4s to 5s and 4p photoexcitation dynamics of K atoms from the surface of helium nanodroplets: a theoretical study

Maxime Martinez a, François Coppens a, Manuel Barranco abc, Nadine Halberstadt *a and Martí Pi bc
aLaboratoire Collisions Agrégats Réactivité (LCAR), IRSAMC, Université de Toulouse, CNRS UMR 5589, Toulouse, France. E-mail: Nadine.Halberstadt@irsamc.ups-tlse.fr
bDepartament FQA, Facultat de Física, Universitat de Barcelona. Diagonal 645, 08028 Barcelona, Spain
cInstitute 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 2S or 4p 2P 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 2S ← 4s 2S excitation the difference is rather large, and we use a quantum description for the ensuing direct dissociation dynamics. In the case of the 4p 2P ← 4s 2S absorption spectrum, the difference is much smaller, hence a classical description of K is used to describe 4p 2P excitation dynamics. Excitation to the 4p 2Σ1/2 state leads to the direct dissociation of the K atom, while the 4p 2Π3/2 state initially leads to the formation of an exciplex and the 4p 2Π1/2 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 2P3/2 state and to the formation of a bound, ring excimer for the 2Π1/2 state.

1 Introduction

Helium nanodroplets exhibit many fascinating properties, ranging from superfluidity, very low temperature (∼0.4 K in the usual experimental conditions), high heat conductivity, to the chemical neutrality and the ability to pickup and solvate any atom or molecule that can be put in the gas phase.1 Most of these dopants tend to solvate near the center of the droplets,2 except alkali and the heavier alkali-earth atoms Sr and Ba. Because of the electron-helium repulsion, the extended and diffuse valence orbital of alkali atoms only allows for stabilization in a dimple at or near the droplet surface. Upon photoexcitation, the valence orbital becomes even more diffusive and the alkali atom usually desorbs, either bare or as an exciplex (a complex formed with the excited atom and one or a few helium atoms), with the notable exception of Rb and Cs excited to the low energy range of the np 2Π1/2 state.3–6 This process has been extensively studied, both experimentally and theoretically.7

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 2Π1/2, 2Π3/2 pseudo-diatomic states on top of the 4p 2P1/2, 2P3/2 ← 4s 2S1/2 (D1, D2) atomic lines, and a shoulder corresponding to absorption to the 4p 2Σ1/2 pseudo-diatomic state to the blue (higher photon energies) of the spectrum. Preliminary dispersed fluorescence spectra8 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 13[thin space (1/6-em)]046 cm−1 (close to the 4p 2P3/2 atomic line) than for excitation at 13[thin space (1/6-em)]001 cm−1 (close to the 4p 2P1/2 atomic line).

Dispersed fluorescence spectra upon excitation near the D1 or D2 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 D1 and D2 lead to population in both the 2Π1/2 and 2Π3/2 bound levels, which implied 2Π3/2 to 2Π1/2 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 2Π1/2; +11 cm−1 for 2Π3/2; and +8 cm−1 for 2Σ1/2. Excitation to the 2Σ1/2 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 2Π3/2 absorption.

By monitoring the time evolution of the fluorescence following 4p 2P ← 4s 2S 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 D1 line; and a rise time of 50 ps when exciting at the frequency of the D2 line. The results were interpreted as different formation mechanisms for the K*–He excimer. Exciplexes can be formed directly in the 2Π3/2 state whereas they have to tunnel through a barrier in the 2Π1/2 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 Yamashita13 have calculated the structure and simulated the 4p 2P ← 4s 2S 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-He300 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 Shiga14 have studied the 4p 2P ← 4s 2S 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 2Σ1/2 and 15–30 ps for 2Π1/2 or 2Π3/2 excitation. In the adiabatic dynamics excitation to 2Σ1/2 or to 2Π1/2 was found to lead to desorption of K atoms. Only in the excitation to the 2Π3/2 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 4He 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 2S ← 4s 2S 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 2P ← 4s 2S 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.

2 Method

Due to their inherently quantum behavior and large number of atoms, the theoretical description of the dynamics of doped helium nanodroplets is a real challenge. During the past decade, density functional theory (He-DFT) and its time dependent counterpart (He-TDDFT) have emerged as the best compromise between accuracy and the ability to treat a large number of atoms.7 The version used here is that of the 4He-DFT-BCN-TLS code.15 Details on the method can be found in ref. 16. We recall here the essential features to help understand the differences between quantum and classical treatment of the potassium atom.

