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Separable approximation for diatomic transition operators: a variational approach
Volume
CHEMICAL PHYSICS LETTERS
54, number 2
SEPARABLE
APPROXIMATION
A VARIATIONAL
Z.C. KURUOCLU
17 November
OPERATORS:
APPROACH*
and D-4. MICHA
Quunruml71cot-y i?wj~c~,Depnrtnents GainesnIle. Fi’onda 3.261 I, USA. Rcceivcd
FOR DIATOMIC TRANSITION
1 March 1978
of Clremrstry and Physics. (Inivemity of Horida,
1977
A gencr.11 timte-rank separable approximation for the two-body transition operator ha, been obtained from the Lrppmann- Schwingcr vxicltmnal principle. The method involves cup.mGon of the off-shell wavefunction in d finrte set of square integrable function\. and is applied to I x and 3X potcntial5 of 112 for thermal energies and momenta.
I_ Introduction Recent developments [ I,?] f m few-body dynamics suggest ;I new approach to the study of reactive molecular collisions c, which we stlall treat as manybody processes. The motivation behind the manybody approach, whcrc the bodies are atoms or ions, is to take advantage of our knowledge on the simpler atom-atom problem. The input to the theory of three-atom dynamics [5,6] arc the atom-atom transition operators. Specificially we need the so-called “off-energy-shell” matrix elements [7] of tnesc two-body operators in momentum space. After an angular momentum analysis, the Faddccv equations in momentum space become coupled equations in two continuous variables. USC of finite-rank separable expansions for atom--atom transition operators leatis to ;I larger set of coupled intcyral equations in one variable [3] , which ace much nx.xe convenient for computational work. The aim of the present contribution is to obtain separable approximations to atom--atom transition operators via the Lippmann-Schwinger variational principle [8,9] at the cncrgy ranges required in the study of thermal atom- -diatom collisions for quite * Pxtly supported
by NSF Gr.mt CHC-01077A02. $ For rcv~ews see ret. [3]. i-L For J recent review see ref. [4] _
300
general &atomic potentials. The method involves the expansion of two-body “off-shcli wavefunctions” [7] in terms of a (in principle complete) set of two-body states. Although we have chcsen to work with a set of square integrable functions, other basis sets may also he employed and lead to various approximations proposed previously by other authors [7,10] _
2. Variationa
expression for the trzmsition opentor
WC consider two particles interacting via a local and central potential V. The two-body T-operator, a function of the complex energy parameter z, is defined by
7’(z) = V + YGo(z) T(z) ,
(1)
where GO(Z) = (z -Ho)-’ with Ho being the kinetic energy operator. Matrix elements of the T-operator in momentum space satisfy the following integral equation: WI T(z)Ik) +
= WI
&)
d&“
It is convenient to formulate the variational principle for the T-operator in operator form. For this purpose we make use of the wave operator W(z), defined by W(z)=
1 +(z -Ho
- V)-‘v=(l
in terms of which we obtain
-G&v)-‘,
(3)
the identity
T(z) = VW(z) + Wt(z*) Y
H2 at energics, E < 0, and relative momenta k, k’, in the ranges required in the study of thermal II f H, collisions z_ For the i Z potential a Morse curve v(r) = D [e-20-r,)
- VG, V)W(z)
.
(4)
If the exact wave operator W(z) in cq. (4) is replaced by a trial wave operator FV&) we obtain a stationary functional Tt [ W,, z] , which is the well-known Schwinger variational principle written in operator form. We shall utilize cq. (4) by choosing a trial wave operator of the form rut (2) z c
n
Ian )
I = I *)
(9
where (I@,)) iq a linearly independent set in two-particle space. Unknown coefficients
(6)
where
[J-%)1 nm = ODnlV- VG,(Z)V@~).
(7)
We note that most of the extant separable expansions [7j for the T-operator can be put into the form (6) with appropriate basis functions. In performing the calculations reported in this letter, for z real and ncgative, we have chosen our basis set to be square integrable functions in coordinate space. This choice can be justified by noting that in the variational functional W, appears always multiplying V, which we consider to be a rapidly decreasing potential at large distances.
