- Email: [email protected]
Integrals for electron-atom scattering calculations
JOURNAL
OF COMPUTATIONAL
PHYSICS
(1969)
4, 579-593
FRANK
E.
H-HARRIS
AND H.
H.
MICHELS
United Aircraft Research Laboratories,
East Har
Received June 16. 1969
L&~STRACT
The integrals needed for expansion identified and classified. Methods are Slater-type orbitals for the bound-state spherical Bessel functions and spherical
calculations of electron-atom scattering are described for evaluating ail integrals, using functions and free-wave functions built from harmonics.
I. lNTRoDUcTroN Recent interest in the application of expansion methods to elastic and inefastic scattering of electrons by atoms has led to a need for methods of eva~~at~~g the integrals arising in such an approach. The classical studies of KuIthCn [I] and Kohn [2], and more recent work by Nesbet [3] and by the present authors [CF], all involve matrix elements connecting the functions used in the expansion of the wavefunctions. The functions of primary interest for expansion a~~rQx~rnat~on to scattering wavefunctions are many-electron functions built from bound-state orbitals and free-wave functions of appropriate asymptotic behavior. The formalism to be described assumes the use of the partial wave expansion, and the free-wave functions therefore have a radial dependence given by spherical Bessel functions. 1 Supported in part at the University of Utah by a grant from the University Research Committee, and by National Science Foundation Grant CP-11170, and in part at the United Aircraft Research Laboratories by the U.S. Air Force Office of Scientific Research under contract No. AF 49(638)-1711.
579
580
HARRIS AND MICHELS
The bound-state orbitals to be considered here are those most appropriate to atomic problems, the well-known Slater-type orbitals (STO’s). The integral evaluation problems to be encountered here differ in some respects from those involving only bound states. First, integrals involving free-wave functions may not converge unless grouped into suitable combinations. Second, some of the integrals require evaluation techniques rather different from those encountered for bound-state functions. Third, some free-wave matrix elements are not invariant under Hermitian conjugation. Some of the quantities needed for these integrals have received previous study, as they are equivalent to Fourier transforms of STO’s. Geller [5] and others [6] have investigated these transforms, and the existent work can be of use here. In addition, portions of the material here reported have been previously presented in limited-distribution reports [7]. Valuable extensions of many of the methods here described have recently been made by Lyons and Nesbet [8].
II. CLASSIFICATION
OF INTEGRALS
The matrix elements entering into expansion formulations of electron scattering from an n-electron atom are of the form (?P’ j H - E / Y), where !P and Y are antisymmetrized (n + 1) electron space-spin functions, H is the Hamiltonian operator, and E is the total energy. For an atom in an initial state of energy E, and a scattering electron incident at momentum of magnitude k (and energy in atomic units &k2), E = E,, + $k2. The (n + 1) electron functions K and Y’ are regarded as built from one-electron functions and, if electron detachment processes are excluded, can be of two types. Either y1of the electrons are in bound-state orbitals and one electron has a free-wave form, or all IZ + 1 electrons are in bound-state orbitals. The bound-state orbital3 are, as previously stated, assumed to be STO’s. Orbital xa , of screening parameter 5, and quantum numbers n a , I, , m, , has in spherical polar coordinates the form
where YLmis a normalized spherical harmonic and N, is the radial normalization factor
N, = [ (2(y;;+1]1’2.
(2)
The free-wave functions are chosen to be of pure angular symmetry and can describe free waves of arbitrary phase asymptotically at large r. One such function,
INTEGRALS
FOR
ELECTRON-ATOM
SCATTERING
CALCULATIONS
which following Nesbet [3] we denote S to indicate it asymptotically sin(kr - @r)/kr, we take as
581 a~~roa~~~s
where j, is a spherical Bessel function of definition
Note that S depends upon a free-wave momentum of magnitude k and has angular quantum numbers 1 and m. When two sine-like free waves occur in expression, we refer to the second as s’, with implied dependence upon k’ A second asymptotic function could be given by replacing j, by a s Bessel function of the second kind, but we would then have fur&i troublesome singularities at r = 0. Instead, following Armstead f!?], we take Ike asymptotically equivalent form
where C varies asymptotically as cos(kr - !&r)/kr and depends upon k and quantum numbers 1 and m. A second cosine-like function will be denoted C’, with implied dependence upon k’, I’, ml. If Y and Yy’ are both built from bound-state orbitals, ( only the bound-state one and two-electron integrals comm calculations. If either Y or Y’, but not both, contain a free-wa use the Hermitian property to transform, if necessary, so that contains the free wave. Hermiticity is assured in this case by the exponential factor in at wavefunction for every electron, thereby giving for each electron a s approach to zero for large r. e now write N - E in the form H - E = H,, - E, + (Tf - +k2) + where HO is the Hamiltonian and E, its expectation value for the n-electron atomic wavefunction included as part of !?‘, the term Tf - +k2 refers to the kinetic ene operator and the asymptotic energy of the free electron of ?P, and the summat includes all potential energy terms involving the free electron “‘f “. The interaction between electron “:f” and the nucleus (of charge +n> is written as in to nt indicate its asymptotic cancellation against the electron repulsions. It
582
HARRIS
AND
MICHELS
from Eq. (6) that the decomposition of (!?“I H - E IY} into one- and two-electron integrals will lead, if !P contains S, to free-wave integrals of types
(XaIm, (xalT- WIS>, (xJ;i+
and (X.Xb~~~X.S).