2.1 Generalities on He-DFT and He-TDDFT

Within He-DFT, all the properties are fully determined by the helium density ρ(r).17 The dopant-helium system is described using the mean-field approximation. The equilibrium characteristics and other time-independent properties are obtained by minimizing the total energy functional with respect to image file: c8cp05253k-t1.tif (pseudo-wave function or order parameter). The dynamics is determined by solving equations obtained by minimizing the action of the system with respect to Ψ(r,t). The resulting equations differ depending on the treatment (classical or quantum) of K.

2.2 Classical treatment of K

If the K atom is treated classically, minimizing E[ρ] with respect to the effective wave function image file: c8cp05253k-t2.tif leads to the Schrödinger-like equation
image file: c8cp05253k-t3.tif(1)
where mHe4 is the atomic mass of 4He; [scr E, script letter E]c is the helium density functional; V4sK–He is the K–He pair potential interaction for K in the ground electronic state 4s 2S1/2; and μ is the chemical potential.

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:

image file: c8cp05253k-t4.tif(2)
image file: c8cp05253k-t5.tif(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 2P 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

image file: c8cp05253k-t6.tif(4)
where s stands for the electronic spin. The following electronic basis set is also used for analysis purposes:
|p0,s〉 = |pz,s〉, s = ± stands for ±1/2(5)
image file: c8cp05253k-t7.tif(6)

The time evolution of |λ〉 is governed by the electronic Hamiltonian Hel written as

Hel = HDIM + HSO(7)

In this equation HDIM is the diatomics in molecules (DIM)18 Hamiltonian which describes the potential interaction energy of the excited np potassium atom and HSO 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〉.

image file: c8cp05253k-t8.tif(8)
image file: c8cp05253k-t9.tif(9)
image file: c8cp05253k-t10.tif(10)

In eqn (8) and (9), VλK–He = 〈λ|Hel|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 rK [eqn (3) or (9)], and of the electronic wave packet coefficients λis 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.

2.3 Quantum treatment of K

In order to treat the potassium atom quantum mechanically, we introduce its wave function ϕ(rK). Minimizing the total energy functional with respect to Ψ and ϕ gives two coupled equations for the static properties
image file: c8cp05253k-t11.tif(11)
image file: c8cp05253k-t12.tif(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 trajectories20 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(rK,t) ≙ φ(rK,t)ei[scr S, script letter S](rK,t)/ħ, where φ(rK,t) = |ϕ(rK,t)| is the probability density and [scr S, script letter S] the phase of ϕ. Introducing this form in the time-dependent Schrödinger equation leads to

image file: c8cp05253k-t13.tif(13)
image file: c8cp05253k-t14.tif(14)
where we have introduced v ≙ (1/m)∇rK[scr S, script letter S], so that the current density j(rK,t) is simply j = φ2v. In eqn (14) [scr Q, script letter Q](rK,t) is the so-called quantum potential.

Eqn (14) is then solved by writing the density as a histogram of M (fictitious) test-particles with trajectories {Ri(t)}Mi=1 such that Ri(t) = R(rKi,t) and Ri(0) = rKi

image file: c8cp05253k-t15.tif(15)
image file: c8cp05253k-t16.tif(16)

The continuity equation, eqn (13), is automatically fulfilled if i(t) = v[Ri(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

mK[R with combining umlaut]i(t) = −∇[[scr Q, script letter Q](rK,t) + V(rK,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

2.4 Spectrum calculation

The 4p ← 4s spectrum of a potassium attached to helium nanodroplets has been measured in several experiments, as already mentioned in the Introduction. Even the 5s 2S1/2 ← 4s 2S1/2 absorption spectrum could in principle be measured experimentally, since the weak perturbation induced by the helium environment on the potassium electronic states induces a small transition dipole moment.12 Hence the absorption spectra constitute an important observable to test the model and the potential energy curves involved. Born–Oppenheimer separation of nuclear and electronic motion is assumed, and the dependence of the transition dipole moment on the nuclear coordinates is neglected (Franck–Condon approximation). We then follow the same method as Mateo et al.21 Briefly, the kinetic energy is neglected compared to the potential energy in the excited state. The spectrum is obtained as the histogram of the interaction potential energy difference between the excited and the ground state for M atomic-like He1000 configurations. If K is treated quantum mechanically, the sampling also applies to its wave function.

