Time-Dependent Density Functional Theory TDDFT

Time-Dependent Density Functional Theory (TDDFT) 

A set of one-electron time-dependent Kohn-Sham (KS) equations (me = l,h= l,e = 1) 

The integrals are computed by Chebyshev interpolation in the time domain. 

A second and more widely used approach for the computation of excitation energies within DFT is based on the linear-response formulation of the time-dependent perturbation of the electronic density. The basic quantity in linear response TDDFT (LR-TDDFT) is the time-dependent density-density response function [33] 

In terms of the set of KS orbitals / (r), 4 a(r) (we use indices i and j for occupied orbitals (occupation/ = 1) and indices a and b for virtual orbitals (occupation f = 0), respectively), the matrix elements for the single particle density response function induced by a perturbation with frequency co become 

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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. 

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. 

In the Time Dependent Density Functional Theory (TDDFT) [16], the correlated many-electron problem is mapped into a set of coupled Schrodinger equations for each single electronic wavefunctions (o7 (r, t),j= 1, ), which yields the so-called Kohn-Sham equations (in atomic units)... 

The fluorescence properties of free 2AP are simple. AJablonski diagram of 2AP (Fig. 13.IB) computed with time-dependent density functional theory (TDDFT) finds a dominant singlet excited state transition from S() to at 292 nm (Jean and Hall, 2001). In solution, the free nucleobase has a fluorescence excitation maximum of 305 nm and an emission maximum of 360 nm at pH 7. Its quantum yield is not high 0.68 at pH 7.0 in 100 mM NaCl, 25 °C. Its fluorescence lifetime in aqueous solution is 10 ns at 22 °C and is described by a single exponential decay. 

By contrast, the alternative PCM-LR approach [15-17] determines in a single step calculation the excitation energies for a whole manifold of excited states. This general theory may be combined with the Time-Dependent Density Functional Theory (TDDFT) as QM level for the solute. Within the PCM-TDDFT formalism, the excitation energies are obtained by proper diagonalization of the free energy functional Hessian. 

Since DFT calculations are in principle only applicable for the electronic ground state, they cannot be used in order to describe electronic excitations. Still it is possible to treat electronic exciations from first principles by either using quantum chemistry methods [114] or time-dependent density-functional theory (TDDFT) [115,116], First attempts have been done in order to calculate the chemicurrent created by an atom incident on a metal surface based on time-dependent density functional theory [117, 118]. In this approach, three independent steps are preformed. First, a conventional Kohn-Sham DFT calculation is performed in order to evaluate the ground state potential energy surface. Then, the resulting Kohn-Sham states are used in the framework of time-dependent DFT in order to obtain a position dependent friction coefficient. Finally, this friction coefficient is used in a forced oscillator model in which the probability density of electron-hole pair excitations caused by the classical motion of the incident atom is estimated. 

Time-dependent density functional theory (TDDFT) as a complete formalism [7] is a more recent development, although the historical roots date back to the time-dependent Thomas-Fermi model proposed by Bloch [8] as early as 1933. The first and rather successful steps towards a time-dependent Kohn-Sham (TDKS) scheme were taken by Peuckert [9] and by Zangwill and Soven [10]. These authors treated the linear density response of rare-gas atoms to a time-dependent external potential as the response of non-interacting electrons to an effective time-dependent potential. In analogy to stationary KS theory, this effective potential was assumed to contain an exchange-correlation (xc) part, r,c(r, t), in addition to the time-dependent external and Hartree terms ... 

Such large enhancement factors for localized and isolated hot spots from few atom Ag clusters arising from only the chemical enhancement under certain conditions are supported by calculations. Zhao working with Jensen and Schatz used time-dependent density functional theory (TDDFT) to investigate the adsorption and Raman response of pyrazine molecules [21]. Figure 10.6 shows the Raman response of (a) isolated pyrazine compared to that of pyrazine complexed to the vertex of a (b) one and (c) two 20 Ag atom clusters with enhancements of 10 and 10 predicted, respectively. Small clusters of Ag atoms have little or no plasmon response, suggesting that the chemical enhancement can be quite significant and certainly may allow for enhancement hot spots. 

Banerjee and Harbola [69] have worked out a variation perturbation method within the hydrodynamic approach to the time-dependent density functional theory (TDDFT) in order to evaluate the linear and nonlinear responses of alkali metal clusters. They employed the spherical jellium background model to determine the static and degenerate four-wave mixing (DFWM) y and showed that y evolves almost linearly with the number of atoms in the cluster. 

The treatment in terms of induced current is in the mainstream of modem development of the time-dependent density functional theory (TDDFT). Moreover, the current density formalism has been proposed [4] as a variant of TDDFT. The evolution of current density presents properly the response of electrons on an external field. In general words, such a strong basis is promising for a theoretical treatment of many aspects of ion interactions with atoms, molecules and solids. 

The theory in which the susceptibility is formally defined for jellium surfaces is the time-dependent density functional theory (TDDFT). In this theory, the susceptibility for interacting electrons (also called screened susceptibility) x(q, z, z ) is related to the susceptibility for non-interacting (independent) electrons Xo(q, ta, q z, z ) via the integral equation... 

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. 

In the first part of this work, a brief overview over several strategies to combine such time domain transport simulations with first principles electronic structure theory is given. For the latter, we restrict ourselves to a discussion of time dependent density functional theory (TDDFT) only. This method is by far the most employed many body approach in this field and provides an excellent ratio of accuracy over computational cost, allowing for the treatment of realistic molecular devices. This digest builds on the earlier excellent survey by Koentopp and co-workers on a similar topic [13]. Admittedly and inevitably, the choice of the covered material is biased by the authors interests and background. 

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