Na 2 Vibrating in the Double-Well Potential of State 2 1 Σ u + (JM = 00): A Pulsating Quantum Bubble with Antagonistic Electronic Flux

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1 Articl Cit This: J. Phys. Chm. A 08,, pubs.acs.org/jpca a Vibratig i th Doubl-Wll Pottial of Stat Σ u + (JM = 00): A Pulsatig Quatum Bubbl with Atagoistic Elctroic Flux Publishd as part of Th Joural of Physical Chmistry virtual spcial issu Maul Yańẽz ad Otilia Mo Fstschrift. D. J. Distlr,, D. Jia, J. Maz,*,,, ad Y. Yag, Istitut fu r Chmi ud Biochmi, Fri Uivrsitaẗ Brli, 495 Brli, Grmay Uivrsity of braska Licol, Licol, braska 68583, Uitd Stats Stat Ky Laboratory of Quatum Optics ad Quatum Optics Dvics, Istitut of Lasr Spctroscopy, Shaxi Uivrsity, Taiyua , Chia Collaborativ Iovatio Ctr of Extrm Optics, Shaxi Uivrsity, Taiyua , Chia *S Supportig Iformatio ABSTRACT: Th thory of cocrtd lctroic ad uclar flux dsitis associatd with th vibratio ad dissociatio of a multilctro orotatig homouclar diatomic molcul (or io) i a lctroic stat S+ Σ + g,u (JM = 00) is prstd. Th lctroic populatio dsity, uclar probability dsity, ad uclar flux dsity ar isotropic. A thorm of Barth, prstd i this issu, shows that th lctroic flux dsity (EFD) is also isotropic. Hc, th volvig systm appars as a pulsatig, or xplodig, quatum bubbl. Applicatio of th thory to a vibratig i th doubl-miimum pottial of th Σ + u (JM = 00) xcitd stat rvals that th EFD cosists of two atagoistic compots. O ariss from lctros that flow sstially cohrtly with th ucli. Th othr, which is oppositly dirctd (i.., atagoistic) ad mor its, is du to th trasitio i lctroic structur from Rydbrg to ioic typ as th ucli travrs th pottial barrir btw ir ad outr pottial wlls. This trasitio compot of th EFD riss ad falls sharply as th ucli cross th barrir.. ITRODUCTIO Th most complt charactrizatio of a lctroically adiabatic molcular procss (th vibratio of a diatomic molcul, for a lmtary istac) should tll us ot oly whr th lctros ad ucli ar at ay istat (i.., probability (populatio) dsitis) but also how thy gt thr (i.., flux dsitis). Howvr, xcpt i spcial cass, th lctroic flux dsity, i particular, is byod th rach of stadard computatioal tchiqus basd o th Bor Opphimr approximatio (BOA). Idd, it is a w spcial cas that this work dals with: vibratig or dissociatig, but orotatig (JM = 00), multilctro homouclar diatomic molculs, such as a. A prvious articl rports th rsults of a study of cocrtd + lctro uclar dyamics i th o-lctro systm H prpard i th spcial stat Σ + g (JM = 00), for which th corrspodig wav fuctio is xprssd i th BOA as Ψ (, rr,) t =Ψ(; rr) Ψ ( R,) t (.) I q. r stads for th positio of th lctro rlativ to th uclar ctr of mass (CM) ad R for th distac vctor from o proto to th othr, Ψ (r;r) is th groud-stat lctroic rgy igfuctio, ad th uclar wav packt is assumd to b sphrically symmtric Ψ ( R, t) = Y00( θϕ, ) R χ( R, t) / = (4 π) R χ( R, t) (.) Th subscript 00 o th sphrical harmoic fuctio corrspods to JM = 00. Th cosquc is that th uclar probability ad flux dsitis ar isotropic (i.., sphrically symmtric) about th CM. Th lctroic probability ad flux dsitis ar likwis isotropic. Hc, wh th total rgy is blow th thrshold for dissociatio, th volvig systm appars as a pulsatig quatum bubbl : all probability dsitis ad flux dsitis rmai sphrically distributd about th CM. At rgis abov thrshold th quatum bubbl xplods.,3 I sctio w outli th xtsio of th thory for th olctro orotatig homouclar diatomic to th gral cas of Rcivd: ovmbr 9, 07 Rvisd: Jauary 4, 08 Publishd: Jauary 4, Amrica Chmical Socity 50 DOI: 0.0/acs.jpca.7b73 J. Phys. Chm. A 08,, 50 59

2 Th Joural of Physical Chmistry A th multilctro orotatig homouclar diatomic i th stat S+ + Σ g,u (JM = 00). W show that th rotatioal quatum umbrs JM = 00 imply isotropy of th uclar probability ad flux dsitis, as wll as that of th lctroic populatio dsity ad w ifr from this agrmt with th o-lctro cas that th lctroic flux dsity is also isotropic. (A rigorous proof of this proprty is prstd i this volum by Igo Barth. 4 ) Th isotropy of all dsitis ad fluxs mas that th orotatig multilctro homouclar diatomic molcul volvs as a pulsatig quatum bubbl, irrspctiv of th umbr of lctros. Our choic of th spcific systm, a vibratig i th doublwll pottial of th Σ + u stat, is motivatd by svral prior ivstigatios of its lctroic structur, 5 7 ad by th sarch for th proprtis ad dyamical cosqucs of th doubl miimum, 8 8 both thortically 0 8 ad xprimtally. 8,9,5 7 Valac ad guy Tua, who first discovrd th doubl-wll, otd that th ir ad outr wlls hav diffrt lctroic cofiguratios. 5 Th quatum chmical aalysis of Jug attributs th phomo to th avoidd crossig of two ighborig lctroic stats with th sam symmtry but diffrt lctroic structurs. 6 Accordig to Vrg s t al., 7 th rsultig ir ad outr pottial wlls of th Σ + u stat hav domiat (though ot prfct 8 ) Rydbrg ad ioic charactr, rspctivly. Bcaus ths labls hav b adoptd i th subsqut litratur, 9 7 w us thm i th prst cotxt. Th mai purpos of Vrg s t al. 7,8 was to dtrmi th shap of th doubl-miimum pottial accuratly by ivrsio of Fourir trasform spctra. Subsqutly, fforts to tas out th implicatios of th trasformatio of th lctroic structur for dyamical proprtis hav b mad. For xampl, Arasaki t al. prdictd that th lctroic-structur chag should iduc sigificat chags i dipol trasitios from th Σ + u stat to th groud stat ad to xcitd ioic stats, which i tur should b rflctd i th rgy-, agl-, ad (fmtoscod) tim-rsolvd photolctro spctra. 