Time-dependent Kohn-Sham

Note that in all current implementations of TDDFT the so-called adiabatic approximation is employed. Here, the time-dependent exchange-correlation potential that occurs in the corresponding time-dependent Kohn-Sham equations and which is rigorously defined as the functional derivative of the exchange-correlation action Axc[p] with respect to the time-dependent electron-density is approximated as the functional derivative of the standard, time-independent Exc with respect to the charge density at time t, i. e.,... 

Because of the separation into a time-independent unperturbed wavefunction and a time-dependent perturbation correction, the time derivative on the right-hand side of the time-dependent Kohn-Sham equation will act only on the response orbitals. From this perturbed wavefunction the first-order response density follows as ... 

The effective Time Dependent Kohn-Sham (TDKS) potential vks p (r>0 is decomposed into several pieces. The external source field vext(r,0 characterizes the excitation mechanism, namely the electromagentic pulse as delivered by a by passing ion or a laser pulse. The term vlon(r,/) accounts for the effect of ions on electrons (the time dependence reflects here the fact that ions are allowed to move). Finally, appear the Coulomb (direct part) potential of the total electron density p, and the exchange correlation potential vxc[p](r,/). The latter xc potential is expressed as a functional of the electronic density, which is at the heart of the DFT description. In practice, the functional form of the potential has to be approximated. The simplest choice consists in the Time Dependent Local Density Approximation (TDLDA). This latter approximation approximation to express vxc[p(r, /)]... 

Finally we also expand the time-dependent Kohn-Sham Hamiltonian as ... 

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 arrive at Eq. (180) we have used the definitions (145), (148), (171) and (175) of the density response functions. Furthermore, we have abbreviated the kernel of the (instantaneous) Coulomb interaction by w(x, x ) = 3(t — t )/ r — r. Finally, by inserting Eq. (180) into (168) one arrives at the time-dependent Kohn-Sham equations for the second-order density response ... 

The Berry phase is not explicitly found at first sight in a time-dependent density functional theory. From Eq. (A.5) we find it quite generally resides in the action functional of the theory. It lies buried as the sum over the occupied states of the individual phases associated with the time-dependent Kohn-Sham orbitals. From Eqs. (A.1) and (A.5), the Berry phase may be identified by writing the physical action functional as a sum of two quantities ... 

Craig, C.F., Duncan, W.R. and Prezhdo, O.V. (2005) Trajectory surface hopping in the time-dependent Kohn-Sham approach for electron-nuclear dynamics. Phys. Rev. Lett., 95, 163001-1-163001-4. 

Mentioned should also be the recent work of Zhou and Chu [29] who formulate the time-dependent Kohn-Sham equations in reciprocal space ... 

Castro A, Marques M, Rubio A (2004) Propagators for the time-dependent Kohn-Sham equations. J Chem Phys 121 3425... 

Table 20.10 Calculated static isotropic polarizabilities by time-dependent Kohn-Sham theory in atomic unit...

As an alternative to Hartree-Fock semiempirical and ab initio calculations, density functional theory has been used to obtain nonlinear optical properties in both the finite field and TDHF > (or time-dependent Kohn-Sham) approaches. 

Equation (4.27) using the effective potential in Eq. (4.28) is called the time-dependent Kohn-Sham equation (Runge and Gross 1984). 

Substituting Eq. (4.59) into the first derivative of Eq. (4.57) leads to an equation similar to the time-dependent Kohn-Sham equation,... 

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)...

So far, the current density functional has attracted attention, not in the context of the response to a magnetic field, as mentioned above, but to an electric field. The time-dependent Kohn-Sham equation in Eq. (4.27) incorporating the time-dependent vector potential, Aeff, is written as... 

The basic quantity in TDDFT is the time-dependent electron density. Many important spectroscopic properties can be derived from it. As in ordinary (ground-state) DFT, the density is obtained from an auxihary system of independent electrons which move in an effective potential, the time-dependent Kohn Sham (KS) potential. With the exact KS potential, Vs[p](r, i), the density of the noninteracting electron system will be equal to the true electron density. In practice, the KS potential contains the exchange-correlation (xc) potential, Vxc(r, i), for which approximations are needed. 

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] ... 

In this section, we discuss briefly the generalized Floqnet formnlation of TDDFT [28,60-64]. It can be applied to the nonperturbative stndy of mnltiphoton processes of many-electron atoms and molecules in intense periodic or qnasi-periodic (multicolor) time-dependent fields, allowing the transformation of time-dependent Kohn-Sham equations to an equivalent time-independent generalized Floquet matrix eigenvalue problems. 

Time-dependent density-functional theory (TDDFT) extends the basic ideas of ground-state density-functional theory (DFT) to the treatment of excitations and of more general time-dependent phenomena. TDDFT can be viewed as an alternative formulation of time-dependent quantum mechanics but, in contrast to the normal approach that relies on wave-functions and on the many-body Schrodinger equation, its basic variable is the one-body electron density, n(r,t). The advantages are clear The many-body wave-function, a function in a 3A-dimensional space (where N is the number of electrons in the system), is a very complex mathematical object, while the density is a simple function that depends solely on the 3-dimensional vector r. The standard way to obtain n r,t) is with the help of a fictitious system of noninteracting electrons, the Kohn-Sham system. The final equations are simple to tackle numerically, and are routinely solved for systems with a large number of atoms. These electrons feel an effective potential, the time-dependent Kohn-Sham potential. The exact form of this potential is unknown, and has therefore to be approximated. 

The density of the interacting system can be obtained from the time-dependent Kohn-Sham orbitals... 

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