Bound and resonance states of the dipolar anion of hydrogen cyanide: competition between threshold effects and rotation in an open quantum system
K. Fossez, N. Michel, W. Nazarewicz, M. Płoszajczak, Y. Jaganathen
BBound and resonance states of the dipolar anion of hydrogen cyanide: competitionbetween threshold effects and rotation in an open quantum system
K. Fossez, N. Michel, W. Nazarewicz,
2, 3, 4
M. P(cid:32)loszajczak, and Y. Jaganathen
5, 6 Grand Acc´el´erateur National d’Ions Lourds (GANIL),CEA/DSM - CNRS/IN2P3, BP 55027, F-14076 Caen Cedex, France Department of Physics and Astronomy and NSCL/FRIB Laboratory,Michigan State University, East Lansing, Michigan 48824, USA Physics Division, Oak Ridge National Laboratory,P. O. Box 2008, Oak Ridge, Tennessee 37831, USA Institute of Theoretical Physics, Faculty of Physics,University of Warsaw, ul. Ho ˙ z a 69, PL-00-681 Warsaw, Poland Department of Physics & Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA NSCL/FRIB Laboratory, Michigan State University, East Lansing, Michigan 48824, USA (Dated: September 10, 2018)Bound and resonance states of the dipole-bound anion of hydrogen cyanide HCN − are studiedusing a non-adiabatic pseudopotential method and the Berggren expansion technique involvingbound states, decaying resonant states, and non-resonant scattering continuum. We devise analgorithm to identify the resonant states in the complex energy plane. To characterize spatialdistributions of electronic wave functions, we introduce the body-fixed density and use it to assignfamilies of resonant states into collective rotational bands. We find that the non-adiabatic couplingof electronic motion to molecular rotation results in a transition from the strong-coupling to weak-coupling regime. In the strong coupling limit, the electron moving in a subthreshold, spatiallyextended halo state follows the rotational motion of the molecule. Above the ionization threshold,electron’s motion in a resonance state becomes largely decoupled from molecular rotation. Widthsof resonance-band members depend primarily on the electron orbital angular momentum. PACS numbers: 03.65.Nk, 31.15.-p, 31.15.V-, 33.15.Ry
I. INTRODUCTION
Dipolar anions are one of the most spectacular exam-ples of marginally bound quantum systems [1–11]. Wavefunctions of electrons coupled to neutral dipole molecules[12, 13] are extremely extended; they form the extremequantum halo states [14–19]. Resonance energies of dipo-lar anions, including those associated with rotationalthreshold states, can been determined in high resolutionelectron photodetachment experiments [20–25]. Theoret-ically, however, the literature on the unbound part of thespectrum of dipole potentials, and multipolar anions inparticular, is fairly limited [26–34].The breakdown of the adiabatic approximation indipolar molecules possessing a supercritical moment [35–39] caused by coupling of electron’s motion to the rota-tional motion of the molecule, is expected to profoundlyimpact the properties of rotational bands in such systems[25, 30, 31, 36], such as the the number of rotationallyexcited bound anion states.In this study, we address the nature of the unboundpart of the spectrum of dipolar anions. In particular, weare interested in elucidating the transition from the ro-tational motion of weakly-bound subthreshold states tothe rotational-like behavior exhibited by unbound reso-nances. The competition between continuum effects, col-lective rotation, and non-adiabatic aspects of the problemmakes the description of threshold states in dipole-boundmolecules both interesting and challenging. Our theoretical framework is based on the Bergggrenexpansion method (BEM) – a complex-energy resonantstate expansion [40–42] based on a completeness rela-tion introduced by Berggren [43] that involves bound,decaying, and scattering states. In the context ofcoupled-channel method, BEM was successfully appliedto molecules [39] and nuclei [44–49]. The advantage ofthis method, which is of particular importance to theproblem of dipole-bound anions when the rotational mo-tion of the molecule is considered [39, 50], is that theBEM is largely independent of the precise implementa-tion of boundary conditions at infinity. This is not thecase for other techniques such the direct method of inte-grating coupled-channel equations.The calculations have been carried out for the ro-tational spectrum of dipole-bound anions of hydrogencyanide HCN − , which has long served as a prototype ofa