Quasiclassical trajectry study f D+HH and H+HD Glen J. McNamara B.Sc, University f Nrthern British Clumbia, 2004 Thesis Submitted in Partial Fulfillment f The Requirements fr the Degree f Master f Science in Mathematical, Cmputer and Physical Sciences (Chemistry) The University f Nrthern British Clumbia June 2010 Glen J. McNamara, 2010
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11 Abstract An examinatin f the dynamical behaviur f H + HD and D + H 2 in the grund electrnic state is perfrmed n the BKMP2 ptential energy surface [J. Chem. Phys. 104, 7139 (1996)] using the quasiclassical trajectry methd. The cmplete set f state-t-state energy transfer and state-specific dissciative crss sectins and thermal rate cefficients has been btained fr bth systems. Cmparisns are made t the H + H 2 system t investigate istpic effects n reactivity and energy transfer. Cllisin-induced dissciatin, exchange reactins, and nn-reactive energy transfer are analyzed and cmpared t previus results n this system, when such are available. As a prttypical three-bdy reactive system, H + H 2 and its istpic analgues are interesting as benchmarks fr bth theretical and experimental methds. The results f this wrk can be applied t the general field f mlecular reactin dynamics, t interstellar physics and chemistry, t mdels f planetary atmspheres and stellar system frmatin, and t studies f the effects f istpes n reactin rates. The wrk is mtivated by astrphysical applicatins: in particular, the data may be used as inputs fr master equatin calculatins fr interstellar gases.
Cntents 1 Intrductin 1 1.1 Mlecular reactin dynamics 1 1.1.1 The Brn-Oppenheimer apprximatin 4 1.1.2 Theretical methds 6 1.1.3 The quasiclassical trajectry methd 7 1.1.4 2-bdy ptentials fr the grund state f H + H 2 and its istpic analgues 9 1.2 H + H 2 and its istpic analgues 10 1.3 Existing wrk n H + H 2,H + HD, and D + H 2 13 1.3.1 LSTH ptential energy surface 15 1.3.2 DMBE ptential energy surface 16 1.3.3 Calculatins n the LSTH and DMBE ptential energy surfaces... 17 1.3.4 BKMP ptential energy surface 18 1.3.5 BKMP2 ptential energy surface 19 1.3.6 Mielke, Garrett, and Petersn's ptential energy surfaces 21 1.3.7 Other 3-bdy ptentials fr H + H 2 22 2 Methd 24 2.1 Ptential energy surface 24 2.2 Trajectries 25
iv 2.3 Outcmes 26 2.4 Opacity functins 28 2.5 Crss sectins 29 2.5.1 Assignment f final results t quantum states 30 2.5.2 Calculatin 30 2.5.3 Micrscpic reversibility 30 2.6 Rate cefficients 32 2.6.1 Calculatin 32 2.6.2 Temperature dependence and threshlds 33 2.6.3 Micrscpic reversibility 33 2.7 Reference states 34 2.8 Relative cllisin times 35 3 Cllisin-induced dissciatin 40 3.1 Opacity functins fr CID 42 3.1.1 Opacity functins fr CID frm (0,0) states 43 3.1.2 Opacity functins fr CID frm (0,j) states 46 3.1.3 Opacity functins fr CID frm (u, 0) states 49 3.1.4 Opacity functins fr CID frm (v ^ 0, j ^ 0) states 50 3.2 Threshlds t dissciatin 50 3.3 Crss sectins and rate cefficients fr dissciatin frm states with lw internal energy 51 3.4 Crss sectins and rate cefficients fr dissciatin frm states with internal energy near 55 kcal ml -1 54 3.5 Crss sectins and rate cefficients fr dissciatin frm states with internal energy near 109.5 kcal ml -1 55 3.6 Overall trends fr CID 58
V 3.6.1 v-dependence f CID 59 3.6.2 j-dependence f CID 61 3.7 CID frm states with energy less than 1 ev 62 4 Nnreactive energy transfer 75 4.1 Threshlds t nnreactive energy transfer 76 4.2 Micrscpic reversibility 78 4.3 State-t-state behaviur f nnreactive transitins 78 4.3.1 Nnreactive rtatinal transitins frm grund state 78 4.3.2 Nnreactive rtatinal transitins frm lw energy states 80 4.3.3 Nnreactive vibratinal transitins frm lw energy states 83 4.3.4 Nnreactive rtatinal transitins frm states with internal energy near 55 kcal mr 1 84 4.3.5 Nnreactive vibratinal transitins frm states with internal energy near 55 kcal ml" 1 86 4.3.6 Nnreactive rtatinal transitins frm states with internal energy near 109.5 kcal ml" 1 88 4.3.7 Nnreactive vibratinal transitins frm states with internal energy near 109.5 kcal mr 1 90 4.3.8 Mixed vibratinal and rtatinal transitins fr nnreactive energy transfer 91 4.3.9 Summary f state-t-state nnreactive energy transfer 93 4.4 Ttal nnreactive energy transfer 94 4.5 Relative imprtance f different transitin types t nnreactive energy transfer. 98 5 Exchange reactins 124 5.1 Threshlds t the exchange reactin 124 5.2 State-t-state transitins fr exchange 125
vi 5.2.1 Cmparisn f state-t-state exchange crss sectins t prir results. 126 5.3 Overall energy transfer in the exchange reactin 127 5.3.1 Cmparisn f ttal exchange rate cefficients t previus results... 132 5.4 Average energy transfer in exchange and nnreactive cllisins 134 5.5 Summary f exchange reactins 139 6 Cmpetitin amng prcesses 155 6.1 States with lw internal energy 156 6.2 States with internal energy near 55 kcal ml -1 158 6.3 States with energy near 109.5 kcal ml -1 159 7 Future directins 185 Appendix A Energy levels f H 2 and HD 194 B Outcme cunts f selected batches f trajectries 199
Tables 4.1 Cmparisn f rate cefficients fr rtatinal transitins in H + HD 100 5.1 Cmparisn f crss sectins fr state-t-state exchange f D + H L > 140 5.2 Cmparisn f rate cefficients fr ttal exchange 140 6.1 Cmparisn f crss sectins fr state-t-state exchange 166 6.2 Fitted rate cefficient parameters frm Martin and Mandy, 1995 166 6.3 Cmparisn f mdels fr nnreactive energy transfer f high energy states.. 167 A. 1 Internal energies f selected states 195 A.2 Internal energies f states belw 1 ev 195 A.3 Inner and uter turning pints fr selected states 196 A.4 Estimated cllisin times 196 B.l Trajectry cunts frm grund state 200 B.2 Trajectry cunts frm (5,0) 201
