-electron atoms and molecules

PHYSICAL REVIEW A VOLUME 30, NUMBER 5 NOVEMBER 1984

High-energy asymptotic behavior of the dipole-oscillator-strength densityand a new "modified" dipole-oscillator-strength sum rule for nonrelativistic

N-electron atoms and molecules

Michael C. StruenseeDepartment of Physics, Uniuersity of Texas at Austin, Austin, Texas 78712

(Received 25 June 1984)

The high-energy asymptotic behavior of the oscillator-strength distribution of nonrelativistic N-

electron atoms and molecules is examined. A derivation is presented which leads to the coefficients

of the first two asymptotic terms of the oscillator-strength density in terms of expectation values ofthe wave function. Furthermore, it is shown that truncating the contribution of these first two

terms beyond some cutoff energy leads to a sum rule for the third moment of the resulting modified

distribution. Numerical calculations have been performed on atomic hydrogen which illustrate the

sum rule for the third moment of the modified distribution.

I. INTRODUCTION

The high-energy asymptotic behavior of the dipole-oscillator-strength density has been examined for two-electron atoms by Salpeter and Zaidi' for use in a Lamb-shift calculation of helium and by Dalgarno and Ewartfor the photodetachment cross section of the negative hy-drogen ion. A Lamb-shift calculation by Schwartz wasperformed using a different approach from that of Sal-peter and Zaidi, and the attention was directed toward ob-taining J(k), a sum over states expression parametrizedby k, the virtual photon energy. The present work showsthat the high-k asymptotic expansion of J(k) given bySchwartz leads to the coefficients of the first two asymp-totic terms of the dipole oscillator strength density for anynonrelativistic ¹lectron atom in terms of expectationvalues of the wave function. Moreover, an additionalterm appears in the expansion of J(k) which is related tothe third moment (with respect to the energy) of the oscil-lator strength distribution with the contributions of thefirst two asymptotic terms of the oscillator strength densi-

ty subtracted out beyond some "cutoff" value of the ener-

gy. This leads to a "sum rule" which gives the third mo-ment of this modified distribution in terms of an expecta-tion value of the wave function and the cutoff parameter.Note that the third moment of the (unmodified) dipole os-cillator strength distribution diverges due to the contribu-tions from the first two asymptotic terms, so the subtrac-tion scheme involving these terms is required in order toobtain finite results. Finally, it is shown that the aboveresults obtained for atoms can be generalized to apply tomolecules in the fixed nucleus approximation.

The organization of the paper is as follows. Section IIexamines the evaluation of the third moment of the oscil-lator strength distribution and indicates the reason for thedivergence of thi. s moment. The divergence of the thirdmoment is then shown to lead to a rather weak result con-cerning the asymptotic behavior of the oscillator strengthdensity at high energies. Section III contains a derivationof the high-k asymptotic expansion of J(k) in terms of

expectation values of the wave function of interest. Sec-tion IV contains a derivation of the high-k asymptotic ex-

pansion of J(k) in terms of the oscillator strength mo-

ments and the high-energy asymptotic behavior of the os-cillator strength density. Finally, the expansion of J(k)obtained in Sec. III is compared to the expansion of J(k)obtained in Sec. IV. Equating the coefficients of the twoexpressions term by term leads to the new results. SectionV indicates how the results derived for atoms may be ex-tended to apply to molecules. In Sec. VI, numerical calcu-lations have been performed on atomic hydrogen whichconfirm that the third moment of the modified distribu-tion obtained from the sum rule is in agreement with thethird moment of the modified distribution computed ex-

plicitly by summing over the spectrum, as expected.

II. DIVERGENCE OF S(3) FOR ATOMICSTATES WITH NONZERO ONE-ELECTRON

DENSITY AT THE NUCLEUS

The familiar oscillator-strength sum rules for S(—1),S(0), S(1), and S(2) express moments of the oscillator-strength distribution in terms of expectation values ofoperators for the wave function under consideration. It isalso straightforward to calculate the operators required toobtain sum rules for S(3),S(4), . . . (see, e.g. , Jackiw ) butthese are of little interest for atomic ground states sincethey diverge. The divergence of S(3) for atomic stat&swith nonzero one-electron density at the nucleus is shownin the following. The divergent terms may easily be iso-lated and the remaining convergent terms will be seen toappear in the sum rule derived in Sec. IV for the modified

,distribution.For the following discussion, attention shall be restrict-

'

ed to nonrelativistic N-electron atoms so the Hamiltonianis given by

The oscillator strength f„»between state~

n ) and state

~b) is defined by

30 2339 Qc1984 The American Physical Society

2340 MICHAEL C. STRUENSEE 30

f.b= —'(n g x„b) + (n gy„b) + (n xz„b)' ro., (2)

where p ranges over the N electrons of the atom andco„b Eb——E„—. The dipole oscillator-strength sum of or-der k is defined by

S«)= +fob(~ob)', (3)

where IO) has been taken to denote the ground state.The following relations are easily shown to exist betweenthe length, velocity, and acceleration forms of the dipolematrix element:

p;(r)= f «; f dU~; IyoI', (12)

(Po g, 00)=X J ', dr

where the integral is evaluated over the configurationspace of all the electrons except that indexed by i, then anintegration is performed over the solid angle associatedwith electron i. Then the first matrix element may bewritten

Zn, b = —n V b co~b~p;(r) dp;(r)+ lnr

dr p

where

=(nI

rI&)~'b

Using the cusp condition,

d p;(r)—f lnr drdr' (13)

and

r= ip,P

v=gv„,P

(7)

rr 3

rp3

P P

Using relations (4) and (5), along with the Hermiticity ofthe length and' acceleration operators (6) and (8) and theanti-Hermiticity of the velocity operator (7), readily leadsto the usual oscillator-strength sum rules for S(—1), S(0),S(1), and $(2).

The sum rule for the third moment of the oscillator-strength distribution, $(3), may easily be obtained by writ-ing the oscillator strength in the acceleration-accelerationform and one finds

$(3)= ufo. (~o. )'

dp;(r)lim = —2Z limp;(r)r~0 dr r~0

(from Kato ), allows Eq. (13) to be rewritten as

(N

1N

fo ——lim g —+2Zlnr p;(r)/

r . r~o1

r

d p;(r)lnr

dr

Therefore S(3) is given byr

2 2N

S(3)= —,Z lim g —+2Z lnr p;(r)1

r~p.1

r

m d p;(r)—-', Z'y f lnr ', drdr

3 3 3 0ri rj

(14)

3 3 (10)

It is convenient to separate the one-electron terms fromthe two-electron terms which yields

(11)The first of the two matrix elements in responsible for thedivergence and the second matrix element is well behaved.The first matrix element may be expressed in a formwhere the divergence becomes even more transparent bycarrying out two successive integrations by parts. The ra-dial one-electron density shall be defined according to

&~&~=~z'(0o X, 4a)+~z'(PD X ', ,' 4o).

