Missing Rung Problem in Vibrational Ladder Climbing

Missing Rung Problem in Vibrational Ladder Climbing Takahiro Horiba, Soichi Shirai, and Hirotoshi Hirai∗ Toyota Central Research and Development Labs., Inc., 41-1, Yokomichi, Nagakute, Aichi 480-1192, Japan (Dated: August 18, 2021) We observed vanishing of the transition dipole moment, interrupting vibrational ladder climbing (VLC) in molecular systems. We clarified the mechanism of this phenomenon and present a method to use an additional chirped pulse to preserve the VLC. To show the effectiveness of our method, we conducted wavepacket dynamics simulations for LiH dissociations with chirped pulses. The results indicate that the efficiency of LiH dissociation is significantly improved by our method compared to conventional methods. We also revealed the quantum interference effect behind the excitation process of VLC. Controlling molecular reactions as desired is one of such as plasma systems [21] and superconducting circuits the ultimate goals of chemistry. Compared to conven- [22]. The excitation condition proposed by Duan et al. tional macroscopic reaction control methods using tem- and the characterization parameters proposed by Fried- perature and pressure, proposed methods that directly land et al. depend on the physical properties of the sys- control the quantum state of molecules using the electric tem such as the potential energy surface (PES) and the field of a laser are expected to achieve dramatically im- dipole function. Thereby, these properties determine the proved efficiency and selectivity of reactions [1–3]. Such characteristic of the VLC. However, most of the theo- techniques, first proposed by Brumer and Shapiro in the retical treatments were based on simple model functions 1980s [1], are called “coherent control”, and with the ad- of these properties such as Morse potentials and linear vent of femtosecond pulsed lasers and the development dipole moments [12, 15, 23–25]. of laser shaping technology have been the subject of nu- In this study, we performed wavepacket dynamics sim- merous studies and continue to attract considerable at- ulations of VLC based on the PES and the dipole moment tention [4–6]. A promising coherent control method is computed by a highly accurate ab-initio quantum chem- vibrational ladder climbing (VLC) [7], which employs a istry method. The results of the simulations revealed cascade excitation process in molecular vibrational levels a problem that has not previously been recognized in under chirped infrared laser pulses with time-dependent VLC: VLC is interrupted by the existence of a level with frequencies corresponding to the vibrational excitation a nearly zero adjacent transition dipole moment (TDM), energy levels. VLC can focus an input laser pulse energy i.e. vanishing of the TDM. We call this problem the miss- on a specific molecular bond. Thereby, it is expected to ing rung problem (MRP) as it is like a rung missing in realize highly efficient bond-selective photodissociation the middle of a ladder. To clarify the issue in more detail, [8]. VLC has been studied intensively both experimen- the absolute values of TDMs for the diatomic molecules tally and theoretically. Experimentally, the vibrational LiH and HF are shown in the lower panels of Fig. 1(b) excitation by VLC in diatomic molecules (NO [9], HF and (c), respectively. It can be seen that the TDMs of [10]) and amino acids [11] have been reported. Recently, the 16th level of LiH and the 12th level of HF have nearly Morichika et al. succeeded in breaking molecular bonds zero values, which indicate “missing rungs” in the vibra- in transition metal carbonyl WCO6 with a chirped in- tional levels. frared pulse enhanced by surface plasmon resonance [8]. To understand the cause of the MRP, we start by con- Unfortunately, there are few experimental reports of suc- sidering a harmonic oscillator. The vibrational wavefunc- cessful bond breaking by VLC, which remains a chal- tions of the harmonic potential φi are described by Her- arXiv:2108.00797v2 [quant-ph] 17 Aug 2021 lenge. In early theoretical works, Liu et al. and Yuan mite polynomials, and their parities| i are different between et al. studied the classical motion of driven Morse oscil- adjacent levels, as described in Fig. 1(a). Assuming an lators [12–14]. They analyzed the excitation and disso- odd function for the dipole moment, the TDM between ciation dynamics of diatomic molecules under a chirped adjacent levels µi,i+1 = φi µ φi+1 becomes an integral electric field using the action-angle variable, and found of an even function, whichh yields| | ai nonzero value (i.e. an the condition for efficient excitation [12, 13]. Regarding allowed transition). The well-known infrared selection the quantum aspects, a pioneering work by Friedland et rule ∆ν = 1 corresponds to this result [26]. On the al. [15] focused on quantum and classical excitation pro- other hand,± if we assume