2.5 Excitation process

In order to define the initial conditions for the dynamics, electronic excitation of the K atom is described as an instantaneous process. This corresponds to femtosecond laser pulse excitation in which the laser pulse is short enough that the nuclei do not have time to move, while long enough that the associated frequency width only covers the target electronic transition. Hence for the photodissociation dynamics the system is initially at equilibrium in its ground electronic state, and at time zero it is taken to the excited state by simply changing the 4s K–He pair potential interaction to the one for K in the excited electronic state. In other words, the electronic state instantly changes while the helium density and the potassium atom remain frozen. In the case of 4p excitation one dynamics is run for each of the three electronic states resulting from diagonalization of the electronic Hamiltonian at time zero (in increasing energy order: 2Π1/2, 2Π3/2, 2Σ1/2). The initial conditions for the dynamics are then the helium density and the potassium position (classical) or wave function (quantum) resulting from the static configuration. In the classical description for K, its initial velocity is set to zero.

2.6 Numerical details

Both the quantum and the classical dynamics have been solved using the 4He-DFT BCN-TLS code15 with the so-called solid density version of the functional.22 As explained in ref. 7, the solid density functional is the version of the Orsay–Trento functional that is best adapted to treat very attractive He-dopant interactions such as the ones encountered in the p Π excited states of alkalis. The He–K(4s) pair potential is taken from Patil23 and the He–K(5s) and He–K(4p) Π and Σ pair potentials from Pascale.24 The number of test particles used in the quantum treatment of K was M = 106.

3 Results for 5s ← 4s photoexcitation

3.1 Potentials

The He–K interaction in the ground electronic state is only slightly attractive, as can be seen from the He–K(4s) potential energy curve in the left plot of Fig. 1. The well depth is 1.0 cm−1 (1.4 K) at a distance of 7.2 Å. Because of this weak attraction, the potassium atom does not get solvated inside the droplet. Like all the other alkalis it sits in a dimple near the surface, as already determined in e.g.ref. 25. As can be seen from the lower right plot in Fig. 1, its binding energy is 9.7 cm−1 (13.9 K) at 26.3 Å from the helium center of mass (about 6.7 Å from the droplet surface).
image file: c8cp05253k-f1.tif
Fig. 1 Left plot: He–K interaction potential for K(4s)23 and (5s).24 Right plot: Corresponding He1000–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 one26). More likely, as proposed by Kleimenov et al.27 for Ne(1S0)–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 He1000 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 22[thin space (1/6-em)]200 and 22[thin space (1/6-em)]900 cm−1 (32[thin space (1/6-em)]000 and 33[thin space (1/6-em)]000 K), out of scale for the figure.

3.2 Static calculations

The equilibrium structures resulting from the solution of eqn (1) or eqn (11) and (12) by imaginary time propagation are displayed in Fig. 2 for the classical and quantum description of the potassium atom. Due to the zero point motion of the impurity atom, the expected position of K is further away from the center of mass of the He1000 droplet in the quantum case: 26.4 Å instead of 25.8 Å for the classical equilibrium position. In addition, the surface profile is smoother in the quantum case compared to the classical one because of the convolution of the K-droplet interaction with the K atom wave function: it decays slightly less abruptly to zero and the oscillations are somewhat damped.
image file: c8cp05253k-f2.tif
Fig. 2 He density profiles for a classical (blue lines) and quantum (red lines) description of K. In the case of the quantum description the wave function of K is represented, while in the classical description its position is represented by a straight vertical line. The origin is at the center of mass of the He1000 droplet.

