H02(g2A ) Potential Energy Surface from the Double ManyBody
Transcript Of H02(g2A ) Potential Energy Surface from the Double ManyBody
3732
J. Phys. Chem. 1988, 92, 31323742
nuclei interacting with Ln3+ions (in which case y in eq 3 refers to the nucleus and r to the electronnuclear distance). As part of an NMR study of lanthanidebound micelles,we have measured the proton Ti relaxation times for SDS micelles (0.07 M surfactant) to which a variety of Ln3+ ions (0.002 M) had been added.22 The quantity of interest, plotted as hollow squares in Figure 1, is the relaxation enhancement for the CH, group in SDS
bound directly to the sulfate, defined as (l/Tl(L)) (l/Tl(’)),
where TI(La)nd are the proton spinlattice relaxation times in the presence and absence of lanthanide, respectively. (Gd3+ was not included because it produced line widths too broad to measure properly.) For Ln3+ions the dipolar interaction typically dominates the Fermi contact term.19,23 Note that the pattern
~~~~~~
~
(22) In this concentration regime TI’ was not linear with Ln” concentration. Therefore, all measurements were carried out at the same concentration.
(23) Alsaadi, B.; Rossotti, F.; Williams, R. J . Chem. SOC.,Chem. Commun. 1980, 2147.
of dipolar Ti enhancements (hollow squares) does not match the pattern of k, values (filled squares). This argues against a significant dipolar contribution to k,.
Conclusion
We have measured bimolecular quenching rate constants k, for interaction of lanthanide ions with the 1,9biradical 2. The evidence so far suggests that spin exchange is the principal quenching mechanism. The dipolar mechanism does not appear to have a major influence on the quenching. Further investigations, including the magnetic field dependence and chain length dependence of k,, and lanthanide effect on intramolecular product ratios, are in progress.
Acknowledgment. The authors thank the National Science Foundation and Air Force Office of Scientific Research for support. C.D. thanks the N S F (CHE8421140) for support. K.C.W. thanks the National Institutes of Health for a postdoctoral fellowship, NCINIH No. CA07957.
A Realistlc H02(g2A”)Potential Energy Surface from the Double ManyBody Expansion Method
A. J. C. Varandas,* J. BrandHo, and L. A. M. Quintalest
Departamento de Quimica, Universidade de Coimbra. 3049 Coimbra Codex, Portugal (Received: July 21, 1987; In Final Form: December 29, 1987)
+  + A double manybody expansion potential energy surface reported previously for H02(R2A”)and referred to here as DMBE
I is modified to produce thermal rate coefficients for the reaction 0 OH O2 H in good agreement with experiment. This new potential energy surface will be referred to as DMBE 11. By the further imposition that the potential function should reproduce the experimentalspectroscopicforce field data for the hydroperoxylradical, another potential energy surface has been obtained, DMBE 111. Both of these improved DMBE I1 and DMBE 111 potential energy surfaces preserve the functional form used previously for DMBE I except for the longrange 0..OH electrostatic interaction, which is defined in the spirit of a more satisfactory adiabatic theory.
1. Introduction
The potential energy surface fo_r the electronic ground state of the hydroperoxyl radical, H02(X2A”), is important in under
+  + standing the chainbranching reaction (i) H O2 OH 0 +  of many combustion processesi and its reverse (ii) 0 OH
+ 0, H, both of which are also important in the HO, cycle of
atm are
ospheri studies
c c of
hemi~ isotope
try exc
. ~C. ~onnected hange4in 0
+with reactions OH and the
( v
i i
) b
an rat
d (ii) ional
relaxation5 of O2in collisions with H. It is also important for
theoretical studies of the vibrationalrotational spectroscopy of H02(R2A”) and, as a building block, for construction of the
potential energy functions of larger polyatomics (which have
groundstate H 0 2as a dissociation fragment) from the manybody expansion (MBE)6 and double manybody expansion (DMBE)’*
methods. Thus, it is not surprising that there has been a considerable theoretical effort to arrive at a reliable potential energy surface for the electronic ground state of the hydroperoxyl radical, with use of both a b initiog14and semiempiri~al’~’m’ ethods.
On the experimental side, the dissociation en erg^,'^,'^ the ge
ometry,6s202a1nd the quadratic force constants620of the hydroperoxyl radical have also been reported. An extensive list of references to spectroscopic studies by a variety of techniques
covering a wide range of the spectra can be found in ref 14. Once a reliable threedimensional (3D) working potential energy
surface is available, it can be used for dynamics calculations by
‘Permanent address: Facultad de Quhica, Universidad de Salamanca, 37008 Salamanca, Spain.
00223654/88/20923732$01.50/0
using either the classical trajectory method4,5,222o9r approximate quantum mechanical t h e o r i e ~ ~ ’a.n~d~for variational transition
(1) Benson, S . W.; Nangia, P. S. Acc. Chem. Res. 1979, 12, 223. (2) Rowland, F. S.; Molina, M. Rev. Geophys. Space Phys. 1975, 13, 1. (3) Lee, Y. P.; Howard, C. J. J . Chem. Phys. 1982, 77, 756. (4) Miller, J. A. J . Chem. Phys. 1981, 75, 5349. (5) Miller, J. A. J. Chem. Phys. 1981, 74, 5120. (6) Murrell, J. N.; Carter, S.; Farantos, S.C.; Huxley, P.; Varandas, A. J. C. Molecular Potential Energy Functions; Wiley: Chichester, 1984. (7) Varandas, A. J. C. Mol. Phys. 1984, 53, 1303. (8) Varandas, A. J. C. THEOCHEM 1985, 120, 401. (9) Melius, C. F.; Blint, R. J. Chem. Phys. Lett. 1979, 64, 183. (10) Langhoff, S. R.; Jaffe, R. L. J . Chem. Phys. 1979, 71, 1475. (11) Dunning, T. H., Jr.; Walch, S. P.; Wagner, A. C. In Potential Energy Surfaces and Dynamics Calculations; Truhlar, D. G.,Ed.; Plenum: New York, 1981; p 329. (12) Dunning, T. H., Jr.; Walch, S. P.; Goodgame, M. M. J . Chem. Phys. 1981, 74, 3482. (13) Metz, J. Y.; Lievin, J. Theor. Chim. Acta 1983, 62, 195. (14) Vazquez, G.J.; Peyerimhoff,S. D.; Buenker, R. J. Chem. Phys. 1985, 99, 239. (15) Farantos, S. C.; Leisegang, E. C.; Murrell, J. N.; Sorbie, K. S.; Teixeira Dias, J. J. C.; Varandas, A. J. C. Mol. Phys. 1977, 34, 947. (16) Gauss, A., Jr. Chem. Phys. Lett. 1977, 52, 252. (17) Varandas, A. J. C.; Brandlo, J. Mol. Phys. 1986, 57, 387. (18) Howard, C. J. J. A m . Chem. SOC.1980, 102, 6937. Foner, S . N.; Hudson, R. L. J. Chem. Phys. 1962, 36, 2681. (19) Ogilvie, J. F. Can. J. Spectrosc. 1973, 19, 171. (20) Beers, Y.; Howard, C. J. J . Chem. Phys. 1976, 64, 1541. (21) Tuckett, R. P.; Freedman, P. A,; Jones, W. J. Mol. Phys. 1979, 37, 379, 403. (22) Gauss, A,, Jr. J . Chem. Phys. 1978, 68, 1689.
0 1988 American Chemical Society
A Realistic HO2(WZA") Potential Energy Surface
state3, and adiabatic ~ h a n n e ltr~ea~tm. ~en~ts; ref 34 reports also
simplified 1D shortrange/longrange switching models for the
radial and angular interaction 0OH potential in the context of
the adiabatic channel model.
Ab initio electronic structure calculations of the groundstate
H 0 2potential energy have been carried out by Melius and Blint:
Langhoff and Jaffi5,'O Dunning et al.,l'*lzMetz and Lievin,13and
Vazquez et al.I4; for references to earlier work see ref 9. Some
of these a r t i ~ l e s ' ~ *h'a~veJ ~also considered excited states of HO,.
Melius and Blintgcarried out MCSCF C I calculations and defined
a global HOzsurface by fitting the calculated points to an ad hoc
funct mol'
i
ona for
l t
fo he
rm H
. +
This O2a
s d
u d
rface ition
gives reacti
a on
b .
arrier Using
of a
abo mor
u e
t 2.3 kcal extensive
C I wave function, Langhoff and JaffE'O reported no barrier for
this addition step. Dunning et a1.1'q12carried out generalized
valence bond calculations and concluded that by taking the
computational deficiencies into account, it is likely that there is
no barrier to the addition reaction. Metz and LievinI3 and
Vazquez et al.I4 concentrate on the electronic spectra and UV
photodissociation of HOzand hence do not address this barrier
problem on the groundstate potential surface of H02.
In a recent articleI7 (hereafter referred to 5s article I), we
reported a potential energy surface for HOz(X2A") using the
DMBE method. This HOZDMBE I potential surface conforms
with the a b initio data of Melius and Blintg for the threebody
energy and shows reasonable agreement with available spectro
+  + scopic data for the energy and geometry of the minima referring
to the equilibrium triatomic. In agreement with the best ab initio estimates, it also shows no barrier for the H O2 OH 0 reaction. In addition, the H02DMBE I potential energy surface
predicts two secondary minima that have chemical interest. One
refers to a Tshaped (C2J HO2 weakly bound complex, while
the other is related to a linear (CJ weak hydrogenbonded 0.6HO
structure. Moreover, it shows the appropriate longrange behavior
of the HO2 and 0OH asymptotic channels being the electro
static energy defined in the spirit of an adiabatic theory recently
proposed by Clary and Werner.31 This is particularly significant since Clary30$3h1as shown that the longrange forces are major
+  factors in determining the rate constant for the 0 O H O2
+ H reaction. Such importance has most recently been stressed
by T r ~ ine hi~s s~tatistical adiabatic channel model. Along the
+ same direction, Wagner35has explicitly considered coupling be
tween the longrange 0 OH spinorbit potential curves. Despite the fact that the HO, DMBE I potential energy surface
+  + shows some definite improvement over previous functions, it is
not completely satisfactory. For example, exploratory trajectory resultsz9for the 0 OH O2 H reaction have yielded thermal rate coefficients that significantly underestimate the best available
experimental measurements (for a critical review, see ref 36). In
contrast, similar calculations carried out on the MeliusBlint fitg
agree wellz8with the experimental results. The aim of this work
is ther_eforeto report a new DMBE potential energy surface for
H02(X2A") that overcomes such difficulty. In addition we suggest
an alternative, perhaps more satisfactory, adiabatic description
of the longrange OOH electrostatic interaction. Finally, we show
how to make the potential reproduce the complete quadratic force
field of the hydroperoxyl radical as surveyed in ref 6 and 21, while
maintaining a good description of the rate coefficient measure
(23) Blint, R. J. J . Chem. Phys. 1980, 73, 765. (24) Bottomley, M.; Bradley, J. N.; Gilbert, J. R. Int. J . Chem. Kinet. 1981. 13. 957. (25) Gallucci, C. S.; Schatz, G. C. J . Phys. Chem. 1982, 86, 2352. (26) Brown, N. J.; Miller, J. A. J . Chem. Phys. 1984, 80, 5568. (27) Kleinermanns, K.; Schinke, R. J . Chem. Phys. 1984, 80,1440. (28) Miller, J. A. J. Chem. Phys. 1986, 84, 6170.
(29) Quintales, L. A. M.; Varandas, A. J. C.; Alvarifio, J. M. J . Phys. Chem., in press.
(30) Clary, D. C. Mol. Phys. 1984, 53, 3. (31) Clary, D. C.; Werner, H. J. Chem. Phys. Lett. 1984, 112, 346.
(32) Rai, S. N.; Truhlar, D. G. J . Chem. Phys. 1983, 79, 6046. (33) Cobos, C. J.; Hippler, H.; Troe, J. J . Phys. Chem. 1985, 89, 342. (34) Troe, J. J. Phys. Chem. 1986, 90, 3485. (35) Wagner, A,, private communication.
(36) Cohen, N.; Westberg, K. R. J . Phys. Chem. ReJ Data 1983, 12, 531.
The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 3733
H
R
n U
v n
+  + Figure 1. Coordinates used to define the HOzpotential energy surface.
ments for the 0 O H Oz H reaction. This article is organized as follows: Section 2 outlines the
mathematical form of the energy terms in the DMBE and describes the approach used to obtain the numerical values of the parameters they contain from available ab initiogand spectroscopic6,21data. Section 3 presents a discussion of the results. The conclusions are gathered in section 4.
