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

9

LCAO Calculations of Perfect-crystal Properties

In the next three chapters we illustrate the possibilities of LCAO methods in the calculations of di erent crystalline-solid properties.

We do not discuss here the electron–phonon and isotope e ects on the optical spectra of solids. In [564, 565] these e ects were studied in the framework of the tight-binding (TB) LCAO approach for several semiconductors with diamond and zincblende structure. The results reproduce the overall trend of the available experimental data for the bandgap as a function of temperature, as well as give correctly the mass dependence of the bandgap.

The TB approach represents a conceptual bridge between ab-initio simulations and model-potential ones. In particular, the coupling of TB with the molecular dynamics (MD) generates the TBMD method as a valuable tool for atomic-scale materials modeling [566]. A description of the TB method can be found, for example, in [6,567].

In this chapter the perfect-crystal properties are considered: the analysis of chemical bonding on the basis of population analysis, the one-electron properties and properties, defined by the total energy and its derivatives and the magnetic ordering in crystals.

In Chap. 10 we discuss the models of crystals with point defects and calculation of their properties by LCAO methods.

In Chap. 11 the possibilities of LCAO methods in surface modeling are illustrated by numerous results of recent LCAO calculations.

We refer the reader to the very informative review article [568], where it is possible to find many illustrations of ab-initio simulation possibilities in the area of solid-state chemistry, physics, materials and surface science, and catalysis. This publication is especially useful as an introductory overview for the reader who is not acquainted with solid-state simulation. All the examples discussed in [568] have been generated with the CRYSTAL code and implemented by the authors and collaborators. CRYSTAL was the first periodic ab-initio code to be distributed to the scientific community beginning in 1989. Now, several ab-initio LCAO codes are available to users (see Appendix C), but the CRYSTAL code remains the most user-friendly and fast-developing code being applied now in more than 200 research groups in the world.

328 9 LCAO Calculations of Perfect-crystal Properties

9.1 Theoretical Analysis of Chemical Bonding in Crystals

9.1.1 Local Properties of Electronic Structure in LCAO HF

and DFT Methods for Crystals and Post-HF Methods for Molecules

Crystals, like molecules, are made up of atoms interacting with one another, which gives rise to electronic-density localization along bonds (covalent solids), around atomic nuclei (ionic crystals), or to the more complex and most widespread pattern of electronic-density distribution. Considered in Chap. 4, the density matrix of molecules and crystals in the LCAO approximation permits one to derive the local characteristics of the electronic structure employed usually when describing chemical bonding in molecules or crystals (the electronic configuration of an atom, atomic charges QA, atomic bond orders WAB and covalency CA, and the total and free atomic valences). In molecular quantum chemistry such calculations are known as population analysis. In molecular theory the local electronic-structure characteristics were introduced at first for the orthogonal atomic basis used in semiempirical ZDO calculations (see Chap. 6), with subsequent generalization to the case of a nonorthogonal basis employed in nonempirical HF and DFT calculations [96, 569, 570]. The first attempts at a theoretical determination of local electronic-structure characteristics of periodic systems were made later: for the orthogonal basis set used in semiempirical versions of the HF method in [571, 572] and for the nonorthogonal AO basis – in [573, 574]. Note that determination of local characteristics of electronic structure by traditional methods of solid-state physics, which make use, as a rule, of the plane-wave basis, requires special projection techniques (see Sect. 9.1.4), involving additional approximations to calculate electronic-density-matrix elements in an atomic basis.

The local properties of the electronic structure of a periodic system are defined by the density matrix ρ(R, R ),see (4.91); the electron position vectors R and R vary within the basic domain of a crystal consisting of N primitive unit cells. For a onedeterminant wavefunction the density matrix can be expressed through Bloch-type spin orbitals ψiσk(R)(σ = α, β):

				occ
ρ(R, R ) = ρα(R, R ) + ρβ (R, R ) =					ψiσk(R)ψiσk (R )	(9.1)
				σ i	k
In the LCAO approximation
	ψiσk(R) = Ciµσ kχµk(R)					(9.2)
				µ
1
χµk(R) =	√			exp(ikRn)χµ(R	− RA − Rn)	(9.3)
		N	n

where χµ(R − RA − Rn) = χAnµ (R) is the atomic orbital centered on atom A in the primitive unit cell with a translation vector Rn. In the basis of Bloch sums χµk(R) the overlap matrix Sµν (k) and density matrix Pµν (k) are introduced:

