Equations of motion for natural orbitals of strongly driven two-electron systems

Natural orbital theory is a computationally useful approach to the few- and many-body quantum problems. While natural orbitals have been known and applied for many years in electronic structure applications, their potential for time-dependent problems started being investigated only recently. Correlated two-particle systems are of particular importance because the structure of the two-body reduced density matrix expanded in natural orbitals is known exactly in this case. However, in the time-dependent case the natural orbitals carry time-dependent phases that allow for certain time-dependent gauge transformations of the first kind. Different phase conventions will, in general, lead to different equations of motion for the natural orbitals. A particular phase choice allows us to derive the exact equations of motion for the natural orbitals of any (laser-) driven two-electron system explicitly, i.e., without any dependence on quantities that, in practice, require further approximations. For illustration, we solve the equations of motion for a model helium system. Besides calculating the spin-singlet and spin-triplet ground states, we show that the linear response spectra and the results for resonant Rabi flopping are in excellent agreement with the benchmark results obtained from the exact solution of the time-dependent Schrödinger equation.

Figure 1

Singlet (a) and triplet (b) linear response spectra for a different numbers of RNOs $N_{\mathrm{o}}$, compared to the exact TDSE result. For comparison, bare (i.e., with ground-state frozen Hamiltonian) TDRNOT results following the phase convention of Sec. 2e1 are shown with dashed lines. To guide the eyes, vertical lines indicate some of the distinct peaks in the exact TDSE spectrum.

Figure 2

ONs $n_k\left(t\right)$ vs time $t$ for the spin singlet in a laser field of frequency $\omega =0.5337$ resonantly tuned to the first excited state. The four most significant ONs $n_1\left(t\right),n_3\left(t\right),n_5\left(t\right),n_7\left(t\right)$ obtained by TDRNOT with $N_{\mathrm{o}}=16$ RNOs (solid lines) correctly reproduce more than two Rabi cycles of the exact TDSE propagation (dotted lines). Due to the truncation to a finite number of RNOs in TDRNOT, fewer significant orbitals are missing the proper coupling to lower orbitals, leading to erroneous behavior of small ONs over time. For longer propagation times, also higher ONs are affected because the RNOs are coupled.