Bound states of dipolar molecules studied with the Berggren expansion method
aa r X i v : . [ phy s i c s . a t m - c l u s ] M a r Bound states of dipolar molecules studied with the Berggren expansion method
K. Fossez, N. Michel,
2, 3
W. Nazarewicz,
2, 3, 4 and M. P loszajczak Grand Acc´el´erateur National d’Ions Lourds (GANIL),CEA/DSM - CNRS/IN2P3, BP 55027, F-14076 Caen Cedex, France Department of Physics & Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Institute of Theoretical Physics, University of Warsaw, ul. Ho˙za 69, 00-681 Warsaw, Poland Grand Acc´el´erateur National d’Ions Lourds (GANIL),CEA/DSM - CNRS/IN2P3, BP 55027, F-14076 Caen Cedex, France (Dated: September 26, 2018)Bound states of dipole-bound negative anions are studied by using a non-adiabatic pseudopotentialmethod and the Berggren expansion involving bound states, decaying resonant states, and non-resonant scattering continuum. The method is benchmarked by using the traditional techniqueof direct integration of coupled channel equations. A good agreement between the two methodshas been found for well-bound states. For weakly-bound subthreshold states with binding energiescomparable with rotational energies of the anion, the direct integration approach breaks down andthe Berggren expansion method becomes the tool of choice.
PACS numbers: 03.65.Nk, 31.15.-p, 31.15.V-, 33.15.Ry
I. INTRODUCTION
Weakly-bound many-body systems are intensely stud-ied in different domains of mesoscopic physics [1, 2], in-cluding nuclear [3–7], molecular [8–13], and atomic [14–16] physics. In this context, dipolar anions are one of themost spectacular examples of marginally bound quantumsystems [17–38].The mechanism for forming anion states by the long-range dipolar potential has been proposed by Fermi andTeller [39], who studied the capture of negatively chargedmesons in matter. They found that if a negative mesonis captured by a hydrogen nucleus, the binding energy ofthe electron becomes zero for the electric dipole momentof a meson-proton system µ cr = 1 .
625 D. Later this resultwas generalized to the case of an extended dipole with aninfinite moment of inertia [40]. Lifting the adiabatic ap-proximation by considering the rotational degrees of free-dom of the anion [17–22] turned out to be crucial; it alsoboosted the critical value of µ to about 2.5 D. For anionswith µ > µ cr , the number of bound states of the electronbecomes finite, and the critical electric dipole moment µ cr depends on the moment of inertia of the molecule.In the non-adiabatic calculations, the pseudo-potentialwas used to take into account finite size effects, repulsivecore, polarization effects, and quadrupolar interaction.The pseudo-potential method has provided a convenientdescription of binding energy of the electron bound by anelectric dipolar field. Recently, this method was appliedto linear electric quadrupole systems [41]. Some recenttheoretical studies of dipole-bound anions also employedthe coupled cluster technique [42–44].The unbound part of the spectrum of multipolar an-ions has been discussed theoretically in Refs. [45, 46]and Refs. quoted therein. Resonance energies ofdipolar anions have been determined experimentally bylow energy electron scattering off the dipolar molecules [25, 26, 29, 31, 33].Both the long-range dipole potential and the weakbinding of dipolar anions provide a considerable chal-lenge for theory. The impact of the molecular rotation ona weakly-bound electron can be represented by coupled-channel (CC) equations that can be solved by means ofthe direct integration. While this approach correctly pre-dicts the number of bound states of polar anions, it isless precise for treatment of weakly-bound excited states.Moreover, it cannot be used for studies of dipolar anionresonances because the exact asymptotics for a dipolarpotential in the presence of a molecular rotor cannot bedetermined.In this paper, we apply the complex-energy configura-tion interaction framework based on the Berggren ensem-ble [47] to the problem of bound states in dipole-boundnegative anions. The Berggren completeness relation is aresonant-state expansion; it treats the resonant and scat-tering states on the same footing as bound states. Wehave successfully applied this tool to a variety of nuclearstructure problems pertaining to weakly-bound and un-bound nuclear states [48–51] (for a recent review see Ref.[52]). The nuclear many-body realisation of the complexenergy configuration interaction method is known underthe name of the Gamow Shell Model.Resonances do not belong to the Hilbert space, sothe mathematical apparatus of quantum mechanics inHilbert space is inadequate for Gamow states [53], whichare not square-integrable. It turned out that the mathe-matical structure of the Rigged Hilbert Space (RHS) [54–56] can accommodate time-asymmetric processes, such asparticle decays, by extending the domain of quantum me-chanics. The mathematical setting of the resonant stateexpansions follows directly from the formulation of quan-tum mechanics in the RHS [54, 55], rather than the usualHilbert space [56–58].The Berggren ensemble provides a natural generaliza-tion of the configuration interaction for the descriptionof the particle continuum. The complex-energy Gamow-Siegert states [53, 59] states have been used in variouscontexts in nuclear, atomic, and molecular physics [60–73]. Some recent applications of Gamow-Siegert states,also in the context of a CC formalism relevant to theproblem of dipole anions, can be found in, e.g., Refs.[74–78].This paper is organized as follows. The Hamiltonianof the pseudo-potential method is briefly discussed inSec. II. The CC formulation of the Schr¨odinger equationfor dipole-bound negative anions is outlined in Sec. III.Section IV discusses the direct integration method (DIM)for solving the CC problem with a focus on difficul-ties in imposing proper boundary conditions when therotational motion of the molecule is considered. TheBergggren expansion method (BEM) is introduced inSec. V. Section VI specifies the coupling constants ofthe pseudo-potential and other calculation parameters.Salient features of DIM and BEM solutions are com-pared in Sec. VII. The predictions of DIM and BEM forlow-lying energy states and r.m.s. radii of LiI − , LiCl − ,LiF − , and LiH − anions are collected in Sec. VIII. Finally,Sec. IX contains the conclusions and outlook. II. HAMILTONIAN
