0 r . . . J urn

UCIO-17169

Lawrence Liv&more Laboratory

USER'S GUIDE TO PROGRAM RCN

R. D. Cowan, K. Rajnak, and P. Renard

June 1976

f

This is «n Inform*) report intended primarily 'or internal or limited external distribution. The opinions •nd condusiom Hated ere thcss of the author and may or may not be those ol the faboratory.

Prepared for U.S. Energy Research & Development Administration under contract No. W-7405-Eng-48.

"""03T1T '; I

MSTEH OtSTRtRUT'OM OF TH<LS mCUVF^IT >S tWMMITTD

USER'S GUIDE TO PROGRAM RCN R. 0. Cowan

This is a set of three Fortran IV programs, RCN29, HFM0D7 and RCN229 based on the Herman-Skillman and Charlotte ^roese Fischer programs, with extensive modifications and additions by R. D. Cowan, D. C. Griffin and K. L. Andrew at LASL. Modifications for running at LLL were made by Paul Renard.

The programs ompute self-consistent-~ie1d radial wave functions and the various radial integrals involved in the computation of atomic energy levels and spectra. The codes RCN29 and RCN229 as obtained from LASL (March, 1975) were versions with seme dimensions which were too small for continuum functions. These were changed as per Cowan's instructions so that the current LLL code should be the same as the large LASL version. Responsibility for any errors involving these dimension changes must rest with LLL. Other changes have been limited to those necessary to make the codes run on the LLL system. Two of us (Rajnak and Renard) make no pretense of understanding the detailed workings of the codes and Cowan should be contacted (Ext. 5139 at LASL) if any major changes are contemplated. The codes are also running at LBL and John Conway (Ext. 5H1) mav be able to help with some difficulties.

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Program Descriptions 2 A. Overview 2 B. RCN29 7

1. Description 7 2. Input 10 3. Calculational Procedure 16 4. Input/Output Units 18 5. Output 19 6. Continuum Functions 20 7. Re la t i v i s t i c Corrections 21

C. HFM0D7 24

1. Description 24 2. Input 25 3. Input/Output Units 27 4. Miscellaneous 27

D. RCN229 23 1. Description 28 2. Input 29 3. Calculational Procedure 32 4. Input/Output Units 34 5. Output 34 6. Autoionization Calculations 34

£. Storage 38 F. Conversion to Other Computers 39 G, Comments, Discussion and Warnings 40 H. References 44 I. Appendix , 46

1. Small-r so lut ion in the HXR Method 46 2. Correlation Corrections 47

Running the Codes 50 A. Availability and Compilation 50 8. Running 50

1. Separately 50 2. Under ORDER 51

C. Sample Input 52 D. Sample Output 55

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' • '1.r.0iLrA"i -les<:r iptions A. Overview

These three programs can be run in a chain in any one of the following four combinations:

RCN29 ftCH29/RC(229 RCH29/HFK0D7 RCN29/HFH0D7/RCM229

The primary input information is always to RC:<29, and each program automatically provides input information to tne succeeding program of the chain. Details of t'ie programs w i l l be described l a t e r ; ' j re we out l ine only tne basic ourpose of each program.

Program RCI!29 calculates s ingle-conf igurat ion radial wavefjnc-tions for a spherically-symmetrized atom via any one of the fal lowing homogeneous-differential-equation approximations to the Hartree-Fock method.

(1) Hartree (H); (2) Hartree-Fock-Slater (HFS), with any desired value of the

coef f i c ien t for S later 's approximate exchange term, and ei ther w i th ­out or with (HFSL) La t te r ' s t a i l cutof f in the c e n t r a l - f i e l d potent ia l -energy funct ion;

(3) Hartree-plus-stat ist ical-exchanqe (HX); (4) Hartree-Slater (HS).

Normally, the HX method is the only one used. (HX and HXR (see below] are the only methods which have been used at LLL. A'd others should be regarded as untested). Also calculated in each case are various radial integrals (< r" > , F , G , .-.) and the total energy E a v of the atom, including approximate relativistic and correlation enernv cor­rections. Relativistic terms can be included in the differential eQuation (HXR) to give approximate relativistic wavefunction correc-

2 tions for heavy atoms. (See B 7 below). (Important for outer orbitals only if Z > 54, for inner orbitals if Z • 35.) Ootions are available to provide radial wavefunctions as innut to either RCN229 or HFM0D7.

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Proqram HfMOD? calculates S ingle-co-if i'j<ir.it ion Hdiln.'e-i -V • ra.li a i w^vefunctions corresponding to c i ther the cen te r -a f -q r j v i t v oncrci* of the conf igurat ion, or to any specif ied LS tern of the configuration (provided the LS value of interest appears only once in that conf ig­ura t ion) . (We have computed only center-of -grav i ty enernie^ at L!.L but the LS term feature has. been used at LBL). Radial intoij. -als .mi r are calculated as in RCH. except that spin-orb i t parameters a)-c

a v 3 computed by the Blume-Hatson method instead of via the simple onp-c lec t ron , cen t r a l - f i e l d formula. Input is en t i re l y from RCN29; output wavefunctions provide input to RCN329. Re la t i v i s t i c terms can :-.•> i n c i t e ! in tne HF equations if desm 'KFR),

Program RC.IZ21) accepts radial wavefunctions ( fo r one or nor? d i f fe ren t configurations of one or more atons or ions) f'-oc: ei trier RCN29 or HFK0D7 and calculates mil t ip le-conf i rwrat ion radial m t e i r a U : conf igurat ion- in teract ion Coulomb integrals (R ) anj sp in-orb i t inte­grals ' . , -> . and radial e lec t r i c -d ipo le arid e le t t rk -auadrupo le in tegrals . In i t s most commonly used opt ion, i t automatically computes a l l ntMn-t i t l e s required for calculat ing energy levels an! spectra of an .non. and creates a f i l e PUHCHA which gives input in tne enact f om re­quired by program SCO MOO 5 or 6, which performs these calculat ions. ' "

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Computing ?inies

The .foHawSnq approximate computing times for RCN29 (HX or HXR) and HFM0.i)7 were obtained on the CDC 7600 at LASL.

Tine [sec)

•' j^onfjg HX HXR HXa + HF HXRb + HFR

Ar I 3p 6 5 6 1 + 3 2 + 4 Kr I 4p 6 9 9 7 + 6 3 + 3 X* I 5p 6 1) \2 9 + ?? 4 + 15 ny I 4 f 1 0 6 s 2 13 2-1 14 + 17 7 + 24 Rn I op 6

U 1 5f 3 6d7s ?

17

20

21

27 13 + 26 15 + 51

7 + 33

a + 62

! aTime for 5CF calc (TOLENO = 5 • 10" 3 ) + ZETA1 bTiine for SCF ealc (TOLEIID = 5 • 10" ? ) only

Running iitider OfOtR at LLL the fol lowing times were obtaineJ.

HUR * HFR ("iin)

f t i 4f s sc^Gs 4 f H 5 d 3 r , s 2

Au ![ Af]"b(i06s 4 f l a b d 8 6 s 2

14 9 Hg III 4f '*5.T6s

4f , 45d'V J

.69 + 4.4 5

Pt I as above

U I 4 configurations

Fe I . Mb H . Gd 1. U I 3 configurations each HXR(RCN2) + HFR(HFHOD7) + RCN229

RCN229 .02 n in .

HXR + HFR + RCII229 7.10 min.

10 minutes

-6-Computing time for RCN229 is usually only a small fraction of

the RCN23 or HFM0D7 time. However, if there are a large numcer of configurations on unit 2 (say, » 20) and the configurations in question are the ones with high serial number (i.e., near the end) read and re­wind times can be uncoivforably long, especially with a tape .-ather than disk unit.

File Sizes The 1st data card gives decimal sizes for the disk files OUTP

(the output file), PUNCHA (input for RCG), TAPE4 (a scratch file), TAPE2 (input to RCN229), TAPC7 (input to HFM0U7) and EHC (an encode/ decode). When running under ORDER the size of OUTP is really deter­mined by the COPY instruction on the ORDER card, PUNCHA contains essen­tially only values of radial integrals and is quite small.

3 2 For running U I 5f 6d7s through all three programs, the follow­

ing file sizes are necessary. TAPE4 19000 TAPE2 5000 TAPE7 5000

TAPE4 rewinds between configurations so its size is determined by the configuration containing the largest number of shells. Ths njrnber of wavefunctions on TAPE2 is controlled by tne value of N0R3PT or. tie control card. (See B2). As usually run only the S outermost orbita's of each configuration art written on TAPE2 so it is smaller tnan TAPE7. TAPE7 contains wavefunctions for each shell of each configuration so for actinides its size ^ 5000 /. * of configurations.

If Hr'M0u7 is not used, TAPE7 isn't needed. The size of TAPE4 increases to 30000 because a different write statement is jsed.

For U I f 4 s 2 an-l f 3ds 99p (continuum p) 1APE4 is 72QOO, TAPE2 is 7000.

ENC needs only a size of 10.

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B. PROGRAM RCN29

1. Description

Thi i is a Fortran IV program for the COC 7600 for the calculat ion of radial wavefunctions of a spherically-symmetrized atom, corresponding to the center-of-gravi ty energy £ of an electron conf igurat ion

av (n^)" 1 (n 2i 2) W ? • • • • ( n q y W q • (D

Computed from these radial wavefunctions are the Coulomb integrals F and (5 and the spin-orbit integrals r., along with radial integrals required to give kinetic and electron-nuclear energies, and rough rela-tivistic and correlation energy corrections to both the one-electron binding energies and total electronic binding energy E .

av The program was originally (January, 1964) based on the Share

program 1417 (ML HFSS) described by Herman and Skillman5, and this reference may he used for a description of the integration mesh and basic numerical methods used in RCN29. However, except for the sub­routine CROSYH and most of SC.1EQ, the original program has been changed beyond recognition (including r.otational changes from SNL(M.I) and SNLO(I) to PNL(!,M) and °HL0(I), and re-definition of N N U from lOOn + ]Qi + const to lOOn + ».) • Options are still available for making calcu­lations via the Hartree-Fock-Slater approximation used by Herman and Skillraan either with or without Latter's tail-cut-off in the one-electron potential, using either Slater's original exchange coefficient j or Kohn and Sham's modified value of 1 (or any other value). However, these approximations are all considered to be unsatisfactory, and the program is intended primarily for calculations via the Kartree-plus-statistical-exchange (HX) approximation described in Phys. Rev. 163. 54 (1967), which should be consulted for theoretical discussions (except that the method of calculating correlation corrections has recently been modified). Still a further option is the HS approximation of Lindgren and Rose*n.

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Program Outline

The program is all in Fortran IV and consists of a main program and ten subroutines.

RCN29 RCN29 is the main program. It handles all input, most output,

controls the self-consistent-field iteration, and does much of the detailed calculation.

SCHEq SCHEQ integrates the one-electron radial wave equation, and

controls the iteration on the one-electron eigenvalue E=EE(M) to give a radial wavefunction F n {(r) = rH Ar) = PIILO(I) = PNL(l.M) which satisfies the boundary conditions ar r = 0 and =••. It also properly normalizes the function for either a bound electron (E < 0} or a free electron (E > 0); it will not handle the car.e E=0. The normalization for a free electron is likely to be inaccurate if E -0.5 rydberq (because of too coarse an integration mesh) unless 0.) EMX = E 5 EMX and very large dimensions are used for PNL, H, and other variables (see remarks below regarding continuum-wavefunction calculations).

SUBCOR SUBCQR calculates a nodified free-elictron correlation energy for

a specified electron density, CROSYM

CROSYM solves a system of linear equations to provide a value of

A Z ( r ' 4 " 1 ^ r . o m POWER

POWER computes values of < r f f l > fo r -3 = m = 6 (except m = 5) for each o r b i t a l .

ZETA1 ZETA1 computes sp in-orb i t parameter values r, from the cent ra l -

potent ial formula

o

using two d i f f e ren t forms of the one-electron potent ia l V; the r i gh t -hand one of the two values printed out is the one used in pract ice. Also computed and printed are the one-electron k inet ic-p lus-nuclear energy I , and the r e l a t i v i s t i c mass-velocity and Darvjin correct ions. (Actual ly , I is computed in RCN29, and only pr inted in ZETA1.)

SLIT k k SL11 computes and prints all Slater integrals F and G (in units

of both rydbergs and cm ); it also computes overlap integrals and calls RCN3S.

RCH3i; RCN3S prints the overlap integrals, and calculates (from the F

I,

and G , with the aid of subroutine S3J0SQ) and prints the Coulomb inter­action energy between each pair of electrons. It then calculates corre­lation-energy corrections, and (using values of I from 2ETA1) calculates and prints one-electron binding energies and the total binding energy of the atom. (See Ref. 6 for equation). It also prints the sinale-

L k configuration parameter values E , F (ii), :,., F (ij}, and G (ij) required for the calculation of atomic energy levels; E (total binding energy) is given is rydberqs, all other parameter values are in KK (units of 10QT en"').

S3J0Sq S3JOS0 calculates the value of

•i J J N ?

i-h ]2 J3'i where (m, m' m^! is a 3-j symbol.

FCTRL FCTRL calculates the factorial of an integer A, required by S3J0SQ.

HFWRTP HFWRT? interpolates the RCN wavefunotions to points on the logari­

thmic mesh used by HF, and writes on TAPE? these wavefunctions and other input data for a Hartree-Fock calculation.

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2. Input

Input consists of a card giving f i l e sizes (see IA). a 13-car.i deck providing a Start ing estimate for the one-electron potential RIJ, a control card, and one configurat ion card for each confiqurat ion to be computed. (Addit ional cards are required i f an LS-term HF calc^lat ior , is to be done; see below). Normally, the potent ial cards (aitd to some extent the control cards] are universal , and are never changed from one run to the next.

(a) F i le s izes: The f i r s t card give$ the sizes for the f i l e s OUTP. PUNCHA, TAPE4, TAPE2. TAPE7 and ENC. The format is FILE SIZL=6llO (b) Potential Cards: Read at statement 195 of main proqram HCN2-*. These cards are ident ical (except for format) with Herman-Ski 1Iman's corresponding input , and contain values of the i r potent ial t;(Xj

for some appropriate value of Z ( e . g . , 1 - 18, 3G, or 54: the v^lue used is not pa r t i cu la r l y important). (A l l LLL runs nave begun with the Ar po ten t ia l ) .

