The double basis set method in Hylleraas coordinates described previously [
Nucl. Instrum. Methods B31,
7 (
1988)] is applied to the calculation of high-precision eigenvalues for the 1s2p1P and 3P states of helium. Convergence to a few parts in 1014 is obtained for the nonrelativistic energies. The new wave functions are used to calculate mass-polarization, relativistic, relativistic reduced mass, and quantum-electrodynamic corrections. A comparison of the P-state energies, together with the previously calculated S- and D-state energies, with the experimental energy levels tabulated by
Martin [
Phys. Rev A 36,
3575 (
1987)] yields well-defined discrepancies of the order of 0.001 cm−1, which can be attributed to higher-order relativistic and quantum-electrodynamic effects.
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Values of the Optimized Nonlinear Parameters (in units of Z/a0)
N
2 1P
2 3P
α1
β1
α2
β2
β1
β1
α2
β2
Infinite Nuclear Mass
4
0.8132
0.4348
1.2691
0.9855
0.8737
0.3957
1.0731
0.9483
5
0.8500
0.4573
1.2170
1.1702
0.9601
0.4843
1.1307
1.0051
6
0.8170
0.4363
1.3242
1.2932
0.8370
0.4095
1.2160
1.3002
7
0.8040
0.4527
1.3872
1.4554
0.7627
0.3597
1.3228
1.2168
8
0.7988
0.4335
1.3679
1.5165
0.8174
0.4081
1.3786
1.4448
9
0.7636
0.4668
1.5936
1.7152
0.8070
0.4282
1.4777
1.6807
10
0.7863
0.4731
1.7324
1.8809
0.8232
0.4327
1.7656
2.0173
11
0.7979
0.4653
1.9190
2.0705
0.8461
0.4369
2.0199
2.2724
12
0.7921
0.4786
2.1403
2.2933
0.8362
0.4290
2.2859
2.5432
Finite Nuclear Mass
4
0.8087
0.4336
1.2720
9.9749
0.8707
0.3949
1.0702
0.9498
5
0.8424
0.4573
1.2063
1.1567
0.9577
0.4852
1.1264
0.9955
6
0.7989
0.4278
1.3447
1.2804
0.8273
0.4073
1.2100
1.2959
7
0.7891
0.4581
1.4194
1.3861
0.7499
0.3538
1.3361
1.1840
8
0.7500
0.4111
1.3970
1.4636
0.7664
0.3938
1.3905
1.3922
9
0.7560
0.4715
1.6095
1.6980
0.7626
0.3935
1.4554
1.6332
10
0.7523
0.4685
1.7148
1.8994
0.7649
0.4214
1.7416
1.9917
11
0.7061
0.4707
1.9384
2.0500
0.8178
0.4450
2.0199
2.3185
12
0.7845
0.4980
2.1520
2.3398
0.8210
0.4495
2.4511
2.6489
Table 3
Contributions to the Matrix Elements of 〈P14〉/4 = 〈F0〉 + (μ/M)[〈F1〉 + 〈G〉] as Defined by Eqs. (24) and (25) (in atomic units)
State
〈F0〉
〈F1〉
〈G〉
H− 1s2 1S
0.61563964(3)
−0.00063343(3)
−0.0238770(2)
He 1s2 1S
13.5220168(1)
−0.1758280(1)
−0.9082723(1)
1s2s1S
10.2796689(1)
0.0200726(2)
−0.0797993(1)
1s2s3S
10.45888519(1)
0.01062391(4)
−0.01558432(1)
1s2p1P
10.0292513215(1)
0.3393(1)
−0.0767530999(1)
1s2p3P
9.912093697(4)
−0.5923(1)
0.11975737(1)
Table 4
Values for Various Matrix Elements Required to Calculate Relativistic and Quantum-Electrodynamic Corrections to the Energya
State
Matrix Element
〈T0〉
〈T1〉
He 1s2p1P
〈p14〉/4
10.0292513215(1)
0.2625(1)
〈H2〉/α2
−0.02033047408(2)
0.1045056(3)
π 〈δ(r1) 〉
4.00362331908(2)
0.12376(3)
π 〈δ(r12)〉
0.002309601(1)
−0.010853(2)
Δ2/(α2m/M)
−16.286503967(1)
Q
0.003374498(1)
He 1s2p3P
〈P14〉/4
9.912093697(4)
0.4728(1)
〈H2〉/α2
0.03508088684(1)
0.1523726(2)
π 〈δ(r1) 〉
3.954827224(1)
−0.22525(3)
〈3P‖ Hso‖ 3P〉/α2
0.207955244(1)
0.69869(1)
〈3P‖Hsoo‖3P〉/α2
−0.3088684536(5)
−0.959469(3)
〈3P‖Hss‖3P〈/α2
−0.1351209953(1)
−0.3244130(1)
〈3P‖Δ1‖3P〉/(α2m/M)
−0.59732701(1)
Δ2/(α2m/M)
−15.75868217(1)
Q
0.00381391791(1)
He 1s2p3P–1s2p1p
〈3P‖Hso‖1P〉/α2
0.1073194807(3)
−0.054970(1)
〈3P‖Hsoo‖1P〉/α2
0.03876204224(3)
−0.0320412(3)
〈3P‖Δ1‖1P〉/α2m/M
−0.07646528(2)
Each quantity is expressed in the form 〈T〉 = 〈T0〉 + 〈T1〉 (μ/M (arbitrary units), where 〈T1〉 (μ/M)is the change in the matrix element when the mass-polarization term p1 · p2(μ/M) is included explicitly in the Hamiltonian H = −(∇12 + ∇22)/2 − Z/r1 − Z/r2 + 1/r12.
