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A semiclassical approach to collisional ionization with application to the ArHe system
Chemical Physics 33 (1978) 219-226 0 North-Holland Publishing Company
A SEMICLASSICAL APPROACH TO COLLISIONAL IONIZATION WITH APPLICATION TO THE Ar-He SYSTEM* Kai-Shue LAM, John C. BELLUM* and Thomas F. GEORGE* Depariment of Chemistry, University of Rochester, Rochester, New York 14627, USA Received 17 February 1978
A semiclassical approach based on the propagation of classical trajectories on potential surfaces analytically continued into the complex plane, together with a discretization procedure, has been developed for the problem of collisional ionization. Based on Franck-Condon considerations the formalism is reduced to that of the two-state approximation. Preioniza-
tion loss and tunnel& beyond turning points have not been considered. Calculated partial ionization cross sections for the Ar-He system show good agreement with a fully quantum mechanical treatment.
1. Introduction Collisional ionization processes of the type A*+B+A+B++e-,
(1.1)
usually
referred to as Penning ionization (PI), have been amply studied on the basis of clas’sical and semiclassical [ 1,2], and quantum mechanical theories [1,3-7] _In the former approaches, cross sections are usually computed in terms of the autoionization width which may be obtained as a product of Landau-Zener transition probabilities. The quantum mechanical theoties fali into two classes. The first [1,3-S] makes use of a local complex adiabatic potential to account for transitions into the continuum of electronic states, whereas the second [6,7] is a coupled-channels treatment based on discretization of the continuum. *This research was sponsored by the National Aeronautics and Space Administration under Grant No. NSG2198, the Air Force Office of Scientilic Research (AFSC), United States Air Force, under Grant No. F49620-78-C-000.5 and the National Science Foundation under Grant Nos. CHE75 06775 A01 and CHE77-27826. The United States Government is authorized to reproduce and distriiute reprints for governmental purposes notwithstanding any copyright notation hereon. ’ Alfred P. Sloan Research Fellow, Camille and Henry Dreyfus Teacher-Scholar. * Resent address: Fachbereich Physik, Postfach 3049, Universitit Kaiserslautem, D-6750 Kaiserslautem, West Germany.
The practicability of the discretization procedure [6,7] has prompted us to test another semiclassical approach to inelastic collisions on the problem of colIisional ionization. This approach [8], based on the propagation of classical trajectories on potential surfaces treated as functions of complex coordinates, has been applied successfully to a variey of both field-free [9,10] and laser-enhanced [11,12] discrete-state collisional processes. The ionization problem, however, involves bound-continuum interactions which render the discrete-state sem&lassical formalism not directly applicable. We have previously extended this formalism to treat the problem of collision-induced emission [ 131, which shares in common with ionization the features of bound-continuum interactions. Here we wish to demonstrate that yet another variant of this semiclassical formalism is suitable for the description of ionization processes. Indeed, the present approach lends itself much more readily to dynamical interpretation than its quantum mechanical counterparts; and while it is as yet difficult to judge their relative usefulness, these semiclassical approaches are expected to play a major role in future studies of bound-continuum problems. In section 2 we present an account of the synthesis of the discretization procedure with the semic!assical formalism_ Section 3 deals with the application of this formalism to the He-Ar system and displays comparisons of computed results with the previous coupled-
220
K.-S. Lam et aL/Semi&ssical
approach to collisinnl
ionization
channels treatment [6,7] -The final section discusses some limitations and prospects of the present work.
2. Theory The main feature in all ionization problems is the interaction between a bound electronic state with the continuum of free electronic states in which it is embedded. The description of this is provided by the following expansion of the stationary electronic wavefunction:
(2.1)
I@> is the discrete state, and IQ+, e) is the state representing the ionization threshold with a free electron of energy E_I$> is to satisfy the Schrijdinger equation
HlJ/, =ElG).
(2.2)
In eq. (2.2)H is that part of the total hamiltonian representing all interactions except-the nuclear kinetic motion, and E represents the eigenvahres for the electronic energy of the system parametrized by the interuuclear distance R. Eqs. (2.1) and (2.2) lead to the following infinite set of coupled equations for the coefficients od and b,:
‘d’d
+s
deb, V& = Ead ,
(2.3a)
0
The quantities vd, V,, V& and E can ab be viewed as functions ofR. Vd and V+ represent the bound (excited-state) potential curve and the ionization threshold curve respectively, and V& is the bound-continuum coupling [having units of (energy)1/2]. We now apply the discretization procedure [6,7] to eq. (2.3) and write
s
deb,Vde
=c
b7 V& ,
(2.9)
7
0
where F represents a discretized electronic energy. V& is the coupling between the bound state and a discretized state (thus having units of energy) and is defined by V& =@IHI$J+, 23 -
(2.10)
The set of complete discretized states I&, 7) satisfy the orthonormal condition (2.11)
G$+,Z’I@l+,F> = SE ?’ ) and as before, we write (~+,~‘IHl9+,F’>
=(V++96677’
T
(2.12)
which follows from eq. (2.6). Making use of the closure relationship
s
d&3+. W+,
(2.13)
4 = 1,
0
(2.3b)
q&f(V+fe)bE=EbE,
for the continuum states I@+,E), the transformation between Vda and V,, is given by
where Vd--@i&#J>,
(2.4)
VdF = 1
(2.14)
deaF (E) Vde ,
0
V,, = wfm+,
E)-
(2.5)
Once the electron is far from the molecukir ion: G#Q,E’IHI@*, E) *(V+
f E)8(E - d) ,
a,- (e) = G#J;,EIfat, ?> -
(2.6)
where we have ignored direct coupling between continuum states. In arriving at eqs. (2.3) we have made use of the Facts that <@+,il@+, E) =6(c - i) , and
where
(2.7)
(2.15)
Eqs. (2.11) and (2.13) then lead to the following orthonormality condition for the coefficients a7 (E): s 0
deas(&+(e)
=t&
_
(2.16)
Eq. (2.16) can be satisfied by a large variety of com-
221
K.-S_ Lam et al_/Semiclassi~al approadz to collisional ionization
plete sets of basis functions parametrized by the discrete variable ‘i. However, a particularly simple set of basis functions, satisfying eq. (2.16) approximately, is given by: a; (E) = (AE)-~/~,
for
v(L) =
where
Z-$AEGEGF+$AE;
vy
= 0,
otherwise,
= P-d +L(L + 1)/Z/.&
)
(2.23)
,
(2.24)
(2.17)
where the stepsize AE can be chosen to be arbitrarily small. This set has been used successfully in a coupledchannels treatment of the ionization problem [6,7] and will be used also in the present work. We will now return to eq. (2.3) and extract the potential surfaces for the propagation of classical trajectories leading to ionization. Eqs. (2.9) and (2.14) imply that be =
(2.22) ’
V(L) f = v, + L(.L + 1)/2&
and ,u is the reduced mass of the collision system.
