Theoretical Investigation of the Odd Configurations of Ni ii

Abstract

Two groups of odd levels in Ni ii were investigated: those belonging to the complex 3d⁸4p + 3d⁷4s4p+3d⁸5p and those belonging to the configuration 3d⁸4f. In the first group the calculated positions of the levels were fit to the positions of the 174 observed levels with an rms error of 133 cm⁻¹. The fit for the second group was based on 60 observed levels and had an rms error of 25 cm⁻¹. The predictions of this investigation helped in the discovery of many of the observed levels.

1. Introduction

The configuration 3d⁸4p has been well known, and many of its observed levels have been reported in AEL [1].¹ Theoretical interpretations of the 3d⁸4p level structure were performed by various investigators [2–4].

About four years ago Professor A. G. Shenstone informed us of some newly discovered odd levels, presumably belonging to the configurations 3d⁷4s4p, 3d⁸5p, and 3d⁸nf (n = 4, 5, 6, 7). This paper is the result of his suggestion that a theoretical investigation of these configurations be made to help him with his experimental investigation. The companion paper containing Shenstone’s experimental results has already been published [5].

Our calculations involved the diagonalization of the energy matrices of the 3d⁸4p, 3d⁷4s4p, and 3d⁸5p configurations calculated as one complex, and the energy tions calculated as one complex, and the energy matrices associated with the 3d⁸4f configuration. In the case of the 3d⁸4p + 3d⁷4s4p + 3d⁸5p complex, we were able to fit the 174 observed levels to the calculated ones with an rms error of 133 cm⁻¹. For the 3d⁸4f configuration, the 60 observed levels could be fitted to the calculated ones with an rms error of 25 cm⁻¹. All the levels were designated in a well defined coupling scheme.

2. Notations and Definitions

In the text and tables Slater parameters and spin-orbit parameters are designated in the usual way. Other symbols and abbreviations used in the text have the following meanings:

B, C= linear combinations of Slater parameters F₂(dd) and F₄(dd); (see, for example, ref. [7]).

α, β, T= effective interactions among d electrons; [9].

H, J, K= parameters of configuration interaction which are appropriate linear combinations of Slater integrals; (see, for example, refs. [4] and [8]).

Δ = root mean square error (“rms error”).

“Diag.”, “L.S.”= abbreviations for “Diagonalization” and “Least-squares calculation,” respectively.

In cases where several configurations have analogous parameters, the configuration is also explicitly specified.

3. The Theoretical Interpretation of the Configurations 3d⁸4p + 3d⁷4s4p + 3d⁸5p

We shall use the following abbreviations:

$$d^{8}p=3d^{8}4p,$$

$$d^{7}sp=3d^{7}4s4p,$$

$$d^8{p}^{\prime }=3d^{8}5p.$$

In his first letter, Professor Shenstone supplied us with 17 levels belonging to the d⁷sp configuration; 9 of them were low and were assumed to be based on d⁷s(⁵F); 8 of them were high and it was supposed that they were based on d⁷s (³P).

In the first stage of our calculations only the two configurations d⁸p + d⁷sp were included. The interaction parameters of d⁸p are well known [2–4]. For an estimate of initial parameters for d⁷sp, we were able to use analogous calculations performed by C. Roth [4] on the Cu ii and Zn ii spectra and by A. Schwimmer [6] on Sc ii, Ti ii and V ii. It is well known from works on the even configurations in the iron group [7, 8] and from reference (4) that the electrostatic interaction parameters change linearly along sequences of spectra with constant ionization. The behavior of D’, defined as the separation between the centers of the configurations dⁿ⁻¹sp and dⁿp, and the behavior of the spin-orbit interaction parameters are also approximately linear.

Hence we could interpolate the values of D’ and the d-p interaction parameters from the spectra of Sc ii, Ti ii, V ii, Cu ii, and Zn ii mentioned above. For the G₁ (sp) parameter we could rely upon the spectra of the right hand side of the period. Values for the parameters B, C, G₂ (ds) and H were simply taken from the even configurations of Ni ii [10]. J and K were also extrapolated from the right hand side of the period.

The first diagonalization was performed using the described above parameters. The nine low observed levels of Shenstone’s first list were quartets. It also became evident that the correct coupling for the d⁷sp configuration is the following: first the s and p electrons are coupled and then the resulting term is coupled with a term of d⁷. Such a coupling was already used by C. Roth [4]. Using this scheme we found that all the nine low-lying levels reported by Shenstone are based on the combination d⁷(⁴F)sp(³P).

