Time-Dependent Density Functional Theory

Instead, the time-dependent density-functional theory, first formulated by Runge and Gross (see also refs. 86 and 87), provides a mathematically exact scheme with which excitation energies can be calculated within density-functional theory. We shall here briefly describe its main foundations and subsequently give a few recent examples of its applications. Although the theory was formulated almost two decades ago, it is only within the last few years that it has become of practical use for a larger class of systems. 

Runge and Gross proved that under certain conditions the total electron density for a general system subject to some time-dependent interaction can be written as 

The single-particle orbitals are calculated from time-dependent Schrodinger- or Kohn-Sham-like equations. 

This expression is very similar to the standard Kohn-Sham expression (except for an extra time-dependence) with, however, the important difference that the last term contains not an energy but an action. For a general quantum-mechanical system with the wavefunction P(t) (the dependence on all other coordinates is not shown) the action is defined as 

The exchange-correlation kernel is thus the functional derivative of the exchange-correlation potential which in turn can be expressed in terms of the functional derivative of the exchange-correlation energy density All these 

TD-DFT is the extension of the conventional ground-state DFT formulation to a time-dependent domain. The TD-DFT formalism is thus suited for the description of an electronic system under the influence of a time-dependent external potential and for the treatment of excitations. 

The Runge-Gross theorem states that for a many-body system evolving from a fixed initial state there is a one-to-one correspondence between the external time-dependent potential and the (time-dependent) electron density p(r) = p(r, t). Therefore, the behavior 

The variational principle for the action integral is derived starting from the observation that, if the time-dependent wave function 4/(r, 0 is a solution of the time-dependent Schrodinger equation, then it corresponds to a stationary point of the quantum-mechanical action integral  

because of the Runge-Gross theorem, the wave function is a functional of the electron density, the action integral of Eq. (4.47) is also a unique functional of the density  

Note that the action integral of Eq. (4.48) can be partitioned into an universal part  

A rather different approach compared to the ones discussed above is to the use of density functional theory (DFT) in conjunction with an all-electron method. Because of the more favourable computational scaling DFT is often used to describe large complexes in which only one or a few atoms belong to the class of f elements. Because the performance of the various classes of density functional approximations is well documented for light elements, we will focus our attention on the electronic structure of the f element. 

In many cases, just applying ground state DFT is not sufficient because one would also like to investigate the relative energies of the excited states. This is nowadays 

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Other Forms of Time-Dependent Density Functional Theory... 

Time-Dependent Density Functional theory (TDDFT) has been considered with increasing interest since the late 1970 s and many papers have been published on the subject. The treatments presented by Runge and Gross (36) and Gross and Kohn (37) are widely cited in the discussion of the evolution of pure states. The evolution of mixed states has been considered extensively by Rajagopal et al. (38), but that treatment differs in many aspects from the form given here. 

Chandra and his coworkers have developed analytical theories to predict and explain the interfacial solvation dynamics. For example, Chandra et al. [61] have developed a time-dependent density functional theory to predict polarization relaxation at the solid-liquid interface. They find that the interfacial molecules relax more slowly than does the bulk and that the rate of relaxation changes nonmonotonically with distance from the interface They attribute the changing relaxation rate to the presence of distinct solvent layers at the interface. Senapati and Chandra have applied theories of solvents at interfaces to a range of model systems [62-64]. 

Adamo, C., Scuseria, G. E., Barone, V., 1999, Accurate Excitation Energies from Time-Dependent Density Functional Theory Assessing the PBE0 Model , J. Chem. Phys., Ill, 2889. 

Bauernschmitt, R., Ahlrichs, R., 1996b, Treatment of Electronic Excitations Within the Adiabatic Approximation of Time Dependent Density Functional Theory , Chem. Phys. Lett., 256, 454. 

Burke, K., Gross, E. K. U., 1998, A Guided Tour of Time-Dependent Density Functional Theory in Density Functionals Theory and Applications, Lecture Notes in Physics, Vol. 500, Joubert, D. (ed.), Springer, Heidelberg. 

Casida, M. E., Casida, K. C., Salahub, D. R., 1998, Excited-State Potential Energy Curves from Time-Dependent Density-Functional Theory A Cross Section of Formaldehyde s A Manifold , hit. J. Quant. Chem., 70, 933. 

Van Caillie, C., Amos, R. D., 2000, Raman Intensities Using Time Dependent Density Functional Theory , Phys. Chem. Chem. Phys., 2, 2123. 

Roewer G, Herzog U, Trommer K, Muller E, Friihauf S (2002) Silicon Carbide - A Survey of Synthetic Approaches, Properties and Applications 101 59-136 Rosa A, Ricciardi G, Gritsenko O, Baerends EJ (2004) Excitation Energies of Metal Complexes with Time-dependent Density Functional Theory 112 49-116 Rosokha SV, Kochi JK (2007) X-ray Structures and Electronic Spectra of the n-Halogen Complexes between Halogen Donors and Acceptors with jc-Receptors. 126 137-160 Rudolf P, see Golden MS (2004) 109 201-229... 

Levine BG, Ko C, Quenneville J, Martinez TJ (2006) Conical intersections and double excitations in time-dependent density functional theory. Mol Phys 104 1039... 

Introduction to Time-dependent Density-functional Theory (TDDFT)... 

The study of behavior of many-electron systems such as atoms, molecules, and solids under the action of time-dependent (TD) external fields, which includes interaction with radiation, has been an important area of research. In the linear response regime, where one considers the external held to cause a small perturbation to the initial ground state of the system, one can obtain many important physical quantities such as polarizabilities, dielectric functions, excitation energies, photoabsorption spectra, van der Waals coefficients, etc. In many situations, for example, in the case of interaction of many-electron systems with strong laser held, however, it is necessary to go beyond linear response for investigation of the properties. Since a full theoretical description based on accurate solution of TD Schrodinger equation is not yet within the reach of computational capabilities, new methods which can efficiently handle the TD many-electron correlations need to be explored, and time-dependent density functional theory (TDDFT) is one such valuable approach. 

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