Resonance theory for three body recombination kinetics for calculating recombination rate constant of deuterium ato

Bernstein, R. B.

Roberts, R. E.

English

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THE UNllVfRSITY OF w ISCOW SIH 
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RATE CONSTANT FOR DEUTERIUM ATOM 
RECOMBINATION CALCUTED BY THE ORBITING RESONANCE THEORY* 
R. E. Roberts 
Department of Cl~emis try, Indiana University 
Bloomington, Indiana 47401 
and 
R. B. Bernstein 
Theoret ica l  Chemistry I n s t i t u t e  and Chemistry Department 
University of Wisconsin, Madison, Wisconsin 53706 
ABSTRACT 
The resonance theory of termolecular recombination k i n e t i c s  
is applied t o  the  react ion D + D + D 2 - - t  2D2. The isotope e f f e c t ,  
i.e., t h e  r a t e  of deuterium recombination compared t o  thaf of hydrogen, 
is computed over the  range 50 - 3 0 0 ~ ~ .  From 100' t o  room temperature 
the  r a t i o  f o r  D/H is 0. 71 (-+I. 02) , i n  agreement with the  experimental 
value of Amdur (1935) and a very recent  determination by Kaufman, Ham 
and Trainar.  
- - - - - 
I 
This work received p a r t i a l  f i n a n c i a l  support from National Science 
Foundation Grant GP-7409 and NASA Grant NGL 50-002-001.- Acknowledg- 
ment is a l s o  made t o  t h e  Petroleum Research Fund administered by 
the  American Chemical Society f o r  p a r t i a l  support of t h i s  research. 
An o r b i t i n g  resonance r h e o q  f o r  rhree-body recombina~ion  
k i n e t i c s  has been preselzted [I] and a p p l i e d  t o  t h e  r e c o m b i ~ a t i u n  of 
hydrogen atoms. Although agreement was obtained wi th  tho. a v a i l a b l e  
experimental  evidence ac room temperature,  Ehe d iscrepancies  among t h e  
var ious  experimental  vaLhaes did not  ph3~lfiit a c r i t i c 8 Q  c k e ~ k  u f  the  
theory.  However, a recent  i n v e s t i g a t i o n  by Raufman et  a l .  [ 2 ]  on 
t h e  r e l a t i v e  r a t e s  of rscelxblnaeiun f o ~  d e u c e ~ i w i ~  and hydrogen stnggcscs 
t h a t  an accu ra t e  experimental deter'rnl.ztatirsn of t h e  r a z i o  w i l l  sooa  
b e  a v a i l a b l e ,  which could be  compared wi th  theory.  Such a compaci.son 
would b e  advantageous s ince  s y s t a r n a ~ l c  errors involved in eke eomputaccioa 
s f  t h e  deuterium and t h e  hydrogen r a t e s  should be approximately the same. 
I n  t h i s  comnunication w e  presenz r e s u l t s  of t h e  resonance theory 
c a l c u l a t i o n  f o r  t h e  re-mbinatisn of deueerium atoms, compatible w i t h  
t h e  e a r l i e r  hydrogen work. Since t h e  resonance theory of te rmolecular  
recombination has been f u l l y  descr ibed  [l] , only t h e  resu1.t~ p e r t i n e n t  
t o  t h e  present  case  w i l l  be given.  
The assumptions i n  t h e  theory have been discussed i n  d e t a i l  
i n  connection wi th  t h e  H 4- H recombina'cion, It tu rns  out  that: t h e  
q u a n t i t a t i v e  c r i t e r i a  used t o  a s se s s  these assumptions do not  change 
appreciably f o r  t h e  D $. D case .  
Deta i led  c a l c u l a t i o n s  have been c a r r i e d  out [ 3 ]  , using t h e  'X ' 
!3 