(c) Control Card: Read at statement 200 of RCN20- and containing the fol lowing quant i t ies

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cols format 1 i l ITPO'-J 2 11 IPTVU 3 n IPTEB

4-5 12 NORBPT b n 1ZHXBM

7-B 12 JPHFWF 9-10 12 IHF

11-15 F5.1 TOLS TB 16-20 F5.1 T0LKM2 ?l-30 E10.5 TOLEND 31-4:1 E10.5 THRESH 41-4? I? KUT!) 43-45 13 KUT1 4o 11 I RL'L 47-48 12 MAXIT 40-50 K' :;PR 51-55 F5.5 ExFlO 56-60 F5.5 EXFH1 61-65 F5.5 EMX 66-70 F5.5 CAO 71-7-3 F5.5 CORRF 76-so F5.5 CA1

"W 5 E-OS and 1 . E - l l i f IRE t ions to E a v are desired.

noma! val je - -tt on iy HX + HF HF only

2 2 0 j 0 0 0 0 0

- 5 -5 -5 0 0 0 0 0 0 0 2 1

1.0 1.0 1.0 1.0 1.0 K 0 a 5.E- 08 6.E-0C 5.E-02* l .E- 11 l.E-11 l .E-05 a

-2 -2 Oh °h

o b

30 30 30 0 0 0

1.0 1.0 1.0 0.65 0.65 0.65 0.0C 0.0 C 0.0 C

0.5 0.5 0.5 0.0 0.0 0.0 0.7 0.7 0.7

•11 i f IREL = 0 and HX r e l a t i v i s i t i c enemy ccrrec-ions to E are df

bUse IREL = 2 for HXR, IREL = 1 or 2 for HFR.

See notes regarding continuum-wavefunction calculat ions, section Do.

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The f i r s t seven quant i t ies and MPR control the amount of printed output, as fo l lows:

iTonu I 5 1 o r = 3 » p r i n t SCF i te ra t ion information I IFUW 1 >g ( c a l l p 0 W E R t 0 p r j n t ^ i j

IPTVU

IPTEB

N O W T

1ZHXBW

IPHFWF

=1, p r in t F and GN

=2, p r in t Coulomb interact ion energies ; 3 , p r in t wavefunction overlap integrals =4, p r in t HFS potent ials (RLI, RUEE, etc)

. =5, p r in t HX potent ia ls {V™) \ =6, in SCHEQ, p r i n t r e l a t i v i s t i c contr ibut ion f potent ial

f =1, p r i n t EE, JJJ, R(JJJ), AZ I =2, p r i n t E L Lkin' =tc

>0,

any,

FlPR

do not print wavefunctions print first two and last NORBPT wavefunctions

at everv fifth mesh point print continuum wavefunctions at every mesh point for

configurations 1, 5, 10, 15, ... If IHF=0 write on TAPE 2 the las; [NOBBPT| wavefunctions

(at least A; if |N0RBPT|=9, write all wavefunctions) / >0, in HF, calculate and print all quantities using HX input I wavefunctions (primarily to give c, calculated by

Blume-Watson method).

>0, in HF, print last llPHFUF j wavefunctions at every mesh point

<0, same, except at only every fourth point ±9, print all wavefunctions =0, Signifies RCN calculation only; do not load HF program 7*0, RCN output wavefunctions are for input to HF (via TAPE7)

rather than for RCN229 (via TAPE2) =1, skip all RCN calculations of radial integrals if

IREL > 1; if IREL = 0 and TOLEND < 1.0E-4, call only ZETA1 (to calculate relativistic energy correction for HF E ) . .

=2, carry out all RCN calculations ( of F K, G K , C, E a„) .. av (^0, diagnostic potentials and diagnostic wavefunctions are I printed during the course of the RCN SCF iteration

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Tne significance of the other input quantities is as follows:

TOLEND is the maximum permissible value of DELTA (change in the ab­solute value of RU} for ending the SCF iteration (RCN29. state­ments number 530-550, 705).

THRESH is the maximum permissible fractional change in the value of the eigenvalue E to end the eigenvalue iteration (SCKEQ, 305 and 205).

MAXIT is the maximum allowable number of SCF iterations; if conver­gence is not obtained, the calculation is continued for 4 more cycles with diagnostic printout (NPR.GT.Q) (710-729).

EXF10 is the coefficient of Slater's exchange term for a HFS calcula­tion with no tail cutoff (KUT = 1) or with tail cutoff (KUT = 0) (411-417). EXF10 = 1.5 is Slater's original value, EXF10 = l.-J is Kohn and Sham's modified value.

EXFMl, CAO, and CA1_ are values of k,, k,, k_, respectively, for KUT = -1 [HX calculation; Reference 6, eos. (13) - (14)]. EXFMl = 0.0 will give a Hartree calculation for KUT = -1. KUT = -2 gives a HS calculation.

If IREL = 0, relativistic terms are omitted from the differential equa­tions (HX and HF); if IREL = 1 or 2. they are included (HXR and HFR); if IREL > ?., some diagnostic printouts will occur. (IREL = 2 is supposedly better for HX, but sometimes leads to instabili­ties present in HXR that are not present with IREL = 1; the difference lies only in numerical methods used in calculating dV/dr.

(d) Configuration Card: read at statement 213 of RCN29 and con­taining the following information:

- 1 4 -

cols format variable

1 11 NHFTRH

2-5 14 IZ 6-10 15 NSPECT

signif icance

11-28

31 32

33-35

36-37

3A6 COHF

IPfl.O II

ALFMIN ) ALFHAX /

A3 NLBCD8

A2 WWNL8

Ho. of LS terms for which HF calculat ions are to be ma-ie

atomic number ( f ixed-point )

1 + degree of ionizat ion (1 = neut ra l , 2 = s ingly-ionized, etc)

BCD configurat ion label ( e . g . , CA I 4S 3D), for i den t i f i ca t i on purposes only

/ i t e r a t i o n control variables ({normally l e f t blank)

o rb i ta l speci f icat ion ( e . g . , " " « " , "12D")

occupation number (0" or "0 = 0, blank = 1 , r or " 1 = 1 , l" or -Z = 2, etc)

68-70 A3 71-72 A2

NLBCD8 WWNL8

73-74 12 KUT Overrides value of KUT1 • om control card for th is cor-' igura-t ion

75-30 F6.5 £E8 Eigenvalue of outermost orbital, for a free electron only. (EE8 must be > 0, and last punched value of NLBC03 should be "995", "99P", "99D", etc., and last value of WWNL8 must be blank or one.

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For a continuum wavefunction eH, punch 99£ ( e . g . , "99S", "99P". "99D", etc) at the appropriate place in cols 33-7'i, and punch the de­si red value of the k inet ic energy E ( i n rydbergs) i n cols 75-80 ( f o r ­mat F6.5).

For heavy elements (Z > 70), i t may be necessary to decrease by an order o f magnitude the values of TOLEND and THRESH punched on the control card. (This card is read a t statement 200 of RCN29 main). (See Sect. E).

I f NHFTRM (col 1 of the conf igurat ion card) i s non-zero, then NHFTRM (or more, as required) addi t ional cards must f o l l ow , containing the fo l lowing informat ion:

co]s_

1-7

8-10

11-15

21-29

30

31

33-35 37-39

format variable

7X ...

A3 TERM(I)

15 NFG

signi f icance

unused i d e n t i f i c a t i o n information ( e . g . , the conf igurat ion "PD", " P ^ D " , etc)

I.S term i den t i f i ca t i on ( e . g . , " IP" fo r ' P , "3D" fo r 3 D , etc)

k k No. of Slater parameters (F and G ) involved in the diagonal energy matrix element fo r t h i s LS term (excluding

FK or G \

and " A 4D"

Eav'-F9.8 CFG(J,I) Coef. f k or g k of the j t h

Al FG(J.I) "F" or "G", as appropriate

11 KFG(J.l) Value of k.

A3 A3

IFG(J-I) JFG(JJ)

{Orbital codes; e.g., "MP" lfor F^Op^d) and GR(3p,4d

41-59; 61-79J

k k same as 21-39 fo r second and t h i r d F and G .

Code dimensions l i m i t th is opt ion to 9 terms/conf igurat ion k k

and a to ta l of 10 F and G in tegra ls in the energy expression.

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If mors than three Slater parameters are involved in the expression for the energy of this LS term (relative to E ) , then the remaining coeffi-

av cients and parameters are punched four to a card in cols 1-19, 21-39, 41-59, and 61-79. ( I f an HF calcu lat ion for the center -o f -grav i ty energy is de­s i red in addi t ion to one or more calculat ions for speci f ic LS terms, i t is only necessary to fake an LS-term card with TERM = C-G, NFG = 1 , CFG(1,I) = 0 .0 , IFG and JFG equal any two o rh i ta l s of the conf igurat ion. ) Hone of t h i s information is used in RCN29, but is simply passed on unchanged to the HFM0D7 program.

3. Calculat ional Procedure

Af ter reading the input potent ia l and control card, the fo l lowing is done f o r each conf igurat ion card:

(a) I f 11 > 0 but columns 11-16 are blank ( f i r s t conf igurat ion card o n l y ) , the output (see l a te r ) from IZ completed configurat ions done on previous runs i s skipped over on TAPE2.

I f IZ - 0, go back and read a new control card. I f 11 < 0, wr i te EOF on TAPE2 and TAPE7 and e * i t . I f IZ > 0 and columns 11-16 are not a l l blank make a ca lcu la t ion as

fo l lows: (b) Calculate the number of electrons in the atoms from

N = IZ + 1 - NSPECT ;

set up the heaviest rare gas core (grojnd configuration of Ne, Ar, Kr, Xe, Rn, or Z = 118) having no more than N electrons; modify or add to this core accordinq to any values of NLBCD3/WWNL8 punched on the configuration rard so as to obtain the final desired configuration (1); estimati. a value of the eigenvalue for each orbital (unless EE8 is non-ze.-o, in which case this value is used for the outermost orbital). [Decoding of WWILS (statements 230-233) and of NLBCU8 (237-242) *s done partly via table look-up].

-,7-

The ion izat ion state punched in columns 6-10 of a conf igurat ion card is used only to f ind the rare-gas conf igurat ion from which to s ta r t in sett ing up the desired conf igurat ion. Negative numbers sometimes can be used to s impl i fy the card punching. Fox example, the fo l lowing two cards w i l l produce exactly the same resul ts for neutral germanium:

32 1 GE I 4P 4D 3D10 4S2 4P 4D 3Z -3 GE I 4P 4D 4P 4D

The former s ta r ts with Ar I 3p and adds on 14 more e lectrons; the l a t t e r s tar ts wi th Kr I 4p , changes 4p to 4p , and adds on 4d . Note that i f

10 2 the previous rare gas did not include a f i l l e d shqll (such as 3d and 4s above or 4f ) i t must be included on the conf igurat ion card. I f a f i l l e d shell which should have been included is omitted i t is most easi ly seen bv looking at the f i r s t l i ne of the output. E0N=(degree of i on i za t ion ) . I f ION *'s not the expected value, a wrong conf igurat ion card i s usual ly the source of the d i f f i c u l t y .

(c) A sui table s ta r t i ng atomic potent ia l fo r the current value of Z is obtained by scaling the input potent ial ( f i r s t conf igurat ion only) or the f i na l SCF potent ia l from the preceeding conf igura t ion , and a radial in te ­grat ion mesh is constructed.

(d) A SCF i t e ra t i on is carr ied out by repeatedly (up to MAXIT times) (1) in tegrat ing the d i f f e r e n t i a l equation to obtain a radial

funct ion PNL(I,M) for each o rb i t a l H, (2) recalculat ing the potent ia l RU from the PNL, (3) comparing the PNL and/or RU with values from the preceding

cycle, and revis ing as appropriate. Arv HX calcu lat ion is started by f i r s t making an HFS calculat ion

(KUT = 0) for at least two cycles to obtain approximate wavofunction? ( re­quired for the calculat ion of HX potent ial functions for use in the d i f f e r ­en t ia l equat ion), and then automatical ly switching to HX (KUT = -1) wnen

-18-

ULLTA T0LKM2 (729-736). ITo calculate via HFS only, make T0LKM2 = 0 on the control card; to calculate for two or more values of KUT in succession (in the order +1, 0, -1, -2), make T0LKM2 = TOLEND on the control card, KUT1 = 1 (or 0, if KUT = 1 is not desired), and KUTC = 0 if KUT = 0 is not desired (cf, statements 996- ).

Stabilization of the iteration is via RU on early cycles (600-660), and on the PNL on later cycles (454-490). The control involves quantities ALFM(M), recomputed each cycle within the range ALFMIN to ALFMAX (normally O.E to 1.0);if the SCF iteration should fail to converge because of oscilla­tory instabilities (very unlikely), three things can be tried;

(1) decrease the valjes of ALFMIH and ALFMAX read in on the configura­tion card (see code near statement 215 for details);

(2) increase the value of TOLEND on the control card; (3) on the card just before statement 708, increase the number

"5.E-06", which is an upper bound on the fractional change allowed in the tail of each wavefunction.

(e) Using the final SCF radial wavefunctions PNL, various one-electron radial integrals (e.g., kinetic energy, electron-nuclear potential energy) are computed, the subroutines POWER, ZETA1, SLI1, and RCN3S are called, and output is written on the output file 0UT1, and on output TAPE2, for 'jse by the computer program RCN229, or (if IHF > 0) HFWRTP is called to convert from a linear to a logarithmic mesh and write wavefunctions on TAPE7 for use by program HFM0D7.

(f) If the values cf T0LKM2, TOLEND, KUT, and KUTD so indicate, additional calculations are automatically made for the same configuration, but with different values of KUT [see under (d) above).

4. Input/Output Units

Jata input and printed output are via files INP1 and 0UT1, respec­tively. In addition, disk file TAPE2 or TAPE7 is used for output wave-functions (input to RCN229 or HFM0D7, respectively), and disk file TAPE4 is used for temporary storage.

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<>. Output

The amount of printed output, is controlled by the quantities ITPOW, IPTV'U, IPTEU, NORBPT, IHF, and NPR punched en the control card, in ways described under Ti?. Provided flORBPT.GT.O, the first two anj the last NORBPT radial wavefunctions are printed out. For each function the value is printed at every fifth mesh point in a two-column format (mesh points 1, 11, 21, 31, in the first column, and mesh points 6, 16, 26, 36 in the second column); the radius is printed only at mesh points 1, 11, 21, il If NORBPT.GT.5, continuum wavefunctions are printed in the above manner, and also at every mesh point in a ten-column format.

Interaction energies between two electrons, and one-electron and total binding energies (without corrections, and with relativistic anJ/or correlation corrections) are printed out on the final page; ALL VALUES ARE IN RYDBERGS.

