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

9.2 Electron Properties of Crystals in LCAO Methods

377

The electronic structures of both compounds have been well studied experimentally. The experimental data show that the MgO crystal is a wide-bandgap insulator (Eg = 7.8 eV ); titanium dioxide TiO2 in the rutile structure is a semiconductor with an experimental bandgap of approximately 3 eV. These di erences are reproduced in the band strucrure of these two binary oxides, calculated in [623] by HF and LDA LCAO methods and shown in Figures 9.5 and 9.6, respectively. The details of the AO basis-set choice and BZ summation can be found in [623].

a

Energy, a.u.

2.00

1.50

1.00

0.50

0.00

EF

-0.50

Energy, a.u.

L		X W Mg:s O:s O:p total
	k	DOS

b

2.00

1.50

1.00

0.50

0.00

EF

-0.50

L		X W Mg:s O:s O:p total
	k	DOS

Fig. 9.5. Band structure and DOS of MgO crystal, [623]: (a) HF LCAO method; (b) DFT(LDA) LCAO method

In MgO (in accordance with the results of other calculations), the two highest valence bands are the oxygen s- and p-like bands, respectively, whereas the conduction bands are more complicated. The upper valence bands in TiO2 are also oxygen s-

378 9 LCAO Calculations of Perfect-crystal Properties

a

Energy, a.u.

1.00

0.50

0.00

EF

-0.50

-1.00

0 ;		5 = $ Ti:G O:V O:S total
b	k	DOS

0.50

Energy, a.u.

0.00

EF

-0.50

0 ;	5 = $ Ti:G O:V O:S total
k	DOS

Fig. 9.6. Band structure and DOS of rutile TiO2 crystal, [623]: (a) HF LCAO method; (b) DFT(LDA) LCAO method

and p-like bands; however, they consist of 4 and 12 sheets, respectively, because the primitive cell of titanium oxide contains four oxygen atoms.

The lowest conduction band in TiO2 consists of 10 branches formed by 3d-states of two titanium atoms and is noticeably separated in energy from the upper conduction bands. The symmetry of the one-electron states can be found using the BR theory of space groups and data on the crystalline structures (see Chap. 3).

Evidently the symmetry of band states does not depend on the basis choice for the calculation (LCAO or PW); the change of basis set can only make changes in the relative positions of one-electron energy levels.

9.2 Electron Properties of Crystals in LCAO Methods

379

An important parameter in the band theory of solids is the Fermi-level energy (see Figures 9.5 and 9.6), the top of the available electron energy levels at low temperatures. The position of the Fermi level in relation to the conduction band is a crucial factor in determining electrical and thermal properties of solids. The Fermi energy is the analog of the highest-occupied MO energy (HOMO) in molecules. The LUMO (the lowest unoccupied MO) energy in molecules corresponds to the conduction-band bottom in solids. The HOMO-LUMO energy interval in a solid is called the forbidden energy gap or simply bandgap.

Depending on the translation symmetry of the corresponding Bloch states the gap may be direct or indirect. In HF and LDA calculations of both crystals under consideration the bandgap is direct, i.e. the one-electron energies at the top of the valence band and the bottom of the conduction band belong to the same Γ -point of the BZ (this result agrees both with the experiment and other band-structure calculations). It is seen that the HF bandgap in both crystals is essentially overestimated and decreases in LDA calculations mainly due to the lowering of the conduction-band bottom energy (see Chap. 7). The influence of the correlation e ects on the valence-band states is smaller.

It is also seen that the oxygen 2p bandwidth in MgO is smaller than that in TiO2. The bandwidth is a measure of dispersion in the k-space and depends on the magnitude and the range of interactions within the crystal: for the more covalent rutile crystal the oxygen-oxygen interactions are stronger. The core bands of both crystals (not shown on the figures) are separated by a large energy gap from the valence bands and are completely flat due to the high localization of core states near the atomic nuclei.