3.3 Absorption spectrum

The He1000K (5s ← 4s) absorption spectra obtained as described in Section 2.4 for a classical or a quantum treatment of the potassium atom are displayed in Fig. 3. The K (5s ← 4s) atomic line is also represented for reference. Both spectra are shifted to higher frequencies, as expected from the repulsive K–He1000 interaction in the 5s state (Fig. 1, right plot). The blue shift is smaller when the K atom is treated quantum mechanically (570 cm−1) than when it is treated classically (730 cm−1), and the width of the absorption band is smaller: full width at half maximum (FWHM) = 200 cm−1 compared to 230 cm−1. This is a consequence of the reflection principle29,30 for direct photodissociation. Since the K atom is further away from the droplet in the quantum description, Franck–Condon (vertical) excitation from the equilibrium position reaches the excited potential energy curve at a larger distance, hence at a lower energy and with a smaller gradient than in the classical case. Due to these differences, the photodissociation dynamics upon 5s excitation is treated using a quantum test particle description for the K atom, together with a classical one for comparison.
image file: c8cp05253k-f3.tif
Fig. 3 K@He1000 5s ← 4s absorption spectrum for classical (blue line, maximum at ∼21[thin space (1/6-em)]758 cm−1) or quantum (red line, maximum at ∼ 21[thin space (1/6-em)]592 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.

3.4 Photodissociation dynamics

Upon excitation to the 5s state the potassium atom dissociates very fast (see Fig. 4), as expected from the repulsive interaction in that electronic state. This is true both for the classical and the quantum description of K. The only difference is that in the classical treatment the K atom feels a stronger acceleration, because it is initially closer to the droplet, so that the final velocity is 480 m s−1 (4.8 Å ps−1) compared to 420 m s−1 (4.2 Å ps−1) for the quantum description of K.
image file: c8cp05253k-f4.tif
Fig. 4 K distance to the center of mass of the He1000 droplet, d(K–He1000) (top), and corresponding velocity (bottom) as a function of time following K(5s ← 4s) photoexcitation at time t = 0. The potassium atom is treated classically (blue lines) or quantum mechanically (red lines).

4 Results for 4p ← 4s photoexcitation

4.1 Potentials

The pair potentials for He–K interactions in the 4p excited electronic states are presented in Fig. 5 (left plot), as well as the corresponding 4He1000–K interaction potentials for helium density frozen at the 4He1000K(4s) equilibrium density (right plot). The ground electronic state potentials are also represented (on a different energy scale) in order to help visualize the Franck–Condon region: the ground state equilibrium distance for K–He (7.2 Å) or K–He1000 (26.3 Å) is materialized as a vertical line. The pair potentials are obtained from Pascale's ab initio 4p 2Σ and 2Π curves (depending on whether the orientation of the 4p K orbital is parallel or perpendicular to the interatomic axis), to which the spin–orbit Hamiltonian approximated as the isolated K spin–orbit Hamiltonian is added before diagonalization. The resulting 2Σ1/2 curve is repulsive, except for a shallow van der Waals well (0.4 cm−1 or 0.6 K at 10.8 Å). Both the 2Π3/2 and the 2Π1/2 curves exhibit a well at short distance, 2.80 Å, slightly deeper for 2Π3/2 (245 cm−1 or 352 K) than for 2Π1/2 (226 cm−1 or 325 K). In the case of the 2Π1/2 curve, this well is separated from the van der Waals well (0.4 cm−1 or 0.6 K at 10.2 Å) by a barrier with maximum 5.7 cm−1 (8.2 K) above the dissociation limit at 6.7 Å, whereas there is no barrier in the 2Π3/2 curve. Averaging over the helium equilibrium density is performed using the diatomics in molecules (DIM) model. The resulting 2Π curves have much shallower wells (6.4 cm−1 or 9.2 K at 29 Å for 2Π1/2, 9.1 cm−1 or 13.2 K at 28 Å for 2Π3/2) and the 2Σ curve has a van der Waals well (5.9 cm−1 or 8.5 K at 29.7 Å). All three curves display a repulsive behavior in the Franck–Condon region. The degree of repulsion can be quantified by the amount of energy with respect to dissociation at a ground state equilibrium distance of 26.3 Å for the ground equilibrium helium density. This is only an indicator, since the helium droplet can exchange energy with the K atom, but it helps quantify how impulsive the dissociation will be. With this indicator, the most impulsive process will occur upon excitation to the 2Σ curve (118 cm−1 above the dissociation limit at 26.3 Å) and the least repulsive one to the 2Π3/2 curve (7 cm−1), the behavior upon excitation to 2Π1/2 (21 cm−1) being intermediate.
image file: c8cp05253k-f5.tif
Fig. 5 Potential energy curves for potassium interaction with a 1000-atom helium droplet. Left plot: Ground (bottom part) 4s and excited (top part) 2Π1/2, 2Π3/2 and 2Σ1/2 (in increasing energy order) K–He pair potentials; Right plot: Corresponding 4He1000–K interaction potentials for helium density frozen at the 4He1000K(4s) equilibrium density.

