We describe a new method based on algebraic techniques, which leads to a model of atom-diatom collisions

Frank, Alejandro

Lemus, Renato

Santiago, R. D.

English

NASA Technical Reports Server

/ 5Y, 
N95- 22961
8F
ALGEBRAIC DESCRIPTION OF ATOM-MOLECULE
INTERACTIONS
R.D. Santiago 1, A. Frank 1'2) and R. Lemus 1
I InAituto de Ciencias Nucleates, UNAM
Apdo. Postal 70-543, Circuito Exferior, C.U.
04510 Mdzico, D.F., Mdxico.
2 Instituto de Fisica, Laboratorio de Cuernavaca, UNAM
Apdo. Po_tal 139-B, Cuernavaca, Morelo_, Mdxico
Abstract
We describe a new method based on algebraic techniques, which leads to a model
of atom-diatom collisions.
1 Introduction
One-dimensional atom-molecule collisions have been studied using a combination of differential
and algebraic techniques in conjunction with time evolution operator methods for a variety
of molecular potentials [1]. For three-dimensional systems, however, one needs to resort to a
number of computational integro-differential techniques to describe such interactions [2,3]. In
recent papers we have proposed an algebraic framework based on the vibron model [4], that
leads to a three-dimensional scheme for such interactions [5], which starts from a V(4) x V(4)
description of three-atomic molecules. We then apply a coherent state method to one of the
U(4)'s, thus extracting a coordinate dependence for the interaction between one of the atoms and
the remaining diatom, while the latter is still described algebraically. The resulting Hamiltonian
is formally analogous to the ones used in one dimension [1] and can be solved in principle in the
interaction picture [5]. This is a difficult task, however, particularly with respect to assessing the
different degrees of approximation involved. For this reason we consider here a one-dimensional
version of our model based on a U(2) × U(2) dynamical algebra, which is simpler to analyze and
already incorporates much of the complexity of the full three-dimensional model.
2 One-Dimensional Model
In our one dimensional model, the stretching vibrations of triatomic molecules are described in
terms of the dynamical algebra
U,(2) x /.72(2) (1)
31
https://ntrs.nasa.gov/search.jsp?R=19950016544 2020-06-16T08:04:49+00:00Z
by means of the Hamiltonian
_I = Eo(N,,N2) -4- AJ_, 4- BJ_ -4- CJz13z_ 4- D J1. J2 +..., (2)
where _,, 3,, and _, are the SUi(2) generators and Ni the (fixed) total boson number associ-
ated to Ui(2). The dots at the end of (2) indicate that we may also need other combinations of
generators, such as 3,1 •3,2. Such Hamiltonians are well suited to describe stretching vibrations,
due to the connection that can be established between the U(2) algebra and the one-dimensional
Morse oscillator Schr5dinger equation [6]. We now introduce the coherent state basis associated
to bond number 2 [7],
1
[[Nit >= (s t + rtt)NI 0 > , (3)
x/N!(1 + r 2)N
where s t, t t are two scalar boson creation operators through which the U(2) algebras are realized
[6]. Computing the expectation value of (2) in the basis (3) and carrying out the coordinate
transformation
i e_bx/a °r= 2 -- e-b_/a ' (4)
we arrive at a Hamiltonian of the form
H =/_o + I21 + V2 + I_'3 , (5)
where
/_o= A3_, , (6a)
_, = __-_/:ao(2 _ _-_/-o),/: , (6_)
V2 " -J3(2e-b_/_°- e-2b_°/"°) , (6c)
T/3= _e-b_l'° , (6d)
and
5=-2(C+D)N3z, , (7a)
= -4BN(N- 1) , (7b)
i
= -IoN.I,, (7c)
2
The potentials (6b) and (6c) generalize the typical interactions of exponential form (6d) used
in previous works [1]. For collision energies high compared with the vibrational excitations, we
may "freeze" the molecular coordinates and substitute Jz_ and 3,, in (7a) and (7c) by their
32
expectation values in the ]jm > basis,leading to a well defned potential for which we evaluate
classicaltrajectories, associatedto a classicalLagrangian of the form
L = + V,(x) + V2(x) + V3( )
2
(s)
Substituting these trajectories x(t) back into (5) then leads to a time-dependent potential of
the form