0 4 Thir thortical rsults stimulatd xprimtal two-color fmtoscod pump prob photolctro spctroscopy dsigd to moitor th uclar wav packt dyamics i th doubl-miimum pottial of th Σ + u stat of a. 5 Wollhaupt t al. 5 obsrv boud-to-ioic ioizatio probability four tims gratr at th outr ( ioic ) classical turig poit tha at th ir ( Rydbrg ) turig poit (s also rfs 9, 6, ad7). I a rct articl svral of us aoucd th discovry of a ovl cosquc for th lctro dyamics (amly th occurrc of atagoistic, or oppositly dirctd, lctro fluxs i cotiguous rgios) du to trasformatio of th lctroic structur i th Σ + u (JM = 00) stat of a that occurs as th ucli cross th pottial barrir. 8 Sctio 3 provids a dtaild aalysis of th phomo basd o th thortical dvlopmt of sctio. Th sstial ida is that th rapid chag of lctroic structur at th barrir btw th ir ad outr pottial wlls should b accompaid by sigificat rarragmt of th lctroic populatio dsity, ad this rarragmt, i tur, should b rflctd i th lctroic flux. I othr words, o should obsrv a strog isotropic lctroic flux dsity as th ucli travrs th pottial barrir. W strss that this flux du to th lctroic structural chag is complmtary to th ormal flux of th lctros that travl mor or lss cohrtly with th ucli. 3 Aalogous strog lctroic fluxs du to o-adiabatic trasitios, or surfac hoppig btw diffrt lctroic stats, hav alrady b rportd by Takatsuka ad co-workrs. 9 I cotrast, th prst work documts strog lctroic flux dsity durig adiabatic passag ovr th barrir btw th Rydbrg ad ioic structurs. Th rsults for th Σ u + (JM = 00) stat of a ar prstd ad discussd i sctio 3. Our coclusios ar summarizd i sctio 4.. THEORY W cosidr a isolatd gric homouclar diatomic molcul or molcular io that ca b adquatly dscribd by orlativistic quatum mchaics. I trms of ctr-of-mass coordiats w ca writ th itral Hamiltoia (i.., th Hamiltoia xclusiv of th total ctr of mass) as H = H + T = T + V( r, R) + T (.) whr th lctroic Hamiltoia H is dfid implicitly, V(r,R) is th Coulombic pottial rgy of itractio of all particls, T = ħ r i m i= is th lctroic kitic rgy oprator ad T = ħ μ R (.a) (.b) is th uclar kitic rgy oprator. Th rducd mass of th ucli is μ = M/, whr M is th uclar mass. I qs. ad. r stads for th cofiguratio of th lctros with rspct to th uclar ctr of mass (CM) ad R for th distac vctor btw th ucli. Th drivatio of qs. also ivoks th followig approximatios 3 (s also Appdix A of th Supportig Iformatio): th ctr of mass of th whol systm is rplacd by th CM; th mass-polarizatio cotributio to H is igord; th rducd mass of th lctro is rplacd by its mass, m ; spi is igord, xcpt for th implicit rquirmt that th igfuctios of H oby th Pauli xclusio pricipl. W assum that th molcul is i a stat fully charactrizd by th BOA wav fuctio Ψ ( r, R, t) =Ψ( r ; R) Ψ( R, t) (.3) which is aalogous to th o-lctro wav fuctio i q.. Th BOA is justifid by otig that for th particular systm of itrst, amly, a vibratig i th doubl-wll pottial of th Σ + u stat, th kitic rgy i th viciity of th pottial barrir, whr th lctroic structur udrgos a strog trasformatio, is about a ordr of magitud smallr tha th rgy gaps btw th ighborig pottial curvs. Th cosquc is that th oadiabatic couplig to ighborig lctroic stats is gligibl, i accord with rfs 0 7. Th lctroic wav fuctio Ψ obys th lctroic rgy igvalu quatio H Ψ ( r ; R) = VR ( ) Ψ( r ; R) Articl (.4a) whr th igvalu V(R) is a paramtric fuctio of th itruclar distac R = R that srvs as pottial rgy for th uclar motio. Th uclar wav packt obys th uclar Schro digr quatio (SE) 5 DOI: 0.0/acs.jpca.7b73 J. Phys. Chm. A 08,, 50 59

3 Th Joural of Physical Chmistry A ħ Ψ ( R, t) i = [ T + V( R)] Ψ( R, t) t (.4b) Fially, xploitig th zro agular momtum of th uclar stat, w ca xprss th assumd ormalizd packt covitly by / Ψ ( R, t) = Y ( ΘΦ, ) R χ( R, t) = (4 π) R χ( R, t) 00 (.5) which is th aalogu of q.. Th auxiliary packt satisfis th radial SE χ ħ ( Rt, ) i t = ħ μ R + VR ( ) χ( Rt, ) Th lctroic populatio dsity (EPD) is giv by d (,) rt = Ψ() t δ( r r) Ψ () t i= = dr d r Ψ( r ; R) Ψ* ( R, t) δ( r r) Ψ( r ; R) Ψ( R, t) i= i i = d R Ψ ( R, t) d r [ Ψ ( r ; R)] i= j i j ri= r (.6) (.7) whr dr dr = = dr d r... dr i i. (ot that dr d (r,t) =, i.., d (r,t) is ormalizd to th umbr of lctros (populatio).) Trasformig th uclar variabls of itgratio from Cartsia coordiats to sphrical coordiats ad ivokig th idistiguishability of th lctros, w ca rwrit q.7 as d (,) r t = (4 π) d R χ( R,) t dφ π d(cos Θ) d ( rr ; ) 0, 0 (.8) W dfi th o-lctro probability dsity apparig i q.8 by d (; rr) d r [ Ψ ( r ; R)], j i j ri= r (.9) which is th probability of fidig ay lctro i th volum lmt dr about th poit of obsrvatio r for a fixd sparatio R btw th ucli. Thus, d, (r;r) is th populatio dsity of lctros at r with R fixd (th smicolo dots that d, (r;r) dpds paramtrically o R). If th o-lctro probability dsity posssss cylidrical symmtry, as is th cas for th lctroic stats of itrst hr, th d, (r;r) dpds oly o r, R, ad th cosi of th agl γ btw r ad R. But γ is rlatd to th polar agls of r ad R by th formula cos γ = cos Θ cos θ + si Θsi θ cos( Φ ϕ) (.0) W ow obsrv that bcaus th itgratio idicatd i th basic formula for d (r,t) (q.7) rus ovr all possibl valus of R, th polar axis of th sphrical coordiat fram d ot b spcifid. I othr words, th choic of th polar axis is arbitrary. 