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On the sudden approximation of cross: computational tests and factorization of cross sections and related scattering phenomena
Chemical Physics 47 (1980) 195-208 Q North-Holland,Publishing Company
ON THE SUDDEN APPROXIMATION OF CROSS: COMPUTATIONAL TESTS AND FACTORIZATION OF CROSS SECTIONS AND RELATED SCATTERING PHENOMENA’ D.E. FIT-2 and D.J. KOURI* Departmenr
Received
o(Chemirtry,
14 September
Revised manuscript
tinicrrsity of Houston, Houston, TX 77004. USA
1979
received
23 October
1979
A sudden approximation recently derived by Cross using a semiclassical treatment of the orbital motion is recast into a form which permits factorization of differential and integral degeneracy averaged cross sections, opacities as a function of fmal anaular momentum quantum number, the scattering amplitude, and the phenomenologkd cross section which describes spectral line broadening. Calculations are done using an average of initial and tinal orbital aneular momentum quantum numbers for the partial wave parameter for Ar-N2, Ar-TlF, H+-H, and Lie-Hz. The results indicate that the method is a good approximation for integral cross sections and opacities when the energy sudden approximation is v*alid and when the coupling of the orbital motion is important.
I. Introduction A variety of approximate quantum mechanical procedures have been developed in recent years to treat inelastic scattering in atom-diatom systems and more complicated systems. These approximate procedures have the advantage over the numericaily “exact” procedures in the sense that they significantly reduce computation time and often give reasonably accurate results for detailed scattering phenomena_ As a result of this, many phenomena such as m-dependent rotationally inelastic scattering, pressure broadening of rotational lines in gases, NMR spin-lattice relaxation, and thermal diffusion, which until recently were impractical to treat, can now be examined in detail and the processes which produce these phenomena can be more clearly understood. One direction in which approximate quantum methods have moved involves decoupling of the orbital and/or rotational motion of the system in order to reduce the number of coupled equations which must simultaneously be solved; hence the reduction in computation time. Of these approximations the centrifugal sudden (CS) approximation [l, 2]* has been quite successful in treating scattering in systems where changes in rotational energy compared to total energy may be significant and the inelastic transitions are dominated by the strong short ranged anisotropy. The infinite order sudden (10s) method [44]‘has been shown to give reasonable results when both the energy differences are small and the long range orbital coupling is weak. Furthermore, it is about an order of magnitude faster than the CS. Another method, the energy sudden (ES) [7]$ has been shown to be good when the orbital coupling plays a significant role but the changes in rotatio+ Research
supported by National Science Foundation Grant CHE-77-22911 of Houston. * J.S. Guggenheim Foundation Fellow, 1978-79. * The literature on the CS is quite large. A review containing many references * See also ref. [3] for extensive references to the 10s. i+ Other references are also given in ref. [3].
and by the Computer
on this subject
Center
is ref. [3]_
of the University
196
D.E. Fir-_ rir~llD.J. Korwi/On rhe sudden approxbnution
of Cross
nal energy are small compared to the total energy. This procedure is often not the most practical to apply, though, particularly for large orbital angular momenta where it may become more time consuming to use than the CS and sometimes the CC methods. Because ol the great simplicity of the IOS, it is very desirable to develop an approximation of similar computational requirements but which could be corrected for either centrifugal sudden or energy sudden breakdown and still give reliable results. Recently, an approximation which is addressed to the first of these two cases has been reported by Cross [S]. For convenience we will refer to this method by the abbreviation (SA). This approximation is also attractive in that it cztn be cast into a form which allows for factorization [9] of various scattering quantities as do the 10s and ES [7]_ Th e same thing appears possible with two related approximations [IO]. Basically. the SA is an alternate way to do an ES calculation when the ES becomes too unwieldly to apply The number of systems for which this approximation can be adequately tested is limited, though, due to the large number of rotational states accessible when the SA (and ES) is applicable. In this paper we will test the SA method fo{ Ar-TIF (an ideal system for this method) as well as for Ar-N, and the Iess ideal systems H+-HZ and Lie-HI_ Furthermore. we shall show how the SA method may be used to fztor the m-dependent rotational scattering amplitude. the degeneracy averaged rotational integral and differential cross sections, the opacity as a function of final orbital angular momentum quantum number (as opposed to total angular momentum quantum number which does not lead to factorization), and the phenomenological cross section for describing line broadening and general relaxation phenomena.
2. Formalism
In this paper the following notation will be used to describe the atom-diatom scattering process: j, 1, and _I are the rotational, orbital, and total angular momentum quantum numbers; primed and unprimed variables represent post- and precollision quantities; f; is the initial wavenumber for thej = 0 state; 122is the projection of ion the space-hxed c-axis which is parallel to X-(not to be confused with the z-axis shown in fig. 1); r= (I + l’)/2 is the I-averaged sudden partial wave parameter: Q = (0,0) are the angles of the vector R describing the rotor and are measured with respect to the c-axis used in Cross’s SA [S]; the z-axis is shown in fig. 1 and is perpendicular to k; 0 is the class&t1 orbital angle determined by the spherically symmetric part, V,(r), of the interaction potential, V(r, p’) and equals zero at the turning point F~:o;r is the distance from the c.m. of the
diatom to the atom: ,!Yis the angle between
Fig I.
Axis
r
and R; and .Kis the reduced mass of the atom-diatom
system and description of variables used in SA.
system.
D.E. Fitr and D.J..Kc+/~n
197
the sudden approximarion o/Crosk
The derivation of Cross’s SA is given in suflicient detail in ref. [S] and will not be repeated only the necessary equations needed to use this method will be given. In order to understand these equations it is necessary to recall that the SA allows the coupled equations in j and I to by a simple transformation. This transformation results in an uncoupled differential equation parametrically on the orientation of the rotor in space, R:
here. Instead,the significance be diagonalized which depends
of
(1) The solution to this equation for fixed R is given by u(r, Q). The orbital diatom c.m. is taken into account through p’ which is defined by cosfl,
motion
of the atom relative to the
(2)
= cos [O r O(r)] sin 0.
This relation implies that for any given value of R there are two branches branch for example, the asymptotic solution is zl(r, 52) .F_ exp [ -i(lir where the solution
to eq. (1). If one chases the ” f”
- fxT)] - S:(Q) exp [i(kr - &J)],
to 2?+(Q) is expressed
(3)
in terms of a phase shift according
to
ST+(Q) = exp [ZrlT,(n)].
(4)
We follow Cross‘s suggestion [a] for avoiding the double average of the phase shifts for the two branches:
branch
difficulty
by defining
.ST(Q)in terms of the
ST(Q) = exp {i[rr+(Q) + II’_(Q)]).
(5)
In Cross‘s approach p+ is r-dependent and this dependence is determined by eq. (9), where 0 is found as a function of r by solving eq. (1) with V(r, 8’. Q) replaced by the spherically symmetric part of the potential, V,(r). The phase shift associated with this equation has only a single branch and is represented by rr,(fi. The trajectory which determines 0 is traced out by rO in fig. 1 for the “t” branch. Using these definitions and modifying the phase of the scattering amplitude in eq. (3) so that the asymptotic form of the total wavefunction yields a result for the scattering amplitude as given by Arthurs and Dalgamo [9] [see eqs. (21) and (x)] the S-matrix is S;.l.:jr = i’+‘.-‘rexp
{i[rl,JQ + rl&)
- 2r),(r)] 1
~~‘-“-(n)Sr(n)~‘-‘(n)d~~
(6)
s where ST(Q) is given by eq. (5). For our purposes phase shifts
where rs is’rhe turning defined by
it s&ices
to use the WKB solution
point (at which point the term in brackets
to eq. (1) to obtain the
equals zero). The classical
orbital
angle is
D.E. Fir= and D.J. Kouri/On rhr sudden approximation
198
of Cross
where r,, is the turning point for the effective potential using I’,(r). Because the SA is a Iarge angular approximation, it is convenient to note that in the limit oflarge 1 exp :i[~&
momentum
+ Ml’) - 2s,(4]1 -r-z, 1,
(9)
and we shall use eq. (9) in cq. (6) in all of the calculations in this paper. For degeneracy averaged cross sections this approximate phase is irrelevant. For differential cross sections and Ill-dependent integral cross sections this approximation may be important. We are now ready to cast the S-matrix into a form more amenable to factorization. First, we expand S$2) in terms ofspherical harmonics according to S’(Q) = (4ic)“’ 1 YP(Q)S:2,,. LI, If eqs. (9) and (IO) are substituted
where the summation over .I! Unfortunately, this expression part of the second 3-j symbol. angular momentum limit ofa
L 1-I’ the coupled
(10)
into eq. (6) the integral over Q can be done analytically
[12] to give
in eq. (10) has been done using conservation properties of the 3-j symbols. is not easy to manipulate because of the appearance of!, I’and J in the lower This situation can be rectified by recalling that this 3-j symbol is the large 6-i symbol. Using a symmetrized form of this relationship+
j
(12)
.*
J-l
basis S-matrix
can be given in the more useful form
(13) which will be used in the remainder
The relationship
of this paper.
between the SA and IOS S-matrices
can be obtained
easily by starting
with the inverse of
eq. (10) .S’,.,_,. = (47i)- I”
Y;-“-(Q.S’(Q)dn. s
(14)
In the IOS approximation the angle ,8’does not vary with r but instead is fixed at a value p = 7 defined by D = 0. The two dimensional integral in eq. (14) over (0, (D)QR be reduced to a one dimensional integral over 7 if one notes tha: the spherical harmonic in R defined with respect to the X, Y, 2 axis system in fig. 2 can be defmed correspondingly in terms of spherical harmonics in (7, I$) defined with respect to the X’, I”, Z’ axis system by rotating through the Euler angles (0. -z/2,0) from the X’, Y’, Z’ axis system to the X, Y, Z axis * The ndvanta_ee of symmetrization is that one can permute the two larse angular momenta and still get the same result. We have shown chat this process _rives much better results than not to use it for a reIated case in ref. [13b] and the same holds true here.
