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Double linked Hylleraas–CI for the lowest 4 Pe states of the Li isoelectronic series
27 March 1998
Chemical Physics Letters 285 Ž1998. 417–421
Double linked Hylleraas–CI for the lowest 4 P e states of the Li isoelectronic series Rene´ Barrois, Hans–Peter Merckens, Heinz Kleindienst Institut fur D-40225 Dusseldorf, Germany ¨ Physikalische Chemie und Elektrochemie, Heinrich-Heine-UniÕersitat ¨ Dusseldorf, ¨ ¨ Received 20 November 1997; in final form 15 January 1998
Abstract An accurate upper bound calculation for the lowest 4 P e states of the Li isoelectronic series from Li I–Ne VIII is given using an extensive double linked Hylleraas–CI calculation. We obtained, in infinite mass approximation, E s y5.2454098 0 Eh for the 2 4 P e state of the Li atom. Additionally we discuss the transitions 2 4 P o ™ 2 4 P e and 2 4 P e ™ 3 4 S and give a comparison to other calculations and experimental data. q 1998 Elsevier Science B.V.
1. Introduction
with the three particle antisymmetrizer A, a totally symmetric spinfunction x and the spatial functions
Recently the Hylleraas–CI method has successfully been applied to the S, 2 P and 4 P o states of the Li isoelectronic series w1–11x. As far as the 4 P e states of the Li atom are concerned there are on the one hand a 30-term correlated CI by Holøien and Geltman w4x and on the other hand conventional CIs, e.g., by Bunge and Bunge w11x and Hsu et al. w12x. In this Letter we present results for the lowest 4 P e states of the Li isoelectronic series using extensive double linked Hylleraas–CI basis sets.
l i m i n i yŽ a r 1qb r 2qg r 3 . f i s r 1a i r 2b i r 3c i r 23 r 13 r 12 e
2. Method The ansatz for the wavefunction C of the 4 P e states is given as
C s Ý ci A Ž fi x . , i
Ž 1.
= sinq 2 sinq 3 sin Ž w 2 y w 3 . ,
Ž 2.
with a , b ,g ) 0; a i , l i , m i , n i g N 0 and bi ,c i g N. The angular term ensures the even parity of the wavefunction which is an eigenfunction of the angular momentum operators L2 and L z with the eigenvalues LŽ L q 1. s 2 and ML s 0. The corresponding Hamiltonian H is expressed in terms of a nonorthogonal coordinate set Ž ri ,ri j ,qi , w i . w8x. An efficient method to evaluate the required integrals is given in Refs. w8,9x. The eigenvalues of the resulting large general symmetric eigenproblem are evaluated with a modified Wielandt procedure w14x using Falk’s elevator method w15x. The nonlinear parameters were optimized using the Newton-Raphson and the Nelder-Mead simplex method w16x algorithms. Initial values of the nonlinear parameters were obtained by search on a coarse grid with 49 basic functions for the variational space.
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 1 1 7 - 1
R. Barrois et al.r Chemical Physics Letters 285 (1998) 417–421
418
3. Results All calculations were achieved in IEEE 8 Byte precision on a Convex C3840 computer. The integrals are accurate to 12 digits. In order to obtain optimal values for the energy of the lowest 4 P e state of the Li atom the following strategy was used. First we optimized the nonlinear parameters a / b s g for a small basis set Ž N s 115, N: number of basis functions and v s a i q bi q c i q l i q m i q n i F 6. with the Nelder–Mead w16x and the Newton–Raphson methods respectively. The choice b s g induces the restriction bi - c i
k bi s c i n m i F n i .
Ž 3.
