2354
J. Phys. Chem. 1981, 85,2354-2359
argon than in the gas phase. This could be due to additional repulsion for the large C10 molecule in the argon cage resulting in a sharper potential well. Conclusion The C10 radical has been produced from the matrix reaction of chlorine atoms and ozone. However, there was no evidence for the formation of asymmetric C103 under conditions favorable for its formation. In addition to C10 radical, ClOO radical and ClClO were also formed. The
same molecular species were produced and trapped when both chlorine and oxygen were passed in argon through the microwave discharge and condensed at 15 K. The vibrational frequency for 3sC1160radical at 849.2 cm-' in solid argon is blue shifted 5 cm-l above the 844.217-cm-' fundamental in the gas phase.1° Acknowledgment. We gratefully acknowledge support for this research from the National Aeronautics and Space Administration under NAS 1-14908-TASK 12.
Polar Representation of Complex-Rotated Resonance Wave Functions Bruce K. Holmer,+Nimrod Moiseyev,* and Phillip R. Certain" Theoretical Chemistry Institute, University of Wisconsin-Madison, Madison, Wisconsin 53706 (Received: February 23, 1981;
In Final Form: April 29, 1981)
Resonance wave functions arising in the complex-coordinate approach to atomic and molecular resonance states can be defined by a polar (magnitude-phase) representation. The utility of this representation is investigated for three model problems. It is found that the polar representation can be useful in analyzing the sources of inaccuracy in conventional complex-coordinate calculations and that it can provide an initial estimate of the resonance lifetime with very little computation. It is not clear whether the polar representation is capable of providing any computational advantage over the conventional Cartesian (real-imaginary) representation of rotated wave functions. If basis sets which are appropriate for the Cartesian representation are used for the magnitude, convergence in the polar representation is poor. New types of basis sets are required.
I. Introduction Variational calculations of electron scattering resonances employing complex coordinate techniques1 have been plagued by rather large basis sets2 and by the lack of variational bounding relations with which to judge the reliability of the result^.^ Recently one of us proposed4 that some of these difficulties could be alleviated by approximating the magnitude and phase of the complex resonance wave functions separately rather than by approximating its real and imaginary parts, as is usually done. The purpose of the present contribution is to assess the usefulness of this proposal by studying three model problems which have well-characterized resonance states. For two of the models, the exact resonance wave functions are known, so that their basis-set expansions can be studied in detail. For the third (the 2s2resonance of helium) there have been many previous calculations with which to compare our r e s ~ l t s . ~ In the complex coordinate method, the resonance wave function and energy are eigensolutions of a rotated wave equation6 H0*@ = w\ko (1) where Ho is the rotated Hamiltonian operator and W is the complex resonance energy which contains both the resonance position E and width r (eq 2). The resonance wave W = E - ir/2 (2) function \kois complex and square-integrable and can be defined either by a Cartesian representation in terms of real and imaginary components (eq 3) or by a polar reptNSF Predoctoral Fellow. Department of Chemistry, Technion-Israel Institute of Technology, Haifa, Israel.
iP0 = \kr
+ iiPi
(3) resentation in terms of its magnitude and phase (eq 4). Of iPt = $eis
(4)
course, either representation is valid when the exact wave function is known, but the question under consideration is whether one or the other is more amenable to basis-set expansions in cases which cannot be solved exactly. The results of our study are somewhat negative in that it was found that in most cases the polar representation has little computational advantage over the Cartesian one, although this conclusion could change if new types of basis sets are developed which reflect the proper fmctional form for $ and S. On the positive side, the formalism of the polar representation shows very clearly why the width r of a resonance is a difficult quantity to calculate reliably, and in the polar representation it is possible to obtain an initial estimate of resonance widths with very little computation. Because the polar representation has not been used previously, we present in the next section some of the special features of this approach. In the following sections, we discuss the model problems: the Doolen potential' which may be solved analytically, a one-dimensional predissociating models whose resonance wave function can be (1) See, for example, the special complex-coordinate issue of Int. J. Quant. Chem. 1978,14, No. 4. (2) Winkler, P. Z.Phys. A. 1977, 283, 149. (3) Moiseyev, N.; Weinhold, F. Int. J. Quant. Chem. 1980, 17, 1201. (4) Certain, P. R. Chem. Phys. Lett. 1979, 65, 71. (5) Moiseyev, N.; Certain, P. R.; Weinhold, F. Int. J. Quant. Chem. 1978, 14, 727 and references contained therein. (6) Simon, B. Ann. Math. 1973, 97, 247. (7) Doolen, G. D. Int. J. Quant. Chem. 1978, 14, 523.
