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Integrals of gaussian and continuum functions for polyatomic molecules. An addition theorem for solid harmonic gaussians
Volume 92, number5
CHEMICAL PHYSICS LE-MERS
INTEGRALS OF GAUSSIAN AND CONTINUUM FUNCTIONS FOR POLYATOMC MOLECULES. M ADDITION THEOREM FOR SOLID HARMONIC GAUSSIANS
Rolf SEECER SonderforJchungsbereich 91 der U>rrerncar Karrersburem. D-6750 Karrerxkuem.
Ic’esrGerrrrany
Recerved9 July 1982;IIIiiu~alform 28 July 1982
An ad&Donrhwrem for sobd harmomcgaussianfuncnons m termsof juriace sphcrul hatmomcsISdewed and presentedIIIa form parhculnly unful ior evaluattngmoleculartntenals involvmgBaussIIuI and continuum functions on arrayprocessors.
1. Introduction
For polynuclear molecular bound-state calculahons the evaluation of integrals has achieved a high degree of sophrstication. Partrcularly, basis sets consisting of solid harmonic gaussran functions *
(1) centered at some fixed points A, lead to very compact expressions. Integration can be eartied out throughout. In the theory of electron-molecule
anal~ticd~
scattering, however, where the electronic states have continuum
character, themolecular basisshmldalso con&
appropriate continuum functions with the requtred asymptotic oscillatory behaviour. These functions raise two problems: Frrst, they are themselves not L.2integrable. One method to circumvent this difficulty is given by R-matrix theory [3-51. In contrast to integrals for bound-state systems, radial integration 1scarried out over a finite interval [O,R), R bemg the radius of a suitably chosen R-matrix sphere centered at a point 0. Second, the simple analytic character of multrcenter integrals involving merely gaussian functions is lost. To maintain the advantages of the ptopettres of gaussians, at least to some extent, the multicenter integrals involving continuum functions may be expanded in terms cf single-center ones. This can be achieved applying appropriate addrtion theorems that allow us to expand functions centered at some point A about a Mferent point B. Ruedenberg [6], Sack [7], and Steinborn have developed the theory of such addition theorems and have apphed rt to several types of functions useful m quantum chemistry [8-141. In this paper we formulate an additron theorem for solid harmonic gaussians wirh the intention to use it evahrsting molecular integrals involving contmuum functions. Such a theorem is required for two reasons. Fust, the physical properties of continuum functions do not allow them to factorrze into functions of pure cartesian coordinates as do gaussians. Second, rather than expandrng continuum functions about the gaussian centers, for which ad&ion theoremsare available [6], we are forced toexpand the gaussians about the origtn of the continuum functions that coincide with the center of the R-matrix sphere, 0. This is necessary to avold complications in integrating to theR-matriu boundary.
l
$$2$
denote surfacesphericalharmorucs&sdefiied e p. m ref [ 11.The polaranglesof the vectorr. 8 and UJ.are represented For a comprehensivetreatze on proper&esof gaussIanfuncuons useful in quantum chermsuysee ref. [ 2).
collecnvely by +
0 009-2614/82/0000-0000/S
02.75 0 1982 North-Holland
493
CHEBIICAL
Volume 92, number 5
PHYSICS
5 November 1982
LETTERS
1. Derivation of the theorem
For iuncrrons on R3 which factonze into a pure radul pan and a sphericalharmonrc, like the solid harmomc gaussransof eq (I), Stetnbom and Frlter [I I] have shown that addition theorems m terms of spherical harmonics have, If they exist, the following form:
In this expression $,,I,
are pure radial mtegrals, i.e.
(3)
“’ ,,“‘I ;)f2)Gatmt coefficients **, and 1, and f2 are non-negatne integers Here]&) are spherml Besselfuncttons, (L satafymg the tnangle relatron A(Lr,I,) and pan+ restrrctron of the Gaunt coefticnnts. In order to obtain an addttion theorem for sold harmoruc gaussrans that makes further integrarion feasible. w2 have to fmd an onu~tic expresston for $j2 with
g(f) = rL esp(-c&).
