ComputerPhysicsCommunications74(1993)399—414 Computer PhysicsNorth-Holland Communications

A programfor computingmagneticdipole andelectric quadrupolehyperfine constantsfrom MCHF wavefunctions

PerJönsson,Claes-GöranWahlströmDepartmentof Physics,Lund Instituteof Technology,P.O.Box 118, S-221 00 Lund, Sweden

and

CharlotteFroeseFischerDepartmentof ComputerScience,Vanderbilt University, Nashville, TN37235, USA

Received27 March1992

Given an electronicwave function, generatedby the MCHF program (LS format) or the MCHF_CI program (LSJformat), thisprogramcomputesthe hyperfineinteractionconstants,A

1 and B1. In strongexperimentalmagneticfields, thesplitting is alsoaffectedby theoff-diagonalhyperfineconstantsA1,1_1,B1,11,andB1,1_2andthesearecomputedaswell,dependingon theinput data.

PROGRAM SUMMARY

Titleofprogram: MCHF_HFS Memoryrequiredto executewith typical data:210K words

Cataloguenumber:ACLE No. of bits in a word: 32

Program obtainable from: CPC Program Library, Queen’sUniversityof Belfast, N. Ireland(seeapplicationform in this Peripheralsused:terminal; diskissue)

No. of lines in distributedprogram, including test data, etc.:

Licensingprovisions:none 3291

Computerfor which the program is designedand others onwhich it has been tested: CPCprogram Library subroutinesused:Computer: VAX 11/780, DEC 3100, SUN SPARC-station Cataloguenwnber Title Ref in CPC330; Installation: Lund University,VanderbiltUniversity ABBY NJGRAF 50(1988)375

ABZU MCHF_LIBRARIES 64 (1991)399Operatingsystemunder which theprogram is executed:VMS,ULTRIX, SunUNIX Keywords: hyperfine structure, A factor, B factor, orbital

term, spin-dipole term, Fermi contact term, electricProgramminglanguagesused: FORTRAN77 quadrupoleterm,MCHF calculation,CI calculation

Correspondenceto: C. FroeseFischer, Departmentof Com- Natureofphysicalproblemputer Science,Vanderbilt University, Nashville, TN 37235, Theatomichyperfinesplitting is determinedby thehyperfineUSA. interaction constants A1 :and B1 [1]. In strong external

0010-4655/93/$06.00© 1993 — Elsevier SciencePublishersB.V. All rights reserved

400 P. Jönssonet al. / Computingmagneticdipoleandelectricquadrupolehyperfineconstants

magnetic fields, where J is no longer a good quantum subshellwith I � 3. If 1 � 4, the LS term for the subshellisnumber, the splitting is also affected by the off-diagonal restricted to those allowed for I = 4. Only the subshellshyperfine constantsA

111, B111 and B1,12 [2—4].This outsidea setof closedsubshellscommon to all configurationsprogramcalculatesthehyperfine’constantsusinganelectronic need to be specified.A maximum of 5 (five) subshells(inwavefunctiongeneratedwith the MCHF or MCHF_CI pro- addition to common closedsubshells)is allowed. This restric-gramsof FroeseFischer[5]. tion maybe removedby changingdimensionstatementsand

someformat statements.

Methodof solutionTheelectronicwavefunction,~I’,for a statelabelled yJ canbe Typicalrunning timeexpanded in terms of configuration state functions, ~ The CPU time requiredfor the testcasesis 0.1 secondsfor~ The hyperfineconstantscan then generally the first and 0.19 seconds for the second on a SUNbecalculatedas SPARC-station330.

Lk,kclck coef(j, k)(71L1S1IIT(K) “Yk LkSk) References

where T(K) is a spherical tensor operator of rank K. [1] I. Lindgrenand A. Rosen,CaseStud. At. Phys.4 (1974)Evaluationof thereducedmatrixelementbetweenarbitrarilyLS coupledconfigurationsis done by an extendedversionof [2] G.K. Woodgate,Proc. Roy. Soc. LondonA 293 (1966) 17.the programTENSORoriginally written by Robb[6—8]. [3] J.S.M. Harvey, Proc. Roy. Soc. LondonA 285 (1965)581.

[4] H. Orth, H. Ackermannand E.W. Otten,Z. Phys.A 273(1975)221.

Unusualfeaturesof theprogram [5] C. FroeseFischer,Comput.Phys.Commun.64 (1991)431,The programallows for a limited degreeof nonorthogonalitybetweenorbitals in the configuration stateexpansion. [61W.D. Robb, Comput. Phys. Commun. 6 (1973) 132; 9

(1985) 181.Restrictionson thecomplexityof theproblem [7] R. Glass and A. Hibbert, Comput. Phys. Commun. 11The orthogonality constraints are relaxed only within the (1976) 125.restrictions describedin ref. [9], giving rise to at most two [8] C. Froese Fischer, M.R. Godefroid and A. Hibbert, 64overlap integrals multiplying the one-electronactive radial (1991)486.integral. Any numberof s-, p-, or d-electronsareallowedin a [9] A. Hibbert, C. Froese Fischer and MR. Godefroid,configuration subshell,but no more than two electronsin a Comput.Phys.Commun.51(1988)285.

LONG WRITE-UP

1. Introduction

The presentMCHF....HFS program is closely linked with the MCHF atomic structurepackage.Therefore,it adheresto the structureand constraintsunderlyingthe package.A summaryof the datainput for the various programswas includedwith the descriptionof the package[1]. The role of theprogram is to calculate magnetic dipole and electric quadrupolehyperfine constantsfrom MCHFwavefunctions,eitherin the nonrelativisticLS scheme,or in the Breit—Pauli LSJ scheme.

