J. Phys. Chem. 1993,97, 2092-2096
2092
Tests of an Adiabatic Perturbation Theory for Symmetric Charge Exchange+ R. J. Cross Chemistry Department, Yale University, P.O. Box 6666, New Haven, Connecticut 0651 1 Received: August 4, 1992; In Final Form: September 22, 1992
A recently developed adiabatic perturbation theory for symmetric molecular charge exchange is tested against accurate calculations over a range of energies from 0.5 to 32 eV. Good agreement is found over the whole energy range, and the discrepancies are roughly the same at all energies so that an adiabatic basis set should be good even a t high energies, The total charge exchange cross section is calculated for specific initial vibrational states. It depends strongly on the initial state, and it oscillates as a function of energy, so that this cross section may be a useful way to fit a potential to experimental data. I. Introduction We have recently developed a semiclassical, adiabatic perturbation theory to treat symmetric charge exchange between molecules.' This was tested largely at a single energy for the system 02+02.The test showedgocdagreementwith accurate, previously published state-testate transition probabilities. The purpose of this paper is threefold: to expand the test to a much broader range of energies, to use the adiabatic method to calculate quantities which can be measured experimentally, and to examine the limits of the approximation. One might well ask why an approximate method is needed if the exact results are available. The answer lies in the amount of computer time needed. At the highest energies used here, the exact calculations required over a week of CPU time on a DECstation 3100 RISC workstation. The approximate calculations took only a few minutes. If we were trying to fit the potential to a set of experimental points, we would have to repeat the calculation many times as we varied the potential. With the exact calculatiop, this would take far too much computer time. Furthermore, the'potential used here does not contain any angle-dependent terms, so that rotationally inelastic scattering is ignored. Inclusion of these terms greatly increases the amount of computer time needed so that the exact calculation becomes impossible. To generate the adiabatic representation,we set up the coupled Schrodinger equations in a diabatic representation by using the asymptotic vibrational wave functions of the separated collision partners as the basis functions. The adiabatic representation is then obtained by diagonalizingthe entire coupling matrix except for the kinetic energy term. This is done at each value of R, the separation between the molecules. In the diabatic representation the kinetic energy is diagonal,and the coupling between the states is the result of off-diagonal terms in the potential matrix. In the adiabatic representation the potential energy is diagonal, but since the diagonalizingtransformation matrix depends on R, the kinetic energy is no longer diagonal, and the coupling between states is given by the off-diagonalelements in the kinetic energy. The diagonalizingtransformation is R-dependent, and so its first and second derivatives with respect to R appear in the coupled Schrodinger equations. Traditionally, the adiabatic approximation is thought of as a low-energy approximation to be used when the duration of the collision is long compared with the internal motions of the system. It should therefore fail as the two time scales become comparable. At the opposite limit is the sudden approximation,2 where the internal motion is frozen during the collision. It is used when the collision time is short compared to the period of the internal
+
' This paper is dedicated to Prof. Dudley R. Herschbach on the occasion of his 60th birthday. 0022-3654/93/2091-2092104.00/0
