M. Baer/Electronically nonadiabatic effects. II ... Z.-H. Top, M. Baer/ElectronicaUy nonadiabatic effects. II ...... Professor T.F. George for stimulating discussions on.
ChemicalPhysics25 (1977) l-18 0 North-Holland Publishing Company
INCORPORATIONOFELECTRONICALLYNONADIABATICEFFECTSINTOBlMOLECULAR REACTIVESYSTEMS.H.THECOLLiNEAR(H,+H+,H;+H)SYSTEM .Zvi H. TOP and Michael BAER Department of Theoretical Physics and Applied Mathematics. Soreq Nuciea? Reseamh Centre. Yavne. Israel and Department of Chemical Physics, The Weizmann Institute of Science, Rehovot. Ismel Received 13 April 1977
An exact quantum mechanical study of the collinear reactive (Ha f H+, Hs f H) system with the emphasis on electronic nonadiabatic processes is presented. This system was studied both ignoring nonadiabatic effects (i.e. a single surface calculation) and incorporating them. It was found that two features, i.e. the fact that the three interacting particles have identical masses and the existence of a deep potential well in the interaction region (in the lower surface) make this system very different from anyother,including its isotopic analogs. Electronic transition probabilities were calculated as a function of energy and initialvibrational state. It was established that the charge transfer process takes place due to favorable resonance conditions between high vibrational states in the lower surface and corresponding states in the upper surface. If the initial state is above the threshold for charge transfer, the process is direct (and therefore adiabatic as far as the total intemol energy is concerned), and if it is below then the process takes place in the viciniV of the interaction region where high vibrational states of the lower surface are populated due to the deep potential wellin the interaction region. Applying the information theoretical approach a surprisal analysis of the results was performed. It was found that most of the results for both the oneand the two-surface cases fit the predicted behavior due to this approach very nicely. , I
1. Introduction In the first paper of this series (referred as I) [1] a method was proposed for incorporating electronically nonadiabatic transitions into bimolecular reactive systems. This method was applied to the collinear (H2 + H+; Hi f H) [2,3 ] system, or in brief the Hf system, and the results are reported here. The isotopic analogs of the $ system have been studied both experimentally [4-91 and theoretically [lo-131. On the other hand, the pure Hi system (one proton and two hydrogen atoms) was hardly exposed to any of these treatments. The authors are not aware of any experimental work on electronic transitions in the “few eV rang” (< 10 ev), and only results on vibrational [7,14,15] and rotational [X,16] transitions have been reported. Theoretical treatments were devoted to the study of vibrational [12,17,18] and rotational [19,20]. transitions where only one surface was used, and except for one study [21] in which only a few results were reported, no other work has been published. The reason is due to the fact that alI nu-
merical approaches were semiclassical, and not quantum mechanical. The semiclassical approaches [ 10,221 are based on the classical trajectory method and a necessary requirement for a successful application of this method to electronic transitions is that the trajectories cross the line or the region where electronic transitions are expected to take place. In this study it is shown that the interaction in the single surface Hi system in the “few eV range” is vibrational adiabatic*. Therefore the probability for a classical trajectory calculated with a low initial vibrational state to cross the seam (the line along which the probability for electronic transition is the highest) is very slim. Consequently, only negligibly small electronic transition probabilities are expected from such a treatment WIless the ir&ial vibrational state is high enough (23). This is also the reason why the only published results * Some preliminary results were reported by us before [23] but we found out later that the potential surface used by us is not exactly the same as the DIM20 potential of Preston and TuRy [271 as was claimed.
2
Z.H. Top, M. BaerfElectronically nonadr%atic effects. II
for electronic transitions are from high initial vibra-
tionalstates(ui=5,6) [21]. The present method, used for calculating electronic transitions, was discussed in detail in I. However, due to numerical instabilities some modifications were made and they are described in the next and the third section. The one-surface and two-surface results are discussed in the fourth and fifth sections respectively. In the sixth section, a surprisal analysis [24-261 is performed, in the seventh a comparison with experimental results and other numerical treatments is made and a summary is given in the last section.
2. The potential energy surfaces (PES) In order to obtain the PES which govern the motion of the three particles, the DIM method [27-291 was employed. This method yields the adiabaricpotential energy surfaces (APES) and the electronic nonudiubutic coupling terms (ENCT). In figs. lb and 2b the equipotential contour lines of the two lowest APES are shown as a function of the scaled coordinates [I] randR,andinfigs. laand2athethree dimensional pictures of these surfaces are seen as a function of the internuclear distances rl and r2. The DIM method yields three APES of which the two lower ones are of lZf symmetry and the third - the highest is of 3 Z’ symmetry. From symmetry considerations [30] no coupling exists between singlet and triplet states and therefore the third surface can be ignored. The mahi features of the remaining two surfaces are that in the interaction region (rl =r2) the lower surface exhibits a deep well of 8.14 eV below dissociation and the upper one exhibits a barrier with the saddle point 4.13 eV above the separated H2 + I?. T-he ENCT are in general written: 7x = (t, I ww,
(11.1)
),
where &; i = 1,2, are the two electronic eigenfunctions and x is some nuclear coordinate (the bra and ket notation is used with respect to the electronic coordinates only). The ENCT are expressed either in terms of the scaled coordinates r and R or in terms of polar coordinates p and (Y[ I]. The relation between the two is given as: pcoscc= ‘00 - r,
psim=Roo-R,
(11.2)
Fig. 1. The ground adiabatic potential surface of the collinear Hz system as obtained by the DIMZO method: (a) A threedimensional surface as a function of the interatomic’distances. (b) Equipotential contour lines as a function of the scaled distaxes r and R (the numbers are in eV).
