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The centrifugally decoupled exponential distorted wave (CDEDW) approximation for the calculation of rotationally inelastic molecular collision cross se
Chemical ?hysics 33 (1978) 435-441 0 North-HollandPublishingCompany
THE CENTRIFUGALLY DECOWLED EXPONENTIAL DISTORTED WAVE (CDEDW) APPRhIMATION
FOR THE CALCULATION OF ROTATIONALLY INELASTIC
MOLECULAR COLLISION CROSS SECTIONS L.
EN0 and G.G. BALINT-KURT1
School of Chemistry, Bristol University, Bristol BS8 lTS, UK Received 18 April 1978 A new approximate method is presented for the mpid calculaGon of rotationally inelastic molecular colliiion cross SeCtions. The method is called the centrifugally decoupled exponential distorted wave (CDEDW) approximation and involves the combination OF two well known approximations. The first approximation is the neglect of the off-diagonal cou@lng terms which arise from the orbital angular momentum operator in the coupled differential equations in the body-fixed axis system. The second approximation is to treat the remainiig coupling terms, which arise from the interaction potential, using a unitary perturbation approximation. The CDEDW method is applied to the calculation of total and partial rotationally inelastic cross sections in the Ar-N2 system, and detailed comparisons are made with exact and several other types of approximate calculations. Ageement with exact calculations is good and often comparable with the coupled states and p-helicity decoupled approximations. TheCDEDW method requires a similar amount of computational effort to the infiite order sudden (10s) approximation, and we show that for the present system the CDEDW method gives more reliable results.
1. Introduction The exact calculation of inelastic cross sections involves solving sets of coupled differential equations, using the so-called close coupling method [l] . Even with present-day computers and advanced numerical techniques such calculations remain a formidable task. In an attempt to overcome these difficulties, there has been a renewed effort over the last few years to develop accurate approximate descriptions of the molecular collision dynamics. One approach has been to try to reduce the number of equations which are coupled together, but still to retain the physically important coupling terms [2-51. Even with these reductions in the numbers of coupled equations, the calculation bf rotationally inelastic cross sections remains a problCm. Another set of approximate methods has therefore been developed which involve only the solution of uncoupled differential equations [1,6-P]. The two most widely used methods of this latter type are the exponential distorted wave (EDW) [ 1,7] and the infmite order sudden (10s) [8,9] approximations. These methods are in many ways complementary. The 10s method treats the potential exactly, but
neglects differences in channel energies and orbital angular momenta, and is therefore most reliable for low total angular momentum (J) quantum numbers where the potential has a dominant effect. The EDW method, on the other hand, treats all the coupling in the equations using a unitarised perturbation theory, it is therefore most reliable at large J where the coupling terms become small and a perturbation treatment is valid. The EDW method is in general computationally more time consuming because it requires the calculation of a complete matrix of distorted wave integrals between all the channels in the problem. The number of these integrals rapidly proliferates with increasing rotational basis due to the 2j + 1 projection degeneracy associated with each rotational level. In this paper we combine the coupled states and EDW approximations. This greatly reduces the number of distorted wave integrals which need to be evaluated and makes the computational effort required comparable to that needed in the IOS approximation. The theory is derived in section 2 where we also consider its relationship to the EDW method. In section 3 we present the results of applying the method to the Ar + N, system and make detailed comparisons with pre-
436
L. EIW. G.G. Balint-Kurti/RotatioMol[yinChtiC SCatteriIIg
viously published results using the exact close coupled (CC), coupled states (CS), p-helicity decoupled (PHD) and IOS methods. In section 4 we discuss the method and summa& our findings.
2. lYleoIy 2. I.
l7zeoreticaI development
important thing to notice about eqs. (1) and (3) is that the potential [eq. (3)] is diagonal in the helicity quantum number, while the matrix elements of the square of the orbital angular momentum operator ($$,,) are diagonal in rotational quantum number j_ The boundary conditions on the radial BF wavefunction may be written as [IO,1 l] :
@$(R)Ry
0,
$ff(R)
- &{exp [-i(kp-&+j)n)]
Rz
6jj,6PP,
We consider the time-independent Schrijdinger equation for the scattering of an atom by a linear molecule in a body-fiied coordinate system. The total wavefunction is expanded in terms of body-fmed (BF) molecular states and a set of coupled differential equations for the radial wavefunctions is obtained for each total angular momentum quantum number J:
symptotic behavlour of the radial wavefunctions. The integral cross section, u!?~ I +j, may be written in terms of Smatrix elements as
(d2/dR3 f kj! - Cj;./R2)$j$R)
of?Lj= [r/k; (2j+l)]
+
-(ki/ki.)1’2S’p.jp
exp [i(kjtR-&J-$)x)]
}.
(4)
This equation defmes the S matrix in terms of the a-
(Cf;p._,/R2)$/s$._,(R)
(1)
For use later in the paper we also define the dimensionless partial cross section$+ [16] :
where CJ!” , yJ(J+l)
P/J and
c&*,l
+j’(j’+1) - 21*‘2,
The present CDEDW approximation simplifies eq. (1) in two separate stages. Firstly, in common with other approximations [3,4] , the off-diagonal centrifugal coupling is neglected;
= [J(J+ 1) - !_&‘A I)] 112
x b’(j’+1) -&j&l)]?
