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Neumann Series Solution for the AtomRigid Rotor CollisionT. J. O'Brien and T. T. Holloway Citation: Journal of Mathematical Physics 13, 1485 (1972); doi: 10.1063/1.1665867 View online: http://dx.doi.org/10.1063/1.1665867 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/13/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Body frame close coupling wave packet approach to gas phase atom–rigid rotor inelastic collisions J. Chem. Phys. 90, 241 (1989); 10.1063/1.456526 Wave packet study of gas phase atom–rigid rotor scattering J. Chem. Phys. 89, 2958 (1988); 10.1063/1.455001 Rotationally inelastic collisions of LiH with He: Quasiclassical dynamics of atomrigid rotor trajectories J. Chem. Phys. 81, 1682 (1984); 10.1063/1.447894 Virial theorem for inelastic molecular collisions. Atom–rigid rotor scattering J. Chem. Phys. 73, 3823 (1980); 10.1063/1.440613 Molecular Collisions. XV. Classical Limit of the Generalized Phase Shift Treatment of Rotational Excitation:Atom—Rigid Rotor J. Chem. Phys. 55, 3682 (1971); 10.1063/1.1676649
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Neumann Series Solution for the Atom-Rigid Rotor Collision*
T. J. 0 'Brien and T. T. Holloway Department 0/ Chemistry, Texas Tech University, Lubbock, Texas 79409 (Received 17 February 1972; Revised Manuscript Received 15 May 1972)
The solutions of the coupled differential equations arising in the quantum mechanical discussion of the collision of an atom with a rigid, rotating diatom are written as Neumann series, i.e., expanded in terms of spherical Bessel functions. The coefficients of these series are generated by a set of coupled recursion relations. The formalism is limited to potentials less singular than r- 2 at the origin.
1. INTRODUCTION
The exact quantum mechanical description of the collision of a structureless particle with a rigid rotor has been reduced to solving an infinite set of coupled equations. 1 In the close-coupling approximation, this infinite system of equations is truncated to a finite set which is then solved numerically.2 As the size of the truncated set is increased, the exact solution is approached, but the computational effort becomes enormous.
Recently Gersten3 presented an analytic method for representing, as an expansion in spherical Bessel functions, the solution of the radial Schrodinger equation describing a particle scattered by a spherically symmetric potential. The coefficients in this expansion are generated by a recursion formula. Thus, in the region in which the series converges, the solution can be generated very rapidly in a convenient analytic form.
In this paper we generalize Gersten's method to obtain a solution of the close-coupled equations of rotational excitation. Again the solutions are represented as series of spherical Bessel functions and the coefficients are generated by a set of coupled recursion formulas. In Sec. 2 this more general approach is applied, in detail, to the simple case of spherically symmetric scattering. Without explicit use of a second expansion procedure, we obtain a result entirely equivalent to Gersten's but of a different form. The method is applied to the coupled equations in Sec. 3. Section 4 contains the algorithm for the determination of the phase shift, and Sec. 5 contains a discussion of the significance and usefulness of the method.
2. THE RADIAL EQUATION FOR SPHEmCALLY SYMMETmC SCATTERING
The basic formulas used in this derivation are slight modifications of those discussed by Watson4 : Iff(z) is analytic inside and on the circle I z I = R, and if C denotes the contour formed by this circle, thenf (z) has the representation
( )1/2 00
; z If(z) = nL;;o anjl+n (z), (2.1)
where
an = (27Ti)-1 §c f(z)An.1+1/ 2 (Z)dz (2.2)
and Z is an arbitrary, nonnegative integer. Here h(z) is the regular spherical Bessel function,5 and
[nI2] ) A (z) = 2n+m(n + rn) :6 r(n + rn - v Z2v (2.3)
n,m zn+1 v=O 22vv!
is a Gegenbauer polynomial. 6
The radial Schrodinger equation for the spherically symmetric case is
1485
subject to the boundary conditions tP1(0) = 0 and
tPk) r::;'oo A[sin(kr - iZ7T) + tan1Jlcos(kr - il7T)]. (2.5)
For potentials less singular than r- 2, the origin is a regular singular point and tPl (r) = O(rl+1). The representation
0()
tPk) = kr :6 f/; jl+n(kr) (2.6) n=O
satisfies this condition and hence the boundary condition at the origin.
