Jonathan Tennyson et al- H3^+ near dissociation: theoretical progress

832019 Jonathan Tennyson et al- H3^+ near dissociation theoretical progress

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H+3 near dissociation theoretical progress

B y J o n a t h a n T e n n y s o n M A K o s t i ny

H Y M us sa

O L Polyanskyy a n d R P r o s m i t iz

Department of Physics and Astronomy University College London

Gower Street London WC1E 6BT UK

The observation of an infrared spectrum of the H+

3 molecular ion at its dissociationlimit by Carrington and co-workers has presented a tremendous challenge to theoryTo compute this spectrum it is necessary to model accurately the global potentialenergy surface of H+

3 vibrationally excited states at dissociation rotational exci-

tation lifetimes of rotationally excited `shapersquo resonances and infrared transitiondipoles near dissociation Progress in each of these aspects is reviewed and results arepresented for highly excited vibrational levels using a new ab initio global potentialThe use of massively parallel computers in solving aspects of the problem is discussed

Keywords discrete variable representation

massively parallel computers resonances

1 Introduction

In 1982 Carrington and co-workers announced the observation of a very unusualphotodissociation spectrum for the H+

3molecular ion (Carrington et al 1982) These

workers made H+

3using electron-impact ionization of H2 selected these ions with a

mass spectrometer and photodissociated them using a CO2 laser

H+

3+ hcedil H2 + H+ (11)

Photodissociation was monitored using an electrostatic analyser (ESA) which wasset to monitor the proton current The ESA is also sensitive to the kinetic energy of the ions monitored and the original spectra were obtained by observing those ions

with approximately zero kinetic energy in the frame of the moleculeSimplistically the kinetic energy released in the molecular frame corresponds tothe amount of energy above dissociation contained in the fragmenting ion How-ever the situation is complicated by the fact that some of the excess energy canbe carried away by the molecular hydrogen fragment as vibrational or rotationalexcitation Furthermore the ESA only gives approximate kinetic energy resolutionand protons with a range of kinetic energies are in practice expected to contributeto the spectrum Indeed Carrington et al (1993) showed that the spectrum variesstrongly with kinetic energy release

The experimental set-up of Carrington and co-workers is very sensitive their spec-trum results from between 01 and 1 of the ions in their system (Carrington amp

y Permanent address Institute of Applied Physics Russian Academy of Science Uljanov Street 46Nizhnii Novgorod Russia 603024

z Present address Instituto de Mathematicas y Fisica Fundamental Consejo Superior de Investiga-ciones Cienticas C Serrano 123 28006 Madrid Spain

Phil Trans R Soc Lond A (2000) 358 24192432

2419

creg 2000 The Royal Society



2420 J Tennyson and others

Kennedy 1984) However Carrington et al (1993) observed a very large number of well-dened transitions in the limited range of their laser 8721094 cmiexcl1 As thedissociation energy of H+

3 is ca 35 000 cmiexcl1 (Cosby amp Helm 1988 Lie amp Frye 1992)this experiment monitors only those levels of the ion that lie very close to the disso-ciation limit Prior to this experiment the structure of the rotationvibration energy

levels at dissociation for chemically bound molecules larger than diatomic was com-pletely unknown Indeed H+

3and its isotopomers remains the only strongly bound

polyatomic that has been studied in this fashionThe spectrum of Carrington et al (1993) was unexpected and truly remarkable It

has stimulated a whole host of theoretical studies both (semi-)classical and quantalOf course H+

3 is a highly quantal system indeed the very existence of the spec-trum depending as it does on tunnelling means that classical mechanics can onlyhave limited application However classical studies proved successful in interpretingthe observed behaviour upon isotopic substitution (Gomez Llorente amp Pollak 1987Chambers amp Child 1988)

One feature of the spectrum has provoked particular interest Carrington amp Ken-nedy (1984) convolved their high-resolution spectrum to show that at low resolutionit showed coarse-grained structure four peaks separated by ca 50 cmiexcl1 Classicalcalculations show that H+

3 is highly chaotic in the near-dissociation region Therehave been several classical trajectory studies that tried to identify the regular motionembedded in the chaos that could be responsible for this coarse-grained structureGomez Llorente amp Pollak (1989) identied the `horseshoersquo motion where one Hnucleus passes between the other two as the key to this structure However sub-sequent calculations (Polavieja et al 1994 1996) have suggested at least one other

possible candidateClassical work on the H+

3near-dissociation spectrum has been reviewed by Pollak

amp Schlier (1989) Carrington amp McNab (1989) McNab (1994) and Kemp et al (thisissue) have reviewed the experimental work on this problem

In this paper we address the problem of constructing a full quantum mechan-ical model of the H+

3 near-dissociation spectrum This is a formidable problemthat requires consideration of many aspects of the problem the electronic poten-tial energy surface highly excited vibrational motion rotational excitation lifetimesof the quasi-bound states transition moments between the states and other more-detailed considerations such as the validity of the BornOppenheimer approximation

and modelling the initial conditions of the experiment In the following sections weconsider each of these aspects in turn and report on the latest progress in each case

2 Potential energy surfaces

H+

3is a two-electron system and is therefore amenable to very-high-accuracy treat-

ment of its electronic structure Indeed for spectroscopic studies ab initio electronicstructure calculations have been performed which are accurate in the absolute energyto better than 005 cmiexcl1 (Cencek et al 1998) This has led to ab initio spectroscopic

studies that reproduce experiment to a similar level of accuracy (Polyansky amp Ten-nyson 1999)

The situation is very dinoterent at energies near the H+

3dissociation limit All the

(semi-)classical studies and until very recently all the quantal studies were con-ducted using H+

3potential energy surfaces which cannot be considered reliable let

Phil Trans R Soc Lond A (2000)



H+

3 near dissociation theory 2421

alone accurate The two most popular potentials the diatomics-in-molecule (DIM)surface of Preston amp Tully (1971) and the adjusted ab initio MeyerBotschwinaBurton (MBB) surface (Meyer et al 1986) both sunotered from the same defectwhich would appear to be crucial to modelling near-dissociation behaviour Neithersurface accurately models the potential energy of the system as it dissociates Indeed

the MBB potential used for nearly all quantal studies was only designed for spec-troscopic studies and can only be considered reliable below halfway to dissociationThe long-range near-dissociation behaviour of H+

3is actually relatively easy to

model as the leading terms can be obtained accurately using perturbation theory(Giese amp Gentry 1974) Indeed Schinke et al (1980) produced a surface that isglobally correct using this long-range behaviour However this surface sunotered fromproblems with symmetry and joins between dinoterent regions of the surface

Recently we (Prosmiti et al 1997 Polyansky et al 2000) have attempted to rectifythis problem using two dinoterent approaches In both cases two coupled surfaces wereactually constructed to allow for the crossing of the H+ + H2 and H + H+

2dissociation

channels at large diatomic bond lengthsOur earlier work (Prosmiti et al 1997) although it used some ab initio data from

Schinke et al (1980) was largely constructed from experimental data It used theknown long-range behaviour and spectroscopically determined portions of the surface(Dinelli et al 1995) to create a reliable global surface for the H+

3 system Thissurface proved relatively easy to construct and calculations using it have alreadybeen reported (Prosmiti et al 1998) However the lack of assigned spectroscopicdata on H+

