Odd Configurations in Singly-Ionized Copper

Abstract

Experimental levels of the configurations 3d⁹4p, 3d⁹5p, 3d⁹6p, 3d⁸4s4p, 3d⁹4f, and 3d⁹5f of Cu ii were compared with corresponding calculated values. The electrostatic interactions between the configuration 3d⁸4s4p and the configurations 3d⁹4p, 3d⁹5p, and 3d⁹6p were considered explicitly. It was shown that the configurations 3d⁹4f and 3d⁹5f of Cu ii do not interact strongly with other configurations.

1. Introduction

The configurations (3d + 4s)ⁿ in the second spectra of the iron group were studied by Racah, Shadmi, Oreg, and Stein [1–3].^l The configurations 3dⁿ4p in the second spectra of the iron group as well as the configurations 3dⁿ4p + 3dⁿ⁻¹4s4p for Sc ii, Ti ii and V ii were investigated by the author [4, 5].²

An examination of the spectrum of Cu ii [6], indicates that the experimental data are very abundant. The configuration d⁹p consists of 6 terms splitting into 12 levels. All the predicted levels for the configurations 3d⁹4p and 3d⁹5p are given in AEL [6], whereas for 3d⁹6p only the experimental level 6p ³P₀ is missing. The configuration d⁸sp comprises 38 terms splitting into 90 levels. In AEL, 29 terms splitting into 65 levels are given for the configuration 3d⁸4s4p with definite term designations. In addition the levels $1_1^0$ at 140482? and $3_1^0$ at 144241 are assigned to 3d⁸s4p. The configuration d⁹f comprises 10 theoretical terms splitting into 20 levels. All the predicted levels for the configurations 3d⁹4f and 3d⁹5f are given in AEL. In addition 5 experimental terms splitting into 8 levels are given for the configuration 3d⁹6f. However in the latter configuration 5 levels appear with question marks.

To treat the seven configurations as one problem and consider all the interactions between configurations would involve more electrostatic parameters than the terms available. This method is therefore quite meaningless.

The configuration 3d⁹4p is much lower than the other odd configurations and thus the interaction between configurations is expected to be weak here. This expectation is borne out by treating this configuration individually. The rms error is only 119 cm⁻¹ and the 9 experimental g-factors agree well with the calculated values.

Separate treatments of the configurations 3d⁸4s4p, 3d⁹5p and 3d⁹6p did not yield favorable results (rms error ~ 250 cm⁻¹). In addition the parameters in these three cases were quite unreasonable. The parameter G₃ even assumed negative values for 3d⁸4s4p, 3d⁹5p and 3d⁹6p. These results are not surprising since the configurations 3d⁹5p and 3d⁹6p are in the middle of the configuration 3d⁸4s4p and we may expect these three configurations to be strongly interacting. We thus considered the three configurations 3d⁸4s4p, 3d⁹5p and 3d⁹6p as one problem, inserting the interactions between configurations 3d⁹5p − 3d⁸4s4p and 3d⁹6p − 3d⁸4s4p. The interaction 3d⁹5p −3d⁹6p was neglected as then there would be too many parameters, causing the subsequent results to become meaningless. In addition, since the configurations 3d⁹5p and 3d⁹6p are separated we do not expect the interaction between these configurations to be very strong. For 3d⁹5p + 3d⁸4s4p + 3d⁹6p, the rms error was 136 cm⁻¹.

Separate treatments of the configurations 3d⁹4f and 3d⁹5f yielded excellent results. The rms errors were only 51 and 4.5 cm⁻¹, respectively. We could expect to obtain similar results for 3d⁹6f and can be quite certain that this configuration does not interact strongly with the other configurations. The experimental data for 3d⁹6f is, however, too limited to consider it separately.

Finally, the configurations 3d⁹4p, 3d⁸4s4p, 3d⁹5p, and 3d⁹6p were considered as one problem by inserting the interactions 3d⁸4s4p−3d⁹4p, 3d⁸4s4p−3d⁹5p, and 3d⁸4s4p − 3d⁹6p. The purpose here was to obtain approximate values for the parameters of the interaction between the configurations 3dⁿ4p − 3dⁿ⁻¹4s4p in the second spectra of the iron group for elements on the right side of the periodic table.

2. The Configuration 3d⁹ 4p − Cu ii

The results for Cu ii − 3d⁹4p in the general treatment of the configurations dⁿp of the second spectra of the iron group [4], indicate that the agreement between the observed and calculated values and g-factors of some levels is not very good. In order to ascertain whether these discrepancies are caused by the interaction with 3d⁸4s4p or are due to the fact that the parameters were forced to be linear, it is necessary to refer to the individual treatment of Cu ii − 3d⁹4p, [4]. The parameters with their standard errors are given in table 1.

Table 1.

Parameters for Cu ii − 3d⁹4p

Parameter	Initial value	Final value
A	70.281	69,802 ± 42
F2	383	344 ± 7
G1	306	305 ± 7
G3	45	38 ± 6
a	95	100 (Fix)
ξd	821	802 ± 43
ξp	536	502 ± 82
rms error		119

Whereas in the general treatment the highest deviation for Cu ii − 3d⁹4p is −270, in the individual treatment it is only 167. Furthermore, there is excellent agreement between the observed and calculated g-factors.

As for the general treatment of Cu ii 3d⁹4p, the following changes in designation were made:

$$3d^9{(}^2\text{D})4p^3{\text{D}}_2\leftrightarrow 3d^9{(}^2\text{D})4p^1{\text{D}}_2$$

$${}_3{d}^9{(}^2\text{D})4p^3{\text{D}}_3\leftrightarrow 3d^9{(}^2\text{D})4p^1{\text{F}}_3$$

In both cases there was considerable mixing between the eigenfunctions involved.

