Near-resonant effects in the quantum dynamics of the H+H_2^+ \rightarrow H_2+ H^+ charge transfer reaction and isotopic variants
NNear-resonant effects in the quantum dynamics of the H+H + → H + H + charge transfer reaction and isotopicvariants Cristina Sanz-Sanz, Alfredo Aguado, and Octavio Roncero a)1) Unidad Asociada UAM-CSIC, Departamento de Química Física Aplicada, Facultad de Ciencias M-14,Universidad Autónoma de Madrid, 28049, Madrid, Spain Instituto de Física Fundamental (IFF-CSIC), C.S.I.C., Serrano 123, 28006 Madrid,Spain
The non-adiabatic quantum dynamics of the H+H + → H + H + charge transfer reactions, and some isotopic variants,is studied with an accurate wave packet method. A recently developed 3 × ab initio calculations and includes the long-range interactions for ground and excited states.It is found that for initial H + (v=0), the quasi-degenerate H (v’=4) non-reactive charge transfer product is enhanced,producing an increase of the reaction probability and cross section. It becomes the dominant channel from collisionenergies above 0.2 eV, producing a ratio, between v’=4 and the rest of v’s, that increases up to 1 eV. H+H + → H + +H exchange reaction channel is nearly negligible, while the reactive and non-reactive charge transfer reaction channelsare of the same order, except that corresponding to H (v’=4), and the two charge transfer processes compete below 0.2eV. This enhancement is expected to play an important vibrational and isotopic effect that need to be evaluated. For thethree proton case, the problem of the permutation symmetry is discussed when using reactant Jacobi coordinates.Accepted in J. Chem. Phys. (2021), I. INTRODUCTION
Hydrogen is the most abundant element in Universe, withnearly 73% of the barionic mass, and plays a fundamental rolein the chemistry of the interstellar medium (ISM). It is at theorigin of the chemical cycles of most of the molecules, whichin turn play a fundamental role in the collapse of molecularclouds to form stars and planetary systems. The most abun-dant molecular species are H and H + , while H + , formed byionization of H by cosmic rays, is rapidly destroyed by theexothermic reaction H + H + → H + + H . (1)H + is considered as the universal protonator by producinghydrides when colliding with atoms and molecules . Itscollisions with H produce ortho/para transitions of the twospecies, and/or to its deuteration when colliding with HD iso-topic variant.In local galaxies, the formation of H is attributed to re-actions on cosmic grains and ices , because the gas phaseroutes have too low rate constants to reproduce the observedabundances. In Early Universe, where grains and ices do notexist, one of the key problems is therefore to determine theprocesses, and their related rate constants, giving rise to H .One of this is the charge transfer reactionH + H + → H + H + , (2)where the reactant and product channels of Eq. 2 correspondto the first excited and the ground electronic states of the H + system, respectively.H + plays a fundamental role in astrochemisty and ithas been the subject of many studies summarized in a) Electronic mail: [email protected] reviews and special issues . Its infrared spectrumwas first detected in the laboratory by Oka and later in thespace . Since then, H + has become commonly used toprobe spacial conditions, such as a thermometer and a clockof cold molecular clouds . Its infrared spectrum has beentheoretically characterized with spectroscopic accuracy ,based on highly accurate potential energy surfaces (PESs),local and global .The H + + H exchange reaction has been the subject ofmany experimental and theoretical studies. Thisreaction governs the ortho/para transitions of H and is alsoresponsible of the H deuteration. For energies below ≈ + .In the ground H + electronic state there is a deep insertion well,of ≈ . Above 1.82eV collision energy, the H + + H channel becomes accessible,opening the charge transfer process, inverse to the reaction 2,and has been studied in a broad energy range .Furthermore, the non-adiabatic transitions in H + were studiedrecently in photodissociation experiments by Urbain et al. .In spite of all these studies on H + , reaction 2 has onlybeen studied experimentally by Karpas et al , by McCart-ney et al. , at energies of 30-100 keV, and more recentlyAndrianarijaona et al studied the H + D + isotopic vari-ant in a broader energy range of 0.1-100 eV. The experimentalstudy of reaction 2 have the difficulty of involving two radi-cal reactants, and accurate theoretical simulations are then ofhigh interest to provide realistic rate constants for the modelof Early Universe . Some approximated theoreticaltreatments where applied to reaction 2 . Also,an accurate quantum treatment of this reaction was done veryrecently , during the publication of this work. In this workwe present accurate quantum wave packet calculations of theH+H + charge transfer (CT) reaction, and some isotopic vari-1 a r X i v : . [ phy s i c s . c h e m - ph ] F e b nts, at collision energies below 1 eV using a very accurateset of three coupled PESs recently proposed , which includelong range interactions for the ground and excited electronicstates. The manuscript is distributed as follows. First, in sec-tion 2, a brief description of the PESs and dynamical methodsare shown, presenting a description of the permutation sym-metry problem. In section 3 the theoretical simulations areshown and discussed, and, finally, section 4 is devoted to ex-tract some conclusions, outlining future work. II. METHODOLOGYA. Diabatic electronic representation
The potential used is described by a 3 × . It consists of a zero-order diatomics-in-molecules (DIM) matrix , describing accurately the H andH + diatomic fragments (shown in Fig. 1), plus three-bodyterms added in the diagonal, and non-diagonal matrix, V B ,as it was proposed by Varandas and co-workers . In this po-tential the long range term has been improved and includedin all the electronic states. The dominant long range termsfor H+H + are the charge-induce dipole and charge-inducedquadrupole dispersion interactions, varying as R − and R − ,respectively, For H + + H the dominant long range terms arethe charge-quadrupole and the charge-induced dipole disper-sion energies, varying as R − and R − , respectively. The V B was fitted to very accurate ab initio calculations using multi-reference configuration interaction (MRCI) calculations, per-formed with the MOLPRO suite of programs . A completebasis set (CBS) approximation was done based on the aV5Zand aV6Z basis sets of Dunning . These V B matrix ele-ments change gradually what allows a very accurate fitting,with a small root mean square error , what is an advantageas compared to the fitting of the sharply varying non-adiabaticmatrix elements (NACMEs) required when using an adiabaticrepresentation .Diabatic representations are generally considered to be ap-proximate, due to the impossibility to eliminate all NACMEsas a function of all internal coordinates because of thecurl condition . The quasi-diabatic representation ofthe PES used here is considered as a regularizationrepresentation , in which the singularities present at coni-cal intersections (CI) are removed exactly . In the vicinity ofCIs the calculation of NACMEs presents numerical problemswhat adds some error in the adiabatic representation since theBorn-Oppenheimer is ill-conditioned at CIs . The resultingNACMEs obtained by Aguado et al. showed to be in excel-lent agreement with the ab initio points. Moreover, the resid-ual NACMEs appearing in all diabatic representation are verysmall in the present case, because the three-fold basis is rathercomplete in this system. These residual NACMEs are verysmall, and only appreciable far from CIs, where the energydifference among the different electronic states is so large thatthey can be neglected without loss of accuracy. H D D H (v=0)D (v=0) E ne r g y ( e V ) R (Å)
FIG. 1. Potential energy curves and vibrational levels of H (D )and H + (D + ). The radial part of the nuclear wave functions for H + (D + ) have been represented in green. Quasi-diabatic representations are not unique. In Ref. itwas considered a minimum basis set formed by 3 functions,denoted φ i , in which there is a positive charge of nucleus i ,and one 1s electron in the remaining nuclei. In this basis set,the excited H + ( Σ + g ) is a linear combination of at least two φ i functions. A unitary transformation is done, in order to de-fine properly the H ( Σ + g ) and H + ( Σ + g , Σ + u ) for a particular(reactant) rearrangement channel, as φ = φ φ = √ ( φ − φ ) (3) φ = √ ( φ + φ ) . Transforming the 3 × and H + ground electronic states become diagonal in channel 1, cor-responding to the hydrogen 1 at very long distances fromthe other two, and the corresponding potentials are shown inFig. 1. However, on the other two rearrangements, the poten-tial matrix is non diagonal, and the diabatic couplings are non-zero as can be seen in the right panels of Fig. 2, where theyare represented in Jacobi coordinates, r and R for a collinearconfiguration. As a consequence, the rovibrational states ofthe products are expanded in several electronic states, for H and H + as φ i = , v (cid:48) j (cid:48) ( r (cid:48) ) = ∑ k φ k C v (cid:48) j (cid:48) k ( r (cid:48) ) (4) φ i = , + v (cid:48) j (cid:48) ( r (cid:48) ) = ∑ k φ k C v (cid:48) j (cid:48) + k ( r (cid:48) ) , r (cid:48) denotes the internuclear distance in thecorresponding product Jacobi coordinates r (cid:48) = r or r (cid:48) = r for i = 2 and 3, respectively.
