Hylleraas-configuration-interaction nonrelativistic energies for the 1 S ground states of the beryllium isoelectronic sequence. Supplemental Material James S. Sims National Institute of Standards and Technology Gaithersburg, Maryland 20878-9957, USA Stanley A. Hagstrom Departments of Chemistry and Computer Science Indiana University Bloomington, Indiana 47405, USA 1
In a previous work Sims and Hagstrom [Phys. Rev. A 83,032518 (2011)] reported Hylleraasconfiguration-interaction (Hy-CI) method variational calculations for the 1 S ground state of neutral beryllium with an estimated accuracy of a tenth of a microhartree. The previous calculations have been extended in our current work to higher accuracy and, by simple scaling of the orbital exponents, to the entire Be 2 1 S isoelectronic sequence. Various tables are presented here which supplement this current work. The Hy-CI wave function for four electon states is Ψ = N C K Φ K, (1) K=1 where the Φ K are configuration state functions (CSFs) which can be written as Φ K = Λ(F K (r 1,r 2,r 3,r 4 )Θ K ) (2) in terms of spatial and spin functions F K (r 1,r 2,r 3,r 4 ) and Θ K (Λ is a projection operator which projects out the state of 1 S symmetry). Since only one spin function Θ K = Θ 1 = αβαβ is used in this work, the CSFs are uniquely specified by the spatial part of the wave function F K (r 1,r 2,r 3,r 4 ) given by a particular choice of r ij factor and Hartree orbital product F K (r 1,r 2,r 3,r 4 ) = r ν K ij 4 {φ Ks (r s )}, (3) where φ Ks (r s ) represents the sth basis orbital in the K th term and ν K is either 0 or 1. The basis orbitals are unnormalized Slater-type orbitals (STOs) φ(r) which are defined as s=1 φ i (r) = r n i 1 e α ir Y m i l i (θ,φ), (4) where Y m l (θ, φ) is a normalized spherical harmonic in the Condon and Shortley phase convention [1]. The STOs we use are fully defined in [2]. An s-type STO has l = 0, a p STO has l = 1, a d STO has l = 2, etc. Table I lists the configuration state function (CSF) basis used in the initial calculations in this work. In Table I and elsewhere, E for each CSF block is the amount that that block lowers the nonrelativistic energy when added to the expansion. The K shell exponents used here in Table I and elsewhere, with one exception, for the Be ground state are K = 2.789, K1 = 10.0, K2 = 15.0, Kp = 5.162, Kd = 5.40, Kf = 6.43, Kg = 5.0 and the L shell orbital exponents are L= 1.4675, and Lp = 1.596, Ld = 2.35, Lf = 2.5, Lg = 2.65. The one exception is for the 83,598 Be ground state, where the orbital exponents used were K = 3.75, K1 = 10.0, K2 = 15.0, Kp = 5.55, Kd = 5.74, Kf = 6.468, Kg = 5.0 and the L shell orbital exponents are L= 1.4465, and Lp = 1.69, Ld = 2.41, Lf = 2.5, Lg = 2.65. For the rest of the sequence orbital exponents were 2
obtained by Z/4 scaling, except for the separately optimized C++ calculation, where the orbital exponents used were K = 4.40, K1 = 10.0, K2 = 15.0, Kp = 5.62, Kd = 5.74, Kf = 6.468, Kg = 5.0 and the L shell orbital exponents are L= 1.5473, and Lp = 1.69, Ld = 2.41, Lf = 2.5, Lg = 2.65.. In column one are listed the CSF block specifications used to generate the CSF terms for the various block types, in the order electron 1 (α spin), electron 2 (β spin), electron 3 (α spin), electron 4 (β spin). For example, in the first line 1:8s K means the basis orbitals are 1s K through 8s K orbitals (K denoting an orbital exponent appropriate for a K shell electron). All of the listed basis orbitals are used to generate all of the CSFs that are unique for this basis set selection except that N max, the