Because krj l+n (kr) is a solution of the homogeneous (U= 0) form of Eq. (2. 4), with l replaced by l + n, inserting the expansion (2.6) into Eq. (2.4) yields the expression
0()
:6 [(l + n)(l + n + 1) - l(l + 1)]f/; jl+n(kr) n=O
0()
= :6 f;'r 2U(r)jl+m(kr). (2.7) m=O
Up to this point the derivation is identical to that presented by Gersten. However, his method of obtaining a recursion relation for thef/; is not applicable to the more general case of coupled equations. We will rederive what is essentially the same result but using a method which can also be used in Sec. 3 of this paper.
We note that, by setting kr = z, Eq. (2. 7) is of the form of Eq. (2. 1) with
an = [(l + n)(l + n + 1) - l(l + 1)]f; (2.8) and
f(z) = (:Y/2 k- 2 Eo
f;"Z2-1 U~) h+m(z). (2.9)
But combining Eqs. (2. 2) and (2.9) gives an alternative expression for an:
For potentials which can be expanded as 0()
U(r) = :6 ullrll , 11=-1
(2.11)
the integral in brackets in Eq. (2.10) can be evaluated by the residue theorem (see Appendix). The significant result is that the integral vanishes for rn> n - 1. Thus the rhs of Eq. (2. 10) is a finite sum. Using the explicit result for the integral when rn~ n - 1, Eq. (A3), and equating Eqs. (2.8) and (2.10) yields the previously derived3 recursion relation for thef/;:
J. Math. Phys., Vol. 13, No. 10, October 1972
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1486 T. J. O'BRIEN AND T. T. HOLLOWAY
[(1 +n)(l +n + 1)-1(1 + l)]f'[ n-l
= (n + 1 + 1.) '\' II C I 2 L.J m mn' m=O
(2.12)
where
I<
X~ u=O v! (K - V)! r(m + 1 + K + ~ - v)
(2.13)
and N = [(n - m - 1)/2], the largest integer contained in (n - m - 1)/2. For an arbitrary choice offd, which corresponds to a normalization constant, these equations can be used to generate, recursively, the set of coefficientsf,!.
3. THE COUPLED RADIAL EQUATIONS FOR ASYMMETmC SCATTERING
The generalized method of the preceeding section, which we used to obtain a known result although in a somewhat different form, will be used here to reduce the set of coupled differential equations occurring in the theory of rotational excitation to a system of coupled recursion relations. These equations 1 we write in the form
(~ + k2 _ 1(1 + 1)) lJI. (r) dr2 } r 2 } I
= ~ (jl;Jj U(r) Jj'l';J)lJIj'I,(r), (3.1) j' /I
where
kl = 2J.!n-2 E - J.!j(j + 1)// (3.2)
j,l and J are the quantum numbers specifying, respectively, the angular momentum of the rigid rotor, the orbital angular momentum of the atom and the total angular momentum of the system, J.! is the reduced mass of the atom-diatomic system,1 is the moment of inertia of the rigid rotor, and E is the total center-of-mass energy. The boundary conditions are
lJIj1(0) = 0 (3.3) and
lJIjl(r) r:'oo Ojjo 0110 exp[- i(kjr - ~l1T)]
- (kj/kj )l/2 SJ(jl,jolo) exp[i(kjr - ~11T)]. (3.4)
Because of Eq. (3. 4), lJIj I should also be labeled by J,jo, and 10 , To reduce clutter, however, this extra notation has been suppressed.
