- Email: [email protected]
Use of the atomic fues potential for the excited states of He: energies and oscillator strengths
\‘olume
95. numblx
CHEMICAL
6
PHYSICS
LID-l-ERS
18 March
1983
USE OF THE ATOMIC FUES POTENTIAL FOR THE EXCITED STATES OF He: ENERGIES AND OSCILLATOR STRENGTHS S. f
1-i October
1962: in final form
3 1 December
to the description of the exited states in heliunl. The accuracy in the energies strengths are well described in the model potential formulation, with an accuracy
I’hc 31omic Fucs polcnkl has been applied The oscillaror is bcrrcr lbrrn 4.5 X IO-’ .I”. . ~cncr~liy hcrrcr 111:m 1% _
I. Introduction The c3fcuf3tion moIccuf3r
of3
f3:ge
characteristics 3uthors
wavefunctions.
to the construction
of ~~scudo- 3nd model
of atomic
most of the previous work was carried out on alkalis. These eigenfunctions have also been used as virtual orbitals for many-electron CI calculations in helium [lo] _ This potential is particularly attractive because it al-
and
needs a simple representa-
tion of the v3fcnce-cfcctron led nitny
number
Tflis
of various
has types
[ 11. In the first ap-
potentirlls
pfo3ch “one need represent
esplicitly
only
the valence
cfcclrons.
and need not impose any orthogonality conslrainrs’- (IO the core-electron orbitals) [ I]_ In the second approxfl tfie stomic hamiltonian spectrum contains
Icvcls
I~~~cc~fcct
corresponding
ortfw~onrtliry Alost work voted
to core
ran wavefunction
energies
possesses
to the core
orbitals.
concerning
these
to 3fom.5 or ions with
and the va-
nodes
potentials
one vrtlence
lows exact solution of the single-electron equation.
SchrGdinger
In this paper we consider the application of the AFP to the case of helium. After a brief recall on the AFP (section 2), we present the energies (section 3), and then discuss the use of the AFP in the pseudoand model-potential formulations for the calculation of the oscillator strengths.
realizing
has been electron
de-
2. The atomic Fues potential
out-
siJe a closed sflell. f lowever the Klapisch model potential has been applied to the description of the excited stafcs in f le [2]. Simons (31 was the first to use what he named the atomic Fues potential (AFP), which was employed 1-i ] for thr calculation of oscillator strengths in Na 3nd K. In a slightly different approach, several authors flavc used the eigenfunctions of the AFP with adjustable orbital parameters 3s orbitals for the valence elecIron. 3nd ilaw computed oscillator strengths [5,6], photoionisation cross sec:ions [7] and atomic polarizabilirics [8.9]_ In ref_ [9] this method was successfully spplied to metsstable states in He and Li+, although
564
1982
The potential v(r)
= -Z/r
+
proposed
by Simons
[3] is
Ice. BIkI/r’ ,
(1)
where 2 is the net charge of the core, P, the projection operator over the subspace defined by a given angular momentum I, BI an adjustable parameter_ The radial equation [-;dz/‘dr2
+al&
+ 1)/2r’
-Z/r]&-)
=E&-)
(2)
has the solutions E = -Zz/2(N
- I +#
0 009-26 14/83/0000-0000/S
,
(3)
03.00 0 1983 North-Holland
_. 18 March 1983
CHEMICAL PHYSICS LEl-fERS
Volume 95, number 6 jI(r) = Cr@e--E%$L;+)
)
(4) The AFP appears as a generalization
(4)
of the
hydrogen case, with the introduction of an effective
with
“non-integer
07 = - f + !j [(2Z + 1)2 + MI] 1’2,
eigenfunctions behave “asymptotically” hydrogen_
I”_ With increasing N the energies and
as those of
(5)
E=Z/(N-z+al).