2. Theory

2.1. Hyperfineinteraction

The hyperfinestructureof the atomicenergylevels is causedby the interactionbetweenthe electronsandthe electromagneticmultipole momentsof the nucleus.The contributionto the Hamiltoniancan be

P. Jänssonetal. / Computingmagneticdipoleand electricquadrupolehyperfineconstants 401

representedby an expansionin multipolesof order K,

HhfS= ~ T(K).M(K), (1)K� 1

where ~ and M(K) are sphericaltensoroperatorsof rank K in the electronicand nuclearspace,respectively[2]. The K = 1 term representsthe magneticdipole interactionand the K = 2 term theelectricquadrupoleinteraction.Higherorder termsaremuchsmallerandcan often beneglected.

For an N-electronatom the electronictensoroperatorsare, in atomicunits,

T~’~= ~a2~ [2 1”~(i) r13_g5~ii~[C(2)(i)Xs(1)(i)](

0r13 +g5~’rr6(r~)s(1)(i)1 (2)

and

NT~

2~= — ~ C~(i) r~3, (3)i= 1

where g~= 2.00232is the electronicg factor and 6(r) the three-dimensionaldelta function [3]. C~= ~/4’w/(2k + 1) Y,~,with Y,~being a normalizedsphericalharmonicwith the phaseconventionsofCondonand Shortley.

The magneticdipole operator(2) representsthe magneticfield dueto the electronsatthe site of thenucleus.The first termof the operatorrepresentsthe field causedby orbital motion of the electronsandis called the orbital term.The secondterm representsthe dipole field dueto the spin motionsof theelectronandis calledthe spin-dipoleterm.The last termrepresentsthe contactinteractionbetweenthenucleusandthe electronspin andcontributesonly for s electrons.The electricquadrupoleoperators(3)representsthe electric field gradientat the site of the nucleus.

The conventionalnuclearmagneticdipole moment jx1 andelectricquadrupolemoment Q aredefined

as the expectationvaluesof the nucleartensoroperatorsM~1~andM~2~in the statewith the maximum

componentof the nuclearspin, M1 = I,

(yjHIM0”~lyjII)=p~j, (4)

(y~III M~ = ~Q. (5)

When the hyperfine contributionsare addedto the Hamiltonian,the wavefunctionrepresentationforwhich the total Hamiltonian is diagonalis I -YJYJIJFMF),whereF = I +J.

In this representation,perturbationtheory gives thefollowing hyperfinecorrectionsto the electronicenergy

WM!(J, J’)=(yJyJIJFMFIT~1~•M~’~IyJyJIJ’FMF),J’=J, f—i, (6)

WE2(J,J’)—<yJyJIJFMFIT~

2~•M~2)IyJyJIJ’FMF),J’=J,J—l, J—2. (7)

This can, usingtensoralgebra,alsobewritten as

WM1(J, J’) = (— 1)~’FW(IJiJ; F1)(71J II II y1J’)<711 II M~’~II y~I>, (8)

WE2(J, J’) = (— 1)’~” W(IJ’IJ; F2)(71J II II -y1J’X711 II M~ II yjl>, (9)

402 P. JönssonetaL / Computingmagneticdipoleandelectricquadrupolehyperfineconstants

where W(JJ‘If; Fl) and W(JJ‘If; F2) are W coefficientsof Racah.The reducedmatrix elementsaredefinedthroughthe relation

MIJ K J’\

KyJMIT~j’~Iy’J’M’>=(—1)’ ~—M Q M~)(7JMT~~’)/J> (10)The energiesareusuallyexpressedin termsof the hyperfineinteractionconstants(A and B factors)

1A1= I [J(J + i)(2J+ 1)11/2 Ky1J II T~’~II yjJ>, (11)

1

= — ________________________

~ [J(2J — 1)(2J+ 1)11/2(y1J II II ~1 ~ 1)), (12)

B1 2Q( J(2J—1) 1/2

= (J + 1)(2J+ 1)(2J + ~ (y~J11 T~2~II y

1J), (13)B~11 J(J—1) 1/2

- = - (J+ 1)(2J—1)(2J + 1)) (71~II T~2~II y~(J — 1)>, (14)

lJ(J_ 1)(2J— 1) )1/2

BJJ._2=+Q((2J3)(2J1) Ky~JIIT~

2~IIy1(J—2)>. (15)

The energycorrectionsare thengiven by

WM1(J, J) = ~AJC, (16)

WM1(j, J— 1) = fA1,~1[(K+ 1)(K—2F)(K— 21)(K— 2J+ 1)]l~’2 (17)

~C(C+ 1) —1(1+ 1)J(J+ 1)WE2(J, J) =B1 (18)21(21—1)J(2J—1)

WE2(J, f—i)

[(F +1+ 1)(F— 1)—f2 + i] [3(K+ 1)(K—2F)(K— 21)(K— 2J+ 1)]1~’2

=B1,11 , (19)

21(2!— 1)J(J— 1)

WE2(J, J—2)

= B~,1..2

[6K(K+ 1)(K—21—1)(K—2F — 1)(K— 2F)(K— 21)(K—2J+ 1)(K—2J+ 2)]1~’2x

21(21— 1)J(J— 1)(2J— 1)

(20)

whereC = F(F + 1) — J(J + 1) — 1(1 + 1) and K = 1+ J+ F.The off-diagonal constant,A1,11, can experimentallyby determinedin strong external magnetic

fields,where J is no longera good quantumnumber[4—6].Usingthe I y1y1JIM1M1>representation,thisparametermay also be expressedas

1A1,11=

[j2 — M/] ~“2M

1~y1y1JIM1M1 I . MW I y1y1(J — 1) 1M1M1). (21)

P. JönssonetaL / Computingmagneticdipoleandelectricquadrupolehyperfineconstants 403

2.2. Outlineofmethod

In the MCHF method,the nonrelativisticelectronicwavefunction,~t’, for a state labelled 7LSJ isexpandedin termsof configurationstatefunctionswith the sameLS termvalues.

