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Photoionization of He...Li: A Theoretical Study Sameh Saidi, Hamid Berriche, and Nadine Halberstadt J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b02428 • Publication Date (Web): 07 May 2015 Downloaded from http://pubs.acs.org on May 25, 2015

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The Journal of Physical Chemistry

Photoionization of He...Li2: A Theoretical Study Samah SAIDI,†,‡ Hamid BERRICHE,‡,¶ and Nadine HALBERSTADT∗,† LCAR-IRSAMC, Université Toulouse 3 - Paul Sabatier and CNRS, 31062 Toulouse, France, Laboratoire des Interfaces et Matériaux Avancés, Département de Physique, Faculté des Sciences, Université de Monastir, Avenue de l’Environnement, 5019 Monastir, Tunisia, and Department of Mathematics and Natural Sciences, School of Arts and Sciences, American University of Ras Al Khaimah, Ras Al Khaimah, UAE E-mail: [email protected]

∗

To whom correspondence should be addressed Université Toulouse 3 - Paul Sabatier and CNRS, 31062 Toulouse, France ‡ Laboratoire des Interfaces et Matériaux Avancés, Département de Physique, Faculté des Sciences, Université de Monastir, Avenue de l’Environnement, 5019 Monastir, Tunisia ¶ Department of Mathematics and Natural Sciences, School of Arts and Sciences, American University of Ras Al Khaimah, Ras Al Khaimah, UAE † LCAR-IRSAMC,

1 ACS Paragon Plus Environment


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Abstract Photoionization of the He...Li2 Van der Waals complex to the ground electronic state of the He...Li+ 2 ion is investigated theoretically. The photoionization cross section is computed using existing interaction potentials. Resonances are found on top of a structured continuum. They are assigned to vibrational predissociation of the ion by comparison with Fermi Golden Rule calculations. Because of the differences in potential energy surfaces between the neutral and ionic complexes, only the resonances corresponding to quasibound states with the highest excitation in the Van der Waals modes are visible.The other quasi bound states obtained in the Fermi Golden Rule calculations can give information on vibrational energy relaxation rates in other collisional processes involving the lithium dimer ion and a helium atom.


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1

Introduction

Van der Waals complexes built with a diatomic molecule and rare gas atoms are ideal model systems to study energy transfer. Because of the weakness of the Van der Waals bond, the diatomic molecule retains its spectroscopic characteristics and energy transfers are therefore easily identifiable. This has been extensively studied in a number of experiments and theoretical simulations. 1–3 Studying the ionization of these systems is interesting for several reasons. Interactions between the diatomic ion and rare gas atoms are more energetic than for the neutral diatom, giving access to a new energy range for studying energy transfer. At the same time the ionization potentials of the partners are different enough that the overall picture does not get mixed with charge transfer. Also, resonances appearing in the photoionization spectra can play an important role in collisional energy relaxation of the molecular ion with rare gases. Studying the photoionization spectrum of a diatom-rare gas complex can also contribute to the study of systems in which the diatomic is in interaction with a rare gas environment. This environment can be a cooling buffer gas, such as the ones used in ion traps or in Bose-Einstein condensates, or a rare gas matrix. They can also serve as model systems for studying energy exchange in real solvents. Studying helium complexes exhibit the additional interest that it is related to the study of molecules in helium nanodroplets. Because of their remarkable properties: quantum fluidity, low temperature and weak interactions with dopants, 4–6 superfluid helium nanodroplets are increasingly used as ultimate spectroscopic matrices, 7 in which to study the spectroscopy of various species using the powerful helium nanodroplet isolation (HENDI) spectroscopy. 4 The study of alkali dimers attached to helium clusters by laser induced fluorescence 8,9 has shown that they are located near the surface of the droplet, and that the helium environment only represents a weak perturbation to the electronic states of the molecular probe. This was confirmed by a Helium density functional simulation. 10 Ionization of Li2 can lead to its desorption from the droplet surface. In the similar case of alkali atom photodesorption, departing atoms have been shown to bring along one or a few helium atoms, forming exciplexes. Hence photoionization of Li2 on the surface of a helium droplet could eject ionic complexes formed with Li+ 2 and one or several helium atoms, 3 ACS Paragon Plus Environment


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in particular when Li2 is first excited to an intermediate electronic state, like in the experiment of Schlesinger et al. 11 If Li+ 2 is vibrationally excited, it can relax by evaporating helium atoms. This is related to the process we are studying in the present work. We focus here on the case where Li2 interacts with a single helium atom. This system is simple enough to be amenable to high quality ab initio calculations. In particular, very recent ab initio 14,15 They have been used calculations by Bodo et al. are available for He...Li2 12,13 and He...Li+ 2. 14,15 clusters, rotational 16 and vibrational 17,18 to study the stability of Li2 Hen 12,13 and Li+ 2 -Hen + 2 + 19 cooling of Li2 (1 Σ+ by ultracold collisions with helium. g ) and rotational cooling of Li2 ( Σg )