Table 5
Contributions to the P-State Energies of Helium (in inverse centimeters), Using R∞ = 109737.31569 cm−1, α−1 = 137.03596, μ/M = 1.370745633 × 10−4, and RM = 109722.273495 cm−1a
Parameter
2 1P1
2 3P0
2 3P1
2 3P2
Enr
−27 176.690015
−29 222.155521
−29 222.155521
−29 222.155521
ΔEM(1)
1.385032
−1.942356
−1.942356
−1.942356
ΔEM(2)
−0.000694
−0.000845
−0.000845
−0.000845
ΔErel
−0.467728
1.300852
0.314817
0.237533
ΔEanom
0.0
0.001203
−0.000620
0.000131
ΔEst
0.000158
0.0
−0.000158
0.0
(ΔRR)M
−0.000284
−0.000014
0.000208
0.000132
(ΔERR)X
0.000126
0.000593
0.000263
0.000228
ΔEnuc
0.000002
−0.000026
−0.000026
−0.000026
ΔEL,1
0.002009
−0.041087
−0.041095
−0.041111
ΔEL,2
−0.002096
−0.001518
−0.001518
−0.001518
Total
−27 175.773489
−29 222.838720
−29 223.826851
−29 223.903353
ΔEM(1) and ΔEM(2) are the first- and second-order mass-polarization corrections given by Eqs. (13) and (14), and ΔEst is the singlet-triplet mixing correction.
Table 6
Comparison of Calculated Energy Levels (in Inverse Centimeters) with the Tabulation of Martin of Experimental Values
The calculated energies are adjusted by 198 310.773489 cm−1 to bring the 2 1P state into correspondence. Ref. 4.
Quoted experimental uncertainties are for the 2 3S1–4 3D1 and 2 3S1–5 3D1 separations.
Tables (6)
Table 1
Variational Eigenvalues for the P States of Helium (in arbitrary units)
Values of the Optimized Nonlinear Parameters (in units of Z/a0)
N
2 1P
2 3P
α1
β1
α2
β2
β1
β1
α2
β2
Infinite Nuclear Mass
4
0.8132
0.4348
1.2691
0.9855
0.8737
0.3957
1.0731
0.9483
5
0.8500
0.4573
1.2170
1.1702
0.9601
0.4843
1.1307
1.0051
6
0.8170
0.4363
1.3242
1.2932
0.8370
0.4095
1.2160
1.3002
7
0.8040
0.4527
1.3872
1.4554
0.7627
0.3597
1.3228
1.2168
8
0.7988
0.4335
1.3679
1.5165
0.8174
0.4081
1.3786
1.4448
9
0.7636
0.4668
1.5936
1.7152
0.8070
0.4282
1.4777
1.6807
10
0.7863
0.4731
1.7324
1.8809
0.8232
0.4327
1.7656
2.0173
11
0.7979
0.4653
1.9190
2.0705
0.8461
0.4369
2.0199
2.2724
12
0.7921
0.4786
2.1403
2.2933
0.8362
0.4290
2.2859
2.5432
Finite Nuclear Mass
4
0.8087
0.4336
1.2720
9.9749
0.8707
0.3949
1.0702
0.9498
5
0.8424
0.4573
1.2063
1.1567
0.9577
0.4852
1.1264
0.9955
6
0.7989
0.4278
1.3447
1.2804
0.8273
0.4073
1.2100
1.2959
7
0.7891
0.4581
1.4194
1.3861
0.7499
0.3538
1.3361
1.1840
8
0.7500
0.4111
1.3970
1.4636
0.7664
0.3938
1.3905
1.3922
9
0.7560
0.4715
1.6095
1.6980
0.7626
0.3935
1.4554
1.6332
10
0.7523
0.4685
1.7148
1.8994
0.7649
0.4214
1.7416
1.9917
11
0.7061
0.4707
1.9384
2.0500
0.8178
0.4450
2.0199
2.3185
12
0.7845
0.4980
2.1520
2.3398
0.8210
0.4495
2.4511
2.6489
Table 3
Contributions to the Matrix Elements of 〈P14〉/4 = 〈F0〉 + (μ/M)[〈F1〉 + 〈G〉] as Defined by Eqs. (24) and (25) (in atomic units)
State
