Resonances in rotationally inelastic scattering of OH($X^2Π$) with helium and neon

1.8 MB·English

by  Koos B. Gubbels

X2Π
Resonances in rotationally inelastic scattering of OH( ) with helium and neon
Koos B. Gubbels,1,2,∗ Qianli Ma,3 Millard H. Alexander,4 Paul J. Dagdigian,3 Dick Tanis,2
Gerrit C. Groenenboom,2 Ad van der Avoird,2 and Sebastiaan Y. T. van de Meerakker2
1Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin,
Germany
2 Radboud University Nijmegen, Institute for Molecules and Materials, Heyendaalseweg 135, 6525 AJ Nijmegen,
The Netherlands
3Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218-2685, USA
4Department of Chemistry and Biochemistry and Institute for Physical Science and Technology,
2
University of Maryland, College Park, MD 20742-2021, USA
1
(Dated: January 17, 2012)
0
2 We present detailed calculations on resonances in rotationally and spin-orbit inelastic scattering
n of OH (X2Π,j =3/2,F1,f) radicals with He and Ne atoms. We calculate new ab initio potential
a energy surfaces for OH-He, and the cross sections derived from these surfaces compare favorably
J with the recent crossed beam scattering experiment of Kirste et al. [Phys. Rev. A 82, 042717
6 (2010)]. We identify both shape and Feshbach resonances in the integral and differential state-to-
1 statescatteringcrosssections,andwediscusstheprospectsforexperimentallyobservingscattering
resonances using Stark decelerated beams of OH radicals.

]
h
p I. INTRODUCTION statecrosssections,bothintegralanddifferential,aswell
-
m as their dependence upon the collision energy [11]. The
recently developed Stark deceleration technique, taking
Measurements of state-to-state cross sections pro-
o
advantageoftheinteractionofpolarmoleculeswithtime-
t vide important tests of the reliability of computed po-
a varying electric fields, has allowed continuous tuning of
tential energy surfaces (PES’s) describing the interac-
s. tion of atoms and molecules [1]. Cross sections for thebeamvelocity[12]. Thishasfacilitatedmeasurements
c of the collision energy dependence of state-to-state in-
collision-induced rotational transitions are sensitive to
si the anisotropy of the PES. Since non-bonding interac- tegral cross sections down to energies of 70 cm−1 [13].
y Moreover,the velocity spread in such decelerated beams
tions are relatively weak, the magnitudes of the cross
h
is much smaller than in conventional molecular beams.
p sections are mostly sensitive to the repulsive part of the Thus far, an energy resolution of ≥ 13 cm−1 has been
[ PES, except at very low collision energies. An alterna-
achievedforcollisionsofOHradicalswithraregasatoms
tive, spectroscopic approach to gaining information on
1 [13–15]. This resolutionismainly limited bythe velocity
PES’s is the determination of the energies of the bound
v andangularspreadoftheatomiccollisionpartner,andis
levels of van der Waals complexes of the collision part-
8
too low to experimentally resolve scattering resonances.
6 ners [2, 3]. The energies of the bound levels are mainly
A recentstudy has shownthat the energyresolutioncan
2 sensitive to the attractive part of the PES’s. As we go
3 up higher in the manifold of these weakly bound lev- beimprovedsignificantlybyanappropriatechoiceofthe
1. els, the energies of these levels eventually become higher beam velocities and interaction angle [16]. When these
measuresareputintopracticeinthelaboratory,collision
0 than the dissociation energy of the complex, and such
energy resolutions can be obtained that may enable the
2 levels are quasi-bound. These quasi-bound levels are of-
1 ten described as resonances and can be thought of as a observation of scattering resonances.
v: distortion of the continuum in the collision energy de- Atom-molecule collisions are the simplest type of col-
i pendence of state-to-state cross sections [4]. In inelastic lision process in which rotationally inelastic transitions
X
scattering, resonances are called shape or orbiting res- can be observed. Early calculations [17, 18] on rotation-
ar onances when the quasi-bound levels involve monomer ally inelastic scattering of N2 molecules with He atoms
levels that are the same as in the initial or final level of have shown that resonances occur at low collision ener-
the collision-induced transition, or Feshbach resonances gies, but the experimental verification of these predic-
when they involve different monomer states [1, 4]. Due tions was not yet possible. Collisions of OH(X2Π) with
totheirsensitivitytothePES,resonancescanrevealim- raregaseshaveemergedasparadigmsofscatteringofan
portant information on the PES [5, 6]. So far, however, open-shell molecule with an atom [13, 15, 19–24]. The
resonantstructuresinscatteringcrosssectionshavebeen OH-rare gas systems are good candidates for the obser-
experimentallyobservedonlyinexceptionalcases[7–10]. vationandanalysisofresonancesinrotationallyinelastic
The crossed molecular beam technique has been an collisions because the collision energy can be reduced by
extremely useful tool for the determination of state-to- Stark deceleration of the OH beam. Since OH(X2Π) is
an open-shell molecule with orbital degeneracy, the col-
lision dynamics is governed by two PES’s,[25] and in-
teresting multi-state dynamics can occur. Of particular
∗Electronicaddress: [email protected] interest for the study of resonances are the OH-He and
2
OH-Ne systems, since the dissociation energies of these
systems are smaller than the rotational level spacings of
the OH radical. The resonance features associated with
the variousrotationallevels arethereforewell separated.