The final line of each listing (except for timing information) contains the configuration identification, the number of parameters required for an energy-level calculation in the single-configuration approximation, and a list of the parameter values together with an integer code number:

0 = E = total binding energy (in rydbergs) 1 = Fk(e.iLi) (in units of 1000 cm" 1) 2 = .-.. (in units of 1000 cm" 1) 3 - F k( V.) 4 • GV^J)

The order of the parameters is the same as that required as input to program RCG.

Finally, if IHF = 0, then in the main program (statement 939) most of the important output information is written on TAPF2. The information includes a configuration serial nuttiDer ("1" for the first configuration, and automatically made one greater for each succeeding configuration), the atomic number IZ anJ ionization stage ION (0 = neutral), and a number of Wfvefunctions which for the first configuration of given IZ and ION is Jeter-mineJ by HORisPT, and which for later configurations is such is to Jelete any wavefunctions deleted from the first configuration. (The purpose here

-20-

is to delete most closed-shell wavefunctions, which are not needed by RCN229, so as to decrease required disk read/..rite and storage requirements in RCN229

If 1HF j* 0, then wavefunctions and other information required to start a MF calculation are written on TAPE7. If IHF > 1, the entire RCN calcu­lation described above is carried out. If IH"7 - 1, most radial integrals are not computed—only the call of ZETA1 to calculate a relativistic enerqy correction for carryover to the HF program; if IREL > 0, the HF program calculates its own relativistic energy, and time can be saved by using TOLEND =• 0.05, so that the RCN SCF iteration is not carried to final con­vergence,

6. Continuum Functions

A continuum wavefunction is computed when the last orbital defined on the configuration card is singly occupied, has NLQCD8 - "99S", "99P", "99D", etc., and a positive eigenvalue (kinetic energy) EE8 (in rydbergs) has been punched in columns 75-80.

In columns 61-65 of the control card, a quantity EMX should be punched with a value equal (approximately) to the largest value of EE8 contained on any of the configuration cards. The action of EMX is to change the radial integration mesh R(I) which is set up: instead of AR ; R(I)-R(I-1) being doubled at mesh points 41, 31, 121, 161, ..., no further doubling is done after AR becomes = —j . This insures that the mesh interval will not become so large but what there will still be about 10 mesh points within each half-wavelength of the continuum function, even at large R.

It is of course necessary that the dimension of the integration mesh be great enough, or that EMX be small enough, so that sufficiently large radii can be reached: the value cf "F" printed via Format 97 of SCHE0 should not differ from unity by more than about 0.1. The required dimen­sion is usually auite large: In place of the 611 ooints used for bound functions, 2201 points are used for continuum functions. Storaqe space was obtained by olacinn PNl and 0</(j in RCN and PNL in RCN2 in larrje core memory. (Continuum wavefuncltons cannot be handled by the HFM0D7 program, because the logarithmic radial mesh becomes much too coarse at large R.)

- 2 1 -

7. Rc-lat iv ist ic Corrections

Both RCN2 {the HX nethocl) and HFM0D7 (the Hartree-Fock approach) provide the option of employing a method of approximating r e l a t i v i s t i c ef fects w i th in the format of a non - re la t i v i s t i c ca lcu la t ion . This is done by rewr i t ing the non- re la t i v i s t i c one-electron d i f f e ren t i a l equation for Pn,s,i as

{.d,.^i.,v,.^ r.'-»vr ( dr" r1" L J

. v ^ [ , ^ , . . . ».(r))] • ' ( g L)[^ . i ] }p , ( r ) . . V l

where a is the fine structure constant, r is in Bohr units and all energies are in rydbergs. V ^ r ) is an SCf central field potential such as that com­puted by the HX method. The spin-orbit term of the Pauli equation has been omitted to prevent the introduction of the quantum number j. Condon and Shortley show that the terr.i involvino [? i '/P. - 1/r] contributes onlv for s-states if the potential is nurel.v Coulombic. This term is small for non-s-states in the HX or HF method and it has been neglected in this code. The factor

[i • £ ( < • ' - v V ) ) ] ' 1

has been retained to avoid a 1/r s ingu la r i t y .

In RCN29, resul ts obtained by solving the above equation with the HX metnod are labeled HXR while chose obtained by the HF method in HFM0D7 are labeled HFR. Table I compares HX, HXR, HF and HFR calculat ions for one-electron bindim; energies of U 1 with Dirac-Fock and experimental resu l t s . Table I I Mkos a s imi lar comparison for spin-orbi t coupling constants. Tne deta i ls of the s tar t ing series for the solution to the above equation are given in

2 the appendix.

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Table I One Electron Binding energies, Center of Gravity, for U I 5f 36d7s 2 (ry)

HF HX HXR HFR DFa Exp b HX HXR HFR DFa Exp b

Is -7433 -7434 -8644 -8591 -8559 -8497

2s -1302 -1302 -1630 -1619 -1612 -1599

2p -1258 -1258 -1356 -1357 -1365 -1355

3s -333.3 -333.8 -417.9 -415.0 -413.2 -407.3

3p -311.9 -312.4 -341.1 -340-3 -342.5 -337.9

3d -272.3 -272.8 -272.0 -271.6 -270.1 -266.3

4s - 87.04 - 87.43 -110.2 -109.0 -108.6 -106.0 5s - 20.12 - 20.34 - 25.71 - 25-28 - 25.15 - 23.3

5s - 3.365 - 3.422 - 4.404 - 4.283 - 4.248 - 5.2

5 f - 1.269 - 1.356 - 0.740 - 0.663 - 0.704 — 6d - 0.533 .528 - 0.372 - 0.376 - 0.416 — 7s - 0.333 . 333 - 0.414 - 0,403 - 0.398

a) J- B. Mann, private communication. For «.t0, these values are 2j + 1 weighted averages, except for 5f <ini 6d, where the 5f_ and 6d_ values are listed.

b) K. Sieabahn, et.al.. ESCA; Atomic, Molacular & Solid-State Structure Studied by Means of ETectron Spectroscopy (AlmquTst & '-^kseTTr, il-nsala), 1967. Weighted averages f.'-o used as for OF.

- 2 3 -

3 2 Table I I Spin O r b i t Parameter Values c. . ( r y ) f o r U I , 5 f 6d7s

HF^_ HXa HXRa _ H F R ^ DFb E x p c

2p 132.8 133.5 200.0 199.1 187.8 135.3

3p 30.12 30.09 46.06 46.21 43.72 43.0

3d A.027 4.943 5.333 5.327 5.293 5.17

4p 7.799 7.704 11.90 11.98 11.54 11.2

4d 1 .16G 1.151 1.246 1.264 1.238 1.23

i f 0.237 0.230 0.220 0.221 0.250 0.23

5p 1.761 1.339 2.813 2.701 2.763 3.2

5d 0.216 0.225 0-237 0.230 0.334 0.26

5f 0.0210 0.0206 0.0158 0.0165 — - . . .

5p 0.267 0.291 0.444 0.427 0.539 0.49

6d 0.0210 0.0161 0.0132 D.0139 ___ __ ~

a) Values computed via Blune-Watson method.

b) Values computed from - ^ = ( E r " , ' j + - Laii") • 2/ (2t + ^]

c) Ref b of Table I . Values computed as for DF.

- 2 4 -

C. PROGRAM HFMQD7 [D.C. G r i f f i n and R.D. Cowa.i (November, 1974)]

1. Description This is a Fortran IV program based on the well-known program of C.

Froese-Fischer . I t has been extensively modif ied, to accept input from RCN29 and with the mul t i -conf igurat ion option removed, by D.C. G r i f f i n [Thesis, Purdue Univers i ty , (1970)] . Further modif ications make possible inclusion of r e l a t i v i s t i c terms in the d i f f e ren t i a l equation i f IREL >0 ( G r i f f i n , 1973), and incorporate a new method of s t ab i l i z i ng the SO' i t e ra t i on .

Program Out!ine

The program consists of a main program and 42 subroutines, of which only a few of the more important ones will be described:

HFK0i)7 is the main program, and does little except read a control card and call various subroutines.

DATA reads the input information vv.ich RCN29 has written on TAPE7 and sets up calculation... accordingly.

SCF controls the self-consistent-field iteration for a given con­figuration (or for a given LS tern of a given configuration). One iteration cycle consists of two parts: (1) A considerable number of calculations are made for different o,vitals, in an order that depends on which orbital appears to be farthest from self consistency; (2) each orbital is recalculated once in the order from innermost to outc most subshell. [The iteration cycle in RCN29 consists of only th Js second ; phase and hence convergence requires many more cycles to reach self- I consistency; the total number of orbital calculations is, however, approximately the same in RCN29 as in HFMOD7.]

SOLVE controls the iteration on the orbital eigenvalue to obtain a J normalized radial function having positive initial slope and the correct J number of nodes (= n-S.-l). 1

-25-

2. [njHrt The fi'-st card is blank if HFM0D7 is to be run. A 2 in column 1

will skio HFH0D7. (This feature was inserted when the three codes were overlayed as one and it is still there.)

Most of the irvut has already been discussed in the RCN29 writeup. Here we need describe only the HF control card (actually, two cards): Columns

1- 10 13- 15 38- 40 13- 45 48- 50 53- 55 61- 65 68- 70 76- 80

Co umns 1 -5 6 -10 n -15 16 -20 26 -30

UFTOL is the maximum allowable absolute change in any wavefunction (at any mesh point) from one SCF cycle to the next before ending the iteration.

NITER is the maximum allowable number of SCF cycles. (If exceeded before WFT0L is satisfied, the calculation is ended, and no output written on TAPE2 for this case.)

ACMIN is the minimum allowed value of the acceleration factor (amount of trial function included in trial function for the next cycle).

Format Variable Normal Value

El 0.2 WFTOL l.E-07 13 NITER 15 F3.1 ACMIN (default = 0.2) 13 ISCACC 0 13 1SENCD 0 13 IGRANG 0 F5.0 TlMPM (default = 280. sec) 13 IPRINT 0 F5.3 HXFAC 0.95

Format Variable Normal Value 15 NSLV 51 15 ITERSL 0 F5.3 QNRNG 0.5 F5.3 FACHL 0.65 15 IRCN 0

..,!., U4Ul«ll!iiiiMi*..' • • - - w » ^ i ^ -

-26-

If ISCACC = 0, the acceleration factor ACC(M) for each orbital will be auto­matically modified (between the limits ACMIN = ACC = 0.994) in an attempt to speed the SCF convergence. [Note that ACC in HFM0D7 is equal to 1-ALF of RCN29.]

ISENCD f 0 deletes portion (1) of the calculation described under SCF. IGRANG f 0 excludes the off-diagonal Lagrangian multipliers t. .. TIMPM is the maximum computational time allowed for RCN29 and HFM0D7 (in

seconds) before exiting from HFM0D7 to load RCN229.

This feature was designed to allow one to get integrals from RCN229 between those configurations already completed before time runs out. Since one can either allow a long run time or restart the problem at LLL this really isn't needed. A large number like 1999 will avoid such an exit to RCN229 before the HFM007 calculation is complete. If IPRINT f 0, eigenvalue-iteration information i_- printed (for the outer­

most two orbitals only, if IPRINT = 1). HXFAC determines the magnitude of the term that modifies the inhomoganeous

differential equation to make it approximately homogeneous, tD eliminate convergence difficulties. If HXFAC > 0 and the atom is less than three­fold ionized, this modification is used for the outermost two orbitals if they are singly occupied. (The pertinent code is in subroutine SOLVE following statement 4 and at statement 125.)

IRCN accomplishes the same thing as the variable IZHXBW of RCN29.

The remaining quantities on the second control card deal with calls of SOLVEl and SLVTST (at the end of SCF, if ITERSL t 0 and the above HXFAC is in effect for the outermost orbital) to make a diagnostic calculation of the shape of the normalization curve. (Several variables not listed above are read from the control card, but are no longer used in the code.)

-27-

3. Input/Output Units Control card input is via INP2; printed output via 0UT2. Input

from RCN29 is via TAPE7, and wavefunction output to RCN29 via TAPE2. Printed output is in somewhat different format from that of RCN29, but is basically the same in scope.

4, Miscellaneous If no diagnostic norm curves are to be computed (ITERSI. = 0) then

subroutines SOLVE 1 and SLVTST may be discarded, In any case, the code sections in SLVTST and HXPOT having to do with microfilm plotting (involving library subroutine PLOJB) should be deleted.

All plotting features have been commented out of the LLL version. Dummy subroutines were inserted where necessary. They all contain the state­ment DUMMY = 0 to facilitate finding them with TRIX.

This program was designed to handle only inhomogeneous differential equations, which excludes all configurations for which there are nc exchange terms (namely, one-electron atoms, and s configurations of helium-like ions.) An attempt has been made to overcome this limitation by a section of code at the beginning of subroutine HXPOT, but this feature has never been tested. A simple way of handling these configurations is to use RCN29 only (since HX gives exact KF results in these cases); if He-like configurations are to be passed on to RCN29, a simple procedure would be to let RCN29 write s configurations directly to TAPE2 rather than passing them on to HF (and to delete the TAPE2 rewind at the beginning of HFM0D7).

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D. PROGRAM RCN229

1. Description

This i s a Fortran IV program for the CDC 7600 that reads information from the output TAPE2 of RCN29 or HFM0D7, calculates conf igurat ion-in teract ion radia l Coulomb integrals R and e l ec t r i c dipole and quadrupole

IV\ f k l reduced matrix elements P : ' ; = < Z«rK •»«. > (k = 1 or 2 ) , scales a l l energy-

k k k 12 leve l -s t ruc ture parameters F , G , R , £, and punches a complete set of data input cards fo r making energy-level and spectrum calculat ions via program RCG Hod 5 or Mod 6. [Note: For HF wavefunctions, RCN229 output is useful mainly for center -o f -grav i ty (not LS-term-dependent) ca lcu la t ions . ]

Program Outline The program consists of a main program RCN229 and eight subroutines. RCH229 reads control cards and cal ls various subroutines accordingly. OVER computes an overlap integral between two specif ied radial wave-

functions from the same or d i f fe ren t conf igurat ions.

ZETA2 computes si ogle-configurat ion or conf igurat ion- in teract ion sp in-orb i t parameters. The former are now computed by ZETAl (of RCN), and the l a t t e r are usually so small as to be neg l i g ib le ; i n pract ice th is program is almost never used.

fk l DIP computes e l ec t r i c dipole and quadrupole integrals P : „ , .

SLI2 computes conf igurat ion- in teract ion integrals R' ' . G5INP is a master program which ca l ls SLI2 and DIP as appropriate,

and prepares the input data decks for RCG. (h j2 h\ \m] m 2 m 3 ^ S3J. calculates 3-j symbols

S3J0SQ calculates values of

FCTRL calculates factorials of an argument A for use by S3J and S3JDSQ.