D.E. Fitz and D.J: Kouri/On the sudden approxikacion of Cross
Fig. 2. Axis system and definition of variables used to reduce the SA to the 10s.
system as shown in fig_ 2. (fig. I becomes Y:-“‘(n)
= F Yj’(y, &D&-,.(0,
fig. 2 when 0 = 0). This rotation
(15) cn
Zir Dfl_,.(O,
-n/2,0)
1 The C#integration means that
by
-7T/2,0).
If we chose 0 = 0, S’(Q) in eq. (14) only depends .S’,.!_[. = (47~)~“‘~
is thus described
y
and can be written as
I
d4 s 0
sin ydy Yi-‘(7, +)Si(y).
(16)
s 0
can now be done and gives the product
of 27~and a Kronecker
&function
in i. and 0 which
. 7
s,.,-,.
L’
= D,.,-JO,
--x/2,0)
(2i
i- 1)“”
2
sin 7 dp ST(7)P,(cos y).
(17)
s
0
The D-matrix can be recognized as being the large anguIar written in symmetrized form [ 12, 13b] as %.I-,.(O, Furthermore,
-rr/2,0)
= (- I)![(21 + 1)(21’ + I,] l,,(,
in the iOS approximation
momentum
;
Iimit of the 3-j symbol which can be
;).
(18)
one defines
SrL = (2L + I) xsin p dy P L(cos y)S7(v). 2 s
(19)
0
Thus, substitution gives the factored
of eq. (18) into (17) using the definition IO.5 expression for the S-matrix [IO]
$.r-;jr = i’+“-2T(-
in (19) followed
by substitution
ofeq.
(17) into (13)
I)‘*‘+’ ‘[(2j + 1)(21’ + I)(21 + 1)(21’ + I)]“” (20)
except that here we employ
a general I rather than the final-l labelling
used ‘in ref. [lo].
D.E.Fir2 and D.I. KourifOn the sudden approximiltionoJCross
200
23. Fuctorization of the scatterirrg amplitude For convenience of presentation (and to facilitate actual computations) 3 function g&‘nf; jm) will be introduced which wit1 somewhat simplify the notation. The scattering amplitude quantized along the space fixed :-axis parallel to the incident asymptotic \vdve vector k is then 111 F y,.(Jaz’; jm) x?-m.(x),
f(i’zzz’+ jrzz lx) =
(21)
where
Substitution of eq. (13) into this expression gives a result which appears not to simplify at first glance. However, ifeither of the 3-j symbols in this expression is recombined with the 6-j symbol in eq. (13) to get a summation over three 3-j symbols. the summation over J can be performed using orthogonality relations of the 3-j symbok. The resulting expression allows the s-coefficient to be fdctored in the form
where hi = IN- niand
From this tktorization
reIationship
it follows directly that the scattering
amplitude
fzzctors according
to
in which the definition off{.!& + 00 1~) in terms ofeqs. (2if and (24) is obvious. Before going on to other expressions which factor, it is usefuf to give two expressions which do not factor but which are good starting points for some of the expressions which shall be given later. First, note that the definite polarization differential cross section is defined as
The zrz-dependent Glj’l~z’
t
jnz)
integral cross section is then given by =
dy * s
- dz
fj-rtz’
(27t:
2.4. Facrorkatiot~ of&e degetzeracy aceraged di~re}lt~ui cross secriotr The degeneracy cross section by
averaged
differential
cross section
is defined in terms of the definite polarization
differential
D.E. Fit= and D.J. Kouri/On
the sudden approxirnarion of Cross.
over 1)~and ~1’may be perFormed using 3-j symbol factored form of the differential cross section:
If eq. (26) is substituted into eq. (28), the summations orthogonaiity properties. The result is the following
where (30)
2.5. Factorization
of the
The degeneracy
integrul boss section nnd P-opacity
averaged
integral cross section
can be gotten either by integrating
eq. (29) over z or by (31)
The term inside the summation hence it follows that G~_~
=
be recognized
as the (21’+ I)-weighted
opacity
in I’;
5x(2’ + 1jP,.O_-J). J
P,.(j -j’)
over /’can immediately
(32)
1
can be shown
immediately
to Factor according
to (33)
where (34) It should be noted that no such relationship exists for the more familiar opacity as a function of J. In a similar manner to that above, the degeneracy averaged integral cross section can be shown to factor according to
where the cross sectlon
2.6. Factorization
o-o-L is obtained
from settingj
= 0 in eq. (32) and substituting
cq. (34) into eq. (32).
of phenorneno~ogicai cross sections
In spectral line broadening theory one speaks of a spectral transition i + f between initial and final spectroscopic states. The notation in this section is similar to that in the previous sections except that now all variables will have a subscript “i” or “f’ depending on whether they describe an initial or final spectroscopic state. Thk variabIe K in this section denotes the order of the spectral transition (K = 1 corresponds to pure rotational excitation or deexcitation via a permanent dipole moment interacting with the radiation field, for instance). The generalized phenomenological cross section which is used to describe such processes is [ 131 -’
D.E. firz und D.J. Korfri/Orr rhr szrddrn approsimurion of Cross
101
Once again, if the SA S-matrix expression in eq. (13) is substituted into this expression, can be collapsed by using well known properties olthe 6-j symbols [12]. The resultinS -give
the sums over Ji and Jr expression factors to
(37) where the dynamics t:r..ee
of the phenomena
= ;(-l)“(X 0
are contained
+ 1)-1i2~[sLos,!.
It shotrId be noted that the two S-coefficients
3. Computational
in
- S:..,_I.S~_,_I.]. are not necessarily
(38) evaluated
at the same energy.
procedures
The potentials for the four systems studied here. Ar-N,. Ar-TIF. H’-Hz and Lii-I-I2 will be described briefly to avoid any ambiguity concerning exactly which potentials were used in the calculations. The potentials for Ar-N, and Ar-TIF are Lennard-Jones 6-12 type potentials [14]: L’(i.7) = h:(rO/F)*2[I
+ ZSRP2(COS7)] - Z(r&)6[1
f zLRPI(COS7)]j_
(39)
Two different sets oFrrnisotropies are used for the Ar-NZ potentials so that comparisons cdn be made with the defnite polarization cross sections of Alexander [ 141. These two sets of anisotropies are denoted LR and SR for the stronger und weaker long range anisotropies, respectively. The parameters for both Ar-N2 potentials and the Ar-TlF potential are listed in table 1. The potential which we used for H &--Hz was based on a potential given by Giese and Gentry [ 1.51. Basical!y, Lx--efollowed McGuire-s procedure [16] and did a Legendre polynomial series expansion of their potential for r between 0.5 and 6.0 au. This expansion was truncated at Pi,. The coefficients were found by using a standard 32-point Gauss-Legendre integration of the inversion integral and were tabulated at intervals of 0.1 au. Table 1 Porenthl parameters for Ar-Nza’ and Ar-TIFb’ ----_-System r&W
c(meV)
Ar+ (SR) Ar-N, (LR j Ar-TlF
IO.296 IO.‘96 11.376
I’ ReT.[ 143. ” ReT.[jb].
