TDDFT Linear Response

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). 

In the linear response TDDFT formalism the excitation energies of a molecular system are determined as poles of the linear response of the ground state electron density to a time dependent perturbation [2], After Fourier transformation from the time to frequency domain, and some algebra, the excitation energies can be obtained as eigenvalues of the non-Hermitian eigensystem [23]... 

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]... 

The most common time-dependent perturbation is a long-wavelength electric field, oscillating with frequency eo. In the usual situation, this field is a weak perturbation to the molecule, and one can therefore perform a linear response analysis. From the linear response, we can extract the optical absorption spectrum of the molecule due to electronic excitations. Thus, linear response TDDFT can be used to predict the transition frequencies to electronic excited states (along with many other properties), and this has been the primary use of TDDFT so far, with many applications to large molecules. 

A more sophisticated ground-state approximate energy functional can be constructed using the frequency-dependent response function of linear response TDDFT. We now introduce the basic formula and then discuss some of the systems this method is being used to study. 

In order to calculate nonadiabatic couplings in the framework of the TDDFT method a representation of the wavefunction based on Kohn-Sham (KS) orbitals is required. Since in the linear response TDDFT method the time-dependent electron density contains only contributions of single excitations from the manifold of occupied to virtual KS orbitals, a natural ansatz for the excited state electronic wavefunction is the configuration interaction singles (ClS)-Uke expansion ... 

TDDFT describes a system s response to external perturbations, such as electromagnetic fields and phonons. Linear response TDDFT is frequently used to evaluate electronic excitation energies. The full TDDFT implemented in studies discussed throughout this chapter propagates electron density explicitly in time. Taking full computational advantage of time-independent DFT, its solutions are used as a basis for our TDDFT calculations. ... 

Lately, the CP-MD approach has been combined with a mixed QM/MM scheme [10-12] which enables the treatment of chemical reactions in biological systems comprising tens of thousands of atoms [11, 26]. Furthermore, CP-MD and mixed QM/MM CP-MD simulations have also been extended to the treatment of excited states within a restricted open-shell Kohn-Sham approach [16, 17, 27] or within a linear response formulation of TDDFT [16, 18], enabling the study of biological photoreceptors [28] and the in situ design of optimal fluorescence probes with tailored optical properties [32]. Among the latest extensions of this method are also the calculation of NMR chemical shifts [14]. 

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. 

In what follows, we present in this short review, the basic formalism of TDDFT of many-electron systems (1) for periodic TD scalar potentials, and also (2) for arbitrary TD electric and magnetic fields in a generalized manner. Practical schemes within the framework of quantum hydrodynamical approach as well as the orbital-based TD single-particle Schrodinger-like equations are presented. Also discussed is the linear response formalism within the framework of TDDFT along with a few miscellaneous aspects. 

After this brief discussion of the physical significance of the OR parameter, the role of TDDFT in the description of chiroptical effects becomes clear. Chiroptical effects, on a molecular level, are related to perturbations of the electric or magnetic dipole moment by time-dependent magnetic and electric fields, respectively. Since equations (1) establish that the perturbations are linear in the applied field amplitudes, the computational protocol will typically first involve a (static, no external fields) DFT computation of the molecule s ground state, followed by a linear response computation to determine />(oj) from the elements of the tensor computed for a specific EM field frequency co. These tensor elements are computed from the first-order perturbations of the dipole moments due to the presence of EM fields of a specified frequency. 

Herein lies an opportunity for computing excitation spectra (and the actual CD intensity) from TDDFT linear response Once a response equation for /i(ffl) (or 4>(a>)]) has been derived, circular dichroism can be computed from an equation system that determines the poles of [> on the frequency axis, just like regular electronic absorption spectra are related to the poles of the electronic polarizability a [27]. Details are provided in Sect. 2.3. We call this the linear response route to calculating excitation spectra, in contrast to solving (approximations of) the Schrodinger equation for excited state and explicitly calculating excited state... 

This section summarizes the TDDFT linear response approach to compute optical rotation and circular dichroism. For reasons of brevity, assume a closed shell system, real orbitals, and a complete basis set (see Sect. 2.4 for comments regarding basis set incompleteness issues). From solving the canonical ground state Kohn-Sham (KS) equations,... 

When written with the help of the Tl matrix as in (19), from (20) the OR parameter and other linear response properties are seen to afford singularities where co = coj, just like in the SOS equation (2). Therefore, at and near resonances the solutions of the TDDFT response equations (and response equations derived for other quantum chemical methods) yield diverging results that cannot be compared directly to experimental data. In reality, the excited states are broadened, which may be incorporated in the formalism by introducing dephasing constants 1 such that o, —> ooj — iT j for the excitation frequencies. This would lead to a nonsingular behavior of (20) near the coj where the real and the imaginary part of the response function varies smoothly, as in the broadened scenario at the top of Fig. 1. 

This chapter, therefore, encompasses two extremes of excited-state theories within the single-reference, linear-response framework One that aims at high and controlled accuracy for relatively small gas-phase molecules such as EOM-CC and the other with low to medium accuracy for large molecules and solids represented by CIS and TDDFT. We aim at clarifying the mutual relationship among these excited-state methods including the two extremes, while delegating a more complete exposition of EOM-CC to the next chapter contributed by Watts. 

The time-dependent density functional theory, widely known as TDDFT, is an exact many-body theory [1] in which the ground state time-dependent electron density is the fundamental variable. For small changes in the time-dependent electron density, a linear response (LR) approach can be applied to solve the TDDFT equations. In... 

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