MCBP - A program to calculate angular coefficients of the breit interaction between electrons in the low energy limit

Computer Physics Communications 18 (1979) 245—260 © North-Holland Publishing Company

MCBP A PROGRAM TO CALCULATE ANGULAR COEFFICIENTS OF THE BREtT INTERACTION BETWEEN ELECTRONS IN THE LOW ENERGY LIMIT —

N. BEATHAM, I.P. GRANT, B.J. McKENZIE Theoretical Chemistry Department, University of Oxford, I South Parks Road, Oxford OXI 3TG, UK

and N.C. PYPER University Chemical Laboratories, Lensfield Road, Cambridge, CB2 JEW, UK Received 9 March 1979

PROGRAM SUMMARY Title of program: MCBP—BREIT ANGULAR COEFFICIENTS

ACWE

IN JJ-COUPLING MCP75

Catalogue number: AAAL

14 (1978) 311 11(1976) 397 14 (1978) 312

Program obtainable from: CPC Program Library, Queen’s

Keywords: atomic, configuration interaction, Racah, frac-

University, Belfast, N. Ireland (see application form in this issue)

tional parentage, recoupling coefficients, Slater integrals, complex atoms, relativistic, Dirac equation, wave function, retarded interaction, Breit, fl-coupling, theoretical methods

Computer: IBM 360/195;Installation: Rutherford Laboratory, Chilton, Didcot, Oxfordshire

OX1 1 OQX, UK

Nature of physical problem

Overlay structure: none

The retarded interaction between a pair of electrons makes a non-negligible contribution to atomic energy levels especially in processes such as X-ray emission and absorption and Auger spectroscopy, involving inner shell holes. This program handles the algebraic part of this calculation using the formulation of Grant and Pyper [11and lists the coefficients of radial integrals in a form suitable for use with the wavefunctions produced by multiconfiguration Dirac—Fock programs, such as that of Grant, Mayers and Pyper [2]. The input to this program is identical with that used in MCP75 [3,4].

No. of magnetic tapes used: none

Restrictions on the complexity of the problem

Operating system: OS/360, HASP Programming language used: FORTRAN IV High speed storage required: 176 kbytes (execution only) No. of bits in a byte: 8

The program assumes that all open shell configuration state functions are formed from the same orthonormal set of basis orbitals. The number of electrons in a shell havingj > is restricted to be not greater than 2 by the available c.f.p. tables. The present version allows up to 10 open or closed shell orbitals and 30 configurational states with up to 6000 coefficients. These numbers can be changed by the user. It is advisable to ensure that the library subprograms AAHD and ACRI have been converted to use double precision floating point arithmetic when implementing the package on IBM machines.

Other peripherals used: card reader, line printer, disc file (if available)

No. of cards in combined program and test deck: 1974 Card punching code: EBCDIC CPC Library subprograms used: Cat. no. Title AAHD ACRI

NJSYM CFPJJ—CFP

Ref. in CPC 8 (1974) 151 4 (1972) 377

245

246

N. Beatham eta!. /.4ngular coefficients of the Breit interaction

Typical running time This depends on the problem. The two test problems, one with 3 orbitals and 10 CSFs, the second with 7 orbitals and 7 CSFs, used 2.8 s of execution time on the IBM 360/195.

[2] l.P. Grant, D.F. Mayers and N.C. Pyper, J. Phys. B9 (1976) 2777. [3] 1.P. Grant, Comput. Phys. Commun. 5 (1973) 263. [4] I.P. Grant, Comput. Phys. Commun. 11(1976) 397.

References [1] I.P. Grant and NC. Pyper, J. Phys. B9 (1976) 761.

LONG WRITE-UP 1. Introduction There has recently been growing awareness that relativity affects valence electrons as much as core electrons. This has led us to develop a series of programs for handling the angular momentum algebra involved in dealing with general open shell problems in f/-coupling [1—3].We have used the output from these programs as input to a multiconfiguration Dirac—Fock program to study a wide range of problems [41. We intend to make this last program available through CPC in due course. One important omission from this suite of programs is one to handle the Breit interaction. Its omission is the biggest single source of error in atomic structure calculations of inner shells and it is necessary to include it for high precision calculation of X-ray and Auger energy levels [5], for example. The present contribution provides means to calculate the coefficients of radial integrals in matrix elements of 11-coupled open shell CSFs with a Breit interaction in the low energy limit [6]. This is expected to be adequate for many purposes, though there will almost certainly be some problems for which the more general energy-dependent form of the Breit operator is needed. This has been left for future development.

2. Outline of method 2.1. The basic formula The theory underlying the program is set out in ref. [6] and it is only necessary to quote the results of that paper. The interaction concerned, in Hartree atomic units, is Bo(r1, r2)

—~1 ~2

—

-

V2)R,

(1)

where R = r1 r2j is the distance between the two electrons at positions r1, r2 and ~ are their Dirac operators. We wish to evaluate its matrix elements with respect to a basis of configuration state functions —

,~

IT,M>=~T1,T2,...,TNw;ctJM),

(2)

!T~,M1)=j7~(v1,w~,Jj),M1

(3)

where

is a state ofN1 equivalent electrons coupled to give total angular momentum quantum numbers J~,M1, seniority number v1 any quantum numbers needed to resolve further degeneracy are denoted by w. The angular momenta ~Nw are coupled in standard order ((...(...((Ji,J2)Xl,J3)X2...)Xf...)XNW_2,JNW)J,

where X1 X2 ,

(4)

XNw_2 are intermediate angular momenta whose values are restricted by the usual triangle con-

247

N. Beatham et al. / Angular coefficients of the Breit interaction

ditions. The matrix element of Breit interaction, summed over all N pairs of electrons, N general form, see ref. [6] eq. (31), (T, MI i
N,

=

~ N;, has the

112 = ~JJ’öMM’

A,B,C,D ~

(—l)~{NA (NB— öAB)NC(ND

—

6CD)]

(TA {ITA/A)(TB{I TBJB)(TC/CI} T~)(TDjDI}~

11

X

iA,B,C,D —

= ~

Ce(l

—

~J~,JfCd(1+~SAB~CD)1[JA,jD]h/2XK(A,B,C,D) K

~AB)(i

l/2X~C(A,B,D, C~}.