(2)
The subscripts p’ are the helicity quantum numbers. For derivation and discussion of these equations the reader is referred to the literature [IO,1 13. The mii_ trix elements of the potential are given by [IO] I ~P,,j,>P&)
=$
F V,(R)[(2j”+l)/(2j’+l)]
x c(-j”Aj’lp’op’)c(j’Aj’looo),
“*
(3)
where V,(R) are the coefficients of the potential in the standard expansion in Legendrepolynomialsand Cg"'Aj'lp'Op') are Clebsch-Gordan coefficients. ‘ihe
C$;p.tl
=o,
(7)
whilst the diagonal f?-!’ , terms are retained unaltered [4,12]. The problem’&en reduces, for every J, to the
solution of separate sets of coupled equations for each /.L,where all values of p consistent with IpI< min(jj’,J) must be considered. The problem is further simplified by the conservation of parity which leads to the fact that sets of equations differing only in the sign of fl are identical. The second stage of the CDEPW approximation involves using a unitarised perturbation approximation, i.e. the EDW method, to calculate the S matrix separa-
tely for each set of coupled equations cokesponding to defmite values ofJ and p_ To apply this method we
L. Eno, G.G.Balint-KurtilRotationaQ inelasticscattering
431
must first calculate the distorted waves, which are the regular solutions to diagonal equations in which we have ignored the off-diagonal matrix elements of the potential: Q&.(R) = 0,
(8)
with the boundary conditions: $$ (R) Rz $!P8(R) Rzsin
0, [kj.R-(J+j’~/2
+ qjsPP],
(9)
where qj,r* are phase shifts. By calculating the distorted wave integra!s, which are the matrix elements of the potential between these distorted wavefunctions, we can evaluate the S matrices in either the distorted wave (see ref. [13] p. 116-117) or exponential distorted wave (EDW) [1] approximations. The S matrix is unitary in the EDW approximation, as is required by the conservation ofparticle flux, and furthermore it has been shown [l] that the EDW approximation corresponds to a resummation of the distorted wave Born series and includes, even in the first order approximation, contributions from all of the higher order terms of the distorted wave Born series. The expression for the S matrix elements in the CDEDW approximation is: Sfpt,ip = exp(ivj.p.) [exdiAJ)l j,p,jp exp(iyJ hplrs, (10) where: ifi’ =i;
A]$ iN = 0,
9
AfP j,, = -2(kjljJ1”
I
01) X f Q/p(R) $p,jp(R)$(R) 0
*a
otherwise.
It is interesting to compare the work involved in applying the EDW and CDEDW approximations. Let us consider the calculation of a single partial cross section, P!, _jT fnr an atom-homonuclear diatom collision using a baslsJ = 0,2, __.,jma?tand suppose that Jai,,. In the full EDW method the total number of channels (j, I combinations) NEDw is given by: jmax NED\’ = C (2j+l) = (jmax+l)(jm,/2+l). j=o,2.. _
(14)
These channels may be divided into those of even and those of odd parity, and the two sets are not coupled to each other. For even values of J the number of even parity channels is Neven = (jm,,2+1)2 and the number of odd parity channels isNodd = &.nax/2+l)(jmax/2), while for odd J the numbers of even and odd parity channels are reversed. The total number of distorted wave integralsNfDw which must be evaluated is therefore: NEDW =&._(N,,,-U/2 I
+&&&-U/2
=j mau(imax+2) [(i,~.+Wm,/2+lHl
18. (15)
In the CDEDW case the problem divides up into separate problems for each helicity, p. For even hehcities there are (j,,/2+1--u/2) coupled channels while for odd helicities there are (jm,/2 - (n-1)/2) coupled channels. The total number of distorted wave integrals which have to be evaluated in the CDEDW method is therefore hnax
The expressions for the integral cross section, and for the dimensionless partial cross section in the CDEDW approximation are: u;:j. = [ir$(2j+1)]
2.2. Relationship of CDED W TOED Wapproximation
F (zJ+l) c ls;P,iP - 6#, P 02)
p=. 7 (i,,12tl-~~2)Cim,-lr)/4 *iCDEDW= E,-,. .. tirn&&)
+
c
p=1,3,..
.
(im,-r-rtWm,-~-W (16)
L. Eno, G.G.BalidGuti/Rotationally
438
Hence we find that:
inelastic scattering
Table 1
iVCDEDWIN,EDW = 2&,+1)/3 I
[(i,,+2)2-2].
As an example, if we use a basis set ofj=0,2,4.. then
(17) .12
IPEDw/$mw = l/22. (18) I For this basis, therefore, the number of integrals which we need to calculate in the CDEDW method is a factor of 22 fewer than the number needed in the EDW method. Besides this saving in computational effort there is a further substantial saving arising from the fact that in the CDEDW method we have to exponentiate several small matrices (one for each helicity) while in the EDW method we must exponentiate two much larger matrices, which requires considerably more computational effort.
3. Application to Ar + N, In order to test the validity of the CDEDW method we have applied it to the problem of rotationally inelastic scattering in the Ar + N, system. We chose thii system because a larger number of exact [14,15] and approximate [14-171 calculations have already been reported in the lietrature onit and it is, therefore, possible to make detailed comparisons with them. AU of these calculations used the same empirical potential
Comparison of different methods for the calculation crqss sections in Ar+Nz collisions at Etot = 300 K
of integral
Ui,i’ @*I
j-q
0 2 0 4
cc (ref. [14])
Ef.
22.5 22.4
22.2 22.8
[16])
~~~~~rlcj
pf.
