Time-dependent density functional linear response

Time-dependent Density Functional Linear Response Theory... 

The study of behavior of many-electron systems such as atoms, molecules, and solids under the action of time-dependent (TD) external fields, which includes interaction with radiation, has been an important area of research. In the linear response regime, where one considers the external held to cause a small perturbation to the initial ground state of the system, one can obtain many important physical quantities such as polarizabilities, dielectric functions, excitation energies, photoabsorption spectra, van der Waals coefficients, etc. In many situations, for example, in the case of interaction of many-electron systems with strong laser held, however, it is necessary to go beyond linear response for investigation of the properties. Since a full theoretical description based on accurate solution of TD Schrodinger equation is not yet within the reach of computational capabilities, new methods which can efficiently handle the TD many-electron correlations need to be explored, and time-dependent density functional theory (TDDFT) is one such valuable approach. 

An alternative theory is the popular time-dependent density functional theory [44], in which transition energies are obtained from the poles of dynamic linear response properties. There are several excellent reviews on time-dependent density functional theory. See, for instance, Ref. [45]. 

The time-dependent density functional theory [38] for electronic systems is usually implemented at adiabatic local density approximation (ALDA) when density and single-particle potential are supposed to vary slowly both in time and space. Last years, the current-dependent Kohn-Sham functionals with a current density as a basic variable were introduced to treat the collective motion beyond ALDA (see e.g. [13]). These functionals are robust for a time-dependent linear response problem where the ordinary density functionals become strongly nonlocal. The theory is reformulated in terms of a vector potential for exchange and correlations, depending on the induced current density. So, T-odd variables appear in electronic functionals as well. 

The frequency dependence of SHG at simple metal surface has been the focus of a recent theoretical study of Liebsch [100]. Time-dependent density functional theory was used in these calculations. The results suggest that the perpendicular surface contribution to the second harmonic current is found to be significantly larger than had been assumed previously. He also concludes that for 2 a> close to the threshold for electron emission, the self-consistently screened nonlinear electronic response becomes resonantly enhanced, analogous to local field enhancement in the linear response near the bulk plasma frequency. 

S. Comi, R. Cammi, B. Mennucci, J. Tomasi, Electronic excitation energies of molecules in solution within continuum solvation models Investigating the discrepancy between state-specific and linear-response methods, Formation and relaxation of excited states in solution A new time dependent polarizable continuum model based on time dependent density functional theory. J. Chem. Phys. 123, 134512 (2005)... 

The time-dependent density functional theory, widely known as TDDFT, is an exact many-body theory [1] in which the ground state time-dependent electron density is the fundamental variable. For small changes in the time-dependent electron density, a linear response (LR) approach can be applied to solve the TDDFT equations. In... 

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 date, most applications of TDDFT fall in the regime of linear response. The linear response limit of time-dependent density functional theory will be discussed in Sect. 5.1. After that, in Sect. 5.2, we shall describe the density-functional calculation of higher orders of the density response. For practical applications, approximations of the time-dependent xc potential are needed. In Sect. 6 we shall describe in detail the construction of such approximate functionals. Some exact constraints, which serve as guidelines in the construction, will also be derived in this section. Finally, in Sects. 7 and 8, we will discuss applications of TDDFT within and beyond the perturbative regime. Apart from linear response calculations of the photoabsorbtion spectrum (Sect. 7.1) which, by now, is a mature and widely applied subject, we also describe some very recent developments such as the density functional calculation of excitation energies (Sect. 7.2), van der Waals forces (Sect. 7.3) and atoms in superintense laser pulses (Sect. 8). 

Banerjee and Harbola [69] have worked out a variation perturbation method within the hydrodynamic approach to the time-dependent density functional theory (TDDFT) in order to evaluate the linear and nonlinear responses of alkali metal clusters. They employed the spherical jellium background model to determine the static and degenerate four-wave mixing (DFWM) y and showed that y evolves almost linearly with the number of atoms in the cluster. 

In their recent papers, Tretiak et al. proposed the technique for calculations of TPA properties which is to some extent the combination of the methods described above [45, 74, 108, 109]. The method proposed by Tretiak takes an advantage of the quantities that can be calculated within the linear response theory framework. Tire remaining quantities that appear in the expressions for the two-photon absorption cross section can be evaluated as the functional derivatives based on the time-dependent density functional (TDDFT) method. Although the response theory is involved in their evaluation, it is important to note that the TPA cross section is calculated via SOS formulae. 

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. 