dipole-bound anion [4, 51] and was a subject of ex-perimental and theoretical studies [25, 52, 53]. Here,we extend our previous studies [39] of bound statesof dipolar molecules to the unbound part of the spec-trum. To integrate coupled channel equations, we usethe Berggren expansion method as it offers superior ac-curacy as compared to the direct integration approach forweakly-bound states and, contrary to direct integrationapproach, allows to describe unbound resonant states.This paper is organized as follows. The model Hamilto-nian is discussed in Sec. II. The coupled channel formula-tion of the Schr¨odinger equation for dipole-bound anions a r X i v : . [ phy s i c s . a t m - c l u s ] O c t is outlined in Sec. III. The Berggren expansion method isintroduced in Sec. IV. The parameters of our calculationare given in Sec. V. Section VI presents the techniqueadopted to identify the decaying Gamow states (reso-nances). To visualize valence electron distributions, inSec. VII we introduce the intrinsic one-body density. Thepredictions for bound states and resonances of HCN − arecollected in Sec. VIII. Finally, Sec. IX contains the con-clusions and outlook. II. HAMILTONIAN
The dipolar anions are composed of a neutral po-lar molecule with a dipole moment µ that is largeenough to bind an additional electron. In the presentstudy, the HCN − dipolar anion is described in the Born-Oppenheimer approximation, and the intrinsic spin of anexternal electron is neglected [35], largely simplifying theequations [36]. Within the pseudo-potential method, theHamiltonian of a dipolar anion can be written as: H = p e m e + j I + V (1)where I is the moment of inertia of the molecule, p e isthe linear momentum of the valence electron, and m e itsmass. The electron-molecule interaction V is approxi-mated by a one-body pseudo-potential [35, 54, 55]: V ( r, θ ) = V dip ( r, θ )+ V α ( r, θ )+ V Q zz ( r, θ )+ V SR ( r ) , (2)where θ is the angle between the dipolar charge separa-tion s and electron coordinate; V dip ( r, θ ) = − µe (cid:88) λ =1 , , ··· (cid:18) r < r > (cid:19) λ sr > P λ (cos θ ) (3)is the electric dipole potential of the molecule; V α ( r, θ ) = − e r [ α + α P (cos θ )] f ( r ) (4)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 ) (5)is the potential due to the permanent quadrupole mo-ment of the molecule, and V SR ( r ) = V exp (cid:104) − ( r/r c ) (cid:105) , (6)is the short-range potential, where r c is a radius range.The short-range potential accounts for the exchange ef-fects and compensates for spurious effects induced by theregularization function f ( r ) = 1 − exp (cid:2) − ( r/r ) (cid:3) (7) introduced in Eqs. (4,5) to avoid a singularity at r → r in Eq. (7) defines an effective shortrange for the regularization.The dipolar potential V dip ( r, θ ) is discontinuous at r = s . To remove this discontinuity, in Eq. (3) we replace r > r < −→ (cid:8) rs f a ( r ) + sr [1 − f a ( r )] (cid:9) erf( ar ) (8) r > −→ sf a ( r ) + r [1 − f a ( r )] (9)with f a ( r ) = (1 + exp[( r − s ) /a ]) − . III. COUPLED-CHANNEL EQUATIONS
In the description of dipolar anions with the Hamilto-nian (1), the coupled-channel formalism is well adaptedto express the wave function of the system [1, 35, 55–57]. The eigenfunction of H corresponding to the totalangular momentum J can be written asΨ J = (cid:88) c u Jc ( r )Θ J(cid:96) c j c , (10)where the index c labels the channel ( (cid:96), j ), u Jc ( r ) is theradial wave function of the valence electron, Θ J(cid:96) c j c is thechannel function, and j + (cid:96) = J . Since the Hamiltonianis rotationally invariant, its eigenvalues are independentof the magnetic quantum number M J , which will beomitted in the following.In order to write the Schr¨odinger equation as a setof coupled-channel equations, the potential V ( r, θ ) inEqs. (2 - 6) is expanded in multipoles: V ( r, θ ) = (cid:88) λ V λ ( r ) P λ (cos θ ) , (11)where V λ ( r ) is the radial form factor and P λ (cos θ ) = 4 π λ + 1 Y ( mol ) λ (ˆ s ) · Y ( e ) λ (ˆ r ) . (12)The matrix elements (cid:104) Θ J(cid:96) c (cid:48) j c (cid:48) | P λ (cos θ ) | Θ J(cid:96) c j c (cid:105) are ob-tained by means of the standard angular momentum al-gebra [39]. The resulting coupled-channel equations forthe radial wave functions u Jc ( r ) can be written as: (cid:20) d dr − (cid:96) c ( (cid:96) c + 1) r − j c ( j c + 1) I + E J (cid:21) u Jc ( r )= (cid:88) c (cid:48) V Jcc (cid:48) ( r ) u Jc (cid:48) ( r ) , (13)where E J is the energy of the system and V Jcc (cid:48) ( r ) = (cid:88) λ (cid:104) Θ J(cid:96) c (cid:48) j c (cid:48) | P λ (cos θ ) | Θ J(cid:96) c j c (cid:105) V λ ( r ) (14)is the coupling potential. IV. BERGGREN EXPANSION METHOD