Figures 1.1 Depictin f impact parameter 23 2.1 Cmparisn f BKMP2 and CCIPESs 37 2.2 Trajectry utcmes fr exchange and nnreactive energy transfer 38 2.3 Bltzmann distributin at selected temperatures 39 3.1 Opacity functins fr CID at 205 kcal ml" 1 64 3.2 Opacity functins fr CID at 130 kcal ml" 1 65 3.3 CID crss sectins and rate cefficients fr CID frm lw energy states... 66 3.4 CID crss sectins and rate ceffiecients fr CID frm 55 kcal ml -1 states.. 67 3.5 CID crss sectins and rate cefficients fr CID frm high energy states... 68 3.6 Cntur plt f CID rate cefficients at 1000 K 69 3.7 Cntur plt f CID rate cefficients at 6000 K 70 3.8 Cntur plt f CID rate cefficients at 20000 K 71 3.9 CID crss sectins and rate cefficients vs. v fr j ; = 0 72 3.10 CID crss sectins and rate cefficients vs. j'fr v = 0 73 3.11 Crss sectins and rate cefficients fr CID frm states belw 1 ev 74 4.1 Scatter plts f nnreactive trajectries frm D + H 2 (2,14) 101 4.2 Cmparisn f crrected and uncrrected 7(T) fr nnreactive transitins... 102 4.3 Upward nnreactive a{e) and 7(T) fr (0,0), (0,2), (0,4) 103 4.4 Dwnward nnreactive a(e) and j(t) fr (0,0), (0,2), (0,4) 104
ix 4.5 Rtatinal nnreactive cr(e) vs..e^ans. lw energy states 105 4.6 Rtatinal nnreactive j(t) vs. T, lw energy states 106 4.7 Vibratinal nnreactive transfers, lw energy states 107 4.8 Rtatinal nnreactive a(e) and j(t), states near 55 kcal ml -1, up 108 4.9 Rtatinal nnreactive cr(e) and -y(t), states near 55 kcal ml -1, dwn... 109 4.10 Vibratinal nnreactive (E) and -y(t), states near 55 kcal ml -1, up 110 4.11 Vibratinal nnreactive 7(T) vs. T, states near 55 kcal ml -1, dwn Ill 4.12 Rtatinal nnreactive a(e) and 7(T), high energy states, up 112 4.13 Rtatinal nnreactive a(e) and y(t), high energy states, dwn 113 4.14 Vibratinal nnreactive cr(e) and 7(T), high energy states, up 114 4.15 Vibratinal nnreactive <r(e) and j(t), high energy states, dwn 115 4.16 Vibratinal/rtatinal intercnversin, cr(e) and j(t) frm (2,14/16) 116 4.17 Vibratinal/rtatinal intercnversin, (E) and -y(t) frm (5/8,24) 117 4.18 Ttal nnreactive -( ) frm (0,0), (1,0), (0,8) 118 4.19 Ttal nnreactive a(e) frm (5,0), (2,14/16), (0,20/24) 119 4.20 Ttal nnreactive <r( ) frm (12/14,0), (5/8,24), (0,32/37) 120 4.21 Imprtance f isergic transitins, lw energy states 121 4.22 Imprtance f isergic transitins, states near 55 kcal ml -1 122 4.23 Imprtance f isergic transitins, states near 55 kcal ml -1 123 5.1 Scatter plts f exchange trajectries frm D + H 2 (2,14) 141 5.2 State-t-state exchange crss sectins frm lw energy states 142 5.3 Ttal exchange cr(j?) vs. Strans fr lw energy states 143 5.4 Ttal exchange cr( ') vs. E'trans fr states near 55 kcal ml -1 144 5.5 Ttal exchange a(e) vs. E^ans fr high energy states 145 5.6 Average energy transfer fr cllisins frm (0,0) 146 5.7 Average energy transfer fr cllisins frm (0,8) 147
X 5.8 Average energy transfer fr cllisins frm (1,0) 148 5.9 Average energy transfer fr cllisins frm (5,0) 149 5.10 Average energy transfer fr cllisins frm (2,14/16) 150 5.11 Average energy transfer fr cllisins frm (0,20/24) 151 5.12 Average energy transfer fr cllisins frm (12/14,0) 152 5.13 Average energy transfer fr cllisins frm (5/8,24) 153 5.14 Average energy transfer fr cllisins frm (0,32/37) 154 6.1 Ttal rate cefficients frm (0,0) 168 6.2 Ttal rate cefficients frm (0,0) 169 6.3 Ttal rate cefficients split int up/dwn cmpnents, (0,8) 170 6.4 Ttal rate cefficients frm (1,0) 171 6.5 Ttal rate cefficients split int up/dwn cmpnents, (1,0) 172 6.6 Ttal rate cefficients frm (5,0) 173 6.7 Ttal rate cefficients split int up/dwn cmpnents, (5,0) 174 6.8 Ttal rate cefficients frm (2,14/16) 175 6.9 Ttal rate cefficients split int up/dwn cmpnents, (2,14/16) 176 6.10 Ttal rate cefficients frm (0,20/24) 177 6.11 Ttal rate cefficients split int up/dwn cmpnents, (0,20/24) 178 6.12 Ttal rate cefficients frm (12/14,0) 179 6.13 Ttal rate cefficients split int up/dwn cmpnents, (12/14,0) 180 6.14 Ttal rate cefficients frm (5/8,24) 181 6.15 Ttal rate cefficients split int up/dwn cmpnents, (5/8,24) 182 6.16 Ttal rate cefficients frm (0,32/37) 183 6.17 Ttal rate cefficients split int up/dwn cmpnents, (0,32/37) 184 A.l HD energy levels 197 A.2 H 2 energy levels 198
Chapter 1 Intrductin 1.1 Mlecular reactin dynamics Mlecular reactin dynamics is the study f hw energy is distributed during cllisins between mlecules. Thisfieldemerged frm the field f chemical kinetics, the study f rates f chemical prcesses, as researchers develped its theretical underpinnings. In rder t predict the rate at which a reactin will ccur, it is necessary t knw the prbabilities f the elementary prcesses which cntribute t it. The study f reactin dynamics cnnects micrscpic and macrscpic systems. By determining the prperties f individual mlecular cllisins and averaging ver all pssible cllisins, macrscpic behaviur may be predicted. Reactin dynamics studies ften determine values f imprtance t studies f chemical systems: crss sectins and rate cefficients. Crss sectins are values related t the prbability f simple micrscpic prcesses. In principle, a crss sectin can be calculated directly frm first principles using quantum mechanics. Rate cefficients are bulk statistical quantities, useful in describing hw ppulatins f species change ver time in chemical kinetics calculatins. Rate cefficients are calculated by averaging crss sectins ver an apprpriate distributin f translatinal r internal energy. A crss sectin, as a measure f the prbability f a given prcess, is a functin f the relative mtins and physical prperties f tw clliding species. A crss sectin given as a functin f translatinal r ttal energy may be referred t as an excitatin functin. Tw types
2 f crss sectin are cmmnly cnsidered: differential (r scattering) and integral (r ttal) crss sectins. A differential crss sectin is a measure f the prbability f scattering at a particular angle. It is expressed in units f area per slid angle unit. Differential crss sectins can be determined by crssed beam scattering experiments. Observatins f a particular prduct at each angle give rise t the differential crss sectin fr the prcess leading t that prduct. An integral crss sectin may be determined by integrating a differential crss sectin ver all angles. As an average ver all pssible initial rientatins f the target and cllider, an integral crss sectin crrespnds t the ttal prbability f the ccurrence f a particular prcess. Integral crss sectins are expressed in units f area. An integral crss sectin may be cnsidered as an effective area f the target mlecule thrugh which a clliding species must pass in rder fr the prcess f interest t take place. Many prcesses will nt ccur unless certain cnditins have been met. A transitin frm a state with energy E t a state with energy E' > E will require at least E' E energy t be transferred