3. The Configurations 3d⁹5p + 3d⁸4s4p + 3d⁹6p − Cu ii

3.1. Initial Parameters

The matrix elements of the interactions between configurations 3d⁸4s4p − 3d⁹5p and 3d⁸4s4p − 3d⁹6p were obtained from Rosenzweig [7]. However, now the interaction matrix elements between the cores 3d⁸4s and 3d⁹ vanish. This is due to the fact that since H is the parameter pertaining to the interaction between electrons d and s, the quantum numbers of the electrons p must be the same on both sides of the matrix elements. Thus only the matrices of J and K enter into the electrostatic matrix dⁿ⁻¹sp − dⁿp’, and with the same coefficients as for dⁿ⁻¹sp − dⁿp. The matrices of J and K for d⁸sp − d⁹p’ and d⁸sp − d⁹p” were added to the previously obtained matrices of (d + s)⁹p.

The values of the parameters F₂, G₁ G₃, α, ζ_d, and ζ_p obtained from 3d⁹4p in the variation of the GLS (general least-squares) with β and T eliminated [4], were used as initial values for the configuration 3d⁸4s4p. The parameters B and C were obtained from the same GLS by adding to the values of 3d⁸4p the linear intervals of 65 and 310 respectively. Thus, initially,

$$\begin{gathered} B^{\prime }={1140}^3 \\ {C}^{\prime }=4460 \\ {F}_2^{\prime }=370 \\ {G}_1^{\prime }=300 \\ {G}_3^{\prime }=40 \\ {\alpha }^{\prime }=97 \\ {\zeta }_d^{\prime }=770 \\ {\zeta }_p^{\prime }=460 \end{gathered}$$ (1)

Since $G_{ds}^{\prime }$ is the parameter of the d − s interaction for the core d⁸s its approximate value can be taken from Cu iii=3d⁹ + 3d⁸4s. From Shadmi [8], we obtain

$$G_{ds}^{\prime }=1890$$ (2)

A starting value for the parameter $G_{ps}^{\prime }$ is obtained from the interpolation of $G_{ps}^{\prime }(sp)$ and $G_{ps}^{\prime }(d^{10}sp)$. From AEL, the center of gravity of 4s(²S)4p y³P in Sc ii is 39230 and 4s(²S)4p y¹P in Sc ii is 55716. Thus,

$$G_{ps}^{\prime }(sp)=8243$$ (3)

A similar calculation for Gall − 3d¹⁰4s4p yields

$$G_{ps}^{\prime }\left(d^{10}sp\right)=11212$$ (4)

Hence by interpolation,

$$G_{ps}^{\prime }\left(d^{8}sp\right)=10620$$ (5)

In order to obtain an approximate value for the height of the configuration d⁸sp, it is most reasonable to consider the quintets as they have, of course, no interaction with d⁹p. From an examination of the experimental data it would seem most appropriate to consider the electrostatic interaction matrix of ⁵F as there the Lande interval rule is satisfied well, and unlike ⁵D, in ⁵F there is no level given with a question mark. Then, approximately,

$${\text{F}}_{\text{C}.\text{G}.}=A^{\prime }-8B^{\prime }-2G_{ds}^{\prime }+3F_2^{\prime }-G_{ps}^{\prime }+12{\alpha }^{\prime }=113700$$ (6)

Using values for the parameters obtained previously we get

$$A^{\prime }=134,950$$ (7)

For the configurations 3d⁹5p and 3d⁹6p initial values of the parameters were obtained by using the electro-static matrices of d⁹p (p. 299, TAS [9]), and taking the centers of gravity of the experimental terms [6]. Then

$$\begin{gathered} A^{″}=121360 \\ {F}_2^{″}=114 \\ {G}_1^{″}=115 \\ {G}_3^{″}=61 \\ {A}^{″\prime }=139950 \\ {F}_2^{″\prime }=11 \\ {G}_1^{″\prime }=56 \\ {G}_3^{″\prime }=43. \end{gathered}$$ (8)

Unlike the electrostatic parameters, the spin-orbit interaction parameters obtained in the individual treatments of 3d⁹5p and 3d⁹6p were quite reasonable.

Thus they were adopted as starting values here:

$$\begin{gathered} {\zeta }_d^{\prime \prime }=586 \\ {\zeta }_p^{\prime \prime }=142 \\ {\zeta }_d^{\prime \prime }{}^{\prime }=740 \\ {\zeta }_p^{\prime \prime }{}^{\prime }=27 \end{gathered}$$ (9)

In the initial diagonalization the parameters of the interaction between configurations were not inserted.

From the results of 3dⁿ4p + 3dⁿ⁻¹4s4p for Sc ii, Ti ii, and V ii [5], we note that both J and K are positive and K is almost three times J. However, here the interactions are between 3d⁸4s4p − 3d⁹5p and 3d⁸4s4p − 3d⁹6p, and thus we would expect the parameters to be considerably smaller than for

$$3d^{n}4p-3d^{n-1}4s4p,n⩽3$$

Thus, in the second iteration the following values for the parameters of the interactions between configurations were inserted:

$$\begin{gathered} J\left(3d^{8}4s4p-3d^{9}5p\right)=J\left(3d^{8}4s4p-3d^{9}6p\right)=200 \\ K\left(3d^{8}4s4p-3d^{9}5p\right)=K\left(3d^{8}4s4p-3d^{9}6p\right)=600 \end{gathered}$$ (10)

3.2. Results and Discussion

Of the 90 levels assigned to 3d⁹5p + 3d⁸4s4p + 3d⁹6p in AEL we found it necessary to omit the following five levels:

1. 3d⁸4s(²D)4p” ‘ ¹p at 125400

2. 3d⁸4s(⁴P)4p^{iv 5}S at 128366

3. 3d⁸4s(²D)4p” ‘¹D at 130632

4. 3d⁸4s (⁴P)4p^iv $1_1^{\text{0}}$ at 140482?

5. 3d⁸4s(⁴P)4p^iv $3_1^0$ at 144241.

