Configuration Mixing in the N=4 Electron Isoelectronic Sequence*

Halis Odabasi

doi:10.1364/JOSA.59.000583

Journal of the Optical Society of America
Vol. 59,
Issue 5,
pp. 583-588
(1969)
•https://doi.org/10.1364/JOSA.59.000583

Configuration Mixing in the N=4 Electron Isoelectronic Sequence

Halis Odabasi

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Halis Odabasi, "Configuration Mixing in the N=4 Electron Isoelectronic Sequence*," J. Opt. Soc. Am. 59, 583-588 (1969)

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Abstract

The theoretical central-field-approximation predictions for the ³P and ¹P terms of the 1s² 2s³p configuration in the N=4 electron isoelectronic sequence are in violent disagreement with observation. Correlation effects in the form of configuration interaction have been included by considering the mixing among nine odd configurations. Even though the terms are displaced considerably, they remain quite pure, and after configuration mixing the agreement between theory and observation is good. In order to show the use of such theoretical calculation, weighted oscillator strengths for the 2s^{2 1}S₁–2s3p ¹P₁, ³P₁ transitions are also computed.

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2snp:2snp	$\tilde{E}$ (np)
2pns:2pns	$\tilde{E}$ (ns)
2pnd:2pnd	$\tilde{E}$ (nd)
2snp:2smp	Ē(np,mp)
2snp:npms	Ē(2s,ms)
2snp:n′pms	E″(ms,np)
2snp:2pmd	E″(np,md)
2pns:2pms	Ē(ns,ms)
2pns:2pmd	E″(ns,md)
2pnd:2pmd	Ē(nd,md)
$\tilde{E}$ (nl)=E+E′(nl)+E″(nl)
E=2I(1s)+F⁰(1s,1s)
$E^{'} (n s) = I (2 p) + I (n s) + 2 F^{0} (1 s, 2 p) - \frac{1}{3} G^{1} (1 s, 2 p) + 2 F^{0} (1 s, n s) - G^{0} (1 s, n s)$
$E^{'} (n p) = I (2 s) + I (n p) + 2 F^{0} (1 s, 2 s) - G^{0} (1 s, 2 s) + 2 F^{0} (1 s, n p) - \frac{1}{3} G^{1} (1 s, n p)$
$E^{'} (n d) = I (2 p) + I (n d) + 2 F^{0} (1 s, 2 p) - \frac{1}{3} G^{1} (1 s, 2 p) + 2 F^{0} (1 s, n d) - \frac{1}{5} G^{2} (1 s, n d)$
$E^{″} (n s) = F^{0} (2 p, n s) \pm \frac{1}{3} G^{1} (2 p, n s)$ ^a
$E^{″} (n p) = F^{0} (2 s, n p) \pm \frac{1}{3} G^{1} (2 s, n p)$
$E^{″} (n d) = F^{0} (2 p, n d) + \frac{1}{5} F^{2} (2 p, n d) \pm (1 / 15) G^{1} (2 p, n d) \pm (9 / 35) G^{3} (2 p, n d)$
Ē(nl,ml)=E′(nl,ml)+E″(nl,ml)
E′(ns,ms)=I(ns,ms)+2R⁰(ns,1s; ms,1s)−R⁰(ns,1s; 1s,ms)
$E^{'} (n p, m p) = I (n p, m p) + 2 R^{0} (n p, 1 s; m p, 1 s) - \frac{1}{3} R^{1} (n p, 1 s; 1 s, m p)$
$E^{'} (n d, m d) = I (n d, m d) + 2 R^{0} (n d, 1 s; m d, 1 s) - \frac{1}{5} R^{2} (n d, 1 s; 1 s, m d)$
$E^{″} (n s, m s) = R^{0} (2 p, n s; 2 p, m s) \pm \frac{1}{3} R^{1} (2 p, n s; m s, 2 p)$
$E^{″} (n p, m p) = R^{0} (2 s, n p; 2 s, m p) \pm \frac{1}{3} R^{1} (2 s, n p; m p, 2 s)$
$\begin{array}{l} E^{″} (n d, m d) = R^{0} (2 p, n d; 2 p, m d) + \frac{1}{5} R^{2} (2 p, n d; 2 p, m d) \\ \pm (1 / 15) R^{1} (2 p, n d; m d, 2 p) \pm (9 / 35) R^{3} (2 p, n d; m d, 2 p) \end{array}$
$\begin{array}{l} \bar{Ē} (2 s, m s) = \pm [I (2 s, m s) + 2 R^{0} (2 s, 1 s; m s, 1 s) - R^{0} (2 s, 1 s; 1 s, m s) \\ + R^{0} (2 p, 2 s; 2 p, m s) \pm \frac{1}{3} R^{1} (2 p, 2 s; m s, 2 p)] \end{array}$
$E^{″} (n s, m p) = \pm R^{0} (2 p, n s; m p, 2 s) + \frac{1}{3} R^{1} (2 p, n s; 2 s, m p)$
$E^{″} (n s, m d) = - (\sqrt{2} / 5) R^{2} (2 p, n s; 2 p, m d) \mp (\sqrt{2} / 3) R^{1} (2 p, n s; m d, 2 p)$
$E^{″} (n p, m d) = - (\sqrt{2} / 5) R^{2} (2 s, n p; m d, 2 p) \mp (\sqrt{2} / 3) R^{1} (2 s, n p; 2 p, m d)$

