Y.-K. Kim, W. C. Martin, and A. W. Weiss, "Relativistic and correlation effects in the 2s3p configuration of berylliumlike ions," J. Opt. Soc. Am. B 5, 2215-2224 (1988)
Relativistic and correlation effects cause several inversions of the triplet and singlet J = 1 levels in the 2s3p configuration of berylliumlike ions. We have studied these effects in the range of atomic number Z = 6–54 by using both relativistic and nonrelativistic multiconfiguration wave functions. Theoretical predictions of energies of these levels with respect to the 2s21S0 ground level are compared with experimental data, and we conclude that overall the predictions for the ions having Z ≥ 15 are probably more accurate than the available measurements. Calculated oscillator strengths for these resonance transitions are also reported. Near cadmium (Z = 48), we confirm still another inversion that is due to the complete dominance of relativistic effects, which was reported previously. The 2p3s (1/2,1/2)1 level, a doubly excited state, goes below the 2s3p (1/2,3/2)1 level, owing to relativistic hydrogenic ordering. This type of level crossing due to hydrogenic ordering is anticipated in other sequences as well, contrary to our usual experience in atomic spectroscopy.
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Computed excitation energy including limited configuration interaction 2s2 + 2p2 and 2s3p + 2p3s + 2p3d.
Increment due to residual valence-shell correlation.
Increment due to intershell correlation.
Increment due to core correlation.
Increment due to Pauli relativistic corrections.
Net excitation energy obtained by summing all increments.
Experimental value from Ref. 15.
Table 2
Theoretical Energies, 2s21S0 and 2s3p J = 1 Levels (in cm−1)a
Mg8+
Al9+
Si10+
P11+
S12+
Cl13+
Ar14+
K15+
1S0
E(Total)
−35 636 638
−42 176 445
−49 271 583
−56 923 318
−65 132 791
−73 901 491
−83 230 830
−93 122 193
E(Coulomb)
−35 646 682
−42 189 557
−49 288 354
−56 944 394
−65 158 875
−73 933 347
−83 269 278
−93 168 112
E(Breit)
5 391
6 926
8 728
10 821
13 225
15 963
19 058
22 531
E(Lamb)
4 654
6187
8 043
10 255
12 860
15 893
19 390
23 388
Lower J = 1
σ(Coul)
1 593 504
1 924 041
2 285 618
2 678 313
3 102 194
3 557 344
4 043 850
4 561 814
Δσ(Breit)
−271
−350
−439
−539
−646
−763
−893
−1 041
Δσ(Lamb)
−190
−260
−348
−453
−579
−727
−899
−1 098
Δσ(corr)
619
610
601
592
583
575
567
560
σ(calc)
1 593 662
1 924 041
2 285 432
2 677 913
3 101 552
3 556 429
4 042 625
4 560 235
Upper J = 1
σ(Coul)
1 596 995
1 927 805
2 289 671
2 682 722
3 107 098
3 562 972
4 050 515
4 569 895
Δσ(Breit)
−246
−325
−422
−541
−686
−860
−1 063
−1 295
Δσ(Lamb)
−177
−244
−327
−428
−551
−698
−870
−1 071
Δσ(corr)
595
590
585
580
575
570
565
560
σ(calc)
1 597 167
1 927 826
2 289 507
2 682 333
3 106 436
3 561 984
4 049 147
4 568 089
Ca16+
Sc17+
Ti18+
V19+
Cr20+
Mn21+
Fe22+
1S0
E(Total)
−103 577 448
−114 598 433
−126 186 853
−138 344 756
−151 074 193
−164 377 468
−178 256 818
E(Coulomb)
−103 631 777
−114 662 173
−126 261 062
−138 430 557
−151 172 768
−164 490 065
−178 384 745
E(Breit)
26 406
30 706
35 452
40 670
46 381
52 610
59 381
E(Lamb)
27 923
33 035
38 756
45 131
52 194
59 987
68 545
Lower J = 1
σ(Coul)
5 111 384
5 692 696
6 305 579
6 950 693
7 627 975
8 337 610
9 079 772
Δσ(Breit)
−1 214
−1 398
−1 589
−1 811
−2 062
−2 340
−2 646,
Δσ(Lamb)
−1 325
−1 585
−1 876
−2 205
−2 573
−2 985
−3 441
Δσ(corr)
554
550
547
545