1I/(7LSJ) = ~c1i~P~(y1LSJ). (22)

The configuration state functions are sums of productsof spin-orbitals. The radial parts of thespin-orbitalsandthe expansioncoefficientsaredeterminediterativelyby the~ulticonfigurationself-con-sistent field method [7]. Relativistic correctionsmay be includedby the ~reit—Pauliapproximation,wherethe wavefunctionis expandedin termsof configurationstatefunctionswith differing LS termvalues.

1J~(yf) = ~c1P1(y~L1S1J). (23)

The J dependentexpansioncoefficients are then obtained from a configuration interaction (CI)calculation[8].

The magnetichyperfine constantA is written as a sum of three terms, A0~i~,D~pand A’°’~,correspondingto the orbital, spin-dipole,and contacttermof the Hamiltonian,respectively.Below wederivecomputableexpressionsfor thedifferentmagnetictermsandfor the electricquadrupoleterm B,in termsof reducedmatrixelementsof one-particletensoroperatorsbetweenthe LS coupledconfigura-tion statefunctionsin thewavefunctionexpansion.

2.3. Orbital terms

The orbital termscan,after recouplingof J and(f — 1), be written as

a2~ 2f+1 1/2

A~’= ~ + 1) ~ ~J,k.cfckw(LJs~1f; fLk) (y~L1S~IIE1l~’~(i)r1

3 II ykLkSk), (24)

A~...1= ~ ~cJckW(LJSJ1(J— 1); fLk) (y1L~S1IILIW(i) rr

3 II7kLk~~k). (25)V j,k

Using an extendedversion of the programTENSOR[9,10],originally written by KoDb, the reducedmatrix elementcan be calculatedas

(y~L1S~II ).l~

1~(i)r13 II YICLkSk> v(pJok) (ia. II II lcTk)(flP.1P. I r’3 I nO,klO.k), Pi * ~ (26)

<71L1S1II ~1~

1)(i) r1’’

3 II ykLkSk)= ~v(p1p1) (l~~111(1)11 I r’’

3 I n~J,,), P1=

0k’ (27)

where p and er are the different orbitals. The programTENSOR hasbeenadaptedfor the useofnonorthogonalorbitals [11]. In case of nonorthogonality the one-electronactive radial integral,(n~l~I r3 n,,l~),is tobemultipliedwith anumberof overlapintegral(n,hl,L I ~ (n~o,,I ~The presentversion of TENSOR allows no more than two overlap integrals to multiply the radialintegral[11,12].

404 P. Jönssoneta!. / Computingmagneticdipoleandelectricquadrupolehyperfineconstants

2.4. Spin-dipole terms

The spin-dipoletermscan, afterrecouplingof f and(J — 1), bewritten as

_____________ (21) .a2 j~ 130(2J+1)J L Si ~I

) j,k {2 1 i)A~~”21g5[ f(J+1 ECJCk Lk

5k fky~L1S1II~(CS)(z) r, IIykLkSk>, (28)

a2 130 11~/~2 (L~ Sj f

Adip,1-1

21g5[7j ~CiCk~ Lk5k ~ l}~~L

1~IIECs)~20(i)rI3IIykLkSk>.

i,k ~2 1 1

(29)

The reducedmatrix elementcan becalculatedas

(21) .

(y1L1S1 II ~ (Cs) (i) r II )‘kLkSk> = V(P10k) (ia. II C~

2~II l~J.k)(sII $~1)J~s)(n~.l~. I r3 I n~klUk),

Pi*0k, (30)

<71L1S1 II ~ (Cc)~

20(i)r3 II ykLkSk> = Ev(p~p3) (ia. II C~

2~II l~.)(sII sm II s)(n / I r3 I ~P~P~

p

1

Pj0k• (31)

2.5. Contact ter~ns

Using the relation 4m~r26(r)= 6(r) between the three- and one-dimensionaldelta functions, thecontacttermscan,after recouplingof f and(f — 1), bewritten

A~ont= a2 p.~ 2g51 2f+ 1 1/2~1 ECJCk( 1) W(L1S1f1, fSk)j,k

X<y3L1S~II ~s~’~(i) r126( r.) II ~‘kLkSk), (32)

a2~12g51A~~_1=~ ~CJCk(l)~W(LJSJ(f 1)1; fSk)

x (y~L3S1II ~sW(i) r126(r

1) I ykI~~kSk>. (33)

The reducedmatrix elementcan be calculatedas

K7~L1S1II ~s”~(i) r[26(r,) II ykLkSk)= v(pfcrk)(s swII s)(nl~.lp

1II r’’26(r) I nO.klUk), í * Uk,

(34)

