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R-matrix approach to collinear inelastic collisions: Resonances
Volume
23, number 1
R-MATRIX
CHEMICAL
APPROACH
TO COLLINEAR
PHYSICS
1 November
LETTERS
INELASTIC
COLLISIONS:
1973
RESONANCES
Received 17 June 1973
The quamum mechanical collinear atom-diatom collision problem is treated function in terms of uncoupled. distorted wnve states. using the Wigner-Eisrnbud
via a discrete expansion of R-matrix
the waveformalism. Previously
unreported turrow Feshbach resonances. which reduce the inelastic transition probabi!ity. are easily found and r.wmined. model.
Off resonant
tmilaition probnbiliries are in agreement with the work of Diestlcr
Experience from atomic collision theory has shown discrete, L2 basis set expansion calculations to be superior in many ways to methods, such as direct numerical
integration,
which are tied more closely to approach, the major computational step is often reduced to a single diagonalization, and phenomena such as resonances are easier to handle. Among the L2 methods, one of the oldest is the Wigner-Eisenbud R-matrix formalism
coordinate space. In the expansion
[I-3] Numerical applications of the R-matrix approach ha-le, however, not always met with unqualitied success [4] 1:. Several alternative methods have been proposed [S-S], and some of these have been motivated by a desire to improve an the R-matrix method. Unfortunately, the new methods are often much less efficient and much more cumbersome than the R-matrix approach. As will be seen in the work which follows, the Rmatrix method can be quite efficient and accurate, provided certain conditions are met. These conditions include the availability of eigenfunctions of an uncoupled hamiltonian Ho for use as a basis set, and the stipulation that the coupling potential, rather than the wavefunction be truncated in the basis set (see below). The uncoupled hamiltonian Ho should have the same thresholds and continua as the Full hamil$ f*
Present address: The James Franck Institute, The Univcrsity of Chicago. Chbgo. Illinois 60637, USA. See discussion in refs. [3.5-71.
102
and Feucr using the Y~C
tonian H = Ho + Y. Such conditions
are not met for
reactive collisions but can be enforced for inelastic collisions. The model we wish to study consists of a harmonically bound atom interacting with a projectile atom via a Morse potential. As is well known, this scheme corresponds to a collinear atom-diatom collision in transformed coordinates, and serves also as a crude model of gas-surface interactions. Models similar to this have been well studied, and our main aim is to test the R-matrix method for suitability to more complicated problems. However, the method allows us to extract a new piece of physics from the model: narrotv quasi-bound states or resonances. For a set of parameters used by Diestler and Feuer [9], correspond-, ing to a helium-tungsten collision, we find a resonance of width r = 1Om5 of the tungsten level spacing. We begin with a brief account of the model problem and the R-matrix approach to solving it. Using the standard convenient reduced coordinates, the Schr6dinger equation for an harmonically bound atom interacting with a free atom via a Morse potential is:
+ D(e-2Q(X-~)- 2emQfx-Y))-E
1
$(x,y)
=0 ,
(1)
Volu~-ne23.
number 1
where y = I~I~/II~,, a
CHEMICAL PHYSICS LETTERS = ao(R/tnswS)‘r2,
D = Dg/(fiw&
The energy E is in units of fiw, where ws is the Debye frequency of the target;no and D, are the “physical” inverse length and depth parameters, respectively. l?rs is the mass of the target atom and w, the mass of the free atom. For the parameters we wiil be choosing it as reasonable to rewrite the hamiltonian (I) in the following way:
I -_-a2
I a’
+~(e-24.~_~e-a.r)
+ +??
(2) The last quantity in brackets is the coupling potential V(x,j). The eigenfunctions of&, are products of oscillator target states and Morse functions:: ti;,,t (%Y) = #&9 x,,I 6x1 .
(3)
These are distorted wave states suitable for use in the Born approximation (DWBA). We use them here in the R-matrix formalism to obtain a more accurate description of rhe scattering. Depending on the parameters, the Morse functions may have bound states but there is always a continuum which must be dealt with. If an artificial boundary condition is imposed on the x’s at a sufficiently large but finite radius A, then the basis ($&,) will be discrete and yet complete in the interaction region. At relatively low energy, only a limited range of the variable x and a limited number of oscillator states are important in the coupling caused by Y(x,_Y)_The basic R-matrix idea is to solve the coupled problem in the interaction region and match the resulting boundary values of the wavefunction ro,r(lheknown asymptotic forms. In general aU of the f wcnl 3 are coupled by V, so some approximation must be introduced to solve the problem in the interaction region. Rather than truncate the expansion of the ~~uue~~~cr~~iz in terms of the basis functions as is usually done, we instead truncate the
expansion of the tions:
% See
potetztid
1 November 1973
in terms of the basis func-
ref. (IO] and foiiowing papers for o gccd the propotties of Morse 0sciUator functions.