〈F0〉
〈F1〉
〈G〉
H− 1s2 1S
0.61563964(3)
−0.00063343(3)
−0.0238770(2)
He 1s2 1S
13.5220168(1)
−0.1758280(1)
−0.9082723(1)
1s2s1S
10.2796689(1)
0.0200726(2)
−0.0797993(1)
1s2s3S
10.45888519(1)
0.01062391(4)
−0.01558432(1)
1s2p1P
10.0292513215(1)
0.3393(1)
−0.0767530999(1)
1s2p3P
9.912093697(4)
−0.5923(1)
0.11975737(1)
Table 4
Values for Various Matrix Elements Required to Calculate Relativistic and Quantum-Electrodynamic Corrections to the Energya
State
Matrix Element
〈T0〉
〈T1〉
He 1s2p1P
〈p14〉/4
10.0292513215(1)
0.2625(1)
〈H2〉/α2
−0.02033047408(2)
0.1045056(3)
π 〈δ(r1) 〉
4.00362331908(2)
0.12376(3)
π 〈δ(r12)〉
0.002309601(1)
−0.010853(2)
Δ2/(α2m/M)
−16.286503967(1)
Q
0.003374498(1)
He 1s2p3P
〈P14〉/4
9.912093697(4)
0.4728(1)
〈H2〉/α2
0.03508088684(1)
0.1523726(2)
π 〈δ(r1) 〉
3.954827224(1)
−0.22525(3)
〈3P‖ Hso‖ 3P〉/α2
0.207955244(1)
0.69869(1)
〈3P‖Hsoo‖3P〉/α2
−0.3088684536(5)
−0.959469(3)
〈3P‖Hss‖3P〈/α2
−0.1351209953(1)
−0.3244130(1)
〈3P‖Δ1‖3P〉/(α2m/M)
−0.59732701(1)
Δ2/(α2m/M)
−15.75868217(1)
Q
0.00381391791(1)
He 1s2p3P–1s2p1p
〈3P‖Hso‖1P〉/α2
0.1073194807(3)
−0.054970(1)
〈3P‖Hsoo‖1P〉/α2
0.03876204224(3)
−0.0320412(3)
〈3P‖Δ1‖1P〉/α2m/M
−0.07646528(2)
Each quantity is expressed in the form 〈T〉 = 〈T0〉 + 〈T1〉 (μ/M (arbitrary units), where 〈T1〉 (μ/M)is the change in the matrix element when the mass-polarization term p1 · p2(μ/M) is included explicitly in the Hamiltonian H = −(∇12 + ∇22)/2 − Z/r1 − Z/r2 + 1/r12.
Table 5
Contributions to the P-State Energies of Helium (in inverse centimeters), Using R∞ = 109737.31569 cm−1, α−1 = 137.03596, μ/M = 1.370745633 × 10−4, and RM = 109722.273495 cm−1a
Parameter
2 1P1
2 3P0
2 3P1
2 3P2
Enr
−27 176.690015
−29 222.155521
−29 222.155521
−29 222.155521
ΔEM(1)
1.385032
−1.942356
−1.942356
−1.942356
ΔEM(2)
−0.000694
−0.000845
−0.000845
−0.000845
ΔErel
−0.467728
1.300852
0.314817
0.237533
ΔEanom
0.0
0.001203
−0.000620
0.000131
ΔEst
0.000158
0.0
−0.000158
0.0
(ΔRR)M
−0.000284
−0.000014
0.000208
0.000132
(ΔERR)X
0.000126
0.000593
0.000263
0.000228
ΔEnuc
0.000002
−0.000026
−0.000026
−0.000026
ΔEL,1
0.002009
−0.041087
−0.041095
−0.041111
ΔEL,2
−0.002096
−0.001518
−0.001518
−0.001518
Total
−27 175.773489
−29 222.838720
−29 223.826851
−29 223.903353
ΔEM(1) and ΔEM(2) are the first- and second-order mass-polarization corrections given by Eqs. (13) and (14), and ΔEst is the singlet-triplet mixing correction.
Table 6
Comparison of Calculated Energy Levels (in Inverse Centimeters) with the Tabulation of Martin of Experimental Values
The calculated energies are adjusted by 198 310.773489 cm−1 to bring the 2 1P state into correspondence. Ref. 4.
Quoted experimental uncertainties are for the 2 3S1–4 3D1 and 2 3S1–5 3D1 separations.