A program for computing magnetic dipole and electric quadrupole hyperfine constants from MCHF wavefunctions

Computer Physics Communications

Computer Physics Communications 74(1993)399—414 North-Holland

A program for computing magnetic dipole and electric quadrupole hyperfine constants from MCHF wavefunctions Per Jönsson, Claes-Göran Wahlström Department of Physics, Lund Institute of Technology, P.O. Box 118, S-221 00 Lund, Sweden

and Charlotte Froese Fischer Department of Computer Science, Vanderbilt University, Nashville, TN 37235, USA Received 27 March 1992

Given an electronic wave function, generated by the MCHF program (LS format) or the MCHF_CI program (LSJ format), this program computes the hyperfine interaction constants, A 1 and B1. In strong experimental magnetic fields, the splitting is also affected by the off-diagonal hyperfine constants A1,1_1, B1,11, and B1,1_2 and these are computed as well, depending on the input data.

PROGRAM SUMMARY Title of program: MCHF_HFS

Memory required to execute with typical data: 210K words

Catalogue number: ACLE

No. of bits in a word: 32

Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue) No. Licensing provisions: none Computer for which the program is designed and others on which it has been tested: Computer: VAX 11/780, DEC 3100, SUN SPARC-station 330; Installation: Lund University, Vanderbilt University Operating system under which the program is executed: VMS, ULTRIX, Sun UNIX

Peripherals used: terminal; disk of lines in distributed program, including test data, etc.: 3291 CPC program Library subroutines used: Catalogue nwnber Title Ref in CPC ABBY NJGRAF 50 (1988) 375 ABZU MCHF_LIBRARIES 64 (1991) 399

Programming languages used: FORTRAN77

Keywords: hyperfine structure, A factor, B factor, orbital term, spin-dipole term, Fermi contact term, electric quadrupole term, MCHF calculation, CI calculation

Correspondence to: C. Froese Fischer, Department of Computer Science, Vanderbilt University, Nashville, TN 37235, USA.

Nature of physical problem The atomic hyperfine splitting is determined by the hyperfine interaction constants A1 :and B1 [1]. In strong external

0010-4655/93/$06.00 © 1993

—

Elsevier Science Publishers B.V. All rights reserved

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Computing magnetic dipole and electric quadrupole hyperfine constants

magnetic fields, where J is no longer a good quantum number, the splitting is also affected by the off-diagonal hyperfine constants A 111, B111 and B1,12 [2—4].This program calculates the hyperfine’ constants using an electronic wavefunction generated with the MCHF or MCHF_ CI programs of Froese Fischer [5].

subshell with I 3. If 1 4, the LS term for the subshell is restricted to those allowed for I = 4. Only the subshells outside a set of closed subshells common to all configurations need to be specified. A maximum of 5 (five) subshells (in addition to common closed subshells) is allowed. This restriction may be removed by changing dimension statements and some format statements.

Method of solution The electronic wavefunction, ~I’,for a state labelled yJ can be expanded in terms of configuration state functions, ~ ~ The hyperfine constants can then generally be calculated as

Typical running The CPU time the first and SPARC-station

Lk,kclck coef(j, k)( 71L1S1II T(K) “Yk LkSk)

References

where T(K) is a spherical tensor operator of rank K. Evaluation of the reduced matrix element between arbitrarily LS coupled configurations is done by an extended version of the program TENSOR originally written by Robb [6—8].

[1] I. Lindgren and A. Rosen, Case Stud. At. Phys. 4 (1974)

Unusual features of the program The program allows for a limited degree of nonorthogonality between orbitals in the configuration state expansion. Restrictions on the complexity of theproblem The orthogonality constraints are relaxed only within the restrictions described in ref. [9], giving rise to at most two overlap integrals multiplying the one-electron active radial integral. Any number of s-, p-, or d-electrons are allowed in a configuration subshell, but no more than two electrons in a

time required for the test cases is 0.1 seconds for 0.19 seconds for the second on a SUN 330.

[2] G.K. Woodgate, Proc. Roy. Soc. London A 293 (1966) 17. [3] J.S.M. Harvey, Proc. Roy. Soc. London A 285 (1965) 581. [4] H. Orth, H. Ackermann and E.W. Otten, Z. Phys. A 273 (1975) 221. [5] C. Froese Fischer, Comput. Phys. Commun. 64 (1991) 431, [61 W.D. Robb, Comput. Phys. Commun. 6 (1973) 132; 9 (1985) 181. [7] R. Glass and A. Hibbert, Comput. Phys. Commun. 11 (1976) 125. [8] C. Froese Fischer, M.R. Godefroid and A. Hibbert, 64 (1991) 486. [9] A. Hibbert, C. Froese Fischer and MR. Godefroid, Comput. Phys. Commun. 51(1988) 285.