0022-3654/8~2085-2354$0 1.25/0 0 1981 American Chemical Society
The Journal of Physical Chemistry, Vol. 85, No. 16, 1981 2355
Complex-Rotated Resonance Wave Functions
obtained by numerical integration, and the 2s2 resonance of the helium atom. The paper concludes with a brief summary.
Yzr = (Teff)sin 28 - (Vi)
(14)
(5)
This last relationship gives insight into why it is difficult to make accurate estimates of I?. Both terms on the right are about the same size, which gives rise to large differencing errors in r. As we show in section IV, the magnitude of r is often of th_e same orter as the error in a variational estimate of (Teff)and (Vi). Given these general results, we turn now to applications to the model problems.
where r = e-isis the scale parameter for coordinate rotation through angle 6 and V is the potential energy operator. In the polar representation, the rotated wave equation is equivalent to the two coupled equations4
111. Applications to Model Problems. The Doolen Potential Doolen has introduced' an exactly soluble model potential (atomic units)
11. Polar Representation For a system of particles with position coordinates Fm, the complex-rotated Hamiltonian operator is (in atomic units)
A, = -f/2+C~,~ +V(F/~) rn
or)+
(6)
- (Oi))+= 0
(7)
W Z [ V m 2 - ( ~ r n s+) ~ ~ m
-f/2ZVm.$2VmS m
+ $(Q
where
V(F;n/V) =
Vr + iVi = (or+ ioi)$
(8)
Since the phase is determined only to within an additive constant, eq 6 and 7 involve only the gradient of the phase. The resonance position and width are determined indirectly by eq 6 and 7 and are given by eq 10 and 11. E = (cos 28)Eeff+ sin 28( (10)
oi) Y2Zr = (sin 28)Eeff cos 28( oi)
(11)
-
These are exact relations for eigensolutions of eq 1, and we shall consider_them_todefine E and r for aeproximate wave functions, \ko = exp(iS). For a given \ks, the resonance energies obtained in this way differ from those obtained _from calculating the appropriate expectation value of Hs. Variational functionals for $ and have been given previously,4 but, since there were several misprints in the original article, we repeat the functionals here in a slightly generalized form. The functional for +, assuming S is exact, is
V(r) = l / r - A/(8r2) r = (0,m) (15) which, for values of the strength parameter A > 2, exhibits one or more shape resonances. These resonance states may be obtained by solving the complex-rotated wave equation' or, more simply, by applying the Siegert criterion9 to the unrotated wave equation. In the latter approach, one seeks an eigenfunction &(r) which is purely outgoing for r m, and, for the Doolen potential, one obtains
-
4,(r) = r--leiknrp,(r) (16) where p n is a complex polynomial of degree n and s = [l - i ( A - 1 ) 1 / 2 ] / 2
k, = [2/(A
8, = tan-' [2(2n + l ) ( A - l)1/2/(A- 4n2 - 4n - 2)] (17)
The resonance energies are given by
W , = 1/2k2 n = 0, 1, 2,
...,nmarc s(A)
To obtain the solution of the complex-rotated wave e q ~ a t i o n we , ~ simply make the variable substitution r = peio
p
= (0,m)
28
> 8,
Jc($Vrns)2 d r + 4 1 s + ( 0 : , - (oi))$ d7
(19)
This does not affect the energy, but it converts the divergent Siegert eigenfunction into a square-integrable rotated wave function \ko,(p) = ps-'eikn'0P,(p)
(20)
where
m
P,b) = p,(r) = Pn(peie)
In applications, of course, Jq is used with an approximate phase and J,g is used with an approximate magnitude, which weakens the variational nature of the calculation. It would be of great interest to discover functionals which are stationary with respect to variations in 1+5and S simultaneously, but we have been unsuccessful in this aspect of our research. Equation 6 has true bound (square-integrable) solutions corresponding to resonances. Defining the effective kinetic energy operator Teff
=
-l/ZC[vm - (v~s)~I m
allows the position and the width to be written E = (Teff)cos 28 + (Vr) (8) Moiseyev, N.; Certain, P.
1613.