5 For covic, w choos it to li alog th dirctio of r (i.., w tak r to li alog th polar axis paralll with z, th Cartsia uit vctor). It follows from q.0 that θ = 0 ad thrfor that cos γ = cosθ. Hc, d, (r;r) ca b xprssd as δ, (r;r,cosθ). (ot that to this juctur th symbol d is usd for dsity xprssd grally i trms of th vctors r ad R; hcforth, th symbol δ is rsrvd for dsity xprssd i sphrical coordiats.) Th q.8 ca b rwritt δ( rt,) = (4) π d R χ( Rt,) δ, (; rr) (.) 0 whr, for th sak of covic, w dfi th quatity (static lctroic populatio dsity at f ixd R) δ, π (; rr) dφ d(cos Θ) δ (; rr,cos Θ) 0 = π d(cos Θ) δ ( r; R,cos Θ),, (.) Equatio. implis that th EPD is isotropic, which is a cosquc of th isotropy of th uclar wav packt (q.5). A altrativ, mor dtaild, dmostratio that th EPD is sphrically symmtric is giv i Appdix A of th Supportig Iformatio. W ot that δ, (r;r) ca b rgardd as th rsult of th itgratio ovr all solid agls of o vctor of fixd lgth with rspct to aothr vctor of f ixd lgth. From th origial viwpoit dscribd abov, r is tak to b paralll with z ad th itgratio is tak to ru ovr all solid agls of R with rspct r. Howvr, w ca as wll tak R to b paralll with z ad th itgratio to rag ovr all solid agls of r with rspct to R. Th q. ca b rcast as δ ( rr ; ) = π d(cos θ) δ ( r,cos θ; R),, (.3) Bcaus oly th magituds of r ad R appar i th itgrad, alog with th cosi of th agl btw r ad R, w ca adopt ithr prspctiv i pricipl. Howvr, from a physical stadpoit q.3 provids a illumiatig itrprtatio of th EPD at fixd R, as wll as a formula that ca b radily implmtd withi th framwork of stadard quatum chmistry. Th uclar probability dsity (PD) (i.., th probability pr uit volum of obsrvig th itruclar distac i volum lmt dr about R ) is giv by d ( R, t) = Ψ( t) δ( R R) Ψ ( t) = dr d r Ψ( r ; R) Ψ* ( R, t) δ( R R) Ψ( r ; R) Ψ( R, t) = Ψ ( R, t) = (4 πr ) χ( R, t) (.4) whr th last li follows from q.5 ad th (assumd) ormalizatio of th lctroic rgy igfuctio: dr Ψ (r ;R) =. As show i rf 3 (q 93) (s also rf ), th probability dsity of obsrvig uclus α (=a, b) at distac R α from th CM is rlatd to d (R,t) by d, α( Rα, t) = 8 d( Rα, t) = χ( Rα, t) 4πR α Articl (.5) DOI: 0.0/acs.jpca.7b73 J. Phys. Chm. A 08,, 50 59

4 Th Joural of Physical Chmistry A Articl As xpctd, i ithr fram (obsrvr o ithr a or b or o th CM) th PD is sphrically symmtric (i.., w may st d (R,t) = δ (R,t) ad d,α (R α,t) =δ,α (R α,t)). Elswhr i this issu Barth provs th followig thorm: If th molcular wav fuctio satisfis th SE iħ Ψ(t) / t = H Ψ(t) ad th total agular momtum ad its z-compot vaish (i.., J = 0 ad M = 0), th th lctroic ad uclar flux dsitis ar sphrically symmtric. 4 Though th BOA wav fuctio dos ot oby th SE, it dos satisfy th costraits of th thorm approximatly. Thrfor, w assum that th EFD for our stat of itrst is sphrically symmtric. I that cas, w immdiatly hav th xprssio for th radial (i.., th oly ozro) compot r δ j rt = r r r ( r, t) (,) d,r 0 t (.6) which follows from th rducd radial cotiuity quatio (s q 39 of rf 3). Th uclar flux dsity (FD), i.., th flux dsity of o uclus rlativ to th othr, is giv by j ( R,) t = R{ Ψ() t δ( R R) R Ψ ()} t = ħ Im{ dr d r Ψ( r ; R) Ψ* ( R, t) μ δ( R R) Ψ( r ; R) Ψ( R, t)} R = ħ Im{ Ψ* ( R, t) Ψ R ( R, t)} R= R μ * =ħ /4πμ R Im[ χ ( R, t) χ( R, t)/ R] R= R R (.7) whr th third li follows from th ormalizatio of Ψ (r ;R) ad from q.5. It ca b show (s qs 9 ad 93 of rf 3) that th flux dsity of uclus α with rspct to th CM is j ( R = α α, t) 4 j ( Rα, t), (.8) As idicatd by qs.7 ad.8, th FDs i ithr of th atomic (a or b) frams or th CM fram ar sphrically symmtric (i.., j (R,t) =j (R,t) R ad j,α (R α,t) =j,α (R α,t) R ). 3. ATAGOISTIC ELECTROIC FLUX I THE Σ + U (JM = 00) EXCITED STATE OF a Th applicatio of th thory to a bgis with th computatio of th doubl miimum pottial of th Σ + u xcitd stat. For this purpos w mploy Gaussia 09 3 to solv q.4a for th pottial curv V(R) ad th lctroic igfuctio Ψ (r ;R) usig th symmtry adaptd clustr-cofiguratio itractio (SAC-CI) mthod 4 with Duig s corrlatio cosisttpolarizd valc tripl-ζ basis st augmtd with th diffus fuctios (aug-cc-pvtz) basis st. 5 Th ab iitio doubl-wll pottial is show as a cotiuous li i Figur a. Th miima of th rathr arrow ad shallow ir pottial wll ad of th dpr ad widr outr wll ar locatd at R i = 3.8 Å ad R o = 6.9 Å, rspctivly, ad sparatd by th barrir, whos maximum is at R = 4.9 Å. Th dots i Figur a rprst th xprimtal doubl-wll pottial dtrmid by Coopr t al. by a rfid RKR aalysis of high-rsolutio Fourir trasform spctra. 8 Th agrmt of th thortical curv with th xprimtal data is quit satisfactory. Idd, w chos th SAC-CI/aug-cc-pVTZ mthod, aftr a systmatic ivstigatio of a umbr of othr ab iitio mthods, bcaus it yilds th bst agrmt with Figur. uclar ad lctroic proprtis of a i th lctroic xcitd stat Σ + u. (a) Doubl miimum pottial curv (solid black curv, SAC-CI/aug-cc-pVTZ calculatio; dots, xprimtal data from rf 8). Itruclar distacs R i, R, ad R o at th miimum of th ir wll, at th top of th barrir, ad at th miimum of th outr wll ar markd by vrtical dashd, dash-dottd, ad dottd lis, rspctivly. Th horizotal li idicats rgy. Also show ar sapshots of uclar probability dsity δ (R,t) (wightd by 4πR ) at tims t =0,t i = 66 fs, t = 68 fs, t o = 334 fs, ad t qto = 548 fs, wh ma valus of R ar R qti, R i, R, R o, ad R qto, rspctivly. (b) (d) Rsults of approximat CASSCF-CI(,36)/aug-cc-pVTZ calculatio. (b) Probabilitis of th HOMO-to-k xcitatios, P