(4)
The mtegral over r III eq. (3) is the radral part of the Fourier transform of the solid harmonic gaussran. It can be solved analytically [ 161, I.e.
J
exp(-or?)jL(2f)
drrL+2
= (~/‘)‘/“(20l)-(L+~p)~~
exp(-@/4o).
6)
0
The lntegal
over h- UI eq. (3)
comprises the rati
part oi the translation operator,
represented by1/2(k72)r
and
the backtransform of the translated function into r space. Inserting eq. (5) into eq. (3) we obtam 00
g&*(rt‘J =’
11+31:!-L,1/2~-(L-I).-(L+3/2)
J
dk kL+? exp(-k2j40)
jj,(!o) j,$ATA).
0 Thrs backtransformation functrons
can be carried out analytically, too, applying the recurrence relation for spherical Bessel
II+,(Z) = (21+ 3)z-ljj+t(z)
(7)
-I,(Z).
Thus, adoptmg the following notation for the integral rn eq. (6), a
I;,12 = f dk kL+2 exp(-X.‘/4a)j,,(kr),~2(~~*),
(8)
0
we obtain the following recurrence relations for I, and j2, respectively: l l
The Gaunt coeiticlenrs are driied
mlm2M
as
are Cobsch--Cm& wherethe C~,/,L eken (p~~ty resuxt~on), and 111- 121 G L
494
codislents
[IS].
Gamr
coeiticlenrs are non-zero only iorM
c II + 12 which 1sone of the three representattonsof
= ml
+ m2,
L + 11 + 12
the tmingle relation, NL, It, 121.
Volume
91,number 5
CHEMICAL PHYSICS LETTERS
Note the symmetry wrth respect to simukmeous mterchanges off, with 1? and r with fA. To obtain a complctc set ofIt,, mregrals the following starting points for recurrence based on eqs (9) are sufticrent. (IO) Ii0 = (~n)*lr(I/3)iL~~(~*)-l
exp(--ci(2
+ ri)]
(HL(~“‘R-)
esp(cimA) -HL(u”‘R+)exp(-onA)]
(1 I)
with R+=r+rA,
R- =‘-‘A,
(1 la)
where HL(:) are Hermite polynomtals. Surular expressions can be derived ior I,,, L IL,-,t, andffl Note that fhese integrals may be mterpreted as Fourier transforms of km exp(-k2/4~) ~7th nr 2 0 and hence they all eust. The 1 are real due to the restrIctions that apply to L, I,, and I,. Thus the followmg conclusrons may bc drawn forgl,2 ,z . Frrst, they emt and are real for all vabd combmations ofL, I,, and 12. Second, they may be represented by the following expresuon &Jr,
rA)= i1r+3’a-L (n/o)(2a)-L
esp[-o(G
t ri)] g
C,fPW~(?Q)s~
[exp(arf,)
+ f, 2\p(-ak7.J
(13
with the
summatron parameters {c,p,q,sifi)i,l r”(L*‘r*‘a)being signed integers, 11’fume. In pnnciple, these may be from eqs (9)-(11). However, wee eq. (12) suggests to pre-compute them for some given range of L, a simpler recurrence algorithm, particularly suitable for computer coding, is presented. Expandmg the spherical Bessel functrons of eq. (8) [ 171, and applymg product relations of sure and cosme, Ii!, may be expanded as a lmear combination of the following rntegrals smce the rntegrand of the I Integral is even due to the parity restriction: obtmed
Jr/’
(J--E
2 -w
km
exp(-kz/40)
exp(1X-R’)
dk t j
k-“I
exp(-k2/ja)
eap(rkRI) dt)
,
(13)
E
whereR’ =f krA, e=Oform~Oande>Oform>O.
