Download Adiabaticity of the Nonreactive Bond in Atom

J. Phys. Chem. 1993,97, 6436-6443
6436
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Adiabaticity of the Nonreactive Bond in Atom-Triatom Reactions: A Quantum Mechanical Study
of the H + H20 OH + H2 System?
H.Szichman, I. Last, A. Baram, and M. Baer.
Department of Physics and Applied Mathematics, Soreq NRC, Yaune 70600, Israel
Received: January 19, I993
+
-
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This work is a quantum mechanical study of the four-atom process H HzO
H2 OH and its isotopic
analogs. State-to-state reactive probabilities were calculated for the collinear configuration. The adiabaticity
behavior with regard to the nhreactive bond, as established by Sinha et al. ( J . Chem. Phys. 1991,94,4298),
was confirmed for the H HOD system but not for the H H2O one. Enhancement of the reaction process
due to vibrational excitation as was established by Sinha et al. and Bronikowski et al. ( J . Chem.Phys. 1991,
95, 8645) was fully confirmed for the H HOD system and for the H HzO system. The calculations were
carried out employing negative imaginary potentials to form absorbing boundary conditions.
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I. Iatroduction
Confidence in the reliability of applying negative imaginary
potentials (NIPS)to study time-independent scattering processes
in general and exchange processes in particular has been steadily
growing.l-l0 These potentialswere applied not only t0collinear~9~9~
and other frozen configurations3but to three-dimensional systems
as ell.^.^ In case of the process D + Hz(uj) HD (~7’)+ H,
state-testate reactive probabilitieswere calculated6and the results
compared with the results from more established approaches;’
very encouraging fits were obtained. Other groups have studied
NIPS.~-~O
In a detailed study, Seideman and Miller,loconfirmed
that accurate reactive transition probabilitiesare well reproduced
employing these potentials. These authors reported, moreover,
that different types of potentials, e.g., the WoodsSaxon potential
and/or power-law potentials, also yield accurate results, for a
wide range of parameters.
The present work extends these treatments to four-atom
systems. We report on state-to-state reactive transition probabilities for the three isotopic reactions:
-
H + H,O(u,O,u’)
-
+ OH(u,)
H + DHO(U,O,U’) HD(uH) + OH(U,)
H + HDO(U,O,U’) H~(UH)
+ OD(0,)
H2(u,)
4
(1)
(11)
(111)
These processes were the subject of two interesting ’bondoriented chemistry- experiments by Sinha et al. l2 and Bronikowski
et al.” In fact the possibility of eventually being able to compare
experimental results with theoreticalresults was the main incentive
for the present study. As will be seen, a number of interesting
conclusions were reached.
Our collinear study is, to a certain extent, similar to that carried
out by Bowman and Wang,14 except that they chose to study a
noncollinear frozen configuration. They also used hyperspherical
coordinates whereas, as will be shown, we applied Jacobi
coordinates.
A number of studies on these systems (but less relevant to our
present study) are a~ailable.’~-’~
Among these are the experimental study of Kleinermanns and W01frum’~and the detailed
classical trajectory calculation of Schatz et a1.16 The potential
energy surface employed in our work is a modified versionlg of
the Schatz-Elgersma-Walch-Dunning surface.’”’*
We would like to remind the reader that this is our second
study of a four-atom system employing NIPs.Zo In the previous
Thiswork was supported by the US.-Israel Binational ScienceFoundation,
theIsraelAcademyofSciencesandHumanities,andby ITAMPattheHarvard
Smithsonian Center for Astrophysics, Cambridge, MA.
+
study we considered H2 + H2+system for which we calculated
stateselected probabilities for the reactive process:
+
Hz(ul) Hc(uz)
-
H3++ H
and state-testate probabilities for the charge-transfer process:
The calculationswere carried out for various frozen configuration
and led finally to state-selectedreactive and charge-transfer cross
sections.
The paper is organized in the following way: the theory related
to the four-atom system is presented in section 11, results are
shown in section 111, the analysis is given in section IV,and the
summary in section V.