k (R), for th domiat trms i th CI xpasio (q 3.0). (c) Equidistat cotour plots of th lctroic probability dsity for HOMO (top), LUMO (k =, middl), ad th xcitd Rydbrg -typ MO (k = 4, bottom) i th x z pla at itruclar distacs R i (lft), R (middl), ad R o (right). (d) Equidistat cotour plots of th populatio dsity [d, (x,z;r)] of valc lctros i th x z pla, approximatd by th sum of MO dsitis show (pal c) wightd by occupatio probabilitis P k (R) (pal b). () Equidistat cotour plots of th SAC-CI/aug-cc-pVTZ populatio dsity of valc lctros [d, (x,z;r)] i th x z pla; plots of δ, (r;r)(q.9) vrsus r oritd at th right-had boudaris (abscissa is paralll with th z-axis). Valus of r at maxima ad miimum ar idicatd by dottd lis, which also idicat itrsctios of th x z pla with sphrical surfacs o which δ, (r;r) is valuatd at ths valus of r (q.3). 53 DOI: 0.0/acs.jpca.7b73 J. Phys. Chm. A 08,, 50 59

5 Th Joural of Physical Chmistry A xprimt. 8 This agrmt supports th rliability of th lctroic proprtis that w driv o th basis of th SAC-CI/ aug-cc-pvtz mthod. xt w focus o th tim volutio of th uclar wavpackt. Th two obsrvabls of pricipal itrst ar th PD ad FD. Though th tim dvlopmt of th FD is prstd hr for th first tim, that of th PD has alrady b rportd. 9 8 W iclud th PD hr for svral rasos. First, th rsults for th cocrtd PD ad EPD, as wll as for th FD ad EFD, ar basd o th pottial V(R) ad th lctroic igfuctio Ψ (r ;R)), which ar obtaid by mas of o ad th sam solutio of q.4a. Also V(R) agrs wll with xprimt. 8 I cotrast, prvious quatum simulatios 9 7 mploy th xprimtal pottial, possibly shiftd to yild th kow xcitatio rgy ad ioizatio pottial. Scod, th PD is a idispsabl iput to th subsqut calculatio of th EPD ad EFD (qs. ad.6, rspctivly). Third, th PD srvs as a rfrc for discrimiatio of two brachs of th EPD that giv ris to two diffrt cotributios to th lctroic flux (i.., th ormal flux that travls with th ucli ad th atagoistic trasitio flux du to th chag of lctroic structur wh th ucli cross th barrir). For th quatum simulatio of th tim volutio of th PD, w follow th xprimtal scario of rf 9, which mploys a liarly polarizd Gaussia lasr puls with lctric fild ε(t) = E 0 s(t) cos(ω c (t t c )), whr s(t) = xp[ (t t c ) /τ ] is th Gaussia vlop, τ is th puls duratio, ad ω c is th carrir frqucy. Th costat t c is arbitrary. Th amplitud E 0 is also arbitrary, xcpt that it should b i th wak fild limit (i.., th xcitatio probability should ot xcd a fw prct). This rstrictio avoids itrfrc from such subsqut procsss as multiphoto xcitatio of highr lctroic stats ad ioizatio. Th carrir frqucy is rlatd to th xprimtal wavlgth λ = 34.5 m by ω c =πc/λ, whr c is th spd of light. Th xprimtal full width at half-maximum (FWHM) of th corrspodig itsity I(t) = ε 0 cε(t), whr ε 0 is th prmittivity of th vacuum, is FWHM = l τ = 60 fs. ot that th xprimtal lasr paramtrs of rf 9 diffr from thos mployd i rfs 6 ad 7 (s Figur 4.9 of rf 6, whr λ = 340 m ad FWHM = 35 fs, valus also usd i rf 8). Th prst logr wavlgth ad duratio imply a slightly smallr xcitatio rgy ad a sigificatly arrowr spctral width Δω = l /FWHM tha that i rfs 6 8. As a cosquc, th prst uclar wav packt is broadr, ad th prst priod of th strogly aharmoic uclar motio with rgy just abov th pottial barrir is slightly logr, compard with thos of rfs 6 8. At th d of th lasr xcitatio th tim is st to zro, ad th iitial (t = 0) uclar wav fuctio is dtrmid as i rf 6 (for dtails, s Appdix C of th Supportig Iformatio): χ( R, 0) = cχ ( R) v v v (3.) whr th subscript v dots th vth vibratioal igstat of th lctroic xcitd stat = Σ u +, ad th xpasio cofficits ar giv by c = χ χ xp[ ( ω ω) FWHM /8 l()] v v 00 v c (3.) I q 3. ω v =(E v E 00 )/ħ, whr E 00 is th vibroic groudstat rgy. Th Frack Codo factors χ v χ 00 (= dr χ v (R) χ 00 (R)) for th trasitio from th vibroic groud stat (00) to th xcitd stat (v) is multiplid by a Gaussia filtr that slcts th vibratioal stats χ v with rgy E v clos to th carrir photo rgy ħω c. Th cofficits ar ormalizd by th factor such that v c v =. Th vibratioal igfuctios χ v ad rgis E v ar solutios of th igvalu quatio [ T + V( R)] χ ( R) = E χ ( R) (3.3) v v v which is solvd by mas of th Fourir grid Hamiltoia mthod 6 o a uiform grid with 00 poits i th domai.4 Å R.4 Å. Th solutio of th uclar SE (q.6) is xprssd as χ( Rt, ) = c xp( i E t/ ħ) χ ( R) v v v v Th ma rgy of th rlativ uclar motio, E = c E v v v (3.4) (3.5) is idicatd by th horizotal li i Figur a. Th rgy gap btw E ad th groud-stat rgy E 00 is clos to th carrir photo rgy, i.., ( E E00) ħωc (3.6) Th itrsctios E = V(R ct ) mark th ir ad outr classical turig poits R cti = 3. Å ad R cto = 9. Å, rspctivly. Th ma valu of th itruclar distac is Articl Rt () = d R χ( Rt,) R (3.7) Th ma valus R(0) = R qti = 3.3 Å at t = 0 ad at th quatum-mchaical outr turig poit, R(t qto ) = R qto = 8.9 Å at t qto = 554 fs, ar clos to th classical os, R cti ad R cto, rspctivly. Th dviatios ar du to various ffcts, icludig wav packt tulig from th classical turig poit ito th classically forbidd domai, 7 disprsio, ad itrfrcs. 