Di-
agonalization of VtL) gives _ the “adiabatic representation” o.‘the potential surfaces (ionization surfaces): E(L) =
(2.25)
where
Cb;a;(E).
(2.18)
7 Focusing on a particular electronic energy E = 7 and making use of eq. (2.17), the sum in eq. (2.18) is approximated by the single term bZ ~~(a_ Also vie = @)vd+?
_
(2-19)
section 3). The dynamics
Hence eq. (2.3b) becomes
adV&(V++~b7
=Eb;
The relation between VcL) and EcL) is essentially illustrated in figs. 1 and 2 for the Ar-He system (see
_
of the ionization
process can
(2.20)
At this point we introduce the present
treatment,
a crucial approximation of that of localized electronic cou-
pling. Even though, in principle, ionization resulting in a spread of electronic energies is possible at a particular nuclear configuration, Franck-Condon considerations imply that only localized bound-continuum couplings are significant. Considering an ionization event which results in an ejected electron of energy z, then, the bound-to-discretized-state transition characterized by 7 is taken to occur independently of all others. The sum in eq. (2.9) is replaced by the single term bF Vdz and eq. (2.3a) then reads adVd -tb;,Vdz
=Ead _
(2.21)
Hence discretization has reduced the problem to a form which resembles the usual discrete-state diabatic representation. In this form -the potential surfaces for nuclear motion with angular momentum tlL, describing the ionization event leading to ejected electrons of energy 7, are given, from eqs. (2.20) and (2.21), by:
R Fig. 1. Schematic representation of the diabatic-like surfaces in dL?
Dashed tines represent the free electron continuum.
222
_ -K.-S.Lam et al/Semiclassicalapproachto collisionaliohizatiop
5r
surfaces f$L) and E’2L)respectively and RO is the real part of the complex branch point R, [given by EiL)(R,) = E$L’(R,)] , eq. (2.27) leads to
RI
R2
(2.28a)
and lS~‘l2
= 4p2(1 -pa)
sin $Y-B-S); R1
RO
, (2.28b)
for the Ltb partial wave contribution to the S-matrix element. In eqs. (2.28) p1 andp2 represent the local
transition probabilities corresponding to the two energy regions. We have * R*
dR(k, - kl)
p1 = exp -2 Im j I
Ro
1,
* R* p2 =exp -2 irns Fig. 2. Ionization surfaces 8’) for the Ar-He system obtained for r = 0.151 au. Ra’is the real part of the branch point where E$aa) intersects E,(22). E,- represent incident collision energies corresponding to different dynamical situations.
be described with reference to fig. 2 as follows. The entrance channel is on J?,(~), which corresponds to ViL) asymptotically; and the exit channel on EF’, corre-
sponding to v?) t ?’ asymptotically. Two classical paths, dependmg on whether transition is made on the incoming or outgoing portion of the trajectory, contribute to the scattering matrix element Sz between surfaces 1 and 2. This S-matrix element is given by [9] : s7 = [p(l -p)]l/2(&‘a
- e’@b)_
(2.27)
P is the local
transition probability computed by propagating classical trajectories around the branch (intersection) point of the potential surfaces; and Ga and &,
are the classical actions corresponding to the two classical paths described above. In the present work we will only consider cases in which the total energy of the collision system Et lies in the regions represented by Et1 and Et2 in fig. 2. If RI and R2 represent the classical turning points on
C
(2.29)
R2
dR(kq--1)
dRk2
+2is RO
RO
1 ,
(2.30) RZ a=Re J dR(% - $) , Ro Ro
RO
dRk,,
B=_/
(2.31)
A=$
Rl
dRk;!,
(2.32)
R2
and k&R) = {2&T,
- Ei’L’(R)]}1/2/R
_
(2.33)
6 is a constant phase factor which arises as a consequence of matchingsemiclassical wavefunctions at the classical turning points and the avoided crossing, and assumes the value n/4 for the present case where the diabatic curves cross [9] _ Analogous to eq. (2.18) S-matrix elements for transitions from the bound state to the true continuum are then given by: .SCL) e = c L(e)Sg’ z
_
(2.34)
K-4. Lam et al./SemicIassicaIapproach to colh’ional ionization
The present choice of Qo)
as given in eq. (2.17) then
implies that S(L) =(A~)-l/z@ Z--$AKEG~+$A~. (2.35) E Finally, the partial ionization cross section per unit energy of the emitted electron for transition to the true continuum is given by: (2.36) where
‘g=
(A+
(2.37)
and
The potential parameters are taken to have the same numerical values (ii atomic units) as used in refs. [6,7]. A1 = 44.65270 au;
A2 = 4.39678 au,
(3Sa)
b, = 0.5320 a,-,;
b, = 0.9675 n,, ,
(3Sb)
Gel = 7.94 au L$ ;
Cs2 = 226 au “60,
(3 SC)
C41 = 0.6904 au ai;
C42’0,
(3Sd)
/31= 6.42317;
fi2 =2.81505,
(3Se)
R,,
R,,
= 5.76899 q,;
= 8.73010 Q,, ,
p=6631_406au.
li2K$2p =RIim_ [Et - Fd(R)] .