The ground-state PES of the He...Li+ 2 complex has a deep well in the linear configuration, while the neutral He...Li2 exhibits a shallow well in the T-shaped configuration. The purpose of this study is to investigate the photoionization of the He...Li2 Van der Waals complex to the ground electronic state of the He...Li+ 2 ion. Photoionization spectrum calculations are carried out in the energy-resolved formulation to obtain the photoionization cross section and final product state distributions. The structure appearing in the spectrum is analyzed in terms of vibrational predissociation of the He...Li+ 2 ion by comparing with Fermi golden rule simulations. The resonances identified could play an important role in other dynamical processes involving the dimer ion Li+ 2 and a helium atom, like collisional vibrational relaxation or ionization of He...Li2 from another electronic state. In particular, experimental investigations on atom or molecule photoexcitation on doped helium nanodroplets generally use photoionization to detect the products. Exciplexes formed with the excited atom and one or a few helium attached to it have been detected in experiments on the photo-excitation of alkali-atom doped helium droplets. 20–27 The corresponding exciplexes for Li2 would be characterized by the resonances identified in this work. This paper is organized as follows. Section II presents the methodology used for calculating the photoionization cross sections. Section III introduces the potential energy surfaces used in this work. Section IV presents the results of the photoionization spectrum calculations, which are discussed in Section V by comparing with a Fermi Golden Rule model. Finally, Section VI is devoted to the conclusions.


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2

Methodology

The primary goal of this study is to determine the dissociative photoionization cross section of the He...Li2 Van der Waals complex: hν He...Li2 −→ He + Li+ 2 (v, j)

(1)

The method used here is the same as the one used for studying the photoexcitation spectrum of Van der Waals complexes built with a dihalogen molecule and a rare gas atom. 28–30 The main aspects are summarized here for the sake of completeness. In the framework of the first order perturbation theory for electric dipole transitions, the partial cross section for photo-exciting the system from its initial bound state Ψi with energy Ei to a given final continuum state Ψ f E with energy E is given by 2 σ f E←i ∝ hΨ f E |µ · e|Ψi i

(2)

where Ψi is the initial (discrete) rovibrational wave function of He...Li2 (X 1 Σ+ g ) and Ψ f E is the final 2 + continuum wave function for He + Li+ 2 (X Σg , f ) in the ground electronic state and f rovibrational

state of the ion; e is the photon polarization and hν its energy (hν = E − Ei ); and µ is the transition dipole moment. The selected Jacobi coordinates are r, the distance vector between the Li nuclei, and R, the distance vector from the diatomic center of mass to the Helium atom, with θ the angle between them. Only the case of zero total angular momentum (J = 0) is considered in this work. In the framework of the Born-Oppenheimer approximation, Ψi and Ψ f E are solutions of the nuclear Hamiltonian : h¯ 2 ∂ 2 h¯ 2 ∂ 2 1 1 − + + j2 +VLiε 2 (r) +V ε (r, R, θ ) H =− 2µ ∂ r2 2m ∂ R2 2mR2 2µr2 ε

(3)

where j is the angular momentum operator associated with r; µ = mLi /2 and m = (mHe mLi2 )/(mHe + 5 ACS Paragon Plus Environment


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mLi2 ) are the reduced masses associated with r and R, respectively; VLiε 2 (r) is the potential of the isolated dimer and V ε (r, R, θ ) the intermolecular Van der Waals potential in the neutral (ε = 0) or ionic (ε = 1) state. For the initial bound state, or for the quasibound states used in the discussion of the results, the wave function Ψi , i ≡ (ε, v◦ , n◦ ), is expanded as

Ψi (r, R, θ ) = χvε◦ (r) ψvε◦ n◦ (R, θ ) √ ψvε◦ n◦ (R, θ ) = ∑ ∑ ains j φnεs (R) 2π Y j0 (θ , 0) ns

(4) (5)

j

where χvε◦ (r) is the vibrational wave function of the free lithium dimer with quantum number v◦ (v◦ = 0 for the initial bound state of the neutral dimer), φnεs (R) is the wave function of a harmonic √ oscillator basis set appropriately chosen with quantum number ns , and 2π Y j0 (θ , 0) a spherical harmonic representing a free rotor wave function for J = 0 in the body-fixed system with its z axis parallel to R. The energy and expansion coefficients ains j of the bound or quasibound state are obtained by diagonalization of the Hamiltonian in Eq. (3). The continuum wave function |Ψ f E i in Eq. (2), with f ≡ (ε = 1, v, j), is expanded as:

Ψ f E (r, R, θ ) =

∑0 χv10 (r) ψv1v0 jE (R, θ )

(6)

∑0 ϕv1v0 j0jE (R)

(7)

v

1v jE

ψv0

(R, θ ) =

√ 2π Y j0 0 (θ , 0)

j

with the outgoing asymptotic condition for R → ∞, i.e., √ √ Ψ1v jE ∼ χv1 (r) 2π Y j0 (θ , 0) eikv j R + ∑ Sv∗ j,v0 j0 (E) χv10 (r) 2π Y j0 0 (θ , 0) e−ikv0 j0 R

(8)

v0 j0

where S is the scattering matrix and kv j =

p 2m[E − Ev1 − Bv j( j + 1)] with Ev1 − B1v j( j + 1) being

1 the energy of the rovibrational level (v, j) of Li+ 2 with Bv the rotational constant.