2. New DMBE H 0 2 Potentials
The DMBE of the potential energy for groundstate HOz has the form37,38
I/ = VEHF(R)+ Vwrr(R)
(1)
where E H F denotes an extendedHartreeFocktype energy that includes the nondynamical correlation due to degeneracies or neardegeneracies of the valence orbitals, and corr is the dynamical correlation energy due to the true dynamic correlation of the electrons; R = R,, R2,R3is a collective variable of the internuclear separations, which are defined in Figure 1. In eq 1, both the E H F and corr energy terms are written as a manybody expansion+
3
vEHF(R) = i 1GzAF,i(Ri) + %?lF(R1,R2,R3) (2)
3
Vcorr(R) = fl!!r,i(Ri) + G%(R18Z?R3)
(3)
i=l
where all twobody and threebody fragments are assumed to be
in their groundelectronic states as predicted from the Wigner
Witmer spinspatial correlation rules; i = 1 is taken to label the
homonuclear 00 bond, and i = 2 and 3 label the two OH bonds.
A detailed review of the theory has been given e l s e ~ h e r aend~ ~ ~ ~ ~ ~ ~
will not be repeated here. Instead, we refer only to those aspects
that are essential for this work. The notation is mainly that of
article I; for further clarity, we use the initials EHF37,38rather
than HF7,s,17to represent the extendedHartreeFocktypeenergy.
In article I the twobody energy terms assumed the form of the
realistic EHFACE
and are thus kept unchanged in the
current work. They have the general form
+ 3
V&(R) = DR"(1 C a d ) exp(yr)
(4)
i= 1
C v!;r(R) = 
CnXn(R)R"
(5)
n=6,8.10
where =  Rm is the
''Ordinate
from the
equilibrium diatomic geometry. In eq 5, xnare dispersion damping
(37) Varandas, A. J. C. In Structure and Dynamics of Weakly Bound Complexes; Weber, A., Ed.; D. Reidel: Dordrecht, 1987; p 357.
(38) Varandas, A . J. C. Adv. Chem. Phys., in press. (39) Varandas, A. J. C.;Dias da Silva, J. J . Chem. SOC.F, araday Trans. 2 1986, 82, 593.
3734 The Journal of Physical Chemistry, Vol. 92. No. 13, 1988
Varandas et al.
TABLE I: Coefficients Used for the GroundStateO2and OH Potentials"
coeff
02
OH
0.14291 3.64459 3.92812 2.09867 3.35225 0 5.66169 2.2818 15.40 235.22 4066.24
0.13825 2.65648 1.74505 0.71014 2.54533 0 6.29489 1.8344 10.00 180.45 3685.26
nCalculated from equations 4 and 5; see also text. All values are in atomic units. bCalculated from eq 6a with ( r H 2 )= 3a: and43( r o 2 ) =
2.0043a02.
TABLE I 1 Coefficients for the ThreeBody Correlation Energy"
k6 = 2.46501 (2) 76 = k'6 = 0.68758
OH bondb*'
ks = 5.03696 (2) 78 = k', = 0.82542
klo = 6.29438 (2) 710 = k'lo = 0.94034
k6 = ?.7_8478 (1)
76 = k 6  0.95274
00 bondsbed ks = 4.68155 (1)
78 = k'8 = 0.94148
klo = 1.20507 (0) 710 = k'lo = 0.72379
Calculated from eq 7; units are as in Table I. Given in parentheses
are the powers of 10 by which the numbers should be multiplied, e.g.,
2.46501 (2) = 2.46501 X
b9',, (n = 6, 8, 10) are taken equal to
unity for all diatomic fragments. c R o = 1.8344 ao. d R o = 2.2818 ao.
functions (for recent references to earlier work on dispersion damping see ref 38 and 40) defined bySv4l
XfltR) = [1  exp((A,R/p)  (B"R2/P2))1" (6a)
where
A, =
(6b)
+ B, = POexp(P14
(6c)
p = (R, 2.5Ro)/2
(64
Ro = + 2((rX2)'l2 (rY2)'l2)
(6e)
is the Le Roy42distance for the onset of breakdown of the as
ymptotic R"perturbation series expansion, X and Y label the
two interacting atoms, (rX2)and (ry2) are the corresponding expectation values for the squared radii of the outer orbitals of
X and Y , and ai and p i (i = 0, 1) are dimensionless universal
constants8for all isotropic interactions: a,, = 25.9528, al= 1.1868,
Bo = 15.7381, and = 0.09729. (Unless mentioned otherwise,
all values reported are in atomic units: 1 hartree ( E h )= 1 au of energy = 4.359 821 5 aJ; 1 bohr (ao) = 1 au of length = 0.529 177
A = 0.052 917 7 nm.) Table I define? the numerictl values of
the coefficients in eq 46 for the OH(X211) and 02(X3Z,) dia
tomic fragments.
Also the threebody dynamical correlation energy, which is calculated semiempirically within the DMBE framework, is defined as in article I. It is written as37938944
3
I3 E X R ) = C
CnXn(Ri)(l %kn(Ri+l(mrx13)) X
i=l n=6.8,10
hn(Ri+z(mod3)) + gn(Ri+2(md 3)) hn(Ri+l(mod 3))IRT"I (7a)
where
gn(Ri)= 1 + k,(i) exp[k',(')(R,  R i O ) ] (7b)
h,(Ri) = [tanh (7,,(i)R,)]q'2)
(7c)
and R: is a reference geometry; for the numerical values of the
40) Knowles, P. J.; Meath, W. J. Mol. Phys. 1987, 60, 1143. 41) Varandas, A. J. C.; BrandHo, J. Mol. Phys. 1982, 45, 857.
42) Le Roy, R. J. Mol. Spectrosc. Chem. Soc. (London) 1973, I, 113.
43) Desclaux, J. P. At. Data 1973, 12, 311. 44) Varandas, A. J. C.; Brown, F. B.; Mead, C. A.; Truhlar, D. G.;Blais, C. J. Chem. Phys. 1987, 86, 6258.
TABLE III: Coefficients for the ThreeBody Electrostatic Energy Term"
OH bondsb K4 = K5= 0.O883lc Kb = K; = q4 = v 5 = 2.54533
00 bondb K4 = K , = 0.0
k h = K', = 74 = 7, = 3.35225
Calculated from eq 8, with 6 = 4; all quantities are in atomic units.
bq'n (n = 4, 5) are taken equal to unity for all diatomic fragments. cK4 = K5= ( l / R 0 ) 4 with R o = R, = 1.8344 ao.
coefficients in eq 7, see Table 11.
Modifications that lead to the improved DMBE potentials of
the current work refer therefore only to the threebody E H F
energy term and to the description of the 0OH electrostatic
energy, which is treated" separately from the rest of the threebody
E H F term. We therefore begin our discussion with this elec
trostatic energy term.
2.1. ThreeBody Electrostatic Energy. As in previous ~ o r k , ' ~ , ~ ~
the threebody electrostatic energy term is similarly represented
by
c?j(R1)/2c c 3
=
[CnGn(Ri+l(mod3)) hn(Ri+2(mod3)) +
1=1 fl=4,5
C'~Gn(~i+Z(mo3d)) hn(Ri+l(mod3))I Xn(Ri)Rr" (8a)
where ele stands for electrostatic
G,(R,) = K,R; exp[K',(')(R,  R i O ) ]
(8b)
and h, is defined by eq 7c. Note that C, represents the longrange
+ electrostatic coefficient for the atomdiatom interaction involving
the i l(mod 3) diatomic and the remaining atom of the ith pair
+ (that associated with Ri),and C',, has a similar meaning but refers
to the other atom of the ith pair with the i 2(mod 3) diatomic.
Similarly, x,,denotes the damping function of order n for the ith
diatomic fragment, which has been defined by eq 6a. Since there
is no H00 longrange electrostatic interaction, one must have
G,(Rl) = 0; see Table 111.
F$;i As for the K,, and K',, coefficients, they are determined from
the requirement that given values for 7, and v,',
should
reproduce the longrange electrostatic energy that results from
the interaction between the permanent quadrupole moment (e,)
of the 0 atom and the dipole (&) and quadrupole (0,) moments
of OH; a labels the 0 center, while b stands for the center of mass of OH. Note that the quadrupoledipole interaction (ea&) leads
to an energy contribution that varies as C4r4,while that for the
quadrupolequadrupole (e,+,) interaction varies as CSis,where
C4and C, are defined by4s
c4 = 74/,8,Fb[COS8b(3 COS2 8,  1) 
2 sin 8, sin 8b cos 8, cos dab] (9)
+ cs = 3/168,8b(1  5 cos2 8,  5 cos2 8b 17 cos2 8, cos2 o b +
+, 2 sin28, sin2 8, cos2 
16 sin 8, sin eb cos 8, cos 8, cos +ab) (10)
where r is the S O H centerofmass separation, 8, and 8b are the
angles made by the axis of the electric multipole moments with
r, and +ab is the dihedral angle between those axes; in the present calculations, the electric multipole moments for the 0 atom and O H diatomic are given the values468, = 1.60 eao2and47& = 0.656 eao and480, = 1.37 euo2,respectively. Note especially that the C,and C, coefficients depend on the atomdiatom orientation though they vanish when averaged (for a fixed atomcenter of mass of the diatomic separation) over the angles 8,, Bb, and +ab
= 6, 4b.4s
To define C4and C,, we have, in article I, followed the adiabatic theory of Clary and Werner,31which constrains the angle 8, at 0, = 0 such as to give the lowest value for the dipolequadrupole and quadrupolequadrupole electrostatic energies. However, Clary's approach assumes that the quadrupole axis of the 0 atom always lies along the vector connecting the atom to the center of
(45) Hirschfelder, J. 0.C;urtiss, C. F.; Bird, R. B. The Molecular Theory of Gases and Liquids, 2nd ed.; Wiley: New York, 1964.
(46) Fisher, C. F. At. Data 1973, 12, 87. (47) Meerts, W. L.; Dymanus, A. Chem. Phys. Lett. 1973, 23, 45. (48) Chu, S. I.; Yoshimine, M.; Liu, B. J . Chem. Phys. 1974, 62, 5389.
A Realistic H02(g2A") Potential Energy Surface
r
1
The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 3135

I
c.!
N! , ,
,,.,,.,,,, ,
.,,,,,,,,,,.,,,.,.,
e , / I
0.' ' '36. ' ' 6 6 . ' '3d. ' '!go:
'150.' 'l#O*~ '2i'On' '2d0.' 'Z?Om' ,300.' '33'0: '360.
d e g r e e s
Figure 2. Optimum values of the C, and C, 0OH longrange electrostatic coefficients as a function of the angle Ob (this may be either O2 or
O3 in the notation of Figure l ) , the angle formed by the OH axis with the line connecting the remaining 0 atom with the center of mass of OH. The threeterm Fourier analysis of eq 11 is essentially indistinguishable
within the scale of the figure, and hence it is not shown.
TABLE IV: Values of the Expansion Coefficients in the Fourier
Series Analysis of Eq 11"
uarameter
Cp
bo b, b2 b,
b4
b5
b6
max. dev (three terms), % max. dev (four terms), %
1.987 67 0.393 61 0.195 76
Ob
0.009 78
Ob
0.000 94 1.1 0.1
4.29002
Ob
1.043 70
Ob
0.1 15 02
Ob
0.01672 1.7 0.1
Units are
and
for C., and c,, respectively. Smaller in
absolute value than 0.0001 au.
mass of the OH molecule. As a result, there is a change of sign of the potential as the angle of approach of the 0 atom changes from head on with the 0 end of OH to head on with the H end of OH.
A more satisfactory adiabatic description of the longrange
electrostatic interaction is to let the electronic distribution on the 0 atom instantaneously adjust to the OH electronic charge distribution, whatever the angle of approach, so as to produce the lowest potential energy. We will follow this approach in the
present work. Since a quadrupole can be thought of as having one charge on each comer of a square with the charges alternating in sign around the square, this would correspond to a change in its angle of approach by 180'. Thus, the negatively charged
diagonal axis of the quadrupole square will be coincident with the OH axis when 0 approaches the H end of OH, b u t t h e positively charged diagonal will be coincident if the 0 end is approached. W e choose therefore, for a fixed value of ob, the value of Oa that gives the lowest interaction potential energy. Figure 2 shows the optimum values of C, and C, obtained from this procedure. They can be described in terms of a Fourier expansion:
where the coefficients for N = 6 are given in Table IV. Note
XI;, Figure 3. Contours of the electrostatic energy, eq 8, for an 0 atom
moving around an equilibrium OH with the center of mass fixed at the
origin. Contours are equally spaced by 0.01 E,, starting at A = 0.20
Eh.
0
9
c
T
i <\)\\\ , I / j , c.
~
,
, ,
, I ,\
5.0 4.0 3.0 2.0 !,l0.o ! . O 2,o 3.0 4,O 5.0
X/a,
Figure 4. Same as in Figure 3 but for a H atom moving around an equilibrium 02.
that sine terms in the Fourier series expansion vanish due to symmetry reasons. Note especially that a threeterm Fourier
analysis reproduces the calculated data within 12% and that a conversion to the (R1,R2,R3se)t of coordinates can be made by using the pseudoangular coordinate reported in eq 11 of article
I; see also ref 38. However, rather than using qp'w,e adopt a
simpler approach that is similar to that employed for the dispersion
coefficient^.^^^^^^^ Accordingly, we parametrize eq 8 so as to
reproduce the spherically averaged values of these optimum longrange coefficients for the 0. OHelectrostatic interaction energy. To perform numerically the sineweighted O b averaging,
we have used a GaussLegendre quadrature technique, having
obtained ( e 4=) 0.929Ehao4and (C,) = 1.790Ehao5.