The following changes in designation were found necessary:

1. AEL d⁸s(²F)p” ³F₃ ↔ AEL d⁸s(²F)p” ³G₃

2. AEL d⁸s(⁴D)p” ³D_2,3 ↔ AEL d⁸s(⁴P)p^iv5 P_2,3

3. AEL d⁸s(²D)p” ³D₁ → ³P(³P) ⁵P ₁

4. AEL d⁸s(⁴P)p^{iv 5}P₁ → ¹D(³P) ³P₁

5. AEL d⁸s(²D)p′ ′ ′¹F → ³P(³P) ⁵D₃

6. AEL d⁸s(⁴P)p^{iv 5}P₃→;³P(³P)⁵D₄

7. AEL d⁸s(²P)p^{v 3}P₁ ↔ AEL d⁸s(²P)p^{v 3}D₁

8. AEL d⁸s(²P)p^{v 1}D → ³P(³P) ⁵S

9. AEL d⁸s(²G)p^{vi 1}H→ ¹G(³P) ³H₅

10. AEL d⁸s(²G)p^{vi 3}F₂→ (²D)6p ³P₂

11. AEL d⁸s(²G)p^{vi 3}F₃→ (²D)6p ¹F

12. AEL d⁸s(²G)p^{vi 3}F₄ → ³F(¹P) ³F₄

13. AEL (²D)6p ³D_{2 ,3} → ³F(¹P)³F_{2 ,3}

14. AEL (²D)6p ³P₂→ (²D)6p ¹D

15. AEL (²D)6p ¹D → (²D)6p ³D₂

16. AEL (²D)6p ³F₃ → (²D)6p ³D₃

17. AEL (²D)6p ¹F→ (²D)6p ³F_3.

The following levels showed very strong mixing and the main contribution in each case was not the same as that given in AEL:

1. (²D)5p ¹F and (²D)5p ³F₃

2. (²D)5p^lD, d⁸s(²F)p″ ¹D, and (²D)5p³D₂

3. ³F(¹P)³F_2,3,4 and ¹G(³P)³F_{2, 3, 4}

4. (²D)6p ³P₂ and (²D)6p ¹P.

The 85 experimental levels were fitted by means of 26 final parameters with an rms error of 136. The parameters with their standard errors are given in table 2. The final value of 1430 ± 66 for $G_{ds}^{\prime }$ seems too low when compared with the initial value of 1890. Martin and Sugar [10] resolved a similar problem for Cu i by introducing the Sack correction

$$E_s\left[S(S+1)-S_c\left(S_c+1\right)\right]$$

where S is the net spin of d⁸sp and S_c is the spin of d⁸s, which absorbed the distortion in the d − s interaction.

Table 2.

Parameters for Cu ii − 3d⁹5p 4 + 3d⁸4s4p + 3d⁹6p

Parameter	Initial value	Final value
A′	134,950	134,252 ± 44
A″	121,360	121,591 ± 88
A‴	139,950	139,725 ± 117
B′	1,140	1,210 ± 5
C′	4,460	4,777 ± 34
$G_{ds}^{\prime }$	1,890	1,430 ± 66
$F_2^{\prime }$	370	486 ± 6
$F_2^{\prime \prime }$	114	88 ± 12
$F_2^{\prime \prime \prime }$	11	10 ± 9
$G_1^{\prime }$	300	428 ± 13
$G_1^{\prime \prime }$	115	73 ± 13
$G_1^{\prime \prime \prime }$	56	10 ± 14
$G_3^{\prime }$	40	74 ± 6
$G_3^{\prime \prime }$	61	15 ± 8
$G_3^{\prime \prime \prime }$	43	0(Fix)
$G_{ps}^{\prime }$	10,620	10,836 ± 40
a′	97	72 ± 6
J(3d84s4p ‒ 3d95p)	200	291 ± 110
K(3d84s4p ‒ 3d95p)	600	761 ± 56
J(3d84s4p ‒ 3d96p)	200	150 ± 1l4
K(3d84s4p ‒ 3d96p)	600	674 ± 351
${\zeta }_d^{\prime }$	770	933 ± 25
${\zeta }_d^{\prime \prime }$	856	811 ± 46
${\zeta }_d^{\prime \prime }{}^{\prime }$	740	843 ± 47
$G_p^{\prime }$	460	686 ± 62
${\zeta }_p^{\prime \prime }$	142	184 ± 111
${\zeta }_p^{\prime \prime }{}^{\prime }$	27	48 ± 51
rms error		136

Since $G_{ps}^{\prime }$ is much larger than $G_{ds}^{\prime }$, the p − s interaction is stronger than the d − s interaction. Thus the levels of the configuration d⁸sp are coupled as d⁸ (S₁L₁)_sp(^1,3P)SL and not d⁸_s(S₂L₁)p SL as given in AEL.

For each of the rejected levels there is no corresponding theoretical level predicted in the vicinity of the experimental level given for that particular J.

The closest theoretical level of J equal to 1 for 4p”‘¹P given at 125,400, is the level ¹D(³P)³D₁ at around 129,000. An examination of the original paper by Shenstone [11], reveals that this level has only the three combinations with 3d¹⁰a ¹S, 3d⁹4s ¹D and 3d⁹5s ¹D. We omitted this level from the calculations on the basis of not being relevant to the interactions considered.

The level 4p^iv5S at 128,366 has altogether five combinations with even levels, the J values of which are 1, 2, and 3. Thus, the J value of this level should be 2. Since the nearest theoretically predicted level for J equal to 2 is at 137,190, the level 4p^{iv 5}S was neglected.

The level 4p’“ ¹D only has the two combinations with 3d⁹4s^lD and 3d⁹5s ¹D. Thus, conceivably, this level could be given a J assignment of either 1, 2, or 3. However, even then the smallest deviation would be almost 2000, and hence we also neglected this level.

The level $3d^{8}4s{(}^4\text{P)}4p^{iv}{1}_1^0$, given at 140482, with a question mark, has only the combinations with 3d⁸4s^{2 3}F₂ and 3d⁹4s ³D₁. Thus the value of J for this level should be either 1 or 2. However, the nearest level of J equal to 1 is ³P(³P)³S at 138720. Had there been several combinations of this level with even levels of J equal to 0 and 1, then perhaps the level $1_1^0$, could have been assigned to either ³P(³P)¹S or (²D)6p ³P₀. However, with only the two combinations given by Shenstone [11], the level $1_1^0$ has to be rejected. Similarly the level $3d^{8}4s{(}^4\text{P)}4p^{\text{iv}}{3}_1^0$, has only two combinations, i.e., with 3d⁸4s^{2 3}P₀ and 3d⁹4s³D₂, both given with question marks by Shenstone [11]. As there are no theoretically predicted levels for J equal to either 0, 1, or 2 in that vicinity, this level had to be rejected as well.

