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Nakatsuji H, Nakashima H. Exact Theory Applied to the Lithium Atom. J Chem Theory Comput 2024. [PMID: 39225699 DOI: 10.1021/acs.jctc.4c00884] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 09/04/2024]
Abstract
The free complement (FC) theory for solving the scaled Schrödinger equation (SSE) was applied to the Li atom for calculating the exact wave functions, the energies, and the various properties of the ground doublet S and excited P states. The SSE is equivalent to the Schrödinger equation (SE) but does not have the divergence difficulty of the variational equation of the SE. Because the Li atom is a three-electron system, the variational exact FC calculations for solving the SSE are possible using the function g = 1 - exp(-γr) as the "correct" scaling function of the SSE. The "reasonable" scaling function g = r was also used as comparative calculations. We performed variational calculations to the order eight of the FC theory and could obtain essentially exact solutions of the SSE or SE with the "correct" g function of the FC theory. We report here the values of the exact energy, spin density, electron density, and electron-nuclear and electron-electron cusp values of the doublet S and P states. They agreed very well with the experimental values and the best theoretical values presented by Drake and collaborators. This is a simple example that the exact theory gives the exact solutions.
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Affiliation(s)
- Hiroshi Nakatsuji
- Quantum Chemistry Research Institute, Kyoto Technoscience Center 16, 14 Yoshida Kawaramachi, Sakyo-ku, Kyoto 606-8305, Japan
| | - Hiroyuki Nakashima
- Quantum Chemistry Research Institute, Kyoto Technoscience Center 16, 14 Yoshida Kawaramachi, Sakyo-ku, Kyoto 606-8305, Japan
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2
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Šmydke J. Using Koopmans' theorem for constructing basis sets: approaching high Rydberg excited states of lithium with a compact Gaussian basis. Phys Chem Chem Phys 2023. [PMID: 37486232 DOI: 10.1039/d2cp04633d] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 07/25/2023]
Abstract
For accurate ab initio description of Rydberg excited states, this study suggests generating appropriate diffuse basis functions by cheap variational optimization of virtual orbitals of the corresponding ion core. By following this approach, dozens of converged correlated lithium Rydberg states, namely, all the states up to 24 2S, 25 2P, 14 2D, 16 2F and 16 2G, not yet achieved via other ab initio approaches, could be obtained at the EOM-CCSD level of theory with compact and mostly state-selective contracted Gaussian basis sets. Despite its small size and Gaussian character, the optimized basis leads to highly accurate excitation energies that differ merely in the order of meV from the reference state-of-the-art explicitly correlated Gaussian method and even surpass Full-CI results on the Slater basis by an order of magnitude.
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Affiliation(s)
- Jan Šmydke
- Department of Radiation and Chemical Physics, FZU - Institute of Physics of the Czech Academy of Sciences, Na Slovance 1999/2, 18200 Praha 8, Czech Republic.
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Sims JS. Hylleraas-Configuration Interaction (Hy-CI) Non-Relativistic Energies for the 3 1 S , 4 1 S , 5 1 S , 6 1 S , and 7 1 S Excited States of the Beryllium Atom. JOURNAL OF RESEARCH OF THE NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY 2020; 125:125006. [PMID: 39309134 PMCID: PMC11302959 DOI: 10.6028/jres.125.006] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Accepted: 01/06/2020] [Indexed: 09/25/2024]
Abstract
In a previous work Sims and Hagstrom [J Chem Phys 140,224312(2014)] reported Hylleraas-configuration interaction (Hy-CI) method variational calculations for the 1 S ground states of the beryllium isoelectronic sequence with an estimated accuracy of 10 to 20 nanohartrees (nHa). In this work the calculations have been extended to the five higher states of the neutral beryllium atom, 3 1 S, 4 1 S, 5 1 S, 6 1 S, and 7 1 S. The best non-relativistic energies obtained for these states are -14.4182 4034 6, -14.3700 8789 0, -14.3515 1167 6,-14.3424 0357 8, and -14.3372 6649 96 Ha, respectively. The 6 1 S result is superior to the known reference energy for that state, while for the 7 1 S state there is no other comparable calculation.
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Affiliation(s)
- James S. Sims
- National Institute of Standards and Technology,
Gaithersburg, MD 20899, USA
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4
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Sims JS. Hylleraas-Configuration Interaction study of the 1S ground state of the negative Li ion. JOURNAL OF PHYSICS. B, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS : AN INSTITUTE OF PHYSICS JOURNAL 2017; 50:245003. [PMID: 29379225 PMCID: PMC5785787 DOI: 10.1088/1361-6455/aa961e] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
In a previous work Sims and Hagstrom [J. Chem. Phys. 140, 224312 (2014)] reported Hylleraas-Configuration Interaction (Hy-CI) method variational calculations for the neutral atom and positive ion 1S ground states of the beryllium isoelectronic sequence. The Li- ion, nominally the first member of this series, has a decidedly different electronic structure. This paper reports the results of a large, comparable calculation for the Li- ground state to explore how well the Hy-CI method can represent the more diffuse L shell of Li- which is representative of the Be(2sns) excited states as well. The best non-relativistic energy obtained was -7.500 776 596 hartree, indicating that 10 - 20 nh accuracy is attainable in Hy-CI and that convergence of the r12r34 double cusp is fast and that this correlation type can be accurately represented within the Hy-CI model.