during the cllisin. The energetic requirement E' E is the energetic threshld. Cllisin distance and gemetry may als be factrs in the amunt f energy which must be transferred fr a particular prcess t ccur: fr example, if a cllider can nt apprach the target mlecule clsely enugh t cause the prcess under cnsideratin, then that prcess can nt ccur, even if the energetic requirement has been met. The same is true if a cllider can nt apprach at the crrect cnfrmatin fr the prcess t ccur. (This is perhaps mst cmmnly recgnized in reactins f rganic mlecules, fr which a "steric factr" may be used t mdify rate cefficients t accunt fr gemetrical restrictins.) If E" is the amunt f energy required fr a cllider t apprach clsely enugh and in the crrect gemetry t cause a particular cllisinal utcme, then E" E is the dynamical threshld. Fr a particlar prcess, if E" E > E' E, r equivalently E" > E', then there is said t exist an elevatin f the dynamical threshld fr that prcess. When describing a cllisin, it is helpful t use the variables which describe its initial
apprach. One f these is the impact parameter (Figure 1.1), the distance f what wuld be the clsest apprach, if there were n interactin, frm the cllider's centre f mass t the target mlecule's centre f mass. An pacity functin is the prbability f a cllisinal prcess as a functin f impact parameter b and translatinal energy E, usually dented P(b, E) (r smetimes simply as P(b), when the translatinal energy is held cnstant). Opacity functins are useful in the analysis f excitatin functins because they indicate which types f cllisins are the mst imprtant cntributrs t the excitatin functin. The relatinship between an pacity functin and an assciated integral crss sectin a is / a(e) = 2TT / bp(b,e)db. (1.1) J A rate cefficient describes the rate at which a prcess ccurs as a functin f temperature. Fr instance, the rate f the elementary reactin 3 ma + nb -> prducts (1.2) at temperature T can be described by a rate cefficient j(t) with the frmula il^2l = 7(r)[Ar[B1», as, fr m, n the stichimetric cefficients f A and B in the elementary reactin. Experimentally, crssed beam experiments are the mst cmmn methd f btaining infrmatin abut the dynamics f a cllisin and the assciated crss sectins. In these experiments, tw beams f particles are crssed at a chsen angle, ften with specific relative speeds and initial internal states selected, and the prducts are detected at varius angles relative t the beams. Analysis f the distributin f scattered prducts is a sensitive way t prbe the ptential energy f systems (as in the study f N 2 -N 2 f Aquilanti et al, 2002 [1], r the study f Br-HBr by Meuwly and Hutsn, 2000 [2]). Reactants with specific translatinal energies can be selected t determine the energy dependence f the varius cllisinal utcmes. Sme
4 selectin f specific vibratinal and rtatinal energies is pssible (fr instance, by use f tuned lasers), giving infrmatin abut dynamics fr specific initial and final states [3]. Dynamical data can be used t derive infrmatin abut kinetics, which can then be cmpared t results frm shck tube and ther bulk experiments. Cnsiderable theretical wrk has been dne n a variety f systems, with treatments ranging frm purely classical, thrugh quasiclassical and semiclassical, t detailed quantum mechanical. Increases in cmputing pwer have rendered cmplex calculatins feasible, allwing fr detailed quasiclassical and semiclassical studies and smewhat limited (but mre exact) quantum studies f many systems. The specific treatment used fr any given system depends upn a number f factrs, including the physical prperties f the mlecules invlved, the parameter space t be prbed, and the cmputatinal resurces available. The theretical aspect f the field has been develped t the pint where in sme cases thery is capable f guiding experiment [4]. Because f their simplicity, thse systems with nly three interacting atms are f particular interest in the study f reactin dynamics. Calculatins n small systems are relatively easy t perfrm, and there exists a large bdy f experimental and theretical data pertaining t many f them. H + H 2 is the simplest pen-shell three-bdy interactin; therefre, H +H_> and its istpic analgues frm a basis fr cmparisn f all pen-shell systems. 1.1.1 The Brn-Oppenheimer apprximatin The Brn-Oppenheimer apprximatin is the prpsal that as the nuclei in a mlecular system mve, the electrns adjust immediately t crrespnd t the new cnfiguratin, s that the relative psitins f the nuclei alne determine the ptential energy f a mlecular system. When the apprximatin is made, the nuclear and electrnic mtins f a system are separable, which allws the frces between atms t be calculated as a functin slely f the psitins f the nuclei. The dependence f the ptential energy n the psitins f the nuclei alne allws fr cnsiderable savings in terms f the amunt f time required t perfrm theretical
calculatins, ften with little impact n the verall accuracy f results. If all particles are treated as pint masses, q % are the psitins f electrns, q a are the psitins f nuclei, and tp is the wavefunctin, the Brn-Oppenheimer apprximatin assumes that the nuclear and electrnic prtins f the wavefunctin are separable [5] ^(ft.gc) = ^el(9t.9a)^(9a)- (1-4) This is a gd apprximatin if < 1. (1.5) m a J Given the Brn-Oppenheimer apprximatin, it is pssible t create a ptential energy surface (PES) a single functin f nuclear separatin which returns a ptential energy which may then be used fr each time step f each simulated cllisin. The use f a PES saves the cmputatinal expense f having t slve cstly differential equatins t evaluate the ab initi energy at each pint f the calculatin. PESs are specific t a system and electrnic cnfiguratin: changing the number f prtns in the nuclei, adding r remving nuclei, r changing the number r cnfiguratin f electrns will mean that a different PES is required. Because the nuclear and electrnic mtins are separated, a PES is independent f the nuclear masses: a system and its istplgues may share the same PES. When the Brn-Oppenheimer apprximatin is invked, there may be a discntinuus jump in the PES where the electrnic state is changed. In practice, PESs are derived frm experimental data, frm ab initi quantum calculatins using varius apprximatins, r frm sme cmbinatin f the tw. Fr H + HD and D + H 2, where H is l H and D is 2 H, m,\ ( 3m P ^ m a ) \2mH + m > = 0.142. (1.6) Fr H + H 2 Hh.Y 4 =(^h.y /4 = 0.152. (1.7) It is pssible t invke the Brn-Oppenheimer apprximatin fr these systems, but the results must be treated cautiusly. In mre sphisticated schemes, a crrectin is applied t accunt