It should be noted that the predicted level 4p’“ ¹P, i.e., ¹D(¹P)¹P is at 153778, whereas the predicted level 4p^{iv 5}S, i.e., ³P(³P)⁵S is at 136223. The theoretically predicted level p”’¹D, i.e., ¹D(¹P)¹D is at 150054.

The necessity for the changes 1,2, and 3 was already clearly evident from the initial diagonalization. Later it became apparent that in order to improve the agreement, the level p^{iv 5}P₁ should be assigned to the vacant level ¹D(³P)³P₁.

Also from the initial diagonalization it was found that for J equal to 3 there is only one level in the neighborhood of 131000. As the theoretical level d⁸s(²D)p”‘ ¹F, i.e., ¹D(¹P)¹F is predicted at around 150500, it would seem that the experimental level p”’ ¹F should be neglected. However, an examination of the combinations for the levels p”‘ ¹F and p^iv5D₃ [11], permits an alternate more satisfying possibility. The level p^{iv 5}D₃ has combinations only with J equal to 3 and 4. The level p”‘ ¹F has ten combinations with even levels. Eight of these ten combinations are with triplets and seven of the ten are with J equal to 2. From the above considerations the level p”‘ ^lF must be a valid level and assigned to J equal to 3, but the level p^{iv 5}D₃ could conceivably be assigned to J equal to 4, i.e., to the level ³P(³P)⁵D₄. The level p”‘ ^lF is then assigned to p^{iv 5}D₃.

The exchange 7 was performed in a later iteration. After the exchange, the theoretical splittings of the terms p^{v 3}P and p^{v 3}D correspond more closely to the experimental splittings. It should be noted that there is considerable mixing between the eigenfunctions of the two levels p^{v 3}P₂ and p^{v 3}D₂.

Attempts to fit the level d⁸s(²P)p^{v l}D at 135953 to the theoretical level ³P(³P)¹D gave deviations of the order of 1000. As this level has ten combinations with even levels, it is definitely a valid level. Since eight of the ten combinations are with triplets and since this level fits very nicely to ³P(³P)⁵S, we adopted the change 8.

The changes 9 to 16 were performed after numerous attempts to fit as many levels as possible with the same assignments as given in AEL. These changes are mainly due to the fact that the coupling for the configuration 3d⁹6p is far from LS – probably much closer to jl – and in addition this configuration is very strongly mixed with the terms ³F(¹P)³D, ³F and ¹G(³P)³F of 3d⁸4s4p. The above facts are vividly illustrated in the “PERCENTAGE” column of table 7.

Table 7.