	a₂₁	a₂₂	a₂₃	a₂₄	a₂₅	a₂₆	a₂₇	a₂₈	a₂₉
Z=4,Be i	−0.06191	0.98163	−0.02755	−0.14555	0.01804	−0.06731	0.01041	−0.07351	−0.01589
	0.33660	0.86791	−0.16049	−0.30050	−0.02834	−0.12301	−0.01765	−0.06889	−0.03303
5,B ii	−0.02848	0.98863	−0.05926	−0.08222	0.00884	−0.03943	0.00589	−0.09879	−0.01027
	0.07359	0.95692	−0.25712	−0.07439	−0.04503	−0.03640	−0.02331	0.05744	−0.00759
6,C iii	−0.01586	0.98938	−0.07243	−0.05709	0.00472	−0.02780	0.00369	−0.10737	−0.00525
	0.03008	0.95429	−0.27878	−0.04790	−0.04061	−0.02428	−0.02004	0.07581	−0.00457
7,N iv	−0.00995	0.98941	−0.07882	−0.04370	0.00281	−0.02146	0.00258	−0.11118	−0.00215
	0.01593	0.95133	−0.29036	−0.03675	−0.03585	−0.01887	−0.01720	0.08435	−0.00410
8,O v	−0.00675	0.98935	−0.08230	−0.03536	0.00181	−0.01745	0.00195	−0.11312	−0.00028
	0.00960	0.94913	−0.29781	−0.03018	−0.03180	−0.01559	−0.01497	0.08920	−0.00403
9,F vi	−0.00484	0.98930	−0.08436	−0.02968	0.00124	−0.01470	0.00155	−0.11423	0.00087
	0.00625	0.94749	−0.30311	−0.02574	−0.02846	−0.01333	−0.01321	0.09229	−0.00401
10,Ne vii	−0.00361	0.98926	−0.08565	−0.02555	0.00089	−0.01270	0.00129	−0.11489	0.00160
	0.00428	0.94622	−0.30706	−0.02249	−0.02570	−0.01167	−0.01181	0.09440	−0.00397
11,Na viii	−0.00277	0.98923	−0.08649	−0.02243	0.00066	−0.01117	0.00110	−0.11531	0.00205
	0.00304	0.94520	−0.31015	−0.02000	−0.02341	−0.01038	−0.01067	0.09593	−0.00390
12,Mg ix	−0.00218	0.98922	−0.08707	−0.01998	0.00050	−0.00997	0.00095	−0.11558	0.00233
	0.00221	0.94438	−0.31264	−0.01802	−0.02148	−0.00936	−0.00973	0.09707	−0.00380
13,Al x	−0.00175	0.98921	−0.08748	−0.01801	0.00039	−0.00900	0.00084	−0.11577	0.00251
	0.00163	0.94368	−0.31472	−0.01639	−0.01983	−0.00852	−0.00893	0.09797	−0.00370
14,Si xi	−0.00143	0.98920	−0.08776	−0.01640	0.00031	−0.00820	0.00076	−0.11589	0.00261
	0.00121	0.94309	−0.31644	−0.01505	−0.01841	−0.00782	−0.00826	0.09867	−0.00359
15,P xii	−0.00118	0.98920	−0.08798	−0.01504	0.00025	−0.00753	0.00068	−0.11598	0.00266
	0.00090	0.94258	−0.31794	−0.01391	−0.01718	−0.00723	−0.00768	0.09925	−0.00348
Ψ(2A)=a₂₁Φ(2s2pA)+a₂₂Φ(2s3pA)+a₂₃Φ(2p3sA)+a₂₄Φ(2s4pA)+a₂₅Φ(2p4sA)+a₂₆Φ(2s5pA)+a₂₇Φ(2p5sA)+a₂₈Φ(2p3dA)+a₂₉Φ(2p4dA)