543
542
541
σ(calc)
5 109 399
5 690 264
6 302 661
6 947 222
7 623 883
8 332 827
9 074 226
Upper J = 1
σ(Coul)
5 121 123
5 704 782
6 320 788
6 969 355
7 650 703
8 365 064
9 112 670
Δσ(Breit)
−1 563
−1 852
−2 156
−2 496
−2 871
−3 281
−3 727
Δσ(Lamb)
−1 309
−1 570
−1 866
−2 202
−2 578
−2 999
−3 464
Δσ(corr)
556
552
548
545
542
540
538
σ(calc)
5 118 807
5 701 913
6 317 314
6 965 202
7 645 796
8 359 324
9 106 017
Co23+
Ni24+
Cu25+
Zn26+
Ge28+
Se30+
Kr32+
1S0
E(Total)
−192 714 783
−207 753 818
−223 376 751
−239 586 264
−273 776 978
−310 351 074
−349 335 732
E(Coulomb)
−192 859 412
−207 916 584
−223 559 158
−239 789 877
−274 027 957
−310 656 497
−349 703 215
E(Breit)
66 717
74 644
83 185
92 365
112 743
135 984
162 294
E(Lamb)
77 911
88 122
99 222
111 248
138 236
169 439
205 189
Lower J = 1
σ(Coul)
9 854 659
10 662 468
11 503 402
12 377 686
14 227 223
16 213 036
18 337 254
Δσ(Breit)
−2 980
−3 339
−3 738
−4 169
−5 133
−6 248
−7 528
Δσ(Lamb)
−3 945
−4 497
−5106
−5 771
−7 280
−9 052
−11 111
Δσ(corr)
541
540
540
540
540
540
540
σ(calc)
9 848 275
10 655 172
11 495 098
12 368 286
14 215 350
16 198 276
18 319 155
Upper J = 1
σ(Coul)
9 893 774
10 708 632
11 557 523
12 440 726
14 311 272
16 322 797
18 478 000
Δσ(Breit)
−4 210
−4 722
−5 294
−5 897
−7 224
−8 716
−10 357
Δσ(Lamb)
−3 979
−4 547
−5 172
−5 857
−7 423
−9 285
−11 494
Δσ(corr)
536
534
533
532
531
530
530
σ(calc)
9 886 121
10 699 886
11 547 590
12 429 504
14 297 156
16 305 326
18 456 679
E(Coulomb) is the total energy contributions from the Dirac Hamiltonian and the Coulomb repulsion, E(Breit) is the total energy contribution from the Breit operator, E(Lamb) is the total energy contributions from self-energy and vacuum polarization, σ(Coul) is the excitation-energy contributions from the Dirac Hamiltonian and the Coulomb repulsion, Δσ-(Breit) is the correction due to the Breit operator, Δσ(Lamb) is the correction due to the Lamb shift, Δσ(corr) is the correction due to residual correlation, σ(calc) is the total theoretical excitation energy.
Table 3
Eigenvector Percentages for 2s3p (J = 1) Levels, Wavelengths (λ), and Oscillator Strengths (f) for 2s2–2s3p Transitions in the Berylliumlike Ions
Eigenvector Percentages for the Three Lowest J = 1 Levels of (2s3p + 2p3s), Wavelengths (λ), and Oscillator Strengths (f) for 2s2−(2s3p + 2p3s) Transitions in Berylliumlike Ions
Computed excitation energy including limited configuration interaction 2s2 + 2p2 and 2s3p + 2p3s + 2p3d.
Increment due to residual valence-shell correlation.
Increment due to intershell correlation.
Increment due to core correlation.
Increment due to Pauli relativistic corrections.
Net excitation energy obtained by summing all increments.
Experimental value from Ref. 15.
Table 2
Theoretical Energies, 2s21S0 and 2s3p J = 1 Levels (in cm−1)a
Mg8+
Al9+
Si10+
P11+
S12+
Cl13+
Ar14+
K15+
1S0
E(Total)
−35 636 638
−42 176 445
−49 271 583
−56 923 318
−65 132 791
−73 901 491
−83 230 830
−93 122 193
E(Coulomb)
−35 646 682
−42 189 557
−49 288 354
−56 944 394
−65 158 875
−73 933 347
−83 269 278
−93 168 112
E(Breit)
5 391
6 926
8 728
10 821
13 225
15 963
19 058
22 531
E(Lamb)
4 654
6187
8 043
10 255
12 860
15 893
19 390
23 388
Lower J = 1
σ(Coul)
1 593 504
1 924 041
2 285 618
2 678 313
3 102 194
3 557 344
4 043 850
4 561 814
Δσ(Breit)
−271
−350
−439
−539
−646
−763
−893
−1 041
Δσ(Lamb)
−190
−260
−348
−453
−579
−727
−899
−1 098
Δσ(corr)
619