hydrogenic p o t e n t i a l  of Kelos and Wolraiewicz / 4 1 ,  cha rac t e r i z ing  the 
complete bound and qtaasibourd s ta te  speccrum of D . With t h i s  -infoxma- 2 
t i o n  i t  is  poss ib l e  t o  compute t h e  resonance coneribut ion go t h e  deuterium 
recombination r eac t ion  t.sith ~ h e  same acctlracy as f u r  the  hydrogen case, 
i. Data f o r  t h e  r e l e v a n t  o r b i t i n g  resonances  D , LabeLed by  heir ro"Laz;ioaab 2 
and v i b r a t i o n a l  quantum nun~bers  j and v, are given i!? TaS1e .I, The relevaat 
s t a t e s  r e p r e s e n t  those  coinpLexes wi t11 t ~ ~ z d c h s  I' sttifici-6nt:Sy large 
3. 
- 1 
m t o  meet ~ h c  "second i ~ i r r r i o n "  oi- Xai. [i]. 
Values f o r  cIre E ~ ~ J L ~ L L ~ L \ I ~  constants 1C are obcdra~eii d : ~ r e c t l y  
eq 
from simp1.e scaci,,r iLaL ~noiin a3 i c ~ l  c l ; , i s ~ ~ l ~ ~ ~  d c ~ d ~ a s  [I] , git/-n rile eazekgy 
of t h e  co!np[l.ex r e s a t i v e  co ,iic s~-ps;,:,eed azoios E ,  ARC. she zu~&~Fsc,l 
2 
quantum nttmber . The r o ~ a t i o n a *  d t ; -e~rc i t az ion  c r o s s  seccions 3 h m ~ e  i 
a g a i n  b eeaz computed by the L e~~ormaj .~.aad nnodif i e d  teaf. e raeinber approxi-  
mat ion [I, 51 , us ing  a l s o  el12 ~ Y ~ V L O U S  Iy ces t2d ':LICLIO~~IE : k n , ~ i  G approz1.~- 
mation" [I]. A s  b e f o r e ,  zhe cho ice  u f  ?;he zrhree-body i ta te~actLnf i  to-  
g e t h e r  w i t h  t h e  caLcuLacion of  t h e  r o t a t i o n a l  crovs  s e c c l o n s  u s i n g  p e r t u r -  
b a t i o n  t h e o r y  r e p r e s e n t  t h e  weakest pa rk  sf  t h e  t h e o r y .  The so-call.ed 
v i b r a t i o n a l  r e l a x a t f o n  mechanism, a p p l i e d  s u c c e s s f u l l y  t o  ehe hydrogen 
recombinat ion r e a c t i o n  a t  elevated t empera tu res  (- Z,OOO'K) [ 6 3 shou ld  
n o t  b e  impor tan t  below room cempexazure where r o t a k f o n a l  relaxatioa 
shou ld  p l a y  t h e  predomilit~anc r o l e  [ I ]  . 
Fig .  1 shows t h e  r e s u l t s  of t h e  c a l c u l a r i o n s  of k D2 (D + D t D2) 
S 
compared w i t h  lcrH2 b o t h  sys tems w i t h  c o n s i s t e n t  approx imat ions .  The 
monoenerget ic  approximat ion w o u l d  not  afiec.L the Iiiglaer t empera tu re  
v a l u e s ,  b u t  d i s p l a c e s  t h e  maximurii i ~ a  chi.   ate eo sLsg11;i;iy louer tempera- 
gi 0 
t u r e s .  The r a e i o s  / - 2  s!id-ciS  ^be unai.zaced s b o r e  :-30 K. Zable 
K 
2 summarizes t h e  r e l a t i v e  races and cor~q;ares clze yesonarace thaory wi&h 
exper iment .  
I n  c a l c u l a t i n g  the  deuterium r a t e  cons tan t ,  succes s ive ly  more 
resonance s t a t e s  were added in order  t o  a s c e r t a i n  t h e  convergence 
p rope r t i e s  of k '2. The r e s u l t s  (Table 3) , show ilhat af ter  the add i t i on  
I: 
R 
of t he  14th s t a t e ,  kr-2 ( 3 0 0 ' ~ )  has converged. A t  l a g e r  "oi2perarur.e~ 
fewer s t a t e s  need be  inc luded-  For H 2 ,  the  a d d i t i o r i  of Lour m o m  
quasibound s t a c e s  beymd  the s f x  of Ref.  [ l J  ykzlds an i , icrease en 
kr 
H2 of Les.: t h a n  2% f o r  T .- i 3 0 0 ' ~ .  
The p o t e n t i a l  range paraecear  s: and .el-~,~? s ize prizanietex r 30Q TP 