4,Be i	5,B ii	6,C iii	7,N iv	8,O v	9,F vi	10,Ne vii	11,Na viii	12,Mg ix	13,Al x	14,Si xi	15,P xii
0.299	0.045	0.180	0.279	0.350	0.403	0.446	0.479	0.505	0.526	0.546	0.561
⋯	⋯	0.2	0.3	0.3	⋯	⋯	⋯	⋯	⋯	⋯	⋯
0.299	0.045	0.180	0.279	0.350	0.402	0.443	0.471	0.489	0.496	0.494	0.479
4.68×10⁻⁹ (0.53575)	6.95×10⁻⁶ (1.31213)	3.10×10⁻⁵ (2.36665)	1.37×10⁻⁴ (3.69958)	4.73×10⁻⁴ (5.31121)	1.36×10⁻³ (7.20197)	3.40×10⁻³ (9.3708)	7.67×10⁻³ (11.8172)	1.57×10⁻² (14.5417)	2.97×10⁻² (17.5440)	5.17×10⁻² (20.8234)	8.19×10⁻² (24.3824)

2snp:2snp	$\tilde{E}$ (np)
2pns:2pns	$\tilde{E}$ (ns)
2pnd:2pnd	$\tilde{E}$ (nd)
2snp:2smp	Ē(np,mp)
2snp:npms	Ē(2s,ms)
2snp:n′pms	E″(ms,np)
2snp:2pmd	E″(np,md)
2pns:2pms	Ē(ns,ms)
2pns:2pmd	E″(ns,md)
2pnd:2pmd	Ē(nd,md)
$\tilde{E}$ (nl)=E+E′(nl)+E″(nl)
E=2I(1s)+F⁰(1s,1s)
$E^{'} (n s) = I (2 p) + I (n s) + 2 F^{0} (1 s, 2 p) - \frac{1}{3} G^{1} (1 s, 2 p) + 2 F^{0} (1 s, n s) - G^{0} (1 s, n s)$
$E^{'} (n p) = I (2 s) + I (n p) + 2 F^{0} (1 s, 2 s) - G^{0} (1 s, 2 s) + 2 F^{0} (1 s, n p) - \frac{1}{3} G^{1} (1 s, n p)$
$E^{'} (n d) = I (2 p) + I (n d) + 2 F^{0} (1 s, 2 p) - \frac{1}{3} G^{1} (1 s, 2 p) + 2 F^{0} (1 s, n d) - \frac{1}{5} G^{2} (1 s, n d)$
$E^{″} (n s) = F^{0} (2 p, n s) \pm \frac{1}{3} G^{1} (2 p, n s)$ ^a
$E^{″} (n p) = F^{0} (2 s, n p) \pm \frac{1}{3} G^{1} (2 s, n p)$
$E^{″} (n d) = F^{0} (2 p, n d) + \frac{1}{5} F^{2} (2 p, n d) \pm (1 / 15) G^{1} (2 p, n d) \pm (9 / 35) G^{3} (2 p, n d)$
Ē(nl,ml)=E′(nl,ml)+E″(nl,ml)
E′(ns,ms)=I(ns,ms)+2R⁰(ns,1s; ms,1s)−R⁰(ns,1s; 1s,ms)
$E^{'} (n p, m p) = I (n p, m p) + 2 R^{0} (n p, 1 s; m p, 1 s) - \frac{1}{3} R^{1} (n p, 1 s; 1 s, m p)$
$E^{'} (n d, m d) = I (n d, m d) + 2 R^{0} (n d, 1 s; m d, 1 s) - \frac{1}{5} R^{2} (n d, 1 s; 1 s, m d)$
$E^{″} (n s, m s) = R^{0} (2 p, n s; 2 p, m s) \pm \frac{1}{3} R^{1} (2 p, n s; m s, 2 p)$
$E^{″} (n p, m p) = R^{0} (2 s, n p; 2 s, m p) \pm \frac{1}{3} R^{1} (2 s, n p; m p, 2 s)$
$\begin{array}{l} E^{″} (n d, m d) = R^{0} (2 p, n d; 2 p, m d) + \frac{1}{5} R^{2} (2 p, n d; 2 p, m d) \\ \pm (1 / 15) R^{1} (2 p, n d; m d, 2 p) \pm (9 / 35) R^{3} (2 p, n d; m d, 2 p) \end{array}$
$\begin{array}{l} \bar{Ē} (2 s, m s) = \pm [I (2 s, m s) + 2 R^{0} (2 s, 1 s; m s, 1 s) - R^{0} (2 s, 1 s; 1 s, m s) \\ + R^{0} (2 p, 2 s; 2 p, m s) \pm \frac{1}{3} R^{1} (2 p, 2 s; m s, 2 p)] \end{array}$
$E^{″} (n s, m p) = \pm R^{0} (2 p, n s; m p, 2 s) + \frac{1}{3} R^{1} (2 p, n s; 2 s, m p)$
$E^{″} (n s, m d) = - (\sqrt{2} / 5) R^{2} (2 p, n s; 2 p, m d) \mp (\sqrt{2} / 3) R^{1} (2 p, n s; m d, 2 p)$
$E^{″} (n p, m d) = - (\sqrt{2} / 5) R^{2} (2 s, n p; m d, 2 p) \mp (\sqrt{2} / 3) R^{1} (2 s, n p; 2 p, m d)$