3for anything except the low-energy region means that the potential is

not strongly constrained

More recently we (Polyansky et al 2000) determined the global H+

3 surface entirelyfrom ab initio data For this we used the ultra-high-accuracy but limited data of Cencek et al (1998) augmented by further calculations of our own giving about200 high-accuracy ab initio electronic energies At high energies the surface wasconstrained using the data of Schinke et al (1980) It proved surprisingly dimacrcultto obtain a satisfactory t to all the ab initio data to 5 cmiexcl1 accuracy leading usto present two alternative ts Results presented below used t 2 which removedunphysical features from the potential at the price of giving a slightly poorer tto the ab initio data The dimacrculty of tting the ab initio data to its intrinsicaccuracy suggests that the true surface displays really quite subtle behaviour which

will require further development of potential functions and a denser grid of ab initiopoints to model accurately

It should be noted that Aguado et al (2000) have also recently presented a globalab initio surface for H+

3 These workers performed ab initio calculations at a verylarge number of geometries 8469 which they tted to the DIM form of Preston ampTully (1971) with an accuracy of ca 20 cmiexcl1 Calculating many points denes thewhole surface but at the cost of using lower-accuracy ab initio calculations Perhapsmore disappointingly Aguado et al (2000) did not ensure that their surface has thecorrect long-range behaviour at the near-dissociation limit

3 Highly excited vibrational states

Following early but limited attempts to perform full quantum mechanical nuclearmotion calculations on the H+

3near-dissociation problem (Tennyson amp Sutclinote 1984





Pfeinoter amp Child 1987 Gomez Llorente et al 1988 Tennyson amp Henderson 1989 Ten-nyson et al 1990) H+

3 has become something of a benchmark system for studies of highly excited vibrational states The rst calculations that successfully obtained allthe bound vibrational states of a chemically bound polyatomic molecule were per-formed on H+

3 (Henderson amp Tennyson 1990 Henderson et al 1993) Since then a

number of other workers have addressed this problem using methods of increasingaccuracy The work of Bramley et al (1994) and Mandelshtam amp Taylor (1997) is par-ticularly notable However all these studies were performed on the non-dissociatingMBB surface

A common feature of our recent potentials when compared with the MBB surfaceis the much more attractive nature of the surface in the H2 + H+ dissociation channelIt is more dimacrcult to compute all the bound vibrational states of these new more-realistic surfaces This is because these surfaces support signicantly more boundstates and many of the new states cover a much greater range in the dissociativecoordinate generally denoted R

To help address these problems we have been performing calculations on the highlyexcited vibrational states of H+

3 using the PDVR3D program (Mussa et al 1998Mussa amp Tennyson 2000) This program is an implementation of the DVR3D pro-gram (Tennyson et al 1995) for performing calculations on highly excited states of triatomic species The present calculations were performed on the 576-node Cray-T3E system located at the University of Manchester Computer Centre as part of theChemReact computing consortium

Besides the sheer size of a calculation designed to compute 103 or more statesof a system there are a number of severe technical problems specic to calculating

high-lying states of H+

3 The rst of these involves the correct inclusion of symmetryCoordinates that can easily be fully symmetrized such as hyperspherical coordinates(Wolniewicz amp Hinze 1994) and interparticle coordinates (Watson 1994) have beenused successfully for high-accuracy studies of low-lying states However there are dif-culties with employing any of these at high energy Instead all the near-dissociationcalculations so far have used scattering (or Jacobi) coordinates which are expressedin terms of HH distance r the H+ H2 centre of mass distance R and the anglebetween r and R In these coordinates the only symmetry for the rotationlessJ = 0 problem is given by whether the basis has even or odd parity about = ordm=2(Tennyson amp Sutclinote 1984)

The second problem which is more severe involves the barrier to linearity H+3

can probe linear geometries at energies of approximately one-third the dissociationenergy In Jacobi coordinates these geometries are handled correctly at = 0 or ordm butwith dimacrculty for R = 0 (Henderson et al 1993) One result of this is that a numberof calculations in Jacobi coordinates (Bamicrocic amp Zhang 1992 Tennyson 1993 Hendersonet al 1993) show dinoterential convergence between odd and even symmetries This canbe at least partly cured by using symmetry-dependent basis functions in the R coor-dinate (Henderson et al 1993 Mandelshtam amp Taylor 1997 Prosmiti et al 1998)

However as discussed below calculations on rotationally excited states involve

mixing between calculations with dinoterent `krsquo blocks where k is the projection of the rotational angular momentum onto the body-xed z-axis For the best z-axisembedding dinoterent k blocks employ `evenrsquo and `oddrsquo R basis sets Under thesecircumstances both dinoterential-convergence and k-block-dependent radial basis setsare undesirable




H+


For these reasons we are exploring the use of an alternative coordinate systemfor modelling the H+

3 system Radau coordinates (Smith 1980) are like Jacobi coor-dinates orthogonal and therefore simple to use in discrete variable representation(DVR) calculations They have been used very successfully for a number of chal-lenging calculations on rotationally excited water (Polyansky et al 1997 Mussa amp

Tennyson 1998) Radau coordinates do not give a particularly good physical modelof the H+

3system but numerical studies have demonstrated (Bramley amp Carrington

1994) that Jacobi coordinates do not either Radau coordinates have the importantadvantages that linear geometries can be modelled without any of the radial coordi-nates going to zero and the same functions can be used for even and odd symmetries

Table 1 presents both even and odd J = 0 results for the MBB potential Com-parison is made with the highly converged calculations of Bramley et al (1994)Calculations are presented for both Radau and Jacobi coordinates using the pro-gram PDVR3DRJ (Mussa amp Tennyson 2000) For even symmetry a similar level of convergence is found in both coordinates for a nal Hamiltonian size of N = 8500

However for odd symmetry there are signicant dinoterences between the Radau andJacobi calculations with the Radau calculations again giving reasonable agreementwith those of Bramley et al (1994)

The odd-symmetry Jacobi calculations give energies that are too low and aretherefore non-variational This problem was extensively analysed by Henderson et

al (1993) and attributed to the use of the quadrature approximation standard inDVR approaches (Bamicrocic amp Light 1989) As R 0 integrals over the moment of inertia which behaves as Riexcl2 are not correctly evaluated with this approximationHenderson et al (1993) were able to solve this problem by transforming between

basis set and grid representations but only at considerable computation expenseTheir method would be very hard to implement emacrciently on a parallel computerIt should be noted that N = 8500 is not sumacrcient to fully converge our calculations

and that larger calculations give close agreement with the results of Bramley et al (1994) for the MBB potential However it is of more interest to converge the resultson our more recent global potentials Table 2 presents preliminary results for the ab

initio t 2 of Polyansky et al (2000) These results show uniform convergence forboth even and odd symmetry with the N = 11 000 results converged to ca 1 cmiexcl1As was anticipated from their shape and because of their slightly higher dissociationenergy the new potentials support nearly 50 more vibrational states than the MBB

potential Full results of these calculations will be reported elsewhere

4 Rotational excitation

While nearly all quantal studies of H+

3near dissociation have been performed for J =

0 rotational excitation is actually an essential ingredient of the observed spectrumIt has been established both from the experimental data and from early calculationsthat the states observed are trapped behind rotational barriers or in the standardterminology of scattering theory are shape resonances It is therefore essential to

consider rotational excitation as part of a proper analysis of the observed near-dissociation experiments Indeed it is impossible to correctly model the lifetimeenotects discussed in the next section without considering rotational excitation

There have only been two fully quantal attempts to compute rotationally excitedstates of H+