If Y contains C, we will encounter similar integrals with S replaced by C. For convenience we refer to these two-electron integrals as of “hybrid” type. Evaluation of all integrals of these types follows the scheme previously described by us [7]. If both Y and Y’ contain free waves, we have the two possible cases k = k’ and k f k’. In the application of Eq. (6), if k f k’ the bound-state parts of Y .and Y’ will refer to states of different energy, causing (Y’ 1 Tf - $k2 I!?‘) to vanish due to bound-state orthogonality. For this operator we therefore need consider only (Si T - +k2 IS), (Cl T - ik2 IS), (S/ T - &k2 IC}, and (Cl T - +k2 IC). Whether or not k = k’, the matrix elements of H,, - E,, will vanish, either because (HO) = E,, or from orthogonality. The remaining integrals, from the potential-energy terms of Eq. (6), are of the types
or types obtainable therefrom by replacement of S and/or s’ by C and C’, or are types previously encountered. We call the two-electron integrals in which the same electron is free in both wavefunctions “Coulomb” type, and those in which different electrons are free in different wavefunctions “exchange” type. The Coulomb type integrals cannot be divided into separate terms containing l/u,, and l/r, , as these individual terms would diverge. There are no convergence difficulties with the exchange type integrals. All the integrals arising in this study can be written in terms of a small number of basic integral types. We accordingly introduce the following definitions: J,,(k, 6) = jm r”j,(kr) e-@ dr,
(7)
0
HEv(k, k’, 5) = Jrn r’“j(kr)j,(k’r)
e-@ dr,
(8)
0
KXk i, 5') = j," r y&(kr) A,(<‘r)
e-@ dr,
(9)
YEW, k’,5)=jmCL’j,(kr)M+) 4&i?) dr, 0
Xzz(k, k’, 5, 1;‘) = s,” dr r$(kr)
e-@ Jw ds snj,(k’s) e-4”, T
(11)
INTEGRALS
FOR
ELECTRON-ATOM
SCATTERING
583
CALCULATIONS
where A,(v) stands for
which is the standard definition of the “A” integral arising in mole&ar calculations. I!lEP appear in the evaluation schemes for the other integrals and rst, in Section III. The PC’??r;l”y”,and X,Y are covered in the following sections. e one-electron integrals referred to earlier in this section may now be evaluated directly or identified as expressions involving the JUVdefined in Eq. (7). The results are (13) (xc: ’ S> = h, &in,m,NzJna+dk 5,) = &.z, &rwnaNa IIJn,+wa (k la> + (+I (~~1 T - i-k” is) = 0 (K, I 0
h.&l T - w IO = --6,,ra hwna
JnG,dk,
iye,]
(14 (15) (16)
(Sl T - pz2 is> = 0
(!7)
(G/ T - &k2 IS> = 0
0%
(Sl T - &k” 162) = - & 377 (I + l)(Z i 2)(4Z -+ 9> cc/ T - Sk” 10 = - x (21 + 3)(2~ + 5)(21” 7)
$?ql
The explicit values for the kinetic energy integrals follow from the fact that S satisfies (T - +k2) S = 0 and from material in the Bateman ~a~us~r~~~ [lo]. The two-electron integrals are handled by using ‘the Laplace expansion of 1/rzz . The expression we need is
584
HARRIS
AND
MICHELS
where Y+ and Y- are the greater and lesser, respectively, of r, and r2 . Inserting Eq. (23) we find that a general two-electron integral assumes the form (24) where the coefficients C, arise from the angular integrations and
and where Yi(r) refers to the radial part of Yi . For the Coulomb type integrals, rlil is replaced by rG1 - r$ in Eq. (24) thereby also modifying Eq. (25) if p = 0 by replacement of r;’ by r;’ - r-l For the integrals treated in this “work, Yd is always a free wave, and Y, or Yb (but not both) may be. In evaluation of Eq. (25) the order of integration is manipulated when possible to put all free-wave functions into the outer integral, and to give the inner integral an infinite upper limit. We note that the introduction of r;l - r2-’ for the Coulomb type integrals causes Eq. (25) to have integration limits assuring convergence. The resulting formulas are
[xaxcI XbSL= 4#4 0) [xaxc I XbCl, = L(l,
(26) (27)
I;(2, 1)
0) + (qq
lxaxcI s’a& = 4L’oA 0, 0)
(28)
[xaxc I S’CI, = L’(L 0, 0) + (q+j
1,‘(2,0, 1)
[Xaxc I C’CI, = 4L’(1, 1,0> + (y-j
1,‘(2, 1, 1)
+ (‘g-)
L’(L2,
1) +
(q-qy-j
vxe I XbSlu= m, 0, 60)
1,‘(2,2,2)
(30) (31)
L-%x0 I XaCl,= 1x1,0,0 0) + (+j
1x2,0,
[cicoI xa
1x2, 1, 1,O)
= KU, I,% 0) + p-g, + (qsj
(29)
m,2,0,
1,O)
1) + (q+j(Wj
(32)
1,“(2,2, 1, 1) (33)
IINTEGRALS FOR ELECTRON-ATOM
I:(& jT q, p) = N,N,[X$~~~-”
§CA~TE~I~~
CALCWLATIONS
(k, k’, 5, , LJ + ~;~;%?%‘,
k, Sb , <,>I-
535
C363
In Eqs. (34)-(36), we have adopted the notations 5 = [, f 1;, , TZ= n, + n, s w = nb - p, d = n, - q, n’ = n + v f 1, n’ = n - y + 2. Note that eveaa for integrals involving free waves,
III.
AUXILIARY
FUNCTIONS
In this section we discuss the functions JUd and already mentioned the Juv arise in the Fourier tran e the transform of xa ,
where (k, 8, 4) are spherical coordinates explicit formula for Juv , p > v > 0, is
m-Is of STO’S.
in the transform
space.
eller’s pg
(39) Actually, the Jev are defined beyond the range of Eq. (39), as Eq. (7) is a convergent integral for p + v 2 0, including index values for which p < v or ,U c 0. Tt is convenient to develop the JIly recursively, using relations which folio
586
HARRIS
AND
MICHELS
from the properties of the-j, or from partial integrations. Some such formulas are the following: [7]
Jli,u+l+
Ju,v-1=
(vj
+ v - 1)Jw+l = 5 f+j (P+ v>Jw+x+ (EL CJ,,+ kJim = (k2+ P>Jp+l,v-1 + CP- 4 kJ,v= (k2+ 5”)Ju+~,v + (p + V)(P- v - 1)J&--I,”= (p + v + 1)kJ,v+ (P- v - 1)Ju,v+l=
JG+,
Jm
(P + 4 Ju--l,u (P + v>5Jw-1 &dJw (k2+ 5”)J~+I,~+I .