Sµν (k) = exp(ikRn) χµ(R − RA)χν (R − RB − Rn)dR

(9.4)

n

9.1 Theoretical Analysis of Chemical Bonding in Crystals		329
occ
Pµνσ (k) = i	Ciµσ kCiµσ k	(9.5)

The expansion coe cients Ciµσ k are calculated by solving the matrix equation of the CO LCAO method for crystals, see (4.67):

F σ(k)Cσ(k) = S(k)Cσ(k)Eσ(k)

(9.6)

In (9.6) F σ(k) is the matrix of the Hartree–Fock (HF) or Kohn–Sham (KS) operator. The former operator includes a nonlocal exchange part, depending on the density matrix ρ(R, R ), whereas the latter operator involves the electron density ρ(R) = ρ(R, R), that is, it depends only on the diagonal elements of the density matrix, see Chapters 4 and 7.

In view of the translational symmetry of a crystal, one can introduce density

normalization per primitive unit cell containing n electrons:
Sp(P S) =	1		[P (k)S(k)]		=	(P S)A0,A0	= n	(9.7)
				µµ
	N	k	µ			µµ
						µ

The Mulliken population analysis may be extended to crystalline solids [573,574,576] giving the following definitions for an electronic population NA0 on an atom, atomic charge QA0, bond order WA0,Bn, covalency CA0, and the total valency VA0 (A0, Bn mean atom A in the reference unit cell and atom B in the unit cell with a translation

vector Rn):
NA0 =	PµµA0,A0 +	RA0,B0 +	RA0,Bn	(9.8)
	µ A0	B A0	n =0 B
where the overlap population is defined as
	RA0,Bn =	PµνA0,BnSνµBn,A0		(9.9)
	µ A0 ν Bn
and the atomic charge as
	QA0 = ZA0 − NA0			(9.10)

In (9.10) ZA0 is the nuclear charge for the all-electron calculation or the core charge if the pseudopotential approximation is used. The bond order and covalency are

WA0,B0			= µ A0 ν Bn (P S)µνA0,Bn (P S)νµBn,A0 + (P sS)νµA0,Bn (P sS)νµBn,A0				(9.11)
			CA0 =	WA0,B0 +	WA0,Bn = 2NA0 − WA0,A0		(9.12)
			B =A0		n =0Bn
In (9.11), P = P α + P β				and P s = P α − P β . In deriving (9.12) the2 idempotency
relation for the density matrix in the nonorthogonal AO basis (P S)						= 2 (P S) −
(P sS)	2
		was used (see (4.130). Equations 9.11 and 9.12 are applied in unrestricted HF

(UHF) or spin-polarized (SP)KS calculations. For the closed-shell case (RHF and KS calculations) P α = P β and P s = 0.

The definition of the total valency of the atom accounting for both the ionic (electrovalence) and covalent (covalence) components of chemical bonding was suggested

330 9 LCAO Calculations of Perfect-crystal Properties

in [571] for molecular systems and extended to crystalline solids in [572]. The atomic total valence is of the form

VA0 =	1	CA0 +	6CA2 0 + 4QA2 0		(9.13)
	2

To study the influence of the correlation e ects on the local properties of electronicstructure DFT or post-HF methods can be used. If the KS equations are solved the definitions (9.8)–(9.13) of local properties include the correlated density matrix and therefore take into account the correlation e ects.

We note that there are no rigorous quantum-mechanical definitions for the local characteristics of the electronic structure. The definitions given above require the LCAO approximation, and the numerical results depend on the choice of atomic basis. In attempting to separate an atomic subsystem or a molecular subsystem in systems with a strong coupling, we should refuse to describe this subsystem by using pure states, which leads to considerable conceptual and computational problems. For example, an atomic subsystem in a molecular system, as a rule, cannot be assigned a certain integral number of the electrons involved (the calculated atomic electronic population (9.8) is noninteger), which implies that this subsystem should be represented by an ensemble of states matched by di erent numbers of electrons. However, an approach exists that gets around the above di culties. This approach rests on the analysis of reduced density matrices (as a rule, these are the first-order and secondorder matrices) of a system as a whole, and the separation of a particular subsystem is generally based on some geometric criteria. This makes it possible to considerably simplify computations, but leads to some arbitrariness in the choice of the definitions for local characteristics of the electronic structure.

