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Exponential perturbation theories for inelastic scattering
Chemic$ Physics 48 (!980).237-252
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-. Received I6 November 1979
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.:. ~An Exponential Perturbation Theory (EPT) is derived whereby one calculates a &ase&ift matrix by an nth order per-. terbation theory and then exponentiates it to obtain the scattering matrix. The theory has be&developed to include high: order terms, dosed channels and~reaonanees.The radial wavefunctions used &e WfU3 solutiona &ch are generalized tb. i-.1 cases -wherethere are multiple tornhrg points. The orbital angular momentum may be treated exactlybr in the’&&i&l o$’ andden limits_ Calculationa are done for the rotatidoaliy inelastic sdtering in He + Hi, Ar + Na .&d &-+ RC&.The fii two systems give fair to good agreement with accurate calculations: the last case gives poor agreement: The fmt-oider EF!T is very much beuer than the fti-order distorted-wave approximation.
. 1. Introduction
simplest systems. Since the cost’in time and moneytkr solve a set ofNcoupled equations is rou&ly propor-
Vibrationally and rotationally inelastic collisions tional to b3-, there is much to be gamed inreducing : are involved in a wide range of phenomena ranging N. Rabitz-developed an Effective~Poter$al (EP) api :from transport properties [l], ultrasonics [2], shock proximation [l I] which reduceslVby reducing the : waves [3], and line broadening [4] to the activation dimensionality of the problem. Pack [12]. and &iCGuire of reagents prior to a chemical reaction and the deand Kouii [ 133 have developed the.Coupled States (CS) activation of excited products. The operation of gasmethod which makes approximations to the orbital phase lasers depends on the many vibrational and angular momentum thereby reducingIV_ Generally the rotational transitions involved in the laser action [5]. CS method is more.accurate a&more expensive than There have recently been a wide variety of detailed the EP method. It is not-hard;howeverj to find even, simple systems ,for which the EP and CS appro+raexperimeuts on ineIastic scattering using pulsed beams of ions 163 and neutrals [7], measurements of changes tions are still much too expensive: to be practical; in velocity before and after collision [S] , and by variAt the other extreme-are perturbation theories .: ous spectroscopic techniques [Sb,7]. such as the Born Approximation @A) and-DistortedA large set of theories of vibrationally and rotationWave Approximation @WA) [lo] ;-Unfortunatelyy; ... ally inelastic scattering has been developed in recent most cases require more than a first-order perturbation years [lo]. As is apparent from the range of experitheory so that these methodsare oftenvery inaccurate. ments, no single method is practical for all cases, but There remains a broad-middle ground between these rather a range of theories-is needed. It is simple tore-extremes to be filled m Gne old-method; now enjoywrite SchrGdinger’s equation as a set of coupled. mg a renaissance is the Bxpon&i~Perturbation i- - -..’ second-order differential equations;.one for each qnan~ Theory (EPT) [ 14-191 in.which one uses the BA or : .; turn state involved-m the scattering [iO]i Numerical-’ DWA to~calculate a transitionmatrixandthen.expo;~ IL;. ~nf++-i+ .&is &&to i&t& the,f&d &&$ i j: : 5 solution of these equations is known as the._Close~--.?) Coupled (CC) meth_od~&rd~servestod&_as the pr+ry r+ixr Ti& expbi;entiatior+ &sures that the~s=+tte&& i; starrdard for testing appro~a~~me‘tho~~~~BecauSe_ of ( mat&:& ii&try (particl~~are-~oonseNe;d);ja p&b& :-: ..’ v;ith & B& ~d:DwA9~~~t.the_~~~iti~~i~o~gbil_ _-;_ the large number of quantum states involved;.the~CC~ iti fi~.L.el~~,e~&~~~-$ti~ rTiie‘e;ti6tietiti_i~~Lxaisi, ~; method becomes prohibitive@ expensive forall but--&e ._-.., ,_~ : 1. ... :...~... __ .- -. Z..)Z -: -7.:__,..i-.!__.‘.+,;‘;‘. .. : ;-_.__.__y..’ : .--. _.; ..- _. : -_=._-, ~. 7 r I ;..._..~ _,~_::_i__ ._ : .- ;..< : .. --.~- . ,. i. ; ._ ... .-:.._+:’ -__(... ,._ .~____.,‘._...‘_... _. .,_ /.~ ___~.__.:‘... ..:,I. .:.. :;. . . :.:-- ._-. .i.. -. .i _~=_.)_ ; _-_~-:~.,+-I-,.I_~-,. I‘_..: :.- ~...:<__. 1 .._~ )y ‘...._ :-‘- ::-;-,..,.;..:; _y:;-_.- ._._y._.;. ..;, ~_ z-.-.-‘;__. . .~ y<;:. .-; :-. l.._~ :_ ‘-
238
R.J. Oos.s/Erponentid perturbation theoriesfor
appears to give accurate results for the higher-order transitions where the BA and DWA give zero transition probabilities. Recent calculations on rotationally 116,171 and vibrationally [l&19] Inelastic scattering show very good agreement with CC results in many cases. The method has a good chance of giving the accuracy of the CS and EP methods for some systems while being aImost as cheap as the BA or DWA The basic exponential method has been around for a long time. In the case of time-dependent perturiiation theory, it is known as the Magnus approximation [ZO]. In the case of scattering theory it has been developed by several people including the present author [14,15]_ In most cases it has just been used without justification. In this paper we wish to develop the theory as a rigorous perturbation theory out to high order- We include effects such as closed channels and cases where the classical trajectory has three or more turning points- Several options tie avaiIahIe for treatment of the orbitaI angular momentum. By making some approximations in the theory we obtain a timedependent version where the unperturbed prob!em is elastic scattering on the classicaI trajectory for the sphericaiiy symmetric reference potential. The timedependent theory gives the Magnus approximation pIus additional terms arising from the deviations of the trajectory caused by the perturbing potent&I_ In both the sudden and dassicd limits to the internal motion the second- and higher-order terms in the Magnus series are identically zero. This explains, in part, why the theory works so well. FmalIy, we demonstrate the theory in the case of pure rotationalIy inelastic scattering. We use three systems of varying anisotropy: He + HZ, Ar + N, and Ar f HCl. For He + HZ up to 2-I eV relative energy the first-order theory is accurate to several percent and the second-order theory to a few percent. Above 2.1 eV the EPT diverges from the CC results. For Ar + Na the fmt-order theory is accurate to IO-20%; the secondorder results are similar_ For Ar + HCI, fast-order results are in error by factors of 24.
d*Ui/d?
+-p:(r) Ui(r) = F
qj(r)
Uj(r) i
(1)
where r is the distance between the centers of mass of the two colliding molecules and p?(r) = L+ - U,(r) - Zi(Zi+ 1)f2
,
0)
kf = 21.cR-*(,5tot - .&*i),
(3)
U&) = 2pV&)l@
(4)
U&(r) = 2pZ
,
-‘ ,
(5)
and p is the reduced mass of the two molecules_ Note that i and i are collective indices which also label the orbital angular momentum quantum number L The reference, unperturbed problem is d2u,j/dr”
+p;
uDj(r) = 0 _
(6)
There are two lineady independent solutions to (6), “lj(‘) which is weII behaved at the origin, and u.&)_ Rather than use accurate solutions to (6) which implies a time-consundng numerical integration, we use approximate, WKB sohitions which are much more easiry obtained_ We keep in the theory the deviations due to the approximate solutions and due to Uii(+ We can write the solutions to (I) in terms Of ZfIi(‘) and ZQ-(~) as Z+(T)= U1j(r) Xj(r) + U*j(‘) Yj(r) -
(7)
With two unknowns we can impose an auxiliary condition, and we make the standard choice, Yi”O,
UljXi)+U2j
(8)
so that (I) becomes Xi’ = c (/$& i
i
The fundame+ theory has been derived in a previous paper [15] (referred to as I). We break the potential into two parts, a spherically symmetric V,(r)
scattering
and AL’ = Ir - I’0 - The zero-order problem is the elastic scattering due to V,-,(T).We can write the coupled Schr&liuger equations as
Y; = c 2. Basic theory
inelastic
+ Bii Yi) ,
(CvXj + Dii 5) ,
(94
49b)
where WW (lob)
=(27#*
qi(r)
Ipil’1~2 IZ,I~/~ Ai(
w
OW
u-&)
=(2~#/~ lp$1/2 lzj11J4 Bit&) . -. Ai and Bi are Airy functions 121, p_ 4461 and
(12b)
So far, this procedure is exactly equivalent to (1). The exponential perturbation theory &es an exponential form for S, exp(inci) , ‘...
su = exp(i*ci)lexp(2%tiii
and rcj is the classical turning point where pi(rcj),= 0. For the moment we consider the case where there is only one rc. Near rc, z is linear in r to give the solutions for the linear turning point. Outside rc, ulf -
[2/Pi (r)l 1’2 sM& +
- (Zlkj)“’ uli - [2/pi(r)]!j2
+
where the phase-shift matrix Aq is a generalization of the usual elastic phase shift ~7~.The exponentiation of Aq is carried out by a power-series expansicn or by diagonalization of Aq, exponentiation of the eigenvalues, and transformation back to the original representation. If Aa is hermitian,.then S is unitary. Our procedure for obtaining A9 is simply to obtain E and F by perturbation theory and then to equate (19) and (20) order by order. To second-order,
a/4)
Sin(kir + QO~- li~/2) 2 co&$
Wa)
n/4)
N (2/%)rL2 coS(kir + qOt - ZtX/2) , where nor is the WKB phase shift for V&), Inside rc U]i N I Qj Ibu2 exp(-Sjl T
(1 =I ki
and ZP
eXP(E$ -
A(r’) df
E(r) = j
0 r’ + r&f dr” [A( s s 0 0 F(r) = j
,
(21a)
[C(r’)A(r?) + D(r’) C(r”)] -
@lb)
+ B(r’)C(r”)]
C(r’) dr’
0 JJ CL+
W-4 +
U2f N lPi/21-“2
Wb)
r&r ss
0
0
The wronskian W is easily found to be -2.
The first-order term becomes
As can be seen from (165) U2j becomes inftite at the origin SO that as i+ 0, Yi(r) +O. At r =O, then, X(r) =X0 and Y = 0. We can then integrate (9) out from r = 0 until the potential is negligible. Let
Atl,, =
X(r) = [I + E(r)1 70 a
(20)
WI = F(r) x0 -,
(17)
0
0 .(22)
The
second-order term is
where I is the unit matrix- Then the total wavefunction be&mes
1
.‘;-
,.’
._
‘.