We then extended the basis using a selection procedure described in Ref. w17x to obtain a basis set with v F 11 and N s 982. For this basis we reoptimized the nonlinear parameters. We present a convergence pattern with the new parameters in Table 1. As a second step we extended the basis to ‘‘3 = 982’’ functions using nonlinear parameters obtained by optimizing a ‘‘3 = 115’’ basis set. The corresponding convergence pattern is given in Table 2. In order to get a smaller basis set, we chose b / g and eliminated the restriction in Eq. Ž3. by adding the missing functions to the set. The new unrestricted basis had a dimension of 1881 functions. We used nonlinear parameters optimized for a small basis with v F 6 and N s 203. Table 3 shows the convergence pattern. Comparing the last rows of Tables 2 and 3 one notices that no loss of accuracy occurs by using the smaller basis of 1881 functions. Furthermore we checked the overlap of the corresponding wavefunctions C 2946 and C 1881. We found ²C 2946 N C 1881 : s 1 y 8.4 = 10y9 . Due to these facts we used Table 1 Basis selection procedure with a s 3.122 and b sg s1.233, full basis dimension N, selected basis dimension Nsel and energies E in Eh for the 24 P e state of the Li atom
v
N
Nsel
E
6 7 8 9 10 11
115 238 452 798 1333 2133
115 238 404 563 718 982
y5.245 277 4 3 y5.245 384 9 8 y5.245 406 11 y5.245 408 91 y5.245 409 59 y5.245 409 73
Table 2 Energies in Eh for the 24 P e state of Li calculated with three parameter sets Number of functions
a
b sg
E
1– 982 983–1964 1965–2946
2.802 7.755 3.833
1.039 1.347 1.615
y5.245 409 56 y5.245 409 72 y5.245 409 8 0
Table 3 Energies E in Eh for the 24 P e state of Li with a s 2.921, b s1.415 and g s 0.781 using the unrestricted basis
v
N
E
6 7 8 9 10 11
203 434 752 1061 1365 1881
y5.245 406 30 y5.245 408 8 2 y5.245 409 54 y5.245 409 71 y5.245 409 77 y5.245 409 8 0
Table 4 Energies E in Eh for the 24 P e state of some members of the Li isoelectronic series, difference D in m Eh of our values from these of Holøien and Geltman w4x System
E Žthis work.
E w4x
D
Li I Be II B III C IV NV O VI F VII Ne VIII
y5.245 409 8 0 y9.870 897 1 0 y16.000 433 6 y23.631 809 2 y32.764 164 6 y43.397 101 2 y55.530 410 0 y69.163 971 4
y5.245 92 y9.868 80 y15.996 75 y23.627 16 y32.758 71 y43.391 04 y55.523 88 y69.157 50
0.5102 y2.0971 y3.6836 y4.6492 y5.4546 y6.0612 y6.5300 y6.4714
Table 5 Energies E in Eh for the 34 P e state of some members of the Li isoelectronic series, difference D in m Eh of our values from these of Holøien and Geltman w4x System
E Žthis work.
E w4x
D
Li I Be II B III C IV NV O VI F VII Ne VIII
y5.096 831 33 y9.428 868 46 y15.124 487 3 y22.182 342 2 y30.601 898 5 y40.382 911 2 y51.525 253 5 y64.028 853 9
y5.093 86 y9.423 20 y15.119 00 y22.176 54 y30.596 09 y40.376 96 y51.519 24 y64.022 50
y2.9713 y5.6684 y5.4873 y5.8022 y5.8085 y5.9512 y6.0135 y6.3539
R. Barrois et al.r Chemical Physics Letters 285 (1998) 417–421
419
Table 6 4 e : < < ² : ² : Expectation values ² V :, ²T : and ² == i j in E h and the values 1 y h for the 2 P states with h s y Ž V r 2 T . System
²V :
²T :
² == : i j
<1 y h <
Li I Be II B III C IV NV O VI F VII Ne VIII
y10.490 819 4 y19.741 794 0 y32.000 867 0 y47.263 619 4 y65.528 328 8 y86.794 202 8 y111.060 82 0 y138.327 94 3
5.245 409 6 0 9.870 896 9 7 16.000 433 4 23.631 810 1 32.764 164 2 43.397 101 6 55.530 409 9 69.163 971 6
0.350 691 34 0 0.999 652 78 7 1.970 921 8 6 3.258 400 9 3 4.859 982 6 4 6.774 786 1 2 9.002 382 8 3 11.542 544 8
1.93 = 10y8 6.32 = 10y9 5.66 = 10y9 1.88 = 10y8 6.58 = 10y9 4.99 = 10y9 9.27 = 10y10 1.15 = 10y9
