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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, vKS. As vKS 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 vxc. Kohn & Sham also proposed a simple approximation to vxc, 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.

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