	a₂₁	a₂₂	a₂₃	a₂₄	a₂₅	a₂₆	a₂₇	a₂₈	a₂₉
Z=4,Be i	−0.06191	0.98163	−0.02755	−0.14555	0.01804	−0.06731	0.01041	−0.07351	−0.01589
	0.33660	0.86791	−0.16049	−0.30050	−0.02834	−0.12301	−0.01765	−0.06889	−0.03303
5,B ii	−0.02848	0.98863	−0.05926	−0.08222	0.00884	−0.03943	0.00589	−0.09879	−0.01027
	0.07359	0.95692	−0.25712	−0.07439	−0.04503	−0.03640	−0.02331	0.05744	−0.00759
6,C iii	−0.01586	0.98938	−0.07243	−0.05709	0.00472	−0.02780	0.00369	−0.10737	−0.00525
	0.03008	0.95429	−0.27878	−0.04790	−0.04061	−0.02428	−0.02004	0.07581	−0.00457
7,N iv	−0.00995	0.98941	−0.07882	−0.04370	0.00281	−0.02146	0.00258	−0.11118	−0.00215
	0.01593	0.95133	−0.29036	−0.03675	−0.03585	−0.01887	−0.01720	0.08435	−0.00410
8,O v	−0.00675	0.98935	−0.08230	−0.03536	0.00181	−0.01745	0.00195	−0.11312	−0.00028
	0.00960	0.94913	−0.29781	−0.03018	−0.03180	−0.01559	−0.01497	0.08920	−0.00403
9,F vi	−0.00484	0.98930	−0.08436	−0.02968	0.00124	−0.01470	0.00155	−0.11423	0.00087
	0.00625	0.94749	−0.30311	−0.02574	−0.02846	−0.01333	−0.01321	0.09229	−0.00401
10,Ne vii	−0.00361	0.98926	−0.08565	−0.02555	0.00089	−0.01270	0.00129	−0.11489	0.00160
	0.00428	0.94622	−0.30706	−0.02249	−0.02570	−0.01167	−0.01181	0.09440	−0.00397
11,Na viii	−0.00277	0.98923	−0.08649	−0.02243	0.00066	−0.01117	0.00110	−0.11531	0.00205
	0.00304	0.94520	−0.31015	−0.02000	−0.02341	−0.01038	−0.01067	0.09593	−0.00390
12,Mg ix	−0.00218	0.98922	−0.08707	−0.01998	0.00050	−0.00997	0.00095	−0.11558	0.00233
	0.00221	0.94438	−0.31264	−0.01802	−0.02148	−0.00936	−0.00973	0.09707	−0.00380
13,Al x	−0.00175	0.98921	−0.08748	−0.01801	0.00039	−0.00900	0.00084	−0.11577	0.00251
	0.00163	0.94368	−0.31472	−0.01639	−0.01983	−0.00852	−0.00893	0.09797	−0.00370
14,Si xi	−0.00143	0.98920	−0.08776	−0.01640	0.00031	−0.00820	0.00076	−0.11589	0.00261
	0.00121	0.94309	−0.31644	−0.01505	−0.01841	−0.00782	−0.00826	0.09867	−0.00359
15,P xii	−0.00118	0.98920	−0.08798	−0.01504	0.00025	−0.00753	0.00068	−0.11598	0.00266
	0.00090	0.94258	−0.31794	−0.01391	−0.01718	−0.00723	−0.00768	0.09925	−0.00348
Ψ(2A)=a₂₁Φ(2s2pA)+a₂₂Φ(2s3pA)+a₂₃Φ(2p3sA)+a₂₄Φ(2s4pA)+a₂₅Φ(2p4sA)+a₂₆Φ(2s5pA)+a₂₇Φ(2p5sA)+a₂₈Φ(2p3dA)+a₂₉Φ(2p4dA)

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Equations (31)

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