7.424 7.121 9.800
0.50 0.315 0.50
0.13 0.315 0.30
D.E. Fitz and D.J. Kouri/On the sudden approximation of Cross The Ve coefficient near 6.0 au can be fit to a quadratic potential for this system (in au)
and shown to fit smoothly
Vb_“(r) = -2.6091r-4, but the VZcoefficients
to the long range form bf the
(40)
do not match smoothly
VZLR(r)= -0.60735r-’
203
+ 0.45886r-3.
with the long range form (41)
To remedy this problem we did a cubic spline lit between 4.9 and 6.0 au to get V, and its first derivative smoothly joined in this region. Likewise, a cubic spline tit was done for V’ between 4.7 and 6.0 au with V, and its first derivative set equal to zero at P = 6.0 au. Furthermore, we set Ve = 0 for r 1 3.9 au, Ve = 0 for r 2 4.4 au, and Vi,, = 0 for r L 4.9 au. The new potential was then retabulated at intervals of 0.1 au for r between 0.5 and 6.0 au. Values of the potential needed with this range were obtained via a 3-point Lagrange of each set interpolation_ For r > 6.0 au, eqs. (40) and (41) were used. For r c 0.5 au the functional dependence A exp (- Er), where A and B were determined by setting this of expansion coefficients was fit to an exponential expression equal to the values of the coefficients at r = 0.5 and r = 0.6 au. The Li+-Ha potential has been given in adequate detail by Lester and Schaefer [17]. We have used the rigid rotor form of their expression. 3.2. Nrmerical
techniqtces
The most time consuming part of the calculations using the SA is the calculation of the S-coefficients defined in eq. (14). A reasonably accurate and efficient numerical procedure for evaluating the integral in this expression is to use a Gauss-Legendre method to do the 0 integration and a spherical product procedure [18] for the Q integration. In our calculations no values of L > 8 were ever needed. Because of this and because ST(Q) generally had no more than two oscillations in the range of integration, a 16-point integration in 0, range O-n, and a 32-point integration in Q, range O-28, were found to give at least three figure accuracy. At first glance there appear to be 512 points for which S’(Q) and hence $(Q) must be evaluated. Since thesystems studied here are homonuclear, symmetry considerations immediately reduce the integration ranges to 0 over 0+2 and @ over O--x, leaving only 128 points to be evaluated. The use oTa spherically symmetric reference potential (se? fi_e. 1) allows a further symmetry reduction in @ to be achieved leaving on 64 unique points. However, the delinition of the phase shift in eq. (5) in terms of two branches doubles the required number of phase shifts needed so that 128 separate phase shifts must be calculated. This number of points was used in all of the calculations done here. In order to calculate I?(Q) one first must obtain the orbital angle O(r) in a suitable tabular form. This was done by splitting eq. (8) into three intervals vc,, r,), (r,, r2) and (rz, 03). It is understood that for r I To, Ok) = 0. For a given system at a particular value of L one first determines the turning point of the effective potential using V, and utilizing a Newton method [19]_ All turning points were calculated to at least six figure accuracy lor all systems. Then the interval CF,, ri) was split into four equal subintervals of width 0.1 au each. Within each subinterval the integral was calculated analytically after fitting the term within square brackets in eq. (8) to a quadratic. This technique accurately integrates out the singularity at P,. The integral over the second interval (rlr rJ was done by Simpson’s rule using a step size of 0.1 au. The value of rz was set to about 20 au for Ar-N, and Ar-TIF and about 40 au for H’-Hz and Li+-H,. For the integral over the third interval, the potential was set equal to zero allowing the integration to be performed analytically_ This combined procedure gives B(r) in tabular form in the interval Fe, r2) and in analytic form elsewhere. Values of e(r) within the interval cfo, r2) are obtained using a 3-point Lagrange interpolation [ 191. Using O(r), #(Q) can now be calculated. For a lixed value of fand 0 = (0, O), one calcu!ates the turning point for the full potential using Newton’s method. Once the turning point is known, an efficient Gauss-Mehler
D.E. Fir= md
204
D.J. Kouri/On
rhe sudden approxinmion
oJCross
quadrature c2.203 requiring onIy 8 points for Ar-N, and Ar-TIF and 24 points for H’--H, and Lit-Hz, was used to evaluate eq. (7). The energies for which our calculations were done required that the S-caehicients be calculated for i= O-150 for Ar-N2, i= C-200 for Lie-Hz, i= 30-300 for H’-Hz and i = O-370 for Ar-TIF. Using a Honeywell 6660, the computation times to tabulate all cf these coeflicients in double precision for L = 0.2.4, 6,s and 121= - 8 to 8 were 0.28 hours for Ar-N2, 0.53 hours for Li+-Hz, 0.70 hours for Ar-TlF, and 0.92 hours for H +-Hz. The calculations of In-dependent cross sections from these S-coeffkients generally required about 0.01 to 0.02 hours for all of the permissible transitions.
4. Results and discussion Of the four systems studied here, the system for which the SA is best suited is Ar-TIF. The energy at which we did calculations for this system w;1s 0.1154 eV. The energy differences between low lying rotational states are quite small for this system and one expects the ES approximation to hold. On the other hand. most of the contribution to scattering between the low lying states comes at large values of the orbital angular momentum quantum number: hence one might expect a breakdown of the CS approximation. This is exactly what the results show. Throughout this section it is understood that all 10s and SA calculations were done using the Secrest choice [?I] i= (I t l’)jZ. Furthermore, the SA and 10s opncities which we give are defined with respect to P according to eq. (27) and are weighted by (?I’ l I) whereas the CC opacities are delined in the more usual manner with respect to J, hence the labelling on the figures. For these systems and the transitions studied, an insignilicant amount oferror is introduced by making this comparison [22]. In figs. 3 and 4 the r-weighted IOS and SA opacities are compared with the J-weighted CC opacities of Pack [5b] for the 0 + 2 and 0 -+ 4 transitions, respectively. One can readily see that the SA results are quite good while the IOS results ovcrestimatc the opacities at large P. Although not shown, the ES results of Khare [7] for this system lie almost directly on top of the SA results implying that the SA is a good approximation to the ES. In table 2 comparison is made between the IOS and SA tn-dependent cross sections for Ar-TlF at E = 0.1153 eV. In iable 3 the CC_ S-4 nnd IOS integral rotational cross sections are compared for this system. Unfortunately. no CC results yet exist for the rn-dependent cross sections for this system. Our results are
300 -
Ar- TIF
II- .\.
fit 3. Comparison oTCC. SA and IOS weighted opacities for Ar-TIF at E = 0.1154 eV f:lr rhe 0 4 2 transition.
Fig. 4. Comparison of CC, SA and 10s weighted opacities for Ar-TIF at E = 0.1154 eV for the 0 * 4 transition.
D.E. Firz and D.J. KourilOn Table 2 Comparison oiSA and IOS m-dependent Ar-TlFat E = 0.1154eV
jnr-+Jm’
cross sections
for
the sudden approximation
Table 3 Comparison of CC, SA and IOS integral Ar-TlFatE=O.l154eV
j-j
u~m’ + jm)
205
of Cross
cross sections
for
U(j Ai’) -
SA
IOS
9.45 2.21 21.81 0.85 1.72 1.13 1.77 3.69
14.55 2.50 22.73 2.92 1.67 3.39 1.94 6.00
Cc?
SA
IOS
58.9 16.4 ___-
57.6 17.5
65.0 28.9
00 + 20 00 421 00 -t 22 00+40 00+41 00 -+ 42 oO-r43 00+44
0+2 O-4 =’ Ref. C5b-J.
truncated at r = 370 in order to make a comparison with the CC results. At this point the 0 -+ 4 cross sections are completely converged whereas only the cross sections for which 111= odd are converged for 0 + 2. The difference in the ))I-dependent results for the IOS and SA shows that the way in which the orbital coupling is treated has a significant effect on the rn-dependent cross sections. Furthermore, such effects are quite likely to show up in the calculation of cross sections needed to calculate shear viscosities and viscomagnetic effects in other atom-diatom systems [23]. All calculations for Ar-N, were done at a total energy of E = 0.0258 eV ( zz 300 K) and all cross sections given were converged at r = 150. Calculations for this system were done with two different potentials Iabelled LR (for the one with the stronger long range anisotropy) and SR, whose parameters are given in table 1. In tables 4-6 the SA and IOS m-dependent cross sections for both potentials are compared with the CC results of Alexander [14]. In table 7 the same comparison is made for the integral rotationally inelastic cross sections. It should be noted here that not all of the CC cross sections in table 7 were taken from the same calculations and hence may not be totally consistent. For the SR potential it is clear that both the 10s and SA approximations are good ecenfir rhe rn-dependetzf cross sections. Such is not the case for the LR potential, where both the IOS and SA fail to &e the correct trend in the polarization transitions. In figs. 5 and 6 the I-weighted SP. and 10s opacities are compared with the J-weighted CC opacities of Alexander [24] for the 0 + 2 and 0 - 4
Table 4 Comparison of IOS, SA and CC m-dependent cross sections (in A’) for the 0 + 2 rotational Ar-NZ (SR and LR potentials) m’
G(h’
t
I 2
integral transition
Ar-N2
ni
in
(SR and LR potentials) G(4m’ + 00)
SR LR
CC”
SA
10s
CC”
SA
10s
4.2 1 2.79 6.48
3.92 1.90 6.62
3.79 2.15 7.13
IS.6 12.6 9.67
9.95 7.54 15.89
8.84 7.13 11.45
z’ Ref. [ 141.
integral rransition
in
00)
SR
0
Table 5 Comparison of 10s. SA and CC m-dqxndent cross sections (in AZ) for the 0 -t 4 rorarional
0 1 2 3 4
LR
-
CC“
SA
ios
CC=’
SA
10s
2.86 2.89 2.95 2.95 I -7-i
2.05 2.10
1.94 2.03
3.17 2.57
2.25
2.15
1.46
2.50 2.0 1
2.48 1.91
0.57 0.19
0.70 0.87 1.11 2.19 4.76
1.70 274 1.94 3.45 4.69
z) Ref. [14].
D.E. Firr and D.J. Kouri/On the sudden approximation of Cross
206
~~~
~~~~
0 0
J
100
50 J
Fig. 6. Comparison of CC, SA and 10s weighted opacities at E = 0.0258 eV for the 0 + 4 transition using the LR potential.