This arbitrariness most clearly manifests itself in going beyond the scope of the HF approximation, as evidenced by a wide variety of definitions for molecular systems available in the literature for valences and bond orders in the case of post-Hartree– Fock methods for molecular systems [570,578–580]. In post-HF methods local characteristics of molecular electronic structure are usually defined in terms of the first-order density matrix and in this sense there is no conceptual di erence between HF and postHF approaches [577]. It is convenient to introduce natural (molecular) spin orbitals (NSOs), i.e. those that diagonalise the one-particle density matrix. The first-order density matrix in the most general case represents some ensemble of one-electron states described by NSOs

2n
i	(9.14)
ρ = λi|ψi ψi|
=1
	2n
where λi are the NSO occupation numbers so that	λi = N, λi ≥ 0 for all i =

i=1

1, 2, . . . , 2n and the characteristic feature of the HF case is that exactly N weights λi in this expansion are equal to 1 (due to the normalization condition, the remaining (2n −N ) should be equal to 0). Here, N is the total number of electrons. In geometric terms, this property of the HF density matrix means that it is a projector on the subspace of occupied NSO or, equivalently, that the N -electron state is described by a single-determinant wavefunction. In any post-HF method, based on the wavefunction formalism, the coe cients in expansion (9.14) satisfy the representability conditions:

9.1 Theoretical Analysis of Chemical Bonding in Crystals

331

0 ≤ λi ≤ 1. Some of these coe cients can be close to 1 (strongly occupied NSO) and some close to 0 (weakly occupied NSO), but in any case the post-HF density matrix loses its property ρ2 = ρ. Turning to the basis of natural orbitals (NOs) results in the appearance of two spin components of the density matrix, ρα and ρβ . For RHF, ROHF, and UHF cases, these spin components are idempotent, being projectors on the subspace of the full MO space spanned by the occupied σ-MOs (σ = α, β). The density operator is the Hermitian positive-semidefinite operator with a spur equal to the number of electrons. At the same time, in general, this operator does not possess any other specific properties such as, for example, idempontency. After the convolution over the spin variables, the density operator breaks down into two components whose matrix representation in the basis set of atomic orbitals (AOs) has the form

n

ρˆσχµ =	χν (P σ S)νµ	(9.15)
	ν=1

where S is the AO overlap matrix, and the matrix elements of the P σ matrix are related to the occupation numbers and the coe cients Ciµσ in the expansion of the spatial parts of natural spin orbitals in the AO basis set through the expressions

n
i
Pµνσ = Ciµσ λiσ (Cνiσ )	(9.16)
=1

In principle, it is not di cult to define the σ-occupancy of the atomic subspace (subspace spanned by AOs of atom A). To this end it is su cient to calculate

N σ	= Sp [ρˆσ ρˆ ]	(9.17)
A	A

where ρˆA is the projector on the subspace of atom A. This value gives us a certain measure of immersion of atom A subspace into the subspace of σ-occupied NOs. Such a definition, however, is not satisfactory because due to the nonorthogonality of the AO basis the sum of occupancies will be greater than the total number of electrons. A reasonable approach consists of turning to the so-called biorthogonal basis χ˜µ and introducing the atomic projector as

ρˆA = |χµ χ˜µ|	(9.18)

µ A

where χ˜ = χS−1 and χµ|χ˜µ = δµν . Operator (9.18) is idempotent and Hermitian. Let the atomic basis set be well localized on atoms (or the atomic fragments under

consideration) and be orthonormal. The unit operator can be expanded as

ˆ	(9.19)
I = ρˆA

A

This operator has a unit matrix with respect to the AO basis set and can be used as the right identity in manipulations in AO basis. In particular,

Sp ρˆσ = Sp ρˆσρˆA = N σ	(9.20)

A

332 9 LCAO Calculations of Perfect-crystal Properties

With such definitions, the total electron occupancy of atom A is calculated as

NA = NAα + NAβ = Sp ρˆσρˆA =	(P σ S)µµ		(9.21)
σ	σ	µ A

In order to determine the two-center bond order, it is reasonable to attempt to obtain the expansion similar to (9.20), but for the squares of the density-operator compo-

nents. Unfortunately, in the general case, the spur of the density operator squared