Figure 3 shows contours of the electrostatic energy, as predicted from eq 8, for an 0 atom moving around an equilibrium OH diatomic. A similar contour plot for the H atom moving around an equilibrium O2 is shown in Figure 4. In contrast to the electrostatic energy from article I that shows, as a polar plot in 0OH radial distance and angle of approach, a sign change at the line corresponding approximately to a 90' insertion of 0 into
the middle of OH, the current approach produces longrange electrostatic potentials that are attractive at every angle of approach though they are least attractive for the perpendicular insertion of 0 into OH.
Although the change of sign in C, and C5of article I might affect the dynamics at low collision energies, it is hardly expected
from the magnitude of the energies involved that this sign change
may significantly alter the dynamics at high energies. Indeed, exploratory dynamics studies29on the H 0 2 DMBE I potential
+ + energy surface have shown that a small energy barrier along the
minimum energy path for the 0 OH + O2 H reaction was responsible for the poor agreement between the calculated thermal rate coefficients and e ~ p e r i m e n t . ~ ~ ~T~h' ~is' barrier, which also
(49) Lewis, R. S.;Watson, R. T. J . Phys. Chem. 1980, 84, 3495. (50) Howard, M. J.; Smith, I . W. M. J . Chem. SOC.F, araday Trans. 2 1981, 77, 997. ( 5 1) Deleted in proof.
(52) Frank, P.; Just, Th. Ber. BunsenGes. Phys. Chem. 1985, 89, 181.
3736 The Journal of Physical Chemistry, Vol. 92, No. 13, 1988
appears on the 0OH spherically averaged interaction energy
N
curve, must therefore be removed. For this purpose a further
I
leastsquares fit to the threebody ab initio E H F energies has been
carried out as described in the following subsection.
2.2. Nonelectrostatic ThreeBody EHF Energy. As in article
I, the remainder of the threebody E H F energy assumes the form
Varandas et al.
where the prime denotes that the electrostatic energy is left out, and Qi ( i = 13) are the D3*symmetry coordinates defined by
Also as in article I we define yo(’) =  yl(‘)RYr,where RIref(i =
13) is a reference C, geometry defined through the experimental
equilibrium H 0 2geometry (denoted R: (i = 13); note that the
+ latter differs slightly numerically from that
follows: Rlref= Rleand Rzref= R F f = (R:
used in R3e)/2.
ref 17) as Moreover,
the value of yl(’)has been taken from the nonlinear leastsquares
fit carried out in article I, while those of yl(2)= yl(3)have been
obtained through a trialanderror procedure by carrying out linear
leastsquares fits (the leastsquares parameters being VO and c,,
with i = 133) to the remainder threebody energies which are
obtained by subtracting the sum of the threebody dynamical
correlation energy (see ref 17) and the threebody electrostatic
energy from the threebody correlated ab initio energies of Melius
and B l i ~ ~(tA. l~though the ab initio surface of Melius and Blint
gives inaccurate dissociation energies and hence is questionable
for dynamics studies, we believet7that due to a cancellation of
errors the ab initio threebody energy contributions have greater
reliability. By combining the latter with accurate semiempirical
curves for the diatomic fragments, one should therefore get an
improved description of the complete triatomic potential energy
surface.) Thus, leastsquares fits to the 365 data points so obtained
have been generated until no nonphysical minima appeared in the
final potential energy function, and the corresponding barrier in
+ the 0 OH minimum energy reaction path (or alternatively in
the spherically averaged component Voof the OOH interaction
+ potential; see later) was below the 0 OH dissociation limit; see
Figure 5. Note that this barrier separates the shallow minimum
associated with the 0...HO hydrogenbonded structure from the
deep chemical minimum corresponding to equilibrium H02. In
our previous HO, DMBE I potential the minimum associated with
this 0. .HO hydrogenbonded structure was separated from the
deep H 0 2chemical well by an energy barrier of at least 0.13 kcal
mol’ (relative to 0 + O H ) located at R l = 4.60ao, R2 = 1.84ao,
+ R3 = 4.63ao. A simple way to remove the positive barrier in the 0 OH minimum energy reaction path has been to allow small adjustments
into the poo and DOH rangedetermining parameters of the two
body Morse potentials used by Melius and Blintg (which are then
subtracted from the H 0 2energies to calculate the threebody ab
initio energies) while yl(2)= y1(3w) ere varied freely as discussed
above to yield the best leastsquares fit. To improve the reliability
of our prediction at the equilibrium geometry of the hydroperoxyl
i:I‘I ,
7.
Figure 5. Potential along the minimum energy reaction path as a function of the separation, in angstroms, between the 0 atom and the center of mass of OH: (*) MeliusBlint9 potential surface; () MBE”
potential surface; ( ) DMBE I” potential surface; (. .) this work
DMBE I1 potential surface; () this work DMBE I11 potential surface.
TABLE V Weights Used for the LeastSquares Fitting Procedures
Related to the H 0 2 DMBE Potentials of This Worko
DMBE I1
a
b
initio W, =
+points
1 99
(i = exp
1365): [lOO~
~
.
,
(
R,
R:)’]
special point
no.
Rl
R2
R3
V
wt
366 5.521 810 1.906955 3.614856 0.178 168 100’
DMBE 111 ab initio points (i = 1365): W, = 1  e x p [  l O O ~ ~ ~ , ( RR,:)]’
suecial uoints
no.
R,
R2
R3
V
wt
366 367 368 369 370 371373 374379
5.521 810 2.512 1.85 2.226 492 2.283 606 2.512 2.512
1.906955 1.843 1.8344 4.542 893 7.263427 1.843 1.843
3.614856 3.457 0.99 4.542 893 7.263 427 3.457 3.457
0.178168 0.27 47
0.34 097 0.20 1568 0.19 2700 1st der
2nd der
1 Ob 1 OOOC
Id
1000d 1000d 10000‘ 10/Fz;f
“Units are as in Table I. bOH...O hydrogenbonded structure. Experimental binding energy of H02.6J821dEstimated energy from the H 0 2 DMBE I1 potential surface of the present work. e Condition of zero first derivativeat the equilibriumgeometry of H 0 2 . /Condition to impose the quadratic force constants of H 0 2 : FI1= 13’V/dR2~;F22 = a2v/aR22;F,, = a2v/aff2F; , =~ a2v/aR,aR2;F,, = a2v/aR,affF; ~~ = a2vja~~a~.
radical, we considered a leastsquares weight function in fitting
+ + the nonelectrostatic threebody residual E H F energies, namely, W = 1 + 99 exp[g((Rl  RIe), (R,  R2e))Z (R3  R3’)*)];
see Table V. Note that the diatomic Morse parameters (poo and
POH) were not specified by Melius and Blint’ for the ab initio
energies, and we adopted, somewhat arbitrarily, the values Boo
= 1.45ao’ and POH = 1.38ao’ in article I. To reduce the current
leastsquares fitting procedure to a twovariable trialanderror
search, we have forced the poo/@oH ratio to be equal to thats3
(53) Huber, K. P.: Herzberg, G. Molecular Spectra and Molecular Structure: Constants of Diatomic Molecules: van Nostrand: New York, 1979.
A Realistic HOZ(R2A") Potential Energy Surface
The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 3737
TABLE VI:
Coefficients for the Residual ThreeBody ExtendedHartreeFock Energy Term"sb
VO = 5.8472 (0) CI = 9.8340 (1) c2 = 2.1836 (0) cj = 3.8540 (1) cq = 3.8215 (2) c5 = 1.9349 (0) ~6 = 5.5602 (1) CT = 7.1609 (2) cg = 7.0660 (2) cg = 6.5073 (2)
cIO = 6.2814 (1) c ~ I= 3.8207 (1) c12 = 1.0142 (1) c13 = 6.0124 (3) c14 = 2.5117 (2) CIS = 5.7318 (2) ~ 1 =6 2.0818 (2) c17 = 8.8011 (2)
DMBE I1
c1g = 8.5610 (2) ~ 1 =9 2.7164 (2) ~ 2 =0 1.1228 (2) c2I = 2.7195 (2) ~ 2 =2 1.5328 (4) ~ 2 =3 2.4537 (3) c24 = 1.1601 (2) ~ 2 =5 1.3576 (3)
c26 = 7.5568 (3) ~ 2 7= 4.4659 (3) c2g = 6.0668 (3) c29 = 2.0637 (3) c3g = 1.8701 (3) cjl = 3.9858 (3) ~ 3 =2 2.7786 (3) cj3 = 2.5483 (4)
1(O(I) = 2.46683 (0) yo(') = 1.64300 (0) yl(')= 0.98202 (0) y1(2'= 0.62 (0)
yo(3)= 1.64300 (0) yl(') = 0.62 (0)
VO = 2.0245 (1) CI = 1.0827 (0) ~2 = 1.5532 (0) cj = 4.8201 (1) cq = 9.9026 (2) c5 = 1.2977 (0)
c6 = 4.0758 (1) ~7 = 1.0942 (1) cg 5.1977 (2) ~9 = 6.5829 (4)
~ 1 =0 4.0323 (1) c11 = 2.5285 (1) c12 = 3.3377 (3) ~ 1 =3 1.2622 (2) ~ 1 =4 8.4826 (3) CIS = 1.0826 (2) ~ 1 =6 8.0499 (4) ~ 1 =7 5.5122 (2)
DMBE 111
c1g = 5.1866 (2) ~ 1 =9 1.9565 (3) ~ 2 =0 4.6763 (3) c ~ I = 7.6118 (3) ~ 2 =2 5.8673 (4) ~ 2 =3 3.8808 (4) ~ 2 =4 2.1401 (3) ~ 2 =5 1.3589 (4)
C26 = 1.8273 (3) ~ 2 7= 2.7984 (3) c2g = 3.4617 (3)
~ 2 9 1.8739 (4) c30 = 9.4458 (4) cjl = 1.1727 (3) ~ 3 =2 9.5638 (4) ~ 3 =3 3.7996 (4)
yo(') = 2.46683 (0) yo(') = 1.64300 (0) yl(l)= 0.98202 (0) yl(') = 0.62 (0)
= 1.64300 (0) yl(j)= 0.62 (0)
'Calculated from eq 12; units are as in Table I. bGiven in parentheses are the powers of 10 by which the numbers should be multiplied, e.g., 5.8472 (0) =
5.8472 X 10'.
TABLE VII: Spectroscopic Properties of the Hydroperoxyl Radical (C,Mirima)"
DMBE
property
ab initio9 MBEI5qb
Ill
I1
Rdao Rzlao Rdao
HOO/deg
2.58
1.83 3.51
104.7
2.570 1.861
106.0
2.543 1.893 3.525 104.28
Geometry
2.584 1.870 3.530 103.70
I11
2.512 1.843 3.457 104.02
2.52 1.85
104.1
eXpt16,1821
2.57
2.512 f 0.001
1.86
1.843 f 0.004
106
104.02 iz 0.24
Dissociation Energy
De/Eh
0.2488
0.2747
0.2808
0.2797
0.2745
0.2747 * 0.003
F,lIEhao2 F22/EhaO*
Fl2 I Ehao2
F e u / EhaOl
F1,lEhaol
FZu/ Ehalll
0.452 0.497 0.0172 0.277 0.0941
0.0247
0.375 0.418 0.0064 0.240
0.0482 0.0607
0.522 0.329 0.0264 0.445 0.138 0.0068
Force Constants
0.426 0.273 0.0186 0.393 0.1 16 0.0202
0.375 0.417 0.0060 0.241 0.0482 0.0600
0.375 0.418 0.0064 0.240 0.0482 0.0607
'The dissociation energy, in Eh, is taken relative to the three isolated atoms. R Ilabels the 00 bond distance, in ao. *The force field has been fitted to the experimental data reported in this table.
experimentally found for the isolated diatomic fragments, Le.,
&,o//~oH= 1.15. Thus, only DOH and yl(,)= y1(3w) ere system
+ atically varied until the isotropic component of the 0OH in
teraction potential and the 0 OH minimum energy path were
barrier free. This procedure gave POH = 1.42ao' and Boo =
1.633ao' for the final fit. The DMBE potential energy surface (denoted hereafter as
DMBE 11) obtained from the above procedure is numerically defined in Table VI,while the spectroscopic force field it predicts for the hydroperoxyl radical is reported in Table VII. We note
+  + the reasonably good agreement with the best available empirical
estimates and our own force field data from article I. Moreover, quasiclassical trajectories for the 0 OH 0, H reaction
run on this potential surface have produced thermal rate coefficients in good agreement with the best reported experimental v a l ~ e s ;a~de~tai,le~d ~des~cr~iption of these trajectory calculations
as well as those obtained on the DMBE I11 potential surface reported next will be presented el~ewhere.,~
Although the DMBE I1 potential surface shows a definite improvement over the ad hoc potential of Melius and Blint: which was fitted to their own ab initio data, it would be desirable for spectroscopic studies to have a H 0 2 potential function (DMBE 111) that reproduces the experimental quadratic force field reported
elsewhere6*21for this species. This has been accomplished by
having 10 extra conditions imposed (namely, the well depth, three first derivatives, and six quadratic force constants at the HO, equilibrium geometry) as further items in the leastsquares minimization procedure. For example, a first derivative is con
sidered as an extra data point for which the model function is the gradient of the potential energy surface with respect to a given internuclear coordinate; in the present case, such gradient will
be evaluated at the triatomic equilibrium geometry and hence is equal to zero. Thus, we suggest a leastsquares fitting procedure where the ab initio energies and the experimental data are dealt with on an equal footing. To ensure that the calculated and experimental values agreed within error bars, we attributed special weights to the various items as summarized in Table V. Since the resulting function showed a tendency to develop nonphysical minima at the strong interaction region where ab initio data points are unavailable, a further point had to be given to guide the potential at such regions. We have estimated this point from the HOz DMBE I1 potential energy surface reported above. In ad
dition, two extra points (numbers 369 and 370 of Table V) were added to prevent the DMBE I11 potential surface from developing
a small barrier (<1 kcal mol') at the H + O2asymptotic channel;
the associated energies have also been estimated from the DMBE
3738 T h e Journal of Physical Chemistry, Vol. 92, No. 13, 1988
I DMBE I1
.
