Expectation values of rP for an arbitrary nl hydrogenic state are expressed in terms of state-independent coefficients that are determined by a simple recursion relation. Explicit results are given for negative P down to -16 and, by an algebraic transformation, for positive P up to 13. Closed-form expressions are obtained for the coefficients of the two highest-order terms in l multiplying each power of n. The result is an asymptotic expression valid for each P and sufficiently large l. © 1990 The American Physical Society

Drake, Gordon W. F.

Swainson, R. A.

Scholarship at UWindsor

University of WindsorScholarship at UWindsorPhysics Publications Department of Physics1990Expectation values of rP for arbitrary hydrogenicstatesGordon W. F. DrakeUniversity of WindsorR. A. SwainsonFollow this and additional works at: http://scholar.uwindsor.ca/physicspubPart of the Physics CommonsThis Article is brought to you for free and open access by the Department of Physics at Scholarship at UWindsor. It has been accepted for inclusion inPhysics Publications by an authorized administrator of Scholarship at UWindsor. For more information, please contact scholarship@uwindsor.ca.Recommended CitationDrake, Gordon W. F. and Swainson, R. A.. (1990). Expectation values of rP for arbitrary hydrogenic states. Physical Review A, 42 (3),1123-1126.http://scholar.uwindsor.ca/physicspub/83PHYSICAL REVIE% A VOLUME 42, NUMBER 3Expectation values of r for arbitrary hydrogenic states1 AUGUST 1990G. W. F. Drake and R. A. SwainsonDepartment of Physics, University of Windsor, Windsor, Ontario, Canada N9B3P4(Received 16 March 1990)Expectation values of r for an arbitrary nl hydrogenic state are expressed in terms of state-independent coefficients that are determined by a simple recursion relation. Explicit results aregiven for negative P down to —16 and, by an algebraic transformation, for positive P up to 13.Closed-form expressions are obtained for the coefficients of the two highest-order terms in I multi-plying each power of n. The result is an asymptotic expression valid for each P and sufficiently largeI.I. INTRODUCTIONThe derivation of general expressions for expectationvalues of r with respect to hydrogen-atom wave func-tions has a long history, beginning with the early work ofWailer, ' Van Vleck, and Pasternak. Here, P is a posi-tive or negative integer and r is the radial coordinate.Such general expressions are particularly important inasymptotic expansions for the energy and other atomicproperties of Rydberg atoms consisting of an electron ina high-nl quantum state moving in the field of a polariz-able core. For example, the ionization energy of the Ryd-berg electron is given by the well-known expansion—hE =ai(r )+(a2 —6P&)(r )+ Ry,where e, is the dipole polarizability, ez is the quadrupolepolarizability, P, is the leading nonadiabatic correctionetc. , and the expectation value is with respect to the Ryd-berg electron.The terms in Eq. (1) are currently known in their en-tirety up to ( r ). Recent advances in experimentalprecision for the Rydberg states of helium and the accu-racy that can be achieved in variational calculations callfor an extension of Eq. (1) to higher inverse powers. Themost extensive tabulation of