TheshallowvanderWaalswellssupportonlyoneortwo
stretch vibrational levels [26], resulting in a rather sim-
ple, yet interesting, analysis of the resonances. Shape
resonances in OH-He collisions were previously analyzed
by Dagdigian and Alexander[24] in a study of elastic de-
polarization. Bound states of the OH-He complex have
been investigated spectroscopically by Han and Heaven,
who identified complex features as scattering resonances
in OH(A)-He [27].
Here, we present a detailed and precise study of scat-
tering resonances in the OH-He and OH-Ne systems, in
order to develop insight into the nature and strength of
the resonances and to assist in the experimental search FIG. 1: Energies of the lower rotational levels of OH(X2Π).
for such scattering resonances. For the OH-He system, TheΛ-doubletsplittingisexaggerated forclarity. Theinitial
we have calculated new three dimensional potential en- level for all scattering calculations is the j =3/2,F1,f level.
ergy surfaces. The collision energy dependence of the
relative state-to-state integral scattering cross sections
that are derived from these potentials compares more programs[33], andwith a secondindependent scattering
favorably with recent experiments [15] than the results programforopen-shelldiatom-atomscatteringdescribed
from previous calculations. For the correct assessment in Ref. 34. Care was taken to independently check the
of the resonances, the calculations are performed on a resultswiththetwoscatteringprogramsandtoconverge
very fine grid of collision energies, and particular care is the cross sections. For OH-He the maximum total an-
takento convergethe calculationsto avoidnumericalar- gular momentum was J = 100.5 – 140.5, depending on
tifacts to be interpreted as resonant structures [28]. We the collision energy, and the channel basis consisted of
characterize the resonances with various techniques, in- all rotational levels of OH with j ≤ 6.5, while for OH-
cluding the adiabatic bender model [29, 30] andcollision Ne the channel basis consisted of all rotational levels
lifetime analysis [31]. We investigate how the differen- with j ≤ 7.5. In this paper, we calculate cross sections
tial cross section for several transitions changes as the from fully convergedclose-coupling calculations in order
collision energy is scanned through the resonances, and to study resonances in inelastic collisions between low-
observe dramatic effects. lying rotational states of the OH radical. For reference,
This paper is organized as follows: The details of the the energies of the lower rotational levels of OH(X2Π)
scatteringcalculationsarebrieflypresentedinSectionII. are displayed graphically in Fig. 1.
InSectionIIIwedescribethenewthree-dimensional(3D)
PES’s that are developed for OH-He. Section IV de-
scribes our calculations on the state-to-state scattering III. 3D OH-HELIUM POTENTIAL
cross sections in OH-He collisions. A detailed analysis
of shape and Feshbach resonances is given. Section V
A crucial role in the scattering calculations is played
presents similar results for the OH-Ne system. A dis-
by the interaction potential. In Ref. 34 a detailed ex-
cussion of the prospects for observing these resonances
perimental and theoretical study of inelastic scattering
in crossed beam experiments using a Stark decelerator,
betweenOH radicalsandthe raregas atoms He, Ne, Ar,
by either recording the integral or the differential cross
Kr and Xe was performed. The theoretical results in
sections, follows in Sec. VI.
thatstudy wereshownto be inexcellentagreementwith
experimentally measured inelastic cross sections. The
agreementbetweentheoryandexperimentwas,although
II. SCATTERING CALCULATIONS still very good, the worst for the OH-He system. It was
believedthatthiswasduetothequalityofthePES,since
The theory of scattering between a molecule in a 2Π fortheOH-He systemasmallerbasissetwasusedinthe
electronic state and a structureless atom is well estab- calculations than for the other systems. For this reason,
lished [25]. The interaction can be described by two weconstructhereanewpotentialfortheOH-Hesystem.
PES’s corresponding to states of A′ and A′′ symmetries. We note that in Ref. 34 the experimental resolution was
For OH-He, we have constructed new PES’s, which are unfortunatelynotyethighenoughtoobserveresonances.
explained in Sect. III, while for OH-Ne, we used the In trying to improve the agreement with the experi-
PES’s by Sumiyoshi et al. [32]. Close-coupling calcula- mental results, we first constructed new 2D PES’s for
tions were performed both with the HIBRIDON suite of OH-He collisions. This was done by enhancing the ba-
3
He fitted to inverse powers R−n with n≥6, namely
11
f (βR)
q b Rb R X Ra vll,r0(R,θ)=n=Xn0(l)cl,n nRn , (2)
q q a
Q r where we note that the allowed values for n depend on
H l [38]. For example, for l = 0 we have n (l) = 6 and
0
O
onlyevenvaluesofnarepresent,while forl =1wehave
n (l) = 7 and only odd values of n are present. From
0
FIG. 2: Illustration of the OH molecule and the He atom
thefittedcoefficientsweonlykepttheleadinglong-range
containing the relevant coordinates as used in the fitting of
terms for l = 0 to l = 4. We used the Tang-Toennies
the OH-He potential. R is the length of the vector R that
damping function [39]
connects the He atom and the center-of-mass (Q) of the OH
molecule, while θ is the angle between R and the OH bond n xk
vofectthoervrec(ptoorinRtingthfraotmcoOnnteoctHs)thoeflHenegatthomr.aRnaditshtehHelaentogmth, fn(x)=1−e−xX k! (3)
a k=0
while θ is the angle between R and r. R is the length of
a a b
the vector R that connects the He atom and the O atom, to damp these five long-range terms in the short range
whileθ isthebanglebetweenR and−r. ThepointXmarks withβ =0.6a−1. Intheshortrange(R<5.5a ), the ex-
b b 0 0
thelocation of themidbond orbitals. pansioncoefficientsv (R)werefittedtoanexponential,
l,0
namely
sis set for the coupled-cluster calculations of the interac- vsr(R)=s e−αlR. (4)
l l
tion energy from the augmented triple-zeta correlation-
consistent basis set (AVTZ) used by Lee et al. [26] to The difference between the ab initio interaction energies
the quintuple-zeta basis set (AV5Z). We computed the andtheanalyticlongrangeandshortrangefunctionswas
interaction energies with the open-shell single and dou- fitted with a reproducing kernel Hilbert space (RKHS)
ble excitation coupled cluster method with perturbative method [40]. The RKHS parameter m was chosen such
triples asimplemented inthe molpro package[35]. The that the RKHS fit would decay faster than the leading
interactionenergieswereevaluatedfor288geometrieson long-range term for each l. The RKHS smoothness pa-
a two-dimensional grid with 12 Gauss-Legendre points rameter was set to 2. For the expansion coefficients of
in the Jacobi-angle θ. The OH bond length was fixed the difference potential, vl,2(R), no analytic short range
at the vibrationally averageddistance of r =1.8502 a , and long range fit was performed, so that everywhere
0 0
whereas Lee et al. used the equilibrium distance r . The the RKHS method was used. Using the describedproce-
e
relevant geometry is illustrated in Fig. 2. We included dure, we obtained an accurate fit to the ab initio points.
