Complete set of orthogonal scalar operators for the configuration f<sup>3</sup>

B. R. Judd; M. A. Suskin

doi:10.1364/JOSAB.1.000261

Journal of the Optical Society of America B
Vol. 1,
Issue 2,
pp. 261-265
(1984)
•https://doi.org/10.1364/JOSAB.1.000261

Complete set of orthogonal scalar operators for the configuration f³

B. R. Judd and M. A. Suskin

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B. R. Judd and M. A. Suskin, "Complete set of orthogonal scalar operators for the configuration f³," J. Opt. Soc. Am. B 1, 261-265 (1984)

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Abstract

When Coulomb interactions between f³ and other electronic configurations are taken to third and higher orders of perturbation theory, it becomes necessary to augment the standard set of thirteen electrostatic parameters with eight more. The operators associated with these eight parameters are orthogonalized to the standard set of operators, and a complete set of the matrix elements of all 21 orthogonalized operators is given for f³. The group-theoretical properties of these operators are examined. Each operator corresponds to a unique pair of irreducible representations of R₇ (the rotation group in the seven-dimensional orbital space of an f electron) and of G₂ (Cartan’s first exceptional group), but the associated quasispin ranks and irreducible representations of Sp₁₄ (the symplectic group in the 14-dimensional spin–orbital space of an f electron) are sometimes mixed. The combinations of quasispin ranks are odd for the three-electron operators t and even for the two-electron operators e.

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ψ	ψ′	e₀	$e_{1}^{'}$	F₂e₂	F₃e₃	$F_{α} e_{α}^{'}$	$F_{β} e_{β}^{'}$	$e_{γ}^{'}$	$F_{2}^{'} t_{2}^{'}$	$F_{3}^{'} t_{3}$	F₄t₄	F₆t₆	F₇t₇	F₈t₈	F₁₁t₁₁	F₁₂t₁₂	F₁₄t₁₄	F₁₅t₁₅	F₁₆t₁₆	F₁₇t₁₇	F₁₈t₁₈	F₁₉t₁₉
4S	4S	3	−27	0	0	0	12	−3	0	288	0	0	0	0	18	0	−72	0	0	0	0	0
4D	4D	3	−27	0	231	−77	−2	−3	847	8	−8008	0	0	0	−3	−154	−2	2002	0	0	0	0
4F	4F	3	−27	0	0	0	6	−3	0	−72	0	0	0	0	9	0	18	0	0	0	0	0
4G	4G	3	−27	0	84	−28	−2	−3	308	8	7280	0	0	0	−3	−56	−2	−1820	0	0	0	0
4I	4I	3	−27	0	−147	49	−2	−3	−539	8	−1960	0	0	0	−3	98	−2	490	0	0	0	0
2P	2P	3	12	0	−77	−231	6	3	−154	−48	0	−30030	0	0	−6	−77	−48	0	−462	0	165	0
2D1	2D1	3	12	−858	−33	−209	4	3	−176	32	−1144	12870	10296	0	−4	−33	32	−1144	−66	−1144	165	−1716
2D1	2D2	0	0	−468(33)^½	24(33)^½	12(33)^½	0	0	−12(33)^½	0	468(33)^½	−624(33)^½	156(33)^½	0	0	24(33)^½	0	468(33)^½	48(33)^½	−78(33)^½	12(33)^½	429(33)^½
2D2	2D2	3	12	1131	12	−309	−3	3	−321	−3	3237	−1677	−1833	4641	3	12	−3	3237	171	−1131	−54	−2106
2F1	2F1	3	90	0	0	0	0	−15	0	0	0	0	0	0	0	0	0	0	0	0	0	0
2F1	2F2	0	0	0	−42(22)^½	−21(22)^½	0	0	231(22)^½	0	0	0	0	0	0	63(22)^½	0	0	0	0	0	0
2F2	2F2	3	12	1365	231	−147	−3	3	−378	−3	1365	1365	−1365	−3315	3	231	−3	1365	−189	−1365	−45	0
2G1	2G1	3	12	780	−12	−76	4	3	−64	32	1040	4680	−9360	0	−4	−12	32	1040	−24	1040	60	1560
2G1	2G2	0	0	24(4290)^½	2(4290)^½	(4290)^½	0	0	−(4290)^½	0	−24(4290)^½	−52(4290)^½	−8(4290)^½	0	0	2(4290)^½	0	−24(4290)^½	4(4290)^½	4(4290)^½	(4290)^½	−22(4290)^½
2G2	2G2	3	12	−1683	187	−99	−3	3	−286	−3	−2475	1221	1947	1309	3	187	−3	−2475	−165	1683	−33	792
2H1	2H1	3	12	0	21	63	6	3	42	−48	0	8190	0	0	−6	21	−48	0	126	0	−45	0
2H1	2H2	0	0	84(455)^½	0	0	0	0	0	0	84(455)^½	0	252(455)^½	0	0	0	0	84(455)^½	0	42(455)^½	0	63(455)^½
2H2	2H2	3	12	147	−168	−189	−3	3	−21	−3	−1995	−2709	567	−1071	3	−168	−3	−1995	315	−147	−18	2142
2I	2I	3	12	−210	21	133	4	3	112	32	−280	−8190	2520	0	−4	21	32	−280	42	−280	−105	−420
2K	2K	3	12	−945	−77	84	−3	3	161	−3	1827	−252	21	−1071	3	−77	−3	1827	42	945	21	−2772
2L	2L	3	12	735	−21	252	−3	3	273	−3	−525	1260	−315	945	3	−21	−3	−525	−126	−735	45	1260