sum of the powers of r in the hartree product (HP), must be less than or equal to 16. The choice of terms is highly regular, there having been no attempt to cut down on the number of terms. The number of unique terms (CSFs) in a block can be computed from the listed basis orbitals and the condition that N max <= 16. For example, consider 1:5s K1 1:5s K1 2:6s L 2:6s L R 1 in the last row of Table I. There are 5 * 6 / 2 = 15 unique pairs of orbitals for electrons 1 and 2, and 6 * 7 / 2 = 21 unique pairs of orbitals for electrons 3 and 4. Since the K shell orbital exponent is different from the L shell orbital, there are (15 * 21) = 315 different CSF terms for this block. The 315 terms in this block are combined with R 1 = {1,r 12,r 34 }, which means combine with 1, r 12,and r 34, respectively, to form (315 * 3) = 945 CSF terms. Applying the condition that r-sum, the sum of the powers of r for the four orbitals in a term, has to be 16, the number of terms for this CSF block is reduced to 666 (see Reference [3] for further details). 3
Table I: 38,253 term s, p, d, f Hy-CI expansion for the Be ground state energy in hartrees (see text for the orbital exponents used). In the table, R 1 = {1,r 12,r 34 }, R 2 = {r 13,r 14 }, R = {1,r 12,r 34,r 13,r 14 }, N is the number of terms added and N tot is the cumulative number of terms. All terms are r-sum filtered using N max = 16 (see text). For p 4, the p 0 p 0 p 0 p 0 and p 1 p 1 p 1 p 1 terms were used for each CSF. Terms added N N tot E(N tot ) in hartrees E in µh 1:8s K 1:8s K 1:8s L 1:8s L R 4480 4480-14.6629 2522 1235 2:8p Kp 2:8p Kp 1:8s L 1:8s L R 3150 7630-14.6667 4500 9793 3819.788588 1:8s K 1:8s K 2:8p Lp 2:8p Lp R 3150 10780-14.6673 1467 7621 569.667828 2:7p Kp 2:7p Kp 2:7p Lp 2:7p Lp R 1 1974 12754-14.6673 4850 8548 33.830927 2:6p Kp 2:6p Kp 2:6p Lp 2:6p Lp R 2 836 13590-14.6673 4900 4451 0.495903 1:6s K 2:6p Kp 1:6s L 2:6p Lp R 1 2595 16185-14.6673 5003 9847 1.035396 1:5s K 2:5p Kp 1:5s L 2:5p Lp R 2 800 16985-14.6673 5005 6437 0.016590 1:6s K 2:6p Kp 2:6p Lp 1:6s L R 1 2595 19580-14.6673 5024 5982 0.189545 1:5s K 2:5p Kp 2:5p Lp 1:5s L R 2 800 20380-14.6673 5025 4557 0.008575 1:6s K 1:6s K 3:7d Ld 3:7d Ld R 1330 21710-14.6673 5446 6583 4.212026 3:7d Kd 3:7d Kd 1:6s L 1:6s L R 1330 23040-14.6673 5497 2996 0.506412 1:6s K 2:6p Kp 2:6p Lp 3:6d Ld R 2825 25865-14.6673 5594 4022 0.971026 2:6p Kp 2:6p Kp 1:6s L 3:6d Ld R 1680 27545-14.6673 5599 0154 0.046133 1:6s K 3:6d Kd 2:6p Lp 2:6p Lp R 1680 29225-14.6673 5605 4444 0.064289 2:6p Kp 1:6s K 2:6p Lp 3:6d Ld R 2825 32050-14.6673 5611 2573 0.058129 3:6d Kd 2:5p Kp 1:4s L 2:5p Lp R 1280 33330-14.6673 5612 0756 0.008183 2:5p Kp 3:6d Kd 1:4s L 2:5p Lp R 1280 34610-14.6673 5612 7907 0.007151 1:6s K 1:6s K 4:7f Lf 4:7f Lf 164 34774-14.6673 5612 8601 0.000694 1:6s K 1:6s K 4:7f Lf 4:7f Lf r 12 164 34938-14.6673 5620 0395 0.071794 4:7f Kf 4:7f Kf 1:6s L 1:6s L R 1 492 35430-14.6673 5621 3590 0.013195 3:7d 0Kd 3:7d 0Kd 2:6p 0Lp 2:6p 0Lp R 1 531 35961-14.6673 5625 9579 0.045988 3:7d 1Kd 3:7d 1Kd 2:6p 1Lp 2:6p 1Lp R 1 531 36492-14.6673 5634 5658 0.086080 3:7d 1Kd 3:7d 1Kd 2:6p 1Lp 2:6p 1Lp 177 36669-14.6673 5635 0590 0.004932 2:6p 0Kp 2:6p 0Kp 3:7d 0Ld 3:7d 0Ld 177 36846-14.6673 5635 1501 0.000911 2:6p 0Kp 2:6p 0Kp 3:7d 0Ld 3:7d 0Ld r 12 177 37023-14.6673 5635 2750 0.001248 2:6p 1Kp 2:6p 1Kp 3:7d 1Ld 3:7d 1Ld 177 37200-14.6673 5636 4813 0.012063 2:6p 1Kp 2:6p 1Kp 3:7d 1Ld 3:7d 1Ld r 12 177 37377-14.6673 5636 7506 0.002693 2:6p 1Kp 2:6p 1Kp 4:7f 1Lf 4:7f 1Lf 105 37482-14.6673 5636 7763 0.000257 2:6p 1Kp 2:6p 1Kp 4:7f 1Lf 4:7f 1Lf r 12 105 37587-14.6673 5636 8052 0.000289 1:5s K1 1:5s K1 2:6s L 2:6s L R 1 666 38253-14.6673 5640 7951 0.039899 4