midbond orbitals (3s,3p,2d,1f,1g)with the exponents of Moredetailsofthefitcanbe foundonepaps [41],where
Ref. 36. These midbond functions were centered on the we provide a fortran 77 code for the two-dimensional
vector R that connects the He atom and the center-of- AV5Z potential.
mass of the OH molecule, at a distance from the helium We found that the absolute minimum of the fitted po-
◦ ′
atom that is half the distance of the helium atom to the tentialislocatedatθ =68.7 ,R=5.69a0ontheA PES,
nearest atom of the OH molecule. Also the counterpoise corresponding to an interaction energy of VA′ = −29.8
correction of Boys and Bernardi was applied [37]. The cm−1. The minimum potential energy values for θ = 0◦
◦
grid of atom-molecule separations consisted of 18 points andθ =180 werefoundatR=6.56a0andR=6.09a0,
rangingfromR=3a0 to9a0 atshortrangeand6points giving rise to VA′/A′′ =−27.1 cm−1 and VA′/A′′ =−21.6
onanapproximatelylogarithmicscaleupto25a atlong cm−1, respectively. For comparison, we also mention
0
range. the values obtained by Lee et al. [26], who found that
Asmentionedbefore,twopotentialenergysurfacesbe- the absolute minimum of their potential was located at
longing to states of A′ and A′′ symmetry are involvedin θ =68.6◦, R=5.69a for A′ symmetry,with aninterac-
0
the OH-rare-gas atom scattering. The average Vs and tion energy of VA′ = −30.0 cm−1. The minimum values
◦ ◦
half-difference V of these potentials can be expanded in for θ = 0 and θ = 180 were found at R = 6.54a
d 0
Racah normalized spherical harmonics Cl,m, namely and R = 6.09, giving rise to VA′/A′′ = −27.1 cm−1 and
VA′ +VA′′ lmax VarAe′/Ase′′en=t−o2g1i.v8ecvme−ry1,srimesiplaecrtirveesluyl.tsTfhoerttwhoe plooctaelntaianlds
V = = v (R)C (θ,0),
s 2 X l,0 l,0 globalminima. Moreover,using the new AV5Z potential
l=0
forscatteringcalculations,wefoundonlyaveryslightim-
VA′′ −VA′ lmax provementinthe agreementwith the experimentaldata.
V = = v (R)C (θ,0), (1)
d 2 X l,2 l,2 Therefore,wetriedtoimprovethePESfurtherbytak-
l=2 ing the vibrational motion of the OH radical into ac-
wherewe included allterms upto l =11. Inthe long count. To this end, we computed the interaction ener-
max
range(R>10a ),theexpansioncoefficientsv (R)were gies of the OH-He system on a three-dimensional grid.