3in the near-dissociation region Miller amp Tennyson (1988) looked for the





Table 1 H+3 band origins for the MBB potential

(Vibrational band origins in cmiexcl1 for H+3 calculated in Radau coordinates (E R ) using the MBB

potential (Meyer et al 1986) and a macrnal Hamiltonian of dimension N = 8500 Diregerences witha Jacobi (E J ) coordinate calculation and the results of Bramley et al (1994) (E B ) are given forcomparison)

even parity odd parity

state E R E R iexcl E J E R iexcl E B state E R E R iexcl E J E J iexcl E B

530 32 69839 013 180 442 32 38975 149 iexcl183

531 32 71431 iexcl037 078 443 32 39243 114 070

532 32 71498 iexcl013 086 444 32 41284 589 064

533 32 74822 iexcl040 082 445 32 42508 554 036

534 32 77792 176 240 446 32 43630 898 iexcl035

535 32 80384 049 134 447 32 43857 093 iexcl027

536 32 82348 097 144 448 32 51002 547 116

537 32 83391 095 180 449 32 52510 787 017

538 32 84000 024 111 450 32 54236 1090 087

539 32 84906 iexcl023 049 451 32 54840 173 iexcl015

540 32 86605 iexcl111 iexcl010 452 32 58723 167 018

541 32 89986 iexcl069 021 453 32 61868 428 014

542 32 90144 007 093 454 32 62304 451 012

543 32 90608 iexcl036 016 455 32 64322 162 iexcl111

544 32 91933 iexcl024 138 456 32 64935 374 iexcl013

545 32 94388 040 121 457 32 67480 776 051546 32 96882 044 139 458 32 71507 389 095

547 32 98450 iexcl004 097 459 32 74869 396 129

548 32 98712 028 105 460 32 76947 753 iexcl021

549 32 99481 iexcl176 iexcl096 461 32 77596 105 044

550 33 01758 iexcl028 iexcl035 462 32 80303 262 053

551 33 04358 iexcl069 065 463 32 82009 097 iexcl195

552 33 05401 080 027 464 32 83102 089 iexcl109

553 33 06476 017 173 465 32 84732 723 iexcl125

554 33 08569 iexcl109 131 466 32 86643 451 028

555 33 11870 iexcl004 123 467 32 87500 618 026556 33 13263 iexcl017 068 468 32 89262 1245 032

557 33 15478 044 193 469 32 90219 730 168

558 33 15748 iexcl044 080 470 32 91531 384 038

559 33 18987 iexcl103 023 471 32 91923 400 128

560 33 19961 iexcl018 057 472 32 94322 589 054

561 33 20574 033 179 473 32 96770 348 027

562 33 22069 087 192 474 32 98630 389 023

563 33 23142 iexcl096 070 475 32 99671 466 094

564 33 28287 046 134 476 33 04325 303 032

565 33 29292 iexcl051 170 477 33 05408 238 113566 33 30123 260 544 478 33 05455 215 081

567 33 30494 iexcl211 iexcl239 479 33 08587 775 149

568 33 31423 iexcl012 192 480 33 11772 295 025




H+


Table 1 (Cont)



569 33 32570 iexcl126 033 481 33 13229 424 034570 33 33245 337 464 482 33 13985 1031 052

571 33 34847 iexcl100 034 483 33 15659 352 iexcl009

572 33 35475 iexcl190 iexcl022 484 33 17898 142 077

573 33 39654 iexcl075 028 485 33 18997 266 033

574 33 40144 iexcl055 018 486 33 20442 274 047

575 33 40327 076 188 487 33 21936 289 060

576 33 42295 017 090 488 33 22824 397 iexcl023

577 33 43461 iexcl019 065 489 33 23198 580 126

highest rotational state that was still bound Their results are in good agreementwith a recent semi-classical study by Kozin et al (1999) which used the concept of relative equilibria to map out the behaviour of H+

3as the molecule is rotationally

excited This work has recently been extended to H2D+ and D2H+ (Kozin et al 2000)

The other quantal near-dissociation study was performed by Henderson amp Ten-nyson (1996) who computed states with J = 0 1 and 2 up to dissociation Thesecalculations were limited by the available computer resources some of their calcu-

lations took more than two weeksrsquo computer time and even then were not highlyconvergedRecent studies of rotationally excited states of water in Radau coordinates by

Mussa amp Tennyson (1998) were able to converge all J = 2 up to dissociation inca 1 h using 64 processors on a Cray-T3E As water is both heavier and has a higherdissociation energy than H+

3 there is no reason why Mussa amp Tennysonrsquos method

should not work well for H+

3 Indeed Mussa amp Tennyson (1998) performed calcula-

tions for water with J = 10 J 610 should be sumacrcient to cover the critical portionsof the low kinetic energy release spectrum (Gomez Llorente amp Pollak 1989)

At high kinetic release it seems likely (Carrington et al 1993) that the sparser and

stronger photodissociation spectrum is actually sampling states of much higher rota-tional excitation Experience with water (Viti 1997) and the early H+

3calculations

of Miller amp Tennyson (1988) have shown that because there are fewer bound statesfor high J values such calculations are actually easier However present methods(Mussa amp Tennyson 2000) which scale linearly with J for low J would probablyneed to be adapted for this situation

5 Lifetime eregects

One problem with any theoretical attempt to model the H+

3 spectrum of Carringtonand co-workers is that the experiment is only sensitive to states that fall inside certainlifetime windows Particularly the lower states some or all of which are quasi-bound(Carrington amp Kennedy 1984) must live long enough to enter the portion of theapparatus with the photodissociating laser Conversely the upper states must be





Table 2 Convergence of H+3 band origins

(H+3 vibrational band origins in cmiexcl1 as a function of the size of the macrnal Hamiltonian matrix

N Calculation in Radau coordinates for macrt 2 of the ab initio potential of Polyansky et al (2000)For N lt 11 000 diregerences to the N = 11 000 results are given)


state N = 11 000 10 000 8500 state N = 11 000 10 000 8500

753 35 84993 iexcl084 iexcl205 660 35 84243 iexcl044 iexcl149






759 35 92610 iexcl097 iexcl246 666 35 91811 iexcl043 iexcl130760 35 93888 iexcl080 iexcl182 667 35 95549 iexcl074 iexcl198

































H+


Table 2 (Cont)


state N = 11 000 10 000 8500 state N = 11 000 10 000 8500









short enough lived to decay before the excited H+

3ions leave the apparatus altogether

In addition for a few transitions Carrington et al (1993) were able to determinelifetimes from measurements of linewidths although in most cases the states weretoo long lived for this to be possible

Mandelshtam amp Taylor (1997) have computed positions and widths for resonancesin the H+

3system However these calculations were only performed for vibrationally

excited states with J = 0 It is well established (Pollak amp Schlier 1989 Drolshagenet al 1989) that such Feshbach resonances are much too broad ie decay much too

quickly to be important for the observed near-dissociation spectrumThere are essentially no calculations on quasi-bound rotationally excited statesfor chemically bound systems An exception are the studies by Skokov amp Bowman(1999a b) on the simpler HOCl system However even these calculations employed anapproximation which neglects all coupling between k-blocks ie all the onot-diagonalCoriolis interactions This decoupling approximation is valid for HOCl but not usefulfor H+

3

We have adapted the PDVR3D program suite to study resonances using proce-dures similar to those adopted by both Mandelshtam amp Taylor (1997) and Skokovamp Bowman (1999a b) Initial results as reported by Mussa amp Tennyson (2000) are

only for J = 0 but we have recently generalized this method to deal with fully Cori-olis coupled rotationally excited states The new procedure will be applied to H+

3in

the near future

6 Dipole transitions

To model any spectrum successfully it is necessary to consider transition moments aswell as energy levels Indeed the near-dissociation spectrum of H+

3shows pronounced

variation in line intensities both through the structure in the coarse-grained spectrum

at low kinetic energy release (Carrington amp Kennedy 1984) and the stronger lineswith high kinetic energy release (Carrington et al 1993)

