be metallic instead of insulating ground state. It tends to underestimate the ground state energy, band gap and the ionization energies, but overestimates the bonding energy.
The Generalized Gradient Approximation (GGA)
To promote the accuracy, not only ρ(r) but also the gradient of ρ(r) are considered to ac-count the exchange-correlation energy since the fact that the electron density of a real system is non-uniform. The form of exchange-correlation energy in the generalized gradient approxima-tion [6] can be expressed as
EXCGGA[ρ↑(r), ρ↓(r)] = Z
ρ (r)εXC[ρ↑(r), ρ↓(r), ∇ρ↑(r), ∇ρ↓(r)]. (2.26) The GGA has reduced the errors of LDA, but the variation of electron density still must be slow enough.
2.4 Mott Insulators
Though density functional theory successfully describes various properties of materials, there still exist some systems like Mott insulators, named after N. F. Mott, cannot be explained.
In 1937, J. H. de Boer and E. J. W. Verwey indicated that a variety of transition metal oxides predicted to be conductors by band theory are insulators [8]. Also in 1937, N. F. Mott and R.
Peierls tried to explain that this anomaly can be the result of that band theory did not consider the on-site electron-electron Coulomb repulsion. Since the density functional theory is a mean-field single particle theory, which cannot describe the many-bode correlations, there must be some modifications. The Hubbard model and the improved density functional theory, the so-called DFT+U, will be introduced in the following subsections.
2.4.1 Hubbard Model
The Hubbard model, originally proposed in 1963 [9, 10], is an approximate model which can be used to solve the anomaly of Mott insulators. First, we regard the nuclei in the system as the fixed array, i.e. without considering the lattice vibrations. For simplicity, every site of the array has only one energy level, thus only two electrons can occupy an energy level (spin up and spin down) due to the Pauli principle. Electrons move around the array of nuclei interact via a screened Coulomb interaction, the biggest interaction will be the Coulomb repulsion of the two electrons occupy the same site. For simplicity, There is no interaction between electrons on different sites. The on-site interactions are modeled by a term which is zero if the atom is empty of electrons or has only a single electron on it, but has the value of repulsion energy U if the atom has two electrons. The competitive term, the kinetic term, is introduced allowing electrons to move from one site to another. The energy scale t which governs this ’hopping’ will
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be determined by the overlap of two wavefunctions. Since wavefunctions decay exponentially, we can consider that the hopping only occurs between the neighboring atoms.
Now, we can write down the Hubbard Hamiltonian. To begin with, we recall the creation and annihilation operators which deal with the harmonic oscillator. We will spend some time reviewing their properties, which parallel those of the operators in the Hubbard model in many ways, in this more familiar setting.
ab=r mω these operators obey the commutation relations,
The Hubbard model is also written in terms of ’fermion’ creation and annihilation operators.
However, there are several different respects. The fermion operators in the Hubbard model are not introduced in terms of familiar position and momentum operators. They are distinguished by attaching indices j and σ (spin up or spin down) which can be written asbc+iσ andcbiσ, where bc+iσ is the operator which creates an electron of spin σ on lattice site i,bciσ is the operator which annihilates an electrons of spin σ on lattice site i, and niσ =bc+iσbciσ is the number operator. As a consequence, the occupation number states are no longer characterized by a single number n, as for a single harmonic oscillator, but instead by a collection of occupation numbers niσ. We can write a state as | n1↑, n2↓, n3↓, ... >. Because these operators are used to describe fermions, they must satisfy the following relations to agree with the Pauli principle,
{cbiσ,cb+jσ0} = δi, jδσ ,σ0 {bc+iσ,bc+jσ0} = 0 {bciσ,bcjσ0} = 0,
(2.30)
where { bA, bB} = bA bB+ bB bA(the anticommutation relation). The maximum occupation of a par-ticular site with a given spin is 1. The Hubbard Hamiltonian is then,
H= −t ∑
The first term is the kinetic term. It describes the annihilation of an electron of spin σ on site j and its creation on site i (or vice-versa). The symbol <i, j> emphasizes that hopping is allowed
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only between two neighboring sites. The second term , the correlation term, is the Coulomb re-pulsion energy, an energy U will be added if there is doubly occupied on j-site. The competition between the kinetic term and the correlation term characterize the system. As t/U >> 1, the energy is minimized making the kinetic term as small as possible through delocalization (little price is paid on the occupied atomic sites to overcome repulsion U ). As t/U << 1, the kinetic energy of electrons is not large enough to overcome the on-site repulsion. Electrons undergo a Mott localization. The last term is a chemical potential term which controls the filling. We refer to the situation where the filling is one electron per site as ’half-filling’ since the lattice contains half as many electrons as the maximum number (two per site). Studies of the Hubbard model often focus on the half-filled case because it exhibits a lot of interesting phenomena (Mott insulators, anti-ferromagnetic order, etc.)