6 fr the apprximatin (fr instance, the Brn-Oppenheimer diagnal crrectin in quantum calculatins [6]). It has been estimated that the apprximatin affects the ptential energy f H + H 2 at the saddle pint by as much as 0.21 kcal ml -1 [7]. Fr cmparisn, the ptential fr D+D 2 was estimated in the same study t be affected by 0.11 kcal ml -1. It may be expected that crss sectins and rate cefficients btained with the Brn-Oppenheimer apprximatin invked may differ slightly frm thse btained withut the apprximatin r with crrectins applied. A PES fr a system with mre than tw atms may be cnsidered as the sum f the 2- bdy ptentials f each pair f atms, plus crrectin terms. In the limit f infinite separatin between tw parts f the system, the crrectin terms becme zer, and the ptential energy f the system extraplates t the sum f the internal energies f the tw parts. When ne atm f a 3-bdy system is mved far frm the ther tw, the ptential energy f the system depends nly upn the 2-bdy ptential between the tw clsest atms, and the PES becmes the 2-bdy ptential. 1.1.2 Theretical methds Purely classical methds are easy t implement and require relatively little cmputing time; hwever, they d nt accurately describe systems in which quantum effects are imprtant. Classical methds generate data by simulating trajectries: initial psitins and mmenta fr the atms invlved are selected, then integrated ver time steps until the cllisin is deemed cmplete. A number f trajectries are then used t determine values f interest. Such simulatins are cmputatinally efficient, and are faster fr cllisins with mre translatinal energy. These are statistical methds, whse errrs decrease as the number f trajectries is increased. At the ther extreme, detailed quantum calculatins can in principle give very accurate results fr all systems; hwever, these are prhibitively expensive t perfrm fr all but the lwest energy states f the simplest systems, at best scaling with cmplexity 0(E 6 ). Quantum calculatins generate data (ften crss sectins) thrugh the numerical slutin f Schrdinger's
7 equatin, yielding exact values within the limits f the basis set and apprximatin methd chsen. Quasiclassical and semiclassical methds seek t cmbine classical and quantum methds in rder t increase accuracy but decrease cmputatinal expense. Semiclassical methds treat ne r mre degrees f freedm f a system as purely classical, while using quantum mechanical descriptins fr the rest. Quasiclassical methds set up initial cnditins accrding t quantum mechanical principles, with initial rtatinal and vibratinal energies crrespnding exactly t quantum numbers. The prblem is then slved classically. The final vibratinal and rtatinal energies calculated by a classical methd can take n values ther than thse crrespnding t any quantum state f the prduct mlecule; hwever, quasiclassical and semiclassical trajectry methds can include a methd t cnvert these classically-derived results int integral quantum numbers. These are nt trivial cnversins t perfrm [8]. Quasiclassical and semiclassical methds suffer inaccuracies in the frm f the assumptins they make: fr instance, quantum tunnelling may be imprtant at very lw temperatures, and quantum resnances (which d nt ccur classically) may exist. In a classical methd, unlike a quantum methd, the zer pint energy f a mlecule is nt a cnstraint. Fr bth quantum and classical methds, limitatins t the ptential energy functin used fr the calculatin may exist, bth in the frm f apprximatins made in the calculatin f the ptential energy functin, and in hw intersecting ptentials (fr electrnically excited states, fr example) are treated[9]. 1.1.3 The quasiclassical trajectry methd The quasiclassical trajectry (QCT) methd is useful in that it bth enables a mdel f cllisinal prcesses in terms f quantum states and allws fr cmputatinally efficient slutins. While quantum and quasiclassical trajectry results will ften match very clsely fr high energies, they tend t diverge, ften significantly, at lwer energies (c.f. Mandy and Martin, 1992 [9]). Quantum tunnelling may increase quantum rate cefficients ver classical
8 nes by allwing fr nuclear mtins thrugh ptential energy barriers that can nt be vercme classically. Quantum resnances may cause crss sectins fr sme transitins t be enhanced relative t thers. These resnances d nt ccur in classical calculatins. Zer pint energy leak may cause sme classical crss sectins t be verestimated as a result f the cnsideratin f prduct mlecules with internal energy less than that f the grund state. (Zer pint energy may als be imprtant thermdynamically, in a way which des nt directly depend n the methd: fr example, HD has a lwer zer pint energy than des H 2, suggesting that if nly thermdynamics were taken int cnsideratin, an exchange reactin that prduces HD wuld be favured energetically ver an exchange reactin that prduces H 2.) The QCT methd's accuracy depends n the number f trajectries cmputed, and accuracy increases slwly with the number f trajectries (scaling as 0(l/y/n) fr cllisinal utcmes with prbability p 3> 1/n; and 0(l/n) fr cllisinal utcmes with p ~ \jn see Equatin 2.8). Despite their limitatins, quasiclassical and semiclassical methds are currently the nly feasible methds available fr studies which seek a cmplete set f state-t-state crss sectins. They are excellent fr mderate t high energy cllisins, and since the primary intended applicatin f this wrk is in master equatin studies f the interstellar medium (ISM), where temperatures are ften high (particularly in shcked regins and the warm neutral medium), the lw temperature deficiencies are less significant. Fr cld regins f the ISM, nly the lwest energy states are imprtant, and exact quantum crss sectins exist fr many f these already. Neither tunnelling thrugh the exchange barrier nr tunnelling thrugh the rtatinal barrier is accunted fr in the QCT methd; hwever, the H + H 2 surface is reactive, meaning that the dynamical threshlds fr mst prcesses are nt highly elevated abve their crrespnding energetic threshlds, s that the cases in which tunnelling may have a significant impact are mstly limited t thse exchange prcesses with small prbabilities near the energetic threshld. As all quasibund states f H + H 2 and its istpic analgues have large prbabilities fr cllisin-induced dissciatin (CID) at all translatinal energies, and these states are unlikely t be ppulated at the lwest temperatures studied, the neglect f dissciative tunnelling is