where (rW, Roe) isa fured point in the plateau region (see fig. 3). The coordinates r and p are vibrational coordinates, whereas R and 01are translational coordinates. The vibrational ENCT r,(r, R)has a ridge along the R axis approxirhately at the value r = r, = 2.3 au (which corresponds to r1 = 2.5 au). The function T#, R)ap. proaches zero for any value of r as R + 00, except at the value r = rs where it approaches infiity. In appendix A it is proved that
2.H. Top. M. Baer/Electronically nonadiabatic effects. II
3
(al KG21 /
Fig. 3. The system of coordinates for treating the collinear
system. [n 1 it was proven that the polar ENCT rp(p, a) and I-,@, ol) are related to the Cartesian ENCT T,(I-,R) and rR (r, R) ia the following way: ~,,(p, o) = -[cos~~(r,
R) + sinorR(r, R)] ,
(11.5) 7&j, ol) = /I [Sin&rr(?, R) - cos~~,j (r, R)] .
0001 000
1
I
I
300
600
900
It can be seen that for p = 0 the term r,(p, ol) E 0. For symmetric systems(XZ f X +X f ZX) there is also an important feature regarding T,(P, cx)_It can be verified that for collinear systems in the configuration r, = rz = $ r3 (the rj; i = 1,2,3 are the interatomic distances) are related to each other as follows, T andr ‘2 rl (11.6) ‘rl
R(od
Fig. 2. The hst excited adiabatic potential surface of the collinear H; system as obtained by the DIMZO method: (a) A three-dimensional surface as a function of the interatomic distances. (b) Equipotential contour lines as a function of the scaled distances r and R (the numbers are in eV).
lin_
~&r, R) = $n6(r - rs) .
(11.3)
The translational ENCT TV(I, R) is usually smaller than rr(r, R) and becomes zero as R + m Em T&,R)=O. R-L-
(11.4)
Three dimensional figures of vibrational and translational ENCT are shown in figs. 4 and 5 respectively.
= -‘rz
.
If /3is the angle between the skewed axes in the (r, R) system, i.e.: 0 = arctan(+kf/nzX) 2 1Ql=“IZ+2mx)
(11.7)
where ntZ and nzx are the masses of the atoms, then from eqs. (11.5) and (11.6) it can be seen that for o = C_(j) 1-,~,cr=$r-j3))=0.
(11.8)
Although the information required to perform the calcuIation is given in the adiabatic scheme, the actual computations of the wavefunction and the S matrix elements were made in the diabatic scheme. The reasons for this are threefold: (1) the asymptotic states which correspond to the physical non-interacting systems are the diabatic states; (2) in the adiabatic scheme
Z.-H. Top, M. Baer/ElectronicaUy nonadiabatic effects. II
the diabatic one has been presented several times [ 1,311 and we will therefore discuss only the main points. If A(r, R) is the transformation matrix to go from the adiabatic to the diabatic scheme and ifV(r, R) is the APEM, then the diabatic potential energy matrix (DPEM) is given in’the form: W(r, R) = A*(r,R)V(r,R)A(r,R)
(11.9)
.
For the two-surfaces case A(r, R) can be written cosy
Shy
A= ( -silly
cds7 1
(11.10)
,
where r(r,R)=7(‘0,Rg)+j~r~~,Rg)dr+SR~~~~,R)dR, '0
Ro
(II.1 1)
Fig. 4. A
threedimensional surfaceof the vibrational nonadiabatic coupling term as a function of the interatomic distances.
that is to say 7(r, R) is a path integral over the ENCT. In fig. 6 7(r, R) is presented as a function of r for different values of R. In polar coordinates the value of 7@, or) reduces to a simpler form:
where ~,b, a) is given by eq. (KS). The form of the potential in the asymptotic region is of special interest. In general the four matrix elements ofW(r,R) are Iv, J = VJ COG7 + ~2sin27 ) w** = v, si&
+ v2co&y,
(11.13)
&I = I52 = (VI - ~+n7cos7, Fig. 5. A three-dimensionaJsurface of the tian&tionaJ nonadiabaticcoupling term as a function of the interatomic distanCeS.
the equations also contain fmt order derivatives (due to the ENCl’) which cause some difficulties in applying the existing efficient integration methods for solving such problems; (3) the handling of large ENCT is much easier in this scheme. The transformation from the adiabatic scheme to
where VJ(r, R) and V,(r, R) are the two lowest adiabatic surfaces. In the asymptotic region rR(r, R) = 0 and 7,.(r,R) is given by eq. (11.3). Inserting eq. (11.3) in eq. (II. I 1) for a fmed value ofR = RO, one obtains [assuming r(~, Ro) = 01: 7(r,R)=O;rr,_
Substituting eq. (lI.14) for y(~, R) in eq. (11.13) yields:
Z.H. Top, M. BaerfEIectronically nonadiabaric effecrs. II
rot
rz(0.u)
r2(o.u.)
Fig. 6. u(rl, r2) asfunction of r2 for different rI values. Here r2 is the vibrational coordinate of the diatomic molecule and rl is the distancebetween the incoming atom (or ion) and the nearest atom of the diatomicmolecule. Curves 1-3 refer to three finite MhteS (distances)of rt (curve 1 refers to the smallest) whereas curve 4 is for the asymptotic case (namely rl -, -). The right handside fmre showsan asymptoticcut of the three adiabaticsurfacesas a function of r2. The valuer2 = rat correspondsto the q-value where y varies discontinuously from zero to (742).
