Volume 5 1, number 2 CHEMICAL PHYSICS LETTERS 15 October 1977

ELECTRONIC NONADIABATIC TRANSITIONS

IN THE REACTIVE (Af + Hz, Ar + K;, ArH=- + H) SYSTEM.

NUMERICAL RESULTS FOR THE COLLINEAR CONFIGURATION

M. BAER

Department of Theoretical Physfcs and Applied Mathematics Soreq Nuclear Research Center, Yavne, Israel and Department of Chemica! Physics, Weizmann Institute of Science. Rehovot, Israel

and

J.A. BESWICK $ Department of Chemical Physics, Weizmann Institute of Science, Rehovot. Israel

Received 13 June 1977 Revised manuscript received 15 July 1977

In this communication are presented exact quantum mechanical nonadiabatic electronic transition probabilities for the collinear reaction Af + Hz (vi = 0) - ArH+(vf) + H. The calculations were performed using a potential surface calculated by the DIM method. It is established that large probabilities (= 1.0) can be obtained only if there is enough translational energy to overcome a potential barrier formed due to the crossing between ui = 0 of the Ar* + H2 system and ui = 2 of the Ar + Hz

system. The threshold for the reaction is found to be 0.06 eV.

I_ Introduction

Electronic nonadiabatic transitions are of consider- able interest in the field of molecular reaction dynam- ics. One particular case of these nonadiabatic effects is provided by the ionic collisions where charge and atom transfer are simultaneously present. A typical example for that kind is the (ArH$ system for which at low energies the following reactions are possible:

;fir+ + H, =+ ArH+ f H, AE=-1.45eV, (I a)

Ar +H5 *ArH++H, AE=-1_30eV, (lb)

Ar++H, *Ar+H$ AE=-os5ev. (ic)

At -the asymptotic region where the distance (Ar-HZ)+ is large the two lowest surfaces Ar + Hz and Ar+ + Hz cross along the vibrational coordinate of Hz (see fig. 1) and at the asymptotic region where the distance &H-H)+ is large the two lowest surfaces ArH+ + H

z Permanent address: Laboratoire de Photophysique Molikulaire, Universite de Paris Sud, 91700 Orsay, France.

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5

4 1

3-

2-

I-

O-

z z-t -

g-2-

-3 -

I I I I I 0 I 2

a”_” 61 3 4

Fig. 1. The asymptotic diatomic potential curves (R~_Q = -1 for the @r--H+)+ system.

and ArH + H+ are well separated and only the first leads to a stable molecule. Due to the crossing in the (Ar f Hz)+ region it is evident that in this system are involv- ed at least two surfaces.

Recently a quantum mechanical method for treating

Volume 5 1, number 2 CHEMICAL PHYSICS LETTERS 15 October 1977

electronic nonadiabatic effects in molecular collisions was presented [l-3] and applied to the collinear (Hz + H+, H; + H) system [4-6]_ In the present paper we

report 0; a similar study for the (ArH2)+ system with the emphasis on the total reaction probability and the final vibrational distribution due to reaction (la).

2. The potential surfaces and nonadiabatic coupling terns

In fig. 2 are shown the two last lowest adiabatic po- tential energy surfaces for the collinear configuration as obtained by Kuntz and Roach using the diatomic-in- molecules method with zero-overlap (DIMZO) [7] _ The adiabatic surfaces were obtained by diagonalizing the DIMZO hamiltonian and the non-adiabatic cou- pling terms TX (x is a nuclear coordinate) defined as:

TX L (@llWW@2) (2)

(pi; i = 1,2 are the eigenfunctions of the electronic hamiltonian) were calculated from the expression

T, = qawaXk2w2 - v,), (3)

where VI (I = 1,2) are the two lowest eigen&lues of H (the adiabatic surfaces) and cl (I = 1,2) are the corre- sponding eigenvectors. Since the treatment is done for the collinear configuration only two functions TX are needed one associated with the vibrational coordi- nate r namely, T,. and the other with the translation- al coordinate R namely, TR.

Although the information for performing the cai- culations are given in the adiabatic representation it was found that the diabatic representation is much- more appropriate_ First, in the adiabatic Schrddinger equation one encounters, in addition to the second derivatives, also first derivatives of the wavefunction. This fact prevents one from using the efficient meth- ods of integration developed for treating this kind of problems. Second, in the present case the nonadia- batic coupling term T,(r, R) becomes singular when zatpiproaches infinity. It was shown elsewhere [6]

a

t This form of Tr ensures the fact that the two potentials of Hz and Haare the asymptotic states of the (ArH2)+ sys- tems.

s 600- (a) I I

r

02 0 Z-00 4-00 6-00

Distance Ar -H (81

5 6-00- (b)

s

42

z

2 4-00- .- c3

04

zoo-

I OO

I I , I , ,

Z-00 4-00 6 00 Distance Ar - H (11

Fig. 2. Equipotentiai contour lines as a function of inter-atomic distances for the (ArHz)+ system. (a) The ground adiabatic po- tential surface, (b) the fist excited adiabatic potential surface.