Varandas et al.
x de
Figure 6. Contour plots for an 0 atom moving around an equilibrium OH that lies along the x axis with the 0 end on the negative part of this axis and the center of the bond fixed at the origin. In this figure, as well as in Figure 7 , the upper plot refers to the DMBE I1 potential energy surface, while the other is for the DMBE 111 potential surface. Contours
+ are equally spaced by 0.01 Eh,starting at A = 0.277 Eh. Shown by the
dashed line (contour a) is the energy contour corresponding to the 0 OH dissociation limit.
I1 potential energy surface from the current work. We conclude this section by reporting the unweighted rootmeansquare deviations for the two DMBE potential surfaces with respect to the threebody residual E H F a b initio energies of Melius and Blint: 0.0073 and 0.0121 E, for the DMBE I1 and DMBE 111potential surfaces, respectively. These values may be compared with those for the ad hoc MeliusBlint9 form and our previous DMBE I surface,17respectively 0.0116 and 0.0073 Eh. Note that the larger rootmeansquare deviation for the DMBE 111 potential surface is mainly due to the form chosen for the weighting function that weights least the ab initio points near the equilibrium HO, geometry.
3. Results and Discussion
The HO, DMBE (I1 and 111) potential surfaces from the present work are numerically defined in Tables 1111 and VI. Note that the coefficients for the threebody electrostatic energy term and the residual threebody E H F term (which are reported in Tables 111and VI, respectively) are the only ones that differ from those reported in article I; the coefficients of Tables I and I1 have therefore been given here only for completeness.
Figure 6 displays equipotential energy contours of the final HO, DMBE (I1 and 111) potential surfaces for an 0 atom moving around an equilibrium OH diatomic. Similar equipotential energy contours but for H moving around an equilibrium O2are shown in Figure 7. Note the absence of an energy barrier in the plots of Figures 6 and 7 and hence absent for both 0 approaching O H and H approaching 0,.
In Figure 8 we show equipotential energy contours for the 00 and OH stretching in HO, with the HOO angle kept fixed at the corresponding equilibrium value, while Figure 9 shows contours
 for the stretching of the two O H bonds in linear OHO. Apparent
from Figure 9 are the two minima associated with the OH. 0 and 0...HO equivalent hydrogenbonded structures. Note, however, that the seemingly Dmhminimum at short OH distances is an artifact of the dimensionality of the graphical representation being truly a saddle point for the motion of 0 around the H end of OH. Despite the similar topographies of the DMBE I1 and
7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 l . C 2.0 3.0 4.G 5 ~ 5 6,O /.C
X/a,
Figure 7. Contour plots for a H atom moving around an equilibrium O2
+ molecule with the center of the bond fixed at the origin. Contours are
as in Figure 6 , except for contour a, which refers now to the H 0, dissociation limit.
 + DMBE 111 potential surfaces at these regions of configuration
space, they show some noticeable differences, particularly con
cerning the height of the barrier for the 0 + O H OH 0
H atom exchange reaction. Accurate ab initio calculations for
the H atom migration between the two 0 atoms would therefore
provide an important way of settling this issue.
Figure 10 shows contours for the C2, insertion of H into 0,.
Again, the DMBE I1 and DMBE 111potential surfaces show some
quantitative differences mainly near the collinear 0H0 geom
etries referred to in the previous paragraph.
Figures 11 and 12 show the leading terms in the Legendre
analysis of the 0OH and H0, interaction potentials with the
diatomics kept fixed at their equilibrium geometries, while Figure
13 compares the V, for the 0OH interaction with that obtained
from other treatments. Again, we have reported the results ob
tained from both the H0, DMBE I1 and 111potential surfaces.
Note the absence of a positive barrier in the spherically symmetric
component of the OOH interaction potential for both the DMBE
I1 and MeliusBlint potentials; see also Figure 5. For the HO,
DMBE 111potential energy surface this barrier exists, though it
is small. Note especially that the condition of no barrier in V,
+ (or, if existing, a small barrier) has been considered in the present
work a key feature for a good 0 O H dynamics.
The properties of the C, chemical minima for both the HO,
DMBE I1 and 111potential energy surfaces are gathered in Table
VII. Also shown for comparison in this table are the attributes
of the H 0 2D M B E I potential surface," those of the MeliusBlint9
ad hoc functional form, and the experimental
Par
ticularly striking is the good agreement between the properties
from potential I1 and experiment. Table VI11 summarizes the
properties of other stationary points in the H 0 2 groundstate
potential energy surface. It is seen that all relevant chemical
structures associated with stationary points in article I remain in
the new potential, although they differ somehow quantitatively.
4. Concluding Remarks
The properties of the HO, DMBE potentials derived in the present work have been carefully analyzed by numerical methods and by graphical techniques. Whenever available, experimental information has been used to parametrize the present functions. In addition, a leastsquares fitting procedure has been adopted
A Realistic H 0 2 ( ~ 2 A ”P) otential Energy Surface
0 h
The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 3139
9
In
0 LD
G.“%
\
I T
0 M
0
N
0 ..
.5
2.5
305
4.5
5.5
6.5
5
b/a,
,5
2.5
3.5
4.5
5.5
6.5
7.5
h/a,
Figure 8. Contour plots for the stretching of the 00 and OH bonds in H02(g2A”)with the HOO angle fixed at 103.7’. In this figure, as well as in Figures 912, the plot in the lefthand side refers to the DMBE I1 potential energy surface, while the other is for the DMBE 111potential surface.
Contours are as in Figure 6 .
I
I
1
I
I
I
I
I
1n o
2.0
3.0
4.0
500
600
1.0
2.0
3.0
4.0
5.0
6.0
R2h
R2/ a0
Figure 9. Contours for the stretching of the OH bonds in O H 0 (g2A”)showing the two equivalent OH..O and O.HO hydrogenbonded structures.
Contours are as in Figure 6 .
to derive the H 0 2DMBE I11 potential energy surface that takes
the a b initio energies of Melius and Blint9 and the available
experimental spectroscopic force field data for the hydroperoxyl
radica16J821on an equal footing, perhaps yielding the most realistic
global representation currently available for that surface. The H 0 2DMBE I1 and I11 potentials derived in this work have
+ also been used for extensive quasiclassical trajectory calculations
of the 0 OH + O2+ H reaction with good success. Since a
detailed description of these calculations is presented el~ewhere?~ we refer to only the results for the thermal rate constant at T = 500 and 2000 K based on the H 0 2DMBE I11 potential surface. At T = 500 K one gets k(500) = 1.55 (f0.12) X 1013cm3mol’ sl, a result that is in good agreement with the best available experimental e s t i m a t e ~ ~(1~.4v5~(f~O.05) X lOI3and 1.68 (f0.06) X 1013cm3 molI sl, respectively), while at T = 2000 K one obtains k(2000) = 8.58 (f0.10) X 10l2cm3 mol’ sl, which is
in reasonably good agreement with the experimental correlation
of Cohen and W e ~ t b e r g(k~(2~000) = 1.02 (f0.36) X l O I 3 cm3
mol’ sl) but somewhat too low in comparison with the most
recent estimate of Frank and Justs2 (k(2000) = 1.39 (50.38) X
1013cm3 mol’ s’).
Neither the DMBE I1 nor DMBE I11 potential energy surfaces
+ from the present work explicitly treat all the finestructure states
of O(3P) OH(211). Indeed, no representations so far published
for this system have done so, including ones4that has just recently
appeared. Quantum mechanically, this can be done by a Gen
+ tryGieseSs type analysis of
applied to 0 OH produces
th a
e longrange matrix of 36
force which when states. The diago
nalization of this matrix as a function of angle of approach and
(54) Lemon, W. J.; Hase, W. L. J . Phys. Chem. 1987, 91, 1596. (55) Gentry, W. R.; Giese, C. F.J . Chem. Phys. 1977, 67, 2 3 5 5 .
3740 The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 ? .t
0 W
0 LD
0 U
0
m
\
M
0 N
 0
0
0
1.5
2.5
305
405
505
Rho
Figure 10. Contour plots for the C,, insertion of H into 02.Contours are as in Figure 7 .
ln I
Varandas et al. 6,5
m
ru 
W \
>
m
N
0 id
' 1
, m
1.
'2.
'3.
'4. '5.
'6.
'7.
'8.
ria,
'la
'2. '3.
'4.
'5.
'6.
'7. '8.
3.
r/a,
Figure 11. Isotropic (V,) and anisotropic (VI, V,, V,, V,, and V,) components of the 0OH interaction potential, with the molecule ilxed at the
v,; v,; v,. v,; equilibrium diatomic geometry, for 1 Ir I9 q,. Note that r is now the distance from the 0 atom to the center of mass of OH: () V,; () V,;
(.)
(*..) (    ) (.)
A Realistic HO2(a2A'') Potential Energy Surface 2
 N
DMBE I1
The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 3741
m
.cP
W \
>
N m
0
d
I I I
w I
I
I
1
1
. m I
'2
'3
'4 a
'5
'6
'7 a
'8 a
r/a,
. '2.
'3.
'4 *
'5.
'6.
7. '8.
3.
r/a,
Figure 12. Isotropic (Vo)and anisotropic (V2,V4,V,,V,,and Vlo) components of the H a 2interaction potential, with the molecule fixed at the equilibrium diatomic geometry, for 1 5 r 5 9 ao: () Vo;() V2;() V4;(.a) V,;(    ) V,;(  e  ) VI,,,
. '2.
'3. ' A .
3m '6.
'7.
'8. '9.
r/a,
Figure 13. Comparison of Vofor the 0OH interaction with that ob
tained from other models. Symbols are as in Figure 5.
distance of 0 from OH [email protected] 18 doubly degenerate potential energy surfaces that include the coupling and the quadrupole axis orientation effects mentioned in section 2.1; the lowest doubly degenerate potential energy surface so obtained is therefore the adiabatic potential that should go smoothly to the MeliusBlint electronic structure calculations. Yet, although spinorbit coupling
+  TABLE VIII: Geometries and Energies of the Metastable Minima
+ and Saddle Points hedicted by the DMBE Potentials for the
H O2 OH 0 and Isomerization Reactionsa DMBE
property
I"
I1
I11
TShaped HO2 Structure
Rdao
2.243
2.230
R2/00
4.927
4.932
R3/aQ
4.927
4.932
HOO/deg
76.84
76.94
V/Eh
Rdao
 0.2012
0.205 0
HydrogenBonded OH. e 0 Structure
5.824
5.680
R2/aQ
1.875
1.888
Rdao
3.949
3.792
HOO/deg
0
0
0.1796
0.1784
2.267 4.824 4.824 76.41 0.2039
5.808 1.863 3.945 0 0.1807
Saddle Point Structure for the H + O2Reaction
Rl/%
2.284
2.268
2.289
R2lao
3.529
3.547
3.426
Rdao
4.702
4.730
4.648
HOO/deg
106.00
106.80
107.16
0.1925
0.1967
0.1921
Saddle Point Structure for the H 0 2 Isomerization
Rl/aQ
2.749
2.783
2.641
R2lao
2.192
2.212
2.217
Rdao
2.192
2.212
2.217
HOO/deg
5 1.16
51.01
53.45
0.2257
0.2 332
0.2509
Energies, in Eh, are taken realtive to three isolated atoms.