Of course the potential
would be effectively
truncated
if the basis set were truncated, but the distinction between the two types of truncation lies in the way Ho is treated, If the basis set is restricted to iv” terms, then Ho becomes an N,, X No diagonal matrix, which is only an approximation to the infinite diagonal matrix in the basis of the complete set [$~~,l>_ It is important (and easy) to make no approximation to ffo (see discussion of the R" term below) as so we leave the wavefunction expansion complete. The hamiltonian H, + VT can now be solved exactly in the interaction region, using the complete basis set {SF,,, 1. Only the blocks coupled by VT need be diagonaltied, the remainder of the hamiltonian is already diagonal. The resulting wavefunctions are again discrete and satisfy the prescribed boundary condition at .r = A. These discrete functions can be used to construct the full Green’s function which in turn can be used to construct the scattering matrix at any energy. The Bloch L-operator formalism [I I] is
perhaps the easiest route to expressions for the scattering matrix, and in our case, for the boundary condition W;&,YYaxjx=A
(51
= 0,
we get
s cC.=O,i([l
iiRL]-‘[l
+iRL*]}Cc~Ic.,
(6)
with Lcc’ =(Q&Nx=A
S,& -
The cl,, f, are outgoing, incoming waves in channel c and are taken to be ,
(3, = exp(~~~)/(~~)“z k, = {2&!Y-Ec)yz
1, = exp(-~~~)~(~~)“’
)
, (7)
where E, is the channel threshold energy. Most ~portantIy, the expression for the R-matrix is
discussion of
..
‘.
..I -.
:.
Volume
23,
nomhcr
Ct-iLlt:r~~L
1
LETTERS
PHYSICS
0.06
1 November1973
no2
1
/
401
P 22
/
/’ /
L I,94410
0.04
/
i.94414 /
/
J
/
/
Hrc
0.02 t~-i
Fig. 1. Using the same qrametcrs JS in tig. 2, the unccneztcd Pi:! [that is. as obtained from eqs.(5). t?),ond (S), omittinS thcR” term] is shown alonp’with the correct result, obtained by inclltdjnp the R” iaclor. The resorwnce present in fig. 2 Ins been omitted for cbrity.
with Y,, =(zY)-1’2 5 ‘$;,,,th)X,,,(A). ,,I The E,‘s are the eigenvalues of the blocks requiring diagonalization. The ~ndition (5) is quite arbitrary but is the most convenient; more generally it would read - bc~i~.$=A = 0 1_
a{@,,,(4
c=1,3,....{9]
In any approximate calculation there will be a weak, unimportant dependence of the result on the choice of baundary condition. With these definitions, the transition probability is given by P& = IS,,?.
W)
The finite set {IA)) are eigeniunctions of Ho + VT and are linear combinations of those unperturbed states coupled by Yr; the remaining unperturbed states are accounted for in the (diagonal)R” teim, which is easily determined from the uncoupled Rmatrix 131. The only difference between a variatianalty srabli: t~ncaI~d basis set calculation [2] and a truqcated potential cakulation is the R” term. Fig. 1 104
‘,
‘.,
Fig. 2. R-m;ltris and DWSA calculation using reduced urilsr = 0.021775.a = 0.052671.D = 0.132386; the t-usr three channels iwe included and a boundary radius of A = 10 ~‘3s used; 61 uncoupledsr3reswereincluded in the diqon;lliz3don. The two squaresshow the values calculated by Diesrler and Feuer [91. The inserr gives P detail of the resonance, which shows only as a vertical line on the original energy scale. Ttiese cu~vcswere generated from 120 separate cvaiuwhich required 6 set on the CDC 6400 once been determined, which tookabout
shows that this corrtction term has a crucial effect on the accuracy of the result. Further comment should be made regarding the correction term R” : as the density of states f IX>) in. creases, so does the xed for including Rm. Previous model calculations, in nuclear physics for example, have involved coupling only a relativelyfew, widefy separated zeroth order states; in this case the effec! OF Rw is not so noticeabie [3]. Clearly, for the He-W scattering of fig. 1, i: is essential. Unfortunately, in other R-matrix applications, such as reactive scattering, it may be necessary to truncate the wavefunction; this mai be the reason for the disappointingRmatrix reactive scattering results of Crawford 141. If there are narrow resonances present, an Eh will fall very close to the exact resonance energy. There’ are more Ex’s than resonances, but those corresponding to iesonanccs can easily be determined by jnspection of the Yhc’s.If, :bf a given X,Yxc is small far all open,
channelscomparedto typical nonresonantrkC’s5
Volume
23, number
1
CHEMICAL
PHYSICS LETTERS coupling;
c
tI -L
/’
DWBA
c
0 P
12
0.
