On-the-Fly Ab Initio Semiclassical Evaluation of Electronic Coherences in Polyatomic Molecules Reveals a Simple Mechanism of Decoherence

Irradiation of a molecular system by an intense laser field can trigger dynamics of both electronic and nuclear subsystems. The lighter electrons usually move on much faster, attosecond timescale but the slow nuclear rearrangement damps ultrafast electronic oscillations, leading to the decoherence of the electronic dynamics within a few femtoseconds. We show that a simple, single-trajectory semiclassical scheme can evaluate the electronic coherence time in polyatomic molecules accurately by demonstrating an excellent agreement with full-dimensional quantum calculations. In contrast to numerical quantum methods, the semiclassical one reveals the physical mechanism of decoherence beyond the general blame on nuclear motion. In the propiolic acid, the rate of decoherence and the large deviation from the static frequency of electronic oscillations are quantitatively described with just two semiclassical parameters—the phase space distance and signed area between the trajectories moving on two electronic surfaces. Because it evaluates the electronic structure on the fly, the semiclassical technique avoids the “curse of dimensionality” and should be useful for preselecting molecules for experimental studies.

Figure 1

Electronic coherence measured by the time-dependent overlap ${\chi }_{13}(t)$ of the nuclear wave packets propagated on the first and third cationic states of propiolic acid after removal of a HOMO electron. Dynamics was performed with the full quantum MCTDH method (“quantum”) or with the semiclassical vertical-Hessian thawed Gaussian approximation (“semiclass.”), and either with the vibronic coupling (“VC”) or on-the-fly (“OTF”) Hamiltonian, both obtained at the ab initio many-body Green’s function ADC(3) level of theory. All simulations except the “quantum VC” calculation employed the diabatic approximation (“diab.”), which neglects nonadiabatic couplings between the diabatic states, or the adiabatic approximation (“adiab.”), which neglects nonadiabatic couplings between adiabatic states.

Figure 2

Geometric interpretation of Eq. (5). The reduced action $S_{\mathrm{red}}$ is the gray phase-space area enclosed by the curve $\mathbf{C}$ consisting of the classical trajectory propagated in state $i$ with potential energy $V_i$ and connecting ${\mathbf{X}}_i(t)$ with ${\mathbf{X}}_0$ (blue curve), classical trajectory propagated in state $j$ with potential energy $V_j$ and connecting ${\mathbf{X}}_0$ with ${\mathbf{X}}_j(t)$ (red curve), and a straight (black) line connecting ${\mathbf{X}}_j(t)$ with ${\mathbf{X}}_i(t)$. The correct sign of $S_{\mathrm{red}}$ is obtained by taking the curve integral along $\mathbf{C}$ in the direction indicated by the arrows. The phase space distance between the final points ${\mathbf{X}}_j(t)$ and ${\mathbf{X}}_i(t)$, given by the length $d$ of the black line segment, determines the coherence decay (i.e., the decay of $|{\chi }_{ij}(t)|$), while the reduced action $S_{\mathrm{red}}$, equal to the gray area, affects the frequency of oscillations of the coherence ${\chi }_{ij}(t)$.

Figure 3

Semiclassical analysis of the electronic coherence ${\chi }_{13}(t)$ from Fig. 1. Comparison of the coherence computed with the on-the-fly version of TGA (green dash-dotted line), analytical semiclassical expression (3) (red solid line), factors describing the decay (blue solid line) and oscillations (gray solid line) of coherence in the presence of nuclear motion, and the undamped coherence in the absence of nuclear motion (gray dashed line).

Figure 4

Top panel: time evolution of the electronic coherences ${\chi }_{ij}(t)$ created by the equally weighted coherent superposition of the second and fourth cationic states (ionization from the HOMO-1, $A^{\prime }$ symmetry), and the first and third cationic states (ionization from the HOMO, $A^{\prime \prime }$ symmetry). Bottom panel: comparison of the electronic purity function $\mathrm{Tr}({\rho }^2)$ for the propiolamide and propiolic acid molecules.