TDDFT theory

This review summarizes our efforts up to this time. In the next section an overview of our formulation will be presented. We will start with a summary of the theory of MCD. We will then provide a short summary of TDDFT theory to establish our starting point. Bringing together MCD and TDDFT theory allows us to formulate our approach for calculating MCD spectra. To finish the theoretical section we provide some notes on the steps of a calculation and interpretation of the results. 

Wang, F. and Ziegler T., Excitation energies of some dl systems calculated using time-dependent density functional theory an implementation of open-shell TDDFT theory for doublet-doublet excitations. Mol.Phys (2004) 102 2585 -2595. 

We have reviewed some recent computational methodologies based on the combination of the TDDFT theory with the Polarizable Continuum solvation Model (PCM) to study chromophores in homogenous solutions. In particular we have considered... 

Polarizabilities in the Time-Dependent Density Functional (TDDFT) 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. 

In this chapter we will focus on one particular, recently developed DFT-based approach, namely on first-principles (Car-Parri-nello) molecular dynamics (CP-MD) [9] and its latest advancements into a mixed quantum mechanical/molecular mechanical (QM/MM) scheme [10-12] in combination with the calculation of various response properties [13-18] within DFT perturbation theory (DFTPT) and time-dependent DFT theory (TDDFT) [19]. 

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

There are several possible ways of deriving the equations for TDDFT. The most natural way departs from density-functional perturbation theory as outlined above. Initially it is assumed that an external perturbation is applied, which oscillates at a frequency co. The linear response of the system is then computed, which will be oscillating with the same imposed frequency co. In contrast with the standard static formulation of DFPT, there will be special frequencies cov for which the solutions of the perturbation theory equations will persist even when the external field vanishes. These particular solutions for orbitals and frequencies describe excited electronic states and energies with very good accuracy. 

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. 

From the discussion so far, it is clear that the mapping to a system of noninteracting particles under the action of suitable effective potentials provides an efficient means for the calculation of the density and current density variables of the actual system of interacting electrons. The question that often arises is whether there are effective ways to obtain other properties of the interacting system from the calculation of the noninteracting model system. Examples of such properties are the one-particle reduced density matrix, response functions, etc. An excellent overview of response theory within TDDFT has been provided by Casida [15] and also more recently by van Leeuwen [17]. A recent formulation of density matrix-based TD density functional response theory has been provided by Furche [22]. 

In this short review, a brief overview of the underlying principles of TDDFT has been presented. The formal aspects for TDDFT in the presence of scalar potentials with periodic time dependence as well as TD electric and magnetic fields with arbitrary time dependence are discussed. This formalism is suitable for treatment of interaction with radiation in atomic and molecular systems. The Kohn-Sham-like TD equations are derived, and it is shown that the basic picture of the original Kohn-Sham theory in terms of a fictitious system of noninteracting particles is retained and a suitable expression for the effective potential is derived. 

Quantum mechanics offers a variety of ways to calculate these quantities as should be clear from the variety of methods used to calculate MCD spectra that are mentioned in Section I. In this section we will briefly describe TDDFT, the theory upon which our approach is based, focusing upon the aspects that are relevant for MCD. 

DFT is a ground state theory. In order to calculate MCD spectra it is obviously necessary to know something about excited states. TDDFT allows DFT calculations to provide the necessary information about excited states. 

For chemical applications of the type we are interested in, theories based on the first-order correction to the density from a first-order correction to the potential V(1)(w) (linear-response TDDFT) are typically used. The interested reader is referred to the reviews for the details of how ( ) is calculated (52,54). 

TDDFT Time-dependent density functional theory... 

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)... 

A recent re-investigation of the electronic structure of MeReCU by both DFT/TDDFT and more highly correlated methods leads to the conclusion that the second transition (ca. 240 nm) has both methyl carbon and oxo character (by DFT/TDDFT). Use of CASSCF/MS-CASPT2 calculations shows more oxo and less C contribution, demonstrating that the level of theory is important in interpreting the photochemistry [10]. 

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