2.4.2 Beyond DFT : DFT+U
Due to the insufficiency of density functional theory describing the strong correlated system, several attempts like the self-interaction correction (SIC) method [11], the Hartree-Fock (HF) method [12] and the GW approximation (GWA) [13] for improving the DFT were proposed.
However, any of them has its inadequacy. From 1990 to 1995, the LDA+U method consists in a Hubbard-like correction to the LDA energy functional was introduced and developed by Anisi-mov and coworkers, which gives a better description of electronic correlations. A rotationally invariant version was introduced by Liechtenstein et al. [14], and the simplified approach was introduced by Dudarev et al. [15],
ELSDA+U = ELSDA+(U − J)
2 ∑
σ
(nm,σ− n2m,σ), (2.32)
where U and J are screened Coulomb and exchange parameters.
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Chapter 3
Crystal Field Theory
Crystal Field Theory (CFT) is an ionic theory that describes the breaking of symmetries of orbital states, usually d or f -orbitals, in the presence of surrounding ligands. It was proposed by the physicist Hans Bethe in 1929 [16]. Subsequent modifications were proposed by J. H. Van Vleck [17] to allow for some covalency in the interactions to be successfully used to explain and predict some magnetic, spectral and thermodynamic properties of transition metal complexes.
3.1 Atomic Orbitals
An atomic orbital is the probability distribution of electrons around the nucleus of atom. The shape of the orbital mainly depends on the quantum numbers (n, the principle quantum number, l, the orbital quantum number, and m, the angular momentum quantum number) associated with the particular energy state. Each set of quantum numbers of an orbital can be occupied by two electrons, spin up and down, as listed in Table 3.1. To understand CFT, it is important to have Table 3.1: Quantum numbers of atomic orbitals are listed, where n (n∈N) is the principle quan-tum number, l (l = 0, 1, ..., n−1) is the orbital quanquan-tum number and m (m = 0, ±1, ..., ±l) is the angular momentum quantum number.
orbital s p d f
l=0 l=1 l=2 l=3
n=1 m=0
n=2 m=0 m=0, ±1
n=3 m=0 m=0, ±1 m=0, ±1, ±2
n=4 m=0 m=0, ±1 m=0, ±1, ±2 m=0, ±1, ±2, ±3
a clear picture of the shapes (angular dependence functions) of the d-orbitals. Consider a free transition metal atom. There are five d-orbitals in two spatial groups, three of the five d-orbitals (dxy, dyz and dzx) are collectively referred to as t2g, means that all three (t) orbitals have the
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Figure 3.1: d-orbitals - For dx2−y2, the four lobes lie on the x and y axes, and for dz2, there are two lobes on the z axes and there is a donut shape ring that lies on the xy-plane around the other two lobes. The four lobes of dxyorbital lie in-between the x and the y axes, and so on. [Jmol: an open-source Java viewer for chemical structures in 3D. http://www.jmol.org/]
reflection asymmetry (2) and inversion symmetry (g). Take dxy for example, the four lobes lie in-between the x and the y axes as can be seen in Figure 3.1, and so on. The remaining two (dz2 and dx2−y2) as eg, means that both two (e) orbitals have the reflection symmetry (1) and inversion symmetry (g). For dx2−y2, the four lobes lie on the x and y axes, and for dz2, there are two lobes on the z axes and there is a donut shape ring that lies on the xy-plane around the other two lobes. All d-orbitals are degenerat when the atom is free. However, in the presence of surrounding ligands, symmetries of d-orbitals are broken. Energy of t2g and egsplit into two levels in octahedral and tetrahedral cases. More complex splittings occur in other cases and will be discussed in the following sections.