1 4 7 r (Å)1 4 710 R ( Å ) −10123 1 4 710 −10123 1 4 710 −10123 1 4 7 10 −4 −4 −4 −4 FIG. 2. Contour plots of the diagonal (left panels) and non-diagonal(rigth panels) elements of the 3 × o , as a function of the Jacobi distances r and R , in the diabaticrepresentation of functions φ i . In the diagonal terms, the zero ofenergy is set at the eigen value of H + ( v = , j = ) ground rovibra-tional state, which is at 2.09 eV in Fig. 1. The absolute value ofnondiagonal terms are plotted in logarithmic scale. The diagonal and non-diagonal elements of the potentialmatrix in this diabatic representation are shown in Fig. 2 for acollinear configuration. For long R and short r values, thediagonal terms represent H ( Σ + g ) (bottom left panel), theH + ( Σ + g ) (middle left panel) and H + ( Σ + u ) (top left panel),the last being purely repulsive. The diagonal terms at long r and R values are repulsive, and do not represent the productschannels because the non-diagonal terms (shown in the rightpanels) are non-zero and pretty large. In the reactant channels(long R and short r values) the (cid:104) | V | (cid:105) non-diagonal couplingterm, coupling H + ( Σ + g ) with H ( Σ + g ) , decrease slowly withincreasing R , being of the order of ≈ R =7 Å. Thisclearly shows that this charge transfer coupling extends to-wards very long R distances. B. Quantum wave packet calculations
The reaction dynamics of the H/D collisions withH + (v=0,j=0) and H with D + (v=0,j=0) are studied here us-ing a quantum wave packet method as implemented in theMADWAVE3 program . For each total angular momen-tum, J , the state-to-state S-matrix elements are calculated us-ing a reactant Jacobi coordinate based method , generalizedto consider product rovibrational states expanded in severalelectronic states, as in Eq. 4. Three different processes ( α channels) are distinguishedA + B + ( v = , j = , J ) −→ A + + B ( v (cid:48) , j (cid:48) ) non − reactive charge transfer ( NRCT ) −→ AB + ( v (cid:48) , j (cid:48) ) + Bexchange reaction ( ER ) −→ AB ( v (cid:48) , j (cid:48) ) + B + reactive charge transfer ( RCT ) (5)with A = H, D and B = H , D . Channels α = α = TABLE I. Parameters used in the wave packet calculations in reac-tant Jacobi coordinates: r min ≤ r ≤ r max is the H internuclear dis-tance, R min ≤ R ≤ R max is the distance between H center-of-massand the A atom, 0 ≤ γ ≤ π is the angle between (cid:126) r and (cid:126) R vectors. r min , r max = N r =360 r abs = 16 Å α r =10 − ,n=6 R min , R max = N R =360 R abs = 16 Å α R =10 − ,n= 6 N γ = 100 in [ , π / ] R = 14 Å E , ∆ E = 0.5,0.5 eV R ∞ = 16 Å R (cid:48) ∞ = 10Å The parameters used in the wave-packet calculations arelisted in table I. Briefly, the initial wave packet correspondsto the product of the H + (v=0,j=0) ro-vibrational eigenfunc-tion, a normalized Wigner function for J and a real Gaussianfunction, describing the translation on the R Jacobi distance(between the incomming A atom and the center of mass ofH + ), initially placed at R , with a central energy E and anenergy width ∆ E . The flux on individual rovibrational statesare analysed at R ∞ and R (cid:48) ∞ for reactants and products, respec-tively. The wave packet is absorbed at the edges of the radialgrid by multiplying it by an absorbing function e α ρ ( ρ − ρ abs ) n for ρ ≥ ρ abs ( ρ = r or R ) at each Chebyshev iteration, usinga modified Chebyshev propagator . Using this propagatoronly a real wave packet is propagated , and the corre-sponding kinetic term is evaluated using a sine Fourier trans-form which ensures the proper regular behaviour at R = ,appearing in this reaction. Finally, a Gauss-Legendre quadra-ture is used to describe the Jacobi angle cos γ = r · R / rR , and3he corresponding kinetic terms are evaluated using a DiscreteVariable Representation (DVR) method . For J =0, about 8 × iterations are needed to converge the reactions probabil-ity down to 0.01 eV. This long propagation is needed becauseof the presence of resonant structures that will be commentedbelow. This is also the situation of higher J until the rota-tional barrier push the reactive and NRCT probabilities to-wards higher energy, what happens at J =14 and 17 for H/D +H + cases, respectively. For higher J ’s, the number of itera-tions needed gradually decreases.The calculations have been performed for all J up to J =15and J =20 for A = H and D, respectively. For higher values of J , wave packet calculations are performed every 5 J valuesup to J =60 and J =80, for A = H/D, respectively. A maximumof helicity components Ω max = 23 is considered in the J > J values, where no wave packetcalculations were done, an interpolation method based onthe J -shifting approximation is performed to evaluate theindividual state-to-state S matrix elements .The state-to-state integral cross sections are then evaluatedusing the usual partial wave expansion as σ α v j → α (cid:48) v (cid:48) j (cid:48) ( E ) = π k − α v j ( j + ) ∑ J ΩΩ (cid:48) ( J + ) (cid:12)(cid:12)(cid:12) S J t α vJ Ω → α (cid:48) v (cid:48) J (cid:48) Ω (cid:48) ( E ) (cid:12)(cid:12)(cid:12) (6) C. Symmetry considerations
