Electronic Density Functional Theory pp 199-216 | Cite as

# Current-Density Functional Theory of Linear Response to Time-Dependent Electromagnetic Fields

## Abstract

Local density approximations are known to be very useful in calculating the ground-state exchange and correlation (xc) energy of many-electron systems [1], and local approximations are gaining in importance for the description of xc effects in time-dependent situations also [2,3]. Most of the time-dependent work has dealt with a *scalar* time-dependent xc potential \({v_{xc}}(\vec r,\omega )\) approximated as a local functional of the time-dependent density \(n(\vec r,\omega )\) as described in the chapter “Time-dependent density functional theory” in the introductory section of this Book. Despite much progress in this scalar approach, it will be shown in this Chapter that existing approximations for *v* _{ xc } require modification. In particular, it will be shown that (i) there is no local density approximation for the scalar xc potential at finite frequency (ii) there is, however, a consistent local approximation for a *vector* xc potential \({{\vec a}_{xc}}(\vec r,\omega )\) in terms of the dynamic current density \(\vec j(\vec r,\omega )\) and its space derivatives, as well as the ground-state density and its space derivatives. This approximation is valid, at a given frequency, for sufficiently slow spatial variations of the ground-state density and of the perturbing dynamic potential. The appropriately modified vector xc potential will be derived here in some detail, thus filling out the brief description published recently [4]. New material will also be presented, providing interpretation of the findings in the simple case of one dimensional inhomogeneity.

## Keywords

Local Density Approximation Time Dependent Density Functional Theory Gradient Expansion Accelerate Frame Local Field Factor## Preview

Unable to display preview. Download preview PDF.

## References

- 1.W. Kohn and L. Sham, Phys. Rev.
**140**, A1133 (1965).MathSciNetADSCrossRefGoogle Scholar - 2.E. K. U. Gross and W. Kohn, Phys. Rev. Lett.
**55**, 2850 (1985).ADSCrossRefGoogle Scholar - E. K. U. Gross and W. Kohn, Adv. Quantum Chemistry
**21**, 255 (1990).ADSCrossRefGoogle Scholar - 3.A. Zangwill and P. Soven Phys. Rev. Lett.
**45**, 204 (1980); Phys. Rev. B**24**, 4121 (1981).ADSCrossRefGoogle Scholar - 4.G. Vignale and W. Kohn, Phys. Rev. Lett.
**77**, 2037 (1996).ADSCrossRefGoogle Scholar - 5.E. Runge and E. K. U. Gross, Phys. Rev. Lett.
**52**, 997, (1984).ADSCrossRefGoogle Scholar - 6.In this paper, a function \(f(\vec r\, - \,\vec r)\) is said to be of short range if the integral \(\int {f(\vec r,\,\vec r\, + \vec s)} d\vec s\) is finite. This is equivalent to saying that the Fourier transform of
*f*, with wave vector →k, with respect to the separation \(\vec s\, = \,\vec r\, - \,\vec r\), remains finite in the limit \(\vec k \to 0.\).Google Scholar - 7.H.M. Böhm, S. Conti, and M. P. Tosi, J. Phys.: Condensed Matter
**8**, 781 (1996).ADSCrossRefGoogle Scholar - 8.J. F. Dobson, Phys. Rev. Lett.
**73**, 2244 (1994).ADSCrossRefGoogle Scholar - 9.W. Kohn, Phys. Rev.
**123**, 1242 (1961), L. Brey*et al*, Phys. Rev. B**40**, 10647 (1989); ibid.**42**, 1240 (1990); S. K. Yip, Phys. Rev. B**43**, 1707 (1991).ADSMATHCrossRefGoogle Scholar - 10.J. F. Dobson, M. Bünner, and E. K. U. Gross, Phys. Rev. Lett.
**79**, 1905 (1997).ADSCrossRefGoogle Scholar - 11.J. F. Dobson, Proceedings of the NATO ASI on
*Density Functional Theory*, edited by E. K. U. Gross and R. M. Dreizler (Plenum, New York, 1994), p. 393.Google Scholar - 12.G. Vignale, Phys. Rev. Lett.
**74**, 3233 (1995).ADSCrossRefGoogle Scholar - 13.G. Vignale, Phys. Lett. A
**209**, 206 (1995).ADSCrossRefGoogle Scholar - 14.P. Nozières,
*The Theory of Interacting Fermi Systems*(W. A. Benjamin, New York, 1964), Chapter 6.Google Scholar - 15.Tai Kai Ng, Phys. Rev. Lett.
**62**, 2417 (1989).ADSCrossRefGoogle Scholar - 16.K. S. Singwi and M. P. Tosi, in
*Solid State Physics*, edited by H. Ehrenreich, F. Scitz, and D. Turnbull (Academic, New York, 1981), Vol. 36, p. 177.Google Scholar