V(t) = V,(t)Jz, + V2(t) + V3(t)J m + V4(t)Jx, , (9)
where Vi(t) are obtained from (6), except for V4(t), which is identically zero.
We now use the interaction scheme, which is appropriate to determine time-dependent
solutions for Hamiltonians of the form/2/=/_0 + I?(t), as in (5). The evolution operator in this
scheme satisfies the equation
ihaU!t)-- _ %.U(t) , (10)
Ot
where
_.t(t) = eit_fl°(t-t')V(t)e -ihfl°(t-t_) (11)
A great simplification, arises when lYint is a linear combination of a closed algebra. For the
particular case of SU(2), U(tl, t) can be expressed in the form
U(t) = e -ilhg' (t)_ e-i/h92(t),)w e-i/ttg3(t)), e-i/hg4(t) (12)
If this is the case, substitution of (12) into (10) gives rise to differential equations for gi(t) in
terms of the (known) potentials _(t). Once U(t_, t) is known, the scattering matrix is defined
in terms of it as
,_ = U(-_, oo) , (13)
i.e., the transition probabilities can be obtained through
P(Ijm >_ Ijm' >)= I< JmlSIJ m'> I;
_-i _ ,.-i_,,'.4J_. ,.,,,, (-rr/2)d{.,,(-_r/2)d_,,,,(_2)l 2
lrm 2
(14)
where _ = gi(-c¢, oo)/h and the d-functions are the usual Jacobi functions appearing in the
definition of Wigner's D-functions [8].
3 Calculations
The main stumbling Lblock for the evaluation of the evolution opperator is the calculation of
_,t(t) through (11). We proceed to discuss briefly some approximations tb carry out this task.
33
a) SuddenApproximation
In the suddenapproximation, the interaction time is consideredto be very short. A Taylor
expansion of (11) along tl leads to
i ^
_,,(t) = V(t) + -_[Ho,V(t)lV(t- tg) + ... (15)
which at zeroth order gives _,t(t) = V(t). This represents an approximation where all dy-
namical information of the molecular potential is lost in equation (10). We may then solve for
the classical trajectories using (12), in some ca.ses analytically, which generalize results known
previously for the exponential interaction (6d) [9]. In this limit we are able to exactly reproduce
the analysis of Levine and Wulfman [10,11] and extend them to the more general interactions
in (6). We omit these results here for lack of space and refer the reader to reference [9].
1
b) Averaging Techniques for V/,,,(t)
Since we are interested in comparing our algebraic methods with quantum mechanical ab
initio calculations (when these are available), we should improve the evaluation of V/nt so as to
introduce the dynamical effects of the molecular potential. The sudden approximation of the
previous subsection fails to account for the fine properties required. Fortunately, a quantum
mechanical calculation for the transition probabilities in the scattering of an atom from a one-
dimensional Morse oscillator has been presented by Clark and Dickinson [12], so we may gauge
our approximations by comparing with their results. Returning to eq. (11), we substitute (6a)
and make use of the commutation relations
[J_, 3y] = -Jy- 2iJz3_ ,
[J_, J_] = -J_ + 2i.]z3y (16)
After some algebra we find the expression
[ -'_,,t(t) = 2sin(-_owot) (-cos(2xowotJ_),], + j, cos(2xowotJ,))
+ 2sin(xowot) (-cos(2XowotJ:)J, + Jycos(2xowotJz)) V3(t)
+ V,(t)L, + V2(t) , (17)
where xowo is the standard anharmonicity parameter, which can be expressed in terms of the
Morse potential parameter through xowo = -:_. The problem with eq. (17) is that it no longer
satisfies being linear in the SU(2)^ generators, so (12) is not valid. To de_l with this problem,
we proceed as follows. We apply Vmt(t) to a Morse eigenfunction Ijm > and insert a complete
set of states _t, IJP >< J#l to the resulting expression. Taking into account the selection rules
for .]u, which imply, p = m + 1, we find a state-dependent potential:
Vm(t) = (cos(2XoWotm), cos(2xowot(rn + 1))V3(t)ljm + 1 >< jrn + 1]cos( . ot) v (t) + cos( o.,ot)
cos(2xowot(m - 1))V3(t)ljm- 1 >< jm- 1[))Jy- + V,(t)Jz + V2(t) (18)
+ cos(x0w0t) /
34
The m-dependent cosine functions, however, are practically the same for small values of the
