Solvent states and spectroscopy of doped helium clusters as a quantum-chemistry-like problem

10126 Phys. Chem. Chem. Phys., 2013, 15, 10126--10140 This journal is c the Owner Societies 2013

Cite this: Phys. Chem.Chem.Phys.,2013,15, 10126

Solvent states and spectroscopy of doped heliumclusters as a quantum-chemistry-like problem†

Nestor F. Aguirre,a Pablo Villarreal,a Gerardo Delgado-Barrio,a

Alexander O. Mitrushchenkovb and Marıa Pilar de Lara-Castells*a

The Full-Configuration-Interaction Nuclear-Orbital (FCI-NO) approach [J. Chem. Phys., 2009, 131, 19401],

as the implementation of the quantum-chemistry ansatz, is overviewed and applied to (He)N–Cl2(X)

clusters (N r 4). The ground and excited states of both fermionic 3He and bosonic 4He [see also, J. Phys.

Chem. Lett., 2012, 2, 2145] clusters are studied. It is shown that the FCI-NO approach allows us to

overcome three main difficulties: (1) the Fermi–Dirac (Bose–Einstein) nuclear statistics; (2) the wide

(highly anharmonic) amplitudes of the He–dopant and He–He motions; and (3) both the weakly

attractive (long-range) and the strongly repulsive (short-range) interaction between the helium atoms.

Special emphasis is placed on the dependence of the cluster properties on the number of helium

atoms, and on the comparison between the two helium isotopes. In particular, we analyze the

analogies between quantum rings comprising electrons and 3He atoms. The synthetic vibro-rotational

Raman spectra of Cl2(X) immersed in (3,4He)N clusters (N r 4) are discussed as a function of the cluster

size and the nuclear statistics. It is shown that the Coriolis couplings play a key role in modifying the

spectral dopant profile in 3He. Finally, we point out possible directions for future research using the

quantum-chemistry ansatz.

1 Introduction

During the last few years, superfluid droplets composed of 4Heatoms have become an ideal cryogenic matrix for molecularhigh-resolution spectroscopy.1,2 The droplets are produced byexpansion of 4He into vacuum1,3,4 and they cool very rapidly(via evaporation), reaching a temperature of about 0.37 K,5–7

well below the superfluid transition temperature (2.17 K) inbulk liquid.8 As first observed for sulfur hexafluoride, SF6,6,9 andcarbonyl sulfide, OCS,10 the spectra of molecular impuritiesinside 4He droplets attain a very well resolved profile withsharp rotational lines, which in turn indicated the free-rotor-like behaviour of the impurity with a reduced (renormalized)

rotational constant.10 In contrast, the spectra of OCS in 3Hedroplets10 showed broad (about 1 cm�1) peaks. Since thetemperature of 3He droplets (0.1 K11) is above the superfluidtransition temperature in bulk 3He liquid (Tl = 3 � 10�3 K), thesharp rotational spectrum in 4He droplets was interpreted as amanifestation of their superfluidity, as opposed to dropletscomposed of 3He atoms. Moreover, a well resolved spectralprofile was recovered after adding about 60 4He atoms to 3Hedroplets with a single OCS molecule inside, with the 4He atomsconcentrating around the impurity.10 This experimental findingindicated that a small number of 4He atoms are required for theonset of the so-called ‘‘molecular superfluidity’’.10 This keyresult motivated a large number of further experimental high-resolution spectroscopic probes of single molecules, such asOCS,12 N2O,13,14 CO2,15 CO,16,17 and HCN,18 embedded intosmall clusters composed of up to a few tens of 4He atoms. Oneof the main objectives of many of them has been to address thequestion of how many 4He atoms are required for the onset ofsuperfluidity, as signaled by the change of the trend for thevalue of the molecular rotational constant (from decreasingwith the number of 4He atoms to increasing). Zero-temperatureDiffusion Monte-Carlo (DMC) and finite-temperature Path-Integral Monte-Carlo (PIMC) simulations have been parti-cularly useful in interpreting the non-classical evolution of

a Instituto de Fısica Fundamental, CSIC, Serrano 123, 28006 Madrid, Spain.

E-mail: [email protected]; Tel: +34-915616800b Universite Paris-Est, Laboratoire Modelisation et Simulation Multi Echelle,

MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-la-Vallee, France

† Electronic supplementary information (ESI) available: Tables containing addi-tional information such as the energies of ground and excited solvent states andthe corresponding non-adiabatic Coriolis coupling terms. Details of the fittingprocedure for the He–Cl2 potential energy surface, the FCI Nuclear Orbitalcalculations, the method to obtain the dopant spectra, the vibrational predis-sociation mechanism in broadening the spectral lines, the Coriolis couplingeffects on the spectra, and the effective renormalization of the diatomic rotationalconstant from the Coriolis terms. See DOI: 10.1039/c3cp50282a

Received 21st January 2013,Accepted 7th March 2013

DOI: 10.1039/c3cp50282a

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the rotational constant and their relationship with both theanisotropy of the 4He–molecule potential energy surface (PES)and the magnitude of the molecular rotational constant(whether 4He atoms experience different response to slow orfast rotors), as recently reviewed by Szalewicz with the focus onthe crucial role of the 4He–molecule interaction potential.19 Asfar as slow rotors with a global minimum located around themolecular waist such as OCS are concerned, the underlying ideadates back to the ‘‘donut model’’ introduced by Toennies andcollaborators.20 A donut ring of 4He atoms around the waistwould participate in the end-over-end rotation of the molecule,resulting in the increase of the rotational molecular constant.A theoretical analysis by Whaley and collaborators21 based onDMC and PIMC results identified this first ring as a non-superfluid fraction of 4He atoms that adiabatically follows themolecular rotation, as opposed to the decoupled motion of therest of 4He atoms that execute long-range exchanges amongthemselves and then contribute to the superfluid fraction. Thus,distinct superfluid fractions for rotation parallel or perpendi-cular to the impurity molecular axis were defined within theso-called two-fluid model.22 More recent PIMC simulations23

provided new physical insights into the role played by thesymmetry of the He–molecule interaction potential anisotropy,identifying the long-range exchange of helium atoms betweenthe two ends of the N2O molecule as equivalent to the end-over-end molecular rotation (augmenting the superfluid fraction inthe direction perpendicular to the molecular axis). The model ofa symmetric and quasi-rigid donut ring around the impurity hasalso been used to explain spectral features concerning the axialmotion of the 4He atoms, e.g. the absence of the Q-branch.24 Aspointed out by Szalewicz,19 another explanation from Whaleyand collaborators22 invokes the superfluidity response of the 4Heatoms to rotations around the molecular axis at the completionof the first ring, without resorting to the argument of a sym-metric location of the helium atoms. Actually, as discussed inref. 25, the lowest energy excitations of 4He atoms confined inring-shaped arrangements correspond to the classical energy of arigid body. As will be discussed below, similar arguments applyto the fully spin-polarized fermionic 3He case.

Since 4He atoms are composite spinless bosons while 3Heatoms are fermionic particles with a nuclear spin equal to 1/2,one fundamental question is the influence of the spin and theantisymmetry of the fermionic 3He wave-function in providingcongested spectral profiles, as opposed to the case of 4He. Thenodeless character of the 4He ground-state wavefunctionrenders the application of quantum Monte-Carlo (QMC) methodsrelatively simple so that computational QMC implementationsshow a favorable scaling with the doped cluster size.26 Whenapplied to obtain a wave-function containing nodes (excitedstates and fermionic systems), however, QMC methods face theso-called sign problem so that a particular structure of the trialwave-function is used to guide the DMC calculation.27 Thisfixed-node approximation has been successfully applied tostudy atomic impurities in either 4He, 3He, or mixed 4He–3Heclusters by Navarro and collaborators.28 Modern genetic algo-rithm (GA) based implementations29 allow the calculation of

nodal surfaces on-the-fly also. This strategy has been veryrecently employed to obtain the ground and excited states ofdoped 4He clusters,30 with up to 10 4He atoms. The other mostoften used quantum approaches are PIMC and methods basedon density functional theory (DFT).31 The former allows us toexplore finite temperature effects while the latter is a phenom-enological approach that can be applied to both 4He and 3Heisotopes, either in the bulk liquid or in nanodroplets. It is alsoworth mentioning the multi-configuration time-dependentHartree (MCTDH) method,32 which has been very recentlyapplied to dopant-(4He)4 clusters33 by considering the 4Heatoms as distinguishable particles.