<73L1S1 II Es~’~(i)r1

2&(r,) II ykLkSk> = ~v(p1p1) (s II s”’ II s)(n~1l~1I r

26(r) I ~ Pj = elk.I P

1

(35)

P.Jönssoneta!. / Computingmagneticdipoleand electricquadrupolehyperfineconstants 405

The radial matrix elementis expressedin the radial startingparameterAZ(nl),

(n~l~I r28(r) I n~l~)= r rLo = 61 ,06~~,0AZ(n~/~)AZ(n~,l~). (36)

2.6. Quadrupoleter,ns

The quadrupoletermscan,after recouplingof f, (f — 1), and(f — 2), beWritten

4J(2J—1)(2f+ 1) 1/2B1= (f+ 1)(2J+3) ) ~clckW(LJSj2f; fLk)<YJLJSJII~C~

2~(i)r13 IykLkSk),

(37)

1 ff 1)1~’2

B1,11= _Q~(~ ) ~c~ckW(LJSj2(J—1); fLk) (yJLJSJIIEC~

2~(i)r13IIykLkSk>,

(38)

B1,12= —Q~4-[J(J — 1)(2f— 1)J1/2 ~CICkW(LJSJ2(J—2); fLk)

X(y~L1S1II~C~2)(i)r13IIykLkSk>. (39)

The reducedmatrix elementis calculatedas

<71L1S1 II ~C~

2~(i)r1’’

3II Yk’-’kSk> = V(pjelk) (la. II C~2~II lO,k)(np.l I r3 I ‘~01J0k)’ ~ * ~k, (40)

Ky~L1S1II ~C~

2~(i)r13II ykLkSk> = ~v(p1p1) (/~. II C~

2~II l~1)(n~./~.I r’

3 I ~ p1 = elk. (41)

2.Z Hyperfineparametersin pureLScoupling

For small atoms, in which LS coupling is valid to a good approximation~the hyperfineconstantsAand B areoften expressedin termsof the totally electronicparametersa1, ad, a~and bq,

a1 = <YLSMLMSI ~I~(i) r13 I YLSMLMS>, (42)

ad— <YLSMLMSI ~2C~(i) s~(i)r13IYLSMLMS), (43)

a~= <YLSMLMSI L2s~(i)r126(r1) I YLSMLMS), (44)

bq = <YLSMLMSI ~2C~(i) rI3IyLSMLMS), (45)

where ML = L and M5 = S.

Explicit relationsbetweenthe hyperfmeconstantsA and B and the parametersaboveare given inrefs. [10,131.

406 P.Jönssonet aL / Computingmagneticdipoleandelectricquadrupolehyperfineconstants

2.8. Nonorthogonal orbitals

A totally unrestrictednonorthogonalschemewould lead to a very elaboratecomputerprogram.Thereforethepresentversionof TENSOR[ii] imposesthefollowing restrictionson the orthogonalityofthe orbitals in the left hand side configuration, (Pd, andthe right handside configuration, ~k, of thereducedmatrix element[12].1. The orbitals of a single configurationaremutuallyorthogonal.2. Thereareat most two subshellsin (P~containingspectatorelectronswhoseorbitals arenot orthogonal

to orbitals in ~

3. If all the spectatorelectronswith nonorthogonalorbitals havethesame/-value, thenthereare atmosttwo suchelectronsin eachof cP~and ‘

1~k~

4. KP, I ~Pk> = 6jk evenif nonorthogonalorbitals areincluded.If the useris trying to useexpansionswith configurationsbreakingthe above restrictions,this program,aswell asthe otherprogramsin theMCHF package,will stopandgive an error message,sayingthat thenonorthogonalityis setup incorrectly.

2.9. Generalform of the expressions

The contributionto the differentA factorsfrom a pair of configurationsj andk can in its mostbasicform be written as

CA—1-cfck ~ coef(p, U) rad(p, U) (i.tI,,s) (vli’ ) , (46)

the mostbasicform of the B factor

CBQCJCk ~ coef(p,U) rad(p,U) (~eI ,x!)?hl(p I ,i~)’~2, (47)p,o~,/~,v

wherethe coefficients and the radial matrix elementsrad(p,u) are determinedfrom the expressionsabove.Most oftenthe A and B factorsareexpressedin the experimentallyconvenientunit MHz. If thenuclearmagneticmoment,j~, is given in nuclearmagnetonsandthenuclearquadrupolemoment,Q, inbarns(1028 m

2), the conversionfactorsto MHz areCA = (1ja2)2cRome/Mp= 47.70534and CB = 2cR~10~m2/a~= 234.9647.

3. Programstructure

Many of the subroutinesusedin this programareincludedin theMCHF programpackage[1]. Below

we describethe subroutinesthatarespecific for the hyperfinecode.

3.1. Newsubroutines

READWTReadsthe weights of all configurationsin the wavefunctionexpansion.If the weights havebeen

obtainedfrom a f dependentCI calculationtheyarereadfrom the file Kname>.jotherwisetheyarereadfrom (name).l.

P.Jönssonet a!. / Computingmagneticdipoleandelectricquadrupolehyperfineconstants 407

LSJ

Determines L, S,L + S and I L — S I for a configuration in the wavefunction expansion.

All the subroutinesbelow arerelatedto a pair of configurationsj, k.

LSJFACTCalculatesthe L~,S

1. Lk,5k and f dependent factors in eqs. (24), (25), (28), (29), (32), (33) and

(37—(39).