:~
: i_-....-
.-_ ..-
_
:
:
,_
:
.:.: ,. ::-
Arl,=
J-
-P
r dr
J
- dr’
[C(r) A@‘) + D(r) C(r’)]
-0
Cdr s 0
A~J-
dr’ [D(r) C(r’) - C(r’) A(r)] _ 0
0
where Aq,(r) is given by (22) using r as the upper limit of the integral. Since n@) + 0 as r + 0, the integrals are alI stable_ In appendix A a general form is derived for the nth-order term, An =n(_i
- f&(a)
-I-$&S(m) - $‘@(a) f .._ ,(24)
dQ(r)/dr=C--A+DfiZ-$Zf3Q_
(25)
The solution of (25) by perturbation methods gives (22) and (23) and the various higher-order terms in AlISo far, we have considered cases where onIy open (energetically accessible) channels are included. However, the derivation of (1) requires a complete basis set of internd states includmg the closed channels. Although it is impossible to get a transition (at r = -) to a cIosed channel, they may Influence the scattering. Feshbach resonances are an example of this_ For closed channels u1 and ~2 may be defmed by (16) using any re as Iong as it lies welI beyond the range of the potential_ As before, zcl is well-behaved at the origin. For larger, uli” exp(lc& and uzim exp(--rjr), where K~ is given by K’
= 2~
-*fEim i - E,,,)
_
(26)
The asymptotic totaI wavefunction
is given by
Ui - (2Ki)- IL? eXp(-inui) X F (iQ- + iEii> Xoi exp&r - (2ai)-1/2
exp&&
for some suitably defied complex 17oiand C-. TO 0th tain a well-behaved wavefunction, we must adjust X0 so that the fmt term of (27) is zero. This term plays a roIe simiIar to the first term of (18) for open channels in that it represents input information. We are usually not interested in the second term of (27) since it does not persist at larger. We can make the first terms of (18) and (27) identical by repIacing Fij in tie first term of (18) with Fh defined such that Fh = Fq for i open and Fb = 0 for i closed. This replaces the second F in (19) by F’. We can also replace the first F by F’ without affecting transitions to open channels. The result of this modification is that (22) -(25) are valid provided that i and j are open channels. For closed channels AnV= 0. There are no effects of closed channels on the first-order term and thus no transitions to closed states. In second-order the sum over k in (23) does include closed channels as do the sums over virtud intermediate states implied in(2.5).
- Zrr/2)
Ci
(27)
3. Multipie turning points So far we have examined cases with one classical turning point (open channels) and with no turning points (closed channels). When V,-,(r) has an attractive part, more complicated cases exist. A state with two turning points is closed at r = - but becomes open at finite r due to the attraction in the potential. The most common case is where there are three turning points due to the centrifugal term in (2). If VO is negative there can be three roots of&r,) = 0. Thii situation gives rise to orbiting resonances in eiastic scattering and to similar resonant behavior in inelastic scattering. We distinguish four cases: (i) The minimum in p2 is large and positive. Here there is only one turning point, and we can use (12) as before. (ii) The minimum is smaU and positive. Here again there is only one turning point, but there are errors in (12) near the minimum because p* is not linear near r.?(iii) The minimum is small and negative. There are three turning points and the two near the minimum’ are nonlinear so that (12) cannot be used. (iv) The minimum is large and negative. There are three turning points and aU of them are linear_ Case (i) is already taken care of. In case (iv) we have
.241
RX C?oss/Exponential perturbation theories for inda$iic scattering
solutions of the type (12) for each ofthe three turning points. Let rca
&=-&,+&b &=-$+&*
(28)
29 =-s@l@ul,+cos@rc~c.
(29)
Note
where 'Cf.3 6 ab =
PJr)
s
dr
3
(30)
(38)
that the wronskian of u1 and ui is still -2. -. In regions (ii) and (iii) we must fmd solutiqns-to (6) for a quadratic turning point. We expand p* abont the minimumro, p2 = -Co + C2(r - rO)2 ;
‘ca s,,=
with the quasibound s&e between rca andrcb..%eeffect of this resonance will be a pronou?ced change in All as the region inside rcc is opened,u$Uiing(32) and (33) we can project back the other linearly independent solution
IzPi(r)drl
(331)
-
c The connection between solutions is sin6ab
cos 6,
cos 6,,
-sin Sab
(32) Ulc
u2c
)-
(33)
We need two global solutions u1 and 212which must satisfy two conditions: u1 must be well-behaved at small r so that u1 a ula; u1 and u2 must behave asymptotically as (15) for some phase shift pg. Let R = [4 ex&&.,,_) Cos2sab+ $ exp(-2tibc)
where Co is positive in region (iii) and negative in region (ii). Using x = (4 C2)‘/4(r - ‘0) )
WI.
a = C,(4Cz)-l~2
(41)
)
eq. (6) reduces to
I
)(
(3%
sin26,b]“*,
d*u/dx’ - au + $x2u = 0 _
i42)
The solutions to (42) are the parabolic cyhnder functions 121, p_ 6851 u1 bc = W(G x) f
(434
t+bc = w(a, -x) .
(+b)
We must now connect (43) to the solutions away from r. as before. Of the several asymptotic expansions of W(a,x), the most useful is Darwin’s expansion [Zl, p. 6851. Let
(34) 6 = arctan [$ exp(-26b,)
tan S,,]
_
135)
Using (32) and (33) we -find that K1 =R-kla
=R-‘(sin
&&lb
~=cosQulc+sinQu2,,
u1 +Sin(~
-I-4,. + cp- Z77l2),
x=(X2
COS 6,buZb)
=
(37)
where qcl is the WKB phase shift using only the outermost turning point rc. The correct phase shift is sci + @,a result previously obtained by M$er [22] _By assumption the minimum in p2 is large and negative so that sbc will be large. Usually this means that R and + are small and that we can neglect the region inside rcc_ However, if&,, Z=S (a + &rS c& 6,, is small and R and @become large; Here we have a resonance
xdr,
cos@ f $r) )
(2yKP
Iv(Q, -x) = (2/_YxP (36)
*=5J
(44)
2&Z ?v(a,x)
+
- 41Q1)‘/2,
(45a)
sin@ + $70 ,
Wb)
where y =
(1 +
&a)1/2
_ ena _
(46)
For case (iii) the classical turning points “e at x = &&I12 am-lp z &Q)‘!2 x. Using (13) we’fmd that ,$=e,so that
u1 bc = (2y)-1!2(~;)118
Ulb
: (+b)
zQbC= (2y)-Q2
UIC
:. (rrTcc)-;(47c)~
(ti2)?8
,(+)
242
RX Q.oss/Exponenttit perturhtion theoriesfsr inelasticscattering
t42bc = 0,/2)“*
(4c#*
(r Q rcb) _(47d)
U2,,
The connection between ub and uc is therefore
6;)=
C&
‘; K;;)-
(48)
Using (31) we obtain 6,, = tra. Since, for large Q, y =Z $ exp(-$.& (48) goes smoothly into (33) as we go from region (iii) to region (iv). We repIace (34) and (35) with R = (y-2 cos2Gab+y2 sin26,,J1~* ,
(49)
Q = an%rlCy’tan 6,b) ,
(5@)
and use y = [I+ exp(2$_,)J
Vr - exp(6hc) -
(51)
(47). In region (ii) we let rCt, = rCC= r. and find that 5 = B as before- Using (28) we obtain (4c2)tQqaSin
6,,
+ r&&cos gab)
’ (r-&-n), q uc -(y/2)1/”
(4C*)‘~S zf*e
(52a)
(r Q q$ > WC) (4$)“*
ulc
(Uti - itr*i) =p-t/*
2-ti2
(uu + in*& =p-t12
uqi =
(rS=r 0 ) - (52d)
This gives the connection between u, and u,- Again (36)I(38) are valid. As the barrier b&omeHmore negative, y h 1, and we go smoothly into the case of a singIe turning point with q. = 71el + 6, - This same procedure will work for any number of turning points.
4. Time-dependent formulation A popular set of approximations is based on timedependent perturbation theory using the trajectory corresponding to Vn. The originrd theory In I derived a connection between the EET and this time-dependent approach. We sketch the results here because they give a valuable insight into the errors and interpretation of the EFT. To make the connection we assume that k
eXp(i&
-
inl4) , (53a)
exp(-i& + in/4) .
(53’3 The wronskian of u3 and r~ is -2i so that (10) becomes 9Qii = &p-i
Vi;-exp(-i&- + i$)
$p-l
U& eXp(-i5;
-
L$j)
,
(W
,
Wb)
@ii=~p-tLli;-exp(~i+i~~), ‘oii = k i p-t
Uh
eXp(i&
(54c) - i!$
_
t5W
that 1 and cb oscillate slowly In r whereas % and C! oscillate rapidly. If we keep only the slowly oscillating terms, the first-order term for Aq becomes, Note that ,$ is large so
Aqii=-$
(r s Q) , (=b)
u2bc - (,42)1’2 (4c2)1’8 &la cos 6, - u&sin 6,b)
U2bc -(2y)-li2
Lc3i= 2-l/*
cx3ii’
Then (36)-(38) are valid in regions (iii) and (iv) except near the minimum where we must use (43) and
ut bc - (2y)-I/*
and 1 are large and that A k[k and A@ are smaII for ah transitions. We replace all the rc’s with 2 common averaged r-c and use the asymptotic form given by (IS) for ut and u2 over the whoIe range of integration. We transform (15) to a new basis set,
J
p-1
U&(r)(&As+e-iAE)dr,
(5%
J-C
where 4 = & - $. Now, tip is the classical radial momentum. It is related to the time t and trajectory angle 0 by dr=Gipdrjfi.
(56)
dB = ql+
(57)
;) &l(r2p) ,
where the top sign refers to the incoming half of the trajectory and the bottom sign to the outgoing half; 0 goes from ir at t = -0, r = 0 to the final scattering angIexat t=m,r=w_ For small Ak and AZ [iSI a
= & - gi = *qj(t_ - tJ 7 (Zi - $)(S - 0,) ,
(58)
where wij = (Eat I - Elntl)!tr and tc and Bc are t and 8 evaluated at r = rc. With these substitutions
where Gii(r) = u$ exp[ioii(t
- rJ - i(Zi.-
(60)
w(e - e>l -
,. RJ. CPoss/Gqwnenhid
pertybatiyz
Except for the o-terms in (60), the result is identical to the Usual time-dependent perturbation theory. Most tie-dependent methods do not include orbital angular momentum. we will shortiy give two methods of removing it fm.m the problem. The second-order term can be broken into two parts Aqti and Aq*,,: Aqa =
2 4dt -0D
j dr’ [G(t), G(t’)] _ -m
(61)
This is the second-order term of the Magnus approximation [ZO] [except for the e-dependent part of (60)];
PC
rc
- e (r)S (r’) f (?!(r’)93 (r)] -
(62)
In appendix B it is shown that AqZb = : s
p-l
$ [(p[p’)(oQ 2 + a’)]
dr .
theoriesfor
inelastic scattering
:
243
weakly onj and ti; Here again; we-can diaioxialize G by a time-independent transformation. The detail!:?= given in ref. [23] _ Briefly, we transform fK6ti.a rep+. sentation diagonal in the action variablesi and m to a-: representation diagonal in the conjugate an&e sari- : ables. The.treatment is similar in many respects to the semiclassical mechanics of Mier and Marcus 1241, which also uses an action-a&e transformation to diagonalize the problem. In the sudden liiit the internal motion is frozen during the collision; the internal wavefunctions are Dint delta functions, independent of time. In the classical limit the internal wavefunction is an infmitely narrow wave packet which moves along the classical trajectory for the internal motion. One can easily qonsiruct hybrid approximations where some of the degrees of freedom are sudden and others are classical_ Here too, An& is zero. In particular, the orbital angular momentum is nearly always classical, although the sudden approximation is often used. In many rotational cases, at low j the motion is sudden and at highj classical.