Dealing with two (D+H + ) or three (H+H + ) hydrogenatoms, fermions with nuclear spin i =1/2, some considerationsabout the permutation symmetry should be done. The totalwave function has to be antisymmetric under the exchange ofany hydrogen pair. The total wave function can be factor-ized as a product of electronic, nuclear spin and rovibrationalcomponents as Ψ = Ψ e Ψ I Ψ Rot . The symmetry of the elec-tronic function, Ψ e , is analyzed in the body fixed frame of thenuclei, and the permutation of any pair of nuclei correspondsto a reflection through a plane perpendicular to the molecularplane . In this case, all singlet states of H + are totally sym-metric. The nuclear spin functions, Ψ I , are characterized bythe total nuclear spin, I and I , for 2 and 3 hydrogen systems,respectively. I = 0 and 1, and the corresponding functions Ψ I are antisymmetric and symmetric with respect to the permu-tation of the two hydrogen atoms, respectively. To make thetotal wave function antisymmetric, the corresponding rovibra-tional function have to be symmetric and antisymmetric for I = 0 and 1, respectively. In diatomic molecules, this cor-responds to the usual separation between even and odd ro-tational levels, since the symmetry of spherical harmonic is ( − ) j , denoted as para and ortho, respectively.For three hydrogen atoms system, I = 1/2 and 3/2,and the spin functions can be written as | I M ; i (cid:105) = ∑ σ m ( i , / | m , σ , I M ) | / , σ (cid:105)| i , m (cid:105) , where | i , m (cid:105) isthe spin function of two hydrogen, each one described by | / , σ (cid:105) , and ( | ) are Clebsh-Gordan coefficients. The exist-ing functions are then | / M ; 1 (cid:105) , | / M ; 1 (cid:105) and | / M ; 0 (cid:105) ,with I = 3/2, 1/2 and 1/2, respectively. Considering the D h permutation-inversion group, the spin functions with I =3/2(ortho) belong to the A representation, while with I = E representation. To build antisymmet-ric total wave functions, the corresponding rovibrational com-ponents have to belong to A (for I =3/2, ortho) and E (for I =1/2, para) representations. It should be noted here, that theproduct for I =1/2, the total symmetry is E × E = A + A + E ,and only A is anti-symmetric while the other solutions are notphysically allowed.All this said, and neglecting the weak hyperfine nuclearspin rotation coupling, the spin of each hydrogen atom is con-served, and therefore the transformation from ortho to para,and viceversa, can only take place by hydrogen exchange, i . e . ,by a reaction.When using reactant Jacobi coordinates, the permutationsymmetry of the D+H + and H + D + is fully acount for, whilethis is not the case for H+H + . In the three identical hydrogenatoms case, the A and E representation has to be considered,separately, while the A does not exist. Thus, the diatomicrotational channels (for reactants and products) included ineach representation of the nuclear wave function are E : even and odd j,j’ → I =1/2, I =1,0 (para) of A A : only even j,j’ → does not exist for i =1/2 A : only odd j,j’ → I =3/2 (ortho) of A .Clearly the case of three identical hydrogens, with reac-tants in j =0, contains the E and A representation of the D h inversion-permutation group. For the final reactants or prod-ucts in even j (cid:48) also contains E and A representations and aretherefore difficult to distinguish when using Jacobi coordi-nates. Finally, the initial case j =0 and final odd j (cid:48) (para-to-ortho transition of dihydrogen molecules/cations) clearly be-long to the E representation of H + .Therefore, when using reactant Jacobi coordinates, totalintegral reaction cross sections can not be obtained, whilethe para-ortho state-to-state integral cross section can be ob-tained. However, when referring to H+H + (v=0,j=0) reactiondynamics we shall include all the cases, physical ( E ) and non-physical ( A ), to compare with the D+H + (v=0,j=0). III. RESULTS