23.2 25.2
22.3 18.8
[14])
son is made with close coupling (CC) [14], coupled states (CS) [16] and infinite order sudden (IOS) [14] results from the literature. The partial cross sections for the 0 + 4 transition are shown in fig. 1. From the table we see that the agreement of the CDEDW cross sections with the exact ones is good and is comparable to the IOS results. Fig. 1, however, &LOWS that the CDEDW partial cross section agrees far better with the exact CC partial cross section than does the IOS one. Indeed, the CDEDW partial cross section is of comparable accuracy to the CS. In general the IOS method is expected to break down for sufficiently low collision energies or for sufficiently large Aj transitions. This is because the IOS method assumes that all the channels
[181 V(R$) = E[(R,,/R)‘~ - 2(R0/R$] + da,2(R0/R)‘2
- 2a6(R0/R)61~z(cosQ,
where E = 1.0297 X 10e2 eV,R,-, = 3.929 A, aI2 = 0.5 and a6 = 0.13. Three sets of calculations were performed corresponding to energies of&,, = 300 K, 618 K and 768 K respectively, relative to thej = 0 rotational state:In every case all the open rotational channels were used in the basis set, i_e_jmz = 8,12 and 14, respectively. AU the necessary distorted wave integrals were calculated exactly using a fast numerical technique recently developed at Bristol [19]. In the first set of calcuiations (E,, = 300 K) total and partial cross sections were calculated forj = 0 + j’ = 2 andj = 0 + j’ = 4 transitions. The results for the total cross sections are presented in table 1 where compari-
I Oo
10
20
30
Total Angular Fig. 1. Dimensionless pztial
4.0 Momentum,
50
60
70
J
cross sections for thei = 0 -+I = 4
rotational excitation in At + N2 collisions at Etot = 300 K. Solid tines and filed circles are CC results [ 141; open triangles are
CS[16]; crosses xe CDEDWand open cixIes are IOS [14].
L. Eno, G.G. Bali&Kum~Rotationally
Table 2 Comparison of different methods for the calculation of integsal cross sections aO*_i) (in R*) at EtGt = 618 K. The numbers, in descending order, correspond to CC 1151, CDEDW and IOS[151results. The square bracketed CC results are considered to be outside Pack’s 4% accuracy limits j=6
j=8
(3.31
1.10
3.1
1.19 0.98
0.40 0.34 0.29
5.3 5.3 5.2
2.50 2.20 2.10
4.5
[16-O] 15.0 22.3
10.5 10.4 11.7
13.4 17.9 19.1
15.7 20.4
11.4 12.3 12.7
11.4 11.4 13.5
13.4 13.3 16.8
4.5 3.8 4.9
5.8 5.1 7.1
7.9 7.7 10.5
5.7 5.5 5.6 10.9 10.2 11.8
11.7 11.0 15.4
are degenerate and this assumption breaks down in the two situations just mentioned. In table 2 inelastic integral cross sections are presented at Etot= 618 K. The CDEDW cross sectons are
compared with CC and 10s cross sections calculated by Pack [15]. The higher energy in this case should favour the IOS method. It is, however, clear from an examination of the entire table that the CDEDW results are definitely closer to the exact CC values than are the IOS results. Fimahy we examined the CDEDW method at the highest energy for which there are comparison calculations i.e. E,, = 768 K [IS-171. In table 3, inelastic integral cross sections are presented at this energy and compared with CC [ 151, CS [16], IOS [ 151 and phelicity decoupled (PHD) [17] results. For highj transitions (lower right hand corner of table) the CDEDW and CS results are comparable in accuracy, while for transitions involving low j quantum numbers the CS results are generally somewhat better. Overall the CDEDW cross sections agree better with the CC results
439
inelastic scattering
than do the IOS cross sections. A particular point to notice is that the percentage accuracy of the CDEDW results is symmetric about the diagonal (i.e. the percent error in ~$2~6) is the same as that in ~(6~2)). This is not, on the other hand, the case for the IOS results. The 10s results are significantly better in the top right hand half of the table than in the lower left hand half. It should be noted also that this implies an absence of the detailed balancing requirement from these results, although it would be possible to implement the IOS method in such a way as to satisfy detailed balancing. At this energy (768 K), there are several partial cross section plots available in the literature [ 15,16]_ In figs. 2,3 and 4 we compare our CDEDW partial cross sectionsforthej=6+j’=8,j=lO+j’=6andj= 10 +j’ = 8 transitions with those available from the literature. The CDEDW partial cross sections are, in every case, very similar to the CS and PHD partial cross sections, and share with them a tendency to have an exaggerated maximum and to fall off too sharply at large 1. Where comparison is possible with the IOS method (figs. 3 and 4) the CDEDW partial cross sections are defmitely in better agreement with the CC results. The centrifugal decoupling approximation is expected to work best at small J values while the EDW approximation should work well at large J. The errors in the CDEDW results at large J are, therefore, due to the centrifugal decoupling approximation.
The 10s
approximation, which is expected to work best for low J values, is seen from figs. I, 3 and 4 to have substantial errors in this region.
