Time-dependent density functional theory kernel

Gritsenko, O. and Baerends E.Jan., Asymptotic correction of the exchange - correlation kernel of time-dependent density functional theory for long-range charge-transfer excitations. J.Chem.Phys. (2004) 121 655-660. 

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. 

A HYBRID FUNCTIONAL FOR THE EXCHANGE-CORRELATION KERNEL IN TIME-DEPENDENT DENSITY FUNCTIONAL THEORY... 

Time-Dependent Density Functional Theory 161 4.3.2 The XC Kernel... 

Del Sole, and R. W. Godby, Phys. Rev. B, 69,155112 (2004). Long-Range Contribution to the Exchange-Correlation Kernel of Time-Dependent Density Functional Theory. 

Rev. Lett., 95,253006 (2005). Measuring the Kernel of Time-Dependent Density Functional Theory with X-ray Absorption Spectroscopy of 3d Transition Metals. 

Gritsenko OV, Baerends EJ. Double excitation effect in non-adiabatic time-dependent density functional theory with an analytic construction of the exchange-correlation kernel in the common energy denominator approximation. Phys Chem Chem Phys. 2009 11 4640-4646. KUmmel S, Kronik L. Orbital-dependent density functionals Theory and apphcations. Rev Mod Phys. 2008 80 3-60. 

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. 

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. 

With eqn (7) the time-dependent Kohn-Sham scheme is an exact many-body theory. But, as in the time-independent case, the exchange-correlation action functional is not known and has to be approximated. The most common approximation is the adiabatic local density approximation (ALDA). Here, the non-local (in time) exchange-correlation kernel, i.e., the action functional, is approximated by a time-independent kernel that is local in time. Thus, it is assumed that the variation of the total electron density in time is slow, and as a consequence it is possible to use a time-independent exchange-correlation potential from a ground-state calculation. Therefore, the functional is written as the integral over time of the exchange-... 

---