the unrestricted basis Ž N s 1881. for the following calculations. The energies of the 2 4 P e and 3 4 P e states of some members of the Li isoelectronic series were calculated similarly using the unrestricted basis with nonlinear parameters which we optimized for N s 203 functions Ž v F 6.. Comparing our results for the isoelectronic series to those of Holøien and Geltman w4x ŽTables 4 and 5., the improvement of all the given energies is evident with the exception of the lowest 4 P e state of Li. There is no reasonable explanation for this discrepancy as Bunge and Bunge have already pointed out in Ref. w11x. Bunge and Bunge found E s y5.245308 Eh as a variational upper bound and the extrapolated value y5.245351 E h . The upper bound of Hsu et al. w12x is E s y5.2453903 Eh and their extrapolated value is E s y5.2454045 Eh . Therefore we believe the quoted result of Holøien and Geltman w4x to be a misprint. Considering the patterns in Tables 2 and 3 we think
that the accuracy of our energy values for the Li atom is in the scale of 0.1 m Eh . For the ions we believe eight digits to be significant. Using the extended basis Ž N s 1881, v F 6. and the corresponding eigenvectors we calculated some expectation values for the 2 4 P e and 3 4 P e states of the Li isoelectronic series. The results are summarized in Tables 6 and 7, respectively. We calculated the specific and normal mass shifts D ESM S and : D E NM S from the given expectation values ² == i j :s ²C <Ý i - j== : C and the energies in infinite mass i j approximation `E using the well-known equations D ESM S s y
D E NM S s y
me M q me me M q me
` ²`C < Ý == : i jC
and
i-j `
E,
Ž 4.
where M is the nuclear mass and m e the mass of the
Table 7 4 e : < < ² : ² : Expectation values ² V :, ²T : and ² == i j in E h and the values 1 y h for the 3 P states with h s y Ž V r 2 T . System
²V :
²T :
² == : i j
<1 y h <
Li I Be II B III C IV NV O VI F VII Ne VIII
y10.193 661 1 y18.857 735 7 y30.248 973 6 y44.364 683 5 y61.203 796 0 y80.765 821 2 y103.050 50 6 y128.057 70 6
5.096 829 8 5 9.428 867 2 7 15.124 486 2 22.182 341 2 30.601 897 4 40.382 910 0 51.525 252 7 64.028 852 7
0.312 374 172 0.761 409 40 4 1.406 623 6 0 2.244 988 2 4 3.275 436 1 9 4.497 548 78 5.911 134 4 8 7.516 097 0 2
1.45 = 10y7 6.24 = 10y8 3.78 = 10y8 2.29 = 10y8 1.89 = 10y8 1.43 = 10y8 7.55 = 10y9 1.02 = 10y8
R. Barrois et al.r Chemical Physics Letters 285 (1998) 417–421
420
Table 8 Energy values in Eh for some isotopes of the Li isoelectronic series
electron. The finite mass energies M E to first order were obtained by E s`E q D ESM S q D E NMS .
System
24 P e
34 P e
M
6
y5.244 963 32 y5.245 027 0 2 y9.870 357 0 0 y15.999 664 7 y15.999 734 3 y23.630 877 6 y23.630 949 5 y32.763 071 2 y32.763 143 9 y43.395 844 8 y43.395 919 1 y43.395 984 7 y55.529 066 2 y69.162 390 0 y69.162 465 4 y69.162 533 7
y5.096 394 9 0 y5.096 457 1 7 y9.428 340 77 y15.123 735 6 y15.123 803 7 y22.181 430 6 y22.181 501 0 y30.600 827 7 y30.600 898 9 y40.381 680 1 y40.381 752 9 y40.381 817 2 y51.523 936 1 y64.027 302 9 y64.027 376 9 y64.027 443 9
Table 8 shows the results for M E of some selected isotopes, the values for the nuclear masses are taken from Ref. w18x. In Table 9 we present some calculations for the 4 4 P e state of Li. The accuracy of the energies is better than 2 m Eh . Finally we discuss the transitions 2 4 P o ™ 2 4 P e and 2 4 P e ™ 3 4 S . In Table 10 a comparison of the nonrelativistic energies, including mass polarization of the required states, between Hsu’s et al. w12,13x and ours is given. Table 11 contains the nonrelativistic Žmass polarization included. and the total transition energies. Similar to our previous work w7x we used the relativistic corrections given by Hsu et al. w12,13x. The differences between the experimental data and the theoretical values are on the one hand due to QED corrections, which are estimated by Hsu et al. w12x to be of the order of 0.6 cmy1 and on the other hand to uncertainty of the experiment.