610
601
592
583
575
567
560
σ(calc)
1 593 662
1 924 041
2 285 432
2 677 913
3 101 552
3 556 429
4 042 625
4 560 235
Upper J = 1
σ(Coul)
1 596 995
1 927 805
2 289 671
2 682 722
3 107 098
3 562 972
4 050 515
4 569 895
Δσ(Breit)
−246
−325
−422
−541
−686
−860
−1 063
−1 295
Δσ(Lamb)
−177
−244
−327
−428
−551
−698
−870
−1 071
Δσ(corr)
595
590
585
580
575
570
565
560
σ(calc)
1 597 167
1 927 826
2 289 507
2 682 333
3 106 436
3 561 984
4 049 147
4 568 089
Ca16+
Sc17+
Ti18+
V19+
Cr20+
Mn21+
Fe22+
1S0
E(Total)
−103 577 448
−114 598 433
−126 186 853
−138 344 756
−151 074 193
−164 377 468
−178 256 818
E(Coulomb)
−103 631 777
−114 662 173
−126 261 062
−138 430 557
−151 172 768
−164 490 065
−178 384 745
E(Breit)
26 406
30 706
35 452
40 670
46 381
52 610
59 381
E(Lamb)
27 923
33 035
38 756
45 131
52 194
59 987
68 545
Lower J = 1
σ(Coul)
5 111 384
5 692 696
6 305 579
6 950 693
7 627 975
8 337 610
9 079 772
Δσ(Breit)
−1 214
−1 398
−1 589
−1 811
−2 062
−2 340
−2 646,
Δσ(Lamb)
−1 325
−1 585
−1 876
−2 205
−2 573
−2 985
−3 441
Δσ(corr)
554
550
547
545
543
542
541
σ(calc)
5 109 399
5 690 264
6 302 661
6 947 222
7 623 883
8 332 827
9 074 226
Upper J = 1
σ(Coul)
5 121 123
5 704 782
6 320 788
6 969 355
7 650 703
8 365 064
9 112 670
Δσ(Breit)
−1 563
−1 852
−2 156
−2 496
−2 871
−3 281
−3 727
Δσ(Lamb)
−1 309
−1 570
−1 866
−2 202
−2 578
−2 999
−3 464
Δσ(corr)
556
552
548
545
542
540
538
σ(calc)
5 118 807
5 701 913
6 317 314
6 965 202
7 645 796
8 359 324
9 106 017
Co23+
Ni24+
Cu25+
Zn26+
Ge28+
Se30+
Kr32+
1S0
E(Total)
−192 714 783
−207 753 818
−223 376 751
−239 586 264
−273 776 978
−310 351 074
−349 335 732
E(Coulomb)
−192 859 412
−207 916 584
−223 559 158
−239 789 877
−274 027 957
−310 656 497
−349 703 215
E(Breit)
66 717
74 644
83 185
92 365
112 743
135 984
162 294
E(Lamb)
77 911
88 122
99 222
111 248
138 236
169 439
205 189
Lower J = 1
σ(Coul)
9 854 659
10 662 468
11 503 402
12 377 686
14 227 223
16 213 036
18 337 254
Δσ(Breit)
−2 980
−3 339
−3 738
−4 169
−5 133
−6 248
−7 528
Δσ(Lamb)
−3 945
−4 497
−5106
−5 771
−7 280
−9 052
−11 111
Δσ(corr)
541
540
540
540
540
540
540
σ(calc)
9 848 275
10 655 172
11 495 098
12 368 286
14 215 350
16 198 276
18 319 155
Upper J = 1
σ(Coul)
9 893 774
10 708 632
11 557 523
12 440 726
14 311 272
16 322 797
18 478 000
Δσ(Breit)
−4 210
−4 722
−5 294
−5 897
−7 224
−8 716
−10 357
Δσ(Lamb)
−3 979
−4 547
−5 172
−5 857
−7 423
−9 285
−11 494
Δσ(corr)
536
534
533
532
531
530
530
σ(calc)
9 886 121
10 699 886
11 547 590
12 429 504
14 297 156
16 305 326
18 456 679
E(Coulomb) is the total energy contributions from the Dirac Hamiltonian and the Coulomb repulsion, E(Breit) is the total energy contribution from the Breit operator, E(Lamb) is the total energy contributions from self-energy and vacuum polarization, σ(Coul) is the excitation-energy contributions from the Dirac Hamiltonian and the Coulomb repulsion, Δσ-(Breit) is the correction due to the Breit operator, Δσ(Lamb) is the correction due to the Lamb shift, Δσ(corr) is the correction due to residual correlation, σ(calc) is the total theoretical excitation energy.
Table 3
Eigenvector Percentages for 2s3p (J = 1) Levels, Wavelengths (λ), and Oscillator Strengths (f) for 2s2–2s3p Transitions in the Berylliumlike Ions
Eigenvector Percentages for the Three Lowest J = 1 Levels of (2s3p + 2p3s), Wavelengths (λ), and Oscillator Strengths (f) for 2s2−(2s3p + 2p3s) Transitions in Berylliumlike Ions