(2.38)
3. BesuIts and discussion
(3.la)
pd = p, D E0 =0.149 au,
(3.lb) i =
1,2 .
(3.2)
The short-range repulsion is represented by the first
term and the long-range force is represented by the product of the van der Waals attraction 19~and the Eckart potential 4_ These are given by tJi(R) = Ccl/R6 f C4i{R4 , &i(R) ={ I+ exp[-(R
coupling I& has the form
e) = cu(R)/e114 ,
(3-6)
where
The approach presented in the preceding section has been applied to the case of PI of Ar by He* (ls2s, 3S). The sources and reasons for a choice of functional forms for the potentials Fd and V, and the boundcontinuum coupling I& have been described in the previous work by BeUum and Micha [6,7] (hereafter referred to as BM [6] and BM [7]), which is a coupled-channels treatment based on discretization. In the present work, we wiIl retain the same set of potentials and coupling, together with the discretization scheme based on eq. (2.17). The expressions for the potentials and coupling are given below:
V-(R) =AiemRbf --Bi(R)@i(R); I
wf) (3.%)
The bound-continuum V&R,
V+ = VI - EO,
223
(3.3)
-RoJ/bi])-’
X [l .+&Cl + exp[(R _ Roi)/bi]}-‘1
.
(3.4)
(Y(R) = (2/~r)‘/~A m exp(-R/a m )/2’14 t
(3.7)
with A,
=SOau;
~~ =0_7a,
_
(3.8)
Eqs. (2.14) and (2.17) then imply that Vdz(R) = (AE)~/~,(R)/~~/~ .
(3.9)
In the present calculations, Et is taken to be 23.89 X low4 au (~65 meV) and AE is 2 X 1O-4 au, as has been done in BM [6] _In view of eq. (2.35), discretization implies that computed values for ionization cross sections wiU have an energy resolution of oniy up to Ae. We have chosen representative electronic energies Z corresponding to an ionization region between 7 a and 9 ao. For each F, the ionization surfaces E!LP and the associated R,(?‘) are computed by eq. (2.i6). Cross sections for successive F which lie within one interval of Ae are then averaged to reflect finite energy resolution. The semiclassical curves here reported for do/de versus Ef, the relative collision energy in the final channel (Rt = Et f EO - 3. have been obtained by joining smoothly these averaged values. Figs. 3 to 5 display comparisons between calculations based on the present semiclassical treatment and those based on the previous quantum mechanical treatment reported in BM [6,7] _As mentioned earlier,
224
K.-S. Lam et al.~Semichsical approach to collisional ionization
I
I
I .‘. i
4
3-
-
; P
-------
“0
s*miclossicol
x
% N oz
3
5 2-
5
-
2
0
6
3 8; I I ~
L
LJ
O, 3
0
Ef (X 10m4
I..: -...I:ii; ‘,_
I
5
i
.-_
i
I
IO
15
‘.
.
20
25
au)
Fig. 3. Lth partial ionization cross section per unit energy of the emitted electron for the L = 20 angular momentum component of the heavy particles, calculated for PI of AI by He* (Is?s, 3S) at an incident collision energy ofEt = 23.89 X 10m4 au. The quantum mechanical results are from ref. (61.
for each electronic energy, high L values.for which the dynamical situation corresponds to the total energy lying in regions marked by Et3 and Et4 in fig. 2 have not been considered_ The Et3 vase leads predominantly to associative ionization, where the heavy particles are trapped in the potential well, and multiple reflection and tunnelling need to be considered_ In our present study we are concerned with just Penning ionization. The Et4 case, however, is expected to have only minor contributions to the ionization cross section, since the cost probable configuration for ionization is in the non-classical region of EiL), and, unlike the Et3 case, associative ionzation is energetically forbidden. In addition, we expect pre-ionization loss, where electrons of the energy 7 under consideration may be emitted outside the pseudo-crossing region nearRo, to be negIigibIe on the basis of Franck-Condon consid6rations. Our present treatment ignores such pre-ionization loss, and we discuss this further in section 4. Therefore we
Fig. 4. Lth partial ionization cross section per unit energy of the emitted electron calculated for PI of Ar by He*(ls2s, 3S), at an incident collision energy of Et = 23.89 X 10s4 au and a retative coItision energy in the final channet ofEf = 3.8 X low4 au..The semiclassical calculations for L = 22 give a resonance partial ionization cross section of dc@)/do = 2950.5 a@ (not shown in figure). The quantum mechanical calculations are based on the formalism reported inrefs. [6,7].
r
I
I
I
-0UANTUM
I
I
I
I
MECHANICAL r
-SEMICLASSICAL
L
I
0 40
35
30
25
I
I
I
I
20
I5
IO
5
E&X Fig. 5. Partial ionization
Id40
-I
0
“1
cross section per unit energy of the emittedelectroncalculatedforPIofArbyHe*(ls2s,3S)atan incident collision energy ofEt = 23.89 X low4 au. The quantum mechanical results are from ref. [7 1. and show a pronounced peakinvolvinga resonance contribution from do(24)/de.