A dipole-bound negative anion is composed of a neutralpolar molecule with a dipole moment greater than µ cr and a valence electron. The Hamiltonian of the totalsystem can be written as: H tot = H e + H mol + V (1)where H e is the Hamiltonian of the valence elec-tron, H mol is the Hamiltonian of the molecule, and V is the electron-molecule interaction. The many-bodySchr¨odinger equation for H tot couples all electrons of thesystem; hence, an approximation scheme has to be de-veloped.As a first simplification, we assume that the vibrationalmotion of a molecule is much slower than other modes sothat it can be treated in the Born-Oppenheimer approx-imation. The Hamiltonian (1) simplifies considerably ifone considers anions of closed-shell systems. Moreover,if spin is neglected [22], the molecule can be treated as arigid rotor. Note that the energy scales associated withthe rotational motion of the molecule and the motion ofthe weakly-bound valence electron may be comparable.Consequently, there appears a strong non-adiabatic cou-pling between the molecular angular momentum j andthe orbital angular momentum ℓ of the electron. Eq. (1)thus writes within this approximation scheme: H tot = p e m e + j I + V (2)where I is the moment of inertia of the neutral molecule, p e is the linear momentum of the valence electron and m e its mass. The interaction V is approximated by aone-body pseudo-potential V ( r, θ ) acting on the valenceelectron [22, 79, 80]: V ( r, θ ) = V µ ( r, θ ) + V α ( r, θ ) + V Q zz ( r, θ ) + V SR ( r ) , (3)where θ is the angle between the dipolar charge separa-tion s and electron coordinate; V µ ( r, θ ) = − µe X λ =1 , , ··· (cid:18) r < r > (cid:19) λ sr > P λ (cos θ ) (4)is the dipole potential of the molecule; V α ( r, θ ) = − e r [ α + α P (cos θ )] f ( r ) (5)is the induced dipole potential, where α and α arethe spherical and quadrupole polarizabilities of the linearmolecule; V Q zz ( r, θ ) = − er Q zz P (cos θ ) f ( r ) (6)is the potential due to the permanent quadrupole mo-ment of the molecule; and a short-range potential V SR ( r ) = V exp( − ( r/r c ) ) (7)accounts for the exchange effects and compensates forspurious effects induced by the cut-off function f ( r ) = 1 − exp {− ( r/r ) } (8)introduced in Eqs. (5,6) to avoid a singularity at r → r in Eq. (8) is an effective short-rangecutoff distance for the long-range interactions. III. COUPLED-CHANNEL EXPRESSION OFTHE HAMILTONIAN
The eigenfunctions of the Hamiltonian (2) can be con-veniently expressed in the CC representation:Ψ J = X c u Jc ( r )Φ Jj c ℓ c (9)where the index c labels the channel, u Jc ( r ) is the radialwave function of the valence electron in a channel c , andthe channel function Φ Jj c ℓ c arises from the coupling of j c and ℓ c to the total angular momentum J of the anion: j + ℓ = J . Due to rotational invariance of H tot , its ma-trix elements are independent of the magnetic quantumnumber M , which will be omitted in the following.The potential V ( r, θ ) in Eqs. (3 - 7) can be expandedin multipoles: V ( r, θ ) = X λ V λ ( r ) P λ (cos θ ) , (10)where P λ (cos θ ) = 4 π λ + 1 Y ( mol ) λ (ˆ s ) · Y ( e ) λ (ˆ r ) . (11)The matrix elements of P λ (cos θ ) between the channels c and c ′ are obtained by means of the standard angularmomentum algebra: h Φ Jj c ′ ℓ c ′ | P λ (cos θ ) | Φ Jj c ℓ c i = ( − j c ′ + j c + J (cid:26) j c ′ ℓ c ′ Jℓ c j c λ (cid:27) (cid:18) j c ′ λ j c (cid:19) (cid:18) ℓ c ′ λ ℓ c (cid:19) × p (2 ℓ c ′ + 1)(2 ℓ c + 1)(2 j c ′ + 1)(2 j c + 1) . (12)In the following, we express r in units of the Bohr ra-dius a , I in units of m e a , and energy in Ry. The radialfunctions u Jc ( r ) are solutions of the set of CC equations: (cid:20) d dr − ℓ c ( ℓ c + 1) r − j c ( j c + 1) I + E J (cid:21) u Jc ( r )= X c ′ v Jcc ′ ( r ) u Jc ′ ( r ) , (13)where E J is the energy of the system and v Jcc ′ ( r ) = X λ h Φ Jj c ′ ℓ c ′ | P λ (cos θ ) | Φ Jj c ℓ c i V λ ( r ) . (14) IV. DIRECT INTEGRATION OFCOUPLED-CHANNEL EQUATIONS
The CC equations (13) can be solved by the DImethod. Below we describe the method used to generatethe channel wave functions u c ( r ) (from now on, the quan-tum number J is omitted to simplify notation) obeyingthe physical boundary conditions. Namely, we assumethat u c ( r ) is regular at origin: u c ( r = 0) = 0, and for r → + ∞ it behaves like an outgoing wave u + c ( r ).The central issue of DI lies in the boundary conditionat infinity. Indeed, as we shall see in Sec. (IV B), anasymptotic wave function of a dipole-bound anion is notanalytic in general, so that one cannot exactly imposeoutgoing boundary conditions. This calls for the use ofcontrolled approximations. In the following, we describethe numerical integration of CC equations. While themethod is standard (cf. Sec. 3.3.2 of Ref. [81]), thisparticular application is not; hence key details should begiven. A. The basis method with the direct integration
