Asymptotic-expansion method for the evaluation of correlated three-electron integrals

Drake, G. W. F.; Zhou, Yan

doi:10.1103/physreva.52.3681

Cited by 45 publications

(45 citation statements)

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“…It is interesting to compare this with the relativistic all orders many-body perturbation theory result of Blundell [13] 0.333 3 Blundell et al [14] 0.341 5(11) Experiment Brog et al [3] 0.335 340 (7) Orth et al [4] 0.335 338 1(19) Scherf et al [5] 0.335 345 (7) et al [14]. There, the difference between theory and experiment of 0.0062͑11͒ cm 21 comes mainly from the incomplete treatment of correlation corrections to the lowest order Breit interaction, rather than from QED effects not included in their calculation.…”

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confidence: 84%

“…On the theoretical side, there has been a long-standing problem in precise calculations for the lithium 1s 2 2p 2 P J fine structure, including the lowest-order relativistic terms a 4 mc 2 (or a 2 in atomic units) and ͑m͞M͒ a 4 mc 2 , where m is the electron reduced mass and M is the nuclear mass, and the lowestorder QED terms a 5 mc 2 and ͑m͞M͒ a 5 mc 2 . However, recent advances [6][7][8][9][10] in high-precision variational calculations for the lithium atom, using multiple basis sets in Hylleraas coordinates, now make it possible to perform a well-converged calculation for the 1s 2 2p 2 P J fine structure splitting. The purpose of this Letter is to report the first significant theoretical progress in fully correlated calculations for lithium fine structure studies.…”

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confidence: 99%

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Lithium Fine Structure in the1s22p2P
Yan
¹
,
Drake
²

1997
Phys. Rev. Lett.
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The first fully correlated calculations of fine structure in lithium are presented. For the 1s 2 2p 2 states the fine structure is calculated to a computational accuracy of 1 part in 10 6 , including relativistic and QED terms up to O͑a 4 mc 2 ͒, O͑ ͑ ͑͑m͞M͒a 4 mc 2 ͒ ͒ ͒, O͑a 5 mc 2 ͒, and O͑ ͑ ͑͑m͞M͒a 5 mc 2 ͒ ͒ ͒. A comparison is made with other calculations and experiments. A residual discrepancy between theory and experiment of ͑65 6 2͒ 3 10 26 cm 21 is found due to higher-order QED terms. [S0031-9007(97)03922-7] PACS numbers: 31.15. Pf, 31.30.Jv, 32.10.Fn Atomic fine structure is a fundamental testing ground for relativistic and QED effects. For helium, theoretical calculations for the fine structure splittings in 1s 2p 3 P J states have recently been performed up to O͑a 7 ln a͒ mc 2 [1,2]. With future progress in both theory and experiment, comparisons with high precision measurements will potentially yield an atomic physics determination of the fine structure constant a. For lithium, using the level crossing technique, Brog et al.[3] measured the fine structure splitting in 1s 2 2p 2 P J states with an accuracy of 20 ppm (parts per million). The most precise measurement reported in the literature for this splitting was done by Orth et al. [4], using the optical double resonance method. The precision they achieved is 6 ppm. Very recently, Scherf et al.[5] remeasured the 1s 2 2p 2 P J fine structure using the laser-atomic beam spectroscopy method. The precision they obtained is 20 ppm. All these measurements are in excellent agreement. On the theoretical side, there has been a long-standing problem in precise calculations for the lithium 1s 2 2p 2 P J fine structure, including the lowest-order relativistic terms a 4 mc 2 (or a 2 in atomic units) and ͑m͞M͒ a 4 mc 2 , where m is the electron reduced mass and M is the nuclear mass, and the lowestorder QED terms a 5 mc 2 and ͑m͞M͒ a 5 mc 2 . However, recent advances [6-10] in high-precision variational calculations for the lithium atom, using multiple basis sets in Hylleraas coordinates, now make it possible to perform a well-converged calculation for the 1s 2 2p 2 P J fine structure splitting. The purpose of this Letter is to report the first significant theoretical progress in fully correlated calculations for lithium fine structure studies. The key advance is the development of techniques for handling the more highly singular integrals required to calculate matrix elements of the Breit interaction in Hylleraas coordinates.The spin-dependent fine structure splitting can be written in the form [11,12] (in atomic units throughout)where C is a nonrelativistic wave function and H fs is defined by

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confidence: 84%

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confidence: 99%

Lithium Fine Structure in the1s22p2P
Yan
¹
,
Drake
²

1997
Phys. Rev. Lett.
Self Cite
24
3
15
0
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“…(4) For the numerical evaluation of these integrals, we used an efficient algorithm recently developed by Drake and Yan [16]. All calculations were performed in quadruple precision on an IBM RISC͞6000 580.…”