K.-S. Lam et d/Semiclassicalapproachto collisiod ionization
do not consider the Et4 case to be of enough importance to warrant extension of our treatment to include transitions which require tunnel@. While the treatment of these cases is not beyond the capability of the semiclassical approach [lo], it does require alternate formalisms and its inclusion is not likely to lend extra credence to the main fact that we wish to demonstrate here, namely, the compatibility between the semiclassical and the quantum mechanical treatments of collisional ionization. This fact can be well established by the results presented in figs. 3 and 4. The observable discrepancy, however, between the two approaches may be due to two causes which are as yet quite difficult to disentangle. On the one hand there may be inherent insufficiency in the semiclassical approach as opposed to the more rigorous quantum mechanical treatment. On the other, the present discretization procedure, which characterizes both the semiclassical and the quantum mechanical approaches, sets a limit to the resolution of computed cross sections on the energy scale. A refinement to smaller mesh intervals LIEwould reduce the resolution problem and permit a better comparison between the two approaches. Fig. 5 shows semiclassical partial cross sections du/de in an energy range for which the L-partial contributions corresponding to cases Et3 and Et4 (see fig. 2) are negligible, thereby allowing ;I reasonable comparison between the present semiclassical results and the quantum mechanical results which included contributions for all cases in fig. 2. This demonstrates that, for E,> ==10m3 au, the dynamical situations COTresponding to the neglected L do not contribute significantly to ionization. The semiclassical approach (with reference to the ionization surfaces) calls this fact to attention much more succinctly than its quantum mechanical counterpart. The present calculations also reveal resonance effects for dynamical situations marked by Et2 in fig. 2. In these cases the avoided crossing R is just barely accessible on the entrance surface El z ). The heavy partitle velocity is decreased drastically as the heavy particIes approach this transition region and hence, semiclassically, they spend appreciable time there, very likely resulting in transition_ One case which matches the resonance condition especially well is where L = 22 and E, = 3.8 X 10m4 au, and a value for da(**)/de = 2950.5 $/au is computed. Here it is worthwhile to note that I
da(22)/de >
c
d&“/de
225
.
L-c22
A similar effect is evident in the quantum mechanical results (BM [7]), in which a very pronounced peak exists for do/de, but occurs, however, at a slightly different Ef and involves a resonance contribution from do(24)/de (fig. 5).
4. Concluding remarks We have demonstrated that, using the same discretization procedure, the present semiclassical approach is able to parallel a coupled-channels quantum mechanical treatment of the problem of coliision-induced ionization. This particular application has strengthened our conviction in the general compatibility of these two approaches in the treatment of molecular collision problems. In the context of the ionization problem, however, there is still much room for refinement of these theories to include (after discretization) the spread of electronic energies which can interact with the discrete state. This spread of continuum energies is already partially taken into account by our twostate coupled-channels treatment [6,7], since it involves no Franck-Condon (local transition) restriction, allowing emission at a given electron energy 7 to occur at any R. Further refinement could be attained quantum mechanically by coupling the discrete state to more than one discretized continuum state at a time, leading to multi-channel coupled equations. Semiclassically, a first step toward improvement is to relax the present assumption of localized electronic coupling in a suitable manner. One would have to weigh S-matrix elements for particular ionization events (with specified ?‘) to reflect the uncertainty of the ionization configuration. This problem is not only relevant to collisional ionization, but also underlies the whole gamut of laser-enhanced processes in which the collision system is coupled to photons having a distribution of frequencies. We wish to point out that the isolation of a single discretized electronic energy ? in the construction of ionization surfaces has reduced the present semiclassical formalism to one which resembles that of the two-state approximation. Physically this amounts to ignoring the possibility of ionization at all values of
R other than thi one, say R’, for which Vd(R’) VJR’) = 7, and also the possibility of emitting electrons at R’ with energies 7 for which 7 # ?$(R’) F/+(R’). The former possibility would not only neces-
cesses [ II-131, we expect that the present semiclassical approach to field-free collisional ionization can be readily extended to treat the laser-enhanced cases. Work along this direction is in progress.
sitate a pre-ionization loss factor (Le., “survival factor”) as has been used in previous work [I], but also a relaxation-of the Franck-Condon assumption, for its proper treatment. The latter would require relaxing the Franck-Condon restrictions also, and (within the context of our discretization procedure) the inclusion of multi-surface dynamics_ While these problems raised by the presence of the continuum in the spectrum of electronic energies are in principle important as well as of tremendous interest, we are not aware of any evidence (theoretical
or experimental)
pointing
to-
wards their significance in practice concerning the case of Penning ionization_ Indeed, the two-state quantum mechanical calculations, which relax the FranckCondon restriction, compare well with the semiclassical results which are based on Franck-Condon transitions. Also, our previous study of the laser-enhanced emission problem [ 131 indicated that the FranckCondon approximation can be applied with confidence. Though lacking completeness, then, the present semiclassical treatment may be viewed as a study of the validity of the two-state approximation beyond its normal range of applicability. In recent work [4,15J we have presented an extension of the coupled-channels formalism, also based on discretization, to treat laser-enhanced collisional ionization. Inview ofwork reported here, and our past experiences with semiclassical treatments oflaser-enhanced pro-
References [i] W.H. Miller, J. Chem. Phys. 52 (1970) 3563. [2] W.H. Miller and H. Morgner, J. Chem. Phys. 67 (1977) 4923. [3] H. Nakamura, J. Phys. Sot. Japan 26 (1969)
1473.
[4] A.P. Hickman and H. Morgner, J. Phys. B: Atom. Mol. Phys. 9 (1976) 1765. [S] A.P. Hickman, AD_ Isaacson and W-H. Miller, J. Chem. Phys. 66 (1977) 1483; 1492. [6] J.C. B&urn and D.A. Micha, Chem. Phys. 20 (1977) 121. J.C. BeUum and D.A. hlicha, Phys. Rev. A, to be published.
[ 71
[8 ] W.H. Miller and T.P. George, J. Chem. Phys. 56 (1972) 5637. [9] J.R. Laing, T.F. George, LH. Zimmerman and Y.W. Lin, J. Chem. Phgs. 63 (1975) 842. [IO] J.R. Laing, J.M. Yuan, I.H. Zimmerman, P.L. de Vries
andT.F. George,J. Chem. Phys. 66 (1977) 2801. 1111J.M. Yuan. T.,‘. George and F.J. Mclaffertv. _ Chem. Letters 40 (1976)
Phvs.
16%
[12] J.M. Yuan, J.R. Laingand 66 (1977) 1107.
T.F. George,
[13] K.-S. Lam, LH. Zimmerman,
J. Chem. Phys.
J.M. Yuan, J.R. Laingand
T.F. George,Chem. Phys. 26 (1977) 455. [14] J.C. BeUumand T.F. George,J. Chem. Phys. 68 (1978) 134.