+ may be important and should ultimately be included in any
representation of the 0 OH interaction, it is also possible that for calculations of thermal rate constants and other macroscopic observables an explicit treatment of spinorbit coupling may be
J. Phys. Chem. 1988, 92, 31323742
nuclei interacting with Ln3+ions (in which case y in eq 3 refers to the nucleus and r to the electronnuclear distance). As part of an NMR study of lanthanidebound micelles,we have measured the proton Ti relaxation times for SDS micelles (0.07 M surfactant) to which a variety of Ln3+ ions (0.002 M) had been added.22 The quantity of interest, plotted as hollow squares in Figure 1, is the relaxation enhancement for the CH, group in SDS
bound directly to the sulfate, defined as (l/Tl(L)) (l/Tl(’)),
where TI(La)nd are the proton spinlattice relaxation times in the presence and absence of lanthanide, respectively. (Gd3+ was not included because it produced line widths too broad to measure properly.) For Ln3+ions the dipolar interaction typically dominates the Fermi contact term.19,23 Note that the pattern
~~~~~~
~
(22) In this concentration regime TI’ was not linear with Ln” concentration. Therefore, all measurements were carried out at the same concentration.
(23) Alsaadi, B.; Rossotti, F.; Williams, R. J . Chem. SOC.,Chem. Commun. 1980, 2147.
of dipolar Ti enhancements (hollow squares) does not match the pattern of k, values (filled squares). This argues against a significant dipolar contribution to k,.
Conclusion
We have measured bimolecular quenching rate constants k, for interaction of lanthanide ions with the 1,9biradical 2. The evidence so far suggests that spin exchange is the principal quenching mechanism. The dipolar mechanism does not appear to have a major influence on the quenching. Further investigations, including the magnetic field dependence and chain length dependence of k,, and lanthanide effect on intramolecular product ratios, are in progress.
Acknowledgment. The authors thank the National Science Foundation and Air Force Office of Scientific Research for support. C.D. thanks the N S F (CHE8421140) for support. K.C.W. thanks the National Institutes of Health for a postdoctoral fellowship, NCINIH No. CA07957.
A Realistlc H02(g2A”)Potential Energy Surface from the Double ManyBody Expansion Method
A. J. C. Varandas,* J. BrandHo, and L. A. M. Quintalest
Departamento de Quimica, Universidade de Coimbra. 3049 Coimbra Codex, Portugal (Received: July 21, 1987; In Final Form: December 29, 1987)
+  + A double manybody expansion potential energy surface reported previously for H02(R2A”)and referred to here as DMBE
I is modified to produce thermal rate coefficients for the reaction 0 OH O2 H in good agreement with experiment. This new potential energy surface will be referred to as DMBE 11. By the further imposition that the potential function should reproduce the experimentalspectroscopicforce field data for the hydroperoxylradical, another potential energy surface has been obtained, DMBE 111. Both of these improved DMBE I1 and DMBE 111 potential energy surfaces preserve the functional form used previously for DMBE I except for the longrange 0..OH electrostatic interaction, which is defined in the spirit of a more satisfactory adiabatic theory.
1. Introduction
The potential energy surface fo_r the electronic ground state of the hydroperoxyl radical, H02(X2A”), is important in under
+  + standing the chainbranching reaction (i) H O2 OH 0 +  of many combustion processesi and its reverse (ii) 0 OH
+ 0, H, both of which are also important in the HO, cycle of
atm are
ospheri studies
c c of
hemi~ isotope
try exc
. ~C. ~onnected hange4in 0
+with reactions OH and the
( v
i i
) b
an rat
d (ii) ional
relaxation5 of O2in collisions with H. It is also important for
theoretical studies of the vibrationalrotational spectroscopy of H02(R2A”) and, as a building block, for construction of the
potential energy functions of larger polyatomics (which have
groundstate H 0 2as a dissociation fragment) from the manybody expansion (MBE)6 and double manybody expansion (DMBE)’*
methods. Thus, it is not surprising that there has been a considerable theoretical effort to arrive at a reliable potential energy surface for the electronic ground state of the hydroperoxyl radical, with use of both a b initiog14and semiempiri~al’~’m’ ethods.
On the experimental side, the dissociation en erg^,'^,'^ the ge
ometry,6s202a1nd the quadratic force constants620of the hydroperoxyl radical have also been reported. An extensive list of references to spectroscopic studies by a variety of techniques
covering a wide range of the spectra can be found in ref 14. Once a reliable threedimensional (3D) working potential energy
surface is available, it can be used for dynamics calculations by
‘Permanent address: Facultad de Quhica, Universidad de Salamanca, 37008 Salamanca, Spain.
00223654/88/20923732$01.50/0
using either the classical trajectory method4,5,222o9r approximate quantum mechanical t h e o r i e ~ ~ ’a.n~d~for variational transition
(1) Benson, S . W.; Nangia, P. S. Acc. Chem. Res. 1979, 12, 223. (2) Rowland, F. S.; Molina, M. Rev. Geophys. Space Phys. 1975, 13, 1. (3) Lee, Y. P.; Howard, C. J. J . Chem. Phys. 1982, 77, 756. (4) Miller, J. A. J . Chem. Phys. 1981, 75, 5349. (5) Miller, J. A. J. Chem. Phys. 1981, 74, 5120. (6) Murrell, J. N.; Carter, S.; Farantos, S.C.; Huxley, P.; Varandas, A. J. C. Molecular Potential Energy Functions; Wiley: Chichester, 1984. (7) Varandas, A. J. C. Mol. Phys. 1984, 53, 1303. (8) Varandas, A. J. C. THEOCHEM 1985, 120, 401. (9) Melius, C. F.; Blint, R. J. Chem. Phys. Lett. 1979, 64, 183. (10) Langhoff, S. R.; Jaffe, R. L. J . Chem. Phys. 1979, 71, 1475. (11) Dunning, T. H., Jr.; Walch, S. P.; Wagner, A. C. In Potential Energy Surfaces and Dynamics Calculations; Truhlar, D. G.,Ed.; Plenum: New York, 1981; p 329. (12) Dunning, T. H., Jr.; Walch, S. P.; Goodgame, M. M. J . Chem. Phys. 1981, 74, 3482. (13) Metz, J. Y.; Lievin, J. Theor. Chim. Acta 1983, 62, 195. (14) Vazquez, G.J.; Peyerimhoff,S. D.; Buenker, R. J. Chem. Phys. 1985, 99, 239. (15) Farantos, S. C.; Leisegang, E. C.; Murrell, J. N.; Sorbie, K. S.; Teixeira Dias, J. J. C.; Varandas, A. J. C. Mol. Phys. 1977, 34, 947. (16) Gauss, A., Jr. Chem. Phys. Lett. 1977, 52, 252. (17) Varandas, A. J. C.; Brandlo, J. Mol. Phys. 1986, 57, 387. (18) Howard, C. J. J. A m . Chem. SOC.1980, 102, 6937. Foner, S . N.; Hudson, R. L. J. Chem. Phys. 1962, 36, 2681. (19) Ogilvie, J. F. Can. J. Spectrosc. 1973, 19, 171. (20) Beers, Y.; Howard, C. J. J . Chem. Phys. 1976, 64, 1541. (21) Tuckett, R. P.; Freedman, P. A,; Jones, W. J. Mol. Phys. 1979, 37, 379, 403. (22) Gauss, A,, Jr. J . Chem. Phys. 1978, 68, 1689.
0 1988 American Chemical Society
A Realistic HO2(WZA") Potential Energy Surface
state3, and adiabatic ~ h a n n e ltr~ea~tm. ~en~ts; ref 34 reports also
simplified 1D shortrange/longrange switching models for the
radial and angular interaction 0OH potential in the context of
the adiabatic channel model.
Ab initio electronic structure calculations of the groundstate
H 0 2potential energy have been carried out by Melius and Blint:
Langhoff and Jaffi5,'O Dunning et al.,l'*lzMetz and Lievin,13and
Vazquez et al.I4; for references to earlier work see ref 9. Some
of these a r t i ~ l e s ' ~ *h'a~veJ ~also considered excited states of HO,.
Melius and Blintgcarried out MCSCF C I calculations and defined
a global HOzsurface by fitting the calculated points to an ad hoc
funct mol'
i
ona for
l t
fo he
rm H
. +
This O2a
s d
u d
rface ition
gives reacti
a on
b .
arrier Using
of a
abo mor
u e
t 2.3 kcal extensive
C I wave function, Langhoff and JaffE'O reported no barrier for
this addition step. Dunning et a1.1'q12carried out generalized
valence bond calculations and concluded that by taking the
computational deficiencies into account, it is likely that there is
no barrier to the addition reaction. Metz and LievinI3 and
Vazquez et al.I4 concentrate on the electronic spectra and UV
photodissociation of HOzand hence do not address this barrier
problem on the groundstate potential surface of H02.
In a recent articleI7 (hereafter referred to 5s article I), we
reported a potential energy surface for HOz(X2A") using the
DMBE method. This HOZDMBE I potential surface conforms
with the a b initio data of Melius and Blintg for the threebody
energy and shows reasonable agreement with available spectro
+  + scopic data for the energy and geometry of the minima referring
to the equilibrium triatomic. In agreement with the best ab initio estimates, it also shows no barrier for the H O2 OH 0 reaction. In addition, the H02DMBE I potential energy surface
predicts two secondary minima that have chemical interest. One
refers to a Tshaped (C2J HO2 weakly bound complex, while
the other is related to a linear (CJ weak hydrogenbonded 0.6HO
structure. Moreover, it shows the appropriate longrange behavior
of the HO2 and 0OH asymptotic channels being the electro
static energy defined in the spirit of an adiabatic theory recently
proposed by Clary and Werner.31 This is particularly significant since Clary30$3h1as shown that the longrange forces are major
+  factors in determining the rate constant for the 0 O H O2
+ H reaction. Such importance has most recently been stressed
by T r ~ ine hi~s s~tatistical adiabatic channel model. Along the
+ same direction, Wagner35has explicitly considered coupling be
tween the longrange 0 OH spinorbit potential curves. Despite the fact that the HO, DMBE I potential energy surface
+  + shows some definite improvement over previous functions, it is
not completely satisfactory. For example, exploratory trajectory resultsz9for the 0 OH O2 H reaction have yielded thermal rate coefficients that significantly underestimate the best available
experimental measurements (for a critical review, see ref 36). In
contrast, similar calculations carried out on the MeliusBlint fitg
agree wellz8with the experimental results. The aim of this work
is ther_eforeto report a new DMBE potential energy surface for
H02(X2A") that overcomes such difficulty. In addition we suggest
an alternative, perhaps more satisfactory, adiabatic description
of the longrange OOH electrostatic interaction. Finally, we show
how to make the potential reproduce the complete quadratic force
field of the hydroperoxyl radical as surveyed in ref 6 and 21, while
maintaining a good description of the rate coefficient measure
(23) Blint, R. J. J . Chem. Phys. 1980, 73, 765. (24) Bottomley, M.; Bradley, J. N.; Gilbert, J. R. Int. J . Chem. Kinet. 1981. 13. 957. (25) Gallucci, C. S.; Schatz, G. C. J . Phys. Chem. 1982, 86, 2352. (26) Brown, N. J.; Miller, J. A. J . Chem. Phys. 1984, 80, 5568. (27) Kleinermanns, K.; Schinke, R. J . Chem. Phys. 1984, 80,1440. (28) Miller, J. A. J. Chem. Phys. 1986, 84, 6170.
(29) Quintales, L. A. M.; Varandas, A. J. C.; Alvarifio, J. M. J . Phys. Chem., in press.
(30) Clary, D. C. Mol. Phys. 1984, 53, 3. (31) Clary, D. C.; Werner, H. J. Chem. Phys. Lett. 1984, 112, 346.
(32) Rai, S. N.; Truhlar, D. G. J . Chem. Phys. 1983, 79, 6046. (33) Cobos, C. J.; Hippler, H.; Troe, J. J . Phys. Chem. 1985, 89, 342. (34) Troe, J. J. Phys. Chem. 1986, 90, 3485. (35) Wagner, A,, private communication.
(36) Cohen, N.; Westberg, K. R. J . Phys. Chem. ReJ Data 1983, 12, 531.
The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 3733
H
R
n U
v n
+  + Figure 1. Coordinates used to define the HOzpotential energy surface.
ments for the 0 O H Oz H reaction. This article is organized as follows: Section 2 outlines the
mathematical form of the energy terms in the DMBE and describes the approach used to obtain the numerical values of the parameters they contain from available ab initiogand spectroscopic6,21data. Section 3 presents a discussion of the results. The conclusions are gathered in section 4.
