Time-dependent density functional theory derivation

Fortunately, the same limiting conditions that validate the friction approximation can also be used with time-dependent density functional theory to give a theoretical description of rjxx. This expression was originally derived to describe vibrational damping of molecules adsorbed on surfaces [71]. It was later shown to also be applicable to any molecular or external coordinate and at any location on the PES, and thus more generally applicable to non-adiabatic dynamics at surfaces [68,72]. The expression is... 

A mode coupling theory is recently developed [135] which goes beyond the time-dependent density functional theory method. In this theory a projection operator formalism is used to derive an expression for the coupling vertex projecting the fluctuating transition frequency onto the subspace spanned by the product of the solvent self-density and solvent collective density modes. The theory has been applied to the case of nonpolar solvation dynamics of dense Lennard-Jones fluid. Also it has been extended to the case of solvation dynamics of the LJ fluid in the supercritical state [135],... 

In the following, we will first present a formal derivation of the general PE equations rooted in intermolecular perturbation theory. Next follows a derivation of the PE scheme within the concepts of time-dependent density functional theory, and finally, we present a few illustrative examples. The PE model has been implemented in the Dalton program package [14]. 

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). 

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. 

Kurth S, Stefanucci G, Khosravi E, Verdozzi C, Gross E (2010) Dynamical Coulomb blockade and the derivative discontinuity of time-dependent density functional theory. Phys Rev Lett 104(23) 236801... 

Fry NL, Rose Ml et al (2010) Ruthenium nitrosyls derived from tetradoitate ligands containing carboxamido-N and phenolato-O donms syntheses, structures, photolability, and time dependent density functional theory studies. Inorg Chem 49 1487-1495... 

Guo H-B, He F, Gu B, Liang L, Smith JC (2012) Time-dependent density functional theory assessment of UV absorption of benzoic acid derivatives. J Phys Chem A 116 11870-11879... 

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... 

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. 

Molecular frequencies depend on the second derivative of the energy with respect to the nuclear positions. Analytic second derivatives are available for the Hartree-Fock (HF keyword). Density Functional Theory (primarily the B3LYP keyword in this book), second-order Moller-Plesset (MP2 keyword) and CASSCF (CASSCF keyword) theoretical procedures. Numeric second derivatives—which are much more time consuming—are available for other methods. 

Eqs. (101)-(103) constitute the KS scheme of time-dependent spin-density functional theory. With the xc action functional [ t. i] defined in analogy to Eq. (43), the spin-dependent xc potentials can be represented as functional derivatives ... 

Car and Parrinello in their celebrated 1985 paper [2] proposed an alternative route for molecular simulations of electrons and nuclei altogether, in the framework of density functional theory. Their idea was to reintroduce the expansion coefficients Cj(G) of the Kohn-Sham orbitals in the plane wave basis set, with respect to which the Kohn-Sham energy functional should be minimized, as degrees of freedom of the system. They then proposed an extended Car-Parrinello Lagrangian for the system, which has dependance on the fictitious degrees of freedom Cj(G) and their time derivative Cj (G) ... 

Jonsson et al. review tlie Kohn-Sham density functional theory (DFT) for time-dependent (TD) response functions. They describe the derivation of the working expressions. They also review recent progress in the application of TD-DFT to open shell systems. They reported results on several properties (i) hyperpolarizabilities (e.g. para-nitroaniline, benzene, Cgg fullerene), (ii) excited state polarizabilities (e.g. pyrimidine), (iii) three-photon absorption and (iv) EPR spin Hamiltonian parameters. 

The electronic contribution can be computed using two derivative schemes involving quantum mechanical calculations of the free energy or, alternar tively, of the dipole moment followed by derivatives with respect to the perturbing external field, computed at zero intensity. At Hartree-Fock (HF) or Density Functional Theory level both approaches lead to the use of the coupled HF or KS theory either in its time-independent (CHF, CKS) or time-dependent (TDCHF, TDDFT) version according to the case. 

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