3. Energies
Lq is a generalized Laguerre polynomial [ll], Ca norniillization constant_ P(T) is quadratically integrable provided [7,1 l] al>-;,
(W
Iv-Z>l.
(6b)
Using eq. (3) we determined tire aI parameters by fitting exactly the lowest state of each series with
N - I = 1, for the singlet and triplet states in helium For the S series the 2s state was taken as the lowest and the fundamental 1 1 S state neglected_ The results for S, P, D states are reported in tables 1 and 2. The greatest difference between calculated and experimental energies is 45 X 10m5 au (for the 3 3S state), to be compared with 132 X 10-S au (for the 2 ‘P state), obtained with the Klapisch potential [2] _Moreover the fist level in each series is fitted exactly. The states with I > 2 are quite hydrogen-like (ai = 1) and the agreement is always better than 10m6 au.
Although the properties of the AFP have been studied [3-lo] we briefly recall the most important
points: (1) The solution of the radial equation (2) is analytical. (2) For a given I series, one single parameter (al or SI) determines an orthonormal set of eigenfunctions and the corresponding spectrum_ (3) The choice of al is related to the definition of a quantum defect for a series as it can be seen from
eq. (3). Table 1 Parametersol
I
and ionization
ffl
energies for the sinslet states in atomic units. Experimental
values are taken from ref. [ 121
Energy esp.
CZIlC_
n=2
n=3
n=4
n=5
fi=2
n=3
n=4
n=5
0
0.85087
0.14595
0.06152
0.03372
0.02125
0.14595
0.06126
0.03356
0.02117
1
1.00949
0.12382
0.05521
0.03110
0.01992
0.12383
0.055 14
0.03106
0.01990
2
1.99844
0.05561
0.03127
0.02001
0.05561
0.03128
0.02001
Table 2 AFP
parameters and ionization energies for the triplet states, in atomic units. Experimental
ai
Enegy esp.
CaiC.
0.68928 0.93780 1.99802
values are taken from ref. [ 121
n=2
n=3
n=4
Jl=5
II=2
0.17521 O-13315
0.06913 0.05793 0.05563
0.03674 O-03224 0.03128
0.02274 0.0205 1 0.02002
0.17521 0_13315
I1 =
3
0.06868 0_05808 0.05563
n=4
n=5
0.0365 1 0.03232 0.02002
0.02262 0.02055 0.02002
565
Our fitting procedure proved to be the most satisf~-tov from the viewpoint of the rms deviation.
1. Oscillator strengths
In this section we discuss the application of the AI-l’ to the calculation of atomic oscillator strengths. following the dipole Icngth formulation (7)
which _civcs. after summing plct. tilt wcragc value: j;,,__
,,‘,’
=
;
over all lines in a multi-
AE[m3s(l.i’~l(‘Z
•t I)]
Our values arc compared to precise calculations involving central-field wavefunctions with exchange and wnfipurarion intcrriction [I;]_ The agreement betwccn their results for the S-P transitions and those of rcfs. [ 13.15 1 using f-lyllcraas-type functions. is bcttcr than 2%_ Kcf. [IS ] reviews experimental values for s01iw 0f thcsc tmnsitions. At this point the determination of the parameters is worth reconsidering in greater detail. With the previous choice wc described the first state of each series by a nodclcss wavcfunction. III other words we used the XFl’ 3s 3 pseudo-potcnti;J The use of these radial I‘uncrions
le~rl to catastrophic
trririsitions. as
18 March 1983
CHEMICAL PHYSICS LE-ITERS
Volume 95. number 6
results for the S-P
in table 3 for the singlet case. This fact may bc attributed to the incorrect numbcr of modes (rl - 1) in the IIS wavefunctions. in the C;ISCof Iie it is possible to constrain the wsvcfunctions IO cshibit the right number of modes. using the :\FI‘ as a model potential. This is readily obtsincd with new parameters for the S states: sh.nw
"0’ = -0-14913
for the 1~ states,
a; = -
for the 3S states,
0.31072