– Observed and calculated levels of Cu ii 3d⁹5p + 3d⁸4s4p + 3d⁹6p

Name	J	Percentage	AEL		Obs. Level (em‒1)	Calc. Level (em‒1)	O–C	Calc. g
			Config.	Oesig.
3F(3P)5D	0	94				111,640
	1	93	3d84s(4F)4p	4p′ 5D	111,124?	111,249	‒125	1.482
	2	92	3d84s(4F)4p	4p′ 5D	110,363	110,481	‒118	1.484
	3	91	3d84s(4F)4p	4p′ 5D	109,276	109,392	‒116	1.490
	4	94	3d84s(4F)4p	4p′ 5D	107,942	108,072	‒130	1.496
3F(3P)5G	2	96	3d84s(4F)4p	4p′ 5G	112,424	112,383	41	0.362
	3	89 + 73F(3P)5F	3d84s(4F)4p	4p′ 5G	111,877	111,811	66	0.940
	4	84 + 103F(3P)5F	3d84s(4F)4p	4p′ 5G	111,219	111,122	97	1.167
	5	83 + 133F(3P)5F	3d84s(4F)4p	4p′ 5G	110,632	110,489	143	1.281
	6	100				110,168		1.333
3F(3P)5F	1	98	3d84s(4F)4p	4p′ 5F	114,756	114,672	84	0.021
	2	92	3d84s(4F)4p	4p′ 5F	114,482	114,373	109	0.981
	3	86 + 73F(3P)5G	3d84s(4F)4p	4p′ 5F	114,000	113,859	141	1.223
	4	84 + 93F(3P)5G	3d84s(4F)4p	4p′ 5F	113,303	113,125	178	1.324
	5	86 + 11FF(3P)5G				112,189		1.380
3F(3P)3G	3	74 + 223F(3P)3D	3d84s(2F)4p	4p″ 3F	116,644	116,690	‒46	0.893
	4	81 + 213F(3P)1G	3d84s(2F)4p	4p″ 3G	115,360	115,402	‒42	1.050
	5	94	3d84s(2F)4p	4p″ 3G	115,546	115,611	‒65	1.205
3F(3P)3D	1	88 + 61D(3P)3D	3d84s(2F)4p	4p″ 3D	118,071	118,069	2	0.500
	2	76 + 103F(3P)3F + 71D(3P)3D	3d84s(2F)4p	4p″ 3D	117,130	117,091	39	1.109
	3	60 + 193F(3P)3G + 91D(3P)3D	3d84s(2F)4p	4p″ 3D	116,375	116,376	‒1	1.183
3F(3P)3F	2	83 + 93F(3P)3D	3d84s(2F)4p	4p″3F	119,040	119,081	‒41	0.725
	3	63 + 163F(3P)1F + 83F(3P)3D	3d84s(2F) 4p	4p″ 3G	118,143	118,114	29	1.088
	4	89	3d84s(2F) 4p	4p″ 3F	117,667	117,674	‒7	1.242
3F(3P)1G	4	74 + 213F(3P)3G	3d84s(2F) 4p	4p″ 1G	118,992	119,063	‒71	1.020
(2D)5p 1F	3	47 + 39(2D)5p 3F	3d9(2D5/2)5p	5p 3F	120,685	120,670	15	1.003
(2D)5p 1D	2	43 + 333F(3P)1D + 12(2D)5p 3D	3d84s(2F)4p	4p″ 1D	120,876	120,878	‒2	1.041
3F(3P)1F	3	42 + 35(2D)5p3D + 16(2D)5p 3F	3d84s(2F)4p	4p″ 1F	121,079	121,068	11	1.134
3F(3P)1D	2	40 + 28(2D)5p3D + 24(2D)5p 3F	3d9(2D5/2)5p	5p 3D	121,982	121,974	8	0.991
(2D)5p 3P	0	99			122,224	122,231	‒7
	1	66 + 28(2D)5p 1P			120,920	120,;947	‒27	l..352
	2	94			120,092	120,125	‒33	1.492
(2D)5p 3F	2	69 + 16(2D)5p3D + 7(2D)5p 1D			122,746	122,667	79	0.810
	3	40 + 45(2D)5p1F + 8(2D)5p 3D	3d9(2D3/2)5p	5p 1F	123,017	123,033	‒16	1.090
	4	97			120,790	120,718	72	1.246
(2D)5p 1P	1	60 + 33(2D)5p 3P			122,868	122,848	20	1.172
(2D)5p 3D	1	85+12(2D)5p 1P			123,305	123,343	‒38	0.575
	2	36 + 45(2D)5p 1D+ 123F(3P)1D	3d9(2D3/2)5p	5p 1D	123,557	123,557	0	1.067
	3	53 + 303F(3P)1F			121,525	121,664	‒139	1.204
3P(3P)5P	1	95	3d84s(2D)4p	4p‴ 3D	125,569	125,659	‒90	2.440
	2	89	3d84s(2D)4p	4p‴ 3D	125,248	125,335	‒87	1.784
	3	89 + 81D(3P)3D	3d84s(2D)4p	4p‴ 3D	125,231	125,261	‒30	1.628
1D(3P)3F	2	70 + 141D(3P)3D	3d84s(2D)4p	4p‴ 3F	128,570	128,480	90	0.822
	3	69 + 123P(3P)5D + 81D(3P)3D	3d84s(2D)4p	4p‴ 3F	128,559	128,585	‒36	1.178
	4	63 + 303P(3P)5D	3d84s(2D)4p	4p‴ 3F	128,778	128,731	47	1.327
1D(3P)3D	1	62 + 101D(3P)3P+ 103F(3P)3D				128,751		0.790
	2	59 + 181D(3P)3F + 83F(3P)3D	3d84s(4P)4p	4piv 5P	128,853	128,890	‒37	1.113
	3	65 + 91D(3P)3F + 83P(3P)5P	3d84s(4P)4p	4piv 5P	129,117	129,082	35	1.331
1D(3P)3P	0	63 + 333P(3P)3P				129,001
	1	54 + 211D(3P)3D+ 183P(3P)3P	3d84s(4P)4p	4piv 5P	129,760	129,721	39	1.290
	2	73 + 143P(3P)3P + 71D(3P)3D	3d84s(2P)4p	4piv 3P	130,386	130,375	11	1.490
3P(3P)5D	0	91	3d84s(4P)4p	4piv 5D	131,206	131,045	161
	1	91	3d84s(4P)4p	4piv 5D	130,945	131,021	‒76	1.486
	2	88	3d84s(4P)4p	4piv 5D	130,945	131,012	‒67	1.465
	3	80 + 121D(3P)3F	3d84s(2P)4p	4p‴ 1F	131,044	131,106	38	1.438
	4	65 + 271D(3P)3F	3d84s(4P)4p	4piv 5D3	131,313	131,377	‒64	1.417
3P(3P)3D	1	59 + 183P(3P)3P + 71D(3P)3P	3d84s(2P)4p	4pv 3P	134,360	134,277	83	0.765
	2	42 + 283P(3P)3P	3d84s(2P)4p	4pv 3D	134,676	134,714	‒38	1.288
	3	56 + 273F(1P)3D	3d84s(2P)4p	4pv 3D	133,985	134,013	‒28	1.323
3P(3P)3P	0	63 + 331D(3P)3P	3d84s(2P)4p	4pv 3P	135,484	135,440	44
	1	52 + 181D(3P)3P + 153P(3P)3D	3d84s(2P)4p	4pv 3D	135,136	135,087	49	1.184
	2	50 + 263P(3P)3D + 91D(3P)3D	3d84s(2P)4p	4pv 3P	133,826	133,710	116	1.378
3F(1P)3G	3	68 + 151G(3P)3F	3d84s(4F)4p	4p′ 3G	137,078	137,061	17	0.857
	4	67 + 223F(1P)3F	3d84s(4F)4p	4p′ 3G	135,835	135,925	‒90	1.115
	5	100	3d84s(4F)4p	4p′ 3G	134,111	133,887	224	1.200
3P(3P)5S	2	92	3d84s(2P)4p	4pv 1D	135,953	136,223	‒270	1.958
1G(3P)3H	4	99	3d84s(2G)4p	4pvi 3H	136,694	136,594	100	0.802
	5	100	3d84s(2G)4p	4pvi 1H	137,082	136,925	157	1.034
	6	100				137,359		1.167
3P(3P)1P	1	86 + 73P(3P)3P	3d84s(2P)4p	4pv 1P	137,213	137,118	95	1.039
1G(3P)3F	2	44 + 343F(1P)3F + 143P(3P)1D	3d84s(4F)4p	4p′ 3F	137,649	137,493	156	0.744
	3	26 + 273F(1P)3F + 22(2D)6p3D	3d84s(4F)4p	4p′ 3F	136,442	136,446	‒4	1.158
	4	39 + 49(2D)6p3F + 103F(1P)3F	3d84s(4F)4p	4p′ 3F	134,743	135,017	‒274	1.243
3P(3P)1D	2	59 + 71G(3P)3F + 6(2D)6p3D				137,701		0.985
3F(1P}3D	1	52 + 21(2D)6p3D + 153P(3P)3D	3d84s(4F)4p	4p′ 3D	137,914	137,851	63	0.546
	2	44 + 213P(3P)1D + 14(2D)6p3D	3d84s(4F)4p	4p′ 3D	136,799	136,751	48	1.119
	3	43 + 363P(3P)3D + 111D(3P)3D	3d84s(4F)4p	4p′ 3D	135,734	135,791	‒57	1.320
(2D) 6p 1F	3	34 + 43(2D)6p3F + 141G(3P)3F	3d84s(2G)4p	4pvi 3F	138,402	138,467	‒65	1.048
3P(3P)3S	1	99				138,723		1.992
(2D)6p 1P	1	47 + 39(2D)6p3P + 93F(1P)3D	3d9(2D5/2)6p	6p 3P	139,242	139,199	43	1.138
3F(1P)3F	2	31 + 22(2D)6p3F + 201G(3P)3F	3d9(2D5/2)6p	6p 3D	139,710	139,949	‒239	0.661
	3	39 + 283F(1P)3G + 221G(3P)3F	3d9(2D5/2)6p	6p 3D	139,741	139,861	‒120	0.998
	4	53 + 313F(1P)3G + 111’G(3P)3F	3d84s(2G)4p	4pv 3F	137,939	138,088	‒149	1.187
3P(3P)1S	0	97				140,345
(2D)6p 3P	0	97				140,977
	1	54 + 44(2D)6p1 P	3d9(2D3/2)6p	6p 1P	140,948	141,028	‒44	1.276
	2	76 + 19(2D)6p 1D	3d84s(2G)4p	4pvi 3F	139,028	138,861	167	1.398
(2D)6p 1D	2	36 + 14(2D)6p3P + 13(2D)6p3D	3d9(2D5/2)6p	6p 3P	139,217	139,053	164	1.183
(2D)6p 3D	1	78 + 143F(1P)3D			141,245	141,484	‒239	0.539
	2	53 + 231D(1P)1D + 6(2D)6p3P	3d9(2D5/2)6p	6p 1D	141,542	141,240	302	1.104
	3	56 + 24(2D)6p1F + 83F(1P)3F	3d9(2D5/2)6p	6p 3F	139,331	139,295	36	1.227
(2D)6p 3F	2	58 + 193F(1P)3F + 111G(3P)3F			141,734	141,579	155	0.723
	3	55 + 23(2D)6p1F + 133F(1P)3F	3d9(2D3/2)6p	6p 1F	141,204	141,260	‒56	1.077
	4	49 + 301G(3P)3F + 183F(1P)3F			139,396	139,736	‒340	1.249
1G(3P)3G	3	99				143,346		0.752
	4	99				143,435		1.050
	5	99				143,500		1.200
1D(1P)1D	2	52 + 433P(1P)3P	3d84s(2S)4p	4pvii 3P	150,250	150,054	196	1.220
1D(1P)1F	3	83 + 113P(1P)3D				150,521		1.036
3P(1P)3P	0	98				152,190
	1	75 + 191D(1P)1P				151,298		1.391
	2	55 + 431D(1P)1D				152,38.3		1.278
1D(1P)1P	1	71 + 223P(1P)3P				153,778		1.110
3P(1P)3D	1	93				155,336		0.518
	2	95				154,968		1.165
	3	86 + 91D(1P)1F				154,568		1.293
1G(1P)1H	5	100				158,704		1.000
3P(1P)3S	1	98				159,422		1.978
1G(1P)1F	3	92 + 61D(1P)1F				159,919		1.004
1G(1P)1G	4	100				165,078		1.000
1S(3P)3P	0	99				173,635
	1	99				173,934		1.500
	2	99				174,559		1.500
1S(1P)1P	1	99				195,915		1.000