WI1 = vl;rr,,
W22=V2;rrs, w21=w12=0. (WI2 is identically zero because either y = 0 or y = &, or, V1 = V2 when y varies from zero to frr.) The functions WI1 and It’*‘22 are precisely the Morse potentials of H2 and Hl respectively. A similar procedure yields the relevant diabatic states of the products.
3. The Schriidinger equation (SE) The SE in the diabatic scheme takes the form -(fi2/24
V2$ + WJr = E+ ,
(III. 1)
where W correctly describes the physical asymptotic potential and V2 stands for: V2 = a2/llR2 + a2/ar2 .
(111.2)
The main disadvanrage of using eq. (111.1)as it is written is that the off-diagonal terms in W increase as R decreases almost independently of the ENCT. From eq. (11.13) it can be seen that Wl2 is proportional to the difference between the two surfaces. In many cases (and especially in the Hf case) only the lower APES
leads to a reaction, whereas the upper one increases rather rapidly and becomes strongly classically forbidden. Thus one would expect that in the region where this happens, the upper surface and the ENCT which became negligibly small have a minor effect on the reaction process taking place on the lower surface. This feature is not explicitly expressed in eq. (111.1). On the contrary, for a given region where the ENCT are identically zero, 7 becomes constant (#0) and therefore
the diabatic
coupling
term
WI2 increases
at
APES separate from each other. This fact was found to cause such instabilities in the numerical treatment that the whole procedure became doubtful. To overcome this problem a new procedure was developed which permits a smooth transition from the pure diabatic scheme, which is the appropriate one in the asymptotic region, and the weak interaction region to the “most” adiabatic scheme which governs the system in the interaction region (see fig. 3). Instead of solving eq. (!II.l) as it is written, the procedure is first to rotate the system of coordinates by a constant angle ro, solve the equations in the rotated frame of coordinates and then, if necessary, rotate it back to the original position. This rotation, which usually does not Introduce any significant error, is done in such a way as to reduce the off-diagonal terms of W and therefore is expected to increase the stability of the numerical treatment. The general procedure is as follows. Multiplying eq. (111.1)by a matrix A(ro) and replacing Jl(r. R) by x(r, R) defmed as: the same rate as the two
Z.H. Top. M. BaerjElectronically nonadiabatic effects. II
6
%(r,R)= AhoN+, R)
(111.3)
leads to:
x = xi+1 is completed so that x(‘l(x,+, , y) is known then *(xi+1 ,y) is obtained (111.10)
‘k(~~+~,v) = A*(~j)X(~l(x~+l .Y) + iKfX= 0 )
-(@/2fl)v*X
(111.4)
where (111.5)
fi = A(T,$“JA*(~~) .
,
-
(III.9’)
(11.9’)
X”+‘)(xf+l, _I’)= A(Ti+l - ‘Yi)Kci)(Xi+l 23’) . (III- 11)
(111.6)
This procedure is started at the asymptotic region and is repeated at every step. It can be seen that the diabatic coupling terms Wl2(T - -rj) are large or small according to whether the ENCT are large or small in the region of interest. If, for instance, the ENCT becomes zero from a certain value of x, then y wllI approach a constant value 700. Since each -yi is selected to be one of the 7’~ along the line x = Xi>~ISOri approaches the same value ~~0 and so eq. (111.4)becomes decoupled, namely:
we find tRat fi= A(ro)A*(r)vA(~)A*(ro>
K(‘+‘)(x,i+&) = A(~i+#(xi+l.y)
Combining e IS. (111.10)and (111.9’)we obtain
Recalling now that vu F A*(r)vA(r)
In order to continue to xi+2 a new value of 7 is selected, i.e., Q = ri+l and a new ~(‘~ll(x~+l,y) is formed:
>
Or ti=W(r-70).
The advantage of this procedure canbe seen immediately in case the ENCT become zero in a certain repion. If this is the case then 7 becomes constant, i.e. 7 = rou and therefore ifru is taken to be equal to ~00, El2 =O and I$= Vi;‘= 1,2. In practice this idea is installed in the following way. Let us assume that the solution Jr(x,y) [(x,y) may be either (R, r) or (o, p) where x is the translational coordinate] has been propagated up to the value x = xi so that Jl(xi,y) is known for everyy and the aim is to propagate it to x = xr+l, where xi+1 is not necessarily close to X~ We rotate JI (xi, y) by an angle ri defined as: ri = lu(xjl Ui) -
(111.7)
-(~2/2~)V2Xj t WjjXj =EXj;i=
1,2
(111.12)
where wfi= Vj, j= 1,2 and the transformation (111.11) reduces to the identity transformation. The following two points remain to be considered. (a) The matching of the two wavefunctions, one propagated along the reagent channel and the other along the product channel. In appendix B it is shown that along the line or= ac (see tig. 3) the continuity requirements of the total wavefimction and its fast derivative reduce to:
The equation to be solved is -(@/2&V*
x(i) + W(.y _ y.)j@ =Ex (il 3 I
(111.8)
where x(j) at x = xi is known: X’r’(xi*Y) = A(YJ *(xi, V) *
(111.9)
The value ofyj which determines ri is assumed to be given and we shall refer to it later, but in general it can be written as: Vi =Y(xi) 9
(111.10)
where y =y (x) is a known function. From the deftition of ri it can be seen that W(y - ri) is diagonal at least at one point along the limex = xi and at that point the diagonal term Wjj(xi,yJ is equal to the APE vi(xi.yi). Assuming the propagation to the line
(111.13)
axiRCo,a=ac)taa--aXiP@,n=a,)f~a;i= I,2,
where superscripts R and P stand for “reagent” and “product” respectively. (b) Determination of the function y = y(x) for which Ti = y(xi, _v(xi)). In principle this line can be any line that leads from reagents to products, but in practice it was found that some lines lead to more stable results than others. Since the reason for introducing the whole procedure is to reduce the off-diagonal coupling terms along the main route of the wave packet (in this way the propagation of the wavefimction is performed on surfaces which are as close ,aspossible to the adiabatic surfaces), the line y =y(x) has to
2X Top. M. BaerlElectronically nonadtibatic effects. II
be chosen accordiigly. In the Hi system it was found that the line which follows the minimum energy path along the lower adiabatic surface produced stable and well converged results. The reason is that the diagonal matrix elements Wii(r - vi): i= 1,2 represent two well behaved surfaces: Wll(r - yi) becomes the lower surface on which the reactive process takes place, while Wzz(-y - -yJ becomes the upper inelastic surface.
