Maxime
Martinez
^{a},
François
Coppens
^{a},
Manuel
Barranco
^{abc},
Nadine
Halberstadt
*^{a} and
Martí
Pi
^{bc}
^{a}Laboratoire Collisions Agrégats Réactivité (LCAR), IRSAMC, Université de Toulouse, CNRS UMR 5589, Toulouse, France. E-mail: Nadine.Halberstadt@irsamc.ups-tlse.fr
^{b}Departament FQA, Facultat de Física, Universitat de Barcelona. Diagonal 645, 08028 Barcelona, Spain
^{c}Institute of Nanoscience and Nanotechnology (IN2UB), Universitat de Barcelona, Barcelona, Spain

Received
17th August 2018
, Accepted 10th October 2018

First published on 11th October 2018

We study the photodissociation of the potassium atom from a superfluid helium nanodroplet upon 5s ^{2}S or 4p ^{2}P excitation using the time-dependent helium density functional method (He-TDDFT). The importance of quantum effects is assessed by comparing the absorption spectrum obtained for a classical or a quantum description of the K atom. In the case of the 5s ^{2}S ← 4s ^{2}S excitation the difference is rather large, and we use a quantum description for the ensuing direct dissociation dynamics. In the case of the 4p ^{2}P ← 4s ^{2}S absorption spectrum, the difference is much smaller, hence a classical description of K is used to describe 4p ^{2}P excitation dynamics. Excitation to the 4p ^{2}Σ_{1/2} state leads to the direct dissociation of the K atom, while the 4p ^{2}Π_{3/2} state initially leads to the formation of an exciplex and the 4p ^{2}Π_{1/2} state to a bouncing atom above the droplet surface. Remarkably, electronic relaxation can be observed for the latter two states, leading to spin–orbit relaxation and the binding of the initially departing one-atom excimer as a ring excimer for the ^{2}P_{3/2} state and to the formation of a bound, ring excimer for the ^{2}Π_{1/2} state.

The first experimental work on the spectroscopy of a potassium atom attached to helium nanodroplets has been conducted in the group of Scoles.^{8} The 4p ← 4s fluorescence excitation spectrum was measured and compared to a simulated absorption spectrum obtained in the framework of the pseudo-diatomic model in which the droplet is represented as a pseudo-atom. The spectrum was composed of two peaks corresponding to absorption to the 4p ^{2}Π_{1/2}, ^{2}Π_{3/2} pseudo-diatomic states on top of the 4p ^{2}P_{1/2}, ^{2}P_{3/2} ← 4s ^{2}S_{1/2} (D_{1}, D_{2}) atomic lines, and a shoulder corresponding to absorption to the 4p ^{2}Σ_{1/2} pseudo-diatomic state to the blue (higher photon energies) of the spectrum. Preliminary dispersed fluorescence spectra^{8} revealed both atomic emission and red-shifted, broad band (∼1300 cm^{−1}) emission with some maxima, the relative proportion of atomic emission being larger for excitation at 13046 cm^{−1} (close to the 4p ^{2}P_{3/2} atomic line) than for excitation at 13001 cm^{−1} (close to the 4p ^{2}P_{1/2} atomic line).

Dispersed fluorescence spectra upon excitation near the D_{1} or D_{2} atomic line were analyzed in more detail by Reho et al.^{9} and the red-shifted, broad band emission was assigned to He–K* exciplex fluorescence. Simulations of the He–K* exciplex emission showed a good agreement with experiment, assuming that the vibrational population was not completely relaxed. It was concluded that both D_{1} and D_{2} lead to population in both the ^{2}Π_{1/2} and ^{2}Π_{3/2} bound levels, which implied ^{2}Π_{3/2} to ^{2}Π_{1/2} conversion by helium. The red end of the emission spectrum which could not be reproduced was attributed to higher exciplexes.

The fluorescence absorption spectrum of potassium attached to a helium nanodroplet has been revisited by Bünermann et al.^{10} in a joint experimental and theoretical study. The spectrum was simulated within the pseudo-diatomic model using a frozen droplet density. The simulated spectrum reproduced the main features of the experimental one, although with a small blue shift: +3 cm^{−1} for ^{2}Π_{1/2}; +11 cm^{−1} for ^{2}Π_{3/2}; and +8 cm^{−1} for ^{2}Σ_{1/2}. Excitation to the ^{2}Σ_{1/2} state appeared as an almost separated broad band whereas in the experimental spectrum it was not resolved and looked like an extended tail to the blue of ^{2}Π_{3/2} absorption.