~CD)[fA,iC]

—

(5)

Here,A, B, C, D label the “active” shells of the matrix element; the sum, T, is taken over all parent states TA, TB, TC, TD of these shells that can contribute; X”~(A, B, C, D), to be defmed below, is the effective interaction strength for the Breit interaction (1); and Cd, Ce are recoupling coefficients for direct and exchange contributions, Cd

= (...

(J’a/B) ~B

Ce

= (...

(J~fB)JB... (J~

...

~

(ICK)/A ) J~...;

1(1 CK) IA )JA

Il

...

...; XI ...

~JD~K/B)/D)J~

(JDJD)JJ

...

...

~Jck)J~’ ...;

((JC(KfB)/C)J~...;

X’),

X and X’ being the coupling schemes of Tand T’ as defined in (4). The symbols (TI) T, f)~ (T, / {IT) denote coefficients of fractional parentage. The phase ~ is defined by (7)

C+1

A+1

where N~=Nx—4—~~~,

NN~—~~C—~j.,

and we have assumed that the shell labels are ordered so that A ~ B, C ~ D. The effective interaction strength for the Breit interaction may in general be written, ref. [6], eq. (22): K+ 1

XK(A,B, C,D)=(_1)K(/AlI<~~llfC)(jBlIC~)II/D) ~ vK—1 x ~

S~K(A,B,

C, D)S~(A,B, C, D).

(8)

Here

(/ IIC~IIj’)=(—l)

/2[//~]1/2

(I

K

_~)

(9)

and H’(K, ,~‘, v) is a parity selector, ll’(~,~’,v) I 0

if l+l’+visodd, if

1+1’ +vis even,

(10)

where 1 =/ ~a,,~= —(j + ~)aand a = ±1 distinguishes the two types of spin—orbit coupling of a Dirac central fIeld orbital. In general, there are eight distinct radial integrals, S~(A,B, C, D), for each choice of A, B, C, D and v, —

248

N. Beatham eta!. /Angular coefficients of the Breit interaction

though it is desirable to take account of symmetry to reduce this number in certain cases. Define first the mixed charge density (11)

pAC(r)=PA(r)Qc(r);

we draw attention to the lack of symmetry in the suffixes A, C. Then if

=1 f

S°EACIBD]

pAcfr)UV~r, s)pBD~s)drds,

Uv~,s)r0/svfI

ifr
(12)

with

ifr>s,

(13)

the eight integrals, S~(A,B, C, D), arise from permutation of the shell labels as shown in table 1. With this convention, the coefficients S~~K(A, B, C, D) are given in table 2. Collecting terms, we see that each matrix element (5) can be written as a sum over radial integrals of the form N

(T,M]

~Bo(rj,rj)IT’,M’)=~JJ’~MM’

~

i
A,B,c’,D

~

t~’°(A,B,C,D)T~’°(A,B,C,D),

(14)

v ~s1

where m is an integer distinguishing case for which the full 8 integrals of table I need not be listed. Case m = 1. This is the general case; Pm = 8 and there are 8 distinct nonvanishing terms in (14) for each set of active sheilsA, B, C, D and each value of the tensor index v. Clearly T~(A,B, C, D) Case m

=

2, Pm

=

(15)

= S~(A, B, C, D).

1, A

=

C, B

D (and similarly for B

T~(A,B,A,D)R0(A;BD)ff

=

D; A ~ C).

In

this case

pAA(r)U,,(r,s)~BD(s)drds,

(16)

where U~(r,s) = U0(r, s)

+ U~(s,r),

PBD(s)

=

PB(S)QD(S) + QB(s)PD(s)

and v is an odd integer only. The simplified version of eq. (8) may be written XK(A, B, A, D) su(A;B,D)Rv(A;BD)opK,

Table 1 The definition of S~(A,B, C, D) in terms of SV[~Ikl] [ij~kll

1 2 3

4

[ACIBDJ [BDIAC] [CA1DB] [DBICAI

[z~Ikl] 5

IACIDB]

6

[DBIAC] [CAIBD]

7 8

[BDICA]

(17)

N. Beatham et al. / Angular coefficients of the Breit interaction

249

Table 2 The coefficients

s~K(A,B, C,

D); non-zero contributions arise only from values of v for which ‘A

+

1C + v, ‘B +

+

v are both odd

a) K = v ~K(A

B, C, D)

(~A+ ~C)(KB+ I
=

+

1)

p

=

1, 2

8

b) K = v ±1 p

Kv+l

Kv—1

p

Kv+1

Kv—1

1

(K+H)(b+EH’) (K+H’)(h+cTH) (K—H)(b— EH’) (K — H’)(b — Eli)

(b+cll)(H’—K—l) (b+cH’)(H—K— 1) (b — cH)(—H’ —K—i) (b — cH’)(—H — K — 1)

5 6

—(K+H)(b—EH’) —(K—H’)(b+Eli) —(K—H)(b+~H’) —(K + H’)(b — Eli)

—(b+cH)(—lf—K—1) —(b — cH’)(H—K —1) —(b —cff)(H’ —K—i) —(b + cH’)(—H—K —1)

2 3

4

7

8

Definitions: ~

H’=KJJ-KB;

b = (v + 2)/(4v b = (v

—

+

1)/(2v

+

1)(2v + 2),

v ~ 0;

(v + 2)/2v(2v

+

1),

u ~ 1.