(41) 442) (43) (44) (45) (46)
Some of the above recurrence relations assume simpler forms for certain index values due to the vanishing of particular coefficients. Two extremely useful such cases are JV+l,V= J v+l,v
(47)
=
We also need some explicit values from which to initiate the recursion process. We take Jo,, = k-l tan-l(k/{) (49) J,,, = (k2 + [“)-‘.
(50)
A detailed examination of Eqs. (40)-(48) indicates that the coefficients of various Jpy vanish in such a way that one cannot proceed recursively to Juy with p < v from Juy values all with p > v. However, Eq. (49) gives one Jpy with p < v and it (with J& suffices to permit recursive generation of all Juy . This point has been
discussed more thoroughly by Lyons and Nesbet [8]. Those authors have also developed methods for obtaining the Juy for parameter values such that some of the procedures indicated here may become numerically unsatisfactory. The second set of auxiliary functions, the Hi,, , may also be generated recursively. The key relationships are
INTEGRALS
FOR ELECTRON-ATOM
SCATTERENG ~A~~~~AT~o~s
587
and the symmetry condition
ff:v@, k’,0 = ffW, k,0.
cm
Starting values are provided by writing j&r) or jWI(kk’r) as complex ex~o~~~~t~~~§~ leading to identification of H$ and H,Y-, in terms of integrals S,, of compkx argument: Hi,, = k’-‘Im J,-,Jk, c - ik’) H,f,-, = k’-l Re J,-lJz,
5 - ik’).
Nesbet give additional formulas involving I!!:” ~
IV. HYBRID AND Cou~om
TYPE INTEGRALS
he hybrid type two-electron integrals depend upon the function Wp ~~~.~e~ in Eq. (9). The WF may be generated recursively y making use of the we~~-~~o~~~ relations for the A, : M)
= 4 A,-,(r)
A,(r)
= r-l edT.
+ AO@)
The corresponding formulas for WF are
W;O = <‘-lJD-l,u(k, r: + ,y’).
WI
The Coulomb type integrals depend upon the Y,“y”of Eq. (IO). Using Eqs. (55) and (56), the Y,$ may be recursively related to the auxiliary functions A$. The result is ym?= LLLV
4 5
V. EXCHANGE
yPA*n-s LkY
+
K2
TYPE INTEGRALS
The integrals of exchange type depend upon the functions Xz”y”of the evaluation of these functions leads to more involved procedur
55%
588
HARRIS
AND
MICHELS
have been necessary for the other integrals. The Xl”y”are needed for index values ,u, v 3 0, for p > 1, and with 4 values which may be either positive or negative, but withp + 4 >, 1. The X,“y”satisfy recursion and transposition relationships which are may be used in their evaluation. Interchanging the order of integration, performing partial integrations, and inserting relationships satisfied by the j, , we find many formulas including the following:
RXk k’, 5, 5’) + xE(k’, k 5’25) = J,,(k 5)Jr&‘, 1;‘)
(k2 + 5”) X,f,?
- 2PSXE$
= - SH,F+l+ Xf&
(p + p)(p - p - 1) xi;l*.1.Q
kH,Zl;t,qt’ + (p - p - 1) H,“:
+ x;:1,”
(61)
= ( ““;
l ) x;;lsg.
(63) (64)
To each of Eqs. (62)-(64) corresponds an equation in which the variation of p andp is replaced by variation of v and q, k and < are replaced by k’ and 5’ and the sign of every HE” is reversed. For example, to Eq. (62) corresponds
Additional formulas result from linear combination of those tested. Generation of general X,“v” requires starting values and, just as for the Jgy , there are index boundaries which cannot be crossed in certain directions recursively. We accordingly need starting values in various index regions. Some starting values are provided by setting v = 0, q = 1, following which the inner integral admits of explicit evaluation. We obtain
Xpl WI= -& Im N <, J ik ) Jdk 5 + 6’ - iW].
o-33)
If now we set p = y = 0 and use Eq. (61), we also have
X&’ = - k Im [ (&)
J&k’, 5 + 5’ - ik)] + k,ck2‘+ 52jtan-’ ($-).
(67)
Evaluation of J,,,, for complex arguments requires the identity tan-l(u + iv) = $ tan-l (
2u
1 - 23 - 02 ) + ; log ( ;; +’ g ? ;;: 1.
w9
INTEGRALS FOR ELECTRON-ATOM
~ivale~t starting values are obtainable
SCATTERING CALC~LA~~Q~~
by using IV;
589
of comIslex argu
Even with the starting values represented by Eqs. (6~)-(7~), not all the ~e~~ssar~ X,“e are accessible. We still need additional values in the regions represented by Xi,0 and X,j?!, . These functions appea.r not to be elementary, and we have found it expedient to evaluate them numerically after manipulating them to convenient forms by methods described in Appendix I. Tbe forms to e integrated ~~rne~i~a~ly are 8’ 1 -1 R sin B + R’ sin 4 (71) Xii = m dd s -e, tan RCOS~+R’COS~
where R = ?/kz + iz, R’ = 2/kQ + c12, 6 = tan-*(k/Q 0’ = tani(k’/[‘). The method which leads to Eq. (71) also provides an alternative formula fix more general X integrals. The result is
A$==&jr
8’
J&k,
5 + R’ ei”> d+
ACKNOWLEDGMENTS We thank Mr. J. W. Viers and Mrs. R. Barrett for assistance in developing computer pro~ams for many of the processes reported here. Mr. Viers also checked many phases of the work. We are also happy to acknowledge a discussion with Drs. R. K. Nesbet and J. D. Lyons which led to the discovery of Eq. (73).
APPENDIX
I.