(18)
where
while that for S, assuming $ is exact, is
J,g =
+ 4n2 + 4n)1/2]exp(-i8,/2)
R.;Weinhold, F. Mol. Phys.
k,' = k n / ? It is the wave function \ken(p) whose behavior we wish to examine, and in particular its polar representation in terms of its magnitude $,(p) and phase S,(p) q d p ) = +,b)eisJp)
(21)
Let us first consider the case n = 0. Here the unnormalized complex-rotated wave function is \kkso(p)= ps-leiko'p
(12)
(13)
Wo = ( 2 / W s o (22) Since s and k( are complex, both the real and imaginary parts of \keo(p)are oscillatory. It is these oscillatory terms that have been suspected of causing difficulties in ap-
1978, 36, (9) Siegert, A. J. F. Phys. Rev. 1939, 56, 750.
2356
The Journal of Physical Chemistry, Vol. 85,
No. 16, 198 7
Holmer et 81.
i
i./ 1.5
TABLE I: Convergence of the Lowest Resonance Energy of the Potential Shown in Figure 1 as the Number of Basis Functions is Increased (0 = 17.2")
nA
-5
0
complex energy for functions approximated
-1
5
X
Figure 1. Model potential for predissociation resonances. See eq 24.
proximating complex-rotated wave functions with finite basis sets of exponentially decreasing functions.1° The magnitude and phase of \km(p), on the other hand, are simple nonoscillatory functions" \keo(p) = $0(p)ei50b) $o(p) = p-l/2e-2si"(e-eo/2)p/.\/j:
So(p) = 2 cos (0 - 0 0 / 2 ) p / h - [(A -1)1/2/2] In p
(23)
Thus, q 0 ( p ) and So(p) are smoother functions than \ken(p) and should be amenable to basis-set expansions, for example, in terms of Slater functions for $o and polynomials for So. For resonances corresponding to n 1 1, however, there is a fundamental difficulty with the polar representation of \k,(p). The magnitude function qn(p) contains the factor [P,(p)P,,*(~)ll/~which has n branch points in the complex p plane corresponding to nodes Pn(po)= 0 which occur for complex po. Thus, \ken(p) cannot have a convergent power series representation except for small values of p. While it is possible in principle to obtain an arbitrarily accurate (in a mean-square sense) expansion of the magnitude in a complete basis, convergence may be slow. This point is investigated in the next section for another model potential. Predissociation Model. Another one-dimensional model potential which has been used in previous complex-coordinate studies8 is v(x)= ('/2x2 - J) exp(-Ax2) J x = (-m,m) (24)
2 4 6 8 10 accurate a
1.39767 1.400431.400501.400481.400471.40047-
0.00116i 0.00052i 0.00060i 0.00063i 0.00064i 0.00064i
1.40909 + 0.01108i 1.40080 - 0.00034~' 1.40052 - 0.00060i 1.40049- 0.00064i 1.40048- 0.00064i 1.40047- 0.00064i
Number at basis functions.
precludes accurate variational approximations of its magnitude and phase. Our method of analysis is to generate the exact rotated wave function and then to approximate its two possible representations (real-imaginary or magnitude-phase) by using a size and type of basis set which is appropriate for treating the bound states of the potential. Since the variational principles for rotated wave functions do not give bounds on the complex energies,8 the significance of variationally determined wave functions is not clear and we have avoided this ambiguity by using the least-squares criterion to fit the accurate wave functions with a basis-set expansion. Of course, if a large or carefully tailored basis is used, the wave function can be fit with arbitrary precision. However, one motivation for using the complex-coordinate method is that it allows resonance (scattering) states to be treated on the same footing as bound states. The assumption generally has been that real bound-state basis sets can be used (although a few calculations have used complex "Siegert-type" basis function^'^), so we have adopted a simple harmonic oscillator basis for the model potential. In addition, for a given finite basis, it is possible to compare the relative accuracy of the approximate Cartesian and polar representations of the wave function and in this way learn which representation has the greater promise for variational calculations. The basis set expansion of the Cartesian representation of the rotated wave function is N-1
@&x) =
n=O
c,xne-ax* = Gr + iQi
(25)
where the c, values are complex adjustable parameters and a is a real adjustable parameter. The magnitude and phase functions are expanded
+
For parameter values J = 0.8 and h = 0.12, this potential, which is shown in Figure 1, supports one bound state and two resonance states, each of which closely resembles the corresponding harmonic oscillator states. Although the rotated wave equation for this potential cannot be solved analytically, accurate numerical solutions can be generated by simple modifications12 to the Cooley algorithm13 for numerical integration of one-dimensional wave equations. We have used this model potentia1 to investigate whether the presence of nodes in resonance wave function (10) Junker, B. R. Int. J. Quant. Chem. 1979, 14, 371. (11) The factor p-'lz in $o and In p in Soare the result of the singular nature of the attractive potential r-2 and are not characteristic of true atomic and molecular systems. (12) Atabek, 0.; Lefebvre, R.; Requena, A. Mol. Phys. 1980,40,1107. (13) Cooley, J. W.Math. Comp. 1961, 15, 363.
