Time-dependent current density functional

Berger JA, Snijders JG (2002) Ultranonlocality in time-dependent current-density-functional theory Application to conjugated polymers, Phys. Rev. Lett, 83 694-697... 

Since the current density in the bulk measures the surface charges, the time-dependent current-density functional theory (CDFT) appears to be a way to investigate this problem. At least the results presented by de Boeij et al. [171] for the bulk susceptibility and by van Faassen et al. [172] for the polarizability of linear chains are encouraging, although this may not be the case for second-and third-order effects. 

By the calculations of the time-dependent current density functional theory using this equation, accurate excitation energies are obtained for some n tt excitations (van Faassen and de Boeij 2004). Meanwhile, however, it has been found that quite poor excitation energies are produced for the excitations of some types of molecules. On the other hand, calculations of the adiabatic excitation energy benchmark set, containing 109 molecules, show that the vector potential correction hardly affects the calculated excitation energies (Bates and Furche 2012). 

TDCDFT Time-dependent current density functional theory... 

Lett., 88, 186401 (2002). Ultranonlocality in Time-Dependent Current-Density-Functional Theory Application to Conjugated Polymers. 

Sen and Chakrabarti have employed the CPHF, CPKS/B3LYP, TDDFT/ALDA, and TDCDFT/VK (time-dependent current density functional theory/Vignale Kohn) methods to calculate the polarizability in alkali-doped traws -polyacetylene chains as a function of chain length and as a function of the nature of the dopant. They have evidenced a minimum in the evolution of the average polarizability per unit cell and have attributed it to the charge transfer between the alkali and the chain. 

Time-dependent Current Density Functional Theory... 

The role of relativity in the optical response of gold within the time-dependent current-density-functional theory, P. Romaniello and P. L. de Boeij, J. Chem. Phys., 2005,122, 164303. 

Since RGI establishes that the external potential is a functional of the current density, one could choose to use the current density as the basic variable instead of the density. This is known as time-dependent cmrent density functional theory (TDCDFT) (See discussion below on solids.)... 

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.,... 

This equation defines the TDKS potentials Ajo implicitly in terms of the functionals A[j] and A,[j]. Clearly, Eq. (137) is rather complicated. The external-potential terms 5 and J are simple functionals of the density and the paramagnetic current density. The complexity of Eq. (137) arises from the fact that the density, Eq. (128), and the paramagnetic currents, Eqs. (129), (134), are complicated functionals of j. Hence a formulation directly in terms of the density and the paramagnetic current density would be desirable. For electrons in static electromagnetic fields, Vignale and Rasolt [61-63] have formulated a current-density functional theory in terms of the density and the paramagnetic current density which has been successfully applied to a variety of systems [63]. A time-dependent HKS formalism in terms of the density and the paramagnetic current density, however, has not been achieved so far. 

A totally different point of view is proposed by Time-Dependent Density Functional Theory [211-215] (TD-DFT). This important extension of DFT is based on the Runge-Gross theorem [216]. It extends the Hohenberg-Kohn theorem to time-dependent situations and states that there is a one to one map between the time-dependent external potential t>ea t(r, t) and the time-dependent charge density n(r, t) (provided we know the system wavefunction at t = —oo). Although it is linked to a stationary principle for the system action, its demonstration does not rely on any variational principle but on a step by step construction of the charge current. 

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... 

A special problem where the time-dependent density-functional theory could be useful is that of calculating the polarizability and hyperpolarizability (Section 12). It turned out that although accurate results could be achieved for smaller molecules (partly, however, requiring a careful choice of the approximate density functional), severe problems could turn up (but did not always) when considering extended systems. It might mean that the current density functionals are lacking an explicit dependence on the polarization, but further studies are needed in order to clarify this point. 

The shape of the probability density function, depends on the system. Some examples are shown in Fig. 4-4. This figure also contains probability density of age (see Section 4.2.3). Figure 4-4a might correspond to a lake with inlet and outlet on opposite sides of the lake. Most water molecules will then have a residence time in the lake roughly equal to the time it takes for the mean current to carry the water from the... 

The volume of the metal produced depends on the current density, the electrolysis time, the cathode area, and the electrochemical equivalent. Yt is usually expressed as a function of the current density. The interrelationship among Yt, the specific area of the cathode, As (m2/m3), i.e., the area of the cathode per unitvolume of the reactor (cathode area/volume of the cell), the fractional current efficiency, T, and the current density, I, is given by ... 

Assume that current is passed either through the total nucleus surface area or through part thereof, such as the edge of a two-dimensional nucleus of monoatomic thickness. The transition of the ion Mz+ to the metallic state obeys the equation for an irreversible electrode reaction, i.e. Eqs (5.2.12), (5.2.23) and (5.2.37). The effect of transport processes is neglected. The current density at time t thus depends on the number of nuclei and their active surface area. If there is a large number of nuclei, then the dependence of their number on time can be considered to be a continuous function. For the overall current density at time t we have... 

The function cg(x, t) is the response of the local surface concentration to a uniform stepwise change in current density, given by Soliman and Chambre (SI7c), as a somewhat involved analytic expression that combines space and time dependence in the dimensionless variable of Eq. (18) ... 

The effective potential has a form similar to that for the time-independent case with the density argument replaced by the TD density variable. Also some of the functionals may be current density dependent. 

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