-29-

2. Input Input consists of radial wavefunctions and other information

written on TAPE2 by program RCN29 or by program HFM0D7 and of a G5INP card or one data card for each calculation to be made. The G5INP option is the one usually used. When the first three characters on the data card are "GGI", subroutines are not called directly but only via G5INP. Control is then transferred to subroutine G5IMP, which de-codes the card according to the fallowing format (statement 100}--

Columns Format Variable Normal Value NCMIN 0 NCONF # of configurations NSC0NF(3,2) fln zeros IABG 0 IV 0 EAV11 0.0 FACT(5)( 1.00, 1.00, 1.00, 1.00, 1.00

{or 0.85, 0.95, 0.85, 0.85, 0.85 lor 0.67, 0.67, 0.67, 0.67, 0.67 etc.

IMAG 0 IQUAD 0 ALF 0.0 IPRINT(5) 00000 ICON 0 ISLI 1 or 2 IDIP 0

6-7 12 8-10 13 11-2U 2(11,212)

21 n 22 n

23-30 F8.5 31-55 5F5.2

56-59 14 60 n

61-65 F5.0 66-70 511

73 11 74 11 75 n

-30-

NCMIN is never used in pract ice. NCONF is the number of configurations computed.

M/, IMAG, and IPRINT are quant i t ies not used in RCN229, but simply transmitted unchanged to RCG. ( I f desired, they can be ignored here, and modified by hand repunching of the RCG control card.)

I f IABG.GT.O, i t is a signal that the ef fect ive-operator parameters u, », •<, are to be included in the RCG ca lcu la t ion . Values of these parameters cannot easi ly be computed theore t i ca l l y , hence, RCN229 simply leaves room at the proper places on the s ingle-conf igurat ion parameter cards, where empirical parameter values can be inserted by hand [except that .he number (ALF) punched in columns 61-65 of the control card w i l l be punched for s].

EAV11 is the value desired for the center-of -grav i ty energy E of the f i r s t conf igurat ion [ i n uni ts of 1000 cm" ). I t merely sh i f t s the energy scale of a l l energy levels calculated hy RCG,

FACT(5) are fudge factors to be applied to the theoret ical values of the para­meters Fk(H i ) , c , F k ( e . , i . ) , G k f f . . l , ) , Rk(< l t'/) respect ive ly .

i i • i J ' J i J i J to give scaled-down parameters for input to RCG. I t is known empir ical ly Lhat computed energy-level in tervals w i l l agroe better wi th experiment i f the parameter values are 10-40" smaller than theore t i ca l . Values of FACT of 0.85, 0.95, 0.85, 0.85, 0.50 are assumed by the code i f zeroes are read i n ; fo r HX wavefunctions, values of 0.85, 0.95, 0.35, 0.85, 0.S5 are about r i gh t for ^ 10-fold ionized atoms with Z - 25; values should probably be more l i ke 0.75 for most neutral atoms; Cowan has previously estimated a l l f i ve values = 2/3 as a good f i r s t guess for the act in ides. The experience of Mark Frad and K. Rajnak for U I is that when HFR is used, the factor is closer to .4 for the configurat ion interact ion in te ­gra ls , -v. .9 for £, f, perhaps as low as .06 for c d and ^ 1 . 1 for : . The code requires a single factor fo r a l l c ' s .

IWG • 1 , 2 , or 3 s ign i f ies that RCG is to calculate magnetic dipole spectra for configurat ions of the f i r s t , second, or both p a r i t i e s , respect ively; IHAG is not used by RCN229, but is simply passed on to RCG.

I f iqUAD = 1 , 2, or 3, e lec t r i c quadrupole spectra w i l l be computed by RCG for the f i r s t , second, or both p a r i t i e s , and RCN229 computes and punches values

i2) of P: . ; accordingly.

- 31 -

i f ICON / 3. the RCG-inp'jt control and conf igurat ion cards are not punched.

I f iSLI

I f tOIP

» 0, p r i n t a l l SL32 output

* 1 , omit restore and most pr in tout

• 2. oc i t a l l but f l oa l punctied-card Image (except that parameter values are unsealed)

- 3, omit a l l p r in tout

I * 8, dc not punch dipole cards

I * 9, do not punch nor p r i n t Other values of ID IP, in combination with ISLI - 4 . provide some special-purpose options of no in te res t here.

I f the C5INP option is -iat used, the data c a r l is read as a s t r ing of HO BCD characters at s ia i c iwM i'00 of RCH229, and is then decoded according to a format appropriate to subroutines other than GB1NP. The f i r s t three characters (columns 1-3 on tne data card) specify the subroutine to be ca l l ed , columns l l -2 f l contain the ser ia l nunbers of the two conf igurat ions to be jsed in the ca lcu la t i on , and colunns 2*1 -30 or 21-40 the radial wavefunct' ns to ^e selected. For example, i f one wishes to calculate sp in-orb i t in tegra ls ; . . , between members of the Rydberg ser ies , the subroutine ZETA? must be ca l l ed . The input is then:

-32-

Column

1-3 Z£T 8-10 Atomic # {probably not used)

11-15 Serial ninnbor of f i r s t configurat ion 16-20 Serial number of second conf igurat ion 23-25 2B-30 33-35 3^-40

Orbitals to be usedjin case of ZETA2, only the first two are involved: n£ and n'l in 23-25 and 23-30 respectively.

A separate card is necessary for each nair of configurations. These soin-orbit integrals are not automatically calculated via the G5INP ootion so this is the only way to get them. For configurations involvinq r l n, £

between singly occupied shells T . ,. = r c,„ so this calculation is probably not necessary. 3 The G5INP option will give the required R integrals. If the R5INP option is not used a separate card calling the appropriate sub­routine is necessary for each integral desired.

3. Calculational Procedure In order for 65INP to properly prepare input decks for RCG, it

is necessary that the pertinent configurations be arranged on TAPE2 in a specific order, and hence that the input configuration cards for RCN29 also be arranged in this order:

All configurations of given IZ and ION which are to be included in one RCG calculation must be arranged in succession. Usually the lowest-energy configuration is chosen as the first one; all configurations of this same parity follow, and then come all configurations of the opposite parity (if any). Within a given parity, all configurations which are members of a single Rydberg series (e.g., 3p 3d, 3p 4d, 3p 5d,...) should be placed in succession. For any one configuration, the pertinent input data card for RCN29 should be punched so that all nOn=-filled subshells (other than the outer, most-weakly-bound, singly-occupied subshell) are arranged in order of Increasing n, and for given n in order of increasing i. [If orbitals are not arranged ki::S way, G5INP attempts to rearrange things but is not bug-proof.]

-33-

The calculat ional procedure is as fo l lows:

(1) TAPE2 is searched to determine the number of conf igurat ions of each par i ty w'lich have common values of IZ and ION, and which appear on the disk as successive configurat ions wi th no intervening conf igurat ion of d i f fe ren t IZ nor ION.

(2) A l l subshells !.w which are f u l l (w = 4V + 2) in a l l configurations are ignored. The remaining subshells are arranged according to the requirements of RCG.

(3) The RCG control card is punched, containing the to ta l number of subshells and the number of configurat ions of each par i t y [NSCONF(3,2)], among other th ings,

(4) The I and w values are punched fo r each conf igurat ion.

(5) The s ingle-conf iqurat ion parameter values VPAR supplied by RCN29 or MFM0D7 (via TAPE2) are rearranged ( i f necessary—see parenthetical remark above), scaled by the factors FACT, and punched. [ F i r s t par i t y on l y . ]

(6) A l l possible conf igura t ion- in terac t ion parameter values R

are computed wi th the aid of ca l l s to SLI2, and punched. [ F i r s t p a r i t y . ]

(7) I f the value of IQUAD so ind icates, values of the e lec t r i c [21 quadrupole reduced matrix element P i t , are computed (via DIP) and punched.

[ F i r s t p a r i t y . ]

(8) I f configurat ions of both pa r i t i es are present, ( 5 ) , (6) and (7) are repeated fo r the second p a r i t y , and then values o f the electric dipole reduced matrix element P L , are computed (via DIP) and punched.

(9) A card wi th "-99999999." in columns 21-30 is punched, to signal

the end of an RCG ca lcu la t ion .

(10) TAPE2 is searched to see i f there ex is t addi t ional conf igurat ions of d i f f e ren t IZ and/or ION. I f so, items (2)- (9) are repeated fo r each such set of conf igurat ions.

(11) When TAPE? has been exhausted (or the conf igurat ion with ser ia l number NCMAX has been reached), a return is made to the main program, and a new input card read; a negative number in columns 8-10 resul ts in an ex i t from the program.

-M.

4- Input/Output Units

Data input and printed output are via files I!IP3 and 0UT3. respec­tively. In addition, PUNCKA Is used for output punching, TAPK2 for input wavefunctlons from RCN29 or HFMQD7. TAPE4 is defined on the program card of the current code but it is not actually needed. It is used in RCN29.

5. Output

Output includes f i l e PUNCHA ready to be used d i rec t l y as input to program RCG. (Such f i l e s may be useu without change or they may be modified by hand in various obvious ways. For example, the p r in t options in columns 66-7Q may be modif ied; i f one wishes to add or delete ca lcu la t ion of magnetic dipole spectra, th is imy be done simply by changing column 49 appropr iately; i f configurations of both parl'.y were included 1n the RCN29 and RCN229 run but one wishes to include only the f i r s t par i ty in the RCG ca lcu la t i on , i t i s only necessary to inser t zeroes in columns 16-20, and delete a l l con f igura t ion-de f in i t ion and parameter-value cards fo r the second p a r i t y , and remove a l l e l ec t r i c dipole cards, e t c . ) .

A l l information in PUNCHA is also pr inted / ia the output f i l e 0UT3, as t,

is some addit ional de ta i l regarding the calculat ion of the R and the P ( k )

6. Autoionization Calculations

In certain cases, a calculat ion is also made of the radial contr ibutions to Fano'sratid q fo r autoionized states . The code involved is that in G5INP, statements 600-660, and output is pr inted l i s t i ngs only. This calculat ion is made I f :

(a) there is only one configuration of the f i r s t p a r i t y , (b) there are at least three configurations of the second

p a r i t y , (c) the outermost o rb i ta l of the last conf igurat ion (number

I O ) is a continuum function (c > 0 ) , and (d) the outermost subshell is Singly occupied for a l l configurations

of the second par i t y wi th the possible exception of the f i r s t one (configurat ion no. NCN).

-Jt>-

Under these eondit ionr i t i'- assumed that :

(1) conf igurat ion NCN (the f i r s t conf igurat ion of the second pa r i t y ) involves some levels ly ing above the f i r s t ion izat ion l i m i t , which can autoionlze by conf igurat ion in te rac t ion wi th continuum states;

(2) a l l configurat ions fo l lowing NCN are members of one or more Rydberg ser ies , of which at least the las t conf igurat ion (NCX) is a member of th is series which l i e s in the continuum; and,

(3) that dipole t rans i t ions are possible from the lone conf igurat ion of f i r s t pa r i t y to a l l flydberg-series configurat ions (and perhaps also to the autoionized conf igura t ion) . A simple example i s :

At I 3s23p (odd par i t y ) 3s3p (even pa r i t y ) 3sZ3d 3s24d 3 s Z

E l d

3s 3 r ,d ?

3s e 3d ? 2

Absorption t rans i t ions from 3s 3p are possible to 3s3p and to 2 2

a l l 3s nd and 3s Ed; a l l o s c i l l a t o r strengths are grossly 2

modified as a resu l t of conf igurat ion mixing ot the 3s3p with 2

each 3s d. Fano's r and q are measures of the width and shape of the autoionized discrete l ines

3s23p 2P - 3s3p2 2 D .

The f u l l width of an autoicnized l i ne at half-maximum { in the l i m i t of large q or of q = 0) is

r = Zir < • L S IH | * E > ' : , (8)

where •?> is a basis function for one of the autoionized states and i|v is a basis function for a state of the continuum having energy E. For pure LS coupling conditions and Coulomb interactions,

-36-

r = 2v <\ S 1H|^ E > 2

"li r» "7 (9)

where R 1s the appropr ia te r a d i a l i n t e g r a l {computed by SLI2) and r. i s the corresponding angular f a c t o r (computed l a t e r by RCG). Assuming there i s only one value of k, or tha t one term in the sum is dominant and dropping a l l smal ler terms,

r - 2n < r k ) 2 < R k ) ' ( 1 0 )

The quantities tabulated in the computer listings are the eigenvalue EE or c of the valence electron; 1f EE < 0, the effective quant™ number

n* - — ^ 7 7 , (Z* » I0N • 1) ; (11)

a weighting factor

WT - f ^ l = | £ d (12) 3 ' #- - . - 1 / ?

•m -M equal to the inverse square root of the energy i n t e r v a l per uound

conf igura t ion ;

"RKSC" WT.Rk (bound) R k (free) ( rydbergs) 1 / Z , (13)

t. where R 1s a con f igura t ion - in te rac t ion in tegra l between conf igurat ion

NCH and one of the Rydberg-series conf igurat ions; and

"GAMMA/RKSQ" ' r / ( r . ) 2 = 2ir (RKSC2) (14)

-37-

Typica l ly , values o f r k are l i ke t0 .5 , SO that values o f r are l i k e 1/4 of the numbers tabulated.

The value of q for an absorption t r ans i t i on from an I n i t i a l state 1 to the autoionized discrete state ( s the unperturbed discrete state i) is

- (dipole mx. ulem. to discrete state) q " ^(d lpole elera. to cont . ) ( in te rac t ion elem.)

- < l ' r t > nci ' iT< 1[ rJ* E > <*E]H|ii/> U 3 '

e ____l± ' l i i i l - T _ T _ TT < 1 | r |v E > < * E I H 1 ; - -

For pure LS coupl ing, and again neglecting a l l but the largest term of the Coulomb-Interaction matrix element,

" T < 1 L S j ' r | * E L ' S ' J ' > < * E L 1 S , i H ' * L ' S 1 > (16)

( " " " " ' O < i L S J r ! ^ L - S l J : > a n g

= T. < n " r » r u n " • ! ' " , > < LSJ ' r ) *EL 'S"J '> r k < R R > m a x * ang

The quant i t ies tabulated are

..pnrpsr" JWT-^ i i l r l in 'J i 1 ) (bound) PDIPSC - ^ • • t " i n n « ' i f ' ) ( f r e e )

(17) 2 2 1/2 = radial dipole integrals (units of e a / ryd )

and

1 "V < i | ^ ; r ! ' ^ i • s • j • >. > ang

< 1 L S j l r l < * L ' S , J , > a (18)

< nairiin'z'> / j i . s „ j „ j „ r i r— Idiraensioniess),

IT <n"rnrlln"'i'"> RK

-38-

The quantity r, is again like +1/2, so that inclusion of this factor would roake q perhaps a factor 2 greater in magnitude than the number "Q*RK" tabulated in the computer listing. The values of<-| r | •"!>-,._ can be anything from very small to something likeV2; the ratio of the two values can thus be most any nagnitude and either sign. However, for a strong discrete line and a strong continuum, the magnitude of the ratio probably would lie somewhere between 1/2 and 2, so that the value of q would be within a factor 2 or so of the computer output value.