0 l,0
4
At short and intermediate range we used a step size of (a) r = 0.7500 a
0
∆R = 0.25a for 3a ≤ R ≤ 12.5a and ∆r = 0.25a 12
0 0 0 0
feoqrui0d.i7s5taan0t≤gridr o≤f◦146.5pao0in.tsFionrcltuhdeinagn0glaenθdw18e0◦uswedithana 180 −0.−205.5 −−0.05.25
spacingof∆θ =12 . Atlongrangeweused4equidistant a 0 −1 −
porfo=i∆ntr4s.5=baet00w.5weaean0s f1ao4lrso≤0.i7nR5clau≤0de2≤d0,rinw≤htihl4ee.2wl5oena0gu.-sreaTdnhgaeesdfitietsp.tasFniozceer /yHe 246 −2−4−8−16 −16111000000000−20 −28−36 −8−4−21
the angle θ we used an equidistant grid of 9 points with
◦ 0
a spacing of ∆θ = 22.5 . On this grid we computed the
interaction energies with a triple-zeta basis set (AVTZ) 12 (b) r = re= 1.8324 a0
apeTpnlneooehiTdnicgenteonhutrn,bosptbiwonsenyuiregc[fr4omoimsn2rbtme]girt.dauagbnctirnohEtsiuenedsodrdpfiefpetotocchirnaiioebnatlfcitleetlsturynah.plelaesofrtolgwsiaroyutinmtilfohsaonrdriggnietodetohermnerraeoeccxattottnrariryordlawe-npdssapoepmyplooasaenttcnlediloondinnntReivgna,eftlrgrtoeVrghxmiesde-,. /y a He0124680 −0.2−50.5−1−2−4−8−16−20 −22 −2114100000−000206−28 −27 −16−8−4−2−−10.−50.25
we proceed in the following way. We represent the po- 0
tential as a sum of three terms, namely (c) r = 3.0000 a
0
12
wThheisrereptVVhrssees(srRe(dRni,fftθaae,,trrθieo)ann,=tri)sc+ocooVnrdvssiern(naRiteben,stθbba,errec)ad+uesfieVnstlrhe(deRc,ionθo,rFrd)ii,gn.a(t5e2)s. / yaHe014680 −0.25−0.5−1−2−4−8 −20 −24 1−10200800 −14 −−−8−−4021.5
of the first and second term of Eq. (5) are ideally suited 2 −16 10000
to describe the short-range behavior near the H and O
0
atom, respectively, while the coordinates of the third !" # 0 5 10
term are convenientto describe the long-rangebehavior. xHe / a0
The short-range terms are fitted by
FIG. 3: A′ potential energy surface. The OH radical lies on
l0max thehorizontalaxis,withthecenter-of-massofthemoleculeat
Vssr(Ri,θi,r)= Xe−βiRiPl(cosθi)s(li) the origin. The O atom lies left of the origin, the H atom to
l=0 the right. For each geometry of the complex, defined by the
limaxkmiaxnimax OHbondlengthrandtheposition(xHe,yHe)oftheHeatom,
+X X X Rine−βiRiPl(cosθi)rke−αir3sl(ni)k, (6) tthheeiunnteitraoctfiocnme−n1e.rgyThisecathlcrueleatpeldo,trsesduifflteinrginintchoentOouHrsbwointhd
l=0 k=0 n=0 length, namely in panel a) we have r = 0.75a , in panel b)
0
r=1.8324a , and in panel c) r=3.00a .
where i = a,b, while P(x) are Legendre polynomials 0 0
l
corresponding to the functions C (θ,0) of Eq. 1. We
l,0
used the values l0 = 1, na = 3, ka = 8, la = 7,
nb = 3, kb =ma8xand lb max= 5. Thmealxong ranmgaexterm now the Legendre polynomials Pl(x) are replaced by as-
max max max sociated Legendre functions P2(x) corresponding to the
is fitted by l
Racah spherical harmonics C (θ,0) of Eq. 1, so that
l,2
13 n−4 all sums start with l = 2. Moreover, we use l0 = 2,
f (βR) max
Vlr(R,θ,r)= n P (cosθ)c (r), (7) na = 5, ka = 5, la = 6, nb = 5, kb = 4 and
s XX Rn l nl max max max max max
n=6l=0 lmbax = 6. The linear and the nonlinear fit parameters
were determined by minimizing a weighted least-squares
where f is the damping function of Eq. (3). Nonzero
n error.
valuesofc occuronlyforevenvaluesofl+n, andthen
nl Byevaluatingtheanalyticrepresentationofthepoten-
they are given by tial on the ab initio grid, we were able to compare the
fitted energy values with the ab initio values. We found
3
c (r)=c0 + rke−αnr3c . (8) that we only obtained a reliable fit for OH bond lengths
nl nl X nlk r ≤3a . AtsmallervaluesofRthe largestrelativeerror
0
k=0 of an analytic value compared to an ab initio value for
We use two different values for α , namely α = α for r ≤ 3a was 6.67% for the sum potential and 1.10% for
n n I 0
6 ≤ n ≤ 9, and α = α for 10 ≤ n ≤ 13. For the dif- the difference potential. At large R, again considering
n II
ference potential V similar fit functions are used, only only r ≤ 3a , the largest relative error was 3.23 % for
d 0
5
(a) r = 0.7500 a0 % a) b)
12
n /
y/ a He01024608 −0.−205.−5−12−4−8−16 101100000000 −16−36−20 −8−4−2−−−010.5.25 relative cross sectio500 200
1
c) d)
(b) r = re= 1.8324 a0 %
12
y/ a He01002468 −0.25−0.5−1−2−4−8−20−16 110100000000 −16−20−24−8−4−2−−10−.50.25 relative cross section / 0.05100 200 300 40005100 200 300 400
(c) r = 3.0000 a Collision energy / cm-1 Collision energy / cm-1
0
12
y/ a He010468 −0.25−0.−51−2−4−8 −16 −14 110010000000 −14 −−−482−1−0.5 tFtrTwieeIohhsnGnueitlsil.eeatxs5lot:phfoweefROirtLtiehHmhelaeetetoh(nieXrevtteae2taaliΠdscld.tia,aaal[jblt2tyaea6=-t]pctioaaco3-lricsp/netu2otails,tnateFefctnr1eloutid,nimdfaece)llrRdaaorssesatafsidsd.cisa3scess4oacchlltaiaseditrdotewncecsuisruthirhnrwovvgweieHtssnch,e.raaotOansshtsdnedosompttethhocssee--.,
2 −20 vertical axes of the plots, 100 % refers to the total inelastic
0 !" # 0 5 10 cross section. Relative cross sections for inelastic collisions
xHe/ a 0 populating the (a) j =3/2,F1,e (black), j =5/2,F1,e (pur-
ple),andj =5/2,F ,f (orange) states; (b)thej =1/2,F ,e
1 2
(brown)andj =3/2,F ,f (pink)states;(c)thej =1/2,F ,f
2 2
FIG.4: SimilartoFig. 3,butshowingtheA′′potentialenergy (red) and j = 3/2,F2,e (blue) states; (d) the j = 7/2,F1,e
surface. (green) and j =7/2,F1,f (cyan) states.