An alternative to Quantum Monte-Carlo methods has beenprovided by Quantum-Chemistry(QC)-like Nuclear-Orbital (NO)approaches, which are the focus of this work. The general ideaconsists in exploiting the fact that both an atomic nucleus anda heavy dopant impurity act as an attractive center, bindingelectrons around the former and 3He atoms about the latter,with the 3He atoms otherwise unbound if their number isbelow a certain critical size (B3034–36). This approach was firstproposed by Jungwirth and Krylov,37 and applied to the tripletstate of the 3He2–SF6 complex at the Hartree–Fock (HF) level.Since this pioneering work, methods based on the Quantum-Chemistry(QC) ansatz using a one-particle (‘‘nuclear’’) orbitalpicture have become well established with implementations atdifferent levels of theory such as HF,38 Full-Configuration-Interaction (FCI),39 Kohn–Sham (KS) DFT level of theory,40

and even using explicitly correlated functions41 (ECFs). Ascheme considering the different approaches and illustratingtheir evolution over the time is shown in Fig. 1. As overviewedbelow (Section 2), theoretical studies using the QC ansatzcomplement those based on DMC and PIMC approaches,extending the familiar concepts to the quantum chemist thathave served molecular structure theory, and enabling thedescription of excited states with a similar accuracy to that ofthe ground-state. Thus, early implementations using multi-orbital Hartree38 (MO-H) and Hartree–Fock42 approximationsto calculate wave-function for 4He and 3He atoms, respectively,allowed us to obtain either the Raman42–44 or the infrared44–46

spectra of a heavy dopant molecule as perturbed by the

Fig. 1 Scheme illustrating the evolution over time of methodological andcomputational implementations using the quantum-chemistry-like ansatz.

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helium environment. More recently, aimed at establishing aparameter-free QC-like benchmark approach, an efficient Full-Configuration-Interaction Nuclear Orbital (FCI-NO) treatmentwas developed to calculate ground and excited fermionicsolvent wave-functions. It has been applied to small 3Heclusters with molecular impurities inside.25,39,47–52 Thisapproach was first applied to describe 3He atoms in a highlydelocalized ring. A high degree of degeneracy for the lowestspin-states was also obtained at the FCI-NO level,39,47,48 leadingto broadened spectral profiles.49,51,52 Also, the adequacy of thenuclear orbital approach was demonstrated, showing that a fewnatural orbitals are able to account for the most of the heliumpopulation.39,48 New insight was also provided into energeticand structural aspects, demonstrating that a molecular impurityis able to hold together the otherwise unbound 3He atoms andto induce their structural pairing to benefit from the weakminimum of the He–He potential.47 Short-range correlationeffects were also shown to be responsible for a quasi-linearscaling of ground-state total energies with the number of 3Heatoms despite the Pauli’s exclusion principle. The spin multi-plicity was found to be ultimately responsible for this beha-viour. An extension of the FCI-NO implementation to thebosonic wave-function was further developed, with the focuson the collective excited states of para-H2 molecules confined ina ring.25 It was also found25,49 that the energy gap to the lowestrotational excited states of 3He and 4He atoms in ring-shapedarrangements scales very differently with the angular momen-tum projection. The microscopic origin of this behaviour as afunction of the cluster size is analyzed here by considering Cl2

as the guest molecule (Section 3). This choice is motivated bythe quasi-degeneracy of the two minima attained by theHe–Cl2(X) ab initio potential energy surface (PES), enabling theanalysis of the critical size at which a transition from ring-likestructures to a regime where the region along the molecular axisstarts to be populated. In particular, we will explore the analogieswith an electronic structure problem: quantum rings confiningelectrons by an external electrostatic potential.53,54

As mentioned above, our quantum-chemistry approach alsoallows calculating the spectra of the embedded molecularchromophore. As an illustrative application, the vibro-rotationalRaman spectra of Cl2(X) surrounded by up to four 4He or 3Heatoms are discussed here (Section 4), with the emphasis on therole of non-adiabatic Coriolis couplings and the evolution ofthe spectral profiles with the number of helium atoms and theirfermionic or bosonic quantum statistics. At a variance with the4He counterpart, spectroscopic probes of molecular chromo-phores in 3He are rather sparse.2 To date, experimentalmeasurements have been accomplished in doped 3He droplets.11

Albeit spectroscopic probes in small 3He clusters are notdirectly comparable with those in 3He droplets, they still enablethose to complement them. Actually, according to the lastanalysis of the experimental spectra of OCS in 3He nanodroplets11

and KS-DFT calculations,40 the measured spectral profiles aremainly determined by the structuring of about eleven 3He atomsthat are strongly attached to the dopant while the otheratoms become decoupled. In contrast with the 4He counterpart,

this shell comprises not only the first ring around the dopant(about four 3He atoms) but additional 3He atoms that are,however, not enough to fully wrap up the impurity. Besides thespectral congestion in 3He droplets,11 another remarkabledifference with the 4He counterpart lies in the moment ofinertia: it was found to be about two times larger in 3He.Importantly, both KS-DFT calculations40 and experimentalmeasurements provided support to a symmetric top clusterstructure that predicts the presence of Q branches and thus theexcitation of the axial rotational motion of the 3He atoms.By considering Cl2 as the molecular impurity and differentcluster sizes, before and after filling the first ring around thedopant, we will show that the vibro-rotational Raman spectra ofthe molecular chromophore are consistent with those mea-sured in doped 3He nanodroplets.11 Moreover, the key role ofCoriolis couplings in 3He, inducing an effective reduction of thedopant rotational constant (an increase of the momentof inertia) and then aiding the experimental findings, will bedemonstrated.11

Finally, the last section will close with a summary and futureperspectives. In particular, we will point out possible directionsfor future applications and methodological extensions of theFCI-NO method and, more generally, the quantum-chemistryansatz.

2 The quantum-chemistry ansatz

As in conventional electronic structure calculations, thequantum-chemistry (QC) approach to doped helium clustersstarts by fixing the position of the molecular dopant that is thentreated at the infinite dopant mass approximation within theBorn–Oppenheimer (BO) approach. The next step consists inselecting the basis set to expand the ‘‘nuclear’’ orbitals (thehelium single-particle wave-functions). The earliest implemen-tation employed the same strategies and basis-type functions asin the standard electronic structure calculations: Gaussian-typefunctions, GTFs, centered on the dopant.37 In addition, thepairwise additive approximation was assumed for all the inter-actions. The further expansion of the He–He and He–dopantpotentials into a series of GTFs37 rendered the analyticalevaluation of one and two-particle integrals rather simple, asin standard electronic structure implementations. Due to thedifferent nature of the He–dopant interaction, however, theconvergence with respect to the number of dopant-centeredGTFs was less favourable than in the electronic structure case(see Fig. 2).