E~—(v

2),

1)/(4v + 2),

c

—

where ?(A;BD)

(1AIIC~~ II/A)(1BIIC~II/D)4KA(KB~

1).

Casem3,pm 1,AC,BD. This can be expressed in terms of the magnetic F-integrals, ref. [7], eq. (8.15) T(A,B,A,B)=F~(A,B)=

~R”(A;B,B)ff

pAA(r)U~(r,s)pBB(s)drds.

(18)

The effective interaction strength is XK(A,B,A,B)_~SV(A;B)F~ag(A,B)5V,K,

where u is an odd integer only and where sV (A; B) Casem 4,Pm 4,A D,BC,A ~B.

(19)

= 2?(A; BB).

Table 3 Coefficients of magnetic G and H integrals Kv—i -(K - li)(b- ER)

-(b

2(bK—EH2)

—2[b(K+i)+cH2]

t~’f’(A,B)

—(K+H)(b+EH)

—(b+clf)(H—K— 1)

~~“~(A B)

211(1,

Definitions: H = KB

—

~‘f(A,B)

—

c7C)

-

—2H[b

cH)(-H

+

-

K

-

c(K + 1)]

KA, b, ë b, c and parity selection as in table 2.

1)

250

N. Beat/zam Ct al.

/ Angular coefficients of the Breit

interaction

The effective interaction strength may be written XK(A, B, B, A)

= (_l)K+iA~JB(jA

Id~IIlB)2

~

~

,K~ B) G~g~A, B)

p=K±1 y=—I

~ s~(A,B)T~(A,B,B,A).

+~~K(A,B)H~ag(A,B)}=

(20)

I)

In the notation of(14), T~’°(A, B, B, A)

= 2S°[BAIBA]

= G~’(A,

B),

T~~V(A,B,B,A)SV[ABIBA]+S0[BAIAB]G~g(A,B), T~’°(A, B, B, A)= 2SV[ABIAB]

=

G~(A, B),

T~(A,B,B,A)=SV[ABIBA]_SV[BA lAB] (ref. [6],

eq. (26), (27)). The coefficients ~

H~nag(A,B),

(21)

(A, B), ifK (A, B) are given in table 3.

2.2. Implementation of the basic formula in MCBP The program computes and lists groups of coefficients t~mv(A,B, C, D) (p = I Pm). These are stored sequentially in the array XSLDR, omitting groups, all of whose Pm coefficients vanish, unless this happens because the appropriate recoupling coefficient vanishes. The program requires the NJSYM recoupling coefficient package [8] and the fractional parentage coefficient table search [1] together with 15 unmodified subroutines from MCP75 [2]. These subroutines are: CARDRH, SETUP, VIJOUT, CFIN, CFOUT, ORTHOG, OCON, TMSOUT, LTAB, BKDATA, MODJ23, GAM, MUMDAD, D3J, ITRIG.

One further subroutine, SETJ, which sets the input arrays to the NJSYM recoupling package, has been slightly modified to ensure consistency with formula (5). The uncorrected routine gives a phase error in this calculation because of the way the Breit interaction couples large and small components of the wave functions but makes no difference in MCP75. 3. Program structure and subroutines 3.1. MCBP master segment This drives the package. Its flow diagram, fig. l,is essentially the same as that of MCP75. The bracketed labels below refer to the numbered boxes in the figure. (1.0). The parameters IREAD, IWRITE may be set by the user at compilation time to the local-device numbers for card reader and line printer, respectively. The parameter NSTAT, which can be set by the user at compilation time, is used to construct an identifying label in the array ISLDR. Each entry in ISLDR relates to a coefficient t~°~(A, B, C, D) in the corresponding position in XSLDR. This entry is equivalent to p

+

NSTAT*(A

+

NSTAT*(B + NSTAT*(C

+

NSTAT*(D

+

NSTAT*ITYPE)))),

for which v, A, B, C, D and ITYPE can be extracted by modular arithmetic. ITYPE = I for m > I, ITYPE 2 for m = 1. All coefficients t~v(A, B, C, D), p = 1, ..., Pm share the same identifier. The number of coefficients and the arrangement of orbital lables suffice to identify the type of radial integral by which the coefficient must be multiplied.

N. Beat/sam et al. /Angular coefficients of the Breit interaction

[1.0

251

S.t device sod otoraga psr...t.r.

2.0

‘F

Read data and debag parameters

I

‘(

__________________________ 3.0 ThSOUT

I~1G4:i>

ar

of

Print table ]

—

tate

•~ ~N

as

inBLOCKDAT&

eetl ~j

Next problem CPDI — Read problem data CFOUT — Print preblem datal

1

5.0

Next cans 4 all pairs T,T’ SETUP — Define configuratiems T and T’ (_m~i~~pr

J5.IVIJOtP1’-PrintT,T’j


I

ORTHOG — Look for “Ovj~”

set

orthogonality ; LET 0 if found.

