Response kernel correlation

Fig. 4.2.3 [Bliil] Time conventions for three-pulse excitation. In 3D correlation spectroscopy, the pulse seperations t/ are used as parameters. In nonlinear system theory, the parameters are the time delays at of the cross-correlation function corresponding to the arguments r, of the response kernels.

It has recently been shown [ 12] that time-dependent or linear-response theory based on local exchange and correlation potentials is inconsistent in the pure exchange limit with the time-dependent Hartree-Fock theory (TDHF) of Dirac [13] and with the random-phase approximation (RPA) [14] including exchange. The DFT-based exchange-response kernel [15] is inconsistent with the structure of the second-quantized Hamiltonian. 

On the other hand, functional derivatives of the Bethe-Salpeter equation allows to evaluate the nonlinear responses using the interaction kernels h only (which depend on the Hartree and exchange-correlation energies). The relations between the screened nonlinear responses and the bare ones are derived by using nonlinear PRF [32],... 

In this equation g(t) represents the retarded effect of the frictional force, and /(f) is an external force including the random force from the solvent molecules. We see, in contrast to the simple Langevin equation with a constant friction coefficient, that the friction force at a given time t depends on all previous velocities along the trajectory. The friction force is no longer local in time and does not depend on the current velocity alone. The time-dependent friction coefficient is therefore also referred to as a memory kernel . A short-time expansion of the velocity correlation function based on the GLE gives (fcfiT/M)( 1 — (g/M)t2/(2r) + ), where r is the decay time of g(t), and it therefore does not have a discontinuous first derivative at t = 0. The discussion of the properties of the GLE is most easily accomplished by using so-called linear response theory, which forms the theoretical basis for the equation and is a powerful method that allows us to determine non-equilibrium transport coefficients from equilibrium properties of the systems. A discussion of this is, however, beyond the scope of this book. 

When extended to include electronic correlation, for which an exact but implicit orbital functional was derived above, the TDHF formalism becomes a formally exact theory of linear response. In practice, some simplified orbital functional Ec[ 4>i ] must be used, and the accuracy of results is limited by this choice. The Hartree-Fock operator Ti is replaced by G = Ti + vc. Dirac defines an idempo-tent density operator p whose kernel is JA i(r) i (r/)- The Did. equations are equivalent to [0, p] = 0. The corresponding time-dependent equations are itijtP = [Q(t), p(t)]. Dirac proved, for Hermitian G, (hat the time-dependent equation ih i(rt) implies that p(l) is idempotent. Hence pit) corresponds to a normalized time-dependent reference state. 

A second problem with the GME derived from the contraction over a Liouville equation, either classical or quantum, has to do with the correct evaluation of the memory kernel. Within the density perspective this memory kernel can be expressed in terms of correlation functions. If the linear response assumption is made, the two-time correlation function affords an exhaustive representation of the statistical process under study. In Section III.B we shall see with a simple quantum mechanical example, based on the Anderson localization, that the second-order approximation might lead to results conflicting with quantum mechanical coherence. 

The many-body ground and excited states of a many-electron system are unknown hence, the exact linear and quadratic density-response functions are difficult to calculate. In the framework of time-dependent density functional theory (TDDFT) [46], the exact density-response functions are obtained from the knowledge of their noninteracting counterparts and the exchange-correlation (xc) kernel /xcCf, which equals the second functional derivative of the unknown xc energy functional ExcL i]- In the so-called time-dependent Hartree approximation or RPA, the xc kernel is simply taken to be zero. 

Recent studies by Wotton and Strange (8) provided circumstantial evidence for phytoalexin involvement in the resistance of peanuts to Aspergillus flavus. Their results indicated that resistance of peanut kernels to invasion by A. flavus was correlated with their capacity to synthesize phytoalexins as an early response to wounding. Also, conditions that promoted invasion of peanuts by A. flavus inhibited phytoalexin production. Thus, kernels of drought stressed plants, which are more susceptible to A. flavus than kernels of nondrought stressed plants, produced less phytoalexin in response to wounding by slicing than kernels from non-stressed plants. 

Dasgupta, L., Lin, S. M., Carin, L. 2002. Modeling Pharmacogenomics of the NCI-60 Anticancer Data Set Utilizing kernel PLS to correlate the Microarray Data to Therapeutic Responses. In Methods of Microarray Data Analysis II (ed. S. Lin and K. M. Johnson). Kluwer Academic Publishers. 

The derivative of the exchange-correlation potential in terms of electron density, /xc, is called the exchange-correlation integral kernel. Define the response function of the electron density, /ks, for the infinitesimal change in the Kohn-Sham potential,... 

They were interested in the performance of different density-functional approaches. In all cases they used the ALDA for the exchange-correlation kernel, but for the response function Xs of Eq. (96) they used different density functionals. Since most of the current density functionals give a wrong ionization threshold (cf. the discussion of the preceding section), they rigidly shifted the single-particle energies so that the experimental ionization potential was obtained. 

The second approach is used by Baerends and co-workers. They use linear response theory, but instead of calculating the full linear response function they use the response function of the noninteracting Kohn-Sham system together with an effective potential. This response function can be calculated from the Kohn-Sham orbitals and energies and the occupation numbers. They use the adiabatic local density approximation (ALDA), and so their exchange correlation kernel, /xc (which is the functional derivative of the exchange correlation potential, Vxc, with respect to the time-dependent density) is local in space and in time. They report frequency dependent polarizabilities for rare gas atoms, and static polarizabilities for molecules. 

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