nt a prblem in the study f CID. (Rate cefficients fr lw temperature exchange reactins calculated using the QCT methd may suffer due t the existence f an energetic barrier t exchange.) 1.1.4 2-bdy ptentials fr the grund state f H + H 2 and its istpic analgues Because the species differ nly by a neutrn and the Brn-Oppenheimer apprximatin is being made, the 2-bdy ptential energy curves f H 2 and HD are the same. Hwever, the difference in mass means that the spacing and number f internal energy levels differs. HD has 464 available energy levels, cmpared t 349 fr H 2. These energetic states are dented by their vibratinal quantum number v and rtatinal quantum number j, as in H 2 (i>, j). The atmic harmnic scillatr slutins t the Schrdinger equatins fr H 2 and HD can be used t illustrate why there are mre (v,j) states fr HD than fr H 2. EH 2 = (v + z)hu H2,E }JD = (v + -)/I^HD, (1.8) where E is the energy f the state, v is the vibratinal quantum number, h is Planck's cnstant, and the vibratinal frequency u is 1 [k "up (ḻ 9) fr which k is a frce cnstant and the mass f the diatm is /J,. Because /^HD > A t H 2^ there are mre values f v which satisfy v + -)hu<e dlss (1.10) fr the HD mlecule than fr the H 2 mlecule, where Ed iss is the dissciatin energy. The rtatinal cntributin t internal energy is given by _ j(j + i)h 2 Er ~ 2»Rl (U1) fr equilibrium nuclear separatin R e, reduced mass ji, and rtatinal quantum number j. Increasing the [i in Equatin 1.11 means that there are mre j which satisfy j(j + l)ft 2 2/xflg < E d - (L12)
10 Thus, HD has mre rtatinal levels in each vibratinal manifld than des H 2. A cmmnly used 2-bdy ptential fr H 2 is that f Schwenke [10], which is used explicitly in the BKMP2 ptential energy surface. It is used in this study t calculate the energies at which t initiate trajectries. Internal energies fr all {v,j) states f HD and H 2 btained frm the Schwenke ptential are displayed in Figure A.l and Figure A.2 respectively in Appendix A. Energies fr selected states are given in Table A.l and Table A.2, while turning pints fr selected states are given in Table A.3. 1.2 H + H 2 and its istpic analgues The first electrnically excited state f H 2 (and HD) has electrnic energy apprximately tw-thirds that f the dissciatin energy f the grund state. The transitin between electrnic states n the H + H 2 surface is difficult t deal with theretically, as withut relaxing the Brn- Oppenheimer apprximatin, the electrns must make a sudden jump t a new energy surface, which makes the psitin and mmentum derivatives discntinuus. Under thermal cnditins at temperatures typical fr neutral interstellar gas (under apprximately 10000 K), H 2 and HD mlecules will have sufficiently lw internal energy that studies n the grund electrnic state prvide a reasnable descriptin f the behaviur f hydrgen gas. The spacings between lw level vibratinal and rtatinal states in H 2 and HD are large relative t thse f mst ther diatmics. These spacings make experiments n H 2 relatively easy t perfrm and interpret. Extensive theretical and experimental studies have been perfrmed n H + H 2 and its istpic analgues. Studies f these systems prvide a pint f departure fr mre cmplex studies. Accrding t Aiz et al. [4], "all majr theretical advances in the field f gas-phase kinetics and reactin dynamics have used the hydrgen-atm exchange reactin as a benchmark." Hydrgen atm exchange in the H + H 2 reactin can nt be measured directly in experiment, as the incming and utging atms are indistinguishable. Therefre, the related systems D + H 2 and H + HD, in which ne f the J H atms has been substituted with a 2 H
11 atm, are imprtant fr experimental wrk. (As in Equatin 1.6, *H and 2 H are written as H and D, respectively.) Thus, measurements f exchange reactins in these istpic systems may prvide insights int the behaviur f H atms in the H + H 2 reactin. These measurements als prvide a valuable link between thery and experiment. The dynamical behaviur f H + H 2 and its istpic analgues are als f particular imprtance t astrphysical research, as mlecular hydrgen is the majr cmpnent f cld interstellar cluds, and atmic hydrgen is the dminant species in the interstellar medium. In nebulae, mlecular hydrgen may act as a radiatin shield, and the time evlutin f a nebula depends in part upn the cllisinal behaviur f H 2. Bth mlecular and atmic hydrgen are imprtant cntributrs t interstellar chemistry. Mdels f stellar utflws, galaxy frmatin, interstellar gas cluds and circumstellar disks all require knwledge f the dynamical behaviur f H 2 and its istpic analgues [11, 12, 13, 14]. Amng the differences between H + H 2 and its istpic analgues, the mst imprtant difference frm the perspective f interstellar chemistry is that f the selectin rules fr radiative transitins. HD has a (small) permanent diple mment fj, = 0.00059 D, allwing electric diple transitins with the rtatinal selectin rule A J = ±1 [15]. Due t the lack f a permanent electric diple, the mst prbable radiative prcesses fr H 2 are due t quadruple transitins, with rtatinal selectin rule A J = ±2. The radiative transitin prbability frm the lwest excited state f HD is abut a thusand times that f H 2, meaning that the cling rates f lw density interstellar regins with deuterium may be sensitive t the HD ppulatin [16]. The energy levels f HD (Appendix A, Figure A.l) differ frm thse f H 2 (Figure A.2). Nt nly des a grund state HD mlecule require mre energy t underg dissciatin than des a grund state H 2 mlecule, but the HD mlecule has a larger energetic barrier t exchange. The lwer zer pint energy f HD cmpared t H 2 and the difference in mment f inertia als affect the quantum tunnelling rates. In additin, because the D/H mass rati is large, there are significant differences in cllisin time between H + H 2, D + H 2 and H + HD at