TENSOR2To fit the structureof the hyperfineprogramthe subroutineTENSORhasbeenmodified. TENSOR2

outputs the following information: IIRHO(m) — the interacting shells in the initial configuration;IISIG(m) — the interactingshells in thefinal configuration;NNOVLP — the mimberof overlapintegrals;MMU(m)and MMUP(m)— the shells in the first overlapintegral;NNU(m) andNNUP(m) — The shellsin the second overlapintegral; IIHSH andVVSHELL(m) aredefinedas follows,

IIHSH

<y1L3S1 II EQ~~iII Yk’-’kSk> = ~ VVSHELL(m) (p IIQ(A) II U), (48)

m=1

wherethe two casesp * U and p = U arehandledon the samefooting. In caseof nonorthogonality,thereducedone-electronmatrix elementis to be multiplied with at most two overlapintegrals.

MULTW7’Multiplies the weightsof configurationj and k.

ORBITAL

When p3 * ~k, ORBITAL calculates the following quantity in eq. (26)

V(pjok)(i~~111(1)1110k). (49)

When p3 =

1~Tk’the following quantitiesin eq. (27) are calculated:

v(p1p1)(l~~11(1) II l~). (50)

DIPOLE

When p3 * elk. DIPOLE calculatesthe following quantity in eq. (3)):

v(pielk) (l~~IIC2IIc5)(sIIs~IIs). (51)

When p3 = ~k, the following quantitiesin eq. (31) are calculated:

v(p~p3) (la. II C~2~II l~

1)(sII ~~1) II s). (52)

CONTACT

When p3 * elk, CONTACT calculatesthe following quantityin eq. (34):

v(p)elk)(sIIs~’~IIs). (53)

408 P. Jönssonet aL / Computingmagneticdipoleandelectricquadrupolehyperfineconstants

When p3 = elk, the following quantitiesin eq. (35) arecalculated:

v(p~p3)(sIIs”~IIs). (54)

QDRPOLE

When p * elk, QDRPOLEcalculatesthefollowing quantity in eq. (40):

v(pfok)(1P II C~2~II ‘O’k)~ (55)

When p3 = elk, the following quantitiesin eq. (41) arecalculated:

v(p1p1)(l~.II C~2~II la.). (56)

RADIALICalculatesthe one-electronactive radial integral (n~l~Ir3 In

010). In caseof nonorthogonalitytheoverlapintegralsarecalculatedandmultiplied with the active radial integral.

RADIAL2Calculatesthe one-electronactive radial integral(n~l~Ir

26(r) 1n010).In caseof nonorthogonalitythe

overlapintegralsarecalculatedandmultiplied with the active radial integral.

PRINTDependenton the responsesto the prompts in the beginningof the program,this subroutineprints

out parametersfrom the matrix elementcalculationsbetweenconfigurationj and k.

3.2. New common blocksA seriesof COMMON blocksarisingfrom theoriginal cfp packagesandfrom Robb’stensorprogram

havebeenredefinedin thewrite-up of the MCHF- LIB - COM, MCHF LIB ANG, MCHF - RAD3 andNJGRAF libraries. The commonblocks OVRINT and OVRLAP arerelatedto the non-orthogonalityfeatureandaredescribedin the write-up of MCHFNONH [14].

COEFORBF(n),DIPF(n), CONTF(n),QUADF(n)containsthe L3, S~,Lk,

5k’ f dependentcoefficientsineqs.(24), (25), (28), (32), (33) and(37—39), wheren is a parameterorderingthe differentcombinationsoff and f’.

J(n),JP(n)areused in the printout to give J and f’ asrationalnumbers.JJMAX, JJMIN aretwicethe maximum and minimum f quantumnumbers,respectively,for the wavefunction‘P(y

1LSf) in eqs.(22) and (23). LL1, SS1,JJ1MAX, JJ1MIN correspondto 2L3, 2S3, 2 I L3 + S~I, 2 I L3 — S~I. LL2, SS2,JJ2MAX,JJ2MIN correspondto

2Lk, 25k’ 2 I Lk + 5k I, 2 I Lk — 5k I.

TENSORThis commonblock holdsthe output from TENSOR2.

HYPERQuantitiesin this commonblockarecloselylinked to the output from TENSOR.After thesubroutine

ORBITAL hasbeencalledthe quantitiesin the commonblockwill be definedas follows:NHY = IIHSH,

P. Jönssoneta!. / Computingmagneticdipoleand electricquadrupolehyperfineconstants 409

\THY(m) = VVSHELL(mX/~Il,(~Il/Ok),

IRHY(m) = IJFUL(IIRHO(m)),ISHY(m) = IJFUL(IISIG(m)),JRHY(m) = IAJCMP(IRHY(m)),JSHY(m)= IAJCMP(ISHY(m)),IMUHY(m) = IJFUL(MMU(m)),IMUPHY(m) = IJFUL(MMUP(m)),JMUHY(m) = IAJCMP(IMUHY(m)),JMUPHY(m)= IAJCMP(JMUHY(m)),INUHY(m) = IJFUL(NNU(m)),INUPHY(m) = IJFUL(NNUP(m)),JNUHY(m)= IAJCMP(INUHY(m)),JNUPHY(m)= IAJCMP(INUPHY(m)),

wherem = 1,..., IIHSH.In the sameway the quantitiesabove can be inferred from eqs. (51)—(56) after the subroutines

DIPOLE, CONTACT andQDRPOLEhavebeencalled.

