Hylleraas-configuration-interaction nonrelativistic energies for the 1 S ground states of the beryllium isoelectronic sequence. Supplemental Material

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1 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 , USA Stanley A. Hagstrom Departments of Chemistry and Computer Science Indiana University Bloomington, Indiana 47405, USA 1

2 In a previous work Sims and Hagstrom [Phys. Rev. A 83, (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= , and Lp = 1.596, Ld = 2.35, Lf = 2.5, Lg = 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= , and Lp = 1.69, Ld = 2.41, Lf = 2.5, Lg = For the rest of the sequence orbital exponents were 2

3 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= , and Lp = 1.69, Ld = 2.41, Lf = 2.5, Lg = 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

4 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 :8p Kp 2:8p Kp 1:8s L 1:8s L R :8s K 1:8s K 2:8p Lp 2:8p Lp R :7p Kp 2:7p Kp 2:7p Lp 2:7p Lp R :6p Kp 2:6p Kp 2:6p Lp 2:6p Lp R :6s K 2:6p Kp 1:6s L 2:6p Lp R :5s K 2:5p Kp 1:5s L 2:5p Lp R :6s K 2:6p Kp 2:6p Lp 1:6s L R :5s K 2:5p Kp 2:5p Lp 1:5s L R :6s K 1:6s K 3:7d Ld 3:7d Ld R :7d Kd 3:7d Kd 1:6s L 1:6s L R :6s K 2:6p Kp 2:6p Lp 3:6d Ld R :6p Kp 2:6p Kp 1:6s L 3:6d Ld R :6s K 3:6d Kd 2:6p Lp 2:6p Lp R :6p Kp 1:6s K 2:6p Lp 3:6d Ld R :6d Kd 2:5p Kp 1:4s L 2:5p Lp R :5p Kp 3:6d Kd 1:4s L 2:5p Lp R :6s K 1:6s K 4:7f Lf 4:7f Lf :6s K 1:6s K 4:7f Lf 4:7f Lf r :7f Kf 4:7f Kf 1:6s L 1:6s L R :7d 0Kd 3:7d 0Kd 2:6p 0Lp 2:6p 0Lp R :7d 1Kd 3:7d 1Kd 2:6p 1Lp 2:6p 1Lp R :7d 1Kd 3:7d 1Kd 2:6p 1Lp 2:6p 1Lp :6p 0Kp 2:6p 0Kp 3:7d 0Ld 3:7d 0Ld :6p 0Kp 2:6p 0Kp 3:7d 0Ld 3:7d 0Ld r :6p 1Kp 2:6p 1Kp 3:7d 1Ld 3:7d 1Ld :6p 1Kp 2:6p 1Kp 3:7d 1Ld 3:7d 1Ld r :6p 1Kp 2:6p 1Kp 4:7f 1Lf 4:7f 1Lf :6p 1Kp 2:6p 1Kp 4:7f 1Lf 4:7f 1Lf r :5s K1 1:5s K1 2:6s L 2:6s L R

5 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 :9p Kp 2:9p Kp 1:8s L 1:8s L R :8p Kp 2:8p Kp 1:8s L 1:8s L R :8s K 1:8s K 2:9p Lp 2:9p Lp R :8s K 1:8s K 2:8p Lp 2:8p Lp R :7p Kp 2:7p Kp 2:7p Lp 2:7p Lp R :6p Kp 2:6p Kp 2:6p Lp 2:6p Lp R :6s K 2:6p Kp 1:6s L 2:6p Lp R :5s K 2:5p Kp 1:5s L 2:5p Lp R :6s K 2:6p Kp 2:6p Lp 1:6s L R :5s K 2:5p Kp 2:5p Lp 1:5s L R :7s K 1:7s K 3:8d Ld 3:8d Ld R :8d Kd 3:8d Kd 1:7s L 1:7s L R :6s K 3:6d Kd 1:6s L 3:6d Ld R (r-sum 14) :6s K 3:6d Kd 3:6d Ld 1:6s L R (r-sum 14) :6s K 2:6p Kp 2:6p Lp 3:6d Ld R :6p Kp 2:6p Kp 1:6s L 3:6d Ld R :6s K 3:6d Kd 2:6p Lp 2:6p Lp R :6p Kp 1:6s K 2:6p Lp 3:6d Ld R :5d Kd 2:6p Kp 1:5s L 2:6p Lp R :6p Kp 3:7d Kd 1:5s L 2:5p Lp R :6s K 1:6s K 4:7f Lf 4:7f Lf :6s K 1:6s K 4:7f Lf 4:7f Lf r :6s K 1:6s K 4:7f Lf 4:7f Lf R