(63)
rC
5. Orbital angnlar momentum
In the
case of a spherically symmetric potential V(r) = V,(r) f AV(r) one can expand the WKB phase shift in a power series in AV to give p=~~+A~,+Arl,,+o(Av~)_
(64)
Aql involves the integration of AV over a trajectory given by V,-,; AqZb is a correction due to the deviations of the true trajectory from the one given by V,-,. There are two important limiting cases where Aqh and all higher terms of the Magnus expansion are zero. These are the sudden and classical limits. In the sudden limit the energy difference between quantum states is neglected; kjisreplaced by some averaged ko. If all degrees of freedom are sudden, then u1 and ~2, given by (12) are independent of state and wii s 0. This means that G(t) Q U’. We can then simply transform U’ back to the coordinate representation where it is diagonal. The transformation “matrix” is the set of internal wavefunctions Ii> and is independent oft. In the diagonal representation the commutator in (61) is obviously zero. In the classical limit A k and U’ depend on (i-j) but only parametrically on i. In the rotational case, for large j the relevant potential matrix depends strongly on Aj and Am but depends only
Of all the degrees of freedom the orbital angular momentum is usually the least interesting. It is not directly measurable and therefore summed or averaged in any calculation. It is no surprise, therefore, that this is where many approximations begin. Both the sudden and classical approximations are used, and we describe how these can be used in the framework of the EPT. The orbital sudden approximation is the replacement of Ziby an averaged 10 in (2). This is the Coupled States (CS) approximation [12,13] and results in a reduction of the number of coupled equations to be solved- For an atom and a linear rotor the-system is described by the coordinates r = (r, 0, @)).the separation of the atom and the center of mass of the rotor and R = (R, 0, a)), where (0, *) is the orientation of the rotor andR is one or more vibrational coordkates. The usual basis set is lnZjJ&f>,where n is one or more vibrational quantum numbers, 2 quantizes the orbita angular momentum, j the rotational angular momentum and J and M the total angular moinentum- If1 is large and j < I, then one can show that. [25] Oz’Z~‘JiV(~,R,~)lnljJ)=(~j.P’IV(~,R,P)lnjS)
,&
244
R. J_ CzossfExponential
perturbation
where-yistheangIebetweenrandR,6’=J-1’,6= J - 2, and /3is the angle between (0, @) and (r&O). Note that 6 behaves like an m-quantum number -j d 6 Gj. In the orbital sudden approximation u1 and 29 depend parametrically on lo but are otherwise independent of 2. Using a basis set Inj6) and (60) one can compute the scattering matrix as described above. Ref. [25] gives a compendium of formulas to obtain differentiaI and integral cross sections. In the classical limit we see from (60) that the effect of changes in 2 is to muhiply the potential matrix by a factor of exp{-i(2i - 2#l(r) - 0,_]- If this is done. the effect [25] is to modify (65) so that
theories for inelnslic scattering
The potential matrix is G2’j’S’IV(r,R,P’)lnjS)
=
g(4)
(g$ 11’2
(n’i?JJn)
X C(j2j’; 000) Cbitj’; 6 -
~5’) Y&r,
(68) Te(r)) ,
where [j] = 2j + 1, C is a Clebsch-Go&n coefficient [26], and Y is a spherical harmonic. The first-order
phase-shift matrix is given by (22); we take the average of the incoming and outgoing halves of the trajectory [the f sign in (68)]_ We can define a reduced phase-shift matrix,
(n’ZyJJIV(r, R, $lnri.r) + bij%‘l V(r, R, p’)&5>, (66)
where /3’ is the angIe between (0, a) and [7r/2, ‘(e(r) - @,=)I_As before, the top sign refers to the incoming branch of the trajectory and the bottom sign to the outgoing branch. The use of (66) is not restricted to a timedependent formuiation. Several methods for adapting it to the more accurate theory are given in ref_ 1254 _ The one used here is to pick some average state in the middle of the manifold being used; 0 is set
X Re[Y,,,(d-% WII
ulin(rI * -
(69)
The full phase-shift matrix is given by,
to Be E 0 Inside me_Outside r,, B is caIcuIated by integrating (57). The two approximations are easiIy visu&.zed in terms of action-angle variables. The angle variable conjugate to I is c?(r). In the orbital sudden approximation 0(r) is frozen at 0. In the classical limit B varies
in time over the classical trajectory_ For a hard coIIision at smaI1 impact parameter, B(t) does not vary much during the collision so that the sudden approximation may be used. For a grazing collision at large impact parameter, e(r) varies by ISO”, and the classical approximation must be used. For several reasons the sudden approximation is faster. In the calculations which follow we compare the two limits.
6. Rotational angular momentum Depending on the system there are some shortcuts that can save computer time. Again considering the case of an atom and a linear moIecuIe, we can expand
If the number of terms required in (67) is small, then one may well have many fewer integrals to evaluate using the reduced phase-shift matrix. We note two symmetry reiationships which further reduce the
amount of calculation required, , .,_ A,$,$ = (-I)’ An:;“, ,--,
Arm,“,~~
=(-I)’
(72)
For the case of the orbital sudden approximation 0(r) s 0, and we can remove the spherical harmonic from the integral in (69). This permits a further reduc-
tion in Aq _ The cakulation of the scattering matrix aIso requires the exponentiation of Aq. We can often achieve an increase in speed here by using a reduced A?. We can defme a similar reduction for Aqk,
the potential
where (67)
(71)
, .,_ nnII_ AqOA!L - Qn
sj 7 6,O $0
2
R. J. Oossf.&&wial
pertwbation
and the case of k = 1 is given by (69) The recursion relation is
where { ] is a 6-j coefficient [27] _The scattering matrix is given by a similar reduction, S* n njj”6.6
= gC(jhj’;6
-&Y)Sfz’,
(74)
where (77)
These reduced forms will be useful for cases involving large j where the reduced AR eliminates the many combinations of 6 and 6’ needed.
7.
Results
The exponential perturbation theory has been applied to three systems of varying anisotropy. He + Hz was one of the fmt systems studied by the CC and CS methods. The potential has a very small anisotropy, and the large energy gap in Hz further reduces the rotationally inelastic cross section. Ar + N2 has a higher anisotropy and much smaller energy spacings. Finally,Ar + HC1has a very anisotropic po?ential. In all three cases, calculations were done to fit-order [es_ (22)] and to second-order [including eq. (23)]_ Both the sudden and cIassical approximations for the orbital angular momentum were used. No attempt was made to use the .theory in the case of mult$e turning points. Instead, the outermost turning point was used with the usual WKB wavefunctions. The calculations were done on an LSI-1 I microcomputer and required lo-100 mm per energy depending on the system. For He + H2 we used the Kraus-Mies potential surface 1281, averaged over the vibrational coordinate
PI
theories for inelastic scattering V. = (214.5
eV)
exp(-3.49
V, = (40.47 eV) exp(-3.47
245
A-1 r) ,
0’8b)
A-l r) . -.
.(78c)
Table 1 gives the results for the integral elastic and inelastic cross sections. Also included are results for the fmt-order distorted-wave approximation obtained by taking only the first two terms in the exponentiation of Art_ There seem to be two errors involved. The classical limit to the orbital angular momentum used in (65) requires that I %-j. At low energies for a light system such as He + Hz, part of the inelastic sc%ttering occurs for small I where this is not true- As the energy is raised still further, this error decreases since more and more of the scattering occurs at large 1. At still higher energies the results for the BPT become low by several percent. This is probably due to large deviations in the trajectory from that given by the averaged V,. In this region the assumption of a rigid rotor is also seriously in error since vibrationally inelastic scattering is important. Below 2.1 eV the second-order results for the classical treatment of the orbital angular momentum are better than 10% in accuracy for the 0 + 2 transition. In many cases, this is better than the CS results. The results for the OS approximation converge to the CS results, and those for the classical approximation to the CC results The results of the DWA are considerably worse than those of the EPT except at 0.1 eV where the two are equivaient because Aq is very small- Above 0.9 eV the transition probability for 0 + 2 exceeds one for small impact parameters. The potential used for Ar + N2 is 1331 V(r, r) = ~g(‘) + V,(r) Q(cos r) , V&l
=E&&p
~2(')=+&&)12
VW
- 2(',/r)61 1
-2+&/d61
Pb) ,
(794
with E = 0.010297 eV, r, = 3.929 & al2 = 0.50 and 4 = 0.13. Table 2 gives the results for Ar + N2_ The orbital and classical approximations are almost indistinguishable which is not surprising since the CS and CC results are also in close agreement. The agreement of the first-order EPT is better than 20% for all t&sitions except 0 + 8. Note that the selections rules for AR are Aj = 0, *2 so that.all the higher-order cross sections come from the exponentiation. The agreement of the second-order theory iscomparable. Also shown are the results of Eno and Bali&Kurti [ 171 who did a
R_.J_0ass/.0ponenriaI
246
sartteriig
perturbation theories for in&sic
Table 1 Results for He + Hz al cc b,
E(eY)
eIasticj=O+O 0.10 0.65 0.90 150 2.10 X00 4.20 inelasticj=O-2 0.10 0.65 0.90
I.50 210 3.00 4.20
Second
First order d,
Cs =)
order
d)
DWA
Cld)
OS d)
Cld)
osd)
55.4 42.7 40.3 35.5 34.6
55.4 43.3 41.0 37.6 35.7
55-4 43.1 40.7 37.1 35.1
559 c) 43.3 f) 41.1 g) 37.8g) 35.9 g) 34.1 g] 32.6 g)
55.9 e) 43.2 0 41.0 g) 37.6g) 35.5 g)
55.4 42.9 4o.5 35.8 35.1
55.5 46.1 44.4 38.8 41.6
33.7g)
34.3
33.6
34.9
34.1
42.1
32.3 g)
33.2
32.4
33.6
32.8
41.0
0.205 =) 2.69 f,
0.190 =) 2.80 ft
0.207 3.08
0.209 328
0.173 2.69
= 0.175 2.88
0.206 3.80
3.33d
3_40g)
3.62
3.90
3.23
3.49
5.02
4.13 4-48 46 I 451
4.40 4.go 4.96 4-81
4.08 4.01 3.67 328
4-41 4.3s 397 354
3.77 3.84 3.65 3.3s
4.13 4.22 4.01 3.61
7.36 9.23 11.5 13.9
0.113 0.260 0.717 1.04 1.38 1.51
0.118 0.280 0.791 1.17 1.57 1.72
0.087 0.202 0.565 0.84 1.17 l-41
0.091 0.217 0.625 0.95 1.35 1.63
0 0 0 0 0 0
g) g] g) g)
g) g) g) g)
inelastic j = 0 - 4 0.65 090 150 210 3.00 4.20
0.095 0
0.089 0
a) Potentialis the Kmus-b&s surface [ZS]. see eq. (78). Units arc ,k2_ b, Closed-Coupled. Cl Coupled-States: The sudden approximation is used for the orbital angular momentum. ‘] fiponential perturbation theory using fust- or sxondonbx theory for A-. Cl uses a classiral treatment of the orbital angular momentum; OS is the Orbital Sudden. simitar toCS_ e)Ref_ [30]. f)Ref. ]31]. d Ret [32]_
Table 2 Results for Ar f N2 a)
lians.
cc b,
cs c)
ECS d)
10s e)
First order fJ Cl
Second order f) OS
CI
OS
18.5 21.9 15.8 4.18
o-2
22s
23.2
23.2
22.3
19.3
18.7
19.2
o-4
22.4
22.8
252
18.8
219
22.8
21.1
12.8 2-49
13.2 2.53
15.3 4.10
O-6 O-8
IO.5 1.17
a)TotaI energy= 0.02585eV=300#=208_5cm~1. b,CloseCoupkd. ref. 1331. c) Coupled States, ref.