The Berggren expansion method for studies of thebound states of dipolar anions has been introduced inRef. [39]. In this method, the Hamiltonian is diagonal-ized in a complete basis of single-particle (s.p.) states,the so-called Berggren ensemble [41–43] which is gener-ated by a finite-depth spherical one-body potential. TheBerggren ensemble contains bound ( b ), decaying ( d ), andscattering ( s ) single-particle states along the contour L + (cid:96),j for each considered partial wave ( (cid:96), j ). For that rea-son, the Berggren ensemble is ideally suited to deal withweakly-bound and unbound structures having large spa-tial extensions, such as halos, Rydberg states, or decayingresonances. For more details and recent applications ofBEM in the many-body context, see Ref. [58] and refer-ences cited therein.While the finite-depth potential generating theBerggren ensemble can be chosen arbitrarily, to improvethe convergence we take the diagonal part of the channelcoupling potential V cc (cid:48) ( r ). The basis states Φ k,c ( r ) areeigenstates of the spherical potential V cc ( r ), which areregular at origin and meet outgoing ( b, d ) and scattering( s ) boundary conditions. Note that the wave number k characterizing eigenstates Φ k,c ( r ) is in general complex.The normalization of the bound states is standard, whilethat for the decaying states involves the exterior complexscaling [39, 58, 59]. The scattering states are normalizedto the Dirac delta function.To determine Berggren ensemble, one calculates firstthe s.p. bound and resonance states of the generatings.p. potential for all chosen partial waves ( (cid:96), j ). Then,for each channel ( (cid:96), j ), one selects the contour L + (cid:96),j ina fourth quadrant of the complex k -plane. All ( (cid:96), j )-scattering states in this ensemble belong to L + (cid:96),j . Theprecise form of L + (cid:96),j is unimportant providing that all se-lected s.p. resonances for a given ( (cid:96), j ) lie between thiscontour and the real k -axis for R ( k ) >
0. For each chan-nel, the set of all resonant states and scattering states on L + (cid:96) c ,j c forms a complete s.p. basis.In the present study, each contour L + (cid:96),j is composed ofthree segments: the first one from the origin to k peak inthe fourth quadrant of the complex k -plane, the secondone from k peak to k middle on the real k -axis ( R ( k ) > k middle to k max also on the real k -axis. In all practical applications of the BEM, each con-tour L + (cid:96)j is discretized and the Gauss-Legendre quadra-ture is applied. The cutoff momentum k = k max shouldbe sufficiently large to guarantee the completeness to adesired precision. The discretized scattering states | Φ n,c (cid:105) are renormalized using the Gauss-Legendre weights. Inthis way, the Dirac delta normalization of the scatteringstates is replaced by the usual Kronecker delta normal-ization. In this way, all | Φ i,c (cid:105) states can be treated onthe same footing in the discretized Berggren complete- ness relation: N (cid:88) i =1 | Φ i,c (cid:105) (cid:104) Φ i,c | (cid:39) , (15)where the N basis states include bound, resonance, anddiscretized scattering states for each considered channel c . Finally, since the V cc (cid:48) ( r ) decreases at least as fast as r − , all the off-diagonal matrix elements of the couplingpotential can be computed by the means of the complexscaling. V. PARAMETERS OF THE BEMCALCULATION
The parameters of the pseudo-potential for the HCN − anion are taken from Ref. [36]. These are: α = 15 . a ,α = 1 . a ,Q zz = 3 . ea ,I = 7 . × m e a ,r = 4 . a ,r c = 3 . a ,V = 4 . ,s = 2 . a , and a = a . The value of r c has been adjusted to re-produce the experimental ground state ( J π = 0 + ) energy[25]: E exp (0 +1 ) = − . × − Ry. For the dipolarmoment of the molecule, we take the experimental value µ = 1 . ea . In the following, we express r in unitsof the Bohr radius a , I in units of m e a , and energy inRy. The J π = 1 − band head energy is also known ex-perimentally, E exp (1 − ) = − . × − Ry, but noadjustment of the model parameters has been attemptedto fit the experimental value.To achieve stability of bound-state energies, the BEMcalculations were carried out by including all partialwaves with (cid:96) ≤ (cid:96) max = 9 and taking the optimized num-ber of points ( N C = 165) on the complex contour with k max = 6 a − for each J π . For all ( (cid:96), j ) channels andall J π -values, the complex contour L + (cid:96),j is taken close tothe real axis ( k peak = 0 . − i − , k middle = 1 .
