Nonadiabatic effects in electronic and nuclear dynamics

Abstract

Due to their very nature, ultrafast phenomena are often accompanied by the occurrence of nonadiabatic effects. From a theoretical perspective, the treatment of nonadiabatic processes makes it necessary to go beyond the (quasi) static picture provided by the time-independent Schrödinger equation within the Born-Oppenheimer approximation and to find ways to tackle instead the full time-dependent electronic and nuclear quantum problem. In this review, we give an overview of different nonadiabatic processes that manifest themselves in electronic and nuclear dynamics ranging from the nonadiabatic phenomena taking place during tunnel ionization of atoms in strong laser fields to the radiationless relaxation through conical intersections and the nonadiabatic coupling of vibrational modes and discuss the computational approaches that have been developed to describe such phenomena. These methods range from the full solution of the combined nuclear-electronic quantum problem to a hierarchy of semiclassical approaches and even purely classical frameworks. The power of these simulation tools is illustrated by representative applications and the direct confrontation with experimental measurements performed in the National Centre of Competence for Molecular Ultrafast Science and Technology.

FIG. 1.

Criteria of nonadiabaticity of quantum dynamics. (a) The static energy-gap criterion does not take into account the dynamics of the wave packet. (b) The population transfer criterion measures the actual decay of probability density on the initial surface. It is more sensitive than the energy-gap criterion. “Adiabaticity,” i.e., the fidelity (overlap) of the adiabatic and exact wavefunctions can capture the population transfer (b) as well as other nonadiabatic effects such as displacement (c) or interference (d) on a single surface, which would be undetected by the population transfer criterion. H_adiab is the decoupled adiabatic Born-Oppenheimer Hamiltonian, whereas H_nonad is the fully coupled nonadiabatic Hamiltonian. Adapted with permission from Zimmermann and Vaníček, J. Chem. Phys. 136, 094106 (2012). Copyright 2012 AIP Publishing LLC.

FIG. 2.

Quasistatic picture of strong-field tunnel ionization.

FIG. 3.

Multiphoton ionization for the case of γ ≫ 1.

FIG. 4.

Nonadiabatic picture of strong-field tunnel ionization: the adiabatic approximation is shown in grey for comparison. For the nonadiabatic case, we take into account the dynamics of the laser field during the tunneling process, shown here with the dashed lines of the potential barrier. This allows the electron to gain energy through photon absorption which leads to a shorter exit radius (Ref. , Fig. 3 and Ref. 97) For elliptical polarization, the most probable electron trajectory exhibits a transverse momentum at the tunnel exit, tangential to the laser field rotation in the polarization plane (shown as a red arrow).

FIG. 5.

Left: asymptotic photoelectron momentum distribution for elliptically polarized ionizing field (major axis of polarization along x, minor axis along y). Adapted from Ref. . Right: prediction of p_max in the adiabatic (blue dashed) or nonadiabatic (red solid) framework, depending on the peak field strength. Reproduced with permission from Hofmann et al., Phys. Rev. A 90, 043406 (2014). Copyright 2014 American Physical Society.

FIG. 6.

Jacobi coordinates used in the three-dimensional model of iodomethane.

FIG. 7.

Population dynamics of iodomethane induced by an excitation with a pump laser pulse shown in the bottom panel. Time dependence of the populations of three electronic states in the different models: symbols 1D, 2F, and 3D stand for one-, two-, and three-dimensional models, while the acronyms NoNACs and NACs denote, respectively, models without and with nonadiabatic couplings.

FIG. 8.

Dissociation dynamics of iodomethane induced by a pump laser pulse. Time-dependence of the expectation value of the dissociative coordinate R averaged over the full molecular wavepacket (top panel) and restricted to the vibrational wavepackets in individual electronic states (bottom three panels).

FIG. 9.

Influence of nonadiabatic dynamics on the time-resolved stimulated emission spectrum of pyrazine. Comparison of numerically evaluated spectra in which the nonadiabatic couplings were included [panel (a)] or neglected [panel (b)]. (a) Due to nonadiabatic dynamics, the signal decays with an increasing delay time τ. (b) In the absence of nonadiabatic couplings, the signal does not decay. Both spectra were computed with the MSDR combined with the fewest-switches surface hopping dynamics. Reproduced with permission from Zimmermann and Vaníček, J. Chem. Phys. 141, 134102 (2014). Copyright 2014 AIP Publishing LLC.

FIG. 10.

(a) Sketch of the experimental setup: the high order harmonic radiation is recombined with the fundamental IR (probe) pulses through a drilled mirror. (b) extreme ultra violet (XUV) excitation spectrum: a tin filter allows us to switch between the full (solid magenta) and a filtered (dashed orange line) spectrum. (c) Left: cationic states (blue) reachable with the XUV photon energies. Right: initial population of the cationic states. (d) D0, D1, D2, and D3 potential energy surfaces along the C-H stretch coordinate and the twist angle, calculated at the aLR-TDDFT/PBE0 level of theory with a 6–31** basis set on the ground-state geometry optimised geometries of the cation. The reference zero energy is the ground state energy of the neutral molecule.

FIG. 11.

(a) Measured ion spectrum resulting from XUV photoionization of ethylene with the pump pulses of Fig. 1(b). (b) Delay-dependent IR-induced relative charge in the yields. For improved readability, to some curves a vertical offset is applied.

FIG. 12.

Calculated population of the four lowest cationic states as a function of the time, after selective excitation to D3, D2, and D1 [panel (a)] and averaged over all the initial excitations [panel (b)].

FIG. 13.

Hopping geometries for transition between different pairs of states. (a) Maximum C-H distance distributions and (b) torsion angle distributions.

FIG. 14.

(a) Diabatic potential energy surfaces of the two local OH stretch vibrations of water ($|1,0,0⟩$ in red and $|0,1,0⟩$ in green) and the fundamental ($|0,0,1⟩$ in magenta) and the first overtone of the HOH bending vibration ($|0,0,2⟩$ in blue) as well as the ground state ($|0,0,0⟩$ in black). The inter-molecular structures of the complex in the minima of the $|0,0,0⟩$ and $|1,0,0⟩$ surfaces are shown. (b) Vibrational relaxation after a “vertical” excitation into the OH stretch modes, using the same color code as in panel (a) for the various states. For symmetry reasons, the population of state $|0,1,0⟩$ is the same as that of $|1,0,0⟩$ and is not shown. Adapted with permission from Hamm and Stock J. Chem. Phys. 143, 134308 (2015). Copyright 2015 AIP Publishing LLC.

FIG. 15.

(a) Interaction energy at the inner turning point (r = 1.072 Å) of the v = 0 vibration for the three lowest electronic states of (N₂⁺-Ar) (the angle between r and R is 168°). The curves were computed at the MRCI+Q/avtz level of theory in the ²A' symmetry. Note that the separation between ground state and first excited state is only about 700 cm⁻¹ at around R = 4.5 Å.