To integrate CC equations, we introduce the matchingradius r m that defines the internal region [0 : r m ], wherethe centrifugal potential is appreciable, and the externalzone [ r m : + ∞ ]. An internal basis function u b ; c ( r ) in[0 : r m ] is regular at r = 0: u b ; c b ( r ) ∼ r ℓ cb +1 (15) in one channel c b = ( j c b , ℓ c b ). The CC equations implythat when r → u b ; c ( r ) with c = c b must behave as:2 m e V cc b (0) ~ × r ℓcb +3 ℓ cb +5 ln( r/r m ) for ℓ c = ℓ c b + 2 , r ℓcb +3 ( ℓ cb +2)( ℓ cb +3) − ℓ c ( ℓ c +1) otherwise.Note that is it necessary to pay attention when integrat-ing CC equations close to r = s , as the potential (4) isnot differentiable therein.In the external region [ r m : + ∞ ], the basis wave func-tions are denoted u + b ; c b ( r ). By construction, at very largedistances of the order of hundreds of a (asymptotic re-gion), u + b ; c b ( r ) = 0 for c b = ( j c b , ℓ c b ) and u + b ; c ( r ) = 0 forother channels c = c b . The asymptotic behavior of exter-nal channel functions is discussed in Sec. IV B below.Both sets of internal and external basis functions areused to expand the channel function u c ( r ): u c ( r ) = (cid:26) P b C b u b ; c ( r ) for r ≤ r m , P b C + b u + b ; c ( r ) for r ≥ r m (16)The matching conditions at r = r m X b h C b u b ; c ( r m ) − C + b u + b ; c ( r m ) i = 0 , (17) X b " C b du b ; c dr ( r m ) − C + b du + b ; c dr ( r m ) = 0 , (18)form a linear system of equations: AX = 0. The condi-tion of det A = 0 determines the energy of a bound or res-onant state. (One can thus see that det A is thus the gen-eralization of the Jost function for CC equations.) Oncethe eigenenergy has been found, the amplitudes C b , C + b are given by the eigenvector X of A . The overall normis determined by the condition: X c Z + ∞ | u c ( r ) | dr = 1 . (19) B. The coupled-channel equations in theasymptotic region
At large distances, V cc ′ ( r ) can be written as: V cc ′ ( r ) = ~ m e h χ cc ′ r + V ( r ) i , (20)where χ cc ′ is a constant and V ( r ) decreases for r → + ∞ as r − . In the following, we shall assume that V ( r ) = 0in the asymptotic region. As the numerical integration upto r ∼ a is stable, the error made by neglecting V is around 10 − a − , which is sufficiently small to insurethat the asymptotic zone has been practically reached.Let us first consider the case of an infinite moment ofinertia I → + ∞ . Here, Eq. (13) becomes: u ′′ c ( r ) = ℓ c ( ℓ c + 1) r u c ( r )+ X c ′ χ cc ′ r u c ′ ( r ) − k u c ( r ) , (21)where k = √ E . The outgoing solution of (21) in a basischannel b can be written in terms of spherical Hankelfunctions: u + b ; c ( r ) = g ( b ) c H + ℓ ( b ) eff ( kr ) , (22)where ℓ ( b ) eff is an effective angular momentum given byeigenvalues of the eigenproblem ℓ c ( ℓ c + 1) g ( b ) c + X c ′ χ cc ′ g ( b ) c ′ = ℓ ( b ) eff ( ℓ ( b ) eff + 1) g ( b ) c . (23)Indeed, it immediately follows from Eqs. (21) and (23)that: u + b ; c ( r ) ′′ = ℓ ( b ) eff ( ℓ ( b ) eff + 1) r − k ! u + b ; c ( r ) , (24)so the physical interpretation of ℓ ( b ) eff in terms of an ef-fective angular momentum is justified.If I is finite, however, solutions of Eq. (13) are nolonger analytical at large distances. Nevertheless, itis possible to construct an adiabatic approximation for u c ( r ) in the asymptotic region. To this end, one de-fines the linear momentum k c = p E − j c ( j c + 1) /I for achannel c . In the asymptotic region, Eq. (13) becomes: u ′′ c ( r ) = ℓ c ( ℓ c + 1) r u c ( r )+ X c ′ χ cc ′ r u c ′ ( r ) − k c u c ( r ) , (25)where, compared to Eq. (21), k is replaced by the chan-nel momentum k c . This approximation can be appliedif | E | ≫ j c ( j c + 1) /I for all channels of importance. Inthose cases, one can introduce an ansatz for u c ( r ) by re-placing k by k c in Eq. (22). The relative error on a basisfunction u ( b ; c )+ ( r ) associated with this approximation is X c ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ cc ′ r g ( b ) c ′ g ( b ) c H + ℓ ( b ) eff ( k c ′ r ) H + ℓ ( b ) eff ( k c r ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (26)i.e., is of the order of | k c − k c ′ | /r .In practical calculations, I ∼ and j max ∼ j max ( j max + 1) /I ∼ − . Consequently, if | E | > − Ry, the error | k c − k c ′ | /r < − a − for r ∼ a is close to that associated with the neglect of V ( r ). On can thus see that the proposed ansatz accountsfor the coupling term (20) in many cases. However, thisapproximation breaks down for weakly-bound/unboundstates with | E | < − Ry; hence, a more adequate theo-retical method based on a resonant state expansion needsto be introduced.
V. DIAGONALIZATION WITH THEBERGGREN BASIS
Another way to find eigenstates of the CC problem(13) is to diagonalize the associated Hamiltonian in a complete basis of single-particle states. Since our goalis to describe weakly-bound or unbound states, specialcare should be taken to treat the asymptotic part ofwave functions as precisely as possible. A suitable ba-sis for this problem is the one-body Berggren ensemble[47, 63, 82]. This basis is generated by a finite-depthspherical potential and contains bound ( b ), decaying ( d ),and scattering ( s ) one-body states. Fot that reason, theBerggren ensemble is ideally suited to deal with struc-tures having large spatial extensions (such as halos orRydberg states) or outgoing behavior (such as decayingresonances). Some recent applications, in a many-bodycontext, have been reviewed in Ref. [52]. A. The Berggren basis