[34]d] Exponential Coupled States, ref- [ 17]e, tnfinie-Order Sudden Approximation (the sudden approximation for both orbital and rotational motion], ref. 1333. f, ~ponentkl Perturbation Theory using first- or second-order terms in Aq_ Cl uses the classical limit to the orbital angular momentum and OS the sudden limit.
R.J. (>ossjE.qunzentfal pekrbation theoiiesfor inelasticscitteritig~ _’
fmt-order exponential calculation using the orbital sudden approximation. There are some minor differences between their calculation and ours such as the use of exact rather than WKB wavefunckons. Their I~SU!~S should be equivalent to the fnst-order OS results. I have no explanation for the diierence between their results and mine. The Inftite-Order Sudden ([OS) approximation uses the sudden approximation for both orbital and rotational motion [33,35] and, unlike the EPT, is not a perturbation theory. The potential for Ar + HCI is given by NielsonCordon potential 52 [36], V(r, 7) = V,(r) [ 1+ 0.35 P,(cos 7) + 0.65 P2 (cos $1’ +v.(r)[l
+o.3(rmlr)P~(cos~)+o.09q(cos~)],
(8W V,(r) = e(6/a)(l-
6/0)-~ exp @(I-
V,(r) = -e(l - 6/~$-‘(~~/r)~
r,/r)]
,
,
(80b) (804
where e is the well depth, e = 202 K = 0.0174
eV, rrr, is the position of the mlnlmum, r, = 3.805 & and (Y= 13.5. The potential is highly anisotropic, and HCI has wide energy-level spacings which makes this a difficult test case. As can be seen from table 3, the agreement is marginal at best. The second-order results are worthless; the exponentiation of A9 was done by a power-series expansion and, at most energies, did not converge even after 20 terms. Some improvement might be obtained by using a larger basis set and by using matrix diagonalization to exponentiate Aq, but it seems unlikely to give accurate results. The failure of EPT for a system like Ar + HCI should not come as a surprise since it is, after all, a perturbation theory which must become less accurate as the perturbation increases. Nevertheless, it is encouraging that, even in this system, which has a very anisotropic potential and large energy-level spacings, a fust-order perturbatiion treatment gives results which are in error by only a factor of 24.
8. Diiussion As is shown by the calculations both here and elsewhere [lo-125 the Exponential Perturbation Theory is a fast and versatile theory which is accurate for a broad range of problems. with some improvement it
.. .
Table 3 ReSukSforAr
~0.005 0.010.
.0.018
0.030
0 -+ 1. ...
o-+1 O-2
247 :
.
..
_.
.- .-.
3122
: 24.8
.1g_1
42.4 22.7 9.6
27.7 5.7 1.6 29.3 24.3 2.7 2.1 2.1 8.2 27.1 26.1 4.3 0.42
;I:
-
0+2 o-3 l-+2
25.9 1.2 14.5
41.3 22.8 9.4 459 25.7 0.5 14.1
:I; 0 -1 o-2 O-3 o-4
6.3 1.9 42.7 21.8 3.4 0.18
5.9 1.9 42.5 21.9 2.7 0.13
:I: l-4 2-3 2+4 3-+4
14.8 3.2 0.36 8.2 0.75 3.3
15.5 3.5 0.34 7.9 0.88 35
1 -.:
;:; 0.66. 3.3 3.5 2.3
a) The potential is given by the Nielsorr-Gordon potential 52 (ref. 1361, eq. (75)). Crosssections are h A*b, Ref- [37]. might be more accurate than the CS approximation. With this in
mind we have given a derivation which broadens the theory to including high-order terms, closed channels, problems with multiple turning points, and a classical treatment of the orbital angular momentum. AS a perturbation theory it is very much better than the DWA. It gives more accurate results for the fmt-order tran&ions, and it gives reasonable answers for second-order transitions where the DWA gives zero. The EPT gives a unitary scattering matrix unlike the DWA, which frequently gives transition probabilities exceeding one- In a typical calculation, roughly half the time is spent calculating Aq and half in exponentiating it. Thus the great improvement of the EKT is accomplished at a cost of doubling the corn-putation time. The EPT is useful for systems of small or moderate anisotropy. It should be excellent for treating small-angle scattering. Here-the class&l limit to the orbital angular momentum should be used sir& 0(r) changes by 180°.during the collision.. .. ; The E&“l!is one member of a large &a of per&_ bation theories which express the s&e& ma&in -. ~-
R..L CtossfExponentiaZperturbation theoriez for inehmicsmtreting
248
obtained
the general form
Sji = exP(Noi)YA);,-editloj) ,
W
where A is an appropriate perturbation series and f is some functional form. The simpIest case is the distorted-wave series where f(A) = A. The exponential case, [eq- (20)], uses a more complicated f and evidently converges faster. A third alternative was obtained in deriving the higher-order terms in Aal (see appendix A), ~~~=eXp(i?&)[(l
+in)(t
- in)-*]iiexp(i_rlo$,
(82)
where R is given by (25). The a matrix is similar to the conventional reactance or X-matrix except that it does not include the eiastic part of the phase shiftLike the exponentiaI method, S is unitary if R is hermitian. From (24) and (25) it appears that calculating R may be simpler than calcuIating An but this does not imply that (82) is more or less accurate than (20). The best possibk test of these approximations is to do accurate calcularions of S for several types of potentiaIs and compare the two approximations as a function of asymmetry. There are several ways that the EPT can be used in conjunction with a more accurate method such as the CC or CS method. The most obvious is to use the CC or CS method at small I and the EPT at large 1. One of the probIems in doing a CC or CS caiculation is that the scattering matrix oscillates rapidIy as a function of I so that one must repeat the calculation at many different Z’s_The modified S-matrix Sr e = exp(-i~o;)~~
exp(-in,-,$
= [exp(2iArl)lV, 033)
is a much smoother function of Z so that fewer values need to be calculated: the intermediate values can be
by interpolating St_ By diagonalizing S, one can obtain A11which is usually a slowly varying function of Zand E. There is a possible probIem in that the eigenvalues of S1 determine the eigenvalues of Aq onlyto within a multiple of IT.If the EET is also used to obtain Aq. this ambiguity can be removed. Going further, we can use the EF’T to obtain the matrix b= &cc
- Atl~p~ .
w
Even if the EPT is only crudely accurate, S will be
small and a slowly varying function of I; 6 will rapidly go to zero at large I where the EPT becomes more accurate. One can then get Sfj = exp(@gi) [exp(2iaqEpT
+ 2iC)lv exp@&
(85)
which is the exact S-matrix_ High accuracy can be ob-
tained by calculating 6 at only a few Z’sand interpolating to obtain the rest. The advantage of this method is increased if the calculations are done at several energies, since one can then interpolate in both E and 1. As derived here, the EPT is clearly related to timedependent theories such as the Magnus approximation [20] _The time-dependent formulation has a computational advantage in that (59), (61), and (63) are more easily evaluated than (22) and (23), sincethere are no Airy functions to calculate and since the integrands do not oscillate as rapidly. However, the approximations inherent in the time-dependent methods may degrade the accuracy_ The time-dependent methods all refer to a single classical trajectory, which implies a single energy. In contrast, the general theory uses a separate energy and trajectory for each quanturn state- Unless the energy defect is small, a large part of the transition probability is determined by the region near the classical turning points where the time-dependent formalism is least accurate.
Appendix A: Infinite-order expansion of Aq From (19) we have S, = (I + E + iF)(l + E - iF)-l
= exp(2iAqj.
(A-1)
Lkfme Q such that S-&(l+E)=F.
(A-2)
Then St= (I + in) (I - X&)-t = 2(1- ifi)-l
- I.
w-3
RJ. CkossfExponentiniperturbntion t&o&
249
for inelastic scattering
A comparison of the power-series expansion of (A-3) and exp(2iAq) gives (24). A quick and dirty proof isto solve (A-3) for R in terms of S1, Q = -i&r
- I) (St i 1)-r = -i[exp(2iA)
Aa = arctanQ = &I- $Z3 + @i
-I ] [ex&iiq)
(A-4)
+ I] -I= tan Aq ;
- ... .
(A-9
Some of these operations are questionable for matrices. If Q, is the nth prder tenrr in C&,then (A-2) gives Q, = F, -&I,E,_,
-Q2En_2
---XL&
(A-6)
_
From (9) we have En = i [A@‘) E,.-r(r’) + D(r’) Fn-t@‘)l e’ , 0
(A.7a)
F, = j C@(g) %_r(r’) •r-D(r’) F&r’)] 0
(A-7b)
*’ ,
and E. = I, Fo= 0. Note that (A-7) gives (21) directly. We wish to prove by induction that
n,(r)
=j
[-q_tA
+ DRn_r -BrBl-&_,
--h2Ban_3
- ..- -ZL,J%fir]
dr ,
(A-8)
0
where n,=Jcdr. 0
(A-9)
Using (A.7b) and (A-9) we have, + DF,] dr=fh,~,
Fn+L = j [n;E, 0 =fi,En
+j
]+AE,_r
+ j [-f21~E,_, 0
-.(t,BF,_,
+ Da,
-PIBF,_,
i- DfirE,+r
+ DF,] dr
f ... + D%_,E,]
(A_&
(A.11)
dr ,
0 where the last equation was obtained by using (A.6) for F,. Substituting (A.1 1) into (A.6) gives, an+r = 1 [Dn, 0
-a,DF,_,
-BtAE,_,
+ Dfi,E,_,
+ ...+ Dfk,_,E,]
dr -SQE,_,
- ___-S&El
_
(A-12)
Next, we note that,
sJ-
[+A+
Df$] E+t
0
do.= j “;E,+r 0
dr
tn+E,_2 +fi,BF,_,l dr -
(A.13)
(A.14)
0 -We continue the iteration in this fashion. At each.step we use (A-6) to give Fjrr hi terms of n,
and lower-order
RJ. CkqsfErponentid pertuibationtheoriesforinelaitic.scattering
Referencesr _._
:
i
K_ Bergmann,-W:Dem~~~er,-M.
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-
-.
[Z’]
kR Edmonds. Angular momentum in quantum mechanics (P&c&on University Press. Princeton. 1957). 1281M- ?%aus and F, bfies_J. Chem Phys. 42 (1965) 270% WI P. McGuire and DA Micha, Intern. J. Quantum Chem. 6 (1972) LLL-
1353 U. Buck and V.. Kbare. C&m. Phys. 26 (1977j 215;
[3OJ P- XcGuireand D.J. Kouri, J. Chem. Phys. 60 (1974) 2488. [3S 1 P_ McGuire. C&m. Phys. 8 (1975) 231. 1321 P- McGuire, J- Chem Phys. 62 (1975) 52%
GA- l+ker and R-T Pack, J. Chem. Phys. 69 (1978) 1585. 1361 W-B_ NieIsuMnand RG.Gordon, XCkem. Phys. SS (1973) 4149.