the sum potentialand3.32%for the difference potential.
andthensubtractedthevaluesofthispotentialatr =r
0
We also calculated the average relative error, which for
and added the two-dimensional potential calculated for
the sum potential was 0.32% at short range and 0.49%
r = r at the AV5Z level. This implies that the depen-
0
at long range, while for the difference potential it was
dence of the intermolecular potential on the most rele-
0.04% in the short range and 0.90% in the long range.
vant coordinates for the scattering calculation, namely
In Fig. 3, we show two-dimensional contour plots of the
Randθ, iscomputedatthe AV5Z levelforr =r ,while
′ 0
fitted OH-He A PES for r = 0.75a , r = 1.8324a and
0 0 the variation of the potential with the OH bond length
r = 3.00a , while in Fig. 4 the same plots are shown
0 r is taken into account at the AVTZ level. We solved
′′
for the fitted A PES. For the equilibrium bond length
for the OH vibrational motion in the full 3D potential
r =1.8324a ,ourfitofthe3-dimensionalpotentialisin
e 0 generated by taking the intermolecular potential energy
verycloseagreementwiththePESofLee et al.[26],asit
V(R,θ,r) and adding the free OH monomer potential
should, since both PES’s were calculated with the same
V (r) [43]. For fixed R and θ this leads to an effec-
ab initio method using the same basis set. The absolute OH
tively one-dimensionalproblemthat can be easilysolved
potentialenergyminimumforr =1.8324a islocatedat
0 bystandardnumericalmethods,suchasthediscretevari-
◦ ′
θ = 69.2 , R = 5.69a for A symmetry, leading to an
0 able representation based on sinc-functions (sinc-DVR)
interaction energy of VA′ = −30.0 cm−1. The minimum [44]. Taking the resulting ground state energy for each
◦ ◦
valuesforθ =0 andθ =180 werefoundatR=6.55a
0 R and θ and subtracting the v =0 monomer vibrational
and R = 6.09, giving rise to VA′/A′′ = −27.2 cm−1 and energy in the absence of the He atom then results in an
VA′/A′′ =−21.7cm−1, respectively. A fortran 77 code adiabatictwo-dimensionalPES.Wefound,actually,that
to generate the interaction potential V(R,θ,r) is made this adiabatic potential is very similar to the ‘diabatic’
available as an EPAPS document [41]. potential obtained by first calculating the lowest vibra-
To use the three dimensional potential for scattering, tionalstate of OHin the monomerpotential V (r) and
OH
we started with the three-dimensional AVTZ potential, then averaging the interaction potential V(R,θ,r) over
6
this groundstate. The twomethodsareexpectedtogive
similar results since the vibrationallevels of OH are well
separatedin energy,so that the weak OH-He interaction
gives only a slight admixture of the higher vibrational
states of OH.
TheinelasticOH-HecrosssectionswithOHinitiallyin
thej =3/2,F ,f levelwerecalculatedwiththeadiabatic
1
potential. The results areshown in Fig.5, where the ex-
perimental data of Ref. 34 is also shown, as well as the
scatteringresultsobtainedwiththepotentialofLeeetal.
[26]. Thetheoreticaldataareconvolutedwiththeexper-
imentalenergyresolution. TothisendaGaussianenergy
distribution is taken with a standard deviation that is a
function of the energy. The value of the standard devi-
ation ranges from 24 cm−1 at low collision energies to
59 cm−1 at the highest collision energies. We note that
the relative cross sections are plotted, rather than the
absolute cross sections, because these relative cross sec-
tions are experimentally measured. More details can be
found in Refs. 13, 34. We see from Fig. 5 that the over-
all the agreement with experimental data has improved
noticeably with the adiabatic potential.
IV. OH-HELIUM COLLISIONS
A. State-to-state integral cross sections
For OH-He collisions, the state-to-state scattering
cross sections were calculated with the adiabatic poten-
tial described in the previous section. In Fig. 6, the en-
ergy dependence of the state-to-state integral cross sec-
tionsforseveraltransitionsoutofthej =3/2,F ,f level
1
of OH are shown. This level, which is the higher Λ-
doublet component of the ground rotational level (see
Fig. 1), can be selected with the Stark decelerator since
it is low-field seeking in an inhomogeneous electric field FIG. 6: State-to-state integral cross sections vs. collision en-
[13]. The crosssections arecomputed ona veryfine grid ergyfortransitionsoutoftheOHj =3/2,F1,f levelincolli-
sionswithHe. Thefinallevelsareindicatedforeachtransition
ofenergiestobeabletostudyresonantfeaturesindetail.
for which thecross section is plotted.
Away from the resonances, these results are in good
agreement with those previously reported by Kl os et al.