The finite-depth potential generating the Berggren en-semble can be chosen arbitrarily. To improve the conver-gence, however, it is convenient in practical applicationsto use a one-body potential, which is as close as possibleto the Hartree-Fock field of the Hamiltonian in question.Therefore, in the case of the one-body problem (13), themost optimal potential to generate the Berggren basis isthe diagonal part of v cc ′ ( r ). This means that the basisstates Φ k,c ( r ) are eigenstates of the spherical potential v cc ( r ):Φ ′′ k,c ( r ) = (cid:18) ℓ c ( ℓ c + 1) r + v cc ( r ) − k (cid:19) Φ k,c ( r ) (27)that obey the following boundary conditions:Φ k,c ( r ) ∼ C r ℓ c +1 , r ∼ , (28)Φ k,c ( r ) ∼ C + H + ℓ c ( kr ) , r → + ∞ ( b, d ) , (29)Φ k,c ( r ) ∼ C + H + ℓ c ( kr ) + C − H − ℓ c ( kr ) , r → + ∞ ( s ) , (30)where the boundary conditions at r ∼ r → + ∞ for scattering states ( s ) are standard, and for bound anddecaying states ( b, d ) one imposes the outgoing boundarycondition. Note that k in Eq. (27) is in general complex.The scattering states are normalized to the Dirac delta,which results in a condition for the C − and C + ampli-tudes in (30) [47]: h Φ k,c | Φ k ′ ,c i = δ ( k − k ′ ) ⇔ πC + C − = 1 . (31)The normalization of bound states is standard as well,but that of decaying resonant states is not. Indeed, reso-nant states rapidly oscillate and diverge exponentially inmodulus along the real r -axis; hence, one cannot calcu-late their norm in the same way as for the bound states.The solution of this problem is provided by the exteriorcomplex scaling [83], i.e., one calculates the norm of theresonant state using complex r radii: h Φ k,c | Φ k,c i = Z R Φ k,c ( r ) dr + Z + ∞ Φ k,c ( R + xe iθ ) e iθ dx, , (32)where R is a radius taken sufficiently large so that con-dition (29) is fulfilled. In the above formula, θ is anangle of rotation chosen so that Φ k,c ( R + xe iθ ) → x → + ∞ , which is always possible provided θ is largerthan a critical value depending on k [84]. Note that nomodulus enters Eq. (32). This arises from the finite-lifetime character of resonant states, which requires us touse the biorthogonal scalar product [52, 84–87]. It can beshown [85–87] that the norm defined in Eq. (32) is indeedindependent of R and θ , as expected from a norm. Sincethe expression (32) is also valid for bound states, boundand decaying states enter the Berggren ensemble as onefamily of resonant states.The exterior complex scaling can be used to calculatematrix elements of a one-body operator O ( r ) as well,provided it decreases faster than 1 /r along the complex r -contour: h Φ k ′ ,c ′ | O | Φ k,c i = Z R Φ k ′ ,c ′ ( r ) O cc ′ ( r ) Φ k,c ( r ) dr + Z + ∞ Φ k ′ ,c ′ ( z ( x )) O cc ′ ( z ( x )) Φ k,c ( z ( x )) e iθ dx, (33)where z ( x ) = R + xe iθ , and | Φ k,c i and | Φ k ′ ,c ′ i can herebe bound, decaying, or scattering states. B. The Berggren completeness relation a L + Re(k) I m ( k ) b d s Berggren ensemble
FIG. 1. Berggren ensemble in the complex momentum plane.The bound ( b ) and antibound ( a ) states are distributed alongthe imaginary k -axis at Im ( k ) > Im ( k ) <
0, respec-tively. The decaying resonant states ( d ) are located in thefourth quadrant ( Re ( k ) > , Im ( k ) <
0. The Berggren com-pleteness relation involves bound states, scattering states ( s )on the L + contour, and decaying states lying between thereal- k axis and L + . If antibound states are included, the L + contour has to be slightly deformed [52, 88]. If the contour L + lies on the real k -axis, the Berggren completeness relationreduces to the Newton completeness relation [52, 89] involvingbound and real-energy scattering states. Figure 1 shows a distribution the Berggren ensemblein the complex momentum plane. To determine the ba-sis, one first chooses a L + contour in the fourth quad- rant containing the decaying eigenstates. The scatteringstates of the ensemble lie on this contour. The resonantpart of the ensemble contains the bound states lying onthe imaginary- k axis and those decaying states of (27)that are found in the region between the real- k axis and L + . The Berggren basis is built from all those states: X n ∈ ( b,d ) | Φ k n ,c i h Φ k n ,c | + Z L + | Φ k,c i h Φ k,c | dk = 1 . (34)This completeness relation corresponds to a given chan-nel c ; hence, one has to construct Berggren ensembles forall the channels considered in Eq. (13).In order to be able to use (34) in practice, one needsto discretize L + . Our method of choice is to applythe Gauss-Legendre quadrature to each of the segmentsdefining L + in Fig. 1. The last segment, chosen alongthe real- k axis, extends to the large cutoff momentum k = k max that is sufficiently large to guarantee complete-ness to desired precision. It is then convenient to renor-malize scattering states using the corresponding Gauss-Legendre weights ω k n : | Φ n,c i = √ ω k n | Φ k n ,c i . (35)The discretized Berggren completeness relation, used inpractical computations, reads: N X i =1 | Φ i,c i h Φ i,c | ≃ , (36)where the N basis states | Φ i,c i include all bound, decay-ing, and discretized scattering states of the channel c . Byusing Eq. (35), the Dirac delta normalization of scatter-ing states has been replaced by the usual normalizationto Kronecker’s delta. in this way, all | Φ i,c i states can betreated on the same footing in Eq. (36), as in any basisof discrete states. C. Hamiltonian matrix in the Berggren basis
As the basis states | Φ i,c i are generated by v cc ( r ), theHamiltonian matrix within the same channel c is diago-nal: h Φ i ′ ,c | h | Φ i,c i = (cid:18) k i + j c ( j c + 1) I (cid:19) δ ii ′ (37)Matrix elements between two basis states belonging todifferent channels c and c ′ are: h Φ i ′ ,c ′ | h | Φ i,c i = h Φ i ′ ,c | v | Φ i,c i = Z R Φ i ′ ,c ′ ( r ) v cc ′ ( r ) Φ i,c ( r ) dr + Z + ∞ Φ i ′ ,c ′ ( z ( x )) v cc ′ ( z ( x )) Φ i,c ( z ( x )) e iθ dx, (38)where the complex scaling (33) can be used, because v cc ′ ( r ) decreases at least as fast as r − .As the off-diagonal matrix elements are present onlyfor c = c ′ , the Berggren basis generated by Eq. (27) isoptimal. The channel wave functions u c ( r ) can be ex-pressed in the Berggren basis by diagonalizing the matrixof h (37,38). VI. CALCULATION PARAMETERS