where the a,, b,, and 0values are all real adjustable parameters. The adjustable parameters are chosen by a nonlinear regression algorithm to least-squares fit the basis-set expansions to the corresponding accurate numerical functions. Figure 2 shows \ke,$, and dS/dx for the lowest resonance (W = 1.40 - 6.45 X 10-4i, 8 = 1 7 . 2 O ) . Even though there is a node at x = 0, it can be treated by a suitable boundary condition, so that the bound portion of the resonance wave function is essentially nodeless. Table I shows the convergence properties of the Cartesian and polar repre(14) Junker, B. R.; Huang, C. L. Phys. Rev. A 1978,18,313. Junker, B. R. Ibid. 1978, 18, 2437.
The Journal of Physlcal Chemlstty, Vol. 85, No. 16, 1981 2357
Complex-Rotated Resonance Wave Functions
TABLE 11: Analysis of Errors in the Calculation o f the Lowest Resonance Energy for the Potential Shown in-Figure 1(e = 17.2", N = 4) functions (dz/dXz) ( ' / z ( d S / & ) * ) ( VJ (Vi, W 0.133 0 2 0.133 46 0.133 47 0.133 36 0.133 37
0.512 23 0.512 08 0.512 08 0.512 13 0.512 13
'Tr, T i
5,dS ldx 5,dS/dx i , d,Y Idx i , dS/dx
0.867 0.868 0.868 0.867 0.867
0.363 8 1 0.364 17 0.364 17 0.363 83 0.363 83
88 01 01 72 72
1.40043 1.40080 1.40080 1.40047 1.40047 -
0.00052i 0.00034i 0.00034i 0.00064i 0.00064i
TABLE 111: Covergence of the Second Lowest Resonance Energy for the Potential Shown in Figure 1 as the Number of Basis Functions Is Increased (e = 17.2" )
A
complex energy for functions approximated i,
Qr, i i
NU
2.0269 2.0289 2.0286 2.0283 2.0284 -
4 6 8 10
accurate
2.1466 2.0382 2.0397 2.0345 2.0284 -
0.05231' 0.05351' 0.05391' 0.0540i 0.0541i
0.09331' 0.05401' 0.0603i 0.05791' 0.05411'
Number of basis functions. IMAGINARY
sentations of \ke. Both representations are satisfactory and have similar convergence behavior. Table I1 gives an analysis of the energy errors for a small size basis. It is seen that the absolute errors in some of the energy components are a significant fraction of the absolute value of the width. Also, notice that nearly all of the error in the magnitude-phase representation is due to the approximate magnitude. Turning now to the second resonance (W = 2.0284 - 5.41 X low2$,we see in Figure 3B a sharp minimum in the magnitude which is the remnant of the node which occurs in the unrotated resonance wave function. Corresponding to this minimum, dS/dx has a sharp minimum which 0. Since these sharp becomes a delta function for 8 minima in the magnitude and dS/dx lie within the bounded region, scaling the coordinates of a bound-state wave function will give some insight into this behavior. The unnormalized wave function for the second excited state of a harmonic oscillator is
2.5
5.0
7.5
X
-
B
-
-
\ k ( x ) = (1 -
Letting x then
-+
(28)
xeie, the magnitude and phase derivative are
~ c / ( x )= (1- 4x2 cos 28
dS/dx = -
+ 4X4)1/2e-(1/2)x2c~28
4x sin 28 1 - 4x2 cos 20
+ 4x4
-x
sin 28
(29) 2.5
(30)
Thus, the sharp in dS/dx can be represented by a rational function. This is the motivation for the particular form of dS/dx used. Table I11 shows the convergence properties of the Cartesian and polar representations of \k8 for the second resonance. For this resonance state, it is seen that the Cartesian representation is more favorable. The accuracy
5.0
7.5
X
Figure 2. Complex-rotated wave function for the lowest resonance of the potential shown in Figure 1 (0 = 17.2'): (A) real and imaginary components; (B) magnitude and phase derivative.
