Time-dependent linear response theory

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 time-dependent linear response principle can be applied to the CC theory [22-24], We begin with the Hamiltonian operator H that is a sum of the usual time-independent one 7/"1 and a time-dependent perturbation g(1) ... 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

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]. 

Figure 5 Typical velocity relationship of kinetic friction for a sliding contact in which friction is from adsorbed layers confined between two incommensurate walls. The kinetic friction F is normalized by the static friction Fs. At extremely small velocities v, the confined layer is close to thermal equilibrium and, consequently, F is linear in v, as to be expected from linear response theory. In an intermediate velocity regime, the velocity dependence of F is logarithmic. Instabilities or pops of the atoms can be thermally activated. At large velocities, the surface moves too quickly for thermal effects to play a role. Time-temperature superposition could be applied. All data were scaled to one reference temperature. Reprinted with permission from Ref. 25. 

The last method used in this study is CCSD linear response theory [37]. The frequency-dependent polarizabilities are again identified from the time evolution of the corresponding moments. However, in CCSD response theory the moments are calculated as transition expectation values between the coupled cluster state l cc(O) and a dual state... 

We denote the fluctuations of the number density of the monomers of component j at a point r and at a time t as pj r,t). With this definition we have pj(r,t))=0. In linear response theory, the Fourier-Laplace transform of the time-dependent mean density response to an external time dependent potential U r,t) is expressed as ... 

Linear response theory is reviewed in Section II in order to establish contact between experiment and time-correlation functions. In Section III the memory function equation is derived and applied in Section IV to the calculation of time-correlation functions. Section V shows how time-correlation functions can be used to guess time-dependent distribution functions and similar methods are then applied in Section VI to the determination of time-correlation functions. In Section VII a succinct review is given of other exact and experimental calculations of time-correlation functions. 

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. 

To get excitation properties of the plasma-embedded atomic system, usually a time dependent perturbation approach within linear response theory is used. The response of the systems is studied using an external harmonic perturbation... 

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. 

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)... 

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies. 

The fermion creation and destruction operators are defined such that apa +a ap = Spq. In analogy to relativistic theory, and more appropriate to the linear response theory to be considered here, the elementary fermion operators ap can be treated as algebraic objects fixed in time, while the orbital functions are solutions of a time-dependent Schrodinger equation... 

Time-dependent theory and linear response 6.4.2 Exact linear response theory... 

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... 

The next set of open-shell cluster expansion theories to appear on the scene emphasized the size-extensivity feature (al), and all of them were designed to compute energy differences with a fixed number of valence electrons. Several related theories may be described here - (i) the level-shift function approach in a time-dependent CC framework by Monkhorst/56/ and later generalizations by Dalgaard and Monkhorst/57/, also by Takahasi and Paldus/105/, (ii) the CC-based linear response theory by Mukherjee and Mukherjee/58/, and generalized later by Ghosh et a 1/59.60.107/,(iii)the closely related formulations by Nakatsuji/50,52/ and Emrich/62/ and (iv) variational theories by Paldus e t a I / 54/ and Saute et. al /55/ and by Nakatsuji/50/. 

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). 

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