3. Application
to atom-atom
systems
We have applied the method outlined in the previous section to the lowest 1 C and 3C radical potentials of
_ 2e-afr-r,)]
wasusedwithD=0_1759au,Q= 1.4006 ao. For the 3Z potential potential V(r) = V. (e””
- Wt(z+)(V
1 hlarcll 1978
CHEhlICAL PHYSICS LEl-l-ERS
Volumr 54, number 2
- 1)--l,
(8) 1.1178~~~,~e= we used the IIulth@n s2% (9)
with V0 = 0.739 au. and a = 0.942 ao_ For potentials supporting bound states, matrix clements of T(E) with E < 0 will have poles .it bound state energies. It is important to correctly Iocatc the pole structure, at least, in the cncrgy region of interest. For low-energy H f HZ collisions WCneed the two-body T-operator only below, say, E = -0.1 au. Kestrrcting our intcrcst to this energy domain, WC need a basis set which is capable of describing the first few vibrational levels for each partial wave. We used a basis of harmomc oscillator wavefunctions 1123 with parameters 1131 chosen to accurately reproduce the first few vibrational levels of the Morse potential. That is WC take the basis functions lb&r) for each partial wave 1 to be @,(r) = l@,Jr)/rl
Y,,(r^)
,
en(r) = N,,H,,(arx) .-~Z~2/2
(10) ,
(11)
where N, is a normalization constant, H,z the rlth Hcrmitc polynomial, and x = r - ro_ Here 01and r. are two non-linear parameters at our dispo\aI. Although the radial functions e,,(r) should, m general, be idependent, we useti the same (sufficiently large) basis set for all 1 in the calculations reported below. WC took a=3.2Qg1 and r. = 1.8 no_ With this choice the boundary condition at the origin ic satisfied to a good approximation. For the Iiulthdn potential we agam used a set of harmonic oscillator functions. This time, however, we do not have the kind of criteria, namely bound states, which we used for Morse potential to fix the values of (ILand ro. Hcncc we took r. = 0 and Itsed only oddparity functions to satisfy the boundary condition at the origin. The remaining parameter (Yhas been optimized by comparison to numerical calcuiations [ 141
$ A summnry oi; the prehminary results appears in ref. [ 111. 301
Volume
54, number
CIIEMCAL
2
PHYSICS
of the T-matrix. A small amount of experimentation has icd*us to the choice 01= 1.Oa~'. In actual calculations we resorted to a partial wave decomposition as follows
(k'16(E)lk) =E
Yr,n(~')Yj,(i;)Ol(k',k;E),
(12)
--__ -___ -. _ - ._ :‘.~ r----I-7
1.m r--r--
1.00 -
-3m
- 7.00
wberc 0 (c) is any rotationally invariant two-body operator. The variational calculatzons of the partial wave T-matrix tl(k, k';E) then involves the evaluation of three types of integrals:
1 March 1978
LETTERS
-
- 9.00 - 5.m _uOJ
I._.0.0
---ml
I
1-------
4.00
6.m
R
8.m
I0.m
(au)
Fig. 1. The off-shell wavefunction Ws(r, k; E)/r for the Morse potentid with E = -0.15au,k = 4.5 au. The solid line is the numcricd
(14)
ban I VGor(E’) Vl4,) 0
=
s
k2 dk i,&)(K
- &)+,&k)
,
(15)
0
where &is the spherical Bessel function. Integrals GI and V,,,, have been evaluated numcritally. The matrix ctements given in eq. (15) are more difficult to compute. These can be evaluated in momentum representation, as given on the right hand side of cq. (I S), with the use of quadrature techniques, but this is not very practical since the integrand itseIf contains integrals I,,)_ Instead we have evaluated these integrals by approximating Gur as
result. and the dashed line the wriationni
result.
-0.15 au and k = 4.5 au. The variational result was obtained with N = 30 and is seen to coincide with the exact result up to about r = 4.0 no_ For r > 4 (16 the variational off-shell wavcfunction is vanishingly small whereas the exact one assumes its oscillatory asymptotic form. However, having a good representation of Ft$(r, k;E) within the inner region suffices to obtain accurate results for tr(k’, k;E) as long as the contribution from the tail of the potential is relatively small. Tables 1 and 2 give a comparison of numerical versus variational results for t5 (k’, k;E) with E = -0.15 au, k = 4.0 au, at several values of k' for Morse and Hulthen potentials, respectively. Even when 10 basis functions are used, variational results are rather satisfactory, except for some T-matrix elements small in magnitude. These disagreements arise from the inability of the basis sets to represent the off-shell wavefunctions at large distances. The situation can be remedied by using different basis functions for each partial wave. The present basis sets are optimal for I= 0 and hence yield better results for the first few partial waves.
