5-7.8. Density Functional Theory (DFT)
The density-functional theory (DFT) is a specific technique to solve the
many-body Schrödinger equation, in terms of single-particle equations and
an effective potential. Today, density functional theory is widely employed
in the description of bulk materials and molecules in physics, chemistry, and
biology. The beginning of the DFT was in the papers of Thomas and Fermi
in the 1927 and its extension due to Dirac in 1928, but only during the
1960s the theory became complete and accurate in the work of Kohn, Sham
and Hohenberg. In much the same way as the Hartree equation, the DFT
provides a way to decouple the electron-electron interactions, and solve the
one electron equations by a self-consistent method. However, the solution is
performed starting from the ground-state electron density, like the selfconsistent field method. Therefore, the many-electron Schrödinger equation
is replaced by the so-called Kohn-Sham equation:
 2 2
' 
 2m   Veff  i (r )  Ei i (r ),


i  1,2,...N
(5-63a)
where the effective potentials, Veff, is defined as follows:
Veff = V(r) + VH + Vxc
(5-63b)
The first term in equation V(r) is the external potential and includes the
potential originating from the nuclei as well as an external applied electric
field if present. The term VH corresponds to the classical Coulomb potential
(Hartree potential) for an electron density n(r). This term is obtained by
solving the Poisson equation for the scalar potential
2VH(r) = e.( /= - (e2/) n(r)
(5-63c)
Finally, the exchange-correlation potential Vxc takes into account the manybody effects, and is defined as the functional derivative of the exchange
energy, EXC with respect to the electron density Vxc =∂Exc /∂n. The last term
contains the remaining contributions to the potential that we do not know
exactly. There exist N eigenfunctions i for each electron, from which the
electron density can be calculated using the Thomas-Fermi model
(5-63d)
Here, he sum is taken over all the occupied states and fi = f(Ei - EF) is the
occupation probability (the Fermi-Dirac distribution). Also, the total energy
is given by:
ET =  Ei . f(Ei - EF)
(5-63e)
where f is the Fermi-Dirac distribution and the sum is taken over the total
number of electrons (N). The solutions of the Kohn-Sham equations in
principle give the ground state properties of the system. However, the term
Exc is not known exactly and some kind of approximation is needed. The two
types of approximations which are generally used are the local density
approximation (LDA) and the generalized gradient approximation (GGA).
Note 5-2. Hohenberg-Kohen (HK) Theorem
The Hohenberg-Kohen (HK) theorem states that: for any system constituted
by interacting particles, for a fixed the interaction, there exists a bijective
relation among the external potential (up to an arbitrary additive constant),
the ground–state many–body wave–function and the ground–state density of
the system, Vext ⇐⇒ 0 ⇐⇒ ρ0 ;
ρ0 = <0|ρ|0>. Thereforte, the
ground–state energy Eo and density o can be determined by minimizing a
functional of the charge–density n. This functional has a minimum when ρ =
ρ0. As a consequence of the HK theorem, the knowledge of the ground–state
wave–function 0 gives access to all physical observables. Moreover the
external potential fixes the Hamiltonian and knowing the Hamiltonian we
can in principle access the excited states of the system. It is surprising that
given the ground–state density we can access all this physical information.
However DFT is in practice used only for ground–state properties, as there is
no practical scheme able to describe excited stated starting from the ground–
state density. Excited state properties can be obtained from the extension to
the time domain of DFT: this is Time–Dependent DFT (TDDFT). δρ(1) =
χKS (1, 1′)δvKS(1′) where χKS(1, 2) is the response function of the KS non
interacting system, while δvKS(1) is the variation of the TDKS potential
needed to follow the evolution of the density when an external potential
δVext(1) is applied. The analogous version to the Hohenberg-Kohn theorem
which includes time-dependent potentials is known as the Runge-Gross
theorem.
Fig. 5-21. Illustration of the Kohen-Sham method.
In both LDA and GGA, the exchange-correlation potential Vxc is a local
functional of the charge density. They are accurate when the charge density
n(r) is a smoothly varying function. However, for systems where strong
correlation effects are important, i. e., where the charge density n(r) changes
sharply, these approximations most likely fail.
5-7.9. Local Density Approximation (LDA)
The local density approximation (LDA) was proposed by Kohn and Sham.
They showed that it could be applied to the limiting case of a slowly varying
electron density. Therefore, one considers exchange energy EXC to be that of
a uniform electron gas of the partial density n(r). In this case, the KohnSham equation (5-63a) may be written as follows:
 2 2

n( r ' )
'
 2m   V (r )   | r  r '|dr '  VLDA   i (r )  E (r ),


i  1,2,...N
(5-63f)
where VLDA = ∂E/∂n is the local approximation to the Kohn-Sham exchangecorrelation potential and V(r) is the external potential energy.
The Kohn-Sham-LDA can be further extended to the spin dependent case by
replacing the scalar external potential V(r) by a spin dependent potential and
replacing the electron charge density n(r) by the density matrix. The DFT in
the local density approximation (DFT-LDA) has been very successful in the
analysis of interacting-electron ground states. Many efforts have also been
devoted to DFT-based schemes for excited states.
Note 5-3. Ab initio Methods
Ab initio quantum methods are computational methods based on quantum
physics. Ab initio electronic structure methods have the advantage that they
can be made to converge to the exact solution, when all approximations are
sufficiently small. In this case, configuration interaction, where all possible
configurations are included, tends to the exact solution of the electronic
Schrödinger equation (in the Born-Oppenheimer approximation). The
convergence, however, is usually not monotonic, and sometimes the smallest
calculation gives the best result for some properties. The dark side of ab
initio methods is their computational cost. They often take enormous
amounts of computer time, memory, and disk space. The Hartee-Fock, (HF)
technique, which is the simplest Ab inito method, scales as N4 (N being the
number of basis functions). However in practice it can scale closer to N3 as
the program can identify and neglect zero and small integrals. Density
functional theory (DFT) methods using functionals which include Hartree–
Fock exchange scale in a similar manner to Hartree–Fock but with a larger
proportionality term and are thus more expensive than an equivalent
Hartree–Fock calculation. DFT methods that do not include Hartree–Fock
exchange can scale better than Hartree–Fock