The Fermi energy of a crystal with n electrons in the primitive unit cell is defined from the condition

M	εF				εF			εF
		1	BZ
n = 2 i=1		VB		θ(ε − εi(k))dk =		n(ε)dε =	i		ni(ε)dε	(9.66)
	−∞				−∞			−∞

where n(ε) is the total density of states (DOS) per unit energy. DOS is an important quantity calculated for crystals. The total DOS can be expressed as the sum of contributions ni(ε) from individual energy bands, see (9.66). The total DOS definition is independent of the basis set choice (PW or LCAO); the product ni(ε)dε defines the number of states with energy in the interval dε. Each energy band spans a limited energy interval between the minimal and maximal one-electron energies ε , ε

so that / εmax n(ε)dε gives DOS in the corresponding energy interval. min max

εmin

The connection between the electronic structure of a crystal and the one-electron states of the constituent atoms is given by the projected density of states (PDOS), associated with the separate AOs, their shells or individual atoms.

Let us rewrite the total DOS as

	M		M
		1	BZ
n(ε) =	i=1 ni(ε) =	VB	i=1	f i(k)δ(ε − εi(k))dk	(9.67)
The weighting function f (i)(k) in (9.67) chosen as
	f (i)(k) = Ciµ(k)Ciν (k) exp(−ikRn)				(9.68)

380 9 LCAO Calculations of Perfect-crystal Properties

defines PDOS associated with AO µ in the reference cell and AO ν in the cell with the translation vector Rn. After summation over band index i and integration up to εF PDOS (9.68) gives the density matrix elements (9.65).

According to Mulliken population analysis, the DOS projected onto a given set of AOs {λ} (belonging to the given shell or to the whole atom), is defined by the weight

function	f (i) (k) =		(k)C	(k)S		(k)	(9.69)
		C			µν
	{λ}	iµ	iν
	µ{λ}	ν

where the overlap integral Sµν (k) is defined by (4.123).

Let the set {λ} consist of one AO from the reference cell. The summation of (9.69) over the direct lattice translations Rn and integration over BZ gives the orbital DOS

	1	BZ
nµ(ε) =	VBZ i ν Rn			Ciµ(k)Ciν (k) exp (ikRn) Sµν (k)δ(ε − εi(k))dk	(9.70)

The DOS of an atom A nA(ε) and the total DOS ntot(ε) are caluculated from the or-

bital DOS:
	nA(ε) =	nµ(ε); ntot(ε) =	nA(ε)	(9.71)
	µ A		A

The total DOS and PDOS give rich information on the chemical structure of a system, connecting the calculated band structure with the atomic states. We demonstrate this by considering total and projected DOS for binary oxides MgO, TiO2 and ternary oxides SrTiO3 and SrZrO3 with cubic perovskite structure (see Figures 9.5–9.8).

a			b
Total			Total
O: 2p			O: 2p
O: 2s			O: 2s
Ti: 3d (x 5.00)			Zr: 3d (x 5.00)
Sr: 4p			Sr: 4p
-20.0 -15.0 -10.0 -5.0	0.0	5.0	-20.0 -15.0 -10.0 -5.0	0.0	5.0
Energy, eV			Energy, eV

Fig. 9.7. Full and partial DOS in (a) SrTiO3 and (b) SrZrO3 crystals

The analysis of total and partial DOS demonstrates that in all the oxides under consideration the upper valence band is predominantly formed by the O 2p states. It

9.2 Electron Properties of Crystals in LCAO Methods

381

		a
	12.0
	8.0
	4.0
eV	0.0
Energy,	-4.0
	-12.0
	-8.E0F
	-16.0
	-20.0
	-24.0
	-28.0
		;	0	5
			k
		d
	12.0
	8.0
	4.0
eV	0.0
Energy,	-4.0
	-12.0
	-8.E0F
	-16.0
	-20.0
	-24.0
	-28.0
		;	0	5
			k

b c

EF EF

; 0 5 ; 0 5

k	k
e	f

EF EF

; 0 5 ; 0 5 k k

Fig. 9.8. Band structure of cubic crystals: (a),(b),(c)–SrTiO3, (d),(e),(f)–SrZrO3. HF LCAO method–(a),(d); hybrid HF-DFT(PBE0) LCAO method–(b),(e); DFT(PBE) LCAO method– (c),(f).

can also be seen that Mg 3s, Ti 3d and Zr 4d states make the dominant contribution to the bottom of the conduction band.