E. Storage 1. RCN29

Storage requirements for an integration mesh of 2201 points and up to 25 orbitals is, on the CDC 7600, about as follows (number of 60-bit words, octal):

Fortran (including local variables 102 K and system/library)

common 44.5 K LCM storage 160 K 1/0 buffers 6 K

Total LCM image size 334.5 K 2. HFN007

Storage requirements for an integrat ion mesh of 200 points (561 in HXWRTP) anJ up to 20 o rb i ta l s i s , on the CDC 7600 (60-b i t words, oc ta l ) :

Fortran Code (local var iables, system/l ibrary) 76 K Common (Storage) 33.1 K 1/0 Buffer 6 K Total LCM Image 137.3 K

3. RCN229 Storage requirements for up to 50 configurations, eight orbitals

per configuration (so far as the infornation written on TAPE2 is concerned), and 2201 mesh points for the CDC 7600 is about as follows (number of 60- bit words, octal):

-39-

Fortran Code (local variables; system/library) 56 K Common (storage) 70 K LCM (storage) 42.4 K I/O Buffers 6 K

Total LCM Image 21 c 4 K

LASL storage is a few K less in all cases since I/O buffers are not in SCM at LASL.

F. Conversion to Other Computers Conversion problems to other computers having a Fortran IV compiler

should be minor but because the LLL time-sharing requires file-manipulation statements not required by other systems the LASL version of the code is probably an easier one to convert. The ENCODE/DECODE changes discussed below have been made in the LLL code.

In RCN29 (main program), thers are ENCODE and DECODE statements at "• 223 and v. 237. These are concerned with memory-to-memory conversion of two words with format (12, Al) to three characters of BCD information [stored in NLBCD(M)], or vice versa. With compilers not having ENCODE/DECODE statements it may be necessary to accomplish the conversion in each case via formated Write and Read statements involving tape or disk.

The SCF iteration must be carried out to very tight tolerance if one wishes to compute meaningful excitation energies by differencing values of E for two configurations, because many significant figures are lost in this difference ( 6 or 7 significant figures for an actii.ide). This means that most calculations must be done to 9 or 10 significant fiqures. On machines using 60- or 64-bit basic floating-point word lengths, this is no problem. With the IBM 360 32-bit words (7 significant figures?), it may be necessary to define as double precision most variables, includinq RSATOM, QO, PSQ, R, RU, RUEE, RUEXCH, PNL, EE, EI, EVEL, EDAR, V, EK.IN, EEN, UEE, UEX, "SCORE, RSVALE, PNLO, XI, XJ, T, Q, RRYD, VPAR, Fl, F2, XA, XB, EHN, RKP1, AO, A, B, ET2, ET3CR, and probably many otl ;rs.

Conversion problems for RCN229 should be minor except for ENCODE/ DECODE statements. These include format 19 (which was chosen for the 10-charac-ter word length of the CDC) and the following statement in RCN229 main,

laiiiiiBHiim niiMm>4*i

-40-

and statements 100 and •v 178 of 65INP. No double-precision calculations should be necessary.

An end-of-file test on unit 2 at statement 111 of G5INP may have to be altered for non-CDC compilers.

G, Comments, Discussion and Warnings 1. Output

a. Continuum output from RCN229 The output from RCN229 gives R integrals involving a continu urn

-1 11? funct ion in both " ry " and "cm ". The un i ts are r ea l l y ry ' and the conver­sion to cm is made as i f they were ry . Divide resul ts in "cm " b y V • The spectrum code RCG is designed to take output from RCN229 and make the appropriate cor rec t ion. I f t h i s code i s corrected, RCG must be changed also,

b. RCN2/SLI2 Output fo r Configuration Interact ions of the General Form

w. w , w (Unprimed conf . ) • • • e . J • • • l * • • • s, y • • • ( 1 )

j x y

{Primed conf . ) • - - S,.J - - - £ x - - - fc y ' ^' J x y

w. with I = i , where l-^ is any un f i l l ed subshell ( l y ing e i the r before the x subshel l , as shown, between x and y , or fo l lowing y ) .

elements In th is case, conf igurat ion in terac t ion Hamiltonian matrix

( • [ H | t /> (3)

include not only the usual e lectron-electron Coulomb contr ibut ions

O I ?— I *'> , (4)

but also contr ibut ions from the one-electron terms of the Hamiltonian. The spin-r b i t operator leads to contr ibut ions involv ing the sp in-orb i t CI radial parameter

-41-

Vr/(jJ)7y^ 0 ^ '

The angular coefficients will not be discussed here. A lengthy derivation shows that the effect of the kinetic-energy and the

electron-nuclear contributions to (3) is (for Hf radial functions) to replace the ordinary R. integral(s) in (4) by linear combinations of radial integrals k k similar to the ordinary R. and R , except employing one radial function from (1) and three functions from (2) or vice versa, instead of two radial functions from each configuration. Consequently, for interactions of this type, SLI2 computes (and prints) four sets of radial integrels:

(A) R^ab.c'd'), R*(ab,c'd') = Rj{ab,d'c'),

(B) R^ab, ad'), R k(ab,ad'),

(C) RJjtc'b.c'd'), R*(c'b,Cd'),

(D) average of (B) and (C),

where either (ab,c'd') - (xx.x'y') or (yx.y'y1) or (jx.j'y 1). [The set (A) is the normal set; the sets (B) - (D) are these modified integrals].

The replacement for the normal R., made automatically by SLI2 in the output written on file PUNCHA, is as follows {Z-i =« ):

(I) In place of R^xx.x'y').

2 l

1 ' k>o k

(II) In place of Rrf(yx,y'y'),

2 9 1 f 2£+l \ ,- (I k i.\ nV., . . , . , 1 ,. (t k 1 \ „k, ,.

-42-

( I H ) In place of R H ( j x , j V ) ,

k x o J o o ri i ? r R . ( J x , j y ' ) + M j ' x . j ' y ' )

In the Rydberg-series case (w = w, = 0 ) , P. = P., so that th is * y J J

l as t expression i s approximately

1 r / l , k l \ n k , . W V M R^Ux.j'y'); ' t \ o o o / e

in this case, SLI2 replaces R*}(jx,j'y') by zero, and RCG (Mod 5 or 6 or 7} adds

2

(w. is the occupation number of the j subshell to the anqular coeffic"—t J k r

calculated fron {&/ for R . [This makes the square CI block of the Hamil-e L tonian matrix identical with each single-configuration block, except with F k k k reolaced by R., G redaced by R , and no analoa of E "1. a e a V J

c. Orthogonality integrals from HFR aro ,06 for <1 s 12s=" for U I bw. much smaller in HF. The code calculates the total binding enerq;' E assuming these orthogonality integrals to be exactly zero.

is not used.

d. Energies HFM0D7 prints out 6 total energies when the relativistic option

ET = total nonrelativistic energy ETR = total energy with relativistic correction ETCO = total energy with correlation (method 1) ETRCO = total energy with relativistic and correlation corrections ETCN = total energy with correlation (method 2) ETRCN = ETCN with addition of relativistic correction.

Method 1 is that described in Ref. 6, a modified free-electron-gas model, and method 2 is described in Appendix (T2).

-43-

e. Convergence in RCN29 Calculations If TOLEND is too small a diagnostic print will result on the

last 4 iteration cycles. (See pp£3a,b). The last line of such a print contains the iteration number (35 in eg.), timing information and DELTA, the change in the potential from the previous iteration cycle. If this is .000000 in all cases convergence is at least that good and probably sufficient for any­thing except differencing total binding energies.

T..1S problem occurs with TOLEND = 5 x 10 for HXR runs on some, but not all, actinide configurations. Since the starting potential is derived from that of the previous configuration computed it sometimes even depends on the order in which the configurations are computed. TOLEND and THRESH can be increased by a factor of 10 or the number of SCF iterations can be increased. The increased output when the diagnostic print occurs may overflow the oi'tput file.

A 3 in col. 1 Of the RCN29 Control Card will print the line containing DELTA for every iteration cycle so one can tell if oscillations are

5 occuring. For some actinide configurations, particularly 5f 7s of (J I, oscilla­tions are large and DELTA is still > 1 instead of 10" after 30 iteration cycles. In this case better convergence can be obtained by narrowing the range of allowed changes in the potential from one cycle to the next. This is accom­plished by punching values of ALFMIN and ALFMAX (col. 31 & 32) on the configura­tion card instead of using the default values. (See statement 213 of RCN29 MAIN). Experimentation shows that the best results for f s are obtained with ALF'ir.*! = 9 and ALFMAX = 5. DELTA is still .000006 on iteration cycle 34 but oscillations have been eliminated. An increase in the number of iterations (MAXIT) on the control card should achieve convergence. For most purposes A = 6 x 10" is sufficient.

There were several other actinide configurations where DELTA was i* 5 x 10 on the 34tb iteration cycle when the default values of ALFMIN and ALFMAX were used. Using values of 9 and 5 resulted in convergence (TOLEND = 5 x 10" ) after only 30 iteration cycles.

Using IREL = 1 instead of IREL = 2 will speed convergence. The results are not the same in the 2 cases, however. For 5f 7p of CF II the difference is % 1% in AZ for the Is orbital, At in F (f,f) and .3% in ^ 5 f. IREL = 1 results in the larger values. Because of the larger AZ, IREL = 1 night be preferable but further comparisons are necessary for a clear cut choice. Both cases are approximations to Dirac-Fock results and any calculated values will require some kind of scaling to get agreement with empirical results.

135 RU.RUEE.RUEXCH -192 .000000 -179 .337026 - 1 6 8 . 9 0 7 3 8 6 -151 ,804158 -137 .927864

-29 .783122 -17 .163693 - 1 0 . 9 0 9 2 1 3 -5 .449101 - 3 . 3 1 9 7 1 5

164! 000000 751227 13-591042

177.319951 24.226628 183.397751 41. 186.

844177 566236 55.859697

190.28767 7

-2. .0 534349 -.928067

-2.503644 -1.134014 -2.306964 -2.'. 648334 015397 -1.787551

-1.607392

035 , MOD. s'.

IS HVFN) 000000 000010

1.0000052 1.564281 2.000010

-9609.466143 1.917790 1. 2.000010 2.

R»VSELF, 997417 000010

R-VTOTAL, 1.999934 2.000010

•S936. -30.

.000000

.877676 -180.060191 -16.279279 -170.138531 -12.057688 -153 -6,

.203462 • ,660078 •139.297512 -4.537966

.0 .000000

7,553505 .0 4.912757 .0

1 , ,277973 .0 .272637

.0

.0 .000000 7.553505 .0

4,91275? .0 1. ,277973 .0 .272637

.0

035 , MOD 2

25 . WVFN1 . 000000 .000002

1.0000009 .475145

2.00D002

-1828.449315 .655583 1 2.000002 2

R«V/SELF, .507119 .000002

*9936 -30 . 000000 .886934 -179.460096 -18.293222 -169.347808 -12.079502 -152 -6 ,522054 • .700512 -138,903670 -4,670981

" .0 .000000 1.716763 -.000000 -1.188121 -.OOOOOO

.330082 .0 -4.052753 .0

" .0 .000000 1 .71S763 -.000000 -1.186121 -.000000 -A .330082 .0

-4.002753 ,0

035 , MOD

2

2P . WVFN) . 000000 .000000

1.OOOOOOO -1502.504595 R*V/SELF , R»VTOTAL. 035 , MOD

2

2P . WVFN) . 000000 .000000 .485373 2.OOOOOO .934830 2.OOOOOO

1 2 .565812 ,OOOOOO 1.852325 2.OOOOOO

16. -1 . 197034 771264 -99,739199 -1.236898 -76.194956 -.715218 -60. 140746 423995

-JO 938821 153317

77, 91, ,988443 405056 94.663906 191.780255

118.290355 191.970362 134. 191 , .731908 .995834 153 191

588977 999938

-2, -1 , , 1854 77 , 176320 -2.403105 -1.017153 -2.485311 -.665580

-2, ,872656 .424829 -2 527798

1532 55

R*\ /EFF, (IF DIAONOSTIC PRINT OUT R»»/3, R»VT, CALC. rfVFN 2 2 ,000010 .OOOO10 2.000010

2.00001O 2.000010 2.000011

2. 2 ,000010 .000010 2 2

000010 000010

117 -3 .428136 .047477 -100.893505 -2.492489 -77.318333

-2.085453 -61 -2

.128410 ,01'051 -42 -2

.040465 .000152

.010167 .0

.000340 .0 .OOOOOO 0 .000000

-0 .OOOOOO ,0

.010167 .0

.000340 .0 .OOOOOO ,0

.OOOOOO .0

,OOOOOO .0

( I F DIAGNOSTIC PRINT OUT-R-VS, R»VT, CALC. rf\/FN

R*VEFF, ( I F DIASNOSTIC PRINT OUT-R-VS, R'VT, CALC. W\/FN 1.987422 1.999139 1.999997 2 , 0 0 0 0 0 0 2 OOOOOO 2 .000000 2 .000000 2 .000000 2 . 0 0 0 0 0 0 2 OOOOOO

936.000000 -29.240774 - 179.017701 --16.821391 168.521666 -10.775613

-151 -5 249993 726S60 -137

-3 336385 -965150 115.489134 -2.695767 -98

-2 977887 343860

75.449880 -2 124103 -59

-2 305148 031,195

40 -2 311754 000582

.0 .112482 .023695 -.195241 -.026735 -.243629

055096 103703 " 002961 409580 .081549 .5562 78 038111 435135 -.106263 .168738 -068169 050931 145096

003445

.0 .112462 .023895

-. 195241 - .026736 -.243629 055098 103703 -002961 409560

.081549 .536278 036111 435135 -.106263 .168738 068169 050931

145096 003445

5 26.31 000000 226 1

DELTA 28 .68 1 .00 .67 67 44 .44 .44 .30 .30 .30 .20 .30 .30 .30 .30 30 44 .3

\ 000000 226 1

DELTA Iteration number

-44-

The scale factors will just be different in the two cases. It is important to use the same value of IREL for comparison of different configurations, however.

f. AZ Values from HFR When the relativistic correction is added to the HF equations

S.+1 P no longer goes like r near the origin. Consequently, the initial slope AZ is not a measure of the electron density at the nucleus and the numbers in that column are meaningless. HXR solutions have been fit by a polynomial

5 + 1 and in this case AZ is the coefficient of r

2. Miscellaneous Comments a. As written, continuum functions can be computed only in the

HX code (RCN29). There's sufficient uncertainty in the continuum functions that doing HF on the bound states of the configuration is probably not worthwhile, although it could be done with relatively minor modifications.

b. It's best to keep the order of the shells on the configuration cards in order of increasing n and within a given n, increasing I. V'hen several configurations are being computed, empty shells need not be punched even though they occur in only part of the configurations, i.e., one can use

5F3 601 7S1 7P1 and 5F4 7S2 instead of 5F4 6D0 7S2 7P0

H. References 1. R. D. Cowan, "Programs RCN M0D29/HFH0D7/RCN2 M0D29", LASL, Nov.

1974, Unpublished. 2. R. D. Cowan and D. C. Rriffin, J. Oot. Soc. Am., in press.