Operator	WU	Multiplicative Factor
e₀	(000)(00)	(F₀ = 1)
$e_{1}^{'}$	(000)(00)	(F₁ = 1)
e₂	(400)(40)	F₂ = 7
e₃	(220)(22)	F₃ = 1
$e_{α}^{'}$	(220)(22)	F_α = 7
$e_{β}^{'}$	(111)(00)	F_β = 2
$e_{γ}^{'}$	(000)(00)	(F_γ = 1)
$t_{2}^{'}$	(220)(22)	$F_{2}^{'} = 770 {(2)}^{1 / 2}$
t₃	(222)(00)	$F_{3}^{'} = {(6720)}^{1 / 2}$
t₄	(222)(40)	F₄ = 56(15015)^1/2
t₆	(420)(22)	F₆ = 924(455)^1/2
t₇	(420)(40)	F₇ = 168(5005)^1/2
t₈	(420)(42)	F₈ = (16336320)^1/2
$t_{11}^{'}$	(111)(00)	F₁₁ = (660/7)^1/2
t₁₂	(220)(22)	F₁₂ = 49(110/3)^1/2
t₁₄	(222)(00)	F₁₄ = (1680)^1/2
t₁₅	(222)(40)	F₁₅ = 14(60060)^1/2
t₁₆	(311)(22)	F₁₆ = 252(77/15)^1/2
t₁₇	(311)(40)	F₁₇ = 84(1001)^1/2
t₁₈	(321)(22)	F₁₈ = (41580)^1/2
t₁₉	(321)(40)	F₁₉ = 42(15015)^1/2

ψ	ψ′	$t_{12}^{'} {}^{7}{〈 1111000 〉}$	$t_{12}^{″} {}^{3}{〈 1111000 〉}$	$t_{12}^{‴} {}^{3}{〈 2211000 〉}$
4S	4S	0 + 0	0 + 0	0
4D	4D	154 − 154/3	154 − 77/4	−847
4F	4F	0 + 0	0 + 0	0
4G	4G	56 − 56/3	56−7	−308
4I	4I	−98 + 98/3	−98 + 49/4	539
2P	2P	−154 + 154/3	−154 + 77/4	−77
2D1	2D1	−110 + 110/3	−110 + 55/4	33
2D1	2D2	(24 − 8)(33)^1/2	(24 − 3)(33)^1/2	60(33)^1/2
2D2	2D2	−114 + 38	−114 + 57/4	219
2F1	2F1	0 + 0	0 + 0	0
2F1	2F2	(−42 − 14)(22)^1/2	(−42 + 21)(22)^1/2	0
2F2	2F2	126 − 42	126 − 63/4	735
2GI	2G1	−40 + 40/3	−40 + 5	12
2G1	2G2	(2 − 2/3)(4290)^1/2	(2 − 1/4)(4290)^1/2	5(4290)^1/2
2G2	2G2	110 − 110/3	110 − 55/4	583
2H1	2H1	42 − 14	42 − 21/4	21
2H1	2H2	0 + 0	0 + 0	0
2H2	2H2	−210 + 70	−210 + 105/4	−357
2I	2I	70 − 70/3	70 − 35/4	−21
2K	2K	−28 + 28/3	−28 + 7/2	266
2L	2L	84 − 28	84 − 21/2	−210