motions. Here one starts in the diabatic representationand finds a matrix which diagonalizes the potential. The transformation then goes from the energy representation to the position representation. Then one neglects the energy differences between the quantum states, and the whole problem becomes diagonal. There are two differences between the adiabatic and the sudden approximations. The diagonalizing transformation is R-dependent in the adiabatic case and R-independent in the sudden case. The diagonalizationtransformation in the sudden approximation does not include the internal energies whereas it does in the adiabatic case. Thus, the sudden approximation is merely the high-energy limit to the more general adiabatic case. We might therefore expect that the adiabatic approximation would be generally good at all energies rather than just at low energies. The results shown below seem to substantiate this, but unfortunately, wecannot extend the test to higher energies because the exact calculations take too much computer time. 11. Review of the Theory
The basic theory has been derived and will only be reviewed here. We start by expandingthe total wave function in terms of the diabatic electronic wave functionsof the separated molecules to get a set of coupled Schrodinger equations in the nuclear motion. Since we are dealing with symmetric charge exchange, the electronic states come in symmetric pairs. Interchanging the ion and neutral gives a second electronic state degenerate with the original one. We keep only the two lowest states: D corresponding to A+ A and X corresponding to A A+. We note that the two states are not orthogonal, and this complicates the algebra. The coupled equations are
+
+
[ p ( R )+ Vo(R)+
+ A R (J')
[ p ( R )+ Vo(R)
- E ] $ D ( R Y ~ A , ~+B ) $X(R,RA,RB) = 0 (1)
+ fi"(RA) + fiii(RB) - E]$X(R,RA,RB) +
A V R ) $D(RRA,&) = 0 (2) Here Hi and finare the internal Hamiltonians of the ion and neutral and F(R) is the kinetic energy operator, including the centrifugal term. In Becker's original treatment3of the problem, the direct part of the potential V, and the exchange part AYare assumed to be independent of the vibrational coordinatesRAand Rg. The generalization of the theory to include this dependence is straightforward. More importantly, Becker's theory does not include the orientation dependence of the potential. This, however, requires the inclusion of rotational wave functions. At the high energiesof these calculations, we would need to include hundreds of rotational wave functions, and this would multiply the computation time by millions. f! we n?w expand $0 and $x in terms of the eigenfunctions of Hi and H,,we get Q 1993 American Chemical Society
Perturbation Theory for Symmetric Charge Exchange
The Journal
01Physical Chemistry, Vol. 97, No. 10, 1993 2093
where E(a)are the adiabatic‘edcrgied ohained in the diagonalization. It was also shown that
+
(jla2/aR2b) = aDyj/aR ( D ~ ) ~ ~ (14) The coupled differential equations become
Z{u,”6yj + 2Dyju,’
+ [aDyj/aR + (D2)j,j]uj(R) +
i
Here ub,,(R) is the coefficient in +D of the vibrational wave functionscorrespondingto ionicvibrationalstate m and the neutral vibrational state n and
kmn2= (2p/h2)(E - Em - E,) (5) (mln) is the overlap integral between the ionic vibrational state Im) and the neutral state In). Equations 3 and 4 are equivalent to Becker’s coupled differential equation^.^ We can use the symmetryof the problem to simplifythe coupled equations. We define u*,,(R) = 2-’’2[u,mn(R) Then, eqs 3 and 4 become
=F
uxm,(R)I
(6)
[k;(R)
- (2p/h2)Vo(R) - / ( I + 1)/R2]6y,u,(R)) = 0 (15)
As the energy increases, k2increases; Vo increases because the collision takes place at smaller R. The radial wave function oscillates more rapidly so that u = 0(1), u ’ = O(k),and ut’= 0 ( k 2 ) .The termsinvolving Dbecomelessimportant as theenergy is raised. Thus, adiabatic theory goes smoothly into the sudden approximation at high energy. At the energies we are using the second term in eq 15 is much larger than the term in 02. To construct the adiabatic perturbation theory, we ignore the terms in eq 15 involving D2.The resulting coupled equations are then diagonal and easily solved using the WKB approximation. Then we treat the terms in D as a perturbation. The resulting S matrix is given by