Li I Li I 9 Be II 10 B III 11 B III 12 C IV 13 C IV 14 NV 15 NV 16 O VI 17 O VI 18 O VI 19 F VII 20 Ne VIII 21 Ne VIII 22 Ne VIII 7
Table 9 Summarized results for the 44 P e state of Li ` E ²T : ²V : ² == : i j N1yh N EŽ6 Li. EŽ7 Li.
y5.0640891 Eh 5.064076 6 E h y10.128166 E h 0.30795272 Eh 1.29=10y6 y5.0636553 Eh y5.0637172 E h
Acknowledgements
Table 10 Upper bounds for the nonrelativistic energies including mass polarization of the 24 P o , 24 P e and the 34 S states of 7 Li in E h State
Enr 4
o
1s2s2p P 1s2p2p 4 P e 1s2s3s 4 S a b
a
y5.367 605 8 y5.245 027 0 y5.212 339 1
Enr
We thank the Deutsche Forschungsgemeinschaft for financial support.
b
y5.367 594 8 y5.245 007 7 y5.212 329 1
Refs. w5–7x & this work. Refs. w12,13x.
References w1x A. Luchow, H. Kleindienst, Int. J. Quantum Chem. 51 Ž1994. ¨ 211. w2x F.W. King, Phys. Rev. A 40 Ž1989. 1735.
Table 11 Comparison of experimental transition energies D EŽ24 P o ™ 24 P e . and D EŽ24 P e ™ 34 S. in cmy1 Transition
This work
D Enr Ž24 P o ™ 24 P e . D Etot.Ž24 P o ™ 24 P e . D Enr Ž24 P e ™ 34 S. D Etot.Ž24 P e ™ 34 S.
26902.9 26916.1 7174.2 7155.2
a b c
Ž 5.
c
Ref. w12,13x & this work. Ref. w19x. Including relativistic correction of Hsu et al. w12,13x.
Theor. a 26904.7 26916.3 7172.1 7156.2
Exp. b 26915.16 7156.8
R. Barrois et al.r Chemical Physics Letters 285 (1998) 417–421 w3x D.K. Mc Kenzie, G.W.F. Drake, Phys. Rev. A 44 Ž1991. R6973. w4x E. Holøien, S. Geltman, Phys. Rev. 153 Ž1967. 81. w5x A. Luchow, R. Barrois, H. Kleindienst, Chem. Phys. Lett. ¨ 216 Ž1993. 359. w6x R. Barrois, A. Luchow, H. Kleindienst, Chem. Phys. Lett. ¨ 249 Ž1996. 249. w7x R. Barrois, S. Bekavac, H. Kleindienst, Chem. Phys. Lett. 268 Ž1997. 531. w8x R. Barrois, A. Luchow, H. Kleindienst, Int. J. Quantum ¨ Chem. 61 Ž1997. 107. w9x R. Barrois, A. Luchow, H. Kleindienst, Int. J. Quantum ¨ Chem. 62 Ž1997. 77. w10x Z.-C. Yan, G.W.F. Drake, Phys. Rev. A 52 Ž1995. 3711. w11x C.F. Bunge, A.V. Bunge, Phys. Rev. A 17 Ž1978. 822.