6.0

I~C0 — main calculation

forl

(11B

01T’)

17.0

SPEAK — outpnt resultel

~aT~)

Fig. i.

The array NDIR is used to point to the fIrst coefficient pf the set relating to a given matrix element in the arrays ISLDR and XSLDR. The number of non-zero coefficients is stored in NNLD. (2.0). The input and output of the package is controlled by parameters on two data cards. The punching conventions appropriate to the free format card reading subroutine CARDRH are described in ref. [2a], p. 274. Cols card

2

Label

Data

l-~4

5-~72

PARM BUG

NPROB NOUT IBUG1 IBUG2 IBUG3 IBUG4 IBUG5 IBUG6

where:

NPROB NOUT IBUG1

= number of problems to be run; =

output option to be selected;

= 0,

minimal printing, printing in MCBP, VIJOUT, ORTHOG, CFP, = 2, additional printing in VIJOUT, RKCO, BREIT;

= 1,

252

N. Beat/sam et al.

IBUG2

/ Angular coefficients of the Breit interaction

no printing, 1, c.f. ps in MUMDAD; IBUG3 = 0, no printing, = I, printing intermediate results in NJSYM, GENSUM [81; IBUG4 = 0, no printing, = 1, print table of quantum numbers of terms 0fJN using TMSOUT; =

0,

=

IBUG5

0, no printing, 1, printing intermediate results in BREIT, BREID; IBUG6 = not in use. (3.0). Subroutine TMSOUT see ref. [2a], p. 269. (4.0). Subroutines CFIN, CFOUT The first subroutine reads iii problem data, the second prints it out. The data punching conventions of CARDRH, ref. [2a], p. 274, are in use. =

=

—

Cols card

Label I 4

Data 5 -÷72 (or I

1

ICHD

ISR Z NW NCF INCOR

IWHD(J)

J NP(J) KAP(J)

—~

—~

72 for unlabelled cards)

2 }

NW+ I NW+2 I J NCQ(J,I) (JQS(K,J,I), K

2*NW+ I 2*NW+2

where ICHD

=

ISR

=

Z NW NCF INCOR

=

=

1, 3)

(JCUP(J,I), J=l, NPEEL(I) —I)

a heading, usually the chemical symbol for the atom; a serial number for the problem; if set negative it ensures that contributions from a kernel of subshells closed in all CSFs are retained; see ref. [2b], p. 399;

atomic number of orbitals defmed by the user; = number of CSFs defined by the user; = 0, ignore interactions with core shells, = 1, generate all coefficients including core—core, core—valence contributions. For each J = 1, NW, the orbitals to be used are designated: IWHD(J) = spectroscopic symbol (e.g. 2P+, 3D—); NP(J) = principal quantum number; KAP(J) = angular quantum number, K = —(I + ~)a, a ±1. (This is immediately doubled and stored as =

number

=

—(2j+ I)a). The next set of NW+l cards must be repeated for each of the NCF configurational states (CSF) of the problem:

NCQ(J,I) JWS(K,J,I)

=

the number of electrons assigned to orbital J in the CSF I;

=

shell quantum numbers; for each pair (J,I), the three entries are the quantum numbers (v. w, 2J+ 1), eq. (3).

The last card needed to defme a problem specifies the

angular momentum

coupling of the open shells. The inter-

N. Beatham et al. / Angular coefficients oft/se Breit interaction

253

mediate angular momenta X, eq. (4) and (6) are stored in: JCUP(J,I) = value of 2X+l for the Jth open shell in CSF I. The list has NPEEL(I)—1 entries for each CSF, where NPEEL(I) is the number of shells which are defined but not filled in the CSF I. This is not needed if there is only one peel shell, NPEEL(I) = 1, and in this case the card giving coupling data must be omitted. The following errors are trapped and cause termination of the run; it is important to realize that errors in the data preceding that to which the diagnostic print points may be involved: (a) Card punched with a character not recognized by the CARDRH subroutine, ref. [2a], p. 274; (b) Wrong number of items on a data card (due to mispunching or shuffling). (c) multiple definition of orbitals. (5.0). Subroutine SETUP see ref. [2a], p.27l; ref. [2b], p. 399. (5.1). Subroutine VIJOUT see ref. [2a], p. 271. (5.2). Subroutine ORTHOG see ref. [2a],p. 27l;ref. [2b],p. 399. (6.0). Subroutine RKCO this subroutine, which controls the main arithmetic part of the calculation is virtually identical with the one in MCP75, ref. [2a], p. 271; ref. [2b]. The only difference is in the replacement of the subroutines CUR and CORD, for the Coulomb coefficients, with BREIT and BREID. (7.0). Subroutine SPEAK this output results to the line printer. Each coefficient is labelled by the six parameters embodied in the identifier in ISLDR, —

—

—

—

—

TYPE NU A C B D. The last four are orbital labels distinguishing the charge clouds pAC(r), pBD(S) in the radial integral which the coefficient is to multiply; NU is the order, v, of the operator; and TYPE along with the four parameters classifies the type of radial integral to be used. TYPE

Meaning integrals involving U~(r,s): a)A C,BD. F~ag(A,B),eq.(18) b) A =D, B = C, A ~‘B(magnetic G integrals). T~(A,B, B, A), p in order of increasing ~i)

=

1,2,3. Eq. (20) (listed

c)A =C,BD,Rv(A;BD),eq.(16) or

A ~C,BD, 2

R” (B;AC), eq.(l6)

integrals involving U 0(r, s): SV[ACIBD] eq. (12)