Finally, the predicted level ¹S(³P)³P₂ is at around 175000 and thus the experimental level d⁸s(²S)p^vii3P₂ must be fitted with different assignment. The agreement is very good if this level is assigned to ¹D(¹P)¹D, which is mixed with ³P(¹P)³P₂.

The final parameters seem very reasonable, although most of the parameters pertaining to the configuration 3d⁹6p are not well defined. This is especially true for the parameter $G_3^{\prime \prime \prime }$, which had a value 1 ± 9, and thus was fixed at 0 in the final variation. The parameters β and T were eliminated as they have no significance here because no levels based on d^{8 1}S are known experimentally.

4. The Configurations 3d⁹4p + 3d⁹5p + 3d⁸4s4p + 3d⁹6p – Cuii

Initially the parameters for the configurations 3d⁹5p + 3d⁸4s4p + 3d⁹6p were taken from table 2. The starting values for the parameters of 3d⁹4p were obtained from table 1. Initial values for the parameters of the interaction between the configurations 3d⁹4p and 3d⁸4s4p were estimated by considering the values obtained for the interaction 3dⁿ4p − 3dⁿ⁻¹4s4p in Se ii, Ti ii, and V ii, as well as the results of table 2 for the interactions 3d⁹5p − 3d⁸4s4p and 3d⁹6p − 3d⁸4s4p. The following starting values were used for the parameters of the interaction 3d⁹4p − 3d⁸4s4p:

$$\begin{gathered} H\left(3d^{9}4p-3d^{8}4s4p\right)=50 \\ J\left(3d^{9}4p-3d^{8}4s4p\right)=500 \\ K\left(3d^{9}4p-3d^{8}4s4p\right)=1500 \end{gathered}$$ (11)

In AEL, 102 levels are assigned to the four configurations 3d⁹4p, 3d⁹5p, 3d⁸4s4p, and 3d⁹6p. Omitting the same levels as in the previous section and performing the same changes in designation as well as the changes

$${(}^2\text{D})4p^3{\text{D}}_2\leftrightarrow {(}^2\text{D})4p^1\text{D}$$

$${(}^2\text{D})4p^3{\text{D}}_3\leftrightarrow {(}^2\text{D})4p^1\text{F}$$

we fitted 97 experimental levels with an rms error of 117. The final parameters are given in table 3.

Table 3.