2. New DMBE H 0 2 Potentials
The DMBE of the potential energy for groundstate HOz has the form37,38
I/ = VEHF(R)+ Vwrr(R)
(1)
where E H F denotes an extendedHartreeFocktype energy that includes the nondynamical correlation due to degeneracies or neardegeneracies of the valence orbitals, and corr is the dynamical correlation energy due to the true dynamic correlation of the electrons; R = R,, R2,R3is a collective variable of the internuclear separations, which are defined in Figure 1. In eq 1, both the E H F and corr energy terms are written as a manybody expansion+
3
vEHF(R) = i 1GzAF,i(Ri) + %?lF(R1,R2,R3) (2)
3
Vcorr(R) = fl!!r,i(Ri) + G%(R18Z?R3)
(3)
i=l
where all twobody and threebody fragments are assumed to be
in their groundelectronic states as predicted from the Wigner
Witmer spinspatial correlation rules; i = 1 is taken to label the
homonuclear 00 bond, and i = 2 and 3 label the two OH bonds.
A detailed review of the theory has been given e l s e ~ h e r aend~ ~ ~ ~ ~ ~ ~
will not be repeated here. Instead, we refer only to those aspects
that are essential for this work. The notation is mainly that of
article I; for further clarity, we use the initials EHF37,38rather
than HF7,s,17to represent the extendedHartreeFocktypeenergy.
In article I the twobody energy terms assumed the form of the
realistic EHFACE
and are thus kept unchanged in the
current work. They have the general form
+ 3
V&(R) = DR"(1 C a d ) exp(yr)
(4)
i= 1
C v!;r(R) = 
CnXn(R)R"
(5)
n=6,8.10
where =  Rm is the
''Ordinate
from the
equilibrium diatomic geometry. In eq 5, xnare dispersion damping
(37) Varandas, A. J. C. In Structure and Dynamics of Weakly Bound Complexes; Weber, A., Ed.; D. Reidel: Dordrecht, 1987; p 357.
(38) Varandas, A . J. C. Adv. Chem. Phys., in press. (39) Varandas, A. J. C.;Dias da Silva, J. J . Chem. SOC.F, araday Trans. 2 1986, 82, 593.
3734 The Journal of Physical Chemistry, Vol. 92. No. 13, 1988
Varandas et al.
TABLE I: Coefficients Used for the GroundStateO2and OH Potentials"
coeff
02
OH
0.14291 3.64459 3.92812 2.09867 3.35225 0 5.66169 2.2818 15.40 235.22 4066.24
0.13825 2.65648 1.74505 0.71014 2.54533 0 6.29489 1.8344 10.00 180.45 3685.26
nCalculated from equations 4 and 5; see also text. All values are in atomic units. bCalculated from eq 6a with ( r H 2 )= 3a: and43( r o 2 ) =
2.0043a02.
TABLE I 1 Coefficients for the ThreeBody Correlation Energy"
k6 = 2.46501 (2) 76 = k'6 = 0.68758
OH bondb*'
ks = 5.03696 (2) 78 = k', = 0.82542
klo = 6.29438 (2) 710 = k'lo = 0.94034
k6 = ?.7_8478 (1)
76 = k 6  0.95274
00 bondsbed ks = 4.68155 (1)
78 = k'8 = 0.94148
klo = 1.20507 (0) 710 = k'lo = 0.72379
Calculated from eq 7; units are as in Table I. Given in parentheses
are the powers of 10 by which the numbers should be multiplied, e.g.,
2.46501 (2) = 2.46501 X
b9',, (n = 6, 8, 10) are taken equal to
unity for all diatomic fragments. c R o = 1.8344 ao. d R o = 2.2818 ao.
functions (for recent references to earlier work on dispersion damping see ref 38 and 40) defined bySv4l
XfltR) = [1  exp((A,R/p)  (B"R2/P2))1" (6a)
where
A, =
(6b)
+ B, = POexp(P14
(6c)
p = (R, 2.5Ro)/2
(64
Ro = + 2((rX2)'l2 (rY2)'l2)
(6e)
is the Le Roy42distance for the onset of breakdown of the as
ymptotic R"perturbation series expansion, X and Y label the
two interacting atoms, (rX2)and (ry2) are the corresponding expectation values for the squared radii of the outer orbitals of
X and Y , and ai and p i (i = 0, 1) are dimensionless universal
constants8for all isotropic interactions: a,, = 25.9528, al= 1.1868,
Bo = 15.7381, and = 0.09729. (Unless mentioned otherwise,
all values reported are in atomic units: 1 hartree ( E h )= 1 au of energy = 4.359 821 5 aJ; 1 bohr (ao) = 1 au of length = 0.529 177
A = 0.052 917 7 nm.) Table I define? the numerictl values of
the coefficients in eq 46 for the OH(X211) and 02(X3Z,) dia
tomic fragments.
Also the threebody dynamical correlation energy, which is calculated semiempirically within the DMBE framework, is defined as in article I. It is written as37938944
3
I3 E X R ) = C
CnXn(Ri)(l %kn(Ri+l(mrx13)) X
i=l n=6.8,10
hn(Ri+z(mod3)) + gn(Ri+2(md 3)) hn(Ri+l(mod 3))IRT"I (7a)
where
gn(Ri)= 1 + k,(i) exp[k',(')(R,  R i O ) ] (7b)
h,(Ri) = [tanh (7,,(i)R,)]q'2)
(7c)
and R: is a reference geometry; for the numerical values of the
40) Knowles, P. J.; Meath, W. J. Mol. Phys. 1987, 60, 1143. 41) Varandas, A. J. C.; BrandHo, J. Mol. Phys. 1982, 45, 857.
42) Le Roy, R. J. Mol. Spectrosc. Chem. Soc. (London) 1973, I, 113.
43) Desclaux, J. P. At. Data 1973, 12, 311. 44) Varandas, A. J. C.; Brown, F. B.; Mead, C. A.; Truhlar, D. G.;Blais, C. J. Chem. Phys. 1987, 86, 6258.
TABLE III: Coefficients for the ThreeBody Electrostatic Energy Term"
OH bondsb K4 = K5= 0.O883lc Kb = K; = q4 = v 5 = 2.54533
00 bondb K4 = K , = 0.0
k h = K', = 74 = 7, = 3.35225
Calculated from eq 8, with 6 = 4; all quantities are in atomic units.
bq'n (n = 4, 5) are taken equal to unity for all diatomic fragments. cK4 = K5= ( l / R 0 ) 4 with R o = R, = 1.8344 ao.
coefficients in eq 7, see Table 11.
Modifications that lead to the improved DMBE potentials of
the current work refer therefore only to the threebody E H F
energy term and to the description of the 0OH electrostatic
energy, which is treated" separately from the rest of the threebody
E H F term. We therefore begin our discussion with this elec
trostatic energy term.
2.1. ThreeBody Electrostatic Energy. As in previous ~ o r k , ' ~ , ~ ~
the threebody electrostatic energy term is similarly represented
by
c?j(R1)/2c c 3
=
[CnGn(Ri+l(mod3)) hn(Ri+2(mod3)) +
1=1 fl=4,5
C'~Gn(~i+Z(mo3d)) hn(Ri+l(mod3))I Xn(Ri)Rr" (8a)
where ele stands for electrostatic
G,(R,) = K,R; exp[K',(')(R,  R i O ) ]
(8b)
and h, is defined by eq 7c. Note that C, represents the longrange
+ electrostatic coefficient for the atomdiatom interaction involving
the i l(mod 3) diatomic and the remaining atom of the ith pair
+ (that associated with Ri),and C',, has a similar meaning but refers
to the other atom of the ith pair with the i 2(mod 3) diatomic.
Similarly, x,,denotes the damping function of order n for the ith
diatomic fragment, which has been defined by eq 6a. Since there
is no H00 longrange electrostatic interaction, one must have
G,(Rl) = 0; see Table 111.
F$;i As for the K,, and K',, coefficients, they are determined from
the requirement that given values for 7, and v,',
should
reproduce the longrange electrostatic energy that results from
the interaction between the permanent quadrupole moment (e,)
of the 0 atom and the dipole (&) and quadrupole (0,) moments
of OH; a labels the 0 center, while b stands for the center of mass of OH. Note that the quadrupoledipole interaction (ea&) leads
to an energy contribution that varies as C4r4,while that for the
quadrupolequadrupole (e,+,) interaction varies as CSis,where
C4and C, are defined by4s
c4 = 74/,8,Fb[COS8b(3 COS2 8,  1) 
2 sin 8, sin 8b cos 8, cos dab] (9)
+ cs = 3/168,8b(1  5 cos2 8,  5 cos2 8b 17 cos2 8, cos2 o b +
+, 2 sin28, sin2 8, cos2 
16 sin 8, sin eb cos 8, cos 8, cos +ab) (10)
where r is the S O H centerofmass separation, 8, and 8b are the
angles made by the axis of the electric multipole moments with
r, and +ab is the dihedral angle between those axes; in the present calculations, the electric multipole moments for the 0 atom and O H diatomic are given the values468, = 1.60 eao2and47& = 0.656 eao and480, = 1.37 euo2,respectively. Note especially that the C,and C, coefficients depend on the atomdiatom orientation though they vanish when averaged (for a fixed atomcenter of mass of the diatomic separation) over the angles 8,, Bb, and +ab
= 6, 4b.4s
To define C4and C,, we have, in article I, followed the adiabatic theory of Clary and Werner,31which constrains the angle 8, at 0, = 0 such as to give the lowest value for the dipolequadrupole and quadrupolequadrupole electrostatic energies. However, Clary's approach assumes that the quadrupole axis of the 0 atom always lies along the vector connecting the atom to the center of
(45) Hirschfelder, J. 0.C;urtiss, C. F.; Bird, R. B. The Molecular Theory of Gases and Liquids, 2nd ed.; Wiley: New York, 1964.
(46) Fisher, C. F. At. Data 1973, 12, 87. (47) Meerts, W. L.; Dymanus, A. Chem. Phys. Lett. 1973, 23, 45. (48) Chu, S. I.; Yoshimine, M.; Liu, B. J . Chem. Phys. 1974, 62, 5389.
A Realistic H02(g2A") Potential Energy Surface
r
1
The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 3135

I
c.!
N! , ,
,,.,,.,,,, ,
.,,,,,,,,,,.,,,.,.,
e , / I
0.' ' '36. ' ' 6 6 . ' '3d. ' '!go:
'150.' 'l#O*~ '2i'On' '2d0.' 'Z?Om' ,300.' '33'0: '360.
d e g r e e s
Figure 2. Optimum values of the C, and C, 0OH longrange electrostatic coefficients as a function of the angle Ob (this may be either O2 or
O3 in the notation of Figure l ) , the angle formed by the OH axis with the line connecting the remaining 0 atom with the center of mass of OH. The threeterm Fourier analysis of eq 11 is essentially indistinguishable
within the scale of the figure, and hence it is not shown.
TABLE IV: Values of the Expansion Coefficients in the Fourier
Series Analysis of Eq 11"
uarameter
Cp
bo b, b2 b,
b4
b5
b6
max. dev (three terms), % max. dev (four terms), %
1.987 67 0.393 61 0.195 76
Ob
0.009 78
Ob
0.000 94 1.1 0.1
4.29002
Ob
1.043 70
Ob
0.1 15 02
Ob
0.01672 1.7 0.1
Units are
and
for C., and c,, respectively. Smaller in
absolute value than 0.0001 au.
mass of the OH molecule. As a result, there is a change of sign of the potential as the angle of approach of the 0 atom changes from head on with the 0 end of OH to head on with the H end of OH.
A more satisfactory adiabatic description of the longrange
electrostatic interaction is to let the electronic distribution on the 0 atom instantaneously adjust to the OH electronic charge distribution, whatever the angle of approach, so as to produce the lowest potential energy. We will follow this approach in the
present work. Since a quadrupole can be thought of as having one charge on each comer of a square with the charges alternating in sign around the square, this would correspond to a change in its angle of approach by 180'. Thus, the negatively charged
diagonal axis of the quadrupole square will be coincident with the OH axis when 0 approaches the H end of OH, b u t t h e positively charged diagonal will be coincident if the 0 end is approached. W e choose therefore, for a fixed value of ob, the value of Oa that gives the lowest interaction potential energy. Figure 2 shows the optimum values of C, and C, obtained from this procedure. They can be described in terms of a Fourier expansion:
where the coefficients for N = 6 are given in Table IV. Note
XI;, Figure 3. Contours of the electrostatic energy, eq 8, for an 0 atom
moving around an equilibrium OH with the center of mass fixed at the
origin. Contours are equally spaced by 0.01 E,, starting at A = 0.20
Eh.
0
9
c
T
i <\)\\\ , I / j , c.
~
,
, ,
, I ,\
5.0 4.0 3.0 2.0 !,l0.o ! . O 2,o 3.0 4,O 5.0
X/a,
Figure 4. Same as in Figure 3 but for a H atom moving around an equilibrium 02.
that sine terms in the Fourier series expansion vanish due to symmetry reasons. Note especially that a threeterm Fourier
analysis reproduces the calculated data within 12% and that a conversion to the (R1,R2,R3se)t of coordinates can be made by using the pseudoangular coordinate reported in eq 11 of article
I; see also ref 38. However, rather than using qp'w,e adopt a
simpler approach that is similar to that employed for the dispersion
coefficient^.^^^^^^^ Accordingly, we parametrize eq 8 so as to
reproduce the spherically averaged values of these optimum longrange coefficients for the 0. OHelectrostatic interaction energy. To perform numerically the sineweighted O b averaging,
we have used a GaussLegendre quadrature technique, having
obtained ( e 4=) 0.929Ehao4and (C,) = 1.790Ehao5.