	$t_{12}^{'}$	$t_{12}^{″}$
¹S	0 + 0	0 + 0
³P	154 + 0	154 + 0
¹D	−154 + 0	−154 + 0
³F	0 + 0	0 + 0
¹G	−56 + 0	−56 + 0
³H	−42 + 0	−42 + 0
¹I	98 + 0	98 + 0

ψ	ψ′	e₀	$e_{1}^{'}$	F₂e₂	F₃e₃	$F_{α} e_{α}^{'}$	$F_{β} e_{β}^{'}$	$e_{γ}^{'}$	$F_{2}^{'} t_{2}^{'}$	$F_{3}^{'} t_{3}$	F₄t₄	F₆t₆	F₇t₇	F₈t₈	F₁₁t₁₁	F₁₂t₁₂	F₁₄t₁₄	F₁₅t₁₅	F₁₆t₁₆	F₁₇t₁₇	F₁₈t₁₈	F₁₉t₁₉
4S	4S	3	−27	0	0	0	12	−3	0	288	0	0	0	0	18	0	−72	0	0	0	0	0
4D	4D	3	−27	0	231	−77	−2	−3	847	8	−8008	0	0	0	−3	−154	−2	2002	0	0	0	0
4F	4F	3	−27	0	0	0	6	−3	0	−72	0	0	0	0	9	0	18	0	0	0	0	0
4G	4G	3	−27	0	84	−28	−2	−3	308	8	7280	0	0	0	−3	−56	−2	−1820	0	0	0	0
4I	4I	3	−27	0	−147	49	−2	−3	−539	8	−1960	0	0	0	−3	98	−2	490	0	0	0	0
2P	2P	3	12	0	−77	−231	6	3	−154	−48	0	−30030	0	0	−6	−77	−48	0	−462	0	165	0
2D1	2D1	3	12	−858	−33	−209	4	3	−176	32	−1144	12870	10296	0	−4	−33	32	−1144	−66	−1144	165	−1716
2D1	2D2	0	0	−468(33)^½	24(33)^½	12(33)^½	0	0	−12(33)^½	0	468(33)^½	−624(33)^½	156(33)^½	0	0	24(33)^½	0	468(33)^½	48(33)^½	−78(33)^½	12(33)^½	429(33)^½
2D2	2D2	3	12	1131	12	−309	−3	3	−321	−3	3237	−1677	−1833	4641	3	12	−3	3237	171	−1131	−54	−2106
2F1	2F1	3	90	0	0	0	0	−15	0	0	0	0	0	0	0	0	0	0	0	0	0	0
2F1	2F2	0	0	0	−42(22)^½	−21(22)^½	0	0	231(22)^½	0	0	0	0	0	0	63(22)^½	0	0	0	0	0	0
2F2	2F2	3	12	1365	231	−147	−3	3	−378	−3	1365	1365	−1365	−3315	3	231	−3	1365	−189	−1365	−45	0
2G1	2G1	3	12	780	−12	−76	4	3	−64	32	1040	4680	−9360	0	−4	−12	32	1040	−24	1040	60	1560
2G1	2G2	0	0	24(4290)^½	2(4290)^½	(4290)^½	0	0	−(4290)^½	0	−24(4290)^½	−52(4290)^½	−8(4290)^½	0	0	2(4290)^½	0	−24(4290)^½	4(4290)^½	4(4290)^½	(4290)^½	−22(4290)^½
2G2	2G2	3	12	−1683	187	−99	−3	3	−286	−3	−2475	1221	1947	1309	3	187	−3	−2475	−165	1683	−33	792
2H1	2H1	3	12	0	21	63	6	3	42	−48	0	8190	0	0	−6	21	−48	0	126	0	−45	0
2H1	2H2	0	0	84(455)^½	0	0	0	0	0	0	84(455)^½	0	252(455)^½	0	0	0	0	84(455)^½	0	42(455)^½	0	63(455)^½
2H2	2H2	3	12	147	−168	−189	−3	3	−21	−3	−1995	−2709	567	−1071	3	−168	−3	−1995	315	−147	−18	2142
2I	2I	3	12	−210	21	133	4	3	112	32	−280	−8190	2520	0	−4	21	32	−280	42	−280	−105	−420
2K	2K	3	12	−945	−77	84	−3	3	161	−3	1827	−252	21	−1071	3	−77	−3	1827	42	945	21	−2772
2L	2L	3	12	735	−21	252	−3	3	273	−3	−525	1260	−315	945	3	−21	−3	−525	−126	−735	45	1260

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Equations (20)

Journal of the Optical Society of America B