0, and k max = 6 .
0; all in a − ). Its precise form has been ad-justed by looking at the convergence of bound state en-ergies when changing the imaginary part of k peak . Eachsegment of any contour L + (cid:96),j is discretized with the samenumber of points ( N C / VI. IDENTIFICATION OF THE RESONANCES
The diagonalization of a complex-symmetric Hamilto-nian matrix in BEM yields a set of eigenenergies whichare the physical states (poles of the resolvent of theHamiltonian) and a large number of complex-energy scat-tering states. The resonances are thus embedded in adiscretized continuum of scattering states and their iden-tification is not trivial [60, 61].The eigenstates associated with resonances should bestable with respect to changes of the contour [60, 61].Moreover, their dominant channel wave functions shouldexhaust a large fraction of the real part of the norm. Thenorm of an eigenstate of the Hamiltonian is given by: (cid:88) c (cid:88) i (cid:104) Φ k,c | u c (cid:105) = (cid:88) c n c = 1 , (16)where n c the norm of the channel wave function. In gen-eral, the norms of individual channel wave functions forresonances are complex numbers and their real parts arenot necessarily positive definite. It may happen that ifa large number of weak channels { c i } with small nega-tive norms R ( n c i ) < c can have a norm n c >
1. This does not come as a surprise as the channelwave functions have no obvious probabilistic interpreta-tion.To check the stability of BEM eigenstates, we var-ied the imaginary part of k peak from 0 to − . a − in all partial-wave contours. Resulting contour varia-tions change both real ∆ (cid:60) ( E ) (cid:28) (cid:60) ( E ) and imaginary∆ (cid:61) ( E ) parts of the eigenenergies. The precision of the TABLE I. Relative variation of the real part δ (cid:60) ( E ) =∆ (cid:60) ( E ) / (cid:60) ( E ) (in percent) and imaginary part δ (cid:61) ( E ) =∆ (cid:61) ( E ) / (cid:61) ( E ) (in percent) of energies of twenty J π = 2 + res-onances with the change of k peak . All energies are in Ry. Thenumbers in parentheses denote powers of 10.resonance (cid:60) ( E ) δ (cid:60) ( E ) (cid:61) ( E ) δ (cid:61) ( E )1 2.51(-5) 2.47(-1) -9.68(-6) 2.09(-1)2 2.69(-4) 1.29(-4) -3.45(-10) 1.32(+1)3 2.77(-4) 1.37(-5) -3.58(-9) 1.56(+1)4 3.55(-4) 5.61(-4) -7.20(-7) 1.605 3.67(-4) 3.70(-4) -1.21(-6) 1.786 3.96(-4) 3.52(-3) -2.34(-6) 4.55(-1)7 3.98(-4) 2.07(-2) -5.05(-5) 6.19(-2)8 4.25(-4) 6.02(-3) -1.04(-4) 3.02(-2)9 6.48(-4) 9.70(-5) -6.72(-7) 1.4210 6.60(-4) 6.86(-4) -8.32(-7) 2.5211 6.81(-4) 6.77(-3) -1.19(-5) 7.41(-1)12 6.86(-4) 9.86(-4) -1.60(-6) 1.5513 7.40(-4) 5.05(-3) -6.68(-5) 3.85(-2)14 9.80(-4) 7.89(-4) -7.86(-7) 1.45(+1)15 1.05(-3) 4.80(-5) -6.22(-7) 1.3916 1.06(-3) 1.87(-4) -8.54(-7) 2.6617 1.07(-3) 1.82(-3) -5.60(-6) 1.1018 1.09(-3) 4.00(-4) -4.89(-7) 7.6719 1.11(-3) 8.05(-4) -1.66(-6) 9.6120 1.14(-3) 2.28(-3) -2.71(-5) 1.31(-1) resonance-identification method is assessed by looking atthe ratio ∆ (cid:61) ( E ) / (cid:61) ( E ), which is in the range [0 . , .
3] for the resonance states. As an example, the eigenvaluesof J π = 2 + resonant states are listed in Table I. It is seen (10 -3 Ry) − 1.2− 1.0− 0.8− 0.6− 0.4− 0.20.0 ( - R y ) HCN − FIG. 1. (Color online) Illustration of the stability of the en-ergies of the J π = 2 + resonant states of HCN − listed in Ta-ble I (large dots) when the non-resonant scattering contour isshifted. Here, the imaginary part of k peak was varied from 0to − . a − . As a comparison, non-resonant eigenenergiesare marked with tiny dots and exhibit significant shifts. (10 -3 Ry) − 1.2− 1.0− 0.8− 0.6− 0.4− 0.20.0 ( - R y ) HCN − FIG. 2. (Color online) Similar as in Fig. 1 but zoomed in onthe two threshold resonances (states 2 and 3 in Table I). Here,the real part of k peak is also varied from 0.14 a − to 0.16 a − . that the relative variations of (cid:60) ( E ) are always smallerthan 1%, while the relative variations of (cid:61) ( E ) can reach ∼ (cid:61) ( E ) / (cid:61) ( E ) for differentresonant states can differ by three orders of magnitude.In general, a better stability of the BEM eigenstates and,i.e., smaller values of ∆ (cid:61) ( E ) / (cid:61) ( E ), is found for thoseeigenstates, which have several channel wave functionscontributing