Time-dependent density functional theory: Derivation of gradient-corrected dynamical exchange-correlational potentials

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.

Figure 1

The zero-frequency bulk modulus $K_{\mathrm{xc}}^0$ (in units of $2{\omega }_p{n}_0$) as a function of $r_s$ for several values of the reduced gradient $s=\mid \mathbf{\nabla }{n}_0\mid ∕\left[2{\left(3{\pi }^2\right)}^{1∕3}\right]$. $s=0$ corresponds to the uniform electron gas.

Figure 2

The high-frequency bulk modulus $K_{\mathrm{xc}}^{\infty }$ (in units of $2{\omega }_p{n}_0$) as a function of $r_s$ for several values of $s$.

Figure 3

The high-frequency shear modulus ${\mu }_{\mathrm{xc}}^{\infty }$ (in units of $2{\omega }_p{n}_0$) as a function of $r_s$ for several values of $s$.

Figure 4

The bulk viscosity $\zeta \left(\omega \right)$ (in units of $\hslash n_0$) for several values of $s$ at $r_s=3$. The same RPA value of ${\mu }_{\mathrm{xc}}\left(0\right)$ for all $s$ is taken from Table I of Ref. 60.

Figure 5

The shear viscosity $\eta \left(\omega \right)$ (in units of $\hslash n_0$) for several values of $s$ at $r_s=3$. The same ${\mu }_{\mathrm{xc}}\left(0\right)$ for all $s$ is employed as in Fig. 4.

Figure 6

The real part of the finite-frequency bulk modulus $\mathrm{Re}{K}_{\mathrm{xc}}\left(\omega \right)$ (in units of $2{\omega }_p{n}_0$) for several values of $s$ at $r_s=3$.

Figure 7

The real part of the finite-frequency shear modulus $\mathrm{Re}{\mu }_{\mathrm{xc}}\left(\omega \right)$ (in units of $2{\omega }_p{n}_0$) for several values of $s$ at $r_s=3$.