The term in the large square brackets clearly yields adivergent result upon evaluation at the limit r=0 (i.e., atthe nucleus). Hence S(3) diverges for wave functions withnonzero one-electron density at the nucleus. The tworemaining terms are finite and will appear in the sum rulefor the modified distribution to be derived in Sec. IV.

The convergence of S(2) and the divergence of S(3) leadto a rather weak result concerning the asymptoticbehavior of the oscillator strength distribution, apparentlyfirst observed by Dirac and Harding. The fact that thesecond moment converges and that the third moment doesnot evidently implies that the oscillator-strength densitymust behave asymptotically as df /d E ~ e ", where3 & n &4. For atomic states with zero one-electron densi-ty at the nucleus (e.g. , n-p, n d, n-f, etc. states of a-tomichydrogen) $(3) is finite, hence df/de~@ ", where 4& n

HIGH-ENERGY ASYMPTOTIC BEHAVIOR OF THE DIPOLE-. . . 2341

III. DERIVATION OF THE HIGH-kASYMPTOTIC EXPANSION OF J(k) IN TERMSOF WAVE-FUNCTION EXPECTATION VAI UES

In this section the asymptotic expansion of J(k) forhigh values of k is derived in terms of expectation values

of the wave function of interest. The derivation essential-

ly follows the presentation by Schwartz, although manyadditional details are included in this treatment. The firstfive terms of the asymptotic expansion of J(k) are ob-tained with the coefficients determined in terms of expec-tation values of operators for the wave function of in-

terest. Of the five coefficients, the first two reflect well-

known properties of the oscillator-strength distribution,while the last three yield information not previouslyknown (as will be shown in Sec. IV).

The sum over states expression J (k) is defined by

J(k)= —', gQ)p~ +k

(17)

The disappearance of the asymptotic terms which behave

as e " with 3 & n (4 for atomic states with zero nuclearone-electron density tends to indicate that the coefficientsof these terms involve the one-electron density at the nu-

cleus. This suspicion will be confirmed in Sec. IV, whichis not surprising since high-energy scattering phenomenais frequently highly sensitive to the behavior of the wave

function at the nucleus.

It will prove not only useful, but in fact crucial, to in-troduce the functional (apparently first suggested bySchwartz")

w(g, k) = &g i EQ H ——ki g& —2&(

iV

i1(Q& (22)

which is associated with the differential Eq. (20). If equa-tion (20) has a solution, then this functional is stationary(i.e., the first variation 5w of w vanishes) for

i g& =i8&.

Conversely, a function for which this functional is sta-tionary satisfies Eq. (20). These claims are proved as fol-lows. To obtain the variations of the functional, '

w (8+eQ, k) must be expanded in a Taylor series in e,

w(8+EQ k) =w(8)k)+ewr(8 Q k)

+2Ew (OQk)+''' (23)

(O+ Q, I)=(&8i+ &Qi)(Z, —H —k)(i8&+ in&)

—2(&8 i+e&Qi

)Vi P &, (24)

= &8i(EQ —H —k)

i8& —2&8 V

i1(tQ&

+e[2&Qi(EQ —H —k)

i8&

Note that the expansion defines the functionals wi andThe first variation 5m of m is defined by

5w=ew, (8,Q, k). The second variation is defined by5 w = —,

'e wz(8, Q, k). Performing the expansion in order

to make the identifications leads to

1&0. IV lr('Q& I'

E„—Ep+ k(18)

—2&Qi

Vi PQ&]

+ —,e [2&Qi (EQ —H —k)

i Q&] . (25)

As is well known, ' J(k) may be expressed in the alter-nate form

J(k)= —&8i Vi qQ&,

Therefore

5w =26& Qi[(EQ H —k)

i8 &

——Vi qQ & ] (26)

where 8=8(r,k) is the solution to the inhomogeneousdifferential equation

(H —Z, +k) iO&+V iy, &=0. (20)

~ &P. IV I@Q&

iE„—Eo+k

Hence, inserting Eq. (21) into Eq. (19) yields

1&0. VIA I'E„EQ+k—

(21)

3

2

On~On

No~ +k

which verifies the claim.

When k&0 the operator H —Eo+k is positive on thespace of functions orthogonal to the ground-state wave

function; therefore if a solution to Eq. (20) exists in this

space, it is unique. ' The equivalence of the forin of J(k)defined by Eqs. (19) and (20) with that given in Eqs. (17)and (18) is seen by noting that Eq. (20) may be formallysolved in terms of the complete set of eigenfunctions of Hby

5 w =e &Qi

(EQ H —k)i

Q&—and all higher variations vanish. Inserting Eq. (20) intoEq. (26) yields

5w =2@&Qi[(EQ H —k)

i8 &

—V—i gQ & ]=0, (28)

therefore the functional is stationary fori g &

= 8 &. Theconverse is similarly shown by performing the expansionabout an undetermined function for which the functionalis required to be stationary. The fact that e

iQ& is arbi-

trary requires that for the first variation to vanish, the un-determined-function must be a solution to Eq. (20). Alsonote that a variational calculation of w will lead to a valuesuch that w &J(k). This follows from Eq. (27) which in-

dicates that for k &0 the second variation is always nega-tive. Therefore any deviation from the stationary positionof w [where w =J(k)] will tend to decrease w, so the sta-tionary position is indeed a maximum. Since all varia-tions beyond the second vanish, the maximum is not onlylocal but in fact global.