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Affiliation(s)
- James S Sims
- National Institute of Standards and Technology, Gaithersburg, Maryland 20878-9957, USA
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5
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A benchmark study of Li 2 + , Li 2 − , LiH + and LiH − : Quantum Monte-Carlo and coupled-cluster computations. COMPUT THEOR CHEM 2017. [DOI: 10.1016/j.comptc.2017.05.015] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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6
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Sharkey KL, Adamowicz L. Lower Rydberg 2Fstates of the lithium atom: finite-nuclear-mass calculations with explicitly correlated Gaussian functions. Mol Phys 2014. [DOI: 10.1080/00268976.2013.863404] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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7
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Bubin S, Adamowicz L. Prediction of (1)P Rydberg energy levels of beryllium based on calculations with explicitly correlated Gaussians. J Chem Phys 2014; 140:024301. [PMID: 24437871 DOI: 10.1063/1.4858275] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Benchmark variational calculations are performed for the seven lowest 1s(2)2s np ((1)P), n = 2...8, states of the beryllium atom. The calculations explicitly include the effect of finite mass of (9)Be nucleus and account perturbatively for the mass-velocity, Darwin, and spin-spin relativistic corrections. The wave functions of the states are expanded in terms of all-electron explicitly correlated Gaussian functions. Basis sets of up to 12,500 optimized Gaussians are used. The maximum discrepancy between the calculated nonrelativistic and experimental energies of 1s(2)2s np ((1)P) →1s(2)2s(2) ((1)S) transition is about 12 cm(-1). The inclusion of the relativistic corrections reduces the discrepancy to bellow 0.8 cm(-1).
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Affiliation(s)
- Sergiy Bubin
- Department of Chemistry, University of Rochester, Rochester, New York 14627, USA
| | - Ludwik Adamowicz
- Department of Chemistry and Biochemistry and Department of Physics, University of Arizona, Tucson, Arizona 85721, USA
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Komasa J, Słupski R, Jankowski K, Wasilewski J, Teale AM. High accuracy ab initio studies of electron-densities for the ground state of Be-like atomic systems. J Chem Phys 2013; 138:164306. [PMID: 23635137 DOI: 10.1063/1.4800766] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023] Open
Abstract
Benchmark results for electron densities in the ground states of Li(-), Be, C(2+), Ne(6+), and Ar(14+) have been generated from very accurate variational wave functions represented in terms of extensive basis sets of exponentially correlated Gaussian functions. For Ne(6+), and Ar(14+), the upper bounds to the energies improve over previous results known from the literature. For the remaining systems our bounds are from 0.1 to 1.1 μhartree higher than the most accurate ones. We present in graphical and, partially, numerical form results both for the radial electron densities and for the difference radial density distributions (DRD) (defined with respect to the Hartree-Fock radial density) that highlight the impact of correlation effects on electron densities. Next, we have employed these DRD distributions in studies of the performance of several broadly used orbital-based quantum-chemical methods in accounting for correlation effects on the density. Our computed benchmark densities for Be have been also applied for testing the possibility of using the mathematically strict result concerning exact atomic electron densities, obtained by Ahlrichs et al. [Phys. Rev. A 23, 2106 (1981)], for the determination of the reliability range of computed densities in the long-range asymptotic region. The results obtained for Be are encouraging.
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Affiliation(s)
- J Komasa
- Faculty of Chemistry, A. Mickiewicz University, Poznań, Poland
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Bubin S, Sharkey KL, Adamowicz L. Prediction of 2DRydberg energy levels of 6Li and 7Li based on very accurate quantum mechanical calculations performed with explicitly correlated Gaussian functions. J Chem Phys 2013; 138:164308. [DOI: 10.1063/1.4801855] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/31/2023] Open
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Sharkey KL, Kirnosov N, Adamowicz L. An algorithm for quantum mechanical finite-nuclear-mass variational calculations of atoms with L = 3 using all-electron explicitly correlated Gaussian basis functions. J Chem Phys 2013; 138:104107. [PMID: 23514465 DOI: 10.1063/1.4794192] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023] Open
Abstract
A new algorithm for quantum-mechanical nonrelativistic calculation of the Hamiltonian matrix elements with all-electron explicitly correlated Gaussian functions for atoms with an arbitrary number of s electrons and with three p electrons, or one p electron and one d electron, or one f electron is developed and implemented. In particular the implementation concerns atomic states with L = 3 and M = 0. The Hamiltonian used in the approach is obtained by rigorously separating the center-of-mass motion from the laboratory-frame all particle Hamiltonian, and thus it explicitly depends on the finite mass of the nucleus. The approach is employed to perform test calculations on the lowest (2)F state of the two main isotopes of the lithium atom, (7)Li and (6)Li.