iRESHoi_D
1 0.0, .o
t
1.16
/
1.32
1 I.46
I
I
t.64
I.00
-E
Fig. 3. R-matrix and DWBA calcuIation =X05445,0 =0.10S34,0=0.59jFi,A and 64 basis functions.
then
that
We
X corresponds
t
1.5b6
I
21 Iz
usingreduced units T = IO, rhrcc channels
to a resonance.
are now in a position to discuss the results
shown in figs. 2 and 3. Fig. 2 displays results of a 4He-W calculation usingunreduced parametersan = LOA-‘$0 = 100 cal/mole, OD = 380°K. We find the same ratio P~~vBA/P~~ct = I .04-l .05 found by Diestler and Feuer 191, except at the resonance occurring at E = 1,944 134 and possessing a width of 1Op5 (Em,). The resonance is of the classic Feshbach closed channel type [ 121: the uncoupled Morse potential.hasa single bound state, and the resonant state consists of the target excited into its (closed) third vibrational level (c = 3), and the projectile trapped in the Morse bound state. This picture explains the depression ofP,, at resonance. since a particle incident in c = 2 say, and trapped in c = 3, must induce a twoquan turn transition to exit in c = 1, which in this weak coupling case is far fess likely than a c = 2 exit. Off resonance, c = 1 is a singte quantum away from c = 2, and c = 3 is not invoived. Since P,, =Pzl 1we need not consider the case of incident channel c = 1. Fig. 3 shows a case involving considerably stronger
The Morse
I November the
DWBA is correspondingly
potential
has two bound
1973
less accurate.
states
(causing
the
resonances) and almost a third (causing the anomalous threshold behavior). As it must, P,z does return to zero at E = 1.0, though this cannot be seen on the energy scale of fig. 3. When a distorted wave basis set is available, the limiting factor in executing a successful R-matrix calculation becomes the size of the hamiltonian matrix to be diagonalized. The dimension is the number of channels multiplied by the number of basis functions per channel. This tends to be large in molecular calculations, but low energy inelastic scattering in polyatomic systems with IO-15 open channels seems to be attainable. The rich resonance structure to be expected and the Facts to be learned concerning energy transfer in these larger systems should make these studies worthwhile. The encouragement and support of Professor William P. Reinhardl is gratefully acknowledged. We thank Professor R-C. Gordon for providing useful computer matrix routines. This work was partially supported by a grant from the National Science Foundation.
References [ 11 A.W
Lane and R.G. Thorna\. Rev. Wed. Phys_ 30 (I 958) 257. [2) AX. Lnnennd D. Robson,Phys. Rev. 178 (1959) 1715. j3j P.J.A. Buttlc.Phys. Rw. l&(1967) 719. [4[ O.H. Crawford, J. Chcm. Phys. 55 (1971) 2563. [5] R-F. Barrett ct al., Rev. Mod. Phys. 45 (1973) 44: C. Mahaus and H.A. Weidenmlillcr. Phys. Rev. 170 (1968) 847. 161 O.H. Crawford, J. Chem. Phys. 5.5 (1971) 257 1. [7] L. Gzrside and W. Tobocman, Phys. Rev. 173 (1968) 1047. IS] J.H. Wearc and E. The%. Phys. Rev. 167 (1968) 11. [Q] D.J. Dies&r and P. Feucr. J. Chem. Phyr. 54 (1971)
4626. [tOi J.E. Lcnnnrd-lone&and C. Scrxhnn, Proc. Roy_ Sot. 150 (1935) 150. [I 11 A.M. Lane and D. Robson, Phys. Rev. 15 1 f1966) 774. (12) H. Feshbach. Ann. Phys. 19 (1962) 287.