Note that w(8, k)=J(k), so ifi8& is known exactly

then either Eq. (19) or Eq. (22) may be used to evaluate

2342 MICHAEL C. STRUENSEE

J(k). However if only a close approximationI

8'& toI0& is known, it is better to use Eq. (22) and evaluate

w(O', k) rather than

—&O'

since the functional associated with Eq. (19),

first five terms of the asymptotic expansion, while usingthe alternate expression yields correct values of only thecoefficients of the first three terms. In addition, in thiscase, use of the stationary functional helps lead to an ap-proximation to

I8 & which would appear to be difficult to

achieve using other techniques.The leading term of the asymptotic expansion of

I8&

for large k is clearly seen from Eq. (20) to be

will not be stationary about /=8. Inserting a close ap-proximation to

I8& into the stationary functional should

lead to a more accurate approximation to J(k) than usingthe other expression due to the fact that the first variationof a stationary functional vanishes. Indeed, inserting theapproximation to

I8 & (which will be obtained below) into

the stationary functional gives correct coefficients for the

1-——~ IPo&kso

I8 & may be written as

I8&=-—~ Iy. &+ IU&.k

Substituting Eq. (29) into Eq. (22) then leads to

(29)

& 4o I'7k(&o If —k)—~

I 4o &——

& Po I

()'I fo &

——& U

I(Eo H k)—7)

I0—o &

k k k

+&UI(Zo —a —k)

IU& —2{U

IV

I yo&,

= —k &Col ~'l@o& —

k, &Pol ~l:~,H]I 0o& &U—

I [—% I @o&+& I(Eo —If —k)I U&

1 2Z N j.= ——&Vol~'Ifo& —,4o ~ g ', 0o —

kU g ', go +&Ul(Eo II k)l U—&. —

k k Pg i(30)

Note that V is anti-Hermitian and g ( r; Ir; ) is Hermitian and the reality of the matrix element implies

N0V, p= —p V, 4o

N

0 6 ri 0 ~ (31)

Therefore to(8, k) reduces to

(32)1 2' N

w(k), k)= ——(()(V'(() )—,P g E(F) o)o — U g, () )+(U ((E H —k)

(

U)—k k k i, y,.

3

= —k &@I~ Ifo& —„,4o g@r ) 4o +tJ(U, k),1 2 2~Z

i=1(33)

where a new functional u) (g, k) has been defined by property implies thatI

U& satisfies the differential equa-tion

2Z$(o) k)=(o)

l(Eo H —k)l o)) — o) X, Po) . —

k ',.1 yi3 z(H Eo+k)

IU&=———g,'

lgo& .k, , y,.' (35)

(34)

Since to(g, k) was stationary with respect to variationsabout

I g& =I8&, the functional io(g, k) must be station-

ary with respect to variations aboutIg&=

IU&. This

An approximate solution toI

U & must be constructedsuch that the function behaves properly for high k andsmall rj. The importance of the wave function near thenucleus for high-energy phenomena has already been es-

30 HIGH-ENERGY ASYMPTOTIC BEHAVIOR OF THE DIPOLE-. . . 2343

where

X[(1 Zrj—)+ aj rj+O(rj )],

The small r behavior of 8& may now be deduced by

examining Eq. (21),

V. I~

I AE —Ep+k

Evidently the wave functions may be expanded according

to(ifI r; I

) Ir

I )

Wn = gcnvlm(ri r2 ~ rj —i rj+i. ~, r~)vs

tablished and this is essentially why it is necessary for spe-

cial care to be taken thatI

U& has the correct behavior

for small rz. Before proceeding with the derivation it is

necessary to introduce the version of the cusp condition

due to Bingel, ' which states that for small rj and

Ir; I & I r, I

far all i~j(()'j(ri, r2, . . . , rz i, rz, rz+i, . . . , rx)

Note that the correct small- rj behavior of Vjgo cannot beobtained from an expansion in the basis which describes

the g„wave functions. Each of the three components of

r.J go(ri rg, ~ ~ ~, rj i 0 rj+i, ~ ~ ~, rN)rJ.

has P symmetry in the jth electron coordinate, yet is

nonzero at the nucleus. All the terms in the Ig,i(rj ) j set

behave as rj, so use of a finite number s of these functions

to expand a P state which is finite at the nucleus must

lead to a function which behaves as rj near the nucleus.

Taking the limit s ~ ao would most likely lead tobehavior similar to Gibbs phenomena' ' but it is un-

necessary to pursue this further. Note that the secondterm in the expansion of Vjgo

aojfo( r 1, r2 . . . , rj i,0, rj+i, . . . , rg)

is also nonzero at the nucleus, but has components with Ssymmetry, therefare can be obtained from an expansion interms of the set I(I)~(rj) j.

The above observations may now be applied to the ex-

pansion af 18&. Assuming k is very large, then awayfrom the nucleus it is reasonable to expect

X0 I(rJ)l'im(~, 0;) (37)Io& =- —g

V o

n

where the elements of Ip~(rj)j (associated with one-electron "S states") behave as 1 —Zrj near the nucleusand the elements of I p, (rj ) j (associated with one-electron"P states") behave as rj near the origin. Using Eq. (36) itis clear that near the origin, V jPo behaves as

V jQo(ri, r2, ~, rj i, i j, i j~i, . . . , r~)

= o( i, rp, . . . , r, i,0, rJ+i, . . . , rg)

=——k X I W. & &4. 1~1&o&

n

Hence, away from the nucleus, completeness may be usedand

rjX —Z + aoj+ &(rj )

ri

(38) which is the first term in the expansion for 18&, abtainedbefore. However, near the nucleus

+o(r;) g~l—= —

k X I @.& 0. g ao; Wo—g E„—Ep+ k

—k g ao. Ifo& + go(r, )

i =1. i=1(39)

MICHAEL C. STRUENSEE

is the expected behavior from the arguments presentedpreviously. Note that in the summation over n involving

(Z—r;/r;), the E„E—p term in the denominator

must be retained, since it provides a built in cutoff for thehigh E„statecontributions. This prevents the occurrenceof Gibbs phenomena with associated undefined behaviorof

~8& near rj ——0.