MANIPULATIQN
OF X$’ AND X&
The purpose of the rearrangements leading to Eqs. (71) and (72) is to obtain forms which do not exhibit rapid oscillation over the region of integration, so that standard numerical integration procedures will converge rapidly. Cionsi first X$‘. Writing the inner integration in the form
590
HARRIS
AND
MICHELS
we introduce in the first integral the complex variable z = (5’ - ik’)s, and in the second integral z = (5’ + ik’)s. The inner integration is thus equivalent to the contour integral
where r consists of the two rays marked I’, and r, in Fig. 1.
FIG. 1. Contours used in the integration and c - ik’.
of Eq. (74). Points A and B are respectiveIy t;’ + ik
Applying the Cauchy integral theorem, noting that z-%9 is analytic except at the origin, the integral over r = r, + J’, is seen equivalent to an integration over r’ = r, + I’, . Contour section r, is a circular arc of radius YRI, running between arguments -0’ and +O’, where R’ = 2/c’” + k’2 and 0’ = tan-l&‘/c’). Now, we observe that the integral over I’, vanishes due to the large positive real part of z, and write the integral over I’, in polar coordinates leading to (75)
Using Eq. (75), and interchanging Xc0 009 we have Xi; = &
the order of the 4 and r integrations
sIo, d+ s,” dr j,,(kr) e-(c+R’ei6)T.
in
(76)
Next, the r integral in Eq. (76) is processed as was the s integral, with the significant difference that the two separate pieces of the integrand become singular at r = 0.
INTEGRALS
FOR
ELECTRQN-ATOM
SCATTERING
CA~C~LATI~~S
591
However, the r integrand as a whole is bounded at r = 0 so that the lower iiim~t may be changed from r = 0 to a small positive value P = E. e thus have
re we have also made the change of variable 4 --P -c# in the first ter nge the sign of sin q5). Making changes of variable to z = (-5 firsttermandtoz=(-l-R’cos+-ikarrive at
where T consists of two rays between infinity and a circle about the origin of radius E’ = [(< + R’ cos 4)” + (k f R’ sin r~5)~]~k Using the Cauchy integral theorem, we replace I’by an arc of radius E’ from tan-l
--k - R’ sin (b ( 5 + R’ eos + 1
to
k + R’ sin 5, tan-1 ( 5 + 3’ cos 4 1’
plus an arc of large radius on which the integrand vanishes. Since e-# = i on the arc of radius E’, Eq. (77) integrates to
which is clearly equivalent to the result in the main text at Eq. (71). To handle A$?, , we start by interchanging the s and r integrations, f~l~o~~~~ which we process the r integration by the techniques used for Xi: . We obtain x,“pwl =
m ds j-,(k’s) e-c’s s ’ drj,(kr) s0 0
= &
j,” dsj-,(k’s)
6’s
je +j(l 4
e-CT - e-Re’+s),
where 6 = tan-l(k/[) and R = Y’k2 + 5”. Next, noting that the integrand of the s integral is regular at s = 0 we change the s = 0 kimit to a small positive value E. Then we write j-,(k’s) in terms of 581/4/4-1r
592
HARRIS AND MICHELS
exponentials and bring the $ integration outside the s integral. Finally, we make changes of variable from s to z so that Eq. (80) becomes
where T’ consists of four rays as indicated in Fig. 2. Invoking the Cauchy integral theorem, and noting that ehz = 1 near z = 0 and e-a = 0 for large positive Re(z), we have (82)
FIG. 2. Contours used in the integration of Eq. (81). Points A, B, C, D are respectively (1’ + ik’)c, (5’ - ik’)~, (5’ $ R cos #J + ik’ + iR sin (b)~, and (5’ + R cos 4 - ik’ f iR sin&, where R = (kz + 1;2)1/2and e + 0.
where r’ consists of radial and circular pieces as shown in detail in Fig. 3. Performing these elementary integrations on r’, we reach (5’ + R cos 4),)”+ (-k’
+ R sin 4)”
1
k’2 + p +$og( +itan-l(
(5’ + R cos 4)” + (k’ + R sin #I)>” 1 k12 + 1;‘2 k’ -I- R sin $ )+itan-‘( 5’fRcos+
-k’
+ R sin $
c’+Rcos+
The imaginary terms in this expression vanish upon 4 integration, terms are equivalent to Eq. (72).
)I’
(83)
and the real
INTEGRALS
FOR
ELECTRON-ATOM
SCATTERING
\
CALCULATIONS
533
\
FIG. 3. Contours used in the integration of Eq. (82). Not shown are connections at Re(z) -+ fao which make no contribution to the integral. Points A, B, C, D are as &he% in Fig. 2.
The process leading to Eq. (71) also yields Eq. (73), as may be seen by exa~~~~i~g Eq. (76). That equation shows that Xg is of the Eq. (73) form, and an anaI~4s identical to that for X$’ would lead, for X,” , to the given result. REFERENCES Kgl. Fysioguaf. StilIskap Lund, F&h. 14 (1944). Fhys. Rev. 74, 1763 (1948). R. K. NESBET, Phys. Rev. 175, 134 (1968); Fhys. Rev. 179, 60 (1969). F. E. HARRIS, Whys. Rev. Letters 19, 173 (1967); F. E. HARRIS and H. H. h/iac~az~s,P/zyz Rev. Letters 22, 1036 (1969). M. GELLER, J. Chem. Whys. 39, 84 (1963). See, for example, J. LINDERBERG and F. W. BYSTRAND, Ark. Fysik 26, 383 (19 F. E. HARRIS, Technical Report TXVII, Quantum Chemistry group, Uppsala University, Uppsala, Sweden, 1967. 1. D. LYONS and R. K. NESBET, J. Computational Whys. 4, 499-520 (1969). R. L. ARMSTEAD, Phys. Rev. 171, 91 (1968). “Higher Transcendental Functions,” Bateman Manuscript Project (A. Erdelyi, Ed.), Vol. II, p. 51. McGraw-Hill, New York (19.53).
1. L. HULTHEN, 2. W. KOHN,
3. 4.