After our initial diagonalization, additional observed levels were provided by Professor Shenstone to bring the total number of observed levels which were fitted to calculated levels of d⁸p + d⁷sp up to 78. In the final least-squares calculation based on the previously described interaction parameters, 43 levels were found to belong to the d⁸p configuration and 35 to the new configuration d⁷sp. The rms error was 100 cm⁻¹ for the calculated levels.

The observed levels in the range 110,000–120,000 cm⁻¹ could not be satisfactorily fitted in the previous diagonalization. The reason for this was rather clear: in the same energy range some levels belonging to the d⁸p’ configuration were also observed. This means that in order to obtain good results, one has to include the interaction between the configurations d⁷sp and d⁸p’ as well. Because such an extended calculation requires quite a number of additional new parameters, we first performed a separate calculation on the d⁸p’ configuration in which we included only the levels which we believed not to be strongly perturbed by d⁷sp. This auxiliary calculation provided initial values for the 3d– 5p interaction parameters and for ζ_5p; for B and C we used the same values as for the d⁸p configuration.

The extended energy matrix of the three configurations was diagonalized using the approximated values for the parameters found in the previous calculations and estimates for the initial values for J’ and K’. In the subsequent least-squares calculations we gradually fitted more and more observed levels to the calculated ones and reached a stage at which 109 observed levels were fitted unequivocally to the calculated ones with an rms error of 70 cm⁻¹. A considerable improvement of the fit between the observed and calculated levels was achieved in the previously problematic range 110,000–120,000 cm⁻¹.

These results enabled Professor Shenstone to supply us with an improved and extended list of observed levels. This new list of levels was used in a new series of iterated diagonalizations. In the final least-squares calculation of this stage, which included the effective-interaction parameters β and T, 132 levels were fit with an rms error of 112 cm⁻¹.

We were not able to include in the fit ten levels of the experimental list. Of these levels, seven were considered by Professor Shenstone to be of doubtful identification. The inclusion of any of the remaining three in the least-squares calculation increases the rms error considerably and forces some of the parameters to assume unreasonable values.

In subsequent correspondence with Professor Shenstone a final level list was constructed. The ten problematical levels were reassigned – some to other configurations and others to different J values. Also, the list was amended by addition of 40 new levels to bring the total up to 174. A new iteration was performed, and the observed levels were fitted to the calculated values with an rms error of 133 cm⁻¹. The parameters of this final calculation are given in table 1, Column L.S. 1a.

Table 1.

Parameters of the configurations d⁸p + d⁷sp+ d⁸p. All values are in units of cm⁻¹

p	Diag. 1	L.S. la	L.S. Ib	Diag. 2	L.S. 2
A(374s4p)	120990	120980± 80	1.20775 ±11O	120830	120815± 85
A(d7sp)-A(3d84p)	58350	58335 ± 105	58170 ±145	58540	58500 ±120
A(d7sp)-A(3d85p)	7785	7755 ± 95	7550 ±130	7950	7940±100
B-d7sp	1113	1112± 3	1108± 4	1130	1131 ± 3
B-d8p	1046	1047± 4	1046± 5	1073	1075± 4
B-d8p’	1046			1073
C-d7sp	4875	4888± 23	4928± 31	4550	4550± 14
C-d8p	4535	|4547± 27	4549 ± 37	4170	4172±28
C-d8p’	4535			4170
G2(ds)- d7sp	1755	1756 ± 30	1752 ± 42	1770	1751 ± 32
F2(dp)-d7sp	478	477 ± 6	455 ± 8	478	481 ± 6
F2(dp)- d8p	352	353 ± 5	353 ± 8	352	351 ± 7
F2(dp)- d8p’	87	85 ± 7	87± 10	87	87± 8
G1(sp) — d7sp	10220	10192± 31	10076± 42	10290	10306 ± 34
G1(dp)—d7sp	359	362± 9	353 ± 13	346	352 ± 12
G1(dp)- d8p	295	293± 6	296 ± 8	300	299 ± 7
G1(dp)- d8p’	88	88± 7	90± 10	97	98± 9
G3(dp)- d7sp	42	42±3	43± 3	40	41±3
G3(dp)- d8p	42			40
G3(dp)- d8p’	7	8± 4	8± 5	8	9± 4
A	29	27± 3	28± 4	78	76± 2
B	−900	−1230±J60	−1370 ± 230
T	−5.7	Fixed	Fixed
H d7sp - d8p	185	220± 40	175 ± 55	120	135 ± 55
J d7sp — d8p	1650	1830±330	1755 ±465	1300	1290 ±370
J d7sp — d8p’	400	410± 80	475±110	485	465± 85
K d7sp — d8p	2850	2620 ± 370	2730 ±525	2965	2875 ± 425
K d7sp - d8p’	1035	1005± 75	1015±l10	1070	1050 ± 85
ζp - d7sp	749	744± 21	748± 29	749	748± 23
ζp - d8p	663	662± 18	664± 25	663	657± 20
ζp - d8p’	663			663
ζp - d7sp	630	640± 70	720 ±100	595	605± 75
ζp - d8p	455	450± 50	450 ± 75	455	450± 55
ζp - d8p’	140	130± 55	130± 75	155	135± 60
Δ		133 cm−1	188 cm−1		145 cm−1
Number of levels		174	178		174