interpretingN in eqs. (3)-(S) as the principal quantum number. (The CZ~ parameters for I> 1 remain unchanged, because there is no electron with I B 1 in the core, so that the pseudo- and model-potential formulations are equivalent _) When this choice is compatible with the condition in cq. (6a) the solution of the radial equation includes virtual energies and eigenfunctions corresponding to the core, which will automatically by orthogonal to the valence state eigenfunctions. The energies corresponding to the real states remain unchanged_ With these new parameters we have computed the Ivalues for different transitions in helium, using the experimental energies in eq. (8). These results are given in table 4, and compared to precise calculations [ 13]_ Table 4 also includes length and velocity values from ref. [IO]. where the eigenfunctions of the AFP were used in a Cl calculation. In ref. [lo] the orbital parameters were adjusted for each level. The calculated oscillator strengths for the singlet spectrum agree to better than 10% with precise calculations in ref. [ 131. In the triplet spectrum, apart from the 3 3S-4 3P transition, agreement is better than 12% We thus note that the AFP. used as a model potential. gives an accuracy comparable to that obtained with the Klapisch potential, which is better than 10% for oscillator strengths [Z]. The better accuracy for the singlet case may be due to the values of the parameters aI: these are closer to the I values for the singlet states [cf. tables 1 and 2 and eq. (1 I)], which thus appear to be “more hydrogenie” than the triplet states.
5. Conclusion
Tranzition
This calcuhlion
3)
0.3 149 0.0127 0.011
0.3173 0.1513 0.0493
2 ‘s-2 2 ‘s-3 1 ‘s-4 --.-___
‘1’ ‘P ‘1) -__ ..___
~2)SW rd.
1131. “lcnprh” values.
566
We have examined the use of the atomic Fues potential for the description of the excited states of helium_ This potential has some interesting properties: (a) The solution of the radial equation is analytical. The eigenvalues are given by a simple quantum defect formula, and the calculation of the eigenfunctions reduces to that of generalized Laguerre polynomials
illI-
Vblume 95, number 6
Table 4 Oscillator strengths (absolute values) calculated in the model potential used Transition
formulation
(see text). The esperimental
energies were
this work
a)
b)
this work
a)
ZS-2P zs-3P zs-4P
0.3748 0.1666 0.0538
0.3773 0.1513 0.0493
0.383 (0.319) 0.215 (0.153) 0.0737 (0.0462)
0.5636 0.0576 0.0233
0.5398 0.0644 0.0259
3s-2P 3S-3P 3s-4P
0.1385 0.6238 0.1483
0.1457 0.6279 0.1429
0.636 (0.599) 0.352 (0.129)
0.2228 0.8960 0.038 1
0.2087 0.8922 0.0499
4s-2P 4s-3P IS-4P
0.0243 0.3024 0.8547
0.0260 0.3081 0.8603
0.0335 0.4741 1.1085
0.0318 0.4363 12164
ZP-3D 2P-4D
0.7027 0.1200
0.7 106 0.1203
0.6501 0.1288
0.6105 0.1232
3P-3D 3P-4D
0.02 13 0.6354
0.0218 0.6490
0.1118 0.5 140
0.1130 0.4766
4P-3D 4P-4D
0.0162 0.0403 .-
0.0152 0.04 15
0.0327 0.2009
O-037 1 0.2033
a) See rrf‘. ] I 31. “length” values. b, See ref. [ lo] _The values in parentheses
1983
Triplet spectrum
Sinslet spectrum
___-
---
18 hluch
CHEMICAL PHYSICS LETTERS
represent
the “velocity”
(b) The model may be extended to the continuum spectra [7] _ The Fues potential is studied both as a pseudo-potential and as a model potential. Different choices of the empirical parameters are involved and the resulting radial functions have quite different characteristics (e-g_ modes and masima). The accuracy in the energies and oscillator strengths that we have obtained in the model potential formulation is better or comparable to that given by the Klapisch potential. We conclude that the Fues potential well describes the interaction of the excited electron with the core although the inner electron shell is open. The AFP is therefore a reasonable model for the calculation of various atomic and molecular properties involving the excited states of helium.