The (Be) wave function 52,405 term expansion was obtained by expanding the atomic orbitals (AOs) used in the 38,253 term calculations to both include higher powers, e.g., 9p K and 9p L, and a different orbital exponent in the case of the p K2 AOs. Table II: 52,405 s, p, d, f Hy-CI expansion for the Be ground state energy in hartrees (see text for the orbital exoonents used). In the table, R 1 = {1,r 12,r 34 }, R 2 = {r 13,r 14 }, R = {1,r 12,r 34,r 13,r 14 }, N is the number of terms added and N tot is the cumulative number of terms. The terms are r-sum filtered using N max = 16 (see text) unless an explicit r-sum N max value is given. For p 4, the p 0 p 0 p 0 p 0 and p 1 p 1 p 1 p 1 terms were used for each CSF. Terms added N N tot E(N tot ) in hartrees E in µh 1:8s K 1:8s K 1:8s L 1:8s L R 4480 4480-14.6629 2522 1235 2:9p Kp 2:9p Kp 1:8s L 1:8s L R 1 2100 6580-14.6665 0040 7031 3575.185797 2:8p Kp 2:8p Kp 1:8s L 1:8s L R 2 1260 7840-14.6668 2979 0964 329.383933 1:8s K 1:8s K 2:9p Lp 2:9p Lp R 1 2100 9940-14.6672 7899 9130 449.208166 1:8s K 1:8s K 2:8p Lp 2:8p Lp R 2 1260 11200-14.6673 1478 4863 35.785733 2:7p Kp 2:7p Kp 2:7p Lp 2:7p Lp R 1 1974 13174-14.6673 4858 3809 33.798946 2:6p Kp 2:6p Kp 2:6p Lp 2:6p Lp R 2 836 14010-14.6673 4907 8060 0.494251 1:6s K 2:6p Kp 1:6s L 2:6p Lp R 1 2595 16605-14.6673 5008 9908 1.011848 1:5s K 2:5p Kp 1:5s L 2:5p Lp R 2 800 17405-14.6673 5010 6181 0.016273 1:6s K 2:6p Kp 2:6p Lp 1:6s L R 1 2595 20000-14.6673 5028 3700 0.177519 1:5s K 2:5p Kp 2:5p Lp 1:5s L R 2 800 20800-14.6673 5029 2463 0.008763 1:7s K 1:7s K 3:8d Ld 3:8d Ld R 1890 22690-14.6673 5458 6926 4.294462 3:8d Kd 3:8d Kd 1:7s L 1:7s L R 1890 24580-14.6673 5504 1306 0.454381 1:6s K 3:6d Kd 1:6s L 3:6d Ld R (r-sum 14) 2300 26880-14.6673 5512 3483 0.082176 1:6s K 3:6d Kd 3:6d Ld 1:6s L R (r-sum 14) 2300 29180-14.6673 5514 9227 0.025744 1:6s K 2:6p Kp 2:6p Lp 3:6d Ld R 2825 32005-14.6673 5601 9040 0.869813 2:6p Kp 2:6p Kp 1:6s L 3:6d Ld R 1680 33685-14.6673 5605 4646 0.035606 1:6s K 3:6d Kd 2:6p Lp 2:6p Lp R 1680 35365-14.6673 5611 1186 0.056539 2:6p Kp 1:6s K 2:6p Lp 3:6d Ld R 2825 38190-14.6673 5615 7056 0.045871 3:5d Kd 2:6p Kp 1:5s L 2:6p Lp R 1 1110 39300-14.6673 5616 2231 0.005174 2:6p Kp 3:7d Kd 1:5s L 2:5p Lp R 1 1455 40755-14.6673 5616 6259 0.004028 1:6s K 1:6s K 4:7f Lf 4:7f Lf 164 40919-14.6673 5616 6710 0.000451 1:6s K 1:6s K 4:7f Lf 4:7f Lf r 12 164 41083-14.6673 5623 7429 0.070719 1:6s K 1:6s K 4:7f Lf 4:7f Lf R 2 328 41411-14.6673 5623 9991 0.002562 5
Terms added N N tot E(N tot ) in hartrees E in µh 4:8f Kf 4:8f Kf 1:7s L 1:7s L R 1160 42571-14.6673 5625 2720 0.012729 1:6s K 4:7f Kf 1:5s L 4:7f Lf 412 42983-14.6673 5625 3071 0.000351 1:6s K 4:7f Kf 1:5s L 4:7f Lf r 12 412 43395-14.6673 5625 3710 0.000639 1:6s K 4:7f Kf 1:5s L 4:7f Lf r 13 412 43807-14.6673 5625 4134 0.000424 1:6s K 4:7f Kf 4:7f Lf 1:5s L 412 44219-14.6673 5625 5063 0.000930 1:6s K 4:7f Kf 4:7f Lf 1:5s L r 14 412 44631-14.6673 5625 5756 0.000693 3:8d 0Kd 3:8d 0Kd 2:7p 0Lp 2:7p 0Lp R 1220 45851-14.6673 5630 8442 0.052686 3:8d 1Kd 3:8d 1Kd 2:7p 1Lp 2:7p 1Lp R 1 732 46583-14.6673 5638 9953 0.081510 3:8d 1Kd 3:8d 1Kd 2:7p 1Lp 2:7p 1Lp r 13 244 46827-14.6673 5639 2156 0.002203 2:7p 0Kp 2:7p 0Kp 3:7d 0Ld 3:7d 0Ld R 1 630 47457-14.6673 5639 7423 0.005267 2:7p 1Kp 2:7p 1Kp 3:7d 1Ld 3:7d 1Ld R 1 630 48087-14.6673 5641 6796 0.019373 2:7p 1Kp 2:7p 1Kp 3:7d 1Ld 3:7d 1Ld r 13 210 48297-14.6673 5641 7179 0.000383 2:5p 1Kp 2:7d 1Kd 2:6p 1Lp 3:7d 1Ld (r-sum 12) 180 