An approximate solution~

U&, to Eq. (35),where

rl +rl'

(Po)a

Ir

1 2——, g —,V1;„——0, (47)l

can now be determined and a homogeneous solution canbe chosen which gives

~

8 & the appropriate behavior near

rj =0. Writing~

U& as the sum of the particular solu-tion

~U1 & and the homogeneous solution

~Uz &, it is ob-

vious that~

U1 &= —(Zlk ) g, (r;I.r; ) ~1(p& is a valid

particular solution for large k if rj is not close to the nu-

cleus. However the validity of the solution near the nu-

cleus is necessary and must be checked. The radial com-ponent of the Laplacian of a vector expressed in sphericalcoordinates is required and is given by

(V A), =V A„—r

2 ~Aer2 BO

2A cotO 2 BA&

rz rzsin8 Bp(40)

(Ep H)i

U1&=—0 (42)

near rj =0. Recall that~

U1 & has been assumed to be

U1 & =V1I Po& (43)

where

All vector-valued functions that the Laplacian acts uponin what follows are radial vectors and angle independent(to the order of accuracy desired) so Eq. (40) becomes

r

BA, 2A,(V A)„= r —

z (41)r Br Br r

For~

U1 & to be a valid particular solution to Eq. (35) nearthe nucleus, it is necessary that

and

(49)

1V,„—g z V„„=o.

l ri

It is easily confirmed that Eq. (50) is satisfied.An asymptotic solution

~Uz & (to the homogeneous dif-

ferential equation) which gives~

8& the correct behaviorat the nucleus must now be found. The homogeneousequation is given by

(Ep H —k}~

U—z & =0 (51)

Assume a product solution for~

Uz & of the form

~Uz&=Vz

~$0&. Substituting this expression into Eq.

(51), then dividing by 1(0 results in the equation

—k Vp ——0. (52)

Changing variables from r to s =k' r leads to the equa-tion

l 0

Near rj ——0, retaining only the most important terms inthe equation yields

r

1 8 z (50)r,zBr; 'Br;

(44)

Assuming

V, =V', +I -'~'V,'+k-'V'+

(53)

r;Vi. ———

k r(45) and inserting this form for V2 into the differential equa-

tion yieldsSubstituting the above form for

~

U1& into Eq. (42) andthen dividing by pp leads to the condition on V1 to be ver-ified Q -,' V,', Vz' -V,' k

(46)

V~ is the sum of one-electron, angle-independent, radial-vector-valued functions, hence

1VzV1 Vl+ V i t(0~ V„.V, k

0

=0 . (54)


g —,V,gVz —Vz ——0, (55)

Equating the coefficients of the powers of k to zero leadsto the system of differential equations

Therefore the desired solution is given by

N

Vz=cp g 3 exp( —V2S )(1+V2S )l=1 Sl'

(62)

1 Pz V 1 V 1+ V;go V'„V2——0,4o

or, in terms of rj,

C0 N rVz —— g 3 e '(1+pr;),k,.

1 r,.(63)

~ ~ ~ (56)

All that is needed for the present work is the solution to0 + 0V2. Assuming V2 is a sum of one-electron, angle in-

dependent, radial-vector-valued functions, Eq. (55) may bewritten as

0 ] 2 0—1 Vz„——, g —z Vz;„——0,

l l (57)

where Vz„——g,. Vz;, . Since each Vz~„has been assumedto be dependent upon only sj, the above equation may beseparated into solving the equation

1 c)

BSJ+- a

SJ Bsj.

1———1 Vz„——02 JPJ

—V 2s,.The e ' choice for large sj is required since the homo-geneous piece is needed only to correct the solution forsmall rj; away from the nucleus the solution previouslyobtained is valid and must not be affected by the homo-geneous solution. For small sj the sj behavior must bechosen in order to cancel the g,.(r;/r; ) singularity of theparticular solution

IU& &. Writing

Vzj, ——sJ exp( v2sj )f— (59)

for each j=1,2, . . . , N. This equation has a regularsingularity at sJ ——0 and an irregular singularity at sJ = oo.The solution (from the indicial equation) near sj =0 iseasily checked to behave as sz or sj . For large sJ thesolution clearly behaves as

J+~2 .

where p=(2k)'~ . Hence the homogeneous solution is

I Uz & = X,e '(1+v~;)I go& .k, , r,.

(64)

Consequently, for smallbehaves as

I0&= —k

~I Po& k,

—

rf, the approximation toI8&

,'

[1—e" '(1+pr; ) j I Pp&

—Z +ao+0(r)I Po&k,. 1 r;

Z "r+O(r )

I Qo&k; 1r;N

apI 1(p& + g O(r;)

k .i=1 i=1

which is precisely what was needed.It is appropriate at this point to note that a "cancella-

tion of errors" has occurred between the approximate par-ticular solution and the approximate homogeneous solu-tion which makes the total solution even better than ex-pected. Recall that in the particular solution the term

In order to cancel the singularity associated with the par-ticular solution, the undetermined constant c0 must bechosen to be (Z/k). Thus the desired solution to

IU& is

given by

z, [1—e '(1+sr;)/

I fo&k l 1rl

leads to the differential equation for f given by

f"+ ———~2 f—'+ f=0.1 „1,v22 s~

(60)

( o)Bl';

Br;

Z1 2Zk r; k

(1t p)a

1

0o

Attempting a power-series expansion leads to the solution

f, =co(1+v 2sj )

and a second linearly independent solution is easily foundto be

fz =dp( 1 —W2s& )exp(2W2SJ )

in Eq. (47) was ignored for small r~. Hence the homo-geneous equation which was solved afterwards reallyshould have been the inhomogeneous equation

—k 'V2

hence

f=cp( 1+V 2S& )+dp( 1 —v'2SJ )exp(2v 2sJ )

(—2ZWo

p)rl.

r,4 (66)

f =cp(1+ V2SJ ) . (61)

The behavior of fz for large sj will, of course, lead tounwanted behavior of V2J.

„

for large sJ. Thus, d0 ——0 and Now replacing U2 by the solution which was obtainedgives, in the radial component version of Eq. (66), the "er-ror equality"


a

a Ze ""',( p)

p &&; k r;

2Z N r;w(Uk)= — U X o' ))o)+(U(Eo H ——k (U),k

(l(p)—2Z [)r] 1(67)k~; p r

where

Z N

2 2 5 ~1 —e '()+V'))I Pp& .k'; ]r

where the terms on both sides were previously ignoredwhen the homogeneous equation was solved. As can beseen the rJ singularity is canceled, resulting in an evenbetter solution for sma11 rJ than was realized, because Eq.(67) is in error for small r& to the order of r~, while theerror in the original homogeneous equation was of order

—3rJ.