The reaction probabilities obtained for the three reactionsunder study for J =0 are shown in Fig. 3, separated for thethree different processes of Eq. 5 and summing over all rota-tional states of products. The reactions for H+H + (left pan-els), D+H + (middle panels) and H+D + (right panels) showsimilar patterns, and each mechanism is discussed separatelybelow.For the three reactions, the NRCT process (in the top pan-els of Fig. 3) is open from zero collision energy. As dis-played in Fig. 1, the H + (v=0) classical turning points appear atshorter distances than the crossing point between the H + andH potential curves. However, the vibrational ground stateof H + (v=0) and D + (v=0) have significant amplitude at theregion of the crossing, see Fig. 1. This situation allows theelectronic charge transfer in the same channel for long R dis-tances, because the electronic coupling becomes effective (see4 RCT: H+H -> H (v’)+H + P r obab ili t y Collision energy (eV)v’=0v’=1v’=2v’=3v’=4v’=500.1
ER: H+H -> H (v’)+H v’=0v’=1v’=2v’=3 0.2 NRCT: H+H -> H + +H (v’) v’=0v’=1v’=2v’=3v’=4v’=5 0.4 0.8 RCT: D+H -> HD(v’)+H + v’=0v’=1v’=2v’=3v’=4v’=5v’=6 ER: D+H -> HD + (v’)+H v’=0v’=1v’=2v’=3v’=4 NRCT: D+H -> D + +H (v’) v’=0v’=1v’=2v’=3v’=4v’=5v’=6 0.4 0.8 RCT: H+D -> HD(v’)+D + v’=0v’=1v’=2v’=3v’=4v’=5v’=6 ER: H+D -> HD + (v’)+D v’=0v’=1v’=2v’=3v’=4 NRCT: H+D -> H + +D (v’) v’=0v’=1v’=2v’=3v’=4v’=5v’=6 FIG. 3. Vibrationally resolved reaction probability for H+H + (v=0,j=0) (left panels), D+H + (v=0,j=0) (middle panels), and H +D + (v=0) (rightpanels), towards the inelastic charge transfer, NRCT (H + +H , D + +H and H + +D , in the top panels), the exchange reactive channel, ER(H+H + , H+HD + and D +HD + , in the middle panels) and reactive charge transfer, RCT, channels (H + +H , H + +HD and D + +HD, in thebottom panels). Fig. 2).This is particularly evident for H (v’=4) which is nearlyresonant with H + (v=0) (just 0.064eV above), presenting alarge vibrational overlap, what yields to a strong electronictransition to H (v’=4). The ratio between the final probabil-ity in H (v’=4) and all other channels increases with collision energy. This clearly explains the experimental results , whofound that charge transfer is the dominant mechanism, ratherthan hydrogen exchange or complex formation with scram-bling. These two mechanisms, associated to ER and RCT, cancompete at the lower energies considered here, as discussedbelow.5or D + (v=0), however, the energy differences withD (v’=5,6) are larger, of ≈ + (v=0) at the electronic crossing is lower (see Fig. 1). Allthese make less effective the electronic coupling. As a con-sequence, all the final vibrational channels have lower prob-ability. In spite of the larger energy spacing, the D (v’=6)level shows to be considerably more populated that any othervibrational level of D , in clear analogy with the situation ofH .The exchange reaction mechanism, ER in the middle pan-els of Fig. 3, are also non zero, even when it is less proba-ble than the other two charge transfer mechanisms. The adi-abatic potential for the first excited electronic state presents arather high reaction barrier for the exchange. Therefore, ERproceeds in a two step mechanism. First, there is a transi-tion to the ground electronic state, with a deep well, explain-ing the resonance structure present in the ER probabilities.This is followed by a second transition back to the excitedH + H + /H+DH + /D+DH + channel, in each case respectively,giving rise to non-zero ER probabilities, lower than the RCTprobabilities because there is a lower density of final statesfor H + /HD + channels. The probability for the ER channelpresents a clear decreases when the mass of reactants increase,from H+H + , to D+H + and H +D + .The RCT mechanism (see bottom panels of Fig. 3) consistsof two steps, a transition to the ground electronic state and thereactive exchange of a hydrogen atom. Except for H (v’=4)(or D (v’=6)), the RCT probabilities are very similar to thoseof the NRCT channel. This could be explained by the nearstatistical mechanism of the reaction in the ground adiabaticstate , that is, the reaction is mediated by resonancesoriginated by the deep insertion well of the ground electronicstate of H + . At collision energies below 1 eV these reso-nances are long lived and the reaction cross section and thefinal distribution of products are well described by statisticalmethods . At the higher energies considered here (about1.8 eV above the ground H + +H threshold), the resonancepersist and a pseudo-statistical behaviour may be expected.This explains the resonance structure of all reaction probabil-ities in Fig. 3.Three main differences are found with the previous resultsrecently reported by Ghosh et al. . First, the results of Ref. do not show the dense manifold of narrow resonances shownhere. Also their probabilities do not show a progressive in-crease with decreasing collision energy, as the present onesdo, associated to the long range interactions. Finally, the re-sults of Ghosh et al. do show a much lower increase ofthe NRCT for H (v’=4), and instead their probabilities forH (v’=3) are of the same order of v’=4. All these differencesare attributed to the use of different PESs, and, in particular,to the electronic couplings producing the charge transfer. Itis worth noting here, the coupling at relatively long distanceplays an important role in the nearly resonant enhancement ofthe CT for H (v’=4) obtained here. On the contrary, Ghosh et al. used a relatively short initial distance of ≈ R =14 Å used here.The total reaction probabilities for each of the mechanismsand different values of total angular momentum, J , are shown in Fig. 4 for H+H + ,D+H + and H+D + reactions. All theprobabilities shift towards higher energies with increasing J .However, there are important differences. First, D+H + showssmaller shifts, simply because the effective rotational barrierdecreases, because the reduced mass is larger, µ ≈
1, 4/5 and2/3 amu for D+H + , H+D + and H+H + , respectively. TheNRCT channel also shows a smaller shift with J , what isattributed to the long-range character of the electronic cou-plings, as described above. Moreover, this also explains whythe NRCT probabilities increase near the rotational thresholdwhile for ER and RCT the probabilities clearly decrease with J at the threshold. As the rotational barrier increases, it closesthe access to reaction, while the NCR process is still possibledue to long range non-adiabatic couplings, which are effectivebecause the quassi-degeneracy of the H + (v=0) and H (v’=4)levels, or equivalently D + (v=0) and D (v’=6). This behav-ior also produces an enhancement of the NRCT mechanism ascompared to ER and RCT channels, except for low collisions,where RCT mechanism dominates.The opacity function, i.e. the reaction probabilities versus J , reported by Ghosh et al. shows a sudden drop between 0.3and 0.4 eV at J =12. However, in the present results the rota-tional threshold at 0.3 eV is approximately at J =40. A contin-uous and gradual increase of the rotational threshold is foundhere at all collision energies of Fig. 4. This difference withprevious results is attributed to the long-range behaviour ofthe electronic couplings and the longer distances consideredin the dynamical calculations, as discussed above.The total integral cross section for each of the processesare shown in Fig. 5 for the H+H + (v=0,j=0) (bottom panel),D+H + (v=0,j=0) (middle panel) and H+D + (v=0,j=0) (toppanel) collisions. At 1 eV, σ NRCT > σ RCT > σ ER , but theratio changes with collision energy. The exchange reaction,ER, is always lower, since it involves 2 electronic transitionsand the H + /HD + products have a lower density as comparedto H /HD. The NRCT/ER ratio is nearly constant with col-lision energy. On the contrary NRCT/RCT varies consider-ably, being nearly 1 for collision energies near 0 . + / D+H + / H+D + respec-tively. This ratio seems to increase with energy, in agreementwith the experiments performed by Karpas et al. , who foundthat the CT dominates the H+H + reaction.At low collision energies, before the dominant NRCT res-onant H (v’=4) or D (v’=6) channel opens, RCT and NRCTcross sections are of the same order, and in most cases theRCT process dominates. In this region the dynamics is dom-inated by a statistical process: After a first electronic transi-tion, the system gets trapped in the long-lived resonances orig-inated by the deep H + well of the ground electronic state. Atthese long-lived resonances, energy transfer among all inter-nal degrees of freedom becomes very effective and the prod-ucts are formed proportionally with the density of states. Thusfor H + , with three identical H + + H charge transfer products,the NRCT/RCT reaches a factor between 0.5 and 1. For D + + H , this factor decreases, since HD + RCT products havelarger density of states than H + + D. Finally, for H+D + theratio of the density of products states is reversed becomingdenser for the inelastic D channel than for the reactive HD6 -> H +H + P r obab ili t y Collision energy (eV)00.1 ER: H+H -> H +H -> H + +H -> HD+H + J=0J=10J=20J=30J=40J=50J=60J=70J=80