The quantum manifestation of the classical chaos found for highly excited H+

3

might be expected to display itself in spectra that show little or no structure How-ever the vibrational band strength calculations of Le Sueur et al (1993) suggest that





transitions in the near-dissociation region may well show strong propensity rules dueto transition-moment enotects The only attempt to compute actual dipole transitionstrengths at the frequencies covered by Carrington et al rsquos CO2 laser was by Hender-son amp Tennyson (1996) This necessarily simplied calculation could not reproducethe observed coarse-grained structure observed in the H+

3 photodissociation spectra

Henderson amp Tennyson (1996) concluded that it is necessary to combine lifetime andtransition-moment enotects in any attempt to model the near-dissociation spectrumof H+

3

7 Other considerations

The sections above discuss a series of issues that have to be addressed in any attemptto model the H+

3 spectra of Carrington and co-workers However even if all the abovesteps are accomplished one would still not expect perfect agreement between theoryand experiment Before this can be achieved at least two other aspects of the problem

have to be consideredAll the discussion above has been presented in terms of the BornOppenheimerseparation between nuclear and electronic motion H+

3is a light molecule for which

the BornOppenheimer approximation is only known to be valid to an accuracyof ca 1 cmiexcl1 for the well-studied low-energy high-resolution spectra of H+

3and

its isotopomers Any attempt to give a line-by-line interpretation of the H+

3 near-dissociation spectrum will therefore inevitably have to address the BornOppen-heimer problem Although the possibility of doing this is still some way onot it isencouraging to note that Polyansky amp Tennyson (1999) were able to perform ab initionon-BornOppenheimer calculations that reproduced the high-resolution spectrum

of H+3 to within a few hundredths of a wavenumber nearly two orders of magnitude

better than the best possible BornOppenheimer calculation Furthermore it wouldnot be dimacrcult to implement Polyansky amp Tennysonrsquos (1999) procedure in a near-dissociation calculation

A second consideration is experimental As in all laboratory spectroscopic exper-iments on H+

3 the ions are produced in a hydrogen discharge This is an emacrcient

method of creating H+

3 and one which by cooling the discharge can tune the level of

H+

3excitation produced However the resulting population of H+

3vibration and rota-

tion states is not well characterized Indeed any thermal or quasi-thermal population

would be unlikely to yield the near-dissociation states accessed in the experimentsdiscussed here Clearly any thorough theoretical model of these experiments willneed to make some assumptions about the initial population

8 Conclusion

It is now a considerable time since Carrington et al (1982) reported the detectionof a very detailed near-dissociation spectrum of H+

3 This work and its subsequentrenements (Carrington amp Kennedy 1984 Carrington et al 1993) have presenteda considerable challenge that theory has yet to fully meet However advances in

computer technology matched with algorithmic developments suggest that the rstfully quantal models of the spectrum if not the rst quantum number assignmentsshould not be very far onot

There is one aspect of the H+

3 problem that merits comment There has been con-siderable work both experimental and theoretical on the high-resolution spectrum




H+

3near dissociation theory 2429

of H+

3 and its isotopomers using standard spectroscopic techniques (see contributionsby McCall and by Watson in this issue) The H+

3system is well characterized at ener-

gies extending maybe as far as 10 000 cmiexcl1 above its ground state There is thena gap of some 25 000 cmiexcl1 to the region about dissociation probed by Carringtonand co-workers As yet there is no experimental information on this region This isdespite theoretical predictions of relatively high-intensity spectra with a pronounced

structure (Le Sueur et al 1993) and the availability of high-accuracy calculations(Cencek et al 1998 Polyansky amp Tennyson 1999) that could be extended to givegood frequency predictions for a possible experiment There is no doubt that exper-imental information on this intermediate-energy regime would be of great help tothose trying to perform reliable calculations for energies near dissociation

This work has been supported by the UK Engineering and Physical Sciences Research Council

via the ChemReact Computing Consortium and other grants The work of OLP was partly

supported by the Russian Fund for Fundamental Studies RP acknowledges a TMR Fellowship

under contract ERBFMBICT 960901

References

Aguado A Roncero O Tablero C Sanz C amp Paniagua M 2000 Global potential energy

surfaces for the H+3 system Analytic representation of the adiabatic ground state 11

A0 poten-

tial J Chem Phys 112 12401254

Bamiddotciparac Z amp Light J C 1989 Theoretical methods for the ro-vibrational states of degoppy

molecules A Rev Phys Chem 40 469498

Bamiddotciparac Z amp Zhang J Z H 1992 High-lying rovibrational states of degoppy X3 triatomics by a

new D3 h symmetry adapted method application to the H+3 molecule J Chem Phys 96

37073713

Bramley M J amp Carrington Jr T 1994 Calculation of triatomic vibrational eigenstates product

or contracted basis sets Lanczos or conventional eigensolvers What is the most eplusmncient

combination J Chem Phys 101 84948507

Bramley M J Tromp J W Carrington Jr T amp Corey G C 1994 Eplusmncient calculation of

highly excited vibrational energy levels of degoppy molecules the band origins of H+3 up to

35 000 cmiexcl1 J Chem Phys 100 61756194

Carrington A amp Kennedy R A 1984 Infrared predissociation spectrum of the H+3 ion J

Chem Phys 81 91112

Carrington A amp McNab I R 1989 The infrared predissociation spectrum of H+3 Acc Chem

Res 22 218222

Carrington A Buttenshaw J amp Kennedy R A 1982 Observation of the infrared spectrum of

H+3 ion at its dissociation limit Mol Phys 45 753758

Carrington A McNab I R amp West Y D 1993 Infrared predissociation spectrum of the H+3

ion II J Chem Phys 98 10731092

Cencek W Rychlewski J Jaquet R amp Kutzelnigg W 1998 Sub-micro-Hartree accuracy

potential energy surface for H+3 including adiabatic and relativistic eregects I Calculation of

the potential points J Chem Phys 108 28312836

Chambers A V amp Child M S 1988 Barrier eregects on the vibrational predissociation of HD+2

Mol Phys 65 13371344

Cosby P C amp Helm H 1988 Experimental determination of the H+3 bond dissociation energy

Chem Phys Lett 152 7174

Dinelli B M Polyansky O L amp Tennyson J 1995 Spectroscopically determined Born

Oppenheimer and adiabatic surfaces for H+3 H2 D

+ D2 H

+and D

+3 J Chem Phys 103

10 43310 438





Drolshagen G Gianturco F A amp Toennies J P 1989 Vibrational breakup of triatomic colli-sion complexes a model study of H +

iexcl H2 Israel J Chem 29 417425

Giese C F amp Gentry W R 1974 Classical trajectory treatment of inelastic scattering in thecollisions of H+ with H2 HD and D2 Phys Rev A 10 21562173

Gomez Llorente J M amp Pollak E 1987 Order out of chaos in the H +3 molecule Chem Phys

Lett 138 125130

Gomez Llorente J M amp Pollak E 1989 A classical trajectory study of the photoionisationspectrum of H+

3 J Chem Phys 90 54065419

Gomez Llorente J M Zakrzewski J Taylor H S amp Kulander K C 1988 Spectra in thechaotic region a quantum analysis of the photodissociation of H+

3 J Chem Phys 89 59595960

Henderson J R amp Tennyson J 1990 All the vibrational bound states of H +3 Chem Phys Lett

173 133138

Henderson J R amp Tennyson J 1996 Calculated spectrum for near-dissociation H+3 a macrrst

attempt Mol Phys 89 953963

Henderson J R Tennyson J amp Sutclirege B T 1993 All the bound vibrational states of H

+

3 a reappraisal J Chem Phys 98 71917203

Kozin I N Roberts R M amp Tennyson J 1999 Symmetry and structure of H +3 J Chem

Phys 111 140150

Kozin I N Roberts R M amp Tennyson J 2000 Relative equilibria of D 2 H+ and H2 D+ Mol