of the approximate width using the polar representation with N = 6 is the result of cancellation of errors in the energy components. An analysis of the energy errors for N = 10 is given in Table IV. Again, nearly all of the error
TABLE IV: Analysis of Errors in the Calculation of the Second Lowest Resonance Energy of the Potential Shown in Figure 1 (e = 17.2", N = 10) functions
-
iLr,
5i
5,dg/dx 5,dSldx i , dS1d.X $, dS/dx
(-
d2/dx2) 0.3179 0.3118 0.3118 0.3178 0.3178
P/,(dSldx)*) 0.4413 0.4556 0.4556 0.4415 0.4415
(V,, 1.4017 1.4012 1.4012 1.4016 1.4016
(Vi,
W
0.3746 0.3753 0.3753 0.3747 0.3747
2.0283 - 0,05402' 2.0345 - 0.05791' 2.0345 - 0.05792' 2.0284 - 0.0541i 2.0284 - 0.0541i
2358
The Journal of Physical Chemistry, Vol. 85, No. 16, 1981
Holmer et ai.
TABLE V: Resonance Position and Width of the
A
Helium IS ResoKance Function of 6 for the Variational Wave Function @ = $eCzSa
n
- E , , au
2 2 3 3 4 5 6 9 18
0.774983 0.775959 0.777 579 0.779837 0.782723 0.786 227 0.790341 0.795063 0.800 409
2 2 2 3 3 3 4 5 4
0.774990 0.775 979 0.777 623 0.779908 0.782819 0.786 339 0.790446 0.795 124 0.800 357
r, au
6, rad
For the E Basis
0.001 522 0.002941 0.004 158 0.005078 0.005 621 0.005721 0.005343 0.004507 0.003 7
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18
For the I: I Basis
2.5
5.0
7.5
X
0.001 518 0.002916 0.004070 0.004 868 0.005 201 0.004976 0.004 111 0.002 547 0.000248
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18
a n denotes the number of iterations which were required t o achieve the convergence of eq 33 and 34.
0 -1.0
-2.0
-3.0
- 4.0 8
2.5
5.0
r
7.5
X
Figure 3. Complex rotated wave function for the second resonance of the potential shown in Figure 1 (6 = 17.2'): (A) real and imaginary components; (6)magnitude and phase derivative.
in the polar representation arises from the approximate magnitude. Referring back to eq 29, the difficulty lies in approximating the square root of a polynomial by another polyn_omial. It appears as if the rational polynomial basis for dS/& works well, and a similar basis for $ might prove effective,15 although incorporating a function of this form into a variational calculation is difficult. 2s2Helium Resonance. For this example, the accurate resonance wave function is not known and we must resort to variational approximations. Previous accurate variational calculations5employing the Cartesian representation have given values for the width of r = 0.0043 hartree. In our calculations we used polar trial functions with real basis sets ai,and u,
9,= $& where
(31)
Flgure 4. Width of the lowest helium 'S resonance as a function of rotation angle 6. The variational basis sets E,and Z2are described in the text.
The optimum variational coefficients a,, and b, are solutions to coupled algebraic equations
( H - E,&')a = 0
(33)
Fb=d
(34)
Here 2 Hnnl
=
-C
1=1
2
f/
[email protected]) + (anlf/2C (Vi&21@nt) + i=l
cos e(@,,lQpnf)(35)
2
Fmmf
=
($1 i=l C
d, = -2 sin
Viam*Viamtl$)
e($lam(Q - (Q))l$)
(38)
and Q is the usual Coulomb potential. These equations may be solved by an iterative procedure. The resonance position and width are obtained from eq 10 and 11, where
(Oi) = (+~QI$) sin e ~~~
(37)
~~
(15) Bhattacharjee,R. S. Int. J. Quant. Chem. 1980,18, 1485.
(35)
The basis anwas chosen to consist of 30 Hylleraas-type6 functions 9, = (1+ P12)rl~r~rl$ exp[(-3rl - 5r2)/2], where the maximum values of z, J , and k were 3, 3, and 1, re-
2359
J. Phys. Chem. 1981, 85,2359-2363
spectively. For the phase, two basis sets were used. The El basis consists of simple polynomials, 6, = rltri r2r;; i = 1, 2, 3; j = 0, 1; while the basis 2, consists of basis functions with an exponential, urn.= 6, exp[-O).l(rl+ rz)l. The results are presented in Table V and Figure 4. The first stsp in the iterative-solution of eq 29 and 30 is to assume S is a constant (VS = 0), and J/ is obtained as the solution of the secular equation (eq 29). This equation differs from an ordinary bound-state secular equation only in that the potential energy matrix is multiplied by cos 8. The results of this calculation for J? are shown in Figure 4, and, if the resonance width is estimated by the value for which dI'ld8 = 0, the estimate is r = 0.0052 hartree. This compares favorably with the accurate values quoted above, especially considering that the gradient OS was completely neglected. The results of including the phase gradient are also shown in Figure 4 and Table V. Although the instabilities noted in the model problem do not occur here, neither is there a dramatic improvement in the stability of the results. In fact, a true stationary point, aWla8 = 0, was not observed. The estimates of the width obtained by including the phase gradient in a self-consistent manner are less accurate than when it is neglected altogether. This is indicative of
+
the deficiencies in the basis set for the magnitude. The change in the width upon including the phase gradient was not great, however, which suggests that, if better basis sets for the magnitude can be developed, the neglect of the phase gradient may result in reasonable estimates of the width. This point requires further investigation.