4. Results and discussion - the ground state In order to discuss the electronic nonadiabatic transitions (ENT) between two surfaces in a more comprehensive way, we first present some results that were obtained from a single-surface calculation where the upper surface and the ENCT were ignored. Part of these results are somewhat different from those published before [23]. The previous calculations were performed on a modified potential surface but it was later established that these modifications had an effect on the final results. The present calculations were made on the original (adiabatic) surface without any changes and are therefore considered to be the most relevant. The $ single surface transition probabilities are chaiacterized by two features: (i) As seen in fig. 7, the total reactive transition probabilities for different initial vibrational states converge towards unity as a function of ET_
‘hi 0-o c--l e-2 v-3 ‘-4
vf final vibrational state Fig. 8. Final vibrational distributions (singlesurface cahlation) for different total energies and different initial Librational s~&s: (a) pi= 0; (b) “i= 1; (c) ui= 2; (d) “i= 3.
a-5
(ii) From fig. 8, where the final vibrational distribution of the reaction: H2(Ui) + H: ~ HHI(u~) + H’ ) Fig. 7. Total reactive transition probabilities for the single (lower) surface case as a function of total energy for different
initial vibrational states (Ui).
(IV_1)
for different values Of Ui and ET are presented, it can be seen that all reactions are vibrationally adiabatic.
ZH. Top.
M. BaerjEIectronicaliy nonadiabatic effects. II 5. Results and discussion - the two surface case
L
v=oa
v-o
Considering the reactions: H2(Ui)+H’+$(uf) H2(Ui)+ $+
i-H(reactive + nonreactive),(V.l)
Hz&) + H+(reactive +-nonreactive) ,(V.2)
v--p_
Le
v=-v,
v=o
R
V=m
Fii. 9. The idealized potential energy surface for which the classical transition probabiity equation [eq. (Ii1.2)] is derived.
Part of the results can be explained by a classical model with a simple idealized potential (see fig. 9). This model potential is composed of three parts; the reagents channel, the products channel and the interaction region. In each of the channels (which due to the symmetry are assumed to be identical) the free particle does not interact with the two other particles which are assumed to be bound by a square well potential_ In the interaction region a well of depth Vu is assumed. In order to calculate the classical reactive transition probabilities P(ui, uf), where ui and vf are the initial and fmal vibrational states, the kaleidoscope method [32,33] is used. It can be shown that;
the function Pc(Ui, vf, ET) is defmed as the probability for charge transfer [eq. (V-l)] and the function Pd(ui, uf, ET) is defmed as the probability for the electronically adiabatic process [eq. (V.2)]. We also introduce the probability functions Pc(ui, ET) “ndPd(ui,ET) defmed as:
Pd(oiT ET) = C Pd(ui. Vf so
of, ET) 3
(V-4)
that:
pc(ui, ET) ‘Pd(uil ET) = 1 -
(v.3
In fig. 10 Pc(Ui, U@T) is shown as a function of uf for given Ui and ET and in fig. 11 Pc(Ui, ET) is shown as a function of ET for different u,. The branching ratio I’(+ ET) defmed aa: r(Ui, ET) = Pc(ui>&)IPd(Ujr ET) 9
(V.6)
is shown infig. 12 as a function Of Ui for various values of ET and the fmal average vibrational energy Evf(ui, E,,) of Hi defmed as: where Evi is the initial vibrational energy and ET is the total kinetic energy. 6, is equal to either 1 or zero de-
E,f(viJtran)
pending on whether or not the final vibrational energy is equal to the initial vibrational energy. The model
yields the main features found in the calculations; fmt, the fact that the reaction is vibrational adiabatic and second, the large reactivity. From eq. (IV.2) it can be seen that ifJ+ or Vo approach infinity the probability for reaction becomes unity. The model fails only in one respect; the rate at whichP(+ uf) approaches unity as a function of ET is slower than that found in the calculations. This difference could well be due to the fact that the model is classical and not quantum mechanical.
is shown in fig. 13 as a function of E,, for different values of ur where Etranis the translational energy of the reagents: Ebm= ET - Evi _
t-V.81
In all four figures as well as in eqs. (V.3) and (V.7) uf refers to a vibrational state in the upper surface. All four figures indicate that transition probabilities (TP) from q G 3 behave differently than TP’s from Ui> 4. Three features are noticed:. (i) Trarisitions from ui > 4 are-adiabatic with respect to the total internal energy (although electronic
2.H. Top. hf. BaerjElectronicallynonadiabaticeffects. II
9
Tot01 enerqy IN)
Fig. 11. Total charge transfer probabilities Pc(ui. ET) as a function of total energy ET for different initial vibrational states.