8 Th tims wh th ma valu of th itruclar sparatio passs th valus R(t i ) = R i, R(t ) = R, ad R(t o ) = R o for th first tim ar t i = 66 fs, t = 68 fs, ad t o = 334 fs, rspctivly. Th xpctd valus R i, R, ad R o ad th corrspodig tims t i, t, ad t o ar idicatd i Figurs a,b, ad a. Th tim volutio of th PD is illustratd i Figur a by a slctio of sapshots at th tims t =0,t i, t, t o, ad t qto, wh th uclar wav packt is ctrd at th quatum-mchaical ir turig poit R qti, at th miimum of th ir pottial wll R i, at th pottial barrir R, at th miimum of th outr pottial wll R o, ad at th quatum-mchaical outr turig poit R qto, rspctivly. For covic Figur a shows δ (R,t) tims th factor 4πR (i.., χ(r,t) = 4πR δ (R,t); q.4). Th corrspodig probability dsity of obsrvig uclus α (=a, b) at distac R α = R/ from th CM is show i Figur bby color-codd cotours of 4πR α δ,α (R α,t) (q.5). Th ma valu R α (t) = R(t) / is idicatd by th cotiuous gr li i Figur b. I accord with th Ehrfst thorm, this li is clos to th classical trajctory that starts out with zro vlocity from R cti at t = 0. It is appart that, aftr a iitial rathr short priod of acclratio, th ucli sparat at approximatly costat spd util thy dclrat at th outr turig poit R qto at t qto = 554 fs. Scrutiy of R α (t) shows that th ucli slow dow slightly as thy cross th barrir. Aftr th tur at R qto, th ucli procd toward th ir turig poit R qti. Th quatummchaical tim volutio of th backward motio from R qto to 54 DOI: 0.0/acs.jpca.7b73 J. Phys. Chm. A 08,, 50 59

6 Th Joural of Physical Chmistry A Figur. Color-codd cotour plots of uclar ad lctroic probability ad flux dsitis for a i th lctroic xcitd stat Σ u + (JM = 00). (a) Static lctroic populatio dsity δ, (r;r/) (q.3) wightd by factor 4πr as a paramtric fuctio of th itruclar sparatio (R/) (i uits of /Å). Profils alog cuts at R i, R, ad R o corrspod to curvs show i Figur. (b) Tim volutio of uclar probability dsity δ,α (R α =R/,t) wightd by factor 4πR α (i uits of /Å). Its ma valu is idicatd by a cotiuous gr li. (c) Tim volutio of radial uclar flux dsity j,α (R α =R/,t), wightd by factor 4πR α (i uits of /fs). (d) Tim volutio of lctroic populatio dsity δ (r,t), wightd by factor 4πr (i uits of /Å). () Tim volutio of radial lctroic flux dsity j,r (r,t) j (r,t), wightd by factor 4πr (i uits of /fs). R qti is approximatly th rvrs of th forward motio from R qti to R qto. Th corrspodig classical trajctory is prfctly tim rvrsibl with rspct to th tim t cto, wh it coicids with th outr turig poit, but i quatum mchaics th corrspodig uclar wav packts at th aalogous tims bfor ad aftr t qto diffr from ach othr bcaus of disprsio ad itrfrc ffcts. 8 Th rsultig radial flux dsity j,α (R α,t) R =4j (R α,t) R (q.8) of th ucli at distac R α = R/ from th CM is illustratd i Figur c by a color-codd cotour map. For rfrc th ma valu R α (t) is also show as a cotiuous gr li. Th FD follows th PD, of cours with a switch of sig (i.., th FD is positiv as th bod strtchs from R qti to R qto, ad it is gativ as th bod cotracts from R qto to R qti ). Although this obsrvatio appars to cofirm ituitio, it is vrthlss otrivial i viw of th subsqut coutrxampl (s blow), whr part of th lctroic flux dsity (EFD) follows ithr th PD or th EPD. Clos ispctio rvals local maxima i th absolut valus of th FD at th pottial miima ad local miima at th top of th pottial barrir ad th quatum-mchaical turig poits. This obsrvatio may b ratioalizd by rcallig that th corrspodig classical flux Articl dsity is proportioal to th classical probability dsity tims th vlocity. If w assum that th classical dsity is rathr localizd, as suggstd by th quatum mchaical PD show i Figur b, th th tim volutio of th classical flux dsity is domiatd by th vlocity, which has its local maxima at th pottial miima ad its local miima at th pottial barrir ad at th classical turig poits. W tur xt to th computatio of th EPD, which bgis with th calculatio of th o-lctro probability dsity d, (r;r) (q.9) from th SAC-CI/aug-cc-pVTZ wav fuctio Ψ (r ;R). Th EPD is dividd ito two parts3 corrspodig to th 0 cor lctros ad th two valc lctros. It is kow that th cor lctros td to travl with th ucli. 3 Hr w ctr atttio o th valc lctros ( = ). Hcforth, th otatio d, (r;r) (ad likwis th otatio for quatitis drivd from d, (r;r)) rfrs to th olctro probability dsity of th valc lctros. For covic w tak R to b aligd with th z-axis (i.., R = R z ). Th d, (r;r) has cylidrical symmtry about th z-axis (itruclar axis). Th o-lctro probability dsity (multiplid by = ) is show i Figur as cotour plots i th x z pla for thr charactristic itruclar distacs: R = R i, R, ad R o. At th shortst distac R i (i.., i th domai of th ir pottial wll) d, (r;r) cosists of thr parts: a domiat ir o clos to th CM ad two quivalt, much lss its, outr lobs wll rmovd from th CM. Th ir part has two ctrs at th ucli a ad b. Away from th ucli th closd, approximatly llipsoidal cotours idicat molcular compactss. Th outr lobs may b itrprtd as sigaturs of th Rydbrg charactr of th lctroic structur i th domai of th ir pottial wll. Hc w shall call thm Rydbrg lobs. I cotrast, at th largst distac R o (i.., i th domai of th outr pottial wll) th Rydbrg lobs ar abst, whras th ir part of d, (r;r) sparats ito two fragmts that rmai ctrd at th ucli, thus idicatig a highly strtchd a a bod. At th itrmdiat sparatio R (i.., at th pottial barrir btw th ir ad outr wlls) th cotour plot (Figur ) illustrats th trasitio from th charactr of d, (r;r) atr i to that at R o. Appartly, th domiat ir ctrs of d, (r;r) aroud th ucli mov outward as th bod strtchs (i.., thy travl with th ucli). At th sam tim th Rydbrg lobs mov iward util thy ar absorbd by th ir part ad disappar approximatly at th barrir. It turs out that th proprtis of th o-lctro