phonon number v = j - m (large m values), so we can simplify (18) by averaging the excitation
to the two possible states # = m 4- 1, so we arrive at the simple m-dependent potential
f'm(t) = (cos(2xowotm)cos(xo¢oot)V3(t)Ju
+ V,(t)L + V (t) , (19)
which is linear in the :_:U(2) generators, so (12) is valid. Since we are interested in comparing
our approach with the calculations of ref. [12] where only V3(t) _ 0, the potential (19) simplifies
further and only Jy remains. The evolution operator is then given by
(7,.(t) = e-ig'_'''O4/n , (20)
where
g2(m,t) = V3 cos(xowot)cos(2xowomt) (21)
The differential equation for tile evolution operator then lead to exactly solvable forms for the
classical trajectories and to the asymptotic value of 9_(m,t) (see below (14))
E cos(x0w0t) cos(2mx0w0) sech 2 E t (22)
where E is the collision energy. From (14) this leads to the simple expression for the transition
probabilities
P(Ijm >4 Ijm' >) J ' (23)= dm,m,(fl2(m )) ,
where
#wo_a_ [ (2m'- 1):Co (2m' _+__l)xo ] (24)
fl_(rn') = b_h [sinh(_7:--l_oko) + sinh((2m'+ 1)xoko)J '
with k0 -- _b X/_2_,. In Table I we compare the results of our calculation, using (23), (24),
with those of Clark ahd
TABLE I. Comparison between the algebraic model and the Clark-Dickinson Model 12)
Energy ( hWo /2 units): 16
Final state
1
2
3
4
Initial State = 0
Clark-Dickinson
Probability
0.245000
3.38000E(-02)
2.72000E(-03)
1.61000E(-04)
Model
Probability
0.157837
1.45652E(-02)
8.65153E(-04)
3.71652E(-05)
Initial State = 1
Clark-Dickinson Model
Probability Probability
0.318000
5.72000E(-02)
5.16000E(-03)
0.253196
3.62444E(-02)
2.86219E(-03)
35
7.44000E(-06) 1.22993E(-06) 3.24000E(-04) 1.50963E(-04)
Final state
Energy ( hwo/2 units): 12
Initial State = 0 Initial State = 1
Clark-Dickinson
Probability
Model
Probability
Clark-Dickinson
Probability
Model
Probability
0.129000
6.65000E(-03)
1.49000E(-04)
1.48000E(-06)
4.95000E(-09)
0.124019
8.57432E(-03)
3.81572E(-04)
1.22807E(-05)
3.04486E(-07)
0.167000
8.41000E(-03)
1.38000E(-04)
7.01000E(-07)
0.208828
2.19917E(-02)
1.28983E(-03)
5.07090E(-05)
Energy ( hwo/2 units): 10
Initial State = 0 Initial State = 1
7.39000E(-02)
1.66000E(-03)
1.17000E(-05)
1.95000E(-08)
0.
0.105782
6.09133E(-03)
2.25776E(-04)
6.05219E(-06)
1.24982E(-07)
8.15000E(-02)
1.22000E(-03)
5.12000E(-06)
2.95000E(-10)
0.182403
1.58582E(-02)
7.71373E(-04)
2.51949E(-05)
Energy ( hwo/2 units): 8
Initial State -- 0 hfitial State -- 1
3.15000E(-02)
1.85000E(-04)
1.18000E(-07)
0.
0.
8.66178E(-02)
3.98810E(-03)
1.18193E(-04)
2.53330E(-06)
4.18294E(-08)
2.28000E(-02)
3.07000E(-05)
1.37000E(-11)
0.
0.152903
1.05375E(-02)
4.08112E(-04)
1.06318E(-05)
Energy ( hwo/2 units): 6
Initial State = 0 Initial State = 1
7.31000E(-03)
2.30000E(-06)
0.
0.
6.64920E(-02)
2.29488E(-03)
5.09821E(-05)
8.19113E(-07)
1.25000E(-03)
0.
0.
0.120126
6.15327E(-03)
1.77900E(-04)
36
5 0. 1.01384E(-08) 0. 3.46550E(-06)
Dickinson [12] for different collision energies and for two different initial states. Although our
results differ from the exact ones, particularly for higher final states where the probabilities are
very small, the general trend is reproduced remarkably well, taking into account our semiclassical
method.
4 Conclusions
The algebraic model seems to provide an attractive alternative to integro-differential techniques
for the description of atom-molecule collisions. The approximation methods developed for the
one-dimensional test model can be readily extended to the three dimensional case and applied to
real systems [2,3]. We are currently exploring the generalization of these techniques to include
reaction channels, which would represent an important development because of the relevance of
these processed in atmospheric interactions [13].
Acknowledgments
This work was supported in part by CONACyT, Mexico, under project 400340-5-3401E.
References
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38


Algebraic description of atom-molecule interactions

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