421
w12x J.J. Hsu, K.T. Chung, K.N. Huang, Phys. Rev. A 49 Ž1994. 4466. w13x J.J. Hsu, K.T. Chung, K.N. Huang, Phys. Rev. A 44 Ž1991. 5484. w14x A. Luchow, H. Kleindienst, Computers and Chem. 17 Ž1993. ¨ 61. w15x R. Zurmuhl, ¨ S. Falk, Matrizen und ihre Anwendung, vol. 2, Springer Verlag, Berlin, 1986, p. 326. w16x J.A. Nelder, R. Mead, Computer J. 7 Ž1965. 308. w17x A. Luchow, H. Kleindienst, Chem. Phys. Lett. 197 Ž1992. ¨ 105. w18x Handbook of Chemistry and Physics, CRC Press Inc., Boca Raton, 74th ed., 1993r1994, Section 11–35. w19x S. Mannervik, H. Cederquist, Phys. Scr. 27 Ž1983. 175.
Chemical Physics Letters 285 Ž1998. 417–421
Double linked Hylleraas–CI for the lowest 4 P e states of the Li isoelectronic series Rene´ Barrois, Hans–Peter Merckens, Heinz Kleindienst Institut fur D-40225 Dusseldorf, Germany ¨ Physikalische Chemie und Elektrochemie, Heinrich-Heine-UniÕersitat ¨ Dusseldorf, ¨ ¨ Received 20 November 1997; in final form 15 January 1998
Abstract An accurate upper bound calculation for the lowest 4 P e states of the Li isoelectronic series from Li I–Ne VIII is given using an extensive double linked Hylleraas–CI calculation. We obtained, in infinite mass approximation, E s y5.2454098 0 Eh for the 2 4 P e state of the Li atom. Additionally we discuss the transitions 2 4 P o ™ 2 4 P e and 2 4 P e ™ 3 4 S and give a comparison to other calculations and experimental data. q 1998 Elsevier Science B.V.
1. Introduction
with the three particle antisymmetrizer A, a totally symmetric spinfunction x and the spatial functions
Recently the Hylleraas–CI method has successfully been applied to the S, 2 P and 4 P o states of the Li isoelectronic series w1–11x. As far as the 4 P e states of the Li atom are concerned there are on the one hand a 30-term correlated CI by Holøien and Geltman w4x and on the other hand conventional CIs, e.g., by Bunge and Bunge w11x and Hsu et al. w12x. In this Letter we present results for the lowest 4 P e states of the Li isoelectronic series using extensive double linked Hylleraas–CI basis sets.
l i m i n i yŽ a r 1qb r 2qg r 3 . f i s r 1a i r 2b i r 3c i r 23 r 13 r 12 e
2. Method The ansatz for the wavefunction C of the 4 P e states is given as
C s Ý ci A Ž fi x . , i
Ž 1.
= sinq 2 sinq 3 sin Ž w 2 y w 3 . ,
Ž 2.
with a , b ,g ) 0; a i , l i , m i , n i g N 0 and bi ,c i g N. The angular term ensures the even parity of the wavefunction which is an eigenfunction of the angular momentum operators L2 and L z with the eigenvalues LŽ L q 1. s 2 and ML s 0. The corresponding Hamiltonian H is expressed in terms of a nonorthogonal coordinate set Ž ri ,ri j ,qi , w i . w8x. An efficient method to evaluate the required integrals is given in Refs. w8,9x. The eigenvalues of the resulting large general symmetric eigenproblem are evaluated with a modified Wielandt procedure w14x using Falk’s elevator method w15x. The nonlinear parameters were optimized using the Newton-Raphson and the Nelder-Mead simplex method w16x algorithms. Initial values of the nonlinear parameters were obtained by search on a coarse grid with 49 basic functions for the variational space.
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 1 1 7 - 1
R. Barrois et al.r Chemical Physics Letters 285 (1998) 417–421
418
3. Results All calculations were achieved in IEEE 8 Byte precision on a Convex C3840 computer. The integrals are accurate to 12 digits. In order to obtain optimal values for the energy of the lowest 4 P e state of the Li atom the following strategy was used. First we optimized the nonlinear parameters a / b s g for a small basis set Ž N s 115, N: number of basis functions and v s a i q bi q c i q l i q m i q n i F 6. with the Nelder–Mead w16x and the Newton–Raphson methods respectively. The choice b s g induces the restriction bi - c i
k bi s c i n m i F n i .