(19
For m < 0 Jm are Fourier transformsofk-m
exp(-k2/&)
which may be wrrtten as
[lrnl/21 Jm =Jg nqO b3ii)-(“+2”)(2a)(m+n),
(15)
with b; = 1,
b,m= bF_I (1 -m
-
2n)(2 - nz - 2n)/32
(Isa)
and Jo = (n~$t/~ exp(--0R2).
06)
Note that for clanty the f signs 0f.I; and R' are omitted. Form > 0 the following recurrence relation can be established:
495
5 November
CHEhtICAL PHYSICS LmERS
Volume 92. number 5
1982
Here ~~(~~co~t~s the leading terns of the pale, e approachingzero. Recurrence starts with Jo given by eq. (16), and Jt = I erf(&%). As a consequence of the analytic properties of the g mtegrals, as expressed in eq. (121, contributlons from J, and P,(e) to the linear combmation forming If;, 1, must cancel exactly and hence may be omitted. Thus, wlthout affecting the result, we may replace J,,, for nz > 0 by ?jj = (nt - l)!-IJ;
(lga)
wrth the recurrence relatron for JI:, I!1-2 J’!I1 = --(2c~)-~
Z.
(iR)“(m - 2 -n)J&Z_n
.
(18b)
Thus we have established simple recurrence relations for all J terms requtred to obtain rt”,la avoidmg the problems unrent in explicit ~t~gra~on fo~ulated in eq. (13). The starting points are given by eq_ (16) and by J; = 0. Perhaps a spectal case of the addition theorem, for L = 0, is worth being noted. The sums overt, and t2 m eq. (2) then reduce to one single sum, the real analogon of Rayleigh’s plane wave expansion, also a special case of the Gegenbauer addition theorem [ 161. Our adrhtion theorem then assumes the form exp[-o/(r
- rA)12J =4n exp[-a(?+
ri)]
[G i$ (i2arrA) Y;l*(S2,) Y;“(nrA).
WI
Note that the argument of the spherical Bessel funitron is rmaginary and compare this expression with eq. (10). Since the representation in eq. (19) 1sdiagonal with respect to powers of r and rA, compared to the general expansion in eq. (2) with &$ ,a mtegmls formulated in eq. (12), the number of terms is mimmized. Thus, eq. (19) may prove useful in molecular calculations involving floatmg gaussran lobe functions and possibly compensate for one of their deficiencies, namlsly the relatwrly Iarge number of primitives required to represent &her angular momen-
tum basis functions3. Conclusions
We have denved the following addition theorem for solid harmonic gaussian functions Ir - rA IL Y~(~~~~) xc
c
exp]-aI@ - rA)i”) = (~~~)(2~)-~ exp[-a(r2
+ $1 (20)
c/f
r$(2cu)S [exp(arrA) + fr exp(-amA)]
1112 mlm2
with the summation parameters fclp14,s,r,}‘~~~~~~~~~) being accessible by simple recurrence relations. We have applied these relations to produce a computer-coded Fortran block data ior the ovation parameters with 0 G L G 6 and 0 G (It, Z2) 4 9. Thus the last sum of eq. (20) can be obtained very efficiently, pickily on modem array processorsthat take full advantage of the fact that all data required for the entire range of this sum can be made available at once, even for several sets of (1t, I,). It should be noted that, in contrast to addition theorems for Bessel functions [8], no partitioning of r is r&red.
496
CHEMKAL.
Volume 92. number 5
PHYSICS LETTERS
Acknowledgement for encouragmg disand I am mdebted to Professor W. hleyer and the Sonderforschungs~reich 91
I would l&e to thank Professor H. Klar, Dr. R.K. Nesbet, and Professor K. Ruedenberg
cussionsand valuable co~ents, at K~sersIautem.
References [ 11 E.U. Condon
and G.H.
