Time-dependent density functional theory developments

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]. 

A mode coupling theory is recently developed [135] which goes beyond the time-dependent density functional theory method. In this theory a projection operator formalism is used to derive an expression for the coupling vertex projecting the fluctuating transition frequency onto the subspace spanned by the product of the solvent self-density and solvent collective density modes. The theory has been applied to the case of nonpolar solvation dynamics of dense Lennard-Jones fluid. Also it has been extended to the case of solvation dynamics of the LJ fluid in the supercritical state [135],... 

Addressing large molecular systems is the aim of Chapter 3, which reviews a recently developed model based on the combined use of quantum mechanics and molecular mechanics (QM/MM). This approach uses a fully self-consistent polarizable embedding (PE) scheme described in the paper. The PE model is generally compatible with any quantum chemical method, but this review is focused on its combination with density functional theory (DFT) and time-dependent density functional theory (TD-DFT). The PE method is based on the use of an electrostatic embedding potential resulting from the permanent charge distribution of the classically treated part of... 

Till recently, computations of vibronic spectra have been limited to small systems or approximated approaches, mainly as a consequence of the difficulties to obtain accurate descriptions of excited electronic states of polyatomic molecules and to computational cost of full dimensional vibronic treatment. Recent developments in electronic structure theory for excited states within the time-dependent density functional theory (TD-DFT) and resolution-of-the-identity approximation of coupled cluster theory (R1-CC2) and in effective approaches to simulate electronic spectra have paved the route toward the simulation of spectra for significantly larger systems. 

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 ... 

To date, most applications of TDDFT fall in the regime of linear response. The linear response limit of time-dependent density functional theory will be discussed in Sect. 5.1. After that, in Sect. 5.2, we shall describe the density-functional calculation of higher orders of the density response. For practical applications, approximations of the time-dependent xc potential are needed. In Sect. 6 we shall describe in detail the construction of such approximate functionals. Some exact constraints, which serve as guidelines in the construction, will also be derived in this section. Finally, in Sects. 7 and 8, we will discuss applications of TDDFT within and beyond the perturbative regime. Apart from linear response calculations of the photoabsorbtion spectrum (Sect. 7.1) which, by now, is a mature and widely applied subject, we also describe some very recent developments such as the density functional calculation of excitation energies (Sect. 7.2), van der Waals forces (Sect. 7.3) and atoms in superintense laser pulses (Sect. 8). 

The purpose of this review is to highlight the recent theoretical development of density functional response theory, or more specifically Kohn-Sham time-dependent density functional theory, where we also address some recent progress in time-dependent DFT for open-shell systems. 

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. 

As the basis sets developed for glycine produced excellent results for fhat amino acid (see Table 5.1 of Ref. [31] for details), we chose to use the same basis also for fhe presenf calculations [68]. Preliminary calculations were also carried out at the level of time-dependent density functional theory (TD-DFT) using the PBE functional [56]. However, the results differed by less fhan 1% from fhe RPA result and are thus not reported here. 

As discussed in section 2.4, several new developments are in progress in order to make the EFP-QM interface fully viable. In addition, EFP interfaces are being built with methods that can treat excited electronic states. In addition to the existing MCSCF interface, these include Cl singles and time-dependent density functional theory in the short term and more sophisticated Cl and coupled cluster methods in the longer term. 

Using a method analogous to that for static case, the linear response theory can be developed within the LDA for the case when the external electric field, characterized by the potential Vext r o) = E r Yie is time-dependent. This leads to the time-dependent density functional theory (TDLDA) [55]. 

Consequently, DFT is restricted to ground-state properties. For example, band gaps of semiconductors are notoriously underestimated [142] because they are related to the properties of excited states. Nonetheless, DFT-inspired techniques which also deal with excited states have been developed. These either go by the name of time-dependent density-functional theory (TD-DFT), often for molecular properties [147], or are performed in the context of many-body perturbation theory for solids such as Hedin s GW approximation [148]. 

Abstract We review and further develop the excited state structural analysis (ESSA) which was proposed many years ago [Luzanov AV (1980) Russ Chem Rev 49 1033] for semiempirical models of r r -transitions and which was extended quite recently to the time-dependent density functional theory. Herein we discuss ESSA with some new features (generalized bond orders, similarity measures etc.) and provide additional applications of the ESSA to various topics of spectrochemistry and photochemistry. The illustrations focus primarily on the visualization of electronic transitions by portraying the excitation localization on atoms and molecular fragments and by detaiUng excited state structure using specialized charge transfer numbers. An extension of ESSA to general-type wave functions is briefly considered. 

Time was of the essence in our struggle. The two interlinked bottlenecks in the electron density approach were time dependence and excited states. We first developed a rigorous time-dependent density functional theory for a certain class of potentials by utilizing QFD. Since this version of density functional theory was not exact for all potentials, we also developed a similar approach in terms of natural orbitals which are exact in principle. This approach yielded an equation for the ground state density whose accuracy was very good. Using this, we calculated the frequency-dependent multipole (2 -pole, Z = 1, 2, 3, 4) polarizabilities of atoms. Some of these computed numbers still await experimental verification. 

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]. 

The goal of quantum mechanical methods is to predict the structure, energy and properties for an A-particle system, where N refers to both the electrons and the nuclei. The energy of the system is a direct function of the exact position of all of the atoms and the forces that act upon the electrons and the nuclei of each atom. In order to calculate the electronic states of the system and their energy levels, quantum mechanical methods attempt to solve Schrodinger s equation. While most of the work that is relevant to catalysis deals with the solution of the time-independent Schrodinger equation, more recent advances in the development of time-dependent density functional theory will be discussed owing to its relevance to excited-state predictions. 

The calculations of CD spectra have been carried out by using semiempirical " as well as nonempirical " theories, but the trend has been shifting to nonempirical methods for its accuracy and software availability. Especially, time-dependent density functional theory (TDDFT) calculation, a recently developed nonempirical method, has become one of the most popular methods for small-to medium-sized molecules as it provides highly accurate CD predictions, though it is rather computationally demanding. ... 

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