By monitoring the time evolution of the fluorescence following 4p ^{2}P ← 4s ^{2}S excitation of K on helium droplets, Reho et al.^{11} have obtained an exciplex rise time of ∼8 ns when exciting at the frequency of the D_{1} line; and a rise time of 50 ps when exciting at the frequency of the D_{2} line. The results were interpreted as different formation mechanisms for the K*–He excimer. Exciplexes can be formed directly in the ^{2}Π_{3/2} state whereas they have to tunnel through a barrier in the ^{2}Π_{1/2} state.

Only a few studies have explored the importance of quantum effects of the dopant on the spectrum and dynamics of potassium attached to a helium droplet. They are detailed in the next paragraphs.

Hernando et al.^{12} have investigated the desorption dynamics of Li and Na atoms upon (n + 1)s ← ns photoexcitation on the droplet surface in a joint experimental and theoretical study. Ion imaging detection revealed that the mean kinetic energy of the desorbed atoms was linear with the excess energy of the exciting photon. This was interpreted as the fact that the alkali atom was in direct interaction with a small number of helium atoms at the time of the (impulsive) photodissociation. Theoretical simulations combined the well-established time-dependent version of the helium density method (He-TDDFT)^{7} with a quantum treatment based on Bohm trajectories for the light alkali atoms. They reproduced well the key experimental results and showed that the excited alkali desorption created highly non-linear density waves and solitons in the droplet.

Nakayama and Yamashita^{13} have calculated the structure and simulated the 4p ^{2}P ← 4s ^{2}S absorption spectrum of lithium, sodium, or potassium attached to a 300-atom droplet at 0.5 K using the path integral Monte Carlo method. It was found that alkali atoms were trapped in a dimple on the helium cluster surface and that the Ak-He_{300} cluster semiquantitatively reproduced the local structure of experimentally produced large helium clusters in the vicinity of alkali atoms. The main absorption peaks of the simulated absorption spectrum were in the right position, but they were too narrow compared to the experimental spectra.^{8}

Takayanagi and Shiga^{14} have studied the 4p ^{2}P ← 4s ^{2}S photoexcitation dynamics of a K atom bound to the surface of a 300-atom helium cluster using the semiclassical path integral centroid molecular dynamics method for helium and quantum wave packet dynamics for the electronic state of K. The two descriptions were combined either in a mean field approach, or selecting one of the adiabatic states of K. In the first case the K atom was found to desorb from the helium cluster for any of the three possible excitations, which was interpreted as a possible shortcoming of the mean-field approach. The desorption times averaged over 10 trajectories were 10–20 ps for ^{2}Σ_{1/2} and 15–30 ps for ^{2}Π_{1/2} or ^{2}Π_{3/2} excitation. In the adiabatic dynamics excitation to ^{2}Σ_{1/2} or to ^{2}Π_{1/2} was found to lead to desorption of K atoms. Only in the excitation to the ^{2}Π_{3/2} adiabatic state was some HeK* exciplex formation observed, although K desorption was still dominant. The time range for exciplex formation was 10–20 ps when it was formed.

Within this context, potassium is a very interesting test case to study quantum effects. Its behavior is expected to be intermediate between the very quantum one for lighter alkalis (Li and Na) and the very classical one for heavier alkalis (Rb and Cs). In this study we simulate the 5s ← 4s and the 4p ← 4s photoexcitation of a potassium-doped ^{4}He droplet using He-TDDFT. We assess the importance and nature of quantum effects by comparing a classical and a quantum treatment of the potassium atom in the simulation of the absorption spectrum. In the case of the 5s ^{2}S ← 4s ^{2}S absorption spectrum the difference is significant, hence we use a quantum description for the ensuing K atom dynamics and compare it to the classical one. In the 4p ^{2}P ← 4s ^{2}S absorption spectrum the difference is small, hence we go on to study 4p ← 4s photoexcitation of K on a helium nanodroplet using only a classical description of the K atom. The paper is organized as follows. Section 2 recalls the essential features of the quantum and classical trajectory methods in combination with He-TDDFT. The results are presented and discussed in Section 3 for 5s ← 4s, and Section 4 for the 4p ← 4s absorption spectrum and photodissociation dynamics. Finally, Section 5 is devoted to conclusions.