3

magnetic H-integral: T~(A,B, B, A), eq. (20)

The various possible combinations will become clear with study of the test run output. The present version of this subroutine is a simplified one for test pruposes. The user will wish to write his own version of SPEAK to direct output, e.g. to a discfile or to tape as necessary. The parameter NOUT has been included for such use. 3.2. Subroutine BREIT The flow diagram of this routine, which computes the general coefficient t~v(A,B, C, D), is shown in fig. 2. Its structure is close to that of the subroutine COR of MCP75 [2]. The subroutines called by BREIT which differ from their counterparts in MCP75 are described below. (4.2). Subroutine SNCR this determines the limits within which the integer v may vary: NU = ND1 + 1, —

)

254

iV. Beatham et al. /Angular coefficients oft/se Breit interaction [o

initializatioul

2.0 Tabulate quantum numbers, T,!~, of spectator shells

3.0 OCON

J~

3.1 LPAB

—

Compute factor (i)~[ equation (5).

—

...

Locate rows of NTAB listi~]

parents of active shells A,B,C,D. Define parent CSF

4.1

MiEIDAD

—

4.2

SSTJ

—

Set angulal- momenta for NJSYM

4.3

~05C

—Set

4.3.1 MOD.J23, NJSYI4

L

4 TA’ TB, T~,

4.0

CXK

—

Comput, product of cfp’n,eq.(5)

—

range of tensor index NU

1

~

calculate Cd

,

eq. (5)

calculate a’~(&B,C,D), eq. (8)

4.3.2 M00J23, NJSTM — calculate C , eq. (5) CXK —calculate s~~K(A,B,C,0),eq. (8

L

[4.4

Asoembl: coefficients

( 5.0

Output

to

tm(A,Bj~5l

~)No

Yes buffer

~

j

arrays

XSLDR

and ISLDR (NU,A,B,C,D,u’ypE)

(tA,B~~j~lL I

Fig. 2.

NDI + ND2 for “direct” terms, NU = NE1, NE1 + 1, NE1 + NE2 for “exchange” terms. At the same time, it classifies the coefficients according to the type parameter, m = 1, 2, 3, 4, setting IBRD = m for “direct” cases, IBRE = m for “exchange” cases. The indicators IBRD, IBRE are set equal to —I if the contribution for this value of v vanishes. This subroutine implements parity selection rules. (4.3.1, 4.3.2). Subroutine CXK This subroutine computes the coefficients s~K(A, B, C, D) defined in eq. (8). It uses the function subprogram CRB to compute reduced matrix elements (JIIC
3.3. Subroutine BREID This handles the closed shell type coefficients of types m = 3, 4. In other cases, the general formula is used. It is known that the direct interaction, m = 3, of a single electron in orbital B with a closed subshell A gives a null con-

N. Beatham et al. /Angular coefficients of the Breit interaction

tribution unlessB = A ref. [7], section 8.2. Thus the case m = being fully occupied.

3

255

need only be implemented for A

=

B, the shell

4. Common blocks and arrays These are essentially identical with those of MCP75 [2] and need not be described here. Because of the larger number of integrals per matrix element, the arrays ISLDR, XSLDR in the block RESULT have been increased in dimension to 6000.

5. Test run Thestructure and coding of MCP75 have, as far as possible, been retained intact. Apart from the obvious checks to ensure that this has been done correctly, the tests employed aimed: (i) to ensure that both subroutines BREIT and BREID were tested; (ii) to ensure that all types of integral were treated correctly; (iii) to verify that all values of v were present, and that the corresponding sum over K values was correct; (iv) to confirm that the integrals were all listed in the correct order. Test 1. This gives results for the 10 two-electron configurational states 2s2P)i, (2s2p)2,(2P2)o, (2~2p)1,(2~2p)2,(2p2)

(2s2)~, (2s2p)~,(2s2~)

1, (

2)2, 0, (2p

in order. The results confirm that off-diagonal matrix elements are non-zero only if the CSFs concerned have the same overall parity and total angular momentum. The use of IBUG5 = 1 allows checks to be made on various intermediate quantities: values of v and K, Cd and Ce [eq. (6)], and the coefficients ~ (A, B, C, D), p = 1, Pm~For the case m = 1, the print just lists the 8 generals-coefficients in order; form = 2, 3, the s-coefficients are those of eqs. (17) or (19), respectively; for m = 4, the s-coefficients are those of eq. (20). As a specific example, consider the matrix element (31 B 0

4)

=

(~/9)

2p) + R ‘(2s; 2~,

~

t~’~(2s, 2~,2s, 2p) T~’°(2s, 2~,2s, 2p).

v0 ~j1

For the type 1(c) term, the recoupling coefficient Cd = 2~J~/3, a hand calculation gives s = 4s./~/3,from eq. (17), and [IA, ID]”2 = l/2~J~, whose product is 4../~/9.For the exchange terms with v = 0,2, the recoupling coefficient Ce = [i.~,IC] 1/2 = and all the coefficients s~’1~ were checked according to the formulas of table 2b. The diagonal matrix element ~,

(5 B~I5) —

—

~,

f~Fi~iag(2s, 2p) + G~~(2s, 2p) — ~G~g(2s, 2p) + 411~ag(25,2p)

~G~~’(2s, 2p)

+

4~G~g(2s,2p)

+

~G~g’(2s, 2p)

—

~H~ag(2s, 2p)

checks the correot evaluation of coefficients for these types of magnetic F, G and H integrals. Test 2. This treats the J = 0 CSFs of 4~2,4~2, 4p2, 4d2, 4d2, 4T~,4f2 which introduce further special cases where BREID is needed as well as BREIT. For example, (1lBol1>(s2,J0lBols2,J~’0>~F~ag(4s,4s), in agreement with ref. [9]. Similarly (71B 0I 7) which have been hand checked.