4. Discussion ‘Ibe centrifugally decoupled exponential distorted wave (CDEDW) method, developed in this paper, substantially reduces the computational effort required in the application of centrifugally decoupled approximations (such as the coupled states (CS) and p-helicity decoupled (pHD) approximations). Our results demonstrate that, for the Ar + N2 system, the CDEDW cross sections are, nevertheless, of comparable accuracy to those calculated using the more laborious CS and PHD methods. The computational labour needed to calculate a cross section using the CDEDW method is comparable to that required using the 10s method. The comparisons pre-
440
L. Eno, G.G. BaIint-Kurh~Rotctbnally
inelastic scattering
Table 3 Comparison of diffe:ent methads ior the calculation of b&rat cross sections a(jj+) (in A’) at f&t = 768 K. The numbers in descending order correspond to CC [IS] ; CS [ 161, CDEDW and 10s [ 15 ] results. The numbers in parentheses next to the CS re suits are PHD 1171 results whet&available. S&re bracketed CC results are considered to be outsidePack’s 4% accuracy limits i=O
I
-j&
2
j=4
t3.21 3.2 2.9 3.5
0
[15.7] 15.2 14.2 17.5
2
&lo
j=8
1.37 1.43 1.84 1.64
0.99 1.02 1.18 0.95
0.46 0.47 0.53 0.36
0.094 0.094 0.066 0.093
8.4 8.7 8.4 9.3
4.7 4.9 4.7 4.6
2.59 2.62 2.31 2.22
0.74 0.74 0.56 0.76
9.7 9.88 9.4 9.8
5.3 5.2 5.2 4.9
2.35 2.35 2.04 2.48
9.7 9.5 9.4 9.8
5.2 4.9 (4.7) 4.7 5.1
4
11.4 11.9 ,153 14.8
14.4 14.8 14.3 16.7
6
10.8 11.2 129 12.4
105 11.0 10.5 11.9
12.8 13.0 12.3 14.1
8
5.7 5.8 5.0 6.1
6.6 6.6 5.9 7.5
7.9 7.7 7.8 9.2
1.15 1.17 G.81 1.96
I.86 1.87 l-44 3.2
3.5 3.3 3.0 5.8
10
j=6
11.0 10.7 10.7 12.8
10.9 9.6 (10.2) 9.5 9.9
5.9 5.5 (5.3) 5.3 8.2
10.8 9.5 (10.1) 9.4 12.2
‘d
70 lx-
CCL 0
$30. k
A -l?HD. x -CRERW.
a
ZS r B20. 8 e *g 10. Px a
0
.
2 0
20
40 Total Angular
60 00 M0mentum.J
100
Fig. 2. Dimensionlesspartial cross sections for the j = 6 +j’ = 8 rotational excitation in Ar + Nz collisions at Etot = 768 K. Solid lines and filled circles are CC results [ 151; open triangles are CS [16] and crosses are CDEDW.
20 40 Total Angular
d 60 Momentum,
80 J
Fig. 3. Dimensionless partial cross sections for the j = 10 +j’ = 6 rotational transition in Arc Nz~coUisions at E,, = 768 K. Solid Lines and fiied cricles are CC results [ 15 ] ; open triangles are PHD [l?‘]; crosses are CDEDW and open circles are IOS[lS].
L. Eno. G.G. Bolint-KurrilRototionaNy
inelastic scattering
441
Acknowledgement The authors thank the S.R.C. for the provision of computer time on the Rutherford Laboratory computer and L.E. acknowledges the financial assistance of an S.R.C. studentship. They would like to thank Dr. R. Saktreger for the use of his computer program and are grateful to Dr. M. Shapiro for some helpful discussions.
References
60
60
100
Total Angular Mcmentum,J Fig. 4. Dimensionlesspartial crosssectionsfor the j = 10 +I’ = 8 rotational transition in AI + Na colIisions at Etot = 768 K. Solid line and f&-d circles are CC results [15], open triangles are PHD [ 171, crosses are CDEDWand open circlesare 10s [ 151,
sented in the paper show that the CDEDW cross sections for Ar + N2 are in general more accurate than the 10s ones, this comment being especially true with respect to partial cross sections. In other papers [20,21] we have presented approximate and entirely analytic methods for the calculation of distorted wave integrals. In suitable cases, these analytic methods could be readily combined with the present approach to provide an even more efficient method which would be applicable to situations requiring the use of very large basis sets. In summary the CDEDW method has been shown to be a reliable and efficient method for the computation of rotationally inelastic cross sections. In terms of computational effort and accuracy it is certainly competitive with the IOS method and at low collision energies should be much more reliable. Following suggestions made above an entirely analytic form of the method could be implemented. ‘This form is expected to be of great utility in handling intractable problems requiring the use of very large basis sets.
[ 11 G.G. Balint-Kurti, In: International review of science, Phys. Chem. Ser. II, Vol. 1, eds. A.D. Buckingham and C.A. Coulson (Butterworths, London, 1975). [2] H. Rabitz, in: Modern theoretical chemistry HI, ed. W.H. Miller (Plenum Press, New York, 1976).
[3] P. Mffiuire and D.J. Kouri, J. Chem.F’hys.60 (1974) 2488. [4] M.Shapiroand M.Tamir, Chem.Phys. 13 (1976) 215. [S] A.E. DePristo and M.H. Alexander. J. Chem. Phys. 63 (1975) 3552. [6] G.G. Balint-Kurti and R.D. Levine, Chcm. Phys. Letters 7 (1970) 107. [7] R-D. Levine and G.G. Balint-Kurti, Chem. Phys. Letters 6 (1970) 101. [S] T.P. Tsein and R-T Pack, Chem. Phys. Letters 6 (1970) 54,400; 8 (19713579. [9] D. Secrest, J. Chem. Phys. 62 (197.5) 710. [IO] R.T Pack, J. Chem. Phyr 60 (1974) 633. [11] R.B.Walkerond J.C. Light,Chem.Phyr 7 (1975) 84. [12] J-M. Launay, J. Phys. B9 (1976) 1823_ [13] MS. Child, Molecular collision theory (Academic Press, New York, 1974). [ 141 T.P. Tsein, G.A. Parker and R.T Pack,J. Chem.Phys.59
(1973)5373. [151 R.T Pack,J. Chem. Phys. 62 (1975) 3143. [I61 P. McGuire,Chem.Phys. 13 (1976)81. [171 M. Tamir and M. Shapiro, Chem. Phys. Letters 39 (1976) 79. 1181 M.D.Pattengill, R.A. LaBudde, R.B. Bernstein and C.F. Curtiss, J. Chem. Phys. 55 (1971) 5517. [19] R. Saktreger’and G.G. Balint-Kurti, to be published. [20] L. Eno, G.G. Balint-Kurti and R. Saktreger, Chem. Phys. 29 (1978) 453. [21] L. Eno and R. Saktreger, Mol. Phys. 35 (1978) 601.