an even function for the dipole cesses, specifically quantum ladder climbing and classical moment, the adjacent TDM µi,i+1 becomes an integral autoresonance, and proposed parameters to characterize of an odd function, which yields zero (i.e. a forbidden the quantum and classical phenomena [15, 16]. Based transition), as shown in the lower panel of Fig. 1(a). on the characterization parameters, quantum and clas- Based on the above discussions, we consider two real- sical comparisons were made in various systems [17–20] istic molecular systems, LiH and HF, where the shapes 2 (a) (b) (c) FIG. 1: Upper panels: PESs and dipole moments for (a) the harmonic model, (b) LiH molecule, and (c) HF molecule. For the harmonic model, the model dipole moments are shown. The orange and light blue lines shows the vibrational wavefunctions. Lower panels: absolute values of the corresponding TDMs. of the vibrational wavefunctions of molecules are similar Since the position of the missing rung sensitively depends to those of a harmonic potential except for a distortion on the shape of the PES and dipole function (see Sup- due to the anharmonicity of the PESs, as shown in Figs. plemental Material), highly accurate quantum chemistry 1(b) and (c). Therefore, the nature of the parity of the methods are required to identify its exact position. wavefunctions is approximately conserved, whereas the To verify the existence of the missing rung by numeri- parity of molecular dipole moments is not as simple as cal experiments, we conducted wavepacket dynamics sim- in the above discussion for the harmonic potential. Near ulations for the LiH molecule. The time evolution of the equilibrium distance, the dipole moment can be re- the wavepacket Ψ(x, t) under a laser electric field E(t) is garded as being linear (i.e. odd function-like). Thereby, expressed by the following time-dependent Schrödinger the TDM µi,i+1 increases monotonically as the vibra- equation tional level increases, as for the harmonic model. As the bond length increases, however, nonlinearity of the dipole ∂ i~ Ψ(x, t) = H(x, t)Ψ(x, t), (1) moment appears. The dipole function takes maxima (i.e. ∂t is even function-like) and then decreases asymptotically to zero. This behavior results from the relaxation of the where the Hamiltonian H(x, t) is defined by molecular polarization, which always occurs for charge H(x, t) = H (x) µ(x)E(t) (2) neutral heteronuclear diatomic molecules. As the edge of 0 − the distribution of the ith vibrational level of the wave- ~2 ∂2 H0(x) = 2 + V (x). (3) function reaches the maxima of the dipole function, the −2Mred ∂x TDM µi,i+1 begins to decrease due to the even-function nature of the dipole function. Then the TDM has a near- Mred in H0(x) is the reduced mass of LiH. The potential zero value, as can be seen in Fig. 1(b) and (c) for LiH energy V (x) and the dipole moment µ(x) were calculated and HF, respectively. No missing rung will appear in the by the multireference averaged quadratic coupled-cluster VLC simulations based on the assumption of the linear (MR-AQCC) method [27]. Details of these computations dipole function and is likely to give an unrealistic result. are described in the Supplemental Material. We employ a Gaussian pulse as the electric field E(t) of the pulsed 3 (a) (b) (c) FIG. 2: Color maps of the results of the single pulse method and the DSP method (explained later in the paper). (a) Dissociation probabilities and (b) vibrational state with the largest occupation number for the single pulse method. (c) Dissociation probabilities for the DSP method. The blue dots represent the grid points where the wavepacket simulations were conducted. The yellow boxed area in (a) and (b) represents the parameter range chosen for the DSP method. laser: quantum ladder climbing (see Supplementary Material for details). The parameter range was determined to be α(t t )2 E(t) = E exp− − 0 cos ω(t)(t t ), (4) 8 7 2 0 − 0 E0 = 2.0–20[MV/cm] and α = 10− –10− [fs− ]. For this parameter range of E0 and α, a grid of 20 20 points where E0 is the maximum electric field intensity, α is on logarithmic scales was defined and wavepacket× dy- the Gaussian spreading parameter, t0 is the center time namics simulations were carried out for each grid point. of the pulse, and ω(t) is the time-dependent frequency. The chirp parameters γ1, γ2 were optimized by Bayesian The time variation of ω(t) is assumed to be a linear chirp, optimization so as to maximize the degree of excitation and parametrized by the two dimensionless parameters and the dissociation probability. The degree of excitation γ1 and γ2 as follows, was evaluated by the expectation value of the occupied states, and the dissociation probability was evaluated by t t0 γ1 γ2 ω(t) = ω0 (γ1 + γ2) − + 1 + − , (5) a time-integration of the probability density flux of the − · 4σ 2 wavepacket that reaches the dissociation limit.

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