2, = (f

J

= 0IB

2, 0jf

J

0) involves F~ag(4f,40,

V =

1,3,5,7,

N. Beat/jam et al. / Angular coefficients of the Breit interaction

256

2, J 0~B 2,J 0), checks summation over K for each value A more complicated example, (6jB = (Sf 0lf that the permissible values ofK and V are of v. Only “direct” terms appear and an0~7) easy calculation shows

K

5)

K

I

1

3

3

2

1,3

4

3,5

K

v

5 6

5 5,7

Thus the integrals for 5) = 5, say, incorporate contributions from K = 4, 5, 6. A hand check gives an agreement with the printed results. This matrix element gives 32 coefficients, only 4 of which are non-zero; the remaining 28 coefficients occur in pairs.

Segments of the test print are appended to this paper.

Acknowledgements This work was made possible by an S.R.C. research grant, combined with access to the SRC’s IBM 360/195 computer at Atlas Computing Division, Rutherford Laboratory, Chilton, Oxfordshire, both of which are gratefully acknowledged.

References 1] l.P. Grant, Comput. Phys. Commun. 4 (1972) 377; 14 (1978) 311. 121 (a) I.P. Grant, Comput. Phys. Commun. 5 (1973) 263; (b) I.P. Grant, Comput. Phyl. Commun. 11(1976) 397; 13 (1978) 429; 14 (1978) 312. [3] N.C. Pyper, I.P. Grant and N. Beatham, Comput. Phys. Commun. 15 (1978) 387. [4) l.P. Grant, D.F. Mayers and NC. Pyper, J. Phys. B9 (1976) 2777; l.P. Grant and NC. Pyper, Nature 265 (1977) 715; NC. Pyper and l.P. Grant, J. Phys. BlO (1977) 1803; NC. Pyper, I.P. Grant and R.B. Gerber, Chem. Phys. Lett. 49 (1977) 479; NC. Pyper and I.P. Grant, Proc. Roy. Soc. A359 (1978) 525; S.J. Rose, N.C. Pyper and I.P. Grant, J. Phys. Bl 1 (1978) 755. [5] L. Asplund, P. Kelfve, P. Blomster, H. Siegbahn and K. Siegbahn, Phys. Scripta 16 (1977) 268; K.N. Huang, M. Aoyagi, M.S. Chen, B. Crasemann and H. Mark, At. Data Nuci. Data Tables 18 (1976) 243; J.P. Desclaux, Ch. Briançon, J.P. Thibaud and R.J. Walen, Phys. Rev. Lett. 32 (1974) 447; K. Schreckenbach, HG. Börner and J.P. Desclaux, Phys. Lett. 63A (1977) 330; Ch. Briancon and J.P. Desclaux, Phys. Rev. A13 (1976) 2157. [61 I.P. Grant and NC. Pyper, J. Phys. B9 (1976) 761. [7] I.P. Grant, Adv. Phys. 19 (1970) 747. [8] K.A. Berrington, PG. Burke, J.J. Chang, AT. Chivers, W.D. Robb and K.T. Taylor, Comput. Phys. Commun. 8 (1974) 151. [9] l.P. Grant, Proc. Roy. Soc. A262 (1961) 555, Proc. Phys. Soc. 86 (1965) 523.

N. Beat/sam et al. / Angular coefficients of the Breir interaction

257

TEST RUN OUTPUT CASE 3 ORRITAIS AND

1 FOR ATOMIC NUMBER

10 CONFIs.,LJRATICNS HAVE BEEN DEFINED.

C1.flSED/EMPTY SHELL CONFIGURATION

25

I 7 3

2 2

?P— 2P

CONFIGURATION

1.

NP

2 NP

2S 7 ?P— 3 2P 2 ELECTRONS

2

INTERACTIONS HILL BE INCLUDED J

K

0

V

W

1/2 1/2 3/2

—I 1. —2

2 0 0

0 0 U

0 0 0

K

0

V

~

1 1 0 OPEN

U 2 0 2 0 1 SHELLS.

.1

2

1/2 —1 2 1/2 l 2 3/2 —2 OUT OF A TOTAL OF

1

5(TST1I

1 1 0 2 ARE IN

2J+l 1 1 1

*F* *E* *E*

V V V

*E*

V V V

*E*

V V V

2.1+1

COUPLING OFFINFC BY I CONFIGURATION

3

NP

.1

K

1 2 1/2 —1. 2 2P— 2 1/2 1 3 ?P 2 3/2 —2 2 ELFCT~ONS ~UT OF A TOTAL OF

0 1 1 0 2 ARE IN

V

U

1 1 0 OPEN

U 2 0 2 0 1 SHELLS.

2.1+1

V

W

1 0 1 OPEN

0 2 0 1 C 4 SHELLS.

V

U

1 Li 1 OPEN

0 2 0 1 0 4 SHELLS.