significantly to the total norm. A typical ac-cumulation of eigenenergies when changing the contouris shown in Fig. 1. One can see that the non-resonantstates do not exhibit the degree of stability that is typ-ical of resonant states. It is interesting to notice thatseveral resonant states are found fairly away from theregion of non-resonant eigenstates. The stability of reso-nant eigenstates persists if the real part of k peak is variedfrom 0.14 a − to 0.16 a − . In this case, the relative varia-tions of the real part of the eigenstate energies dominateas can be seen in Fig. 2 for the two near-threshold reso-nances labeled 2 and 3 in Table I.In order to demonstrate that the identified resonancesare stable with respect to (cid:96) max , in Fig. 3 we show theenergy convergence for states 1-3 of Table I. In general, (cid:61) ( E ) is significantly more sensitive than (cid:60) ( E ) with re-spect to the addition of channels with higher (cid:96) - and j -values. It is seen that (cid:61) ( E ) for resonances with the dom-inant channels ( (cid:96) = 4 , j = 4) and ( (cid:96) = 3 , j = 1) are con-verged already for (cid:96) max ≥
6. The convergence for the nar-row resonance with the dominant channel ( (cid:96) = 2 , j = 4)shown in Fig. 3(a) is also excellent, considering that inthis case (cid:61) ( E ) is of the order of 10 − Ry, which is closeto the limit of a numerical precision of our BEM calcu-lations. −3.47−3.46−3.45−3.44−3.43−3.56−3.52−3.48−3.443 4 5 6 7 8−9.68−9.67−9.66−9.65−9.64 ( E ) ( - R y )( E ) ( - R y )( E ) ( - R y ) (a)(b)(c) FIG. 3. (Color online) The convergence of (cid:61) ( E ) for J π = 2 + resonances 2 (a), 3 (b), and 1 (c) of Table I as a function of (cid:96) max . The quantum numbers ( (cid:96), j ) of the dominant channelare indicated. VII. INTRINSIC DENSITY
It is instructive to present the density of the valenceelectron in the body-fixed frame. This can easily be donein the strong coupling scheme of the particle-plus-rotormodel [62–64], which is usually formulated in the K -representation associated with the intrinsic frame. Here, K J = K (cid:96) + K j is the projection of the total angular mo-mentum on the symmetry axis of the molecule. Of par-ticular interest is the adiabatic limit of I → ∞ , where all J π members of a rotational band collapse at the band-head, i.e., they all can be associated with one intrinsic configuration. The K -representation is useful to visual-ize wave functions, group states with different J -valuesinto rotational bands, and interpret the results in termsof Coriolis mixing [47, 50, 65–68].In the body-fixed frame, the density of the valence elec-tron in the state J π is axially-symmetric and can be de-composed as: ρ J ( r, θ ) = (cid:88) K J ρ JK J ( r, θ ) , (17)where ( r, θ ) stand for the polar coordinates of the electronin the intrinsic frame, and the K J -components of thedensity are: ρ JK J ( r, θ ) = (cid:88) (cid:96),(cid:96) (cid:48) (cid:88) j j + 12 J + 1 (cid:104) (cid:96)K J j | JK J (cid:105) (cid:104) (cid:96) (cid:48) K J j | JK J (cid:105)× u J(cid:96)j ( r ) ∗ r u J(cid:96) (cid:48) j ( r ) r Y K J ∗ (cid:96) ( θ, Y K J (cid:96) (cid:48) ( θ, . (18)If all K J -components except one vanish in Eq. (17),the adiabatic strong-coupling limit is reached and K J be-comes a good quantum number. In this particular case, ρ JK J can be identified as the intrinsic electronic den-sity in the dipole-fixed reference frame. To quantify thedegree of K J -mixing, it is convenient to introduce thenormalization amplitudes: n JK J = (cid:88) (cid:96),j j + 12 J + 1 (cid:104) (cid:96)K J j | JK J (cid:105) (cid:90) | u J(cid:96)j ( r ) | dr. (19)Due to (16), n JK J fullfil the normalization condition: (cid:88) K J n JK J = 1 . (20) VIII. RESULTS OF BEM CALCULATIONS
Predicted energy spectra of HCN − with J π =0 + , − , + , − , + and 5 − are shown in Table II. Onemay notice that the calculated energy of the 1 − bandhead, E (1 − ) = − . × − Ry, is close to the exper-imental value E exp (1 − ) = − . × − Ry. Moreover,consistently with earlier Refs. [25, 36], we do not find a J π = 3 − bound state.The states listed in Table II are plotted in Fig. 4 inthe complex energy plane. These states can be assembledaccording to their decay widths into five groups labelled g - g . The group 4 contains bound states and very nar-row threshold resonances of the dipolar anion. Narrowresonances are contained in groups 3 and 2 while broaderstates form groups 1 and 0. The characterization of theresonance spectra