Phys 98 295307

Le Sueur C R Henderson J R amp Tennyson J 1993 Gateway states and bath states in thevibrational spectrum of H+

3 Chem Phys Lett 206 429436

Lie G C amp Frye D 1992 Vibrational analysis of a Hylleraas-conmacrguration interaction potentialfor H+

3 J Chem Phys 96 67846790

McNab I R 1994 The spectroscopy of H+

3 Adv Chem Phys 89 187Mandelshtam V A amp Taylor H S 1997 The quantum resonance spectrum of the H+

3 molecularion for J = 0 An accurate calculation using macrlter diagonalisation J Chem Soc Faraday

Trans 93 847860

Meyer W Botschwina P amp Burton P G 1986 Ab initio calculation of near-equilibrium poten-tial and multipole moment surfaces and vibrational frequencies of H +

3 and its isotopomers J

Chem Phys 84 891900

Miller S amp Tennyson J 1988 Calculation of the high angular momentum dissociation limit forH

+3 and H2 D+ Chem Phys Lett 145 117120

Mussa H Y amp Tennyson J 1998 Calculation of rotationvibration states of water at dissoci-

ation J Chem Phys 109 10 88510 892Mussa H Y amp Tennyson J 2000 Bound and quasi-bound rotationvibrational states using

massively parallel computers Computer Phys Commun (In the press)

Mussa H Y Tennyson J Noble C J amp Allan R J 1998 Rotationvibration calculationsusing massively parallel computers Computer Phys Commun 108 2937

Pfeireger R amp Child M S 1987 Towards an understanding of the predissociation excitationspectrum of H+

3 Mol Phys 60 13671378

Polavieja G G Fulton N G amp Tennyson J 1994 Coarse grained spectra dynamics andquantum phase space structures of H

+3 Mol Phys 53 361379

Polavieja G G Fulton N G amp Tennyson J 1996 Quantum transport recurrence and locali-

sation in H+3 Mol Phys 87 651667Pollak E amp Schlier C 1989 Theory of unimolecular dissociation of small metastable molecules

and ions as exemplimacred by H+3 Acc Chem Res 22 223229

Polyansky O L amp Tennyson J 1999 Ab initio calculation of the rotationvibration energylevels of H+

3 and its isotopomers to spectroscopic accuracy J Chem Phys 110 50565064




H+


Polyansky O L Zobov N F Viti S Tennyson J Bernath P F amp Wallace L 1997 Waterin the Sun line assignments based on variational calculations Science 277 346349

Polyansky O L Prosmiti R Klopper W amp Tennyson J 2000 An accurate global ab initio

potential energy surface for the H+3 molecule Mol Phys 98 261273

Preston R K amp Tully J C 1971 Eregects of surface coupling on the H +3 system J Chem Phys

54 42974304

Prosmiti R Polyansky O L amp Tennyson J 1997 A global potential energy surface for H +3


Prosmiti R Mussa H Y amp Tennyson J 1998 Quantum states of triatomic molecules atdissociation In Molecular quantum states at dissociation (ed R Prosmiti J Tennyson ampD C Clary) pp 1216 Daresbury CCP6

Schinke R Dupius M amp Lester Jr W A 1980 Proton-H 2 scattering on an ab initio CIpotential energy surface I Vibrational excitation at 10 eV J Chem Phys 72 39093915

Skokov S amp Bowman J M 1999a Variation of the resonance widths of HOCl(6cedil O H ) with totalangular momentum comparison between ab initio theory and experiment J Chem Phys

110 97899792Skokov S amp Bowman J M 1999b Complex L2 calculation of the variation of resonance widths

of HOCl with total angular momentum J Chem Phys 111 49334941

Smith F T 1980 Modimacred heliocentric coordinates for particle dynamics Phys Rev Lett 4511571160

Tennyson J 1993 Rotational excitation with pointwise vibrational wavefunctions J Chem

Phys 98 96589668

Tennyson J amp Henderson J R 1989 Highly excites states using a discrete variable represen-tation the H

+3 molecular ion J Chem Phys 91 38153825

Tennyson J amp Sutclirege B T 1984 On the rovibrational levels of the H+3 and H2 D+ molecules

Mol Phys 51 887906Tennyson J Brass O amp Pollak E 1990 Quantum mechanics of highly excited states of the

H+3 molecular ion a numerical study of the two degree of freedom C 2 v subspace J Chem

Phys 92 30053017

Tennyson J Henderson J R amp Fulton N G 1995 DVR3D programs for fully pointwisecalculation of ro-vibrational spectra of triatomic molecules Computer Phys Commun 86175198

Viti S 1997 Infrared spectra of cool stars and sunspots PhD thesis University of London

Watson J K G 1994 Vibrationrotation calculations for H+3 using a Morse-based discrete

variable representation Can J Phys 72 238249

Wolniewicz L amp Hinze J 1994 Rotationvibrational states of H+3 computed using hyperspher-ical coordinates and harmonics J Chem Phys 101 98179829

Discussion

B Sutcliffe (Universitparae de Bruxelles Libre Belgium ) It is well known that it isextremely dimacrcult to get potential energy surfaces without `holesrsquo in them particu-larly in the asymptotic regions Are you condent that your surface for H+

3 is withoutholes

J Tennyson It is indeed dimacrcult to construct potential energy surfaces free fromspurious features of which deep unphysical minima `holesrsquo is the worst exampleIn the particular case of our H+

3 potentials (Prosmiti et al 1997 Polyansky et al 2000) we have taken particular measures to stop this happening in the asymptoticregion by using the known behaviour of the potential when one atom or ion is well





separated from the rest of the molecule We believe this method is good for dealingwith the asymptotes but is not a cure for all problems with unphysical behaviourIn constructing our most recent and most accurate H+

3potential (Polyansky et al

2000) we had considerable dimacrculty with what appears to be an unphysical shoulderin the potential at certain near-linear geometries





Kennedy 1984) However Carrington et al (1993) observed a very large number of well-dened transitions in the limited range of their laser 8721094 cmiexcl1 As thedissociation energy of H+

3 is ca 35 000 cmiexcl1 (Cosby amp Helm 1988 Lie amp Frye 1992)this experiment monitors only those levels of the ion that lie very close to the disso-ciation limit Prior to this experiment the structure of the rotationvibration energy

levels at dissociation for chemically bound molecules larger than diatomic was com-pletely unknown Indeed H+

3and its isotopomers remains the only strongly bound

polyatomic that has been studied in this fashionThe spectrum of Carrington et al (1993) was unexpected and truly remarkable It

has stimulated a whole host of theoretical studies both (semi-)classical and quantalOf course H+

3 is a highly quantal system indeed the very existence of the spec-trum depending as it does on tunnelling means that classical mechanics can onlyhave limited application However classical studies proved successful in interpretingthe observed behaviour upon isotopic substitution (Gomez Llorente amp Pollak 1987Chambers amp Child 1988)

One feature of the spectrum has provoked particular interest Carrington amp Ken-nedy (1984) convolved their high-resolution spectrum to show that at low resolutionit showed coarse-grained structure four peaks separated by ca 50 cmiexcl1 Classicalcalculations show that H+

3 is highly chaotic in the near-dissociation region Therehave been several classical trajectory studies that tried to identify the regular motionembedded in the chaos that could be responsible for this coarse-grained structureGomez Llorente amp Pollak (1989) identied the `horseshoersquo motion where one Hnucleus passes between the other two as the key to this structure However sub-sequent calculations (Polavieja et al 1994 1996) have suggested at least one other

possible candidateClassical work on the H+

3near-dissociation spectrum has been reviewed by Pollak

amp Schlier (1989) Carrington amp McNab (1989) McNab (1994) and Kemp et al (thisissue) have reviewed the experimental work on this problem