Parameters for Cu ii − 3d⁹4p + 3d⁹5p + 3d⁸4s4p + 3d⁹6p

Parameter	Initial value	Final value
A	69,802	70,333 ± 173
A′	134,252	134,295 ± 110
A″	121,591	121,679 ± 176
A‴	139,725	139,739 ± 129
B′	1,210	1,210 ± 10
C′	4,777	4,760 ± 107
$G_{ds}^{\prime }$	1,430	1,503 ± 63
F2	344	347 ± 11
$F_2^{\prime }$	486	484 ± 5
$F_2^{\prime \prime }$	88	91 ± 12
$F_2^{\prime \prime \prime }$	10	11 ± 12
G1	305	291 ± 18
$G_1^{\prime }$	428	393 ± 20
$G_1^{\prime \prime }$	73	73 ± 12
$G_1^{\prime \prime \prime }$	10	23 ± 16
G3	38	30 ± 8
$G_3^{\prime }$	74	69 ± 5
$G_3^{\prime \prime }$	15	12 ± 7
$G_3^{\prime \prime \prime }$	0	0 (Fix)
$G_{ps}^{\prime }$	10,836	10,799 ± 44
a′	72	77 ± 14
H(3d84s4p ‒ 3d94p)	50	183 ± 74
J(3d84s4p ‒ 3d94p)	500	795 ± 301
K(3d84s4p ‒ 3d94p)	1,500	3,007 ± 542
J(3d84s4p ‒ 3d95p)	291	427 ± 253
K(3d84s4p ‒ 3d95p)	761	1,013 ± 307
J(3d84s4p ‒ 3d96p)	150	398 ± 143
K(3d84s4p ‒ 3d96p)	674	776 ±163
ξd	802	816 ± 48
${\zeta }_d^{\prime }$	933	938 ± 22
${\zeta }_d^{\prime \prime }$	811	817 ± 34
${\zeta }_d^{\prime \prime }{}^{\prime }$	843	829 ± 41
ξp	502	525 ± 87
${\zeta }_p^{\prime }$	686	630 ± 53
${\zeta }_p^{\prime \prime }{}^{\prime }$	184	152 ± 88
${\zeta }_p^{\prime \prime }{}^{\prime }$	48	34 ± 41
rms error		117

The final parameters seem very reasonable. Although the standard errors especially for the parameters of the interactions between configurations are very high, a fair estimate is obtained for them. When left free, the parameter $G_3^{\prime \prime \prime }$ had a value of 0.5 ± 8, and thus in the final variation we considered it fixed at zero.

Whereas the rms error for 3d⁹5p + 3d⁸4s4p + 3d⁹6p is 136 and the rms error for 3d⁹4p is 119, here the rms error is reduced to 117. Thus, the interaction between the configurations 3d⁹4p and 3d⁸4s4p improves the agreement by only a very small amount especially when compared with the large improvements in Sc ii, Ti ii and V ii, due to the insertion of the interactions between the configurations 3dⁿ4p − 3d^{n − 1} 4s4p, n ⩽ 3, [5].

5. The Configuration 3d⁹4f Cu ii

The electrostatic matrices of d⁹f are given on p. 299 TAS [9]. The spin-orbit matrices can be obtained from those of df by changing the sign of the matrix of ζd. These matrices are given on p. 206, TAS.

Since the coupling here is definitely not Russell-Saunders, we try to find initial parameters by writing down the separate matrices of d⁹f for each of the seven J values. By making use of the fact that the trace of a matrix equals the sum of its eigenvalues, we obtain seven equations for the eight parameters A, F₂(df), F₄(df), G₁(df), G₃(df), G₅(df), ζ_d, and ζ_f. We further make the initial approximation that G₅(df) equals zero.

By solving the resulting seven equations we obtained for F₄ and G₃ very small negative values. Thus, approximately,

$$\begin{gathered} A=136,580 \\ {F}_2=6 \\ {F}_4=0 \\ {G}_1=2 \\ {G}_3=0 \\ {G}_5=0 \\ {\zeta }_f=10 \\ {\zeta }_d=860 \end{gathered}$$ (12)

From an energy diagram of 3d⁹4f it is evident that the coupling is close to j − l. As explained by Racah [12], it is possible, by means of the diagonalization routine, to obtain the j − l assignment of each level by taking ζ_d ⪢ F₂ > 0, and all other parameters equal to zero.

The j − l notation used in table 8 of the observed and calculated levels of 3d⁹4f is that of Racah as illustrated on p. 116 AEL, Vol. II, [6]. The final parameters obtained are given in table 4.

Table 8.

Observed and calculated levels of Cu ii 3d⁹4f

Name j ‒ l		J	AEL	Obs. level (em‒1)	Calc. level (em‒1)	O–C	Calc. g
Config.	Desig.
3d9(2D5/2)4f	4f [0½]	0	3P	135,902	135,838	64
		1	1P	135,958	135,962	–4	0.756
3d9(2D5/2)4f	4f [1½]	1	3P	135,864	135,864	0	1.362
		2	3P	135,9ll	135,929	–18	1.279
3d9(2D5/2)4f	4f [2½]	2	3D	136,014	136,037	–23	0.914
		3	3D	135,990	136,042	–52	1.230
3d9(2D5/2)4f	4f [3½]	3	3F	136,036	136,128	–92	0.964
		4	3G	136,270	136,135	135	1.168
3d9(2D5/2)4f	4f [4½]	4	3F	136,133	136,125	8	1.018
		5	3G	136,161	136,133	28	1.174
3d9(2D5/2)4f	4f [5½]	5	3H	135,934	135,951	–17	1.016
		6	3H	135,931	135,959	–28	1.167
3d9(2D3/2)4f	4f [1½]	1	3D	138,029	138,024	5	0.882
		2	1D	138,003	137,997	6	1.324
3d9(2D3/2)4f	4f [2½]	2	3F	138,177	138,157	20	0.816
		3	1F	138,131	138,165	–34	1.149
3d9(2D3/2)4f	4f [3½]	3	3G	138,262	138,234	28	0.824
		4	1G	138,220	138,242	–22	1.082
3d9(2D3/2)4f	4f [4½]	4	3H	138,074	138,067	7	0.832
		5	1H	138,064	138,076	–12	1.044

Table 4.

Parameters for Cu ii − 3d⁹4f

Parameter	Initial value	Final value
A	136,850	136,870 ± 12
F2 (fd)	6	8.3 ± 1.0
F4 (fd)	0	0.6 ± 0.4
G1 (fd)	2	1.7 ± 1.3
G3 (fd)	0	0 (Fix)
G5 (fd)	0	0 (Fix)
ξf	10	5.0 ± 8.3
ξd	860	837 ± 9
rms error		51

As the parameters G₃ and G₅, when left to vary freely, assume small negative values with standard errors larger than their actual values, the meaningful variation to consider in the least-squares is the one with G₃ and G₅ fixed at their initial values of zero.