4. Results and discussion Acknowledgement It is of interest to see how well does an expansron in square integrable functions represent the off-shell radial wavefunction IV&j-,k;E).Fig. I shows a comparison of numerical [141 and variational calculations of FVs(f, k; E)/r for the Morse potential with E =
302
We thank Lynthis Beard for providing us with the numerical results. The Northeast Florida Regional Computing Center provided partial support for the computational work.
Volume
54, number
2
CHEBIICAL
Table 1 rs(k’. k;E) fortha Morse potential with E = -0.15 dlfterent number, N. of basis function --_-__-------k’
Numerical
--_--_--------_ 0.358 0.818 0.344 0.693 0.870 ---.------
------_--
T,tblc 2 ts(k’, k;E) br the Hulthcn --------_--~--_---__~ k’
Numcricdl
x x x x x
10-s 1o-2 10-l 10-t 10-l ____
Potential.
----
0.178 0.791 0.344 0.697 0.871 ---____-_-
E = -0.15
x x x X x
N=
0.140 0.542 0.104 0.394 0.810
0.959 0.147 0.978 0.388 0.806
10
x x x x x
IO-’ 1O-4 10-z IO-’ 1o-3
x x x x x
IO4 10-4 10” 10-z 10-3
_--_-__
x x x x X
10-4 1O-4 10-3 1(F2 IO-’
References II? L.D. F.~ddeev, Sowct Phy\. JETL’ 12 (1961) 1014. 121 C. Lavelace. Phyc. Rev. 135 (1964) B 1225; E.O. Alt, I?. Grassberger and W. Sandhas. NucI. Phys. HZ (1967) 167. 131 Y.E. Kim and A. Tubis, Ann. Rev. Nucl. Sci. 24 (1974) 69; J.C.Y. Chen, Case Studtes Atomic Physics 3 (1974) 307; K.M. WArson and J. Nuttall, Topics in several particle clynmks (Holden-Day. San rranci%co, 1967).
versus vartational
results
-__----_--___~~~~
------
with
-__-------
N= 15 -----__-____-_0.516 0.933 0.336 0.703 0.867
N=
x lo-’ x x x x
__---_--__-
________-~-.--------.------1 0.264 2 0.480 3 0.104 4 0.396 5 0.818 ----------------
values of k’. Numerical
10-3 10-z 10-t 10-t 10-t
.IU, k = 4 au
VarIationsI -----___-----N= 15
1 March 1978
LE-ITERS
an, k = 4 au at scvcral
Variational _--__--N= 30 --___--
____ 1 2 3 4 5
PHYSICS
1o-2 10-t 10-l 10-t ---.-
0.573 0.971 0.335 0.704 0.867 _--____------
10 x x x x x
--
---
1(F3 10-z 10-t 10-t 10-t
D.A. Mocha, Advan. Chcm. Phys. 30 (1975) 7. D.A. hlichd, J. Chcm. Phys 57 (1972) 2184; 60 (1974) 2480. D.A. hficha. J. Chem. Phys. 66 (1977) 1255. M.K. Srlva\tava and D.W. Sprung. in: Advances tn nuclcdr physss, Vcl. 8, eds. M. Barangcr and E. Vogt (Plenum Press, New York, 1976) p. 121. Phys. Rev. 79 (1950) [Sl B.A. Lippmann and J. Schwinger, 469. 191 1-H. Sloan and T.J. Brady, Phys. Rev. C6 (1972) 701; G.L. Payne and J.D. Perez, Phys. Rev. Cl0 (1974) 1584.
[lo1 S. Wcmberg,
Phys. Rev. 131 (1963) 440; E. lI.uma. l’hys. Rev. Cl (1970) 1667; D J. Ernst, C.hl. Shakin and R.M. Thalcr, Phys. Rev. CH (1973) 46; S.K. Adhtkari and I.H. Sloan, Nucl. Phys. A 241 (1975) 429; V.B. B*aJyaev and AL. Zubarcv, Soviet J:hiucl. Phy\. 14 (1972) 30-5. of ~irn Xl11 1111 hf. Bxat and J. Remhardt. cds.. Proceeding ICPEAC (C.E.A., Par&, 1977) p. 1120. [121 B.W. Shor,:,J. Chem. Phys. 63 (1975) 3835. C.S. I_m and G.W.1’. Drake, Chem. Phyr. Letters 16 (1972) 3s. t131 hl. Alcuandtr’, J. Chcm. Phys. 6i (1974) 5167: 1141 L.H. Heard and D.A. Micha, Chcm. Phy\. Letters 53 (1978) 329.