3. H. Blume & R.E. Watson, Proc. Roy Soc. London, A270, 127, (1962). 4. RCG is a code written by Cowan to compute angular matrix elenents and,

given the appropriate radial integrals, diagonalize the energy matrices and compute the spectrum. RCGM0D5 is available at LLL as GFIVE and RCGM0D6 is available as RCGM6.

-45-

5. Herman and Skillman, Atomic Structure Calculations, Prentice Hall, 1963. 6. R. D. Cowan, "Atomic Self-Consistant-Field Calculations using

Statistical Approximations for Exchange 8 Correlation," Phys. Rev. 162 54, 1967.

7. Lindgren & Rosen, Int. J. Quant. Chem. S5_, 411 (1971); Phys. Scrip. 6, 109 (1972).

8. Bethe & Sal peter, Quantum Mechanics of One-and Two-Electron Atoms, Springer-Verlag, Berlin, 1957, Sect. 13, especially eq. (13.6) and paragraph following (13.11).

9. E. U. Condon & G. H. Shortley, The Theory of Atomic Spectra (Cam­bridge Univ. Press, New York, 1970) p.130.

10. C. Frose-Fischer, Can. J. Phys. 41_, 1895 (1963); ANL-7404 (1968); Comp. Phys. Commun. 1_, 151 (1969).

11. R. D. Cowan & J. B. Mann, Jr., J. Comput. Phys., Ji, 160, (1974). 12. All numbers hereafter referred to as punched go into a file called

PUNCHA which can then be punched or stored as desired. 13. This was added at LLL because an end of file was sometimes not

written on TAPE2. The default is to 50 if NC0NF = 0. If the EOF gets written the code will run properly with the default but putting in NC0NF will ensure running even if the EOF is missing.

14. Mark Fred, ANL, private Communication. 15. W. C. Martin, Jack Sugar & Jack L. Tech, Phys. Rev. A6. 2022 (1972). 16. U. Fano, Phys. Rev. 124_, 1866 (1961). 17. The version obtained from LASL required 4 configurations but this

has been changed to 3. 18. E. Clementi, J. Chem. Phys. 38, 2248 (1963); 39, 175 (1963);

42, 2783 (1965).

-&6-

I. Appendix: 1. Small-r Solution for the HXR Method

In trying to find a series solution of the differential equation for

5,

P . ( r ) , we run in to d i f f i c u l t y with the Darwin factor

1 0 1 * (F - V)c 2 /4 Z 1

10. I ( 1 )

1 + (e + 4/ct2)r/2Z J

where we have taken V = - 2Z/r for small r. The quanti ty e + 4/a" - Z + 274 * 274 is so large that a series expansion of the factor in brackets in (1) is inva l id at r~ (on an integrat ion mesh r. = jh) for any pract ica l value of h; fo r example, on a mesh commonly used for non-rela-t i v i s t i c problems,' h = 0.0022 Z " ' / 3

1 the l im i t a t i on ZrJaH « 1 requires P 3/4 '

Z » (0.0022 • 274 £) = 46. In order to be able to handle a l l values of Z ( inc luding Z as small as u n i t y ) , we t r y using a f ixed value

d x d i h ^—? • < 2 )

i n \ + (c * 4/0! )jh/2Z Because of this approximation, there is no point in carrying series

expansions beyond a couple of terms and so we assume

P * r A + 1 * a / + 2 (3)

for the {unnormalized) radial funct ion. D i f fe ren t ia t ing th is expression we f ind

In the approximation V = -2Z / r , the Darwin term is

and the d i f f e r e n t i a l equation for P.(r) becomes to second order

P" = (A + l l A r * " 1 + a (k + 2){\ + 1 ) r A

"L r 2 " r _ [r*+ 1

+ a ,r l t 2 ]

-47-

Equating coef f ic ients of r " gives

[\ + 1)A = JZ.CE. + 1) - a 2Z - dA

A + 1 -l i + JlU * 1) M"' h Z ^/?.

and equating coefficients of r gives

a l = - -( 2 •>• a E ) Z

2,2

(4)

(A + 2 + d)(A + 1) - SL{1 + 1) + o Z ? < 2 2 In the small-Z l i m i t , d = 0 and |a 'e | = a Z - 0, and these expressions

reduce to the usual non - re la t i v i s t i c resul ts

(5)

A + 1 = 5, + 1 2Z "1 " " 2{Jt + 1)

On the other hand, for large Z and I - 0 (s electrons), d s 1 and (4) becomes

A + 1 = [1 - u2l2] 1/2

(6)

(7)

showing firstly that A < 0 (= l) and secondly that solutions are possible only for Z = a purposes.

• 1 , 137--which is however not a serious restriction for our

2. Correlation Correction - Method 2 Electrons in atoms cannot move around as readily as free electrons

because they are localized by their attraction to the nucleus; we would therefore expect correlation energies in atoms to be smaller in magnitude than those in a free-electron gas, especially at high densities. This is

18 born out by the empirical observation that in atoms

E a„«> - (E. HF av + E„ N 0.08 Ryd/elec, (?)

Instead of using the method of Reference 6, '.ve here compute E„ as though we were adding electrons to the atom one at a time (in the order from most-strongly bound, Is, to least-strongly bound), and compute the correlation energy of the i electron not from the average free-electron value e , but from the incremental value e upon adding one more electron (Method 2).

Thr incremental corre lat ion energy per electron is

d{Ne c) dN 1 «" Jvol

r s 4 ^ c ' 3 dr .

d ( r 5 ' 3 )

(9)

where r is the radius of a sphere whose volume is the average volume per e lec t ron:

[too) 1/3

In the low-density limit this yives

e c 3 V l . H Z I d r s 3

and in the high-density l i m i t e z const we have

/ -1 \ ^ s l . 4 / -1 \ V 1.142 ) d r s 3 l l T U 2 r s J

4 -7 e , 3 c (10)

e_ = e c c (ID

As a semi-empirical interpolation formula between (10) and the hiqh-density limit (11) + (8), we assume

:(r s, - - [4(r5 • 9 ) 1 ' 2 • J (l.l«]rj •1 -C v's' I 5 '' 4 * Ll' si • 0 2 )

We then take the corre la t ion correct ion to the to ta l binding energy Ea

N to be av

i=2 " (r) e (. 1(r s) dr, 13)

i .. where e is given by (12), and r is in turn given as a function of r by

4n 3 1-1

j = i

-1/3 £'/" -1/3 (14)

j=l

-49-

It may be noted that this final method (12) - ( u ) of computing E is more laborious than the method of Reference 5 because an integral has to be evaluated for N-l electrons rather than only q subshells. How­ever, the computing time required to obtain E is still small compared with the time required to obtain the SCF radial wavefunctiens, and the gain in accuracy of computed energies is great enough to make the effort worthwhile.

-SO­

I L Running the Codes A. Availability and Compilation

The program segments, sample input and sample output are available in private files in photostore in the following directory sequences.

. 049625: KR: COWAN: HF :C0DE -. RCN29S

.049625:KR:COWAN:HF:CODE:HFM0D7S

.049625: KR: COWAN:HF :CODE:RCN229S

.049625:KR:COWAN:HF:C0DE:RCN

.049625:KR:COWAN:HF;SAMPLEIN:FENB

. 049625: KR-.CCWAH-.HF: SAMPLE IN :CONT

. 049625: KR:C0VIW:HF;SAWLE0'JT:FENB0U7

.049625:KR:COWAN:HF:SAMPLEOUT:CONTOUT

The first group contains the source codes for the three progrjms. Compilation produces the controlees. These are put into the "LlX" produced library file RCN which is the code to be run under ORDER. The SAMPLEIM and SAMPLEOUT directories contain sample input and output respectively.

The source codes are compiled and loaded via "PUTT". No system libraries other than the standard CLIB are needed. A sample execute line for c_,npi 1 ation is

PUTT RCN29S LIST S ?1 S Z / t * where "LIST" will contain the compiler listing. Suitable choices for t and v are .4 and .5 respectively. "S" indicates the compile mode. The controlee RCN29 will be produced. Appropriate values for *1 and if 2 for the current versions of the three ;odes are

ffl n RCN29S 112000 155000 HFM0D75 106000 140000 RCN2293 72000 155000

B. Running the Codes 1. Running Separately

The codes may be run individually, constrained to the sequences in Section IA. RCN29 expects to find an input file named INPl and a (blank)

-01-

output file named 0UT1. It will generate, internally, additional output files FLINCHA, TAPE4, TAPE2, ENC and TAPE7. PUNCHA will contain data that was to be punched in the LASL version. Cards will not automatically be punched, however. This can be done via the utility routine PUNCH if desired. ENC is a short file not really used but expected to be present for the ENCODE/ DECODE statement. The other files are used to output the data needed for HFM0D7 and/or PXN229, upon completion 0UT1 may be sent to the printer via ALLOUT. Execution is accomplished by

RCN29 / t v

HFM0D7 and RCN229 are run similarly, expecting input files IUP2 and INP3 respectively and (blank) output files 0UT2 and 0UT3. However, TAPE4, TAPE2 and TAPE7 must also be present, with names switched to DAT4,DAT2, DAT7.

It is generally easier to run any sequence of the codes under ORDER. (See following section). Essentially what is done is that all TTY operation for creating files, switching names, etc., become commands to the ORDER controller.

2. Running under ORDER RCN is currently a "LIX" produced library containing binary

codes for RCN29, HFMOD7, and RCN229 and designed to be run under ORDER. The input file, which can have any name, contains both the ORDER cards and input to the codes being run. Control cards contain a * in column 1. The execute line is

ORDER input file / t v

RC.'l must also he on the machine. Any sequence of the codes can be run by concatenating one or

more of the following typical modules.

*ID (first card in entire deck) *XEQ X 1 Retrieves the controlee *XEQMES RCN RCH29 DR. J from library file. *NXT (card betwi.en separate jobs) *X£Q COPY 1 Create output file 0UT1 of length 50000 *XEQME5 COPY 0UT1 L 50000 I octal by copying COPY *NXT *XEQ RCM29 1 Execute controllee identified *DATA 1NP1 A Input file IHP1 is generated via the DATA card

-52-

Oata cards follow. An additional sequence of cards is inserted to switch file

names. (See sample input). The last job in the execute (*XEQ) stream should be ALLOUT to print output files 0UT1, 0UT2, and 0UT3.

*HXT *XEQ ALLOUT *XEqMES HSP 0UT1 0UT2 OUT3 BOX & ID

MCT can be used to store the output i f desired. (See input f i l e C0NT1. DESTROY is the nice-guy clean up stage.

C. Sample input The next page lists a sample input, FENB, which runs (under ORDER)

an HFR calculation for 3 configurations each of Fe, MB and U. The only changes necessary to run some other set of configurations not involving continuum functions are:

1. Account number, Box and ID on ORDER cards 2. Configuration cards 3. Replace 12, the first number, on G51NP card, by the number of

configurations being computed. 4. If a large number of configurations are included in one calcula­

tion file sizes on the first card might nted to be increased. Note that the output file size is really determined by COPY statement on ORDER card.

The file CONT (see p. 54) lists a sample input for an ORDER run which includes continuum functions. ,ince HFM0D7 cannot handle continuum functions only RCN29 and RCN229 are used. Note the following differences from the previous input.

1. On the RCN29 control card IHF=0 instead of 2 since HF calculation is not being done; TOLENO and THRESH have been decreased to carry the HX cal­culation to convergence rather than just to provide starting wavefunctions for HFM0D7; IREL=1 instead of 2. The relativistic correction is included in either case. EMX, the maximum continuum energy, has been added to control card.