48477-14.6673 5641 8440 0.001261 2:5p 1Kp 2:7d 1Kd 2:6p 1Lp 3:7d 1Ld r 12 (r-sum 12) 180 48657-14.6673 5641 8701 0.000261 2:5p 1Kp 2:7d 1Kd 2:6p 1Lp 3:7d 1Ld r 14 (r-sum12) 180 48837-14.6673 5642 0195 0.001494 2:5p 1Kp 2:7d 1Kd 3:7d 1Ld 2:6p 1Lp r 14 (r-sum 12) 180 49017-14.6673 5642 0453 0.000258 4:8f 0Kf 4:8f 0Kf 2:7p 0Lp 2:7p 0Lp r 34 140 49157-14.6673 5642 1397 0.000944 4:8f 1Kf 4:8f 1Kf 2:7p 1Lp 2:7p 1Lp 140 49297-14.6673 5642 1811 0.000414 4:8f 1Kf 4:8f 1Kf 2:7p 1Lp 2:7p 1Lp r 34 140 49437-14.6673 5642 7187 0.005376 2:7p 0Kp 2:7p 0Kp 4:7f 0Lf 4:7f 0Lf r 12 84 49521-14.6673 5642 9131 0.001943 2:7p 1Kp 2:7p 1Kp 4:7f 1Lf 4:7f 1Lf r 12 105 49626-14.6673 5644 1876 0.012745 1:6s K1 1:6s K1 2:6s L 2:6s L R 1 897 50523-14.6673 5645 9929 0.018053 1:5s K 3:6d Kd 3:6d Ld 3:6d Ld 189 50712-14.6673 5646 0979 0.001051 1:5s K 2:6p Kp 3:6d Ld 4:7f Lf 365 51077-14.6673 5646 1930 0.000951 1:5s K 2:6p Kp 3:6d Ld 4:7f Lf r 34 365 51442-14.6673 5646 2449 0.000519 3:6d Kd 4:7f Kf 2:6s L 2:6p Lp r 14 332 51774-14.6673 5646 3069 0.000620 2:6s K 2:6p Kp 4:7f Lf 3:6d Ld 332 52106-14.6673 5646 4140 0.001071 2:6p K2 2:6p K2 1:6s L 1:6s L r 34 299 52405-14.6673 5646 4842 0.000701 The 79,137 and 80,073 term expansions are given next in Table III, then 38,253, 52,405, 79,137, and 80,073 term nonrelativistic energies are compared in Table IV. 6
Table III: Hy-CI calculation of the Be ground state energy in hartrees (see text for the orbital exponents used) 79,137 s,p,d,f expansion; 80,073 term s,p,d,f,g expansion. In the table, R 1 = {1,r 12,r 34 }, R 2 = {r 13,r 14 }, R = {1,r 12,r 34,r 13,r 14 }, N is the number of terms added and N tot is the cumulative number of terms. The terms are r-sum filtered using N max = 16 (see text) unless an explicit r-sum N max value is given. For p 4, the p 0 p 0 p 0 p 0 and p 1 p 1 p 1 p 1 terms were used for each CSF. Terms added N N tot E(N tot ) in hartrees E in nh 1:8s K 1:8s K 1:8s L 1:8s L R 4480 4480-14.6629 2522 1235 2:9p Kp 2:9p Kp 1:8s L 1:8s L R 1 2100 6580-14.6665 0040 7031 3575185.796 2:8p Kp 2:8p Kp 1:8s L 1:8s L R 2 1260 7840-14.6668 2979 0964 329383.933 1:8s K 1:8s K 2:9p Lp 2:9p Lp R 1 2100 9940-14.6672 7899 9130 449208.166 1:8s K 1:8s K 2:8p Lp 2:8p Lp R 2 1260 11200-14.6673 1478 4863 35785.733 2:7p Kp 2:7p Kp 2:7p Lp 2:7p Lp R 1 1974 13174-14.6673 4858 3809 33798.946 2:6p Kp 2:6p Kp 2:6p Lp 2:6p Lp R 2 836 14010-14.6673 4907 8060 494.251 1:7s K 2:7p Kp 1:7s L 2:7p Lp R 1 4338 18348-14.6673 5013 9083 1061.023 1:6s K 2:6p Kp 1:6s L 2:6p Lp R 2 1730 20078-14.6673 5015 4933 15.850 1:7s K 2:7p Kp 2:7p Lp 1:7s L R 1 4338 24416-14.6673 5031 5402 160.469 1:6s K 2:6p Kp 2:6p Lp 1:6s L R 2 1730 26146-14.6673 5032 7409 12.007 1:7s K 1:7s K 3:8d Ld 3:8d Ld R 1890 28036-14.6673 5462 3895 4296.486 3:8d Kd 3:8d Kd 1:6s L 1:6s L R 1 987 29023-14.6673 5481 7208 193.313 3:8d Kd 3:8d Kd 1:7s L 1:7s L R 2 756 29779-14.6673 5506 2782 245.574 1:6s K 3:7d Kd 1:6s L 3:7d Ld R (r-sum 14) 3050 32829-14.6673 5514 7344 84.562 1:6s K 3:7d Kd 3:7d Ld 1:6s L R (r-sum 14) 3050 35879-14.6673 5517 5600 28.256 1:6s K 2:6p Kp 2:6p Lp 3:6d Ld R 2825 38704-14.6673 5603 5225 859.625 2:6p Kp 2:6p Kp 1:6s L 3:6d Ld R 1 1008 39712-14.6673 5606 3669 28.444 2:5p Kp 2:5p Kp 1:5s L 3:5d Ld R 2 300 40012-14.6673 5606 8372 4.703 1:6s K 3:7d Kd 2:6p Lp 2:6p Lp R 1 1212 41224-14.6673 5611 8091 49.719 1:5s K 3:5d Kd 2:5p Lp 2:5p Lp R 2 300 41524-14.6673 5612 4021 5.930 2:6p Kp 1:5s K 2:5p Lp 3:6d Ld R 1 1185 42709-14.6673 5615 0879 26.858 2:6p Kp 1:5s K 2:5p Lp 3:7d Ld R 2 970 43679-14.6673 5616 5404 14.525 3:5d Kd 2:6p Kp 1:5s L 2:6p Lp R 1850 45529-14.6673 5617 1418 6.014 2:6p Kp 3:7d Kd 1:5s L 2:5p Lp R 1 1455 46984-14.6673 5617 4827 3.409 2:6p Kp 3:6d Kd 1:6s L 2:6p Lp R 2 1130 48114-14.6673 5617 5430 0.603 1:7s K 1:7s K 4:7f Lf 4:7f Lf R 925 49039-14.6673 5624 8887 73.457 4:8f Kf 4:8f Kf 1:7s L 1:7s L R 1160 50199-14.6673 5625 8177 9.29 1:6s K 4:7f Kf 1:5s L 4:7f Lf R 2060 52259-14.6673 5626 1836 3.659 1:6s K 4:7f Kf 4:7f Lf 1:5s L R 2060 54319-14.6673 5626 3970 2.134 7