[21]. As notedby these authors,there isa propensityfor
transitionspreservingthetotalparity. Thecrosssections
are found to be smaller for transitions with large energy transitions, the Feshbach resonances are not significant
gaps. The initial and final levels of the two transitions compared with the background continuum. The j =
shown in Fig. 6(a) have a rather large energy separation 3/2,F ,f → j = 3/2,F ,e transition dominates at low
(>100cm−1),andthetotalparityisinvertedduringthe collisio1n energies and als1o gives rise to shape resonances
transitions. Hence,thecrosssectionsforthesetransitions withcrosssectionspeaking above10˚A2. However,these
are small. shaperesonancesoccuratcollisionenergiesofonlyafew
Resonances can be observed in Fig. 6 near the colli- wavenumbers.
sion energies corresponding to thresholds for excitation
oftheOHradicaltohigherrotationalandspin-orbitlev- Inthe followingsubsections,we analyzethe shaperes-
els. Both shape resonances, which appear right above onances in the j = 3/2,F ,f →j = 5/2,F ,e transition
1 1
the threshold energies for the final levels, and Feshbach andthe Feshbachresonancesin the j =3/2,F ,f →j =
1
resonances,which appear nearthe energieswhere higher 1/2,F ,f transition. The former transition has a large
2
rotationallevelsthantheconsideredoutgoingchannelbe- cross section; the latter transition exhibits strong reso-
come open, are present. Except for the j =3/2,F ,f → nances that show the largest enhancement compared to
1
j = 3/2,F ,e and the j = 3/2,F ,f → j = 1/2,F ,f the background.
1 1 2
7
FIG.8: PlotsoftheOH-Headiabaticbendercurvesthatcor-
relate with the OH j = 5/2,F ,e level, obtained from close-
1
couplingcalculations. Curvesarelabeled with J(p),whereJ,
n
p,andnarethetotalangularmomentum,thetotalparityof
the scattering wavefunction, and the cardinal index, respec-
tively.
ergy excluded is diagonalized as a function of R. The
eigenvaluesdefine a set ofadiabatic bender potentialen-
ergy curves, which are labeled by the total angular mo-
mentum J,the totalparityp ofthe scatteringwavefunc-
FIG. 7: (a) State-to-state integral cross sections vs. collision tion, and the cardinal index n of the eigenvalue. In this
energyforthej =3/2,F1,f →j =5/2,F1,etransitionofOH paper, we will use the symbol Jn(+) and Jn(−) to label
incollisionswithHe(thickline),andtheintegralcrosssection close-coupling adiabatic bender curves with p = +1 and
convoluted with Gaussian energy distributions of FWHM of p=−1, respectively.
1 and 5 cm−1 (thin lines). (b)–(e) Differential cross sections
Figure 8 shows several adiabatic bender curves that
dσ/dΩ of the above transition at several energies marked as
dashed lines with Roman numerals in (a), together with the correlate with the OH j = 5/2,F1,e level. The curves
differential cross sections convoluted with Gaussian energy marked with 5/2(+) and 3/2(+) are the two lowest ly-
1 1
distributions as in (a). ingadiabaticbender curves,eachofwhichsupports only
oneboundstretchlevel,withenergiesof77.47cm−1 and
78.15cm−1,respectively. AsJ andnincrease,thecurves
B. Shape resonances move up in energy and the well depths become smaller.
Asaconsequence,someoftheboundlevelsbecomequasi-
bound, and for the high lying curves (for example, the
Shape resonances result from quasi-bound states of
the van der Waals complex formed by the collision part- 15/2(+) curve shown in Fig. 8) the wells are too shallow
2
ners at energies just above the threshold for the final to support any quasi-bound levels.
level. All integral cross sections plotted in Fig. 6 dis- To compute the energies of the shape resonances, we
playshape resonances. Inthis subsectionwe analyzethe treat the adiabatic bender curves in conventional one-
shape resonances associated with the j = 3/2,F1,f → dimensional scattering problems and calculate the phase
j =5/2,F1,etransitionsinceithasalargeintegralcross shift. We should be able to observe rapid changes by
section. Figure 7(a) displays these resonances on an ex- π, signifying resonances, in the collision energy depen-
panded energy scale. Several maxima, with increasing dence of phase shift [4]. Fig. 9 shows the phase shift as
peak width vs. energy, can be observed, as was also pre- afunction ofcollisionenergyfor allthe adiabaticbender
viouslyfound[24]forOH-He andother He-moleculesys- curves that have such a feature. We see fromFig. 9 that
tems [5, 45, 46]. resonances in six adiabatic bender curves contribute to
To gain more insight, we employ the adiabatic ben- each of the three peaks shown in Fig. 7(a) (labeled as I,
der model [29, 30] to analyze the shape resonances. The II, and III), which occur at 84.8, 87.6 and 91 cm−1, re-
method is similar to the previous analysis of OH-He col- spectively. TheresonancefeaturesinFig.7(a)andFig.9
lisions by Dagdigian and Alexander [24], except that we match well both in energy and width. Note that in or-
used a close-coupling channel basis instead of a coupled- der to distinguish the phase shift in different adiabatic
states one. The full Hamiltonian with the inclusion of bender curves, Fig. 9(b) and (c) do not show the whole
Coriolis coupling and only the radial nuclear kinetic en- range of the resonances, and thus the resonant changes
8
FIG.9: PhaseshiftsasafunctionofcollisionenergyforOH(j=5/2,F ,e)-Hecollisions,obtainedfromclose-couplingadiabatic
1
bendercurvesdescribed in thetextandshown in Fig. 8. Curvesarelabeled with J(p),whereJ,p,andnare thetotalangular
n
momentum, thetotal parity of thescattering wavefunction, and thecardinal index, respectively.
in phase shift shown are less than π.
It is also interesting to compare the differential cross
sectionsforcollisionenergiesonandoffaresonance. Fig-
ures 7(b)–(e) display differential cross section for several
energiesmarkedin Fig.7(a) with Romannumerals. The
center of the two major peaks are marked as I and II,
while III and IV correspond to non-resonant energies.