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Introducing Eq. (?? 6) into the time-independent Schrödinger equation using the Hamiltonian 1v jE

H 1 of Eq. (3) leads to the usual close-coupled equations for ϕv0 j0 (R), which are solved using the De Vogelaere algorithm. 31,32 Finally, the (J = 0) total cross section at energy E for dissociative photoionization of He...Li+ 2 is given by σ1,E←i = ∑ ∑ σ1v jE←i v

(9)

j

and the rovibrational distributions of the Li+ 2 fragments are given by Pv j (E) =

σ1v jE←i σ f E←i

(10)

Technical details In the calculation of the bound or quasi-bound states an intermediate DVR×bending product basis was used. The DVR (discrete variable representation) 33–35 was obtained by diagonalizing the R operator in the initial harmonic oscillator basis set. In the calculation of the quasibound states, the frequency and equilibrium distance for the harmonic basis set in R were selected to correspond to a harmonic fit of the average potentiel Vvε◦ v◦ (R, θ ) = hχvε◦ |V ε (r, R, θ )|χvε◦ i, for the vibrational level v◦ at the equilibrium angular geometry. For the ground vibrational state of the neutral complex, the DVR basis set obtained this way exhibited negative eigenvalues of the position operator R, hence the equilibrium distance and the frequency were taken larger and lower, respectively, than the ones ε=0 (R, θ ). The bending basis set was obtained by diagonalizing given by the harmonic fit of V00

Vvε◦ v◦ (R◦ , θ ) in a basis set of spherical harmonics with R◦ fixed at the equilibrium value of the harmonic oscillator basis set. No basis reduction was implemented, but the bending basis set was of help in analyzing the stretching/bending character of the states. The photo-ionization cross section was calculated from Eq. (2). The bound state Ψi for He...Li2 (X, v = 0, n = 0) was calculated with 50 harmonic oscillator wave functions (frequency = 15 cm−1 , equilibrium distance = 12 Å), and 20 spherical harmonics. The interaction potential was expanded



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using 20 Legendre polynomials and its matrix elements in the DVR basis were then analytic. The photo-ionization cross section for He...Li2 were calculated in the vicinity of He...Li+ 2 (X, v = 2 and 9) absorption. Integration of the close-coupled equations was performed from 2.8 to 24.0 Å in steps of 0.01 Å. The converged calculation for v = 2 excitation included 5 vibrational channels (v0 = 0 - 4) and 20 rotational channels for each vibrational state in ?? 6-Eq. (8). In the vicinity of v = 9 excitation we conducted a more approximate calculation including only 2 vibrational channels (v0 = 8 - 9) and 20 rotational channels for each vibrational state. In order to check which resonances in the spectrum could be assigned to quasibound states and hence to a vibrational predissociation process (section Section 5.1), we have determined all the 0 0 quasibound states of He...Li+ 2 (v = 2, n ) by diagonalizing the Hamiltonian in Eq. (3). 60 harmonic

oscillator wave functions (frequency = 15 cm−1 , equilibrium distance = 5.6 Å) and 20 spherical harmonics were used in the expansion [?? 4] of the wave function. The interaction potential was expanded using 40 Legendre polynomials. The calculations were conducted for the most abundant isotope of lithium, with M(7 Li) = 7.01600455. 36

3

Potential energy surfaces

The potential energy surfaces (PES) describing the interaction of a He atom with a neutral or ionic lithium molecule have been taken from the previous works of Bodo et al. 14,16 We have used their analytical fit to their ab initio points. In what follows, (r, R, θ ) are the Jacobi coordinates already introduced, and (Rα , θα ) are the polar coordinates of the α th Li atom to He vectors with respect to the r axis. With these coordinates, V ε (r, R, θ ) from Eq. (3) is written as 2

V ε (r, R, θ ) =

∑ V ε (Rα , θα ) +VLRε (R, θ ) α=1


(11)

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The V ε (Rα , θα ) terms in the sum represent the Li-He atom-atom anisotropic interactions at short range, which are expanded in Legendre polynomials: nmax

V ε (Rα , θα ) = exp(−β ε Rα ) ∑ Rnα n=0

`max

∑ Cn`ε P`(cos θα ),

α = 1, 2

(12)

`=0

ε (R, θ ) in Eq. (11), is expressed in Jacobi cordinates as The long range part, VLR

ε VLR (R, θ ) = ∑ N

fN (β ε R) N−4 LRε ∑ CNL PL (cosθ ) RN L=0

(13)