[ 21 1.Shavttt, ~1: ~ie~o~
Shortley, The theory of atonue spectra (C3mbrtdge Uruv. Przsr, London, 1970). VoL 2, eds. B. Alder, S. Fernbxh and hl. Rotcnberg (Ac3demIc Press. NCW
in compu~tion~phydcs,
York, 1963) p 1. [3]
B.I. Schneld+r. in: Electron--molecule and (Plenum Press, New York, 1979) p. 77.
photon-molecule
[4] B.D. Buckley and P.G. Burke, III- EJection-molecule
coU~~~ons. eels
and photon-molcculc
T Rescqzno. V. AlcKoy and B Schncadcr col~~s~ans,
eds.
T.
Resnpx~.
v. hIcKo> md
B. Schneider (Plenum Press, New York. 1979) p. 133 [S] R K. Nesber. Vanatlonal methods tn electron-atom scattenng theory (Plenum Press. New York. 1980). [6] K. Rueden~~, 17leoret. Chim. Acta 7 (1967) 359 [7] R A. Sack, J. Math. Phys 5 (1964) 260. (81 E 0. Srembom and E. Filter, Intern. J. Quantum Chem S9 (1975) 435. [9] E.O. Stelnbom and E. F&e,, Theoret Chun. Acta 38 (1975) 217. [lo] E.O. Steinborn and E. Fdter, Phys. Rev. Al8 (1978) 1. [ll] E.O. Stembom and E. Fdter, Theoret. Chim. Acta 52 (1979) 189. [ 121 E. Ftiter and E 0. Steinborn, J. M&I. Phys. 21(1980) 2725. (131 H.H. Kranr and E-0. Ste~bom, Phys Rev. A25 (1982) 66. 1141 Ii P. Trive& and E.O. Stemborn. Phys. Rev. A25 (1982) 113. [I51 ME. Rose, Elementary theory of anguti momentum (Wey, New York, 1957) [ 161 W. Magnus,F. Oberhetnnger and R.P. Soni, Formulas and theorems for Ihe spensl tinwions oi m~thenwical Gprmger. Berlm, 1966) (171 A. hfessti, Quantum mechanics, Vol 1 (North-Holland, Amsterdam, 1967) p. 488
physics
497
CHEMICAL PHYSICS LE-MERS
INTEGRALS OF GAUSSIAN AND CONTINUUM FUNCTIONS FOR POLYATOMC MOLECULES. M ADDITION THEOREM FOR SOLID HARMONIC GAUSSIANS
Rolf SEECER SonderforJchungsbereich 91 der U>rrerncar Karrersburem. D-6750 Karrerxkuem.
Ic’esrGerrrrany
Recerved9 July 1982;IIIiiu~alform 28 July 1982
An ad&Donrhwrem for sobd harmomcgaussianfuncnons m termsof juriace sphcrul hatmomcsISdewed and presentedIIIa form parhculnly unful ior evaluattngmoleculartntenals involvmgBaussIIuI and continuum functions on arrayprocessors.
1. Introduction
For polynuclear molecular bound-state calculahons the evaluation of integrals has achieved a high degree of sophrstication. Partrcularly, basis sets consisting of solid harmonic gaussran functions *
(1) centered at some fixed points A, lead to very compact expressions. Integration can be eartied out throughout. In the theory of electron-molecule
anal~ticd~
scattering, however, where the electronic states have continuum
character, themolecular basisshmldalso con&
appropriate continuum functions with the requtred asymptotic oscillatory behaviour. These functions raise two problems: Frrst, they are themselves not L.2integrable. One method to circumvent this difficulty is given by R-matrix theory [3-51. In contrast to integrals for bound-state systems, radial integration 1scarried out over a finite interval [O,R), R bemg the radius of a suitably chosen R-matrix sphere centered at a point 0. Second, the simple analytic character of multrcenter integrals involving merely gaussian functions is lost. To maintain the advantages of the ptopettres of gaussians, at least to some extent, the multicenter integrals involving continuum functions may be expanded in terms cf single-center ones. This can be achieved applying appropriate addrtion theorems that allow us to expand functions centered at some point A about a Mferent point B. Ruedenberg [6], Sack [7], and Steinborn have developed the theory of such addition theorems and have apphed rt to several types of functions useful m quantum chemistry [8-141. In this paper we formulate an additron theorem for solid harmonic gaussians wirh the intention to use it evahrsting molecular integrals involving contmuum functions. Such a theorem is required for two reasons. Fust, the physical properties of continuum functions do not allow them to factorrze into functions of pure cartesian coordinates as do gaussians. Second, rather than expandrng continuum functions about the gaussian centers, for which ad&ion theoremsare available [6], we are forced toexpand the gaussians about the origtn of the continuum functions that coincide with the center of the R-matrix sphere, 0. This is necessary to avold complications in integrating to theR-matriu boundary.