lim T,(r, R) = + 7a o- - rc) , (4) R+-

where r, corresponds to the crossing point of the two surfaces Ar+-H, and Ar + H2+(see fig. 1). The trans- formation from the adiabatic representation to the diabatic one is done through a rotation matrix A@, R)

in such a way that if V(r, R) is the diagonal potential matrix with the elements VI and V2 then the corre- sponding potential matrix W(r, R ) is given in the

form [l] :

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Volume 5 1, number 2 CHEMICAL PHYSICS LETTERS 15 October 1977

WG-,R) = W(7) = A’@, R)V(r, R)A(r, R) ,

where

(5)

A= (6)

and f R

r=r(ro,R~)+_I‘T,(r,Ro)dr I- f TR(r,R)dR . To Ro

(7) The elements of W(y) are then:

W,,(7) = VI cos27 f V2 sin27,

Doing the transformation (Q-(7) eliminates the first derivatives from the Schriidinger equation and yields the correct asymptotic states of the (A&&)+ system ??, Consequently the Schriidinger equation takes the form:

--(fi2/2pj(a2faiz2 + iP/aG)J/ f WJI=E\CI _ (9)

It should be emphasized that the new diabatic elements W11, Fv22 and W12 are not equal to the original clia- batic elements of l-l.

Eq. (9) is yet not in the final form as is used for the numerical treatment. From eq. (8) it is seen that the new diabatic coupiing term W,,(r) increases with the difference between the two potential surfades. Since in the present case one surface is attractive and the other repulsive the difference becomes very large and therefore causes instabilities in the numerical treatment. One way to avoid it is to perform once in a while an addition transformation of a similar kind given in eq. (5) with a constant angle (= -_ro)LThis transformation yields a new potential matrix W(r, R) given as:

4N=W(7-7& (10)

?t This is the case because ior R + m, TR (r. R) becomes zero and with the help of eq. (8) one can show that WLt and Wz, become the potentials of Hz nnd Hz and W12 be-

comes zero.

If +yo is chosen to be equal to various 7% afong the propagation coordinate one notices that this procedure decreases the off diagonal term such that:

%2 = $(EL -E2) sin 2(y - ro) _ (11)

In particular this procedure is very efficient if from a certain value of the translational coordinate y becomes constant (when T, = T, =O), then r. can be assumed to be equal to the same constant value and so @,, = 0. More details about this procedure and how it is im- plemented is given in ref. 163 . In what follows, when- ever is mentioned the diabatic surfaces or the diabatic coupIing term the reference will be to Wt 1 (7 - +yo), W&y - yo) and W12(y - yo) respectively.

3. Results and discussion

We have calculated transition probabilities in the (total) energy range (0.502-0.7 ev). Here, E = 0.502 eV is the energy just above the threshold for the co&- srbn of AP f H2(u = 0) and for E = 0.7 eV the folIow- ing asymptotic srates are open:

AOHZ, v=o, E. = 0.50 I4 eV;

Ar+H& u = o-2, E, = O-1 56,0.42 1,0.679 eV;

I-I + ArH+, u = 04,

F -V = -1.144, -0.841, -0.550, -0.272,

-0.05 1,0.250,0.493 eV.

In this paper we present and discuss results related only to the reaction

Ar++H2(uui=0)+ArH+(uf)+I~. (12)

which involves a surface crossing along two lines i.e. r = rc (in the asymptotic region) and at R = R, (in the weak interaction region) both in the initial (re- agents) channel. in fig. 3 are presented the total transi- tion probabilities for the reaction as a function of the energy and in fig_ 4a is shown the vibrational distribu- tion of the products for the energy E = 06 eV.

Two features characterize the probability function as given in fig. 3. First the relatively high threshold energy of the reaction (-0.06 eV) a feature very un- common for a system with no potential barriers and

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Volume 5 1, number 2 CHEMICAL PHYSICS LETTERS 15 October 1977

TOTAL ENERW kV1

Fig. 3. The transition probability as a function of total energy for the reaCtiOn Ar+ + H2 (ui = 0) - ArH+ (x u f) + H. - Ex- act calculations, --- Landau-Zener results.

I 1 1 I t t

0 1 2 3 4 5 6 FINAL VIBRATION v,

0 I 2 3 4 5 6

FINAL VIBRATION v,

Fig. 4. Normalized vibrational distribution of the products_ (a) Results for the reaction Af + Hz (ur = 0) - ArH+(u f) + H. (b) Results for the ground adiabatic potential surface: Ar + H~(~~)~.4rH~(~f)~H._._v~~O,------vi~1,-vi~2_

the second the sharp dip at E = 0.61 eV which is not typical for heavy particle systems.

