Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Accuracy limits on the description of the lowest S excitation in the Li atom using explicitly correlated Gaussian basis functions

Monika Stanke, Jacek Komasa, Dariusz Kędziera, Sergiy Bubin, Ludwik Adamowicz

Published Nov 14, 2008• 28 citations
Publisher

Abstract

We have performed very accurate quantum-mechanical calculations for the two lowest $S$ states of the lithium atom in order to determine the transition energy. In the nonrelativistic part of the calculations performed using the variational method, we explicitly included the nuclear motion. The nonrelativistic wave function was expanded in terms of explicitly correlated Gaussian functions. Next, this wave function was used to calculate the leading ${\ensuremath{\alpha}}^{2}$ relativistic correction ($\ensuremath{\alpha}$ is the fine-structure constant) and the ${\ensuremath{\alpha}}^{3}$ QED correction. We also estimated the ${\ensuremath{\alpha}}^{4}$ QED correction by calculating its dominating component. The results obtained with Gaussians are compared with the most accurate results obtained recently with the Hylleraas-type basis functions.

DOI
10.1103/physreva.78.052507

Related papers

Rydberg levels of lithium

Richard J. Drachman, A. K. Bhatia

The electrostatic fine structure for singly excited lithium is accurately calculated using an effective potential method that has proven to be very useful for helium. Relevant properties of the two-electron ${\mathrm{Li}}^{+}$ ionic core with finite nuclear mass are determined from variational wave functions. From these, non- relativistic level shifts are obtained analytically, and the leading relativistic correction is applied. A comparison with recent experimental results is made.

Physical Review A•Apr 1, 1995•45 citations

Relativistic corrections to the ground-state energy of the positronium molecule

Sergiy Bubin, Monika Stanke, Dariusz Kędziera +1 more

The leading-order relativistic corrections to the ground-state energy of the positronium molecule $({\mathrm{Ps}}_{2})$ have been computed within the framework of perturbation theory. As the zero-order wave function we used a highly accurate nonrelativistic variational expansion in terms of 6000 explicitly correlated Gaussians that yielded the lowest variational upper bound for this system to date. We also report some expectation values representing the properties of ${\mathrm{Ps}}_{2}$.

Physical Review A•Jun 11, 2007•28 citations

Fine and hyperfine splitting of the low-lying states of Be9

Mariusz Puchalski, Jacek Komasa, Krzysztof Pachucki

We perform accurate calculations of energy levels as well as fine and hyperfine splittings of the lowest $^{1,3}P_{J}, ^{3}S_{1}, ^{3}P_{J}^{e}$, and $^{1,3}D_{J}$ excited states of the $^{9}\mathrm{Be}$ atom using explicitly correlated Gaussian functions, and we report on the breakdown of the standard hyperfine structure theory. Because of the strong hyperfine mixing, which prevents the use of common hyperfine constants, we formulate a description of the fine and hyperfine structure that is valid for an arbitrary coupling strength and may have wide applications in many other atomic systems.

arXiv:2106.16161•Physical Review A•Aug 31, 2021•12 citations

Ab initioproperties of the NaLi molecule in the a3Σ+ electronic state

Marcin Gronowski, Adam M. Koza, Michał Tomza

Ultracold polar and magnetic $^{23}\mathrm{Na}^{6}\mathrm{Li}$ molecules in the rovibrational ground state of the lowest triplet ${a}^{3}{\mathrm{\ensuremath{\Sigma}}}^{+}$ electronic state have been recently produced. Here, we calculate the electronic and rovibrational structure of these 14-electron molecules with spectroscopic accuracy ($<0.5\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$) using state-of-the-art ab initio methods of quantum chemistry. We employ the hierarchy of the coupled-cluster wave functions and Gaussian basis sets extrapolated to the complete basis set limit. We show that the inclusion of higher-level excitations, core-electron correlation, relativistic, QED, and adiabatic corrections is necessary to accurately reproduce scattering and spectroscopic properties of alkali-metal systems. We obtain the well depth, ${D}_{e}=229.9(5)\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$, the dissociation energy, ${D}_{0}=208.2(5)\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$, and the scattering length, ${a}_{s}=\ensuremath{-}{84}_{\ensuremath{-}41}^{+25}\phantom{\rule{4pt}{0ex}}\mathrm{bohr}$, in good agreement with recent experimental measurements. We predict the permanent electric dipole moment in the rovibrational ground state, ${d}_{0}=0.167$(1) D. These values are obtained without any adjustment to experimental data, showing that quantum chemistry methods are capable of predicting scattering properties of many-electron systems, provided relatively weak interaction and small reduced mass of the system.