fdl
0.32
0.16 vi - initial vibration
o.ooo L!!r!LL ,
2
3
4
5
v, -final vibrational state Fig. 10. Final vibrational distriiution in the upper surface due to the reaction: Ha(oi) + H*+ H:(q) + H. The different ewes correspond to various total energiesand each square corresponds to a different initial vibrational state: (a) Ui= 0; (b) y r 1: (c) I = 4; (d) Ui= 5. The adiabatic feature of the trmslhon is seen from figs. (c) and (d).
nonadiabatic processes are involved). From fig. 10 it is seen that the main transition from Ui= 4 is to uf =O artdfromUi=Stou~=l,Z.fttu~outthatallfour lowest vibrational states ui = 0, 1,2,3 are below thresh-
Fig. 12. Branching ratios for charge transfer r(q, ET) = Pc(ui, ET)/~d(Ui. ET) as a function of initial vibrational state for different values of total energy.
charge transfer, whereas ui = 4 is energetically close to uf = 0 of the upper surface and Ui= 5 is in between uf = 1 and vf = 2 (see fig. 14). (ii) TP’s with ui G 3 are much more enemy dependent than those with ui > 4. From fig. 11 it is seen that Pc(ui, ET) ;br vi < 3 increases with energy from zero up to a certain relatively large value (sometimes as h@ as 0.4). On the other hand, TP’s with ui > 4 are much less energy dependent and at most oscillate around some fured value (the case ui = 3 can be considered as old for
10
.
2.00:
2.H. Top. M. Beer/Electronically nonadiabatic effects. II
I I Em
I 2 -
I
3
Imsfatimolenergy
Fig. 13. Final average vibrational ener_g of the H$(uf) molecule due to the reaction H.z(u$ + H+-- H;(vf) + H as a function of initial trunslurionel kinetic energy for different initial vibrational states.
an intermediate case). The same feature is also presented in fig. 13 where E”f(Ui, En,) is shown as a function of the translational energy. Hem the fmal average vibrational energy is strongly energy dependent for Ui Q 3, whereas it is almost constant foq24. (iii) The TP’s with Ui > 4 are in general larger than those with Ui < 3 and oscillate around a fixed value of PC_ This can be seen from the TP’s curves in fig.1 1 and from fig. 12 where the branching ratios I’(ui, ET) are presented_ These three features indicate that the mechanism
which yields the nonadiabatic transitions from initial vibrational states y*;2 4 is different from the mecha-. r&m which yields the transitions from Ui< 3 [3]. The electronic TP’s from high vibrational states, can be interpreted by applying results from pre%ious studies [34] _In this work a model system is described in which two reactive surfaces were coupled by a vibrationalNCT. By varying the energy gap between the two surfaces we were able to show [34] that Pc(ui, uf, ET) is strongly dependent, not on the energy gap between the two surfaces, but on the vibrational energy _ gap Aqf= I.!?,, - EVfI between the two vibrational states under consideration. This result was then supported by applying the Born approximation from which it was shown thatP,.(Ui, uf,E+,-) is inversely proportional to the square of A.Q. Thus the largest TP’s are expected when A~if * 0, narrely when the vibrational states of the two surfaces are in resonance.
Fig. 14. A schematic picture of the asymptotic vibrational levels for the two electronic surfaces. The two high vibrational states ui = 4,s are, in contrast to the lower vibrational states Ui = O-3, close to the resonance situation and therefore the TP’s are large and adiabatic with respect to total internal
energy. Whereas the mechanism which leads to electronic transitions for Vi2 4 is quite clear, the mechanism for ui < 3 is more involved. It certainly cannot be tiuenced by any simple resonance conditions because, as stated previously, the states ui = O-3 are below threshold for charge transfer. In fig. 15 we present the minimum energy paths of each of the surfaces and the corresponding vibrational states as a function of the reaction coordinate. It can be seen that these states are below threshold for charge transfer not only in the asymptotic region but also along the entire reaction coordinate. In a’previous publication [3] we ruled out one model and suggested another. The model that was ruled out suggested that transitions to the upper surface are performed via vibrational excited species which should become excited due to the strong interaction in the reaction zone (around s = 0 in fig. 16). This possibility was discarded because from onesurface calculations (see figs. 7 and 8) as welI as from two-surface calculations (see fig. 16 where pd(ui, Uf, ET) is presented as a function of q) it is seen that the outcome of the close interaction of the three atoms is
2.H. Top, M. Baer/Etectronically nonadiabatic effects. II
0.8
11
E,os, (OV) as0
(.I
.?
22s
-
JW
0
325 W 150 .
0.4
0.0 0.8
0.4 Fig. 15. The minimum energy paths in the two electronic surfaces (dashed lines) and the correspondiigvibrationallevels (full limes)as a function of the reaction coordinates.
vibrational adiabatic products_ This unexpected result, which is as explained in the previous chapter due to the symmetry of the Hi system, forces us to look for a different explanation. The model that was then suggested [3] attributed these strong transitions to the translational nonadiabatic coupling terms. However, we were recently able to estimate the strength of these couplings terms and found them to be about 0.1 eV. Since the vibrational energy gap between the low vibrational states uj < 2 of the lower surface and the ground state of the upper surface is about 1 eV or more (the only exception is Ui= 3 where the energy gap is about 0.5eV) these coupling terms could only partly be responsible for such high transition probabilities. Consequently we return to our first -model but with some modifications. But, before discussing the twosurface case let us fust refer again to the one-surface case: the basic assumption we want to make is that in the vicinity of s = 0 (and only there) the system is vibrationally excited. As can be seen from fig. 15 many vibrational states @lo) are energetically accessible in this region and due to the strong interaction in the potential well at s = 0, one would expect most of them to be populated to a certain degree. Now, whiIe the H’ approaches the H, molecule these vibrational states gradually become populated_ However, due to the symmetry built into the f$ system, the mechanism that populates these states while the two reagents
0.0
0.8
0.4
0.0
0.0
01234567
v,- final vibraknal state Fig. 16. Fiial (normalized) vibrational distributions on the lower surface (two-surface calculation) for different toti energies and different initial vibrational states: (a) Vi= 0; (b) Ui= 2; Cc)9 = 4: (d) ui = 5.