[15] J.C. Bellurn, K--S. Lam and T.F. George, J. Chem. Phys. 69 (1978) August.
A SEMICLASSICAL APPROACH TO COLLISIONAL IONIZATION WITH APPLICATION TO THE Ar-He SYSTEM* Kai-Shue LAM, John C. BELLUM* and Thomas F. GEORGE* Depariment of Chemistry, University of Rochester, Rochester, New York 14627, USA Received 17 February 1978
A semiclassical approach based on the propagation of classical trajectories on potential surfaces analytically continued into the complex plane, together with a discretization procedure, has been developed for the problem of collisional ionization. Based on Franck-Condon considerations the formalism is reduced to that of the two-state approximation. Preioniza-
tion loss and tunnel& beyond turning points have not been considered. Calculated partial ionization cross sections for the Ar-He system show good agreement with a fully quantum mechanical treatment.
1. Introduction Collisional ionization processes of the type A*+B+A+B++e-,
(1.1)
usually
referred to as Penning ionization (PI), have been amply studied on the basis of clas’sical and semiclassical [ 1,2], and quantum mechanical theories [1,3-7] _In the former approaches, cross sections are usually computed in terms of the autoionization width which may be obtained as a product of Landau-Zener transition probabilities. The quantum mechanical theoties fali into two classes. The first [1,3-S] makes use of a local complex adiabatic potential to account for transitions into the continuum of electronic states, whereas the second [6,7] is a coupled-channels treatment based on discretization of the continuum. *This research was sponsored by the National Aeronautics and Space Administration under Grant No. NSG2198, the Air Force Office of Scientilic Research (AFSC), United States Air Force, under Grant No. F49620-78-C-000.5 and the National Science Foundation under Grant Nos. CHE75 06775 A01 and CHE77-27826. The United States Government is authorized to reproduce and distriiute reprints for governmental purposes notwithstanding any copyright notation hereon. ’ Alfred P. Sloan Research Fellow, Camille and Henry Dreyfus Teacher-Scholar. * Resent address: Fachbereich Physik, Postfach 3049, Universitit Kaiserslautem, D-6750 Kaiserslautem, West Germany.
The practicability of the discretization procedure [6,7] has prompted us to test another semiclassical approach to inelastic collisions on the problem of colIisional ionization. This approach [8], based on the propagation of classical trajectories on potential surfaces treated as functions of complex coordinates, has been applied successfully to a variey of both field-free [9,10] and laser-enhanced [11,12] discrete-state collisional processes. The ionization problem, however, involves bound-continuum interactions which render the discrete-state sem&lassical formalism not directly applicable. We have previously extended this formalism to treat the problem of collision-induced emission [ 131, which shares in common with ionization the features of bound-continuum interactions. Here we wish to demonstrate that yet another variant of this semiclassical formalism is suitable for the description of ionization processes. Indeed, the present approach lends itself much more readily to dynamical interpretation than its quantum mechanical counterparts; and while it is as yet difficult to judge their relative usefulness, these semiclassical approaches are expected to play a major role in future studies of bound-continuum problems. In section 2 we present an account of the synthesis of the discretization procedure with the semic!assical formalism_ Section 3 deals with the application of this formalism to the He-Ar system and displays comparisons of computed results with the previous coupled-
220
K.-S. Lam et aL/Semi&ssical
approach to collisinnl
ionization
channels treatment [6,7] -The final section discusses some limitations and prospects of the present work.
2. Theory The main feature in all ionization problems is the interaction between a bound electronic state with the continuum of free electronic states in which it is embedded. The description of this is provided by the following expansion of the stationary electronic wavefunction:
(2.1)
I@> is the discrete state, and IQ+, e) is the state representing the ionization threshold with a free electron of energy E_I$> is to satisfy the Schrijdinger equation
HlJ/, =ElG).
(2.2)
In eq. (2.2)H is that part of the total hamiltonian representing all interactions except-the nuclear kinetic motion, and E represents the eigenvahres for the electronic energy of the system parametrized by the interuuclear distance R. Eqs. (2.1) and (2.2) lead to the following infinite set of coupled equations for the coefficients od and b,:
‘d’d
+s
deb, V& = Ead ,
(2.3a)
0
The quantities vd, V,, V& and E can ab be viewed as functions ofR. Vd and V+ represent the bound (excited-state) potential curve and the ionization threshold curve respectively, and V& is the bound-continuum coupling [having units of (energy)1/2]. We now apply the discretization procedure [6,7] to eq. (2.3) and write
s
deb,Vde
=c
b7 V& ,
(2.9)
7
0
where F represents a discretized electronic energy. V& is the coupling between the bound state and a discretized state (thus having units of energy) and is defined by V& =@IHI$J+, 23 -
(2.10)
The set of complete discretized states I&, 7) satisfy the orthonormal condition (2.11)
G$+,Z’I@l+,F> = SE ?’ ) and as before, we write (~+,~‘IHl9+,F’>
=(V++96677’
T
(2.12)
which follows from eq. (2.6). Making use of the closure relationship
s
d&3+. W+,
(2.13)
4 = 1,
0
(2.3b)
q&f(V+fe)bE=EbE,
for the continuum states I@+,E), the transformation between Vda and V,, is given by
where Vd--@i&#J>,
(2.4)
VdF = 1
(2.14)
deaF (E) Vde ,
0
V,, = wfm+,
E)-
(2.5)
Once the electron is far from the molecukir ion: G#Q,E’IHI@*, E) *(V+
f E)8(E - d) ,
a,- (e) = G#J;,EIfat, ?> -
(2.6)
where we have ignored direct coupling between continuum states. In arriving at eqs. (2.3) we have made use of the Facts that <@+,il@+, E) =6(c - i) , and
where
(2.7)
(2.15)
Eqs. (2.11) and (2.13) then lead to the following orthonormality condition for the coefficients a7 (E): s 0
deas(&+(e)
=t&
_
(2.16)
Eq. (2.16) can be satisfied by a large variety of com-
221
K.-S_ Lam et al_/Semiclassi~al approadz to collisional ionization
plete sets of basis functions parametrized by the discrete variable ‘i. However, a particularly simple set of basis functions, satisfying eq. (2.16) approximately, is given by: a; (E) = (AE)-~/~,
for
v(L) =
where
Z-$AEGEGF+$AE;
vy
= 0,
otherwise,
= P-d +L(L + 1)/Z/.&
)
(2.23)
,
(2.24)
(2.17)
where the stepsize AE can be chosen to be arbitrarily small. This set has been used successfully in a coupledchannels treatment of the ionization problem [6,7] and will be used also in the present work. We will now return to eq. (2.3) and extract the potential surfaces for the propagation of classical trajectories leading to ionization. Eqs. (2.9) and (2.14) imply that be =
(2.22) ’
V(L) f = v, + L(.L + 1)/2&
and ,u is the reduced mass of the collision system.