Results of the direct integration method (DIM) dependboth on the parameters of the pseudo-potential (3) andon the cutoff value of the electron orbital angular mo-mentum ℓ max considered in the CC problem. They arefixed to reproduce the experimental value of the groundstate energy of the LiCl − anion: E exp = − . · − Ry[27].The most important term in (3) is the dipole poten-tial V µ , which depends only on the dipole moment µ and the size s of the neutral molecule. The remain-ing parameters of the pseudo-potential are taken fromRef. [22], namely: α = 15 . a , α = 1 . a , r = 2 . a , r c = 2 . a , Q zz = 3 . ea , and V = 2 . I = 150 , m e a for LiCl − , 240 , m e a for LiI − , 82 , m e a for LiF − ,and 26 , m e a for LiH − . The dipole moment of eachmolecule considered in this work is known experimentallyand has been taken from the NIST database. r (a ) DIMBEM | u c (r) | ( a ) - / J =0 +2 LiCl ! r = s FIG. 2. The modulus of the channel wave function u j =0 ,ℓ =0 near r = 0 for the first excited J π = 0 +2 state of LiCl − calculated in DIM (solid line) and BEM (dotted line) with ℓ max = 9. The charge separation s of LiCl has been adjustedin both approaches to the experimental ground state energyin the limit ℓ max → ∞ . For ℓ max = 9 the ground state energy of the LiCl − anion is reproduced by taking the charge separation s (9)DIM = 0 . a . To remove the dependence of resultson ℓ max in the DIM, the ground state energy of LiCl − is extrapolated for ℓ max → ∞ , and the size of the chargeseparation s is adjusted to reproduce the experimentalbinding energy. In this case, s ( ∞ )DIM = 0 . a . Thematching radius was taken as r m = a . This value wasfound to optimize the DIM procedure.Anion spectra in the BEM depend sensitively on thecutoff parameter k max of the single-particle basis. How-ever, as we shall see in Sec. VII, for a chosen value of k max they are practically independent of ℓ max . In this study,we have chosen k max = 1 . a − for each partial wave inorder to attain both a good numerical precision and ap-proximately the same value of the dipole size parameter s as in DI. In this case, s (9)BEM = s ( ∞ )BEM = 0 . a . We haveused complex contours with straight segments connectingpoints: k = (0 , k = (0 . , − i . k = (1 , k = k max in units of a − . Each scattering contour hasbeen discretized with 220 points. The precise form of thecontour does not change results; since the applicationscarried out in this work pertain to bound states only, wecould have used real scattering contours, i.e., the Newtoncompleteness relation [52, 89]. VII. NUMERICAL TESTS ANDBENCHMARKING
Along with the asymptotic behavior of channel wavefunctions, treated approximately with the DIM and ex-actly within BEM, the Hamiltonian (2) cannot be iden-tically represented in both approaches. Indeed, sincethe potential V µ ( r ) (4) is not differentiable at r = s , itcannot be treated exactly in BEM because the channelwave functions expanded in the Berggren basis are ana-lytic by construction. In practice, this translates into anode beyond r = s in DIM channel wave functions, whichis absent in BEM. This is illustrated in Fig. 2 for a( j = 0 , ℓ = 0) channel function corresponding to the firstexcited J π = 0 +2 state of LiCl − . It is to be noted, how-ever, that beyond this point the channel wave functionscalculated with both methods are very close and – as willbe discussed later – this near-origin pathology has a verysmall impact on the total energy as the contribution fromthis region is small.As discussed in Sec. IV B, DIM is inadequate for stateswith very small energies, while BEM has been shownto be very precise in this case. On the other hand, forstates with binding energies typically greater than 10 − Ry, BEM yields channel wave functions that exhibit spu-rious low-amplitude oscillations. Figure 3 illustrates suchwiggles in the tail of the channel wave function u j =0 ,ℓ =0 of the J π = 0 +1 ground state of LiCl − . For such well-bound states that quickly decay with r , the standard sizeof the Berggren basis (measured in terms of contour dis-cretization points and k max ) is not sufficient. The DIMis thus preferable for such cases, as the asymptotic be-havior of well-bound states is treated almost exactly (seeSec. IV B).The direct integration becomes numerically unstable DIMBEM r (a ) | u c (r) | ( a ) - / r |uc(r)| J =0 +1 LiCl ! FIG. 3. The modulus of the channel wave function u j =0 ,ℓ =0 for the J π = 0 +1 ground state of LiCl − calculated in DIM(solid line) and BEM (dotted line) with ℓ max = 9. At largedistances, spurious wiggles appear in BEM results (see theinset) due to basis truncation. when the channel orbital angular momentum becomeslarge, around ℓ c = 10, even for the states with relativelylarge binding energies. In this case, the matrix of basischannel wave functions u b ; c ( r m ) and u + b ; c ( r m ) and theirderivatives, introduced in Sec. IV A in the context ofmatching conditions at r = r m , is ill-conditioned and itseigenvector of zero eigenvalue becomes imprecise. Thisresults in a discontinuity at r m and spurious occupa-tion of channels with large orbital angular momentum ℓ c >
10. This is illustrated in Fig. 4 for the J π = 0 +1 ground state of LiCl − . A a result, the energy and spa-tial extension of the electron cloud distribution of the CCeigenstate become incorrect.The convergence of the LiCl − ground state energy withrespect to ℓ max is shown in Fig. 5. One may noticean exponential convergence of calculated DIM energieswith ℓ max for 6 ≤ ℓ max ≤
10 and a clear deviation for ℓ ≥
11, which is related to the discontinuity of channelwave functions for ℓ c >
10. The energy calculated inBEM is perfectly stable with ℓ max .The rapid converge of BEM with ℓ is due to k max -truncation of the single-particle basis that suppressescontributions from large- ℓ configurations. This is illus-trated in Fig. 6, which displays the average modulus ofthe off-diagonal matrix element of the channel-channelcoupling in BEM: A c,c ′ = 1 N N X n,n ′ |h Φ n ′ ,c ′ | V | Φ n,c i| (39)between the first channel c = ( j = 0 , ℓ = 0) and higher- ℓ channels c ′ . Only the channels with ℓ c ≤ | ℓ c − ℓ c ′ | ≤ r (a ) | u c (r) | ( a ) - / max DIM J =0 +1 LiCl ! FIG. 4. The modulus of the channel wave function u j =0 ,ℓ =0 for the J π = 0 +1 ground state of LiCl − calculated in DIM withseveral values of ℓ max . For ℓ max ≥