(1) |

The static solution is obtained by solving eqn (1) for Ψ(r) using the imaginary time propagation method.^{16}

Minimizing the action in order to describe the dynamics leads to the following coupled equations:

(2) |

(3) |

Eqn (2) is a time-dependent Schrödinger-like equation for the helium pseudo wave function Ψ (now a complex function such that ρ = |Ψ|^{2}) in an external field created by the interaction with K, while eqn (3) is the classical equation of motion for K in the mean field of the helium density.

For excitation to the ^{2}P states an additional degree of freedom is required to describe the electronic state of the potassium atom. It is described as a time-dependent wave packet |λ〉 which is expanded in the basis set of p orbitals of K as

(4) |

|p_{0},s〉 = |p_{z},s〉, s = ± stands for ±1/2 | (5) |

(6) |

The time evolution of |λ〉 is governed by the electronic Hamiltonian H_{el} written as

H_{el} = H_{DIM} + H_{SO} | (7) |

In this equation H_{DIM} is the diatomics in molecules (DIM)^{18} Hamiltonian which describes the potential interaction energy of the excited np potassium atom and H_{SO} the spin–orbit Hamiltonian taken as the atomic one.^{19}

There are 3 coupled equations to solve, since one equation has to be added to describe the time evolution of the electronic wave packet |λ〉.

(8) |

(9) |

(10) |

In eqn (8) and (9), V^{λ}_{K–He} = 〈λ|H_{el}|λ〉 is the (6 × 6) electronic Hamiltonian averaged over the electronic wave packet.

The coupled equations are solved by simultaneous time-propagation of Ψ(r,t) on a Cartesian grid [eqn (2) or (8)], of r_{K} [eqn (3) or (9)], and of the electronic wave packet coefficients λ_{is} of eqn (4) using eqn (10). The reader interested in getting more details about the method and its achievements can find them in ref. 7.

(11) |

(12) |

Upon 5s excitation, the potassium atom is ejected from the droplet with a high velocity (about 4 Å ps^{−1} = 400 m s^{−1}, see Fig. 4). The resulting fast oscillations in the potassium wave packet would require a very fine spatial grid for the time propagation, which is computationally unaffordable. We use the Bohmian quantum trajectories^{20} as in the work of Hernando et al.^{12} in order to circumvent this problem.

The K atom wave function ϕ is written in the polar form as ϕ(r_{K},t) ≙ φ(r_{K},t)e^{i(rK,t)/ħ}, where φ(r_{K},t) = |ϕ(r_{K},t)| is the probability density and the phase of ϕ. Introducing this form in the time-dependent Schrödinger equation leads to

(13) |

(14) |

Eqn (14) is then solved by writing the density as a histogram of M (fictitious) test-particles with trajectories {R_{i}(t)}^{M}_{i=1} such that R_{i}(t) = R(r_{Ki},t) and R_{i}(0) = r_{Ki}

(15) |

(16) |

The continuity equation, eqn (13), is automatically fulfilled if Ṙ_{i}(t) = v[R_{i}(t)] (the time-derivative of the test-particle position is equal to the value of the velocity field at that position). The equation for the time-propagation of the test-particle positions is obtained by taking the gradient of eqn (14) and rewriting it in the Lagrangian reference frame (d/dt = ∂/∂t + v·∇_{rK}). This gives the so-called quantum Newton's equation

m_{K}_{i}(t) = −∇[(r_{K},t) + V(r_{K},t)]_{|rK=Ri(t)} | (17) |

All the observables can then be obtained by averaging over the M test-particles or by using histograms.^{7,16}

Fig. 1 Left plot: He–K interaction potential for K(4s)^{23} and (5s).^{24} Right plot: Corresponding He_{1000}–K interaction potentials for the equilibrium helium density of the ground electronic state 4s. The ground state wave function for the quantum treatment of K is also represented. The vertical dotted lines help visualize the potential minima in the ground electronic state and the horizontal ones the asymptotic dissociation energies. |