___-formulation_
References [ l] J-N. Bardsley. Case Studies Atom. Phys. 4 (1974) 299. 121 M. Aymar and M. Crance. J. Phys. B13 (1980) 2527. [3] G. Simons. J. Chem. Phys. 55 (1971) 756. [4] 1-V. Avilot- and L-1. Podlubny, Opt. i Spektroskopi_v3
38 (1975) 613.1059. 15 ] G. Simons, J. Chem. Phys. 60 (1974) 645. 161 P.F. Gruzdev and A.I. Sherstyuk. Opt. i Spektroskopiya 40 (1976) 353.617. 171 I_ Martin and G. Simons. J. Chenx Phyr 62 (1975) 4799. 181 S-A. Adelman and A. Szabo, Phys. Rev. Letters 28 (1972) 1427. 191 N.L. ilanakov. V-D. Ovsyannikov and L.P. Rapoport, Opt_ i Spektroskopiya 38 (1975) 115. [ lo] E. Yurtsever and F.H.M. Faisal, Chem. Phys. Letters 66 (1979) 104. [ 111 M. Abmmovitz and LA. Stegn, eds_. Handbook of mathematical functions Applied mathematics series No. 55 (NBS, Washington. 1964). 1121 W.C. Martin, J. Res. Natl. Bur. Std. 64X (1960) 19. [ 131 L.C. Green, N-C. Johnson and E.K. Kolchin. Astrophys. J. 144 (1966) 369. [ I41 B. Schiff, C-L. Pekeris and Y. Accad. Phys. Rev_ A4 (1971) 885. [ 151 M.T. Anderson and F. Weinhold. Phys. Rev. A9 (1974) 118.
567
95. numblx
CHEMICAL
6
PHYSICS
LID-l-ERS
18 March
1983
USE OF THE ATOMIC FUES POTENTIAL FOR THE EXCITED STATES OF He: ENERGIES AND OSCILLATOR STRENGTHS S. f
1-i October
1962: in final form
3 1 December
to the description of the exited states in heliunl. The accuracy in the energies strengths are well described in the model potential formulation, with an accuracy
I’hc 31omic Fucs polcnkl has been applied The oscillaror is bcrrcr lbrrn 4.5 X IO-’ .I”. . ~cncr~liy hcrrcr 111:m 1% _
I. Introduction The c3fcuf3tion moIccuf3r
of3
f3:ge
characteristics 3uthors
wavefunctions.
to the construction
of ~~scudo- 3nd model
of atomic
most of the previous work was carried out on alkalis. These eigenfunctions have also been used as virtual orbitals for many-electron CI calculations in helium [lo] _ This potential is particularly attractive because it al-
and
needs a simple representa-
tion of the v3fcnce-cfcctron led nitny
number
Tflis
of various
has types
[ 11. In the first ap-
potentirlls
pfo3ch “one need represent
esplicitly
only
the valence
cfcclrons.
and need not impose any orthogonality conslrainrs’- (IO the core-electron orbitals) [ I]_ In the second approxfl tfie stomic hamiltonian spectrum contains
Icvcls
I~~~cc~fcct
corresponding
ortfw~onrtliry Alost work voted
to core
ran wavefunction
energies
possesses
to the core
orbitals.
concerning
these
to 3fom.5 or ions with
and the va-
nodes
potentials
one vrtlence
lows exact solution of the single-electron equation.