10, one may notice thedevelopment of a discontinuity at the matching point r m = a . In such cases, the channel wave function becomes ill-conditioned, introducing serious errors in CC eigenenergy andeigenfunction. -0.044812-0.044677-0.0445434 5 6 7 8 12DIMBEM109 11 ℓ max E ( R y ) J =0 +1 LiCl ! FIG. 5. The dependence of the LiCl − ground state energy on ℓ max in DIM (dots) and BEM (triangles). The DIM resultsconverge exponentially (red line). This allows us to determinethe asymptotic value of energy at ℓ max → ∞ . numerically stable results. In this case, the energies ofwell-bound states ( | E | > − Ry) agree in both meth-ods.The numerical instability of DIM at large ℓ max leadsto a collapse of calculated radii. Figure 7 shows the de-pendence of the ground state r.m.s. radius of LiCl − on ℓ max . This result, together with discussion of Fig. 6, sug-gests that the BEM can provide practical guidance on theminimal number of channels in the CC approach.In practical applications, spurious oscillations in BEMchannel wave functions for well-bound states can be takencare of by extrapolating wave functions from the inter- A , c ( - R y ) BEM J =0 +1 LiCl ! ℓ c ′ FIG. 6. Average off-diagonal matrix element A ,c (39) of thechannel-channel coupling in BEM between the channel ( j =0 , ℓ = 0) and c ′ for the J π = 0 +1 ground state of LiCl − . ℓ max r r m s ( a ) J =0 +1 LiCl ! DIM BEM
FIG. 7. The dependence of the LiCl − ground state r.m.s.radius on ℓ max DIM (dots) and BEM (dotted line). The DIMresults are stable up to ℓ max = 10. mediate region of r , where they are reliably calculated,into the asymptotic region. This can be done by applyingthe analytical expression:˜ u c ( r ) ≡ lim r ≫ u c ( r ) = e ik c r M X j =1 α ( c ) j r j , (40)where k c is the channel momentum and α ( c ) j are param-eters to be determined by the fit. The precision of thisprocedure can be assessed by computing the norm of theeigenstate. Using this procedure, one obtains perfectlystable r.m.s. radii in BEM for different values of ℓ max ,as can be seen in Fig. 7.Figures 8-10 compare the four most important chan-nel wave functions ( ℓ, j ) of DIM and BEM corresponding r (a ) | u c (r) | ( a ) - / (0,0)(1,1)(3,3)(2,2) DIMBEM J =0 +1 LiCl ! FIG. 8. Most important channel wave functions u c ( r ) with c = ( j, ℓ ) for the J π = 0 +1 ground state of LiCl − , as calculatedin DIM (solid line) and BEM (dashed line) with ℓ max = 9. r (a ) | u c (r) | ( a ) - / (0,0)(1,1)(3,3)(2,2) J =0 +2 LiCl ! FIG. 9. Similar to Fig. 8 but for the first excited J π = 0 +2 state of LiCl − . to the three lowest J πi = 0 + i eigenstates of LiCl − . Forthe ground state, both approaches predict the same en-ergy E = − . · − Ry and the channel functionsare practically identical. For the first excited state, theagreement is still reasonable. Here, the energy in DIMis E = − . · − Ry while BEM gives slightly morebinding: E = − . · − Ry. Consequently, the BEMwave functions decay faster than those computed withDIM. For a second excited 0 +3 state, both methods differmarkedly. This state has a sub-threshold nature, with E DIM = − . · − Ry and E BEM = − . · − Ry.For this extremely diffused state, the direct integrationmethod fails completely. This is manifested by the verydifferent nodal structure of channel wave functions inDIM seen in Fig. 10.A stringent test of the computational framework to de-scribe dipolar molecules is provided by the analytic result µ cr = 0 . ea for the fixed dipole ( I → ∞ ) [40]. To this (0,0)(1,1)(3,3) (2,2)(0,0)(1,1)(2,2) r (a ) | u c (r) | ( a ) - / J =0 +3 LiCl ! FIG. 10. Similar to Fig. 8 but for the second excited J π = 0 +3 state of LiCl − . end, we performed BEM calculations for a dipolar systemat steadily decreasing moments of inertia [18, 19]. Foreach value of I , the dipolar anion energies have been cal-culated for 1080 values of µ in the interval 0 . ≤ µ ≤ . E < E lim = − − Ry were retained to minimize thenumerical error. These energies correspond to an inter-val ∆ µ ≃ .
377 of the dipole moment. We checked, thatin this energy interval, µ cr can be obtained by using theexpression E ( µ ) = ( µ + b ) aµ e c (41)to extrapolate the calculated energy down to E = 0. Oneshould stress however, that an excellent energy fit in thesubthreshold region does not guarantee an excellent es-timate of the critical dipole moment. The values of µ cr extracted by this extrapolation procedure can be consid-ered reliable only if ∆ µ , which depends on the chosenprecision E lim , is close to the critical dipole moment. Inthe cases studied, this criterion is approximately satisfiedonly for the ground state and the first excited 0 + state.The critical dipole moments for these states in anionswith the dipole length s = 4 a are shown in Table I forvarious moments of inertia. The agreement with the ana-lytic limit is excellent for the ground state configuration,and is fairly good for the first excited 0 + state. This isvery encouraging, considering the slow convergence with I and various sources of numerical errors in the E → VIII. RESULTS FOR SPECTRA AND RADII OFDIPOLAR ANIONS