II. Theory
As has been frequently shown, the relevant equation to be
solved
where E is the total energy, !POAis the A unperturbed part of the
total wave function (here A and later also v designatearrangement
channels (AC)), VAis the relevant perturbation potential and HI
is the full Hamiltonian (which containsthe relevant NIPS)defined
in the v AC. The aim of the numerical treatment is to obtain the
following stateto-state S matrix elements?
where 90,is the v-unperturbedwave function and V,is the relevant
perturbation potential2in this AC. In the present treatment the
A AC is taken to be the (H,H20) side whereas the v AC is the
(H2,OH) side. Consequently the Hamiltonian, HI,relevant for
our case is
where and ri;i = 1,2 are the reduced masses and the vibrational
coordinates, respectively, of the two diatomics, R is the translational coordinate, M,is the v reduced mass of the system:
(4)
U(rlr2R) is the full potential energy surface, -iDRI(Rv) is the
translational NIP, and -iurI(rlv) is the NIP along the bond to be
0022-3654/93/2097-6436$04.00/0 Q 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 24, 1993 6437
Atom-Triatom Reactions
broken, namely, the HZ(or the HD) vibrational coordinate. The
two imaginary potentials are assumed to be linear ramp potentials
of the kindz2
x - XI
XIIx
I
+ AX,
XI
(5)
otherwise
whereu,,~is the height of the potential, Ax1 its width, and XI some
x value outside the interaction region.
Since the u AC is surrounded by absorbing potentials, the wave
function XA can be expanded in terms of L2 basis sets.2g21The
ones we chose are the Gaussian functionsz3for the translational
component and the adiabaticZ3vz4
(contracted) basis sets for the
internal coordinates (in the following all the coordinates R, and
rip;i = 1, 2 are from the Y AC and therefore the index Y will be
omitted):
TABLE I:
Eigenstates and Comesponding Eigenvalues for
the Interacting Systems H HOH,H2 OH, H HOD/
DOH,and HD + OH
+
+
+
H + HOH
Hz + OH
energy
energy
state
(eV) state (eV)
H+E;
HD + OH
energy
energy
state
(eV) state (eV)
(OOO)
(000)
(100)
(001)
(200)
(101)
(002)
(001)
(100)
(002)
(101)
(200)
0.381
0.780
0.817
1.167
1.191
1.240
(00) 0.481
(01)
(10)
(02)
(11)
(20)
0.906
0.978
1.305
1.403
1.440
0.322
0.627
0.743
0.920
1.046
1.139
(00)
(01)
(10)
(02)
(20)
(11)
0.447
0.872
0.885
1.271
1.295
1.310
The function *oA(rpR) is characterized by two quantum
numbers ( n ~ mwhich
~ ) describe the asymptotic eigenstate of the
triatomic system. Thus, writing
QoA(vRI~PJ = ~ ( v I ~ A ~ A ) ~ A ( R I (16)
~A~J
the function f$(rplnAmA) is a solution of the equation
Here nA and mi designate eigenstates of the water molecule, g(Rlq)
is a Gaussian function of the form
where R,; q = 0, ..., N are equidistant division points along the
interval [0, RI M I ]a, is a dimensionless parameter, and u is
the grid size:
g=Rq-Rp-l
(8)
The functions f(rlr2ltq) are eigenfunctions of the equation
(17)
and the function S‘x(RlnAmA) is the solution of the equation
+
fulfilling the relevant asymptotic form. Accordingly the X AC
perturbation potential Vx(rpR) (see eq 1) is defined as
VA(rpR) = U(rpR)
where f(tlRq) are the corresponding eigenvalues ( f is an integer
quantum number related to the tth eigenstate of eq 9).
The next function to be treated is *o~(rpR) which describes
the asymptotic function in the X AC. This function is assumed
to be a solution of the unperturbed SE (all variables and masses
refer to the X AC):
- wA(rpR)
(19)
As was mentioned above (see eq 2), the aim of the calculation
is to obtain the stateto-state S-matrix element, namely, S(n~,nz,
nAmA) where ni,; i = 1,2 are the vibrational quantum numbers
related to the two interacting diatomics. The perturbed potential
V”(rlr2R) (again all coordinates refer to the Y AC) is written in
the form
-
V,hr,R) = U(V2R) - w”(rlr’R)
( 19‘)
where V,is the v-unperturbed potential (assumed to be separable
with respect to the three coordinates rl, r2, and R):
(20)
w”(rlr*R) = Y l A r l ) + uzv(r2) +
Consequently the unperturbed Y function, *o,(r~rzR), takes the
form
p
* 0 ” ( ~ 1 ~ 2 ~ I ~ l f l 2=
, )4l(rll~l”)
42(r,ln,,) r”(RJnl”n2”) (21)
Here &(rilniv) (i = 1, 2) is an eigenfunction of the ith diatomic
and S;(1plnl~n~”)
is the corresponding translational component
fulfilling an equation similar to eq 18.