The 1/I's can be represented as 00
1/Ijl(r) = kjr ~ fjl iz+n(kjr). n=O
(3.5)
This is a valid representation for the regular solution [Eq. (3. 3)] in some region about the origin, provided the potential matrix elements are less singular than r- 2 • Inserting Eq. (3. 5) into Eq. (3. 1) yields
~ (1 + n)(l + n + 1) -l(l + l)]f~liz+n(kjr) n=O
J. Math. Phys., Vol. 13, No. 10, October 1972
By setting !?jr = z, this equation is of the form of Eq. (2.1) with
an = [(1 +n)(l +n + l)-l(l + 1)lt!1 (3.7) and
f(z) = (2/1T)1/ 2z- 1 ~ ~ k, k.-3f~'I' j'I'm=O J J
xz2(jl;Jj U(z/kj)jj'l';J)jl'+m(kj,z/kj ). (3.8)
Inserting Eq. (3. 8) into the integral expreSSion for an' Eq. (2. 2), gives
00
an = (2/1T)1/2 'B ~ k.,k-. 3f!'I' j'l' m={) J } m
X ~21Ti)-1 § C z2-~(il;JI U(~) Ij'l';J)
( k.,Z) J X jl'+m ~j An,I+1/2(z)dz. (3.9)
As before, using the expansion 00
(jl;Jj U(r)jj'l';J) = ~ ul1
(jl,j'l';J)rI1, 11=-1
(3. 10)
the contour integral in Eq. (3.9) can be evaluated (see Appendix). In this case the integral vanishes for m> 1 - l' + n - 1. For m -s, 1 -1' + n - 1, an explicit result can be obtained, Eq. (A 7). Thus the sum over m in Eq. (3.9) is finite and can be readily evaluated. Equating this expression for an to Eq. (3.7) yields the set of coupled recursion relations for thefj/:
[(1 + n)(l + n + 1) - 1 (1 + 1) ]J~I l-I'+n-l
= (n + l + ~ )k~n-I-1 ~ L; ft l' Cmn(jl,j'l';J), J j'l' m=O
where
C (jl j'l"J) = 2/- I'+n- mkl'+m+1 mn , , j'
X l~o (k~') 21< U l-I'+n-m-21<-2 (jl,j '1'; J)
X t (-1)K-U(k/~,)2ur(n +1 +}- v)
u=0 v! (K - v)! r(m + l' + K + 1 - v) •
Here K = [(1 - l' + n - m - 1)/2] and M = min{K, [n/2]).
(3.11)
(3. 12)
At first sight Eq. (3.11) does not appear to be a truly recursive relation for thefjl of channel (ji). The maximum value of m attained on the rhs is 1 - l' + n - 1, so that, for 1 > 1', the values of rn can become greater than n, the index of the lhs of Eq. (3.11). Thus, for 1 > 1', f/,;l' is required for m > n in order to calculate f~l. However ft,;l' is a coefficient of the wave function of a different channel (1' '" l). The coefficients in this (j'l') channel are also generated by a recursion relationship obtainable from Eq. (3.11) by interchanging (jl) with (j'l'). In order to obtain ftl ', fl/:. is needed only up to m = l' -l + n - 1, which is less than n since 1 > 1'. Thus the series for the f~ 'I' can be "run ahead" to larger values of n. On each recursion of the system, by always using Eq. (3.11) for the smallest value of 1 first, f/:. 1' will be available when needed.
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NEUMANN SERIES SOLUTION 1487
It should also be noted that closed channels (kj imaginary) present no formal difficulty aside from the requirement of complex arithmetic. Since the analysis was performed in the complex plane there is no requirement that kj be real. In particular, the series expansion for jz(kjr), which was used in the Appendix, is valid for complex values of the argument.
4. EVALUATION OF THE PHASE SHIFTS
From the representation (2.6) and the boundary condition (2.5), since jl+n (kr) ~ sin(kr - ~l1T - ~n1T)/(kr),
tan1]z =-(Eo (-I)ni~n+l)·C~ (- l)ni~n)-1. (4.1)
As it must be, this result is independent of the choice ofid. However, Eq. (4.1) is only formal. In fact (see Sec. 5), the radius of convergence of Eq. (2. 6) may be smaller than the range of the potential, in which case the method of this paper could not be used to determine the phase shift at all. However, even if the radius of convergence is large enough so that the wavefunction has assumed its asymptotic form [Eq. (2.5)] before the series diverges, Eq. (4. 1) makes the further assumption that jl+n (kr) has assumed it asymptotic form for all values of n which contribute.