arXiv:2003.12051•Physical Review A•Aug 10, 2020•28 citations

Computing accurate forces in quantum Monte Carlo using Pulay’s corrections and energy minimization

Mosè Casalegno, Massimo Mella, Andrew M. Rappe

In order to overcome the difficulty of optimizing molecular geometry using quantum Monte Carlo methods, we introduce various approximations to the exact force expectation value. We follow Pulay’s suggestion [Mol. Phys. 17, 153 (1969)] to correct the Hellmann–Feynman estimator by introducing the contributions due to the changes in the wave function with respect to the nuclear positions. When used in conjunction with energy-optimized explicitly correlated trial wave functions for H2 and LiH, these approximations appear to yield accurate forces using both the variational and diffusion Monte Carlo methods. Also, the accuracy of the second-order estimate of the Hellmann–Feynman force estimator was investigated employing our energy-optimized trial wave functions, and an erratic behavior was uncovered for some of the studied bond lengths. The additional computational cost required to compute the corrections to the Hellmann–Feynman estimator was found to be only a small fraction of the cost for a simple mean energy calculation. The same approach could be exploited also in computing the derivative of other energy-dependent quantum-mechanical observables.

The Journal of Chemical Physics•Apr 11, 2003•51 citations

Ground-State Energy Solutions of the Lithium Atom: Zeroth-, First-, and Second-Order Perturbation Theory and the Variational Method

Afraa Mahboubi, Büşra Gökçe Zolmaz, Devrim Karayer +3 more

In this work, the ground-state energy of the lithium atom is systematically investigated using both time-independent perturbation theory and the variational method to provide a comprehensive pedagogical analysis of many-body atomic systems. The unperturbed Hamiltonian is initially constructed by neglecting electron-electron interactions, treating the system as three independent hydrogen-like electrons to yield a zeroth-order energy baseline of -275.51 eV. The antisymmetric fermionic nature of the exact wave function is rigorously enforced through the Slater determinant formalism. First-order perturbation theory is applied to evaluate static inter-electronic repulsion using exact Coulomb and exchange integrals, refining the energy state to -192.01 eV. To account for dynamical electronic correlation, second-order perturbation theory is computed numerically for virtual single-electron s-orbital transitions, leading to a total perturbative energy of -196.36 eV. A brief discussion of two-electron excitations is also included to encapsulate further physical realism within the framework. Furthermore, a non-orthogonal two-parameter variational approach is employed to model the shell-specific shielding effect. By optimizing the effective nuclear charges, the variational method establishes a superior upper bound energy of -201.187 eV. The results of both methods are comprehensively contrasted against each other and the reference baseline to provide critical insights into the nature of electron correlation and screening in multi-electron atoms.