Energies and r.m.s. radii of the lowest bound 0 + and1 − states of LiI − , LiCl − , LiF − , and LiH − dipolar anionspredicted in this study are listed in Table II. TABLE I. Critical dipole moments for dipolar anions in thetwo lowest 0 + states calculated in this work (BEM) and inRef. [19] for the charge separation s = 4 a and different mo-ments of inertia I . The analytic result at I → ∞ [39, 40] is µ cr = 0 . ea . I ( m e a ) µ (0) cr ( ea ) µ (1) cr ( ea )BEM Ref. [19] BEM Ref. [19]10 + and 1 − boundstates of selected dipolar anions obtained in DIM ( ℓ max = 9)and BEM. The parameters of the calculation are given in Sec.VI. The numbers in parentheses denote powers of 10.Anion state E (Ry) r rms ( a )DIM BEM DIM BEMLiI − +1 -5.079(-2) -5.023(-2) 7.569(0) 7.620(0)0 +2 -9.374(-4) -1.037(-3) 5.112(1) 4.759(1)0 +3 -1.502(-5) -1.797(-5) 3.719(2) 3.308(2)1 − -5.079(-2) -4.995(-2) 7.569(0) 7.641(0)1 − -9.291(-4) -1.023(-3) 5.112(1) 4.886(1)1 − -1.261(-7) -1.099(-5) 3.423(3) 3.464(2)LiCl − +1 -4.483(-2) -4.483(-2) 7.885(0) 7.894(0)0 +2 -7.374(-4) -8.241(-4) 5.632(1) 5.017(1)0 +3 -7.051(-6) -9.907(-6) 5.124(2) 4.106(2)1 − -4.482(-2) -4.458(-2) 7.885(0) 7.915(0)1 − -7.241(-4) -8.067(-4) 5.633(1) 5.337(1)1 − -3.062(-7) -8.159(-7) 2.066(3) 8.831(2)LiF − +1 -2.795(-2) -2.983(-2) 9.117(0) 8.991(0)0 +2 -3.022(-4) -3.525(-4) 8.098(1) 7.501(1)0 +3 — -6.101(-8) — 3.363(3)1 − -2.793(-2) -2.968(-2) 9.117(0) 9.010(0)1 − -2.782(-4) -3.277(-4) 8.124(1) 7.520(1)LiH − +1 -2.149(-2) -2.370(-2) 1.011(1) 9.698(0)0 +2 -1.491(-4) -1.922(-4) 1.058(2) 9.297(1)1 − -2.142(-2) -2.353(-2) 1.011(1) 9.717(0)1 − -7.942(-5) -1.231(-4) 1.146(2) 9.591(1) One can see that for each total angular momentum J π there are at most three bound eigenstates in eachsystem. The r.m.s. radius of an electron cloud shows aspectacular increase with decreasing the binding energyof the state. For the subthreshold states, such as 0 +3 and1 − , the radius is of the order of hundreds to thousands a .Energy spectra and radii of dipolar anions do notchange significantly in the limit ℓ max → ∞ . Usually,the extrapolated results for both E and r rms agree verywell with those in Table II ( ℓ max = 9). For instance,the extrapolated values for the 1 +2 state in LiH − are E = − . · − Ry and r rms = 1 . · a .0The DIM and BEM results are generally consistent forboth energy and radii though significant quantitative dif-ferences persist for excited, weakly-bound states of anionswhere the DIM is not expected to work. In the case ofLiF − , the BEM predicts the existence of the third 0 +3 state at an energy − . · − Ry, which is absent inDIM.It is instructive to compare our DIM results with thosefound in Ref. [22] using a similar approach. Table III listsenergies of the lowest 0 + bound states of LiI − , LiCl − ,LiF − , and LiH − dipolar anions obtained in both studies,and Table IV shows the adopted values of dipole mo-ments. TABLE III. Energies for 0 + bound states of selected dipo-lar anions obtained in DIM in this work ( ℓ max = 9) and inRef. [22]. The numbers in parentheses denote powers of 10.Anion state E (Ry)This work Ref. [22]LiI − +1 -5.079(-2) -4.998(-2)0 +2 -9.374(-4) -1.022(-3)0 +3 -1.502(-5) -1.999(-5)LiCl − +1 -4.483(-2) -4.483(-2)0 +2 -7.374(-4) -7.497(-4)0 +3 -7.051(-6) -9.775(-6)LiF − +1 -2.795(-2) -2.793(-2)0 +2 -3.022(-4) -3.366(-4)0 +3 — -8.746(-7)LiH − +1 -2.149(-2) -2.352(-2)0 +2 -1.491(-4) -1.926(-4) The two calculations agree reasonably well for thelowest-lying states; some difference stems from slightlydifferent dipole moments used in Ref. [22] and here. In-deed, while the charge separation in both studies was ad-justed to reproduce the experimental ground state energyof LiCl − , the fitted values of s in both calculations aredifferent: s = 0 . a in Ref. [22] and s DIM = 0 . a here.The largest deviations, seen for weakly-bound states,can be traced back to the cutoff value of the electronorbital angular momentum when solving CC equations.In Ref. [22], adopted ℓ max was small, typically ℓ max = 4[35], whereas it is fairly large, ℓ max = 9, in our work.As seen in Fig. 5 and discussed in Sec. VII, energies ofweakly-bound states obtained in DIM do converge slowlywith ℓ max . Therefore, calculations employing low ℓ max values cannot be useful when performing extrapolation ℓ max → ∞ . IX. CONCLUSIONS
In this study, we applied the theoretical open-systemframework based on the Berggren ensemble to a problemof weakly-bound states of dipole-bound negative anions.
TABLE IV. Dipole moments of selected dipolar anionsadopted in this work and in Ref. [22].Anion µ ( ea )This work Ref. [22]LiI − − − − The method has been benchmarked by using the tradi-tional technique of direct integration of CC equations.While a fairly good agreement between the two methodshas been found for well-bound states, the direct inte-gration technique breaks down for weakly-bound stateswith energies | E | < − Ry, which is comparable withthe rotational energy of the anion. For those subthresh-old configurations, the Berggren expansion is an obvioustool of choice.The inherent problem of the DIM is the lack of stabil-ity of results when the number of channels is increased.Indeed, the method breaks down when the channel or-bital angular momentum becomes large; around ℓ c = 10.This can be traced back to the applied matching con-dition. We demonstrated that this pathology is absentin BEM. Here, the rapid converge with ℓ is guaranteedby an effective softening of the interaction through themomentum cutoff k max , which suppresses contributionsfrom high- ℓ partial waves.The future applications of BET will include the struc-ture of quadrupole-bound anions [41, 90–92] and the con-tinuum structure of anions, including the characteriza-tion of low-lying resonances. ACKNOWLEDGMENTS
Stimulating discussions with and helpful suggestionsfrom R.N. Compton and W.R. Garrett, who encour-aged us to apply the complex-energy Gamow Shell Modelframework to dipolar anions, are gratefully acknowl-edged. This work was supported by the U.S. Departmentof Energy under Contract No. DE-FG02-96ER40963. [1] K. Riisager, D. V. Fedorov, and A. S. Jensen, Europhys.Lett. , 547 (2000). [2] V. Rotival, K. Bennaceur, and T. Duguet, Phys. Rev. C , 054309 (2009). [3] P. G. Hansen and B. Jonson, Europhys. Lett. , 409(1987).[4] I. Tanihata, J. Phys. G , 157 (1996).[5] A. Cobis, A. S. Jensen, and D. V. Fedorov, J. Phys. G , 401 (1997).[6] A. S. Jensen, K. Riisager, D. V. Fedorov, and E. Garrido,Rev. Mod. Phys. , 215 (2004).[7] I. Mazumdar, A. R. P. Rau, and V. S. Bhasin, Phys. Rev.Lett. , 062503 (2006).[8] T. K. Lim, S. K. Duffy, and W. C. Damer, Phys. Rev.Lett. , 341 (1977).[9] N. Moiseyev and C. Corcoran, Phys. Rev. A , 814(1979).