PRINTAThis commonblock supplies the subroutinePRINT with many of the quantitiesneededfor the

printout.For everypair of configurationsj, k the following quantitiesaresupplied:ORB= ORBF(n)VHY(N),whereVHY(N) is the output from the subroutineORBITAL.DIP= DIPF(n)VHY(N), whereVHY(N) is the output from the subro~.itineDIPOLE.CONT= CONTF(n)VHY(N), whereVHY(N) is the output from the subroutineCONTACT.QUAD = QUADF(n)VHY(N), whereVHY(N) is the output from the subroutine QDRPOLE.RADINT = QUADR(IRHY(N), ISHY(N), -3).AZSQR= AZ(IRHY(N))AZ(ISHY(N))).INTGR = 1 if radial integrationhasbeenperformedand0 otherwise.OVLINT1 = QUADR(IMUHY(N), IMUPHY(N), 0)**IOVEP(1).OVLINT2 = QUADR(INUHY(N), INUPHY(N), 0)**IOVEP(2).CONTRI the contribution to the A and B factors.IREPRI,ICOPRI andIDOPRI areparameterscontrolling the write-out.

4. Input data

To describehow the calculationsareto be done, a few promptsfor datahaveto be answered.

1. Name of stateThis programassumesthenamingconventionsdescribedin thepaperon the atomic-structurepackage

[1], andthe (name>of the statehasto be specified.

2. TypeThis prompt asksyou to specifythe differentconstantsthat areto be calculated.The optionsare

• only diagonalA and B factors.• both diagonalandoff-diagonal A and B factors,• only the coefficientsof the radial matrix elements.

For the last option all neededinput datais in the configurationinput file (name>.c.

410 P. JönssonetaL / Computingmagneticdipoleand electricquadrupolehyperfineconstants

3. HyperfineparametersThe hyperfineparameters a1, aa, a~andbq can beprintedout. This is only meaningfulfor pure LS

terms.

4. InputThis prompt inquires if the wavefunctions expansionhas been obtainedfrom an MCHF or CI

calculation. In caseof an MCHF calculation,configurationweightsarereadfrom he configurationinputfile (name).c.If the expansionhasbeenobtainedfrom a CI calculation, weights are readfrom theeigenvectorfile (name>.jor (name>.l,dependenton whetherthe CI Hamiltonian is f dependentor not.

5. NumberIn general the eigenvectorfiles (name).jor (name>.l from a CI calculationcontain wavefunction

expansionsfor manydifferentstates.Thus,the particularstatesfor which the A and B factorsareto becalculatedhaveto be specified.Suppose,in a f dependentCI calculation, that the expansionin the file(name>.cconsistsof the following four CSFs

ls( 2) 2p( 1)iSO 2P1 2P

ls( 1) 2s( 1)2p( 1)

2s1 2S1 2P1 iS 2P

ls( 1) 2s( 1) 2p( 1)2S1 2s1 2P1 3S 2P

ls( 1) 2s( 1) 2p( 1)

2S1 2S1 2P1 2S 4P

andthat the hyperfinestructureis to be calculatedfor the f = ~ and f = ~ statesof the third CSF

ls( 1) 2s( 1) 2p( 1)2S1 2S1 2P1 3S 2P

Then,by replying 3 to the numberprompttheprogramsearchesfor the two wavefunctionexpansions(one for the f = f stateandone for the f = ~ state)in the file Kname).j,having the third CSF as thedominantcomponent.If no statehasthe third CSFas the dominantcomponentthe programstopsandan error messageis written out, telling the userhow to proceed.This is alsothe casewhen morethanonestatefor a given f value has the third CSF as the dominant component in the expansion.

6. Print-outIn acalculationof the A and B factors,valuesof the A0~b,AdIP, Acont, and B termsarewritten to the

file (name).h.If in this casea full print-out is requested,values of coefficients and radial matrixelementsas well as the contribution to the different A and B factors are printed for every pair ofconfigurations.

7. Tolerancefor printingIn the case of a full print-out it is possible to set a tolerance for printing.If the contributionto a term

is less than the tolerance,then it will not be printed.

8. Nuclear dataThe last promptasksfor the following nucleardata:

• 21,• j.~,in nuclearmagnetons,• Q in barns.

P. JönssonetaL / Computingmagneticdipoleand electricquadrupolehyperfineconstants 411

A recentcompilationof experimentaldataon nuclearmagneticdipole andelectricquadrupolemoments

for ground andexcitedstatescan be found in ref. [15].

5. Output data

When the program is running Ja= 1, Ja = 2,.. . , Ja = N, where N is the number of CFSs in thewavefunctionexpansion, is written out on the screen,indicating how far the programhasexecuted.Output from the programis written to a file (name>.h.If a full print-out is requested,the programoutputsthe following datafrom eqs.(46) and(47) for a pair of configurationsj and k.1. f f’ — the J values for which the matrix element hasbeencalculated.2. Weight — the configuration weight clck.3. Coeff— the coefficient coef(p,el).

4. Radialmatrix elements— values of the radial matrix elementsrad(p,U), (.t I ~i’)~’ and(v I v’)n2

5. A(MHz), B(MHz) — the contribution in MHz to the different A and B factorsfrom this pair ofconfigurations.

6. Examples

The TestRun Output containsfull print-outsfrom two different hyperfinestructurecalculationsof2p2P in 7Li. DiagonalA and B factorsarecalculatedaswell as thehyperfineparametersa

1, ad, a~andbq. The wavefunctions have been obtained from a HF and a non-orthogonalMCHF calculation,respectively. Nuclear data has been supplied by the user.