quant-pharXiv:2606.24238•Jun 24, 2026
We have performed very accurate quantum-mechanical calculations for the two lowest SSS states of the lithium atom in order to determine the transition energy. In the nonrelativistic part of the calculations performed using the variational method, we explicitly included the nuclear motion. The nonrelativistic wave function was expanded in terms of explicitly correlated Gaussian functions. Next, this wave function was used to calculate the leading \ensuremathα2{\ensuremath{\alpha}}^{2}\ensuremathα2 relativistic correction (\ensuremathα\ensuremath{\alpha}\ensuremathα is the fine-structure constant) and the \ensuremathα3{\ensuremath{\alpha}}^{3}\ensuremathα3 QED correction. We also estimated the \ensuremathα4{\ensuremath{\alpha}}^{4}\ensuremathα4 QED correction by calculating its dominating component. The results obtained with Gaussians are compared with the most accurate results obtained recently with the Hylleraas-type basis functions.
The electrostatic fine structure for singly excited lithium is accurately calculated using an effective potential method that has proven to be very useful for helium. Relevant properties of the two-electron Li+{\mathrm{Li}}^{+}Li+ ionic core with finite nuclear mass are determined from variational wave functions. From these, non- relativistic level shifts are obtained analytically, and the leading relativistic correction is applied. A comparison with recent experimental results is made.
The leading-order relativistic corrections to the ground-state energy of the positronium molecule (Ps2)({\mathrm{Ps}}_{2})(Ps2​) have been computed within the framework of perturbation theory. As the zero-order wave function we used a highly accurate nonrelativistic variational expansion in terms of 6000 explicitly correlated Gaussians that yielded the lowest variational upper bound for this system to date. We also report some expectation values representing the properties of Ps2{\mathrm{Ps}}_{2}Ps2​.
We perform accurate calculations of energy levels as well as fine and hyperfine splittings of the lowest 1,3PJ,3S1,3PJe^{1,3}P_{J}, ^{3}S_{1}, ^{3}P_{J}^{e}1,3PJ​,3S1​,3PJe​, and 1,3DJ^{1,3}D_{J}1,3DJ​ excited states of the 9Be^{9}\mathrm{Be}9Be atom using explicitly correlated Gaussian functions, and we report on the breakdown of the standard hyperfine structure theory. Because of the strong hyperfine mixing, which prevents the use of common hyperfine constants, we formulate a description of the fine and hyperfine structure that is valid for an arbitrary coupling strength and may have wide applications in many other atomic systems.
Ultracold polar and magnetic 23Na6Li^{23}\mathrm{Na}^{6}\mathrm{Li}23Na6Li molecules in the rovibrational ground state of the lowest triplet ${a}^{3}{\mathrm{\ensuremath{\Sigma}}}^{+}$ electronic state have been recently produced. Here, we calculate the electronic and rovibrational structure of these 14-electron molecules with spectroscopic accuracy ($<0.5\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$) using state-of-the-art ab initio methods of quantum chemistry. We employ the hierarchy of the coupled-cluster wave functions and Gaussian basis sets extrapolated to the complete basis set limit. We show that the inclusion of higher-level excitations, core-electron correlation, relativistic, QED, and adiabatic corrections is necessary to accurately reproduce scattering and spectroscopic properties of alkali-metal systems. We obtain the well depth, ${D}_{e}=229.9(5)\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$, the dissociation energy, ${D}_{0}=208.2(5)\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$, and the scattering length, ${a}_{s}=\ensuremath{-}{84}_{\ensuremath{-}41}^{+25}\phantom{\rule{4pt}{0ex}}\mathrm{bohr}$, in good agreement with recent experimental measurements. We predict the permanent electric dipole moment in the rovibrational ground state, d0=0.167{d}_{0}=0.167d0​=0.167(1) D. These values are obtained without any adjustment to experimental data, showing that quantum chemistry methods are capable of predicting scattering properties of many-electron systems, provided relatively weak interaction and small reduced mass of the system.
In order to overcome the difficulty of optimizing molecular geometry using quantum Monte Carlo methods, we introduce various approximations to the exact force expectation value. We follow Pulay’s suggestion [Mol. Phys. 17, 153 (1969)] to correct the Hellmann–Feynman estimator by introducing the contributions due to the changes in the wave function with respect to the nuclear positions. When used in conjunction with energy-optimized explicitly correlated trial wave functions for H2 and LiH, these approximations appear to yield accurate forces using both the variational and diffusion Monte Carlo methods. Also, the accuracy of the second-order estimate of the Hellmann–Feynman force estimator was investigated employing our energy-optimized trial wave functions, and an erratic behavior was uncovered for some of the studied bond lengths. The additional computational cost required to compute the corrections to the Hellmann–Feynman estimator was found to be only a small fraction of the cost for a simple mean energy calculation. The same approach could be exploited also in computing the derivative of other energy-dependent quantum-mechanical observables.
In this work, the ground-state energy of the lithium atom is systematically investigated using both time-independent perturbation theory and the variational method to provide a comprehensive pedagogical analysis of many-body atomic systems. The unperturbed Hamiltonian is initially constructed by neglecting electron-electron interactions, treating the system as three independent hydrogen-like electrons to yield a zeroth-order energy baseline of -275.51 eV. The antisymmetric fermionic nature of the exact wave function is rigorously enforced through the Slater determinant formalism. First-order perturbation theory is applied to evaluate static inter-electronic repulsion using exact Coulomb and exchange integrals, refining the energy state to -192.01 eV. To account for dynamical electronic correlation, second-order perturbation theory is computed numerically for virtual single-electron s-orbital transitions, leading to a total perturbative energy of -196.36 eV. A brief discussion of two-electron excitations is also included to encapsulate further physical realism within the framework. Furthermore, a non-orthogonal two-parameter variational approach is employed to model the shell-specific shielding effect. By optimizing the effective nuclear charges, the variational method establishes a superior upper bound energy of -201.187 eV. The results of both methods are comprehensively contrasted against each other and the reference baseline to provide critical insights into the nature of electron correlation and screening in multi-electron atoms.