2. Continuum d orbitals are indicated by 99D on configuration cards and continuum energy has been added.

3. The number of configurations on the G5IMP card has been changed from 12 to 4.

-53-

FENB

•ID 199YLD • XEQ X •XEQMES RON RCN29 DR. • NXT 199YLD • XEQ COPY •XEQMES COPY OUT1 L 50000 • NXT 199YLD • XEO RCN29 •DATA INP1 A FILE SIZE* 00000 1.00000 0.99094 0.9S164 0.91549 0.90634 0.86644 0.77785 0.76393 0.7S044 0.58416 0.56614 0.39963 0.37916 0.22926 0.21645 0. 11053 0. 10094' 0.042 0.037 0.009 0.007

RAJNAK BOX zea M X Z68

BOX Z69

60097 41304 24349 12152 046 Oil 000

60000 0.97216 67109 72473 66266 36377 20466 09259 033 006

150000 0,66266 80429 70066 S2427 34927 16976 06617 030 0.009

a 1.0 1.0 8.E-02 l.E-05-2 1FEI 306 4S2 3D6 432 1FE1 3P4 3DB 4S2 3P4 306 1FEI 3S1 3D? 432 331 307 8NB2 404 4D4 2NB2 4P4 4D6 4P4 4D6 2NB2 431 4DS 461 4D6 1O0I 4F7 501 632 4F7 5D1 • - w>4 4F7 5D3 6S2 5P4 4F7 SSI 4F7 9D2 6SS 5S1 4F7 5F2 6D2 732 6F2 SD2 6P4 SF2 6D4 732 6P4 5F2 6S1 5F2 603 752 63! 5F2

leoi (GDI 1U 1 1U I 1U I

2 "B 26 26 26 41 41 41 64 64 64 92 92 92 -1 • NXT 199YLD «XEQ SWITCH •XEOMES TAPE2 DAT2 • NXT 199YLD • XEQ SWITCH •XEQHES TAPE4 DAT4 • NXT 199YLD • XEQ SWITCH •XEQMES TAPE7 DAT7 -NXT 199YLD <XEQ X •XEOMES RCN HFM0D7 DR. •NXT 199YLD •XEQ COPY •XEQMES 0UT1 0UT2 •NXT 199YL0 •XEQ Hr«007 • DATA 1NPS A

l.E-07 10 SI 0.9 0.6S •NXT 199YLD • XEQ X •XEOMES RCN RCN229 OR. •NXT 199YL0 •XEQ COPY •XEQMES dUTI OUTS •NXT 199YLD •XEQ RCN229 •DATA INP3 A Q51NP ISO 0 00 0 000 «NXT 1B9YLD •XEOMES HSP 0UT1 0UT2 0UT3 BOX Z69 CACL, ALPHA •NXT 199YLD •XEQ DESTROY i _ •XEQMES INP1 JNP2 INP3 DAT2 DAT4 0AT7

I 80000 96311 0 •3S02 67616 49613 32252 16364 07666 028 004

432 432

90000 10 94359 0.93413 0.92476 82227 0.80700 0.79220 65710 0.63732 0.61666 47416 0.49214 0.43184 29667 0.27789 0.26903 18493 0.14653 0.13412 07269 0.08311 0.09596 021 0.017 0.014 003 0.002 0.001

1.0 0.65 0.0 0.60 0,0

1 2 3 4 5 6 7 a 9 10 11 12 13 7

8S2 BD3 652 6D2 6S2 7*2 6D4 7S2 603 752 RAJNAK

RAJNAK

RAJNAK

RAJNAK

RAJNAK

RAJNAK

BOX Z69

BOX Z69

BOX Z69

BOX zea SOX 269

BOX ZBB

996.

100 100 100

RAJNAK BOX Z6B

RAJNAK BOX Z6B

RAJNAK BOX Z69

I 00 100 00 00000 RAJNAK BOX zea 0 RAJNAK BOX Z69

-54-

CONT

«IO 199YAM "XEO X "KEQMES RCN RCN29 OR. "NXT 199VAM "XEQ COPY "XEQMES COPY UUT1 L 1! • NXT 199YAM "XEQ RCN29 RAJNAK

BOX 269

BOX 269

BOX 269 •DATA INP1 FILE SIZE = 1.OOOOO 0. .91519 .77765 .60097 .41304 .24346 .12152 . 046 .01 1 .000

90C0D 99094 0.9B\64 0. 0. 0. 0. o. 0. 0. 0.

90634 76393 36416 39553 22926 1 1 0S3 042 009

.63844

.75044

.56614

.37916

.21645

.10094

.037

.007

50000 0.97219 .67109 .72473 .5S2B5 .36377 . 20468 . 09255 .033 .006

150000 O.96266

65429 70066 52427 34927 19376 03517 030 005

150000 95311 O 33802 ~ 67816 49813 32262 16354 07866 025 004

50000 94359 0.93413

" 0, 0. o. 0, 0. o. o. 0.

62227 65710 47416 29667 16493 07289 021 003

,80700 .63732 .45214 .27769 .14652 .06311 .017 .002

10 92475 79220 6166S 43184 25953 13412 03556 014 001

2 -5 0 1.0 1.0 92 1UI 5F3 6DI 92 1UI SF3 601 92 1UI 5F3 6D1 92 1Ul Sr3 6D1 -1 "NXT 199YAM «XEQ SWITCH •XEQMES TAPE2 DAT2 • NXT 199YAM "XEQ SWITCH "XEQUES TAPE4 0AT4 "NXT 199YAM "XEQ SWITCH "XEQMES TAPE7 0AT7 "NXT I99YAM "XEQ X •XEQMES RCN HFM007 DR. «NXT 199YAM "XEO COPY "XEQMES 0UT1 0UT2 • NXT 199YAM •XEO HFM0D7 •DATA INP2 A 2 I.E-07 ID 51 C.5 0.65 • NXT 199YAM "XEO X "XEQMES RCN RCN229 DR. "NXT 199YAM «XEQ COPY iXEOFlES OUT! 0UT3 "NXT 199YAM "XEQ RCN229 "DATA 1NP3 A 051NP 40 0 00 O 000

5.E-08 1.E-11-2 130 1.0 0,690.02 0 50 0. 7S1 7P1 5F3 6D1 7S1 7P1 7P2 5F3 6D1 7P2 7S1 99D 5F3 6D1 7S1 99D 7S1 99D 5F3 6D1 7S1 99D RAJNAK

RAJNAK

RAJNAK

RAJNAK

RAJNAK

RAJNAK

RAJNAK

RAJNAK

RAJNAK

100 100 100 100 100 00

1999.

-1 "NXT 199YAM »XEQ ALLOUT "XEQMES HSP 0UT1 0UT3 BOX Z69 U "NXT 199YAM •XEQ SWITCH •XEQMES 0UT1 UC0NT1 •NXT 199YAM "XEQ SWITCH "XEQMES 0UT3 UCONT "NXT 199YAM "XEQ MCT "XEQMES ELF WRS L .T:COWAN:HF:01 •NXT 199YAM "XEQ DESTROY "XEQMES INP1 INP3 DAT2 DAT4 0AT7

RAJNAK BOX ZS9 CONTINUUM

RAJNAK BOX 269

RAJNAK BOX 269

RAJNAK BOX 269 TPUT FILE UCONT UC0NT1 RAJNAK END BOX 269

I 10 I 11 I 12 I 13

BOX 269

BOX 269

BOX 269

BOX Z69

BOX 269

BOX 269

BOX 269

BOX Z69

BOX Z69

OOOOO

.COS .02

0. Sample Output Typical sections from the output for the Fe configurations from the

run of KENB follow. The entire Output is in photostore as FENBOUT. Output from the continuum calculation CONT is stored as C0NT0UT and

those sections which are different from the previous case are on pp 62-66. (Note section Gla to avoid errors in interpretation of R k integrals).

The cancellation factor FRAC beside R integrals is defined as the ratio of the R integral to the same integral computed using the modulus of each wave function. A large factor indicates that only a small part of the integral has disappeared due to cancellation. Integrals with small cancella­tion factors are expected to be much more sensitive to the exact form of the wavefunction than those with large cancellation factors.

The lines in the RCN29 output beginning with ***** are printed when the continuum function is close to convergence. The second number is the energy. I is the number of the mesh point and Bl is a normalization factor for the continuum function to correct for the fact that r does not go to °°. It should be close to 1.

NOTICt "Tim report was prepared as alt ai.cuum ol *vork sponsored, by the United Slates Government. Neither the United Slates nor the United Staler Knrtty Research & Development Administration, nur ant of their employee.*, nor any or Iheir toolrjtlors. subcot.tractors, or their employees, makes any warranty, express or implied, or assumes arvy legal liability or responsibility Tor the accural-}, completeness or usefulness of any information, appararus. product or process disclosed, or represents, thai its use would nos infringe privately-oivned rights."

"Reference lo a company or product name does not imply approval or recommendation of the product by the University of California or the U.S. (inergy Research & Development Administration to the exclusion of others ihat may be suitable."

....INPUT 1.00000 .99004 .98164 .97216 .96266 95311 94359 .93413 92475 .91949 .90634 .68844 .67109 .65429 63602 82227 .80700 .79220 . 77768 .7S393 .75044 .72473 .70066 .67916 65710 . 53732 .01666 .B0097 .56416 .58814 .05265 .52427 .49613 47416 .45214 .43184 .41304 .39BB3 . 37916 .36377 .34927 .32262 29887 .27789 .26953 . 24348 .22926 .21845 .20468 .19376 .16354 16493 . 1 4852 .T3412 .05556 .12152 .11003 . 10094 . 03700 .092BB .08517 ,07866 07269 .05311

.T3412 .05556 . 04800 .04200

. 10094 . 03700 .03300 .03000 .02500 .02100 .01700 .01400 .01100 .00900 . 00700 .00600 .00500 .00400 .00300 .00200 .00100 .0 -.0 -.0 -.0 -.0 -.0 -.0 -.0 -.0 -.0 -.0 -.0 -.0 -.0 -.0 -.0 -.0 -.0

-.0 -.0 -.0 -.0 -.0 -.0 -.0 -.0 -.0 -.0 -.0 200-50-0 .... 1NPUT

2 1 .0 .O«.00O0OE-O2» 00000E-05 -2 -0230 -0 1.00 .65 . 0 .50

0 26 1FEI 306 4S2 >s 306 4S2

OUT! - from RCN2 l

PA3B

FEI 3D6 4S2 NCONF" 1 Z= 26 NCORES= 6 WALES. 1 ION' 0 RIN/WI)" .29684661 CA1* .700 TOLSTB-1.000 TOLK-2-1,00000 TSLENO" 5.0E-02 THRESH' l.OE-06 KUT" 0 EXF"I.000 CORRF" .0 CAO" 600

ORCN MOO 29 ICOCJ MESH* 601 )DB° 601 RDB> 979.241 EMK- .00 R(riE8H>» 979.241 1 RELo 2

OTIME- 5.431 SEC, .... INPUT. ... 0 26 1FEI 3P4 306 4S2

5.745, 5.757

*> 3P4 306 4S2

FEI 3P4 306 4S2 NCONF" 2 2- 26 T0LSTB"1.000 TOLK-2-1.00000 TOLEND" B.OE-02 ORCN MOD 29 [COO MESH" 801 IOB" 601

l » U > / X ( l ) = . 2 9 8 6 4 9 6 1

THRESH" l .OE-00 KUT" 0 EXF-1 .000 CORRF. RDB- 979 .241 EMX" . 0 0 RI MESH)- 9 7 9 . 2 « '

PAOE B

CAT . 7 0 0 0 CAO" .500 1REU- 2

OTIME" 6 . 1 3 1 SEC,

. . . . INPUT. . . .

0 26 1FEI 3S1 3D7 492

0UT1 - from RCN2

6 . 4 4 4 , 12 .203

>* 3S1 307 4S2

0UT2 - from HFM0D7 HARTREE-FOCK WAVE FUNCTIONS FOR FEI 306 432

HAVE FUNCTION INITIAL ESTIMATES ECNL1 526.6147 62.6868 64.661?

7.6410 5.1295 1 .1467 .6607

AZCNL) 265.043 67, 101 571.696 33.004 211.566 156,675 6.244

ENER3Y = E[AVERAGE)*

RHO= -3.OOO H= .0625 HXFACi .95 WFTOL- 1.CE-07 TIMPM* 999.0

WFTOL l.OE-07 1 .OE-07 1 .OE-07 1. OE-07 1.OE-07 1.OE-07

DPM(J) JDPM 2.7E-03 109 3.2E-04 109 4.1E-05 106 B.IE-06 106 6.2E-07 106 6.6E-06 106

. 40

.40

.40

.40

.50 .50 .60 .50

MATRIX OF ENERQV PARAMETERS 527.41723177

64 .91430361 55.12606767

6.53116445 .0 .0 .0

.0

.0

.0 .52700692

GRTH0Q0NAL1TY INTEGRALS FOR 26 FEI 3D6 462

(NL1 INL1 INTEGRAL

26 16 -.00360140 -.00131215 -.00122991 36 IS -.00360140 -.00131215 -.00122991 33 23 -.00360140 -.00131215 -.00122991

3P 2P -.00091846 43 IS -.00027904 43 2S -.00024792 46 3S -.00009911

VALUES OF F AND 0 INTEQRALS FOR FE! 305 482

( ZP, ( 2P, 3S) 3S)

( 2P, t 2P, 3P) 3P)

( 2P, ( 2P,

3D) 3D)

( 2P. ( 2P, 4S> 4S>

( 3S, 35) ( 33. i 3S,

3P] 3P) [ 3S, I 3S, 3D) 3D) ( 3S, ( 3S,

4S) 4S) I 3P, 3P) < 3P, < 3P,

3D) JO)

I 3P. ( 3P,

4S> 4S>

I 3D, 3D) I 3D, ( 30, 4S) 4S)

F0= 852896.0 F2= 402289.2 FO" 01 =

3241 OS. 1 26733.6 FRAC" .624 F0= 00= 313036.2 23121.2 FRAC« .281 F2"

02= 60733.8 24261.4 FRAC" .797

FO" 81 =

269290.8 33670.8 FRAG»1 .000 F2" S3"

47302.3 19104.3 FRAC"1 .000

F0= 01 = 94670.5 1126.7 PRAC-" .807

F0= 243040.9 F0= 01 = 236806.6 157819.8 FRAC" .997 F0= 02= 211488.7

91176.S FRAC" .967 FO" Q0= 81492.8

2390. 5 FRAC" .227 F0= 230477.3 F2= U5702.6 F0« 01 =

206287.9 120470,8 FRAC» .969 F2=

63" 96844.7 72801.2 FRAC" .991

PO-QI'

81227,5 3468.9 FRAC" .462

F0 = 187342.1 F2= 89786.8 F0= 02= 79822.9

8139.0 FRAO" .774

26 PEI 3D6 482

IS 001597 2S 063320 2P 067780 3S 757169 3'

87327) 31 552446 41 998042

3P

E(NL)

527.417232 64.914304 55.126068 B.531164 5.553602 1 . 1 89271 .527010

.082346

.080727

.080990

.076953

.076486

.074191

.052396

AZINL)

234.615262 71 .641209

512.549012 26.563236

185.925612 147.210373

B.763581

.044

.124

.135

.342

.424

.917

.377

.044

. 124

. 135

.341

.421

.890

.314

NCONF' 1

1/R"»3

.0 474,8071

.0 56.2787 4.8888

.0

MEAN VALUE OF 1/R««2 1/R

1378.5473 130.6302 40.8299 18.6872 5.5200 2.0525 .9864

5.9536 .058464 6.5727 .265781 5.4541 .235015 1 , 7306 .810633 1.5883 .861139 1.2239 1.091156 .3965 3.206820