Sp(ρˆ2) = Sp(ρˆσ)2 depends on both the AO basis set and the computational tech-

σ

nique and can be treated as a certain characteristic of a system only with the very large basis sets and full configuration interaction. Furthermore, even in this limiting case, the physical meaning of the given characteristic is not quite clear. However, instead of the σ-components of the density operator, it is possible to consider their combinations ρˆα+β = ρˆα + ρˆβ and ρˆα−β = ρˆα − ρˆβ that are referred to as the total electron and spin density operators, respectively, and to examine the contributions to the squares of these operators. Both approaches are consistent with each other and with the standard analysis within the Hartree–Fock approximation for occupied shells when one uses the expansion of the doubled sum of the density-operator σ-components squared

Sp (ρˆσ )2 =		Sp ρˆσρˆAρˆσρˆB =			(P σS)µν (P σS)νµ		(9.22)
σ	σ	A,B		σ A,B	A
				µ ν B
The quantities
		WAB = 2		(P σS)	(P σ S)	νµ	(9.23)
				µν
		σ	A
		µ ν B

at B =A can be interpreted as certain characteristics of bond A − B order. Their sum

CA = WA

(9.24)

B =A

may be treated as the covalence of A atom. The one-center terms FA = WAA in (9.23) are termed the free (nonrealized) valence of atom A.

The results obtained in post-HF methods for solids refer mainly to the energy of the ground state but do not provide the correlated density matrix. The latter is calculated for solids in the one-determinant approximation. The density matrix calculated for crystals in RHF or ROHF one-determinant methods describes the many-electron state with the fixed total spin (zero in RHF or defined by the maximal possible spin projection in ROHF). Meanwhile, the UHF one-determinant approximation formally corresponds to the mixture of many-electron states with the di erent total spin allowed for the fixed total spin projection. Therefore, one can expect that the UHF approach partly takes into account the electron correlation. In particular, of interest is the question to what extent UHF method may account for correlation e ects on the chemical bonding in transition-metal oxides. An answer to this question can be obtained in the framework of the molecular-crystalline approach, proposed in [577] to evaluate the correlation corrections in the study of chemical bonding in crystals.

Conceptually, the method is as follows. The local characteristics of a crystal electronic structure are calculated within the periodic model by the Hartree–Fock method

9.1 Theoretical Analysis of Chemical Bonding in Crystals

333

to choose the molecular cluster that adequately describes the local features of the electronic structure. In the majority of cases, the number of atoms comprising this cluster is appreciably less than the number of atoms contained in the clusters simulating one-electron states in a crystal. At the next stage of the advanced approach, the chosen cluster is calculated in the framework of the multiconfigurational approximation using the molecular programs providing the expansion of wavefunctions into determinants. Then, by applying the UHF method to the calculations of the cluster and the crystal, one can obtain the one-determinant solutions, which correspond to the generalized valence bonds. Simple rearrangements of the multiconfigurational cluster function permit one to explicitly separate the obtained valence bonds in its expansion. If these valence bonds turn out to be dominant in the expansion, then, in order to construct the many-electron function of a crystal, it is su cient to replace the cluster valence bonds by the crystal valence bonds. Thus, the constructed wavefunction explicitly includes the electron-correlation e ects. By convoluting the many-particle density matrix into twoand one-particle density matrices, it is possible to calculate the local characteristics of chemical bonding in a crystal by the known formulas.

In Sect. 9.1.3 we discuss the results of such an approach for application to the Ti2O3 crystal with the open-shell configuration d1 of the Ti3+ ion. It is common knowledge that the electron correlation can play an important role in compounds of transition metals with an unfilled d electronic shell. The density-functional method (LDA or GGA), which has been widely employed for these crystals, often appears to be unsatisfactory because of an incorrect description of the self-interaction, see Chap. 7.

9.1.2 Chemical Bonding in Cyclic-cluster Model: Local Properties of Composite Crystalline Oxides

The above formalism for calculation of local electronic-structure characteristics of crystals was at first applied in the cyclic-cluster CNDO semiempirical calculations [571] of composite crystalline oxides with the metal atom oxidized to various degrees. In chemistry the oxidation state is a measure of the degree of oxidation of an atom in a chemical compound. It is the hypothetical charge that an atom would have if all bonds to atoms of di erent elements were 100% ionic. Metal oxidation states are positive and denoted by I, II, III for oxidation states one, two, three, respectively. It should be remembered that the oxidation state of an atom does not represent the “real” charge on that atom: this is particularly true of high oxidation states, where the ionization energies required to produce a multiply positive ion are far greater than the energies available in chemical reactions. The assignment of electrons between atoms in calculating an oxidation state is purely a formalism, albeit a useful one for the understanding of many chemical reactions. It was shown [306] that it is the full atomic valence (9.13) that correlates and in many cases is numerically close to the oxidation state. Usually, composite oxides of heavy-metal atoms crystallize in structures with low-symmetry space groups and contain many atoms in their unit cell. Due to the large atomic (or ionic) radii of the metallic atoms these compounds have both nontypical metal–oxygen bond lengths and atomic coordination numbers. As a rule, the constituent heavy-metal elements exhibit a di erent oxidation state. The