Fig 5. Comparison of CC, SA and IOS weighted opacities for Ar-N, at E = 0.0258 eV for the 0 -t 2 transition using rhr LR poteniiol.
for Ar-NI
lransitions in Ar-N2 using the LR potential. Although neither the IOS nor the S.4 methods are able to completely describe this system. the SA method is nonetheless a considerable improvement over the 10s method. The problem with the SA appears lo be caused by a breakdown of the energy sudden approximation due to the long range part of the potential. Another system for.which an IOS calculation has been done [25] but is expected to break down is H+-HL at E = 3.7 eV. In fig. 7 the r-weighted SA and IOS opacities are compared with the J-weighted CC opacities of McGuire and Schinke [25] for the 0 --t 2 transition. It should be noted that our 10s opacities do not agree exactly with [heir 10s results (not shown) and this may be due to a difference in matching procedures used to joint the tabulated form of the potential to the long range part. Despite this difiicully, the SA results show a marked improvement over the 10s results. The system with the strongest long range anisotropy studied in this paper is Lie-Hz at E = 1.2 eV, an energy for which the energy sudden approximation is expected to fail. In figs. 8 and 9 the r-weighted SA and 10s opacities are compared to the J-weighted CC opacities of Schaefer and Lester [26] for the 0 + 2 and 0 -+ 4 transitions. Although the SA method still overestimates the opacities at large I’, it is nonetheless a signifinnt Table 6 Comparison of IOS, SA and CC rn-dcpendenr 1LR potential) at E = 0.020158 eV
integral
cross
sections (in AI)
for the 2 - 4 rotational transition in Ar-NI
c(4ni + 2m)
III’
01 =
-2
,,I
=
-
m=O
1
-
--
-.-4 -3 -1 -1 0 1 1 3 1
cc=’
SA
Lo6
2.15
2.02 0.95 0.47 0.30 0.30 0.M) 0.95
5.41 329 i .oa 0.54
0.13
=’Ref. [14-j.
10s
cc=’
-
SA
10s
CC”
SA
10s 3.08 0.39 4.39 5.96
0.65 CL50
15.70 5.34 3.18 1.75 1.20 1.31 1.11 1.72
0.22 1.59 3.35 2.41 0.82 1.16 0.95 0.22
! .07 13.45 6.68 6.24 2.61 1.41 1.11 2.77
I .74 9.69 7.34 5.86 299 1.83 2.59 3.09
0.39 0.82 0.56 2.32 4.30 2.32 0.56 o.sz
3.04 0.1 I 5.54 4.85 7.56 4.85 5.54 0.1 i
5.96 4.39 0.39
1.40
2.82
0.09
0.25
0.86
0.39
3.04
3.08
0.69
8.55
D.E. Firz and D.J. KourifOn the sudden appro.Gmtion
Table 7 Comparison of CC, IOS and SA integral inelastic cross sections for Ar-N2 at E = 0.0258 eV for both the SR and LR potentials j-+f
100 H+- H2 o-2
o(j+j) SR
O-2 O-4 O-6 244
207
of Cross
LR
cc
IOS
SA
CC
10s
SA
22.5”’ 22.4”’ 114C’ 24.gb’
22.3 19.1 20.4
21.0 19.8
61b’ 13b’
46.0 27.3
56.8 18.6 3.6 I 35.2 -
” Ref. [5b].
O.OP
12.9
20.0
b, Ref. [14].
9.P
-
35.3
5 Fig. 7. Comparison of CC, SA and 10s weighted opacities for H--H, at E = 3.7 eV for the 0 + 2 transition.
c, Ref. [27].
improvement over the IOS method. Furthermore, this system shows a phenomenon not seen in the other results. Assume one defined both a J-scaling factor and a magnitude scaling factor such that the position and magnitude of the maximum in the IOS opacity matched that of the SA opacity for the 0 + 2 transition of Lii-Hz shown in fig. 8. If one applied this position and magnitude scaling to both the rest of the 0 --) 2 opxity curve in Cg. 8 and to the whole 0 -+ 4 opacity curve in fig. 9, one would tind remarkably good agreement between the IOS and SA results for this system. Two problems arise in applying the SA which may severely limit its applicability for certain systems. The first problem occurred at i = 15 for the H+--H, system. Because of the extremely deep well in the spherically symmetric part of the potential, the orbital angle O(r) varies quite rapidly near the turning point due to a near orbiting situation. As a result of this rapid change, the angle p’+ changes sufficiently rapidly that I+‘, pi, Q) + h’(T + $)‘/(2Rr’*) becomes greater than the energy, at which p&t eq. (7) no longer applies. Fortunately, for H’-H, the contributions to the low-lying transitions coming from small values of ido not contribute significantly to the cross section and can be neglected. Hence, we started our calculation at i = 30. There may, however, be systems for which this solution simply will not work. In this region of orbital angular momentum, though, the IOS approximation itself would probably be sufficiently accurate and could be used instead of the SA approximation. 150 Lie-H2 I
.-._
100 l-i+-HZ
Fig. 8. Comparison of CC, SA and IO3 weighted opacities for Li+-Hz at E = 1.2 eV for the 0 -t 2 transition.
Fig. 9. Comparison of CC, SA and 10s weighted opacities for Li’-H, at E = 1.2 eV for the 0 -+ 4 transition.
D.E. Fir,- utrd D.J. Kouri/On
208
rhe sudden upproXinltXi0n
Of cross
The second problem is that for Lii-H, the long range anisotropy is sufliciently strong that the collision can no Ionger be considered as “sudden” and that the spherically symmetric orbital anglk also inadequately corrects for the I-coupling. Work is currently underway to correct the energy sudden defect. Furthermore, work is currently being done in this laboratory [ZS] to correct for the orbital coupling using orbital angles calcu!ated from the whole potential, not just the spherically symmetric part. Despite these problems, the SA method should be very useful in treating a variety of systems where the IOS method fails due to difliculties with the CS approximation. We are carrying out additional test cltlculations of differentia! cross sections and magnetic transition cross sections and will report the results at a later date. We are also carrying out calculations to assess the error introduced in differential and m-dependent cross sections by the use of the asymptotic phase as in eq. (9).
Acknowledgement We would especially like to thank M. AIexander for sending us his unpuclished which appear in figs. 5 and 6. Very helpful comments by the referees are gratefully
CC opacities for Ar-N, acknowledged.
References and DJ. Kouri. J. Chem. Phys. 60 (1974) 248% R. T Pack. J. Chem. Phys. 60 (1974) 633. D. Kouri, in: Atom-molecule collision theory: a guide for the experimentalist. ed. R.B. Bernstein (Plenum, 1979). C.F. Curtiss. J. Chem. Phys. 48 (1968) 1725; 49 (1968) 1952. (a) T.P. Tsien and R. T Pack, Chem. Phys. Letters 6 (1970) 54; (b) G.A. Parker and R. T Pack. J. Chem. Phys. 59 (1973) 5373. D. Secrest, J. Chem. Phys. 62 (19753 710. V. Khare. J. Chem. Phys. 68 (1978) 4631. RJ. Cross Jr., J. Chem. Phys. 69 (1978) 4495. A.M. Arthurs and A. Da&no. Proc. Roy. Sot. London A256 (1960) 540. R. Coldflam. S. Green and DJ. Kouri. J. Chem. Phys. 67 (1977) 4149: R. Goldflnm, D-l. Kouri and S. Green, J. Chem. Phys. 67 (1977) 5661. AS. Dickinson and D. Richards, J. Phys. B11 (19783 1085.3513; Y.B. Band. J. Chem. Phys. 70 (1979) 4. and orivate communicntion. AR. Edmonds. An.&& momentum in quantum mechanics (Princeton Univ. Press, Princeton. 1960). (a) R. Shafer and R.G. Gordon, J. Chem. Phys. 58 (1973) 5422; (b) D.E. Fitz and R.A. Marcus, J. Chem. Phys. 62 (1975) 3788. M. Alexander, J. Chem Phys. 67 (1977) 2703. C.F. Giese and W.R. Gentry, Phys. Rev. A10 (1974) 2156. P. McGuire, J. Chem. Phys. 65 (1976) 3275. W.A. Lester Jr. nnd J. Schaefer, J. Chem. Phys. 59 (1973) 3676. of multiple integrals (Prentice-Hall, Engtewood Cliffs, 1971) p. 40. A.H. Srroud. Approximnre calculations F-S. Acton, Numericai methods that (usually) work (Harper and Row, New York, 1970). 2. Kopal. Numerical analysis (Wiley. New York. 1955) p. 383. D. Strrest. unpublished research. Y. Shimoni and D-l. Kouri, J. Chem. Phys. 65 (1976) 3372.3958; 66 (1977) 2841; G.A. Parker and R. T Pack, J. Chem. Phys. 66 (1977) 2850. W.-K. Liu. ER. McCourt. DE Fitz and DJ. Kouri, J. Chem. Phys. 71 (1979) 415; W.-K. Liu. F.R. McCourt and W.E. Kshler, 5. Chem. Phys. 71 (1979) 2566. M. Alexander, private communication. R. Schinke and P. McGuire, Chem. Phys. 28 (1978) 129. J. Schaefer and W.A. Lesster Jr, J. Chem. Phys. 62 (1975) 1913. R. T Pack. J. Chem. Phys. 62 (1975) 3 143. V. Khare and D. Kouri, in preparation.