Terms added N N tot E(N tot ) in hartrees E in nh 3:8d 0Kd 3:8d 0Kd 2:7p 0Lp 2:7p 0Lp R 1220 55539-14.6673 5631 8189 54.219 3:8d 1Kd 3:8d 1Kd 2:7p 1Lp 2:7p 1Lp R 1 732 56271-14.6673 5639 6855 78.666 3:7d 1Kd 3:7d 1Kd 2:6p 1Lp 2:6p 1Lp R 2 354 56625-14.6673 5639 9318 2.463 2:7p 0Kp 2:7p 0Kp 3:7d 0Ld 3:7d 0Ld R 1050 57675-14.6673 5640 4933 5.615 2:7p 1Kp 2:7p 1Kp 3:7d 1Ld 3:7d 1Ld R 1050 58725-14.6673 5642 4294 19.361 2:5p 0Kp 3:7d 0Kd 2:6p 0Lp 3:7d 0Ld R (r-sum 12) 900 59625-14.6673 5642 5166 0.872 2:5p 1Kp 3:7d 1Kd 2:6p 1Lp 3:7d 1Ld R (r-sum 12) 900 60525-14.6673 5642 7849 2.683 2:5p 0Kp 3:7d 0Kd 3:7d 0Ld 2:6p 0Lp R (r-sum 12) 900 61425-14.6673 5642 8343 0.494 2:5p 1Kp 3:7d 1Kd 3:7d 1Ld 2:6p 1Lp R (r-sum 12) 900 62325-14.6673 5643 0194 1.851 4:8f 0Kf 4:8f 0Kf 2:7p 0Lp 2:7p 0Lp R 700 63025-14.6673 5643 2316 2.122 4:8f 1Kf 4:8f 1Kf 2:7p 1Lp 2:7p 1Lp R 700 63725-14.6673 5644 7818 15.502 2:5p 0Kp 2:5p 0Kp 4:7f 0Lf 4:7f 0Lf R 1 252 63977-14.6673 5644 9047 1.229 2:5p 1Kp 2:5p 1Kp 4:7f 1Lf 4:7f 1Lf R 1 252 64229-14.6673 5645 6608 7.561 1:6s K1 1:6s K1 2:6s L 2:6s L R 1 897 65126-14.6673 5647 1886 15.278 1:5s K 3:6d Kd 3:6d Ld 3:6d Ld R 1 567 65693-14.6673 5647 3219 1.333 3:6d Kd 3:6d Kd 2:6s L 3:6d Ld R 1 528 66221-14.6673 5647 3797 0.578 1:5s K 2:6p Kp 3:6d Ld 4:7f Lf R 1 1095 67316-14.6673 5647 5850 2.053 3:6d Kd 4:7f Kf 2:6s L 2:6p Lp R 1660 68976-14.6673 5647 7924 2.074 2:6s K 2:6p Kp 4:7f Lf 3:6d Ld R 1660 70636-14.6673 5647 9152 1.228 3:6d Kd 4:7f Kf 2:6p Lp 2:6s L R 1660 72296-14.6673 5648 0069 0.917 2:6s K 3:6d Kd 2:6p Lp 4:7f Lf R 1 1095 73391-14.6673 5648 0506 0.437 2:6s K 4:7f Kf 2:6p Lp 3:6d Ld R 1 1095 74486-14.6673 5648 0952 0.446 2:6s K 4:7f Kf 3:6d Ld 2:6p Lp 365 74851-14.6673 5648 1101 0.149 2:6s K 3:6d Kd 4:7f Lf 2:6p Lp 365 75216-14.6673 5648 1373 0.272 4:7f Kf 4:7f Kf 2:5s L 3:7d Ld r 14 130 75346-14.6673 5648 1474 0.101 2:5p Kp 1:5s K 2:5p Lp 1:5s L r 14 400 75746-14.6673 5648 1590 0.116 2:5p Kp 1:5s K 1:5s L 2:5p Lp r 13 400 76146-14.6673 5648 1830 0.240 3:7d Kd 1:5s K 1:6s L 3:7d Ld r 14 680 76826-14.6673 5648 1964 0.134 4:7f Kf 1:5s K 4:7f Lf 1:6s L r 14 412 77238-14.6673 5648 2130 0.166 2:5p 1Kp 3:7d 1Kd 2:6p 1Lp 3:7d 1Ld r 13 431 77669-14.6673 5648 2186 0.056 2:5p 1Kp 3:7d 1Kd 3:7d 1Ld 2:6p 1Lp r 12 431 78100-14.6673 5648 2243 0.057 4:8f 1Kf 4:8f 1Kf 2:7p 1Lp 2:7p 1Lp r 13 140 78240-14.6673 5648 2383 0.140 2:6p K2 2:6p K2 1:6s L 1:6s L R 1 897 79137-14.6673 5648 3557 1.174 5:7g Kg 5:7g Kg 1:5s L 1:5s L r 34 (r-sum 20) 90 79227-14.6673 5648 5072 1.515 1:5s K 1:5s K 5:7g Lg 5:7g Lg r 12 (r-sum 20) 90 79317-14.6673 5649 0497 5.425 1:5s K 1:5s K 5:7g Lg 5:7g Lg r 13 (r-sum 20) 90 79407-14.6673 5649 0530 0.033 1:5s K 1:5s K 5:7g Lg 5:7g Lg r 14 (r-sum 20) 90 79497-14.6673 5649 0593 0.063 1:5s K 5:7g Kg 5:7g Lg 1:5s L R 1 576 80073-14.6673 5649 0771 0.178 8