˘ where fN are Tang-AToennies damping functions: 37 (β R)k k=0 k! N

fN (β R) = 1 − e−β R ∑

(14)

All the coefficients appearing in Eq. (12) and (Eq. (13) are taken from Bodo et al. 14,16 0 For the neutral Li2 (X 1 Σ+ g ) interaction, VLi2 in Eq. (3), we have used the Morse Long Range

function (MLR) of Le Roy et al. 38 given by : 2 uLR (r) −β (r)yeq (r) p e VMLR (r) = De 1 − uLR (re )

(15)

where De is the well depth and uLR (r) defines the (attractive) long range interaction [limr→∞ VMLR (r) = De − uLR (r)], with uLR (r) =

C6 C8 C10 + + R6 R8 R10

(16)

The function β (r) in Eq. (15) is written as h i N=17 h ii f re f re f (r)β + 1 − y (r) β y (r) β (r)yre ∞ ∑ i q p p i=0


(17)


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eq

re f

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re f

The variables y p (r) in Eq. (15) and y p (r) and yq (r) in Eq. (17) are defined as yeq p (r) =

p r p − req p r p + req

f yre p (r) =

p r p − rre f p r p + rre f

q rq − rre f re f yq (r) = q q r + rre f

(18)

The parameters defining the MLR potential for the (X 1 Σ+ g ) state of Li2 are taken from Le Roy et al. 38 (the Born-Oppenheimer breakdown correction was omitted). 2 + 39 computed for a large and For Li+ 2 (X Σg ) we used the ab initio potential of Bouzouita et al.

dense grid of intermolecular distances from 3 to 200 a.u.

4

Results: Photo-ionization spectra

The neutral bound state energy was −0.222440 cm−1 with respect to the He+Li2 (X, v = 0) dissociation limit. This value is in agreement with the results of Bodo et al. 13 using the BOUND program of Hutson 40 which yielded a value of -0.225 cm−1 and using the diffusion Monte Carlo method (DMC) which gave −0.237 ± 0.014 cm−1 . The Li2 (X, v = 0) bound state energy was 175.0317 cm−1 with respect to the minimum of the Li-Li interaction potential. 0 The Franck-Condon factors for Li+ 2 (X, v ) ←Li2 (X, v = 0) absorption are collected in Table 1.

From this table, the maximum intensity is expected to be around v = 2 excitation of the ground state ion.

4.1

0 Photo-ionization spectrum in the vicinity of He...Li+ 2 (X, v = 2) absorp-

tion The left part of Figure 1 displays the photoionization spectrum of He...Li2 in the region of v = 2 excitation, which is the region with the largest intensity (see Table 1).It was simulated in the limit of T = 0 K, i.e. from the ground vibrational level of the neutral He...Li2 complex. The spectrum exhibits a continuous background which is increasing with photon energy. In addition, a structure is


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Figure 1: Photo-ionization spectrum in the vicinity of the He...Li2 (X, v = 0, n = 0) −→ + 0 0 0 0 [He...Li+ 2 (X, v , n )] −→ He + Li2 (v f , j f ) transition, left: v = 2, right: v = 9. The zero en+ + ergy corresponds to the dissociated He + Li2 ionized complex with Li2 at the minimum of its potential energy. With this energy reference the threshold for dissociation to He + Li+ 2 (v, j = 0) is −1 −1 648.6 cm for v = 2, and 2359.8 cm for v = 9. Upper plots: linear intensity scale in arbitrary units. Lower plots: decimal logarithmic intensity scale, black curve: converged spectrum including the v00 = 0 to 4 (left), v00 = 8 and 9 (right) channels, see ?? 6 and Eq. (8); red curve: v00 = 1 and 0 (left) and v00 = 8 (right) continuum contribution to the spectrum. The differences between the black 0 0 and the red curves are due to resonances, or quasibound states of the type He...Li+ 2 (v = 2, n ).



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2 + 1 + Table 1: Franck-Condon factors (FCF) for Li+ 2 (X Σg , v) ← Li2 (X Σg , v = 0), and energies E(v) of the ionic vibrational levels. The zero for energies is the bottom of the Li + Li+ interaction potential. Searching for the minimum of the ab initio points of Bouzouita and Berriche 39 gives Te = −53962.286 cm−1 where the reference for energies is Li+ + Li+ and where the energy for Li + Li+ is -43486.922 cm−1 . Hence the Li + Li+ well depth is 10475.364 cm−1 .).

v 0 1 2 3 4 5 6 7 8 9

FCF E (cm−1 ) 0.060 131.352 0.146 391,575 0.195 648.621 0.191 902,513 0.153 1153,237 0.106 1400.786 0.067 1645.225 0.039 1886.533 0.021 2124.720 0.011 2359.824

superimposed, and is more clearly seen in the bottom part of the plot which displays two versions of the spectrum in logarithmic scale: one is the converged spectrum and the other is calculated using only the open channels, i.e. corresponding to direct dissociation. The difference between the two is due to Feshbach resonances or quasibound states, which will be identified in section Section 5.1 below. In order to determine the energy positions and widths of the resonances contributing to the spectra, they were fitted locally to a Lorentzian line shape:

σ1,E←i =

Γ/π Ir (E − Er )2 + Γ2

;

τ = h¯ /2Γ

(19)

where Γ is the half width at half maximum (HWHM) and τ the quasibound state lifetime. The results of the fits are presented in Table 2. 0 There are two distinct resonances (the other resonances below 400 cm−1 correspond to He...Li+ 2 (v =

1) excitation). The most intense one is centered at 648.06 cm−1 , i.e. 0.5 cm−1 below the dissoci−1 ation threshold to He + Li+ 2 (v = 2), with a half width at half maximum (HWHM) of 0.66 cm .


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32% of the Li+ 2 fragments are produced in the v = 1 state and 68% in the v = 0 state. The other resonance is 4.2 cm−1 lower in energy and about 10 times less intense. The fragment vibrational distribution is 46% (v = 1) and 54% (v = 0). The nature of these resonances will be analyzed in the discussion section, by comparison with quasi-bound states and the Golden Rule study of their dissociation. The important proportion of (v − 2) fragments shows that the vibrational coupling is not very weak. This could be expected since the Van der Waals well is about 380 cm−1 deep, which −1 between v = 2 and v = 1 from Table 1). is comparable to the Li+ 2 vibrational spacing (257 cm

Table 2: Energy Er , intensity Ir , linewidth Γ (HWHM) and related life time τ resulting from Lorentzian analysis [Eq. (19)] of the spectrum resonances in the vicinity of + 2 + 0 0 0 He...Li+ 2 (X Σg )(v = 2) photoionization. The final Li2 fragment (v − 1) and (v − 2) vibrational state distribution (%) at the resonance maximum is also given. The zero for en+ ergy is at the minimum of the Li+ 2 potential for the dissociated He+Li2 complex. The He + + 2 + −1 Li2 (X Σg )(v = 2) dissociation limit is at 648.6 cm . Er (cm−1 ) 643.9 648.1

4.2

Ir (arb. u.) 434 4111

Γ (cm−1 ) 1.09 0.66

τ (ps) 4.9 8.0

(v0 − 1) (%) 46 32

(v0 − 2) (%) 54 68

0 Photo-ionization spectrum in the vicinity of He...Li+ 2 (X, v = 9) absorp-

tion The photoionization spectrum of He...Li2 in the region of v = 9 excitation is displayed in the right part of Figure 1.The intensity scale in the top plot is arbitrary, however it is the same scale as on the left top plot. The results of the Lorentzian fits of the resonances are summarized in Table 3. Three resonances could be characterized. They are centered at E = −20.4, −3.3, and −0.2 cm−1 −1 below the He + Li+ 2 (X, v = 9) dissociation limit, with a HWHM Γ = 1.04, 2.99, and 0.09 cm

respectively. Additional structures are visible in the logarithmic plot but they are too low intensity and could not be analyzed.



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Table 3: Energy Er , intensity Ir , linewidth Γ (HWHM) and related life time τ resulting from Lorentzian analysis [Eq. (19)] of the spectrum resonances in the vicinity of + 2 + 0 He...Li+ 2 (X Σg )(v = 9) photoionization. The zero for energy is at the minimum of the Li2 + 2 + potential for the dissociated He+Li+ 2 complex. The He + Li2 (X Σg )(v = 9) dissociation limit is at 2359.8 cm−1 . Er (cm−1 ) 2339.36 2356.46 2359.56

5

Ir (arb. u.) 0.2 4.9 102.4

Γ (cm−1 ) 1.03 2.19 0.09

τ (ps) 5.2 2.4 0.6

Discussion

5.1

resonance assignment to quasibound states

The structure in the photoionization spectrum of He...Li2 can be attributed to Feshbach resonances corresponding to a vibrational predissociation (VP) process similar to what was observed in rare gas-dihalogen Van der Waals complexes 1,2 VP hν 0 0 + He...Li2 (v◦ = 0, n◦ = 0) −→ He...Li+ 2 (v , n ) −→ He + Li2 (v, j)

(20)

Because of the weak He – Li+ 2 interaction, most of the oscillator strength is carried by the lithium molecule. Hence photon absorption excites a quasibound state of the ion with a well defined Li+ 2 vibrational quantum number v0 , the additional quantum number n0 describing the intermolecular modes. This state will eventually decay to a continuum corresponding to a lower vibrational exci+ 0 tation v of Li+ 2 . This can be interpreted as the transfer of ∆v = v − v vibrational quanta of Li2 to

the intermolecular modes, leading to dissociation. In order to check which resonances in the spectrum could be assigned to quasibound states and hence to a vibrational predissociation process, we have determined all the quasibound states of 0 0 He...Li+ 2 (v = 2, n ) by diagonalizing the Hamiltonian in Eq. (3). The resulting energies and Franck

Condon factors are collected in Table 4, and the wave functions are displayed in Figure 2.