Writing Eq. (2. 5) more precisely as
(4.2)
where a is the range of the potential and Yz is the spherical Bessel function of the second kind,5 the phase shift is given by the more complicated expression
kRj/kR) - D(d/dr)[krit(kr)] Ir =R
tan1]z = kRyz (kR) - D(d/dr)[krYz(kr)]l r =/
where
(4.3)
00 (00 d I )-1 D = kR P/~jz+n(kR). Eoi~ dr [krjz+n(kr)] FR '
(4.4)
and R is some value of r outside the range of the potential but inside the radius of convergence of the series (2.6).
This same conSideration applies to the coupled equations. As in any close-coupled calculation,7 N linearly independent sets of solutions to the N equations must be combined to satisfy the boundary condition (3.4). These sets of solutions tJ;/j' i = 1, ... ,N, are obtained by choosing N linearly independent sets of values for the ig. For large r, each function of these N sets has a form similar to Eq. (4. 2) or (2.5):
tJ;h(r) rC::oo AL [sin(kjr - ~l1T) + tan1]1j cos(kjr - ~l1T)]. (4.5)
The tJ;jj must then be combined7 in the usual way to satisfy Eq. (3.4) or, equivalently,A}j and 1]~j must be combined to form the SJ(jl,jolo).
5. DISCUSSION
We have presented a method for generating analytically the solutions of the coupled, differential equations appearing in the rotational excitation problem,
Eq. (3.1). The coefficients of an expansion of the solutions in terms of spherical Bessel functions can be generated recursively. Thus the close-coupled solution can be obtained rapidly and in a convenient form.
The method is of the nature of a power series expansion in terms of the radial variable r. However, the series is rearranged to display the spherical Bessel functions, a natural function for the problem. Because of this the convergence properties of the series should be improved. The rate and radius of convergence will determine the utility of the method. We are analyzing the convergence properties of the series for several model potentials. Unfortunately it does not seem that any simple conclusions can be drawn. We will present a detailed discussion of this complicated question in another paper.
From preliminary work, it appears that the method can also be extended to the vibrational excitation problem, both collinear and three dimensional. We plan to investigate this possibility further.
APPENDIX
According to the residue theorem,8 the value of the contour integral in Eq. (2. 10) is the sum of the residues of the integrand at its Singularities within the contour. Since z-Zjz+m (z)is entire and, under the assumptions on the potential, z 2 U(z/k) is analytic in some region containing the origin, the only singularities are due to the pole of An.Z+1/2 at z = O. Thus by expanding the integrand in powers of z, the residue (and hence the integral) can be found as the coefficient of z-l. This coefficient can be obtained by repeated application of Cauchy's formula for the product of two series. 9 Proceeding in this way, we obtain
I(z) == z-Z+2 U(z/k)jz+m(z)A n.Z+1/ 2 (z)
[n/2] r(n + l +.! - v)z2v X B 2
v=o 22vv!
(1f)1/2 = 2 (n + l + ~) 2n-mzm-n+l
x (~Z2Jl K~ s~:,: + Jl~ Z2Jl-\~ S~;~1.K)' (AI)
where
SZnm = uO - 2K is (- I)K- v r(n + l + ~ - v)
a,K 22Kka-2K v-O V!(K - v)!r(l + m + K + ~ - lJ) (A2)
and M = min(K, [n/2]).
Setting m - n + 1 + 2/J = - 1 or m - n + 1 +211 - 1 = - 1 yields the value of fJ. corresponding to the z-l
J, Math. Phys., Vol. 13, No. 10, October 1972
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1488 T. J. O'BRIEN AND T. T. HOLLOWAY
term: fJ. = [(n - rn - 1)/2] == N. Since fJ.?: 0, there is no residue for rn > n - 1. Further, N satisfies the condition N:::: [n/2] so that, in all cases, M = K. Thus
(27Ti)-1 j I(z)dz C N
= (7T/2)1/2(n + l + ~) 2n- m 6 slnm (A3) ",0 n-m-2,K
for rn :::: n - 1 and vanishes identically otherwise.