approach each other will depopulate them when the two recede, and consequently the fmal result is a vibrational adiabatic process. This situation is supported by numerical evidence. In performing the one-surface calculation it was found that although the system
12
Z.H.Top, M. BaerfEiectronically nonadiabatic effects. II
seemed to behave in a smooth way, a relatively large number ofvibrational states (=40) was needed to obtain converged results. This number is much larger than any other used in previous collinear calculations. The fact that sucha large number of states was needed to attain convergence indicates that all these states take part in the collision process and that the number of states that play a significant task in it is much larger than ten. Adopting this model the transition to the upper surface can be explained in the following way. Assuming that various vibrational states in the vii cinity of s = 0 are populated, these states may interact for a finite time with the states of the upper surface while the products are receding and in this way populate them. This process is strongly energy dependent (at least around the threshold for charge transfer) because the higher the energy the closer are the turning points of the open states of the upper surface to the interaction region (see fe_ 15) so that the overlap between them and the states of the lower surface is larger and the interaction time is longer. Some support for this picture is found in fig. 17a where the branching ratio l’,(NR, RI+ ET), defiiedas zc
is shown as a function of ET for different initial vibrational states. HerePc(NRIUi, ET) staridsfor the electronic probability of the nonreactive process (V-10) + l!$ c + HI (nonreactive) , 0“f and P,(RlUi, ET)* stands for the electronic probability of the reactive process _ Hz(Ui) + q
+ H (reactive) _ Hz(q) + HF + HHf c ( Uf )
V.ll)
The fact that f’,_(R,NRIq,&.) oscillates around 1 (except when the energies are very low where the charge transfer probability is small, > 1W2) could indicate that the electronic transition mainly takes place at the interaction region and not along the entrance and/or the exit channels. A somewhat different picture is seen in fig. 17b where r,(NR, RlU, ET) is presented for high initial vibrational states (Ui > 4). There, as explained before a different mechanism governs the electronic transition process which seems to favor the reactive channel slightly yet consistently.
(a)
6. Information theoretic approach - surprisal analysis If P(Ui,U& is defied as the probability of having a product molecule in vibrational state uf when the state of the reagent inolecule is ui, then the surprisal [24,25] is defmed as: ‘(ui* uf) = -ln[P(ui,
u&lp,(V,)] ,
w.0
whereP,-,(r+) is tire statistical transition probability: fl(Uf) =ft(uf)-1/2/cft(u3-‘/2 -
W.2)
Uf
ml20
’
2.4
’
28
I
32
hOi WJ Fig; 17. Branching ratios for reactive versus non-reactive processes on the upper surface as a function of total energy and for various initial vibrational states in the ground state.
Here &(uf) is the fraction of final total kinetic energy given as translational energy when the product molecule is in vibrational state uf. It has been sugggested [26] on the basis of information theory to approximate I(Ui, Uf) as: I(ui, Uf)= h:i + xhil&(“f) -.f,(vi)i * From the d&Aio~
i-
($1.3)
of PJNRI ui, ET) ZU@P,(R Iq, ET) ii is clear that Pc(ui, E$ =P&NR Iv;, ET) +P,(R 1”~ I?$, see eq. (V-3).
Z.H. Top, M. BaerfE!ectronically nonadiabaticeffects.II
where ft(ui) is the fraction of the initial total kinetic energy given as translational energy when the reagent molecule is in vibrational state ui_
“1
13
2840
0.
6.1. Results for one-surface case
t
r
In fig. 18 I(+ I+) is shown as a function off&, where 8t
f”, = 1 -f&f)
W.4)
for different total energy values and for Ui = 0. The fit to eq. (VI.3) is seen to improve as the energy increases. In fig. 19 I(+ vf) is shown as a function of fu, for
12co-
I
I
I
4 0
.r6* L
a 3
_‘q
lYf!ll 4108
8008 4
4OQ-
0
0
I200
;F
, 1
0
6
:
D
.
m
0
05
0
05
JO
‘”
, b
Fig. 19. Linear surprisal plots - single-surface calculation. The digit on the right hand side of each square stands for the initial vibrational state. All points are for ET = 3.2 eV. The bold num hers are the vibrational temperatures associated with each plot.
different values of ui where the total energy .!+ is 3.2 eV. The fit with eq. (VI.3) is excellent. The different temperatures r 4 it increases to ~4000 K and becomes only weakly dependent on ui. 6.2. The two-surface case
h
Fig. 18. Linearsurprisalplots - single-surface calculation. (a) ET = 1.5 eV; (b) J!?T = 2.5 eV; (c) ET = 3.2 ev. AU points are for Ui= 0.