THE CENTRIFUGALLY DECOWLED EXPONENTIAL DISTORTED WAVE (CDEDW) APPRhIMATION
FOR THE CALCULATION OF ROTATIONALLY INELASTIC
MOLECULAR COLLISION CROSS SECTIONS L.
EN0 and G.G. BALINT-KURT1
School of Chemistry, Bristol University, Bristol BS8 lTS, UK Received 18 April 1978 A new approximate method is presented for the mpid calculaGon of rotationally inelastic molecular colliiion cross SeCtions. The method is called the centrifugally decoupled exponential distorted wave (CDEDW) approximation and involves the combination OF two well known approximations. The first approximation is the neglect of the off-diagonal cou@lng terms which arise from the orbital angular momentum operator in the coupled differential equations in the body-fixed axis system. The second approximation is to treat the remainiig coupling terms, which arise from the interaction potential, using a unitary perturbation approximation. The CDEDW method is applied to the calculation of total and partial rotationally inelastic cross sections in the Ar-N2 system, and detailed comparisons are made with exact and several other types of approximate calculations. Ageement with exact calculations is good and often comparable with the coupled states and p-helicity decoupled approximations. TheCDEDW method requires a similar amount of computational effort to the infiite order sudden (10s) approximation, and we show that for the present system the CDEDW method gives more reliable results.
1. Introduction The exact calculation of inelastic cross sections involves solving sets of coupled differential equations, using the so-called close coupling method [l] . Even with present-day computers and advanced numerical techniques such calculations remain a formidable task. In an attempt to overcome these difficulties, there has been a renewed effort over the last few years to develop accurate approximate descriptions of the molecular collision dynamics. One approach has been to try to reduce the number of equations which are coupled together, but still to retain the physically important coupling terms [2-51. Even with these reductions in the numbers of coupled equations, the calculation bf rotationally inelastic cross sections remains a problCm. Another set of approximate methods has therefore been developed which involve only the solution of uncoupled differential equations [1,6-P]. The two most widely used methods of this latter type are the exponential distorted wave (EDW) [ 1,7] and the infmite order sudden (10s) [8,9] approximations. These methods are in many ways complementary. The 10s method treats the potential exactly, but
neglects differences in channel energies and orbital angular momenta, and is therefore most reliable for low total angular momentum (J) quantum numbers where the potential has a dominant effect. The EDW method, on the other hand, treats all the coupling in the equations using a unitarised perturbation theory, it is therefore most reliable at large J where the coupling terms become small and a perturbation treatment is valid. The EDW method is in general computationally more time consuming because it requires the calculation of a complete matrix of distorted wave integrals between all the channels in the problem. The number of these integrals rapidly proliferates with increasing rotational basis due to the 2j + 1 projection degeneracy associated with each rotational level. In this paper we combine the coupled states and EDW approximations. This greatly reduces the number of distorted wave integrals which need to be evaluated and makes the computational effort required comparable to that needed in the IOS approximation. The theory is derived in section 2 where we also consider its relationship to the EDW method. In section 3 we present the results of applying the method to the Ar + N, system and make detailed comparisons with pre-
436
L. EIW. G.G. Balint-Kurti/RotatioMol[yinChtiC SCatteriIIg
viously published results using the exact close coupled (CC), coupled states (CS), p-helicity decoupled (PHD) and IOS methods. In section 4 we discuss the method and summa& our findings.
2. lYleoIy 2. I.
l7zeoreticaI development
important thing to notice about eqs. (1) and (3) is that the potential [eq. (3)] is diagonal in the helicity quantum number, while the matrix elements of the square of the orbital angular momentum operator ($$,,) are diagonal in rotational quantum number j_ The boundary conditions on the radial BF wavefunction may be written as [IO,1 l] :
@$(R)Ry
0,
$ff(R)
- &{exp [-i(kp-&+j)n)]
Rz
6jj,6PP,
We consider the time-independent Schrijdinger equation for the scattering of an atom by a linear molecule in a body-fiied coordinate system. The total wavefunction is expanded in terms of body-fmed (BF) molecular states and a set of coupled differential equations for the radial wavefunctions is obtained for each total angular momentum quantum number J:
symptotic behavlour of the radial wavefunctions. The integral cross section, u!?~ I +j, may be written in terms of Smatrix elements as
(d2/dR3 f kj! - Cj;./R2)$j$R)
of?Lj= [r/k; (2j+l)]
+
-(ki/ki.)1’2S’p.jp
exp [i(kjtR-&J-$)x)]
}.
(4)
This equation defmes the S matrix in terms of the a-
(Cf;p._,/R2)$/s$._,(R)
(1)
For use later in the paper we also define the dimensionless partial cross section$+ [16] :
where CJ!” , yJ(J+l)
P/J and
c&*,l
+j’(j’+1) - 21*‘2,
The present CDEDW approximation simplifies eq. (1) in two separate stages. Firstly, in common with other approximations [3,4] , the off-diagonal centrifugal coupling is neglected;
= [J(J+ 1) - !_&‘A I)] 112
x b’(j’+1) -&j&l)]?