Ž 3.
We then extended the basis using a selection procedure described in Ref. w17x to obtain a basis set with v F 11 and N s 982. For this basis we reoptimized the nonlinear parameters. We present a convergence pattern with the new parameters in Table 1. As a second step we extended the basis to ‘‘3 = 982’’ functions using nonlinear parameters obtained by optimizing a ‘‘3 = 115’’ basis set. The corresponding convergence pattern is given in Table 2. In order to get a smaller basis set, we chose b / g and eliminated the restriction in Eq. Ž3. by adding the missing functions to the set. The new unrestricted basis had a dimension of 1881 functions. We used nonlinear parameters optimized for a small basis with v F 6 and N s 203. Table 3 shows the convergence pattern. Comparing the last rows of Tables 2 and 3 one notices that no loss of accuracy occurs by using the smaller basis of 1881 functions. Furthermore we checked the overlap of the corresponding wavefunctions C 2946 and C 1881. We found ²C 2946 N C 1881 : s 1 y 8.4 = 10y9 . Due to these facts we used Table 1 Basis selection procedure with a s 3.122 and b sg s1.233, full basis dimension N, selected basis dimension Nsel and energies E in Eh for the 24 P e state of the Li atom
v
N
Nsel
E
6 7 8 9 10 11
115 238 452 798 1333 2133
115 238 404 563 718 982
y5.245 277 4 3 y5.245 384 9 8 y5.245 406 11 y5.245 408 91 y5.245 409 59 y5.245 409 73
Table 2 Energies in Eh for the 24 P e state of Li calculated with three parameter sets Number of functions
a
b sg
E
1– 982 983–1964 1965–2946
2.802 7.755 3.833
1.039 1.347 1.615
y5.245 409 56 y5.245 409 72 y5.245 409 8 0
Table 3 Energies E in Eh for the 24 P e state of Li with a s 2.921, b s1.415 and g s 0.781 using the unrestricted basis
v
N
E
6 7 8 9 10 11
203 434 752 1061 1365 1881
y5.245 406 30 y5.245 408 8 2 y5.245 409 54 y5.245 409 71 y5.245 409 77 y5.245 409 8 0
Table 4 Energies E in Eh for the 24 P e state of some members of the Li isoelectronic series, difference D in m Eh of our values from these of Holøien and Geltman w4x System
E Žthis work.
E w4x
D
Li I Be II B III C IV NV O VI F VII Ne VIII
y5.245 409 8 0 y9.870 897 1 0 y16.000 433 6 y23.631 809 2 y32.764 164 6 y43.397 101 2 y55.530 410 0 y69.163 971 4
y5.245 92 y9.868 80 y15.996 75 y23.627 16 y32.758 71 y43.391 04 y55.523 88 y69.157 50
0.5102 y2.0971 y3.6836 y4.6492 y5.4546 y6.0612 y6.5300 y6.4714
Table 5 Energies E in Eh for the 34 P e state of some members of the Li isoelectronic series, difference D in m Eh of our values from these of Holøien and Geltman w4x System
E Žthis work.