The He–K interaction in the excited 5s electronic state looks unusual, as can be seen in the top left plot of Fig. 1. The usual repulsive behavior in the Franck–Condon region turns into an attractive one at short range and then repulsive again. This can be due to a mixing with a higher, attractive state of Σ symmetry (the 3d Σ interaction is a good candidate since the atomic 3d levels are only about 510 cm^{−1} above the 5s one^{26}). More likely, as proposed by Kleimenov et al.^{27} for Ne(^{1}S_{0})–Ne(4p′) interaction based on an explanation by Lipson and Field,^{28} it is due to a repulsive interaction between the Rydberg 5s electron and the neutral He atom around 7 Å. It is followed by an attractive interaction between the ionic core of the Rydberg atom and He, which produced a well 186 cm^{−1} (268 K) deep at 2.9 Å. The two regions are separated by a barrier with an energy maximum of 88.8 cm^{−1} (127.7 K), i.e. 275 cm^{−1} (396 K) above the potential minimum, at 6.6 Å (in the region of the maximum density for the 5s electron). There is still a very shallow and flat van der Waals well of 0.23 cm^{−1} at 15.7 Å. The Franck–Condon region is still in the repulsive part, so that a continuous dissociative absorption spectrum is expected.

In the interaction of the 5s excited K with a He_{1000} droplet there is no sign of the well and barrier of the He–K(5s) curve (the right plot of Fig. 1). The potential energy curve is repulsive everywhere, except for a shallow van der Waals attraction of 13.1 cm^{−1} (18 K) at long distance (around 34.8 Å) corresponding to the shallow and flat van der Waals well of the pair potential from the dimple (at about 20 Å) to the K(5s) atom. The only reminiscence of the well in the 5s pair potential is a decrease followed by an increase of the slope of the repulsive well between 22200 and 22900 cm^{−1} (32000 and 33000 K), out of scale for the figure.

Fig. 3 K@He_{1000} 5s ← 4s absorption spectrum for classical (blue line, maximum at ∼21758 cm^{−1}) or quantum (red line, maximum at ∼ 21592 cm^{−1}) treatment of the potassium atom. The atomic absorption line is represented by a vertical line at 21026.55 cm^{−1}.^{26} |

There is no published experimental spectrum for K (5s ← 4s) transition. There are comparisons between the experimental and simulated spectra using a similar He-DFT technique but a slightly different sampling for Na and Li (n + 1)s ← ns spectra.^{12} They show a good agreement for the spectral width, with a simulated shift in the correct direction but too small. This was attributed to the fact that the shift is droplet size dependent. Since the simulation was conducted for a 1000-atom droplet whereas the average experimental size was estimated to be ∼6100, the calculated shift was smaller. The same kind of agreement can be expected if the experiment is performed for K.

Fig. 6 4p ← 4s photoabsorption spectrum for K–He_{1000}: comparison between the ^{4}He-DFT simulation with the potassium atom treated classically (blue line) or quantum mechanically (red line) and the experimental spectrum (green line).^{8} The atomic lines for ^{2}P_{1/2} (12985.185724 cm^{−1}) and ^{2}P_{3/2} (13042.896027 cm^{−1})^{27} are represented as vertical lines. |

The 4p ← 4s photoabsorption spectrum of K on a helium droplet has already been investigated using path integral Monte Carlo calculations for a ^{4}He_{300} droplet at 0.5 K,^{13} obtaining a moderate agreement with experiment. The differences with our results can be attributed to the different pair-potentials used, in addition to the different method and the smaller cluster size. The He-DFT simulation of the 4p ← 4s photoabsorption spectrum of K on a ^{4}He_{1000} nanodroplet has already been published using a classical treatment for K.^{10} The spectrum calculated here gives a slightly better agreement with experiment, thanks to the sampling used for the He_{1000} configurations (instead of a simple, vertical calculation from the equilibrium configuration, convoluted with a Gaussian profile with a full width at half maximum of 5 cm^{−1}). The agreement would be even better if ^{2}Π_{1/2} and ^{2}Π_{3/2} were slightly more repulsive and ^{2}Σ_{1/2} was slightly less repulsive.