(2)
The subscripts p’ are the helicity quantum numbers. For derivation and discussion of these equations the reader is referred to the literature [IO,1 13. The mii_ trix elements of the potential are given by [IO] I ~P,,j,>P&)
=$
F V,(R)[(2j”+l)/(2j’+l)]
x c(-j”Aj’lp’op’)c(j’Aj’looo),
“*
(3)
where V,(R) are the coefficients of the potential in the standard expansion in Legendrepolynomialsand Cg"'Aj'lp'Op') are Clebsch-Gordan coefficients. ‘ihe
C$;p.tl
=o,
(7)
whilst the diagonal f?-!’ , terms are retained unaltered [4,12]. The problem’&en reduces, for every J, to the
solution of separate sets of coupled equations for each /.L,where all values of p consistent with IpI< min(jj’,J) must be considered. The problem is further simplified by the conservation of parity which leads to the fact that sets of equations differing only in the sign of fl are identical. The second stage of the CDEPW approximation involves using a unitarised perturbation approximation, i.e. the EDW method, to calculate the S matrix separa-
tely for each set of coupled equations cokesponding to defmite values ofJ and p_ To apply this method we
L. Eno, G.G.Balint-KurtilRotationaQ inelasticscattering
431
must first calculate the distorted waves, which are the regular solutions to diagonal equations in which we have ignored the off-diagonal matrix elements of the potential: Q&.(R) = 0,
(8)
with the boundary conditions: $$ (R) Rz $!P8(R) Rzsin
0, [kj.R-(J+j’~/2
+ qjsPP],
(9)
where qj,r* are phase shifts. By calculating the distorted wave integra!s, which are the matrix elements of the potential between these distorted wavefunctions, we can evaluate the S matrices in either the distorted wave (see ref. [13] p. 116-117) or exponential distorted wave (EDW) [1] approximations. The S matrix is unitary in the EDW approximation, as is required by the conservation ofparticle flux, and furthermore it has been shown [l] that the EDW approximation corresponds to a resummation of the distorted wave Born series and includes, even in the first order approximation, contributions from all of the higher order terms of the distorted wave Born series. The expression for the S matrix elements in the CDEDW approximation is: Sfpt,ip = exp(ivj.p.) [exdiAJ)l j,p,jp exp(iyJ hplrs, (10) where: ifi’ =i;
A]$ iN = 0,
9
AfP j,, = -2(kjljJ1”
I
01) X f Q/p(R) $p,jp(R)$(R) 0
*a
otherwise.
It is interesting to compare the work involved in applying the EDW and CDEDW approximations. Let us consider the calculation of a single partial cross section, P!, _jT fnr an atom-homonuclear diatom collision using a baslsJ = 0,2, __.,jma?tand suppose that Jai,,. In the full EDW method the total number of channels (j, I combinations) NEDw is given by: jmax NED\’ = C (2j+l) = (jmax+l)(jm,/2+l). j=o,2.. _
(14)
These channels may be divided into those of even and those of odd parity, and the two sets are not coupled to each other. For even values of J the number of even parity channels is Neven = (jm,,2+1)2 and the number of odd parity channels isNodd = &.nax/2+l)(jmax/2), while for odd J the numbers of even and odd parity channels are reversed. The total number of distorted wave integralsNfDw which must be evaluated is therefore: NEDW =&._(N,,,-U/2 I
+&&&-U/2
=j mau(imax+2) [(i,~.+Wm,/2+lHl
18. (15)
In the CDEDW case the problem divides up into separate problems for each helicity, p. For even hehcities there are (j,,/2+1--u/2) coupled channels while for odd helicities there are (jm,/2 - (n-1)/2) coupled channels. The total number of distorted wave integrals which have to be evaluated in the CDEDW method is therefore hnax
The expressions for the integral cross section, and for the dimensionless partial cross section in the CDEDW approximation are: u;:j. = [ir$(2j+1)]
2.2. Relationship of CDED W TOED Wapproximation
F (zJ+l) c ls;P,iP - 6#, P 02)
p=. 7 (i,,12tl-~~2)Cim,-lr)/4 *iCDEDW= E,-,. .. tirn&&)
+
c
p=1,3,..
.
(im,-r-rtWm,-~-W (16)
L. Eno, G.G.BalidGuti/Rotationally
438
Hence we find that:
inelastic scattering
Table 1
iVCDEDWIN,EDW = 2&,+1)/3 I
[(i,,+2)2-2].
As an example, if we use a basis set ofj=0,2,4.. then
(17) .12
IPEDw/$mw = l/22. (18) I For this basis, therefore, the number of integrals which we need to calculate in the CDEDW method is a factor of 22 fewer than the number needed in the EDW method. Besides this saving in computational effort there is a further substantial saving arising from the fact that in the CDEDW method we have to exponentiate several small matrices (one for each helicity) while in the EDW method we must exponentiate two much larger matrices, which requires considerably more computational effort.
3. Application to Ar + N, In order to test the validity of the CDEDW method we have applied it to the problem of rotationally inelastic scattering in the Ar + N, system. We chose thii system because a larger number of exact [14,15] and approximate [14-171 calculations have already been reported in the lietrature onit and it is, therefore, possible to make detailed comparisons with them. AU of these calculations used the same empirical potential
Comparison of different methods for the calculation crqss sections in Ar+Nz collisions at Etot = 300 K
of integral
Ui,i’ @*I
j-q
0 2 0 4
cc (ref. [14])
Ef.
22.5 22.4
22.2 22.8
[16])
~~~~~rlcj
pf.