expectation values (r ) byBockasten gives general expressions as a function of nand I up to P=8. However, his method, which involvessums over triple products of binomial coefficients, is alge-braically tedious and difficult to extend to higher powers.The purpose of this paper is to derive a simple recursionrelation which allows one sequentially to obtain generalexpressions for any (r ) with a minimum of effort. Ex-plicit results are given for all cases up to P=16.where N = [P/2] —1 and f„(1)= (1 + r)!/( I —r )! is a func-tion only of I. The notation [ ] denotes "greatest integerin." The functional dependence on n and I in Eq. (2) fol-lows from a theorem stated by Pasternak. The quanti-ties Fp, Gp(n, l), and C' ' are chosen for convenience tobe(P —3)!(P/2 1)!(P/2——2)!(P —2)! (3)zfor P odd,[(P /2 —3/2)! ]G ( I) 2 Z (2I P+2)!— (4)n +'(21+P —1)!Ni2lv —q 1 q N—1Cj"= g (2P —2. —5),(P —2q —2)!q! s =qwith Ctv '=( —1) for all P. The above choices are sug-gested by the known results up to P= 8. They yield a use-ful partial factorization of the final coefficients.The remaining coefficients d' „'are determined by useof the Kramers-Pasternak recursion relationFp= '„—P —2 4(2l +P + 1)(21 P+1)—2P 1 ( p I )Patomic units)N q(nti!r ~nl ) =FpGp(n, l) g Cq n q g dq„f„(l),r=0(2)II. MATHEMATICAL DERIVATION P —1~( p)n P(6)We begin by observing that a general nonrelativisticexpectation value ( r ) for a one-electron atom withnuclear charge Z depends on n and I according to (inISubstituting Eqs. (2)—(5) into (6) and equating coe%cientsof equal powers of n and I yield the recursion relationdq„+ '= ((P —2q)(2P —l)d' „''+2q(P—1)tdI '& „1+[r(r+1)—,'P(P —2)]d' ', „I')—1P (2P —2q —1)for q =0, 1, . . . , X and r =0, 1, . . . , q, starting with the values1123 1990 The American Physical Society1124 G. W. F. DRAKE AND R. A. SWAINSON 42d(2) =1 d'3) =10,0 ~ 0,0and coefficients with negative subscripts are taken to be zero. With these starting values, the recursion relation (7) hasthe properties thatd'"=&q, q(8)d' ' = — (P —2q —2)(P +2q —3)q, q —1 (9)for all P ~ 2 and q =0, 1, . . . , [P/2] —1. The derivation of Eq. (7) requires at an intermediate step the use of the identi-ty(I P/2—+1)(l +P/2)f„(1)=f„+,(I)+ [r (r +1)—,'P(P ——2)]f„(1). (10)Since the coefficients given in Eqs. (8) and (9) multiply the highest powers of I for each n, they determine the high-I(and therefore necessarily high n) dependence of the integrals for all P. The result is(nl~r ~nl ) =FpGp(n, l) g C (P)n 'i [f (I)— (P 2q —2—)(P+2q —3)f i(1)],q=0with Fp, Gr(n, l), and C' ' as defined by Eqs. (3)—(5).One could of course apply directly the recursion rela-tion (6) to obtain sequentially the complete integrals forparticular values of n and l. However, the aim of thecore-polarization mode is to obtain (as far as possible) theasymptotic potential in a state-independent form. Fromthis point of view, it is preferable to display and calculateseparately the contribution from each inverse power of rIn addition, the q=O and 1 terms of the asymptotic Eq.(11) completely determine the coefficients of the leadingI /n and I /n terms in a 1/n expansion of the integrals.This, together with Eq. (1), gives the corresponding 1/nexpansion of the polarization energies and associatedquantum defects for Rydberg states.The connection between matrix elements of positiveand negative powers of r has been discussed by many au-thors. ' ' For diagonal matrix elements, the result is(nI!r ~nl)= (21 P —1)!