Acknowledgements

The authorswish to acknowledgetheassistancegiven to this projectby K.M.S. Saxenawho sharedanearlyversionof a hyperfineprogramwith us, basedon partsof Alan Hibbert’s CIV3 program.

The presentresearchhasbeensupportedby the US Departmentof Energy,Office of Basic EnergyScience,andby the SwedishNaturalScienceResearchCouncil.

References

[1] C. FroeseFischer,Comput.Phys.~ommun. 64 (1991)399.[2] C. Schwartz,Phys.Rev.97 (1955)380.[3] 1. Lindgrenand A. Rosen,CaseStud. At. Phys. 4 (1974)93.[4] J.S.M.Harvey, Proc. Roy. Soc. LondonA 285 (1965)581.[5] H. Orth, H. AckermannandE.W. Otten,Z. Phys.A 273 (1975)221.[6] K.H. Liao, L.K. Lam, R. GuptaandW. Happer, Phys.Rev. Lett. 32 (1974) 1340.[7] C. FroeseFischer,Comput.Phys.Rep. 3 (1986)275.[8] C. FroeseFischer,Comput.Phys.Commun.64 (1991)473.[9] W.D. Robb, Comput.Phys.Commun.6 (1973) 132; 9 (1985)181.

[10] R. GlassandA. Hibbert, ComputerPhys.Commun.11(1976)125.[11] C. FroeseFischer,M.R. Godefroidand A. Hibbert,Comput.Phys.Commun.64 (1991)486.[12] A. Hibbert, C. FroeseFischerand M.R. Godefroid,Comput.Phys.Commun.51(1988)285.[13] A. Hibbert,Rep.Prog. Phys.38 (1975)1217.[14] A. Hibbert andC. FroeseFischer, Comput.Phys.Commun.64 (1991)417.[15] P. Raghavan,At. Data Nuci. DataTables42 (1989)189.

TE

ST R

UN

OU

TP

UT

h

.==

....>

ca

se

#I.

=======> Display cfg.inp

>>

cat

cig.inp

is( 2) 2pl( I)

IS0

PPI

2PO

>a

nor&

h

FULL PRItiT-OUT

? (Y/X)

>Il

Au. IITSruCTIoYs ? (Y/I)

'J

=r=====> Obtain Bnesgy Exprassion

>> mchf

ATOH, TERM. 2 in MRMATLT(A.A,F)

: >Li,ZP,J.

Thor4 are

2 orbital8 as folloos:

1s 2pl

===I===> Obtain Radial Functions

Enter orbitala to be varied: (A11.,IOIE,SOIIB,IIT=,comna

delimited list)

>ALL

Default electron parameters ? (Y/I)

'I Default values (IO,RJ3L,STRDIC)

? (Y/I)

'y Default values for other parameters ?

(Y/I

I)

‘J Do you wish to continue along the sequence ?

>n

>>

mv

cfg

.ou

t 1il.c

>> IV afn.out 1il.w

>> his

calculation

=======> Move cfg.out to 1ii.c

=s==s==> kwe sfn.out to 1il.v

I======> liyperfine structure

==

=I=

==

==

==

==

==

==

==

==

BYPERFIXE

==============ii======

lam

e of state

>lil

Indicate the type of calculation

0 => diagonal A and B factors only;

1 => diagonal and off-diagonal A and B factors;

2 => coefficients of the radial matrix elements only:

>o iiypsrfins

parameters al,ad.ac and bq ? (Y/P)

'J Input from an HCEF (H) or CI (C) calculation ?

htll print-out ? (Y/I)

'Y Tolerance for printing (in ItlIz)

so.0

Give 2*1 and nuclear dipole and quadrupols moments (in n.m. and barns)

>3,3.266,-0.040

The configuration set

STATE (YITE 1 COllFIGURATIDIS):

____________________~~~___~~~__

THWU?, ARE 2 ORSITALS AS FOLLOW:

1s 2pl

COIFIfXJFtATIOE

1 ( OCCUPIgD ORSITALS= 2 ): ls( 2) 2pl( 1)

CouPLIllG scIiEMB:

IS0

2Pl 2PO

Ja= 1

>> cat 1il.h

=======> Display 1il.h file

liyperfino structure calculation

1il.c

Tolerance for printing

0.000000 na

z Jnclsar dipole moment

3.256000 n.m.

nuclear quadrnpole moment

-0.040000 barns

2*1=

3

Li

2P

-7.3650696

(config llhfslconfig 1 )

Orbital term

matrix element

<ZpllR**(-3)12pl>

J

J'

weight

coei

i matrix element values

l/2 l/2 1.000000 2.666667

0.0586

312 J/2 1.000000 1.333333

0.0688

Spin-dipole term

matrix element

eplIR**(-3)12pi~

J

J'

weight

coeff

matrix element values

l/2 i/2 1.000000 2.669769

0.0686

3/2 3/2 1.000000 -0.266976

0.0686

Quadrupole term

matrix element

<2plIR**(-3)12pl>

J

J'

weight

coeii

matrix element values

J/2 3/2 1.000000 0.400000

0.0686

Aorb(lIliz)

16.173963

8.086976

Adip0lliz)

16.192707

-1.619271

Bo4

Ez)

-0.220196

l *

A

fact

ors

in N

Bz

J

J'

Orbital

Spin-dip

l/2

l/2

16.17386

16.19271

3/2

3/2

6.08698

-1.61927

B iactors in naz

Cont

Total

0.00

000

32.36666

0.00000

6.46771

Th

ere

are

6 orbital8 as folloss:

1s 2pl 282 lp2 283 2p3

Enter orbitala to be varied: (ALL,IO~,SO~,IIT=,co~a delimited list)

>2pl,2~2.?p2,2~3,2p3

Default electron parameters ? (Y/I)

'Y Default values (IO.RKL,STROIG) ? (Y/I)

'lr

Default values for other parametera 7 (Y/B)

'J Do you wish to continue along the sequence ?

