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Approach based on basic model study allows to understand electronic states
in strongly correlated systems and to plot phase diagrams as a function of
model parameters. However, in order to use these results for interpretation of
real compounds experimental data one needs to know parameter values characteristic
for particular material. For this purpose there are two approaches:
empirical and first principles.
The first of them, empirical, is based on using experimental data to determine
model parameters. Usually that is spectral data (optical, photoemission,
and X-Ray spectra) that can give information about band width. In the case
of strongly correlated systems on the edge of the metal–insulator transition
(or already in Mott insulator state) it is possible to estimate also value for
Coulomb interaction parameter U determined from the energy separation of
the spectral peaks interpreted as Hubbard bands.
More often the approach is used where model parameters (such as Coulomb
interaction parameter U and band width W) are varied to achieve the best
agreement between experimental data for some characteristic (resistivity, susceptibility,
etc.) and the corresponding value calculated in model solution. In
such approach model parameters are considered to be adjustable parameters.
However, the most justified way to determine model parameters is using
results of electronic structure calculations from first principles methods (or
in other words ab-initio methods). The name first principles is used as opposite
to empirical approach. First principles methods are not formally exact
ones, because a lot of various approximations were done in their derivation.
The key issue is the fact that in these methods construction only the most
basic (first) Quantum Mechanics principles were used such as many-electron
Schroedinger equation. For atoms and small molecules many-electron methods
could be used where wave function is represented as linear combination of Slater determinants. However for solids the most practical approach is Density
Functional Theory (DFT) [45,46] that reduces many-particle problem to oneelectron
one. In Sect. 2.1.1 the basics of DFT are presented and in Sect. 2.1.2
the main calculation methods based on DFT are described.
In DFT one solves differential equations for one-electron wave functions
with effective potential that is in its turn determined by the equation solutions.
That results in very effective and practically realized iterative calculation
scheme. However that means explicit neglect of correlation effects (all electrons
feel the same time and energy independent potential). In the result for Mott
insulators (for example transition metal oxides) DFT methods gave wrong
metallic ground state (see Sect. 2.1.3).
There were proposed many corrections to DFT overcoming this defect and
resulting in correct ground state for Mott insulators (see Sect. 2.1.4). However
those correction were not derived from general enough equations and have
rather restricted area of applicability.
Better perspectives has approach where electronic structure calculation
results by DFT methods are used to calculate parameters for general Hamiltonian
and then the problem defined by this Hamiltonian is solved by one
of the methods developed in basic models theory (see Sect. 1.2). In Sect. 2.2
we describe how it is possible using Wannier functions formalism to calculate
Hamiltonian parameters from eigenfunctions and eigenvalues obtained in
DFT. Using constrain DFT (see Sect. 2.2.5) one can calculate also the value
of Coulomb interaction parameter U.
The problem defined by this Hamiltonian can be solved in the simplest
static mean-filed approximation (unrestricted Hartree-Fock). The resulting
calculation scheme is called LDA + U method and is described in Sect. 2.3.
This method was found to be very successful when applied to systems with
long-range spin and orbital order.
However, more general problems of paramagnetic insulator and strongly
correlated metal can be solved only by DMFT method that is described in
details in Chaps. 3–5. Approach based on solving in DMFT the problem
defined by the Hamiltonian with parameters calculated in DFT calculations
is called LDA + DMFT method and is described in Chap. 6.