Calculations of Configurations of Doubly Ionized Copper (Cu III)

The energy levels belonging to the configurations 3d74s2 and 3d8nℓ (nℓ = 4s, 5s, 4p 5p, 4d 5d 4f, and 5g) have been calculated. The radial energy integrals were treated as parameters and adjusted to give a least-squares fit to the observed levels. Two- and three-body effective electrostatic interactions for equivalent electrons were included, as well as two-body effective interactions for inequivalent electrons. Strong configuration interaction between 3d74s2 and 3d84d was taken into account. Values of the parameters are given for all the above configurations, and the calculated levels are given for all except 3d84s and 3d84p (for which essentially equivalent results have been published). Leading eigenvector percentages are given in appropriate coupling schemes.


. Introduction
A great exte nsion of th e a nalysi s of C u III has rece ntly been achi e ved b y She nston e [1]. 1 In thi s work he de te rmin ed nearly all th e le vels of th e co nfi gura tions 3d 7 4s 2 a nd 3d 8 n f' for n f' = 4d, 5d, 5s, 6s , 5p , 4/ , and muc h of 3d 8 5g. In th e course of thi s analys is we provid ed calcula tions of the level stru ctures a nd continu ally refin ed th e m as new data were obtained . The fin al res ult is a se t of calcula ti ons for all th e known confi gura tions of thi s ion (with the exception of 3d 7 4s4p ) that are intern ally consistent so far as common radial integrals (parame ters) are concerned and that include all the effectiv e electrostatic interactions , as well as the usual Slate r and spinorbit interactions , that have so far bee n considered in the iron group.

Method
Calculation s of the e ne rgy matri ces for th ese confi guration s, as well as th e matrix di ago nalizati ons a nd le vel fittin g, we re carri ed out on th e NBS Univac 1108 computer. The co mpute r progr a ms we re ori ginally obtained from the Labora toire Aim e Cotton (Orsay, France). Successive diagonali zati ons a nd variations of the radial parame ters were perform ed until a leas tsqu a res fit of th e en ergy le vels was ac hi e ved. Final valu es for th e pa ra me ters , the s ta ndard e rror for each param e te r , and th e rm s e rror of th e least-squ ares fit for eac h co nfi gura ti on are give n in ta bl e 1.
(See re f. [2], for exa mple, for more de ta il s of th e ge neral procedure_ ) Th e rm s error is de fin ed as [ " ] 1/ 2 wh ere 8 is th e diffe re nce betw ee n th e expe rim ental and calculated position s for a le vel, n is the numbe r of le vels used in th e fittin g, and m is th e number of free parameters. The standard error for a paramete r value (in pare ntheses followin g the valu e) indicate s how well the value is "de fin ed" by the equations and the experime ntal leve ls. In our initial calculations we were guided by th e theoretical study of the eve n configurations of the third s pectra of the iron group by Shadmi , Cas pi , and Oreg r3] and by a similar work on the odd configurati ons by Roth [2]. These papers included clac ulati ons of th e almost comple tely known 3d 8 4s and 3cf84p co nfi gurations of Cu III. Most of th e paramete rs we use, includin g th e e ffective two-body inte racti ons (ex a nd (3) a nd threebod y interactions (T and Tx) for equi vale nt electrons, a re defin ed in th ese pa pe rs or in reference [4] . In add ition we introdu ced th e two-bod y effecti ve inte racti ons for ine quiv ale nt electrons , de noted here as D" a nd X"  for the direct and exchange parts [5). 2 We found th ese to be significant for the 3d B 4p, 3d85p and 3d B 4d confi gurations.

