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Canadian Journal of Physics

A time-dependent spin density functional theory for the dynamical spin susceptibility

K. L. Liu, S. H. Vosko

Published Nov 1, 1989• 40 citations
Publisher

Abstract

Within the framework of the time-dependent spin density functional formalism, we study the dynamical spin susceptibility [Formula: see text] of a general system of interacting electrons in the presence of a static external scalar potential. Based on the stationary property of the action functional, a variational expression for the real part of [Formula: see text] is derived: [Formula: see text]. The result is valid for a general exchange-correlation (XC) action functional. In particular, the XC factor 1(q,ω) is manifestly dependent on q and ω. The imaginary part Im [Formula: see text] is then determined through the Kramers–Kronig relation. Practical approximations for the required XC kernel are also discussed

DOI
10.1139/p89-178

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Within the framework of the time-dependent spin density functional formalism, we study the dynamical spin susceptibility [Formula: see text] of a general system of interacting electrons in the presence of a static external scalar potential. Based on the stationary property of the action functional, a variational expression for the real part of [Formula: see text] is derived: [Formula: see text]. The result is valid for a general exchange-correlation (XC) action functional. In particular, the XC factor 1(q,ω) is manifestly dependent on q and ω. The imaginary part Im [Formula: see text] is then determined through the Kramers–Kronig relation. Practical approximations for the required XC kernel are also discussed
We have recently proposed an approximation for the dynamical exchange-correlation (XC) potentials of time-dependent current-density functional theory beyond the local density approximation [Phys. Rev. Lett. 97, 036403 (2006)]. The novel feature of the approximation is that the dependence of the dynamical XC potentials upon the density gradient and other inhomogeneity parameters (e.g., the Laplacian of the density and the kinetic energy density) is introduced by applying the generalized gradient approximation (GGA) and meta-GGA to the calculation of the XC stress tensor. The scheme may allow a more accurate treatment of the dynamical properties of an inhomogeneous system, while reducing to the exact form for slowly varying densities and slowly varying external potentials. In this work, we present in detail the derivation of this XC potential, spell out the underlying assumptions, and explain their physical basis.
The scalar f(xc) and tensor f(xc) exchange-correlation (xc) kernels are key ingredients of the time-dependent density functional theory and the time-dependent current density functional theory, respectively. We derive a comparatively simple relation between these two kernels under the assumption that the dynamic xc can be considered "weak." A calculation of the frequency-dependent dielectric function of silicon using this relation in conjunction with Vignale-Kohn f(xc) demonstrates a potential of our method to account for the dynamic many-body effects within the rigorous scheme of time-dependent density functional theory. Our formula provides a bridge between the scalar f(xc), which directly enters many applications, and the tensor f(xc) which, due to its locality in space, is much easier to approximate.
The Runge-Gross [E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984)] action functional of time-dependent density-functional theory leads to a well-known causality paradox; that is, a perturbation of the electronic density in the future affects the response of the system in the present. This paradox is known to be caused by an inconsistent application of the Dirac-Frenkel variational principle. In view of the recent solutions to this problem, the action functional employed by Runge and Gross in their formulation of time-dependent density-functional theory is analyzed in the context of the Keldysh contour technique. The time-dependent electronic density and the concept of causality are extended to the contour. We derive a variational equation that obeys causality and relates the exchange-correlation potential with its kernel and the functional derivative of the exchange-correlation action functional with respect to the density. It is shown that the adiabatic local-density approximation is a consistent solution of this equation and that the time-dependent optimized potential method can also be derived from it. The formalism presented here can be used to find new approximation methods for the exchange-correlation potential and to avoid the causality dilemma.
The key quantity in the formulation of linear-response theory within time-dependent density-functional theory is the so-called exchange-correlation (xc) kernel. This chapter discusses the properties of the xc kernel. In general, it is a complex and frequency-dependent object, and a functional of the ground-state density. Several exact properties such as the zero-force theorem, static and high-frequency limit, and the long-range behaviour are discussed. The xc kernel can be related to a time-dependent variational principle provided one pays attention to certain time-dependent boundary conditions to ensure causality. The adiabatic approximation of the xc kernel is discussed. Finally, the third section of the chapter reviews the xc kernel of the homogeneous electron liquid. Many of its exact properties are covered, such as the high- and low-frequency limit and the long-wavevector limit. Three widely used parametrizations are presented and the analytical behaviour in the complex frequency plane is investigated.
We present some details of our recently-proposed Time-Dependent Density-Functional Theory (TDDFT) for strongly-correlated materials in which the exchange-correlation (XC) kernel is derived from the charge susceptibility obtained using Dynamical Mean-Field Theory (the TDDFT + DMFT approach). We proceed with deriving the expression for the XC kernel for the one-band Hubbard model by solving DMFT equations via two approaches, the Hirsch–Fye Quantum Monte Carlo (HF-QMC) and an approximate low-cost perturbation theory approach, and demonstrate that the latter gives results that are comparable to the exact HF-QMC solution. Furthermore, through a variety of applications, we propose a simple analytical formula for the XC kernel. Additionally, we use the exact and approximate kernels to examine the nonhomogeneous ultrafast response of two systems: a one-band Hubbard model and a Mott insulator YTiO3. We show that the frequency dependence of the kernel, i.e., memory effects, is important for dynamics at the femtosecond timescale. We also conclude that strong correlations lead to the presence of beats in the time-dependent electric conductivity in YTiO3, a feature that could be tested experimentally and that could help validate the few approximations used in our formulation. We conclude by proposing an algorithm for the generalization of the theory to non-linear response.
We propose a nonadiabatic time-dependent spin-density functional theory (TDSDFT) approach for studying single-electron excited states and the ultrafast response of systems with strong electron correlations. The correlation part of the nonadiabatic exchange-correlation (XC) kernel is constructed by using exact results for the Hubbard model of strongly correlated electrons. We demonstrate that the corresponding nonadiabatic XC kernel reproduces the main features of the spectrum of the Hubbard dimer and the 2D, 3D and infinite-dimensional Hubbard models, some of which are impossible to obtain within the adiabatic approach. The formalism may be applied for ab initio examination of strongly correlated electron systems in- and out-of-equilibrium within the TDSDFT, extending it beyond the metallic and semiconductor structures with plasmons, excitons and other excitations.