Having obtained an approximate solution to~

U) withthe desired behavior, all that remains is to insert the ap-proximation to

~U) into the functional [T] and to evaluate

the resulting expression in the limit of large k, i.e., evalu-ate

The functional u]( U, k) may be written

w( U, k) =]7]](U, k)+ u]2( U, k) (68)

8](U,k)= g ]L)];(U,k),

where

with $](U, k) containing contributions from one-electronoperators

w, ,(U, k)= () —[l —e '( +(r,))o] () l2Z 1 —pr.k' r4

J

Z21 —e ' 1+pr E0—H —k 3 1 —e ' [email protected]

J J(69)

and with wq( U, k) containing contributions from two-electron operators

2Z2 N

w, (Uk)=, , X 4o ', ', [l—e ( w+r)r, ))eo]o]lk i~J l ri

Z2 N

+ X ()o o [l —e '( +)r;))e] (Eo H —k) o [l —e —'(1~)er )] o)o) .k 'I+J l rJ rJ

(70)

The expression 8](U, k) shall now be evaluated by determining the contribution from the ]p]~(U, k) term. In the fol-lowing, the indices on the rj s shall be dropped for notational convenience. Defining

g =g(p, r)=1 e~"(1—+pr),a straightforward calculation yields

2Z 1 Z2w&j(»k)= k, oko eE 'ko + ke Ao, k'(Eo —H k), (( ()ol—

2Z' 1 Z' 1 2Z'

WO, R eo —,()0 „E()0 +, Wo, g (Eo H) g eo)'—k' r4 k' r' k' r' r'

3 0 48 0 3 0 4g 0

Z2 r r Z2 1 1 2 14 0p 3g 5g Wp1 + 4 0p 2gr r k r 2 r ()o)+ k, (()o , g (' —,E & —4o)

3 0 4g 0 '

3 0 4g 0 4 0 38 5g 0 + 4 0 3g 58 0

, ()o,((,e o'()+) r) Wo +, 4o ek r e "'——g ()o),k3 r3 r3 k4 r4 r Br

(71)

where the angle-dependent terms of the Laplacian have been dropped because they lead to terms in k less significant thar

the order which will be of interest. Collecting terms results in considerable simplification, yielding

S~J(U,k)= 3 (3+8),z'k

A= 0-- I —e &" j.+p~ o1

B=—(), [(—e "'((+pr)] rp'e "' ](—e "'((—+—pr)] -((,).1 1 „p„2 „8k p~ r Br

I

The matrix element A can easily be written as the integral

A = I —8 I+pl' p P' f'

p.(r) =fd QJ fdu+J.~ $0 (

dr, rI [1—eI'"(1+pr )j .p pZ

~'

e I'" I oe p -P~-- p(r)Io" f-r p 0 p'Pdr1 d

dp'

e "" dp " 1+r dr ~h re "" dp dr

dp'=]Mp(0)+ f

Hence 1ntcgrat10n by parts may bc applied Rnd, dropping thc subscript J from pj (r) for notational convcmcncc, 1t follows

that

=pp(0)+y „0—ln — „0— lnrq dr+Wk(&dp 1 dp ~ d p

dr "]M dr " ]~a dr~

=]Mp(0)+y „o+ln]M „0— . lnr dr+W odp dp d pdl' dr' o dy" 2

=(2k)'~ p(0)+y P „(]+—,' ln(2k) P „(]—f lnr P dr+Wko,

r dr 0 dr(76)

where Wko represents terms in k less significant than k (=1). Note that above, the Euler-Mascheroni constant,

y=0.57721. . . is obtained by applying Eq. 8.367.12 from the integral tables by Gradshteyn and Ryzhik. ' Next, thematrix element 8 may be written as the integral

r

QO8= .f [1—e "'(1+pr)] rp e ""——[1—e ""(1+pr)] dr2k 0 I 2 r dr

(7&)

(79)


and

P= ——f —[1—e ""(1+pr)]—[1—e &"(1+pr)] dr .1 1, 1 „dpk o r2 r dr

(80)

First u will be expanded

oo P~a =f [1—e "'(1+pr)] dr

r dr

e "'—e "' dp —2pr dpdr —p e &" dr

J

~~1 ~2~1

0 r dr dr(81)

dp 1 dpn2 .=0——„,.=0+~„o2 dr

(82)

Note that the first of the two integrals appearing in Eq. (81) was evaluated using Eq. 3.434.2 from Gradshteyn andRyzhik. 's Next examine P. Let

P=Pi+P»where

(83)

Pi= f———[1—e "'(1+sr)] ——oo 1

k o r2 re "' dp dr

r dr(84)

P2 —— f [1—e "'(1+@,r)] dr .k 0 r2 dr

The integral P& may be expanded as follows:

(85)

P = ——f —[1—e ""(1+pr)]1 1 1

k 0 r2 re l"' dp dr

r dr

1 1 e

2k r rdpdr

21 f" 1 e "" dp dr

2k o r r dr 2

'2dp 1 ~ 1 e "" d2p

dr " 2k 0 r r dr

Next, Pz may be expanded

oo e dpP2

——+f [1—e "'(1+pr)] drk o r2 dr

dpr 0+~ko ~

dr .

(86) .

p e

k r

oo

1 —pr dp p ~ e

r dr 0 k 0 rJ

—pe "" p+e ~" drdr

=2 r =0—21n2 r =0+~kodp . dpdr dr

(87)

Finally, the asymptotic expansion of tv( U, k) is obtained by summing the integrals

8~~(U k) = (A +a+P&+P2)Z'

Z 22 1 dPJ 1 dP~(2k) p~ (0)+—ink „0——ln2, o

1 dpj dpj d pj+ y —0++ y p— 1Ilr dr+& odr " dr

The cusp condition allows one to eliminate dpj /dr, o because

(88)


dpj.~p=o= —2Zpj(0)

r

hence,

Z2] (U, k)= p (0)[(2k)'~ —Zlnk+Z(ln2 —1 —2y)] — f lnr dr+Wk (89)

Therefore, to order k, 8](U, k) is given by

w](U, k) = g p;(0)[(2k)]~2—Z ink +Z(ln2 —1 —2y)] —f lnr drk dr

(90)

Equation (90) may be further simplified by noting that for an antisymmetric wave function 1(2o in the defining Eq. (12)for the one-electron density it follows that

p](r) =p2(r) = =p~(r) =p(r),hence

Z2 m 2

u)](U, k) = N p(0)[(2k)]~ —Z lnk+Z(ln2 —1 —2y)] —f lnr drk dr

(91)

Fortunately the task of computing u)z(U, k) is much simpler. The terms which contribute to the order in k of interest

are simply

2Z 2 rv z2 lV

mr(V, k)=, g ()o,' ', 4o +, X ()o o ( —k)o (ko)

k ;~) r; r& k4,~, ri rJ

X (4o, ', 0o)k', ~j r) rj(92)

Therefore, adding u)](U, k) and wq(U, k) gives u)(U, k) with

=Z' d2 Z2w(()k)=, )(r P(0)[(2k)' ' —Zlnk+Z(ln2 —l —22)]—I lnr dr +, g ()o, ', 0o) .

k dr k )~j & r) rj

Substituting this expansion for 8 into the expansion previously obtained for w leads to the final result for J(k),N Z2 Z3

2(k)= ——((ko(o2 (0o)—,(0o g 5(r;) go +, (2k)'r )olP(0) —,ln(k)krP(0)i=i

(93)

Z2 00 d pA

Z(ln2 —1 2y)Np—(0) N f l—nr2 dr+

k dr(9&)

valid to order k in k.

