Time-dependent density functional theory many-body system

The many-body ground and excited states of a many-electron system are unknown hence, the exact linear and quadratic density-response functions are difficult to calculate. In the framework of time-dependent density functional theory (TDDFT) [46], the exact density-response functions are obtained from the knowledge of their noninteracting counterparts and the exchange-correlation (xc) kernel /xcCf, which equals the second functional derivative of the unknown xc energy functional ExcL i]- In the so-called time-dependent Hartree approximation or RPA, the xc kernel is simply taken to be zero. 

The photoabsorption spectrum a(co) of a cluster measures the cross-section for electronic excitations induced by an external electromagnetic field oscillating at frequency co. Experimental measurements of a(co) of free clusters in a beam have been reported, most notably for size-selected alkali-metal clusters [4]. Data for size-selected silver aggregates are also available, both for free clusters and for clusters in a frozen argon matrix [94]. The experimental results for the very small species (dimers and trimers) display the variety of excitations that are characteristic of molecular spectra. Beyond these sizes, the spectra are dominated by collective modes, precursors of plasma excitations in the metal. This distinction provides a clear indication of which theoretical method is best suited to analyze the experimental data for the very small systems, standard chemical approaches are required (Cl, coupled clusters), whereas for larger aggregates the many-body perturbation methods developed by the solid-state community provide a computationally more appealing alternative. We briefly sketch two of these approaches, which can be adapted to a DFT framework (1) the random phase approximation (RPA) of Bohm and Pines [95] and the closely related time-dependent density functional theory (TD-DFT) [96], and (2) the GW method of Hedin and Lundqvist [97]. 

Time-dependent density-functional theory (TDDFT) extends the basic ideas of ground-state density-functional theory (DFT) to the treatment of excitations and of more general time-dependent phenomena. TDDFT can be viewed as an alternative formulation of time-dependent quantum mechanics but, in contrast to the normal approach that relies on wave-functions and on the many-body Schrodinger equation, its basic variable is the one-body electron density, n(r,t). The advantages are clear The many-body wave-function, a function in a 3A-dimensional space (where N is the number of electrons in the system), is a very complex mathematical object, while the density is a simple function that depends solely on the 3-dimensional vector r. The standard way to obtain n r,t) is with the help of a fictitious system of noninteracting electrons, the Kohn-Sham system. The final equations are simple to tackle numerically, and are routinely solved for systems with a large number of atoms. These electrons feel an effective potential, the time-dependent Kohn-Sham potential. The exact form of this potential is unknown, and has therefore to be approximated. 

Based on the Runge-Gross theorem, it is shown that the three-dimensional density of a many body quantum system is sulScient to describe the TD response of the system to an external perturbation, such as electromagnetic field and vibrational motion, R t). This is known as time dependent density functional theory (TDDFT). In the linear response approximation, TDDFT is frequently used to evaluate electronic excitation energies. Here, we use full TDDFT where the density is explicitly propagated in time. Application of the TD (Dirac) vibrational principle to KS energy generates density evolution ... 

To obtain errors of 1 kcal/mol or better, it is essential to treat many-body effects accurately and, we believe, directly. Although commonly used methods such as the density functional theory within the local density approximation (LDA) or the generalized gradient approximation (GGA) may get some properties correctly, it seems unlikely that they, in general, will ever have the needed precision and robustness on a wide variety of molecules. On the other hand, methods that rely on a complete representation of the many-body wavefunction will require a computer time that is exponential in the number of electrons. A typical example of such an approach is the configuration interaction (Cl) method, which expands the wavefunction in Slater determinants of one-body orbitals. Each time an atom is added to the system, an additional number of molecular orbitals must be considered, and the total number of determinants to reach chemical accuracy is then multiplied by this factor. Hence an exponential dependence of the computer time on the number of atoms in the system results. 

In this review, we begin with a treatment of the functional theory employing as basis the maximum entropy principle for the determination of the density matrix of equilibrium ensembles of any system. This naturally leads to the time-dependent functional theory which will be based on the TD-density matrix which obeys the von Neumann equation of motion. In this way, we present a unified formulation of the functional theory of a condensed matter system for both equilibrium and non-equilibrium situations, which we hope will give the reader a complete picture of the functional approach to many-body interacting systems of interest to condensed matter physics and chemistry. 

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