Table IV: Hy-CI calculations of the Be isoelectronic sequence ground state nonrelativistic energies (in hartrees) for 38,253, 52,405, 79,137, and 80,073 terms in the expansion. All energies are variational except for those labelled (P), which are predicted from least squares fits to energy differences (see Section III), and one labelled (I), an interpolated value (see text). The 80,073 energies are estimated to be accurate to ten nanohartrees. Z System 38,253 result 52,405 result 79,137 result 80,073 result 4 Be -14.6673 5640 7951-14.6673 5646 4842-14.6673 5648 3557-14.6673 5649 0770 5 B + -24.3488 8438 1902-24.3488 8443 8277-24.3488 8445 6700-24.3488 8446 3615 6 C ++ -36.5348 5228 5202-36.5348 5233 7877-36.5348 5235 5149-36.5348 5236 1036 7 N 3+ -51.2227 1261 6143-51.2227 1266 4728 (P) -51.2227 1268 0445-51.2227 1268 5352 8 O 4+ -68.4115 4165 7589-68.4115 4170 2346-68.4115 4171 6662-68.4115 4172 0832 9 F 5+ -88.1009 2767 6354-88.1009 2771 7968 (P) -88.1009 2773 0990-88.1009 2773 4599 10 Ne 6+ -110.2906 6107 0069-110.2906 6110 9160-110.2906 6112 1055 (I) -110.2906 6112 4111 11 Na 7+ -134.9806 2460 4257-134.9806 2464 1280 (P) -134.9806 2465 2270-134.9806 2465 4977 13 Al 9+ -191.8609 8633 8262-191.8609 8637 2264-191.8609 8638 1768 (P) -191.8609 8638 3868 (P) 14 Si 10+ -224.0513 1029 8012-224.0513 1033 0917 (P) -224.0513 1033 9874-224.0513 1034 1733 16 S 12+ -295.9321 3928 8646-295.9321 3931 9881-295.9321 3932 7989-295.9321 3932 9531 18 Ar 14+ -377.8131 3186 6050-377.8131 3189 6149-377.8131 3190 3586-377.8131 3190 4835 23 V 19+ -626.2660 1315 3662-626.2660 1318 2130-626.2660 1318 8628 (P) -626.2660 1318 9454 (P) 24 Cr 20+ -683.4566 3218 2920-683.4566 3221 1273-683.4566 3221 7638 (P) -683.4566 3221 8406 (P) 25 Mn 21+ -743.1472 6096 9064-743.1472 6099 7244 (P) -743.1472 6100 3518-743.1472 6100 4233 28 Ni 24+ -937.2191 9419 9135-937.2191 9422 7008-937.2191 9423 3023 (P) -937.2191 9423 3518 (P) 32 Ge 28+ -1230.9818 5105 1250-1230.9818 5107 8890-1230.9818 5108 4487 (P) -1230.9818 5108 4933 (P) 33 As 29+ -1310.6725 2563 1290-1310.6725 2565 8896-1310.6725 2566 4410-1310.6725 2566 4884 38 Sr 34+ -1746.6259 4289 5674-1746.6259 4292 3196-1746.6259 4292 8544 (P) -1746.6259 4292 8910 (P) 43 Tc 39+ -2245.0794 1366 0177-2245.0794 1368 7708-2245.0794 1369 2816 (P) -2245.0794 1369 3116 (P) 48 Cd 44+ -2806.0329 1998 0111-2806.0329 2000 7697-2806.0329 2001 2803-2806.0329 2001 3026 53 I 49+ -3429.4864 5111 1168-3429.4864 5113 8833-3429.4864 5114 3876 (P) -3429.4864 5114 4113 (P) 58 Ce 54+ -4115.4400 0023 2990-4115.4400 0026 0886-4115.4400 0026 5917 (P) -4115.4400 0026 6096 (P) 63 Eu 59+ -4863.8935 6281 1636-4863.8935 6283 9484-4863.8935 6284 4412-4863.8935 6284 4614 68 Er 64+ -5674.8471 3571 4924-5674.8471 3574 2867-5674.8471 3574 7778 (P) -5674.8471 3574 7942 (P) 73 Ta 69+ -6548.3007 1671 1996-6548.3007 1674 0033-6548.3007 1674 4923 (P) -6548.3007 1674 5077 (P) 78 Pt 74+ -7484.2543 0417 0504-7484.2543 0419 8633-7484.2543 0420 3508 (P) -7484.2543 0420 3654 (P) 83 Bi 79+ -8482.7078 9686 8863-8482.7078 9689 7080-8482.7078 9690 1946 (P) -8482.7078 9690 2086 (P) 9
Z System 38,253 result 52,405 result 79,137 result 80,073 result 88 Ra 84+ -9543.6614 9387 4822-9543.6614 9390 3125-9543.6614 9390 7997-9543.6614 9390 8138 92 U 88+ -10437.4243 7408 9092-10437.4243 7411 7555-10437.4243 7412 2416 (P) -10437.4243 7412 2559 (P) 113 Uut 113+ -15785.9295 2282 1037-15785.9295 2284 9728-15785.9295 2285 4605-15785.9295 2285 4723 To test how well the convergence of the r 12 r 34 term type ( double cusp problem ) is treated in Hy-CI, the calculations presented in Table V were done. Table V: Convergence of r 12 r 34 as represented by Hy-CI. N is the number of terms added and N tot is the cumulative number of terms. The terms are filtered using N max = 16 (see text). Terms added N N tot E(N tot ) in hartrees E in µh 1:8s K 1:8s K 1:8s L 1:8s L 896 896-14.5916 8499 6456 2:7p Kp 2:7p Kp 1:8s L 1:8s L 539 1435-14.6297 