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The n0 = 0 level corresponds to a collinear configuration, with a rather large amplitude of the He bending motion as expected for a light atom. The next five levels could easily be assigned in terms of bending and stretching excitation from their nodal pattern. These assignments are reported in Table 4. Higher levels have a more complicated structure. The Franck Condon factors with the neutral bound state function are very small for the lowest levels. This was expected since the neutral ground state has a T-shape configuration. They become appreciable when the wave function extends into the region of perpendicular configuration. This is the case for the n0 = 11 level, with a Franck Condon factor of 0.47, and to a lesser extent for the n0 = 10, 8, and 7 levels (0.07, 0.05, and 0.02, respectively). We have also determined the Li+ 2 (v, j) final rovibrational state distributions using the Fermi Golden Rule approximation applied to vibrational predissociation. Within this approximation the vibrational coupling responsible for vibrational predissociation is treated as a perturbation. The linewidth of the initial quasibound level v0 decaying to the continua of the different final rovibrational levels (v, j) below it is given as

2 Γv j (Ev0 n ) = π Ψ1v jE (r, R, θ ) | V | Ψ1,v0 ,n0 (r, R, θ ) D E 2 1v jE = π ∑ ψv00 (R, θ ) | Vv100 v0 | ψv10 n0 (R, θ ) v00

Vv100 v0 (R, θ ) = χv100 (r) | V 1 (r, R, θ ) | χv10 (r)

(21)

0 0 where Ψ1,v0 ,n0 (r, R, θ ) is the quasibound state wave function of [He...Li+ 2 (v , n )] in Eq. (20) calcu-

lated using ?? 4-?? 5, and Ψ1v jE (r, R, θ ) is the final continuum wave function calculated as in ?? 6 but restricting the expansion to vibrational levels below v0 . This approximation is valid if Vv100 v0 is indeed a perturbation, i.e. if the linewidth of the level is small compared to the energy difference between resonances (quasibound levels). If this is the case, the spectrum resonances can be assigned by comparing their spectral characteristics (energy, relative intensity, linewidth and final rovibrational state distribution) to the ones of the quasibound states. From intensity arguments, the most intense resonance observed in Figure 1 for v0 = 2 and 15 ACS Paragon Plus Environment


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Table 4: Energy, Franck-Condon factors (F.C.), linewidths Γ (FWHM) and final vibrational ... + 2 + 0 state distributions of the Li+ 2 fragments for the quasibound states of He Li2 (X Σg )(v = 2, n0 ) as calculated by the Fermi Golden Rule approximation. The reference for energy is 2 + 0 0 0 the He + Li+ 2 (X Σg )(v = 2, j = 0) dissociation limit. When possible, the (ns , nb ) (stretching,bending) assignment is also indicated. n0

(n0s , n0b )

0 1 2 3 4 5 6 7 8 9 10 11

(0,0) (0,1) (1,0) (0,2) (1,1) (2,0) (2,1) — — — — —

Energy (cm−1 ) -227.827 -158.781 -108.201 -85.070 -58.138 -40.139 -17.174 -11.986 -9.251 -6.703 -4.186 -0.512

F.C. 0.5 × 10−10 0.5 × 10−10 0.2 × 10−5 0.2 × 10−9 0.8 × 10−5 0.7 × 10−3 0.004 0.016 0.053 0.006 0.073 0.470

Γ (cm−1 ) 100.70 6.67 9.57 3.97 5.28 5.73 1.96 1.93 2.12 2.07 0.74 0.44

(v − 1) (%) 76 80 72 84 77 71 77 80 75 82 73 72

(v − 2) (%) 24 20 28 16 23 29 23 20 25 18 27 28

0 tabulated in Table 2 corresponds to n0 = 11. It is located 0.50 cm−1 below the He + Li+ 2 (v = 2)

dissociation limit, which is very close to the quasibound state energy of −0.512 cm−1 . The second most intense resonance at 643.86 cm−1 , i.e. 4.70 cm−1 below the He + Li+ 2 (v = 2) dissociation limit, can be assigned to n0 = 10. The Franck Condon factor of the n0 = 8 quasibound state is close to the one for n0 = 10. However, the associated linewidth is larger, hence the corresponding resonance would be broader and its maximum intensity too low to be identified in the spectrum.

5.2

Rotational distributions of Li+ 2 fragments

0 ... + The final rovibrational distributions of the Li+ 2 (v, j) fragments arising from He Li2 (X, v = 2) dis-

sociation using the Golden rule calculation are shown in Figure 3. Inspection of this figure reveals that the rotational distributions corresponding to v = 1 (∆v = −1) and v = 0 (∆v = −2) are similar. They exhibit the same number of maxima. However, in the case of ∆v = −2 the distributions are broader and shifted to higher j values. Resonances corresponding to pure stretching excita16 ACS Paragon Plus Environment

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2 + 0 0 Figure 2: Contour diagram of the quasibound state wave functions of He...Li+ 2 (X Σg )(v = 2, n ). The plots are labelled with the quasibound state energies En0 in cm−1 from Table 4.



