the chemical bonding in the crystals under study to be largely covalent, with the total valences of the titanium and oxygen atoms di ering substantially from the expected stoichiometric values.

Table 9.5. Local electronic-structure characteristics of titanium oxides in the restricted Hartree–Fock method

Crystal		Mulliken	L¨owdin
		Pd QTi VTi VO	Pd QTi VTi VO
TiO(hex)		2.38 1.62 2.14 2.04	2.70 1.27 2.30 2.15
Ti2O3		1.82 2.26 3.61 2.05	2.50 1.43 3.98 2.30
TiO2	(r)	1.46 2.66 3.94 2.08	2.18 1.73 4.18 2.36
TiO2	(a)	1.47 2.65 3.98 2.08	2.19 1.72 4.23 2.36
TiO2	(b)	1.49 2.63 3.97 2.09	2.21 1.70 4.23 2.38

It was shown [602] that the calculations made using a valence-atomic basis without polarizing functions in the L¨owdin population analysis agree better with the expected values of atomic valencies. The population analysis by Mulliken made in a nonorthogonal basis was found to be less sensitive to the inclusion of polarizing functions into the

344 9 LCAO Calculations of Perfect-crystal Properties

calculation. The results obtained with the restricted and unrestricted HF methods for TiO2 in formal titanium configuration d0 do not di er practically from one another. The results of a calculation of the local properties of the three TiO2 modifications show them to be only weakly sensitive to structural changes.

Table 9.6. Local properties of chemical bonding in titanium oxide crystals in the restricted and unrestricted HF methods: titanium d-orbital populations Pd, atomic charges QA, covalencies CA, atomic valences VA, and bond-orders WAB for nearest-neighbor atoms*

Local	TiO (hex)		Ti2O3		TiO2(r)
properties
	RHF UHF		RHF UHF		RHF
Pd	2.38	2.34	1.82	1.77	1.46
QTi	1.63	1.67	2.26	2.32	2.66
CTi	0.90	0.69	2.19	1.23	2.15
VTi	2.14	2.05	3.61	3.01	3.94
QO	–1.63 –1.67		–1.51 –1.55		–1.33
CO	0.74	0.66	0.94	0.88	1.24
VO	2.04	2.03	2.05	2.05	2.08
WTi−Ti	0.03	0.01	0.89	0.03	0.01
	0.00	0.00	0.00	0.01	0.01
WTi−O	0.11	0.10	0.20	0.19	0.36
	0.00	0.00	0.21	0.19	0.30

*For the TiO2 crystal, the RHF and UHF results coincide.

As seen from Table 9.6 the largest di erence between the calculations made in the RHF and UHF approximations was found for Ti2O3 in corundum structure, see Fig. 2.19, with a formal titanium atom configuration d1, where one can expect substantial spin-polarization and correlation e ects. In the Ti2O3 case the RHF method predicts a high bond order between titanium atoms, which, in its turn, results in an overestimated titanium valence. This pattern of chemical bonding is not borne out by experiments. As follows from Table 9.6, an increase in the degree of Ti oxidation from II to IV gives rise to an increase in the charge on the titanium atom, but the relative ionicity and the absolute value of the charge on the oxygen atom decrease. UHF calculations show that, in all the above insulator oxygen compounds of titanium, there is no strong covalent interaction among the Ti atoms.

An appreciable role in transition-metal compounds is played by electron-correlation e ects, which may be taken into account in periodic DFT calculations or within the framework of multiconfigurational methods employing the cluster model of the crystal. In particular, of interest is the question to what extent the single-determinant UHF method (applied in the periodic LCAO calculations of solids) may account for the correlation e ects. In the general case only a qualitative answer to this question is possible. The comparison of the whole set of local properties for rutile TiO2 and corundum Ti2O3 structures as seen from results of calculations [575, 577, 600] demonstrates the di erent role of correlation e ects on the chemical bonding in these two solids. In Tables 9.7 and 9.8 results are given for RHF, UHF, KS (Kohn–Sham Hamil-

9.1 Theoretical Analysis of Chemical Bonding in Crystals

345

tonian with BLYP exchange-correlation functional – nonspin-polarized) and KSS (the same KS Hamiltonian with inclusion of spin polarization). Also given are results for hybrid spinless and spin-dependent schemes using HF exchange and LYP correlation in the Hamiltonian (HB and HBS, respectively).

As follows from the results of the calculations, the e ects of electronic correlation on local properties of the electronic structure of two crystals under consideration have both similarities and di erences. Their explanation can be given if the peculiarities of the structure are taken into account. In both crystals the essentially covalent Ti–

Table 9.7. DFT and HF results for rutile TiO2 crystal (Mulliken population analysis)