The surprisals for the two surfaces for three different energies where the initial state is the ground vibrational state in the lower surface are shown in fig. 20. The following features are to be noticed: (i) Four points which correspond to the four lowest
14
-4
0.0
050
00
okio
IO
fv Fig. 20. Linear sorprisal plots - two-surface cdculation. The left squares refer to lower surfaceresults (circles)and the
n&t squaresto the UppersurfaceFZsubs(mm@es).(a) 5~ = 3.125 eV; (b) ET = 3.25 eV; (C)ET= 3.50 eV. AU pointsare for “i = 0. vibrational states in the lower surface are distributed in a satisfactory way around a straight line. The points that stand for the higher vibrational states (in the same surface) are removed from this line and do not seem to form a new line unless at least one point (the “transition” point for uf = 4) is ignored. (ii) The points which correspond to fmal vibrational states in the upper surface tend to arrange themselves along a straight lime. (There are two exceptions, one for& = 3.25 eV and one for ET = 3.50 eV which are quite far from the corresponding line. However, since the corresponding probabilities are very small (&l.Ol), the deviation could also be due to numerical inaccuracies). With regard to the lower surface results, a oneparameter model is not sufficient to describe the distribution. This could well be due to two mechanisms that are affecting the vibrational transitions in this surface
Fig. 21. Linear surprisal plots - two-surface calculation. AU plots refer to the lower surface results. ET = 3.5 eV_ Other notations see fig. 19.
and therefore more than two constraints are needed to defme the process. The surprisrdsI(ui, uf), all for the same total energy ET = 3.5 eV but for different initial vibrational states, are presented in figs. 21 and 22. As far as the lower surface results are concerned (see fig. 21) the quality of the fit to the proposed model varies from one distribution to another but is good enough to justify the basic assumptions involved in the information theoretic approach (in particular very encouraging results were obtained for ui = 2,3). As for the upper surface results (fig. 22) there the fit to the predicted lines seems to be even better (with the exception of those with Ui= 3). However, the fact that the temperatures associated with these distributions are so strongly dependent on the initial vibrational states is probably connected with-the different mechanisms that govern the electronic transitions from these states.
Z.H. Top, M. BaerfEiectronicaNy nonadiabatic effects. II
15
0.6 $ r g =. 0.4P E B z h 02g El e 2 a0
1 00
ICI
2D
30
J
EtololCevJ Fig. 23. Final average vib~tioual energy transfer (Ui= 0) as B function of total energy (lower surface results) 0 Single-surface
calculation; A two-surface calculation.
f”
Fig. 22. Linear surprisalplots - two-Surfacecalculation. All plots refer to vibrational distributionson the upper surface
which ara due to electronic transitions. ET = 3.5 eV. For other notations see fig. 19. 7.
Comparison with experimental results and other calculations For various reasons electronic transitions in the low energy range for this particular system were not studied experimentally
and only a limited amount of
work was devoted to vibrational [7,14,15]
and rota-
tional [16] transitions. Udseth et al. [7] reported on vibrational transitions in this system for several energies but only one, I.e. E,, = 4 eV is close to our range of energies. Since the present study is only collinear, no quantitative comparison can be done, but nevertheless some qualitative statements can be made. It was argued in the conclusions of their first paper [7] and in the introduction of another paper [12] that the main mechanism that leads to vibrational transition originates from the nonadiabatic coupling terms. To check this we calculated the fmal average vibra-
tional energy transferzf once ignoring the nonadiabatic coupling terms A& and once incorporating them-E& The results for transitions from the vibrational ground state are shown in fig. 23. As can be seen that no difference between the two is encountered as long as the energies are below threshold for charge transfer and both are very small ( 4), the charge transfer process takes place mainly in the exit channel and is a result of favorable resonance conditions between the initial state in the lower surface and the corresponding state in the upper surface. A surprisal analysis was performed by applying the theoretic information approach. It was found that for a given initial total energy the one-surface vibrational transition probabilities P(Ui + I+) are well described within this theory. Somewhat less satisfactory was the analysis of the two-surface case. For pure vibrational transitions the analysis indicated that at least two mechanisms are involved and therefore a theory based on one parameter (i.e. two constraints) is too simple to be able to describe them. For vibrational transitions accompanied by electronic transitions we found the analysis to be more encouraging. Most of the transitions were quite well described within this model(with the exception of those from ui = 3). Comparison with other calculations was limited because very little has been published on this particular
effects. II
system. The only few reported results based on a semiclassical approach seemed to be similar to ours.
Acknowledgement We would like to thank Professor D.J. Kouri and Professor T.F. George for stimulating discussions on various aspects of this system and Professor R.D. Levine and Dr. H. Kaplan for helpful discussions regarding the application of the theoretic information approach.
Appendix A: The asymptotic form of r,.(r. R) The starting point is the asymptotic potential matrix U(r,R) in the diabatic representation:
(A-1) In general many diabatic representations exist, all related to each other by orthogonal transformations, but there is only one which yields the correct physical asymptotic behavior. In this unique representation Uii, i= 1,2 become (asR + m) the respective diatomic Morse potentials and Ur, becomes zero. If V(r,R) is the potential matrix in the adiabatic representation:
v=
Vl
0
0
V2
(
1’
(A.21
then the relation between U and V can be written as: V = A(7) UA*(7) ,
(A.3)
where cos7 A(7)
=
( -sin7
sin7 (A-4)
3 cos7 )
and 7 is given by: 7 = &r + $ tan-’ [&
- rr,,)/2Q]
.
(A.3
If rs is the r value where (for a givenR) the two diabatic surfaces U22 and UI1 cross (for R + 0~rs is the crossing point of the two diatomic potentials) then AU defmed as
.
Z.H. Top, M. BaerfElectronicaUy nonadiabatic effects. II
AU = U12 - lJI1
(A-6)
AU@, R) = AU&, R) + (r - r,) AU’&, R) ,
rs)/2U12] .