ER: D+H -> HD + +H NRCT: D+H -> D + +H -> HD+D + ER: H+D -> HD + +D NRCT: H+D -> H + +D FIG. 4. Total reaction probability at different total angular momentum, J , for H+H + (v=0,j=0) (left panels), D+H + (v=0,j=0) (middle panels)and H+D + (v=0,j=0), towards the inelastic charge transfer, NRCT (H + +H , D + +H and H + + D products in the top panels), the reactivechannel, ER (H+H + , H+HD + and D+HD + in the middle panels) and reactive charge transfer, RCT, channels (H + +H , H + +HD and D + +HDin the bottom panels). one. This explains why here the RCT is nearly identical toNRCT in this case.The NRCT for the H + D + (v=0,j=0) reaction is comparedwith the experimental work of Andrianarijaona et al. inthe top panel of Fig. 5. In these measurements of the H + D + reactions, only H + products are detected, so that their cross section only corresponds to the NRCT process. The goodagreement obtained with the present results, specially at en-ergies of ≈ + reactants are produced with some vibrational excitation.This could explain why at lower energies the agreement is7 ER (H ) RCT (H )NRCT (H ) C r o ss s e c t i on ( Å ) Collision energy (eV) 0.01 0.1 1 10 100 D+H ER (HD + ) RCT (HD)NRCT (H ) ER (D ) RCT (HD)NRCT (D ) Exp. FIG. 5. Total integral reactive cross sections for H+H + (v=0,j=0)(bottom panel), D+H + (v=0,j=0) (middle panel) and H+D + (v=0,j=0)(top panel), towards the inelastic charge transfer, NRCT (H + +H ,D + +H and H + +D products), the reactive channel, ER (H+H + ,H+HD + and D+HD + products) and reactive charge transfer, RCT,channels (H + +H , H + +HD and D + +HD). In the top panel thepoints correpond to the experimental values from Ref. worse. Some studies on the vibrational effects are now un-der way.The NRCT dominates the high energy region becausethe nearly resonant conditions between H (v’=4) and theH + (v=0) levels (or D + (v=0) and D (v’=6)), as discussedfor the reaction probabilities. The vibrationally resolvedNRCT cross sections, in Fig. 6, also show this effect. Inthe first two reactions, below the H (v’=4) channel opens,at 0.064 eV, H (v’) are formed progresively in ascending or-der, v’=0,1,2 and 3. There are small isotopic effects, but ingeneral the NRCT shows a typical decreasing behaviour asso-ciated to exothermic reactions. At energies above 0.064 eV,H (v’=4) opens, and the cross section for this channels in-creases becoming the dominant channel at energies above 0.2eV. A similar situation holds for H+D + (v=0), but replacingH (v’=4) by D (v’=6). −>H + +H (v’) C r o ss s e c t i on ( Å ) Collision energy (eV)0.1110 NRCT:D+H −>D + +H (v’) −>H + +D (v’) v’=0v’=1v’=2v’=3v’=4v’=5v’=6v’=7 0 0.4 0.8RCT: H+H −>H (v’)+ H + RCT: D+H −>HD(v’)+H + RCT: H+D −>HD(v’)+D + FIG. 6. Vibrationally resolved charge transfer cross sections forH+H + (v=0,j=0) → H + + H (v’)/ H (v’)+ H + (bottom panels)D+H + (v=0,j=0) → D + + H (v’)/HD(v’)+H + (middle panels) andH+D + (v=0,j=0) → H + + D (v’)/HD(v’)+D + (top panels) , for theNRCT (left panels) and RCT (right panels) processes. The vibrational resolved cross sections of Fig. 6 are inqualitative agreement with the results in Fig. 7 by Krstic ,which includes both NRCT and RCT processes calculated in abroader energy range using a close coupling method based onthe infinite order sudden approximation (IOSA) with Delveshyperspherical coordinates. In Krstic work, v (cid:48) =4 cross-sectionis always the most important, but it becomes of the same or-der than that for v (cid:48) =3 at collision energies of 0.2 eV. Below0.2 eV all σ v = → v (cid:48) CT cross sections are of the order of 10Å , and for all v (cid:48) (cid:54) = v (cid:48) = ,similar to the present results of NRCT+RCT. The differencesarise about the position of the crossing between v (cid:48) =4 and theother v (cid:48) , and on the relative magnitude of the cross section as-sociated to different v’s. Since the method used here is moreaccurate, it may be concluded that at energies below 1 eV thepresent results are more accurate.The calculations of Last et al. were done from 0.06 to0.21 eV using a quantum method based on negative imaginary8otential combined with a variational quantum method in a L basis set. Their NRCT final vibrational distributions