7998 1630 38094.985175 1:8s K 1:8s K 2:7p Lp 2:7p Lp 539 1974-14.6589 6443 5905 29184.454275 2:7p Kp 2:7p Kp 2:7p Lp 2:7p Lp 658 2632-14.6610 8625 6532 2121.820627 1:8s K 1:8s K 1:8s L 1:8s L r 12 896 3528-14.6665 2350 2244 5437.245712 1:8s K 1:8s K 2:7p Lp 2:7p Lp r 12 539 4067-14.6669 3690 3188 413.400944 1:8s K 1:8s K 3:7d Ld 3:7d Ld r 12 345 4412-14.6669 5344 8529 16.545342 1:8s K 1:8s K 4:7f Lf 4:7f Lf r 12 198 4610-14.6669 5615 7867 2.709337 1:8s K 1:8s K 5:7g Lg 5:7g Lg r 12 97 4707-14.6669 5729 2953 1.135086 1:8s K 1:8s K 1:8s L 1:8s L r 34 896 5603-14.6669 6704 8825 9.755872 2:7p Kp 2:7p Kp 1:8s L 1:8s L r 34 539 6142-14.6669 6756 8180 0.519356 3:7d Kd 3:7d Kd 1:8s L 1:8s L r 34 312 6454-14.6669 7305 3737 5.485557 4:7f Kf 4:7f Kf 1:8s L 1:8s L r 34 185 6639-14.6669 7405 1162 0.997425 5:7g Kg 5:7g Kg 1:8s L 1:8s L r 34 94 6733-14.6669 7410 8532 0.057369 10
Finally Table VI gives the coefficients for several piecewise Z 1 least square polynomial fits for the 80,073 CSF energy values in Table IV to full precision and Table VII gives both calculated and predicted energies for the whole Z range [4,113]. Table VI: Coefficients for several piecewise Z 1 least square polynomial fits for the 80,073 CSF energy values in Table IV. R max is the maximum Residual. Z=[4,113] Z=[4,18] Z=[18,48] Z=[48,113] n a n a n a n a n 0-0.8771 1915 7812-0.8771 1949 3421-0.8771 1919 3534-0.8771 1914 3410 1-0.0423 7633 8909-0.0423 4710 9006-0.0423 7082 9682-0.0423 8063 1335 2-0.1799 0117 0558-0.1809 9525 1280-0.1802 3231 1479-0.1794 4031 5937 3-0.2069 4606 3242-0.1838 4290 8391-0.1972 9617 8566-0.2279 2379 6446 4-0.3269 2049 0237-0.6304 2234 0463-0.4645 8120 9545 5-0.8248 7322 5406 +1.7497 2452 3077 6 +1.7840 7641 9222-12.3336 5730 9091 7-16.9370 6615 2722 +31.3668 0282 8326 8 +47.0175 2102 5683-46.6461 6775 5127 9-78.4839 4815 7714 R max 0.88 10 10 0.62 10 10 0.74 10 10 0.10 10 9 Table VII: Calculated 80,073 CSF nonrelativistic energy values and predicted energy values obtained from the least squares fit of the whole Z range [4,113]. Energies are in hartrees, energy differences are in nanohartrees. Maximum residual is 0.88 10 10. Z System Data E(Z) Predicted E(Z) Difference 4 Be -14.6673 5649 0770-14.6673 5649 0770 0.000 5 B + -24.3488 8446 3615-24.3488 8446 3616 0.001 6 C ++ -36.5348 5236 1036-36.5348 5236 1030-0.006 7 N 3+ -51.2227 1268 5352-51.2227 1268 5380 0.028 8 O 4+ -68.4115 4172 0832-68.4115 4172 0790-0.042 9 F 5+ -88.1009 2773 4599-88.1009 2773 4584-0.015 10 Ne 6+ -110.2906 6112 4111-110.2906 6112 4179 0.068 11 Na 7+ -134.9806 2465 4977-134.9806 2465 4960-0.017 12 Mg 8+ -162.1707 4795 4594 13 Al 9 + -191.8609 8638 3868-191.8609 8638 3863-0.005 11
Z System Data E(Z) Predicted E(Z) Difference 14 Si 10+ -224.0513 1034 1733-224.0513 1034 1722-0.011 15 P 11+ -258.7416 9946 9303 16 S 12+ -295.9321 3932 9531-295.9321 3932 9484-0.047 17 Cl 13+ -335.6226 1941 4817 18 Ar 14+ -377.8131 3190 4835-377.8131 3190 4866 0.031 19 K 15+ -422.5036 7086 4684 20 Ca 16+ -469.6942 3171 2615 21 Sc 17+ -519.3848 1085 7841 22 Ti 18+ -571.5754 0544 7935 23 V 19+ -626.2660 1318 9454-626.2660 1318 9488 0.034 24 Cr 20+ -683.4566 3221 8406-683.4566 3221 8365-0.041 25 Mn 21+ -743.1472 6100 4233-743.1472 6100 4321 0.088 26 Fe 22+ -805.3378 9827 9863 27 Co 23+ -870.0285 4298 6476 28 Ni 24+ -937.2191 9423 3518-937.2191 9423 3479-0.039 29 Cu 25+ -1006.9098 5126 6195 30 Zn 26+ -1079.1005 1344 1073 31 Ga 27+ -1153.7911 8020 6054 32 Ge 28+ -1230.9818 5108 4933-1230.9818 5108 4941 0.008 33 As 29+ -1310.6725 2566 4884-1310.6725 2566 4865-0.019 34 Se 30+ -1392.8632 0358 6132 