S = e x p [ i ~ ( ~exp[2iA$)] )] exp[iq(’)]
(16) where ~ ( 0 is) the diagonal matrix of the elastic phase shifts for the unperturbed problem and The equations for the symmetric and antisymmetric parts of the problem are completely uncoupled. These are the coupled differentialequations in the diabatic representation. We integrate them using the program Molscat, kindly provided by Dr. Sheldon Green. We use the same potential that Becker3 used. He defined a Morse and an anti-Morse potential Vm(R) = De{exp[-2B(R - RJ1- 2 exp[-B(R - R,Jll
(8)
V,W = De(’/,exp[-28(R - 4 1 1 + exp[-B(R - RJ11 (9) The parameters are D, = 1.129625 X 10-13 erg = 0.070509 eV, /3 = 1.7010 A-1, and R, = 3.3322 A. The two curves intersect at R, = Re - (In 6)/0 = 2.2788 A. The two potentials are
Here IL,!O)(R)is the radial wave function corresponding to the zero-order, unperturbed problem. As mentioned above, the last term in eq 17 is much smaller than the others, and we neglect it. If we use the WKB approximation for do), we get
where R, is the classical turning point, and the classical phase is given by
c;!”)(R ) = J RC
[k;( R ) -% ’ (R)- w ] dR h2 R2
( 19)
O
The calculation must be done twice, once for the plus sign in cq 7 and once for the minus sign. The two differ only in the sign of AV. 111. Results
To obtain an adiabatic basis set,’ we find the unitary matrix which diagonalizes
We use b) to represent an adiabatic state. It should be noted that a larger basis set is normally used in the diagonalization than in the final stages of the problem. In the adiabatic representation, the potential matrix is diagonal, but since (mnb)depends on R , the coupled Schrodinger equations in the adiabatic representation contain both the first and secondderivativesof the transformation matrix. It has been shown previously’ that
The exact calculations were done using the program Molscat. Table I gives the number of basis functions used at each energy. The most convenient way to compare the exact and approximate theories is to plot the state-testate transition probability Mzas a function of the orbital angular momentum quantum number I . Transitions fall into three categories. The most prominent transition is the elastic charge exchange, D X,preserving the vibrational quantum numbers for both the ion and the neutral. In the 02+ + 02 system the vibrational energy spacings in 0 2 are only slightly less than those in 02+.The next most probable transitions, therefore, are those to states where the sum of the vibrational quantum numbers is conserved. These states have energies close to the initial energy. Finally, there are transitions to states with a different number of quanta. We designate the states by the pair (m,n) [02+(v=m)+ 0 2 ( u = n ) ] . For most of the calculations we choose (1.0) as the initial state. In an
-
Cross
2094 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
,
TABLE I: Size of the Basis Set Used by Molscat energy (eV)
basis functions
0.5 1 .o 2.0 4.0 8.0 16.0 32.0
15 21 21 21 28 28 28
0.02
9
0,o
max vib quantum no.
I
0,l
1
----------0.15
0.0
A
1
Ot2
i
h
v Y
a
i
0
1
1
0.0 0
500
1000
1
Direct Scattering 0
0
500
Figure 2. Scattering probability Nzis plotted against the orbital angular momentum quantum number I for the direct process 02+(u=1) + 0 2 (u=O) Oz+(u=m) + 02(u=n) for various final states (n,m). The translational energy is 8.0 eV. The solid curves are the exact results, and the dashed curves are the approximate results. The arrow indicates the approximate upper limit for orbiting collisions.
-
1000
1
Figure 1. Total scattering probability for charge exchange is plotted against the orbital angular momentum quantum number I for the reaction 02+(u=l) + 02(u=O) 0 2 + 0 2 + . Data are shown for translational energies of 0.5, 1.0,2.0,4.0,and 8.0 eV. The solid curves are the exact calculations using Molscat, and the dashed curves are the results of the adiabatic perturbation theory. The arrows indicate the approximate 1 below which orbiting collisions would occur if the ion-induced-dipole interaction were included in the potential.