303
CHEMICAL PHYSICS LETTERS
54, number 2
SEPARABLE
APPROXIMATION
A VARIATIONAL
Z.C. KURUOCLU
17 November
OPERATORS:
APPROACH*
and D-4. MICHA
Quunruml71cot-y i?wj~c~,Depnrtnents GainesnIle. Fi’onda 3.261 I, USA. Rcceivcd
FOR DIATOMIC TRANSITION
1 March 1978
of Clremrstry and Physics. (Inivemity of Horida,
1977
A gencr.11 timte-rank separable approximation for the two-body transition operator ha, been obtained from the Lrppmann- Schwingcr vxicltmnal principle. The method involves cup.mGon of the off-shell wavefunction in d finrte set of square integrable function\. and is applied to I x and 3X potcntial5 of 112 for thermal energies and momenta.
I_ Introduction Recent developments [ I,?] f m few-body dynamics suggest ;I new approach to the study of reactive molecular collisions c, which we stlall treat as manybody processes. The motivation behind the manybody approach, whcrc the bodies are atoms or ions, is to take advantage of our knowledge on the simpler atom-atom problem. The input to the theory of three-atom dynamics [5,6] arc the atom-atom transition operators. Specificially we need the so-called “off-energy-shell” matrix elements [7] of tnesc two-body operators in momentum space. After an angular momentum analysis, the Faddccv equations in momentum space become coupled equations in two continuous variables. USC of finite-rank separable expansions for atom--atom transition operators leatis to ;I larger set of coupled intcyral equations in one variable [3] , which ace much nx.xe convenient for computational work. The aim of the present contribution is to obtain separable approximations to atom--atom transition operators via the Lippmann-Schwinger variational principle [8,9] at the cncrgy ranges required in the study of thermal atom- -diatom collisions for quite * Pxtly supported
by NSF Gr.mt CHC-01077A02. $ For rcv~ews see ret. [3]. i-L For J recent review see ref. [4] _
300
general &atomic potentials. The method involves the expansion of two-body “off-shcli wavefunctions” [7] in terms of a (in principle complete) set of two-body states. Although we have chcsen to work with a set of square integrable functions, other basis sets may also he employed and lead to various approximations proposed previously by other authors [7,10] _
2. Variationa
expression for the trzmsition opentor
WC consider two particles interacting via a local and central potential V. The two-body T-operator, a function of the complex energy parameter z, is defined by
7’(z) = V + YGo(z) T(z) ,
(1)
where GO(Z) = (z -Ho)-’ with Ho being the kinetic energy operator. Matrix elements of the T-operator in momentum space satisfy the following integral equation: WI T(z)Ik) +
= WI
&)
d&“
It is convenient to formulate the variational principle for the T-operator in operator form. For this purpose we make use of the wave operator W(z), defined by W(z)=
1 +(z -Ho
- V)-‘v=(l
in terms of which we obtain
-G&v)-‘,
(3)
the identity
T(z) = VW(z) + Wt(z*) Y
H2 at energics, E < 0, and relative momenta k, k’, in the ranges required in the study of thermal II f H, collisions z_ For the i Z potential a Morse curve v(r) = D [e-20-r,)
- VG, V)W(z)
.
(4)
If the exact wave operator W(z) in cq. (4) is replaced by a trial wave operator FV&) we obtain a stationary functional Tt [ W,, z] , which is the well-known Schwinger variational principle written in operator form. We shall utilize cq. (4) by choosing a trial wave operator of the form rut (2) z c
n
Ian )
I = I *)
(9
where (I@,)) iq a linearly independent set in two-particle space. Unknown coefficients
(6)
where
[J-%)1 nm = ODnlV- VG,(Z)V@~).
(7)
We note that most of the extant separable expansions [7j for the T-operator can be put into the form (6) with appropriate basis functions. In performing the calculations reported in this letter, for z real and ncgative, we have chosen our basis set to be square integrable functions in coordinate space. This choice can be justified by noting that in the variational functional W, appears always multiplying V, which we consider to be a rapidly decreasing potential at large distances.