is the reduced mass of the triatomic molecule:
m3(ml
’= m,
+ m2)
+ m, + m3
(12)
R is the corresponding translational coordinate, and M Ais the
reduced mass of the whole X system:
The unperturbed potential WA(rpR)is assumed to be separable
with respect to the internal (rp) and the translational R
coordinatt~,2~
i.e.
wA(rpR) = uA(rp) + wA(R)
(14)
Here ~ ~ ( rispthe
) potential of the triatomic system and WA(R)
is a potential which decreases as R increases, defined as
wA(R)
= U(r&t$)
where (repe)is the equilibrium configuration for R
-
(15)
OJ.
III. Results
With this method wecalculated reactive transition probabilities
for the three processes 1-111. Since we considered only the
collinear configuration, it is possible that the outcomes are not
always be relevant to the Crim12 and Zare” experiments.
Nevertheless, we found a number of interesting results which
certainlyjustify publication. The maindifficultywith the collinear
configuration is the nonlinearity of the reagent molecule (XYO;
X,Y = H,D),which we force to be linear. This causes reactions
1-111 to be endothermic, not by -0.7 eV1>I6but by only -0.1
eV (see Table I and Figure 1). The reduction of the endothermicity detracts from the importance of the vibrational excitation
to the reactive process, a feature emphasized in the experimental
studies. However, part of the “bond-oriented chemistry” is
maintained and we shall deal with this in particular.
6438 The Journal of Physical Chemistry, Vol. 97, No. 24, 1993
to
Szichman et al.
,
I
a
*
.
0.0
.
*
0.6
10
EJeV)
t e
TOTAL
.-x
4
U
I)
e
-6
.-E
U
u
a
Q,
&
0.0
0.0
-
0.6
10
+
+
Figure 1. Rcactive transition probability as a function of the translational energy for the processes H DOH(u,O,v’) -P HD(u1) OH(v2). u‘refm
to the OH mode and u to the OD mode. (a) Results for v = 0, u’ = 0. (b) Results for u = 0, u‘ = 1. (c) Results for u = 1, v’ = 0. (d) Results for
v = 0, u t = 2. (e) Results for v 1, u t = 1.
We start by considering the energy-dependent state-to-state
and reactive probabilities for the processes
H + DOH(u,O,u’)
-
HD(u,) + OH(u,); (u, u’= 0, 1,2)
(’”)
Here and in what follows the bending mode is zero, the first digit
in the notation (u,O,u’) refers to the OD bond and the second to
the OH bond.
The probability curves for (u = u’ = 0); (u = 0, u’ = l), (u =
l,u’=O), (u=O,u’= 2),and(u= l , u ’ = 1)areshowninFigure
la-e, respectively. Inspecting the figures it is noticed that as a
rule the OH bond does not contribute to nor affect in any way
the reaction process. In other words the reaction is adiabatic
with respect to this mode of vibration. Thus it is most unlikely
that an excited OH starting from (O,O,O) (Figure la) or (1,O.O)
(Figure lb) or a deexcited OH starting from (0,0,1) (Figure IC)
can be met. Consequently: the various four-atom reactive
The Journal of Physical Chemistry, Vol. 97, No. 24, I993 6439
Atom-Triatom Reactions
0.0
l.0
0.6
0.0
0.6
l.0
0.0
0.6
l.0
Etr(eV)
1.0
I
I C
t
I
.:?.
. .
... *..
. .