We observesignificantbackwardscatteringfor energiesI
and II, likely because of an increased time delay of colli-
sionduetotheformationanddecayofquasi-boundlevels
of the van der Waals complex. Backward peaks are in-
significantforenergiesIIIandIV.Wewillfurtherdiscuss
this topic in the next subsection.
C. Feshbach resonances
InFeshbachresonances,quasi-boundlevelsofthecolli-
sioncomplexassociatedwithagivenrotationalleveldis-
sociateto yieldthe moleculeinalower-energyrotational
level. We consider here Feshbach resonances associated
with the j =3/2,F ,f →j =1/2,F ,f transition. This
1 2
transition was chosen for detailed study since the reso-
nancefeaturesshowaca. 4-foldincreaseoverthe contin-
uumbackground(seeFig.6). Figure10(a)displaysthese
resonances on an expanded energy scale.
It is seen that a rich set of Feshbach resonances exists
inacollisionenergyrangeofseveralcm−1belowtheener-
geticthresholdforopeningoftheF ,j =3/2levelat188
2
cm−1. Figure11displaysthecontributiontotheintegral
crosssectionforthej =3/2,F ,f →j =1/2,F ,f tran-
1 2
sitionfromvariousvaluesofthetotalangularmomentum
J (partial cross sections). The individual partial cross
sections exhibit one or more peaks, and their energies FIG. 10: (a) State-to-state integral cross section vs. collision
shift toward higher collision energy as J increases. For energy for the j = 3/2,F ,f → j = 1/2,F ,f transition of
1 2
J ≥ 13/2, no significant resonances can be found in the OH in collisions with He (black line), and the integral cross
energy dependence of partial cross sections. section convoluted with Gaussian energy distributions with
We performed an adiabatic bender analysis similar to FWHMof1(redline)and5cm−1 (blueline). (b)–(j) Differ-
ential cross sections dσ/dΩ of the above transition at several
that described in subsection IVB. We calculated adia-
energies marked as dashed lines and Roman numerals in (a),
baticbenderpotentialsbydiagonalizingtheHamiltonian
together with the convoluted differential cross sections with
expressedinaclose-couplingchannelbasis. Sinceallpos-
Gaussian energy distribution as in (a).
sible values of l (the orbital angular momentum of the
9
FIG. 11: Partial cross sections vs. collision energy for the FIG. 12: Collision lifetime ∆tJ(E) as a function of collision
j =3/2,F ,f →j =1/2,F ,f transition of OH in collisions energy for the OH j =3/2,F ,f → j = 1/2,F ,f transition
1 2 1 2
with He for total angular momentum J ≤ 11/2. The dotted in collisions with He, as defined in Eq. (9) of the text, for
vertical lines denote the computed energies of the van der total angular momenta J = 1/2 – 11/2.
Waalsstretchlevelssupportedbytheclose-couplingadiabatic
bendercurves.
tatively explain backward scattering appearing in differ-
ential cross sections is to calculate the collision lifetime,
van der Waals complex) are included in the channel ba-
which is the difference between the time that the colli-
sis, there are multiple adiabatic bender curves for each
sion partners spend in each other’s neighborhood with
value of J. These adiabatic bender curves look similar
and without the interaction [31, 49, 50]. For a direct
to those shown in Fig. 8 and are not plotted here. The
comparisionwiththe partialcrosssectionsshowninFig.
energiesofthe vanderWaalsstretchlevelssupportedby
11, we compute the collision lifetime from initial state γ
these curves were derived using a fixed step-size discrete ′
to final state γ for individual total angular momenta J,
variable representation (DVR) method[47, 48]. To treat
defined as
levels that might be quasi-bound, an infinite barrier was
placedatthemaximumofthecentrifugalbarrieroneach
aladrigaebaJt,icthbisenadpeprropxoitmenattiiaoln. wFiollrlecaudrvteos caaslscoucliaatteeddewnietrh- ∆tJγγ′(E)=Re−i¯h X δpp′(cid:0)SγJ,γ′,l,l′(cid:1)∗ dSγJd,Eγ′,l,l′
 l,p,l′,p′ 
gieshigherthanthey shouldbe andcouldleadtosignifi-
(9)
canterror. Thesecomputedenergiesareshownasdotted ′ ′
where l, p and l, p denotes the orbital angular momen-
linesinFig.11. Thereisareasonablematchbetweenthe
tum and parity of initial and final levels, respectively,
energiesofthe resonancesandofthe bend-stretchlevels, and SJ denotes S-matrix elements for total angu-
γ,γ′,l,l′
especially for small J.
lar momentum J from close-coupling calculations. The
Fig. 10(b)–(j) display the differential cross section for lifetimes vs. J for the j = 3/2,F ,f → j = 1/2,F ,f
1 2
the OH j =3/2,F1,f →j =1/2,F2,f transitionatsev- transition are plotted in Fig. 12. Clearly, the resonance
eralenergiesmarkedonFig.10(a)withRomannumerals. peaks in Fig. 11 are well reproduced, with collision life-
The energies at I and IX are not at a resonance,and the times of a few picoseconds. The most intense resonance
differentialcrosssectionsshowlittlebackwardscattering, peak lies at 186.4 cm−1, which was largely due to the
whileII –IVcorrespondtoresonanceenergies,forwhich J = 9/2 partial cross section. From Fig. 12 we see that
some backward scattering can be observed. The shapes the corresponding lifetime is about 6 ps. We can com-
ofthedifferentialcrosssectionsarequitedifferentatres- pare this collision lifetime with the rotational period of
onanceenergiescomparedtocollisionenergiesawayfrom the OH-He van der Waals complex. We estimate the ro-
the resonances. tationalconstantofthecomplextobe6.4×10−24Jfrom
A simple way to analyze the resonances and to quali- the expectation value of 1/R2 computed with the wave
10
brational motion was included. In that reference it was
10
(a)
o2n / A j = 5/2 F1 e smheonwtnbethtwateetnhitsheporroyceadnudrheiaglhr-epardeycigsiiovnessceaxtcteelrleinngteaxgpreeer--
o
ecti 5 iments for OH-Ne collisions. Since we found in Section
oss s j = 3/2 F1 e IIIthatthevibrationalmotionofOHcanbeofquantita-
cr tive influence, we also calculated an adiabatic potential
from the three-dimensional potential of Sumiyoshi et al.