Later, this QC Nuclear-Orbital (NO) approach was reformu-lated by separating the center-of-mass (CM) motion from theHamiltonian (using internal coordinates), employing basisfunctions tailored to account for the wide amplitudeHe–dopant mode, and considering both 3He and 4He isotopes(see Fig. 1). Within this framework, Barletta et al.41 obtainedthe nuclear orbitals by numerically solving the Schrodingerequation of the He–dopant pair, resulting in a lowering of thetotal energy as compared to the dopant-centered basis set.37

Parallel computational and methodological developments

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(see, for example, ref. 38 and 42) allowed us to extend this QCNuclear-Orbital ansatz to anisotropic He–diatomic interactionsand to consider up to a few tens of 4He and/or 3He atoms. Usingthis approach, the diatomic position is fixed and satellitecoordinates R are used to refer the helium atoms positions tothe center-of-diatomic-mass. The one-particle operator h (R;r)then reads

hðR; rÞ ¼ ��h2

2m@2

@R2þ l2

2mR2þ vextðR; y; rÞ (1)

with r, m and l being the diatomic internuclear distance, thehelium–dopant reduced mass and the orbital angular momen-tum associated to R, respectively (y is the angle between thedopant axis and R). By solving this equation, independent-particle (IP) orbitals are obtained. For this purpose, a basis setcomprising numerical radial functions Fn(R) and sphericalharmonics Ylm (y,f) is employed, with f as the azimuthal angleof the reference helium atom. Explicit expressions of the one-and two-particle integrals are provided in ref. 38. An optimizedbasis set for highly anisotropic helium–dopant interaction,comprising two-dimensional Fn(R, y) orbitals and complexexponentials for the azimuthal degrees of freedom e�imf wasfurther designed, implementing it to deal with the 4He iso-tope.25 Since this work focuses on the comparison between theresults obtained for the 4He and 3He isotopes, we have adoptedthe Fn(R)Ylm(y,f) basis set.

Once the set of nuclear orbitals {fi} is selected, theHamiltonian in the second quantized form of the pair-interactinghelium system confined by the helium–dopant potential vext

can be written as,

H ¼Xij

hijcyi cj þ

1

2

Xijkl

Vijklcyi cykclcj

where hij and Vijkl are the one- and two-particle integralsevaluated on the {fi} basis. Explicitly,

hij ¼Z

dRf�i ðR; rÞvextðR; y; rÞfjðR; rÞ

and omitting the parametric dependence on r,

Vijkl ¼Z

dRdR0f�i ðRÞf�j ðR0ÞVðR12ÞfkðRÞflðR0Þ;

where the two-particle operator V(R12) is a pair-wise interactionpotential between the helium atoms (R12 = |R � R0|).

We use the Fock space notation in order to uniquely labelthe many-body configuration state functions (CSFs) of N bosonicspinless 4He atoms (fermionic half-integer spin 3He atoms),|Li = |n1n2. . .nMi, with M as the number of orbitals and ni as theoccupation number of the ith orbital, with ni r N for 4He. Forfermionic 3He atoms, spin-orbitals are used (ni = 1 or 0). Byemploying bosonic (fermionic) annihilation and creation

operators cicyj � c

yi cj ¼ dij cic

yj þ c

yi cj ¼ dij

� �, the Hamiltonian

matrix elements on the CSF basis can be easily obtained.In fact, one appealing advantage of QC-like approaches consistsin using nuclear orbitals so that symmetry (bosonic or/andfermionic) effects can be easily included on an equal footingthrough simple commutation or anti-commutation rules ofannihilation and creation operators. Also in contrast withearlier variational (configuration-interaction) approaches,55,56

the rotational dopant motion is considered sequentially throughthe QC ansatz,57 allowing exploitation of the full symmetry of theHe–dopant potential.

Let us now focus on the role of many-body correlation effectsas compared with electronic structure problems. Due to thehard-core wall of the He–He interatomic interaction (e.g., the12–6 Lennard-Jones potential raises as (1/R12)12 when R12 - 0)and their larger masses, helium atoms are much more stronglycorrelated than electrons (with a repulsive 1/R12 electrostaticinteraction). This hard-core repulsion is the origin of unphysical,non-bound, Hartree–Fock solutions.41 An exception is the fullypolarized case for 3He where all atoms have the same spin andare then constrained to occupy different spatial orbitals. There-fore, Pauli’s exclusion principle already forbids the 3He atomsto stay in the contact region whatever the short-range repulsionmay be. The non-applicability of Pauli’s principle for thebosonic 4He atom makes the hard-core problem even morepronounced. As originally proposed in the application of theHartree approach to quantum 4He crystals,58 the hard-coreproblem can be dealt with by softening the hard-core inter-action (e.g., multiplying V(R12) by e�gV(R12)) or simply truncatingthe potential at short distances.38,59 As demonstrated in ref. 38,both approaches work rather well to describe ground-stateproperties at HF and MO-H levels of clusters containingstrongly attractive dopants. Still, the standard self-consistentfield (SCF) method has to be replaced by a direct minimizationprocedure60,61 to ensure convergence to the global minimum.A more rigorous treatment was applied by Barletta et al.41 toSF6-(3,4He)2 trimers. Similar to explicitly correlated R12 electronicstructure methods, the zeroth-order wave-function (e.g., theindependent-particle wave-function41) is multiplied by a singleshort-range correlation factor ef (R12). As the truncation schemes,this factor effectively softens the hard-core of the V(R12) potentialbut the kinetic energy operators of the helium atoms are also

Fig. 2 Schematic illustration of the attractive electrostatic interaction V(r)between a reference electron and an atomic nucleus (highlighted in red) andthe van der Waals interaction between a helium atom and a heavy dopant(highlighted in blue). The lowest-energy single-particle wave-functions (electro-nic and nuclear orbitals) in both scenarios are also represented.

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modified to account for dynamic (short-range) correlationeffects. On the other hand, the potential truncation parameters(either g or the healing distance) are fitted to match an esti-mate of the exact energy for the He–dopant trimer,38,42,43 whilethe correlation factor is determined by energy minimization.41

The KS-DFT method developed very recently by Mateo et al.40

to handle doped 3He clusters also employs a He–Hepotential screened at short distances but in conjunction witha correlation energy functional. The parameters enteringboth terms are fitted to reproduce bulk helium experimentalproperties.62

2.1 Full-configuration-interaction nuclearorbital implementation

As a benchmark parameter-free QC-like treatment of groundand excited helium states,25,39,47 the FCI-NO approach wasdeveloped and applied25,48–52 to dopant-(4,3He)N clusters (N r 5)using the above-described Fn(R)Ylm(y,f) and Fn(R,y)e�imf basissets. By employing the Fn(R)Ylm(y,f) basis, up to two hundredmillion CSFs are necessary to account for the strong He–Hecorrelation in dopant-(4,3He)4 clusters. Due to the large sizeof such FCI configuration space, it is only possible to use theso-called direct-CI iterative methods, where the matrix todiagonalize is never evaluated and stored, but rather its actionon a given trial vector is provided. In electronic structurecalculations the Davidson method63 is the most commonlyused iterative eigen-solver. Its advantage is that no additionalmatrix–vector operation is needed for a given iteration. TheDavidson algorithm belongs to the family of the Krylov-spacemethods. In these approaches, the current approximation -

v tothe eigenvector of Hamiltonian H is obtained by diagonalizingthe Hamiltonian in the Krylov subspace {-v1,-v2 . . .

-vn}. The

current approximation to eigenvalue is given by E = h-v|H|-vi.

Further, the Krylov subspace is expanded by adding the vectorD

-r where -

r is a residual vector, -r = (H � E)-n, and D is a pre-

conditioning operator. In the standard Davidson approach D istaken in a simple diagonal form,

D = Q(Hd � E)�1Q, (2)

where Q is the orthogonal projector, Q = l � |-nih-n|, and Hd isthe diagonal of the Hamiltonian in the basis used toexpress the vectors, Hd = diag(H). It can be noticed that if itwould be possible to solve a similar equation with the fullHamiltonian instead of its diagonal, -nn+1 = (Q(H � E)Q)�1-r or,equivalently

Q(H � E)Q-nn+1 = -r, (3)

this would result in a method similar to the inverse iterationmethod, which is known to have extremely fast convergencerate for nearly any type of matrix, and independently on thevector basis used. Unfortunately, the solution of this linearequation is as difficult as the solution of the eigenvalueproblem itself. Due to the hard-core He–He interaction, verylarge off-diagonal Hamiltonian elements appear and, as aresult, it is not possible to achieve convergence through theDavidson algorithm.39,48 However, this problem is overcome

once the Davidson algorithm is replaced by a Jacobi–Davidsonmodification. In the Jacobi–Davidson method, one solvesthe updated equation, eqn (3), iteratively for each Davidsonmacro iteration. It converges quickly, the incorporation of theincreasing-orbital-space technique providing even more rapidconvergence. In the FCI program, the iterative SYMMLQsolver without any pre-conditioner was used.64 As reported inref. 25, the further development of an optimized basisset allowed reduction of the FCI space by more than two ordersof magnitude.