1331 T-P. Tsien. GA, Parkerand RJ Pack, J. Chem. Phys. 59 (1973) 5373. j34) P. McGuire,them- Phys- 13 11976) 81.
1371 S.M. Tax, H. Rabitz, D.E. F&z and RA. Marcus,J. Chem. Phys. 66 (197?) 2854.
R Goldffam, S. Green and DJ. &xxi. J. Che&. Pliys. 67 (1977)4149; DJ. Kouri, R Goldtlam and Y- SSimord. J- C&em. Phya; 67 (1973) 4534;
‘.
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-. Received I6 November 1979
: _.
.:. ~An Exponential Perturbation Theory (EPT) is derived whereby one calculates a &ase&ift matrix by an nth order per-. terbation theory and then exponentiates it to obtain the scattering matrix. The theory has be&developed to include high: order terms, dosed channels and~reaonanees.The radial wavefunctions used &e WfU3 solutiona &ch are generalized tb. i-.1 cases -wherethere are multiple tornhrg points. The orbital angular momentum may be treated exactlybr in the’&&i&l o$’ andden limits_ Calculationa are done for the rotatidoaliy inelastic sdtering in He + Hi, Ar + Na .&d &-+ RC&.The fii two systems give fair to good agreement with accurate calculations: the last case gives poor agreement: The fmt-oider EF!T is very much beuer than the fti-order distorted-wave approximation.
. 1. Introduction
simplest systems. Since the cost’in time and moneytkr solve a set ofNcoupled equations is rou&ly propor-
Vibrationally and rotationally inelastic collisions tional to b3-, there is much to be gamed inreducing : are involved in a wide range of phenomena ranging N. Rabitz-developed an Effective~Poter$al (EP) api :from transport properties [l], ultrasonics [2], shock proximation [l I] which reduceslVby reducing the : waves [3], and line broadening [4] to the activation dimensionality of the problem. Pack [12]. and &iCGuire of reagents prior to a chemical reaction and the deand Kouii [ 133 have developed the.Coupled States (CS) activation of excited products. The operation of gasmethod which makes approximations to the orbital phase lasers depends on the many vibrational and angular momentum thereby reducingIV_ Generally the rotational transitions involved in the laser action [5]. CS method is more.accurate a&more expensive than There have recently been a wide variety of detailed the EP method. It is not-hard;howeverj to find even, simple systems ,for which the EP and CS appro+raexperimeuts on ineIastic scattering using pulsed beams of ions 163 and neutrals [7], measurements of changes tions are still much too expensive: to be practical; in velocity before and after collision [S] , and by variAt the other extreme-are perturbation theories .: ous spectroscopic techniques [Sb,7]. such as the Born Approximation @A) and-DistortedA large set of theories of vibrationally and rotationWave Approximation @WA) [lo] ;-Unfortunatelyy; ... ally inelastic scattering has been developed in recent most cases require more than a first-order perturbation years [lo]. As is apparent from the range of experitheory so that these methodsare oftenvery inaccurate. ments, no single method is practical for all cases, but There remains a broad-middle ground between these rather a range of theories-is needed. It is simple tore-extremes to be filled m Gne old-method; now enjoywrite SchrGdinger’s equation as a set of coupled. mg a renaissance is the Bxpon&i~Perturbation i- - -..’ second-order differential equations;.one for each qnan~ Theory (EPT) [ 14-191 in.which one uses the BA or : .; turn state involved-m the scattering [iO]i Numerical-’ DWA to~calculate a transitionmatrixandthen.expo;~ IL;. ~nf++-i+ .&is &&to i&t& the,f&d &&$ i j: : 5 solution of these equations is known as the._Close~--.?) Coupled (CC) meth_od~&rd~servestod&_as the pr+ry r+ixr Ti& expbi;entiatior+ &sures that the~s=+tte&& i; starrdard for testing appro~a~~me‘tho~~~~BecauSe_ of ( mat&:& ii&try (particl~~are-~oonseNe;d);ja p&b& :-: ..’ v;ith & B& ~d:DwA9~~~t.the_~~~iti~~i~o~gbil_ _-;_ the large number of quantum states involved;.the~CC~ iti fi~.L.el~~,e~&~~~-$ti~ rTiie‘e;ti6tietiti_i~~Lxaisi, ~; method becomes prohibitive@ expensive forall but--&e ._-.., ,_~ : 1. ... :...~... __ .- -. Z..)Z -: -7.:__,..i-.!__.‘.+,;‘;‘. .. : ;-_.__.__y..’ : .--. _.; ..- _. : -_=._-, ~. 7 r I ;..._..~ _,~_::_i__ ._ : .- ;..< : .. --.~- . ,. i. ; ._ ... .-:.._+:’ -__(... ,._ .~____.,‘._...‘_... _. .,_ /.~ ___~.__.:‘... ..:,I. .:.. :;. . . :.:-- ._-. .i.. -. .i _~=_.)_ ; _-_~-:~.,+-I-,.I_~-,. I‘_..: :.- ~...:<__. 1 .._~ )y ‘...._ :-‘- ::-;-,..,.;..:; _y:;-_.- ._._y._.;. ..;, ~_ z-.-.-‘;__. . .~ y<;:. .-; :-. l.._~ :_ ‘-
238
R.J. Oos.s/Erponentid perturbation theoriesfor
appears to give accurate results for the higher-order transitions where the BA and DWA give zero transition probabilities. Recent calculations on rotationally 116,171 and vibrationally [l&19] Inelastic scattering show very good agreement with CC results in many cases. The method has a good chance of giving the accuracy of the CS and EP methods for some systems while being aImost as cheap as the BA or DWA The basic exponential method has been around for a long time. In the case of time-dependent perturiiation theory, it is known as the Magnus approximation [ZO]. In the case of scattering theory it has been developed by several people including the present author [14,15]_ In most cases it has just been used without justification. In this paper we wish to develop the theory as a rigorous perturbation theory out to high order- We include effects such as closed channels and cases where the classical trajectory has three or more turning points- Several options tie avaiIahIe for treatment of the orbitaI angular momentum. By making some approximations in the theory we obtain a timedependent version where the unperturbed prob!em is elastic scattering on the classicaI trajectory for the sphericaiiy symmetric reference potential. The timedependent theory gives the Magnus approximation pIus additional terms arising from the deviations of the trajectory caused by the perturbing potent&I_ In both the sudden and dassicd limits to the internal motion the second- and higher-order terms in the Magnus series are identically zero. This explains, in part, why the theory works so well. FmalIy, we demonstrate the theory in the case of pure rotationalIy inelastic scattering. We use three systems of varying anisotropy: He + HZ, Ar + N, and Ar f HCl. For He + HZ up to 2-I eV relative energy the first-order theory is accurate to several percent and the second-order theory to a few percent. Above 2.1 eV the EPT diverges from the CC results. For Ar + Na the fmt-order theory is accurate to IO-20%; the secondorder results are similar_ For Ar + HCI, fast-order results are in error by factors of 24.
d*Ui/d?
+-p:(r) Ui(r) = F
qj(r)
Uj(r) i
(1)
where r is the distance between the centers of mass of the two colliding molecules and p?(r) = L+ - U,(r) - Zi(Zi+ 1)f2
,
0)
kf = 21.cR-*(,5tot - .&*i),
(3)
U&) = 2pV&)l@
(4)
U&(r) = 2pZ
,
-‘
(5)
and p is the reduced mass of the two molecules_ Note that i and i are collective indices which also label the orbital angular momentum quantum number L The reference, unperturbed problem is d2u,j/dr”
+p;
uDj(r) = 0 _
(6)
There are two lineady independent solutions to (6), “lj(‘) which is weII behaved at the origin, and u.&)_ Rather than use accurate solutions to (6) which implies a time-consundng numerical integration, we use approximate, WKB sohitions which are much more easiry obtained_ We keep in the theory the deviations due to the approximate solutions and due to Uii(+ We can write the solutions to (I) in terms Of ZfIi(‘) and ZQ-(~) as Z+(T)= U1j(r) Xj(r) + U*j(‘) Yj(r) -
(7)
With two unknowns we can impose an auxiliary condition, and we make the standard choice, Yi”O,
UljXi)+U2j
(8)
so that (I) becomes Xi’ = c (/$& i
i
The fundame+ theory has been derived in a previous paper [15] (referred to as I). We break the potential into two parts, a spherically symmetric V,(r)
scattering
and AL’ = Ir - I’0 - The zero-order problem is the elastic scattering due to V,-,(T).We can write the coupled Schr&liuger equations as
Y; = c 2. Basic theory
inelastic
+ Bii Yi) ,
(CvXj + Dii 5) ,
(94
49b)
where WW (lob)
=(27#*
qi(r)
Ipil’1~2 IZ,I~/~ Ai(
w
OW
u-&)
=(2~#/~ lp$1/2 lzj11J4 Bit&) . -. Ai and Bi are Airy functions 121, p_ 4461 and
(12b)
So far, this procedure is exactly equivalent to (1). The exponential perturbation theory &es an exponential form for S, exp(inci) , ‘...
su = exp(i*ci)lexp(2%tiii
and rcj is the classical turning point where pi(rcj),= 0. For the moment we consider the case where there is only one rc. Near rc, z is linear in r to give the solutions for the linear turning point. Outside rc, ulf -
[2/Pi (r)l 1’2 sM& +
- (Zlkj)“’ uli - [2/pi(r)]!j2
+
where the phase-shift matrix Aq is a generalization of the usual elastic phase shift ~7~.The exponentiation of Aq is carried out by a power-series expansicn or by diagonalization of Aq, exponentiation of the eigenvalues, and transformation back to the original representation. If Aa is hermitian,.then S is unitary. Our procedure for obtaining A9 is simply to obtain E and F by perturbation theory and then to equate (19) and (20) order by order. To second-order,
a/4)
Sin(kir + QO~- li~/2) 2 co&$
Wa)
n/4)
N (2/%)rL2 coS(kir + qOt - ZtX/2) , where nor is the WKB phase shift for V&), Inside rc U]i N I Qj Ibu2 exp(-Sjl T
(1 =I ki
and ZP
eXP(E$ -
A(r’) df
E(r) = j
0 r’ + r&f dr” [A( s s 0 0 F(r) = j
,
(21a)
[C(r’)A(r?) + D(r’) C(r”)] -
@lb)
+ B(r’)C(r”)]
C(r’) dr’
0 JJ CL+
W-4 +
U2f N lPi/21-“2
Wb)
r&r ss
0
0
The wronskian W is easily found to be -2.
The first-order term becomes
As can be seen from (165) U2j becomes inftite at the origin SO that as i+ 0, Yi(r) +O. At r =O, then, X(r) =X0 and Y = 0. We can then integrate (9) out from r = 0 until the potential is negligible. Let
Atl,, =
X(r) = [I + E(r)1 70 a
(20)
WI = F(r) x0 -,
(17)
0
0 .(22)
The
second-order term is
where I is the unit matrix- Then the total wavefunction be&mes
1
.‘;-
,.’