SchrGdinger
In this paper we consider the application of the AFP to the case of helium. After a brief recall on the AFP (section 2), we present the energies (section 3), and then discuss the use of the AFP in the pseudoand model-potential formulations for the calculation of the oscillator strengths.
realizing
has been electron
de-
2. The atomic Fues potential
out-
siJe a closed sflell. f lowever the Klapisch model potential has been applied to the description of the excited stafcs in f le [2]. Simons (31 was the first to use what he named the atomic Fues potential (AFP), which was employed 1-i ] for thr calculation of oscillator strengths in Na 3nd K. In a slightly different approach, several authors flavc used the eigenfunctions of the AFP with adjustable orbital parameters 3s orbitals for the valence elecIron. 3nd ilaw computed oscillator strengths [5,6], photoionisation cross sec:ions [7] and atomic polarizabilirics [8.9]_ In ref_ [9] this method was successfully spplied to metsstable states in He and Li+, although
564
1982
The potential v(r)
= -Z/r
+
proposed
by Simons
[3] is
Ice. BIkI/r’ ,
(1)
where 2 is the net charge of the core, P, the projection operator over the subspace defined by a given angular momentum I, BI an adjustable parameter_ The radial equation [-;dz/‘dr2
+al&
+ 1)/2r’
-Z/r]&-)
=E&-)
(2)
has the solutions E = -Zz/2(N
- I +#
0 009-26 14/83/0000-0000/S
,
(3)
03.00 0 1983 North-Holland
_. 18 March 1983
CHEMICAL PHYSICS LEl-fERS
Volume 95, number 6 jI(r) = Cr@e--E%$L;+)
)
(4) The AFP appears as a generalization
(4)
of the
hydrogen case, with the introduction of an effective
with
“non-integer
07 = - f + !j [(2Z + 1)2 + MI] 1’2,
eigenfunctions behave “asymptotically” hydrogen_
I”_ With increasing N the energies and
as those of
(5)
E=Z/(N-z+al).
3. Energies
Lq is a generalized Laguerre polynomial [ll], Ca norniillization constant_ P(T) is quadratically integrable provided [7,1 l] al>-;,
(W
Iv-Z>l.
(6b)
Using eq. (3) we determined tire aI parameters by fitting exactly the lowest state of each series with
N - I = 1, for the singlet and triplet states in helium For the S series the 2s state was taken as the lowest and the fundamental 1 1 S state neglected_ The results for S, P, D states are reported in tables 1 and 2. The greatest difference between calculated and experimental energies is 45 X 10m5 au (for the 3 3S state), to be compared with 132 X 10-S au (for the 2 ‘P state), obtained with the Klapisch potential [2] _Moreover the fist level in each series is fitted exactly. The states with I > 2 are quite hydrogen-like (ai = 1) and the agreement is always better than 10m6 au.
Although the properties of the AFP have been studied [3-lo] we briefly recall the most important
points: (1) The solution of the radial equation (2) is analytical. (2) For a given I series, one single parameter (al or SI) determines an orthonormal set of eigenfunctions and the corresponding spectrum_ (3) The choice of al is related to the definition of a quantum defect for a series as it can be seen from
eq. (3). Table 1 Parametersol
I
and ionization
ffl
energies for the sinslet states in atomic units. Experimental
values are taken from ref. [ 121
Energy esp.
CZIlC_
n=2
n=3
n=4
n=5
fi=2
n=3
n=4
n=5
0
0.85087
0.14595
0.06152
0.03372
0.02125
0.14595
0.06126
0.03356
0.02117
1
1.00949
0.12382
0.05521
0.03110
0.01992
0.12383
0.055 14
0.03106
0.01990
2
1.99844
0.05561
0.03127
0.02001
0.05561
0.03128
0.02001
Table 2 AFP
parameters and ionization energies for the triplet states, in atomic units. Experimental
ai
Enegy esp.
CaiC.