—2Z2P+3( nl~I /r +~n I ) .(12)Substituting Eq. (2) into the right-hand side of (12) givesP —1 N( nI~r ~!nl ) =F g C' + 'nP+' PZP q=0qX g d"+"f (I)r=O(13)with N=[(P+1)/2]. This is identical in form to Eq. (2)except that the factor Gp+2(n, l) has been replaced byn '/(2Z) . Equation (13) is valid for all I, even thoughthe right-hand side of (12) diverges for 1(X—1. Theasymptotic result corresponding to Eq. (11) isP —1 N(nl!r ~nI ) =Fp+& p g C~ + 'n &+'[f (I)— (P 2q+1)(P+2q—)f i(I)] .q=0 (14)III. RESULTSTo save manual labor and ensure accuracy, it is now a straightforward matter to program the recursion relation (7)and identify the numerical coemcients as rational fractions. There is never a problem of numerical cancellation becausethe three terms on the right-hand side of (7) are always of the same sign (or sum to zero for the termsdz „',r =0, 1, . . . , X —1 with P even). In addition, one knows that the product of the three coefficients Fp, C ', andd' „'in (2) must always be an integer. With this in mind, the results are(1/r ) =G2(n, l)/2,(1/r ) =nG3(n, l),(1/r ) = G(4n, 1)[3n f, (l)], —( 1/r ) =6G&(n, l) Pn —n [fi(l) ——,']],(1/r6) =3G6(n, l)[ ,5n —10n [—fi(I)——,']+f2(l) j,(1/r ) =30G7(n, l){", n ——',4n [fi(I) ———,']+n [f2(I)——', fi(l)+ —', ]),42 EXPECTATION VALUES OF r FOR ARBITRARY. . . 1125(1/rs) =10G8(n, l)[ 231n —63n [fl(l) ——', ]+21n [f?(l)—3f, (1)+—", ]—f3(1)I,(1/r ) =140G9(n, l)I 4,", n —",—n [fl(1)——', ]+9n [f?(l)—5f1(I)+—', ,'] n—[f3(I)—3f?(1)+—', fl(l) ——", ]I,(1/r' ) =35G10(n, l)I ""n —'"'n [f1(I)——']+198n [f?(l)—"f—,(l)+ —"]—36n [f3(l)——", f?(1)+=", ,' f1(1)——",,' ]+f4(l) I,(1/r" ) =63QG„(n,l) j '4" n' —'"—n'[f, (l) ——"]+—"'n'[f, (l)—10f1(l)+—"']"n 3—[f3 ( I ) —"f? ( —I) +48f 3 ( I ) —""+ n [f4( I ) "f—3 (—I) +—'"f? ( I ) —7'f, ( I ) +64]I,(, 1/r' ) =126G, (n l)[4'189n o 121ssn [f (I) —"]+1430n [f (I)—13f (I)+ ""]1430n 4[f ( I) 15f ( I )+ 177f ( I) 3124 ]+55n [f4(I) —"f3(—l)+84f?(l) —""f?(l)+ ""']—f,(l)I,(1/r' ) =2272G&&(n, l)I "", n "—""'n [f,(l) —9]+""n [f?(l)——", f, (l)+ —"']—130n s[f3(l) —2Qf?(l)+ 887 f 1 (I)—'873 ]+ "n [f (I)—18f (I)+174f (I) ""'f,—(1)+""']n[f—8 ( I ) ——"f4( I ) +80f3 ( I ) —" f? ( I ) + 1600f, ( I ) —"' ]],(1/r'4) =462G, 4(n, l) I ""'n '? —8398n '0[f, (l) ——"]+""'n [f?(1)—20f, (l)+ '"' ]—4420n [f3 — f? +, f-1, ]+975n 4[f (I) 76f3(l)+ ls41f (I) 3048f (I)+ 606164]78n?[f (I) 33f (I)+ 680f (I) 12310f (I)+ 46732f (I) 329360 ]+f (l)I(, 1/r' ) =12012G»(n, l)I '""'n' —"'"n "[f,(l) ——"]+1615n [f?(l)—24f, (l)+133]—64 n 7[f (I)—63f (I)+ 3423f (I)—64421]+255n [f4(l) —'~ f3(I)+497f?(l)—",'"f, (l)+ "'"]—3Qn 3[f (I)—33f (I)+ 1372f (I) 13058f (I)+ 96888f (I) 1338036 ]+n[f, (l) 12f,(1)+180—f,(l) —",'~f,(I)+14400f?(l)—3'8400 fl(l)+ ' 4' ]I,(1/r ' ) =1716G,6(n, l) I '"'"'n ' —"~"'n ' [f,(l) —15]+52003n '0[f?