0
1.5 in the same way as we did for OH-He. The resulting
(b)
o2n / A 1 j = 7/2 F1 e acedlilaebnattaicgrpeoetmenetnitalwwitahsftohuenedxtpoerimimpernotvaelsrleigsuhtltlys tfohretehxe-
o
ecti j = 5/2 F1 f scattering of OH and Ne. In the present study, we
cross s0.5 j = 1/2 F2 e j = 3/2 F2 f ucrsoestshseecatdioianbsaotnicapomteuncthiafilntehrrogurigdhothuatnanindtchoemsptuutdeytbhye
0 Scharfenberg et al. [34] in order to study scattering res-
0.2
(c) onances. In Fig. 13 we show the energy dependence of
o2n / A j = 3/2 F2 e state-to-state integral cross sections for collisions of the
ectio0.1 OH radical with Ne atoms, where the OH radicals are
cross s j = 1/2 F2 f iionritioaflltyheinintehleasjtic=c3ro/s2s,Fse1c,tfiolnesveals. aOfvuenrcatlilo,nthoefbeneheragvy-
j = 7/2 F1 f is rather similar to what was observed for the OH-He
0
100 200 300
system in Section IV. For example, we again observe
collision energy / cm-1
a propensity for transitions preserving the total parity.
However, in the OH-Ne system none of the channels ap-
pears to have particularly strong Feshbach resonances,
as was the case for the j = 3/2,F ,f → j = 1/2,F ,f
1 2
FIG. 13: State-to-state inelastic scattering cross sections of transition of the OH-He system. The most pronounced
OH(X2Π ,j =3/2,f)radicalswithNeatomsasafunction resonantfeaturesobservedforOH-Necollisionsareshape
3/2
of the collision energy. Cross sections for inelastic collisions resonances in the j = 3/2,F ,f → j = 5/2,F ,f transi-
1 1
populatingthe(a)j =3/2,F1,e(black)andthej =5/2,F1,e tion. In Fig. 14 we show these shape resonances in more
(purple)states; (b) thej =5/2,F1,f (orange), j =1/2,F2,e detail.
(brown), j =3/2,F ,f (pink), and the j =7/2,F ,e (green)
2 1
states; (c) the j =1/2,F ,f (red), j =3/2,F ,e (blue), and Looking closely at Fig. 14, we see several resonance
2 2
thej =7/2,F1,f (cyan) states. peaks that correspondto an increasein the cross section
byaboutafactoroftwocomparedtothenonresonanten-
ergies. Arelativelystrongresonanceoccursatacollision
function obtained from the DVR method on the lowest energy of 99.23 cm−1; this resonance increases the cross
lying J =9/2 adiabatic bender curve. This corresponds section by a factor of four compared to the background.
to a rotational period 14.9 ps, assuming l=3. The latter resonanceis indicated by the Romannumeral
We thus conclude that the collision lifetime has the V. The main contributions to this resonance originate
same order of magnitude as the rotational period of the from partial cross sections with total angular momenta
OH-He complex. Itis thereforenotsurprisingto observe of J = 37/2 and J = 39/2. In Fig. 14, we also show
significant backward scattering at some resonance ener- thedifferentialcrosssectionsforseveralenergiesthatare
gies. At off-resonance energies the collision lifetime will marked by Roman numerals in panel (a). For the res-
be ≪1 ps, which is much smaller than the OH-He rota- onances at collision energies of 86.83, 94.90 and 99.23
tionalperiod. Hence, backwardscattering is expected to cm−1, the cross sections are shownin the panels (b), (d)
be barely observable. and (f). In these plots, large amplitudes for backscat-
tering are observed. To compare, the differential cross
sections were also calculated away from the resonances
V. OH-NEON COLLISIONS attheenergies93.00,98.00and101.00cm−1,andthere-
sultsareshowninthe panels(c),(e)and(g). Inthecase
To describe the interaction between OH and Ne, ofnonresonantscattering,the observedbackscatteringis
we used the PES of Sumiyoshi et al. [32]. This PES significantly reduced. The differential cross sections at
was calculated using an explicitly correlated, spin- these resonances look similar to those at the shape reso-
unrestrictedcoupled-clusterapproach[UCCSD(T)-F12b] nances for the j = 3/2,F ,f → j = 5/2,F ,e transition
1 1
with a quintuple-zeta basis set (AV5Z). Although of the OH-He system (see Fig. 7), where also an in-
Sumiyoshi et al. calculated a three-dimensional poten- creaseinbackscatteringwasfound. Withameasurement
tial, we used in Ref. [34] their interaction potential eval- of the differential cross sections, the strong increase and
uated at the equilibrium distance r = 1.832a for the decrease in the backscattering might help in experimen-
e 0
scattering calculations, so that no effect of the OH vi- tally identifying the shape resonancesat94.90and99.23