The residue of the integrand in Eq. (3. 9) can be evaluated in much the same way:
I (z) == z2-1(jl;JI U(z/kj )Ij'I';J)jz'+m(kj ,z/kj )A.,1+1/2(z)
= (7T/2) 1/2 21- 1'+n-m (n + I + ~)(kj,jkj)l'+m
X Z 1'-I+m-n+1( ~ z 21l t S;Il~K (jl,j'I';J) jFO 1(=0
(A4)
where, setting M = min{K, [n/2]1,
• Work supported in part by a Robert A. Welch Foundation grant. 1 A. Arthurs and A. Dalgarno, Proc. Roy. Soc. (London) A256, 540
(1960), 2 W.A. Lester, Jr., in Methods in Computational Physics (Acade
mic, New York, 1971), Vol. 10, p. 211; B. R. Johnson and D. Secrest, J. Chem. Phys. 48,4682 (1968); W. N. Sams and D. J. Kouri, ibid. 53,496 (1970); R. G. Gordon, ibid. 51, 14 (1967).
3 A. Gersten, J. Math. Phys.12, 924 (1971). 4 G. N. Watson, Theory of Bessel Functions (Cambridge U.P.,
Cambridge, 1966), Chap.16. S M. Abramowitz and 1. A. Stegun, Handbook of Mathematical Func-
UO
- 2K (jl ,j 'I'; J) (j1,j'I';J) =
ko-2K22K J
M (- 1)K-V(kJ'/kj )2K-2u r(n + I + ~ - Il)
x6 u=O (K - Il)! Il!r(l' + rn + K + % - Il)
(A 5)
According to Eq. (A4), the value of fJ. corresponding to the z-1 term is given by [' - I + m - n + 1 + 2fJ. = - 1 or I' - I + rn - n + 1 + 2fJ. - 1 = - 1, that is,
fJ. = [([ -[' + n - m - 1)/2] == K. (A6)
The condition fJ. ?: 0 implies that the residue vanishes for rn ?: [ - I' + n. Thus
(27Ti)-1 f I (z )dz u C
= (7T/2)1/2 21- 1'+n-m(n + I + ~)(kj,/kj)l'+m k
X 6 S~~m+ 1-1'-2 K (j[ ,j 'I'; J) K=O '
(A7)
for rn :::: 1- l' + n - 1 and vanishes identically otherwise. Insertion of this result in Eq. (3.9) gives the coupled recursion relation (3. 11).
lions (U.S. Department of Commerce, Natl. Bur. Stds., Washington, D.C., 1964), p. 437.
6 Reference 4, p. 283. N.B.: this is a different polynomial from the orthogonal Gegenbauer polynomial found in most texts.
7 W. A. Lester, Jr. and R. B. Bernstein, J. Chem. Phys. 48,4896 (1968); K. Smith, Argonne National Laboratory Report ANL-6095 (1960).
8 R. V. Churchill, Complex Variables and Applications (McGrawHill, New York, 1960), p.155.
9 Reference 5, p.15, Eq. 3. 6. 21.
A Geometrical Theory of the Electromagnetic Field and the Gauge Transformation Antonio F. Raiiada
De/Jar/amenlo de Fisica TeoYica, Uniuersidad de Madrid, Madrid 3, SPain (Received 14 March 1972)
A modification of Weyl's theory or the electromagnetic field is presented, such that the usual gravitational Lagrangian becomes invariant under the corresponding definition of gauge transformation. As another advantage the problem of the weight of the tensors in the construction of Lagrangians is eliminated.
I. INTRODUCTION n. MIXED THEORY OF THE GRAVITATIONAL FIELD
This paper proposes a geometrical interpretation of the electromagnetic field, closely related to Weyl's theory,1 though it uses a different geometrical structure. The model includes a new geometrical definition of gauge transformation, which makes gauge invariance a weaker requirement than in Weyl's theory. Some important advantages follow. For instance, the usual gravitational Lagrangian becomes gauge invariant and the problem of the weights of the tensors is eliminated.
In order to describe the affine properties of spacetime, we consider the 4-vector fields Xi:
The present model has some common pOints with a theory presented in 1958 by Sciama. 2 He needed, however, a complex space-time of the EinsteinSchrodinger type. It seems that this is not necessary, the present interpretation being, therefore, much simpler.
J. Math. Phys., Vol. 13, No. 10, October 1972
and the differential forms
such that
(1)
(2)
(3)
(4)
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