In this paper we address the problem of constructing a full quantum mechan-ical model of the H+

3 near-dissociation spectrum This is a formidable problemthat requires consideration of many aspects of the problem the electronic poten-tial energy surface highly excited vibrational motion rotational excitation lifetimesof the quasi-bound states transition moments between the states and other more-detailed considerations such as the validity of the BornOppenheimer approximation

and modelling the initial conditions of the experiment In the following sections weconsider each of these aspects in turn and report on the latest progress in each case

2 Potential energy surfaces

H+

3is a two-electron system and is therefore amenable to very-high-accuracy treat-

ment of its electronic structure Indeed for spectroscopic studies ab initio electronicstructure calculations have been performed which are accurate in the absolute energyto better than 005 cmiexcl1 (Cencek et al 1998) This has led to ab initio spectroscopic

studies that reproduce experiment to a similar level of accuracy (Polyansky amp Ten-nyson 1999)

The situation is very dinoterent at energies near the H+

3dissociation limit All the

(semi-)classical studies and until very recently all the quantal studies were con-ducted using H+

3potential energy surfaces which cannot be considered reliable let




H+







2dissociation



































H+































530 32 69839 013 180 442 32 38975 149 iexcl183

531 32 71431 iexcl037 078 443 32 39243 114 070

532 32 71498 iexcl013 086 444 32 41284 589 064

533 32 74822 iexcl040 082 445 32 42508 554 036

534 32 77792 176 240 446 32 43630 898 iexcl035

535 32 80384 049 134 447 32 43857 093 iexcl027

536 32 82348 097 144 448 32 51002 547 116

537 32 83391 095 180 449 32 52510 787 017

538 32 84000 024 111 450 32 54236 1090 087

539 32 84906 iexcl023 049 451 32 54840 173 iexcl015

540 32 86605 iexcl111 iexcl010 452 32 58723 167 018

541 32 89986 iexcl069 021 453 32 61868 428 014

542 32 90144 007 093 454 32 62304 451 012

543 32 90608 iexcl036 016 455 32 64322 162 iexcl111

544 32 91933 iexcl024 138 456 32 64935 374 iexcl013

545 32 94388 040 121 457 32 67480 776 051546 32 96882 044 139 458 32 71507 389 095

547 32 98450 iexcl004 097 459 32 74869 396 129

548 32 98712 028 105 460 32 76947 753 iexcl021

549 32 99481 iexcl176 iexcl096 461 32 77596 105 044

550 33 01758 iexcl028 iexcl035 462 32 80303 262 053

551 33 04358 iexcl069 065 463 32 82009 097 iexcl195

552 33 05401 080 027 464 32 83102 089 iexcl109

553 33 06476 017 173 465 32 84732 723 iexcl125

554 33 08569 iexcl109 131 466 32 86643 451 028

555 33 11870 iexcl004 123 467 32 87500 618 026556 33 13263 iexcl017 068 468 32 89262 1245 032

557 33 15478 044 193 469 32 90219 730 168

558 33 15748 iexcl044 080 470 32 91531 384 038

559 33 18987 iexcl103 023 471 32 91923 400 128

560 33 19961 iexcl018 057 472 32 94322 589 054

561 33 20574 033 179 473 32 96770 348 027

562 33 22069 087 192 474 32 98630 389 023

563 33 23142 iexcl096 070 475 32 99671 466 094

564 33 28287 046 134 476 33 04325 303 032

565 33 29292 iexcl051 170 477 33 05408 238 113566 33 30123 260 544 478 33 05455 215 081

567 33 30494 iexcl211 iexcl239 479 33 08587 775 149

568 33 31423 iexcl012 192 480 33 11772 295 025




H+


Table 1 (Cont)



569 33 32570 iexcl126 033 481 33 13229 424 034570 33 33245 337 464 482 33 13985 1031 052

571 33 34847 iexcl100 034 483 33 15659 352 iexcl009

572 33 35475 iexcl190 iexcl022 484 33 17898 142 077

573 33 39654 iexcl075 028 485 33 18997 266 033

574 33 40144 iexcl055 018 486 33 20442 274 047

575 33 40327 076 188 487 33 21936 289 060

576 33 42295 017 090 488 33 22824 397 iexcl023

577 33 43461 iexcl019 065 489 33 23198 580 126













3calculations


5 Lifetime eregects











state N = 11 000 10 000 8500 state N = 11 000 10 000 8500








































H+


Table 2 (Cont)


state N = 11 000 10 000 8500 state N = 11 000 10 000 8500
















3



3in

the near future



3shows pronounced




3









3







3and









H+





8 Conclusion









H+


of H+









References



A0 poten-






37073713








Chem Phys 81 91112


Res 22 218222









Mol Phys 65 13371344





+ D2 H

+and D

+3 J Chem Phys 103

10 43310 438









Lett 138 125130


3 J Chem Phys 90 54065419


3 J Chem Phys 89 59595960


173 133138




+



Phys 111 140150


Phys 98 295307




3 J Chem Phys 96 67846790




Trans 93 847860



Chem Phys 84 891900








3 Mol Phys 60 13671378


+3 Mol Phys 53 361379









H+






54 42974304










Phys 98 96589668






Phys 92 30053017






Discussion


3 is withoutholes













H+







2dissociation



































H+































530 32 69839 013 180 442 32 38975 149 iexcl183

531 32 71431 iexcl037 078 443 32 39243 114 070

532 32 71498 iexcl013 086 444 32 41284 589 064

533 32 74822 iexcl040 082 445 32 42508 554 036

534 32 77792 176 240 446 32 43630 898 iexcl035

535 32 80384 049 134 447 32 43857 093 iexcl027

536 32 82348 097 144 448 32 51002 547 116

537 32 83391 095 180 449 32 52510 787 017

538 32 84000 024 111 450 32 54236 1090 087

539 32 84906 iexcl023 049 451 32 54840 173 iexcl015

540 32 86605 iexcl111 iexcl010 452 32 58723 167 018

541 32 89986 iexcl069 021 453 32 61868 428 014

542 32 90144 007 093 454 32 62304 451 012

543 32 90608 iexcl036 016 455 32 64322 162 iexcl111

544 32 91933 iexcl024 138 456 32 64935 374 iexcl013

545 32 94388 040 121 457 32 67480 776 051546 32 96882 044 139 458 32 71507 389 095

547 32 98450 iexcl004 097 459 32 74869 396 129

548 32 98712 028 105 460 32 76947 753 iexcl021

549 32 99481 iexcl176 iexcl096 461 32 77596 105 044

550 33 01758 iexcl028 iexcl035 462 32 80303 262 053

551 33 04358 iexcl069 065 463 32 82009 097 iexcl195

552 33 05401 080 027 464 32 83102 089 iexcl109

553 33 06476 017 173 465 32 84732 723 iexcl125

554 33 08569 iexcl109 131 466 32 86643 451 028

555 33 11870 iexcl004 123 467 32 87500 618 026556 33 13263 iexcl017 068 468 32 89262 1245 032

557 33 15478 044 193 469 32 90219 730 168

558 33 15748 iexcl044 080 470 32 91531 384 038

559 33 18987 iexcl103 023 471 32 91923 400 128

560 33 19961 iexcl018 057 472 32 94322 589 054

561 33 20574 033 179 473 32 96770 348 027

562 33 22069 087 192 474 32 98630 389 023

563 33 23142 iexcl096 070 475 32 99671 466 094

564 33 28287 046 134 476 33 04325 303 032

565 33 29292 iexcl051 170 477 33 05408 238 113566 33 30123 260 544 478 33 05455 215 081