probability dsity d, (r;r) dscribd just abov ar sstial for th EPD ad for th EFD. Bfor procdig, w look i som dpth at th origi of ths charactristic proprtis. Bcaus th SAC- CI/aug-cc-pVTZ implmtatio i th Gaussia suit 3 dos ot ld itslf to th prst purpos, w prformd complmtary stat ( Σ + u ) slctiv complt activ spac slf-cosistt-fild (CASSCF-CI(,36)) calculatios, which prmit a rady itrprtatio of th approximatio d,casscf, (r;r) tod, (r;r) obtaid by th SAC-CI/aug-ccpVTZ mthod. Ths CASSCF-CI(,36) calculatios for th a molcul aligd alog th z-axis R = R z, ar carrid out by mas of th MOLPRO suit 9 with th sam aug-cc-pvtz basis st ad with iactiv spac cosistig of th lowst t molcular orbitals (MOs) φ k (r;r) doubly occupid by all 0 cor lctros ad a activ spac of 36 MOs, labld k = (HOMO), (LUMO), 3,..., 46 for th two valc lctros. Th fial CASSCF wav fuctio is xpadd i trms of th psudocaoical MOs, as implmtd by MOLPRO. Sic th optimizd CASSCF psudo-caoical MOs k = ad k = 55 DOI: 0.0/acs.jpca.7b73 J. Phys. Chm. A 08,, 50 59

7 Th Joural of Physical Chmistry A ar similar to th corrspodig HOMO ad LUMO, for covic w hcforward rfr to thm as such. It turs out that cotributios of doubl xcitatios from th two valc lctros ar gligibl. Th CASSCF-CI(,36)/aug-cc-pVTZ lctroic wav fuctio ca thrfor b writt approximatly as a cofiguratio-itractio (CI) xpasio rstrictd to sigl xcitatios: k k, β k, α, α, β Ψ ( r ; R) D ( R)[ Ψ ( r ; R) + Ψ ( r ; R)],CASSCF k (3.8) Hr Ψ k,β,α (r ;R)ad Ψ k,α,β (r ;R) ar th Slatr dtrmiats with all 0 cor lctros occupyig MOs k =,..., 0, ad th two valc lctros occupyig th molcular spi orbitals φ (r;r) α ad φ k (r;r) β, or vic vrsa. Th CI-xpasio cofficits D k (R) ar ormalizd accordig to k P ( R) = k> whr k k P ( R) = [ D ( R)] (3.9) (3.0) is th CI-probability of th HOMO-to-k xcitatio. Itgratio of [Ψ,CASSCF (r ;R)] ovr th coordiats of all lctros but o (q.9) yilds, to a vry good approximatio, th CASSCF-CI/ aug-cc-pvdt o-lctro probability dsity of th valc lctros ( = ), k d R φ + φ,casscf,(; r ) (; r R) P ( R) (; r R) k k (3.) I pricipl, cross trms from MOs labld ad k also cotribut to th sum i q 3., but thy tur out to b gligibl. Th bauty of this approximatio (q 3.) is that it allows o to aalyz th o-lctro probability dsity i trms of th (squars of th) CI cofficits D k (R) ad of th HOMO φ (r;r) togthr with th xcitd MOs φ k (r;r), k =, 3, 4,... This is illustratd i Figur b d. Figur b shows th domiat populatios of th HOMO (k = ) ad of th xcitd MOs (=P k (R), k = ad 4) togthr with th xt highst populatios (<0.0) of MOs φ k (r;r), k = 8 ad 4. All othr MOs ar occupid with lowr probability. Appartly, oly thr MOs (k =,, ad 4) mak th domiat cotributios to d,casscf, (r;r). Th corrspodig orbital dsitis d,casscf,k (r;r) = φ k (r;r) ar illustratd i Figur c for th charactristic itruclar distacs R = R i, R, ad R o. Th wightd sum of ths thr orbital dsitis (i.., q 3.) with th sum rstrictd to th two domiat cotributios, k = ad 4, is show i Figur d. Though th agrmt with th SAC- CI/aug-cc-pVTZ o-lctro probability dsity (Figur ) is ot yt prfct, it is clar that d,casscf, (r;r) alrady xhibits th mai faturs of d, (r;r). This suggsts th followig itrprtatio. Th o-lctro probability dsity d, (r;r) (multiplid by = ) of th valc lctros of a ( Σ + u )is sstially th wightd sum of thr orbital dsitis, amly, of th HOMO (k = ), th LUMO (k = ), ad th xcitd orbital k = 4. Th thr orbital probability dsitis maitai thir domiat topologis as th itruclar distac icrass from R i to R o. Thus, o may say that th probability dsity of th HOMO rprsts molcular compactss, xcpt wh th a a bod is highly logatd at R o. I cotrast, th 56 Articl probability dsity of th LUMO displays th topology of sparatd atoms, ot oly at R = R o but also alrady at R = R i ad R. Fially, th probability dsity of th xcitd MO k =4 rprsts typical faturs of a Rydbrg -typ MO, xtdig from th ucli to its lobs far from th ucli. What rally mattrs ar th wightigs of ths MO dsitis. I th domai of th ir pottial wll, d, (r;r) is domiatd by just two MO dsitis, amly, thos of th HOMO ad of th Rydbrg -typ MO (k = 4). This xplais its charactristic faturs (i.., th ir part ad th two outr Rydbrg lobs ). I cotrast, i th domai of th outr pottial wll, d, (r;r)is domiatd by th orbital dsitis of th HOMO ad th LUMO, whras th Rydbrg -typ MO (k = 4) is gligibl. This xplais th topology of th cotours of d, (r;r)atr = R o, idicatig th ost of sparatio of th a atoms. Fially, at th trasitio R = R from th domai ar R = R i to that ar R o, th populatio of th Rydbrg -typ MO (k = 4) is dpltd to th bfit of th HOMO. This xplais th disapparac of th Rydbrg lobs of d, (r;r) through a mrgig with th ir part of th o-lctro probability dsity at th pottial barrir. Th abov itrprtatio is ot quatitativ, but it xplais th most importat faturs of th volutio of d, (r;r) as th itruclar sparatio icrass from R = R i to R = R o.w mphasiz that subsqut calculatios ar basd o d, (r;r) dtrmid through th SAC-CI/aug-cc-pVDT mthod, rathr tha o th smiquatitativ d,casscf, (r;r). W also rmid th radr that d, (r;r) dpds oly o th distacs r ad R ad th agl θ btw th z-axis ad th vctor distac r from th CM to th lctro. To mphasiz th dpdc of d, (r;r) o sphrical coordiats, w hcforth dot it δ, (r,cosθ;r) (q.3). Accordig to qs. ad.3, w rquir th quatity δ, (r;r) to comput th EPD. W accomplish this task by fixig r ad R ad itgratig δ, (r,cosθ;r) ovr th agl θ. Th plots