Parameters of the 3d 8 Core
All configurations treated except 3d 7 4s 2 are built on the 3d B core. It is eviden t from th e parameter values (table 1) that the core parameters are little affected by the additional outer electron of th ese configurati ons. The electrostatic parameters B , C, and a were freely varied in all cases except 3d B 5g; th eir fitted values a re nearly identi cal for each co nfi guration . The seniority parameter {3 could be meanin gfully evaluated only for 3d B 4s and 3d B 4p, for which the double t built on the 3d B I S core state is now known [1]. Its value was nearly th e same in both cases and was also close to th e value deri ved by Shadmi et al. [3] in th eir ge neral treatm e nt of th e third spec tra. It was therefore fixed at the value -496 c m-I , derived from 3d~4s, in all configuration s of C u III. Since most of th e 3d B 5g levels know n with certainty are based on th e 3F core term, all core parameters except the s pin-orbit parameter ~(3d) were fix ed at average values derived from the other co nfiguration s.
Th e fitted valu e of ~(3d) for eac h of th e 3d B nl configurations was practically unchan ged.
The effective 3-body parameter T includes th e interaction of 3s 2 3d B with 3s3£lJ and has a non -zero matrix eleme nt only for th e ID state of 3d B .This parameter cannot be freely de termin ed for 3d8 because the number of core parameters exceeds the number of core terms; we therefore fixed it at the value -5.63 cm-I deduced by Shadmi e t al. [3]. The significance of T is demonstrated by th e fact that th e omi ssion of this parameter leads to values for the parame ters C, a, and {3 of th e 3d8nl confi gurations that are totally inconsistent with th ose derived from th e general treatme nt of third spectra in references [2] and [3). 3 The second three-body parameter [4], Tx , had no effect on the d8nl configurations and was omitted. This parameter was included for 3c:f74s 2 , where (along with T) it is an inde pendent inte raction. It was not possible to obtain meaningful fitt ed values for T and Tx in 3d 7 4s2, probably because their e ffect is small and this configuration is strongly distorted by near-confi guration interaction. The fixed values used for them were estimated from the results of Shadmi e t al. [3].

The Two-Body Effective Interaction for Inequivalent Electrons
The interaction is represented here by the parameters Dh' a nd Xh' calc ulated according to the formulas of Goldschmidt a nd Starkand [5).2 They are the coefficie nts of th e scalor products of unit operators, Dk for the direct and X" for the exchange interaction. The 2: These au th ors use th e notations -F' and -0 for the parameters here designated Dk and Xk", respectiv ely (k = l).
3With T om itt ed, th ese parame te rs take the fo ll ow ing valu es for eu III 3dfl4s: allowed parame ters for d8p are DI and X2 , and for d 8 d they are DI , D3, XI, and X3. In the case of 3d84p and 3d85p both effectiv e parame ters are well defined by a least-squares fit. Only DI and D3 were defined for 3d 8 4d, pe rhaps because th e far-co nfigurati on effects were partly mask ed by th e inte raction with 3c:f74s 2 • Our attempts to include thi s type of inte racti on by least-s quares fit s in th e other co nfigurations of C u III were unsuccessful , but it is s urely importa nt for all 3d ll 4p and 3dl/4d co nfiguration s of the iron period.