E w4x
D
Li I Be II B III C IV NV O VI F VII Ne VIII
y5.096 831 33 y9.428 868 46 y15.124 487 3 y22.182 342 2 y30.601 898 5 y40.382 911 2 y51.525 253 5 y64.028 853 9
y5.093 86 y9.423 20 y15.119 00 y22.176 54 y30.596 09 y40.376 96 y51.519 24 y64.022 50
y2.9713 y5.6684 y5.4873 y5.8022 y5.8085 y5.9512 y6.0135 y6.3539
R. Barrois et al.r Chemical Physics Letters 285 (1998) 417–421
419
Table 6 4 e : < < ² : ² : Expectation values ² V :, ²T : and ² == i j in E h and the values 1 y h for the 2 P states with h s y Ž V r 2 T . System
²V :
²T :
² == : i j
<1 y h <
Li I Be II B III C IV NV O VI F VII Ne VIII
y10.490 819 4 y19.741 794 0 y32.000 867 0 y47.263 619 4 y65.528 328 8 y86.794 202 8 y111.060 82 0 y138.327 94 3
5.245 409 6 0 9.870 896 9 7 16.000 433 4 23.631 810 1 32.764 164 2 43.397 101 6 55.530 409 9 69.163 971 6
0.350 691 34 0 0.999 652 78 7 1.970 921 8 6 3.258 400 9 3 4.859 982 6 4 6.774 786 1 2 9.002 382 8 3 11.542 544 8
1.93 = 10y8 6.32 = 10y9 5.66 = 10y9 1.88 = 10y8 6.58 = 10y9 4.99 = 10y9 9.27 = 10y10 1.15 = 10y9
the unrestricted basis Ž N s 1881. for the following calculations. The energies of the 2 4 P e and 3 4 P e states of some members of the Li isoelectronic series were calculated similarly using the unrestricted basis with nonlinear parameters which we optimized for N s 203 functions Ž v F 6.. Comparing our results for the isoelectronic series to those of Holøien and Geltman w4x ŽTables 4 and 5., the improvement of all the given energies is evident with the exception of the lowest 4 P e state of Li. There is no reasonable explanation for this discrepancy as Bunge and Bunge have already pointed out in Ref. w11x. Bunge and Bunge found E s y5.245308 Eh as a variational upper bound and the extrapolated value y5.245351 E h . The upper bound of Hsu et al. w12x is E s y5.2453903 Eh and their extrapolated value is E s y5.2454045 Eh . Therefore we believe the quoted result of Holøien and Geltman w4x to be a misprint. Considering the patterns in Tables 2 and 3 we think
that the accuracy of our energy values for the Li atom is in the scale of 0.1 m Eh . For the ions we believe eight digits to be significant. Using the extended basis Ž N s 1881, v F 6. and the corresponding eigenvectors we calculated some expectation values for the 2 4 P e and 3 4 P e states of the Li isoelectronic series. The results are summarized in Tables 6 and 7, respectively. We calculated the specific and normal mass shifts D ESM S and : D E NM S from the given expectation values ² == i j :s ²C <Ý i - j== : C and the energies in infinite mass i j approximation `E using the well-known equations D ESM S s y
D E NM S s y
me M q me me M q me
` ²`C < Ý == : i jC
and
i-j `
E,
Ž 4.
where M is the nuclear mass and m e the mass of the
Table 7 4 e : < < ² : ² : Expectation values ² V :, ²T : and ² == i j in E h and the values 1 y h for the 3 P states with h s y Ž V r 2 T . System
²V :
²T :
² == : i j
<1 y h <
Li I Be II B III C IV NV O VI F VII Ne VIII
y10.193 661 1 y18.857 735 7 y30.248 973 6 y44.364 683 5 y61.203 796 0 y80.765 821 2 y103.050 50 6 y128.057 70 6
5.096 829 8 5 9.428 867 2 7 15.124 486 2 22.182 341 2 30.601 897 4 40.382 910 0 51.525 252 7 64.028 852 7
0.312 374 172 0.761 409 40 4 1.406 623 6 0 2.244 988 2 4 3.275 436 1 9 4.497 548 78 5.911 134 4 8 7.516 097 0 2
1.45 = 10y7 6.24 = 10y8 3.78 = 10y8 2.29 = 10y8 1.89 = 10y8 1.43 = 10y8 7.55 = 10y9 1.02 = 10y8
R. Barrois et al.r Chemical Physics Letters 285 (1998) 417–421
420
Table 8 Energy values in Eh for some isotopes of the Li isoelectronic series
electron. The finite mass energies M E to first order were obtained by E s`E q D ESM S q D E NMS .