of δ, (r;r) vrsus r displayd i Figur for th thr charactristic itruclar distacs R = R i, R, ad R o dpd strogly o R. AtR = R i (th miimum of th ir pottial wll), δ, (r;r i ) is bimodal. Th valus of r at th two maxima ad th miimum btw thm ar idicatd i Figur by dottd lis, which also idicat th itrsctio of th x z pla with th sphrical surfacs o which δ, (r;r i ) is valuatd at ths valus of r. Th ir maximum at th smallst r corrspods to th maximum of δ, (r,cosθ;r i ) at th ucli. Th outr maximum at th largst r is du to th Rydbrg lobs of δ, (r,cosθ;r i ). I cotrast, δ, (r,cosθ;r o ) at th miimum R o of th outr wll dos ot possss Rydbrg lobs. As a cosquc, δ, (r;r o ) has just a sigl maximum, which corrspods to th maximum of δ, (r,cosθ;r o ) at th ucli. Th trasitio from a bimodal to a uimodal δ, (r;r), which occurs clos to th barrir at R = R (Figur ), is illustratd i dtail i Figur a, a color-codd cotour plot of δ, (r;r α =R/). Appartly, δ, (r;r/) has two brachs, a domiat ir brach ad a lss promit outr o at small ad larg r, rspctivly. Th ir brach accouts for th valc lctros that travl with th ucli from small to larg valus of R. Th outr brach, du to th Rydbrg lobs, is sigificat oly i th domai of th ir pottial wll. ar th pottial barrir it coalscs with th ir brach, which prsists i th domai of th outr pottial wll. Th EPD δ (r,t) is giv by q. as a itgral ovr R of th product of th tim-idpdt lctro populatio dsity δ, (r;r) ad th tim-dpdt uclar probability dsity DOI: 0.0/acs.jpca.7b73 J. Phys. Chm. A 08,, 50 59

8 Th Joural of Physical Chmistry A (4π) χ(r,t). Th EPD is xhibitd i Figur d as a colorcodd cotour plot. O immdiatly obsrvs th similarity of th topology of δ (r,t) i th tmporal domai 0 < t < t qto (Figur d) to that of δ, (r;r α =R/) i th corrspodig spatial domai R qti / < R α < R qto / (Figur a). This corrlatio may b udrstood as follows. W ot from Figur b that th PD is xtrmly wll localizd. Hc w ca mak th approximatio χ( Rt, ) δ[ R Rt ( ) ] (3.) whr δ hr dots th Dirac distributio. Substitutio of q 3. ito q. yilds δ( rt,) (4) π δ (; r Rt ()), (3.3) Thus, w s that th EPD at tim t is proportioal to δ, (r;r)at th ma valu of R at that tim. As a cosquc, all th proprtis of δ, (r;r) also hold for δ (r,t). I particular, th EPD cosists of two brachs. Th domiat brach accouts for th valc lctros that flow with th ucli. This ormal brach is ctrd at th ma distac of uclus α from th CM, which is idicatd i Figur d by th cotiuous gr li. Th othr brach is du to th Rydbrg lob i th domai of th ir pottial wll. This lss its Rydbrg brach mrgs with th ormal brach at th tim t as th ucli cross th pottial barrir. Equatio 3.3 also holds for tims t > t qto, wh R(t) dcrass from th outr quatum mchaical turig poit R qto to th ir o R qti. As show i Figur d, th EPD i this tmporal domai is approximatly th mirror imag of th EPD i th domai t qti < t < t qto. Dviatios from prfct symmtry ar du to uclar wav packt disprsio ad itrfrc ffcts. 8 Fially, w us q.6 to comput th EFD, which is displayd i Figur as a color-codd cotour plot. Th EFD compriss two brachs. O is du to th valc lctros that travl with th ucli. (To guid th y alog this brach, Figur shows th corrspodig tim volutio of th ma valu R α (t) as a cotiuous gr li). Accordigly, this ormal brach is positiv as R icrass from R qti to R qto, gativ as R dcrass from R qto to R qti, ad zro at th quatum-mchaical turig poits. Th scod brach cosists of a trasit flux of lctros that riss ad falls as th ucli cross th barrir at R. Ispctio of Figur d shows that this trasit flow is du to th rapid chag of th Rydbrg brach of th EPD as it mrgs with th ormal brach (i.., th lctroic structur chags from Rydbrg to ioic as th ucli travrs th pottial barrir). Th coalscc is actually supportd by two ffcts. O o had, as R icrass i th domai R i < R < R th ormal brach bds upward (toward icrasig r). O th othr had, th Rydbrg brach bds dowward. As a cosquc, th two compots of th EFD (i.., th ormal compot du to th valc lctros that travl with th ucli, ad th trasitio compot that accouts for th chag of th lctroic structur at th barrir) ar oppositly dirctd (i.., thy ar atagoistic). W mphasiz that this mchaism of productio of atagoistic fluxs dos ot ivolv itrfrc of th two compots. A altrativ mchaism, which is idd du to itrfrc of diffrt partial wavs, is documtd i rf 3. Figur shows that th trasitio cotributio to EFD is v strogr tha th ormal cotributio. This sms surprisig bcaus Figur d suggsts a opposit trd (i.., th EPD i th Rydbrg brach appars gligibl compard with that of th ormal brach). Th appart paradox ca b 57 Articl xplaid by th corrspodig classical xprssio for th flux dsity, which is th product of th probability dsity, ad th vlocity. Hc, high (low) probability dsity dos ot cssarily imply high (low) flux dsity, rspctivly. Istad, high flux dsity may aris from low probability dsity that flows with high vlocity. I th prst cas, th lctros i th lowdsity Rydbrg lob rarrag so rapidly durig th trasitio from th Rydbrg to th ioic structur that th absolut valu of th trasitio compot of th EFD is v largr tha th ormal compot. Th pricipal rsults ar summarizd i Figur 3, which shows arrow plots of th FD ad EFD suprposd o color-codd Figur 3. Color-codd cotour plots combid with arrow plots of lctroic populatio ad uclar probability dsitis ad flux dsitis wightd by factor 4πr or 4πR α for a i th xcitd stat Σ u + (JM = 00) at thr charactristic tims t