334 9 LCAO Calculations of Perfect-crystal Properties

variable valence of the metal atoms makes the formation of oxides with a di erent chemical compositions (simple, composite and miscellaneous oxides) possible. The simple oxides contain metal atoms with constant valence (for example, Cu2O, La2O3 etc.). However in miscellaneous oxides the metal atoms of the same chemical elements occur in the di erent oxidation states (for example, Fe3O4, Cu4O3). The heavy-metal oxides may lose or acquire oxygen atoms forming a defective crystalline structure and nonstoichiometric compounds.

Unlike light-metal oxides, the investigation of the electronic structure of crystalline heavy-metal oxides is a di cult task because of their composite crystal structure. Thus, it is necessary to use approximations both for the crystal-structure description and for the choice of one-electron Hamiltonian.

It is necessary to investigate the electronic structure of defect-containing metal oxides to understand their physical and chemical properties. However, the first step of such an investigation is connected with the calculation of the electronic structure of perfect nondefective crystal. The analysis of ionic and covalent components of the chemical bonding in perfect crystal allows one to propose realistic models describing the defects in these compounds.

The calculations of local properties of metal-oxide electronic structure [571, 581– 583] were made in the cyclic-cluster model, in the CNDO approximation. As in the CNDO approximation AOs are supposed to be orthogonalized by the L¨owdin procedure (see Chap. 6), the definitions of local properties given in Sect. 9.1.1 for nonorthogonal basis, have to be modified. In particular, the overlap population (9.9) becomes zero in the CNDO approximation, so that the electronic population is defined only by diagonal density matrix elements PµµA0,A0. In (9.6) and in the bond-order definition (9.11) the overlap matrix has to be replaced by an identity matrix. The matrix elements of the cyclic-cluster Hamiltonian are given by (6.61),(6.62) and depend on the semiempirical one-center parameters βM , Uµµ, γM M and two-center Coulomb integrals γMN . The one-center parameters for the light elements with s and p valence electrons have been taken from [584], where these parameters were calibrated for molecular calculations. The one-center parameters for copper, lead, nickel and lanthanum atoms were calibrated so that the results of the electronic-structure calculations correlate well with the experimental data for the simple oxides of these elements. The Coulomb integrals γMN were estimated from the one-center γM M and γNN Coulomb integrals using Ohno’s approximation [239]. The numerical values of the bonding parameters βMN are connected with the choice of the atomic basis functions. As the doubledzeta atomic functions of the neutral atoms were used the bonding parameters βM for the light elements were recalibrated as their values in [584] correspond to Slater

single-dzeta atomic functions. The bonding parameter β was taken to be equal

√ MN

to βMM βN N as is done in many molecular calculations. The numerical values of CNDO parameters of atoms are given in Table 9.1. The details of their choice for actual atoms can be found in [571]. The crystal structures of the considered composite metal oxides are given in Table 9.2 (the references to the corresponding experimental data can be found in [571]). The calculated atomic charges and full atomic valencies are given in Table 9.3 for nonequivalent (not connected by the symmetry transformations) atoms, both with variable valence (Cu,Pb,Ni) and constant valence (oxygen and metal atoms Me in di erent oxidation states).

9.1 Theoretical Analysis of Chemical Bonding in Crystals

335

Table 9.1. CNDO semiempirical parameters of atoms used in crystalline metal oxides calculations (in eV)

Element	γM M	Uss	Upp	Udd	βM
Li	3.47	–4.99			–1.8
O	13.63	–101.31	–84.287		–17.0
K	3.70	–4.15			–1.8
Cu	17.20	–162.23		–171.46	–8.4,–14.6
Sr	3.75	–9.43			–1.8
Y	8.50	–20.80		–22.80	–5.8
Ba	4.20	–10.30			–1.8
La	7.30	–15.08		–19.08	–5.0
Pb	4.30	–26.30	–21.500		–7.7
Ni	17.00	–140.90		–149.00	–5.0,–10.0

For comparison the crystallographic valencies in copper–oxide compounds, calculated for the experimental bond lengths, are given in brackets.

