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##### Response Calculations within Time-Dependent Density Functional Theory

**Authors**: M.A.L. Marques

**Ref.**: Habilitation à Diriger des Recherches, Université Claude Bernard - Lyon 1 (2009)

**Abstract**: It was in 1964 that Hohenberg & Kohn discovered
that to fully describe a stationary electronic system it is sufficient
to know its ground-state density. From this quantity all observables
(and even the many-body wave function) can, in principle, be obtained.
The density is a very convenient variable: it is a physical
observable, it has an intuitive interpretation, and it only depends on
three spatial coordinates. This is in contrast to the many-body wave
function, which is a complex function of 3N spatial coordinates.
Hohenberg & Kohn also established a variational principle in terms of
the density by showing that the total energy can be written as a
density functional whose minimum, the exact ground-state energy of the
system, is attained at the exact density. In this way, they put on a
sound mathematical basis earlier work by Thomas,
Fermi, and others, who had tried to write
the total energy of an interacting electron system as an explicit
functional of the density.

Another breakthrough occurred when Kohn & Sham
proposed the use of an auxiliary noninteracting system, the Kohn-Sham
system, to evaluate the density of the interacting system. Within the
Kohn-Sham system, the electrons obey a simple, one-particle,
Schrodinger equation with an effective external potential,
v_{KS}. As v_{KS} is a functional of the electronic
density, the solution of this equation has to be performed
self-consistently.

In this equation, all the complex many-body effects are contained in the
(unknown) exchange-correlation (xc) potential v_{xc}. Kohn &
Sham also proposed a simple approximation to
v_{xc}, the local density approximation (LDA). This functional,
that uses the knowledge of the xc energy of the homogeneous electron
gas, turned out to be quite accurate for a number
of applications, and is still widely used, especially in solid-state
physics.

The use of the density as the fundamental variable, and the
construction of the Kohn-Sham system form the basis of what became
known as density functional theory
(DFT). The original
formulation assumed an electronic system at zero temperature with a
nondegenerate ground state, but has been extended over the years to
encompass systems at finite temperature,
superconductors,
relativity, etc.

An extension of somewhat different nature is time-dependent DFT
(TDDFT). The
foundation of modern TDDFT was laid in 1984 by Runge &
Gross, who derived a Hohenberg-Kohn like theorem for
the time-dependent Schrodinger equation. The scope of this
generalization of DFT included the calculation of photoabsorption
spectra or, more generally, the interaction of electromagnetic fields
with matter, as well as the time-dependent description of scattering
experiments (which was actually the original motivation of Runge &
Gross). Again, the rigorous theorems of Runge & Gross put on a firm
basis earlier work by Ando and by Zangwill &
Soven who had performed the first time-dependent
Kohn-Sham calculations.
Presently, the most popular application is the extraction of
electronic excited-state properties, especially transition
frequencies. By applying TDDFT after the ground state of a molecule
has been found, we can explore and understand the complexity of its
spectrum, thus providing much more information about the
system. TDDFT is having especially strong impact in the
photochemistry of biological molecules, where the molecules are too
large to be handled by traditional quantum chemical methods, but are
too complex to be understood with simple empirical frontier orbital
theory.

Today, the use of TDDFT is continuously growing, in all areas where
interactions are important but the direct solution of the Schrodinger
equation is too demanding. New and exciting applications are beginning
to emerge, from ground-state energies extracted from TDDFT to
transport through single molecules, to high-intensity laser and
non-equilibrium phenomena, to non-adiabatic excited-state dynamics, to
low-energy electron scattering. In each case, the present
approximations were applied, and found to work well for some
properties, but occasionally fail for others. Thus the search for
more accurate, reliable approximations will continue, and over time,
should attain the same maturity as present ground-state DFT.

The present manuscript contains a fairly condensed overview of TDDFT, and
some of its applications to the fields of nanotechnology and
biochemistry. These have constitute the main research topic of the
Author for the past years. The manuscript is organized as follows.

Before entering the realm of TDDFT, we give a brief overview of
the basic ideas of ground-state DFT, that allows us to fix some
basic notation that will be used in the rest of this article, and to
introduce some key concepts that will be developed later.
Chapter 2 deals with formal theory. We start by stating the
major theorems and proofs, which is followed by an discussion on the
available time-dependent density functionals that we now have at our
disposal. We then present linear response theory within TDDFT, with
its several different flavors.

The next chapter deals with the numerical problems of solving the
time-dependent Kohn-Sham equations. The technique we chose consists on
the use of real-space grids to discretize the Hamiltonian, and
real-time propagation to evolve the time-dependent equations. Other
very important issues like parallelizarion or scalability will also be
discussed. Note that all the numerical developments that are here
described are available in the open-source code octopus.

Chapters 4 and 5 deal with applications of
TDDFT in the real of linear response. The main objective of these
calculations is to obtain reliable spectra (usually absorption) from
calculations. By comparing these spectra with experimental curves, one
is usually able to deduce important information that is not directly
available from experiment. This can include overall geometries,
protonation states, relative abundances, etc. Also the basic knowledge
of the excitation properties of the systems contributes to the better
understanding of these systems.

Chapter 6 is concerned with the important van der Waals
interactions, and how to extract, from TDDFT calculations, relevant
parameters to describe. We will discuss both the interaction between
two finite systems, and between a finite system and a semiconducting
surface.

The final chapter is devoted to applications of TDDFT in the
non-linear regime. The non-linear regime is much less studied than its
linear counterpart, as it is both more complicated and numerically
involved. Also, the existing time-dependent xc functionals that
perform so well in the linear regime often fail in these
circumstances. Nevertheless, we present several exploratory studies
for simple systems.