i = 66 fs, t = 68 fs, ad t o = 334 fs. (a) uclar probability dsity [δ,α (R α =R/,t)] ad flux dsity [j,α (R α =R/,t)] wightd by factor 4πR α, i uits of /Å ad /fs, rspctivly. (b) Elctroic populatio dsity (of valc lctros) [δ (r,t)] ad lctroic flux dsity [j (r,t)] wightd by factor 4πr,i uits of /Å ad /fs, rspctivly. Vrtical lis idicat itruclar sparatios corrspodig to th thr charactristic tims: R i / (log dash); R / (dash-doubl dot); R o / (short dash). Magituds of flux dsity ar idicatd by arrows at th bottom lft of th plots. cotour plots of th PD ad EPD at th thr charactristic tims t i, t, ad t o wh th ma itruclar distac R(t) coicids with th miimum of th ir pottial wll at R i, th top of th pottial barrir at R ad th miimum of th outr pottial wll at R o, rspctivly. Ths sapshots dmostrat sigificat disprsio of th PD; still, th PD of th havy ucli ar always far mor localizd tha th EPD of th light lctros. Th FD is always uidirctioal, with th ucli flowig outward from R qti toward R qto. I cotrast, th EFD at R i DOI: 0.0/acs.jpca.7b73 J. Phys. Chm. A 08,, 50 59

9 Th Joural of Physical Chmistry A ad at R ar atagoistic, for th rasos that hav b xplaid abov i dtail. Th highst magituds of th EFD ar causd by th trasitio of th lctroic structur from Rydbrg to ioic as th ucli cross th pottial barrir. At largr itruclar distacs (.g., at R o ) th EFD is uidirctioal, lik th FD, bcaus thr ar o furthr sigificat chags of th lctroic structur. Th valc lctros simply flow with th ucli. 4. COCLUSIO Th prst work xtds th prior thory of cocrtd lctroic ad uclar fluxs i th o-lctro, orotatig, + diatomic H Σ + g (JM = 00) to th multilctro, orotatig homouclar diatomic i th stat S+ Σ + g,u (JM = 00). Th rsults for th may-lctro systm ar tirly aalogous to thos for th o-lctro systm. I particular, th symmtry of th stat implis that th EPD, PD, ad FD ar isotropic. A thorm du to Barth, 4 basd i part o rf 30, prmits us to dduc that th EFD is also isotropic. Th rducd radial cotiuity quatio yilds a xprssio for th EFD i trms of th EPD. Bcaus all volvig quatitis ar isotropic, th systm ca b viwd as a pulsatig or xplodig quatum bubbl. As a first applicatio of th thory w cosidr a vibratig i th doubl-miimum pottial of th xcitd stat Σ + u. Th most rmarkabl phomo is th two atagoistic cotributios to th EFD. O th o had, thr is a compot du to lctros that travl with th ucli. This ormal compot is positiv as th bod strtchs ad gativ as it cotracts. O th othr had, thr is aothr, mor its, compot apparig at gratr distacs from th CM ad opposig th ormal compot. W rfr to this scod cotributio, which is trasit (i.., it riss ad dcays as th ucli cross th barrir from o pottial wll to th othr), as th trasitio compot, bcaus it is du to th chag of lctroic structur from Rydbrg i th domai of th ir wll to ioic i th domai of th outr wll. Th prst work should stimulat systmatic ivstigatios of cocrtd lctroic ad uclar fluxs associatd with adiabatic vibratio ad dissociatio i othr homouclar diatomic molculs i thir lctroic groud ad xcitd S+ Σ + g,u (JM = 00) stats. Th isotropy of th PD ad EPD suggsts xprimtal masurmts with high radial ad tmporal rsolutios, whr o ds to moitor th probability (or populatio) dsity oly alog o dgr of frdom (s, for xampl, rfs 3 ad 3). Th FD ad EFD ca th b dtrmid from th xprimtal PD ad EPD by mas of th rducd radial cotiuity quatio. 3 Th prst applicatio of th thory should also promot systmatic sarchs for atagoistic lctroic fluxs i mor complicatd systms ad procsss, such as adiabatic chmical ractios. ASSOCIATED COTET *S Supportig Iformatio Th Supportig Iformatio is availabl fr of charg o th ACS Publicatios wbsit at DOI: 0.0/acs.jpca.7b73. Appdix A, dmostratio of sphrical symmtry of uclar probability dsity, uclar flux dsity, ad lctroic populatio dsity for a may-lctro homouclar diatomic molcul i a stat S+ Σ + g,u (JM = 00); Appdix B, ma radial flux dsity from th cotiuity quatio; Appdix C, prparatio of th iitial uclar wav packt by a wak lasr puls (PDF) 58 AUTHOR IFORMATIO Corrspodig Author *J. Maz. addrss: jmaz@chmi.fu-brli.d. ORCID D. J. Distlr: J. Maz: Articl ots Th authors dclar o comptig fiacial itrst. ACKOWLEDGMETS Th authors thak Dr. Igo Barth (Hall), PD Dr. Dirk Adra (Brli) ad Profssor Bat Paulus (Brli) for hlpful discussios ad Dr. Yasuki Arasaki (Tokyo), Profssor Volkr Egl (Wu rzburg), ad Profssor Matthias Wollhaupt (Oldburg) for valuabl advic o th litratur. Grous fiacial support from th Dutsch Forschugsgmischaft (projct Ma 55/7-), th atioal Ky Rsarch ad Dvlopmt of Chia (Grat o. 07YFA030403), th Program for Chagjiag Scholars ad Iovativ Rsarch Tam (IRT3076), th atioal atural Scic Foudatio of Chia (Grat os , , 65755) ad th Shaxi 33 Projct is gratfully ackowldgd. REFERECES () Maz, J.; Pŕz-Torrs, J. F.; Yag, Y. Vibratig H + [ Σ g + (JM = 00)] Io as a Pulsatig Quatum Bubbl i th Laboratory Fram. J. Phys. Chm. A 04, 8, () Pŕz-Torrs, J. F. Dissociatig H + [ Σ g + (JM = 00)] Io as a Explodig Quatum Bubbl. J. Phys. Chm. A 05, 9, (3) Brdtma, T.; Distlr, D. J.; Li, S.-D.; Maz, J.; Pŕz-Torrs, J. F.; Tia, W.-J.; Wu, Y.-B.; Yag, Y.; Zhai, H. J. Quatum Thory of Cocrtd Elctroic ad uclar Fluxs Associatd with Adiabatic Itramolcular Procsss. Phys. Chm. Chm. Phys. 05, 7, (4) Barth, I. Probability ad Flux Dsitis i th Ctr-of-Mass Fram. J. Phys. Chm. A 08, DOI: 0.0/acs.jpca.7b754. (5) Valac, A.; guy Tua, Q. 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