pick at v (cid:48) =3, instead of 4. However, they used the helicity decouplingapproximation, i.e. they did not considered the coupling be-tween different helicities Ω . In this case, this approximationis not appropriate, because there are strong couplings betweenthe different resonances with different Ω value originated bythe deep well in the ground electronic state. However, the rea-son why their results are picked at v (cid:48) =3 and not v (cid:48) = and H + fragments or to the extension of non-adiabatic couplings. −> H + + H (v’=4,j’)j’=0 j’=2 j’=4j’=6 j’=8 C r o ss s e c t i on ( Å ) Collision energy (eV) 0.001 0.01 0.1 1 10 NRCT: D+H −> D + + H (v’=4,j’)j’=0 j’=2 j’=4j’=6 j’=8 −> H + +D (v’=6,j’)j’=0 j’=2 j’=4j’=6 j’=8 FIG. 7. Rotationally resolved NRCT cross sectionsfor H+H + (v=0,j=0) → H + + H (v’=4,j’) (bottom panel),D+H + (v=0,j=0) → D + + H (v’=4,j’) (middle panel) andH+D + (v=0,j=0) → H + + D (v’=6,j’) (top panel). The rotationally resolved NRCT cross sectionsfor H/D+H + (v=0,j=0) → H + /D + + H (v’=4,j’) andH+D + (v=0,j=0) → H + + D (v’=6,j’) collisions, in Fig. 7,clearly show that j’=0 is the dominant final rotational state,which is the closer to the H + (v=0,j=0) or D + (v=0,j=0) initialstate. j’=2/j’=0 ratio is about a factor of 1/4 for collisionenergies above 0.3 eV, and this ratio decreases with rota-tional excitation of CT products. Last et al. also reportedthe maximum of the state-to-state cross section for finalH (v’=4,j’=0) state, but less than a factor of 2 as compared tov’=3,j’=4.Finally, it should be noted that while the results are accu-rate for D+H + and H+D + , for H+H + we are including alsothe non-physical A irreducible representation. While somework is now in progress to include the full permutation sym-metry for H+H + , we can already use as exact the para-to-orthostate-to-state cross sections. Moreover, because of the highpropensity of the NRCT to final H (v’=0,j’=0), some conclu-sions can be extracted. Thus, the present results for H+H + asses that the NRCT channel is dominant for energies above0.2 eV up to 1eV, in clear agreement with the results of Karpas et al. . IV. CONCLUSIONS
In this work accurate quantum calculations have been pre-sented for the H/D+H + and H + D + non-adiabatic chargetransfer reactions using an accurate 3 × , which includes long range inter-actions. It is found that the dominant channel correspondsto the resonant non-reactive charge transfer from H + (v=0) toH (v’=4) and D + (v=0) to D (v’=6), which is enhanced bytheir nearly resonant energies and the long-range dependenceof the non-adiabatic couplings. This enhancement is based onrather general features of the H + asymptotic features, namelythe electronic crossing and the energy difference of the vi-brational states of the neutral and cation systems, and are notexpected to depend on the accuracy of the PESs used in thiswork. The only requirement is to include the electronic cou-pling existing at rather long distances between the reactants,which is described very accurately by the PESs used in thiswork.For D+H + and H+D + state-to-state cross sections are pre-sented for the first time. For H+H + some caution must bepaid because the permutation symmetry is not fully accounted,while some work is now-a-days in progress to include thepermutation among the three fermions. In spite of that, thepresent results show a good qualitative agreement with the ex-perimental data of Karpas et al. , who also reported that theCT channels dominat at ≈ + by An-drianarijaona et al. already demonstrates the adequacy ofthe simulations presented here for E > + reactants shouldbe done to understand the behavior at lower energies. It isalso necessary to extend the present calculations to other rovi-brational states and isotopic variants to provide reaction rate9onstants of interest in astrophysical models of the Early Uni-verse. V. ACKNOWLEDGEMENTS
We acknowledge Dr. Andrianarijaona for providing uswith the experimental data and very interesting discussionsof their results. The research leading to these results hasreceived fundings from MICIU (Spain) under grant FIS2017-83473-C2. We also acknowledge computing time at Finisterre(CESGA) and Marenostrum (BSC) under RES computationalgrants ACCT-2019-3-0004 and AECT-2020-1-0003, and CCC(UAM).
VI. DATA AVAILABILITY
The data that support the findings of this study are availablefrom the corresponding author upon reasonable request.
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