Fig. 7 Dynamics following 4p ^{2}Σ_{1/2} ← 4s excitation of K on a He_{1000} droplet: Top plot: Distance between K and the He_{N} center of mass d(K–He_{1000}) (green curve, left axis) and K-atom velocity (brown curve, right axis) as a function of time. Bottom plot: Square modulus of the projection of the electronic wave packet on the basis functions |p_{0}, +〉 and |p_{1}, −〉 [eqn (5) and (6)] as a function of time. |

Experimental work conducted upon ^{2}Π_{1/2} excitation of K attached to a helium nanodroplet have revealed other interesting features. Dispersed fluorescence experiments upon ^{2}Π_{1/2} excitation conducted by Stienkemeier et al.^{8} have revealed atomic emission at 13001 cm^{−1} due to dissociated K atoms, and red shifted, broad band emission with some maxima. The latter was attributed to the deeper well in the excited electronic state, which produces an increase of the He density in the region of the nodal plane of the p orbital. Reho et al.^{9} have assigned it to fluorescence from He_{n}K* exciplexes, mostly with n = 1, with the most red-shifted part due to higher values of n. In a following publication, Reho et al.^{11} have analyzed the time-resolved fluorescence signal and concluded that an exciplex could be formed by tunneling through the barrier observed in the diatomic curve.

Inspection of the diatomic curves in Fig. 5 reveals that the Franck–Condon region could extend towards the top of the barrier. Hence we examined the dynamics that would result from an initial position of the K atom shifted by ±0.5 Å. This value corresponds to a potential energy increase of about 0.4 K, the temperature of the helium droplets. It is also about the half width of the bound wave function of the potassium atom on the nanodroplet, represented in the 4s curve on the right hand side of Fig. 1.

When the initial position is shifted by +0.5 Å, the dynamics is not qualitatively affected. The situation is quite different when it is shifted by −0.5 Å. As shown in the top left plot of Fig. 9, the K atom is first pushed away to a new position at ∼27.5 Å in a few picoseconds. This is due to the formation of an exciplex which remains bound to the droplet, as illustrated in the top right plot of Fig. 9. As can be seen in the bottom left plot of that figure, the corresponding electronic wave packet is then 100% |p_{1}, −〉, the so-called “apple-shaped” orbital,^{31–33} corresponding to the well in the ^{2}Π_{1/2} diatomic interaction potential curve in the left plot of Fig. 5. The exciplex lasts for about 12 ps. After that, there is a sudden change in character for the electronic wave packet. As shown in the bottom left plot of Fig. 9, it suddenly turns from |p_{1}, −〉 to |p_{y}, −〉 (with ). This sudden symmetry breaking is accompanied by a shift of the K atom position further away from the droplet, while it forms a ring-type exciplex around the node of the p_{y} orbital as illustrated in the bottom right plot of Fig. 9. Finally, after ∼25 ps the extra density between the droplet and the newly formed exciplex starts relaxing and the bound exciplex slowly tends to its equilibrium distance (top left plot in Fig. 9). The asymptotic (t = 30 ps) integrated helium density forming the exciplex around the K atom is then about 5.8, while it was about 0.4 (and not completely formed) just before symmetry breaking occurred (t = 7.5 ps). This is consistent with the results of Takayanagi and Shiga,^{14} who found that stable exciplexes could be formed with up to 6 helium atoms on the lowest electronic state of K(4p), while the second highest state could only accommodate 2 helium atoms.

The red-shifted fluorescence emission with broad peaks observed upon ^{2}Π_{1/2} excitation^{8} was assigned to exciplex fluorescence. Here we conclude to the possible formation of exciplexes, which would remain bound to the droplet. In this case broad band, red shifted fluorescence would also be expected. The time-resolved study detecting fragments in a mass spectrometer^{34} seems to show a detachment from the droplet, but some doubts have been raised as to their interpretation in terms of exciplex formation times.^{35}