P. XlcGuire
New York,
ON THE SUDDEN APPROXIMATION OF CROSS: COMPUTATIONAL TESTS AND FACTORIZATION OF CROSS SECTIONS AND RELATED SCATTERING PHENOMENA’ D.E. FIT-2 and D.J. KOURI* Departmenr
Received
o(Chemirtry,
14 September
Revised manuscript
tinicrrsity of Houston, Houston, TX 77004. USA
1979
received
23 October
1979
A sudden approximation recently derived by Cross using a semiclassical treatment of the orbital motion is recast into a form which permits factorization of differential and integral degeneracy averaged cross sections, opacities as a function of fmal anaular momentum quantum number, the scattering amplitude, and the phenomenologkd cross section which describes spectral line broadening. Calculations are done using an average of initial and tinal orbital aneular momentum quantum numbers for the partial wave parameter for Ar-N2, Ar-TlF, H+-H, and Lie-Hz. The results indicate that the method is a good approximation for integral cross sections and opacities when the energy sudden approximation is v*alid and when the coupling of the orbital motion is important.
I. Introduction A variety of approximate quantum mechanical procedures have been developed in recent years to treat inelastic scattering in atom-diatom systems and more complicated systems. These approximate procedures have the advantage over the numericaily “exact” procedures in the sense that they significantly reduce computation time and often give reasonably accurate results for detailed scattering phenomena_ As a result of this, many phenomena such as m-dependent rotationally inelastic scattering, pressure broadening of rotational lines in gases, NMR spin-lattice relaxation, and thermal diffusion, which until recently were impractical to treat, can now be examined in detail and the processes which produce these phenomena can be more clearly understood. One direction in which approximate quantum methods have moved involves decoupling of the orbital and/or rotational motion of the system in order to reduce the number of coupled equations which must simultaneously be solved; hence the reduction in computation time. Of these approximations the centrifugal sudden (CS) approximation [l, 2]* has been quite successful in treating scattering in systems where changes in rotational energy compared to total energy may be significant and the inelastic transitions are dominated by the strong short ranged anisotropy. The infinite order sudden (10s) method [44]‘has been shown to give reasonable results when both the energy differences are small and the long range orbital coupling is weak. Furthermore, it is about an order of magnitude faster than the CS. Another method, the energy sudden (ES) [7]$ has been shown to be good when the orbital coupling plays a significant role but the changes in rotatio+ Research
supported by National Science Foundation Grant CHE-77-22911 of Houston. * J.S. Guggenheim Foundation Fellow, 1978-79. * The literature on the CS is quite large. A review containing many references * See also ref. [3] for extensive references to the 10s. i+ Other references are also given in ref. [3].
and by the Computer
on this subject
Center
is ref. [3]_
of the University
196
D.E. Fir-_ rir~llD.J. Korwi/On rhe sudden approxbnution
of Cross
nal energy are small compared to the total energy. This procedure is often not the most practical to apply, though, particularly for large orbital angular momenta where it may become more time consuming to use than the CS and sometimes the CC methods. Because ol the great simplicity of the IOS, it is very desirable to develop an approximation of similar computational requirements but which could be corrected for either centrifugal sudden or energy sudden breakdown and still give reliable results. Recently, an approximation which is addressed to the first of these two cases has been reported by Cross [S]. For convenience we will refer to this method by the abbreviation (SA). This approximation is also attractive in that it cztn be cast into a form which allows for factorization [9] of various scattering quantities as do the 10s and ES [7]_ Th e same thing appears possible with two related approximations [IO]. Basically. the SA is an alternate way to do an ES calculation when the ES becomes too unwieldly to apply The number of systems for which this approximation can be adequately tested is limited, though, due to the large number of rotational states accessible when the SA (and ES) is applicable. In this paper we will test the SA method fo{ Ar-TIF (an ideal system for this method) as well as for Ar-N, and the Iess ideal systems H+-HZ and Lie-HI_ Furthermore. we shall show how the SA method may be used to fztor the m-dependent rotational scattering amplitude. the degeneracy averaged rotational integral and differential cross sections, the opacity as a function of final orbital angular momentum quantum number (as opposed to total angular momentum quantum number which does not lead to factorization), and the phenomenological cross section for describing line broadening and general relaxation phenomena.
2. Formalism
In this paper the following notation will be used to describe the atom-diatom scattering process: j, 1, and _I are the rotational, orbital, and total angular momentum quantum numbers; primed and unprimed variables represent post- and precollision quantities; f; is the initial wavenumber for thej = 0 state; 122is the projection of ion the space-hxed c-axis which is parallel to X-(not to be confused with the z-axis shown in fig. 1); r= (I + l’)/2 is the I-averaged sudden partial wave parameter: Q = (0,0) are the angles of the vector R describing the rotor and are measured with respect to the c-axis used in Cross’s SA [S]; the z-axis is shown in fig. 1 and is perpendicular to k; 0 is the class&t1 orbital angle determined by the spherically symmetric part, V,(r), of the interaction potential, V(r, p’) and equals zero at the turning point F~:o;r is the distance from the c.m. of the
diatom to the atom: ,!Yis the angle between
Fig I.
Axis
r
and R; and .Kis the reduced mass of the atom-diatom
system and description of variables used in SA.
system.
D.E. Fitr and D.J..Kc+/~n
197
the sudden approximarion o/Crosk
The derivation of Cross’s SA is given in suflicient detail in ref. [S] and will not be repeated only the necessary equations needed to use this method will be given. In order to understand these equations it is necessary to recall that the SA allows the coupled equations in j and I to by a simple transformation. This transformation results in an uncoupled differential equation parametrically on the orientation of the rotor in space, R:
here. Instead,the significance be diagonalized which depends
of
(1) The solution to this equation for fixed R is given by u(r, Q). The orbital diatom c.m. is taken into account through p’ which is defined by cosfl,
motion
of the atom relative to the
(2)
= cos [O r O(r)] sin 0.
This relation implies that for any given value of R there are two branches branch for example, the asymptotic solution is zl(r, 52) .F_ exp [ -i(lir where the solution
to eq. (1). If one chases the ” f”
- fxT)] - S:(Q) exp [i(kr - &J)],
to 2?+(Q) is expressed
(3)
in terms of a phase shift according
to
ST+(Q) = exp [ZrlT,(n)].
(4)
We follow Cross‘s suggestion [a] for avoiding the double average of the phase shifts for the two branches:
branch
difficulty
by defining
.ST(Q)in terms of the
ST(Q) = exp {i[rr+(Q) + II’_(Q)]).
(5)
In Cross‘s approach p+ is r-dependent and this dependence is determined by eq. (9), where 0 is found as a function of r by solving eq. (1) with V(r, 8’. Q) replaced by the spherically symmetric part of the potential, V,(r). The phase shift associated with this equation has only a single branch and is represented by rr,(fi. The trajectory which determines 0 is traced out by rO in fig. 1 for the “t” branch. Using these definitions and modifying the phase of the scattering amplitude in eq. (3) so that the asymptotic form of the total wavefunction yields a result for the scattering amplitude as given by Arthurs and Dalgamo [9] [see eqs. (21) and (x)] the S-matrix is S;.l.:jr = i’+‘.-‘rexp
{i[rl,JQ + rl&)
- 2r),(r)] 1
~~‘-“-(n)Sr(n)~‘-‘(n)d~~
(6)
s where ST(Q) is given by eq. (5). For our purposes phase shifts
where rs is’rhe turning defined by
it s&ices
to use the WKB solution
point (at which point the term in brackets
to eq. (1) to obtain the
equals zero). The classical
orbital
angle is
D.E. Fir= and D.J. Kouri/On rhr sudden approximation
198
of Cross
where r,, is the turning point for the effective potential using I’,(r). Because the SA is a Iarge angular approximation, it is convenient to note that in the limit oflarge 1 exp :i[~&
momentum
+ Ml’) - 2s,(4]1 -r-z, 1,
(9)
and we shall use eq. (9) in cq. (6) in all of the calculations in this paper. For degeneracy averaged cross sections this approximate phase is irrelevant. For differential cross sections and Ill-dependent integral cross sections this approximation may be important. We are now ready to cast the S-matrix into a form more amenable to factorization. First, we expand S$2) in terms ofspherical harmonics according to S’(Q) = (4ic)“’ 1 YP(Q)S:2,,. LI, If eqs. (9) and (IO) are substituted
where the summation over .I! Unfortunately, this expression part of the second 3-j symbol. angular momentum limit ofa
L 1-I’ the coupled
(10)
into eq. (6) the integral over Q can be done analytically
[12] to give
in eq. (10) has been done using conservation properties of the 3-j symbols. is not easy to manipulate because of the appearance of!, I’and J in the lower This situation can be rectified by recalling that this 3-j symbol is the large 6-i symbol. Using a symmetrized form of this relationship+
j
(12)
.*
J-l
basis S-matrix
can be given in the more useful form
(13) which will be used in the remainder
The relationship
of this paper.
between the SA and IOS S-matrices
can be obtained
easily by starting
with the inverse of
eq. (10) .S’,.,_,. = (47i)- I”
Y;-“-(Q.S’(Q)dn. s
(14)
In the IOS approximation the angle ,8’does not vary with r but instead is fixed at a value p = 7 defined by D = 0. The two dimensional integral in eq. (14) over (0, (D)QR be reduced to a one dimensional integral over 7 if one notes tha: the spherical harmonic in R defined with respect to the X, Y, 2 axis system in fig. 2 can be defmed correspondingly in terms of spherical harmonics in (7, I$) defined with respect to the X’, I”, Z’ axis system by rotating through the Euler angles (0. -z/2,0) from the X’, Y’, Z’ axis system to the X, Y, Z axis * The ndvanta_ee of symmetrization is that one can permute the two larse angular momenta and still get the same result. We have shown chat this process _rives much better results than not to use it for a reIated case in ref. [13b] and the same holds true here.