23.2 25.2
22.3 18.8
[14])
son is made with close coupling (CC) [14], coupled states (CS) [16] and infinite order sudden (IOS) [14] results from the literature. The partial cross sections for the 0 + 4 transition are shown in fig. 1. From the table we see that the agreement of the CDEDW cross sections with the exact ones is good and is comparable to the IOS results. Fig. 1, however, &LOWS that the CDEDW partial cross section agrees far better with the exact CC partial cross section than does the IOS one. Indeed, the CDEDW partial cross section is of comparable accuracy to the CS. In general the IOS method is expected to break down for sufficiently low collision energies or for sufficiently large Aj transitions. This is because the IOS method assumes that all the channels
[181 V(R$) = E[(R,,/R)‘~ - 2(R0/R$] + da,2(R0/R)‘2
- 2a6(R0/R)61~z(cosQ,
where E = 1.0297 X 10e2 eV,R,-, = 3.929 A, aI2 = 0.5 and a6 = 0.13. Three sets of calculations were performed corresponding to energies of&,, = 300 K, 618 K and 768 K respectively, relative to thej = 0 rotational state:In every case all the open rotational channels were used in the basis set, i_e_jmz = 8,12 and 14, respectively. AU the necessary distorted wave integrals were calculated exactly using a fast numerical technique recently developed at Bristol [19]. In the first set of calcuiations (E,, = 300 K) total and partial cross sections were calculated forj = 0 + j’ = 2 andj = 0 + j’ = 4 transitions. The results for the total cross sections are presented in table 1 where compari-
I Oo
10
20
30
Total Angular Fig. 1. Dimensionless pztial
4.0 Momentum,
50
60
70
J
cross sections for thei = 0 -+I = 4
rotational excitation in At + N2 collisions at Etot = 300 K. Solid tines and filed circles are CC results [ 141; open triangles are
CS[16]; crosses xe CDEDWand open cixIes are IOS [14].
L. Eno, G.G. Bali&Kum~Rotationally
Table 2 Comparison of different methods for the calculation of integsal cross sections aO*_i) (in R*) at EtGt = 618 K. The numbers, in descending order, correspond to CC 1151, CDEDW and IOS[151results. The square bracketed CC results are considered to be outside Pack’s 4% accuracy limits j=6
j=8
(3.31
1.10
3.1
1.19 0.98
0.40 0.34 0.29
5.3 5.3 5.2
2.50 2.20 2.10
4.5
[16-O] 15.0 22.3
10.5 10.4 11.7
13.4 17.9 19.1
15.7 20.4
11.4 12.3 12.7
11.4 11.4 13.5
13.4 13.3 16.8
4.5 3.8 4.9
5.8 5.1 7.1
7.9 7.7 10.5
5.7 5.5 5.6 10.9 10.2 11.8
11.7 11.0 15.4
are degenerate and this assumption breaks down in the two situations just mentioned. In table 2 inelastic integral cross sections are presented at Etot= 618 K. The CDEDW cross sectons are
compared with CC and 10s cross sections calculated by Pack [15]. The higher energy in this case should favour the IOS method. It is, however, clear from an examination of the entire table that the CDEDW results are definitely closer to the exact CC values than are the IOS results. Fimahy we examined the CDEDW method at the highest energy for which there are comparison calculations i.e. E,, = 768 K [IS-171. In table 3, inelastic integral cross sections are presented at this energy and compared with CC [ 151, CS [16], IOS [ 151 and phelicity decoupled (PHD) [17] results. For highj transitions (lower right hand corner of table) the CDEDW and CS results are comparable in accuracy, while for transitions involving low j quantum numbers the CS results are generally somewhat better. Overall the CDEDW cross sections agree better with the CC results
439
inelastic scattering
than do the IOS cross sections. A particular point to notice is that the percentage accuracy of the CDEDW results is symmetric about the diagonal (i.e. the percent error in ~$2~6) is the same as that in ~(6~2)). This is not, on the other hand, the case for the IOS results. The 10s results are significantly better in the top right hand half of the table than in the lower left hand half. It should be noted also that this implies an absence of the detailed balancing requirement from these results, although it would be possible to implement the IOS method in such a way as to satisfy detailed balancing. At this energy (768 K), there are several partial cross section plots available in the literature [ 15,16]_ In figs. 2,3 and 4 we compare our CDEDW partial cross sectionsforthej=6+j’=8,j=lO+j’=6andj= 10 +j’ = 8 transitions with those available from the literature. The CDEDW partial cross sections are, in every case, very similar to the CS and PHD partial cross sections, and share with them a tendency to have an exaggerated maximum and to fall off too sharply at large 1. Where comparison is possible with the IOS method (figs. 3 and 4) the CDEDW partial cross sections are defmitely in better agreement with the CC results. The centrifugal decoupling approximation is expected to work best at small J values while the EDW approximation should work well at large J. The errors in the CDEDW results at large J are, therefore, due to the centrifugal decoupling approximation.
The 10s
approximation, which is expected to work best for low J values, is seen from figs. I, 3 and 4 to have substantial errors in this region.
4. Discussion ‘Ibe centrifugally decoupled exponential distorted wave (CDEDW) method, developed in this paper, substantially reduces the computational effort required in the application of centrifugally decoupled approximations (such as the coupled states (CS) and p-helicity decoupled (pHD) approximations). Our results demonstrate that, for the Ar + N2 system, the CDEDW cross sections are, nevertheless, of comparable accuracy to those calculated using the more laborious CS and PHD methods. The computational labour needed to calculate a cross section using the CDEDW method is comparable to that required using the 10s method. The comparisons pre-
440
L. Eno, G.G. BaIint-Kurh~Rotctbnally
inelastic scattering
Table 3 Comparison of diffe:ent methads ior the calculation of b&rat cross sections a(jj+) (in A’) at f&t = 768 K. The numbers in descending order correspond to CC [IS] ; CS [ 161, CDEDW and 10s [ 15 ] results. The numbers in parentheses next to the CS re suits are PHD 1171 results whet&available. S&re bracketed CC results are considered to be outsidePack’s 4% accuracy limits i=O
I
-j&
2
j=4
t3.21 3.2 2.9 3.5
0
[15.7] 15.2 14.2 17.5
2
&lo
j=8
1.37 1.43 1.84 1.64
0.99 1.02 1.18 0.95
0.46 0.47 0.53 0.36
0.094 0.094 0.066 0.093
8.4 8.7 8.4 9.3
4.7 4.9 4.7 4.6
2.59 2.62 2.31 2.22
0.74 0.74 0.56 0.76
9.7 9.88 9.4 9.8
5.3 5.2 5.2 4.9
2.35 2.35 2.04 2.48
9.7 9.5 9.4 9.8
5.2 4.9 (4.7) 4.7 5.1
4
11.4 11.9 ,153 14.8
14.4 14.8 14.3 16.7
6
10.8 11.2 129 12.4
105 11.0 10.5 11.9
12.8 13.0 12.3 14.1
8
5.7 5.8 5.0 6.1
6.6 6.6 5.9 7.5
7.9 7.7 7.8 9.2
1.15 1.17 G.81 1.96
I.86 1.87 l-44 3.2
3.5 3.3 3.0 5.8
10
j=6
11.0 10.7 10.7 12.8
10.9 9.6 (10.2) 9.5 9.9
5.9 5.5 (5.3) 5.3 8.2
10.8 9.5 (10.1) 9.4 12.2
‘d
70 lx-
CCL 0
$30. k
A -l?HD. x -CRERW.