Figure 3 shows contours of the electrostatic energy, as predicted from eq 8, for an 0 atom moving around an equilibrium OH diatomic. A similar contour plot for the H atom moving around an equilibrium O2 is shown in Figure 4. In contrast to the electrostatic energy from article I that shows, as a polar plot in 0OH radial distance and angle of approach, a sign change at the line corresponding approximately to a 90' insertion of 0 into
the middle of OH, the current approach produces longrange electrostatic potentials that are attractive at every angle of approach though they are least attractive for the perpendicular insertion of 0 into OH.
Although the change of sign in C, and C5of article I might affect the dynamics at low collision energies, it is hardly expected
from the magnitude of the energies involved that this sign change
may significantly alter the dynamics at high energies. Indeed, exploratory dynamics studies29on the H 0 2 DMBE I potential
+ + energy surface have shown that a small energy barrier along the
minimum energy path for the 0 OH + O2 H reaction was responsible for the poor agreement between the calculated thermal rate coefficients and e ~ p e r i m e n t . ~ ~ ~T~h' ~is' barrier, which also
(49) Lewis, R. S.;Watson, R. T. J . Phys. Chem. 1980, 84, 3495. (50) Howard, M. J.; Smith, I . W. M. J . Chem. SOC.F, araday Trans. 2 1981, 77, 997. ( 5 1) Deleted in proof.
(52) Frank, P.; Just, Th. Ber. BunsenGes. Phys. Chem. 1985, 89, 181.
3736 The Journal of Physical Chemistry, Vol. 92, No. 13, 1988
appears on the 0OH spherically averaged interaction energy
N
curve, must therefore be removed. For this purpose a further
I
leastsquares fit to the threebody ab initio E H F energies has been
carried out as described in the following subsection.
2.2. Nonelectrostatic ThreeBody EHF Energy. As in article
I, the remainder of the threebody E H F energy assumes the form
Varandas et al.
where the prime denotes that the electrostatic energy is left out, and Qi ( i = 13) are the D3*symmetry coordinates defined by
Also as in article I we define yo(’) =  yl(‘)RYr,where RIref(i =
13) is a reference C, geometry defined through the experimental
equilibrium H 0 2geometry (denoted R: (i = 13); note that the
+ latter differs slightly numerically from that
follows: Rlref= Rleand Rzref= R F f = (R:
used in R3e)/2.
ref 17) as Moreover,
the value of yl(’)has been taken from the nonlinear leastsquares
fit carried out in article I, while those of yl(2)= yl(3)have been
obtained through a trialanderror procedure by carrying out linear
leastsquares fits (the leastsquares parameters being VO and c,,
with i = 133) to the remainder threebody energies which are
obtained by subtracting the sum of the threebody dynamical
correlation energy (see ref 17) and the threebody electrostatic
energy from the threebody correlated ab initio energies of Melius
and B l i ~ ~(tA. l~though the ab initio surface of Melius and Blint
gives inaccurate dissociation energies and hence is questionable
for dynamics studies, we believet7that due to a cancellation of
errors the ab initio threebody energy contributions have greater
reliability. By combining the latter with accurate semiempirical
curves for the diatomic fragments, one should therefore get an
improved description of the complete triatomic potential energy
surface.) Thus, leastsquares fits to the 365 data points so obtained
have been generated until no nonphysical minima appeared in the
final potential energy function, and the corresponding barrier in
+ the 0 OH minimum energy reaction path (or alternatively in
the spherically averaged component Voof the OOH interaction
+ potential; see later) was below the 0 OH dissociation limit; see
Figure 5. Note that this barrier separates the shallow minimum
associated with the 0...HO hydrogenbonded structure from the
deep chemical minimum corresponding to equilibrium H02. In
our previous HO, DMBE I potential the minimum associated with
this 0. .HO hydrogenbonded structure was separated from the
deep H 0 2chemical well by an energy barrier of at least 0.13 kcal
mol’ (relative to 0 + O H ) located at R l = 4.60ao, R2 = 1.84ao,
+ R3 = 4.63ao. A simple way to remove the positive barrier in the 0 OH minimum energy reaction path has been to allow small adjustments
into the poo and DOH rangedetermining parameters of the two
body Morse potentials used by Melius and Blintg (which are then
subtracted from the H 0 2energies to calculate the threebody ab
initio energies) while yl(2)= y1(3w) ere varied freely as discussed
above to yield the best leastsquares fit. To improve the reliability
of our prediction at the equilibrium geometry of the hydroperoxyl
i:I‘I ,
7.
Figure 5. Potential along the minimum energy reaction path as a function of the separation, in angstroms, between the 0 atom and the center of mass of OH: (*) MeliusBlint9 potential surface; () MBE”
potential surface; ( ) DMBE I” potential surface; (. .) this work
DMBE I1 potential surface; () this work DMBE I11 potential surface.
TABLE V Weights Used for the LeastSquares Fitting Procedures
Related to the H 0 2 DMBE Potentials of This Worko
DMBE I1
a
b
initio W, =
+points
1 99
(i = exp
1365): [lOO~
~
.
,
(
R,
R:)’]
special point
no.
Rl
R2
R3
V
wt
366 5.521 810 1.906955 3.614856 0.178 168 100’
DMBE 111 ab initio points (i = 1365): W, = 1  e x p [  l O O ~ ~ ~ , ( RR,:)]’
suecial uoints
no.
R,
R2
R3
V
wt
366 367 368 369 370 371373 374379
5.521 810 2.512 1.85 2.226 492 2.283 606 2.512 2.512
1.906955 1.843 1.8344 4.542 893 7.263427 1.843 1.843
3.614856 3.457 0.99 4.542 893 7.263 427 3.457 3.457
0.178168 0.27 47
0.34 097 0.20 1568 0.19 2700 1st der
2nd der
1 Ob 1 OOOC
Id
1000d 1000d 10000‘ 10/Fz;f
“Units are as in Table I. bOH...O hydrogenbonded structure. Experimental binding energy of H02.6J821dEstimated energy from the H 0 2 DMBE I1 potential surface of the present work. e Condition of zero first derivativeat the equilibriumgeometry of H 0 2 . /Condition to impose the quadratic force constants of H 0 2 : FI1= 13’V/dR2~;F22 = a2v/aR22;F,, = a2v/aff2F; , =~ a2v/aR,aR2;F,, = a2v/aR,affF; ~~ = a2vja~~a~.
radical, we considered a leastsquares weight function in fitting
+ + the nonelectrostatic threebody residual E H F energies, namely, W = 1 + 99 exp[g((Rl  RIe), (R,  R2e))Z (R3  R3’)*)];
see Table V. Note that the diatomic Morse parameters (poo and
POH) were not specified by Melius and Blint’ for the ab initio
energies, and we adopted, somewhat arbitrarily, the values Boo
= 1.45ao’ and POH = 1.38ao’ in article I. To reduce the current
leastsquares fitting procedure to a twovariable trialanderror
search, we have forced the poo/@oH ratio to be equal to thats3
(53) Huber, K. P.: Herzberg, G. Molecular Spectra and Molecular Structure: Constants of Diatomic Molecules: van Nostrand: New York, 1979.
A Realistic HOZ(R2A") Potential Energy Surface
The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 3737
TABLE VI:
Coefficients for the Residual ThreeBody ExtendedHartreeFock Energy Term"sb
VO = 5.8472 (0) CI = 9.8340 (1) c2 = 2.1836 (0) cj = 3.8540 (1) cq = 3.8215 (2) c5 = 1.9349 (0) ~6 = 5.5602 (1) CT = 7.1609 (2) cg = 7.0660 (2) cg = 6.5073 (2)
cIO = 6.2814 (1) c ~ I= 3.8207 (1) c12 = 1.0142 (1) c13 = 6.0124 (3) c14 = 2.5117 (2) CIS = 5.7318 (2) ~ 1 =6 2.0818 (2) c17 = 8.8011 (2)
DMBE I1
c1g = 8.5610 (2) ~ 1 =9 2.7164 (2) ~ 2 =0 1.1228 (2) c2I = 2.7195 (2) ~ 2 =2 1.5328 (4) ~ 2 =3 2.4537 (3) c24 = 1.1601 (2) ~ 2 =5 1.3576 (3)
c26 = 7.5568 (3) ~ 2 7= 4.4659 (3) c2g = 6.0668 (3) c29 = 2.0637 (3) c3g = 1.8701 (3) cjl = 3.9858 (3) ~ 3 =2 2.7786 (3) cj3 = 2.5483 (4)
1(O(I) = 2.46683 (0) yo(') = 1.64300 (0) yl(')= 0.98202 (0) y1(2'= 0.62 (0)
yo(3)= 1.64300 (0) yl(') = 0.62 (0)
VO = 2.0245 (1) CI = 1.0827 (0) ~2 = 1.5532 (0) cj = 4.8201 (1) cq = 9.9026 (2) c5 = 1.2977 (0)
c6 = 4.0758 (1) ~7 = 1.0942 (1) cg 5.1977 (2) ~9 = 6.5829 (4)
~ 1 =0 4.0323 (1) c11 = 2.5285 (1) c12 = 3.3377 (3) ~ 1 =3 1.2622 (2) ~ 1 =4 8.4826 (3) CIS = 1.0826 (2) ~ 1 =6 8.0499 (4) ~ 1 =7 5.5122 (2)
DMBE 111
c1g = 5.1866 (2) ~ 1 =9 1.9565 (3) ~ 2 =0 4.6763 (3) c ~ I = 7.6118 (3) ~ 2 =2 5.8673 (4) ~ 2 =3 3.8808 (4) ~ 2 =4 2.1401 (3) ~ 2 =5 1.3589 (4)
C26 = 1.8273 (3) ~ 2 7= 2.7984 (3) c2g = 3.4617 (3)
~ 2 9 1.8739 (4) c30 = 9.4458 (4) cjl = 1.1727 (3) ~ 3 =2 9.5638 (4) ~ 3 =3 3.7996 (4)
yo(') = 2.46683 (0) yo(') = 1.64300 (0) yl(l)= 0.98202 (0) yl(') = 0.62 (0)
= 1.64300 (0) yl(j)= 0.62 (0)
'Calculated from eq 12; units are as in Table I. bGiven in parentheses are the powers of 10 by which the numbers should be multiplied, e.g., 5.8472 (0) =
5.8472 X 10'.
TABLE VII: Spectroscopic Properties of the Hydroperoxyl Radical (C,Mirima)"
DMBE
property
ab initio9 MBEI5qb
Ill
I1
Rdao Rzlao Rdao
HOO/deg
2.58
1.83 3.51
104.7
2.570 1.861
106.0
2.543 1.893 3.525 104.28
Geometry
2.584 1.870 3.530 103.70
I11
2.512 1.843 3.457 104.02
2.52 1.85
104.1
eXpt16,1821
2.57
2.512 f 0.001
1.86
1.843 f 0.004
106
104.02 iz 0.24
Dissociation Energy
De/Eh
0.2488
0.2747
0.2808
0.2797
0.2745
0.2747 * 0.003
F,lIEhao2 F22/EhaO*
Fl2 I Ehao2
F e u / EhaOl
F1,lEhaol
FZu/ Ehalll
0.452 0.497 0.0172 0.277 0.0941
0.0247
0.375 0.418 0.0064 0.240
0.0482 0.0607
0.522 0.329 0.0264 0.445 0.138 0.0068
Force Constants
0.426 0.273 0.0186 0.393 0.1 16 0.0202
0.375 0.417 0.0060 0.241 0.0482 0.0600
0.375 0.418 0.0064 0.240 0.0482 0.0607
'The dissociation energy, in Eh, is taken relative to the three isolated atoms. R Ilabels the 00 bond distance, in ao. *The force field has been fitted to the experimental data reported in this table.
experimentally found for the isolated diatomic fragments, Le.,
&,o//~oH= 1.15. Thus, only DOH and yl(,)= y1(3w) ere system
+ atically varied until the isotropic component of the 0OH in
teraction potential and the 0 OH minimum energy path were
barrier free. This procedure gave POH = 1.42ao' and Boo =
1.633ao' for the final fit. The DMBE potential energy surface (denoted hereafter as
DMBE 11) obtained from the above procedure is numerically defined in Table VI,while the spectroscopic force field it predicts for the hydroperoxyl radical is reported in Table VII. We note
+  + the reasonably good agreement with the best available empirical
estimates and our own force field data from article I. Moreover, quasiclassical trajectories for the 0 OH 0, H reaction
run on this potential surface have produced thermal rate coefficients in good agreement with the best reported experimental v a l ~ e s ;a~de~tai,le~d ~des~cr~iption of these trajectory calculations
as well as those obtained on the DMBE I11 potential surface reported next will be presented el~ewhere.,~
Although the DMBE I1 potential surface shows a definite improvement over the ad hoc potential of Melius and Blint: which was fitted to their own ab initio data, it would be desirable for spectroscopic studies to have a H 0 2 potential function (DMBE 111) that reproduces the experimental quadratic force field reported
elsewhere6*21for this species. This has been accomplished by
having 10 extra conditions imposed (namely, the well depth, three first derivatives, and six quadratic force constants at the HO, equilibrium geometry) as further items in the leastsquares minimization procedure. For example, a first derivative is con
sidered as an extra data point for which the model function is the gradient of the potential energy surface with respect to a given internuclear coordinate; in the present case, such gradient will
be evaluated at the triatomic equilibrium geometry and hence is equal to zero. Thus, we suggest a leastsquares fitting procedure where the ab initio energies and the experimental data are dealt with on an equal footing. To ensure that the calculated and experimental values agreed within error bars, we attributed special weights to the various items as summarized in Table V. Since the resulting function showed a tendency to develop nonphysical minima at the strong interaction region where ab initio data points are unavailable, a further point had to be given to guide the potential at such regions. We have estimated this point from the HOz DMBE I1 potential energy surface reported above. In ad
dition, two extra points (numbers 369 and 370 of Table V) were added to prevent the DMBE I11 potential surface from developing
a small barrier (<1 kcal mol') at the H + O2asymptotic channel;
the associated energies have also been estimated from the DMBE
3738 T h e Journal of Physical Chemistry, Vol. 92, No. 13, 1988
I DMBE I1
.