23, number 1
R-MATRIX
CHEMICAL
APPROACH
TO COLLINEAR
PHYSICS
1 November
LETTERS
INELASTIC
COLLISIONS:
1973
RESONANCES
Received 17 June 1973
The quamum mechanical collinear atom-diatom collision problem is treated function in terms of uncoupled. distorted wnve states. using the Wigner-Eisrnbud
via a discrete expansion of R-matrix
the waveformalism. Previously
unreported turrow Feshbach resonances. which reduce the inelastic transition probabi!ity. are easily found and r.wmined. model.
Off resonant
tmilaition probnbiliries are in agreement with the work of Diestlcr
Experience from atomic collision theory has shown discrete, L2 basis set expansion calculations to be superior in many ways to methods, such as direct numerical
integration,
which are tied more closely to approach, the major computational step is often reduced to a single diagonalization, and phenomena such as resonances are easier to handle. Among the L2 methods, one of the oldest is the Wigner-Eisenbud R-matrix formalism
coordinate space. In the expansion
[I-3] Numerical applications of the R-matrix approach ha-le, however, not always met with unqualitied success [4] 1:. Several alternative methods have been proposed [S-S], and some of these have been motivated by a desire to improve an the R-matrix method. Unfortunately, the new methods are often much less efficient and much more cumbersome than the R-matrix approach. As will be seen in the work which follows, the Rmatrix method can be quite efficient and accurate, provided certain conditions are met. These conditions include the availability of eigenfunctions of an uncoupled hamiltonian Ho for use as a basis set, and the stipulation that the coupling potential, rather than the wavefunction be truncated in the basis set (see below). The uncoupled hamiltonian Ho should have the same thresholds and continua as the Full hamil$ f*
Present address: The James Franck Institute, The Univcrsity of Chicago. Chbgo. Illinois 60637, USA. See discussion in refs. [3.5-71.
102
and Feucr using the Y~C
tonian H = Ho + Y. Such conditions
are not met for
reactive collisions but can be enforced for inelastic collisions. The model we wish to study consists of a harmonically bound atom interacting with a projectile atom via a Morse potential. As is well known, this scheme corresponds to a collinear atom-diatom collision in transformed coordinates, and serves also as a crude model of gas-surface interactions. Models similar to this have been well studied, and our main aim is to test the R-matrix method for suitability to more complicated problems. However, the method allows us to extract a new piece of physics from the model: narrotv quasi-bound states or resonances. For a set of parameters used by Diestler and Feuer [9], correspond-, ing to a helium-tungsten collision, we find a resonance of width r = 1Om5 of the tungsten level spacing. We begin with a brief account of the model problem and the R-matrix approach to solving it. Using the standard convenient reduced coordinates, the Schr6dinger equation for an harmonically bound atom interacting with a free atom via a Morse potential is:
+ D(e-2Q(X-~)- 2emQfx-Y))-E
1
$(x,y)
=0 ,
(1)
Volu~-ne23.
number 1
where y = I~I~/II~,, a
CHEMICAL PHYSICS LETTERS = ao(R/tnswS)‘r2,
D = Dg/(fiw&
The energy E is in units of fiw, where ws is the Debye frequency of the target;no and D, are the “physical” inverse length and depth parameters, respectively. l?rs is the mass of the target atom and w, the mass of the free atom. For the parameters we wiil be choosing it as reasonable to rewrite the hamiltonian (I) in the following way:
I -_-a2
I a’
+~(e-24.~_~e-a.r)
+ +??
(2) The last quantity in brackets is the coupling potential V(x,j). The eigenfunctions of&, are products of oscillator target states and Morse functions:: ti;,,t (%Y) = #&9 x,,I 6x1 .
(3)
These are distorted wave states suitable for use in the Born approximation (DWBA). We use them here in the R-matrix formalism to obtain a more accurate description of rhe scattering. Depending on the parameters, the Morse functions may have bound states but there is always a continuum which must be dealt with. If an artificial boundary condition is imposed on the x’s at a sufficiently large but finite radius A, then the basis ($&,) will be discrete and yet complete in the interaction region. At relatively low energy, only a limited range of the variable x and a limited number of oscillator states are important in the coupling caused by Y(x,_Y)_The basic R-matrix idea is to solve the coupled problem in the interaction region and match the resulting boundary values of the wavefunction ro,r(lheknown asymptotic forms. In general aU of the f wcnl 3 are coupled by V, so some approximation must be introduced to solve the problem in the interaction region. Rather than truncate the expansion of the ~~uue~~~cr~~iz in terms of the basis functions as is usually done, we instead truncate the
expansion of the tions:
% See
potetztid
1 November 1973
in terms of the basis func-
ref. (IO] and foiiowing papers for o gccd the propotties of Morse 0sciUator functions.