D.E. Fitz and D.J: Kouri/On the sudden approxikacion of Cross
Fig. 2. Axis system and definition of variables used to reduce the SA to the 10s.
system as shown in fig_ 2. (fig. I becomes Y:-“‘(n)
= F Yj’(y, &D&-,.(0,
fig. 2 when 0 = 0). This rotation
(15) cn
Zir Dfl_,.(O,
-n/2,0)
1 The C#integration means that
by
-7T/2,0).
If we chose 0 = 0, S’(Q) in eq. (14) only depends .S’,.!_[. = (47~)~“‘~
is thus described
y
and can be written as
I
d4 s 0
sin ydy Yi-‘(7, +)Si(y).
(16)
s 0
can now be done and gives the product
of 27~and a Kronecker
&function
in i. and 0 which
. 7
s,.,-,.
L’
= D,.,-JO,
--x/2,0)
(2i
i- 1)“”
2
sin 7 dp ST(7)P,(cos y).
(17)
s
0
The D-matrix can be recognized as being the large anguIar written in symmetrized form [ 12, 13b] as %.I-,.(O, Furthermore,
-rr/2,0)
= (- I)![(21 + 1)(21’ + I,] l,,(,
in the iOS approximation
momentum
;
Iimit of the 3-j symbol which can be
;).
(18)
one defines
SrL = (2L + I) xsin p dy P L(cos y)S7(v). 2 s
(19)
0
Thus, substitution gives the factored
of eq. (18) into (17) using the definition IO.5 expression for the S-matrix [IO]
$.r-;jr = i’+“-2T(-
in (19) followed
by substitution
ofeq.
(17) into (13)
I)‘*‘+’ ‘[(2j + 1)(21’ + I)(21 + 1)(21’ + I)]“” (20)
except that here we employ
a general I rather than the final-l labelling
used ‘in ref. [lo].
D.E.Fir2 and D.I. KourifOn the sudden approximiltionoJCross
200
23. Fuctorization of the scatterirrg amplitude For convenience of presentation (and to facilitate actual computations) 3 function g&‘nf; jm) will be introduced which wit1 somewhat simplify the notation. The scattering amplitude quantized along the space fixed :-axis parallel to the incident asymptotic \vdve vector k is then 111 F y,.(Jaz’; jm) x?-m.(x),
f(i’zzz’+ jrzz lx) =
(21)
where
Substitution of eq. (13) into this expression gives a result which appears not to simplify at first glance. However, ifeither of the 3-j symbols in this expression is recombined with the 6-j symbol in eq. (13) to get a summation over three 3-j symbols. the summation over J can be performed using orthogonality relations of the 3-j symbok. The resulting expression allows the s-coefficient to be fdctored in the form
where hi = IN- niand
From this tktorization
reIationship
it follows directly that the scattering
amplitude
fzzctors according
to
in which the definition off{.!& + 00 1~) in terms ofeqs. (2if and (24) is obvious. Before going on to other expressions which factor, it is usefuf to give two expressions which do not factor but which are good starting points for some of the expressions which shall be given later. First, note that the definite polarization differential cross section is defined as
The zrz-dependent Glj’l~z’
t
jnz)
integral cross section is then given by =
dy * s
- dz
fj-rtz’
(27t:
2.4. Facrorkatiot~ of&e degetzeracy aceraged di~re}lt~ui cross secriotr The degeneracy cross section by
averaged
differential
cross section
is defined in terms of the definite polarization
differential
D.E. Fit= and D.J. Kouri/On
the sudden approxirnarion of Cross.
over 1)~and ~1’may be perFormed using 3-j symbol factored form of the differential cross section:
If eq. (26) is substituted into eq. (28), the summations orthogonaiity properties. The result is the following
where (30)
2.5. Factorization
of the
The degeneracy
integrul boss section nnd P-opacity
averaged
integral cross section
can be gotten either by integrating
eq. (29) over z or by (31)
The term inside the summation hence it follows that G~_~
=
be recognized
as the (21’+ I)-weighted
opacity
in I’;
5x(2’ + 1jP,.O_-J). J
P,.(j -j’)
over /’can immediately
(32)
1
can be shown
immediately
to Factor according
to (33)
where (34) It should be noted that no such relationship exists for the more familiar opacity as a function of J. In a similar manner to that above, the degeneracy averaged integral cross section can be shown to factor according to
where the cross sectlon
2.6. Factorization
o-o-L is obtained
from settingj
= 0 in eq. (32) and substituting
cq. (34) into eq. (32).
of phenorneno~ogicai cross sections
In spectral line broadening theory one speaks of a spectral transition i + f between initial and final spectroscopic states. The notation in this section is similar to that in the previous sections except that now all variables will have a subscript “i” or “f’ depending on whether they describe an initial or final spectroscopic state. Thk variabIe K in this section denotes the order of the spectral transition (K = 1 corresponds to pure rotational excitation or deexcitation via a permanent dipole moment interacting with the radiation field, for instance). The generalized phenomenological cross section which is used to describe such processes is [ 131 -’
D.E. firz und D.J. Korfri/Orr rhr szrddrn approsimurion of Cross
101
Once again, if the SA S-matrix expression in eq. (13) is substituted into this expression, can be collapsed by using well known properties olthe 6-j symbols [12]. The resultinS -give
the sums over Ji and Jr expression factors to
(37) where the dynamics t:r..ee
of the phenomena
= ;(-l)“(X 0
are contained
+ 1)-1i2~[sLos,!.
It shotrId be noted that the two S-coefficients
3. Computational
in
- S:..,_I.S~_,_I.]. are not necessarily
(38) evaluated
at the same energy.
procedures
The potentials for the four systems studied here. Ar-N,. Ar-TIF. H’-Hz and Lii-I-I2 will be described briefly to avoid any ambiguity concerning exactly which potentials were used in the calculations. The potentials for Ar-N, and Ar-TIF are Lennard-Jones 6-12 type potentials [14]: L’(i.7) = h:(rO/F)*2[I
+ ZSRP2(COS7)] - Z(r&)6[1
f zLRPI(COS7)]j_
(39)
Two different sets oFrrnisotropies are used for the Ar-NZ potentials so that comparisons cdn be made with the defnite polarization cross sections of Alexander [ 141. These two sets of anisotropies are denoted LR and SR for the stronger und weaker long range anisotropies, respectively. The parameters for both Ar-N2 potentials and the Ar-TlF potential are listed in table 1. The potential which we used for H &--Hz was based on a potential given by Giese and Gentry [ 1.51. Basical!y, Lx--efollowed McGuire-s procedure [16] and did a Legendre polynomial series expansion of their potential for r between 0.5 and 6.0 au. This expansion was truncated at Pi,. The coefficients were found by using a standard 32-point Gauss-Legendre integration of the inversion integral and were tabulated at intervals of 0.1 au. Table 1 Porenthl parameters for Ar-Nza’ and Ar-TIFb’ ----_-System r&W
c(meV)
Ar+ (SR) Ar-N, (LR j Ar-TlF
IO.296 IO.‘96 11.376
I’ ReT.[ 143. ” ReT.[jb].
7.424 7.121 9.800
0.50 0.315 0.50
0.13 0.315 0.30
D.E. Fitz and D.J. Kouri/On the sudden approximation of Cross The Ve coefficient near 6.0 au can be fit to a quadratic potential for this system (in au)
and shown to fit smoothly
Vb_“(r) = -2.6091r-4, but the VZcoefficients
to the long range form bf the
(40)
do not match smoothly
VZLR(r)= -0.60735r-’
203
+ 0.45886r-3.