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tion (n0 = 0, 2, 5) exhibit only one peak, whereas two to three peaks can be seen when bending excitation is involved.

Figure 3: Rotational distributions of the Li+ 2 (v, j) fragments from vibrational predissociation of the + 2 + 0 ... quasibound states of He Li2 (X Σg )(v = 2, n0 ) using Fermi’s Golden Rule calculations. Black curves with asterisks: v = 1; green curves with square symbols: v = 0.

5.3

Comparison between spectrum and Fermi Golden rule calculations

Figure 4 displays the v = 1 and v = 0 rotational distributions of Li+ 2 from the most and the second most prominent resonances in the photoionization spectrum (Figure 1) of He...Li2 in the vicinity of v = 2 excitation. They are compared with the final state rotational distributions obtained with Fermi 18 ACS Paragon Plus Environment

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... + Figure 4: Rotational distribution of the Li+ 2 (v) fragment from photoionization of He Li2 in the vicinity of v0 = 2 excitation: v = 1 (black curve with squares) and v = 0 (green curve with squares ) from spectrum calculation at the resonance energy E (see Eq. Eq. (10), and comparison with the v = 1 (black curve) and v = 0 (green curve) rotational distributions from Golden rule calculation. Left plot: E = 643.86 cm−1 , n0 = 10 quasibound state; Right plot: E = 648.06 cm−1 , n0 = 11 quasibound state. Golden Rule calculations from the quasibound states assigned at the end of section Section 5.1 based on intensity and energy position. For the most intense resonance, which was assigned to the n0 = 11 quasibound state, the Golden Rule calculation reproduces very well the Li+ 2 fragment rotational distributions obtained from spectrum calculation, in particular, the main peak at j = 4 and the secondary peak for higher values of j. However, the ∆v = −2 channel is more populated in the complete calculation (68%) than in the Golden Rule one (28%). We believe that this is due to a situation of intermediate coupling. The coupling from ?? 21 is weak enough for the Golden Rule to be approximately valid, with separate, assignable resonances. But it is strong enough that the quasibound state already includes a non negligible contribution of v0 − 1, inducing a larger v0 − 2 population in the products, which is accounted for by the complete spectrum calculation and not by the Golden Rule one. For the less intense resonance assigned to n0 = 10, the agreement is not as good. Both calculations give single maximum distributions, and the rotational distributions for v = 1 are almost identical, but the v = 0 rotational distribution obtained from the Golden Rule calculation has a maximum between j = 10 and 12 while it is around 4 and broader for the spectrum calculation. This could be due to the fact that this resonance is not as intense as the n0 = 11 one, and its vicinity 19 ACS Paragon Plus Environment


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with other resonances (n0 = 11 and 8) can induce some mixing.

5.4

Consequences for other dynamical processes involving Li+ 2 and He

As seen in the preceding sections, the direct photoionization spectrum of He...Li2 can be interpreted in terms of vibrational predissociation of the He...Li+ 2 ion, by assigning the resonances observed in the spectrum to quasibound states obtained in Fermi Golden Rule calculations. Because of the large difference between the neutral (perpendicular) and ionic (collinear) equilibrium configurations, only the most excited quasibound states have a significant intensity. The other resonances obtained from the Fermi Golden Rule calculations can give information on vibrational energy relaxation rates of the Li+ 2 molecular ion in collisions with helium atoms. They can also give information on the vibrational relaxation rate of the ionized exciplexes in the case of photodesorption of Li2 from a helium droplet. If the He...Li∗2 photoexcited state equilibrium configuration is similar to that of the ground neutral state (perpendicular), photoionization from there will be similar to direct photoionization: only the most excited intermolecular levels will be populated in the ion, and the lifetime will be between 2 and 10 ps (from the lifetimes in Table 4).In the other extreme, if the He...Li∗2 photoexcited state equilibrium configuration is similar to that of the ionic complex (i.e. collinear), only the most excited intermolecular levels will be accessed in the intermediate, electronically excited state, and they will have a good Franck-Condon factor with the states of the ionic complex. In this case again, only the most excited intermolecular levels will be populated in the ion and the lifetime will be between 2 and 10 ps. This conclusion is based on the J = 0 calculations presented in this work. It is justified since the temperature in 4 He droplets has been determined to be 0.37 K. 4,41,42 In the case of an exciplex built with more than one helium atom, the coupling responsible for vibrational relaxation scales as the number of helium atoms n. Depending on the coherence between the helium atoms during the process, the resulting linewidth is expected to be multiplied by a factor between n and n2 . Hence the expected vibrational relaxation times should be shorter than 10 ps. This means that the exciplexes have ample time to relax and eject helium atoms during 20 ACS Paragon Plus Environment

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their flight time between photoionization and detection in the mass spectrum. Hence the detected exciplexes can originate from larger ones that will have vibrationally relaxed by ejecting helium atoms.