The parameters β and T could not both be derived directly from the least squares calculation. Instead, the value of T was fixed at a value obtained from our calculations on the even third spectra of the iron group [8]. When T and β were not included in the calculation, the rms error increased to 145 cm⁻¹ as indicated in table 1, column L.S. 2.

Table 2 contains the list of observed and calculated levels of the configurations d⁸p + d⁷sp+d⁸p’. The spectral purities of the reported assignments are given only when at least one level of the term has a purity of less than 60 percent. In a few cases the parent term was strongly mixed and was not included in the designation. Four observed levels which were assigned to these configurations by Professor Shenstone could not be fit into the scheme on calculated levels. They are:

1. The level d⁷sp²D_5/2 observed at 135258.88 cm⁻¹ with a deviation of about 600 cm⁻¹ from the calculated value;

2. the levels d⁷sp ⁴P_5/2 131834.94 cm⁻¹

   ⁴P_3/2 132225.15 cm⁻¹

   ⁴P_1/2 132120.70 cm⁻¹

which differ from the calculated values by about 1,200 cm⁻¹. When these levels are included in the iterative fitting procedure, the ⁴P levels disagree by about 700 cm⁻¹ and the mean error increases to 188 cm⁻¹. (See table 1, column L.S. 1b; these levels are those in table 2 which are enclosed by parentheses.)

Table 2.

Ni ii – Observed and calculated energy levels 3d⁸4p + 3d⁸5p + 3d⁷4s4p in units of cm⁻¹