The use of 3D Cartesian coordinates to describe the helium density might render the desorption process not strictly cylindrically symmetric, hence the symmetry breaking. However, this is not necessarily a defect: it reveals changes in the dissociation dynamics that could occur under any kind of fluctuations, be it of thermal or of quantum origin. In this case, it amounts to spin–orbit relaxation. The different behavior from that of heavier alkalis may be due the higher spin–orbit splitting. For potassium it is only 57.7 cm^{−1}. Spin–orbit relaxation added to the He-DFT simulation has been shown to lead to the dissociation of an exciplex which would otherwise remain bound, in the case of ^{2}Π_{3/2} excitation of Rb attached to a He_{1000} droplet.^{37} Here spin–orbit relaxation occurs spontaneously upon symmetry breaking, and the effect is to bind a ring exciplex whereas the K atom would otherwise dissociate as a simple exciplex. The difference is that the kinetic energy gained by the K atom upon spin–orbit relaxation is not enough to compensate for the added binding energy obtained from switching to a ring exciplex.

For the 5s ← 4s excitation the difference between the spectra was significant. Hence we have used both a quantum and a classical description of the K atom in the ensuing dynamics. Dissociation is fast because the He–K interaction in the 5s state is very repulsive in the Franck–Condon region. The final K-atom velocity is found to be smaller in the quantum treatment than in the classical one for low photon excess energies, as expected from the larger equilibrium distance from the droplet in the ground state.

Given the rather small difference between the 4p ← 4s spectra simulated using a quantum or a classical description of the K atom, we have studied the subsequent dynamics using only a classical treatment for K. Excitation to the highest 4p state, the ^{2}Σ_{1/2} state, gives an expected fast, direct dissociation since He–K interaction is repulsive in that state. Excitation to the 4p ^{2}Π_{1/2} state leads to the bouncing of the K atom at a larger distance from the droplet compared to the ground electronic state distance. However, we show that a small displacement of the initial K atom position such as the one that could be induced by thermal excitation at 0.4 K or from quantum delocalization can affect the dynamics. It can lead to the formation of a bound, linear exciplex corresponding to a π, apple-shaped orbital. In turn, fluctuations can then make it switch to a ring-shaped one.

The influence of helium density fluctuations is also important in the excitation to the 4p ^{2}Π_{3/2} state. In this case a linear exciplex is formed within the first few picoseconds and starts dissociating from the droplet. The electronic state of the potassium atom is expected to remain fixed (|p_{1}, +〉) if cylindrical symmetry is strictly maintained. However, small fluctuations can break this symmetry, and the excited |p_{1}, +〉 apple-shaped orbital suddenly switches to a |p_{y}, +〉 dumbbell orbital: in this case the exciplex becomes ring-shaped and remains bound at a larger distance from the surface than in the ground electronic state. If the electronic state is fixed so as to correspond to strict cylindrical symmetry, then the K exciplex remains linear and leaves the droplet. The symmetry-breaking fluctuation thus spontaneously induces spin–orbit relaxation, and the effect is to bind a ring exciplex whereas the K atom would otherwise dissociate as a simple exciplex.

Symmetry breaking for both ^{2}Π_{3/2} and ^{2}Π_{1/2} states can be explained as follows. The K atom spin–orbit coupling is weak enough that when helium gets close to it the two Π states are quasi-degenerate (the energy difference between the Σ and Π potential energy curves being much larger than the spin–orbit splitting, see the region of the well in the left plot of Fig. 5). Hence the ^{2}Π_{3/2} and ^{2}Π_{1/2} states can easily mix to form an electronic state corresponding to a p_{y} orbital. Once this is done a ring exciplex starts building up and the process is irreversible, the ring exciplex being much more bound because it involves more helium density in the Π well. Hence although we cannot disregard that the appearance of this symmetry breaking is eased by the finite accuracy of the method, it is rooted on a real physical ground.

Symmetry breaking has also been proposed to induce spin–orbit relaxation for Ba^{+} in helium exciplexes.^{38} The behavior of the potassium atom differs from that of heavier alkalis, like the one recently observed for Rb upon ^{2}Π_{3/2} excitation in which combining spin–orbit relaxation to He-DFT simulations allowed reproduction of the experimental observation that an exciplex could dissociate.^{37} The difference in behavior is attributed to the smaller spin–orbit splitting. It would be very interesting to confirm this explanation by running simulations for ^{2}Π_{3/2} excitation on lighter alkalis (Li and Na) attached to helium nanodroplets. They require a quantum description of the alkali atom dynamics, which is currently underway.

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