3. Application
to atom-atom
systems
We have applied the method outlined in the previous section to the lowest 1 C and 3C radical potentials of
_ 2e-afr-r,)]
wasusedwithD=0_1759au,Q= 1.4006 ao. For the 3Z potential potential V(r) = V. (e””
- Wt(z+)(V
1 hlarcll 1978
CHEhlICAL PHYSICS LEl-l-ERS
Volumr 54, number 2
- 1)--l,
(8) 1.1178~~~,~e= we used the IIulth@n s2% (9)
with V0 = 0.739 au. and a = 0.942 ao_ For potentials supporting bound states, matrix clements of T(E) with E < 0 will have poles .it bound state energies. It is important to correctly Iocatc the pole structure, at least, in the cncrgy region of interest. For low-energy H f HZ collisions WCneed the two-body T-operator only below, say, E = -0.1 au. Kestrrcting our intcrcst to this energy domain, WC need a basis set which is capable of describing the first few vibrational levels for each partial wave. We used a basis of harmomc oscillator wavefunctions 1123 with parameters 1131 chosen to accurately reproduce the first few vibrational levels of the Morse potential. That is WC take the basis functions lb&r) for each partial wave 1 to be @,(r) = l@,Jr)/rl
Y,,(r^)
,
en(r) = N,,H,,(arx) .-~Z~2/2
(10) ,
(11)
where N, is a normalization constant, H,z the rlth Hcrmitc polynomial, and x = r - ro_ Here 01and r. are two non-linear parameters at our dispo\aI. Although the radial functions e,,(r) should, m general, be idependent, we useti the same (sufficiently large) basis set for all 1 in the calculations reported below. WC took a=3.2Qg1 and r. = 1.8 no_ With this choice the boundary condition at the origin ic satisfied to a good approximation. For the Iiulthdn potential we agam used a set of harmonic oscillator functions. This time, however, we do not have the kind of criteria, namely bound states, which we used for Morse potential to fix the values of (ILand ro. Hcncc we took r. = 0 and Itsed only oddparity functions to satisfy the boundary condition at the origin. The remaining parameter (Yhas been optimized by comparison to numerical calcuiations [ 141
$ A summnry oi; the prehminary results appears in ref. [ 111. 301
Volume
54, number
CIIEMCAL
2
PHYSICS
of the T-matrix. A small amount of experimentation has icd*us to the choice 01= 1.Oa~'. In actual calculations we resorted to a partial wave decomposition as follows
(k'16(E)lk) =E
Yr,n(~')Yj,(i;)Ol(k',k;E),
(12)
--__ -___ -. _ - ._ :‘.~ r----I-7
1.m r--r--
1.00 -
-3m
- 7.00
wberc 0 (c) is any rotationally invariant two-body operator. The variational calculatzons of the partial wave T-matrix tl(k, k';E) then involves the evaluation of three types of integrals:
1 March 1978
LETTERS
-
- 9.00 - 5.m _uOJ
I._.0.0
---ml
I
1-------
4.00
6.m
R
8.m
I0.m
(au)
Fig. 1. The off-shell wavefunction Ws(r, k; E)/r for the Morse potentid with E = -0.15au,k = 4.5 au. The solid line is the numcricd
(14)
ban I VGor(E’) Vl4,) 0
=
s
k2 dk i,&)(K
- &)+,&k)
,
(15)
0
where &is the spherical Bessel function. Integrals GI and V,,,, have been evaluated numcritally. The matrix ctements given in eq. (15) are more difficult to compute. These can be evaluated in momentum representation, as given on the right hand side of cq. (I S), with the use of quadrature techniques, but this is not very practical since the integrand itseIf contains integrals I,,)_ Instead we have evaluated these integrals by approximating Gur as
result. and the dashed line the wriationni
result.
-0.15 au and k = 4.5 au. The variational result was obtained with N = 30 and is seen to coincide with the exact result up to about r = 4.0 no_ For r > 4 (16 the variational off-shell wavcfunction is vanishingly small whereas the exact one assumes its oscillatory asymptotic form. However, having a good representation of Ft$(r, k;E) within the inner region suffices to obtain accurate results for tr(k’, k;E) as long as the contribution from the tail of the potential is relatively small. Tables 1 and 2 give a comparison of numerical versus variational results for t5 (k’, k;E) with E = -0.15 au, k = 4.0 au, at several values of k' for Morse and Hulthen potentials, respectively. Even when 10 basis functions are used, variational results are rather satisfactory, except for some T-matrix elements small in magnitude. These disagreements arise from the inability of the basis sets to represent the off-shell wavefunctions at large distances. The situation can be remedied by using different basis functions for each partial wave. The present basis sets are optimal for I= 0 and hence yield better results for the first few partial waves.