LONG WRITE-UP

1. Introduction The present MCHF....HFS program is closely linked with the MCHF atomic structure package. Therefore, it adheres to the structure and constraints underlying the package. A summary of the data input for the various programs was included with the description of the package [1]. The role of the program is to calculate magnetic dipole and electric quadrupole hyperfine constants from MCHF wavefunctions, either in the nonrelativistic LS scheme, or in the Breit—Pauli LSJ scheme.

2. Theory 2.1. Hyperfine interaction

The hyperfine structure of the atomic energy levels is caused by the interaction between the electrons and the electromagnetic multipole moments of the nucleus. The contribution to the Hamiltonian can be

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represented by an expansion in multipoles of order K, ~

HhfS=

(1)

T(K).M(K),

K 1

where ~ and M(K) are spherical tensor operators of rank K in the electronic and nuclear space, respectively [2]. The K = 1 term represents the magnetic dipole interaction and the K = 2 term the electric quadrupole interaction. Higher order terms are much smaller and can often be neglected. For an N-electron atom the electronic tensor operators are, in atomic units, [2 1”~(i)r13 _g

2~ T~’~ = ~a

0r13 +g

5~ii~[C(2)(i)Xs(1)(i)](

5~’rr6(r~)s(1)(i)1

(2)

and 2~ =

—

T~

~N i= 1

(3)

C~(i) r~3,

where g~= 2.00232 is the electronic g factor and 6(r) the three-dimensional delta function [3]. C~ = ~/4’w/(2k + 1) Y,~,with Y,~ being a normalized spherical harmonic with the phase conventions of Condon and Shortley. The magnetic dipole operator (2) represents the magnetic field due to the electrons at the site of the nucleus. The first term of the operator represents the field caused by orbital motion of the electrons and is called the orbital term. The second term represents the dipole field due to the spin motions of the electron and is called the spin-dipole term. The last term represents the contact interaction between the nucleus and the electron spin and contributes only for s electrons. The electric quadrupole operators (3) represents the electric field gradient at the site of the nucleus. The conventional nuclear magnetic dipole moment jx 1 and electric quadrupole moment defined 1~ and M~2~ in the state with Q theare maximum as the expectation values of the nuclear tensor operators M~ component of the nuclear spin, M 1 = I, (yjHIM0”~lyjII)=p~j,

(4)

(y~III M~

(5)

=

~Q.

When the hyperfine contributions are added to the Hamiltonian, the wavefunction representation for which the total Hamiltonian is diagonal is I -YJYJIJFMF), where F = I + J. In this representation, perturbation theory gives the following hyperfine corrections to the electronic energy 1~•M~’~IyJyJIJ’FMF), J’=J, f—i, WM!(J, J’)=(yJyJIJFMFIT~ WE 2~•M~2)IyJyJIJ’FMF),J’=J,J—l, J—2. 2(J, J’)—
(6) (7)

This can, using tensor algebra, also be written as WM 1(J,

J’)

=

(— 1)~’FW(IJiJ; F1)(71J II

II y1J’)<711 II M~’~ I y~I>,

(8)

WE2(J,

J’)

=

(— 1)’~”

II -y1J’X711 II M~II yjl>,

(9)

W(IJ’IJ; F2)(71J

II

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where W(JJ ‘If; Fl) and W(JJ ‘If; F2) are W coefficients of Racah. The reduced matrix elements are defined through the relation K

MIJ

J’\

KyJMIT~j’~Iy’J’M’>=(—1)’ ~—M Q M~)(7JMT~~’)/J> (10) The energies are usually expressed in terms of the hyperfine interaction constants (A and B factors) 1

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

—

=

~ [J(2J B1

(11)

—

1)(2J+ 1)11/2 (y1J II

II ~1

~

1)),

(12)

~

2Q(

J(2J— 1) 1/2 2~ II y = (J + 1)(2J + 1)(2J + (y~J11 T~ 1J), B~1 1 J(J—1) 1/2 2~ II y~(J — 1)>, = (J+ 1)(2J— (71~ II T~ lJ(J_ 1)(2J—1)(2J 1) )1/2+ 1)) BJJ._2=+Q(( 2~IIy 2J3)(2J1) Ky~JIIT~ (J—2)>. 1 -

(13) (14)

-

(15)

The energy corrections are then given by J)

WM1(J,

=

~AJC,

WM1(j, J— 1)

=

WE2(J, J) =B1 WE2(J,

(16)

fA1,~1[(K+ 1)(K— 2F)(K— 21)(K— 2J+ ~C(C+ 1) —1(1+ 1)J(J+ 1)

1,11

(18)

21(21—1)J(2J—1)

f—i)

=B

(17)