l
$$2$
denote surfacesphericalharmorucs&sdefiied e p. m ref [ 11.The polaranglesof the vectorr. 8 and UJ.are represented For a comprehensivetreatze on proper&esof gaussIanfuncuons useful in quantum chermsuysee ref. [ 2).
collecnvely by +
0 009-2614/82/0000-0000/S
02.75 0 1982 North-Holland
493
CHEBIICAL
Volume 92, number 5
PHYSICS
5 November 1982
LETTERS
1. Derivation of the theorem
For iuncrrons on R3 which factonze into a pure radul pan and a sphericalharmonrc, like the solid harmomc gaussransof eq (I), Stetnbom and Frlter [I I] have shown that addition theorems m terms of spherical harmonics have, If they exist, the following form:
In this expression $,,I,
are pure radial mtegrals, i.e.
(3)
“’ ,,“‘I ;)f2)Gatmt coefficients **, and 1, and f2 are non-negatne integers Here]&) are spherml Besselfuncttons, (L satafymg the tnangle relatron A(Lr,I,) and pan+ restrrctron of the Gaunt coefticnnts. In order to obtain an addttion theorem for sold harmoruc gaussrans that makes further integrarion feasible. w2 have to fmd an onu~tic expresston for $j2 with
g(f) = rL esp(-c&).
(4)
The mtegral over r III eq. (3) is the radral part of the Fourier transform of the solid harmonic gaussran. It can be solved analytically [ 161, I.e.
J
exp(-or?)jL(2f)
drrL+2
= (~/‘)‘/“(20l)-(L+~p)~~
exp(-@/4o).
6)
0
The lntegal
over h- UI eq. (3)
comprises the rati
part oi the translation operator,
represented by1/2(k72)r
and
the backtransform of the translated function into r space. Inserting eq. (5) into eq. (3) we obtam 00
g&*(rt‘J =’
11+31:!-L,1/2~-(L-I).-(L+3/2)
J
dk kL+? exp(-k2j40)
jj,(!o) j,$ATA).
0 Thrs backtransformation functrons
can be carried out analytically, too, applying the recurrence relation for spherical Bessel
II+,(Z) = (21+ 3)z-ljj+t(z)
(7)
-I,(Z).
Thus, adoptmg the following notation for the integral rn eq. (6), a
I;,12 = f dk kL+2 exp(-X.‘/4a)j,,(kr),~2(~~*),
(8)
0
we obtain the following recurrence relations for I, and j2, respectively: l l
The Gaunt coeiticlenrs are driied
mlm2M
as
are Cobsch--Cm& wherethe C~,/,L eken (p~~ty resuxt~on), and 111- 121 G L
494
codislents
[IS].
Gamr
coeiticlenrs are non-zero only iorM
c II + 12 which 1sone of the three representattonsof
= ml
+ m2,
L + 11 + 12
the tmingle relation, NL, It, 121.