2.2 Vibro-rotational Raman spectra within the linear rotorapproximation for the diatomic dopant

In this work, we have employed the FCI-NO method to calculatethe solvent helium states and the linear rotor approximation toobtain the vibro-rotational Raman spectra of the diatomicdopant. The FCI-NO states are labeled following the conven-tional notation of electronic states for diatomic molecules,n2S+1L(�)

(g/u), with L and S being, respectively, the projection ofthe total He orbital angular momenta L =

PNlk on the mole-

cular axis and the spin angular momentum quantum number.Within the linear rotor approximation for the diatomic dopant,the effective cluster Hamiltonian reads, Heff = H + Bvj,2 with jand Bv being the diatomic rotational angular momentum andthe rotational constant in the vibrational level |vi, respectively.For a total angular momentum J = j + L + S with a projectiononto the body-fixed (BF) Z-axis O = L + S (S being the projectionof S on Z), the vibro-rotational (Hund’s case a) basis functionscan be written as

jIi � jJM; n2Sþ1Lð�Þðg;uÞðOÞiv ¼ jvijn2Sþ1Lð�Þðg;uÞ;L;SijJ;O;Mi

where |J, O, Mi are Wigner rotation matrices depending on ther polar components in the space-fixed (SF) frame. The diagonalmatrix elements of the cluster Hamiltonian read

hI jHeff jIi ¼ ev þ EFCI-NO n2Sþ1Lð�Þðg=uÞ� �

þ Bv hn2Sþ1Lð�Þðg=uÞjL?2jn2Sþ1Lð�Þðg=uÞi � O2

h�S2 þ SðS þ 1Þ þ JðJ þ 1Þ

�(4)

where ev is the energy of the diatomic in the vibrational level |viand hL>2i = hLx

2 + Ly2i (see ESI†). Further symmetry considera-

tions allow us to define parity-adapted basis functions,65

jJM; n2Sþ1Lð�Þðg;uÞðp� ;OÞiv ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2� dL0dS0p jvi

� n2Sþ1Lð�Þðg;uÞ;L;Si�� J;O;Mi�

� n2Sþ1Lð�Þðg;uÞ;�L;�Si�� J;�O;Mi�;

(5)

where the p� blocks have parity, �(�1)J�S+s with s = 1 for S�

helium states and s = 0 otherwise.One of the main approximations involved in earlier QC-like

approaches is the decoupling of the diatomic rotation from the

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He motion. Importantly, the FCI-NO implementation has beenenhanced so that the Coriolis non-adiabatic couplings betweenthe ground and excited solvent states can be now incorpo-rated.51 The inclusion of these couplings makes the diatomicrotational term, Bvj2RBv[(J � S) � L]2, no longer diagonal inthe Hund’s case a basis. This way, the two Coriolis termsBv(L+S� + L�S+) and �Bv(J+L� + J�L+) (the so-called L-decouplingoperator) of j2 induce a mixing between L and L � 1 FCI-NOstates. On the other hand, the term �Bv(J+S� + J�S+) causes theenergy splitting of spin-rotational components arising from thesame fermionic FCI-NO state with S a 0 (see ESI† for explicitexpressions of the Hamiltonian matrix elements).

In order to calculate the spectral line positions, intensitiesand broadenings, the same approach as in ref. 42 was adoptedwith a few modifications due to the inclusion of Corioliscouplings. Briefly (see ESI† for details), a Boltzmann distribu-tion of the ground and the excited states at a given temperatureis used and the average over the initial rotational states isperformed. Therefore, for a fixed energy of the incident photon%ho0, a line of intensity,

IfiðTÞ /e�ðei=kTÞPi

e�ðei=kTÞ1

2Ji þ 1

XMi

jhf jm0jiij2 (6)

appears at an energy %hofi = %ho0 � (ei � ef) of the scatteredphoton, i(f) being a collective index denoting the quantumnumbers of the initial (final) state.

Finally, a vibrational predissociation (VP) process isassumed to be responsible for the spectral broadenings (seealso ref. 43). First, angular dependent VP widths for theHe–dopant dimer are obtained using the adiabatic angularmodel from Beswick and Delgado-Barrio, as described inref. 66. Hereafter, the angular helium densities of the differentquantum states are used to average the angular dependent VPrates, and the stick lines were broadened with Lorentzians ofthe calculated widths (see ESI† for the details).

3 Bosonic versus fermionic many-bodystates of helium clusters with molecularchlorine inside

As an illustrative application of the quantum-chemistry ansatz,let us consider the many-body helium states of (3,4He)N–Cl2

(N r 4) with the emphasis on their fermionic or bosonic quantumnature as the number of helium atoms increases. The presentresults provide a natural supplement to our previous letterpaper on the same system with the number of helium atomsfixed to four.51 They also complement a previous study,49

without including the 4He isotope and the Coriolis couplingsbetween helium many-body states. In addition, we use here anab initio He–dopant PES more accurate than that employed inref. 49 (see ESI†). This PES comprises two minima (see Fig. 3):a global minimum of B �45.8 cm�1 at a linear configuration(Re = 4.16 Å) and a secondary minimum at a T-shaped configu-ration of B�44.2 cm�1 (Re = 3.39 Å). The analytical form of this

PES was obtained by fixing the internuclear Cl–Cl distance tothe equilibrium value, as described in the ESI.† The three-dimensional analytical form reported by Takayanagi et al.67 wasused to obtain the VP rates (see ESI†). More details of theFCI-NO calculations, total energies, and hL>2i expectationvalues for ground and excited many-body states are providedas ESI.† An index Nr has been added to the state notations

(n 2Sþ1Nr

Lð�Þðg=uÞ) to indicate the approximate number of helium

atoms on the equatorial belt around the dopant.

3.1 Ground states

The cluster size evolution of the ground-state (GS) energies isschematically illustrated in Fig. 4 for 3,4HeN–Cl2 (N r 4).Natural orbitals and associated occupation numbers for theground state with N = 3 and 4 are represented in Fig. 5,indicating the values of the angular momentum projectionalong the molecular axis l. It can be noticed from Fig. 4 thatalthough the global minimum is located at a linear He-moleculeconfiguration (see Fig. 3), zero point energies reverse the orderof stabilities so that ring-shaped densities are obtained for thelowest energy states of (3,4He)N–Cl2 (N r 3) clusters. Therelevant orbitals are also ring-shaped with the number of lobesgiven by the l value (see Fig. 5). From this figure, it can also benoticed that natural orbital occupation numbers follow aGaussian-type dependence on l for N = 3 and the 4He isotopewhile the 3He counterparts are essentially either singly occu-pied or unoccupied.

Despite the delocalized nature of the helium wave-function,many aspects can be easily understood by considering that thehelium atoms behave as if they were localized in an N-membered(finite-width) quantum ring. The one-particle energies and wave-

functions of this ring are e1sg+ B||

effl2 and f1sgðR; yÞ1=

ffiffiffi2p

eilf,

respectively, where e1sgis the energy of the lowest-energy

ring-like 1sg orbital and B||eff is an effective rotational constant

for the axial rotation of one helium atom.25,47,49 It is equal to

Fig. 3 Contour plot of the He–Cl2(X) PES as a function of the distance of the Heatom from the Cl2 center of mass (R) and y, the angle that R forms with the bonddirection. The minimum energy path of the PES along y is also shown (high-lighted in red).