def ined  i n  [I] were choset~, t o  b e  i d e n t i c a l  f o r  the ~ ~ 5 7 ~ s  i s o c s p i ,  s y s t ~ o . ~  ., 
The angular  a n i s o ~ r o p y  parameters, a and b ,  of Ref ,  [la , difzexeht f o r  
each complex, were obtai.ated from t h e  r m s  average int,erriucleamr separakions 
It is i n t e r e s t i n g  t o  no te  t h a t  although t h e  resonance theory involves 
a d e t a i l e d  c a l c u l a t i o n  over a r e l a t i v e l y  smal l  ( m o n s t a t i s t i d )  s e t  s f  
complexes, t he  ca lcu la t ion  gives t h e  same r e s u l t  f o r  t h e  i s o t o p i c  r a t i o  
of r a t e s  a s  a s imple s r a t l s t i c a l  e s t ima te ,  namely llE . A t  l e a s t  f o r  
t h i s  system i t  appears t h a t  the  i so tope  e f f e c t  is predominately determined 
by the  r e l a t i v e  speed f a c t o r ,  i . e .  , t h a t  above ~ O O O K  t h e  sum, 2 ~ : ~  o i9 
is ( e s s e n t i a l l y )  i so topica lLy i n v a r i a n t .  A genera l  t h e o r e t i c a l  proof 
of t h i s  conjec ture  is nor immediately apparent .  
Ackno~~~~edgrnenk 
The au thors  g r a t e f u l l y  acknowledge t h e  kind cooperat ion of 
R. 3 .  LeRoy, who performed che ca l cu la t ions  cha rac te r i z ing  t h e  q ~ a s i -  
bound and bnund s t a t e  spec t rum of deukarium, 
TABLE 1. Cha~acteris t i c s  of the  most im~3ortdnf resanxdce 
s ta tes  f o r  deuterium recombination. 
-- --- ---- - --- .--"---- 
-T----------- --- 
Skate &~d.soLrupjr 
psramefr~cs  [I$ 
TABLE 2 ,  Relative recombinaCion cons tants  f o r  
deuterium and hydrogen. 
Reference k D a / l ,  Ha 
r C 
T C'K> 
Kaufman e t  ale ( b j  
Present  calcuLations 
v i a  resonance theory ('j 
(a) Ref .C7]1-  
(b) Ref . [2 ] .  
-1 (c) P ~ t e n t i a l .  parameters [l] a r e  a = 2.6 a.u, 300 and r = 4.5 a .u ,  TP 
TABLE 3.  Convergence of t h e  ca l cu la t ed  temnolecular 
0 ( 4  recombination coqs tan t  f o r  deuterium at  300 K 
i s t a t e s  included 10-l5 k D2(m6m01e-2sec-1) 
r 
-------- ---- 
-. E 300 (a) P o r e n r i a l  paramerers [I] are s = 2 .0  a .u .  and iTP = +" 5 a e 3 ~ .  
same as f a r  . 
References 
--- 
[ E l  R. E ,  Roberts,  R,  B ,  Berns te in ,  C.  P,  Cur tkss ,  Chem, Phys, L e t t e ~ s  
7 ,  
the  experimenzal i /aL~~? f o r  k nz dai! go Ec;nnoc~ and BLaclzii~or~? 
'r 
-2 -1 
c i t e d  in Table V should b e  1.3 X 1015 cm6 mole sec 
[ 3 ]  R. J ,  LeRoy and B. B ,  Bernstein,  'Un5versity of Wiseonsii.. The~ret3,::al 
Chemistry ' I n s t i t u t e  Report  MIS-TCI-3E57 (1970). 
143 V. Ralss m d  E. ~ ~ s E u f d : ~ ~ c z ,  9 .  @hem, Phys ,  -- 4 3 ,  2423 (1365) . 
[ 5 ]  8. Takayaaagi,  Prag* Theor, Phys.  (Kyoto) 8, 497 ('L952), 
[6] D.  I,. S. McElw%in and B, 0 .  P r i t c l l a r d ,  J. Am. Chem. Ssc ,  - 91 ,  2,693 
(1969).  See also I. R ,  Hurle,  P .  Mackey, and J. It. 3. Roseafeld. 
Daumchen Bunsen-Cesellschaf t f'irr Physika_Eische Chemie - 7 2 , 991 (1968) . 
[ 7 ]  I. Amdur, J. An. Cham. Soc. - 57, 856 (1935). 
P i g u r e  Capt ion 
F ig .  1. Calculated recombination r a t e  cons tants  f o r  H2 and D 2 ' 
Figure 1 


Rate constant for deuterium atom recombination calculated by the orbiting resonance theory

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