2.1 Density Functional Theory
2.1.4 The philosophy of DFT
The density functional theory (DFT) considers the electron density n(r) as the central variable that describes all physical properties of the system, rather than the many-body wavefunction. This conceptual difference with the wave function based methods leads to a remarkable reduction in difficulty. Whereas the many- body electronic wavefunction is a function of 3N variables, the electron density is a function of only three variables x, y, z - three Cartesian coordinates. An early version of the density functional theory was proposed by Thomas [186] and Fermi [187] where the kinetic energy was represented by one electron density. The electron-electron interactions was incorporated via a mean field approximation. However, the semi- classical Thomas Fermi approximation, in spite of pointing out the importance of
2.1 Density Functional Theory one electron density to obtain the ground state properties in a solid, failed to address the effects of exchange and correlation. A subsequent proposal by Dirac [188] that incorporates the exchange energy in terms of the electron density, failed to improve the method significantly. In 1964, Hohenberg and Kohn [147] provided the required breakthrough.
2.1.4.1 Hohenberg-Kohn theorems
Hohenberg and Kohn’s approach is the backbone of current day DFT. It is based on the following remarkable simple theorems:
1. The external potential Vext(r) is uniquely determined by the electron density n(r) so the total energy is a unique functional of the density E[n] [149].
2. The true ground state densityn(r) minimizes the total energy functionalE[n]
[149].
According to the above theorem, the ground state energy of interacting electrons in an external potential Vext(r) is described by an energy functional
E[n] =F[n] + Z
Vext(r)n(r)dr (2.9)
The first term F[n] is a universal functional of the electron density n(r) which includes the kinetic (Te) and electron-electron interaction (Vee) energies in equa- tion 2.4. The second term represents interaction energy with the external potential.
The total energy in the ground state (E0) is realized through the minimization of E[n]. Therefore, the ground state density, in principle, uniquely determines the ground state properties of an interacting electronic system. This approach offers a perspective different than the conventional methods where many body wave func- tions are to be computed, which is often a difficult task.
2.1.4.2 The Kohn-Sham ansatz
The two theorems proved the existence of a universal functional, though they do not give any idea about the nature of the functional. Therefore, the Hohenberg-Kohn theorems first appeared not to be helpful for practical applications. The success of the DFT comes through using the Hohenberg and Kohn theorem in conjunction with the Kohn-Sham formalism [148]. The basic idea of Kohn and Sham is the introduction of a fictitious auxiliary non-interacting electron system in which the
Chapter 2. Methodology
electrons are moving within an effective Kohn-Sham potential, VKS(r). The single- particle Kohn-Sham orbitals are constrained to yield the same ground state density as that of the fully-interacting system, so that the Hohenberg-Kohn-Sham theorems are still valid. This mapping by Kohn and Sham resulted in a Schr¨odinger like equation for single particle which provides a variational total energy and thus the ground state single particle density to a good approximation.
(Hks−ǫi)ψi(r) = 0 (2.10)
Hks =−1
2∇2+VKS (in Hartree units) (2.11) where,
VKS =Vext(r) +VHatree(r) +Vxc(r) (2.12) The second term is called Hartree potential,
VHartree= 1 2
Z n(r)n(r′)
|r−r′| drdr′ (2.13)
and the last term is exchange-correlation potential, defined as, Vxc= δExc[n]
δn(r) (2.14)
which includes electron-electron interaction beyond the Hartree term. The density is calculated from single-electron Kohn-Sham orbitals according to
n(r) =
N
X
i=1
|ψi(r)|2 (2.15)
The total number of electron is obtained as N =
Z
n(r)dr (2.16)
The self-consistent equations (2.9-2.16) are used to compute the ground state energy of an electronic system with
Eks =Ts[n] + Z
Vext(r)n(r)dr +EHartree[n] +Exc[n] (2.17) Addressing magnetism by incorporating spin dependent density is not too com-
2.1 Density Functional Theory plicated in DFT. In this case, the Kohn-Sham equations are solved separately for each spin channel. The charge density is obtained by adding up the densities of the two spin channels (n↑ + n↓), while the spin density is obtained as the difference of the electron densities between the two spin channels (n↑ - n↓).
2.1.4.3 Approximations for the exchange-correlation energy functional In principle, the Kohn-Sham ansatz is exact, except for the exchange-correlation functional. The challenging part of the solution of Kohn-Sham equations lies in approximating the exchange-correlation functionals in which all of the complicated many-body effects are considered.
The simplest approach to obtain an approximateExcis the local density approx- imation (LDA) [148]. The LDA assumes that the variation of density in a solid is locally insignificant, and thus the electron density in a solid can be mimiced using that of the homogeneous electron gas, the exchange and correlation terms of which are known exactly. The Exc, under LDA, is given by
ExcLDA[n] = Z
n(r)ǫhomxc (n(r))d3r (2.18) where ǫhomxc is the sum of the exchange and correlation energies of the homo- geneous electron gas of density n(r). Consideration of spin degrees of freedom of electrons modifies LDA to the local spin density approximation (LSDA).
LDA, however, ignores the exchange-correlation energy at pointr due to nearby inhomogeneities in the electron density. In spite of this, the success of LDA is remarkable due to the fact that it gives the correct sum rule for the exchange- correlation hole. However, it failed upto the extent of producing wrong electronic ground state in some important cases.
An attempt to improve the LDA was made by introducing the gradient correc- tions on the electron density, the generalized gradient approximation (GGA) [189– 191]. The GGA functional is given by
ExcGGA[n] = Z
d3r n(r)ǫGGAxc (n(r),|∇n(r)|) (2.19) For systems where the charge density varies slowly, the GGA functional improves the results. Calculations presented in the thesis are done with the GGA functionals.
For strongly correlated systems, LDA, as well as the GGA, fail. For example,
Chapter 2. Methodology
band gaps of semiconductors are underestimated [192]. The reason for this behavior is found in the mean-field treatment of the Coulomb repulsion between the electrons, which does not take into account strong electronic correlations. In the formulation of LDA exchange-correlation functional, the potential of the Kohn-Sham orbitals does not depend on the occupancy of the orbitals. However, in case of strong on-site correlations, the addition of an electron to a localized site, already containing an electron, requires an additional energy U [193]. The “Hubbard U” in the Hubbard model, introduced by J. Hubbard in 1963, takes into account the on-site Coulomb repulsion by an additional term in the Hamiltonian [194]. The introduction of the
“Hubbard U” in the energy functional of LDA gives rise to the so-called LDA+U method [195–197]. In a similar manner, a term that takes into account correlation effects can also be added to GGA exchange-correlation functional, which gives rise to the GGA+U method. The basic idea behind this method can be understood in the comprehensive review by Anisimov et al.[198]. The absolute value of the U can either be estimated from experimental observations [199, 200] or through different computational approaches [201–204].