6 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 :6s K 4:7f Kf 1:5s L 4:7f Lf :6s K 4:7f Kf 1:5s L 4:7f Lf r :6s K 4:7f Kf 1:5s L 4:7f Lf r :6s K 4:7f Kf 4:7f Lf 1:5s L :6s K 4:7f Kf 4:7f Lf 1:5s L r :8d 0Kd 3:8d 0Kd 2:7p 0Lp 2:7p 0Lp R :8d 1Kd 3:8d 1Kd 2:7p 1Lp 2:7p 1Lp R :8d 1Kd 3:8d 1Kd 2:7p 1Lp 2:7p 1Lp r :7p 0Kp 2:7p 0Kp 3:7d 0Ld 3:7d 0Ld R :7p 1Kp 2:7p 1Kp 3:7d 1Ld 3:7d 1Ld R :7p 1Kp 2:7p 1Kp 3:7d 1Ld 3:7d 1Ld r :5p 1Kp 2:7d 1Kd 2:6p 1Lp 3:7d 1Ld (r-sum 12) :5p 1Kp 2:7d 1Kd 2:6p 1Lp 3:7d 1Ld r 12 (r-sum 12) :5p 1Kp 2:7d 1Kd 2:6p 1Lp 3:7d 1Ld r 14 (r-sum12) :5p 1Kp 2:7d 1Kd 3:7d 1Ld 2:6p 1Lp r 14 (r-sum 12) :8f 0Kf 4:8f 0Kf 2:7p 0Lp 2:7p 0Lp r :8f 1Kf 4:8f 1Kf 2:7p 1Lp 2:7p 1Lp :8f 1Kf 4:8f 1Kf 2:7p 1Lp 2:7p 1Lp r :7p 0Kp 2:7p 0Kp 4:7f 0Lf 4:7f 0Lf r :7p 1Kp 2:7p 1Kp 4:7f 1Lf 4:7f 1Lf r :6s K1 1:6s K1 2:6s L 2:6s L R :5s K 3:6d Kd 3:6d Ld 3:6d Ld :5s K 2:6p Kp 3:6d Ld 4:7f Lf :5s K 2:6p Kp 3:6d Ld 4:7f Lf r :6d Kd 4:7f Kf 2:6s L 2:6p Lp r :6s K 2:6p Kp 4:7f Lf 3:6d Ld :6p K2 2:6p K2 1:6s L 1:6s L r 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

7 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 :9p Kp 2:9p Kp 1:8s L 1:8s L R :8p Kp 2:8p Kp 1:8s L 1:8s L R :8s K 1:8s K 2:9p Lp 2:9p Lp R :8s K 1:8s K 2:8p Lp 2:8p Lp R :7p Kp 2:7p Kp 2:7p Lp 2:7p Lp R :6p Kp 2:6p Kp 2:6p Lp 2:6p Lp R :7s K 2:7p Kp 1:7s L 2:7p Lp R :6s K 2:6p Kp 1:6s L 2:6p Lp R :7s K 2:7p Kp 2:7p Lp 1:7s L R :6s K 2:6p Kp 2:6p Lp 1:6s L R :7s K 1:7s K 3:8d Ld 3:8d Ld R :8d Kd 3:8d Kd 1:6s L 1:6s L R :8d Kd 3:8d Kd 1:7s L 1:7s L R :6s K 3:7d Kd 1:6s L 3:7d Ld R (r-sum 14) :6s K 3:7d Kd 3:7d Ld 1:6s L R (r-sum 14) :6s K 2:6p Kp 2:6p Lp 3:6d Ld R :6p Kp 2:6p Kp 1:6s L 3:6d Ld R :5p Kp 2:5p Kp 1:5s L 3:5d Ld R :6s K 3:7d Kd 2:6p Lp 2:6p Lp R :5s K 3:5d Kd 2:5p Lp 2:5p Lp R :6p Kp 1:5s K 2:5p Lp 3:6d Ld R :6p Kp 1:5s K 2:5p Lp 3:7d Ld R :5d Kd 2:6p Kp 1:5s L 2:6p Lp R :6p Kp 3:7d Kd 1:5s L 2:5p Lp R :6p Kp 3:6d Kd 1:6s L 2:6p Lp R :7s K 1:7s K 4:7f Lf 4:7f Lf R :8f Kf 4:8f Kf 1:7s L 1:7s L R :6s K 4:7f Kf 1:5s L 4:7f Lf R :6s K 4:7f Kf 4:7f Lf 1:5s L R