5. 6. 7. 8. 9.
10.
OF COMPUTATIONAL
PHYSICS
(1969)
4, 579-593
FRANK
E.
H-HARRIS
AND H.
H.
MICHELS
United Aircraft Research Laboratories,
East Har
Received June 16. 1969
L&~STRACT
The integrals needed for expansion identified and classified. Methods are Slater-type orbitals for the bound-state spherical Bessel functions and spherical
calculations of electron-atom scattering are described for evaluating ail integrals, using functions and free-wave functions built from harmonics.
I. lNTRoDUcTroN Recent interest in the application of expansion methods to elastic and inefastic scattering of electrons by atoms has led to a need for methods of eva~~at~~g the integrals arising in such an approach. The classical studies of KuIthCn [I] and Kohn [2], and more recent work by Nesbet [3] and by the present authors [CF], all involve matrix elements connecting the functions used in the expansion of the wavefunctions. The functions of primary interest for expansion a~~rQx~rnat~on to scattering wavefunctions are many-electron functions built from bound-state orbitals and free-wave functions of appropriate asymptotic behavior. The formalism to be described assumes the use of the partial wave expansion, and the free-wave functions therefore have a radial dependence given by spherical Bessel functions. 1 Supported in part at the University of Utah by a grant from the University Research Committee, and by National Science Foundation Grant CP-11170, and in part at the United Aircraft Research Laboratories by the U.S. Air Force Office of Scientific Research under contract No. AF 49(638)-1711.
579
580
HARRIS AND MICHELS
The bound-state orbitals to be considered here are those most appropriate to atomic problems, the well-known Slater-type orbitals (STO’s). The integral evaluation problems to be encountered here differ in some respects from those involving only bound states. First, integrals involving free-wave functions may not converge unless grouped into suitable combinations. Second, some of the integrals require evaluation techniques rather different from those encountered for bound-state functions. Third, some free-wave matrix elements are not invariant under Hermitian conjugation. Some of the quantities needed for these integrals have received previous study, as they are equivalent to Fourier transforms of STO’s. Geller [5] and others [6] have investigated these transforms, and the existent work can be of use here. In addition, portions of the material here reported have been previously presented in limited-distribution reports [7]. Valuable extensions of many of the methods here described have recently been made by Lyons and Nesbet [8].
II. CLASSIFICATION
OF INTEGRALS
The matrix elements entering into expansion formulations of electron scattering from an n-electron atom are of the form (?P’ j H - E / Y), where !P and Y are antisymmetrized (n + 1) electron space-spin functions, H is the Hamiltonian operator, and E is the total energy. For an atom in an initial state of energy E, and a scattering electron incident at momentum of magnitude k (and energy in atomic units &k2), E = E,, + $k2. The (n + 1) electron functions K and Y’ are regarded as built from one-electron functions and, if electron detachment processes are excluded, can be of two types. Either y1of the electrons are in bound-state orbitals and one electron has a free-wave form, or all IZ + 1 electrons are in bound-state orbitals. The bound-state orbital3 are, as previously stated, assumed to be STO’s. Orbital xa , of screening parameter 5, and quantum numbers n a , I, , m, , has in spherical polar coordinates the form
where YLmis a normalized spherical harmonic and N, is the radial normalization factor
N, = [ (2(y;;+1]1’2.
(2)
The free-wave functions are chosen to be of pure angular symmetry and can describe free waves of arbitrary phase asymptotically at large r. One such function,
INTEGRALS
FOR
ELECTRON-ATOM
SCATTERING
CALCULATIONS
which following Nesbet [3] we denote S to indicate it asymptotically sin(kr - @r)/kr, we take as
581 a~~roa~~~s
where j, is a spherical Bessel function of definition
Note that S depends upon a free-wave momentum of magnitude k and has angular quantum numbers 1 and m. When two sine-like free waves occur in expression, we refer to the second as s’, with implied dependence upon k’ A second asymptotic function could be given by replacing j, by a s Bessel function of the second kind, but we would then have fur&i troublesome singularities at r = 0. Instead, following Armstead f!?], we take Ike asymptotically equivalent form
where C varies asymptotically as cos(kr - !&r)/kr and depends upon k and quantum numbers 1 and m. A second cosine-like function will be denoted C’, with implied dependence upon k’, I’, ml. If Y and Yy’ are both built from bound-state orbitals, ( only the bound-state one and two-electron integrals comm calculations. If either Y or Y’, but not both, contain a free-wa use the Hermitian property to transform, if necessary, so that contains the free wave. Hermiticity is assured in this case by the exponential factor in at wavefunction for every electron, thereby giving for each electron a s approach to zero for large r. e now write N - E in the form H - E = H,, - E, + (Tf - +k2) + where HO is the Hamiltonian and E, its expectation value for the n-electron atomic wavefunction included as part of !?‘, the term Tf - +k2 refers to the kinetic ene operator and the asymptotic energy of the free electron of ?P, and the summat includes all potential energy terms involving the free electron “‘f “. The interaction between electron “:f” and the nucleus (of charge +n> is written as in to nt indicate its asymptotic cancellation against the electron repulsions. It
582
HARRIS
AND
MICHELS
from Eq. (6) that the decomposition of (!?“I H - E IY} into one- and two-electron integrals will lead, if !P contains S, to free-wave integrals of types
(XaIm, (xalT- WIS>, (xJ;i+
and (X.Xb~~~X.S).