COUPLING OEFINEO BY 3 CONFIGURATION

NP

4

.1

K

2 1/2 —1 2 1/2 1 2 3/2 —2 (UT OF A TOTAL OF

I

75 2 2P— I 7P 7 ELFCTPIINS

0 1 0 1 2 ARE IN

2.1+1 *E*

V V V

*E*

V V V

*E* *F* *E*

V V V

*E*

V V V

CIBJPL IN~DEFINFO BY I

CONFiGURATION

NP

2S 7 ?P— I 7P 7 FLECTRONS

5

J

K

2 2 2

1/2 —1 1/2 1 3/2 —2 OUT OF A TOTAL OF

I

0 1 0 1 2 ARE IN

2J+1

COUPLING OFFINED BY S CONFIGURATION

I 2 3

NP 2 2 2

2S

2P— 2P

CONFIGURATION

6

NP

I PS 2 7p— I 2P 7 ELECTRONS

7

2 2 2

K

0

V

U

2.1+1

—1 1 —2

0 2 C

0 0 0

0 0 0

1 1 1

J

K

0

V

U

2.1+].

0 1 1 OPEN

I) 1 0 2 0 4 SHELLS.

1/2 —1 1/2 1 3/2 —2 OUT OF A TOTAL OF

COUPI IN(, E)EFINFD BY

3

J 1/2 1/2 3/2

0 1 1 2 ARE IN

N. Beat/iam et aI. / Angular coefficients of the Breit interaction

258 CONFIGURATION

NP

~

J

K

0

I 7$ 2 1/2 —1 7P— 7 1/2 1 3 2P 2 3/2 —2 7 ELFCT~CNS cUT OF A TOTAL OF

V

W

2J+1

U L 1 1 1 0 2 1 1 0 4 2 ARE IN OPEN SHELLS. sJ

2

*E*

V V V

*E* *E*

V V V

COLJPI INC~ DEFINFO B’V ‘5

CONFIGURATION 1 2

3

1

2

9

J

2

2S 2P— 7P

CI1NFIGLIRATION

3

NP

2 2

1/2 1/2 3/2

10 NP

J

7S 2P— 2P

2

1/2 1/2 3/2

2

2

K

0

V

W

2J+1

—1 1 -2

0 0 2

0 L ~

0 1

1 1 1

K

0

V

W

—1 1 —2

0 0 2

0 C 2

0 C ~

2J+1 1 1 5

WARM NG

THF COUPE INC, ASSIGNMENTS HAVE NOT BFFN CHECKFC AT THIS STAGE. ~CflNFiG 3/V/CO~IG TYPF 1

2

7 7 2 2 7 2 7 2 2 2 7 7 2 7 7

NIJ 1 0 u 0 Ci

0 ~ 0 J 2 2 2 2 2 2 2 2

4>

A

C

B

0

COEFFICIENT

2S 2S 2P— 2P 2S 2S 2S 2P 7P— 2S 2P— 2P 2S

2S 2P 2S 2S 2P— 2P 2P— 2$ 2S 2P

2P— 2P— 2S 2S 2P 2S 2S 2P— 2P 2P— 2S 2S 2P 2S 2S 2P— 2P

2P 25 2P 2P— 2S 2P— 2P 2S 2S 2S 2P 2P— 2S 2P— 2P 2S 2S

0.62~539361D+0D 0.0 —J.157134t400+Q0 O.1257~7872D+01 u.1414213560+C1 0.0 —0.4714L~4521D+00 0.0 0.4714045210+00 0.1257078720+00 0.2828421120+00 0.0 —0.1571348400+03 0.0 0.47P4045210+00 u.377123617D.00 —0.9428090420—01

2$

2S 2P— 2P

2S

2P—

2S 2P 2P—

2S 2S


5>

NU

A

C

B

0

1 0 0 D 0 2 2 2 2

7S 2P 2S 7S 2S 2P 2S 2S 2$

2S 2S 2P 2P 2P 2S 2P 2P 2P

2P 2P 2P 2S 2P 2P 2P 2S 2P

2P 2S 25 2P 2S

2S 2S 2P 2S

COEFFICIENT —tj.106666667D+0l 0.1000000000+01 —0.3333333330+00 0.0 u.3333333330+00 —0.4000000000—01 0.3866666670+00 U.1600300000.cjo —0.3333333330+00

*[*

*E*

V

V V

259

N. Beat/sam et al. /Angular coefficients of the Breit interaction 2 FOR ATOMIC NUMBER

CASE 7 ORRITALS AND

7 CONFIGURATICNS HAVE BEEN DEFINED. INTERACTIONS WILL BE INCLUDED

CLOSED/EMPTY SHELL CONFIGURATION 1 2 3 4 5 6 7

45 4P— 4P 40— 40 4F— 4F

CONFIGURATION I 2 I

4 5 A 7

4S 4P— 4P 40— 40 4F— 4F

CONFIGURATION

I 2 3 4 5 6 7

4S 4P— 49 40— 40 4F— 4F

CONFIGURATION I 2 I

4 5 6 7

4S 4P— 4P 40— 40 4F— 4F

CONFIGURATION I 2 3 4 5 6 7

4S 49— 49 40— 40 4F— 4F

CONFIG(IRATION

2 3

1

45 49— 4P

4

40—

5 6 7

4F—

40

4F

1 NP 4 4 1

4 4 4 4

30(TST2)