THEORETICAL ASSIGNMENT		J	OBS.	CALC.	O-C	CALC. g
MAIN COMPONENT	ADDITIONAL
3d8(3F)4p4D		7/2	51557.85	51701	−143	1.423
		5/2	52738.45	52846	−108	1.359
		3/2	53634.62	53721	−86	1.187
		1/2	54176.26	54252	−76	.003
3d8(3F)4p4G		11/2	53496.49	53364	132	1.273
		9/2	53365.17	53367	−2	1.180
		7/2	54262.63	54205	58	1.021
		5/2	55018.71	54931	88	.620
3d8(3F)4p4F		9/2	54557.05	54523	34	1.288
		7/2	55417.83	55342	76	1.185
		5/2	56075.26	55990	85	.987
		3/2	56424.49	56352	72	.423
3d8(3F)4p4G		9/2	55299.65	55315	−15	1.148
		7/2	56371.44	56478	−107	.936
3d8(3F)4p4F		7/2	57080.55	57103	−22	1.119
		5/2	58493.21	58471	22	.934
3d8(3F)4p4D		5/2	57420.16	57376	44	1.132
		3/2	58705.95	58647	59	.797
3d8(3p)4p4P		5/2	66571.35	66599	−28	1.487
		3/2	66579.71	66584	−4	1.564
		1/2	67031.02	66998	33	2.267
3d8(1D)4p 2F		5/2	67694.64	67666	29	.938
		7/2	68131.21	68053	78	1.179
3d8(1D)4p 2D		3/2	681054.3l	68235	−81	1.050
		5/2	68735.98	68796	−60	1.258
3d8(1D)4p 2P		1/2	68281.62	68118	164	1.070
		3/2	68965.65	68831	135	1.264
3d8(3P)4p 4D		7/2	70778.11	70759	19	1.390
		5/2	70635.55	70626	10	1.334
		3/2	70706.74	70672	35	1.189
		1/2	70748.66	70718	31	.012
3d8(3P)4p 2D		5/2	71770.83	71909	−138	1.206
		3/2	72375.42	72449	−74	.850
3d8(3P)4p 2P		3/2	7298.5.65	72963	23	1.309
		1/2	73903.2.5	73886	17	.926
3d8(3P)4p 2S		1/2	74283.33	74399	−116	1.721
3d8(3P)4p 4S		3/2	74300.93	74304	−3	1.968
3d8(1G)4p 4H		9/2	75149.55	75190	−40 17	.910
		11/2	75721.71	75705		1.091
3d8(1G)4p 2F		7/2	705917.61	75977	−59	1.143
		5/2	76402.04	76395	7	.858
3d8(1G)4p 2G		7/2	79823.03	79874	−51	.890
		9/2	79923.88	79977	−53	1.110
3d7(4F)4sp(3P) 6F		11/2	86343.21	86645	−302	1.450
		9/2	86870.03	86956	−86	1.458
		7/2	87538.09	87599	−61	1.417
		5/2	88128.56	88170	−41	1.332
		3/2	88582.01	88608	−26	1.084
		1/2	88881.59	88884	−2	−.619
3d7(4F)sp(3P) 6D		9/2	88171.88	88272	−100	1.519
		7/2	89100.47	89213	−113	1.541
		5/2		89900		1.590
		3/2		90374		1.742
		1/2		90654		3.285
3d7(4F)sp(3P) 6G		13/2		88787		1.384
		11/2	89460.35	89327	133	1.345
		9/2	89918.47	89795	123	1.281
		7/2	90275.30	90164	111	1.164
		5/2	90526.18	90428	98	.903
		3/2		90595		.106
3d7(4F)sp(3P) 4F		9/2	94283.94	94262	22	1.295
		7/2	94705.93	94701	5	1.214
		5/2	95332.53	95324	9	1.001
		3/2	95893.76	95878	16	.423
3d7(4F)sp(3P) 4G		11/2	94396.74	94363	34	1.274
		9/2	95017.71	94989	29	1.208
		7/2	95573.39	95572	1	1.027
		5/2	96052.48	96056	−4	.625
3d7(4F)sp(3P) 4D		7/2	96535.87	96629	−93	1.407
		5/2	97273.83	97376	−102	1.346
		3/2	97799.66	97917	−117	1.178
		1/2	98122.63	98250	−127	.003
3d7(4F)sp(3P) 2G		9/2	98276.70	98301	−24	1.115
		7/2	99844.13	99857	−13	.908
3d7(4P)sp(3P) 6S		5/2		98759		1.997
3d7(4F)sp(3P) 6F		7/2	99418.61	99222	196	1.130
		5/2	100609.01	100430	179	.876
3d7(4P)sp(3P) 6D		5/2	101754.80	101718	37	1.186
		3/2	102742.74	102733	10	.803
3d8(3F)5p 4D		7/2	103653.03	103741	−88	1.414
		5/2	104503.22	104590	−87	1.310
		3/2	105439.85	105478	−38	1.125
		1/2	106022.79	106086	−63	.013
3d8(1S)4p 2P		1/2		103459		.667
		3/2		104087		1.329
3d8(3F)5p 2G		11/2	104147.29	104066	81	1.273
41%	38% 2G	9/2	105588.89	105496	93	1.164
55%	24% 2G	7/2	105499.05	105429	70	1.006
		5/2	106283.16	106228	55	.701
3d8(3F)5p 2G 60%	32% 4G	9/2	104081.04	104045	36	1.181
58%	24% 4G	7/2	106620.53	106525	96	.964
3d8(3F)5p 4F		9/2	104298.23	104285	13	1.271
39%	33% F	7/2	104646.52	104646	1	1.157
53%	24% 4G	5/2	105668.78	105673	− 4	.931
		3/2	106369.30	106399	−30	.463
3d8(3F)5p 2D		5/2	105861.19	106017	−156	1.178
		3/2	107142.12	107313	−171	.822
3d8(3F)5p 2F 46%	32% 4F	7/2	105838.06	105825	13	1.144
		5/2	107082.21	107080	2	.913