If Y contains C, we will encounter similar integrals with S replaced by C. For convenience we refer to these two-electron integrals as of “hybrid” type. Evaluation of all integrals of these types follows the scheme previously described by us [7]. If both Y and Y’ contain free waves, we have the two possible cases k = k’ and k f k’. In the application of Eq. (6), if k f k’ the bound-state parts of Y .and Y’ will refer to states of different energy, causing (Y’ 1 Tf - $k2 I!?‘) to vanish due to bound-state orthogonality. For this operator we therefore need consider only (Si T - +k2 IS), (Cl T - ik2 IS), (S/ T - &k2 IC}, and (Cl T - +k2 IC). Whether or not k = k’, the matrix elements of H,, - E,, will vanish, either because (HO) = E,, or from orthogonality. The remaining integrals, from the potential-energy terms of Eq. (6), are of the types
or types obtainable therefrom by replacement of S and/or s’ by C and C’, or are types previously encountered. We call the two-electron integrals in which the same electron is free in both wavefunctions “Coulomb” type, and those in which different electrons are free in different wavefunctions “exchange” type. The Coulomb type integrals cannot be divided into separate terms containing l/u,, and l/r, , as these individual terms would diverge. There are no convergence difficulties with the exchange type integrals. All the integrals arising in this study can be written in terms of a small number of basic integral types. We accordingly introduce the following definitions: J,,(k, 6) = jm r”j,(kr) e-@ dr,
(7)
0
HEv(k, k’, 5) = Jrn r’“j(kr)j,(k’r)
e-@ dr,
(8)
0
KXk i, 5') = j," r y&(kr) A,(<‘r)
e-@ dr,
(9)
YEW, k’,5)=jmCL’j,(kr)M+) 4&i?) dr, 0
Xzz(k, k’, 5, 1;‘) = s,” dr r$(kr)
e-@ Jw ds snj,(k’s) e-4”, T
(11)
INTEGRALS
FOR
ELECTRON-ATOM
SCATTERING
583
CALCULATIONS
where A,(v) stands for
which is the standard definition of the “A” integral arising in mole&ar calculations. I!lEP appear in the evaluation schemes for the other integrals and rst, in Section III. The PC’??r;l”y”,and X,Y are covered in the following sections. e one-electron integrals referred to earlier in this section may now be evaluated directly or identified as expressions involving the JUVdefined in Eq. (7). The results are (13) (xc: ’ S> = h, &in,m,NzJna+dk 5,) = &.z, &rwnaNa IIJn,+wa (k la> + (+I (~~1 T - i-k” is) = 0 (K, I 0
h.&l T - w IO = --6,,ra hwna
JnG,dk,
iye,]
(14 (15) (16)
(Sl T - pz2 is> = 0
(!7)
(G/ T - &k2 IS> = 0
0%
(Sl T - &k” 162) = - & 377 (I + l)(Z i 2)(4Z -+ 9> cc/ T - Sk” 10 = - x (21 + 3)(2~ + 5)(21” 7)
$?ql
The explicit values for the kinetic energy integrals follow from the fact that S satisfies (T - +k2) S = 0 and from material in the Bateman ~a~us~r~~~ [lo]. The two-electron integrals are handled by using ‘the Laplace expansion of 1/rzz . The expression we need is
584
HARRIS
AND
MICHELS
where Y+ and Y- are the greater and lesser, respectively, of r, and r2 . Inserting Eq. (23) we find that a general two-electron integral assumes the form (24) where the coefficients C, arise from the angular integrations and
and where Yi(r) refers to the radial part of Yi . For the Coulomb type integrals, rlil is replaced by rG1 - r$ in Eq. (24) thereby also modifying Eq. (25) if p = 0 by replacement of r;’ by r;’ - r-l For the integrals treated in this “work, Yd is always a free wave, and Y, or Yb (but not both) may be. In evaluation of Eq. (25) the order of integration is manipulated when possible to put all free-wave functions into the outer integral, and to give the inner integral an infinite upper limit. We note that the introduction of r;l - r2-’ for the Coulomb type integrals causes Eq. (25) to have integration limits assuring convergence. The resulting formulas are
[xaxcI XbSL= 4#4 0) [xaxc I XbCl, = L(l,
(26) (27)
I;(2, 1)
0) + (qq
lxaxcI s’a& = 4L’oA 0, 0)
(28)
[xaxc I S’CI, = L’(L 0, 0) + (q+j
1,‘(2,0, 1)
[Xaxc I C’CI, = 4L’(1, 1,0> + (y-j
1,‘(2, 1, 1)
+ (‘g-)
L’(L2,
1) +
(q-qy-j
vxe I XbSlu= m, 0, 60)
1,‘(2,2,2)
(30) (31)
L-%x0 I XaCl,= 1x1,0,0 0) + (+j
1x2,0,
[cicoI xa
1x2, 1, 1,O)
= KU, I,% 0) + p-g, + (qsj
(29)
m,2,0,
1,O)
1) + (q+j(Wj
(32)
1,“(2,2, 1, 1) (33)
IINTEGRALS FOR ELECTRON-ATOM
I:(& jT q, p) = N,N,[X$~~~-”
§CA~TE~I~~
CALCWLATIONS
(k, k’, 5, , LJ + ~;~;%?%‘,
k, Sb , <,>I-
535
C363
In Eqs. (34)-(36), we have adopted the notations 5 = [, f 1;, , TZ= n, + n, s w = nb - p, d = n, - q, n’ = n + v f 1, n’ = n - y + 2. Note that eveaa for integrals involving free waves,
III.
AUXILIARY
FUNCTIONS
In this section we discuss the functions JUd and already mentioned the Juv arise in the Fourier tran e the transform of xa ,
where (k, 8, 4) are spherical coordinates explicit formula for Juv , p > v > 0, is
m-Is of STO’S.
in the transform
space.
eller’s pg
(39) Actually, the Jev are defined beyond the range of Eq. (39), as Eq. (7) is a convergent integral for p + v 2 0, including index values for which p < v or ,U c 0. Tt is convenient to develop the JIly recursively, using relations which folio
586
HARRIS
AND
MICHELS
from the properties of the-j, or from partial integrations. Some such formulas are the following: [7]
Jli,u+l+
Ju,v-1=
(vj
+ v - 1)Jw+l = 5 f+j (P+ v>Jw+x+ (EL CJ,,+ kJim = (k2+ P>Jp+l,v-1 + CP- 4 kJ,v= (k2+ 5”)Ju+~,v + (p + V)(P- v - 1)J&--I,”= (p + v + 1)kJ,v+ (P- v - 1)Ju,v+l=
JG+,
Jm
(P + 4 Ju--l,u (P + v>5Jw-1 &dJw (k2+ 5”)J~+I,~+I .
