Propagating two-particle reduced density matrices without wave functions

Describing time-dependent many-body systems where correlation effects play an important role remains a major theoretical challenge. In this paper we develop a time-dependent many-body theory that is based on the two-particle reduced density matrix (2-RDM). We present a closed equation of motion for the 2-RDM by developing a reconstruction functional for the three-particle reduced density matrix (3-RDM) that preserves norm, energy, and spin symmetries during time propagation. We show that approximately enforcing $N$-representability during time evolution is essential for achieving stable solutions. As a prototypical test case which features long-range Coulomb interactions we employ the one-dimensional model for lithium hydride (LiH) in strong infrared laser fields. We probe both one-particle observables such as the time-dependent dipole moment and two-particle observables such as the pair density and mean electron-electron interaction energy. Our results are in very good agreement with numerically exact solutions for the $N$-electron wave function obtained from the multiconfigurational time-dependent Hartree-Fock method.

Figure 1

Dynamical purification applied after each time step to project the propagated ${\tilde{D}}_{12}(t+\Delta t)$ onto the set of 2-RDMs that satisfy the $D$ condition and the $Q$ condition, schematically.

Figure 2

Electron density of the 1D LiH molecule. The equilibrium bond length $a=2.3$ between the Li nucleus and the proton is depicted. The electron cloud is predominantly located near the Li core.

Figure 3

(a) The pair-density distribution $\rho (z_1,z_2)$ in coordinate space for the ground state of the LiH molecule. The density distribution shows distinct peaks for interatomic pairs with one electron close to the Li core while the other one is close to the proton (marked by arrows). (b) The imaginary diagonal elements of the exact collision operator $C_{12}$. The positive contribution for pairs with positive total momentum shows that the particle interaction creates pairs moving collectively toward the proton while pairs with negative momentum moving towards the Li core are destroyed.

Figure 4

The laser pulse [Eq. (87)] with $I={10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ ($F_0=0.053$), $\lambda =750$ nm, $N_c=3$. Distinct points in time are marked by numbers for later reference.

Figure 5

Dipole moment of LiH subject to the laser pulse as depicted in Fig. 4 with $I={10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ using the reconstruction functionals $D_{123}^{\mathrm{V}}\left[D_{12}^{\mathrm{exact}}\right]$ and $D_{123}^{\mathrm{C}}\left[D_{12}^{\mathrm{exact}}\right]$ with the exact 2-RDM $D_{12}^{\mathrm{exact}}$ as input at each time step obtained from a concurrent MCTDHF calculation. The high-frequency oscillations near $\tau =3$ originate from superpositions between the ground state and excited states (see right inset). Both $D_{123}^{\mathrm{V}}$ and $D_{123}^{\mathrm{C}}$ can markedly reproduce the MCTDHF result. A close up shows that $D_{123}^{\mathrm{C}}$ performs better than $D_{123}^{\mathrm{V}}$.

Figure 6

(a) Dipole moment of LiH subject to the laser pulse (Fig. 4) with $I={10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ employing the reconstruction functionals $D_{123}^{\mathrm{V}}\left[D_{12}\right]$ and $D_{123}^{\mathrm{C}}\left[D_{12}\right]$ with $D_{12}$ propagated by Eq. (24) while the orbitals are calculated via Eq. (28) using $D_{12}^{\mathrm{exact}}$ from a parallel MCTDHF calculation. The violation of $N$-representability leads to divergence of the dipole moment. The point of divergence is marked by vertical lines for each reconstruction. Note that contraction consistency postpones but does not prevent the divergence. (b) The smallest and largest eigenvalue of the 2-RDM. The violation of $N$-representability clearly visible from eigenvalues outside the allowed range $0\le g_i\le 4$ (marked by dashed horizontal lines) occurs in close temporal proximity to the divergence of the dipole moment.

Figure 7

(a) Dipole moment for the same parameters as in Fig. 6 but with dynamical purification. (b) The smallest eigenvalue of the 2-RDM for the propagation employing $D_{123}^{\mathrm{C}}$ without purification [green line, compare Fig. 6] and with purification. For the latter the smallest eigenvalue is shown before and after each dynamical purification step [Eq. (66)]. The negative occupation number after purification with ten iterations is in general smaller than ${10}^{-9}$ indicating rapid convergence.

Figure 8

Dipole moment of LiH subject to the laser pulse (Fig. 4) with (a) $I={10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and (b) $I=8\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ employing the reconstruction functionals $D_{123}^{\mathrm{V}}\left[D_{12}\right]$ and $D_{123}^{\mathrm{C}}\left[D_{12}\right]$ within a fully self-consistent propagation of the TD-2RDM compared with the exact (MCTDHF) reference.

Figure 9

Dipole moment of LiH subject to the laser pulse (Fig. 4) with (a) $I={10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and (b) $I=8\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ for a fully self-consistent propagation of the TD-2RDM method compared with the exact (MCTDHF) reference, the TDHF, and the TDDFT calculations.

Figure 10

Time-dependent electron-electron interaction energy $E_{\mathrm{int}}\left(t\right)$ of LiH subject to the laser pulse (Fig. 4) with $I={10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ (a) and $I=8\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ (b) for a fully self-consistent propagation of the TD-2RDM method compared with the exact (MCTDHF) reference, the TDHF, and the TDDFT calculations.

Figure 11

Pair density $\rho (z_1,z_2,t)$ of the LiH molecule in the strong laser pulse with $I=8\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ left column exact (MCTDHF); middle column self-consistent propagation of the 2-RDM using the reconstruction functional $D_{123}^{\mathrm{C}}\left[D_{12}\right]$; right column difference $\Delta \rho (z_1,z_2,t)={\rho }^{\mathrm{exact}}(z_1,z_2,t)-\rho (z_1,z_2,t)$ between the exact (MCTDHF) and the TD-2RDM pair density. The pair density is shown at four times depicted in Fig. 4 [rows (1)–(4)]. The stretched-out arms in the pair density are signatures of single-particle ionization. The approximate distributions are in very good agreement with the exact result. Small differences appear at times (3) and (4).