IV. DERIVATION OF THE HIGH-kASYMPTOTIC EXPANSION OF J{k)

IN TERMS OF THE MOMENTSAND ASYMPTOTIC BEHAVIOR

OF THE OSCII.I.ATOR-STRENGTH DISTRIBUTION

In this section it is shown that the form of the sum-over-states expression J(k) allows an expansion in k withcoefficients given in terms of global (i.e., moment) andasymptotic properties of the oscillator-strength distribu-tion. The expansion is similar in spirit ta the Cauchy ex-pansion of the dynamic polarizability which yields a

power series in the frequency, with coefficients given interms of the negative even moments of the oscillator-strength distribution. The coefficients in the expansion ofJ(k) obtained in this section (in terms of properties of theoscillator-strength distribution) may then be identifiedwith those obtained in the previous section (which wereobtained as expectation values of the wave function). Theidentification immediately leads to "sum rules" for thecoefficients of the first and second asymptotic terms ofthe oscillator-strength density, along with the third mo-ment of a modified" oscillator-strength distribution.

The sum-over-states expression J(k) has been definedin Eq. (17) to be


&n —&0+k

where the sum ranges over the bound discrete states andimplies integration over the continuum of scatteringstates. The above summation may be split into (1) g', asummation over all the bound states with an implicit in-

te ration up to some energy b in the continuum, and (2)d», an integration over the continuum from»=b to

bao. Then the above equation may be written as

, fo (» —»0) 3 "df (»»0)J(k)= —,

' + —,»„—»peak'

b d» (»—»0+k)

(95)

An asymptotic expansion in k is now desired for large k.This can be achieved if b is chosen such that b & k overthe range of interest of k. Assuming b &k, it is found(following the same method used for the Cauchy expan-sion of the dynamic polarizability) that

, fo (»n —»o) S'(l, b) S'(2,b) S'(3,b)»„—»0+k k k k

(96)

where S'(i,b) denotes the ith moment of the portion ofthe oscillator-strength distribution below energy b.

An expansion for the integral is also required. Thismay be accomplished by repeated partial integrations. Inthe following, as before, b is taken such that b & k. Also,it shall be assumed that -asymptotically the oscillatorstrength behaves as

df- =a» '"+-p» 4+m-„, (97)

where W 4 represents terms which fall off faster than

e . This choice for the asymptotic behavior is consistentwith Dirac and Harding's observation and it is of thesame form as obtained for atomic hydrogen and atomichelium. After completing the following derivation it willbecome obvious that this is the only choice for the firsttwo asymptotic terms which will yield terms in the expan-sion of J(k) of the same form as that obtained in Sec.III. Note that in the following, S"(i,b) will denote the ithmoment of the portion of the oscillator strength distribu-tion above energy b. Carrying out the partial integrations,

(»—»0)dE

b d» (»—»0+k)r

00 1, (» —»p)d»d»' (» —»0+ k)

r

oa ao dff —f, (» —», )d»'b e

12

d6'(» »0+k)—

r

S ( la)bdf( )d

(b —»0+ k ) b o' d»' (» »0+ k )—S"(j,b)

(b —»0+ k )

ao df » —»p

, (» »0)d»—d»' (»—»p+k) b

(»' —»p)d»' (» »p)d»—(»—»p+k)3f aa df (» »p) —'ao df'

2+2b d» (»»oak)2 o d»'

S"(1,b) S"(l, b)(b»0+k) + (b»p+k)~ +

b

df (» —»0) ~ df, , (» —»o)—2 , (»' —», )d»'d» (» —»0+k) o d» (»—»p+k)

S"(1,b) S"(l,b)b

S"(2,b)

(k+b —»p) (k+b —»,)' (k+b —»p)'

—2 ' (b —»p)+2 '3 (b —»p)

S"(2,b) S"(l,b)

(k +b —»0) (k+b —»0)

( —o)'z (»—»0)+f d»+6f„ f, (»' ») d»' — d»

b d» (»—», +k)' (» —»0+ k )

ao ao df2(»—»0)

3f f— , (»' »,)d»'—d»' (» », +k)'— (98)

At this stage, the above expression may be reduced by noting that to order k the sum of the first five terms in Eq. (98)is equal to

30 HIGH-ENERGY ASYMPTOTIC BEHAVIOR OF THE DIPOLE-. . .

S"(l, b) S"(2,b)k

So, to order k

(e—ep)

b de (e—eo+k)

S"(l, b)k

S"(2,b) "df (e' —eo) ~ df 2 (e—Eo)+ de+6 , (e' —eo) de' de

k' b de (~ ep+—k)3 (&—&o+k)'

df (e—ep)3—f f, (e' —eo)de' 4 de .

b o de' (e—eo+k )

(99)

The three integrals appearing on the right-hand side of Eq. (99) must now be dealt with. The standard procedure cannotbe used because asymptoticaHy

—7/2~—7/2+ p~

—4+E'

so the usual technique would lead to a divergence on the boundary at oo. However, define

df df~

—7/2 p~—4

df' d 6'(100)

and one finds

f co df (e —ep) ce df 2(e—6p) ao df

, +6 , (e' —eo) dE'4

—3,(E' —ep)d6'd& (e eo+k) — o d& (e—ep+k)" ~ de

tt e —e(ae +Pe ) de

b (e —eo+k)

(e—ep)'dE'

(~—~o+k)'

+ f 6 f (aE' 7/2+PE' )(e' ep) de'—. (r—ep+k)

200 (e—ep)—3 f (ae' +/3e' )(e' —ep)de' 4 de+ Mk. (e—ep+k)

where Mk, represents terms of higher order than k and

(101)

S""(3,b) =f" (e eo) de. — (102)

Now only the remain1ng three lntegrals on the nght-hand side of Eq. (101) need be evaluated. These integrals may beperformed analytically and upon expansion to order k 3 it is found that