with the long range form (41)
To remedy this problem we did a cubic spline lit between 4.9 and 6.0 au to get V, and its first derivative smoothly joined in this region. Likewise, a cubic spline tit was done for V’ between 4.7 and 6.0 au with V, and its first derivative set equal to zero at P = 6.0 au. Furthermore, we set Ve = 0 for r 1 3.9 au, Ve = 0 for r 2 4.4 au, and Vi,, = 0 for r L 4.9 au. The new potential was then retabulated at intervals of 0.1 au for r between 0.5 and 6.0 au. Values of the potential needed with this range were obtained via a 3-point Lagrange of each set interpolation_ For r > 6.0 au, eqs. (40) and (41) were used. For r c 0.5 au the functional dependence A exp (- Er), where A and B were determined by setting this of expansion coefficients was fit to an exponential expression equal to the values of the coefficients at r = 0.5 and r = 0.6 au. The Li+-Ha potential has been given in adequate detail by Lester and Schaefer [17]. We have used the rigid rotor form of their expression. 3.2. Nrmerical
techniqtces
The most time consuming part of the calculations using the SA is the calculation of the S-coefficients defined in eq. (14). A reasonably accurate and efficient numerical procedure for evaluating the integral in this expression is to use a Gauss-Legendre method to do the 0 integration and a spherical product procedure [18] for the Q integration. In our calculations no values of L > 8 were ever needed. Because of this and because ST(Q) generally had no more than two oscillations in the range of integration, a 16-point integration in 0, range O-n, and a 32-point integration in Q, range O-28, were found to give at least three figure accuracy. At first glance there appear to be 512 points for which S’(Q) and hence $(Q) must be evaluated. Since thesystems studied here are homonuclear, symmetry considerations immediately reduce the integration ranges to 0 over 0+2 and @ over O--x, leaving only 128 points to be evaluated. The use oTa spherically symmetric reference potential (se? fi_e. 1) allows a further symmetry reduction in @ to be achieved leaving on 64 unique points. However, the delinition of the phase shift in eq. (5) in terms of two branches doubles the required number of phase shifts needed so that 128 separate phase shifts must be calculated. This number of points was used in all of the calculations done here. In order to calculate I?(Q) one first must obtain the orbital angle O(r) in a suitable tabular form. This was done by splitting eq. (8) into three intervals vc,, r,), (r,, r2) and (rz, 03). It is understood that for r I To, Ok) = 0. For a given system at a particular value of L one first determines the turning point of the effective potential using V, and utilizing a Newton method [19]_ All turning points were calculated to at least six figure accuracy lor all systems. Then the interval CF,, ri) was split into four equal subintervals of width 0.1 au each. Within each subinterval the integral was calculated analytically after fitting the term within square brackets in eq. (8) to a quadratic. This technique accurately integrates out the singularity at P,. The integral over the second interval (rlr rJ was done by Simpson’s rule using a step size of 0.1 au. The value of rz was set to about 20 au for Ar-N, and Ar-TIF and about 40 au for H’-Hz and Li+-H,. For the integral over the third interval, the potential was set equal to zero allowing the integration to be performed analytically_ This combined procedure gives B(r) in tabular form in the interval Fe, r2) and in analytic form elsewhere. Values of e(r) within the interval cfo, r2) are obtained using a 3-point Lagrange interpolation [ 191. Using O(r), #(Q) can now be calculated. For a lixed value of fand 0 = (0, O), one calcu!ates the turning point for the full potential using Newton’s method. Once the turning point is known, an efficient Gauss-Mehler
D.E. Fir= md
204
D.J. Kouri/On
rhe sudden approxinmion
oJCross
quadrature c2.203 requiring onIy 8 points for Ar-N, and Ar-TIF and 24 points for H’--H, and Lit-Hz, was used to evaluate eq. (7). The energies for which our calculations were done required that the S-caehicients be calculated for i= O-150 for Ar-N2, i= C-200 for Lie-Hz, i= 30-300 for H’-Hz and i = O-370 for Ar-TIF. Using a Honeywell 6660, the computation times to tabulate all cf these coeflicients in double precision for L = 0.2.4, 6,s and 121= - 8 to 8 were 0.28 hours for Ar-N2, 0.53 hours for Li+-Hz, 0.70 hours for Ar-TlF, and 0.92 hours for H +-Hz. The calculations of In-dependent cross sections from these S-coeffkients generally required about 0.01 to 0.02 hours for all of the permissible transitions.
4. Results and discussion Of the four systems studied here, the system for which the SA is best suited is Ar-TIF. The energy at which we did calculations for this system w;1s 0.1154 eV. The energy differences between low lying rotational states are quite small for this system and one expects the ES approximation to hold. On the other hand. most of the contribution to scattering between the low lying states comes at large values of the orbital angular momentum quantum number: hence one might expect a breakdown of the CS approximation. This is exactly what the results show. Throughout this section it is understood that all 10s and SA calculations were done using the Secrest choice [?I] i= (I t l’)jZ. Furthermore, the SA and 10s opncities which we give are defined with respect to P according to eq. (27) and are weighted by (?I’ l I) whereas the CC opacities are delined in the more usual manner with respect to J, hence the labelling on the figures. For these systems and the transitions studied, an insignilicant amount oferror is introduced by making this comparison [22]. In figs. 3 and 4 the r-weighted IOS and SA opacities are compared with the J-weighted CC opacities of Pack [5b] for the 0 + 2 and 0 -+ 4 transitions, respectively. One can readily see that the SA results are quite good while the IOS results ovcrestimatc the opacities at large P. Although not shown, the ES results of Khare [7] for this system lie almost directly on top of the SA results implying that the SA is a good approximation to the ES. In table 2 comparison is made between the IOS and SA tn-dependent cross sections for Ar-TlF at E = 0.1153 eV. In iable 3 the CC_ S-4 nnd IOS integral rotational cross sections are compared for this system. Unfortunately. no CC results yet exist for the rn-dependent cross sections for this system. Our results are
300 -
Ar- TIF
II- .\.
fit 3. Comparison oTCC. SA and IOS weighted opacities for Ar-TIF at E = 0.1154 eV f:lr rhe 0 4 2 transition.
Fig. 4. Comparison of CC, SA and 10s weighted opacities for Ar-TIF at E = 0.1154 eV for the 0 * 4 transition.
D.E. Firz and D.J. KourilOn Table 2 Comparison oiSA and IOS m-dependent Ar-TlFat E = 0.1154eV
jnr-+Jm’
cross sections
for
the sudden approximation
Table 3 Comparison of CC, SA and IOS integral Ar-TlFatE=O.l154eV
j-j
u~m’ + jm)
205
of Cross
cross sections
for
U(j Ai’) -
SA
IOS
9.45 2.21 21.81 0.85 1.72 1.13 1.77 3.69
14.55 2.50 22.73 2.92 1.67 3.39 1.94 6.00
Cc?
SA
IOS
58.9 16.4 ___-
57.6 17.5
65.0 28.9
00 + 20 00 421 00 -t 22 00+40 00+41 00 -+ 42 oO-r43 00+44
0+2 O-4 =’ Ref. C5b-J.
truncated at r = 370 in order to make a comparison with the CC results. At this point the 0 -+ 4 cross sections are completely converged whereas only the cross sections for which 111= odd are converged for 0 + 2. The difference in the ))I-dependent results for the IOS and SA shows that the way in which the orbital coupling is treated has a significant effect on the rn-dependent cross sections. Furthermore, such effects are quite likely to show up in the calculation of cross sections needed to calculate shear viscosities and viscomagnetic effects in other atom-diatom systems [23]. All calculations for Ar-N, were done at a total energy of E = 0.0258 eV ( zz 300 K) and all cross sections given were converged at r = 150. Calculations for this system were done with two different potentials Iabelled LR (for the one with the stronger long range anisotropy) and SR, whose parameters are given in table 1. In tables 4-6 the SA and IOS m-dependent cross sections for both potentials are compared with the CC results of Alexander [14]. In table 7 the same comparison is made for the integral rotationally inelastic cross sections. It should be noted here that not all of the CC cross sections in table 7 were taken from the same calculations and hence may not be totally consistent. For the SR potential it is clear that both the 10s and SA approximations are good ecenfir rhe rn-dependetzf cross sections. Such is not the case for the LR potential, where both the IOS and SA fail to &e the correct trend in the polarization transitions. In figs. 5 and 6 the I-weighted SP. and 10s opacities are compared with the J-weighted CC opacities of Alexander [24] for the 0 + 2 and 0 - 4
Table 4 Comparison of IOS, SA and CC m-dependent cross sections (in A’) for the 0 + 2 rotational Ar-NZ (SR and LR potentials) m’
G(h’
t
I 2
integral transition
Ar-N2
ni
in
(SR and LR potentials) G(4m’ + 00)
SR LR
CC”
SA
10s
CC”
SA
10s
4.2 1 2.79 6.48
3.92 1.90 6.62
3.79 2.15 7.13
IS.6 12.6 9.67
9.95 7.54 15.89
8.84 7.13 11.45
z’ Ref. [ 141.
integral rransition
in
00)
SR
0
Table 5 Comparison of 10s. SA and CC m-dqxndent cross sections (in AZ) for the 0 -t 4 rorarional
0 1 2 3 4
LR
-
CC“
SA
ios
CC=’
SA
10s
2.86 2.89 2.95 2.95 I -7-i
2.05 2.10
1.94 2.03
3.17 2.57
2.25
2.15
1.46
2.50 2.0 1
2.48 1.91
0.57 0.19
0.70 0.87 1.11 2.19 4.76
1.70 274 1.94 3.45 4.69
z) Ref. [14].
D.E. Firr and D.J. Kouri/On the sudden approximation of Cross
206
~~~
~~~~
0 0
J
100
50 J
Fig. 6. Comparison of CC, SA and 10s weighted opacities at E = 0.0258 eV for the 0 + 4 transition using the LR potential.