6. The Configuration 3d⁹5 f − Cu ii

An energy diagram of 3d⁹5f indicates that the coupling here is almost pure j − l. By performing similar calculations as for 3d⁹4f for the initial parameters with G₅ equal to zero, it is found that F₄, G₃, and ζ_f have very small negative values. Then letting F₄, G₃, and ζ_f equal zero, and using the traces of J equal to 0, 1, 5, and 6, we obtain the following eauations:

$$\begin{gathered} A-24F_2-{\zeta }_d=145,890 \\ 3A-54F_2+70G_1-{\zeta }_d/2=439,873 \\ 3A-5F_2-{\zeta }_d/2=440,007 \\ A-10F_2-{\zeta }_d=145,952 \end{gathered}$$ (13)

Solving (13) yields:

$$\begin{gathered} A=146,812 \\ {F}_2=4.4 \\ {G}_1=1.2 \\ {\zeta }_d=816 \end{gathered}$$ (14)

As for 3d⁹4f the j − l assignments were obtained for each level, as indicated in table 9. The final parameters are given in table 5.

Table 9.

Observed and calculated levels of Cu ii 3d⁹5f

Name		J	AEL	Obs. level (em‒1)	Calc. level (em‒1)	O–C	Calc. g
Config.	Desig.
3d9(2D5/2)5f	5f [0½]	0	3P	145,889.6	145,891.3	–1.7
		1	3P	145,901.1	145,904.0	–2.9	1.360
3d9(2D5/2)5f	5f [1½]	1	1P	145,955.7	145,956.5	–0.8	0.749
		2	3D	145,985.4	145,983.8	1.6	0.913
3d9(2D5/2)5f	5f [2½]	2	3P	145,927.5	145,931.3	–3.8	1.267
		3	3D	145,978.4	145,983.8	–5.4	1.224
3d9(2D5/2)5f	5f [3½]	3	3F	146,021.5	146,026.3	–4.8	0.965
		4	3G	146,029.5	146,026.3	3.2	1.195
3d9(2D5/2)5f	5f [4½]	4	3F	146,024.0	146,025.8	–1.8	0.993
		5	3G	146,032.5	146,025.8	6.7	l.l76
3d9(2D5/2)5f	5f [5½]	5	3H	145,945.8	145,943.8	2.0	1.015
		6	3H	145,951.7	145,943.8	7.9	l.l67
3d9(2D3/2)5f	5f [1½]	1	3D	148,016.3	148,014.6	1.7	0.892
		2	1D	147,987.7	147,989.7	–2.0	1.333
3d9(2D3/2)5f	5f [2½]	2	3F	148,066.3	148,068.4	–2.1	0.820
		3	1F	148,061.7	148,068.4	–6.7	l.l57
3d9(2D3/2)5f	5f [3½]	3	3G	148,103.2	148,104.7	–1.5	0.821
		4	1G	148,105.6	148,104.7	0.9	1.083
3d9(2D3/2)5f	5f [4½]	4	3H	148,033.7	148,026.4	7.3	0.829
		5	1H	148,028.8	148,026.4	2.4	1.042

Table 5.

Parameters for Cu ii − 3d⁹5f

Parameter	Initial value	Final value
A	146,812	146,810 ± 1
F2 (fd)	4.4	3.7 ± 0.1
F4 (fd)	0	0 (Fix)
G1 (fd)	1.2	0.9 ± 0.1
G3 (fd)	0	0 (Fix)
G5 (fd)	0	0 (Fix)
ξf	0	0 (Fix)
ξd	816	828 ± 1
rms error		4.5

The parameters F₄, G₃, G₅, and ζ_f are not significant here. When left free, the standard errors in these parameters are much larger than their actual values. The latter never exceed 0.2.

7. Tables of the Observed and Calculated Levels and g-Factors

In the column “NAME” the calculated designation of the term is given. The terms of d⁸sp are denoted by d⁸S₁L₁ (sp^{1, 3}P)SL. For the configuration 3d⁹4f and 3d⁹5f the j − l. notation of Racah is used (see p. 116 AEL, Vol. II).

The entries in the columns “J”, “OBS. LEVEL cm⁻¹” and “CALC. LEVEL cm⁻¹” are self-evident. In the column “PERCENTAGE” for each calculated level either the three highest contributions or all those contributions exceeding 5 percent are given.

Whenever the experimental and calculated term designations differ, the experimental designation is entered in the column “AEL” using the notation of C. E. Moore, [6].

The column “O-C” gives the difference between the observed and calculated values of the levels.

The columns “OBS. g” and “CALC. g” give the observed and calculated values of the g-factors, respectively.

The entries are in ascending order of magnitude of the calculated terms.

Table 6.

Observed and calculated levels of Cu ii 3d⁹4p, individual treatment

Name	J	Percentage	AEL		Obs. Level (em‒1)	Calc. Level (em‒1)	O–C	Obs. g	Calc. g
			Config.	Oesig.
(2D)3P	0	100			68,850	68,852	‒2
	1	97			67,917	67,976	‒59	1.49	1.480
	2	98			66,419	66,572	‒153	1.49	1.493
(2D)3F	2	94 + 4(2D)3D			69,868	69,718	150	0.67	0.694
	3	69 + 29(2D)1F			68,448	68,412	36	1.06	1.065
	4	100			68,731	68,564	167	1.23	1.250
(2D)1F	3	62 + 19(2D)3D + 18(2D)3F	3d9(2D5/2)4p	4p 3D	70,842	70,858	‒16		1.079
(2D)1D	2	61 + 33(2D)3D + 5(2D)3F	3d9(2D5/2)4p	4p 3D	71,494	71,555	‒61	1.08	1.044
(2D)3D	1	98			73,102	73,137	‒35	0.47	0.517
	2	61 + 37(2D)1D	3d9(2D3/2)4p	4p 1D	73,353	73,381	‒28	0.99	1.103
	3	78 + 12(2D)3F + 9(2D)1F	3d9(2D3/2)4p	4p 1F	71,920	71,919	1		1.272
(2D)1P	1	98			73,596	73,595	1	1.04	1.002