Volume
91,number 5
CHEMICAL PHYSICS LETTERS
Note the symmetry wrth respect to simukmeous mterchanges off, with 1? and r with fA. To obtain a complctc set ofIt,, mregrals the following starting points for recurrence based on eqs (9) are sufticrent. (IO) Ii0 = (~n)*lr(I/3)iL~~(~*)-l
exp(--ci(2
+ ri)]
(HL(~“‘R-)
esp(cimA) -HL(u”‘R+)exp(-onA)]
(1 I)
with R+=r+rA,
R- =‘-‘A,
(1 la)
where HL(:) are Hermite polynomtals. Surular expressions can be derived ior I,,, L IL,-,t, andffl Note that fhese integrals may be mterpreted as Fourier transforms of km exp(-k2/4~) ~7th nr 2 0 and hence they all eust. The 1 are real due to the restrIctions that apply to L, I,, and I,. Thus the followmg conclusrons may bc drawn forgl,2 ,z . Frrst, they emt and are real for all vabd combmations ofL, I,, and 12. Second, they may be represented by the following expresuon &Jr,
rA)= i1r+3’a-L (n/o)(2a)-L
esp[-o(G
t ri)] g
C,fPW~(?Q)s~
[exp(arf,)
+ f, 2\p(-ak7.J
(13
with the
summatron parameters {c,p,q,sifi)i,l r”(L*‘r*‘a)being signed integers, 11’fume. In pnnciple, these may be from eqs (9)-(11). However, wee eq. (12) suggests to pre-compute them for some given range of L, a simpler recurrence algorithm, particularly suitable for computer coding, is presented. Expandmg the spherical Bessel functrons of eq. (8) [ 171, and applymg product relations of sure and cosme, Ii!, may be expanded as a lmear combination of the following rntegrals smce the rntegrand of the I Integral is even due to the parity restriction: obtmed
Jr/’
(J--E
2 -w
km
exp(-kz/40)
exp(1X-R’)
dk t j
k-“I
exp(-k2/ja)
eap(rkRI) dt)
,
(13)
E
whereR’ =f krA, e=Oform~Oande>Oform>O.
(19
For m < 0 Jm are Fourier transformsofk-m
exp(-k2/&)
which may be wrrtten as
[lrnl/21 Jm =Jg nqO b3ii)-(“+2”)(2a)(m+n),
(15)
with b; = 1,
b,m= bF_I (1 -m
-
2n)(2 - nz - 2n)/32
(Isa)
and Jo = (n~$t/~ exp(--0R2).
06)
Note that for clanty the f signs 0f.I; and R' are omitted. Form > 0 the following recurrence relation can be established:
495
5 November
CHEhtICAL PHYSICS LmERS
Volume 92. number 5
1982
Here ~~(~~co~t~s the leading terns of the pale, e approachingzero. Recurrence starts with Jo given by eq. (16), and Jt = I erf(&%). As a consequence of the analytic properties of the g mtegrals, as expressed in eq. (121, contributlons from J, and P,(e) to the linear combmation forming If;, 1, must cancel exactly and hence may be omitted. Thus, wlthout affecting the result, we may replace J,,, for nz > 0 by ?jj = (nt - l)!-IJ;
(lga)
wrth the recurrence relatron for JI:, I!1-2 J’!I1 = --(2c~)-~
Z.
(iR)“(m - 2 -n)J&Z_n
.
(18b)
Thus we have established simple recurrence relations for all J terms requtred to obtain rt”,la avoidmg the problems unrent in explicit ~t~gra~on fo~ulated in eq. (13). The starting points are given by eq_ (16) and by J; = 0. Perhaps a spectal case of the addition theorem, for L = 0, is worth being noted. The sums overt, and t2 m eq. (2) then reduce to one single sum, the real analogon of Rayleigh’s plane wave expansion, also a special case of the Gegenbauer addition theorem [ 161. Our adrhtion theorem then assumes the form exp[-o/(r
- rA)12J =4n exp[-a(?+
ri)]
[G i$ (i2arrA) Y;l*(S2,) Y;“(nrA).