of HCN − in terms of groups g - g willbe provided below. TABLE II. Predicted complex energies (in Ry) of bound and resonance 0 + , − , + , − , 4 + , and 5 − states of the HCN − dipolaranion. Numbers in the parentheses denote powers of 10.state E (0 + ) E (1 − ) E (2 + ) E (3 − ) E (4 + ) E (5 − )1 -1.15(-4) -8.96(-5) -3.69(-5) 3.89(-8) -i 1.06(-8) 2.70(-5) -i 5.55(-9) 8.09(-5) -i 3.08(-9)2 7.62(-5) -i 3.79(-6) 2.70(-5) -i 9.98(-10) 2.51(-5) -i 9.68(-6) 2.63(-4) -i 1.88(-6) 1.84(-4) -i 2.02(-6) 1.33(-4) -i 2.02(-6)3 9.35(-4) -i 9.69(-5) 8.12(-5) -i 7.04(-7) 2.69(-4) -i 3.45(-10) 3.03(-4) -i 9.25(-6) 2.25(-4) -i 2.47(-5) 1.63(-4) -i 3.71(-5)4 1.09(-3) -i 1.24(-5) 1.62(-4) -i 4.77(-10) 2.77(-4) -i 3.58(-9) 4.99(-4) -i 1.28(-6) 3.65(-4) -i 1.40(-6) 2.56(-4) -i 1.87(-6)5 1.11(-3) -i 4.06(-4) 4.88(-4) -i 7.04(-7) 3.55(-4) -i 7.20(-7) 5.32(-4) -i 1.01(-6) 3.99(-4) -i 1.43(-6) 2.91(-4) -i 1.85(-6)6 1.14(-3) -i 1.62(-5) 5.00(-4) -i 1.02(-6) 3.67(-4) -i 1.21(-6) 5.69(-4) -i 1.25(-4) 4.23(-4) -i 1.26(-4) 3.03(-4) -i 1.22(-4)7 1.16(-3) -i 2.19(-4) 5.28(-4) -i 1.65(-6) 3.96(-4) -i 2.34(-6) 8.20(-4) -i 1.17(-5) 6.58(-4) -i 9.78(-7) 4.94(-4) -i 1.03(-6)8 1.19(-3) -i 1.96(-5) 5.34(-4) -i 3.13(-5) 3.98(-4) -i 5.05(-5) 8.80(-4) -i 2.96(-7) 6.91(-4) -i 3.44(-7) 5.28(-4) -i 3.62(-7)9 1.27(-3) -i 2.13(-5) 5.71(-4) -i 9.11(-5) 4.25(-4) -i 1.04(-4) 9.39(-4) -i 9.91(-5) 6.92(-4) -i 1.07(-5) 5.67(-4) -i 9.80(-5)10 1.31(-3) -i 3.45(-4) 6.71(-4) -i 3.31(-4) 6.48(-4) -i 6.72(-6) 1.07(-3) -i 3.55(-4) 7.40(-4) -i 1.01(-4) 5.92(-4) -i 9.87(-6)11 1.43(-3) -i 5.64(-6) 8.37(-4) -i 6.53(-7) 6.60(-4) -i 8.32(-7) 1.16(-3) -i 1.24(-5) 8.66(-4) -i 3.38(-4) 6.82(-4) -i 3.14(-4)12 1.84(-3) -i 1.10(-5) 8.48(-4) -i 8.03(-7) 6.81(-4) -i 1.19(-5) 1.30(-3) -i 7.87(-7) 9.75(-4) -i 1.15(-5) 8.21(-4) -i 1.23(-5)13 3.35(-3) -i 1.42(-4) 8.63(-4) -i 8.45(-6) 6.88(-4) -i 1.60(-6) 1.34(-3) -i 1.09(-7) 1.06(-3) -i 7.83(-7) 8.44(-4) -i 7.67(-7)14 3.68(-3) -i 3.26(-5) 8.76(-4) -i 9.82(-7) 7.40(-4) -i 6.68(-5) 1.41(-3) -i 7.12(-5) 1.09(-3) -i 1.16(-7) 8.78(-4) -i 1.14(-7)15 4.23(-3) -i 3.47(-4) 9.34(-4) -i 5.08(-5) 9.80(-4) -i 7.86(-7) 1.56(-3) -i 3.54(-4) 1.16(-3) -i 7.50(-5) 9.34(-4) -i 7.38(-5)16 4.60(-3) -i 4.45(-5) 1.05(-3) -i 3.13(-4) 1.05(-3) -i 6.22(-7) 1.61(-3) -i 1.41(-5) 1.30(-3) -i 3.37(-4) 1.06(-3) -i 3.13(-4)17 1.17(-3) -i 7.06(-7) 1.06(-3) -i 8.54(-7) 1.65(-3) -i 7.83(-4) 1.37(-3) -i 1.24(-5) 1.16(-3) -i 1.08(-5)18 1.30(-3) -i 3.00(-4) 1.07(-3) -i 5.60(-6) 2.17(-3) -i 1.60(-5) 1.67(-3) -i 4.88(-5) 1.30(-3) -i 6.64(-7)19 1.30(-3) -i 1.41(-6) 1.09(-3) -i 4.89(-7) 2.24(-3) -i 7.85(-4) 1.84(-3) -i 3.41(-4) 1.40(-3) -i 4.89(-5)20 1.62(-3) -i 5.82(-7) 1.11(-3) -i 1.66(-6) 1.88(-3) -i 1.44(-5) 1.55(-3) -i 3.19(-4)21 1.78(-3) -i 2.83(-4) 1.14(-3) -i 2.71(-5) 1.94(-3) -i 7.63(-4) 1.61(-3) -i 1.27(-5)22 2.49(-3) -i 1.64(-5) 1.66(-3) -i 7.36(-4)23 2.58(-3) -i 7.73(-4) 1.96(-3) -i 3.15(-5)24 2.14(-3) -i 3.29(-4)25 2.17(-3) -i 1.46(-5)26 2.25(-3) -i 7.44(-4)27 2.84(-3) -i 1.67(-5)28 2.94(-3) -i 7.61(-4) A. Adiabatic limit
To check the numerical accuracy of the adiabatic ap-proximation, we computed the energies of the loweststates of HCN − in the adiabatic limit of I → ∞ (in prac-tice, I = 10 m e a ). In this limit, which can be asso-ciated with the extreme strong coupling regime, K J be-comes a good quantum number and energies of all bandmembers J = K J , K J + 1 , K J + 2 , . . . collapse at thebandhead E J = K J . In our calculations, the maximumenergy difference between the members of the ground-state band ( J π = 0 +1 , − , +1 , − , +1 , − ) is 1 . × − Ry,which is better than 0.1% of the energy of the 0 + state( E = − . × − Ry). We can conclude, therefore,that the members of the ground-state rotational band arepractically degenerate in the adiabatic limit.Figure 5 illustrates the intrinsic density for the ground-state band in the adiabatic limit ( I → ∞ ; K J = 0).The intrinsic densities for all band members are numeri-cally identical even though the associated wave functionsin the laboratory system are different, see Fig. 6. Thestrongly asymmetric shape of electron’s distribution re-flects the attraction/repulsion between the electron andpositive/negative charge of the dipole (for other illustra-tive examples, see Refs. [5, 7, 11, 25, 69]). We found that the density representation given byEq. (17) can also be useful in the non-adiabatic case,with finite moment of inertia, to assign members of ro-tational bands. This is illustrated in Fig. 7 which showsthe density (17) for the bound states J π = 0 +1 , 1 − , and2 +1 of HCN − . Despite the fact that the strong couplinglimit does not strictly apply in this case, distributionsare practically identical and close to the intrinsic densitydisplayed in Fig. 5. B. Rotational bands