3

f —7/2 4 (~ ~o) 1(ae' +pg ) d& 2ab 1/2+ ak 1/2 6a& b—1/2+ 2 2b 3/2 3b 5/23~ 2

b (e—ep+k)3 k3 8 5

3 —1 2 —2 3 —3P+P ink P 1nb——3Peob +— P~ob eob—2

(103)

6f f (ae' / +Pe' )(e' ep) de' — de=b . (e—ep+k)4 k3 4

"ak'/2+2P +W„, (104)

and

2

—3f f (aE' +Pe' )(e' —ep)de' de= ——ak ——+M—7/2 —4'

( p) 1 7r i 2 Pb o (& &p+k)4 k3 8 2 k (105)


Combining all the above results it is found that, to order k

fon(~n —e'o) S'(l, b) S"(l, b) S'(2, b) S"(2,b) S'(3,b) S""(3,b) k'i ink

(&, —&p+k) k k k' k' k' k' k' k'

g 1/2 6 b—1/2+ 2 ~2g —3/2 2 ~3I —S/2

k0 5 0

—P lnb —3Pepb + 2 Pepb —, P—mob (106)

Noting that S'(i,b)+S"(i,b)=S(i) leads to the furthersimplification that

where el is the first ionization threshold and e is the unitstep function. Also note J(k) is given to order k by

where

fon«. —&o)

(e„Ep+—k)

C+ 3+~k-3k

(107)

S(1) S(2) k'~ ink

k k k 3 S(1) 3 S(2) 3 k' 3 ink 3 cJ(k) =— —— i a7r — i—p2 k 2 k 2 k' 2 k' 2 k'

(110)

c =S'(3,b) iS' (3,b) —2ab'i 6aep—b

+2aepb ——,'

arab P 1nb —3Pepb-

+ 'Pe b ——,'Pepb— (108)

with c given as above.Equating the coefficients of the above J(k) with that

obtained in Sec. III automatically gives the final results,

It is easily verified that c is a constant independent of b,as it must be, since b was only artificially introduced toperform the evaluation. Note that S'(3,b)+S (3,b) isthe third moment of a modified oscillator-strength distri-bution where the

~—7/2 ip —4

contribution to ( df /de) has been subtracted above the en-

ergy b, i.e.,

S'(3,b) iS* (3,b)

= g fpnMpz + (6 6p) dF—'df

~1 dE

S(1)=——', &apl ~'Ifo&

sg)= (p, g 5|;) 0o)

Z=—Np(0),3

2Zv2( )

2V2ZS)

3

(112)

(113)

)((6—Ep) dE,

P= ——,Z Np(0)= —2Z S(2),

S'(3,b) iS* (3,b)

= g fp cop + (e—e'o) dE+ —e(E'—b)(txt +PE)(E'ep) de- '

' df i " df —7/2 —4 3

~l de de

= —,Z Np(0)( —,ln2 ——, —0.5772. . .)i —,Z (QD ~, $0) N f 1nr dr—E+J =1

+2ab'~ +6aEpb '~ 2aepb 3~ + ', aE—ob +plnb+—3pepb ' ——', prob + ,'pepb— (115)

where all quantities are expressed in atomic units. Of the above five results, the first two are well known from the usualsum rules. The third and fourth identities give the coefficients of the first two terms in the asymptotic expansion of the

HIGH-ENERGY ASYMPTOTIC BEHAVIOR OF THE DIPOLE-. . .

oscillator strength density and are in agreement with results obtained by Salpeter and Zaldi for atomic helium Finally

the fifth result is a sum rule for the third moment of the modified distribution.A sum rule which is equivalent to Eq. (115) may be obtained for the third moment of a modified distribution where

the

a(e —eo) '"+P(e—eo)'

contribution to (df /de) has been subtracted above the energy b Th. is form for the sum rule can be obtained using the

same technique followed to obtain Eq. (115). This procedure leads to a more compact result,

S(3,b) = g fo„coo„+f [a—(e eo)—i +P(e eo)—]e(e b) —(e e—o) dede

=Sf (3)+2a(b eo)'~—+I3ln(b —eo}, (116)

2

Sf(3)= —', Z Xp(0)( —,' ln2 ——,

' —y)+ —', Z g fo go Xf —lnr dri&j =1 Ti P'J df'

(117)

V. RESULTS FOR MOLECULES

The previous results obtained for atoms are easily ex-

tended to apply to nonrelativistic X-electron molecules inthe fixed-nucleus approximation. In this case the Hamil-

tonian is. given by

H=g (118)i=1 2

M Z P

@=1 pi+g

igj ij

where p, ranges over the M nuclei of the molecule and r„;denotes the distance between the )Mth nucleus (with charge

Z„)and the ith electron. For this Hamiltonian, thelength, velocity, and acceleration forms of the dipole

operator are given by I., V, and A, respectively, where

S(1}= —3 & Wo I

~'I Po&

MS(2)= g Z„|)0g Sr„,) ))0)p

p, =1 i =1

(122)

The acceleration form of the dipole operator A givenabove for molecules, of course, differs from that used inthe derivation of the expansion of J(k) for atoms. Mak-

ing the necessary replacements for A and H is quitestraightforward and the net effect is the appearance of theindex p (which labels the nuclei) and the associated sum-

mation. Note that the cusp condition applies to bothatoms and rnoleeules so the arguments previously madefor atoms may be easily extended to deal with molecules.The final results obtained are given by

(124)

V=[L,H]= g V';,

MA=[v,a]= g z„g

@=1 i =1 ~pi(121)

M N

a= g Z„'g p„,(0),@=1 i =1I X

P= ——,' g z„'g p„,(0),

with i„l—(1,—r ).

S'(3,b)+S* (3,b)M

g Zp g p~,.(0)( —,ln2 —~—0.5772. . .}

(p, i)+(v,j)=(1,1)Z„z.Wo

M X ~ ~p+Z„'g f lnr ",'dr-rI l rvj dp'

+2ab' +6aeob '~ 2aeob ~ + ', aeob —~ +Pl—nb +3I3eob ' —,'

Prob 2+ ,' Peob——(127)


where p&, (r) in the above is defined by

pp](i) = fd&p] fdU«I fo I

(128)

For atomic hydrogen, the oscillator strengths of thediscrete states and the oscillator-strength density in thecontinuum can be determined analytically. The analyticalresults are used in this section to numerically evaluate thethird moment of the modified distribution. The value isthen compared with the result obtained from the sum rulefor the third moment of the modified distribution derived

in Sec. IV.The total discrete oscillator strength between the

ground state of atomic hydrogen and the np states is givenby5

8 52ri

2 n n —1

3(n —1) (n +1) & +1 (129)

with the "pi" angular integration carried out about thepoint r& and r denotes the separation from the point r„.