12 a given translatinal energy. Sme representative cllisin times are given in Appendix A, Table A.4. Because the prbabilities f the cllisinal utcmes depend in part upn interactin time, especially at relatively large impact parameter, different mass cmbinatins may exhibit different dynamical behaviur. A kinetic istpe effect arises frm the differences f rate cefficients that cme frm changing ne atm in a reactin t a different istpe f the same element. Such a substitutin has the effect f changing the mass f that nucleus withut changing the ptential energy surface (assuming the Brn-Oppenheimer apprximatin). Istpic substitutins may induce changes in equilibrium cefficients; such a change is called a thermdynamic istpe effect. D + H 2, H + HD, and H + H 2 can be cmpared fr kinetic istpe effects. In sme dynamical systems (particularly rganic reactins invlving prtn exchange), because f the increased speed f a lighter atm, rate cefficients may be greater fr reactins invlving the mvement f lighter istpes. In thers, a slwer cllisin may be mre likely t result in a particular cllisinal utcme. The difference f the atmic masses f H and D means that an HD mlecule is nt symmetric abut its centre f mass. The pacity functins fr similar transitins invlving H + H 2 and H + HD may appear different even if the verall crss sectins fr thse transitins are similar. Reactins can ccur at much higher values f the impact parameter in H + HD, because the H nucleus is farther frm the centre f mass. Hwever, the increase in pacity functins at greater impact parameters may be cmpensated fr by a decrease in the pacity functin just belw the maximum impact parameter fr H + HD relative t H + H 2 due t cllisins which "miss" n the (gemetrically smaller) D atm side. HD is interesting in an astrphysical cntext because it can significantly affect the dynamics f interstellar gas cluds. Althugh D exists at small cncentratins in mst f the ISM, reactins n interstellar dust grains may lead t a high ppulatin f deuterated mlecules relative t the ppulatin f atmic D. The clser spacing f energy levels in HD as ppsed t H 2 means that HD can be excited by lwer energy cllisins than can H 2, which in turn may lead
13 t an enhancement f the ppulatin f excited mlecules in HD gas ver H 2 gas. Cmbined with a higher prbability f phtn emissin due t the HD diple, the higher ppulatin f excited mlecules cntributes t increased radiative cling in interstellar regins cntaining HD [17]. In additin t studies f the interstellar medium, temperature-dependent rate cefficients are required fr studies f deuterium chemistry in the Jvian thermsphere [18]. The abundance f atmic D and HD depend in large part upn these rate cefficients, as the main surce f atmic D is via the reactin HD + H > H 2 + D, while the main lss f atmic D is via the reverse reactin. These rate cefficients are used in cnjunctin with deuterium Lyman a bservatins t calculate the H/D rati f the early slar system, which is imprtant in mdels f slar system frmatin [18]. 1.3 Existing wrk n H + H 2, H + HD, and D + H 2 The H + H 2 reactin and its istpic analgues have been studied since the late 1920s [4]. Althugh the majr develpments in H + H 2, D + H 2, and H + HD are discussed here, a full treatment f the histry f these investigatins is beynd the scpe f this wrk. Further details f the wrk n these and ther istpic analgues f H + H 2 can be fund in (fr example) Mayne and Tennies, 1981 [19], Mayne et al, 1990 [20], and a cmprehensive and highly detailed review by Aiz et al, 2005 [4]. The cncept f a ptential energy surface was intrduced by Lndn, and with it the first H + H 2 PES [21]. The functinal frm f the surface was extremely simple, ignring three bdy terms [20]. A lack f cmputing pwer in the 1930s made a large study t difficult t perfrm. The first trajectry attempted was trapped in an unphysical well in the exchange barrier f the ptential, and it was abandned after three years f calculatins. It wuld be anther thirty years befre Karplus, Prter and Sharma fitted experimental data t a LEPS-type ptential energy surface, which had a mre advanced functinal frm than Lndn's surface [22]. (A LEPS surface is a sum f tw-bdy terms, but des nt cntain
14 three-bdy terms and is relatively inflexible cmpared t mdern functinal frms.) They used quasiclassical trajectries t carry ut the first cmprehensive dynamical study. Results frm this study did nt match the predictins f transitin state thery, and the rate cefficients were ver-estimated relative t experimental values [23]. There were qualitative agreements between this study and experiment, hwever, such as the repulsiveness f regins f the surface, the energetic threshld barrier fr the exchange reactin, and the lack f a stable intermediate species in the exchange reactin. Early experiments were perfrmed by LeRy and c-wrkers (als see Aiz et al. 1996 [23], and references 12-19 therein), mstly using fast flw systems [20]. These experiments were limited t temperatures belw 750 K and the grund vibratinal state, but they shwed sme imprtant features. The effects f tunnelling were visible in the differences between H + H 2 rate cefficients and the crrespnding D + H 2 rate cefficients [23]. The first ab initi H + H 2 surface was determined by Liu in 1973 [24]. It was state f the art fr its time, thugh restricted t cllinear gemetries. It has since been discvered that the behaviur f cllinear cllisins is nt representative f the behaviur f cllisins in general, thugh attempts have been made t extend results n restricted dimensinal systems t 3-dimensinal systems [25]. The 2-dimensinal [26] and 3-dimensinal [27] quantum mechanical results f Kuppermann, Schatz and Baer published in 1976 were highly accurate fr their time. The quasiclassical rate cefficients f Karplus et al. were clser t the experimental values than were the QM rate cefficients, and transitin state thery calculatins were the best f all in matching experiment. The accuracy f the TST calculatins was later attributed t gd luck rather than t any superirity in the methd [23], as errrs in creating the PES used in thse calculatins made the barrier height very clse t the true value [20].