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Affiliation(s)
- Keeper L Sharkey
- Department of Chemistry, University of Arizona, Tucson, Arizona 85721, USA
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Bubin S, Pavanello M, Tung WC, Sharkey KL, Adamowicz L. Born–Oppenheimer and Non-Born–Oppenheimer, Atomic and Molecular Calculations with Explicitly Correlated Gaussians. Chem Rev 2012; 113:36-79. [DOI: 10.1021/cr200419d] [Citation(s) in RCA: 115] [Impact Index Per Article: 9.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Sergiy Bubin
- Department
of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235,
United States
| | - Michele Pavanello
- Department
of Chemistry, Rutgers University Newark, Newark, New Jersey 07102,
United States
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12
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Sharkey KL, Bubin S, Adamowicz L. Explicitly correlated Gaussian calculations of the 2DRydberg states of the boron atom. J Chem Phys 2012; 137:064313. [DOI: 10.1063/1.4742819] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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13
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Bubin S, Adamowicz L. Explicitly correlated Gaussian calculations of the 2P(o) Rydberg spectrum of the lithium atom. J Chem Phys 2012; 136:134305. [PMID: 22482550 DOI: 10.1063/1.3698584] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/27/2023] Open
Abstract
Accurate quantum-mechanical nonrelativistic variational calculations are performed for the nine lowest members of the (2)P(o) Rydberg series (1s(2)np(1), n = 2, ..., 10) of the lithium atom. The effect of the finite nuclear mass is included in the calculations allowing for determining the isotopic shifts of the energy levels. The wave functions of the states are expanded in terms of all-electron explicitly correlated Gaussian functions. The exponential parameters of the Gaussians are variationally optimized with the aid of the analytical energy gradient determined with respect to those parameters. The calculated state energies are compared with the available experimental data.
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Affiliation(s)
- Sergiy Bubin
- Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA
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14
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Hättig C, Klopper W, Köhn A, Tew DP. Explicitly Correlated Electrons in Molecules. Chem Rev 2011; 112:4-74. [DOI: 10.1021/cr200168z] [Citation(s) in RCA: 401] [Impact Index Per Article: 30.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Christof Hättig
- Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, D-44780 Bochum, Germany
| | - Wim Klopper
- Abteilung für Theoretische Chemie, Institut für Physikalische Chemie, Karlsruher Institut für Technologie, KIT-Campus Süd, Postfach 6980, D-76049 Karlsruhe, Germany
| | - Andreas Köhn
- Institut für Physikalische Chemie, Johannes Gutenberg-Universität Mainz, D-55099 Mainz, Germany
| | - David P. Tew
- School of Chemistry, University of Bristol, Bristol BS8 1TS, United Kingdom
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Sharkey KL, Bubin S, Adamowicz L. Refinement of the experimental energy levels of higher2DRydberg states of the lithium atom with very accurate quantum mechanical calculations. J Chem Phys 2011; 134:194114. [DOI: 10.1063/1.3591836] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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16
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King FW, Quicker D, Langer J. Compact wave functions for the beryllium isoelectronic series, Li− to Ne6+: A standard Hylleraas approach. J Chem Phys 2011; 134:124114. [PMID: 21456652 DOI: 10.1063/1.3569565] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/31/2023] Open
Affiliation(s)
- Frederick W King
- Department of Chemistry, University of Wisconsin-Eau Claire, Eau Claire, Wisconsin 54702, USA.
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Sharkey KL, Adamowicz L. Exponentially and pre-exponentially correlated Gaussians for atomic quantum calculations. J Chem Phys 2011; 134:094104. [DOI: 10.1063/1.3553177] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023] Open
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18
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Sharkey KL, Bubin S, Adamowicz L. An algorithm for calculating atomic D states with explicitly correlated Gaussian functions. J Chem Phys 2011; 134:044120. [DOI: 10.1063/1.3523348] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023] Open
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Sharkey KL, Bubin S, Adamowicz L. Analytical energy gradient in variational calculations of the two lowest P3 states of the carbon atom with explicitly correlated Gaussian basis functions. J Chem Phys 2010. [DOI: 10.1063/1.3419931] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023] Open
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20
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Bubin S, Komasa J, Stanke M, Adamowicz L. Isotope shifts of the three lowest S1 states of the B+ ion calculated with a finite-nuclear-mass approach and with relativistic and quantum electrodynamics corrections. J Chem Phys 2010; 132:114109. [DOI: 10.1063/1.3358999] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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