VI. CALCULATIONS FOR ATOMIC HYDROGEN

S(3,—, )=8ln2 ——, +8 16v237T

The third moment of the modified distribution shallnow be evaluated numerically. First examine the thirdmoment of the discrete states. Using Eqs. (129) and (130)it can be shown that, for large n,

(136)

32 e 1 1 11 1

3 2 45 4(137)

The following approximation has proven adequate for thepresent needs:

20 cx]

g fn~n = g fn~n+ g gn (138)n=2 n=2 n =21

Note that00 00 20

n =21 n=1 n=1(139)

and g„"]gn may be evaluated in terms of Riemann gfunctions (tabulated by Davis' } as

Also note that the energy difference between the np statesand the 1s state is given by

g g. = '3'e 'lP3) —305)——.'507)) .n=1

(140)

Therefore g„~fnco„can be evaluated using1n=T

n(130)

oo 20 20

g f.~'. =- g f.~'. &g. —For the continuum, the oscillator-strength density is givenby5

n=2 n=1n=2

+—", e [g(3)——,'g(5) ——,", g(7)] . (141)

df 16 1 exp( —4]].arccot]] )

de 3 (e+ —')4 1 —exp( —2m]])

where e denotes the energy of the scattering state and g fnco„=0.036605 188 .n=2

The contribution from the continuum oscillator-strength density between e=0 and 10000 has beenevaluated using numerical integration and above@=10000has been evaluated asymptotically. Using Eqs.(131) and (132), the integrations may be performed usingGaussian quadrature, which leads to

a-=(2e)-]" . (132)

The modified distribution used to check the sum rule isthat for which the

a(e ~ —,'

) 'i +f3(e+ —,'

)

contribution to dflde has been subtracted above e= —,'.For atomic hydrogen it is easily found that

(133)37T

'

]/2 dff (e+ 2 ) de=1.414013074dE'

and

(131)Using this method, the numerical value is found to be

and3

(134)r

df ] —7n—a(e+ —, )1/2

Sf(3)=8 ln2 ——', (135}

Hence, the value of the third moment of the modified dis-tribution may be determined from the sum rule given inEq. (117),

P(e+ —, ) (e+—, ) de=3. 747 760113 . —

Asymptotically, for large 6, MACSYMA has been used toexpand Eq. (131) in terms of e+ —,'. It is found that

df dg 8 2( ]) 35 8( ]) 4 2(2m'+21)v2( ]) 45 ]6( ])de . 31T 9 +2 —

3 +2

(8m —540vr —3075)V 2

(32m —1848~ +65 730m. +265 671)V 245 3607T


The integral

f [dg /de ot(—e+ 2 )' —p(E+'2 ) ](E+ 2 ) de

may be performed analytically and it is found that

—P(e+ —,)" (e+ —, ) de=0. 080975912 .

Summing the individual contributions leads tor

~eton+ I

These identities are, therefore, a supplement to the previ-

ously known sum rules for S(—1), S(0), S(l), and S(2),and provide additional information about the high-energybehavior of the oscillator-strength distribution. As withthe the previous sum rules, these new sum rules may beevaluated using high-accuracy ab initio wave functions.Although applications have not been discussed here, it ispossible to use this additional information to improve theaccuracy of moment theory type calculations of logarith-mic mean excitation energies' (which include the Bethelogarithm for the Lamb shift). This information may alsobe useful for the calculation of the high-energy behavior

of the photoionization cross section of atoms and mole-

cules. '

X e(e—~ ) (e+ —, ) de=5. 279354288 . ACKNOWLEDGMENTS

This value is in agreement with the result from Eq. (136)to ten digits.

VII. DISCUSSION

The results presented in this work allow the determina-

tion of various properties of the oscillator-strength distri-

bution of X-electron atoms and molecules in terms of ex-

pectation values of only the wave function of interest.

I would like to thank Professor J. C. Browne for adviceand encouragement throughout this work. I am gratefulto Dr. E. J. Shipsey for countless helpful conversations

concerning the work. I wish to thank Professor C.Schwartz for some comments concerning his helium

Lamb-shift paper. I would also like to thank Bruce Jen-

sen for help with MACSYMA. This research was supported

by the Robert A. Welch Foundation of Houston, Texasunder Grant No. F-379.

E. E. Salpeter and M. H. Zaidi, Phys. Rev. 125, 248 (1962).A. Dalgarno and R. W. Ewart, Proc. Phys. Soc. London 80,

616 (1962).C. Schwartz, Phys. Rev. 123, 1700 (1961).

4R. Jackiw, Phys. Rev. 157, 1220 (1967).5H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One

and Two-Electron Atoms (Plenum, New York, 1977).6T. Kato, Commun. Pure Appl. Math. 10, 151 (1957).

P. A. M. Dirac and J. W. Harding, Proc. Cambridge Philos.

Soc. 28, 209 (1932).8B. Podolsky, Proc. Natl. Acad. Sci. USA 14, 253 (1928).A. Dalgarno, in Perturbation Theory and its Applications in

Quantum Mechanics, edited by C. H. Wilcox (Wiley, New

York, 1966).~OS. G. Mikhlin, The Problem of the Minimum of a Quadratic

Functional (Holden-Day, San Francisco, 1965).C. Schwartz, Ann. Phys. (N.Y.) 6, 156 (1959).

'2R. Courant and D. Hilbert, Methods of Mathematical Physics,

(Interscience, New York, 1953), Vol. 1.3%. A. Bingel, Z. Naturforsch. 18a, 1249 (1963).

~4P. M. Morse and H. Feshbach, Methods of Theoretical Phys

ics, Part I (McGraw-Hill, New York, 1953).~5H. S. Carslaw, Introduction to the Theory of Fourier's Series

and Integrals (Dover, New York, 1950).~61. S. Grandshteyn and I. M. Ryzhik, Table ofIntegrals, Series,

and Products (Academic, New York, 1980).~7H. T. Davis, Tables ofHigher Mathematical Functions (Princi-

pia, Bloomington, 1935).8M. C. Struensee, Ph.D. thesis, University of Texas at Austin,

1984.

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