Di-
agonalization of VtL) gives _ the “adiabatic representation” o.‘the potential surfaces (ionization surfaces): E(L) =
(2.25)
where
Cb;a;(E).
(2.18)
7 Focusing on a particular electronic energy E = 7 and making use of eq. (2.17), the sum in eq. (2.18) is approximated by the single term bZ ~~(a_ Also vie = @)vd+?
_
(2-19)
section 3). The dynamics
Hence eq. (2.3b) becomes
adV&(V++~b7
=Eb;
The relation between VcL) and EcL) is essentially illustrated in figs. 1 and 2 for the Ar-He system (see
_
of the ionization
process can
(2.20)
At this point we introduce the present
treatment,
a crucial approximation of that of localized electronic cou-
pling. Even though, in principle, ionization resulting in a spread of electronic energies is possible at a particular nuclear configuration, Franck-Condon considerations imply that only localized bound-continuum couplings are significant. Considering an ionization event which results in an ejected electron of energy z, then, the bound-to-discretized-state transition characterized by 7 is taken to occur independently of all others. The sum in eq. (2.9) is replaced by the single term bF Vdz and eq. (2.3a) then reads adVd -tb;,Vdz
=Ead _
(2.21)
Hence discretization has reduced the problem to a form which resembles the usual discrete-state diabatic representation. In this form -the potential surfaces for nuclear motion with angular momentum tlL, describing the ionization event leading to ejected electrons of energy 7, are given, from eqs. (2.20) and (2.21), by:
R Fig. 1. Schematic representation of the diabatic-like surfaces in dL?
Dashed tines represent the free electron continuum.
222
_ -K.-S.Lam et al/Semiclassicalapproachto collisionaliohizatiop
5r
surfaces f$L) and E’2L)respectively and RO is the real part of the complex branch point R, [given by EiL)(R,) = E$L’(R,)] , eq. (2.27) leads to
RI
R2
(2.28a)
and lS~‘l2
= 4p2(1 -pa)
sin $Y-B-S); R1
RO
, (2.28b)
for the Ltb partial wave contribution to the S-matrix element. In eqs. (2.28) p1 andp2 represent the local
transition probabilities corresponding to the two energy regions. We have * R*
dR(k, - kl)
p1 = exp -2 Im j I
Ro
1,
* R* p2 =exp -2 irns Fig. 2. Ionization surfaces 8’) for the Ar-He system obtained for r = 0.151 au. Ra’is the real part of the branch point where E$aa) intersects E,(22). E,- represent incident collision energies corresponding to different dynamical situations.
be described with reference to fig. 2 as follows. The entrance channel is on J?,(~), which corresponds to ViL) asymptotically; and the exit channel on EF’, corre-
sponding to v?) t ?’ asymptotically. Two classical paths, dependmg on whether transition is made on the incoming or outgoing portion of the trajectory, contribute to the scattering matrix element Sz between surfaces 1 and 2. This S-matrix element is given by [9] : s7 = [p(l -p)]l/2(&‘a
- e’@b)_
(2.27)
P is the local
transition probability computed by propagating classical trajectories around the branch (intersection) point of the potential surfaces; and Ga and &,
are the classical actions corresponding to the two classical paths described above. In the present work we will only consider cases in which the total energy of the collision system Et lies in the regions represented by Et1 and Et2 in fig. 2. If RI and R2 represent the classical turning points on
C
(2.29)
R2
dR(kq--1)
dRk2
+2is RO
RO
1 ,
(2.30) RZ a=Re J dR(% - $) , Ro Ro
RO
dRk,,
B=_/
(2.31)
A=$
Rl
dRk;!,
(2.32)
R2
and k&R) = {2&T,
- Ei’L’(R)]}1/2/R
_
(2.33)
6 is a constant phase factor which arises as a consequence of matchingsemiclassical wavefunctions at the classical turning points and the avoided crossing, and assumes the value n/4 for the present case where the diabatic curves cross [9] _ Analogous to eq. (2.18) S-matrix elements for transitions from the bound state to the true continuum are then given by: .SCL) e = c L(e)Sg’ z
_
(2.34)
K-4. Lam et al./SemicIassicaIapproach to colh’ional ionization
The present choice of Qo)
as given in eq. (2.17) then
implies that S(L) =(A~)-l/z@ Z--$AKEG~+$A~. (2.35) E Finally, the partial ionization cross section per unit energy of the emitted electron for transition to the true continuum is given by: (2.36) where
‘g=
(A+
(2.37)
and
The potential parameters are taken to have the same numerical values (ii atomic units) as used in refs. [6,7]. A1 = 44.65270 au;
A2 = 4.39678 au,
(3Sa)
b, = 0.5320 a,-,;
b, = 0.9675 n,, ,
(3Sb)
Gel = 7.94 au L$ ;
Cs2 = 226 au “60,
(3 SC)
C41 = 0.6904 au ai;
C42’0,
(3Sd)
/31= 6.42317;
fi2 =2.81505,
(3Se)
R,,
R,,
= 5.76899 q,;
= 8.73010 Q,, ,
p=6631_406au.
li2K$2p =RIim_ [Et - Fd(R)] .