TOTAL
*
:
0.6
0.0
l.0
c
.-0
U
V
(d
W
&
0.0
0.0
l.0
0.6
E,$eV)
Figwe 2. Reactive transition probability 88 a function of the translationalenergy for the processes H + HOD(u,Op’)
BCC
Figure 1.
probability curves are, essentially, characteristic three-atom
curves. Such a feature was mentioned by Sun and Bowmanz6
who found in their collinear H HCN studies that the two CN
atoms essentially behaved like a single atom. There the adiabaticitv was with r e s w t to the CN bond and was attributed to
the relatively strong kN bond. Here this is not the case since the
OD and OH bonds afe of similar strength. There is therefore
a different reason. The fact that a heaG oxygen is in between
the two light hydrogens seems to decouple the internal collective
motion of the three atoms and to form two independent series of
+
-
Hz(u1)
+ OD(u2). For notation
modes (local modes)of the motions, each related to one of the
light atoms.
A somewhat similar behavior is seen in Figure 2a-e in which
present the reactive probabilities for the Processes:
H + HOD(u,O,u?
-
Hz(u,) + OD(u,)
(111’)
Herein caseof u v’= 0 thereaction product OD( 1) is essentially
missing (very similar to the previous case) but for (u = 1, u’ =
0) and (u = 0, v ’ 5 1) the situation is somewhat different. In the
Szichman et al.
6440 The Journal of Physical Chemistry, Vol. 97, No. 24,1993
LO
,
tb
c
.-0
Y
U
cg
w
0.0
0.6
t.0
0.0
0.0
0.6
l.0
0.0
0.6
LO
Et$eV)
Et,(eV)
-
+
FIpw 3. Reactive transition probability as a function of the translational energy for the procasca H + H20(u,Oy3 Hz(v1) OH(u2). v reands
for the asymmetric mode, and v'for the symmetric mode. (a) Results for v = 0, um = 0. (b) Results for v = 0, v' = 1. (c) Results for u = 1, v ' 0. (d) Results for v = 0, v' = 2. (e) Results for v = 1, v' = 1.
first case (when the OD bond is excited) we got a small amount
(<lo%)of deexcited OD and in the second case a sman amount
of excited OD (<lo%). Thus the results seem to indicate only
a minor intervention of the nonreactive bond in the reaction
process.
Thus far, these findings agreed with the experimental results
of Sinha et al.Iz In those experiments, vibrationally excited
(HDO)molecules were prepared by laser excitation and allowed
to react with slow hydrogen atoms. The study was carried out
with local modes [(O,n)-,0] n = 2,3,4 and [(1,3)-,O]. Reactive
products were obsmed in all four cases but whereas in the first
three no excited OH molecules were recorded in the fourth one
nogroundOHstatewasobservad. Thisindiattedthatthercaction
process was essentially adiabatic with respect to the nonreacting
bond.
To a certain extent the results preacnted also support the
measurements by Bronikowski et al.,l3 who found that the
reactivity of the H + HOD system is enhanced by exciting the
relevant bond. It is difficult to make a quantitative comparison
but in general the following is found (a) Exciting the relevant
The Journal of Physical Chemistry, Vol. 97, No. 24, 1993 6441
Atom-Triatom Reactions
l.3
12
-
l.2l.1
11
-
9-
3I.O
L
to
0.e
0.8
0.0
-
0.8
0.0
1.0
LZ
1.1
la
1.4
p(R)
12-
l.1
-
9I.o 0.e
-
0.8
I
I
I
to
0.e
1 2 1 9
C
la-
f
19-
12
90.e
10
0.8
@@ ii@@
1
-
-
0.8
I
bond (namely the OD bond to form HD or the OH bond to form
H2) results in lowering the threshold for the reaction (see Figures
lb,d,e and 2c,e); (b) Exciting the nonrelevant bond leaves the
threshold almwt unaffected (see Figures IC and 2b).
The situation is unexpectedly different with respect to the
reactions
H H,O(u,O,u’) H,(u,) OH@,)
(111’)
+
0.e
+
(Here u stands for the asymmetric mode and u’for the symmetric
one.)
The relevant energy dependent reactive probabilitiesare shown
in Figure 3a-e. In fact the behavior encounteredfor the reactions
from the ground state is not unusual as per the previous discussion
namely the main products were Hz(0) and OH(0) and to a lesrrer
extent Hz(1) butnoexcitedOHisobtained.Thesituationchangss
once the reactions take place with an excited HzO. An instance
of this is illustrated in Figure 3b which shows results for the
symmetric excitation, namely from an initial state (O,O,l). On
the basis of the previous two HOD cases no excited OH molecules
were to be expected. However not only were the OH molecules
Szichman et al.