Results
An indi cation of th e success of these calc ulati ons is th e low rm s error reac hed for all co nfigurations (table 1) , always less than 100 cm-I a nd us ually mu c h less. This is particularly signifi cant for the hi ghly mixed 3d84d and 3d 7 4s 2 configuration s where large dev iations present in sin gle-confi guration calc ulation s are co nsiderably red uced by th e introd uction of a single parameter R2 (dd' ,ss) for co nfi guration interaction. The inclusion of the effective parameters Dfr a nd X" in the 3d 8 4d co nfi guration appears to be th eir first use for 3d1/4d. As a further test we introd uced the m in a calc ulation 4 of 3d 2 4d of V III anQ found a reduc tion of th e rms error from 117 cm-I reported by Spector [6] to 67 e m-I, W e re peated th e calculation of 3d84p of Cu III by Roth [2] who obtain ed a n rms e rror of 126 cm-I ; with the inclusion of DI and X2 th e rms error was reduced to 72cm-1 (table 1). Tables 2 through 8 [3]. ) All observed levels are from Shenstone [1], th e values being rounded off to the neares t e m-I. Observed levels fo llowed by a question mark were so denoted by Shenstone to indicate that these levels may not be real. The "Leading P ercentages" refe r to s quared eigenvector co mponents given as percentages following the term symbols, and rounded off to the nearest percent.
The "average %" given at the end of a "Leading Percentage" column is the average purity of the levels for the indicated coupling scheme. The 3d8 parent terms for LS-coupling designations are given in parentheses.
The calculated levels for these even configurations are in-tables 2 and 3, respectively. The two leading percentages in LS coupling are given for each level, any second percentage less than 0.5 percent being omitted. In table 2, the two 3d 7 4s 2 2D terms are labeled 1 and 2 as in Nielson and Koster [7].
Shenstone [1] has described his method of assigning LS names to the le vels , beginning with the 3dfJ, 3d84s, and 3d84p levels and proceeding to name th e levels           2 given without a letter in the final column confirm the names assigned by Shenstone. 5 The letters in the final column have the following meanings: A. The leading compon ent of the eigenvector indicates a desi gnation differe nt from that assigned by Shenstone. B. Th e eigenvector yields no th eoretically satis-!> Our conditions for a name are that the lead in g perce ntage be near 50 pe rcent or greate r; a nd for a lead in g percen tage near 50 pe rce nt . th e seco nd pe rce nt age m ust be signi fi cantl y smalle r (second de signations cl earl y less app ropri ate). an d no other eige nvector shou ld have a co mparable lead in g perce nta ge ( -500/0) for the sam e designatio n. factory sin gle-configuration si ngle-term designation. 5 C. Indicates pairs of neighborin g levels whose eigenvectors might possibly be interc ha nged.
The low 3d 8 (3F)4d 4p and 4D term s overlap , but th e 4D term is lower according to our calculations; thi s accounts for the first six "A" notations in the table. Th e several B notations for the} = 9/2 and} = 11/2 levels mainly arise because of the strong admixtures of 3d74s 2 2H components, whi ch are so distributed amongst these levels that no level for either} value can meanin gfully be assigned to this term. The very similar compositions of the}= 11/2 levels at 194033 and 197039 c m-I may be noted; these preve nt a designati on for eithe r level according to our criteria. 3d 8 5p, 3d86s, a nd 3d85d. The results for th ese configura tions are given in tables 4 , 5 , and 6. The leadin g percentage for each level is given in LS couplin g and in a (L IS dJ d coupling sche me. The notations for th e latte r sche me have th e 3dB parent level (in LS coupling) followed by the j value of the outer electron. Most of the levels have meaningful names in either scheme. The average purity in LS coupling is a little higher than the J J purity (3dB5p and 3dB5d) , or the purities in the two schemes are practically equal (3dB6s); the LS names are prob· ably more generally useful.

5.2.
Shenstone assigned some 25 odd levels to the 3d 7 4s4p configuration [1]. We calculated this large configuration, but the results of the level fitting were inconclusive because of the lack of sufficient data. Shenstone assigned a tentative level at 232 990 · cm -I to 3d7 (4F)4s4p (3PO) 2D~/2 and a level at 233286 cm -I to 3d8(3P)5p·iD·/2. The lower of these levels is closer to our prediction for 3d 8 (3P) 5p 2D~/2' but we used neither level in the least-squares calculations. Although the good fit obtained for the 3dB5p levels indicates that the general configuration interaction with 3d74s4p is weak, the closeness of these two 2D~/2 levels might result in significant configuration mixing.

3dB4f and 3dB5g
Shenstone pointed out that the level structure of the 3dB parent configuration could usually be discerned in the pattern of the 3d 8 ni levels. "In 5g the scheme [in which the parent J value is defined] reaches an extreme which makes it possible to identify some, but not all of the levels. In fact, the number of combinations of a level is reduced in most cases to just two ... " A small number of combinations is one effect of pair coupling and, as is evident from tables 7 and 8, the 3d 8 4f and 3d 8 5g configurations are best described by the J IL coupling scheme. The designations for this scheme have the 3d B parent level (L,,sl, andJd preceding the bracketed K value (obtained by coupling J I and the L vector of the outer electron) [8]. Shenstone was able to deduce LS names for the 4f and 5g levels in some accordance with the intensities of their transitions, but many of the LS designations have meaning only in that connection. Only two of the eigenvectors for 3dB4fhave leading percentages less than 50 percent in J Ii coupling, whereas there are 23 such eigenvectors in LS coupling; for 3dB5g, the equivalent numbers are 4 and 26.