Exact nonrelativistic polarizabilities of the hydrogen atom with the Lagrange-mesh method

Baye, Daniel Jean

doi:10.1103/physreva.86.062514

Cited by 17 publications

(11 citation statements)

References 19 publications

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“…[120][121][122]. Strikingly, the LMM calculation is able to provide the exact values of the hydrogen polarizabilities [123]. The interpretation of the numerical calculation requires rounding some easily identifiable digits related to the limited computer accuracy.…”

Section: Static Polarizabilitiesmentioning

confidence: 99%

“…This method is not used here but is described for the hydrogen atom in Ref. [123]. Let me consider the polarization by a multipole operator r λ Y λµ .…”

Section: Static Polarizabilitiesmentioning

confidence: 99%

“…When the LMM version of the Dalgarno-Lewis method presented in Ref. [123] (see Eqs. (C.4) and (C.5) in Appendix C) gives exact results, one easily shows that Eq.…”

Section: Static Polarizabilitiesmentioning

confidence: 99%

“…The polarizability is the response to a perturbation r λ Y λµ (Ω). In the non-relativistic case, the static polarizability of state nlm of a system described by a potential V (r), where n is the radial quantum number or the principal quantum number in the hydrogen case, are given by [123] α (nlm) λµ…”

Section: Appendix C Polarizabilitiesmentioning

confidence: 99%

“…where H l ′ is defined by (5.2) with l replaced by l ′ . The reduced polarizabilities then read [123] α (nll…”

Section: Appendix C Polarizabilitiesmentioning

confidence: 99%

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The Lagrange-mesh method

2015

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The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid thanks to the use of a Gauss-quadrature approximation. The variational basis related to this Gauss quadrature is composed of Lagrange functions which are infinitely differentiable functions vanishing at all mesh points but one. This method is quite simple to use and, more importantly, can be very accurate with small number of mesh points for a number of problems. The accuracy may however be destroyed by singularities of the potential term. This difficulty can often be overcome by a regularization of the Lagrange functions which does not affect the simplicity and accuracy of the method.The principles of the Lagrange-mesh method are described, as well as various generalizations of the Lagrange functions and their regularization. The main existing meshes are reviewed and extensive formulas are provided which make the numerical calculations simple. They are in general based on classical orthogonal polynomials. The extensions to non-classical orthogonal polynomials and periodic functions are also presented.Applications start with the calculations of energies, wave functions and some observables for bound states in simple solvable models which can rather easily be used as exercises by the reader. The Dirac equation is also considered. Various problems in the continuum can also simply and accurately be solved with the Lagrange-mesh technique including multichannel scattering or scattering by non-local potentials. The method can be applied to three-body systems in appropriate systems of coordinates. Simple atomic, molecular and nuclear systems are taken as examples. The applications to the timedependent Schrödinger equation, to the Gross-Pitaevskii equation and to Hartree-Fock calculations are also discussed as well as translations and rotations on a Lagrange mesh.

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Section: Static Polarizabilitiesmentioning

confidence: 99%

“…This method is not used here but is described for the hydrogen atom in Ref. [123]. Let me consider the polarization by a multipole operator r λ Y λµ .…”

Section: Static Polarizabilitiesmentioning

confidence: 99%

“…When the LMM version of the Dalgarno-Lewis method presented in Ref. [123] (see Eqs. (C.4) and (C.5) in Appendix C) gives exact results, one easily shows that Eq.…”

Section: Static Polarizabilitiesmentioning

confidence: 99%

Section: Appendix C Polarizabilitiesmentioning

confidence: 99%

“…where H l ′ is defined by (5.2) with l replaced by l ′ . The reduced polarizabilities then read [123] α (nll…”

Section: Appendix C Polarizabilitiesmentioning

confidence: 99%

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The Lagrange-mesh method

2015

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The generalized maximum hardness principle revisited and applied to atoms and molecules

Grochala

2017

Phys. Chem. Chem. Phys.

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In this perspective contribution, we revisit the Maximum Hardness Principle (MHP), formulated by Pearson in 1987, and an equivalent Minimum Polarizability Principle (MPP) from Chattaraj and Parr, with particular emphasis on the cases where nuclear potential acting on electrons does not remain constant, and where substantial modifications of the nuclear geometry take place (Generalized MHP, GMHP). We first bring basic concepts related to electronic hardness, and then we present an overview of important manifestations of the GMHP for molecular systems such as (i) the tendency of two free radicals to couple, (ii) reduced reactivity of noble gases, (iii) symmetry-breaking distortions related to the Jahn-Teller effect, and/or these connected with (anti)aromatic character of certain organic molecules, (iv) enhanced reactivity of excited states, (v) high-low spin transitions, etc. GMHP is an important qualitative indicator in studies of molecular isomerism and reactivity. Quantitative aspects, traditionally measured by changes of electronic plus nuclear energy, are readily explained by changes of hardness (or polarizability) of a molecular system. Several important exceptions from (G)MHP are discussed.

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Accurate potential energy curve of the LiH+ molecule calculated with explicitly correlated Gaussian functions

Tung

Adamowicz

2014

The Journal of Chemical Physics

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Very accurate calculations of the ground-state potential energy curve (PEC) of the LiH(+) ion performed with all-electron explicitly correlated Gaussian functions with shifted centers are presented. The variational method is employed. The calculations involve optimization of nonlinear exponential parameters of the Gaussians performed with the aid of the analytical first derivatives of the energy determined with respect to the parameters. The diagonal adiabatic correction is also calculated for each PEC point. The PEC is then used to calculate the vibrational energies of the system. In that calculation, the non-adiabatic effects are accounted for by using an effective vibrational mass obtained by the minimization of the difference between the vibrational energies obtained from the calculations where the Born-Oppenheimer approximation was not assumed and the results of the present calculations.

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Exact nonrelativistic polarizabilities of the hydrogen atom with the Lagrange-mesh method

Cited by 17 publications

References 19 publications

The Lagrange-mesh method

The Lagrange-mesh method

The generalized maximum hardness principle revisited and applied to atoms and molecules

Accurate potential energy curve of the LiH+ molecule calculated with explicitly correlated Gaussian functions

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