1)]l~’2

2 + i] [3(K+ 1)(K— 2F)(K— 21)(K— 2J+

1)]1~’2

[(F +1+ 1)(F— 1)—f ,

21(2!— 1)J(J— 1)

(19)

WE2(J, J—2) =

B~,1..2 x

[6K(K+

1)(K— 21— 1)(K— 2F — 1)(K— 2F)(K— 21)(K— 2J+ 1)(K— 2J+ 21(21— 1)J(J— 1)(2J— 1)

2)]1~’2

(20) where C = F(F + 1) — J(J + 1) — 1(1 + 1) and K = 1+ J + F. The off-diagonal constant, A1,11, can experimentally by determined in strong external magnetic fields, where J is no longer a good quantum number [4—6]. Using the I y1y1JIM1M1> representation, this parameter may also be expressed as A1,11

=

[j2

—

1 2M M/] ~“ 1~y1y1JIM1M1

I

.

MW I y1y1(J

—

1) 1M1M1).

(21)

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2.2. Outline of method In the MCHF method, the nonrelativistic electronic wavefunction, ~t’,for a state labelled 7LSJ is expanded in terms of configuration state functions with the same LS term values. 1I/(7LSJ)

=

~c1i~P~(y1LSJ).

(22)

The configuration state functions are sums of products of spin-orbitals. The radial parts of the spin-orbitals and the expansion coefficients are determined iteratively by the ~ulticonfiguration self-consistent field method [7]. Relativistic corrections may be included by the ~reit—Pauliapproximation, where the wavefunction is expanded in terms of configuration state functions with differing LS term values. 1J~(yf)

=

~c1P1(y~L1S1J).

(23)

The J dependent expansion coefficients are then obtained from a configuration interaction (CI) calculation [8]. The magnetic hyperfine constant A is written as a sum of three terms, A0~i~,D~p and A’°’~, corresponding to the orbital, spin-dipole, and contact term of the Hamiltonian, respectively. Below we derive computable expressions for the different magnetic terms and for the electric quadrupole term B, in terms of reduced matrix elements of one-particle tensor operators between the LS coupled configuration state functions in the wavefunction expansion. 2.3. Orbital terms

The orbital terms can, after recoupling of J and (f — 1), be written as 2~ 2f+1 1/2 A~’= ~a + 1) ~ ~J,k.cfckw(LJs~1f; fLk) (y~L 3 II ykLkSk), 1S~ IIE1l~’~(i) r1 A~... 1=

~

3 II7kLk~~k).

V

j,k ~cJckW(LJSJ1(J—

1); fLk)

(y1L~S1IILIW(i) rr

(24) (25)

Using an extended version of the program TENSOR [9,10],originally written by KoDb, the reduced matrix element can be calculated as (y~L

1~(i) r13 II YICLkSk>

<

1)(i) r 3 I ykLkSk)

v(pJok)

1S~ II ).l~

71L1S1 II ~1~

1’’

=

(ia.

II

II lcTk)(flP.1P. I r’3 I nO,klO.k),

~v(p 1p1) (l~~ 111(1)11

Pi *

3 I n~J,,), P

I r’’

~

(26)

0k’

(27)

1=

where p and er are the different orbitals. The program TENSOR has been adapted for the use of nonorthogonal orbitals [11]. In case of nonorthogonality the one-electron active radial integral, (n~l~ I r3 n,,l~),is to be multiplied with a number of overlap integral (n,hl,L I ~ (n~o,,I ~ The present version of TENSOR allows no more than two overlap integrals to multiply the radial integral [11,12].

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2.4. Spin-dipole terms

The spin-dipole terms can, after recoupling of f and (J

130(2J+1)J

2 j~

a

)

A~~”

{2 ECJCk Lk

j,k

f(J+1 130 11~/~2

21g5[ 2

Adip ,1-1

L

_____________

21g5[7j

i,k ~CiCk~

~I

1 5k

i) fky~L

(21)

.

1S1II~(CS) (z) r, f ~

5k

~2 Lk

1), be written as

Si

(L~ Sj

a

—

1

l}~~L

1

IIykLkSk>, (28)

20(i)rI3IIykLkSk>.

1~II ECs)~

(29) The reduced matrix element can be calculated as (21)

(y 1L1S1

II ~

(Cs)

.

