Time-dependent Kohn-Sham method

For this purpose, Runge and Gross proposed the fundamental DFT theorem for periodically time-dependent electronic states, which is called the Runge-Gross theorem (Runge and Gross 1984). This theorem is based on the following two assumptions for the external potential, 

Fext(r, t) consists of a time-independent stationary part,, and a slightly time-dependent perturbation part, 

Under these assumptions, the following four theorems are derived  

The first time-dependent Hohenberg-Kohn theorem. For Fext expansible in terms of time, define that the Fext(r, t) p(r, t) transformation corresponds to solving the time-dependent Schodinger equation. Based on this definition, the inverse transformation, p - Fext, can be performed in the case of the second assumption above. 

The time-derivative of the current density, j, (see Sect. 6.5) is represented as a density functional, i2[p](r, t). 

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Fig. 6.1 Calculated lowest charge transfer excitation energy, lOci, of the ethylene-tetrafluoroethylene dimer with respect to the intermolecular distance, R, in eV. The excitation energy at the distance of 5 A is set to be zero. The DPT (LC-BOP, BOP, and B3LYP) results were obtained by the time-dependent Kohn-Sham method (see Sect. 4.6), while the HP result is given by the time-dependent Hartree-Pock method. Por the SAC-CI method, see Sect. 3.5. Rigorously, the excitation energy should be slightly above the curve of —l/R. The augmented Sadlej pVTZ basis functions are used. See Tawada et al. (2004)...

The Hamiltonian and the coordinates are discretized by means of the generalized pseudospectral (GPS) method in prolate spheroidal coordinates [44-47], allowing optimal and nonuniform spatial grid distribution and accurate solution of the wave functions. The time-dependent Kohn-Sham Equation 3.5 can be solved accurately and efficiently by means of the split-operator method in the energy representation with spectral expansion of the propagator matrices [44-46,48]. We employ the following split operator, second-order short-time propagation formula [40] ... 

Singlet excited-state geometries of a set of medium-sized molecules, including 1,2,4,5-tetrazine, with different characteristic lowest excitations have been studied with two closely related restricted open-shell Kohn-Sham methods and within linear response to time-dependent DFT (TDDFT). The results are compared to wave function-based methods <2003JMT(630)163>. 

Using Eqs. (4.61) and (4.63), matrix U is calculated to give the response properties in terms of the uniform electric field dipole moments, polarizabilities, hyperpolarizabilities, and so forth. Equation (4.61) is called the coupled perturbed Kohn-Sham equation. Other response properties are calculated by solving Eq. (4.61) after setting the first derivative of the Fock operator, F, in terms of each perturbation. Note, however, that this method has problems in actual calculations similarly to the time-dependent response Kohn-Sham method. For example, using most functionals, this method tends to overestimate the electric field response properties of long-chain polyenes. 

A number of different methods have been proposed to introduce a self-interaction correction into the Kohn-Sham formalism (Perdew and Zunger 1981 KUmmel and Perdew 2003 Grafenstein, Kraka, and Cremer 2004). This correction is particularly useful in situations with odd numbers of electrons distributed over more than one atom, e.g., in transition-state structures (Patchkovskii and Ziegler 2002). Unfortunately, the correction introduces an additional level of self-consistency into the KS SCF process because it depends on the KS orbitals, and it tends to be difficult and time-consuming to converge the relevant equations. However, future developments in non-local correlation functionals may be able to correct for self-interaction error in a more efficient manner. 

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. 

For variational methods, such as Hartree-Fock (HF), multi-configurational self-consistent field (MCSCF), and Kohn-Sham density functional theory (KS-DFT), the initial values of the parameters are equal to zero and 0) thus corresponds to the reference state in the absence of the perturbation. The A operators are the non-redundant state-transfer or orbital-transfer operators, and carries no time-dependence (the sole time-dependence lies in the complex A parameters). Furthermore, the operator A (t)A is anti-Hermitian, and tlie exponential operator is thus explicitly unitary so that the norm of the reference state is preserved. Perturbation theory is invoked in order to solve for the time-dependence of the parameters, and we expand the parameters in orders of the perturbation... 

In this chapter we reviewed modern Kohn-Sham time-dependent density functional theory and its applications to linear and non-linear properties. As evident from a variety of application examples, DFT methods undoubtedly hold a prominent... 

The most promising approaches for efficient electronic structure calculations on large molecules are generally based on density functional theory with Kohn-Sham orbitals [32-35]. The most efficient such method for CE-BEs is based on Koopmans theorem, but this approach has quite limited accuracy [36-39]. Better accuracy can be obtained from calculations based on an effective core potential [40-45], an equivalent core approximation [46-48], a fractionally occupied transition state [49-52], or with a ASCF approach [29, 31, 53-57]. Time-dependent density functional theory is also widely used for CEBE calculation [58-62], wherein the best results are usually given with functionals having a long-range correction [63, 64]. 

Following on from the introduction of the Kohn-Sham DFT concept came Runge and Gross DFT for time-dependent systems.Here the properties of molecules are determined as a snapshot in time and, consequently, can be modeled along a timeline to predict the evolution of a chemical species or reaction. The application of such a method to model supramolecular host-guest interactions and other complex phenomena, such as light-harvesting supramolecular complexes, is clear. 

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