(I)—"f, (l)+185]——33915n [f3(I)—38f (I)+ '"'f (I)—12755/7]+ 11 305n [f4( I ) 42f3 ( I ) + ","f—? ( I ) —"","f, ( I ) + 13 718]17854[f (I) 1 I5f ( I) +791f ( I } 26864f ( I ) + 423 260f ( I) 2409320 ]+ 105n [f6(l) —25f5(1)+ '"'f4(l) —6908f, (1)+ -'"'-""f? (I)2372312f (I)+ 224972736 ] f (I) )The above equations agree with those given byBockasten up to P=8. The presentation is slightlydifferent from the usual format in that the powers of nare all even for P even and odd for P odd, with an extrafactor of 1/n being absorbed into Gp(n, l) for P odd.This is done so that the recursion relation assumes thesame form for both P even and P odd. The results showthe usefulness of the factorization that has been achieved,with the denominators being at most products of smallprime numbers.The total coefficients of n 9 'f„(l) have beenchecked by calculating directly the matrix elements of1/r for a sufficiently large triangular array of n and lvalues, and fitting the results of the functional form ofEq. (2). The results agree to within the accuracy of thenumerical fitting (at least seven significant figures for Pup to 13).The corresponding equation for (r ) can be read offdirectly from the above (1/r + ) result, provided thatthe G7, +3(n, I) factor is replaced by n '/(2Z) . For ex-1126 G. W. F. DRAKE AND R. A. SWAINSON 42ample, (1/r ) gives(r ) =6(N/4Z )I ,'n —n—[f,(1)——,']) .In summary, we have considerably extended the rangeof powers for which the diagonal matrix elements of rand 1/r are known as an explicit function of n and I,and have given a recursion relation by which the rangecan easily be extended further. The asymptotic expres-sion (11) completely determines the coefficients of theleading 1/n and 1/n terms in a 1/n expansion of theintegrals. It is hoped that the results presented here willprovide a useful tabulation in the study of Rydbergatoms.ACKNOWLEDGMENTSResearch support by the Natural Sciences and En-gineering Research Council of Canada is gratefully ac-knowledged. We wish to thank Dr. R. J. Drachman forcommunicating to us his equivalent results for the in-tegrals up to P= 12 obtained by use of the symbolic ma-nipulation program MACSYMA.'I. Wailer, Z. Phys. 38, 644 (1926).J. H. Van Vleck, Proc. R. Soc. London, Ser. A 143, 679 (1934).S. Pasternak, Proc. Natl. Acad. Sci. 23, 91 (1937).4R. J. Drachman, Phys. Rev. A 26, 1228 (1982);31, 1253 (1985).5E. A. Hessels, W. G. Sturrus, S. R. Lundeen, and D. R. Cok,Phys. Rev. A 35, 4489 (1987); E. A. Hessels, F. J. Deck, P. W.Arcuni, and S. R. Lundeen, ibid. 41, 3663 (1990).6.W. F. Drake, J. Phys. B 22, L651 (1989).7K. Bockasten, Phys. Rev. A 9, 1087 (1974).The recursion relation (6), often called Kramers relation, wasapparently derived independently in 1937 by Pasternak (Ref.3) and in the same year by H. A. Kramers, Quantum Meehanics (North-Holland, Amsterdam, 1957), p. 251 [translated byD. ter Haar from Quantentheorie des Electrons und derStrahlung (Akademische Verlagesellschaft, Leipzig, 1938)].S. Pasternak and R. M. Sternheimer, J. Math. Phys. 3, 1280(1962).' P. Blanchard, J. Phys. B 7, 993 (1974).R. M. More, J. Phys. B 16, 3859 (1983).'2P. C. Ojha and D. S. F. Crothers, J. Phys. B 17, 4797 (1984).' L. L. Armstrong, J. Phys. (Paris) Colloq. 31, C4-17 (1970).' R. A. Sawainson and G. W. F. Drake, J. Phys. B 23, 1079(1990).

Expectation values of rP for arbitrary hydrogenic states

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Expectation values of rP for arbitrary hydrogenic states

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