6442 The Journal of Physical Chemistry. Vol. 97, No.24, 1993
a
t4
l.2.
13
11
OStO
'
12
0311
.
b
L
0.9
-
0.8
-
tO
0.9
0.8
to
0.9
l.2
l.
0.8
la
to
0.0
tl
12
la
t4
p(8
I
I
13
-
b
1211 -
03
to
-
0.9
-
0.8
-
L
0.8
0.9
to
I
tl
l.2
I
0.8
13
to
0.0
12
l.3
I
I C
f
1.3
12
'
11 .
1.1
n
n
O 3
O S
L
l.2
p(A1
p(1)
1.3
11
L
to -
1.0
0.0
-
0.9
0.8
-
0.8
0.8
0.9
tO
tl
12
13
p(R)
0.0
1.0
tl
12
la
14
p(;l)
Figure 5. Equi-eigenfunction lines of the whnear HzO(u,O.u? molecule. u refers to the asymmetric mode, and u'to the symmetric mode. a-i and
kswnd for the numerical values (4,3,2, 1, l/2, -l/~,-1, -2, -3, -4). (a) Equi-eigenfundtionlines for (O,O,O). (b) Qui-eigenfunctionlinea for (0.0.1).
(c) Equi-eigenfunction lines for (1,O.O). (d) Equi-eigenfunction lines for (0,0,2). (e) Equi-eigenfunctionb e s for (1,OJ). ( f ) Equi-eigenfunctionlines
for (2,0,0).
excited in this case but they were the more abundant product.
The situation was also far from "normal" for the first asymmetric
excitedHz0 (seeFigure 3c). Hereon the basis of what we learned
from the previous caw, we expected all OH molecules to be
vibrationally excited. In fact, ground-state OH molecuks are
formed in iarge percentages: the ratio OH(O)/OH( 1) was 1:2.
Essentiallysimilarresults wereobtained for the (0,0,2) and (1 ,O,l)
states of HzO (see Figure 3d.e).
So far the findings were interesting if not surprising (except
these for (O,O,O) and (l,O,l)) because the HzO molecule behaved
sodifferently fromitsisotopicanalogs. It is important to reiterate
that the results for HOD were all in good qualitative agreement
with both the e ~ p e r i m e n t s ~and
~ J ~with other the quantum
mechanical calculation^.^^ However, the results for the H2O
system are not only novel but also very different and therefore
more insight is needed.
IV. Analysis of the Results
To understand the significant difference in the behavior of
HDO and H20 we present in Figures 4 and 5, respectively the
Atom-Triatom Reactions
eigenfunctions of the two molecules. Each figure shows six
eigenfunctions, corresponding to the (O,O,O), (O,O,l), (1,0,0),
(0,0,2), (l,O,l), (2,0,0)states. In both figures the eigenfunctions
are presented in terms of r, an interatomic coordinate (the OH
distance in case of DOH), and p, the associated Jacobi coordinate
(see eq 11). It is noticed that whereas the eigenfunction (O,O,O)
issimilar for bothmolecules,allothercases(O,O,l),( 1,0,0), (0,0,2),
(l,O,l), (2,0,0) assume different shapes. The HOD molecule is
seen to be a typical local-modetype-molecule(each bond is excited
“separately”)but the HzO is a typical normal-modetype-molecule
(for a more extensivediscussion on the excited stretchingvibrations
of water, see ref 27). These differences have strong implications
on how the systems behave during a reactive collision.
The HDO behaves like a two-atomic molecule, because the
nonreactive bond essentially does not affect the reaction process.
The HzO behaves like a triatomic molecule, where all three atoms
take part in the reaction process. Consequently strong quasiselection rules are encountered in the HOD case but are missing
in the case of the HzO molecule.
v.
Summpry
In this work we have presented a new approach to treat reactive
four-atom systems. The approach is based on the application of
absorbing imaginary potentials (which form absorbing boundary
conditions) and the Kohn variational principle to solve for the
coefficientsof the Lzbasis functions.z1 As a special case we have
treated the reactive interaction between a hydrogen atom and a
water molecule. The four atoms were assumed to be arranged
along a line (collineararrangement). Three cases were considered,
namely, the (H,HOH), (H,DOH), and (H,HOD) systems.