Of course the potential
would be effectively
truncated
if the basis set were truncated, but the distinction between the two types of truncation lies in the way Ho is treated, If the basis set is restricted to iv” terms, then Ho becomes an N,, X No diagonal matrix, which is only an approximation to the infinite diagonal matrix in the basis of the complete set [$~~,l>_ It is important (and easy) to make no approximation to ffo (see discussion of the R" term below) as so we leave the wavefunction expansion complete. The hamiltonian H, + VT can now be solved exactly in the interaction region, using the complete basis set {SF,,, 1. Only the blocks coupled by VT need be diagonaltied, the remainder of the hamiltonian is already diagonal. The resulting wavefunctions are again discrete and satisfy the prescribed boundary condition at .r = A. These discrete functions can be used to construct the full Green’s function which in turn can be used to construct the scattering matrix at any energy. The Bloch L-operator formalism [I I] is
perhaps the easiest route to expressions for the scattering matrix, and in our case, for the boundary condition W;&,YYaxjx=A
(51
= 0,
we get
s cC.=O,i([l
iiRL]-‘[l
+iRL*]}Cc~Ic.,
(6)
with Lcc’ =(Q&Nx=A
S,& -
The cl,, f, are outgoing, incoming waves in channel c and are taken to be ,
(3, = exp(~~~)/(~~)“z k, = {2&!Y-Ec)yz
1, = exp(-~~~)~(~~)“’
)
, (7)
where E, is the channel threshold energy. Most ~portantIy, the expression for the R-matrix is
discussion of
..
‘.
..I -.
:.
Volume
23,
nomhcr
Ct-iLlt:r~~L
1
LETTERS
PHYSICS
0.06
1 November1973
no2
1
/
401
P 22
/
/’ /
L I,94410
0.04
/
i.94414 /
/
J
/
/
Hrc
0.02 t~-i
Fig. 1. Using the same qrametcrs JS in tig. 2, the unccneztcd Pi:! [that is. as obtained from eqs.(5). t?),ond (S), omittinS thcR” term] is shown alonp’with the correct result, obtained by inclltdjnp the R” iaclor. The resorwnce present in fig. 2 Ins been omitted for cbrity.
with Y,, =(zY)-1’2 5 ‘$;,,,th)X,,,(A). ,,I The E,‘s are the eigenvalues of the blocks requiring diagonalization. The ~ndition (5) is quite arbitrary but is the most convenient; more generally it would read - bc~i~.$=A = 0 1_
a{@,,,(4
c=1,3,....{9]
In any approximate calculation there will be a weak, unimportant dependence of the result on the choice of baundary condition. With these definitions, the transition probability is given by P& = IS,,?.
W)
The finite set {IA)) are eigeniunctions of Ho + VT and are linear combinations of those unperturbed states coupled by Yr; the remaining unperturbed states are accounted for in the (diagonal)R” teim, which is easily determined from the uncoupled Rmatrix 131. The only difference between a variatianalty srabli: t~ncaI~d basis set calculation [2] and a truqcated potential cakulation is the R” term. Fig. 1 104
‘,
‘.,
Fig. 2. R-m;ltris and DWSA calculation using reduced urilsr = 0.021775.a = 0.052671.D = 0.132386; the t-usr three channels iwe included and a boundary radius of A = 10 ~‘3s used; 61 uncoupledsr3reswereincluded in the diqon;lliz3don. The two squaresshow the values calculated by Diesrler and Feuer [91. The inserr gives P detail of the resonance, which shows only as a vertical line on the original energy scale. Ttiese cu~vcswere generated from 120 separate cvaiuwhich required 6 set on the CDC 6400 once been determined, which tookabout
shows that this corrtction term has a crucial effect on the accuracy of the result. Further comment should be made regarding the correction term R” : as the density of states f IX>) in. creases, so does the xed for including Rm. Previous model calculations, in nuclear physics for example, have involved coupling only a relativelyfew, widefy separated zeroth order states; in this case the effec! OF Rw is not so noticeabie [3]. Clearly, for the He-W scattering of fig. 1, i: is essential. Unfortunately, in other R-matrix applications, such as reactive scattering, it may be necessary to truncate the wavefunction; this mai be the reason for the disappointingRmatrix reactive scattering results of Crawford 141. If there are narrow resonances present, an Eh will fall very close to the exact resonance energy. There’ are more Ex’s than resonances, but those corresponding to iesonanccs can easily be determined by jnspection of the Yhc’s.If, :bf a given X,Yxc is small far all open,
channelscomparedto typical nonresonantrkC’s5
Volume
23, number
1
CHEMICAL
PHYSICS LETTERS coupling;
c
tI -L
/’
DWBA
c
0 P
12
0.