Method	E (a.u.)	QA CA VA			WTi−O1 WTi−O2
RHF	–69.776	2.66	2.15	3.94	0.36	0.30
		–1.33 1.24 2.08
UHF
SZ = 0	–69.776	2.66	2.15	3.94	0.36	0.30
		–1.33 1.24 2.08
SZ = 1	–69.491	2.80	1.99	3.97	0.32	0.26
		–1.40 1.11 2.07
KS (BLYP)	–71.348	1.88	3.18	4.06	0.50	0.43
		–0.94 1.85 2.25
KSS (BLYP)
SZ = 0	–71.351	1.88	3.18	4.06	0.50	0.43
		–0.94 1.85 2.24
SZ = 1	–71.481	2.24	2.73	3.99	0.43	0.39
		–1.12 1.60 2.17
HB (LYP)	–70.834	2.64	2.17	3.94	0.36	0.30
		–1.32 1.25 2.09
HBS (LYP)
SZ = 0	–70.834	2.64	2.17	3.94	0.36	0.30
		–1.32 1.25 2.09
SZ = 1	–70.554	2.80	1.99	3.97	0.32	0.27
		–1.40 1.11 2.07

O interaction takes place as the two spheres of the Ti atom’s nearest neighbors are formed by oxygen atoms: 1. In TiO2 the Ti atom is surrounded by four oxygens at 1.95 ˚A and the next two oxygens at 1.97 ˚A i.e. oxygens form a distorted octahedron, see Fig. 2.11. 2. In Ti2O3 crystal six oxygens are grouped into two triangles (with close Ti-O distance 2.02 and 2.07 ˚A, respectively), see Fig. 9.1. The Ti–Ti atoms’ configuration in two crystals is di erent: 1. In TiO2 crystal there are two Ti atoms at distance 2.99 ˚A as a third neighbor of the Ti atom. In Ti2O3 crystal each Ti atom interacts with only one Ti atom at the distance 2.58 ˚A, being smaller than that in TiO2 crystal. The results, given in Tables 9.7 and 9.8, may be explained when only the three first neighbors of the Ti atom are considered (from the analysis of calculated bond orders, it follows that the interactions with more distant atoms are su ciently weak). From Tables 9.7 and 9.8 it is seen that both HF and KS calculations demonstrate the essential role of covalence in both crystals due to Ti–O interactions. When comparing atomic charges QTi for the same calculational scheme, one finds that the Ti–O bonding

346 9 LCAO Calculations of Perfect-crystal Properties

Table 9.8. DFT and HF results for Ti2O3 crystal in corundum structure (Mulliken population analysis)

Method	E (a.u.)	QA CA VA			WTi−Ti WTi−O1 WTi−O2
RHF	-108.029	2.26	2.20	3.61	0.87	0.20	0.21
		–1.51 0.94 2.05
UHF
SZ = 0	–108.362	2.32	1.23	3.01	0.03	0.19	0.19
		–1.54 0.88 2.05
SZ = 1	–107.902	2.24	2.37	3.72	0.49	0.20	0.20
		–1.49 0.96 2.05
KS (BLYP)	–110.682	1.65	3.47	4.13	0.34	0.35	0.30
		–1.10 1.63 2.18
KSS (BLYP)
SZ = 0	–110.702	1.69	3.11	3.85	0.10	0.33	0.30
		–1.13 1.60 2.18
SZ = 1	–110.744	1.68	3.04	3.79	0.19	0.33	0.28
		–1.12 1.56 2.15
HB (LYP)	–109.672	2.24	2.22	3.61	0.86	0.20	0.21
		–1.50 0.96 2.05
HBS (LYP)
SZ = 0	–110.027	2.31	1.25	3.01	0.03	0.19	0.20
		–1.54 0.90 2.05
SZ = 1	–109.588	2.24	2.41	3.74	0.51	0.20	0.20
		–1.49 0.97 2.05

in TiO2 crystal is partly more ionic. For TiO2 crystal the Ti–Ti interaction is weak (the corresponding bond orders WTi−Ti are in the range of values 0.01–0.02 for all calculation schemes and are not given in Table 9.7). For Ti2O3 crystal the results of calculations depend on the spin correlation: 1. In the RHF method bond order WTi−Ti appears to be large (0.87) and does not practically change in hybrid HF-KS calculations (HB) without spin; 2. For spin projection Sz = 0 (antiferromagnetic case), WTi−Ti is close to zero in UHF, KSS, and HBS calculations, which means that spincorrelation e ects are mainly important when the Ti–Ti interaction is considered; 3. For spin projection Sz = 1 (ferromagnetic case), not only spin correlation but also the Coulomb one influences WTi−Ti: UHF value WTi−Ti = 0.49 decreases to WTi−Ti = 0.19 in KSS calculation. It is seen also that in UHF and Becke exchange Hamiltonians spin correlation is described in a di erent manner: the WTi−Ti value decreases from 0.49 (UHF) to 0.19 (KSS). Analyzing the total energy E per unit cell, one can draw interesting conclusions from Tables 9.7 and 9.8.

In TiO2 crystal: 1. Spin-correlation e ects are small for Sz = 0 (antiferromagnetic ordering practically does not change E, as seen from a comparison of the RHF-UHF, KS-KSS, and HB-HBS results). 2. The energy of the ferromagnetic state (Sz = 1) is higher than for the antiferromagnetic state (Sz = 0) for UHF and HBS calculations. It again shows the di erence between HF and Becke approximations for exchange. This di erence is due to the nonlocal exchange in the HF Hamiltonian (the one-electron