Solving the Kadanoff-Baym Equations for Inhomogeneous Systems: Application to Atoms and Molecules

We implement time propagation of the nonequilibrium Green function for atoms and molecules by solving the Kadanoff-Baym equations within a conserving self-energy approximation. We here demonstrate the usefulness of time propagation for calculating spectral functions and for describing the correlated electron dynamics in a nonperturbative electric field. We also demonstrate the use of time propagation as a method for calculating charge-neutral excitation energies, equivalent to highly advanced solutions of the Bethe-Salpeter equation.

Figure 1

At $t=0$, the system is in thermal equilibrium, and the Green function is calculated for imaginary times from 0 to $-i\beta$ (left). Describing the system for $t>0$ implies calculating $G(t_1,t_2)$ on an extending time contour (right).

Figure 2

(a) The correlation part of the second-order self-energy, (b) a diagrammatic representation of the Bethe-Salpeter equation, and (c) the corresponding Bethe-Salpeter kernel. The Green function lines represent self-consistent, full Green functions.

Figure 3

The lower figures show the trace of the imaginary part of $G^{<}(t_1,t_2)$ for an ${\mathrm{H}}_2$ molecule in its ground state (left) and in an applied electric field (right). The Green function in equilibrium depends only on $t_1-t_2$. The upper figures show the same quantity for a fixed value of $T=(t_1+t_2)/2$, compared with the same function obtained from TDHF.

Figure 4

The imaginary part of the polarizability $\alpha (\omega )$ of a beryllium atom calculated from the Green function propagated up to a time $T$.

Figure 5

The time-dependent dipole moment of a Be atom, calculated within the HF and the second-order approximation.