(41) 442) (43) (44) (45) (46)
Some of the above recurrence relations assume simpler forms for certain index values due to the vanishing of particular coefficients. Two extremely useful such cases are JV+l,V= J v+l,v
(47)
=
We also need some explicit values from which to initiate the recursion process. We take Jo,, = k-l tan-l(k/{) (49) J,,, = (k2 + [“)-‘.
(50)
A detailed examination of Eqs. (40)-(48) indicates that the coefficients of various Jpy vanish in such a way that one cannot proceed recursively to Juy with p < v from Juy values all with p > v. However, Eq. (49) gives one Jpy with p < v and it (with J& suffices to permit recursive generation of all Juy . This point has been
discussed more thoroughly by Lyons and Nesbet [8]. Those authors have also developed methods for obtaining the Juy for parameter values such that some of the procedures indicated here may become numerically unsatisfactory. The second set of auxiliary functions, the Hi,, , may also be generated recursively. The key relationships are
INTEGRALS
FOR ELECTRON-ATOM
SCATTERENG ~A~~~~AT~o~s
587
and the symmetry condition
ff:v@, k’,0 = ffW, k,0.
cm
Starting values are provided by writing j&r) or jWI(kk’r) as complex ex~o~~~~t~~~§~ leading to identification of H$ and H,Y-, in terms of integrals S,, of compkx argument: Hi,, = k’-‘Im J,-,Jk, c - ik’) H,f,-, = k’-l Re J,-lJz,
5 - ik’).
Nesbet give additional formulas involving I!!:” ~
IV. HYBRID AND Cou~om
TYPE INTEGRALS
he hybrid type two-electron integrals depend upon the function Wp ~~~.~e~ in Eq. (9). The WF may be generated recursively y making use of the we~~-~~o~~~ relations for the A, : M)
= 4 A,-,(r)
A,(r)
= r-l edT.
+ AO@)
The corresponding formulas for WF are
W;O = <‘-lJD-l,u(k, r: + ,y’).
WI
The Coulomb type integrals depend upon the Y,“y”of Eq. (IO). Using Eqs. (55) and (56), the Y,$ may be recursively related to the auxiliary functions A$. The result is ym?= LLLV
4 5
V. EXCHANGE
yPA*n-s LkY
+
K2
TYPE INTEGRALS
The integrals of exchange type depend upon the functions Xz”y”of the evaluation of these functions leads to more involved procedur
55%
588
HARRIS
AND
MICHELS
have been necessary for the other integrals. The Xl”y”are needed for index values ,u, v 3 0, for p > 1, and with 4 values which may be either positive or negative, but withp + 4 >, 1. The X,“y”satisfy recursion and transposition relationships which are may be used in their evaluation. Interchanging the order of integration, performing partial integrations, and inserting relationships satisfied by the j, , we find many formulas including the following:
RXk k’, 5, 5’) + xE(k’, k 5’25) = J,,(k 5)Jr&‘, 1;‘)
(k2 + 5”) X,f,?
- 2PSXE$
= - SH,F+l+ Xf&
(p + p)(p - p - 1) xi;l*.1.Q
kH,Zl;t,qt’ + (p - p - 1) H,“:
+ x;:1,”
(61)
= ( ““;
l ) x;;lsg.
(63) (64)
To each of Eqs. (62)-(64) corresponds an equation in which the variation of p andp is replaced by variation of v and q, k and < are replaced by k’ and 5’ and the sign of every HE” is reversed. For example, to Eq. (62) corresponds
Additional formulas result from linear combination of those tested. Generation of general X,“v” requires starting values and, just as for the Jgy , there are index boundaries which cannot be crossed in certain directions recursively. We accordingly need starting values in various index regions. Some starting values are provided by setting v = 0, q = 1, following which the inner integral admits of explicit evaluation. We obtain
Xpl WI= -& Im N <, J ik ) Jdk 5 + 6’ - iW].
o-33)
If now we set p = y = 0 and use Eq. (61), we also have
X&’ = - k Im [ (&)
J&k’, 5 + 5’ - ik)] + k,ck2‘+ 52jtan-’ ($-).
(67)
Evaluation of J,,,, for complex arguments requires the identity tan-l(u + iv) = $ tan-l (
2u
1 - 23 - 02 ) + ; log ( ;; +’ g ? ;;: 1.
w9
INTEGRALS FOR ELECTRON-ATOM
~ivale~t starting values are obtainable
SCATTERING CALC~LA~~Q~~
by using IV;
589
of comIslex argu
Even with the starting values represented by Eqs. (6~)-(7~), not all the ~e~~ssar~ X,“e are accessible. We still need additional values in the regions represented by Xi,0 and X,j?!, . These functions appea.r not to be elementary, and we have found it expedient to evaluate them numerically after manipulating them to convenient forms by methods described in Appendix I. Tbe forms to e integrated ~~rne~i~a~ly are 8’ 1 -1 R sin B + R’ sin 4 (71) Xii = m dd s -e, tan RCOS~+R’COS~
where R = ?/kz + iz, R’ = 2/kQ + c12, 6 = tan-*(k/Q 0’ = tani(k’/[‘). The method which leads to Eq. (71) also provides an alternative formula fix more general X integrals. The result is
A$==&jr
8’
J&k,
5 + R’ ei”> d+
ACKNOWLEDGMENTS We thank Mr. J. W. Viers and Mrs. R. Barrett for assistance in developing computer pro~ams for many of the processes reported here. Mr. Viers also checked many phases of the work. We are also happy to acknowledge a discussion with Drs. R. K. Nesbet and J. D. Lyons which led to the discovery of Eq. (73).
APPENDIX
I.