(i)

II )‘kLkSk>

r

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

=

(ia. II C~

V(P10k)

(30)

Pi*0k,

20(i)r3 II ykLkSk>

<

71L1S1 II ~ (Cc)~

=

Ev(p~p

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

3)(ia. II C~

p 1

0k•

(31)

Pj

2.5. Contact ter~ns

Using the relation 4m~r26(r)= 6(r) between the three- and one-dimensional delta functions, the contact terms can, after recoupling of f and (f — 1), be written A~ont=a2 p.~2g

~1 26( r.) II

51 2f+ 1

1/2

X
ECJCk( 1) W(L1S1f1, fSk) j,k ~‘kLkSk),

(32)

a2~ A~~_1= ~

12g5 1 ~CJCk(l)~W(LJSJ(f

1)1; fSk)

26(r

x (y~L3S1II ~sW(i) r1

1) I ykI~~kSk>.

The reduced matrix element can be calculated as 26(r,) II ykLkSk) = v(pfcrk)(s K7~L1S1I ~s”~(i) r[ <

2&(r,)

73L1S1

II Es~’~(i)r1 I

II ykLkSk>

=

~v(p

(33)

sw II s)(nl~.lp

26(r)

1II r’’

26(r) I ~

1p1) (s II s”’ II s)(n~1l~1 Ir

P

I nO.klUk),

í

* Uk,

(34) Pj

=

elk.

1

(35)

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The radial matrix element is expressed in the radial starting parameter AZ(nl),

I n~l~)

(n~l~ I r28(r)

r

=

rLo

=

61

, 06~~,0AZ(n~/~) AZ(n~,l~).

(36)

2.6. Quadrupole ter,ns The quadrupole terms can, after recoupling of f, (f — 1), and (f — 2), be Written 4J(2J— 1)(2f+ 1) (f+ 1)(2J+3)

B1=

1 ff

)

1/2

2~(i)r13 IykLkSk), ~clckW(LJSj2f;

fLk)
1)1~’2

)

B

1,11= _Q~( ~

2~(i)r13IIykLkSk>, (38)

~c~ckW(LJSj2(J—1); fLk) (yJLJSJIIEC~

B

1,12 = —Q~4-[J(J

—

1)(2f— 1)J 1/2 ~CICkW(LJSJ2(J —2); fLk) 2)(i)r13IIykLkSk>.

(39)

X(y~L1S1II~C~ The reduced matrix element is calculated as 3II Yk’-’kSk>

=

2~(i)r13 II ykLkSk>

=

2~(i)r

<

71L1S1 II ~C~

Ky~L

V(pjelk)

1’’

1S1II ~C~

(la. II C~2~ II lO,k)(np.l

~v(p

2~ II l~

1p1)(/~. II C~

I r3 I

‘~01J0k)’

3I~ 1)(n~./~. I r’

~ * ~k,

(40)

p 1

=

elk.

(41)

2. Z Hyperfine parameters in pure LS coupling For small atoms, in which LS coupling is valid to a good approximation~the hyperfine constants A and B are often expressed in terms of the totally electronic parameters a1, ad, a~and bq, 3 I YLSMLMS>, (42) a1 =
bq

=

where ML


1) I YLSMLMS),


3IyLSMLMS),

(44) (45)

rI

L and M 5 = S.

Explicit relations between the hyperfme constants A and B and the parameters above are given in refs. [10,131.

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2.8. Nonorthogonal orbitals

A totally unrestricted nonorthogonal scheme would lead to a very elaborate computer program. Therefore the present version of TENSOR [ii] imposes the following restrictions on the orthogonality of the orbitals in the left hand side configuration, (Pd, and the right hand side configuration, ~k, of the reduced matrix element [12]. 1. The orbitals of a single configuration are mutually orthogonal. 2. There are at most two subshells in (P~containing spectator electrons whose orbitals are not orthogonal to orbitals in ~ 3. If all the spectator electrons with nonorthogonal orbitals have the same /-value, then there are at most two such electrons in each of cP~and ‘1~k~ 4. KP, I ~Pk> = 6jk even if nonorthogonal orbitals are included. If the user is trying to use expansions with configurations breaking the above restrictions, this program, as well as the other programs in the MCHF package, will stop and give an error message, saying that the nonorthogonality is set up incorrectly. 2.9. General form of the expressions The contribution to the different A factors from a pair of configurations j and k can in its most basic form be written as CA— 1-cfck

~

coef(p,

U)

rad(p,

U)

(i.tI,,s) (vli’ )

,

(46)

the most basic form of the B factor CBQCJCk

~ coef(p, U) p,o~ ,/~,v

rad(p,

I

U) (~e

,x!)?hl(p

I

,i~)’~2,

(47)

where the coefficients and the radial matrix elements rad(p, u) are determined from the expressions above. Most often the A and B factors are expressed in the experimentally convenient unit MHz. If the nuclear magnetic j~, is given nuclear magnetons and the nuclear =quadrupole moment, Q, in 2), moment, the conversion factorsinto MHz are CA = (1ja2 )2cRome/Mp 47.70534 and CB = 2cR~ barns (1028 m 10~ m2/a~= 234.9647.