3d7(4P) sp (3P) 6D		9/2	105981.50	105888	93	1.554
		7/2		105817		1.585
		5/2		105863		1.654
		3/2		105971		1.837
		1/2		106106		3.296
3d7(4P)sp (3P) 4S		3/2	107737.81	107835	−97	1.851
3d7(4P) sp (3p) 6P		7/2		108783		1.705
		5/2		108873		1.869
		3/2	109038.84	108901	138	.982
3d7(2G) sp(3P) 4F		9/2	109148.05	109136	12	1.324
		7/2	109846.00	109892	−46	1.163
		5/2	110573.36	110573	0	1.022
		3/2	111120.54	111068	53	.518
3d7(2G) sp (3P) 4H		13/2		109796		1.228
		11/2		109673		1.135
		9/2		109780		.977
		7/2		110088		.744
3d7(2P)sp (3P) 4P		1/2		111112		2.458
44%	30% 4D	3/2		111724		1.519
48%	30% 4D	5/2		111917		1.506
3d7(2G)sp (3P) 4G		11/2		111634		1.263
		9/2		111850		1.160
		7/2	111783.79	112087	−303	.975
		5/2		112329		.584
3d7(4P) sp (3P) 4D		7/2		111437		1.427
56%	25% 4P	5/2		111233		1.437
44%	24% 4P	3/2		111271		1.258
		1/2		111497		.166
3d7(4F)sp (1P) 4G		11/2	112422.19	112549	−127	1.272
67%	28% 4F	9/2	113753.04	113728	25	1.214
41%	44% 4F	7/2		114531		1.124
50%	28% 2F	5/2	115108.09	115173	−65	.820
d7sp 4D		7/2		112683		1.421
		5/2		113262		1.380
65%	24% 4P	3/2		113846		1.309
65%	28% 4P	1/2		114523		.843
3d7(2G)sp (3P) 2H		9/2		113082		.904
		11/2		113952		1.085
3d7(4F)sp(1P) 4F 63%	34% 4G	9/2	113321.95	112935	387	1.275
28%	29% 4G + 29% 2G	7/2	114052.21	113788	264	1.056
61%	16% 2F	5/2	115120.00	114836	284	1.046
60%		3/2		115149		.761
3d7(2G) sp(3P) 2G 58%	25%4G +14%4F	7/2		113765		.970
		9/2		114276		1.119
3d7(4P) sp(3P) 2S		1/2		113841		1.925
3d7(4P) sp (3P) 4P 61%		5/2		114043		1.456
53%	40% 4D	3/2		114387		1.495
38%	40% 1D + 20% 2S	1/2		114378		1.459
3d7(2G) sp (3P) 2F 29%	46% 4G	5/2		114229		.831
51%	22% 1D	7/2	115000.25	114866	134	1.209
3d7(2H) sp (3P)4G		11/2	114858.88	114996	−137	1.269
		9/2	116087.38	115720	367	1.139
		7/2	116275.81	116379	−103	1.019
		5/2	116824.15	116833	−9	.643
d7 sp 2D 62%		3/2	114869.35	115018	−149	.923
50%	33% 4D	5/2	116893.98	117046	−152	1.237
d7 sp 4D 78%		1/2		115177		.461
30%	42% 4F	3/2	115592.25	115568	24	.834
32%	25% 4F + 23% 2D	5/2	115565.98	115340	226	1.154
55%	18% 2F	7/2	115209.98	115321	−111	1.298
3d7(2H) sp (3P)I		15/2		115245		1.200
		13/2		115237		1.109
		11/2		115440		.977
		9/2		115784		.785
3d7(4P) sp (3P) 2D 61%	27% 4D	5/2		115868		1.245
43%	25% 2P	3/2		117094		1.057
d7. sp 4D		7/2	116512.06	116603	−91	1.408
50%	23% 4F	5/2		117595		1.242
40%	23% 4S + 21% 2P	3/2		117989		1.384
38%	26% 2P + 23% 2S	1/2		118159		.758
3d7(2P) sp (3P) 4S 37%	12% 2D + 12%4F	3/2	117662.11	117460	202	1.417
d7 sp 2P 41%	2.3% 2S	1/2		117478		.984
27%	29% 1D + 21% 4F	3/2		118284		1.118
3d7(a2D)sp (3P)4F		9/2	117593.68	117552	42	1.333
		7/2	117972.47	117993	−21	1.250
48%	24% 2D	5/2		118627		1.094
32%	32% 2D + 17% 2P	3/2		118786		.874
3d8(1D)5p 2D 36%	12%(1D)2P	3/2	117763.91	1 l 7858	−94	1.165
65%		5/2	117872.78	117995	−122	1.271
3d7(2H)sp(3P)2I		11/2	118248.98	118305	−56	.928
		13/2	]19010.21	118992	18	1.085
3d8(1D)5p 2F		5/2	118379.11	]18389	−10	.979
		7/2	118563.39	118542	21	1.193
3d8(1D)5p 2P 37%	35%(1D)2D	3/2	118442.81	118510	−67	1.158
50%	29%d7sp 4D	1/2	118631.95	118503	128	.606
3d7(2P) sp(3P)2S 36%	44% 2P	1/2		119423		1.438
3d7(2H)sp(3P) 4H		13/2		119729		1.226
		11/2		120027		1.133
		9/2		120280		.976
		7/2		120498		.692
3d8(3P)5p 4P 51%	19%(1D) 2D	5/2	119796.98	119905	−108	1.518
36%	17%(1D)2P	3/2	120166.52	]20221	−54	1.534
		1/2	120316.02	120260	56	2.406
3d7(2P)sp(3P)2p 54%		1/2		119906		.679
31%	36%d85P 2P	3/2		119945		1.385
3d7(a2D)sp(3P) 4P		5/2		120612		1.527
44%	17% 2P	3/2		121766		1.535
		1/2		122378		2.393
3d8(3P)5p 4D 81%		7/2	120903.31	12]052	−149	1.387