15 1.3.1 LSTH ptential energy surface T vercme the inadequacies f earlier PESs, the LSTH surface was develped [28, 29, 30]. It was the first fully 3-dimensinal surface fitted primarily with ab initi quantum pints. (Prter and Karplus had used experimental data fr their surface, in additin t ab initi values.) Cnsidered highly accurate at the time, it was fitted using 25 parameters, 267 ab initi data pints (the 135 cllinear pints used by Liu, plus 132 new nnlinear pints), and 4 empirical data pints (ne at the saddle pint and 3 in the van der Waals well). It has an rms errr f 6.83 kcal ml -1, with a maximum deviatin f 106 kcal ml -1, perfrming abut equally well (r prly) n bth bent and linear cnfiguratins. These errrs are much less when nly cnsidering nn-cmpact gemetries, that is, thse in which nt all internuclear separatins are small: the rms errr is 0.45 kcal ml -1, while the maximum errr is 2.55 kcal ml -1 [31]. In additin t what are nw regarded as high errrs, the LSTH surface has a fundamental defect in that it is nt well fitted arund the regin where the cllider and target mlecule are separated by abut 2 A. The defect is especially apparent at cllinear gemetries, where the cllider clsely appraches ne f the target atms, but it als ccurs at ther gemetries. This defect leads t smewhat suspect results fr sme calculatins [31, 32], althugh experimental rate cefficients fr the exchange reactin D + H 2 > H + HD agreed well with quantum rate cefficients calculated n the LSTH surface at lw temperature [33, 34]. Early theretical studies n the LSTH surface yielded gd results fr the grund vibratinal state. Early transitin state studies by Garrett and Truhlar [35, 36] significantly underestimated the rate cefficients, but thse studies led t crrectins [37], culminating in successful applicatin f the methd [23]. Quasiclassical studies n D + H 2 [38] and H + H 2 [19] managed t accurately predict v 0 exchange rate cefficients abve 500 K, thugh the neglect f quantum tunnelling inherent in the methd caused underestimatin f rate cefficients at lwer temperatures. Despite these successes fr the grund vibratinal state f the target mlecule, the rate
16 cefficients fr the D + H 2 (v=l) exchange reactin remained prblematic. Similarly, rate cefficients fr the H + HD(v=l) exchange reactin did nt match experiments. Quasiclassical trajectry calculatins f bth crss sectins and rate cefficients (c.f. [19, 38, 39]) varied cnsiderably frm the experimental values btained frm studies invlving maser pumping f H 2 int an excited vibratinal state [40, 41]; the QCT crss sectins were t large by several rders f magnitude. T address the difference between thery and experiment, experiments invlving bth better maser tuning f the reactant vibratinal state and better detectin f prducts were perfrmed in the late 1980s (c.f. [20] and references therein, [42]). These experiments measured smaller rate cefficients at lw temperatures than did previus experiments, placing the experimental values clser t the theretical nes. Measurements f the energetic threshld t dissciatin in these experiments were als clser t theretical values. 1.3.2 DMBE ptential energy surface By the time the experiments invlving better maser tuning were carried ut, a new PES was available, the duble many-bdied expansin (DMBE) surface [43, 44]. Calculatins n bth DMBE and LSTH wuld allw fr cmparisns regarding the imprtance f features f the PES n the dynamics f trajectries n that surface. The DMBE methd, in general, was develped with a view t facilitating quantum calculatins that permit strng bnds t be brken. It prduces glbal analytical surfaces with physically mtivated functinal frms. The H + H 2 DMBE surface was fitted using 316 ab initi data pints, 267 f which had been used fr the LSTH surface. The relative errrs n the DMBE surface are cmparable t thse f the LSTH surface in nn-cmpact gemetries and using the entire set f knwn ab initi pints; hwever, DMBE is cnsiderably better than LSTH fr cmpact linear cnfiguratins [31]. The DMBE surface has a drawback similar t that f the LSTH surface in that the regin f the surface in which the separatin between target and cllider is near 2 A is nt well fitted, especially fr cllinear gemetries. These cnfiguratins frm an imprtant part f the surface,
17 especially fr calculatins invlving lw energy cllisins. These pr fits at imprtant gemetries adversely affect bth quantum and quasiclassical calculatins at lw temperature, causing verestimates f crss sectins by as much as a factr f 30 relative t thse btained n mre recent surfaces [31]. 1.3.3 Calculatins n the LSTH and DMBE ptential energy surfaces Even after DMBE was develped, quasiclassical calculatins cntinued t be perfrmed using the LSTH surface [45, 46, 47]. These studies n the LSTH surface investigated the frmatin f cllisin cmplexes, rtatinal distributins f prducts, and what were thught t be pssible resnance effects bserved in earlier quantum mechanical calculatins. Experiments shwed sharp increases f sme crss sectins relative t ther crss sectins with similar vibratinal and rtatinal energies, which lead researchers t believe that they culd be bserving quantum resnance effects. Sme f these were shwn later t be real quantum effects [9]; thers were shwn t exist in classical studies as well [4], and thus culd nt be due exclusively t quantum prcesses. Full quantum mechanical calculatins prved t be challenging. As such, nly very apprximate (and semiclassical) results were initially available. When the first accurate QM results came ut [48,49, 50], it was nted, with sme cnsternatin, that they were further frm the experimental values than were the results f very apprximate methds [51, 52]. QCT and accurate QM rate cefficients were in general clse t each ther, but fr vibratinally excited mlecules, and even fr sme lw rtatinal states in the D + H 2 (v = 0) exchange reactin [53], they differed cnsiderably frm the mst recent and refined experiments [49, 54, 55, 56, 57, 58]. Later, apprximate QM methds wuld prduce results in gd agreement with accurate QM [59], furthering the cntrversy. Fr instance, accurate QM calculatins by Park and Light [33] fr grund state exchange reactins agreed well with experiment up t 1000 K, but were ff by a factr f tw at 1500 K, well utside the experimental errr bars. Shrtly after, the experiments and calculatins f Adelman etal. [60,61] were perfrmed