-
experiment, the most probable state of the neutral is n = 0,and we may be able to select the ionic state (see below). The ground state (0,O) has no other state with zero quantum numbers, so it is, in a sense, atypical, and so we choose (1,O) as the initial state. Calculations were done for translational energies of 0.5, 1,2, 4,8,16,and 32 eV. Below 8 eV most of the scattering is elastic, either to the D or the X channel. Therefore, we plot the total charge transfer probability vs 1. The data are shown in Figure 1, The adiabatic approximate theory (dashed line) agrees well with the exact calculations (solid line) at all energies. For small I the probability for charge transfer oscillates between 0 and 1. This is a familiar feature of time-dependent quantum mechanics. The two eigenstates of the system are the symmetric and antisymmetric combinations of 02++ 02 and 0 2 + 0 2 + . The system starts out in a superposition of the two states, and so the wave function oscillates back and forth between the two linear combinations. The frequency of the oscillation is determined by the energy difference between the symmetric and the antisymmetric combination. As the energy is decreased, the duration of the collision increases, and the number of oscillations increases. It should be noted that the potential does not includethe strongly attractive ion-induced-dipole term. At low energies this term in the potential is responsible for the trapping of trajectories to form a long-lived complex, the Langevin mechani~m,~ which is a simple model for ion-molecule reactions. In the Langevin model all trajectories below a critical impact parameter are trapped. Using a reasonable value for the polarizibility of 0 2 , we find that
the critical I is 200E'/4,where E is the energy in electronvolts. It can be seen that a large portion of the scattering at 0.5 eV is affected. As the energy increases, more of the scattering moves to larger 1, and the Langevin effect becomes less important. In Figures 1-7 the critical I is indicated by an arrow. It is almost certain that the perturbation approximation will break down in the case of long-lived, trapped trajectories. The exact calculations would be extremely difficult here as well, since transitions would occur to almost any vibrational state which is energetically accessible. By 8 eV transitions to states other than elastic charge transfer become important. Figures 2 and 3 show the direct and exchange probabilities as a function of the orbital angular momentum quantum number 1. The largest fractional errors in the approximate treatment are in the probabilities which are fairly small. The largest absolute error is 1 1% in the ( 1,O)exchangeprobability. Figures 4 and 5 show the same probabilities at 16 eV. The agreement is similar. Finally, Figures 6 and 7 show the data at 32 eV. The exact calculations at 32 eV were done with a basis set of 28 states-all states with six or fewer vibrational quanta. Calculations were done at a few points with a larger basis set consisting of 36 states (all states with seven or fewer quanta), and these indicated that the calculation with the smaller basis set was accurate toa few percent. Unfortunately, theamount ofcomputer time required to do the exact calculations with the larger basis set is prohibitively large. At still higher energies, the use of the larger basis set is clearly required, and so 32 eV is about as high as we can go. It is clear that the adiabatic approximation does a fairly good job. For the higher energies the adiabatic theory is faster than the exact calculations by a factor of roughly 1500: 10 min vs 10 days of CPU time on a DECstation 3100. It is also quite clear that the accuracy of the adiabatic theory does not deteriorate markedly as the energy is increased by over 2 orders of magnitude. Thus, the preliminary notion that an adiabatic theory is strictly a low-energy approximation does not appear to be valid.
Perturbation Theory for Symmetric Charge Exchange
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2095
0.2
1 0.0
0.15
0.0
h
0.0
0.1
------------
0
.
0
0
1 1000
500
0
. 500
0
0 1000
1
1 2000
1500
1
Charge Exchange
Charge Exchange
Figure 3. Scattering probability is plotted vs I for the charge exchange process02+(u=l) 02(u=O) +02(u=n) + 02+(u=m). The translational energy is 8.0 eV.
Figure 5. Exchange scattering probabilities at 16 eV. See the caption of Figure 3.
+
I
0.2
,
,
,
I
,
,
,
x
1
,
,
I
: : : : \ : : : : I : :
b
1
,
I
: I :
I
I
I
,0.15
0.0
: : :
h W v
a 0.05
0
t
0,2
h
0.0 0
0.0 0
~
500
1000
1500
2000
1
Direct Scattering Figure 4. Direct scattering probabilities at 16 eV. See the caption of Figure 2.