4.2 Absorption spectrum

The 4p ← 4s photoabsorption spectrum of K on a 4He1000 droplet is shown in Fig. 6 for a quantum and a classical treatment of the K atom, in comparison with the experimental spectrum (average droplet size 1700).8 Both treatments give a good agreement with experiment, the quantum treatment giving a slightly smaller blue shift (with respect to the atomic lines) and narrower peaks than the classical one, as was the case for the 5s ← 4s spectrum, and for the same reason. However, the difference is rather small, with the classical treatment being closer to experiment. Hence the K atom dynamics is treated only classically for all three 4p ← 4s excitations.
image file: c8cp05253k-f6.tif
Fig. 6 4p ← 4s photoabsorption spectrum for K–He1000: comparison between the 4He-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 2P1/2 (12985.185724 cm−1) and 2P3/2 (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 4He300 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 4He1000 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 He1000 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 2Π1/2 and 2Π3/2 were slightly more repulsive and 2Σ1/2 was slightly less repulsive.

4.3 Dynamics upon 4p 2Σ1/2 excitation

The dynamics following 4p 2Σ1/2 ← 4s excitation is presented in Fig. 7. It is quite impulsive, as could be expected from the shape of the 2Σ1/2 potential energy curve in the Franck–Condon region (Fig. 5). The potassium atom has traveled 10 Å from its initial position in about 10 ps, time at which it has almost reached its final velocity (∼ 100 m s−1). As shown in the bottom plot, the wave packet evolves from almost 100% p0 (pz) at short distances to the asymptotic eigenvalue of the corresponding 2P3/2 atomic state (2/3p0 and 1/3p1).
image file: c8cp05253k-f7.tif
Fig. 7 Dynamics following 4p 2Σ1/2 ← 4s excitation of K on a He1000 droplet: Top plot: Distance between K and the HeN center of mass d(K–He1000) (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 |p0, +〉 and |p1, −〉 [eqn (5) and (6)] as a function of time.

4.4 Dynamics upon 4p 2Π1/2 excitation

Upon 4p 2Π1/2 ← 4s excitation, the dynamics is different from the simple, impulsive detachment observed for 4p 2Σ1/2 excitation, as can be observed in Fig. 8. Initially the K atom starts dissociating, albeit more slowly than in the 2Σ1/2 case, as can be seen in the top plot of Fig. 8. But after about 50 ps, when it has travelled about 12 Å, it turns around and starts oscillating around a new equilibrium position some 4 Å further outwards from its initial position. This corresponds to the shallow, van der Waals well in the 2Π1/2 curve which is visible in the right plot of Fig. 5. Hence the K atom is trapped but does not form an exciplex and ends up oscillating above the helium droplet surface at a distance which depends on the van der Waals well and on the surface oscillations due to the nanodroplet excitation upon energy transfer from the K atom (∼25 cm−1). This is illustrated by the snapshot at the bottom of Fig. 8, which shows the helium density and the K atom position at t = 182 ps. The droplet is clearly distorted by the internal multimode excitation resulting from energy transfer during the K atom dynamics.
image file: c8cp05253k-f8.tif
Fig. 8 Dynamics following 4p 2Π1/2 ← 4s excitation of K on a He1000 droplet: Top plot: Distance between K and the HeN center of mass d(K–He1000) as a function of time. Middle plot: Square modulus of the projection of the electronic wave packet on the basis functions |p0, +〉 and |p1, −〉 as a function of time. Bottom plot: snapshot of the helium droplet density and the bouncing K atom at t = 182 ps.