18 at higher energies than previusly pssible, and are interesting nt nly in that they cnfirmed that experimental and theretical studies were still far apart (and suggested that a better PES was needed), but als in that they suggested that the D + H 2 exchange reactin ften cnserves internal energy f the bund mlecule: the prduct HD mlecule may frequently have rughly the same internal energy as did the initial H 2 mlecule. 1.3.4 BKMP ptential energy surface A number f pssible appraches t reslving the difference between thery and experiment were cnsidered. The first was t affirm that the theretical methds used were valid. Since the theretical results came frm a variety f grups, and since the accurate QM and QCT results matched, Kegh et al. [53] discarded the idea that there was any prblem with the calculatins themselves. Because the experiments that led t the difference had been repeated with similar results, experimental artefacts were als discarded as an issue (thugh as discussed in Sectin 1.3.5, there was an unknwn prblem with the experiments which wuld nt be discvered and reslved fr anther decade). Effrt was therefre put int imprving the ptential energy surface, with the hpe that inaccuracies in the existing PESs were the primary cause f the discrepancy between thery and experiment. These effrts resulted in the BKMP surface [62]. The BKMP surface uses the pints used t fit the LSTH surface in additin t newer pints, ttalling 770 ab initi pints and cvering a wider range f separatins, fitted t a highly flexible functin [62]. Like the LSTH and DMBE surfaces, its largest errrs are inside the van der Waals regin (arund 3 A), with the H-H 2 separatin arund 2-2.5 A [34]. Unfrtunately, calculatins f crss sectins (especially quantum calculatins) may be highly sensitive t this part f the ptential. A study f bth QCT and QM results n the BKMP surface [53] fr the D + H 2 exchange reactin shwed that it suffered many f the same prblems as DMBE. A study by Mielke et al. [34] cmpared quantum results frm LSTH, DMBE, and BKMP, cncluding that BKMP was nt gd fr calculatins f crss sectins fr the exchange reactin belw 200 K; this was
19 later shwn t be largely due t an anmalus reactin barrier height [31]. This study by Mielke et al. als shwed that part f the surce f the previusly pr agreement between quantum calculatins and experimental results was the calculatin methd (at least fr high temperature, where mst f the deviatin ccurred), and nt necessarily the ptential energy surfaces. Nevertheless, even after crrecting fr methdlgical issues, there was still a significant difference (as much as 10% fr the states studied) between experimental and theretical results. A similar wrk using quasiclassical trajectries fr the v = 0 and v = 1 states by Aiz et al. [23] presented crss sectins and rate cefficients fr D + H 2 -> H + HD, with v =0 r 1 and j =0-7. This was a cmputatinally difficult prblem at the time, especially since the calculatins were perfrmed n each f the LSTH, DMBE, and BKMP surfaces. Gd agreement with quantum results was btained, even at temperatures as lw as 200 K fr the lwer j states, suggesting that the QCT methd is generally suitable fr the H + H2 prblem. The effect f rtatinal energy n reactin was highlighted, as well. Calculatins f the crss sectins at lw cllisin energy were shwn t be sensitive t the PES chsen, meaning that a very accurate PES is needed fr gd quantitative calculatins [23]. 1.3.5 BKMP2 ptential energy surface The anmalus height f the energetic barrier t exchange n the BKMP surface was crrected, new pints were cmputed t further cnstrain the surface, especially in the van der Waal's well regin, and a new fit was released as the BKMP2 surface [31]. The BKMP2 surface nt nly manages t replicate the symmetry and bnding issues that DMBE was develped t incrprate, but als ensures that the functin has cntinuus derivatives fr high energy cmplexes, where LSTH and DMBE did nt. These derivatives are imprtant fr QCT calculatins, enabling mre efficient integratin f the classical equatins f mtin. The same highly flexible functinal frm used fr BKMP was kept fr BKMP2. The BKMP2 surface was fitted with a ttal f 8701 ab initi pints, yielding a r.m.s. errr f
20 0.17 kcal ml -1, with the wrst abslute errr being 3.9 kcal ml -1. These errrs are an rder f magnitude better than thse f the BKMP and DMBE surfaces, and are crrespndingly better than thse f LSTH. When cmpact cnfiguratins are nt cnsidered, BKMP2 is better still [31]. With the advent f the BKMP2 surface, sme f the extant theretical prblems were reslved, including thse prblems invlving the difference in barrier height between surfaces. Once the BKMP2 surface was in use, mst experimental data matched with bth accurate QM and QCT calculatins, but there were still prblems belw 200 K and abve 1500 K [63]. Fr lw temperatures, quantum D + H 2 rate cefficients calculated using BKMP2 were duble thse measured by experiments; abve 1500 K, they were underestimated by as much as 25% [64]. This discrepancy was attributed mstly t a systematic errr in the experimental methd [53], but the idea remained that the difference was pssibly due t the fact that the van der Waals well n the BKMP2 surface was still t deep [64, 65]. Despite these reservatins, when Mielke et al. checked newly cmputed ab initi pints against BKMP2 (and als BKMP), they fund rms errrs similar t the fitting errrs, indicating that the systematic errrs that arise frm the crrelatin treatment and Lndn basis set crrectins used in these fits largely cancel ut [64]. The remaining discrepancy between thery and experiment was at this time discvered t be due in large part t an issue in the experimental prcedure. In prir experiments, free D (r H) atms were created via flash phtlysis f a DI r DBr mlecule (r HI/HBr), a prcess which interfered with the measurements. A set f shck tube experiments which used instead the thermal decmpsitin f C2H5I r C 2 D 5 I, and crrespnding accurate quantum calculatins n the CCI surface, were perfrmed by Mielke et al. [65] ver the temperature range 167-2112 K fr the H+H 2 > H 2 +H exchange reactin. These changed the experimental values significantly, s that they nw agree "perfectly, within experimental errr, bringing [the reslutin f] this 75-year-ld scientific prblem t cmpletin" [65].