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the average1

2mR2

� �over the 1sg orbital.25 The width of this

quantum ring is determined by the y and R-delocalization of the1sg orbital (about 601 and 3 Å, respectively). The degree of thequantum ring delocalization is provided by the iso-probabilitydensity surface represented in Fig. 5 for N = 3 and l = 0.

From Fig. 4, it can be noticed that ground states are alwaysS-type states (L = 0). For N = 3 and the 3He isotope, the groundstate attains the maximum value for the total spin. For maxi-mum spin states, the antisymmetry condition for the fermionicwave-function is imposed into its spatial part. Therefore, thespin part of the wavefunction can be ignored and fermions in

Fig. 4 Schematic illustration showing the quasi-linear dependence of the ground-state FCI-NO energies with the number of 3He (highlighted with a red line) or 4He(highlighted with a blue line) atoms. The excited-state energy levels are shown together with their correspondence with either ring-shaped densities (represented inred) or ring-shaped densities with two lobes at the poles of the molecules (depicted in green and blue). The density highlighted in blue corresponds to the cluster withtwo 3He atoms both in the ring and at the dopant poles. Single S states are represented with black lines while two-fold degenerated states with L > 0 (P, D, . . .) aredepicted in blue. The quasi degeneration of fermionic states with the same symmetry and different total spin is indicated according to the color palette above. Totalenergy values are tabulated as ESI.†

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maximum spin states with the same sz component are referredto as spinless fermions. Due to the l-dependence of the single-particle energies, the most contributing configuration in thelowest-energy wave-functions is a Slater determinant in whichall the orbitals with l values from �(N � 1)/2 to (N � 1)/2 aresingly occupied, as shown in Fig. 5 for N = 3. In our case, thisconfiguration contributes more than 90%. This is indeed theexact solution in the limiting case of strictly one-dimensional(1D) quantum rings frequently used to characterize electronsconfined in ring-shaped configurations by electromagneticfields.53 In contrast, as can be easily demonstrated for thesimplest case of two spinless bosons in a ring with a repulsivedelta-type interaction,54 the exact wave-function cannot be expressedwith a single term but rather has a multi-configurationalcharacter. As pointed out by Viefers et al.,53 a repulsive deltainteraction for spinless fermions in a 1D ring does not play anyrole because of the Pauli’s principle, and they are termednoninteracting fermions. Contrarily, the repulsive delta inter-action plays an important role for spinless bosons sothat they not only occupy the l = 0 orbital (minimizing thekinetic energy) but are also excited to orbitals with higherl values, as shown in Fig. 5. However, the He–He interactionis highly repulsive at short distances and attractive after the

hard-core wall.68 This is one of the reasons why total GSenergies have a quasi-linear dependence on N for 3He atomsconfined in rings. One-particle energies also scale quasi-linearly. Using the finite width ring model, the total energyfor an odd number of 3He atoms N can be expressed as,69

EGS(N) = N � e1sg+ (B||

eff/12)N � (N2 � 1) + hVHe–Hei,

showing that the quasi-linear behaviour arises from the can-cellation between the second and third terms at the right of thisequation, with hVHe–Hei being the expected value of the He–Heinteraction.47,50 This simple model qualitatively explains theN-dependence of hVHe–Hei values found in our case and pre-vious FCI-NO studies.47,50 It makes an important difference tothe electronic structure case, in which the pairwise Coulombinteraction is purely repulsive.

Considering the case of N = 4 for both isotopes (see Fig. 4),we can notice that a second solvation regime starts so that thefourth helium atom flows towards the ends of chlorine thatbegins to be fully wrapped by helium density. This is alsoreflected by the shape of the natural orbitals (see Fig. 5) offour 4He atoms. The ring-like orbital with l = 0 (the 1sg orbital)combines with the capping orbital located at the moleculepoles (the highest energy independent-particle orbital for 4Hein Fig. 4), forming symmetric and antisymmetric combinationsrepresented at the lower panel of Fig. 5. The increasing numberof helium atoms in the ring makes the effective on-siterepulsion in the 1sg orbital larger so that it combines withthe capping orbital, in which the helium atoms are placedfar away, to reduce the large on-site repulsion in the ring. Thisway, its occupation number increases from 1.60 to 1.70.However, the occupation number of the natural orbital arisingfrom the antisymmetric combination (referred to as 2sg) is themost affected (from almost zero to B0.80). This holds also forthe 3He isotope (see Fig. 5), with the occupation numberbecoming close to unity. The main effect is thus the transferof the fourth helium atom from the ring to the dopant poles. Asextensively discussed in ref. 49 for 3He, the small overlappingbetween helium densities located at the chlorine ends makesthe 2sg orbital nearly degenerate with the 1su orbital (see ESI†),which is composed of two lobes of opposite signs at the dopantpoles. This is the reason for the small energy differencebetween the X3Sg

� ground-state and the ungerade paritycounterpart (the 13Su

�, see ESI†). As the cluster grows insize, however, it can be expected that the increased densityalong the molecular axis favors the stabilization of gerade paritystates.

Due to the lighter mass of the 3He isotope (see Fig. 4), itsflowing towards the Cl2 ends is already apparent for the firstexcited state at N = 3 (see Fig. 4), which is almost degeneratewith the ground state. As can be seen in this figure, theglobal energetic trends for ground-state 4He clusters aresimilar to the fermionic case. Thus, the energies behavequasi-linearly with the number of 4He atoms and, as expecteddue to the heavier mass of the 4He isotope, are lower. Thequasi-linear behaviour of ground-state energies was also foundon (4He)N–Br2(X) clusters (N r 5) and J = 0 by Felker using a

Fig. 5 Upper panel: occupation numbers of the most relevant natural orbitals ofground-state 4He3–Cl2 (depicted with red boxes) and 3He3–Cl2 (represented withgreen boxes) clusters. The occupation numbers are given as a function of thenatural orbital angular momentum projection (l). Gaussian (red line) and box(green line) functions have been superimposed as an approximation for theoccupation number dependence on l. Equiprobability density surfaces of theseorbitals are also represented. The probability values have been selected to beabout half of the maximum value. Red and blue colors indicate positive andnegative lobes of the orbitals, respectively. The positions of the Cl atoms areindicated by green balls. Bottom panel: the same for four 3He or 4He atoms andthe two relevant orbitals with l = 0. The occupation numbers of natural orbitalswith l > 0 are very similar (with differences below 0.01) to those for three heliumatoms (upper panel) and thus are not shown.

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configuration-interaction-like implementation.59 For both iso-topes, the lowest-energy states are of S symmetry (L = 0). Theaverage pair He–He energies are also negative in both cases,but scale with a larger constant rate for 4He. As a result,total energies are a bit lower than N � e1sg

, in contrastwith the fermionic case. As found in ref. 40, one- and two-particle densities are more delocalized for the 3He isotope.As mentioned above, the isotope substitution affects largelythe GS wave-function structure at a variance with the shapesof the natural orbitals (also see Fig. 5). For 4He, the maincontribution to the GS wave-function naturally arises fromthe configuration where all the bosons occupy the samenatural orbital, with the percentage going from 80% for N = 2to 41% and 19% for N = 3 and 4, respectively. This percentagedepends on the interplay between the He–dopant and theHe–He interaction, decreasing as the latter becomes morepronounced.

3.2 Excited states

Excited energy levels are represented in Fig. 4 as a function ofthe cluster size. The following trends can be noticed: (1) as thenumber of helium atoms in the central ring increases, theexcitation energy gap from the ground state monotonicallydecreases for 3He while the opposite holds for 4He; (2) thefirst excited state is of P symmetry for 3He (L = 1); (3) thenumber and degeneracies of 3He excited states lying very closeto the ground state increase with the cluster size so that forN = 4 the formation of different bands with an energy gap inbetween can be distinguished; (4) contrarily, the energiesof 4He states with L a 0 become increasingly higher as thecluster grows in size, with a F state (L = 3) being the lowest-energy one.