IV. Summary The polar representation of complex-rotated resonance wave functions was investigated for three examples. It was found that, for resonance wave functions that are nodeless in the bound regions of configuration space, basis-set expansions of the magnitude and phase can lead to accurate estimates of the resonance position and lifetime. For wave functions with nodes, however, the calculations are unstable and subject to large errors if conventional boundstate basis sets are used. Nevertheless, the polar representation can have utility in analyzing the errors in conventional complex-coordinate calculations and can provide an initial estimate of resonance widths by means of a simple bound-state-like calculation involving only real arithmetic. Acknowledgment. This work was supported by NSF Grant CHE77- 19941.
Mechanism of the Ozone-Ethene Reaction in Dilute N2/02 Mixtures Near I-atm Pressure Charles S. Kan, Fu Su,' Jack G. Calvert,' and John H. Shaw Depattments of Chemistry and Physlcs, The Ohio State University, Columbus, Ohio 43210 (Received: March IO, 1981; In Final Form: Aprll 15, 1981)
-
Kinetic studies of the 03-C2H4reaction (ppm reactant level in 700 torr of O,/N,) were made at several temperatures (9-30 O C ) by using Fourier transform-infrared methods to follow reactants and products in situ. The rate data gave the second-order rate constant for the elementary reaction, O3 + CzH4 [CHzOOOCHz] CH2OOt + CHzO (1);k l (cm3molecule-' s-') = 10-13~69*0~27 exp[(-5.62 f 0.36 kcal/mol)/RT]. The fraction of the CH2OOt species formed in eq 1 which does not fragment, but lives to react with CH20,was found to be 0.37 f 0.02, independent of temperature (9-30 O C ) . The addition of CHzOzto CHzO leads to the transient product X, thought to be HOCH,OCHO, and this produces formic anhydride by some ill-defined path (e.g., HOCHZOCHO (HC0)20 Hz (7); or HOCH20CH0 Oz (HC0)20+ Hz02(7a)). The first-order rate constant k , showed an unexpected, low activation energy and preexponential factor and a sensitivity to wall conditioning and 0 2 pressure which suggested that reaction 7,7a, or some similar reaction forming (HCO)zO occurs heterogeneously.
-
-+
+
+
+
Introduction Recent studies of the ozone-ethene reaction by Fourier transform-infrared spectroscopy have given new insight into the mechanism of the reactions which follow the primary step 1 in gaseous mixtures near 1 atmS2p3 In
-
C2H4+ O3
-
I
[CHzOOOCHz]
CHzO + CH,OOt
(1)
(1) Chevron Research Co., Richmond, CA. (2) F. Su, J. G. Calvert, and J. H. Shaw, J. Phys. Chem., 84,239(1980). (3) H.Niki, P. D. Maker, C. M. Savage, and L. P. Breitenbach, J. Phys. Chem., 85,1024(1981);we are gratefd to the authors for a preprint of this work before publication.
-
particular, the reactions of the highly vibrationally excited CH200t fragment have been elucidated for low reactant concentrations typical of the polluted troposphere. It was estimated in these studies that a significant fraction (38%: 35 f 5Ti3) of the vibrationally rich CHzOOt species formed initially in 1atm of air is thermally equilibrated and may react efficiently with added reactants such as CHzO, CH,CHO, and SOz, even though these compounds are at the ppm level in air. The rates of formation of the major products of the 0 3 - c & reaction in highly dilute gaseous mixtures in Oz/N2 at 700 torr can be rationalized well through the following reaction sequence involving the CH200t and CHzO species formed in reaction 1:
0022-3654/81/2085-2359$01.25/00 1981 American Chemical Society