0.68928 0.93780 1.99802
values are taken from ref. [ 121
n=2
n=3
n=4
Jl=5
II=2
0.17521 O-13315
0.06913 0.05793 0.05563
0.03674 O-03224 0.03128
0.02274 0.0205 1 0.02002
0.17521 0_13315
I1 =
3
0.06868 0_05808 0.05563
n=4
n=5
0.0365 1 0.03232 0.02002
0.02262 0.02055 0.02002
565
Our fitting procedure proved to be the most satisf~-tov from the viewpoint of the rms deviation.
1. Oscillator strengths
In this section we discuss the application of the AI-l’ to the calculation of atomic oscillator strengths. following the dipole Icngth formulation (7)
which _civcs. after summing plct. tilt wcragc value: j;,,__
,,‘,’
=
;
over all lines in a multi-
AE[m3s(l.i’~l(‘Z
•t I)]
Our values arc compared to precise calculations involving central-field wavefunctions with exchange and wnfipurarion intcrriction [I;]_ The agreement betwccn their results for the S-P transitions and those of rcfs. [ 13.15 1 using f-lyllcraas-type functions. is bcttcr than 2%_ Kcf. [IS ] reviews experimental values for s01iw 0f thcsc tmnsitions. At this point the determination of the parameters is worth reconsidering in greater detail. With the previous choice wc described the first state of each series by a nodclcss wavcfunction. III other words we used the XFl’ 3s 3 pseudo-potcnti;J The use of these radial I‘uncrions
le~rl to catastrophic
trririsitions. as
18 March 1983
CHEMICAL PHYSICS LE-ITERS
Volume 95. number 6
results for the S-P
in table 3 for the singlet case. This fact may bc attributed to the incorrect numbcr of modes (rl - 1) in the IIS wavefunctions. in the C;ISCof Iie it is possible to constrain the wsvcfunctions IO cshibit the right number of modes. using the :\FI‘ as a model potential. This is readily obtsincd with new parameters for the S states: sh.nw
"0’ = -0-14913
for the 1~ states,
a; = -
for the 3S states,
0.31072
interpretingN in eqs. (3)-(S) as the principal quantum number. (The CZ~ parameters for I> 1 remain unchanged, because there is no electron with I B 1 in the core, so that the pseudo- and model-potential formulations are equivalent _) When this choice is compatible with the condition in cq. (6a) the solution of the radial equation includes virtual energies and eigenfunctions corresponding to the core, which will automatically by orthogonal to the valence state eigenfunctions. The energies corresponding to the real states remain unchanged_ With these new parameters we have computed the Ivalues for different transitions in helium, using the experimental energies in eq. (8). These results are given in table 4, and compared to precise calculations [ 13]_ Table 4 also includes length and velocity values from ref. [IO]. where the eigenfunctions of the AFP were used in a Cl calculation. In ref. [lo] the orbital parameters were adjusted for each level. The calculated oscillator strengths for the singlet spectrum agree to better than 10% with precise calculations in ref. [ 131. In the triplet spectrum, apart from the 3 3S-4 3P transition, agreement is better than 12% We thus note that the AFP. used as a model potential. gives an accuracy comparable to that obtained with the Klapisch potential, which is better than 10% for oscillator strengths [Z]. The better accuracy for the singlet case may be due to the values of the parameters aI: these are closer to the I values for the singlet states [cf. tables 1 and 2 and eq. (1 I)], which thus appear to be “more hydrogenie” than the triplet states.
5. Conclusion
Tranzition
This calcuhlion
3)
0.3 149 0.0127 0.011
0.3173 0.1513 0.0493
2 ‘s-2 2 ‘s-3 1 ‘s-4 --.-___
‘1’ ‘P ‘1) -__ ..___
~2)SW rd.
1131. “lcnprh” values.