8 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 :8d 1Kd 3:8d 1Kd 2:7p 1Lp 2:7p 1Lp R :7d 1Kd 3:7d 1Kd 2:6p 1Lp 2:6p 1Lp R :7p 0Kp 2:7p 0Kp 3:7d 0Ld 3:7d 0Ld R :7p 1Kp 2:7p 1Kp 3:7d 1Ld 3:7d 1Ld R :5p 0Kp 3:7d 0Kd 2:6p 0Lp 3:7d 0Ld R (r-sum 12) :5p 1Kp 3:7d 1Kd 2:6p 1Lp 3:7d 1Ld R (r-sum 12) :5p 0Kp 3:7d 0Kd 3:7d 0Ld 2:6p 0Lp R (r-sum 12) :5p 1Kp 3:7d 1Kd 3:7d 1Ld 2:6p 1Lp R (r-sum 12) :8f 0Kf 4:8f 0Kf 2:7p 0Lp 2:7p 0Lp R :8f 1Kf 4:8f 1Kf 2:7p 1Lp 2:7p 1Lp R :5p 0Kp 2:5p 0Kp 4:7f 0Lf 4:7f 0Lf R :5p 1Kp 2:5p 1Kp 4:7f 1Lf 4:7f 1Lf R :6s K1 1:6s K1 2:6s L 2:6s L R :5s K 3:6d Kd 3:6d Ld 3:6d Ld R :6d Kd 3:6d Kd 2:6s L 3:6d Ld R :5s K 2:6p Kp 3:6d Ld 4:7f Lf R :6d Kd 4:7f Kf 2:6s L 2:6p Lp R :6s K 2:6p Kp 4:7f Lf 3:6d Ld R :6d Kd 4:7f Kf 2:6p Lp 2:6s L R :6s K 3:6d Kd 2:6p Lp 4:7f Lf R :6s K 4:7f Kf 2:6p Lp 3:6d Ld R :6s K 4:7f Kf 3:6d Ld 2:6p Lp :6s K 3:6d Kd 4:7f Lf 2:6p Lp :7f Kf 4:7f Kf 2:5s L 3:7d Ld r :5p Kp 1:5s K 2:5p Lp 1:5s L r :5p Kp 1:5s K 1:5s L 2:5p Lp r :7d Kd 1:5s K 1:6s L 3:7d Ld r :7f Kf 1:5s K 4:7f Lf 1:6s L r :5p 1Kp 3:7d 1Kd 2:6p 1Lp 3:7d 1Ld r :5p 1Kp 3:7d 1Kd 3:7d 1Ld 2:6p 1Lp r :8f 1Kf 4:8f 1Kf 2:7p 1Lp 2:7p 1Lp r :6p K2 2:6p K2 1:6s L 1:6s L R :7g Kg 5:7g Kg 1:5s L 1:5s L r 34 (r-sum 20) :5s K 1:5s K 5:7g Lg 5:7g Lg r 12 (r-sum 20) :5s K 1:5s K 5:7g Lg 5:7g Lg r 13 (r-sum 20) :5s K 1:5s K 5:7g Lg 5:7g Lg r 14 (r-sum 20) :5s K 5:7g Kg 5:7g Lg 1:5s L R

9 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 B C N (P) O F (P) Ne (I) Na (P) Al (P) (P) 14 Si (P) S Ar V (P) (P) 24 Cr (P) (P) 25 Mn (P) Ni (P) (P) 32 Ge (P) (P) 33 As Sr (P) (P) 43 Tc (P) (P) 48 Cd I (P) (P) 58 Ce (P) (P) 63 Eu Er (P) (P) 73 Ta (P) (P) 78 Pt (P) (P) 83 Bi (P) (P) 9

10 Z System 38,253 result 52,405 result 79,137 result 80,073 result 88 Ra U (P) (P) 113 Uut 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 :7p Kp 2:7p Kp 1:8s L 1:8s L :8s K 1:8s K 2:7p Lp 2:7p Lp :7p Kp 2:7p Kp 2:7p Lp 2:7p Lp :8s K 1:8s K 1:8s L 1:8s L r :8s K 1:8s K 2:7p Lp 2:7p Lp r :8s K 1:8s K 3:7d Ld 3:7d Ld r :8s K 1:8s K 4:7f Lf 4:7f Lf r :8s K 1:8s K 5:7g Lg 5:7g Lg r :8s K 1:8s K 1:8s L 1:8s L r :7p Kp 2:7p Kp 1:8s L 1:8s L r :7d Kd 3:7d Kd 1:8s L 1:8s L r :7f Kf 4:7f Kf 1:8s L 1:8s L r :7g Kg 5:7g Kg 1:8s L 1:8s L r

11 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 R max 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 Z System Data E(Z) Predicted E(Z) Difference 4 Be B C N O F Ne Na Mg Al

12 Z System Data E(Z) Predicted E(Z) Difference 14 Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb

13 Z System Data E(Z) Predicted E(Z) Difference 52 Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac

14 Z System Data E(Z) Predicted E(Z) Difference 90 Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr Rf Db Sg Bh Hs Mt Uun Uuu Uub Uut

15 References [1] E. U. Condon and G. H. Shortley. The Theory of Atomic Spectra. Cambridge U. P., Cambridge, England, [2] J. S. Sims and S. A. Hagstrom. One center r ij integrals over Slater-type orbitals. J. Chem. Phys., 55(10): , [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, [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

1s 1. [He]2s 1. [He]2s 2. [He]2s 2 2p 2. [He]2s 2 2p 1. [He]2s 2 2p 4. [He]2s 2 2p 3. [He]2s 2 2p 6. [He]2s 2 2p 5

1s 1. [He]2s 1. [He]2s 2. [He]2s 2 2p 2. [He]2s 2 2p 1. [He]2s 2 2p 4. [He]2s 2 2p 3. [He]2s 2 2p 6. [He]2s 2 2p 5 Symbol: He Atomic Number: 2 Molar Mass: 4.003 g/mol Electronegativity: - 1s 2 Symbol: H Atomic Number: 1 Molar Mass: 1.008 g/mol Electronegativity: 2.20 1s 1 Symbol: Be Atomic Number: 4 Molar Mass: 9.012

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