As is seen from Fig. 9.8, the inclusion of correlation e ects moves the valence and conduction bands to the higher and lower energies, respectively, which decreases the bandgap. In more ionic stronzium zirconate the energy bands are narrower than in less-ionic strontium titanate.

The ground-state electron-charge density in a crystal can be expressed as

ρ(r) =

ρµν (Rn − Rm)χµ(r − Rn)χν (r − Rm)

(9.72)

µν Rn Rm

and reproduces the essential features of the electron-density distribution near atomic nuclei and along interatomic bonds. For each r the sums over Rn and Rm are restricted to those direct lattice vectors such that the value of χµ(r − Rn) and χν (r − Rm) are nonnegligible.

382 9 LCAO Calculations of Perfect-crystal Properties

The total electron-density maps provide a pictorial representation of the total electronic distribution and are obtained by calculation of the charge density in a grid of points belonging to some planes. More useful information is obtained by considering di erence maps, given as a di erence between the crystal electron-density and a “reference” electron density. The latter is a superposition of atomic or ionic charge distributions.

Fig. 9.9 shows the total and di erence maps for Cu2O crystal, obtained in HF LCAO calculations [622].

Fig. 9.9. Electron-density maps obtained for Cu2O on a (110) plane, [622]. (a) Total electron density. (b) Density di erence maps, bulk minus neutral atom superposition. Values corresponding to neighboring isodensity lines di er by 0.01 e/Bohr3. The full and broken curves in (b) indicate density increase and decrease, respectively.

Two important quantities that require an integration involving ρ(r) are the electrostatic potential and the electric field, [324]. In particular, maps of the electrostatic potential created by electrons and nuclei at a crystal surface may be useful for gathering information about reaction paths and active sites of electrophilic or nucleophilic chemical processes at the surface. As concerns the electric field, it may be of interest to calculate the electric-field gradient at the location of nuclei with a nonzero nuclear quadrupole moment, since comparison with experimental data is possible in such cases.

Three functions may be computed that have the same information content but di erent use in the discussion of theoretical and experimental results [23]: the electron momentum density itself (EMD) ρ(p); the Compton profile (CP) function J(p); the autocorrelation function, or reciprocal space form factor, B(r).

Let χµ(p) be defined as the Fourier transform of AO χµ(r) belonging to atom A

χµ(p) = exp(ipr)χµ(r)dr

(9.73)

and sµ is the fractional coordinate of atom A in the reference cell.

9.2 Electron Properties of Crystals in LCAO Methods

383

EMD is defined as the diagonal element of the six-dimensional Fourier transform of the one-electron density matrix from coordinate to momentum space:

ρ(p) =

ρµν (Rn) exp (−ip(Rn + sµ − sν ))χµ(p)χν (p)

(9.74)

µν Rn

The Compton profile function is obtained by 2D integration of the EMD over a plane through and perpendicular to the direction

J(p) = ρ(p + p )dp

(9.75)

after indicating with p the general vector perpendicular to p.

It is customary to make reference to CPs as functions of a single variable p, with reference to a particular direction <hkl> identified by a vector e = (ha1 + ka2 +

la3)/|ha1 + ka2 + la3|. We have
J<hkl>(p) = J(pe)	(9.76)

The function J<hkl>(p) is referred to as directional CPs. The weighted average of the directional CPs over all directions is the average CP. In the so-called impulse approximation, J<hkl>(p) may be related to the experimental CPs, after correction for the e ect of limited resolution [23].

Once the directional CPs are available, the numerical evaluation of the corresponding autocorrelation function, or reciprocal-space form factor, B(r) is given by the 1D Fourier transform:

		1	∞
	B<hkl>(r) =			J<hkl>(p) exp(ipr)dp	(9.77)
		π

−∞

The structural and electronic properties of Cu2O have been studied in [622] using the HF LCAO method and a posteriori density-functional corrections. The electronic structure and bonding in Cu2O were analyzed and compared with X-ray photoelectron spectroscopy spectra, showing a good agreement for the valence-band states. The Fourier transform of the ground-state charge density of a crystalline system provides the X-ray structure factors of the crystal, which can be determined experimentally by X-ray di raction. To check the quality of the calculated electron density in Cu2O crystal, structure factors have been calculated in [622], showing a good agreement with the available experimental data, see Table 9.22.