._
‘.
:~
: i_-....-
.-_ ..-
_
:
:
,_
:
.:.: ,. ::-
Arl,=
J-
-P
r dr
J
- dr’
[C(r) A@‘) + D(r) C(r’)]
-0
Cdr s 0
A~J-
dr’ [D(r) C(r’) - C(r’) A(r)] _ 0
0
where Aq,(r) is given by (22) using r as the upper limit of the integral. Since n@) + 0 as r + 0, the integrals are alI stable_ In appendix A a general form is derived for the nth-order term, An =n(_i
- f&(a)
-I-$&S(m) - $‘@(a) f .._ ,(24)
dQ(r)/dr=C--A+DfiZ-$Zf3Q_
(25)
The solution of (25) by perturbation methods gives (22) and (23) and the various higher-order terms in AlISo far, we have considered cases where onIy open (energetically accessible) channels are included. However, the derivation of (1) requires a complete basis set of internd states includmg the closed channels. Although it is impossible to get a transition (at r = -) to a cIosed channel, they may Influence the scattering. Feshbach resonances are an example of this_ For closed channels u1 and ~2 may be defmed by (16) using any re as Iong as it lies welI beyond the range of the potential_ As before, zcl is well-behaved at the origin. For larger, uli” exp(lc& and uzim exp(--rjr), where K~ is given by K’
= 2~
-*fEim i - E,,,)
_
(26)
The asymptotic totaI wavefunction
is given by
Ui - (2Ki)- IL? eXp(-inui) X F (iQ- + iEii> Xoi exp&r - (2ai)-1/2
exp&&
for some suitably defied complex 17oiand C-. TO 0th tain a well-behaved wavefunction, we must adjust X0 so that the fmt term of (27) is zero. This term plays a roIe simiIar to the first term of (18) for open channels in that it represents input information. We are usually not interested in the second term of (27) since it does not persist at larger. We can make the first terms of (18) and (27) identical by repIacing Fij in tie first term of (18) with Fh defined such that Fh = Fq for i open and Fb = 0 for i closed. This replaces the second F in (19) by F’. We can also replace the first F by F’ without affecting transitions to open channels. The result of this modification is that (22) -(25) are valid provided that i and j are open channels. For closed channels AnV= 0. There are no effects of closed channels on the first-order term and thus no transitions to closed states. In second-order the sum over k in (23) does include closed channels as do the sums over virtud intermediate states implied in(2.5).
- Zrr/2)
Ci
(27)
3. Multipie turning points So far we have examined cases with one classical turning point (open channels) and with no turning points (closed channels). When V,-,(r) has an attractive part, more complicated cases exist. A state with two turning points is closed at r = - but becomes open at finite r due to the attraction in the potential. The most common case is where there are three turning points due to the centrifugal term in (2). If VO is negative there can be three roots of&r,) = 0. Thii situation gives rise to orbiting resonances in eiastic scattering and to similar resonant behavior in inelastic scattering. We distinguish four cases: (i) The minimum in p2 is large and positive. Here there is only one turning point, and we can use (12) as before. (ii) The minimum is smaU and positive. Here again there is only one turning point, but there are errors in (12) near the minimum because p* is not linear near r.?(iii) The minimum is small and negative. There are three turning points and the two near the minimum’ are nonlinear so that (12) cannot be used. (iv) The minimum is large and negative. There are three turning points and aU of them are linear_ Case (i) is already taken care of. In case (iv) we have
.241
RX C?oss/Exponential perturbation theories for inda$iic scattering
solutions of the type (12) for each ofthe three turning points. Let rca
&=-&,+&b &=-$+&*
(28)
29 =-s@l@ul,+cos@rc~c.
(29)
Note
where 'Cf.3 6 ab =
PJr)
s
dr
3
(30)
(38)
that the wronskian of u1 and ui is still -2. -. In regions (ii) and (iii) we must fmd solutiqns-to (6) for a quadratic turning point. We expand p* abont the minimumro, p2 = -Co + C2(r - rO)2 ;
‘ca s,,=
with the quasibound s&e between rca andrcb..%eeffect of this resonance will be a pronou?ced change in All as the region inside rcc is opened,u$Uiing(32) and (33) we can project back the other linearly independent solution
IzPi(r)drl
(331)
-
c The connection between solutions is sin6ab
cos 6,
cos 6,,
-sin Sab
(32) Ulc
u2c
)-
(33)
We need two global solutions u1 and 212which must satisfy two conditions: u1 must be well-behaved at small r so that u1 a ula; u1 and u2 must behave asymptotically as (15) for some phase shift pg. Let R = [4 ex&&.,,_) Cos2sab+ $ exp(-2tibc)
where Co is positive in region (iii) and negative in region (ii). Using x = (4 C2)‘/4(r - ‘0) )
WI.
a = C,(4Cz)-l~2
(41)
)
eq. (6) reduces to
I
)(
(3%
sin26,b]“*,
d*u/dx’ - au + $x2u = 0 _
i42)
The solutions to (42) are the parabolic cyhnder functions 121, p_ 6851 u1 bc = W(G x) f
(434
t+bc = w(a, -x) .
(+b)
We must now connect (43) to the solutions away from r. as before. Of the several asymptotic expansions of W(a,x), the most useful is Darwin’s expansion [Zl, p. 6851. Let
(34) 6 = arctan [$ exp(-26b,)
tan S,,]
_
135)
Using (32) and (33) we -find that K1 =R-kla
=R-‘(sin
&&lb
~=cosQulc+sinQu2,,
u1 +Sin(~
-I-4,. + cp- Z77l2),
x=(X2
COS 6,buZb)
=
(37)
where qcl is the WKB phase shift using only the outermost turning point rc. The correct phase shift is sci + @,a result previously obtained by M$er [22] _By assumption the minimum in p2 is large and negative so that sbc will be large. Usually this means that R and + are small and that we can neglect the region inside rcc_ However, if&,, Z=S (a + &rS c& 6,, is small and R and @become large; Here we have a resonance
xdr,
cos@ f $r) )
(2yKP
Iv(Q, -x) = (2/_YxP (36)
*=5J
(44)
2&Z ?v(a,x)
+
- 41Q1)‘/2,
(45a)
sin@ + $70 ,
Wb)
where y =
(1 +
&a)1/2
_ ena _
(46)
For case (iii) the classical turning points “e at x = &&I12 am-lp z &Q)‘!2 x. Using (13) we’fmd that ,$=e,so that
u1 bc = (2y)-1!2(~;)118
Ulb
: (+b)
zQbC= (2y)-Q2
UIC
:. (rrTcc)-;(47c)~
(ti2)?8
,(+)
242
RX Q.oss/Exponenttit perturhtion theoriesfsr inelasticscattering
t42bc = 0,/2)“*
(4c#*
(r Q rcb) _(47d)
U2,,
The connection between ub and uc is therefore
6;)=
C&
‘; K;;)-
(48)
Using (31) we obtain 6,, = tra. Since, for large Q, y =Z $ exp(-$.& (48) goes smoothly into (33) as we go from region (iii) to region (iv). We repIace (34) and (35) with R = (y-2 cos2Gab+y2 sin26,,J1~* ,
(49)
Q = an%rlCy’tan 6,b) ,
(5@)
and use y = [I+ exp(2$_,)J
Vr - exp(6hc) -
(51)
(47). In region (ii) we let rCt, = rCC= r. and find that 5 = B as before- Using (28) we obtain (4c2)tQqaSin
6,,
+ r&&cos gab)
’ (r-&-n), q uc -(y/2)1/”
(4C*)‘~S zf*e
(52a)
(r Q q$ > WC) (4$)“*
ulc
(Uti - itr*i) =p-t/*
2-ti2
(uu + in*& =p-t12
uqi =
(rS=r 0 ) - (52d)
This gives the connection between u, and u,- Again (36)I(38) are valid. As the barrier b&omeHmore negative, y h 1, and we go smoothly into the case of a singIe turning point with q. = 71el + 6, - This same procedure will work for any number of turning points.
4. Time-dependent formulation A popular set of approximations is based on timedependent perturbation theory using the trajectory corresponding to Vn. The originrd theory In I derived a connection between the EET and this time-dependent approach. We sketch the results here because they give a valuable insight into the errors and interpretation of the EFT. To make the connection we assume that k
eXp(i&
-
inl4) , (53a)
exp(-i& + in/4) .
(53’3 The wronskian of u3 and r~ is -2i so that (10) becomes 9Qii = &p-i
Vi;-exp(-i&- + i$)
$p-l
U& eXp(-i5;
-
L$j)
,
(W
,
Wb)
@ii=~p-tLli;-exp(~i+i~~), ‘oii = k i p-t
Uh
eXp(i&
(54c) - i!$
_
t5W
that 1 and cb oscillate slowly In r whereas % and C! oscillate rapidly. If we keep only the slowly oscillating terms, the first-order term for Aq becomes, Note that ,$ is large so
Aqii=-$
(r s Q) , (=b)
u2bc - (,42)1’2 (4c2)1’8 &la cos 6, - u&sin 6,b)
U2bc -(2y)-li2
Lc3i= 2-l/*
cx3ii’
Then (36)-(38) are valid in regions (iii) and (iv) except near the minimum where we must use (43) and
ut bc - (2y)-I/*
and 1 are large and that A k[k and A@ are smaII for ah transitions. We replace all the rc’s with 2 common averaged r-c and use the asymptotic form given by (IS) for ut and u2 over the whoIe range of integration. We transform (15) to a new basis set,
J
p-1
U&(r)(&As+e-iAE)dr,
(5%
J-C
where 4 = & - $. Now, tip is the classical radial momentum. It is related to the time t and trajectory angle 0 by dr=Gipdrjfi.
(56)
dB = ql+
(57)
;) &l(r2p) ,
where the top sign refers to the incoming half of the trajectory and the bottom sign to the outgoing half; 0 goes from ir at t = -0, r = 0 to the final scattering angIexat t=m,r=w_ For small Ak and AZ [iSI a
= & - gi = *qj(t_ - tJ 7 (Zi - $)(S - 0,) ,
(58)
where wij = (Eat I - Elntl)!tr and tc and Bc are t and 8 evaluated at r = rc. With these substitutions
where Gii(r) = u$ exp[ioii(t
- rJ - i(Zi.-
(60)
w(e - e>l -
,. RJ. CPoss/Gqwnenhid
pertybatiyz
Except for the o-terms in (60), the result is identical to the Usual time-dependent perturbation theory. Most tie-dependent methods do not include orbital angular momentum. we will shortiy give two methods of removing it fm.m the problem. The second-order term can be broken into two parts Aqti and Aq*,,: Aqa =
2 4dt -0D
j dr’ [G(t), G(t’)] _ -m
(61)
This is the second-order term of the Magnus approximation [ZO] [except for the e-dependent part of (60)];
PC
rc
- e (r)S (r’) f (?!(r’)93 (r)] -
(62)
In appendix B it is shown that AqZb = : s
p-l
$ [(p[p’)(oQ 2 + a’)]
dr .
theoriesfor
inelastic scattering
:
243
weakly onj and ti; Here again; we-can diaioxialize G by a time-independent transformation. The detail!:?= given in ref. [23] _ Briefly, we transform fK6ti.a rep+. sentation diagonal in the action variablesi and m to a-: representation diagonal in the conjugate an&e sari- : ables. The.treatment is similar in many respects to the semiclassical mechanics of Mier and Marcus 1241, which also uses an action-a&e transformation to diagonalize the problem. In the sudden liiit the internal motion is frozen during the collision; the internal wavefunctions are Dint delta functions, independent of time. In the classical limit the internal wavefunction is an infmitely narrow wave packet which moves along the classical trajectory for the internal motion. One can easily qonsiruct hybrid approximations where some of the degrees of freedom are sudden and others are classical_ Here too, An& is zero. In particular, the orbital angular momentum is nearly always classical, although the sudden approximation is often used. In many rotational cases, at low j the motion is sudden and at highj classical.