MANIPULATIQN
OF X$’ AND X&
The purpose of the rearrangements leading to Eqs. (71) and (72) is to obtain forms which do not exhibit rapid oscillation over the region of integration, so that standard numerical integration procedures will converge rapidly. Cionsi first X$‘. Writing the inner integration in the form
590
HARRIS
AND
MICHELS
we introduce in the first integral the complex variable z = (5’ - ik’)s, and in the second integral z = (5’ + ik’)s. The inner integration is thus equivalent to the contour integral
where r consists of the two rays marked I’, and r, in Fig. 1.
FIG. 1. Contours used in the integration and c - ik’.
of Eq. (74). Points A and B are respectiveIy t;’ + ik
Applying the Cauchy integral theorem, noting that z-%9 is analytic except at the origin, the integral over r = r, + J’, is seen equivalent to an integration over r’ = r, + I’, . Contour section r, is a circular arc of radius YRI, running between arguments -0’ and +O’, where R’ = 2/c’” + k’2 and 0’ = tan-l&‘/c’). Now, we observe that the integral over I’, vanishes due to the large positive real part of z, and write the integral over I’, in polar coordinates leading to (75)
Using Eq. (75), and interchanging Xc0 009 we have Xi; = &
the order of the 4 and r integrations
sIo, d+ s,” dr j,,(kr) e-(c+R’ei6)T.
in
(76)
Next, the r integral in Eq. (76) is processed as was the s integral, with the significant difference that the two separate pieces of the integrand become singular at r = 0.
INTEGRALS
FOR
ELECTRQN-ATOM
SCATTERING
CA~C~LATI~~S
591
However, the r integrand as a whole is bounded at r = 0 so that the lower iiim~t may be changed from r = 0 to a small positive value P = E. e thus have
re we have also made the change of variable 4 --P -c# in the first ter nge the sign of sin q5). Making changes of variable to z = (-5 firsttermandtoz=(-l-R’cos+-ikarrive at
where T consists of two rays between infinity and a circle about the origin of radius E’ = [(< + R’ cos 4)” + (k f R’ sin r~5)~]~k Using the Cauchy integral theorem, we replace I’by an arc of radius E’ from tan-l
--k - R’ sin (b ( 5 + R’ eos + 1
to
k + R’ sin 5, tan-1 ( 5 + 3’ cos 4 1’
plus an arc of large radius on which the integrand vanishes. Since e-# = i on the arc of radius E’, Eq. (77) integrates to
which is clearly equivalent to the result in the main text at Eq. (71). To handle A$?, , we start by interchanging the s and r integrations, f~l~o~~~~ which we process the r integration by the techniques used for Xi: . We obtain x,“pwl =
m ds j-,(k’s) e-c’s s ’ drj,(kr) s0 0
= &
j,” dsj-,(k’s)
6’s
je +j(l 4
e-CT - e-Re’+s),
where 6 = tan-l(k/[) and R = Y’k2 + 5”. Next, noting that the integrand of the s integral is regular at s = 0 we change the s = 0 kimit to a small positive value E. Then we write j-,(k’s) in terms of 581/4/4-1r
592
HARRIS AND MICHELS
exponentials and bring the $ integration outside the s integral. Finally, we make changes of variable from s to z so that Eq. (80) becomes
where T’ consists of four rays as indicated in Fig. 2. Invoking the Cauchy integral theorem, and noting that ehz = 1 near z = 0 and e-a = 0 for large positive Re(z), we have (82)
FIG. 2. Contours used in the integration of Eq. (81). Points A, B, C, D are respectively (1’ + ik’)c, (5’ - ik’)~, (5’ $ R cos #J + ik’ + iR sin (b)~, and (5’ + R cos 4 - ik’ f iR sin&, where R = (kz + 1;2)1/2and e + 0.
where r’ consists of radial and circular pieces as shown in detail in Fig. 3. Performing these elementary integrations on r’, we reach (5’ + R cos 4),)”+ (-k’
+ R sin 4)”
1
k’2 + p +$og( +itan-l(
(5’ + R cos 4)” + (k’ + R sin #I)>” 1 k12 + 1;‘2 k’ -I- R sin $ )+itan-‘( 5’fRcos+
-k’
+ R sin $
c’+Rcos+
The imaginary terms in this expression vanish upon 4 integration, terms are equivalent to Eq. (72).
)I’
(83)
and the real
INTEGRALS
FOR
ELECTRON-ATOM
SCATTERING
\
CALCULATIONS
533
\
FIG. 3. Contours used in the integration of Eq. (82). Not shown are connections at Re(z) -+ fao which make no contribution to the integral. Points A, B, C, D are as &he% in Fig. 2.
The process leading to Eq. (71) also yields Eq. (73), as may be seen by exa~~~~i~g Eq. (76). That equation shows that Xg is of the Eq. (73) form, and an anaI~4s identical to that for X$’ would lead, for X,” , to the given result. REFERENCES Kgl. Fysioguaf. StilIskap Lund, F&h. 14 (1944). Fhys. Rev. 74, 1763 (1948). R. K. NESBET, Phys. Rev. 175, 134 (1968); Fhys. Rev. 179, 60 (1969). F. E. HARRIS, Whys. Rev. Letters 19, 173 (1967); F. E. HARRIS and H. H. h/iac~az~s,P/zyz Rev. Letters 22, 1036 (1969). M. GELLER, J. Chem. Whys. 39, 84 (1963). See, for example, J. LINDERBERG and F. W. BYSTRAND, Ark. Fysik 26, 383 (19 F. E. HARRIS, Technical Report TXVII, Quantum Chemistry group, Uppsala University, Uppsala, Sweden, 1967. 1. D. LYONS and R. K. NESBET, J. Computational Whys. 4, 499-520 (1969). R. L. ARMSTEAD, Phys. Rev. 171, 91 (1968). “Higher Transcendental Functions,” Bateman Manuscript Project (A. Erdelyi, Ed.), Vol. II, p. 51. McGraw-Hill, New York (19.53).
1. L. HULTHEN, 2. W. KOHN,
3. 4.
5. 6. 7. 8. 9.
10.