566
We have examined the use of the atomic Fues potential for the description of the excited states of helium_ This potential has some interesting properties: (a) The solution of the radial equation is analytical. The eigenvalues are given by a simple quantum defect formula, and the calculation of the eigenfunctions reduces to that of generalized Laguerre polynomials
illI-
Vblume 95, number 6
Table 4 Oscillator strengths (absolute values) calculated in the model potential used Transition
formulation
(see text). The esperimental
energies were
this work
a)
b)
this work
a)
ZS-2P zs-3P zs-4P
0.3748 0.1666 0.0538
0.3773 0.1513 0.0493
0.383 (0.319) 0.215 (0.153) 0.0737 (0.0462)
0.5636 0.0576 0.0233
0.5398 0.0644 0.0259
3s-2P 3S-3P 3s-4P
0.1385 0.6238 0.1483
0.1457 0.6279 0.1429
0.636 (0.599) 0.352 (0.129)
0.2228 0.8960 0.038 1
0.2087 0.8922 0.0499
4s-2P 4s-3P IS-4P
0.0243 0.3024 0.8547
0.0260 0.3081 0.8603
0.0335 0.4741 1.1085
0.0318 0.4363 12164
ZP-3D 2P-4D
0.7027 0.1200
0.7 106 0.1203
0.6501 0.1288
0.6105 0.1232
3P-3D 3P-4D
0.02 13 0.6354
0.0218 0.6490
0.1118 0.5 140
0.1130 0.4766
4P-3D 4P-4D
0.0162 0.0403 .-
0.0152 0.04 15
0.0327 0.2009
O-037 1 0.2033
a) See rrf‘. ] I 31. “length” values. b, See ref. [ lo] _The values in parentheses
1983
Triplet spectrum
Sinslet spectrum
___-
---
18 hluch
CHEMICAL PHYSICS LETTERS
represent
the “velocity”
(b) The model may be extended to the continuum spectra [7] _ The Fues potential is studied both as a pseudo-potential and as a model potential. Different choices of the empirical parameters are involved and the resulting radial functions have quite different characteristics (e-g_ modes and masima). The accuracy in the energies and oscillator strengths that we have obtained in the model potential formulation is better or comparable to that given by the Klapisch potential. We conclude that the Fues potential well describes the interaction of the excited electron with the core although the inner electron shell is open. The AFP is therefore a reasonable model for the calculation of various atomic and molecular properties involving the excited states of helium.
___-formulation_
References [ l] J-N. Bardsley. Case Studies Atom. Phys. 4 (1974) 299. 121 M. Aymar and M. Crance. J. Phys. B13 (1980) 2527. [3] G. Simons. J. Chem. Phys. 55 (1971) 756. [4] 1-V. Avilot- and L-1. Podlubny, Opt. i Spektroskopi_v3
38 (1975) 613.1059. 15 ] G. Simons, J. Chem. Phys. 60 (1974) 645. 161 P.F. Gruzdev and A.I. Sherstyuk. Opt. i Spektroskopiya 40 (1976) 353.617. 171 I_ Martin and G. Simons. J. Chenx Phyr 62 (1975) 4799. 181 S-A. Adelman and A. Szabo, Phys. Rev. Letters 28 (1972) 1427. 191 N.L. ilanakov. V-D. Ovsyannikov and L.P. Rapoport, Opt_ i Spektroskopiya 38 (1975) 115. [ lo] E. Yurtsever and F.H.M. Faisal, Chem. Phys. Letters 66 (1979) 104. [ 111 M. Abmmovitz and LA. Stegn, eds_. Handbook of mathematical functions Applied mathematics series No. 55 (NBS, Washington. 1964). 1121 W.C. Martin, J. Res. Natl. Bur. Std. 64X (1960) 19. [ 131 L.C. Green, N-C. Johnson and E.K. Kolchin. Astrophys. J. 144 (1966) 369. [ I41 B. Schiff, C-L. Pekeris and Y. Accad. Phys. Rev_ A4 (1971) 885. [ 151 M.T. Anderson and F. Weinhold. Phys. Rev. A9 (1974) 118.
567