Experimental work conducted upon 2Π1/2 excitation of K attached to a helium nanodroplet have revealed other interesting features. Dispersed fluorescence experiments upon 2Π1/2 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 HenK* 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% |p1, −〉, the so-called “apple-shaped” orbital,31–33 corresponding to the well in the 2Π1/2 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 |p1, −〉 to |py, −〉 (with image file: c8cp05253k-t17.tif). 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 py 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.

image file: c8cp05253k-f9.tif
Fig. 9 Dynamics following 4p 2Π1/2 ← 4s excitation of K on a He1000 droplet, with K displaced by −0.5 Å from its ground state equilibrium position. Left, top plot: Distance between K and the He1000 droplet center of mass as a function of time. Left, bottom plot: Square modulus of the projection of the electronic wave packet on the basis functions |p+1, −〉, |p−1, +〉, |px, −〉, |py, −〉 and |pz, +〉, as a function of time. Right: Snapshot of the HeN density and bound K exciplex before and after the symmetry breaking. Top right plot: t = 7.5 ps; bottom right: t = 19.5 ps.

The red-shifted fluorescence emission with broad peaks observed upon 2Π1/2 excitation8 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 spectrometer34 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

4.5 Dynamics upon 4p 2Π3/2 excitation

Exciting the 4p 2Π3/2 ← 4s transition leads without surprise to the formation of an excimer, since there is no barrier to prevent its formation, as can be seen in Fig. 5. It is formed within ∼4 ps and looks like the one observed at short times upon 4p 2Π1/2 excitation when the K position was shifted −0.5 Å from its 4s equilibrium position (top right plot of Fig. 9). As can be seen as the dotted line in the top plot of Fig. 10, the newly formed exciplex bounces once and then starts to leave the rest of the droplet. However, after ∼15 ps, it stabilizes at a new position, 5.5 Å further outwards from the initial atom position, and its subsequent evolution is very similar to that of the 2Π1/2 after the symmetry breaking. A closer look at the time evolution of the electronic wave packet shown in the middle plot of Fig. 10 reveals that something happens at t = 7.7 ps, very much like the symmetry breaking in Fig. 9. The wave packet which was initially almost purely |p+1, +〉 suddenly switches to more than 95% |py, +〉. If cylindrical symmetry was maintained, there is no reason for this transition to occur. In order to check that the stabilization of the exciplex at 5.5 Å from the droplet is due to this symmetry breaking, we have repeated the dynamics by fixing the electronic wave packet to remain in the |p+1, +〉 state. It is shown as a solid line in the top plot of Fig. 10. The linear exciplex (integrated density 1.7 helium at t = 25 ps) does not have time to turn into a ring exciplex and leaves the droplet, as illustrated in the bottom plot of Fig. 10. This is remarkable, since for the heavier alkalis (Rb and Cs) exciplexes formed in the 2Π3/2 state in a He-DFT simulation did not leave the droplet.36
image file: c8cp05253k-f10.tif
Fig. 10 Dynamics following 4p 2Π3/2 ← 4s excitation of K on a He1000 droplet. Top plot: Distance between K and HeN center of mass as a function of time: propagating the electronic wave packet as usual (dotted line) or fixing it as |p1, +〉 (solid line), see text. Middle plot: Square modulus of the projection of the electronic wave packet on the basis functions |p+1, +〉, |p−1, +〉, |px, +〉, and |py, +〉, as a function of time. Bottom plot: Snapshot of the helium droplet density and the dissociating exciplex at t = 18.5 ps.

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 2Π3/2 excitation of Rb attached to a He1000 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.

5 Conclusions

We have studied the 5s ← 4s and 4p ← 4s photodissociation of the potassium atom attached to a helium nanodroplet. In both cases we have tested a quantum and a classical description of the K atom in the spectrum calculation.

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 2Σ1/2 state, gives an expected fast, direct dissociation since He–K interaction is repulsive in that state. Excitation to the 4p 2Π1/2 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 2Π3/2 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 (|p1, +〉) if cylindrical symmetry is strictly maintained. However, small fluctuations can break this symmetry, and the excited |p1, +〉 apple-shaped orbital suddenly switches to a |py, +〉 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 2Π3/2 and 2Π1/2 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 2Π3/2 and 2Π1/2 states can easily mix to form an electronic state corresponding to a py 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 2Π3/2 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 2Π3/2 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.

Conflicts of interest

There are no conflicts to declare.


MB thanks the IDEX of the Université Fédérale Toulouse Midi-Pyrénées for financial support through the “Chaires d'Attractivité 2014” Programme IMDYNHE (IDEX). CALMIP (Grant P1039) is gratefully acknowledged for high performance computing resources. This work has been performed under Grant FIS2017-87801-P (AEI/FEDER, UE)


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