[10] R. E. Grisenti, W. Sch¨ollkopf, J. P. Toennies, G. C.Hegerfeldt, T. K¨ohler, and M. Stoll, Phys. Rev. Lett. , 2284 (2000).[11] D. Bressanini, G. Morosi, L. Bartini, and M. Mella, Few-Body Syst. , 199 (2002).[12] Y. Li, Q. Gou, and T. Shi, Phys. Rev. A , 032502(2006).[13] R. Lefebvre, O. Atabek, M. ˇSindelka, and N. Moiseyev,Phys. Rev. Lett. , 123003 (2009).[14] J. Mitroy, Phys. Rev. Lett. , 033402 (2005).[15] K. Varga, J. Mitroy, J. Z. Mezei, and A. T. Kruppa, Phys.Rev. A , 044502 (2008).[16] F. Ferlaino and R. Grimm, Physics , 9 (2010).[17] W. R. Garrett, Chem. Phys. Lett. , 393 (1970).[18] W. R. Garrett, Phys. Rev. A , 961 (1971).[19] W. R. Garrett, J. Chem. Phys. , 5721 (1980).[20] W. R. Garrett, Phys. Rev. A , 1769 (1980).[21] W. R. Garrett, Phys. Rev. A , 1737 (1981).[22] W. R. Garrett, J. Chem. Phys. , 3666 (1982).[23] L.G. Christophorou, Atomic and Molecular RadiationPhysics , Wiley, New York (1971).[24] R.N. Compton, P.W. Reinhardt, and C.D. Cooper, J.Chem. Phys. , 3305 (1977).[25] S. F. Wong and G. J. Schulz, Phys. Rev. Lett. , 134(1974).[26] K. Rohr and F. Linder, J. Phys. B , 2521 (1976).[27] J. Carlsten, J. Peterson, and W. Lineberger, Chem. Phys.Lett. , 5 (1976).[28] K. D. Jordan and W. Luken, J. Chem. Phys. , 2760(1976).[29] K. R. Lykke, R. D. Mead, and W. C. Lineberger, Phys.Rev. Lett. , 2221 (1984).[30] E. A. Brinkman, S. Berger, J. Marks, and J. I. Brauman,J. Chem. Phys. , 7586 (1993).[31] A. S. Mullin, K. K. Murray, C. P. Schulz, and W. C.Lineberger, J. Phys. Chem. , 10281 (1993).[32] C. Desfran¸cois, H. Abdoul-Carime, and J. P. Schermann,Int. J. Mol. Phys. B , 1339 (1996).[33] J. R. Smith, J. B. Kim, and W. C. Lineberger, Phys.Rev. A , 2036 (1997).[34] H. Abdoul-Carime and C. Desfran¸cois, Eur. Phys. J. D , 149 (1998).[35] S. Ard, W. R. Garrett, R. N. Compton, L. Adamowicz,and S. G. Stepanian, Chem. Phys. Lett. , 223 (2009).[36] R. N. Compton and N. I. Hammer, Advances in GasPhase Ion Chemistry , vol. 4 (Elsevier, Amsterdam,2001).[37] K. D. Jordan and F. Wang, Annu. Rev. Phys. Chem. ,367 (2003).[38] V. E. Chernov, A. V. Danilyan, and B. A. Zon, Phys.Rev. A , 022702 (2009). [39] E. Fermi and E. Teller, Phys. Rev. , 399 (1947).[40] J.-M. L´evy-Leblond, Phys. Rev. , 1 (1967).[41] W. R. Garrett, J. Chem. Phys. , 194309 (2008).[42] J. Kalcher and A. F. Sax, J. Mol. Struct. (Theochem) , 77 (2000).[43] J. Kalcher and A. F. Sax, Chem. Phys. Lett. , 80(2000).[44] P. Skurski, I. Dabkowska, A. Sawicka, and J. Rak, Chem.Phys. , 101 (2002).[45] H. Estrada and W. Domcke, J. Phys. B , 279 (1984).[46] H. R. Sadeghpour, J. L. Bohn, M. J. Cavagnero, B. D.Esry, I. I. Fabrikant, J. H. Macek, and A. R. P. Rau, J.Phys. B , R93 (2000).[47] T. Berggren, Nucl. Phys. A , 265 (1968).[48] N. Michel, W. Nazarewicz, M. P loszajczak, and K. Ben-naceur, Phys. Rev. Lett. , 042502 (2002).[49] N. Michel, W. Nazarewicz, M. P loszajczak, andJ. Oko lowicz, Phys. Rev. C , 054311 (2003).[50] N. Michel, W. Nazarewicz, M. P loszajczak, Phys. Rev.C, , 042501 (2002).[52] N. Michel, W. Nazarewicz, M. P loszajczak, andT. Vertse, J. Phys. G , 013101 (2009).[53] G. Gamow, Z. Phys. , 204 (1928).[54] I. M. Gel’fand and N. Ya. Vilenkin, Generalized func-tions , Vol. 4 , Academic Press , New-York (1961).[55] K. Maurin,
Generalized Eigenfunction expansions andunitary representations of topological groups , Polish Sci-entific Publishers, Warsaw (1968).[56] A. Bohm,
The rigged Hilbert space and quantum mechan-ics , Lecture Notes in Physics 78, Springer, New York(1978).[57] R. de la Madrid, Eur. J. Phys. , 287 (2005).[58] R. de la Madrid, J. Math. Phys. , 102113 (2012).[59] A.F.J. Siegert, Phys. Rev. , 750 (1939).[60] R.E. Peierls, Proc. R. Soc. (London) A 253 , 16 (1959).[61] J. Humblet and L. Rosenfeld, Nucl. Phys. , 529 (1961).[62] B. Gyarmati and A. T. Kruppa, Phys. Rev. A , 2989(1986).[63] P. Lind, Phys. Rev. C , 1903 (1993).[64] C.G. Bollini, O. Civitarese, A.L. De Paoli, and M.C.Rocca, Phys. Lett. B 382 , 205 (1996).[65] R. de la Madrid, and M. Gadella, Am. J. Phys. , 626(2002).[66] E. L. Hamilton and C. H. Greene, Phys. Rev. Lett. ,263003 (2002).[67] E. Kapu´scik, and P. Szczeszek, Czech. J. Phys. , 1053(2003).[68] E. Hernandez, A. J´aregui, and A. Mondragon, Phys. Rev.A , 022721 (2003).[69] E. Kapu´scik, and P. Szczeszek, Found. Phys. Lett. ,573 (2005).[70] R. Santra, J. M. Shainline, and C. H. Greene, Phys. Rev.A , 032703 (2005).[71] J. Julve, and F.J. de Urries, quant-ph/0701213.[72] R. de la Madrid, quant-ph/0607168.[73] K. Toyota, O. I. Tolstikhin, T. Morishita, and S. Watan-abe, Phys. Rev. A , 043418 (2007).[74] A. T. Kruppa, B. Barmore, W. Nazarewicz, andT. Vertse, Phys. Rev. Lett. , 4549 (2000).[75] B. Barmore, A. T. Kruppa, W. Nazarewicz, andT. Vertse, Phys. Rev. C , 054315 (2000).[76] A. T. Kruppa and W. Nazarewicz, Phys. Rev. C , , 062705 (2006).[78] O. I. Tolstikhin, Phys. Rev. A , 032712 (2008), URL http://link.aps.org/doi/10.1103/PhysRevA.77.032712 .[79] W. R. Garrett, J. Chem. Phys. , 2621 (1978).[80] W. R. Garrett, J. Chem. Phys. , 651 (1979).[81] I. J. Thompson, Comput. Phys. Rep. , 167 (1988).[82] T. Berggren and P. Lind, Phys. Rev. C , 768 (1993).[83] B. Gyarmati and T. Vertse, Nucl. Phys. A , 523(1971).[84] N. Michel, W. Nazarewicz, M. P loszajczak, andJ. Oko lowicz, Phys. Rev. C , 054311 (2003).[85] J. Aguilar and J.M. Combes, Commun. Math. Phys. ,269 (1971). [86] E. Balslev, J.M. Combes, Commun. Math. Phys. , 280(1971).[87] B. Simon, Commun. Math. Phys. , 1 (1972).[88] N. Michel, W. Nazarewicz, M. P loszajczak, and J. Ro-tureau, Phys. Rev. C , 054305 (2006).[89] R. Newton, Scattering Theory of Waves and Particles (Springer-Verlag, New York Heidelberg Berlin, 1982).[90] A. Ferron, P. Serra, and S. Kais, J. Chem. Phys. , 18(2004).[91] C. Desfran¸cois, Y. Bouteiller, J. P. Schermann, D. Ra-disic, S. T. Stokes, K. H. Bowen, N. I. Hammer, andR. N. Compton, Phys. Rev. Lett. , 083003 (2004).[92] W. R. Garrett, J. Chem. Phys.136