9.2.2 Magnetic Structure of Metal Oxides in LCAO Methods: Magnetic Phases of LaMnO3 and ScMnO3 Crystals

The crystalline transition metal oxides TiO2, SrTiO3, SrZrO3 considered above are classified as d0 insulators with quite wide bandgap, being diamagnetic with no unpaired electrons. This means that these crystals have no inherent magnetization, but when subjected to an external field develop magnetization that is opposite to the field (the transition atom spins tend to be oriented in the direction opposite to the field).

384 9 LCAO Calculations of Perfect-crystal Properties

Table 9.22. Structure factors for Cu2O calculated with the periodic HF LCAO method, [622]. Experimental values are obtained from X-ray data

hkl	Fcalc	Fexpt
110	13.52	12.42
111	94.78	93.70
200	79.48	78.98
211	9.10	8.53
220	84.34	82.52
222	61.07	59.40
310	6.94	6.56
311	69.22	67.49
330	4.63	4.71
331	55.57	53.78
332	4.39	4.24
333	46.86	45.42
400	65.05	63.09
411	4.80	4.71
420	49.72	48.06
422	53.90	52.18

As we saw in the preceding subsection, the bandgap in d0 insulators is between a filled band of bonding orbitals, with predominately oxygen 2p atomic character, and an empty metal d band of antibonding orbitals.

The transition-metal oxides with dn ions (such as NiO, Cr2O3, LaMnO3) form a class of compounds with the localized “impurity-like” d-levels [624]. These systems are known as Mott–Hubbard or magnetic insulators (exhibit magnetic order in the absence of an external field); their magnetic properties indicate the unpaired electrons expected for the appropriate dn open-shell configuration. d − d transitions in these crystals appear due to the crystal-field splitting and give rise to optical absorptions, with only quite weak perturbations from those expected for isolated dn ions. These oxides have narrow upper valence bands (1–2 eV), because of small overlap between the metal d and oxygen 2p orbitals, so that their local electronic structure can be described in terms of atomic-like states. The fully ionic O2− configuration is not exactly the state of an oxygen atom in a crystal, and O− mobile and correlated configuration (oxygen holes with 2p5 configuration) has to be included. The weak interactions between magnetic moments of dn-ions give rise to magnetic ordering (most commonly antiferromagnetic at low temperatures). At the same time, these compounds are good insulators with bandgaps that may be as large as those in some d0 compounds. Like the latter, however, the magnetic insulators are susceptible to nonstoichiometry, and this may give rise to semiconductive properties and strong optical absorptions. For example, NiO is green and highly insulating when pure, but easily takes up excess oxygen to become black and semiconducting.

The magnetic insulators have an the extraordinary range of structures and properties [625] and provide an excellent case study in quantum chemistry of solids. The existence of a magnetic ground state is a many-body e ect caused by correlated electron–electron interactions. The need for the inclusion of electron-correlation ef-

9.2 Electron Properties of Crystals in LCAO Methods

385

fects has limited the majority of applications to solids by DFT methods. We noted in Chap. 7 that due to the strong Coulomb correlations DFT methods are in serious trouble for systems with localized electrons as these methods overestimate delocalization of the electron density due to nonexact cancelation of the electron self-interaction. DFT results in many cases are in disagreement with experiment. For example, when the LDA is used to describe the magnetically ordered insulating ground states of NiO, a nonmagnetic metallic state is obtained [626]. Initially, this was thought to be a failure of the one-electron approximation itself, meaning that such highly correlated systems could not be described using band theory. It is now apparent that this is not case [627]: the failure of DFT was due to its approximate treatment of the exchange interaction. Analyzing di erent approaches, introducing better descriptions of the onsite exchange interactions we concluded in Chap. 7 that LCAO calculations with hybrid DFT functionals can play the same role in quantum chemistry of solids as DFT+U and SIC DFT calculations (with PW and LMTO basis) in solid-state physics. This is demonstrated by the results of recent numerous LCAO (UHF and hybrid DFT) calculations of the properties of magnetic insulators [568].