63%	27%(3P)2D	5/2	121325.09	121302	23	1.322
54%		3/2	121385.80	121307	79	1.227
83%		1/2	121561.06	121445	116	.064
3d8(3P)5p 2P 35%	18%2D + 17%2S	3/2	121042.57	121091	−48	1.260
		1/2		121917		.751
3d8(3P)5p 2D 57%	20%(3P)4D	5/2	]21050.66	121007	44	1.216
28%	32%(3P)2P	3/2	121800.34	121662	138	1.260
3d8(3P)5p 4S 43%	23%(3P)2D	3/2	121456.30	121409	47	1.509
3d7(2H)sp(3P) 2G		9/2	121692.55	121749	−56	1.108
		7/2	121862.57	121882	−19	.873
3d7(a2D)sp(3P) 2F		5/2		121963		.914
		7/2		122670		1.150
3d8(3P)5p 2S		1/2		122063		1.883
3d7(a2D)sp(3P) 2D		3/2		122131		.930
		5/2		122277		1.204
3d7(a2D)sp(3P) 2P		1/2		124474		.751
		3/2		124771		1.362
3d7(2H)sp(3P) 2H		9/2	124652.00	124787	−135	.911
		11/2	125003.4l	125159	−156	1.092
3d8(1G)5p 2F		7/2	1:27219.57	126938	282	1.142
		5/2	127331.60	1:27071	261	.858
3d8(1G)5p 2H		9/2	126679.98	126895	−215	.910
		11/2	126857.97	127061	−203	1.091
3d7(4P)sp(1P) 4S		3/2	126738.82	126903	− 164	1.990
3d8(1G)5p 2G		9/2	127885.86	127896	−10	1.110
		7/2	127895.33	127888	7	.890
3d7(4P)sp(1P) 4D		7/2	129782.07	129925	−143	1.427
		5/2	129988.05	130121	−133	1.366
		3/2	130331.78	130372	−40	1.200
		1/2	130570.42	130595	−25	.037
3d7(2G)sp(1P) 2H		1l/2	131424.32	131131	293	1.086
		9/2	132311.98	132116	196	.915
3d7(2F)sp(3P) 4G		5/2		132462		.611
		7/2		132685		1.012
		9/2		133035		1.188
		11/2	133625.96	133567	59	1.272
3d7(4P)sp(1P) 4P		5/2	(131834.94)	132957	(−1122)	1.548
		3/2	(132225.15)	133323	(−1098)	1.681
		1/2	(132120.70)	133328	(−1208)	2.490
3d7(2G)sp(1P) 2F		7/2	133169.92	132917	253	1.138
		5/2	134208.30	134110	98	.994
3d7(‘F)sp(3P) 4F		3/2	133190.19	132894	296	.425
		5/2	133209.30	133179	30	1.021
		7/2	133528.02	133541	−13	1.214
		9/2	133853.04	133898	−45	1.303
3d7(2G)sp(1P) 2G		9/2	133445.75	133676	−230	1.116
		7/2	134380.82	134783	−402	.891
3d7(2F)sp(3P) 4D		7/2	133850.83	133888	−37	1.417
		5/2	133973.33	133885	87	1.225
		3/2	134156.28	134113	43	1.198
		1/2	134283.76	134215	69	.047
3d7(2F)sp(3P) 2D		5/2	134783.14	134841	−58	1.205
		3/2	134964.78	135024	−59	.913
3d7(2P)sp(1P) 2P		1/2		135549		.776
		3/2	135382.53	135661	−278	1.169
3d7(2F)sp(3P) 2G		7/2	135746.06	135737	9	.894
		9/2	136076.26	135942	134	1.112
3d7(2P)sp(1P) 2D		5/2	(135258.92)	135900	(−741)	1.200
		3/2		137089		.826
3d7(2H)sp(1P) 2I		13/2		136509		1.077
		11/2		137494		.929
3d7(a2D)sp(1P) 2D		5/2		138244		1.154
		3/2		139402		.864
3d7(2H)sp(1P) 2G		9/2	138495.84	138613	−117	1.109
		7/2		139322		.906
3d7(2F)sp(3P) 2F		7/2		138858		1.138
		5/2		139447		.889
3d7(2P)sp(1P) 2S		1/2		139683		1.911
3d7(a2D) sp (1P) 2F		7/2		139904		1.140
		5/2		141012		.902
3d7(2H) sp (1P) 2H		11/2		141873		1.091
		9/2		142868		.913
3d7(a2D) sp (1P) 2P		3/2		142107		1.332
		1/2		143961		.738
3d7(b2D) sp (3P) 2P		5/2		151277		1.598
		3/2		151257		1.730
		1/2		151281		2.661
3d7(b2D) sp (3P) 2F		3/2		152576		.403
		5/2		152859		1.029
		7/2		153254		1.237
		9/2		153760		1.330
3d8(1S)5p 2P		1/2		153513		.667
		3/2		154114		1.148
3d7(2F) sp (1P) 2G		7/2		154379		.890
		9/2		154810		1.114
3d7(2F) sp (1P) 2D		3/2		154619		.986
		5/2		154998		1.078
3d7(2F) sp (1P) 2F		5/2		155556		.979
		7/2		155892		1.143
3d7(b2D) sp (3P) 2P		3/2		156895		1.330
		1/2		157329		.623
3d7(b2D) sp (3P) 2F		5/2		157904		.869
		7/2		158157		1.166
3d7(b2D) sp (3P) 4D		1/2		158243		.048
		3/2		158398		1.200
		5/2		158717		1.359
		7/2		159298		1.405
3d7(b2D) sp (3P) 2D		5/2		161296		1.200
		3/2		161382		.803
3d7(b2D) sp (1P) 2P		1/2		173566		.667
		3/2		174048		1.332
3d7(b2D) sp (1P) 2F		5/2		174919		.858
		7/2		175397		1.142
3d7(b2D)sp (1P) 2D		3/2		179748		.800
		5/2		180530		1.199