Excitation energies of the lowest-energy resonant (i.e.,bound and resonance) states are plotted in Fig. 8 as afunction of J ( J + 1). The J π = 0 + , 1 − , 2 + bound statesform a K J = 0 rotational band as evidenced by their in-trinsic densities shown in Fig. 7. Another K J = 0 rota-tional band is built upon the 0 +2 resonance. Accordingto Table II, a 1 − member of this band has a decay widththat is reduced by over three orders of magnitude as com-pared to that of the 0 +2 bandhead. We predict other verynarrow resonances as well. Among them, the 2 +4 statehas K J = 2 while 1 − and 2 +3 resonances have a mixedcharacter.As can be judged by results displayed in Fig. 8, ex- (mRy) ( m R y ) (mRy) ( - m R y ) HCN − g g g g g g g FIG. 4. (Color online) Predicted energies of the HCN − dipo-lar anion for J π = 0 + , 1 − , 2 + , 3 − , 4 + , and 5 − states in thecomplex-energy plane. Based on their complex energies, thesestates can be organized into five groups labelled g to g .Bound states and near-threshold resonances belonging to g and narrow resonances of g are shown in the insert. -50 x (a )050 -5 ground-state band HCN - z ( a ) FIG. 5. (Color online) The intrinsic density of the valenceelectron in HCN − in the ground-state rotational band J π =0 +1 , − , +1 , − , . . . (All densities are in a − .) cept for few states with well defined K J -values, majorityof resonances are strongly K J -mixed. Consequently, anidentification of other rotational bands in the continuum,based on the concept of intrinsic density, is not straight-forward. This is true, in particular for the supposedhigher- J members of the ground-state band. Figure 9shows ρ JK J =0 for J π = 3 − , +1 , − resonances, which areexpected – based on energy considerations – to form acontinuation of the ground state rotational band. Onecan see that these densities are not only drastically dif-ferent from those of 0 +1 , 1 − , and 2 +1 states but also changefrom one state to another. It is also worth noting thatthe densities of 3 − , +1 , and 5 − resonances have spatialextensions that are dramatically larger as compared to -0.0400.040.08 (a)(b) -0.0400.040.08 0 500 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) FIG. 6. (Color online) Channel wave functions ( (cid:96), j ) of the J π = 0 +1 (a) and 4 +1 (b) members of the ground-state rota-tional band in HCN − in the adiabatic limit. the three bound members of the ground-state band.As seen in Fig. 4, there appear clusters of resonanceshaving the same total angular momentum J within onegroup g i . In each cluster, dominant channel wave func-tions have the same orbital angular momentum of the va-lence electron (cid:96) , but different rotational angular momentaof the molecule j . Excitation energies of resonances areplotted as a function of the molecular angular momentum j in Fig. 10 for different groups of resonances of Fig. 4.It is seen that these states form very regular rotationalband sequences in j rather than in J . Different mem-bers of such bands lie close in the complex energy planeand have similar densities ρ JK J ( r, θ ). This is illustratedin Fig. 11, which shows ρ JK J ( r, θ ) for the two J π = 5 − resonances marked by arrows in Fig. 10(c); namely 5 − ,having the dominant parentage ( (cid:96), j ) = (6 , − ,having the dominant parentage (6 , (cid:96) - j coupling,whereby the orbital motion of a valence electron is de-coupled from the rotational motion of a dipolar neu-tral molecule. To illustrate the weak coupling better,in Fig. 12 we display the rotational bands of Fig. 10 withrespect to the rigid rotor reference j ( j + 1) / I . In thecase of a perfect (cid:96) - j decoupling, the rescaled energy inFig. 12 should be equal to 1. One can see that this limitis reached in most cases, with deviations from unity be-ing less than 10 %. Larger deviations are found for fewlow- j states in bands with J = 2 in g and J = 5 in g . Consequently, intrinsic densities for resonances inthese two bands exhibit certain differences, whereas theyare almost identical for bands close to the weak-coupling -50 500050050050 -5 HCN − (a)(b)(c) FIG. 7. (Color online) Density (17) of the valence electron inthe bound states J π = 0 +1 (a), 1 − (b), and 2 +1 (c) of HCN − .(All densities are in a − .)