(2.38)
3. BesuIts and discussion
(3.la)
pd = p, D E0 =0.149 au,
(3.lb) i =
1,2 .
(3.2)
The short-range repulsion is represented by the first
term and the long-range force is represented by the product of the van der Waals attraction 19~and the Eckart potential 4_ These are given by tJi(R) = Ccl/R6 f C4i{R4 , &i(R) ={ I+ exp[-(R
coupling I& has the form
e) = cu(R)/e114 ,
(3-6)
where
The approach presented in the preceding section has been applied to the case of PI of Ar by He* (ls2s, 3S). The sources and reasons for a choice of functional forms for the potentials Fd and V, and the boundcontinuum coupling I& have been described in the previous work by BeUum and Micha [6,7] (hereafter referred to as BM [6] and BM [7]), which is a coupled-channels treatment based on discretization. In the present work, we wiIl retain the same set of potentials and coupling, together with the discretization scheme based on eq. (2.17). The expressions for the potentials and coupling are given below:
V-(R) =AiemRbf --Bi(R)@i(R); I
wf) (3.%)
The bound-continuum V&R,
V+ = VI - EO,
223
(3.3)
-RoJ/bi])-’
X [l .+&Cl + exp[(R _ Roi)/bi]}-‘1
.
(3.4)
(Y(R) = (2/~r)‘/~A m exp(-R/a m )/2’14 t
(3.7)
with A,
=SOau;
~~ =0_7a,
_
(3.8)
Eqs. (2.14) and (2.17) then imply that Vdz(R) = (AE)~/~,(R)/~~/~ .
(3.9)
In the present calculations, Et is taken to be 23.89 X low4 au (~65 meV) and AE is 2 X 1O-4 au, as has been done in BM [6] _In view of eq. (2.35), discretization implies that computed values for ionization cross sections wiU have an energy resolution of oniy up to Ae. We have chosen representative electronic energies Z corresponding to an ionization region between 7 a and 9 ao. For each F, the ionization surfaces E!LP and the associated R,(?‘) are computed by eq. (2.i6). Cross sections for successive F which lie within one interval of Ae are then averaged to reflect finite energy resolution. The semiclassical curves here reported for do/de versus Ef, the relative collision energy in the final channel (Rt = Et f EO - 3. have been obtained by joining smoothly these averaged values. Figs. 3 to 5 display comparisons between calculations based on the present semiclassical treatment and those based on the previous quantum mechanical treatment reported in BM [6,7] _As mentioned earlier,
224
K.-S. Lam et al.~Semichsical approach to collisional ionization
I
I
I .‘. i
4
3-
-
; P
-------
“0
s*miclossicol
x
% N oz
3
5 2-
5
-
2
0
6
3 8; I I ~
L
LJ
O, 3
0
Ef (X 10m4
I..: -...I:ii; ‘,_
I
5
i
.-_
i
I
IO
15
‘.
.
20
25
au)
Fig. 3. Lth partial ionization cross section per unit energy of the emitted electron for the L = 20 angular momentum component of the heavy particles, calculated for PI of AI by He* (Is?s, 3S) at an incident collision energy ofEt = 23.89 X 10m4 au. The quantum mechanical results are from ref. (61.
for each electronic energy, high L values.for which the dynamical situation corresponds to the total energy lying in regions marked by Et3 and Et4 in fig. 2 have not been considered_ The Et3 vase leads predominantly to associative ionization, where the heavy particles are trapped in the potential well, and multiple reflection and tunnelling need to be considered_ In our present study we are concerned with just Penning ionization. The Et4 case, however, is expected to have only minor contributions to the ionization cross section, since the cost probable configuration for ionization is in the non-classical region of EiL), and, unlike the Et3 case, associative ionzation is energetically forbidden. In addition, we expect pre-ionization loss, where electrons of the energy 7 under consideration may be emitted outside the pseudo-crossing region nearRo, to be negIigibIe on the basis of Franck-Condon consid6rations. Our present treatment ignores such pre-ionization loss, and we discuss this further in section 4. Therefore we
Fig. 4. Lth partial ionization cross section per unit energy of the emitted electron calculated for PI of Ar by He*(ls2s, 3S), at an incident collision energy of Et = 23.89 X 10s4 au and a retative coItision energy in the final channet ofEf = 3.8 X low4 au..The semiclassical calculations for L = 22 give a resonance partial ionization cross section of dc@)/do = 2950.5 a@ (not shown in figure). The quantum mechanical calculations are based on the formalism reported inrefs. [6,7].
r
I
I
I
-0UANTUM
I
I
I
I
MECHANICAL r
-SEMICLASSICAL
L
I
0 40
35
30
25
I
I
I
I
20
I5
IO
5
E&X Fig. 5. Partial ionization
Id40
-I
0
“1
cross section per unit energy of the emittedelectroncalculatedforPIofArbyHe*(ls2s,3S)atan incident collision energy ofEt = 23.89 X low4 au. The quantum mechanical results are from ref. [7 1. and show a pronounced peakinvolvinga resonance contribution from do(24)/de.