(63)
rC
5. Orbital angnlar momentum
In the
case of a spherically symmetric potential V(r) = V,(r) f AV(r) one can expand the WKB phase shift in a power series in AV to give p=~~+A~,+Arl,,+o(Av~)_
(64)
Aql involves the integration of AV over a trajectory given by V,-,; AqZb is a correction due to the deviations of the true trajectory from the one given by V,-,. There are two important limiting cases where Aqh and all higher terms of the Magnus expansion are zero. These are the sudden and classical limits. In the sudden limit the energy difference between quantum states is neglected; kjisreplaced by some averaged ko. If all degrees of freedom are sudden, then u1 and ~2, given by (12) are independent of state and wii s 0. This means that G(t) Q U’. We can then simply transform U’ back to the coordinate representation where it is diagonal. The transformation “matrix” is the set of internal wavefunctions Ii> and is independent oft. In the diagonal representation the commutator in (61) is obviously zero. In the classical limit A k and U’ depend on (i-j) but only parametrically on i. In the rotational case, for large j the relevant potential matrix depends strongly on Aj and Am but depends only
Of all the degrees of freedom the orbital angular momentum is usually the least interesting. It is not directly measurable and therefore summed or averaged in any calculation. It is no surprise, therefore, that this is where many approximations begin. Both the sudden and classical approximations are used, and we describe how these can be used in the framework of the EPT. The orbital sudden approximation is the replacement of Ziby an averaged 10 in (2). This is the Coupled States (CS) approximation [12,13] and results in a reduction of the number of coupled equations to be solved- For an atom and a linear rotor the-system is described by the coordinates r = (r, 0, @)).the separation of the atom and the center of mass of the rotor and R = (R, 0, a)), where (0, *) is the orientation of the rotor andR is one or more vibrational coordkates. The usual basis set is lnZjJ&f>,where n is one or more vibrational quantum numbers, 2 quantizes the orbita angular momentum, j the rotational angular momentum and J and M the total angular moinentum- If1 is large and j < I, then one can show that. [25] Oz’Z~‘JiV(~,R,~)lnljJ)=(~j.P’IV(~,R,P)lnjS)
,&
244
R. J_ CzossfExponential
perturbation
where-yistheangIebetweenrandR,6’=J-1’,6= J - 2, and /3is the angle between (0, @) and (r&O). Note that 6 behaves like an m-quantum number -j d 6 Gj. In the orbital sudden approximation u1 and 29 depend parametrically on lo but are otherwise independent of 2. Using a basis set Inj6) and (60) one can compute the scattering matrix as described above. Ref. [25] gives a compendium of formulas to obtain differentiaI and integral cross sections. In the classical limit we see from (60) that the effect of changes in 2 is to muhiply the potential matrix by a factor of exp{-i(2i - 2#l(r) - 0,_]- If this is done. the effect [25] is to modify (65) so that
theories for inelnslic scattering
The potential matrix is G2’j’S’IV(r,R,P’)lnjS)
=
g(4)
(g$ 11’2
(n’i?JJn)
X C(j2j’; 000) Cbitj’; 6 -
~5’) Y&r,
(68) Te(r)) ,
where [j] = 2j + 1, C is a Clebsch-Go&n coefficient [26], and Y is a spherical harmonic. The first-order
phase-shift matrix is given by (22); we take the average of the incoming and outgoing halves of the trajectory [the f sign in (68)]_ We can define a reduced phase-shift matrix,
(n’ZyJJIV(r, R, $lnri.r) + bij%‘l V(r, R, p’)&5>, (66)
where /3’ is the angIe between (0, a) and [7r/2, ‘(e(r) - @,=)I_As before, the top sign refers to the incoming branch of the trajectory and the bottom sign to the outgoing branch. The use of (66) is not restricted to a timedependent formuiation. Several methods for adapting it to the more accurate theory are given in ref_ 1254 _ The one used here is to pick some average state in the middle of the manifold being used; 0 is set
X Re[Y,,,(d-% WII
ulin(rI * -
(69)
The full phase-shift matrix is given by,
to Be E 0 Inside me_Outside r,, B is caIcuIated by integrating (57). The two approximations are easiIy visu&.zed in terms of action-angle variables. The angle variable conjugate to I is c?(r). In the orbital sudden approximation 0(r) is frozen at 0. In the classical limit B varies
in time over the classical trajectory_ For a hard coIIision at smaI1 impact parameter, B(t) does not vary much during the collision so that the sudden approximation may be used. For a grazing collision at large impact parameter, e(r) varies by ISO”, and the classical approximation must be used. For several reasons the sudden approximation is faster. In the calculations which follow we compare the two limits.
6. Rotational angular momentum Depending on the system there are some shortcuts that can save computer time. Again considering the case of an atom and a linear moIecuIe, we can expand
If the number of terms required in (67) is small, then one may well have many fewer integrals to evaluate using the reduced phase-shift matrix. We note two symmetry reiationships which further reduce the
amount of calculation required, , .,_ A,$,$ = (-I)’ An:;“, ,--,
Arm,“,~~
=(-I)’
(72)
For the case of the orbital sudden approximation 0(r) s 0, and we can remove the spherical harmonic from the integral in (69). This permits a further reduc-
tion in Aq _ The cakulation of the scattering matrix aIso requires the exponentiation of Aq. We can often achieve an increase in speed here by using a reduced A?. We can defme a similar reduction for Aqk,
the potential
where (67)
(71)
, .,_ nnII_ AqOA!L - Qn
sj 7 6,O $0
2
R. J. Oossf.&&wial
pertwbation
and the case of k = 1 is given by (69) The recursion relation is
where { ] is a 6-j coefficient [27] _The scattering matrix is given by a similar reduction, S* n njj”6.6
= gC(jhj’;6
-&Y)Sfz’,
(74)
where (77)
These reduced forms will be useful for cases involving large j where the reduced AR eliminates the many combinations of 6 and 6’ needed.
7.
Results
The exponential perturbation theory has been applied to three systems of varying anisotropy. He + Hz was one of the fmt systems studied by the CC and CS methods. The potential has a very small anisotropy, and the large energy gap in Hz further reduces the rotationally inelastic cross section. Ar + N2 has a higher anisotropy and much smaller energy spacings. Finally,Ar + HC1has a very anisotropic po?ential. In all three cases, calculations were done to fit-order [es_ (22)] and to second-order [including eq. (23)]_ Both the sudden and cIassical approximations for the orbital angular momentum were used. No attempt was made to use the .theory in the case of mult$e turning points. Instead, the outermost turning point was used with the usual WKB wavefunctions. The calculations were done on an LSI-1 I microcomputer and required lo-100 mm per energy depending on the system. For He + H2 we used the Kraus-Mies potential surface 1281, averaged over the vibrational coordinate
PI
theories for inelastic scattering V. = (214.5
eV)
exp(-3.49
V, = (40.47 eV) exp(-3.47
245
A-1 r) ,
0’8b)
A-l r) . -.
.(78c)
Table 1 gives the results for the integral elastic and inelastic cross sections. Also included are results for the fmt-order distorted-wave approximation obtained by taking only the first two terms in the exponentiation of Art_ There seem to be two errors involved. The classical limit to the orbital angular momentum used in (65) requires that I %-j. At low energies for a light system such as He + Hz, part of the inelastic sc%ttering occurs for small I where this is not true- As the energy is raised still further, this error decreases since more and more of the scattering occurs at large 1. At still higher energies the results for the BPT become low by several percent. This is probably due to large deviations in the trajectory from that given by the averaged V,. In this region the assumption of a rigid rotor is also seriously in error since vibrationally inelastic scattering is important. Below 2.1 eV the second-order results for the classical treatment of the orbital angular momentum are better than 10% in accuracy for the 0 + 2 transition. In many cases, this is better than the CS results. The results for the OS approximation converge to the CS results, and those for the classical approximation to the CC results The results of the DWA are considerably worse than those of the EPT except at 0.1 eV where the two are equivaient because Aq is very small- Above 0.9 eV the transition probability for 0 + 2 exceeds one for small impact parameters. The potential used for Ar + N2 is 1331 V(r, r) = ~g(‘) + V,(r) Q(cos r) , V&l
=E&&p
~2(')=+&&)12
VW
- 2(',/r)61 1
-2+&/d61
Pb) ,
(794
with E = 0.010297 eV, r, = 3.929 & al2 = 0.50 and 4 = 0.13. Table 2 gives the results for Ar + N2_ The orbital and classical approximations are almost indistinguishable which is not surprising since the CS and CC results are also in close agreement. The agreement of the first-order EPT is better than 20% for all t&sitions except 0 + 8. Note that the selections rules for AR are Aj = 0, *2 so that.all the higher-order cross sections come from the exponentiation. The agreement of the second-order theory iscomparable. Also shown are the results of Eno and Bali&Kurti [ 171 who did a
R_.J_0ass/.0ponenriaI
246
sartteriig
perturbation theories for in&sic
Table 1 Results for He + Hz al cc b,
E(eY)
eIasticj=O+O 0.10 0.65 0.90 150 2.10 X00 4.20 inelasticj=O-2 0.10 0.65 0.90
I.50 210 3.00 4.20
Second
First order d,
Cs =)
order
d)
DWA
Cld)
OS d)
Cld)
osd)
55.4 42.7 40.3 35.5 34.6
55.4 43.3 41.0 37.6 35.7
55-4 43.1 40.7 37.1 35.1
559 c) 43.3 f) 41.1 g) 37.8g) 35.9 g) 34.1 g] 32.6 g)
55.9 e) 43.2 0 41.0 g) 37.6g) 35.5 g)
55.4 42.9 4o.5 35.8 35.1
55.5 46.1 44.4 38.8 41.6
33.7g)
34.3
33.6
34.9
34.1
42.1
32.3 g)
33.2
32.4
33.6
32.8
41.0
0.205 =) 2.69 f,
0.190 =) 2.80 ft
0.207 3.08
0.209 328
0.173 2.69
= 0.175 2.88
0.206 3.80
3.33d
3_40g)
3.62
3.90
3.23
3.49
5.02
4.13 4-48 46 I 451
4.40 4.go 4.96 4-81
4.08 4.01 3.67 328
4-41 4.3s 397 354
3.77 3.84 3.65 3.3s
4.13 4.22 4.01 3.61
7.36 9.23 11.5 13.9
0.113 0.260 0.717 1.04 1.38 1.51
0.118 0.280 0.791 1.17 1.57 1.72
0.087 0.202 0.565 0.84 1.17 l-41
0.091 0.217 0.625 0.95 1.35 1.63
0 0 0 0 0 0
g) g] g) g)
g) g) g) g)
inelastic j = 0 - 4 0.65 090 150 210 3.00 4.20
0.095 0
0.089 0
a) Potentialis the Kmus-b&s surface [ZS]. see eq. (78). Units arc ,k2_ b, Closed-Coupled. Cl Coupled-States: The sudden approximation is used for the orbital angular momentum. ‘] fiponential perturbation theory using fust- or sxondonbx theory for A-. Cl uses a classiral treatment of the orbital angular momentum; OS is the Orbital Sudden. simitar toCS_ e)Ref_ [30]. f)Ref. ]31]. d Ret [32]_
Table 2 Results for Ar f N2 a)
lians.
cc b,
cs c)
ECS d)
10s e)
First order fJ Cl
Second order f) OS
CI
OS
18.5 21.9 15.8 4.18
o-2
22s
23.2
23.2
22.3
19.3
18.7
19.2
o-4
22.4
22.8
252
18.8
219
22.8
21.1
12.8 2-49
13.2 2.53
15.3 4.10
O-6 O-8
IO.5 1.17
a)TotaI energy= 0.02585eV=300#=208_5cm~1. b,CloseCoupkd. ref. 1331. c) Coupled States, ref.