The main difficulty with the collinear configuration was that
we were forced to consider a linear HzO. Making the H2O linear
affected the endothermicity of the reaction; the change caused
it to be -0.1 eV rather than -0.7 eV. As a consequence only
limited information regarding the vibrational enhancement of
the reactive process, as was detected experimentally by Sinha et
a1.12 and by Bronikowski et aI.,l3 could be obtained.
Moredetailed and somewhat unexpected results were obtained
regarding to the adiabaticity of the reaction process with respect
to the nonreactive vibrational bond. In this respect we found
differences in the behavior of the (H, Hz0) system and the two
(H, HOD) systems. Whereas the two (H, HOD) systemsreacted
while the nonreactive bond were kept almost adiabatic and
therefore indifferent to the reactive process the (H, HzO) system
reacted with the nonreactive bond taking an active part in the
reaction process and thus getting excited (or deexcited). The
findings regarding the HOD agree with the experiment of Sinha
et al.,lz who established an adiabatic behavior for the (HDO)
system.
The Journal of Physical Chemistry, Vol. 97, No. 24, 1993 6443
All these findings are based on the study of the collinear
configuration. We do not anticipate significant qualitative
changes (in particularly the unexpected behavior of the H20
molecule) while moving to other unfigurations. We are now in
the final stages of modifying our programs, and we hope soon to
report on three-dimensional results.
Acknowledgment. M.B. would like to thank Professor A.
Dalgarno for his kind hospitality during a two-month stay at the
Institute for Theoretical Atomic and Molecular Physics at the
Harvardamithsonian Center for Astrophysics. Part of this
research was done during that visit. M.B. thanks Professor G.
C. Schatz for sendinghim the potential energysurface subroutines
which were used in this study.
References and Notes
(1) Neuhauser, D.; Baer, M. J. Phys. Chem. 1990, 94, 185; J. Chem.
Phys. 1990, 92, 3419.
(2) Baer, M.; Neuhauser, D.; Oreg, Y. J. Chem. Soc., Faraday Trans.
1990,86, 1721.
(3) Baer, M.; Ng, C. Y.; Neuhauser, D. Chem. Phys. Lett. 1990, 169,
534.
(4) Last, I.; Neuhauser, D.; Baer, M. J. Chem. Phys. 1992, 96, 2017.
(5) Last, I.; Baer, M. Chem. Phys. Lett. 1992, 195, 435.
(6) Last, I.; Baram, A,; Baer, M. Chem. Phys. Lerr. 1990, 169, 534.
(7) Last, I.; Baram, A.; Szichman, H.; Baer, M. J. Phys. Chem., in press.
( 8 ) Child, M. S.Mol. Phys. 1991, 72, 89.
(9) Markovic, N.; Billing, G. D. Chem. Phys. Lerr. 1992.195.53. Vibok,
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96, 8712.
(IO) Seideman, T.; Miller, W. H. 1992, 96, 4412.
(11) Zhang, J. 2.H.; Miller, W. H. J. Chem. Phys. 1989,91,1528. Zhao,
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7074.
(12) Sinha, A,; Hsiao, M. C.; Crim, F. F. J . Chem. Phys. 1990,92,6333;
1991, 94, 4928.
(13) Bronikowski, M. J.; Simpon, W. R.; Girard, B.; a r e , R. N. J. Chem.
Phys. 1991,95, 8647.
(14) Bowman, J. M.; Wang, D. J. Chem. Phys. 1992, %, 7852. Wang,
D.; Bowman, J. M. J. Chem. Phys. 1992,96,8906.
(15) Kleinennanns, K.; Wolfrum, J. Appl. Phys. B 1984, 34, 5.
(16) Schatz,G.C.;Elgersma,H. Chem.Phys.Lerr. 1980,73,21. Schatz,
G. C.: Colton. M. C.: Grant. J. L. J . Phvs. Chem. 1988.88. 2971.
(17) Claj,D.C..k ChemlPhys. 1991;95,7298. Truhiar,D.G.;Isaacson,
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