1 3
Theor Chem Acc (2015) 134:113
DOI 10.1007/s00214-015-1705-8
REGULAR ARTICLE
Partial-wave decomposition of the ground-state wavefunction
of the two-electron harmonium atom
Jerzy Cioslowski
1
Received: 3 May 2015 / Accepted: 17 July 2015 / Published online: 5 September 2015
© The Author(s) 2015 . This article is published with open access at Springerlink.com
functions. The choice of these functions is dictated by computational expedience and usually does not refl ect singularities present in the potential energy. Thus, at the one-electron level, the commonly employed Gaussian-type basis
functions do not possess cusps at nuclei, and at the manyelectron level, the Slater determinants do not reproduce the
electron–electron coalescence cusps.
The failure to properly reproduce the particle–particle
coalescence asymptotics bears upon the rates of convergence of the computed energies and other observables
to their complete-basis-set (CBS) limits [ 1 ]. Whereas
in practice this convergence is suffi ciently rapid for the
solutions of the Hartree–Fock equations [ 2 , 3 ], obtaining
accurate approximations to correlated electronic wavefunctions is much more diffi cult [ 4 , 5 ]. In order to alleviate this problem, two distinct strategies have been developed, namely inclusion of a correlation factor in the trial
function [ 6 – 8 ] and extrapolation to the CBS limit [ 9 , 10 ].
Successful implementations of the latter approach hinge
upon understanding how the approximate wavefunction
approaches its exact counterpart as the size of the basis
set increases.
In a seminal paper [ 4 ], Hill carefully analyzed this
asymptotic behavior for singlet ground states of two-electron systems. Results of that investigation, subsequently
rederived [ 11 ] and generalized to excited states [ 5 ] and
explicitly correlated basis functions [ 12 ], have opened an
avenue to a plethora of extrapolation formulae that furnish
approximate CBS limits for the total energy in terms of the
energies computed with sequences of basis sets truncated at
particular values of angular momenta [ 9 , 10 ].
Among Hill’s results, the large- l asymptotics [ 4 ]
(1)
lim
l→∞
l +
1
2
6
ν l =
5 π
4
|( r, r)|
2 r
5 d r,
Abstract An exact formula for the collective occupancy
of natural orbitals with an angular momentum l is derived
for the ground state of the two-electron harmonium atom.
For confi nement strengths ω that correspond to polynomial
correlation factors as well as at the weak ( ω → ∞ ) and
strong ( ω → 0 ) correlations limits, it reduces to closedform expressions. At the former limit, a similar result
obtains for the partial-wave contributions to the groundstate energy. Slow convergence of the collective occupancies to their leading large- l asymptotics provided by Hill’s
formula is uncovered. As the rate of convergence decreases
strongly with ω , a complete breakdown of Hill’s formula
ensues upon the confi nement strength becoming infi nitesimally small. The relevance of these fi ndings to the performance of the extrapolation schemes for the estimation
of the complete-basis-set limits of quantum-mechanical
observables is discussed.
Keywords Partial-wave decomposition · Two-electron
harmonium atom · Electron correlation
1 Introduction
The vast majority of modern quantum-chemical calculations rely on approximating electronic wavefunctions
(or their equivalents) with linear combinations of basis
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan.”
* Jerzy Cioslowski
jerzy@wmf.univ.szczecin.pl
1
Institute of Physics , University of Szczecin , Wielkopolska
15 , 70-451 Szczecin , Poland
149
Reprinted from the journal
Theor Chem Acc (2015) 134:113
DOI 10.1007/s00214-015-1705-8
REGULAR ARTICLE
Partial-wave decomposition of the ground-state wavefunction
of the two-electron harmonium atom
Jerzy Cioslowski
1
Received: 3 May 2015 / Accepted: 17 July 2015 / Published online: 5 September 2015
© The Author(s) 2015 . This article is published with open access at Springerlink.com
functions. The choice of these functions is dictated by computational expedience and usually does not refl ect singularities present in the potential energy. Thus, at the one-electron level, the commonly employed Gaussian-type basis
functions do not possess cusps at nuclei, and at the manyelectron level, the Slater determinants do not reproduce the
electron–electron coalescence cusps.
The failure to properly reproduce the particle–particle
coalescence asymptotics bears upon the rates of convergence of the computed energies and other observables
to their complete-basis-set (CBS) limits [ 1 ]. Whereas
in practice this convergence is suffi ciently rapid for the
solutions of the Hartree–Fock equations [ 2 , 3 ], obtaining
accurate approximations to correlated electronic wavefunctions is much more diffi cult [ 4 , 5 ]. In order to alleviate this problem, two distinct strategies have been developed, namely inclusion of a correlation factor in the trial
function [ 6 – 8 ] and extrapolation to the CBS limit [ 9 , 10 ].
Successful implementations of the latter approach hinge
upon understanding how the approximate wavefunction
approaches its exact counterpart as the size of the basis
set increases.
In a seminal paper [ 4 ], Hill carefully analyzed this
asymptotic behavior for singlet ground states of two-electron systems. Results of that investigation, subsequently
rederived [ 11 ] and generalized to excited states [ 5 ] and
explicitly correlated basis functions [ 12 ], have opened an
avenue to a plethora of extrapolation formulae that furnish
approximate CBS limits for the total energy in terms of the
energies computed with sequences of basis sets truncated at
particular values of angular momenta [ 9 , 10 ].
Among Hill’s results, the large- l asymptotics [ 4 ]
(1)
lim
l→∞
l +
1
2
6
ν l =
5 π
4
|( r, r)|
2 r
5 d r,
Abstract An exact formula for the collective occupancy
of natural orbitals with an angular momentum l is derived
for the ground state of the two-electron harmonium atom.
For confi nement strengths ω that correspond to polynomial
correlation factors as well as at the weak ( ω → ∞ ) and
strong ( ω → 0 ) correlations limits, it reduces to closedform expressions. At the former limit, a similar result
obtains for the partial-wave contributions to the groundstate energy. Slow convergence of the collective occupancies to their leading large- l asymptotics provided by Hill’s
formula is uncovered. As the rate of convergence decreases
strongly with ω , a complete breakdown of Hill’s formula
ensues upon the confi nement strength becoming infi nitesimally small. The relevance of these fi ndings to the performance of the extrapolation schemes for the estimation
of the complete-basis-set limits of quantum-mechanical
observables is discussed.
Keywords Partial-wave decomposition · Two-electron
harmonium atom · Electron correlation
1 Introduction
The vast majority of modern quantum-chemical calculations rely on approximating electronic wavefunctions
(or their equivalents) with linear combinations of basis
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan.”
* Jerzy Cioslowski
jerzy@wmf.univ.szczecin.pl
1
Institute of Physics , University of Szczecin , Wielkopolska
15 , 70-451 Szczecin , Poland
149
Reprinted from the journal