-1.974846 ETR> -2541.651469

EC(NEW)= -1.683880 EREL» .0 ETCO" -2543.626317 ETRCO- -2543.626317 ETON' -2543.335349 ETRCN-

SPIN-ORBIT PARAMETER ZETA< 2P) • 63604.283 CM-1 • ZETA< 3P> s 7519.260 CM-1 • ZETAI 3D) • 408.781 CM-1 •

SPIN-SPIN PARAMETERS .57960496 RVO M0( 2P, 2P) B 137.233 CM-1 .06852073 RYD M0< 3P, 3P> 3.186 CM-1 .00372809 RYD not

M2( 3D, 3D) 1.589 CM-1 .866 CM-1 not

M2( 3D, 3D) • 1.589 CM-1 .866 CM-1

RC9 INPUT PARAMETERS OFEI 306 4S2 4 -2843.3353 0 88.7668 1 83.1904 1 ,4086 2

HF MOD 7 NC0NFQ3 1 TIME* 6.05. 6.06 SEC

0UT2 - from HFM0D7

0UT3 - from RCN229

IRCN2--- INPUT FOR RCO HOD 9

1 9 3 3 D O 000 S 2 P 8 D 6 S 2 P 4 D 8 S I P 6 D 7

100 100 100 100 100 00

FACT" 1 . 00 1 .00 1 . 00

OFE! 3P4 3D8 432 943.66969 1 .00 1 .00 .42670 96.41748 122.10430

116.63198 73.79367

FACT" 1.00 1.00 1 . 00

OFEI 3S1 3D7 4S2 1.00 1,00

FACT" 1 . 00 1 . 00 1 . 00

1RCN MOD 27

EE.EEE- -1.24604 -3.40062

O SLATER INTEGRALS FOR FEI 3D6 482 FEI 3P4 306 4S2

R D M 3P, 3P, 30, 3D)

122103.2060 CM-1 73862.3320 CM-1

REKC 3P, 3P, 3D, 3D)

1 .11Z7046S RYD • 1221 OB.2080 CM-I .958 .$7144288 RVD • 73882 .3320 CM-1 .990

0 EIGENVALUE CONTRIBUTISN OF

OFEI 306 482 -4 3D8 482

0UT3 - from RCN229

. 0 KK INCLUDED IN PARAMETER NUMBER 1

2 122.1002 S 73 .6823 S

IRON HOD 27 EE,EEE= EE,EEE= 0 SLATER 1NTEORALS FOR U I 6P4 SF2 604 7S U I SSI BF2 6D3 ?S

ROK( 6S, 60, 6P, 6P> FRAC .48SI0613 RYO • 83234.2419 CH-1 .966

REM 6S, 6D, 6P, BP) FRAC .4BS10613 RVD c S3234.2419 Cn-I 966

0 EISENVALUE CONTRIBUTION OF .0 KK INCLUDED IN PARAMETER NUMBER

OU I 6P4 0F2 604 -1 5F2 803

-99999999.0

0FIN1SHE0 G8INP ...INPUT. . .

CONT - RCN29

UI 5F3 6DI 7S1 99£> NCONF= 4 2= 92 WCORES-tft NVALES= ION= 1 R d ) / X ( I ) s .19611691

TOLSTBM .000 T0LK-2=1.00000 TOi-END= 5 . 0 E - 0 8 THRESH= ) OE-11 KUT= 0 EXF*1.000 CORRF= .0 CAO" . 800 ORCN MOO 29 (CDC) MESH=2201 1DB= 481 RDB= 9 0 . 3 1 0 EMX= .02 R(MESH)=3534.476 IREL* 1

0 _ • « « " « Z S T , E , R , A 1 , B 1 , F , 1 = 1 .000000 .020000 3516 .401862 .00710954 .90137300 .M302264 2191

2191 »«-««2ST, E.R, Al,61,F,1= 1 .000000 .020000 3516.401662 .00710954 .98137300 .99302289

2191 **»«»ZST,E,R Al ,B ,F, 1 = 1 .000000 .020000 3516.401862 .00710954 .98137300 .99302289

NL WHL EE AZ IR-3) (R-2) (R-1) tR + 1) CR+2) <R*3> <R»4> tR*e> IS 2 -8528.97826 5550 653 .0E*00 4.G43E+04 1.226E+02 1.373E-02 2.643E-04 6.553E-06 1.988E-07 2.999E-10 2S 2 -1601.89293 2208 377 .OE+OO 5.747E+03 2.897E+01 5.750E-02 3.935E-03 3.074E-04 2.709E-05 2.914E-07 2P 6 -1352.84883 381S8 244 4. 221E+04 7.921E+02 2.392E+01 5.355E-02 3.504E-03 2.713E-04 2.429E-05 2.B1SE-07

S.032E-0S 3S 2 -405.96172 1069 046 .0E*0O 1.330E+03 1.066E+01 1.4636-01 2.520E-02 4.706E-03 9.566E-04 2.B1SE-07 S.032E-0S

3P 6 -335.41322 19916 868 9.711E+03 1.835E+02 9. 164E+00 1.522E-01 2.680E-02 5.226E-03 1.117E-03 S.B07E-OS B.2S0E-OB 3D 10 -269.49034 77735 551 1 . 258E+03 9. 173E+01 8.628E+00 1.384E-01 2.236E-02 4.138E-03 8.660E-04 S.B07E-OS B.2S0E-OB

4S 2 -104.30668 556 251 .OE+OO 3.609E+02 4.643E+00 3.217E-01 1.168E-01 4.600E-02 I.949E-02 4.309E-03 7.009E-03 4P 6 -30.87645 10320 491 2.510E.03 4.932E+01 4.036E+00 3.452E-01 1.351E-01 5.759E-02 2.652E-02 4.309E-03 7.009E-03

40 10 -55.73280 42737 251 2.953E.02 2.357E+01 3.672E+00 3.593E-01 1.484E-01 6.768E-02 3.377E-02 1.085E-02 1.616E-02 4F 14 -29.91028 54503 127 6.789E*01 1.356E+01 3.349E+00 3.54SE-01 1.469E-01 7.021E-02 3.826E-02 1.085E-02 1.616E-02

5S 2 -24.04832 281 332 .OE+OO 9.304E+01 2.092E+00 6.706E-01 5.025E-01 4.071E-01 3.537E-01 3.267E-OI B.934E-01 6P 6 -15.64016 5043 324 5.938E+02 I.236E+01 1.796E+00 7.517E-01 6.337E-01 5.802E-01 5.725E-01 3.267E-OI B.934E-01

SD 10 -8.42022 19135 353 5 649E+01 5.I06E+00 1.487E+00 8.888E-01 8.965E-01 9.958E-01 1.211E+00 2.331E+00 4.1»5E«0i 1,8876*02 6S 2 -4.35134 125 531 . OE +00 1.876E+01 8.878E-01 1 . 486E+00 2.465E+00 4.453E+00 8.704E+00 2.331E+00 4.1»5E«0i 1,8876*02 «P 6 -2.50109 2009 596 9.415E+01 2.147E+Q0 7.041E-Q1 1,830E+-00 3.764E+00 8.518E100 2.110E»01 2.331E+00 4.1»5E«0i 1,8876*02

5F 3 -1.10104 16134 591 5.200E+00 1 .439E+00 9.001E-01 1 . 4B4E+-00 2.704E+00 5.962E+00 1,576E*01 1.792E«02 6D 1 -.80371 4978 389 3.827E+00 4.511E-01 4. 169E-01 3.030E+00 1.059E+01 4.194E+01 1.670E+02 •140E+03

T.804E+04 7S 1 -. 85756 44 919 .C2+0C 2.451E+00 3.278E-01 3.829E+00 1 , 639E+01 7.682E+01 3.921E+02 •140E+03 T.804E+04

91 1 .036E+05 7.656E+02 5.433E+01 8.142E+01 2.131E+02 7.877E+02 1.9B4E+04 1

ZETA 2)

1.I39E+03 8.4I3E+00 5.970E-01 8.947E-01 2.342E+00 8.6S6E«00 2.180E+0*

NL UNL R»VI/M*<E-V>/274« ZETA 2) IR'VI) I (RVO) EPS S EVEL EDAR EVEL+EDAR EREU

CP.YD) (CM 1 ) (RVO> (CM-1) IS 2 .0 .3 .0 .0- 9651.43347 -8510.8164013634.0530811722.57313 -1911 .480-10422. m » 2S 2 .0 .0 .0 .0 2490.51892 -1605.76439 -2297.22812 1725.78907 -571.439 2177,2(344 150.7.Oilfj 2P G 169 4503418595046 959 202.584442n228877.016 2257.06976 -1353.79647 -163.25472 5.64613 -153.255 2177,2(344 150.7.Oilfj

3S 2 .0 .0 .0 .0- 1047.SS2O9 -410.97266 -548.60690 399.86809 -148.739 -B5j.7Tl«? 3P 6 37.67015 4133820 894 46.52162 5105157.673 -971.92891 -340.05817 -45.86959 1.36650 -45.370 -3»i.9»777 3D 10 5.32842 584726 864 5.50062 603622.987 -926.80122 -273.130E3 -11 .54422 -.12325 -11.544 -284.67448 -148.50M| 4S 2 .0 .0 .0 .0 -536.20315 -107.83652 -149.62017 107.95827 -41.662 -284.67448 -148.50M|

4P 6 9. 64500 058525 833 12.01049 31.998.975 -495.97916 -84.17957 -12.81411 .35780 -12.814 -».993eS -02.20382 -33.38332 4D 10 1 .23206 135203 125 1.28082 140553.366 -462,15909 -58.79081 -3.41901 -.02604 -3.419 -».993eS -02.20382 -33.38332 IF 14 .232'7 25477 764 .23417 2S696.912 -434.45905 -32.49012 -.B7320 -.02018 -.873 -».993eS -02.20382 -33.38332

5S 2 0 .0 .0 .0 -285.08454 -25.67529 -38.37286 27.59607 -10.777 -36.45208

CONT - RCN229

1RCN MOD 27

EE,l£EE= - . 3 0 3 9 0 - . 15195 EE,EEE= .03500 - . 14945

0 Si_.*TCR INTL^RALS FOR UI 5F3 6D1 7P2 NCONF= 2 Z= 92 ION= 0 EXF= .630 IREL* 1 UI 5F3 6D1 7S1 99i> NCONF= 3 KUT = -1 CORRF= . 0 CLAW 0

0 K RDM 7P, 7P, 7S,99D) FRAC K REK[ 7P, 7P, 7S,99D> FBAO

1 .2034?403 RYD = 22323 .2056 CM-1 .796 1 ,20342418 RYD = S2323.2224 C«- t . 7 f l 0

0 EIGENVALUE CONTRIBUTION OF .0 KK INCLUDED IN PARAMETER NUMBER 1

OU! 5F3 6D1 7P2 - 3 GDI 7S1 1 22 .3232 S

1RC1N MOD 27

FF EEE= .0050C 00250 t h EEE =

1 .O2O0C

934233 21701 0 103193 104219 083934 064776

.01250 1.958142

.217010

.103193

.104219

.083934

.064778

1.946187 .217010 .103193 .104219 .083934 .064778

0 SLATER INTEGRALS FOR Ul 5F3 f ,n i 7S1 99 NCONF* 3 Ul 5F3 6D1 7S1 99 NCONF" 4

0 K RDKt 5F.99D , 5F.99D) FRAC

0 1 94S187I5 RYO 213569.3421 CM-1 .178 2 21700967 RVO 23814 .0571 CM-1 .995 4 10319288 RYD * 11324.1067 CM-1 1 .000

0 1 P34233

.217010

.103193

.104219

.083934

.064778

1.958142 .217010 .103193 .104219 .063934 .064778

1.946187 .217010 .103193 .104219 .063934 .064778

1

0 1 94618715 RYD 213569.3421 CM-1 . 178 2 21700967 RYD 23814 .0570 CM-1 .995 4 10319288 RYD = 11324.1087 CM-1 1.000

0 1 .934233 .217010 .103193 .104219 .083934 .064778

1.958142 .217010 .103193 .104219 .083934 .064778

1.946187 .2170 IC .103193 .104219 . 083934 . 064778

1

0 1 94618715 RYD 213569.3421 CM-1 .178 2 21700967 RYO 23814.0571 CM-1 .995

0 4 10319288 RYD e 11324.1086 CM-1 1 000

0 1 94618715 RYO 213569.3421 CM -1 .178 2 21700967 RYD 20814,0571 CM-1 ,995 4 10319288 RYD = 11324.1068 CM-1 1 .000

I M U i citSS i i

REKC 5F.99D, 5F .99D) FIAO

. 10421917 RYD

.06393357 RYD

.06477770 RYD

11436.7309 CM-1 9 2 1 0 , 6 4 4 8 CM-1 7 1 0 8 . 5 3 0 6 CM-1 as

.10421917 RYD .06393357 RYD .06477770 RYD 11436.7306 CM-1 9210.6446 CM-1 7108.5306 CH-1

.10421917 RYD .08393357 RYD .06477770 RYD

.10421917 RYO .08393357 RYD .06477770 RYD 11436.7309 CM-1 9210.6446 CM-1 7108.3306 CM-1

OVERLAP INTEGRAL

92 - t UI 5F3 6D1 7S1 99 UI 5F3 6D1 7SI 99 (99D/99D>= 15.254966077

3.000'">0 .31266 0,00000 1 .00000 2 .00000 . 0 .01250 15 .25497 3 .00000 .25180 3.00000 3.OO0O0 2 .00000 ,0 .01250 15 ,25497

3 .00000 .19433 3 .00000 5 . 0 0 0 0 0 2.OOOOO . 0 .01250 15 .25497

0 EIGENVALUE CONTRIBUTION OF .0 KK INCLUDED IN PARAMETER NUMBER 1

. 0 5 23 .6141 0 11 .3241 5 11 .4367 5

FRACTION

.003738

R INTEGRAL REDUCED MUPOLE ELEMENT

,60 UI 5F3 6D1 7S1 7P UI 5F3 6D1 7P2 -4.0338*12 ( 7S//R1// 7P) =

5F3 601 7S1 99 -5.116339 ( 7P//R1//9SD1* 5F3 6D1 7S1 99 -4.541084 ( 7P//R1//99D)=

92 -1 .70 .SO UI SF3 6D1 7S1 7P . 073 -.625 92 -1 .70 .50 UI 5F3 6DI 7S1 7P .001 •596

4. .03364 AO -. 9954 ».4t3B*1

7 .23842 «• -.4008 1.74«4«» 6 .42206 AO -.3643 1 .374713

-90999999 0

EE N* RTCO WT= GAMMA/RKSO 5«NST3/ZST2> QxRK P1«(0 iRKS0>/2 PDIPSC RKSCCRYD)

3 6D1 0 3 601 3 6DI

-.303902 .0 . 005000 .020000

.0

.0

.0

.0 .0

.0 .26001 .0 .24540

.0 .872

1 .012

.0 4.03364 1 . 194 7.23842 1.606 6.42206

.0 .203424 .197627

OF IN[SHED GS1NP . . .INPUT.