>IL

J

J'

quadnp010

l/2

l/2

0.00000

J/2

3/2

-0.22020

>> w cfg.out li2.c

>> mv wfn.out li2.u

>> hfa

calculation

**

===================i=

Eyperfine parameters in a.u.

HYPBRFIlE

=======1=1=55=======

al

ad

ac

bq

0.06867

-0.01171

l *

0.00000

-0.02343

lame of state

>li2

Indicate the type of calculation

0 => diagonal A and B factors only;

>> cat cfg.inp

is( 2) 2pl( I)

is0

2Pl

2PO

1r( 1) 2s2( I) 2p2( 1)

2Sl

2.31

2Pl

ls( I) 2.3( I) 2p3( 1)

2Sl

PSI

2Pl

is0

3so

2PO

2PO

>a

noah

FDLL PBIIT-ODT ? (Y/I)

Xl ALL. IITnACTIOYS ? (Y/Y)

'Y

>>cp 1il.s vfn.inp

>>

m&f

=======> Obtain I&dial Functiona

1s 2pi 262 2~2 283 2~3

ATOW, TERM, 2 in FORJIAT(A,A,F)

: >Li,2P,3.

COIFIGURATIOl 1 ( OCCUPIED ORBITALS- 2 ): ls( 2) 2pl( 1)

CODPLIYG SCBEHE:

IS0

2Pl

====I==> k.vo cfg.ont to li2.c

===r===> nave ofn.out to li2.s

=====r=> Elyparfine Itructure

=I=

==

==

>

Cal

ls

12.

=rr====> Display cfg.inp file

1 => diagonal and off-diagonal A and B factors;

2 => coefficients of the radial matrix element8 only:

>I Eyperfins parameters al,ad,ac and bq ? (Y/I)

>]I

Input from an IICEF (II) or CI (C) calculation ?

>Xl

Full print-out ? (Y/X)

'Y Tolerance for printing (in UEz)

>I.0

Give 221 and nuclear dipols and qudrupole moments (in n.m. an&barns)

===t===> Obtain Energy gxpression

>3.3.266.-0.040

The configuration set

STATE (WITH 3 COIFIGURATIOIS):

____________________~~~~~~~~~~~

==I====> Cow old pin. to r,ew vfn.

THERE ARE 6 ORBITALS AS FOLLOWS:

2PO

COlIPIGURATIO8 2 ( OCCUPISD ORBITALS= 3 ): ls( 1) 2a2( 1) 2p2( 1)

COUPLING SCEEIIE:

251

2Sl

2Pl

is0

ZPO

COIPIGURATIOI 3 ( OCCUPISD ORBITALS= 3 ): la( 1) 2s3( I) 2p3( 1)

COUPLIIG SCIIEHB:

251

2Sl

PPI

3so

2PO

Ja- 1

Ja= 2

Ja= 3

>> cat li2.h

=======> Display li2.h file

Byperfino structure calculation

li2.c

Tolerance for printing

1.000000 If82

Iuclear dipole moment

3.266000 II.=.

Iuclear quadrupole moment -0.040000 barns

2*1=

3

Li

2P

-7.3667346

(config llhislconfig 1 )

Orbital term

matrix element

J

J'

weight

1/z i/2 0.990791

3/Z 3/Z 0.999791

3/Z i/2 0.099791

Spin-dipole term

matrix element

J

J'

weight

l/2 i/2 0.999791

3/Z 3/Z 0.999791

3/Z l/2 0.999791 eplIR**(-3)I2pl>

coeii

matrix element values

2.666667

0.0604

1.333333

0.0604

0.666667

0.0604

c2p11n**(-3)I~Lx

element ialuas

COOif

Adip(llEz)

2.669759

0.0604

16.702207

-0.266076

0.0604

-1.670221

-0.333720

0.0604

-2.087776

(cord ig

3lhfslconfig 1 )

Fermi-contact term

matrix element

A?.(Zs3)AZ(

1s) <2p312pl>** 1

J

J'

weight

coefi

matrix element

l/2 i/2 -0.010481 -0.363308 43.1980 0.337

3/Z 3/Z -0.010481 0.363306 43.1980 0.331

3/Z i/Z -0.010481 -0.363308 43.1980 0.337

**

A factors in UEz

valu

es

Aorb(Kiz)

16.682863

8.341432

4.170716

Acont(8Bz)

5.739065

-6.739065

5.739065

J

J'

l/2

1/z

3/Z

3/Z

3/Z

1/z

J

J'

1/z

1/z

3/Z

3/Z

3/Z

1/z

Orbital

Spin-dip

cant

Total

16.64626

16.70147

11.45731

46.08604

8.42313

-1.67816

-11.45731

-4.71233

4.21167

-2.09768

11.45731

13.67119

B factors in 88~

%

3

Quadrupole

0.00000

-0.22935

-0.04966

9 l *

2 B

Eyperfine parameters in a.u.

3

al

ad

ac

bq

1

0.06101

-0.01214

-0.24866

-0.02440

0'

**