Focusing first on the 4He isotope, the main trends can beeasily explained on the basis of the finite width ring modeldiscussed above. As can be seen in Fig. 4, the collectiverotational state in which the L value is equal to the numberof 4He atoms in the ring Nr becomes the lowest state for N Z 3,with the excitation energy scaling linearly with that number asNr B||

eff. This is more clearly noticed in Fig. 6 where the energiesof the lowest many-body states at each L value (the so-calledyrast states YS) are represented for N = Nr = 3. The lineconnecting the yrast states (and then termed the yrastline) shows a strong non-monotonic behaviour with suddenincreases L = 1, Nr � 1.51 The reasons for this behaviour werediscussed in ref. 25 by considering spinless bosonic 4He atomsand para-H2 molecules with CO2 as the dopant species. In thiswork, we put emphasis on the fermionic 3He case, using thenatural orbital picture. For this purpose, we have also plotted inFig. 6 the yrast states of 3He atoms either in the maximum spinstate (the so-called spinless case), or in the lowest-energy spinstates at each L value. It is apparent that the yrast line forspinless 3He atoms shows a very similar profile to the spinless4He case. In the limiting case of 1D quantum rings with apairwise delta interaction, the yrast lines of spinless bosonsand spinless fermions are indeed identical if the number offermions is odd.54 As discussed above, the most contributing

configuration to the GS wave-function of 3He atoms with themaximum total spin can be written as |(�1)1(0)1(1)1i, in whichthe singly occupied natural orbitals are labelled according totheir l values. Due to the quadratic l-dependence of theirenergies (scaling as BB||

effl2), the lowest-energy many-body states

are formed by occupying natural orbitals with consecutive lvalues. Accordingly, the many-body state closer to the GS stateshould bear the dominant configuration |(�1)1(1)1(2)1i withL =

PiNli = 3. As can be seen in Table 1, this configuration

contributes to more than 90%. More generally, increasing thel value of each natural orbital by any integer n,

CL¼nN fif gð Þ ¼ exp inXni

fi

!CGS

L¼0 fif gð Þ; (7)

the GS wave-function part depending on the internal modes isleft unmodified (e.g., the He–He correlation function).Obviously, it acquires the factor eiLF+ with F+ =

PiNfi/N,

corresponding to an overall rotation with an energy increaseof B||

eff/N � L2.25,47,49 For 4He, the states with L = nN (see Fig. 6)are formed by increasing the l value of each natural orbital

Fig. 6 Energies of doped clusters with three helium atoms located on thecentral ring as a function of L. Blue (red) circles indicate the energies of 4Hestates (maximum spin 3He states with S = 3/2) while red bullets show those of3He states with lower spin (S = 1/2). The solid lines connecting the yrast states for4He (highlighted in blue) and maximum spin 3He (depicted in red) have beenshown to indicate the trends. The function BB||

eff/3 � L2 is represented withdashed lines and as shaded areas.

Table 1 Most contributing configurations to the many-body wave-functionswith N = Nr = 3 and 3He

(L,S) Most contributing configurations Weight (%)

(0,3/2) ((�1)1(0)1(1)1) 93(1,3/2) ((�1)1(0)1(2)1)) 90(2,3/2) ((�1)1(1)1(2)1) 87(3,3/2) ((0)1(1)1(2)1) 93(1,1/2) ((0)2(1)1) 55

((�1)1(1)2) 17(2,1/2) ((0)1(1)2) 34

((0)2(2)1) 27

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by n so that the multi-configurational make-up of the GSwave-function is preserved. Hence, the Gaussian-type profilefor the natural orbital occupation numbers as a function of l(see Fig. 5) would be simply shifted by l = n, as expectedfrom the collective nature of low-lying excitations for 4He.Contrarily, for L a nN, the contributing configuration is setup by leaving an empty l value. For example, the lowest-energystate with L = 1 has the dominant configuration |(�1)1(0)1(2)1i,as can be seen in Table 1. Within the 1D ring model, theexcitation energy would be N � B||

eff instead of the valueassigned to an overall rotation (B||

eff/N). This is indicating theinvolvement of the He–He vibrational mode. As can be seen inFig. 1 of ref. 51, it is also reflected in the shape of the two-particle densities as a function of the inter-particle f1–f2

coordinate. Actually, the excitation energies of maximum spinstates with L a nN are higher (see Fig. 6). In fact, the correla-tions effects are not the same in these states, and then thenatural orbitals differ. Therefore, beyond a certain critical size,the lowest-energy excited state has a L value of Nr, as apparentin Fig. 6 for Nr = 3. On the other hand, from the bosonic energylevels represented in Fig. 4, it can be noticed that Nr = 3 isindeed the critical size, with the F-symmetry state (L = 3) as thelowest-energy ring-like state. Accordingly, for Nr = 2, bosonic Pand D-symmetry states are almost degenerate. As mentioned inour previous letter,51 the addition of helium atoms at thedopant poles, leaves the yrast line almost unperturbed, as canbe seen in Fig. 7 by following the position of the levels high-lighted in green. As a result, the F-symmetry state is also thelowest rotational state for four 4He atoms with an excitationenergy of 3 B||

eff.When lower spin states are considered for three 3He atoms

in the ring (see Fig. 6), the yrast line changes completely: overallrotational states are now allowed at each L value with the resultthat the energy of the lowest excited states scales as (B||

eff/3)L2.This holds also for two or four atoms in the ring, resulting inthe (B||

eff/Nr)L2 scaling that can be inferred from Fig. 7. Within

the ring model, it is clear that it is possible to fill naturalorbitals with consecutive l values (at any L) once the spindegree of freedom is accounted for and doubly occupied

orbitals are allowed. Assuming weakly interacting fermions,at L = 1, the most contributing configuration should be|(0)2(1)1i, giving S = 1/2 for the lowest-energy state at L = 1(see Fig. 6). As can be noticed in Table 1, |(0)2(1)1i contributesto more than 55%. The same holds for the yrast state for L = 2,with |(1)2(2)1i as the dominant configuration and S = 1/2. As aconsequence of the strong correlation between 3He atoms,however, the weight of the |(1)2(2)1i configuration is only34%. Similarly to the electronic structure in quantum rings,the analysis of those correlation effects can be further clarifiedon the basis of Hubbard’s Hamiltonian model and the anti-ferromagnetic Heisenberg’s Hamiltonian, as will be shown for3He elsewhere.

To summarize, as can be seen in Fig. 7 for four 3He atoms,the lowest-excitation energies of Nr-membered rings can beexpressed as the sum of the rotational energy corresponding toa single particle with Nr times the reduced mass of a single 3Heatom, B||

eff/Nr � L2, plus a Heisenberg’s term accounting for thesplittings of the possible spin states at each L value. In Fig. 7,we can also discern higher energy levels involving the activationof He–He vibrational modes for Nr = 2 and 3. Since thefrequencies of these modes increase with Nr, no vibrationallevels for the band with Nr = 4 are observed. On the other hand,for four 4He atoms, the rotational energies corresponding toNr-membered rings are given by B||

effNr � L2, as a single particlewith a (1/Nr) the reduced mass of a single 4He atom. The energylevels associated to vibrational He–He modes appear at higherenergies, as in the 3He case. However, as expected from itsheavier mass, the frequencies associated to the He–He modesare smaller for 4He.

So far, our goal has been to analyse short-range correlationeffects between 3He and 4He atoms in ring-shaped arrange-ments (perpendicular to the molecular axis), and the influenceof the spin quantum statistics on the ground- and excitedmany-body states (i.e., associated to axial rotations andHe–He vibrations within the delocalized ring). Also, it has beenshown that the filling of the second global minimum (at thedopant poles) with helium atoms leaves the low edge of themany-body spectra associated to the equatorial Nr-memberedrings almost unperturbed. Making an analogy with electronicstructure problems, they could be identified with the coreelectrons of molecular systems. Concerning the frequenciesof the He–dopant bending mode, opposite trends are alsofound for 4He and 3He isotopes at the completion of theequatorial ring. Thus, for four helium atoms, the activationenergies to these states (shown by red lines in Fig. 7) are largerin 4He by more than a factor of two. The simple occupationscheme of natural orbitals for 3He (with occupation numbersclose to unity, see Fig. 5) renders the low-lying edge of theexcitation spectrum essentially composed of single-particleexcitations. In this effective one-particle picture, the transferof a single 3He atom from the ring to the poles can be mappedto the single-particle excitation between the low-lying 1 pu and1 su orbitals (see ESI†). Contrarily, low-lying excitations for the4He isotope are inherently collective, as discussed for the caseof rotational excitations.