(A-8)
U12, for R + m, becomes zero whereas AU’(r) in the vicinity of rS is different from zero and has a fured sign we may defme E as: Since
e = 2U121AU’(rs) ,
(A.91
such that: Al_ E(R) = 0 -
=9qe,N);x>x,.
(A-7)
where AU’ stands for d(AU)/dr. However AU(r,, R) E 0 and consequently 7 becomes [AU’(r,)(r-
continuous fust derivatives such that: \Ir(e,IV)=W(e,N);x,x,)
As for the derivative known that [ I,3 l]
of A(y,x)
(B.8’) with respect to x it is
3616.
_
aA/ax+T,A=@
(B-9)
Since A(JJ, x) is continuous and assuming that T,b, is continuous it is seen that aAjax is continuous as well, thus:
aA(l)(>r, x
=X,)/ax
Substituting leads to
ai@/ax
=
aAi2)(y,
X = X,)/ax
.
x)
(MO)
eqs. (B.8) and (B.lO) in eqs. (B.7)
.
= axQ)/ax
(B.11)
We have seen that the necessary condition for (B-1 1) to be valid at x = xc is the continuity of A(JJ, x). This condition can be shown to be sufficient as well by starting with eqs. (B.11) and going backwards. Returning to eqs. (111.13), x is identified with (Yand = oe such that: since rhR) and -Qcp) can be chosen at CY #R)
_
(R) =,,(P) TO
this implies that: A(R)@,
[7] H. Udseth, CF. Giese and W.R. Gentry, Phys. Rev. A8 (1973) 2483. [8] J.R. Krenos, R.K. Preston, R. Wolfgangand J.C. Tully, J. Chem. Phys. 60 (1974) 1634. [9] G. Ochs and E. Teloy, J. Chem. Phys. 61 (1974) 4930. [IOP J.C. Tully and R.K. Preston, J. Chem. Phys. 55 (1971) 562. [ll] Y.W. Lin, T.F. George and K. Morokuma, Chem. Phys. Letters 22 (1973) 547; J. Chem. Phys. 60 (1973) 4311. [12] C.F- Giese and W-R. Gentry, Phys. Rev. A10 (1974) 2156. [13] R.K. Preston and R.J. Cross Jr., I. Chem. Phys. 59 (1973)
_&PI
(B-12)
3
_
a! = ac) = A(p) co, (Y= crc)
,
(B.13)
which ensures the validity of eqs. (111.13).
References [l] Z.H. Top and M. Baer, J. Chem. Phys. 66 (1977) 1363. [2] Z.H. Top and M. Baer, J. Chem. Phys. 64 (1976) 3078. [3] Z.H. Top and M. Baer, Chem. Phys. Letters 39 (1976) 134. [4] J. Krcnos and R. Wolfgang, J. Chem. Phys. 52 (1970) 5961. (51 W.B. Maier II, J. Chem. Phys. 54 (1971) 2732. [6] M.G. Holliday, J.T. Muckerman and L. Friedman, J. Chem. Phys. 54 (19713 1058, 3853.
[14] H. Udseth, C.F. Giese and W.R. Gentry, J. Chem. Phys. 54 (1971) 3642. [15] H. Schmidt, V. Hermarm and F. Lindner, Chem. Phys. Letters 41(1976) 365. [16] K. Rudolfand J.P. Toennies. J. Chem. Phys. 65 (1976) 4483. [17] F.S. Collins, R.K. Preston and R.J. Cross, Chem. Phys. Letters 25 (1974) 608: F.S. Collins and R.J. Cross Jr., J. Chem. Phys. 65 (1976) 644. [18] H. Kriiger and R. S&I&, J. Chem. Phys. 60 (1977) 5087. [I91 P. McGuire, K. Rudolf and J.P. Toennies, J. Chem. Phys. 65 (1976) 5522. [20] P. McGuire. J. Chem. Phys. 65 (1976) 3275. [21] Y-W. Lie, T.F. George and K. Morokuma, J. Phys. B8 (1975) 265. [22] W.H. Miller and T.F. George, J. Chem. Phys. 56 (1972) 5637. [23] Z.H. Top and M. Baer, Chem. Phys. Letters 28 (1974) 352. [24] R.B. Bernstein and R.D. Levine, J. Chem. Phys. 57 (1972) 434. [25] A. Be&haul, R.D. Levine and R.B. Bernstein, J. Chem. Phys. 57 (1972) 5427. [26] I. Procaccia and R.D. Levine, Chem. Phys. Letters 35 (1975) 5; J. Chem. Phys. 63 (1975) 4261. [27] R.K. Preston and J.C. Tully, J. Chem. Phys. 54 (1971) 4297. [28] F-0. Ellison, J. Am.Chem. Sot. 85 (1963) 3540,3544. [29] J.K. Cashion and D.R. Henchbach, J. Chem. Phys. 40 (1964) 2358. [30] L.D. Landau and E.M. Lifschitz, Quantum mechanics (Addison-Wesley, Reading, 1958) p. 261. [31] M. Baer, Chem. Phys. Letters 35 (1975) 112;Chem. Phys. 15 (1976) 49. [32] D.W. Jepsen and J.O. Hirschfebler, J. Chem. Phys. 30 (1959) 1032. [33] B. KIeioman and K.T. Tang, J. Chem. Phys. 51(1969) 4587. [34] Z.H. Top and M. Baer, Chem. Phys. 10 (1975) 95. (351 E.E. Niitin. Chemische Eiementarprozesse. ed. H. Hartmann (Springer Verlag, Berlin, 1968). [36] A. Messiah, Quantum mechanics I (North-Holland Publishing Company, Amsterdam, 1964) p. 469.