Fig 5. Comparison of CC, SA and IOS weighted opacities for Ar-N, at E = 0.0258 eV for the 0 -t 2 transition using rhr LR poteniiol.
for Ar-NI
lransitions in Ar-N2 using the LR potential. Although neither the IOS nor the S.4 methods are able to completely describe this system. the SA method is nonetheless a considerable improvement over the 10s method. The problem with the SA appears lo be caused by a breakdown of the energy sudden approximation due to the long range part of the potential. Another system for.which an IOS calculation has been done [25] but is expected to break down is H+-HL at E = 3.7 eV. In fig. 7 the r-weighted SA and IOS opacities are compared with the J-weighted CC opacities of McGuire and Schinke [25] for the 0 --t 2 transition. It should be noted that our 10s opacities do not agree exactly with [heir 10s results (not shown) and this may be due to a difference in matching procedures used to joint the tabulated form of the potential to the long range part. Despite this difiicully, the SA results show a marked improvement over the 10s results. The system with the strongest long range anisotropy studied in this paper is Lie-Hz at E = 1.2 eV, an energy for which the energy sudden approximation is expected to fail. In figs. 8 and 9 the r-weighted SA and 10s opacities are compared to the J-weighted CC opacities of Schaefer and Lester [26] for the 0 + 2 and 0 -+ 4 transitions. Although the SA method still overestimates the opacities at large I’, it is nonetheless a signifinnt Table 6 Comparison of IOS, SA and CC rn-dcpendenr 1LR potential) at E = 0.020158 eV
integral
cross
sections (in AI)
for the 2 - 4 rotational transition in Ar-NI
c(4ni + 2m)
III’
01 =
-2
,,I
=
-
m=O
1
-
--
-.-4 -3 -1 -1 0 1 1 3 1
cc=’
SA
Lo6
2.15
2.02 0.95 0.47 0.30 0.30 0.M) 0.95
5.41 329 i .oa 0.54
0.13
=’Ref. [14-j.
10s
cc=’
-
SA
10s
CC”
SA
10s 3.08 0.39 4.39 5.96
0.65 CL50
15.70 5.34 3.18 1.75 1.20 1.31 1.11 1.72
0.22 1.59 3.35 2.41 0.82 1.16 0.95 0.22
! .07 13.45 6.68 6.24 2.61 1.41 1.11 2.77
I .74 9.69 7.34 5.86 299 1.83 2.59 3.09
0.39 0.82 0.56 2.32 4.30 2.32 0.56 o.sz
3.04 0.1 I 5.54 4.85 7.56 4.85 5.54 0.1 i
5.96 4.39 0.39
1.40
2.82
0.09
0.25
0.86
0.39
3.04
3.08
0.69
8.55
D.E. Firz and D.J. KourifOn the sudden appro.Gmtion
Table 7 Comparison of CC, IOS and SA integral inelastic cross sections for Ar-N2 at E = 0.0258 eV for both the SR and LR potentials j-+f
100 H+- H2 o-2
o(j+j) SR
O-2 O-4 O-6 244
207
of Cross
LR
cc
IOS
SA
CC
10s
SA
22.5”’ 22.4”’ 114C’ 24.gb’
22.3 19.1 20.4
21.0 19.8
61b’ 13b’
46.0 27.3
56.8 18.6 3.6 I 35.2 -
” Ref. [5b].
O.OP
12.9
20.0
b, Ref. [14].
9.P
-
35.3
5 Fig. 7. Comparison of CC, SA and 10s weighted opacities for H--H, at E = 3.7 eV for the 0 + 2 transition.
c, Ref. [27].
improvement over the IOS method. Furthermore, this system shows a phenomenon not seen in the other results. Assume one defined both a J-scaling factor and a magnitude scaling factor such that the position and magnitude of the maximum in the IOS opacity matched that of the SA opacity for the 0 + 2 transition of Lii-Hz shown in fig. 8. If one applied this position and magnitude scaling to both the rest of the 0 --) 2 opxity curve in Cg. 8 and to the whole 0 -+ 4 opacity curve in fig. 9, one would tind remarkably good agreement between the IOS and SA results for this system. Two problems arise in applying the SA which may severely limit its applicability for certain systems. The first problem occurred at i = 15 for the H+--H, system. Because of the extremely deep well in the spherically symmetric part of the potential, the orbital angle O(r) varies quite rapidly near the turning point due to a near orbiting situation. As a result of this rapid change, the angle p’+ changes sufficiently rapidly that I+‘, pi, Q) + h’(T + $)‘/(2Rr’*) becomes greater than the energy, at which p&t eq. (7) no longer applies. Fortunately, for H’-H, the contributions to the low-lying transitions coming from small values of ido not contribute significantly to the cross section and can be neglected. Hence, we started our calculation at i = 30. There may, however, be systems for which this solution simply will not work. In this region of orbital angular momentum, though, the IOS approximation itself would probably be sufficiently accurate and could be used instead of the SA approximation. 150 Lie-H2 I
.-._
100 l-i+-HZ
Fig. 8. Comparison of CC, SA and IO3 weighted opacities for Li+-Hz at E = 1.2 eV for the 0 -t 2 transition.
Fig. 9. Comparison of CC, SA and 10s weighted opacities for Li’-H, at E = 1.2 eV for the 0 -+ 4 transition.
D.E. Fir,- utrd D.J. Kouri/On
208
rhe sudden upproXinltXi0n
Of cross
The second problem is that for Lii-H, the long range anisotropy is sufliciently strong that the collision can no Ionger be considered as “sudden” and that the spherically symmetric orbital anglk also inadequately corrects for the I-coupling. Work is currently underway to correct the energy sudden defect. Furthermore, work is currently being done in this laboratory [ZS] to correct for the orbital coupling using orbital angles calcu!ated from the whole potential, not just the spherically symmetric part. Despite these problems, the SA method should be very useful in treating a variety of systems where the IOS method fails due to difliculties with the CS approximation. We are carrying out additional test cltlculations of differentia! cross sections and magnetic transition cross sections and will report the results at a later date. We are also carrying out calculations to assess the error introduced in differential and m-dependent cross sections by the use of the asymptotic phase as in eq. (9).
Acknowledgement We would especially like to thank M. AIexander for sending us his unpuclished which appear in figs. 5 and 6. Very helpful comments by the referees are gratefully
CC opacities for Ar-N, acknowledged.
References and DJ. Kouri. J. Chem. Phys. 60 (1974) 248% R. T Pack. J. Chem. Phys. 60 (1974) 633. D. Kouri, in: Atom-molecule collision theory: a guide for the experimentalist. ed. R.B. Bernstein (Plenum, 1979). C.F. Curtiss. J. Chem. Phys. 48 (1968) 1725; 49 (1968) 1952. (a) T.P. Tsien and R. T Pack, Chem. Phys. Letters 6 (1970) 54; (b) G.A. Parker and R. T Pack. J. Chem. Phys. 59 (1973) 5373. D. Secrest, J. Chem. Phys. 62 (19753 710. V. Khare. J. Chem. Phys. 68 (1978) 4631. RJ. Cross Jr., J. Chem. Phys. 69 (1978) 4495. A.M. Arthurs and A. Da&no. Proc. Roy. Sot. London A256 (1960) 540. R. Coldflam. S. Green and DJ. Kouri. J. Chem. Phys. 67 (1977) 4149: R. Goldflnm, D-l. Kouri and S. Green, J. Chem. Phys. 67 (1977) 5661. AS. Dickinson and D. Richards, J. Phys. B11 (19783 1085.3513; Y.B. Band. J. Chem. Phys. 70 (1979) 4. and orivate communicntion. AR. Edmonds. An.&& momentum in quantum mechanics (Princeton Univ. Press, Princeton. 1960). (a) R. Shafer and R.G. Gordon, J. Chem. Phys. 58 (1973) 5422; (b) D.E. Fitz and R.A. Marcus, J. Chem. Phys. 62 (1975) 3788. M. Alexander, J. Chem Phys. 67 (1977) 2703. C.F. Giese and W.R. Gentry, Phys. Rev. A10 (1974) 2156. P. McGuire, J. Chem. Phys. 65 (1976) 3275. W.A. Lester Jr. nnd J. Schaefer, J. Chem. Phys. 59 (1973) 3676. of multiple integrals (Prentice-Hall, Engtewood Cliffs, 1971) p. 40. A.H. Srroud. Approximnre calculations F-S. Acton, Numericai methods that (usually) work (Harper and Row, New York, 1970). 2. Kopal. Numerical analysis (Wiley. New York. 1955) p. 383. D. Strrest. unpublished research. Y. Shimoni and D-l. Kouri, J. Chem. Phys. 65 (1976) 3372.3958; 66 (1977) 2841; G.A. Parker and R. T Pack, J. Chem. Phys. 66 (1977) 2850. W.-K. Liu. ER. McCourt. DE Fitz and DJ. Kouri, J. Chem. Phys. 71 (1979) 415; W.-K. Liu. F.R. McCourt and W.E. Kshler, 5. Chem. Phys. 71 (1979) 2566. M. Alexander, private communication. R. Schinke and P. McGuire, Chem. Phys. 28 (1978) 129. J. Schaefer and W.A. Lesster Jr, J. Chem. Phys. 62 (1975) 1913. R. T Pack. J. Chem. Phys. 62 (1975) 3 143. V. Khare and D. Kouri, in preparation.
P. XlcGuire
New York,