3. Program structure Many of the subroutines used in this program are included in the MCHF program package [1]. Below

we describe the subroutines that are specific for the hyperfine code. 3.1. New subroutines READWT

Reads the weights of all configurations in the wavefunction expansion. If the weights have been obtained from a f dependent CI calculation they are read from the file Kname>.j otherwise they are read from (name).l.

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LSJ

Determines L, S, L + S and I L

—

S I for a configuration in the wavefunction expansion.

All the subroutines below are related to a pair of configurations j, k. LSJFACT Calculates the L~,S1. (37—(39).

5k

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

Lk,

TENSOR2 To fit the structure of the hyperfine program the subroutine TENSOR has been modified. TENSOR2 outputs the following information: IIRHO(m) the interacting shells in the initial configuration; IISIG(m) the interacting shells in the final configuration; NNOVLP the mimber of overlap integrals; MMU(m) and MMUP(m) the shells in the first overlap integral; NNU(m) and NNUP(m) The shells in the second overlap integral; IIHSH and VVSHELL(m) are defined as follows, —

—

—

—

—

IIHSH


1L3S1 II EQ~~i II Yk’-’kSk>

~

=

VVSHELL(m) (p

IIQ(A) II U),

(48)

m=1

where the two cases p * U and p = U are handled on the same footing. In case of nonorthogonality, the reduced one-electron matrix element is to be multiplied with at most two overlap integrals. MULTW7’

Multiplies the weights of configuration j and k. ORBITAL When p3

* ~k,

ORBITAL calculates the following quantity in eq. (26)

V(pjok) (i~~ 111(1)1110k). 1~Tk’the

(49)

following quantities in eq. (27) are calculated:

When p3 = v(p 1p1) (l~~ 11(1) I l~).

(50)

DIPOLE

When p3 * elk. DIPOLE calculates the following quantity in eq. (3)): v(pielk) (l~~IIC2IIc5)(sIIs~IIs). When p3

=

~k,

(51)

the following quantities in eq. (31) are calculated: 2~ II l~

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

1)(sII ~~1) I s).

(52)

CONTACT When p3 * elk, CONTACT calculates the following quantity in eq. (34): v(p)elk)(sIIs~’~IIs).

(53)

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the following quantities in eq. (35) are calculated:

v(p~p3)(sIIs”~IIs).

(54)

QDRPOLE

When p

* elk,

v(pfok)(1P

QDRPOLE calculates the following quantity in eq. (40): 2~ II ‘O’k)~

(55)

II C~

When p the following quantities in eq. (41) are calculated: 2~ II la.). v(p1p1)(l~.II C~ 3

=

elk,

(56)

RADIALI

Calculates the one-electron active radial integral (n~l~I r3 In 010). In case of nonorthogonality the overlap integrals are calculated and multiplied with the active radial integral. RADIAL2

26(r) 1n

Calculates the one-electron active radial integral (n~l~I r 010). In case of nonorthogonality the overlap integrals are calculated and multiplied with the active radial integral. PRINT Dependent on the responses to the prompts in the beginning of the program, this subroutine prints out parameters from the matrix element calculations between configuration j and k. 3.2. New common blocks

A series of COMMON blocks arising from the original cfp packages and from Robb’s tensor program have been redefined in the write-up of the MCHF LIB COM, MCHF LIB ANG, MCHF RAD3 and NJGRAF libraries. The common blocks OVRINT and OVRLAP are related to the non-orthogonality feature and are described in the write-up of MCHFNONH [14]. -

-

-

COEF

5k’ f dependent coefficients in QUADF(n) the L3, S~, Lk, the different combinations of eqs.ORBF(n), (24), (25),DIPF(n), (28), (32),CONTF(n), (33) and (37—39), where contains n is a parameter ordering f and f’. J(n), JP(n) are used in the printout to give J and f’ as rational numbers. JJMAX, JJMIN are twice the and minimum f quantum numbers, respectively, for the wavefunction ‘P(y 1LSf) in eqs. (22) and (23). LL1, SS1, JJ1MAX, JJ1MIN correspond to 2L3, 2S3, 2 I L3 + S~ I, 2 I L3 S~ I. LL2, SS2, 2Lk, 25k’ 2 I Lk + 5k I, 2 I Lk 5k I. JJ2MAX, JJ2MIN correspond to

maximum

—

—

TENSOR

This common block holds the output from TENSOR2. HYPER

Quantities in this common block are closely linked to the output from TENSOR. After the subroutine ORBITAL has been called the quantities in the common block will be defined as follows: NHY = IIHSH,

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= 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)),

\THY(m)

JNUHY(m) = IAJCMP(INUHY(m)), JNUPHY(m) = IAJCMP(INUPHY(m)), where m = 1,..., IIHSH.