The crystallographic valence (CV) of the ith atom in a crystal is defined as the

sum	j
)
Vi =		Bij	(9.25)

where the summation is performed over all the atoms with charge opposite in sign to that of the ith atom and Bij is the so-called bond valence of the ith and jth atoms. It has been found that in acid–base networks the bond valence Bij correlates well with the bond length Rij and can be approximated by the inverse power or logarithmic function,

	Bij = (Rij /R0)−N			(9.26)
B		= exp	Rij − R0	(9.27)
			− B
	ij

The fitted constants N, R0 and B depend only on the nature of the bonded atoms (i, j) and are given in the literature [585, 586] or may be calculated by the computer code VALENCE [587] designed to calculate bond valences from bond lengths and vice versa. This code allows also calculation of bond–valence sums and average bond lengths, and can determine bond–valence parameters from the bonding environments of di erent cations.

The numerical values of parameters N, R0 and B are usually found by fitting the crystallographic valence Vi to the mean value of the stoichiometric atomic valence in the row of the simplest crystals containing the ith atom. One has to be careful in valence sums. The drawbacks of the CV definition are evident:

)

interpreting the bond-

1.only the central part of the typical valence–bond length curve may be approximated by (9.26) and (9.27);

2.the real interaction between two atoms in a crystal depends not only on the interatomic distance between atoms, but also on the covalent or ionic character of the formed chemical bonding (in some cases this is taken into account by introducing

336 9 LCAO Calculations of Perfect-crystal Properties

Table 9.2. The crystal structure of metal oxides (Z is the number of formula units in the unit cell)

Compound	Space group					Z	Cu,Pb,Ni	Me	O
Cu2O	P m3m					2	4b		2a
YCuO2	R3m					3	3a	3b	6c
LaCuO2	R3m					3	3a	3b	6c
CuO	C2/c					4	4c	4e
NiO	R3m					3	3a		3b
Cu4O3	I41/adm					4	8c,8d		4a,8e
Li2CuO2	Immm					2	2b	4j	4i
SrCuO2	Cmcm					4	4c	4c	4c,4c
Sr2CuO3	Immm					2	2d	4f	2a,4f
MgCu2O3	P mmm					2	4e	2a	2b,4e
Y2Cu2O5	P n21a					4	4a,4a	4a,4a	4a,4a,4a,4a,4a
NaCuO2	P 1					1	1a	1h	2i
KCuO2	Cmcm					4	4c	4c	8f
LaCuO3	R3c					6	6b	6a	18e
La2O3	P 3m1					3		6d	3a,6d
La2CuO4 orth	Cmca					4	4a	8f	8e,8f
La2NiO4 orth
La2CuO4 tetr	I4/mmm					1	1a	2e	2c,2e
La2NiO4 tetr
YBa2Cu3O6	P 4/mmm					1	1a,2g	Y:1d	4i,2g
								Ba:2h
YBa2Cu3O7	P mmm					1	1a,2q	Y:1h	1e,2s,2r,2q
								Ba:2t
α-PbO	P 4/mnm					2	2c		2a
β-PbO	P bcm					4	4d		4d
α-PbO2	P bcn					2	4c		8d
β-PbO2	P 4/mnm					4	2a		4f
Pb2O3	P 21/a					4	4e,4e		4e,4e,4e
Pb3O4	P 42/mbc					4	4d,8h		8g,8h

additionally the dependence of the fitting parameters on the oxidation states of bonded atoms).

Nevertheless, the crystallographic valence has a number of practical applications, in particular in the determination of crystal structures or identifying elements that cannot be distinguished by X-ray di raction. The fiiting parameters values for CV given in Table 9.3 were taken from [588].

It can be seen that the CVs of atoms correlate on the whole with their full atomic valence V and oxidation state in the compounds considered. However, the dispersion of CV values is large for an atom with the same oxidation state in di erent crystals. For example, the CV of the oxygen atom appears to be in the interval 1.5 to 2.6. The CV approach was used by some authors to find the copper oxidation states in high-temperature superconductors [585]. However, as a rule, these compounds have