012 10 15 20 25 300 5
HCN - threshold FIG. 8. (Color online) Energy spectrum of the HCN − anionfor J π = 0 + , 1 − , 2 + , 3 − , 4 + , and 5 − shown as a function of J ( J + 1). The dominant K J -component (19) is indicated. Ifseveral components are present, the state is marked as “mix”. limit.The variations seen in Fig. 12 can be traced back tothe leading channel components along a j -band. Table IIIdisplays the leading channel wave functions to the reso-nances in different groups g i . Not surprisingly, the res-onances forming j -band structures are associated withhigh orbital angular momentum components (cid:96) = 6 − - + - (a)(b)(c) FIG. 9. (Color online) Similar as in Fig. 7 but for ρ JK J =0 ( r, θ )(in 10 − a ) in (a) 3 − , (b) 4 +1 , and (c) 5 − .TABLE III. Contributions of the two leading channel wavefunctions to the norm of resonances in different groups ofstates in Fig. 4. Only states with dominant channel (cid:96) = 6 for g , g , annd g , and (cid:96) = 8 for g , and g are included.Group (cid:96) of dominant channels6 7 8 9 g – – 60% 40% g
1% – 99% – g
70% 30% – – g
90% 10% – – g in Fig. 12, the (cid:96) -content is almost constant as a functionof j . For instance, for the four J = 5 states in g , the( (cid:96), j ) parentages of the two largest (6, j )/(7, j + 1) compo-nents are: 0.64/0.37 ( j = 5), 0.67/0.36 ( j = 7), 0.69/0.35( j = 9), and 0.70/0.34 ( j = 11). On the other hand, forbands that exhibit stronger j -dependence in Fig. 12 the
50 100 15000120123 50 100 1500 g g g g g (a) (b)(c) (d) FIG. 10. (Color online) Excitation energies of resonancesof the HCN − dipolar anion for various J π as a function of j ( j + 1), where j is the rotational angular momentum of themolecule in the dominant channel wave function for each con-sidered state. Colors are related to groups of states in thecomplex-energy plane identified in Fig. 4. The symbols (cid:4) , • , (cid:72) and + denote states with J π = 2 + , 3 − , 4 + and 5 − , respec-tively. (cid:96) -compositions change.Interesting complementary information about the ar-rangement of resonances in the continuum of HCN − canbe seen in Fig. 13 which shows the decay width for vari-ous j -bands in different groups g i and different total an-gular momenta J within a given group. One can seethat the bands that exhibit largest deviations from theweak-coupling limit in Fig. 12, also show strong in-bandvariations of the decay width. In regular bands belong-ing to g , g , and g , the width stays constant or slightlyincreases with j . On the other hand, the irregular bandsin g and g exhibit a decrease of Γ J with j . Such a be-havior of lifetimes can be traced back to variations of the( (cid:96), j )-content of the resonance wave function with rota-tion. IX. CONCLUSIONS
In this work, we studied bound and resonance states ofthe dipole-bound anion of hydrogen cyanide HCN − usingthe open-system Berggren expansion method. To identifythe decaying resonant states and separate them from thescattering background, we adopted the algorithm basedon contour shift in the complex energy plane. To charac-terize spatial distributions of valence electrons, we intro-duced the intrinsic density of the valence electron. Thisquantity is useful when assigning resonant states into ro-tational bands.Non-adiabatic coupled-channel calculations with a K J =0 K J =1 K J =2 − − (a) (b)(c) (d)(e) (f) FIG. 11. (Color online) Intrinsic densities ρ JK J ( r, θ ) (in10 − a ) with K J = 0 , ,
2, for the two resonances 5 − and5 − belonging to the group g , marked by arrows in Fig. 10(c).For both states, the dominant channel has (cid:96) = 6. g g g g g (a) (b)(c) (d) FIG. 12. (Color online) Similar as in Fig. 10 but for ( E J − E bh ) Ij ( j +1) , where E bh is a bandhead energy at j = 0. pseudo potential adjusted to ground-state properties ofHCN − predict only three bound states of the dipole-bound anion: 0 + , 1 − , and 2 + . Those states are mem-bers of the ground-state rotational band. The lowest 3 − state is a threshold resonance; its intrinsic structure isvery different from that of 0 +1 , 1 − , and 2 +1 states, andthe lowest-energy resonances 4 +1 , and 5 − .The dissociation threshold in the HCN − dipolar anion0
50 100 1500 50 100 15000-1-2-30-1-2-3 g g g g g (a) (b)(c) (d) FIG. 13. (Color online) Similar as in Fig. 10 but for theresonance widths. defines two distinct regimes of rotational motion. Belowthe threshold, rotational bands in J can be associatedwith bound states. Here, the valence electron followsthe collective rotation of the molecule. This is not thecase above the threshold where the motion of a valenceelectron in a resonance state is largely decoupled fromthe molecular rotation with the families of resonancesforming regular band sequences in j . Widths of reso- nances forming j -bands depend primarily on the elec-tron’s orbital angular momentum in the dominant chan-nel and remain fairly constant within each band for regu-lar bands. Small irregularities in moments of inertia anddecay width are predicted for very narrow resonances inthe vicinity of the dissociation threshold.In summary, this work demonstrates the feasibilityof accurate calculations of weakly bound and unboundstates of the dipolar anions using the Berggren expan-sion approach. Our prediction of two distinct modes ofrotation in this open quantum system awaits experimen-tal confirmation. It is interesting to note a similaritybetween the problem of a dipolar anion and a couplingof electrons in high molecular Rydberg states to molec-ular rotations [70, 71]. Namely, in both cases one dealswith non-adiabatic coupling of a slow electron to the fastrotational motion of the core, with no separation in thesingle-particle and collective time scales. ACKNOWLEDGMENTS
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