In the same way the quantities above can be inferred from eqs. (51)—(56) after the subroutines DIPOLE, CONTACT and QDRPOLE have been called. PRIN TA This common block supplies the subroutine PRINT with many of the quantities needed for the printout. For every pair of configurations j, k the following quantities are supplied: ORB = ORBF(n)VHY(N), where VHY(N) is the output from the subroutine ORBITAL. DIP = DIPF(n)VHY(N), where VHY(N) is the output from the subro~.itineDIPOLE. CONT = CONTF(n)VHY(N), where VHY(N) is the output from the subroutine CONTACT. QUAD = QUADF(n)VHY(N), where VHY(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 integration has been performed and 0 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 and IDOPRI are parameters controlling the write-out.

4. Input data To describe how the calculations are to be done, a few prompts for data have to be answered. 1. Name of state

This program assumes the naming conventions described in the paper on the atomic-structure package [1], and the (name> of the state has to be specified. 2. Type This prompt asks you to specify the different constants that are to be calculated. The options are • only diagonal A and B factors. • both diagonal and off-diagonal A and B factors, • only the coefficients of the radial matrix elements. For the last option all needed input data is in the configuration input file (name>.c.

410

P. Jönsson et aL

/

Computing magnetic dipole and electric quadrupole hyperfine constants

3. Hyperfine parameters The hyperfine parameters a1, aa, a~and bq can be printed out. This is only meaningful for pure LS terms. 4. Input This prompt inquires if the wavefunctions expansion has been obtained from an MCHF or CI calculation. In case of an MCHF calculation, configuration weights are read from he configuration input file (name).c. If the expansion has been obtained from a CI calculation, weights are read from the eigenvector file (name>.j or (name>.l, dependent on whether the CI Hamiltonian is f dependent or not. 5. Number

In general the eigenvector files (name).j or (name>.l from a CI calculation contain wavefunction expansions for many different states. Thus, the particular states for which the A and B factors are to be calculated have to be specified. Suppose, in a f dependent CI calculation, that the expansion in the file (name>.c consists of the following four CSFs ls( 2)

2p( 1)

iSO ls( 2s1 1)

2P1 2s( 2S1 1)

ls( 1) 2S1 ls( 1) 2S1

2s(

1) 2s1 2s( 1) 2S1

2p( 2P 1) 2P1 2p( 1) 2P1 2p( 1) 2P1

iS

2P

3S

2P

2S

4P

and that the hyperfine structure is to be calculated for the f ls( 1) 2S1

2s( 1) 2S1

=

~ and f = ~ states of the third CSF

2p( 1) 2P1

3S

2P

Then, by replying 3 to the number prompt the program searches for the two wavefunction expansions (one for the f = f state and one for the f = ~ state) in the file Kname).j, having the third CSF as the dominant component. If no state has the third CSF as the dominant component the program stops and an error message is written out, telling the user how to proceed. This is also the case when more than one state for a given f value has the third CSF as the dominant component in the expansion. 6. Print-out

In a calculation of the A and B factors, values of the A0~b,AdIP, Acont, and B terms are written to the file (name).h. If in this case a full print-out is requested, values of coefficients and radial matrix elements as well as the contribution to the different A and B factors are printed for every pair of configurations.

7. Tolerance for printing In the case of a full print-out it is possible to set a tolerance for printing. If the contribution to a term is less than the tolerance, then it will not be printed. 8. Nuclear data

The last prompt asks for the following nuclear data: • 21, • j.~,in nuclear magnetons, • Q in barns.

P. Jönsson et aL

/

Computing magnetic dipole and electric quadrupole hyperfine constants

411

A recent compilation of experimental data on nuclear magnetic dipole and electric quadrupole moments for ground and excited states can 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 the wavefunction expansion, is written out on the screen, indicating how far the program has executed. Output from the program is written to a file (name>.h. If a full print-out is requested, the program outputs the following data from eqs. (46) and (47) for a pair of configurations j and k. 1. f f’ the J values for which the matrix element has been calculated. 2. Weight — the configuration weight clck. 3. Coeff — the coefficient coef(p, el). 4. Radial matrix elements — values of the radial matrix elements rad(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 factors from this pair of configurations. .