a
ZS r B20. 8 e *g 10. Px a
0
.
2 0
20
40 Total Angular
60 00 M0mentum.J
100
Fig. 2. Dimensionlesspartial cross sections for the j = 6 +j’ = 8 rotational excitation in Ar + Nz collisions at Etot = 768 K. Solid lines and filled circles are CC results [ 151; open triangles are CS [16] and crosses are CDEDW.
20 40 Total Angular
d 60 Momentum,
80 J
Fig. 3. Dimensionless partial cross sections for the j = 10 +j’ = 6 rotational transition in Arc Nz~coUisions at E,, = 768 K. Solid Lines and fiied cricles are CC results [ 15 ] ; open triangles are PHD [l?‘]; crosses are CDEDW and open circles are IOS[lS].
L. Eno. G.G. Bolint-KurrilRototionaNy
inelastic scattering
441
Acknowledgement The authors thank the S.R.C. for the provision of computer time on the Rutherford Laboratory computer and L.E. acknowledges the financial assistance of an S.R.C. studentship. They would like to thank Dr. R. Saktreger for the use of his computer program and are grateful to Dr. M. Shapiro for some helpful discussions.
References
60
60
100
Total Angular Mcmentum,J Fig. 4. Dimensionlesspartial crosssectionsfor the j = 10 +I’ = 8 rotational transition in AI + Na colIisions at Etot = 768 K. Solid line and f&-d circles are CC results [15], open triangles are PHD [ 171, crosses are CDEDWand open circlesare 10s [ 151,
sented in the paper show that the CDEDW cross sections for Ar + N2 are in general more accurate than the 10s ones, this comment being especially true with respect to partial cross sections. In other papers [20,21] we have presented approximate and entirely analytic methods for the calculation of distorted wave integrals. In suitable cases, these analytic methods could be readily combined with the present approach to provide an even more efficient method which would be applicable to situations requiring the use of very large basis sets. In summary the CDEDW method has been shown to be a reliable and efficient method for the computation of rotationally inelastic cross sections. In terms of computational effort and accuracy it is certainly competitive with the IOS method and at low collision energies should be much more reliable. Following suggestions made above an entirely analytic form of the method could be implemented. ‘This form is expected to be of great utility in handling intractable problems requiring the use of very large basis sets.
[ 11 G.G. Balint-Kurti, In: International review of science, Phys. Chem. Ser. II, Vol. 1, eds. A.D. Buckingham and C.A. Coulson (Butterworths, London, 1975). [2] H. Rabitz, in: Modern theoretical chemistry HI, ed. W.H. Miller (Plenum Press, New York, 1976).
[3] P. Mffiuire and D.J. Kouri, J. Chem.F’hys.60 (1974) 2488. [4] M.Shapiroand M.Tamir, Chem.Phys. 13 (1976) 215. [S] A.E. DePristo and M.H. Alexander. J. Chem. Phys. 63 (1975) 3552. [6] G.G. Balint-Kurti and R.D. Levine, Chcm. Phys. Letters 7 (1970) 107. [7] R-D. Levine and G.G. Balint-Kurti, Chem. Phys. Letters 6 (1970) 101. [S] T.P. Tsein and R-T Pack, Chem. Phys. Letters 6 (1970) 54,400; 8 (19713579. [9] D. Secrest, J. Chem. Phys. 62 (197.5) 710. [IO] R.T Pack, J. Chem. Phyr 60 (1974) 633. [11] R.B.Walkerond J.C. Light,Chem.Phyr 7 (1975) 84. [12] J-M. Launay, J. Phys. B9 (1976) 1823_ [13] MS. Child, Molecular collision theory (Academic Press, New York, 1974). [ 141 T.P. Tsein, G.A. Parker and R.T Pack,J. Chem.Phys.59
(1973)5373. [151 R.T Pack,J. Chem. Phys. 62 (1975) 3143. [I61 P. McGuire,Chem.Phys. 13 (1976)81. [171 M. Tamir and M. Shapiro, Chem. Phys. Letters 39 (1976) 79. 1181 M.D.Pattengill, R.A. LaBudde, R.B. Bernstein and C.F. Curtiss, J. Chem. Phys. 55 (1971) 5517. [19] R. Saktreger’and G.G. Balint-Kurti, to be published. [20] L. Eno, G.G. Balint-Kurti and R. Saktreger, Chem. Phys. 29 (1978) 453. [21] L. Eno and R. Saktreger, Mol. Phys. 35 (1978) 601.