System
24 P e
34 P e
M
6
y5.244 963 32 y5.245 027 0 2 y9.870 357 0 0 y15.999 664 7 y15.999 734 3 y23.630 877 6 y23.630 949 5 y32.763 071 2 y32.763 143 9 y43.395 844 8 y43.395 919 1 y43.395 984 7 y55.529 066 2 y69.162 390 0 y69.162 465 4 y69.162 533 7
y5.096 394 9 0 y5.096 457 1 7 y9.428 340 77 y15.123 735 6 y15.123 803 7 y22.181 430 6 y22.181 501 0 y30.600 827 7 y30.600 898 9 y40.381 680 1 y40.381 752 9 y40.381 817 2 y51.523 936 1 y64.027 302 9 y64.027 376 9 y64.027 443 9
Table 8 shows the results for M E of some selected isotopes, the values for the nuclear masses are taken from Ref. w18x. In Table 9 we present some calculations for the 4 4 P e state of Li. The accuracy of the energies is better than 2 m Eh . Finally we discuss the transitions 2 4 P o ™ 2 4 P e and 2 4 P e ™ 3 4 S . In Table 10 a comparison of the nonrelativistic energies, including mass polarization of the required states, between Hsu’s et al. w12,13x and ours is given. Table 11 contains the nonrelativistic Žmass polarization included. and the total transition energies. Similar to our previous work w7x we used the relativistic corrections given by Hsu et al. w12,13x. The differences between the experimental data and the theoretical values are on the one hand due to QED corrections, which are estimated by Hsu et al. w12x to be of the order of 0.6 cmy1 and on the other hand to uncertainty of the experiment.
Li I Li I 9 Be II 10 B III 11 B III 12 C IV 13 C IV 14 NV 15 NV 16 O VI 17 O VI 18 O VI 19 F VII 20 Ne VIII 21 Ne VIII 22 Ne VIII 7
Table 9 Summarized results for the 44 P e state of Li ` E ²T : ²V : ² == : i j N1yh N EŽ6 Li. EŽ7 Li.
y5.0640891 Eh 5.064076 6 E h y10.128166 E h 0.30795272 Eh 1.29=10y6 y5.0636553 Eh y5.0637172 E h
Acknowledgements
Table 10 Upper bounds for the nonrelativistic energies including mass polarization of the 24 P o , 24 P e and the 34 S states of 7 Li in E h State
Enr 4
o
1s2s2p P 1s2p2p 4 P e 1s2s3s 4 S a b
a
y5.367 605 8 y5.245 027 0 y5.212 339 1
Enr
We thank the Deutsche Forschungsgemeinschaft for financial support.
b
y5.367 594 8 y5.245 007 7 y5.212 329 1
Refs. w5–7x & this work. Refs. w12,13x.
References w1x A. Luchow, H. Kleindienst, Int. J. Quantum Chem. 51 Ž1994. ¨ 211. w2x F.W. King, Phys. Rev. A 40 Ž1989. 1735.
Table 11 Comparison of experimental transition energies D EŽ24 P o ™ 24 P e . and D EŽ24 P e ™ 34 S. in cmy1 Transition
This work
D Enr Ž24 P o ™ 24 P e . D Etot.Ž24 P o ™ 24 P e . D Enr Ž24 P e ™ 34 S. D Etot.Ž24 P e ™ 34 S.
26902.9 26916.1 7174.2 7155.2
a b c
Ž 5.
c
Ref. w12,13x & this work. Ref. w19x. Including relativistic correction of Hsu et al. w12,13x.
Theor. a 26904.7 26916.3 7172.1 7156.2
Exp. b 26915.16 7156.8
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421
w12x J.J. Hsu, K.T. Chung, K.N. Huang, Phys. Rev. A 49 Ž1994. 4466. w13x J.J. Hsu, K.T. Chung, K.N. Huang, Phys. Rev. A 44 Ž1991. 5484. w14x A. Luchow, H. Kleindienst, Computers and Chem. 17 Ž1993. ¨ 61. w15x R. Zurmuhl, ¨ S. Falk, Matrizen und ihre Anwendung, vol. 2, Springer Verlag, Berlin, 1986, p. 326. w16x J.A. Nelder, R. Mead, Computer J. 7 Ž1965. 308. w17x A. Luchow, H. Kleindienst, Chem. Phys. Lett. 197 Ž1992. ¨ 105. w18x Handbook of Chemistry and Physics, CRC Press Inc., Boca Raton, 74th ed., 1993r1994, Section 11–35. w19x S. Mannervik, H. Cederquist, Phys. Scr. 27 Ž1983. 175.