.1 1/2 1/2 3/2 3/2 5/2 5/2 7/2

K

0

V

U

—1 1 —2 2 —3 3 —4

2 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 U 0

2.1+1 1. 1 1 1 1 1 1

2 NP

J

K

0

V

.W

4 4 4 4 4 4 4

1/2 1/2 3/2 3/2 5/2 5/2 7/2

—1 1 —2 2 —3 3 —4

0 2 0 0 0 0 Ii

0 0 0 0 0 0 U

0 (1 0 0 0 0 U

1 1 1 1 1 1 1

.1

K

0

V

W

2.1+1

—1 1 —2 2 —3 3 -4

0 U 2 0 0 C 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

1 1 1 1 1 1 1

K

0

V

W

—1 1 —2 2 —3 3 —4

0 0 4) 2 4) 0 0

0 0 0 0 U 0 0

0 0 C) 0 0 0 0

K

0

V

U

—1 3. —2 2 —3 3 —4

0 0 0 0 2 0 0

0 0 0 0 0 0 0

0 0 0 U 0 0 0

1. 1 1 1 1 1 1

K

0

V

W

2.1+1

—1 1 —2 2 —3 3 —4

0 0 0 0 0 2 0

U 0 0 0 3 ~ 0

1) 0 1) 0 0 0 0

1 1 1 1 1 1. 1

NP

3

4 4 4 4 4 4 4 NP

4

4 4 4 4 4 4 4 NP

5

4 4 4 4 4 4 4 NP

6

4 4 4 4 4 4 4

1/2 1/2 3/2 3/2 5/2 5/2 7/2 J 1/2 1/2 3/2 3/2 5/2 5/2 7/2 J 1/2 1/2 3/2 3/2 5/2 5/2 7/2 J 1/2 1/2 3/2 3/2 5/2 5/2 7/2

*F* *E* *E* *E* *E* *E* *E*

V V V V V V V

*E* *F* *E* *E* *E* *E* *E*

V V V V V V V

*E* *E*

V V V V V V V

2.1+1

*E* *E* *E* *E*

2.1+1 1 1 1 1 1 1 1

*E* *E* *E* *E* *E* *E*

V V V V V V V

2J+1 *E* *E* *E* *E* *E* *E*

*E* *E* *E* *E* *E* *E*

V V V V V V V

V V V V V V V

260

N. Beatliam etal. /Angular coefficients of the Breit interaction

C(1NFLGURATTON 1 2 3 4 5 ~ 7

7 NP

J

4 4 4 4 4 4 4

1/2 1/2 3/2 3/2 5/2 5/2 7/2

4S 4P— 4P 40— 40 4F— 4F

K

0

V

W

—1 1 —2 2 —3 3 —4

0

0 0 0 U 0 0 0

C) 1 0 U 0 0 0

0 0 0 0 2

2J+]. 1 1 1 1. 1 1 1

WAR NT NG * ** ** * * THF COUPLING ASSIGNMENTS HAVE NOT BEEN CHECKFD AT THIS STAGE.


TYPF 1

l/V/CONFIG

1>

NUJ

A

C

B

C

1

4S

AS

4S

4S

0 .2666666670+01


TYPF 2 2 7 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 7 2 2 7 7 2 2 2 2 2 2 7 7 2

~r.ONFIG

TYP~ 1 1 1 1

NO 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 5 7 7 7 7~ 7 7 7 7

7>

A

C

B

0

COEFFICIENT

4F— 4F— 4F 4F 4F— 4F 4F 4F— 4F— 4F— 4F 4F 4F— 4f 4F 4F— 4F— 4F— 4F 4F ‘F ‘0: 4F 4F— 4F4F— 4F ‘+F 4F— 4F 4F ‘0:—

4F 4F 4F— 4F— 4F 4F— ‘0:— 4F 4F 4F AF— 4F— 4F 4F— 4F— 4F 4F 4F 4F— 4F-~ 4F 4F— 4F— 4F 4F 4F 4F— 4F4F 4F— 4F— 4F

4F— 4F— ‘0: 4F 4F 4F~ 4F— 4F 4F— ‘,F— 4F 4F ~pF 4F~ 4F— ‘0: 4F— sF—

4F 4F 4F— 4F— 4F‘0: 4F 4F4F 4F 4F 4F— 4F— ‘0: 4F ‘tF4F 4F 4F— 4f— 4F 4F ‘0: 4F 4F AF 4F— 4F— 4F— 4F 4F 4F—

0.1649572200+00 (~.16495722UD+0~ 0.3958973270+00 ~J.3958?7327D+u0 ~ .329914440D+~iO 0.9a974331cD-01 U.989743319D—01 O.32991444CD+00 0.1049727760+UD u.10497277b0+u. ).23094(~1C8Di-~J U.23094u1I~80+L~) .12~9~6554D+~ —0.1649572200+00 —C.16495722L~D+~j t.12B96ó54D+~i 0.13~6579C4D+C,0 ~.1326579U40+oJ u.3749C2772D+Ou 0.374902772D+OD ‘.63’,451;845Du1 —o.824766L99D+~0 —t: .82478ot;99D+~.,) L.63445Ub45D*1 O.282619C130+uO u.282619t,13D+~j) 3.0 0.0 0.0 U.1413095060+0 O.i4i3095060+0 .0

7/V/CUINEIG

NO 1 3 ‘5 7

COEFFICiENT

4I~~

‘0: 4F 4F4F— 4F 4F— 4F— 4F 4F 4F F4F— 4F

7>

A

C

B

0

4F 4F 4F 4F

4F ‘iF 4F 4F

4F 4F 4F 4F

‘iF ‘iF 4F ‘0:

CUEF~ICIENF L.2(~3174603D*01 i~.831168831D+C~i 0.6393606390+00 (~.d7~,24O87CD+tu)

*E* *E* *E* *~* *E* *E*

V V V V V V V