—

—

6. Examples The Test Run Output contains full print-outs from two different hyperfine structure calculations of 2p 2P in 7Li. Diagonal A and B factors are calculated as well as the hyperfine parameters a1, ad, a~and bq. The wavefunctions have been obtained from a HF and a non-orthogonal MCHF calculation, respectively. Nuclear data has been supplied by the user. Acknowledgements The authors wish to acknowledge the assistance given to this project by K.M.S. Saxena who shared an early version of a hyperfine program with us, based on parts of Alan Hibbert’s CIV3 program. The present research has been supported by the US Department of Energy, Office of Basic Energy Science, and by the Swedish Natural Science Research Council. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

C. Froese Fischer, Comput. Phys. ~ommun. 64 (1991) 399. C. Schwartz, Phys. Rev. 97 (1955) 380. 1. Lindgren and A. Rosen, Case Stud. At. Phys. 4 (1974) 93. J.S.M. Harvey, Proc. Roy. Soc. London A 285 (1965) 581. H. Orth, H. Ackermann and E.W. Otten, Z. Phys. A 273 (1975) 221. K.H. Liao, L.K. Lam, R. Gupta and W. Happer, Phys. Rev. Lett. 32 (1974) 1340. C. Froese Fischer, Comput. Phys. Rep. 3 (1986) 275. C. Froese Fischer, Comput. Phys. Commun. 64 (1991) 473. W.D. Robb, Comput. Phys. Commun. 6 (1973) 132; 9 (1985) 181. R. Glass and A. Hibbert, Computer Phys. Commun. 11(1976)125. C. Froese Fischer, M.R. Godefroid and A. Hibbert, Comput. Phys. Commun. 64 (1991) 486. A. Hibbert, C. Froese Fischer and M.R. Godefroid, Comput. Phys. Commun. 51(1988) 285. A. Hibbert, Rep. Prog. Phys. 38 (1975) 1217. A. Hibbert and C. Froese Fischer, Comput. Phys. Commun. 64 (1991) 417. P. Raghavan, At. Data Nuci. Data Tables 42 (1989) 189.

2PO

case #I.

===I===>Obtain Radial Functions

=r=====>Obtain Bnesgy Exprassion

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

.==....>

cfg.out 1il.c

BYPERFIXE ==============ii======

===I=================

=======>Move cfg.outto 1ii.c =s==s==>kwe sfn.out to 1il.v I======>liyperfine structure

lame of state >lil Indicatethe type of calculation 0 => diagonalA and B factorsonly; 1 => diagonaland off-diagonalA and B factors; 2 => coefficientsof the radial matrix elementsonly: >o iiypsrfins parametersal,ad.acand bq ? (Y/P) 'J Input from an HCEF (H) or CI (C) calculation?

>> IV afn.out 1il.w >> his calculation

>> mv

Do you wish to continuealong the sequence? >n

‘J

>> mchf ATOH, TERM. 2 in MRMATLT(A.A,F) : >Li,ZP,J. Thor4 are 2 orbital8as folloos: 1s 2pl Enter orbitalato be varied: (A11.,IOIE,SOIIB,IIT=,comna delimitedlist) >ALL Default electronparameters? (Y/I) 'I Default values (IO,RJ3L,STRDIC) ? (Y/I) 'y Default values for other parameters? (Y/II)

>a nor&h FULL PRItiT-OUT ? (Y/X) >Il Au. IITSruCTIoYs? (Y/I) 'J

is( 2) 2pl( I) PPI IS0

>> cat cig.inp

TEST RUN OUTPUT

liyperfinostructure calculation

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

-0.220196

Bo4Ez)

Adip0lliz) 16.192707 -1.619271

Spin-dipoleterm 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 Quadrupoleterm 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

llhfslconfig 1 ) Orbital term matrix element J J' weight coeii matrix element values l/2 l/2 1.000000 2.666667 0.0586 312 J/2 1.000000 1.333333 0.0688

(config

1il.c Tolerancefor printing 0.000000naz Jnclsar dipole moment 3.256000n.m. nuclear quadrnpolemoment -0.040000barns 2*1= 3 Li 2P -7.3650696

>> cat 1il.h

COIFIfXJFtATIOE 1 ( OCCUPIgDORSITALS=2 ): ls( 2) 2pl( 1) CouPLIllG scIiEMB: IS0 2Pl 2PO Ja= 1

1s 2pl

THWU?,ARE 2 ORSITALSAS FOLLOW:

STATE (YITE 1 COllFIGURATIDIS): ____________________~~~___~~~__

The configurationset

htll print-out ? (Y/I) 'Y Tolerancefor printing (in ItlIz) so.0 Give 2*1 and nuclear dipole and quadrupolsmoments (in n.m. and barns) >3,3.266,-0.040

h

l/2

3/2

J'

l/2

3/2

l/2

3/2

J

l/2

J/2

Spin-dip

NBz

-1.61927

16.19271

in

**

12.

==I====>Cow old pin. to r,ewvfn.

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

>>cp 1il.s vfn.inp

>> m&f

ATOW, TERM, 2 in FORJIAT(A,A,F) : >Li,2P,3.

