Time-dependent Kohn-Sham equations

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 ... 

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 ... 

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... 

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,... 

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 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. 

As mentioned before, the solution of the time-dependent Kohn-Sham equations is an initial value problem. At t = to the system is in some initial state described by the Kohn-Sham orbitals to). In most cases the initial state will be the ground state of the system (i.e., (fi(r,to) will be the solution of the ground-state Kohn-Sham equations). The main task of the computational physicist is then to propagate this initial state until some final time, tf. 

The time-dependent Kohn-Sham equations can be rewritten in the integral form... 

In circumstances where the external time-dependent potential is small, it may not be necessary to solve the full time-dependent Kohn-Sham equations. 

One of the most important uses of TDDFT is the calculation of photoabsorption spectra. This problem can be solved in TDDFT either by propagating the time-dependent Kohn-Sham equations [39] or by using linear-response theory. In this section we will be concerned by the former, relegating the latter to the next section. 

TDDFT is a tool particularly suited for the study of systems under the influence of strong lasers. We recall that the time-dependent Kohn-Sham equations yield the exact density of the system, including all non-linear effects. To simulate laser induced phenomena it is customary to start from the ground-state of the system, which is then propagated imder the influence of the potential... 

ExpKcit propagation of the time-dependent Kohn-Sham equations within the local density approximation has been employed to calculate the static and dynamic polarizabilities, and then the Ce van der Waals coefficients of a large number of polyaromatic hydrocarbons. These calculations have shown that the values of Cg scale approximately with the products of the numbers of atoms in the pair of interacting molecules, with the strongest deviations for the highly anisotropic structures. 

Thus, for a time-dependent wavefunction as a solution to the time-dependent Schrddinger eqn (1) the wavefunction corresponds to a stationary point of the action integral (4). From eqn (5) we finally can derive the time-dependent Kohn-Sham equations. 

As for the time-independent case the time-dependent Kohn-Sham equations are single-particle equations... 

The time-dependent Kohn-Sham equations can now be propagated in real time. This approach is still less common than the linear-response scheme that will be described in the next section. After performing the propagation in some finite time the dipole-strength function can be evaluated as described in section 2.3.2. 

The PCM contributions to the time-dependent Kohn-Sham equations depend [87] on the term... 

To obtain excitation energies and properties within the time-dependent Kohn-Sham framework, it is possible to propagate in time the time-dependent electron density, through the solution of Eq. (4.60), and then extract energies and oscillator strengths from a Fourier analysis of the results [98-102]. Alternatively, the excited-state properties can be determined through the linear response theory. This is an efficient approach which avoids the direct solution of the time-dependent Kohn-Sham equations and is often used in practical applications. 

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