M. Schreiber, M.R. Silva-Junior, S.P.A Sauer, W. Thiel, Benchmarks for electronically excited states CASPT2, CC2, CCSD, and CCS, J. Chem. Phys. 128 (2008) 134110 M.R. Sflva-Junior, M. Schreiber, S.P.A. Sauer, W. Thiel, Benchmarks for electronically excited states Time-dependent density functional theory and density functional theory based multireference configuration interaction, J. Chem. Phys. 129 (2008) 104103 S.P.A. Sauer, M. Schreiber, M.R. Silva-Junior, W. Thiel, Benchmarks for Electronically Excited States A Comparison of Noniterative and Iterative Triples Corrections in Linear Response Coupled Cluster Methods CCSDR(3) versus CCS, J. Chem. Theory Comput. 5 (2009) 555 M.R. Silva-Junior, S.P.A. Sauer, M. Schreiber, W. Thiel, Basis set effects on coupled cluster benchmarks of electronically excited states CCS, CCSDR(3) and CC2, Mol. Phys. 108 (2010) 453 M.R. Silva-Junior, M. Schreiber, S.P.A. Sauer, W. Thiel, Benchmarks of electronically excited states basis set effects on CASPT2 results, J. Chem. Phys. 133 (2010) 174318. 

Using a method analogous to that for static case, the linear response theory can be developed within the LDA for the case when the external electric field, characterized by the potential Vext r o) = E r Yie is time-dependent. This leads to the time-dependent density functional theory (TDLDA) [55]. 

Time-dependent density functional theory (TDDET)i" within the linear response formalismi -" ] is nowadays the most widely used approach to the calculation of electronic excitation energies of molecules and solids. Applied within the adiabatic approximation and with the usual local or semilocal density functionals, TDDFT... 

N. T. Maitra, F. Zhang, R. J. Cave, and K. Burke,/. Chem. Phys., 120, 5932 (2004). Double Excitations in Time-Dependent Density Functional Theory Linear Response. 

Real-Time, Real-Space Implementation of the Linear Response Time-Dependent Density-Functional Theory. 

In the framework of linear and quadratic response theory and time-dependent density functional theory, Jha et have simulated and... 

Time-dependent density-functional response theory. An alternative approach to real-time TDDFT as described above is the application of linear-response theory. If the perturbation to the system in its ground-state—in our case, e.g., the exposure to a time-dependent electric field— is only small, the system will response linearly. The formulation of the resulting time-dependent density-functional response theory (TD-DFRT) has been given by Casida. " ... 

On top of this effective ground-state description a time-dependent extension has been proposed by Niehaus and co-workers, which is usually referred to as a time-dependent density-functional response theory tight-binding (TD-DFRT-TB) scheme. It corresponds to the formulation of Casida s linear-response theory that has been discussed before. The coupling matrix giving the response of the potential with respect to a change in the electron density has to be built as stated in eqn (19), and we use again the adiabatic approximation. 

In addition to the lifetime response methods cited in Section 2.1.2, we would like to mention here that Jensen, Autschbach and coworkers have also presented linear response functions using finite lifetimes at the time-dependent density-functional level of theory [294]. 

An alternative approach is based on the time-dependent density functional theory [40]. From the linear response theory, it can be shown that proper treatment of the excited states can be obtained from the solutions of a non-Hermitian eigenvalue problem [41],... 

This approximation is better known as the time-dependent Hartree—Fock approximation (TDHF) (McLachlan and Ball, 1964) (see Section 11.1) or random phase approximation (RPA) (Rowe, 1968) and can also be derived as the linear response of an SCF wavefunction, as described in Section 11.2. Furthermore, the structure of the equations is the same as in time-dependent density functional theory (TD-DFT), although they differ in the expressions for the elements of the Hessian matrix E22. The polarization propagator in the RPA is then given as... 

Casida, M. E., Ipatov, A., and Cordova, F. (eds) (2006) Linear-response time-dependent density-functional theory for open-shell molecules in Time-Dependent Density-Functional Theory (ed. Marques, M. A. L., Ullrich, C., Nogueira, F., Rubio, A., and Gross, E. K. U.) Springer, Berlin, pp, 243-257. 

Tapavicza, E., Tavernelli, I., 8c Rothlisberger, U. (2007). Trajectory surface hopping within linear response time-dependent density-functional theory. Physical Review Letters, 98(2), 023001. 

The third approach is that used by Salahub and co-workers. They initially used DFT RPA but recendy have reported an implementation of time-dependent density functional response theory (TD DFRT). Their Kohn-Sham linear response function involves a coupling matrix, K, which in the RPA case contains only the response to coulomb terms, but in their present implementation contains exchange and correlation response terms. Their K is time independent as they work within the adiabatic approximation. They calculate the frequency dependent polarizability from a sum-over-states (SOS) formula, and hence have to calculate the excitation spectrum. 

Maitra NT, Zhang F, Cave RJ, Burke K. Double excitations within time-dependent density functional theory linear response. J Chem Phys. 2004 120 5932. 

Besley NA and Asmuruf FA 2010 Time-dependent density functional theory calculations of the spectroscopy of core electrons. Physical Chemistry Chemical Physics 12(38), 12024—12039. Liang W, Fischer SA, Frisch MJ and Li X 2011 Energy-specific linear response tdhf/tddft for calculating high-energy excited states. Journal of Chemical Theory and Computation 7(11), 3540-3547. 

---