Section: (Received 4 March 1998)mentioning

confidence: 99%

Electronic Structure Critical Parameters For the Lithium Isoelectronic Series

1998

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The finite-size scaling method is used to calculate the critical parameters for the lithium isoelectronic series. The critical nuclear charge, which is the minimum charge necessary to bind three electrons, for the ground state was found to be Z c Ӎ 2. Results show that the analytical behavior of the energy as a function of the nuclear charge for lithium is completely different from that of helium. Analogy with standard phase transitions show that for helium, the transition from a bound state to a continuum is "first order," while lithium exhibits a "second order phase transition." [S0031-9007(98)06405-9] PACS numbers: 05.70.Jk The study of the critical parameters of a quantum Hamiltonian and quantum phase transitions have attracted much interest in recent years [1]. These transitions take place at the absolute zero of temperature, where phase transition means that the quantum ground state of the system changes in some fundamental way as some microscopic parameters change in the Hamiltonian. In atomic and molecular physics, it has been suggested that there are possible analogies between critical phenomena and singularities of the energy [2][3][4]. Analogies between symmetry breaking of electronic structure configurations and standard phase transitions have been established for many electron atoms and simple molecular systems by studying the corresponding large dimension limit Hamiltonian [5,6].Recently, we presented the finite-size scaling (FSS) method to calculate the critical parameters for electronic structure [7,8]. In statistical mechanics, the FSS method provides a way to extrapolate information obtained from a finite system to the thermodynamic limit [9,10]. In our applications, the finite size corresponds to the number of elements in a complete basis set used to expand the exact eigenfunction of a given Hamiltonian. In the FSS method we assumed that the two lowest eigenvalues of the quantum Hamiltonian could be taken as the leading eigenvalues of a transfer matrix of a classical pseudosystem. The phenomenological renormalization equation was used to obtain the critical properties of the system. In this approach one has to rely on the analogy to classical statistical mechanics, a more direct finite-size scaling approach without the need to make such an analogy was established by a systematic expansion in a finite (truncated) basis set [11].The analytical behavior of the energy as a function of parameters for a given system has been the subject of study for many years. In particular, the study of the analytical behavior of the energy as a function of the nuclear charge, Z. Morgan and co-workers [12] have performed a 401-order perturbation calculation to resolve the controversy over the radius of convergence of the l 1͞Z expansion for the ground state energy of the heliumlike ions. Such high order calculations were necessary to study the asymptotic behavior of the perturbation series and to determine that the radius of convergence, l ‫ء‬ , is equal to l c , the critical value of l for which the Hamiltonia...

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“…We have recently discovered an asymptotic expansion method to deal with these slowly convergent integrals [10]. This method has proven to be very successful in accelerating the rate of convergence and thus it removes a major obstacle to further progress.…”

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confidence: 99%

Eigenvalues and expectation values for the 1s22s2S, 1
Zhou
¹
,
Drake
²

1995
Phys. Rev. A
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High-precisipn varlatipnal eigenvalues fpr the 1& 2s S, 1+ 2p P, and 1z 3d D states pf hthium are calculated using multiple basis sets in Hylleraas coordinates. Convergence to a few parts in 10' -10" is achieved. The nonrelativistic energies for infinite nuclear mass are -7.478060323 10(31) a.u. for the ls 2s S state, -7.410156 521 8(13) a.u. for the ls 2p P state, and -7.335 523 541 10(43) a.u. for the 1s 3d D state. The corresponding specific isotope shifts due to mass polarization are also calculated with similar accuracy. The 1s 2s S -1s 2p P and 1s 2p P -1s 3d D transition energies for Li and Li, as well as the isotope shifts, are calculated and compared with experiment. The results yield an improved ionization potential for lithium of 43487.167(4) cm '. Expectation values of powers of r; and r; and the delta functions 8(r;) and 6(r;,) are evaluated.PACS number(s): 31.15.Ar, 31.50.+w

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Asymptotic-expansion method for the evaluation of correlated three-electron integrals

Abstract: (1995). Asymptotic-expansion method for the evaluation of correlated three-electron integrals.

Cited by 45 publications

References 21 publications

Lithium Fine Structure in the1s22p2P
Yan
¹
,
Drake
²

1997
Phys. Rev. Lett.
Self Cite
24
3
15
0
View full text Add to dashboard Cite

Lithium Fine Structure in the1s22p2P
Yan
¹
,
Drake
²

1997
Phys. Rev. Lett.
Self Cite
24
3
15
0
View full text Add to dashboard Cite

Electronic Structure Critical Parameters For the Lithium Isoelectronic Series

Eigenvalues and expectation values for the 1s22s2S, 1
Zhou
¹
,
Drake
²

1995
Phys. Rev. A
Self Cite
162
9
139
1
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Asymptotic-expansion method for the evaluation of correlated three-electron integrals

Abstract: (1995). Asymptotic-expansion method for the evaluation of correlated three-electron integrals.

Cited by 45 publications

References 21 publications

Lithium Fine Structure in the1s22p2PYan1, Drake2 1997Phys. Rev. Lett.Self Cite243150View full textAdd to dashboardCite

Lithium Fine Structure in the1s22p2PYan1, Drake2 1997Phys. Rev. Lett.Self Cite243150View full textAdd to dashboardCite

Electronic Structure Critical Parameters For the Lithium Isoelectronic Series

Eigenvalues and expectation values for the 1s22s2S, 1Zhou1, Drake2 1995Phys. Rev. ASelf Cite16291391View full textAdd to dashboardCite

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Lithium Fine Structure in the1s22p2P
Yan
¹
,
Drake
²

1997
Phys. Rev. Lett.
Self Cite
24
3
15
0
View full text Add to dashboard Cite

Lithium Fine Structure in the1s22p2P
Yan
¹
,
Drake
²

1997
Phys. Rev. Lett.
Self Cite
24
3
15
0
View full text Add to dashboard Cite

Eigenvalues and expectation values for the 1s22s2S, 1
Zhou
¹
,
Drake
²

1995
Phys. Rev. A
Self Cite
162
9
139
1
View full text Add to dashboard Cite