Fig. 7 Many-body energy levels of (4He)4–Cl2 (left-hand panel) and (3He)4–Cl2(right-hand panel) clusters for different L values. Levels of states with two, three,and four helium atoms in the ring are shown by red, green and blue lines,respectively.

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4 Isotope and cluster size dependence of thedopant vibro-rotational Raman spectra

To show how the cluster size dependent properties of groundand excited solvent states are reflected in spectroscopic proper-ties, we analyze the vibro-rotational Raman spectra of molecu-lar chlorine. As in our previous letter,51 the spectra simulationswere performed at temperatures of 1.0 and 0.37 K (1.0, 0.37 and0.07 K) for the 4He (3He) isotopes. Temperatures of 0.37 and0.07 K have been measured in doped 4He and 3He droplets (seeref. 11), respectively. Additionally, we present the spectra atT = 1 K for the following two reasons: (1) higher temperatureshave been measured in small doped 4He clusters (0.2–0.80 Kfrom ref. 70); and (2) it allows us to highlight the differencesupon lowering the temperature to that attained by heliumdroplets. Within the chosen temperature range [0.07–1.0 K],the solvent helium states are assumed to follow a Boltzmanntype distribution (see eqn (6)). Fermi–Dirac (Bose–Einstein)type distributions should be considered, however, at even lowertemperatures (in the mK scale) for 3He (4He). A large number ofexcited solvent states was included (36 states for four 3He atoms)to achieve convergence in the line positions and intensities ofthe spectra profile at T = 1 K (see eqn (6)).

The calculated spectra (see Fig. 8) display the scatteredphoton intensity as a function of the energy loss between theincident and the exiting photons, %hw0� %hwfi. These energies aregiven relative to the bare dopant transition (J,v) = (0,1) ’ (0,0),554.37 cm�1 (see also ESI†). Fig. 8 shows the evolution of thespectra profiles as a function of the number of helium atoms N.The spectra of the bare dopant are also represented in theupper panel of this figure. For 4He, the lowest-energy ring-likeS states contribute the most. With N = 4, the FCI-NO energy ofthe S state with one helium atom at the dopant poles (seeFig. 4) is lower by 0.4 cm�1. Nevertheless, the hL>2i values arealways smaller for ring-like states (see ESI†), owing to theirlower density along the molecular axis. Hence, the rotationaldiagonal term (see eqn (4)) reverses the order of stabilities. ForN = 1 and 2, in addition to the Q, R, and S branches from thelowest-energy S state, P and R branches also appear due to thecontribution of P states and the anisotropic polarizabilitycomponent. However, the P states contribution is dampeddown at larger cluster sizes due to the high frequencies of theimplied stretching He–He modes when the symmetric condi-tion for the wave-function is imposed (see Fig. 6). For N > 2, onlyS states with L = 0 provide significant intensities. Still, a weakspectral signal from the collective rotational F state for N = 3can be distinguished (see Fig. 8). Any signature from statesinvolving the axial rotation of the 4He atoms (L > 0), however,drops out at the next cluster size (N = 4). As can be deducedfrom the explicit expression for the dipole matrix elementshf|m0|ii (see ESI†), when only states O = 0, branches with DJ = �1(P and R branches) become forbidden and diatomic-like selec-tion rules, DJ = 0, �2, hold when the ring gets filled with N = 4.This could be interpreted as the parallel superfluidity responseto rotations around the molecular axis at the completion of thering.22 With N = 4, signatures of excited S states implying the

He-bending modes are not observed in the low resolutionspectrum represented in Fig. 8. On the other hand, the negli-gible role of Coriolis couplings (see ESI† for details) indicatesthe adiabatic following of the dopant rotation by the 4He atoms.This is due to both the small value of the dopant rotationalconstant and the high energy of P states.

When the temperature is lowered to that attained by 4Hedroplets (0.37 K), no significant differences are found for N = 4(see Fig. 8). Naturally, the intensities of transitions with Ji > 0are damped down. For N = 3 (not shown for the sake ofsimplicity) only contributions from S states survive. Contrarily,P states contribute to the spectra for N = 2 and 1 at either T = 1or 0.37 K. Therefore, on the basis of our present results, it canbe concluded that the diatomic exhibits a free-rotor like beha-viour at the completion of the first central ring with four 4Heatoms, as apparent by comparing the topmost spectra in Fig. 8.

Moving to the cluster size evolution of the spectra for the3He isotope at T = 1 K, displayed in the bottom panel of Fig. 8,we can summarize the differences with the 4He isotope asfollows. First, the contribution from P states is not lifted asthe cluster size increases. As already mentioned, the spindegree of freedom for the fermionic wave-function rendersthe appearance of pure rotational states with L = 1 possible.The contribution from P states becomes unimportant, how-ever, when lowering the temperature to that attained by 3Hedroplets (T = 0.07 K), as can be noticed in the topmost spectraof the bottom panel in Fig. 4. Second, the fermionic spectra arecomposed of several Q-type branches, with some O, P, R and Sbranches appearing superimposed on them, at T = 1 and (forN > 2) at 0.07 K also. This is due to the major role played byCoriolis couplings and, ultimately, to the spin degree of free-dom in 3He. As a result, the spectra in 3He are composed of alarger number of branches than in 4He and, moreover, thesebranches are broader. This is a common characteristic of all thespectra simulations carried out in 3He with the quantum-chemistry ansatz.42,43,46,49,51 As first remarked in our letter,51

however, the Coriolis couplings render them more complex andtheir role becomes even more important at the very lowtemperature of 3He droplets (0.07 K). Furthermore, theseCoriolis couplings are responsible for an effective reductionof the diatomic rotational constant Bv (see ESI†). Thus, for themost contributing X3Sg

� state with N = 4 (see Fig. 8), areduction of Bv by a factor of 0.7 can be estimated. Lastly, thesecouplings render the (otherwise masked) parity-selection rulesevident in the spectra (see below).

In order to further clarify the Coriolis coupling effect in 3He,let us now focus on the case with N = 3 (see Fig. 9 and ESI† foradditional details). In ref. 9, we can clearly notice the highmixing between spin-rotational levels induced by J-dependentCoriolis couplings. For the lowest J value (1/2), some spin-rotational levels become nearly-degenerate. With increasing J,the energy splittings become clearly apparent, with the wholeenergy band spreading out, and the comprised spin-rotationalstates switching their relative position. As a result, spin-rotationallevels with J = 3/2 turn out to be lower than those arising fromthe energy band with J = 1/2. The spin-decoupling Coriolis

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operator �Bv(J+S� + J�S+) is responsible for this behaviour.Its role is even more evident at the very low temperature of3He droplets (see the bottom panel of Fig. 8). This can be simplyrationalized by using a reduced two-state model with theground- and the first-excited states (the X 4

3Sg� and 1 2

3Pu

states). As can be seen at the left-hand side of Fig. 10, theFCI-NO energies of those states are almost identical (see ESI†).For J = 3/2, when the diagonal Hamiltonian term from the

diatomic rotation Bvhj2i is accounted for (see eqn (4)), one of thespin-rotational levels arising from the 1 2

3Pu state becomes thelowest-energy level (see Fig. 10). On the other hand, the X 4

3Sg�

gives rise to a pair of degenerate levels for either v = 0 or 1 (seeeqn (5)), namely,

f�0��

v¼ 3

2;X4S�

ðgÞ p� ;32

��

v

; f�1��

¼ 3

2;X4S�

ðgÞ p� ;12

��

v

:

Fig. 8 Upper panel: vibro-rotational Raman (v = 1 ’ 0) spectra of (4He)N–Cl2(X) clusters (perturbed Cl2) at T = 1.0 K (lower panel) and 0.37 K (upper panel). Thespectra of the isolated molecule are also shown for comparison purposes. Lower panel: vibro-rotational Raman (v = 1 ’ 0) spectra of (3He)N–Cl2(X) clusters (perturbedCl2) at T = 1.0 K (lower panel) and 0.07 K (upper panel). The energies are given relative to the bare Cl2 transition (J,v) = (0,1) ’ (0,0), 554.37 cm�1. See text for thenotation of the spin-rotational levels.