6

Conclusion

We have conducted quantum coupled channel calculations of the photoionization cross section of the He...Li2 Van der Waals complex in the vicinity of v0 = 2 and v0 = 9 excitation of the ion. The spectrum is characterized by a continuum, with a structure superimposed. This structure is assigned to Feshbach resonances due to a vibrational predissociation process. Their spectral characteristics (position, width, and intensity) are obtained by fitting regions of the spectrum to a weighted sum or Lorentzians. A Fermi Golden Rule treatment of the vibrational predissociation process gives a further analysis of the resonances, by assigning them to quasi bound states of the ionic complex based on their energy and final rovibrational state distribution. Only the most excited (n0 = 10 and 11) quasi bound states are visible as resonances in the spectrum, due to very different neutral (perpendicular) and ionic (collinear) minimum energy structures of the potential energy surfaces. However, these quasi bound state could show up in other processes involving the Li+ 2 ion and a 2 + helium atom, such as He + Li+ 2 ( Σg ) collisions.

Acknowledgments We are particularly thankful to E. Bodo and F. A. Gianturco for providing the potential energy data for He...Li2 and He...Li+ 2 . We acknowledge financial support from the Ministry of Higher Education and Scientific Research of Tunisia. This study has been partially supported through the grant NEXT n◦ ANR-10-LABX-0037 in the framework of the “Programme des Investissements d’Avenir”, and was granted access to the HPC resources of CALMIP supercomputing center under the allocation 2014-P1039.



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(16) Bodo, E.; Gianturco, F. A.; Sebastianelli, F.; Yurtsever, E.; Yurtsever, M. Rotational Cooling of Li2 (1 Σ+ g ) Molecules by Ultracold Collisions with a Helium Gas Buffer. Theor. Chem. Acc. 2004, 112, 263–269. (17) Bodo, E.; Gianturco, F. A.; Yurtsever, E. Vibrational Quenching at Ultralow Energies: Calculations of the Li2 (1 Σ+ g , v 0)+He Superelastic Scattering Cross Sections. Phys. Rev. A 2006, 73, 052715. (18) Bovino, S.; Bodo, E.; Yurtsever, E.; Gianturco, F. A. Vibrational Cooling of Spin-Stretched Dimer States by He Buffer Gas: Quantum Calculations for Li2 (a 3 Σ+ u ) at Ultralow Energies. J. Chem. Phys. 2008, 128, 224312. (19) González-Sánchez, L.; Bodo, E.; Gianturco, F. A. Collisional Quenching of Rotations in + 2 + Lithium Dimers by Ultracold Helium: The Li2 (a3 Σ+ u ) and Li2 (X Σg ) Targets. J. Chem.

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List of Figures 1

0 0 Photo-ionization spectrum in the vicinity of the He...Li2 (X, v = 0, n = 0) −→ [He...Li+ 2 (X, v , n )] 0 0 −→ He + Li+ 2 (v f , j f ) transition, left: v = 2, right: v = 9. The zero energy corre+ sponds to the dissociated He + Li+ 2 ionized complex with Li2 at the minimum of

its potential energy. With this energy reference the threshold for dissociation to He −1 for v = 2, and 2359.8 cm−1 for v = 9. Upper plots: + Li+ 2 (v, j = 0) is 648.6 cm

linear intensity scale in arbitrary units. Lower plots: decimal logarithmic intensity scale, black curve: converged spectrum including the v00 = 0 to 4 (left), v00 = 8 and 9 (right) channels, see ?? 6 and Eq. (8); red curve: v00 = 1 and 0 (left) and v00 = 8 (right) continuum contribution to the spectrum. The differences between the black and the red curves are due to resonances, or quasibound states of the type 0 0 He...Li+ 2 (v = 2, n ).

2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 + 0 Contour diagram of the quasibound state wave functions of He...Li+ 2 (X Σg )(v =

2, n0 ). The plots are labelled with the quasibound state energies En0 in cm−1 from Table 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3

Rotational distributions of the Li+ 2 (v, j) fragments from vibrational predissociation 2 + 0 0 of the quasibound states of He...Li+ 2 (X Σg )(v = 2, n ) using Fermi’s Golden Rule

calculations. Black curves with asterisks: v = 1; green curves with square symbols: v = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4

... + Rotational distribution of the Li+ 2 (v) fragment from photoionization of He Li2 in the vicinity of v0 = 2 excitation: v = 1 (black curve with squares) and v = 0 (green curve with squares ) from spectrum calculation at the resonance energy E (see Eq. Eq. (10), and comparison with the v = 1 (black curve) and v = 0 (green curve) rotational distributions from Golden rule calculation. Left plot: E = 643.86 cm−1 , n0 = 10 quasibound state; Right plot: E = 648.06 cm−1 , n0 = 11 quasibound state.


19

Li2 - American Chemical Society

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