4. Results and discussion Acknowledgement It is of interest to see how well does an expansron in square integrable functions represent the off-shell radial wavefunction IV&j-,k;E).Fig. I shows a comparison of numerical [141 and variational calculations of FVs(f, k; E)/r for the Morse potential with E =
302
We thank Lynthis Beard for providing us with the numerical results. The Northeast Florida Regional Computing Center provided partial support for the computational work.
Volume
54, number
2
CHEBIICAL
Table 1 rs(k’. k;E) fortha Morse potential with E = -0.15 dlfterent number, N. of basis function --_-__-------k’
Numerical
--_--_--------_ 0.358 0.818 0.344 0.693 0.870 ---.------
------_--
T,tblc 2 ts(k’, k;E) br the Hulthcn --------_--~--_---__~ k’
Numcricdl
x x x x x
10-s 1o-2 10-l 10-t 10-l ____
Potential.
----
0.178 0.791 0.344 0.697 0.871 ---____-_-
E = -0.15
x x x X x
N=
0.140 0.542 0.104 0.394 0.810
0.959 0.147 0.978 0.388 0.806
10
x x x x x
IO-’ 1O-4 10-z IO-’ 1o-3
x x x x x
IO4 10-4 10” 10-z 10-3
_--_-__
x x x x X
10-4 1O-4 10-3 1(F2 IO-’
References II? L.D. F.~ddeev, Sowct Phy\. JETL’ 12 (1961) 1014. 121 C. Lavelace. Phyc. Rev. 135 (1964) B 1225; E.O. Alt, I?. Grassberger and W. Sandhas. NucI. Phys. HZ (1967) 167. 131 Y.E. Kim and A. Tubis, Ann. Rev. Nucl. Sci. 24 (1974) 69; J.C.Y. Chen, Case Studtes Atomic Physics 3 (1974) 307; K.M. WArson and J. Nuttall, Topics in several particle clynmks (Holden-Day. San rranci%co, 1967).
versus vartational
results
-__----_--___~~~~
------
with
-__-------
N= 15 -----__-____-_0.516 0.933 0.336 0.703 0.867
N=
x lo-’ x x x x
__---_--__-
________-~-.--------.------1 0.264 2 0.480 3 0.104 4 0.396 5 0.818 ----------------
values of k’. Numerical
10-3 10-z 10-t 10-t 10-t
.IU, k = 4 au
VarIationsI -----___-----N= 15
1 March 1978
LE-ITERS
an, k = 4 au at scvcral
Variational _--__--N= 30 --___--
____ 1 2 3 4 5
PHYSICS
1o-2 10-t 10-l 10-t ---.-
0.573 0.971 0.335 0.704 0.867 _--____------
10 x x x x x
--
---
1(F3 10-z 10-t 10-t 10-t
D.A. Mocha, Advan. Chcm. Phys. 30 (1975) 7. D.A. hlichd, J. Chcm. Phys 57 (1972) 2184; 60 (1974) 2480. D.A. hficha. J. Chem. Phys. 66 (1977) 1255. M.K. Srlva\tava and D.W. Sprung. in: Advances tn nuclcdr physss, Vcl. 8, eds. M. Barangcr and E. Vogt (Plenum Press, New York, 1976) p. 121. Phys. Rev. 79 (1950) [Sl B.A. Lippmann and J. Schwinger, 469. 191 1-H. Sloan and T.J. Brady, Phys. Rev. C6 (1972) 701; G.L. Payne and J.D. Perez, Phys. Rev. Cl0 (1974) 1584.
[lo1 S. Wcmberg,
Phys. Rev. 131 (1963) 440; E. lI.uma. l’hys. Rev. Cl (1970) 1667; D J. Ernst, C.hl. Shakin and R.M. Thalcr, Phys. Rev. CH (1973) 46; S.K. Adhtkari and I.H. Sloan, Nucl. Phys. A 241 (1975) 429; V.B. B*aJyaev and AL. Zubarcv, Soviet J:hiucl. Phy\. 14 (1972) 30-5. of ~irn Xl11 1111 hf. Bxat and J. Remhardt. cds.. Proceeding ICPEAC (C.E.A., Par&, 1977) p. 1120. [121 B.W. Shor,:,J. Chem. Phys. 63 (1975) 3835. C.S. I_m and G.W.1’. Drake, Chem. Phyr. Letters 16 (1972) 3s. t131 hl. Alcuandtr’, J. Chcm. Phys. 6i (1974) 5167: 1141 L.H. Heard and D.A. Micha, Chcm. Phy\. Letters 53 (1978) 329.
303