IV. Charge Exchange of State-Selected IOM While the comparison of scattering probabilities as a function of I may be a good check against an accurate theory, it is clearly not a reasonable check against experiment. We must instead find a cross section that can be measured easily and which is sensitive tovariations in the potential. One possibility is the total cross section for charge exchange for ions in selected vibrational states. In favorable cases one can use resonance-enhanced multiphoton ionization (REMPI) to create a beam of ions in
3
j
, i I
500
,
I
1
1
4
3
1000
1500
i
2000
1
Direct Scattering Figure 6. Direct scattering probabilities at 32 eV. See the caption of Figure 2.
specific vibrational states with very little population in other states.5 To some extent, one can even select a narrow band of rotational states as we11.6 If the neutrals are thermal, they will be automatically state selected with u = 0. By using isotopically labeled 02 for either the ions or the neutrals, one can clearly identify the charge exchange products. The product ions can be easily detected, but they cannot be easily state analyzed, so that we are left with the measurement of the total charge exchange cross section as a function of the initial, relative translational energy. In Figure 8 we show this cross section for ions in states u = 0, 1,2 as a function of translational energy. The calculations
2096 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
Cross alone. A better way to proceed is to start with a set of potential surfaces calculated by a simple theory of electronic structure and then to refine them using the scattering data.
V. Discussion
1.0
t I/+,
________------
0.01. 0
a
.A.
a
500
*
*
. L* I *---
I
#
1000
“
1500
2000
1
Charge Exchange Figure 7. Exchange scattering probabilities at 32 eV. See the caption of Figure 3. 601
I
1
1
1
1
1
I
1
t n
v=o
. -
20 I
I
i
2o
The biggest advantage of the adiabatic approximation is its speed. The calculations required for the 3 1 energies in Figure 7 required about 5 h of CPU time. If they were done using the exact integration of Schrodinger’s equation, they would have taken a few months of CPU time. In a fit of experimental data to a potential, these calculations would have to be repeated a great many times as the potential is varied to get a good fit. A fast approximate treatment would be very useful, at least in the early stages of the fitting. Furthermore, the experimentscan easily be done at energies higher than 32 eV where the exact calculations become far too expensive to do. The problem only gets worse when the orientation dependence of the potential is included. The inclusion of a dozen rotational states for each molecule would make the exact calculation impossible and the approximate calculation very difficult. Fortunately, this is probably unnecessary since the rotational energy level spacing is very small, and the sudden approximation2 is almost certainly very accurate for the rotations. Even with the sudden approximation, one would have to do repeated calculations over a large range of orientations, and this will greatly increase the required computer time. The adiabatic perturbation theory is accurate over a broad range of energies, and this fact provides important insight on the nature of the scattering process. The use of an adiabatic basis set is clearly far superior to the use of a diabatic basis set. It seems reasonable that an exact calculation can be done with an adiabatic basis set, and this many substantiallyreduce the required computer time. It is not clear when the adiabatic approximation will fail. It does not seem to fail at high energies, as the author initially suspected it might. From eq 13 we see that the coupling depends on the derivativeof the potential, and so it may fail in cases where the potential varies rapidly with R. It should also fail at low energies where orbiting complexes occur.
Acknowledgment. Research support is gratefully acknowledged from the National Science Foundation under Grants CHE8901577and CHE-9115967. 1amgratefultoDr.SheldonGreen for sending me a copy of Molscat, and I am grateful to Prof. William Chupka for several discussions on multiphoton ionization.
40Yh
References and Notes
20
Fipn 8. Total charge exchange cross section Q is plotted against the translational energy for ions in vibrational states 0, 1, and 2.
are normally done asa function of total energy, but the experiments will use translational energy. It is immediately obvious that the cross section is quite different for the three states. The cross section also oscillates, and the phase is dependent on the initial quantum state. This situation is fortunate because it indicates that the cross section should be fairly sensitive to the details of the potential and can therefore be used to determine the potential experimentally. If, on the other hand, the cross section were a simple, monotonic function of energy, there would be little hope that we could use it to fit a potential. It is unlikely that a unique fit to the potential can be obtained by using the scattering data
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