Varandas et al.
x de
Figure 6. Contour plots for an 0 atom moving around an equilibrium OH that lies along the x axis with the 0 end on the negative part of this axis and the center of the bond fixed at the origin. In this figure, as well as in Figure 7 , the upper plot refers to the DMBE I1 potential energy surface, while the other is for the DMBE 111 potential surface. Contours
+ are equally spaced by 0.01 Eh,starting at A = 0.277 Eh. Shown by the
dashed line (contour a) is the energy contour corresponding to the 0 OH dissociation limit.
I1 potential energy surface from the current work. We conclude this section by reporting the unweighted rootmeansquare deviations for the two DMBE potential surfaces with respect to the threebody residual E H F a b initio energies of Melius and Blint: 0.0073 and 0.0121 E, for the DMBE I1 and DMBE 111potential surfaces, respectively. These values may be compared with those for the ad hoc MeliusBlint9 form and our previous DMBE I surface,17respectively 0.0116 and 0.0073 Eh. Note that the larger rootmeansquare deviation for the DMBE 111 potential surface is mainly due to the form chosen for the weighting function that weights least the ab initio points near the equilibrium HO, geometry.
3. Results and Discussion
The HO, DMBE (I1 and 111) potential surfaces from the present work are numerically defined in Tables 1111 and VI. Note that the coefficients for the threebody electrostatic energy term and the residual threebody E H F term (which are reported in Tables 111and VI, respectively) are the only ones that differ from those reported in article I; the coefficients of Tables I and I1 have therefore been given here only for completeness.
Figure 6 displays equipotential energy contours of the final HO, DMBE (I1 and 111) potential surfaces for an 0 atom moving around an equilibrium OH diatomic. Similar equipotential energy contours but for H moving around an equilibrium O2are shown in Figure 7. Note the absence of an energy barrier in the plots of Figures 6 and 7 and hence absent for both 0 approaching O H and H approaching 0,.
In Figure 8 we show equipotential energy contours for the 00 and OH stretching in HO, with the HOO angle kept fixed at the corresponding equilibrium value, while Figure 9 shows contours
 for the stretching of the two O H bonds in linear OHO. Apparent
from Figure 9 are the two minima associated with the OH. 0 and 0...HO equivalent hydrogenbonded structures. Note, however, that the seemingly Dmhminimum at short OH distances is an artifact of the dimensionality of the graphical representation being truly a saddle point for the motion of 0 around the H end of OH. Despite the similar topographies of the DMBE I1 and
7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 l . C 2.0 3.0 4.G 5 ~ 5 6,O /.C
X/a,
Figure 7. Contour plots for a H atom moving around an equilibrium O2
+ molecule with the center of the bond fixed at the origin. Contours are
as in Figure 6 , except for contour a, which refers now to the H 0, dissociation limit.
 + DMBE 111 potential surfaces at these regions of configuration
space, they show some noticeable differences, particularly con
cerning the height of the barrier for the 0 + O H OH 0
H atom exchange reaction. Accurate ab initio calculations for
the H atom migration between the two 0 atoms would therefore
provide an important way of settling this issue.
Figure 10 shows contours for the C2, insertion of H into 0,.
Again, the DMBE I1 and DMBE 111potential surfaces show some
quantitative differences mainly near the collinear 0H0 geom
etries referred to in the previous paragraph.
Figures 11 and 12 show the leading terms in the Legendre
analysis of the 0OH and H0, interaction potentials with the
diatomics kept fixed at their equilibrium geometries, while Figure
13 compares the V, for the 0OH interaction with that obtained
from other treatments. Again, we have reported the results ob
tained from both the H0, DMBE I1 and 111potential surfaces.
Note the absence of a positive barrier in the spherically symmetric
component of the OOH interaction potential for both the DMBE
I1 and MeliusBlint potentials; see also Figure 5. For the HO,
DMBE 111potential energy surface this barrier exists, though it
is small. Note especially that the condition of no barrier in V,
+ (or, if existing, a small barrier) has been considered in the present
work a key feature for a good 0 O H dynamics.
The properties of the C, chemical minima for both the HO,
DMBE I1 and 111potential energy surfaces are gathered in Table
VII. Also shown for comparison in this table are the attributes
of the H 0 2D M B E I potential surface," those of the MeliusBlint9
ad hoc functional form, and the experimental
Par
ticularly striking is the good agreement between the properties
from potential I1 and experiment. Table VI11 summarizes the
properties of other stationary points in the H 0 2 groundstate
potential energy surface. It is seen that all relevant chemical
structures associated with stationary points in article I remain in
the new potential, although they differ somehow quantitatively.
4. Concluding Remarks
The properties of the HO, DMBE potentials derived in the present work have been carefully analyzed by numerical methods and by graphical techniques. Whenever available, experimental information has been used to parametrize the present functions. In addition, a leastsquares fitting procedure has been adopted
A Realistic H 0 2 ( ~ 2 A ”P) otential Energy Surface
0 h
The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 3139
9
In
0 LD
G.“%
\
I T
0 M
0
N
0 ..
.5
2.5
305
4.5
5.5
6.5
5
b/a,
,5
2.5
3.5
4.5
5.5
6.5
7.5
h/a,
Figure 8. Contour plots for the stretching of the 00 and OH bonds in H02(g2A”)with the HOO angle fixed at 103.7’. In this figure, as well as in Figures 912, the plot in the lefthand side refers to the DMBE I1 potential energy surface, while the other is for the DMBE 111potential surface.
Contours are as in Figure 6 .
I
I
1
I
I
I
I
I
1n o
2.0
3.0
4.0
500
600
1.0
2.0
3.0
4.0
5.0
6.0
R2h
R2/ a0
Figure 9. Contours for the stretching of the OH bonds in O H 0 (g2A”)showing the two equivalent OH..O and O.HO hydrogenbonded structures.
Contours are as in Figure 6 .
to derive the H 0 2DMBE I11 potential energy surface that takes
the a b initio energies of Melius and Blint9 and the available
experimental spectroscopic force field data for the hydroperoxyl
radica16J821on an equal footing, perhaps yielding the most realistic
global representation currently available for that surface. The H 0 2DMBE I1 and I11 potentials derived in this work have
+ also been used for extensive quasiclassical trajectory calculations
of the 0 OH + O2+ H reaction with good success. Since a
detailed description of these calculations is presented el~ewhere?~ we refer to only the results for the thermal rate constant at T = 500 and 2000 K based on the H 0 2DMBE I11 potential surface. At T = 500 K one gets k(500) = 1.55 (f0.12) X 1013cm3mol’ sl, a result that is in good agreement with the best available experimental e s t i m a t e ~ ~(1~.4v5~(f~O.05) X lOI3and 1.68 (f0.06) X 1013cm3 molI sl, respectively), while at T = 2000 K one obtains k(2000) = 8.58 (f0.10) X 10l2cm3 mol’ sl, which is
in reasonably good agreement with the experimental correlation
of Cohen and W e ~ t b e r g(k~(2~000) = 1.02 (f0.36) X l O I 3 cm3
mol’ sl) but somewhat too low in comparison with the most
recent estimate of Frank and Justs2 (k(2000) = 1.39 (50.38) X
1013cm3 mol’ s’).
Neither the DMBE I1 nor DMBE I11 potential energy surfaces
+ from the present work explicitly treat all the finestructure states
of O(3P) OH(211). Indeed, no representations so far published
for this system have done so, including ones4that has just recently
appeared. Quantum mechanically, this can be done by a Gen
+ tryGieseSs type analysis of
applied to 0 OH produces
th a
e longrange matrix of 36
force which when states. The diago
nalization of this matrix as a function of angle of approach and
(54) Lemon, W. J.; Hase, W. L. J . Phys. Chem. 1987, 91, 1596. (55) Gentry, W. R.; Giese, C. F.J . Chem. Phys. 1977, 67, 2 3 5 5 .
3740 The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 ? .t
0 W
0 LD
0 U
0
m
\
M
0 N
 0
0
0
1.5
2.5
305
405
505
Rho
Figure 10. Contour plots for the C,, insertion of H into 02.Contours are as in Figure 7 .
ln I
Varandas et al. 6,5
m
ru 
W \
>
m
N
0 id
' 1
, m
1.
'2.
'3.
'4. '5.
'6.
'7.
'8.
ria,
'la
'2. '3.
'4.
'5.
'6.
'7. '8.
3.
r/a,
Figure 11. Isotropic (V,) and anisotropic (VI, V,, V,, V,, and V,) components of the 0OH interaction potential, with the molecule ilxed at the
v,; v,; v,. v,; equilibrium diatomic geometry, for 1 Ir I9 q,. Note that r is now the distance from the 0 atom to the center of mass of OH: () V,; () V,;
(.)
(*..) (    ) (.)
A Realistic HO2(a2A'') Potential Energy Surface 2
 N
DMBE I1
The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 3741
m
.cP
W \
>
N m
0
d
I I I
w I
I
I
1
1
. m I
'2
'3
'4 a
'5
'6
'7 a
'8 a
r/a,
. '2.
'3.
'4 *
'5.
'6.
7. '8.
3.
r/a,
Figure 12. Isotropic (Vo)and anisotropic (V2,V4,V,,V,,and Vlo) components of the H a 2interaction potential, with the molecule fixed at the equilibrium diatomic geometry, for 1 5 r 5 9 ao: () Vo;() V2;() V4;(.a) V,;(    ) V,;(  e  ) VI,,,
. '2.
'3. ' A .
3m '6.
'7.
'8. '9.
r/a,
Figure 13. Comparison of Vofor the 0OH interaction with that ob
tained from other models. Symbols are as in Figure 5.
distance of 0 from OH [email protected] 18 doubly degenerate potential energy surfaces that include the coupling and the quadrupole axis orientation effects mentioned in section 2.1; the lowest doubly degenerate potential energy surface so obtained is therefore the adiabatic potential that should go smoothly to the MeliusBlint electronic structure calculations. Yet, although spinorbit coupling
+  TABLE VIII: Geometries and Energies of the Metastable Minima
+ and Saddle Points hedicted by the DMBE Potentials for the
H O2 OH 0 and Isomerization Reactionsa DMBE
property
I"
I1
I11
TShaped HO2 Structure
Rdao
2.243
2.230
R2/00
4.927
4.932
R3/aQ
4.927
4.932
HOO/deg
76.84
76.94
V/Eh
Rdao
 0.2012
0.205 0
HydrogenBonded OH. e 0 Structure
5.824
5.680
R2/aQ
1.875
1.888
Rdao
3.949
3.792
HOO/deg
0
0
0.1796
0.1784
2.267 4.824 4.824 76.41 0.2039
5.808 1.863 3.945 0 0.1807
Saddle Point Structure for the H + O2Reaction
Rl/%
2.284
2.268
2.289
R2lao
3.529
3.547
3.426
Rdao
4.702
4.730
4.648
HOO/deg
106.00
106.80
107.16
0.1925
0.1967
0.1921
Saddle Point Structure for the H 0 2 Isomerization
Rl/aQ
2.749
2.783
2.641
R2lao
2.192
2.212
2.217
Rdao
2.192
2.212
2.217
HOO/deg
5 1.16
51.01
53.45
0.2257
0.2 332
0.2509
Energies, in Eh, are taken realtive to three isolated atoms.
+ may be important and should ultimately be included in any
representation of the 0 OH interaction, it is also possible that for calculations of thermal rate constants and other macroscopic observables an explicit treatment of spinorbit coupling may be