WI
Note that the argument of the spherical Bessel funitron is rmaginary and compare this expression with eq. (10). Since the representation in eq. (19) 1sdiagonal with respect to powers of r and rA, compared to the general expansion in eq. (2) with &$ ,a mtegmls formulated in eq. (12), the number of terms is mimmized. Thus, eq. (19) may prove useful in molecular calculations involving floatmg gaussran lobe functions and possibly compensate for one of their deficiencies, namlsly the relatwrly Iarge number of primitives required to represent &her angular momen-
tum basis functions3. Conclusions
We have denved the following addition theorem for solid harmonic gaussian functions Ir - rA IL Y~(~~~~) xc
c
exp]-aI@ - rA)i”) = (~~~)(2~)-~ exp[-a(r2
+ $1 (20)
c/f
r$(2cu)S [exp(arrA) + fr exp(-amA)]
1112 mlm2
with the summation parameters fclp14,s,r,}‘~~~~~~~~~) being accessible by simple recurrence relations. We have applied these relations to produce a computer-coded Fortran block data ior the ovation parameters with 0 G L G 6 and 0 G (It, Z2) 4 9. Thus the last sum of eq. (20) can be obtained very efficiently, pickily on modem array processorsthat take full advantage of the fact that all data required for the entire range of this sum can be made available at once, even for several sets of (1t, I,). It should be noted that, in contrast to addition theorems for Bessel functions [8], no partitioning of r is r&red.
496
CHEMKAL.
Volume 92. number 5
PHYSICS LETTERS
Acknowledgement for encouragmg disand I am mdebted to Professor W. hleyer and the Sonderforschungs~reich 91
I would l&e to thank Professor H. Klar, Dr. R.K. Nesbet, and Professor K. Ruedenberg
cussionsand valuable co~ents, at K~sersIautem.
References [ 11 E.U. Condon
and G.H.
[ 21 1.Shavttt, ~1: ~ie~o~
Shortley, The theory of atonue spectra (C3mbrtdge Uruv. Przsr, London, 1970). VoL 2, eds. B. Alder, S. Fernbxh and hl. Rotcnberg (Ac3demIc Press. NCW
in compu~tion~phydcs,
York, 1963) p 1. [3]
B.I. Schneld+r. in: Electron--molecule and (Plenum Press, New York, 1979) p. 77.
photon-molecule
[4] B.D. Buckley and P.G. Burke, III- EJection-molecule
coU~~~ons. eels
and photon-molcculc
T Rescqzno. V. AlcKoy and B Schncadcr col~~s~ans,
eds.
T.
Resnpx~.
v. hIcKo> md
B. Schneider (Plenum Press, New York. 1979) p. 133 [S] R K. Nesber. Vanatlonal methods tn electron-atom scattenng theory (Plenum Press. New York. 1980). [6] K. Rueden~~, 17leoret. Chim. Acta 7 (1967) 359 [7] R A. Sack, J. Math. Phys 5 (1964) 260. (81 E 0. Srembom and E. Filter, Intern. J. Quantum Chem S9 (1975) 435. [9] E.O. Stelnbom and E. F&e,, Theoret Chun. Acta 38 (1975) 217. [lo] E.O. Steinborn and E. Fdter, Phys. Rev. Al8 (1978) 1. [ll] E.O. Stembom and E. Fdter, Theoret. Chim. Acta 52 (1979) 189. [ 121 E. Ftiter and E 0. Steinborn, J. M&I. Phys. 21(1980) 2725. (131 H.H. Kranr and E-0. Ste~bom, Phys Rev. A25 (1982) 66. 1141 Ii P. Trive& and E.O. Stemborn. Phys. Rev. A25 (1982) 113. [I51 ME. Rose, Elementary theory of anguti momentum (Wey, New York, 1957) [ 161 W. Magnus,F. Oberhetnnger and R.P. Soni, Formulas and theorems for Ihe spensl tinwions oi m~thenwical Gprmger. Berlm, 1966) (171 A. hfessti, Quantum mechanics, Vol 1 (North-Holland, Amsterdam, 1967) p. 488
physics
497