K.-S. Lam et d/Semiclassicalapproachto collisiod ionization
do not consider the Et4 case to be of enough importance to warrant extension of our treatment to include transitions which require tunnel@. While the treatment of these cases is not beyond the capability of the semiclassical approach [lo], it does require alternate formalisms and its inclusion is not likely to lend extra credence to the main fact that we wish to demonstrate here, namely, the compatibility between the semiclassical and the quantum mechanical treatments of collisional ionization. This fact can be well established by the results presented in figs. 3 and 4. The observable discrepancy, however, between the two approaches may be due to two causes which are as yet quite difficult to disentangle. On the one hand there may be inherent insufficiency in the semiclassical approach as opposed to the more rigorous quantum mechanical treatment. On the other, the present discretization procedure, which characterizes both the semiclassical and the quantum mechanical approaches, sets a limit to the resolution of computed cross sections on the energy scale. A refinement to smaller mesh intervals LIEwould reduce the resolution problem and permit a better comparison between the two approaches. Fig. 5 shows semiclassical partial cross sections du/de in an energy range for which the L-partial contributions corresponding to cases Et3 and Et4 (see fig. 2) are negligible, thereby allowing ;I reasonable comparison between the present semiclassical results and the quantum mechanical results which included contributions for all cases in fig. 2. This demonstrates that, for E,> ==10m3 au, the dynamical situations COTresponding to the neglected L do not contribute significantly to ionization. The semiclassical approach (with reference to the ionization surfaces) calls this fact to attention much more succinctly than its quantum mechanical counterpart. The present calculations also reveal resonance effects for dynamical situations marked by Et2 in fig. 2. In these cases the avoided crossing R is just barely accessible on the entrance surface El z ). The heavy partitle velocity is decreased drastically as the heavy particIes approach this transition region and hence, semiclassically, they spend appreciable time there, very likely resulting in transition_ One case which matches the resonance condition especially well is where L = 22 and E, = 3.8 X 10m4 au, and a value for da(**)/de = 2950.5 $/au is computed. Here it is worthwhile to note that I
da(22)/de >
c
d&“/de
225
.
L-c22
A similar effect is evident in the quantum mechanical results (BM [7]), in which a very pronounced peak exists for do/de, but occurs, however, at a slightly different Ef and involves a resonance contribution from do(24)/de (fig. 5).
4. Concluding remarks We have demonstrated that, using the same discretization procedure, the present semiclassical approach is able to parallel a coupled-channels quantum mechanical treatment of the problem of coliision-induced ionization. This particular application has strengthened our conviction in the general compatibility of these two approaches in the treatment of molecular collision problems. In the context of the ionization problem, however, there is still much room for refinement of these theories to include (after discretization) the spread of electronic energies which can interact with the discrete state. This spread of continuum energies is already partially taken into account by our twostate coupled-channels treatment [6,7], since it involves no Franck-Condon (local transition) restriction, allowing emission at a given electron energy 7 to occur at any R. Further refinement could be attained quantum mechanically by coupling the discrete state to more than one discretized continuum state at a time, leading to multi-channel coupled equations. Semiclassically, a first step toward improvement is to relax the present assumption of localized electronic coupling in a suitable manner. One would have to weigh S-matrix elements for particular ionization events (with specified ?‘) to reflect the uncertainty of the ionization configuration. This problem is not only relevant to collisional ionization, but also underlies the whole gamut of laser-enhanced processes in which the collision system is coupled to photons having a distribution of frequencies. We wish to point out that the isolation of a single discretized electronic energy ? in the construction of ionization surfaces has reduced the present semiclassical formalism to one which resembles that of the two-state approximation. Physically this amounts to ignoring the possibility of ionization at all values of
R other than thi one, say R’, for which Vd(R’) VJR’) = 7, and also the possibility of emitting electrons at R’ with energies 7 for which 7 # ?$(R’) F/+(R’). The former possibility would not only neces-
cesses [ II-131, we expect that the present semiclassical approach to field-free collisional ionization can be readily extended to treat the laser-enhanced cases. Work along this direction is in progress.
sitate a pre-ionization loss factor (Le., “survival factor”) as has been used in previous work [I], but also a relaxation-of the Franck-Condon assumption, for its proper treatment. The latter would require relaxing the Franck-Condon restrictions also, and (within the context of our discretization procedure) the inclusion of multi-surface dynamics_ While these problems raised by the presence of the continuum in the spectrum of electronic energies are in principle important as well as of tremendous interest, we are not aware of any evidence (theoretical
or experimental)
pointing
to-
wards their significance in practice concerning the case of Penning ionization_ Indeed, the two-state quantum mechanical calculations, which relax the FranckCondon restriction, compare well with the semiclassical results which are based on Franck-Condon transitions. Also, our previous study of the laser-enhanced emission problem [ 131 indicated that the FranckCondon approximation can be applied with confidence. Though lacking completeness, then, the present semiclassical treatment may be viewed as a study of the validity of the two-state approximation beyond its normal range of applicability. In recent work [4,15J we have presented an extension of the coupled-channels formalism, also based on discretization, to treat laser-enhanced collisional ionization. Inview ofwork reported here, and our past experiences with semiclassical treatments oflaser-enhanced pro-
References [i] W.H. Miller, J. Chem. Phys. 52 (1970) 3563. [2] W.H. Miller and H. Morgner, J. Chem. Phys. 67 (1977) 4923. [3] H. Nakamura, J. Phys. Sot. Japan 26 (1969)
1473.
[4] A.P. Hickman and H. Morgner, J. Phys. B: Atom. Mol. Phys. 9 (1976) 1765. [S] A.P. Hickman, AD_ Isaacson and W-H. Miller, J. Chem. Phys. 66 (1977) 1483; 1492. [6] J.C. B&urn and D.A. Micha, Chem. Phys. 20 (1977) 121. J.C. BeUum and D.A. hlicha, Phys. Rev. A, to be published.
[ 71
[8 ] W.H. Miller and T.P. George, J. Chem. Phys. 56 (1972) 5637. [9] J.R. Laing, T.F. George, LH. Zimmerman and Y.W. Lin, J. Chem. Phgs. 63 (1975) 842. [IO] J.R. Laing, J.M. Yuan, I.H. Zimmerman, P.L. de Vries
andT.F. George,J. Chem. Phys. 66 (1977) 2801. 1111J.M. Yuan. T.,‘. George and F.J. Mclaffertv. _ Chem. Letters 40 (1976)
Phvs.
16%
[12] J.M. Yuan, J.R. Laingand 66 (1977) 1107.
T.F. George,
[13] K.-S. Lam, LH. Zimmerman,
J. Chem. Phys.
J.M. Yuan, J.R. Laingand
T.F. George,Chem. Phys. 26 (1977) 455. [14] J.C. BeUumand T.F. George,J. Chem. Phys. 68 (1978) 134.
[15] J.C. Bellurn, K--S. Lam and T.F. George, J. Chem. Phys. 69 (1978) August.