4. The Theoretical Interpretation of the Configuration 3d⁸4f

The abbreviation d^sf will be used for the configuration 3d⁸4f. The treatment of this configuration was relatively simple. Initial values for the parameters B, C, and ζ_d were taken from the configuration d⁸ of Ni iii. [8] A rough first estimate for the parameters of the d – f interaction and for ζ_f was done by direct observation of the experimental level values. The final parameters which we obtained for these con figurations are given in table 3. In column L.S.a, all the parameters were set free, and the rms error is 25.6 cm⁻¹. We can see that the parameters F₄(df), G₃(df) and ζ_f are equal to zero within their statistical accuracy. Column L.S.b of table 3 gives the results of a calculation in which the above-mentioned parameters were fixed at zero. In this case the rms error is 25.2 cm⁻¹. The observed and calculated levels of d⁸f are given in table 4.

Table 3.

Parameters of the Configuration d⁸f All values are in units of cm⁻¹

p	Diag.	L.S.a	L.S.B
A — 3d84f	128190	128189. ±	128186 ±5
B	1035	1035.4±0.6	1035.5 ±0.6
C	4080	4086 ±:6	4086 ±6
F2(df)	8.5	8.4 ±0.5	8.3 + 0.4
F4(df)	0	0.1 ±0.1
G1(df)	1	1.6 ±0.8	1.4±0.6
G3(df)	0	0.3 + 0.3
G5(df)	0	0.03 + 0.05
Zd	670	668 ±3	668 ±3
ζf	0	3 ±2
Δ		25.6 cm−1	25.2 cm−1

Table 4.