The explanation for the threshold becomes clear

when one considers the vibrational levels of the two surfaces Wll and W-22 (the diagonal elements of a) as a function of the reaction coordinate along the minimum energy path, as is presented in fig. 5. It is

I 1 I I I _1 -2 -I

Reoctlon Ocoordwate s’IX1 2

Fig. 5. The vibrational levels of the two surfaces as a function of the reaction coordinates. - Viiirational levels of the low-

er surface, --- vibrational levels of the upper surface.

noticed that no reaction from the collision of Arf + H2 is possible unless a transition to the surface Ar + H2+ is performed_ Starting with Ar + + H, (ui = 0) and with low kinetic energy there is a small probability to move to any of the vibrational levels of the lower sur- face. However due to the crossing between the two Ievels Ar+ + H2 (Vi = 0) and Ar + Hs(uUf = 2) a transi- tion becomes possible if the kinetic energy is large enough to reach the crossing point. Although it seems from fig. 5 that in order to reach the crossing point the kinetic energy needed is =O.lS eV it turns out that the threshold is lower, namely, at *O-O6 eV_ The reason is due to the off-diagonal term Iq12 which yields the cou- pling matrix element between the state ui = 0 of Art + Hz and the state vf= 2 of Ar + Hz. We have estimated this matrix element to be 0.090 eV at the crossing and this, therefore, lowers the threshold by the same amount Thus, instead of having a threshold of O-15 eV it be-

comes accordingly 0.060 eV only, which is reasonable close to the threshold found in the exact treatment_

Some more insight can be obtained from fig. 4. In fig. 4a we presented the final vibrational distribution of the products due to the reaction Ar+ + H, and in fig. 4b are shown vibrational distributions ol?final products for the single surface case (calculated using the lowest adiabatic surface only). Three distributions are shown and related to Ui = 0, 1,2. As is noticed the final vibrational distribution which suits best the one of Art + Hz is Ui = 2 as one would expect from the previous discussion.

In fig. 3 we also represent results calculated using

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Volume 51, number 2 CHEMICAL PNYSICS LETTERS 15 October 1977

the Landau-Zener [S] semi classical formula:

pLI 1 _e--alv , (13)

where

u = (2.&/~)~/2, a! = (27dfi)&l IFI - F2 I ,

WI2 =Gwq*m, (14)

F, = afzlfafz; F2 = aeo2/af3 *

HereEk is the kinetic energy measured with respect to the threshold (at threshold& = 0);~ is the reduced mass:

fi = MAr%, f c”A, * MHz ) >

eTi is the second vibrational state of the lower surface and eo2 is the ground vibrational state of the upper surface. To do the calculations the following numeri- cal parameters were used: F, = 0.25 eV A-l ; F2 = -0.6667 eV A-1 ; p = I.905 mu; wI 2 = 0,090 eV. As can be seen the fit between the exact calculations and the Landau-Zener results is rather good.

The sharp dip in the probability function around E = 0.61 eV seems to be due to a quasibound state formed in the interaction region_ As is noticed from fig. 5, certain vibraticnal states can support such a bound state. Estimation of the width yields a lifetime for this (ArHz)* compIex of the order of f O-l2 s which is three orders of magnitude larger than any vibrational period in this system.

4. Summary

fn this study we presented exact electronic non-

adiabatic transition probab~i~es for the collinear re- action:

Ar++Hz (Ui=O)+ArH*(Uf)+H (19

in the energy range E = 0.50-0.70 eV. It is established that this reaction is mainly due to a crossing in the translational coordinate (along the vibrational coordi- nate) between Vi = 0 of the Ar+ f H, system and ui = 2 of the Ar + H2+ system. Although the total kinetic (translational) energy needed for reaching the crossing point is 0.15 eV the threshold for the reaction is about 0.06 eV only. The reason being the off-diagonal dia- batic potential element which lowers the barrier for

the reaction by about 0.09 eV_ Some efforts were made to use the Landau-Zener formula and it seems that for higher energies (15’ > 0.12 eV) the fit is rather good.

In addition it was found that this system supports 2 quasi-bound state for the (ArH2)* complex with a fifetime of lo-I2 s.

References

[I] M. Baer, Chem. Phys. Letters 35 (1975) 112. [2J M. Baer, Chem. Phys. 15 (1976) 49. 131 2-H. Top and M. Baer, 3. Chem. Phys. 66 (1977) 1363. [4] Z-H. Top and MM. Baer, J_ Chem. Phys. 64 (1976) 3078. [Sj 2-H. Top and M. Baer, Chem. Phys. Letters 39 (1976) 137. [6] Z-H. Top and M. Baer, Chem. Phys. 25 (1977) I: [7] P.J. Kuntz and AC. Roach, J. Chem. Sot. Faraday Trans.

1168 (1972) 259. [SJ L.D. Landau, Phys. Z. Sov. 1(1932) 46;

C. Zener, Proc. Roy. Sot. A 137 (1932) 696.

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