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This degeneration is lifted when the Coriolis couplings fromthe �Bv(J+S� + J�S+) term are included. By diagonalizing theHamiltonian matrix, the four states depicted at the right-handside of Fig. 10 are obtained. In Hund’s case (b) basis, which isbetter adapted to S states with S > 0, these four levels corre-spond to the rotational states with N = 1,. . ., 4, with N as thequantum number associated to the total angular momentum

without spin. As apparent from the right-hand side of Fig. 10,the state |c0iv emerges as the lowest-energy level. Due to parity-based selection rules, the transition between spin-rotationallevels with opposite parity are forbidden. As a result, the lowest-energy level |c0iv=0 can only be promoted to the two levels|c1iv=1 and |c3iv=1. Accordingly, two maxima are present in thespectra at energy values corresponding to their O splittings of2.8 and 0.5 cm�1 respectively (De1 and De2 in Fig. 10). In short,the large O-splittings arising from the spin-decoupling operatorrender the parity-based selection rules evident in the spectralprofile, causing the appearance of two Q-type branches atT = 0.07 K (see Fig. 8). At higher temperatures, the X 4

3Sg� and

1 23Pu states clearly compete to dominate the spectrum profile.

Upon lowering the temperature, however, only transitionsbetween levels arising from the lowest-energy S state survive.We also notice that branches forbidden for the isolated dopant(e.g., P and R-types branches) are clearly evident in the spectrawithin the chosen temperature range [0.07–1.0 K]. This makes animportant difference with the 4He isotope. However, at thetemperature attained by 3He droplets (0.07 K), it is also evidentthat this difference is caused by the Coriolis couplings, and notby the involvement of purely rotational states with L a 0 (i.e.,P states). This highlights the importance of considering Corioliseffects and thus the spin state of ground-state doped 3Heclusters. Since the ground-state for N = 2 is a singlet state(S = 0), the spectrum at 0.07 K (see Fig. 8) is very similar to thatof the isolated diatomic. Conversely, ring-like ground states(L = 0) for an odd number of 3He atoms N have the maximumspin so that the Coriolis S-decoupling operator plays a major rolefor N = 3. This way, the spectrum is richer for N = 3 than for N = 2at either T = 1 or 0.07 K. Finally, we notice that the spectrum withN = 4 holds three Q-type branches at T = 0.07 K while only twoQ-type branches are present for N = 3. The third Q-type transitionarises from the ungerade parity counterpart of the ground-state(the 1 4

3Su� state). As already mentioned (see Section 3), these two

states have very similar energies (see ESI†). Hence, in agreementwith the last analysis of experimental measurements11 andKS-DFT calculations,40 the enrichment of the spectral profilesin 3He is not only determined by the first ring around the dopantbut the 3He atoms along the dopant axis also modify themolecular spectrum. Larger cluster sizes are necessary to unravelhow many additional 3He atoms further contribute.

5 Concluding remarks and future perspectives

This work has been focused on the problem of describing thespectroscopy of molecular impurities embedded in heliumclusters as well as the ground and excited helium states,considering fermionic 3He and bosonic 4He isotopes on anequal footing. After motivating the quantum chemist’s view ofthese clusters, different methodologies using the quantum-chemistry ansatz have been briefly outlined. As the benchmarkimplementation, the Full-Configuration-Interaction Nuclear-Orbital (FCI-NO) approach has been applied to (He)N–Cl2(X)clusters (N r 4).

Fig. 9 Graphical representation of spin-rotational energy levels of 3He3–Cl2(X)as a function of the total angular momentum quantum number J, withoutincluding Coriolis couplings (highlighted with red lines) and including them(depicted with black lines). Blue lines have been drawn to follow the correlationbetween both sets of spin-rotational levels, with the coefficients combining thembeing larger than 10�4.

Fig. 10 Schematic representation of the main levels involved in the descriptionof the vibro-rotational Raman spectrum of the 3He3–Cl2(X) cluster at T = 0.07 K.

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The quantum-chemistry ansatz as applied to doped heliumclusters can be viewed as a divide-and-conquer strategy whichallows a systematic approach to the exact solution of the(nuclear) many-body Schrodinger equation within the adiabaticapproach for the dopant vibrational internal modes. It enablesthe use of the familiar orbital molecular picture in order tounravel correlation effects from the many-body 3He states andtheir differences/analogies with the 4He counterparts. Within theframework of the FCI-NO approach, the comparison betweenindependent-particle and natural orbital wave-function represen-tations has been shown to be particularly useful in analysingcorrelation effects. Thus, for bosonic doped 4He droplets, thenatural orbital with the highest occupation number plays the roleof the Bose–Einstein condensate ‘‘macroscopic wave-function’’.71

On the other hand, the patterns identified in the many-bodyexcitation spectrum can be useful in designing the guiding wave-function for QMC calculations. Moreover, the inclusion of theCoriolis couplings between the ground and excited helium stateswith the FCI-NO approach allows the identification of either theadiabatic following of the dopant rotation by the helium atoms ortheir decoupled motion.

As a future perspective, the development of parameter-freeQC-like implementations such as the FCI-NO method, allowinga complete control with respect to the approximations whichare being made (e.g., the cutoff in the basis set), is highlydesirable. Our results demonstrate that the inclusion of short-range correlation effects is crucial in getting physically mean-ingful results. On the other hand, very large basis set sizes andspecial diagonalization techniques are necessary to describethese effects with the FCI-NO method. The design of optimizedbasis sets is essential to render its application to larger clustersizes (up to about 10 helium atoms) possible. The analogiesidentified at the many-body excitation energy spectra of 3Heatoms and electrons confined in ring-shaped arrangementssuggest the application of Hubbard’s Hamiltonian models.A proper description of the strong correlations for even largercluster sizes (a few tens of helium atoms) necessitates explicitlycorrelated methods such as, for instance, the transcorrelatedHamiltonian approach.72 By using a single configurationJastrow ansatz, nuclear correlation is included through HF-likeequations and, therefore, the effective orbital picture ispreserved.38 The problem of describing single molecules insidehelium nanodroplets (with a few hundreds of helium atoms)could be reduced by applying DFT-based embedding techni-ques, as already done in electronic structure problems,73,74

taking advantage of the orbital-free DFT formulation developedby Barranco and collaborators.31 The application of the quan-tum-chemistry ansatz with the FCI-NO approach to a clustercomposed of para-H2 and para-H2O molecules with singlemolecules inside will also lead to future research.

Acknowledgements

We thank Manuel Barranco and Jesus Navarro for fruitfuldiscussions, and for providing us with ref. 40 prior to itspublication, and Tetsuya Taketsugu for sending us a Fortran

code with the He–Cl2(X) potential energy surface. This work hasbeen performed under the grant no. FIS2011-29596-C02-01 fromthe Ministerio de Economıa y Competitividad, Spain, and theCOST Action CM1002 ‘‘Convergent Distributed Environment forComputational Spectroscopy’’ (CODECS). N. F. A. has beensupported by a predoctoral JAE fellowship from the CSIC. Thecalculations were performed at the Cesga Super-ComputerCenter (Galicia) and the Computer Centers at the IFF (CSIC)and the Centro Tecnico de Informatica (CTI, CSIC).

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Solvent states and spectroscopy of doped helium clusters as a quantum-chemistry-like problem