Ni ii – Observed and calculated energy levels 3d⁸4f

THEORETICAL ASSIGNMENT		J	OBS.	CALC.	O-C	CALC. g
d8 PARENT	K
3F4	7	13/2	118803.82	118837	−33	1.086
		15/2	118848.92	118837	12	1.200
	1	3/2	118809.34	118800	9	1.779
		1/2	118774.76	118805	−30	1.523
	2	5/2	118828.61	118833	−4	1.448
		3/2	118877.09	118853	24	1.143
	3	7/2	118874.11	118871	3	1.329
		5/2	118897.94	118900		1.094
	6	11/2	118892.99	118909	−16	1.081
		13/2	118893.24	118909	−16	1.212
	4	9/2	118914.34	118905	9	1.269
		7/2	118923.20	118924	−1	1.082
	5	9/2	118927.02	118924	3	1.081
		11/2	118939.53	118924	16	1.234
3F3	0	1/2	120189.55	120170	20	2.043
	l	3/2	120199.18	120194	5	1.379
		l/2	120203.49	120181	22	.704
	2	5/2	120203.49	120222	−19	1.244
		3/2	120222.89	120224	−1	.861
	6	11/2	120211.30	120205	6	.970
		13/2	120218.22	120205	13	1.117
3F3	3	7/2	120250.17	120249	1	1.176
		5/2	120271.97	120265	7	.909
	4	7/2	120268.81	120281	−12	.938
		9/2	120281.11	120272	9	1.146
	5	11/2	120270.44	120265	5	1.124
		9/2	120272.53	120265	8	.949
3F3	1	3/2	121042.52	120092	−49	1.540
		1/2	121090.71	121092	−1	1.059
	5	11/2	121120.88	121122	−1	.976
		9/2	121125.41	121122	3	.771
	2	5/2	121146.98	121146	1	1.181
		3/2	121161.81	121147	15	.763
	4	7/2	121178.56	121192	−13	.762
		9/2	121180.54	121190	−9	1.008
	3	7/2	121192.32	121183	9	1.063
		5/2	121194.14	121188	6	.753
1D2	4	9/2	132818.16	132855	−37	1.138
		7/2	132846.53	132857	−10	.933
	3	5/2	(132729.48)	132875	(−146)	.878
		7/2	132869.16	132889	−20	1.148
1D2	2	5/2	132912.15	132944	−32	1,197
		3/2	l32927.97	132940	−12	.793
	1	3/2	132982.51	133005	−23	1.283
		1/2	133001.47	133005	−4	.564
	5	11/2	133014.08	132950	64	l.119
		9/2	133031.00	132950	81	.943
3P2	3	7/2	135400.67	135438	−37	1.179
		5/2	135461.55	135452	10	.979
	4	9/2	135435.26	135414	21	1.283
		7/2	135444.47	135430	14	1.127
	2	3/2	135493.26	135485	8	.515
		5/2	135512.92	135501	12	.976
	5	11/2	135538.61	135582	−43	1.243
		9/2	135558.80	135582	−23	l.092
	1	3/2	135652.93	13,5661	−8	1.057
		1/2	135670.49	13,5659	11	.l06
3P1	2	5/2		13,5784		1.2,55
		3/2	135746.13	135784	−38	.88,5
	4	9/2	(135580.35)	135773	(−193)	1.179
		7/2	(135464.86)	13,5776	(−311)	.976
	3	5/2	135849.41	13,5866	− 17	.89.5
		7/2	135879.41	135864	15	1.174
3P0	3	7/2	(135954.09)	136055	(−101)	1.176
		5/2	136122.61	136056	67	.90,5
1G4	1	3/2		140232		1.334
		1/2		140233		.667
	2	5/2		140345		1.200
		3/2		140347		.800
	7	13/2		140355		.9.33
		15/2		140355		1.067
	3	7/2		140491		1.143
		5/2		140492		.857
	4	9/2		140632		1.111
		7/2		140632		.889
	6	11/2		140643		.923
		13/2		140643		1.077
	5	11/2		140708		l.091
		9/2		140708		.909
1S0	3	7/2		171363		1.143
		5/2		171364		.857

We would like to emphasize that the d⁸f configuration was calculated independently without including any interaction with any other configuration. This simple treatment is justified to some extent by the small mean error.

In all, 60 experimental levels of d⁸f were fitted to the calculated ones. Three observed levels could not be fitted. They are:

$$132729.48c{m}^{-1}withJ=5/2,$$

$$135954.09{\text{cm}}^{-1}\text{ with }J=7/2,$$

$$135580.25{\text{cm}}^{-1}\text{ with }J=9/2.$$

The coupling for this configuration is the J–l coupling; that is: the S’ and L’ of the d⁸ parent term first combine to form J”. Then J” is combined with the l = 3 of the f electron forming K and finally the spin of this electron is added to K and the total J is formed. This is the coupling used in table 4.