iRESHoi_D
1 0.0, .o
t
1.16
/
1.32
1 I.46
I
I
t.64
I.00
-E
Fig. 3. R-matrix and DWBA calcuIation =X05445,0 =0.10S34,0=0.59jFi,A and 64 basis functions.
then
that
We
X corresponds
t
1.5b6
I
21 Iz
usingreduced units T = IO, rhrcc channels
to a resonance.
are now in a position to discuss the results
shown in figs. 2 and 3. Fig. 2 displays results of a 4He-W calculation usingunreduced parametersan = LOA-‘$0 = 100 cal/mole, OD = 380°K. We find the same ratio P~~vBA/P~~ct = I .04-l .05 found by Diestler and Feuer 191, except at the resonance occurring at E = 1,944 134 and possessing a width of 1Op5 (Em,). The resonance is of the classic Feshbach closed channel type [ 121: the uncoupled Morse potential.hasa single bound state, and the resonant state consists of the target excited into its (closed) third vibrational level (c = 3), and the projectile trapped in the Morse bound state. This picture explains the depression ofP,, at resonance. since a particle incident in c = 2 say, and trapped in c = 3, must induce a twoquan turn transition to exit in c = 1, which in this weak coupling case is far fess likely than a c = 2 exit. Off resonance, c = 1 is a singte quantum away from c = 2, and c = 3 is not invoived. Since P,, =Pzl 1we need not consider the case of incident channel c = 1. Fig. 3 shows a case involving considerably stronger
The Morse
I November the
DWBA is correspondingly
potential
has two bound
1973
less accurate.
states
(causing
the
resonances) and almost a third (causing the anomalous threshold behavior). As it must, P,z does return to zero at E = 1.0, though this cannot be seen on the energy scale of fig. 3. When a distorted wave basis set is available, the limiting factor in executing a successful R-matrix calculation becomes the size of the hamiltonian matrix to be diagonalized. The dimension is the number of channels multiplied by the number of basis functions per channel. This tends to be large in molecular calculations, but low energy inelastic scattering in polyatomic systems with IO-15 open channels seems to be attainable. The rich resonance structure to be expected and the Facts to be learned concerning energy transfer in these larger systems should make these studies worthwhile. The encouragement and support of Professor William P. Reinhardl is gratefully acknowledged. We thank Professor R-C. Gordon for providing useful computer matrix routines. This work was partially supported by a grant from the National Science Foundation.
References [ 11 A.W
Lane and R.G. Thorna\. Rev. Wed. Phys_ 30 (I 958) 257. [2) AX. Lnnennd D. Robson,Phys. Rev. 178 (1959) 1715. j3j P.J.A. Buttlc.Phys. Rw. l&(1967) 719. [4[ O.H. Crawford, J. Chcm. Phys. 55 (1971) 2563. [5] R-F. Barrett ct al., Rev. Mod. Phys. 45 (1973) 44: C. Mahaus and H.A. Weidenmlillcr. Phys. Rev. 170 (1968) 847. 161 O.H. Crawford, J. Chem. Phys. 5.5 (1971) 257 1. [7] L. Gzrside and W. Tobocman, Phys. Rev. 173 (1968) 1047. IS] J.H. Wearc and E. The%. Phys. Rev. 167 (1968) 11. [Q] D.J. Dies&r and P. Feucr. J. Chem. Phyr. 54 (1971)
4626. [tOi J.E. Lcnnnrd-lone&and C. Scrxhnn, Proc. Roy_ Sot. 150 (1935) 150. [I 11 A.M. Lane and D. Robson, Phys. Rev. 15 1 f1966) 774. (12) H. Feshbach. Ann. Phys. 19 (1962) 287.