[34]d] Exponential Coupled States, ref- [ 17]e, tnfinie-Order Sudden Approximation (the sudden approximation for both orbital and rotational motion], ref. 1333. f, ~ponentkl Perturbation Theory using first- or second-order terms in Aq_ Cl uses the classical limit to the orbital angular momentum and OS the sudden limit.
R.J. (>ossjE.qunzentfal pekrbation theoiiesfor inelasticscitteritig~ _’
fmt-order exponential calculation using the orbital sudden approximation. There are some minor differences between their calculation and ours such as the use of exact rather than WKB wavefunckons. Their I~SU!~S should be equivalent to the fnst-order OS results. I have no explanation for the diierence between their results and mine. The Inftite-Order Sudden ([OS) approximation uses the sudden approximation for both orbital and rotational motion [33,35] and, unlike the EPT, is not a perturbation theory. The potential for Ar + HCI is given by NielsonCordon potential 52 [36], V(r, 7) = V,(r) [ 1+ 0.35 P,(cos 7) + 0.65 P2 (cos $1’ +v.(r)[l
+o.3(rmlr)P~(cos~)+o.09q(cos~)],
(8W V,(r) = e(6/a)(l-
6/0)-~ exp @(I-
V,(r) = -e(l - 6/~$-‘(~~/r)~
r,/r)]
,
,
(80b) (804
where e is the well depth, e = 202 K = 0.0174
eV, rrr, is the position of the mlnlmum, r, = 3.805 & and (Y= 13.5. The potential is highly anisotropic, and HCI has wide energy-level spacings which makes this a difficult test case. As can be seen from table 3, the agreement is marginal at best. The second-order results are worthless; the exponentiation of A9 was done by a power-series expansion and, at most energies, did not converge even after 20 terms. Some improvement might be obtained by using a larger basis set and by using matrix diagonalization to exponentiate Aq, but it seems unlikely to give accurate results. The failure of EPT for a system like Ar + HCI should not come as a surprise since it is, after all, a perturbation theory which must become less accurate as the perturbation increases. Nevertheless, it is encouraging that, even in this system, which has a very anisotropic potential and large energy-level spacings, a fust-order perturbatiion treatment gives results which are in error by only a factor of 24.
8. Diiussion As is shown by the calculations both here and elsewhere [lo-125 the Exponential Perturbation Theory is a fast and versatile theory which is accurate for a broad range of problems. with some improvement it
.. .
Table 3 ReSukSforAr
~0.005 0.010.
.0.018
0.030
0 -+ 1. ...
o-+1 O-2
247 :
.
..
_.
.- .-.
3122
: 24.8
.1g_1
42.4 22.7 9.6
27.7 5.7 1.6 29.3 24.3 2.7 2.1 2.1 8.2 27.1 26.1 4.3 0.42
;I:
-
0+2 o-3 l-+2
25.9 1.2 14.5
41.3 22.8 9.4 459 25.7 0.5 14.1
:I; 0 -1 o-2 O-3 o-4
6.3 1.9 42.7 21.8 3.4 0.18
5.9 1.9 42.5 21.9 2.7 0.13
:I: l-4 2-3 2+4 3-+4
14.8 3.2 0.36 8.2 0.75 3.3
15.5 3.5 0.34 7.9 0.88 35
1 -.:
;:; 0.66. 3.3 3.5 2.3
a) The potential is given by the Nielsorr-Gordon potential 52 (ref. 1361, eq. (75)). Crosssections are h A*b, Ref- [37]. might be more accurate than the CS approximation. With this in
mind we have given a derivation which broadens the theory to including high-order terms, closed channels, problems with multiple turning points, and a classical treatment of the orbital angular momentum. AS a perturbation theory it is very much better than the DWA. It gives more accurate results for the fmt-order tran&ions, and it gives reasonable answers for second-order transitions where the DWA gives zero. The EPT gives a unitary scattering matrix unlike the DWA, which frequently gives transition probabilities exceeding one- In a typical calculation, roughly half the time is spent calculating Aq and half in exponentiating it. Thus the great improvement of the EKT is accomplished at a cost of doubling the corn-putation time. The EPT is useful for systems of small or moderate anisotropy. It should be excellent for treating small-angle scattering. Here-the class&l limit to the orbital angular momentum should be used sir& 0(r) changes by 180°.during the collision.. .. ; The E&“l!is one member of a large &a of per&_ bation theories which express the s&e& ma&in -. ~-
R..L CtossfExponentiaZperturbation theoriez for inehmicsmtreting
248
obtained
the general form
Sji = exP(Noi)YA);,-editloj) ,
W
where A is an appropriate perturbation series and f is some functional form. The simpIest case is the distorted-wave series where f(A) = A. The exponential case, [eq- (20)], uses a more complicated f and evidently converges faster. A third alternative was obtained in deriving the higher-order terms in Aal (see appendix A), ~~~=eXp(i?&)[(l
+in)(t
- in)-*]iiexp(i_rlo$,
(82)
where R is given by (25). The a matrix is similar to the conventional reactance or X-matrix except that it does not include the eiastic part of the phase shiftLike the exponentiaI method, S is unitary if R is hermitian. From (24) and (25) it appears that calculating R may be simpler than calcuIating An but this does not imply that (82) is more or less accurate than (20). The best possibk test of these approximations is to do accurate calcularions of S for several types of potentiaIs and compare the two approximations as a function of asymmetry. There are several ways that the EPT can be used in conjunction with a more accurate method such as the CC or CS method. The most obvious is to use the CC or CS method at small I and the EPT at large 1. One of the probIems in doing a CC or CS caiculation is that the scattering matrix oscillates rapidIy as a function of I so that one must repeat the calculation at many different Z’s_The modified S-matrix Sr e = exp(-i~o;)~~
exp(-in,-,$
= [exp(2iArl)lV, 033)
is a much smoother function of Z so that fewer values need to be calculated: the intermediate values can be
by interpolating St_ By diagonalizing S, one can obtain A11which is usually a slowly varying function of Zand E. There is a possible probIem in that the eigenvalues of S1 determine the eigenvalues of Aq onlyto within a multiple of IT.If the EET is also used to obtain Aq. this ambiguity can be removed. Going further, we can use the EF’T to obtain the matrix b= &cc
- Atl~p~ .
w
Even if the EPT is only crudely accurate, S will be
small and a slowly varying function of I; 6 will rapidly go to zero at large I where the EPT becomes more accurate. One can then get Sfj = exp(@gi) [exp(2iaqEpT
+ 2iC)lv exp@&
(85)
which is the exact S-matrix_ High accuracy can be ob-
tained by calculating 6 at only a few Z’sand interpolating to obtain the rest. The advantage of this method is increased if the calculations are done at several energies, since one can then interpolate in both E and 1. As derived here, the EPT is clearly related to timedependent theories such as the Magnus approximation [20] _The time-dependent formulation has a computational advantage in that (59), (61), and (63) are more easily evaluated than (22) and (23), sincethere are no Airy functions to calculate and since the integrands do not oscillate as rapidly. However, the approximations inherent in the time-dependent methods may degrade the accuracy_ The time-dependent methods all refer to a single classical trajectory, which implies a single energy. In contrast, the general theory uses a separate energy and trajectory for each quanturn state- Unless the energy defect is small, a large part of the transition probability is determined by the region near the classical turning points where the time-dependent formalism is least accurate.
Appendix A: Infinite-order expansion of Aq From (19) we have S, = (I + E + iF)(l + E - iF)-l
= exp(2iAqj.
(A-1)
Lkfme Q such that S-&(l+E)=F.
(A-2)
Then St= (I + in) (I - X&)-t = 2(1- ifi)-l
- I.
w-3
RJ. CkossfExponentiniperturbntion t&o&
249
for inelastic scattering
A comparison of the power-series expansion of (A-3) and exp(2iAq) gives (24). A quick and dirty proof isto solve (A-3) for R in terms of S1, Q = -i&r
- I) (St i 1)-r = -i[exp(2iA)
Aa = arctanQ = &I- $Z3 + @i
-I ] [ex&iiq)
(A-4)
+ I] -I= tan Aq ;
- ... .
(A-9
Some of these operations are questionable for matrices. If Q, is the nth prder tenrr in C&,then (A-2) gives Q, = F, -&I,E,_,
-Q2En_2
---XL&
(A-6)
_
From (9) we have En = i [A@‘) E,.-r(r’) + D(r’) Fn-t@‘)l e’ , 0
(A.7a)
F, = j C@(g) %_r(r’) •r-D(r’) F&r’)] 0
(A-7b)
*’ ,
and E. = I, Fo= 0. Note that (A-7) gives (21) directly. We wish to prove by induction that
n,(r)
=j
[-q_tA
+ DRn_r -BrBl-&_,
--h2Ban_3
- ..- -ZL,J%fir]
dr ,
(A-8)
0
where n,=Jcdr. 0
(A-9)
Using (A.7b) and (A-9) we have, + DF,] dr=fh,~,
Fn+L = j [n;E, 0 =fi,En
+j
]+AE,_r
+ j [-f21~E,_, 0
-.(t,BF,_,
+ Da,
-PIBF,_,
i- DfirE,+r
+ DF,] dr
f ... + D%_,E,]
(A_&
(A.11)
dr ,
0 where the last equation was obtained by using (A.6) for F,. Substituting (A.1 1) into (A.6) gives, an+r = 1 [Dn, 0
-a,DF,_,
-BtAE,_,
+ Dfi,E,_,
+ ...+ Dfk,_,E,]
dr -SQE,_,
- ___-S&El
_
(A-12)
Next, we note that,
sJ-
[+A+
Df$] E+t
0
do.= j “;E,+r 0
dr
tn+E,_2 +fi,BF,_,l dr -
(A.13)
(A.14)
0 -We continue the iteration in this fashion. At each.step we use (A-6) to give Fjrr hi terms of n,
and lower-order
RJ. CkqsfErponentid pertuibationtheoriesforinelaitic.scattering
Referencesr _._
:
i
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