The magnetic properties of di erent oxides have been investigated in the LCAO approximation: MO (M=Ni, Mn), [628], M2O3(M=Cr, Fe), [629, 630], Mn3O4, [631], CuGeO3, Ag2Cu2O3, [632], high-Tc superconductor parent compounds A2CuO2X2 (A=Ca,Sr, X=F, Cl), [633]. The main parameter used to quantify the magnetic properties is the exchange coupling constant between the paramagnetic centers with total spins Si and Sj that is defined through the phenomenological Heisenberg–Ising Hamil-

ˆ
tonian with total spin operators S:
	ˆ ˆ
ˆ		ˆ ˆ	(9.78)
H = −	Jij SiSj = −	Jij SiSj
i>j		i =j

The interactions are generally assumed to be limited to nearest neighbors, but the range of interactions can be extended beyond the nearest neighbors (we refer the reader to Chap. 7 in [9] for a detailed discussion of magnetic behavior of solids and the Heisenberg–Ising model). The comparison of calculated exchange coupling constants with those obtained experimentally is, however, not straightforward since experimental data are in most cases obtained from a fitting of the magnetic susceptibility to the expression obtained using a model that includes simplifying assumptions [632]. The latter are adopted to reduce the number of fitting parameters. In this situation the use of theoretical methods to estimate the exchange coupling constants becomes a valuable tool for the experimentalists.

We discuss here in more detail the recent applications of LCAO methods to the LaMnO3 and ScMnO3 crystals, two representatives of the rare-earth manganese oxides, RMnO3 (R=La, Sc). The structures of these two crystals were described in Chap.2. LaMnO3 crystal has the orthorhombic perovskite structure, found in RMnO3 with a large ionic radius of rare-earth element R (R= La, Pr, Nd, Sm, Eu, Gd, and Tb). ScMnO3 crystal has hexagonal structure found in RMnO3 oxides containing the rare-earth elements with a small ionic radius (R= Ho, Er, Tm, Yb, Lu, and Y). We note that some of the hexagonal manganites can also form in the orthorhombic structure depending on the heat treatment.

Perovskite materials display a wide variety of fundamental properties, from magnetism to ferroelectricity, from colossal magnetoresistance to half-metallicity; they

386 9 LCAO Calculations of Perfect-crystal Properties

are used in a number of important technological applications, such as transducers and memories. They also show electronic and structural peculiarities including orbital and charge ordering, formation of local moments, and Jahn–Teller distortions. Such a richness of properties, combined with their relatively simple structure, makes them ideal materials for investigating the general principles that govern these properties. Unlike the recently extensively studied perovskite manganites, the hexagonal RMnO3 compounds have a ferroelectric transition at very high temperature (e.g. 900 K for YMnO3), and an antiferromagnetic transition at a much lower temperature (e.g., 570 K for YMnO3.7). Such compounds are known as ferroelectromagnets.

There exists a series of LaMnO3 bulk electronic-structure calculations, using a number of first-principles methods e.g. UHF LCAO [634, 635], LDA+U PW [636], and relativistic FPGGA LAPW [637]. LCAO studies with hybrid DFT functionals [638,639] provided a most reliable description of the electronic and magnetic structure of LaMnO3. These studies deal with the energetics of the ferromagnetic (FM) and antiferromagnetic (AF) phases. The experiments show that below 750 K the cubic perovskite phase of LaMnO3 with a lattice constant of a0 =3.95 ˚A is transformed into the orthorhombic phase (four formula units per unit cell). Below TN =140 K (Neel temperature) the A-type AF configuration (AAF) is the lowest in energy. This corresponds to the ferromagnetic-coupling in the basal ab (xy) plane combined with antiferromagnetic coupling in the c (z)-direction in the P bnm setting, see Fig. 9.10.

z

La

Mn

O

y

α

β

x

FM

AAF

GAF

CAF

Fig. 9.10. Magnetic ordering in LaMnO3 crystal. FM–ferromagnetic ordering, AAF, GAF, CAF–antiferromagnetic orderings.

Also, FM, GAF and CAF magnetic states exist: FM corresponds to a fully ferromagnetic material, in GAF all the spins are antiferromagnetically coupled to their nearest neighbors, and in a CAF cell the spins are antiferromagnetically coupled in the basal plane and ferromagnetically between the planes (along the c-axis). In the DFT LCAO calculations [639] Becke three-parameter hybrid functional (B3PW) was