35 Br 31+ -1477.5538 8453 3971 36 Kr 32+ -1564.7445 6823 1758 37 Rb 33+ -1654.4352 5443 5430 38 Sr 34+ -1746.6259 4292 8910-1746.6259 4292 8850-0.060 39 Y 35+ -1841.3166 3351 9936 40 Zr 36+ -1938.5073 2603 7418 41 Nb 37+ -2038.1980 2032 8104 42 Mo 38+ -2140.3887 1625 4564 43 Tc 39+ -2245.0794 1369 3116-2245.0794 1369 3157 0.041 44 Ru 40+ -2352.2701 1253 2356 45 Rh 41+ -2461.9608 1267 1295 46 Pd 42+ -2574.1515 1401 8537 47 Ag 43+ -2688.8422 1649 0997 48 Cd 44+ -2806.0329 2001 3026-2806.0329 2001 3012-0.014 49 In 45+ -2925.7236 2451 5531 50 Sn 46+ -3047.9143 2993 5417 51 Sb 47+ -3172.6050 3621 4819 12
Z System Data E(Z) Predicted E(Z) Difference 52 Te 48+ -3299.7957 4330 0640 53 I 49+ -3429.4864 5114 4113-3429.4864 5114 4055-0.058 54 Xe 50+ -3561.6771 5970 0091 55 Cs 51+ -3696.3678 6892 7258 56 Ba 52+ -3833.5585 7878 7216 57 La 53+ -3973.2492 8924 4485 58 Ce 5 4+ -4115.4400 0026 6096-4115.4400 0026 6181 0.085 59 Pr 55+ -4260.1307 1182 1785 60 Nd 56+ -4407.3214 2388 2936 61 Pm 57+ -4557.0121 3642 3240 62 Sm 58+ -4709.2028 4941 8106 63 Eu 59+ -4863.8935 6284 4614-4863.8935 6284 4595-0.019 64 Gd 60+ -5021.0842 7668 1279 65 Tb 61+ -5180.7749 9090 8128 66 Dy 62+ -5342.9657 0550 6387 67 Ho 63+ -5507.6564 2045 8487 68 Er 64+ -5674.8471 3574 7942-5674.8471 3574 7943 0.001 69 Tm 65+ -5844.5378 5135 9281 70 Yb 66+ -6016.7285 6727 7953 71 Lu 67+ -6191.4192 8349 0277 72 Hf 68+ -6368.6099 9998 3368 73 Ta 69+ -6548.3007 1674 5077-6548.3007 1674 5083 0.006 74 W 70+ -6730.4914 3376 3968 75 Re 71+ -6915.1821 5102 9210 76 Os 72+ -7102.3728 6853 0593 77 Ir 73+ -7292.0635 8625 8457 78 Pt 74+ -7484.2543 0420 3654-7484.2543 0420 3663 0.009 79 Au 75+ -7678.9450 2235 7552 80 Hg 76+ -7876.1357 4071 1924 81 Tl 77+ -8075.8264 5925 8998 82 Pb 78+ -8278.0171 7799 1391 83 Bi 79+ -8482.7078 9690 2086-8482.7078 9690 2094 0.008 84 Po 80+ -8689.8986 1598 4444 85 At 81+ -8899.5893 3523 2106 86 Rn 82+ -9111.7800 5463 9054 87 Fr 83+ -9326.4707 7419 9549 88 Ra 84+ -9543.6614 9390 8138-9543.6614 9390 8126-0.012 89 Ac 85+ -9763.3522 1375 9574 13
Z System Data E(Z) Predicted E(Z) Difference 90 Th 86+ -9985.5429 3374 8926 91 Pa 87+ -10210.2336 5387 1439 92 U 88+ -10437.4243 7412 2559-10437.4243 7412 2588 0.029 93 Np 89+ -10667.1150 9449 8048 94 Pu 90+ -10899.3058 1499 3688 95 Am 91+ -11133.9965 3560 5555 96 Cm 92+ -11371.1872 5632 9870 97 Bk 93+ -11610.8779 7716 3013 98 Cf 94+ -11853.0686 9810 1521 99 Es 95+ -12097.7594 1914 2073 100 Fm 96+ -12344.9501 4028 1488 101 Md 97+ -12594.6408 6151 6716 102 No 98+ -12846.8315 8284 4829 103 Lr 99+ -13101.5223 0426 3018 104 Rf 100+ -13358.7130 2576 8588 105 Db 101+ -13618.4037 4735 8947 106 Sg 102+ -13880.5944 6903 1607 107 Bh 103+ -14145.2851 9078 4174 108 Hs 104+ -14412.4759 1261 4349 109 Mt 105+ -14682.1666 3451 9917 110 Uun 106+ -14954.3573 5649 8748 111 Uuu 107+ -15229.0480 7854 8792 112 Uub 108+ -15506.2388 0066 8075 113 Uut 109+ -15785.9295 2285 4723-15785.9295 2285 4692-0.031 14
References [1] E. U. Condon and G. H. Shortley. The Theory of Atomic Spectra. Cambridge U. P., Cambridge, England, 1963. [2] J. S. Sims and S. A. Hagstrom. One center r ij integrals over Slater-type orbitals. J. Chem. Phys., 55(10):4699 4710, 1971. [3] J. S. Sims and S. A. Hagstrom. Hylleraas-configuration-interaction study of the 1 1 S ground state of neutral beryllium. Phys. Rev. A, 83:032518, 2011. [4] J. S. Sims and S. A. Hagstrom. See supplemental material at [URL will be inserted by AIP] for additional material for Hylleraas-configuration interaction nonrelativistic energies for the 1 S ground states of the beryllium isoelectronic sequence. 15