===t===>Obtain Energy gxpression

2PO

3so

Calls

-0.02343

bq

6.46771

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

=I=====>

0.00000

ac

0.00000

32.36666

Total

>a noah FDLL PBIIT-ODT? (Y/I) Xl ALL.IITnACTIOYS ? (Y/Y) 'Y

2PO

is0

*

-0.01171

0.06867 l

ad

al

Cont 0.00000

Eyperfineparametersin a.u.

-0.22020

0.00000

quadnp010

B iactors in naz

6.08698

16.17386

Orbital

factors

is( 2) 2pl( I) 2Pl 2PO is0 1r( 1) 2s2( I) 2p2( 1) 2Pl 2Sl 2.31 ls( I) 2.3( I) 2p3( 1) PSI 2Pl 2Sl

>> cat cfg.inp

J'

J

A

l*

====I==>k.vo cfg.ontto li2.c ===r===>nave ofn.out to li2.s =====r=>Elyparfine Itructure

COIFIGURATIOl 1 ( OCCUPIED ORBITALS-2 ): ls( 2) 2pl( 1) IS0 CODPLIYG SCBEHE: 2Pl

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

THERE ARE 6 ORBITALS AS FOLLOWS:

STATE (WITH 3 COIFIGURATIOIS): ____________________~~~~~~~~~~~

The configurationset

lame of state >li2 Indicatethe type of calculation 0 => diagonal A and B factors only; 1 => diagonal and off-diagonalA and B factors; 2 => coefficientsof the radial matrix element8 only: >I Eyperfinsparametersal,ad,acand bq ? (Y/I) >]I Input from an IICEF(II)or CI (C) calculation? >Xl Full print-out? (Y/X) 'Y Tolerancefor printing (in UEz) >I.0 Give 221 and nuclear dipols and qudrupole moments (in n.m. an&barns) >3.3.266.-0.040

===================i= HYPBRFIlE =======1=1=55=======

>> w cfg.out li2.c >> mv wfn.out li2.u >> hfa calculation

6 orbital8as folloss: 1s 2pl 282 lp2 283 2p3 Enter orbitalato be varied: (ALL,IO~,SO~,IIT=,co~a delimitedlist) >2pl,2~2.?p2,2~3,2p3 Default electronparameters? (Y/I) 'Y Default values (IO.RKL,STROIG)? (Y/I) 'lr Default values for other parametera7 (Y/B) 'J Do you wish to continuealong the sequence? >IL

There are

Byperfinostructurecalculation

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

A factors in UEz

**

Fermi-contactterm matrix element A?.(Zs3)AZ( 1s) <2p312pl>**1 J J' weight coefi matrix elementvalues 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

3lhfslconfig 1 )

Acont(8Bz) 5.739065 -6.739065 5.739065

Adip(llEz) 16.702207 -1.670221 -2.087776

Spin-dipoleterm matrix element J' weight c2p11n**(-3)I~Lx element ialuas J COOif l/2 i/2 0.999791 2.669759 0.0604 3/Z 3/Z 0.999791-0.266076 0.0604 3/Z l/2 0.999791-0.333720 0.0604

(cordig

Aorb(Kiz) 16.682863 8.341432 4.170716

llhislconfig 1 )

Orbitalterm eplIR**(-3)I2pl> matrix element J' weight coeii J matrix element values 0.0604 1/z i/2 0.990791 2.666667 3/Z 3/Z 0.999791 1.333333 0.0604 3/Z i/2 0.099791 0.666667 0.0604

(config

li2.c Tolerancefor printing 1.000000If82 Iucleardipole moment 3.266000II.=. Iuclearquadrupolemoment -0.040000barns 2*1= 3 Li 2P -7.3667346

>> cat li2.h

COIPIGURATIOI 3 ( OCCUPISDORBITALS=3 ): la( 1) 2s3( I) 2p3( 1) COUPLIIGSCIIEHB: 251 2Sl PPI 3so 2PO Ja- 1 Ja= 2 Ja= 3

COlIPIGURATIO82 ( OCCUPISDORBITALS=3 ): ls( 1) 2a2( 1) 2p2( 1) COUPLINGSCEEIIE: 251 2Sl 2Pl is0 ZPO

2PO

1/z

1/z

3/Z

1/z

3/Z

J'

J

3/Z

1/z

3/Z

3/Z 3/Z

1/z

J'

l/2

J

-2.09768

-1.67816

16.70147

Spin-dip

l

*

11.45731

-0.02440 -0.24866 -0.01214

0.06101

**

bq

ac

ad

al

13.67119

-4.71233

-11.45731

Total 46.08604

cant 11.45731

Eyperfineparametersin a.u.

-0.04966

-0.22935

0.00000

Quadrupole

B factors in 88~

4.21167

8.42313

16.64626

Orbital

1 0'

B 3

9

2

3

%