567 33 30494 iexcl211 iexcl239 479 33 08587 775 149

568 33 31423 iexcl012 192 480 33 11772 295 025




H+


Table 1 (Cont)



569 33 32570 iexcl126 033 481 33 13229 424 034570 33 33245 337 464 482 33 13985 1031 052

571 33 34847 iexcl100 034 483 33 15659 352 iexcl009

572 33 35475 iexcl190 iexcl022 484 33 17898 142 077

573 33 39654 iexcl075 028 485 33 18997 266 033

574 33 40144 iexcl055 018 486 33 20442 274 047

575 33 40327 076 188 487 33 21936 289 060

576 33 42295 017 090 488 33 22824 397 iexcl023

577 33 43461 iexcl019 065 489 33 23198 580 126













3calculations


5 Lifetime eregects











state N = 11 000 10 000 8500 state N = 11 000 10 000 8500








































H+


Table 2 (Cont)


state N = 11 000 10 000 8500 state N = 11 000 10 000 8500
















3



3in

the near future



3shows pronounced




3









3







3and









H+





8 Conclusion









H+


of H+









References



A0 poten-






37073713








Chem Phys 81 91112


Res 22 218222









Mol Phys 65 13371344





+ D2 H

+and D

+3 J Chem Phys 103

10 43310 438









Lett 138 125130


3 J Chem Phys 90 54065419


3 J Chem Phys 89 59595960


173 133138




+



Phys 111 140150


Phys 98 295307




3 J Chem Phys 96 67846790




Trans 93 847860



Chem Phys 84 891900








3 Mol Phys 60 13671378


+3 Mol Phys 53 361379









H+






54 42974304










Phys 98 96589668






Phys 92 30053017






Discussion


3 is withoutholes































H+































530 32 69839 013 180 442 32 38975 149 iexcl183

531 32 71431 iexcl037 078 443 32 39243 114 070

532 32 71498 iexcl013 086 444 32 41284 589 064

533 32 74822 iexcl040 082 445 32 42508 554 036

534 32 77792 176 240 446 32 43630 898 iexcl035

535 32 80384 049 134 447 32 43857 093 iexcl027

536 32 82348 097 144 448 32 51002 547 116

537 32 83391 095 180 449 32 52510 787 017

538 32 84000 024 111 450 32 54236 1090 087

539 32 84906 iexcl023 049 451 32 54840 173 iexcl015

540 32 86605 iexcl111 iexcl010 452 32 58723 167 018

541 32 89986 iexcl069 021 453 32 61868 428 014

542 32 90144 007 093 454 32 62304 451 012

543 32 90608 iexcl036 016 455 32 64322 162 iexcl111

544 32 91933 iexcl024 138 456 32 64935 374 iexcl013

545 32 94388 040 121 457 32 67480 776 051546 32 96882 044 139 458 32 71507 389 095

547 32 98450 iexcl004 097 459 32 74869 396 129

548 32 98712 028 105 460 32 76947 753 iexcl021

549 32 99481 iexcl176 iexcl096 461 32 77596 105 044

550 33 01758 iexcl028 iexcl035 462 32 80303 262 053

551 33 04358 iexcl069 065 463 32 82009 097 iexcl195

552 33 05401 080 027 464 32 83102 089 iexcl109

553 33 06476 017 173 465 32 84732 723 iexcl125

554 33 08569 iexcl109 131 466 32 86643 451 028

555 33 11870 iexcl004 123 467 32 87500 618 026556 33 13263 iexcl017 068 468 32 89262 1245 032

557 33 15478 044 193 469 32 90219 730 168

558 33 15748 iexcl044 080 470 32 91531 384 038

559 33 18987 iexcl103 023 471 32 91923 400 128

560 33 19961 iexcl018 057 472 32 94322 589 054

561 33 20574 033 179 473 32 96770 348 027

562 33 22069 087 192 474 32 98630 389 023

563 33 23142 iexcl096 070 475 32 99671 466 094

564 33 28287 046 134 476 33 04325 303 032

565 33 29292 iexcl051 170 477 33 05408 238 113566 33 30123 260 544 478 33 05455 215 081

567 33 30494 iexcl211 iexcl239 479 33 08587 775 149

568 33 31423 iexcl012 192 480 33 11772 295 025




H+


Table 1 (Cont)



569 33 32570 iexcl126 033 481 33 13229 424 034570 33 33245 337 464 482 33 13985 1031 052

571 33 34847 iexcl100 034 483 33 15659 352 iexcl009

572 33 35475 iexcl190 iexcl022 484 33 17898 142 077

573 33 39654 iexcl075 028 485 33 18997 266 033

574 33 40144 iexcl055 018 486 33 20442 274 047

575 33 40327 076 188 487 33 21936 289 060

576 33 42295 017 090 488 33 22824 397 iexcl023

577 33 43461 iexcl019 065 489 33 23198 580 126













3calculations


5 Lifetime eregects











state N = 11 000 10 000 8500 state N = 11 000 10 000 8500








































H+


Table 2 (Cont)


state N = 11 000 10 000 8500 state N = 11 000 10 000 8500
















3



3in

the near future



3shows pronounced




3









3







3and









H+





8 Conclusion









H+


of H+









References



A0 poten-






37073713








Chem Phys 81 91112


Res 22 218222









Mol Phys 65 13371344





+ D2 H

+and D

+3 J Chem Phys 103

10 43310 438









Lett 138 125130


3 J Chem Phys 90 54065419


3 J Chem Phys 89 59595960


173 133138




+



Phys 111 140150


Phys 98 295307




3 J Chem Phys 96 67846790




Trans 93 847860



Chem Phys 84 891900








3 Mol Phys 60 13671378


+3 Mol Phys 53 361379









H+






54 42974304










Phys 98 96589668






Phys 92 30053017






Discussion


3 is withoutholes



















530 32 69839 013 180 442 32 38975 149 iexcl183

531 32 71431 iexcl037 078 443 32 39243 114 070

532 32 71498 iexcl013 086 444 32 41284 589 064

533 32 74822 iexcl040 082 445 32 42508 554 036

534 32 77792 176 240 446 32 43630 898 iexcl035

535 32 80384 049 134 447 32 43857 093 iexcl027

536 32 82348 097 144 448 32 51002 547 116

537 32 83391 095 180 449 32 52510 787 017

538 32 84000 024 111 450 32 54236 1090 087

539 32 84906 iexcl023 049 451 32 54840 173 iexcl015

540 32 86605 iexcl111 iexcl010 452 32 58723 167 018

541 32 89986 iexcl069 021 453 32 61868 428 014

542 32 90144 007 093 454 32 62304 451 012

543 32 90608 iexcl036 016 455 32 64322 162 iexcl111

544 32 91933 iexcl024 138 456 32 64935 374 iexcl013

545 32 94388 040 121 457 32 67480 776 051546 32 96882 044 139 458 32 71507 389 095

547 32 98450 iexcl004 097 459 32 74869 396 129

548 32 98712 028 105 460 32 76947 753 iexcl021

549 32 99481 iexcl176 iexcl096 461 32 77596 105 044

550 33 01758 iexcl028 iexcl035 462 32 80303 262 053

551 33 04358 iexcl069 065 463 32 82009 097 iexcl195

552 33 05401 080 027 464 32 83102 089 iexcl109

553 33 06476 017 173 465 32 84732 723 iexcl125

554 33 08569 iexcl109 131 466 32 86643 451 028

555 33 11870 iexcl004 123 467 32 87500 618 026556 33 13263 iexcl017 068 468 32 89262 1245 032

557 33 15478 044 193 469 32 90219 730 168

558 33 15748 iexcl044 080 470 32 91531 384 038

559 33 18987 iexcl103 023 471 32 91923 400 128

560 33 19961 iexcl018 057 472 32 94322 589 054

561 33 20574 033 179 473 32 96770 348 027

562 33 22069 087 192 474 32 98630 389 023

563 33 23142 iexcl096 070 475 32 99671 466 094

564 33 28287 046 134 476 33 04325 303 032

565 33 29292 iexcl051 170 477 33 05408 238 113566 33 30123 260 544 478 33 05455 215 081

567 33 30494 iexcl211 iexcl239 479 33 08587 775 149

568 33 31423 iexcl012 192 480 33 11772 295 025




H+


Table 1 (Cont)



569 33 32570 iexcl126 033 481 33 13229 424 034570 33 33245 337 464 482 33 13985 1031 052

571 33 34847 iexcl100 034 483 33 15659 352 iexcl009

572 33 35475 iexcl190 iexcl022 484 33 17898 142 077

573 33 39654 iexcl075 028 485 33 18997 266 033

574 33 40144 iexcl055 018 486 33 20442 274 047

575 33 40327 076 188 487 33 21936 289 060

576 33 42295 017 090 488 33 22824 397 iexcl023

577 33 43461 iexcl019 065 489 33 23198 580 126













3calculations


5 Lifetime eregects











state N = 11 000 10 000 8500 state N = 11 000 10 000 8500








































H+


Table 2 (Cont)


state N = 11 000 10 000 8500 state N = 11 000 10 000 8500
















3



3in

the near future



3shows pronounced




3









3







3and









H+





8 Conclusion









H+


of H+









References



A0 poten-






37073713








Chem Phys 81 91112


Res 22 218222









Mol Phys 65 13371344





+ D2 H

+and D

+3 J Chem Phys 103

10 43310 438









Lett 138 125130


3 J Chem Phys 90 54065419


3 J Chem Phys 89 59595960


173 133138




+



Phys 111 140150


Phys 98 295307




3 J Chem Phys 96 67846790




Trans 93 847860



Chem Phys 84 891900








3 Mol Phys 60 13671378


+3 Mol Phys 53 361379









H+






54 42974304










Phys 98 96589668






Phys 92 30053017






Discussion


3 is withoutholes













H+


Table 1 (Cont)



569 33 32570 iexcl126 033 481 33 13229 424 034570 33 33245 337 464 482 33 13985 1031 052

571 33 34847 iexcl100 034 483 33 15659 352 iexcl009

572 33 35475 iexcl190 iexcl022 484 33 17898 142 077

573 33 39654 iexcl075 028 485 33 18997 266 033

574 33 40144 iexcl055 018 486 33 20442 274 047

575 33 40327 076 188 487 33 21936 289 060

576 33 42295 017 090 488 33 22824 397 iexcl023

577 33 43461 iexcl019 065 489 33 23198 580 126













3calculations


5 Lifetime eregects











state N = 11 000 10 000 8500 state N = 11 000 10 000 8500








































H+


Table 2 (Cont)


state N = 11 000 10 000 8500 state N = 11 000 10 000 8500
















3



3in

the near future



3shows pronounced




3









3







3and









H+





8 Conclusion









H+


of H+









References



A0 poten-






37073713








Chem Phys 81 91112


Res 22 218222









Mol Phys 65 13371344





+ D2 H

+and D

+3 J Chem Phys 103

10 43310 438









Lett 138 125130


3 J Chem Phys 90 54065419


3 J Chem Phys 89 59595960


173 133138




+



Phys 111 140150


Phys 98 295307




3 J Chem Phys 96 67846790




Trans 93 847860



Chem Phys 84 891900








3 Mol Phys 60 13671378


+3 Mol Phys 53 361379









H+






54 42974304










Phys 98 96589668






Phys 92 30053017






Discussion


3 is withoutholes


















state N = 11 000 10 000 8500 state N = 11 000 10 000 8500








































H+


Table 2 (Cont)


state N = 11 000 10 000 8500 state N = 11 000 10 000 8500
















3



3in

the near future



3shows pronounced




3









3







3and









H+





8 Conclusion









H+


of H+









References



A0 poten-






37073713








Chem Phys 81 91112


Res 22 218222









Mol Phys 65 13371344





+ D2 H

+and D

+3 J Chem Phys 103

10 43310 438









Lett 138 125130


3 J Chem Phys 90 54065419


3 J Chem Phys 89 59595960


173 133138




+



Phys 111 140150


Phys 98 295307




3 J Chem Phys 96 67846790




Trans 93 847860



Chem Phys 84 891900








3 Mol Phys 60 13671378


+3 Mol Phys 53 361379









H+






54 42974304










Phys 98 96589668






Phys 92 30053017






Discussion


3 is withoutholes













H+


Table 2 (Cont)


state N = 11 000 10 000 8500 state N = 11 000 10 000 8500
















3



3in

the near future



3shows pronounced




3









3







3and









H+





8 Conclusion









H+


of H+









References



A0 poten-






37073713








Chem Phys 81 91112


Res 22 218222









Mol Phys 65 13371344





+ D2 H

+and D

+3 J Chem Phys 103

10 43310 438









Lett 138 125130


3 J Chem Phys 90 54065419


3 J Chem Phys 89 59595960


173 133138




+



Phys 111 140150


Phys 98 295307




3 J Chem Phys 96 67846790




Trans 93 847860



Chem Phys 84 891900








3 Mol Phys 60 13671378


+3 Mol Phys 53 361379









H+






54 42974304










Phys 98 96589668






Phys 92 30053017






Discussion


3 is withoutholes

















3







3and









H+





8 Conclusion









H+


of H+









References



A0 poten-






37073713








Chem Phys 81 91112


Res 22 218222









Mol Phys 65 13371344





+ D2 H

+and D

+3 J Chem Phys 103

10 43310 438









Lett 138 125130


3 J Chem Phys 90 54065419


3 J Chem Phys 89 59595960


173 133138




+



Phys 111 140150


Phys 98 295307




3 J Chem Phys 96 67846790




Trans 93 847860



Chem Phys 84 891900








3 Mol Phys 60 13671378


+3 Mol Phys 53 361379









H+






54 42974304










Phys 98 96589668






Phys 92 30053017






Discussion


3 is withoutholes













H+


of H+









References



A0 poten-






37073713








Chem Phys 81 91112


Res 22 218222









Mol Phys 65 13371344





+ D2 H

+and D

+3 J Chem Phys 103

10 43310 438









Lett 138 125130


3 J Chem Phys 90 54065419


3 J Chem Phys 89 59595960


173 133138




+



Phys 111 140150


Phys 98 295307




3 J Chem Phys 96 67846790




Trans 93 847860



Chem Phys 84 891900








3 Mol Phys 60 13671378


+3 Mol Phys 53 361379









H+






54 42974304










Phys 98 96589668






Phys 92 30053017






Discussion


3 is withoutholes


















Lett 138 125130


3 J Chem Phys 90 54065419


3 J Chem Phys 89 59595960


173 133138




+



Phys 111 140150


Phys 98 295307




3 J Chem Phys 96 67846790




Trans 93 847860



Chem Phys 84 891900








3 Mol Phys 60 13671378


+3 Mol Phys 53 361379









H+






54 42974304










Phys 98 96589668






Phys 92 30053017






Discussion


3 is withoutholes













H+






54 42974304










Phys 98 96589668






Phys 92 30053017






Discussion


3 is withoutholes











Jonathan Tennyson et al- H3^+ near dissociation: theoretical progress

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Transcript of Jonathan Tennyson et al- H3^+ near dissociation: theoretical progress