# Natural orbitals of the ground state of the two-electron harmonium atom

@article{Cioslowski2018NaturalOO, title={Natural orbitals of the ground state of the two-electron harmonium atom}, author={Jerzy Cioslowski}, journal={Theoretical Chemistry Accounts}, year={2018}, volume={137}, pages={1-11} }

AbstractThe radial components of the natural orbitals (NOs) pertaining to the $$^1S_+$$1S+ ground state of the two-electron harmonium atom are found to satisfy homogeneous differential equations at the values of the confinement strength $$\omega $$ω at which the respective correlation factors are given by polynomials.
Together with the angular momentum l of the NOs, the degrees of these polynomials determine the orders of the differential equations, eigenvalues of which (arising from well…

## 7 Citations

### Universalities among natural orbitals and occupation numbers pertaining to ground states of two electrons in central potentials.

- PhysicsThe Journal of chemical physics
- 2019

Although both the natural orbitals (NOs) {ψnlm(r→)} and their occupation numbers {νnl} pertaining to the ground state of two electrons confined by a central potential are completely determined by the…

### Natural amplitudes of the ground state of the helium atom: Benchmark calculations and their relevance to the issue of unoccupied natural orbitals in the H2 molecule.

- PhysicsThe Journal of chemical physics
- 2019

The numerical trends observed in the present calculations strongly suggest that in the case of the ground state of the helium atom, the only positive-valued natural amplitude is that pertaining to the strongly occupied orbital, i.e., λ10.

### Angular-Momentum Extrapolations to the Complete Basis Set Limit: Why and When They Work.

- PhysicsJournal of chemical theory and computation
- 2021

The leading L-3 dependence of the errors in the energies computed with nuclei-centered basis sets comprising functions with angular momenta not exceeding L is rigorously proven for the 1Σ states of…

### Off-diagonal derivative discontinuities in the reduced density matrices of electronic systems.

- Physics, ChemistryThe Journal of chemical physics
- 2020

The power-law decay of the occupation numbers of the natural orbitals is demonstrated for 1S states of systems composed of arbitrary even numbers of electrons and the implications of the presence of this cusp upon electronic structure calculations of quantum chemistry and solid-state physics are elucidated.

### Electronic structure of first and second row atoms under harmonic confinement

- Physics, ChemistryInternational Journal of Quantum Chemistry
- 2019

Atoms under pressure undergo a series of processes and modification of its electronic structure. Examples are the spontaneous ionization, stabilization of excited-state configurations that result in…

### From Fredholm to Schrödinger via Eikonal: A New Formalism for Revealing Unknown Properties of Natural Orbitals.

- MathematicsJournal of chemical theory and computation
- 2021

The resulting formalism, in which the "on-top" two-electron density ρ2(r⃗, rₗ) solely controls the asymptotic behavior of both ψn(r ⃗) and νn at the limit of the latter becoming infinitesimally small, produces surprisingly accurate values of both quantities even for small n.

## References

SHOWING 1-10 OF 33 REFERENCES

### Partial-wave decomposition of the ground-state wavefunction of the two-electron harmonium atom

- PhysicsTheoretical Chemistry Accounts
- 2015

AbstractAn 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 confinement strengths…

### NATURAL ORBITALS IN THE QUANTUM THEORY OF TWO-ELECTRON SYSTEMS

- Mathematics
- 1956

The wave functions for the singlet and triplet states of a two-electron system in a given nuclear framework are investigated as superpositions of configurations and are shown to be transformationally…

### Solitonic natural orbitals.

- PhysicsThe Journal of chemical physics
- 2018

It is shown that unusual, weakly occupied natural orbitals corresponding to additional positive-valued natural amplitudes emerge upon sufficient weakening of the confinement of the harmonium atom.

### Collective natural orbital occupancies of harmonium.

- MathematicsThe Journal of chemical physics
- 2005

In the harmonium atom, the collective occupancies {n(l)} of natural orbitals with different angular momenta l can be rigorously studied for those values of the confinement strength omega that lead to…

### Wigner molecules: natural orbitals of strongly correlated two-electron harmonium.

- PhysicsThe Journal of chemical physics
- 2006

The occupancies at the omega-->0 limit are vanishingly small and asymptotically independent of the angular momentum, forming a geometric progression with the scale factor proportional to omega(1/3) and the common ratio of 0.0186.

### The ground state of harmonium

- Physics
- 2000

A detailed analysis that benefits from a slate of new approximate numerical and exact asymptotic results produces highly accurate properties of the ground state of the harmonium atom as functions of…

### The weak-correlation limits of few-electron harmonium atoms.

- ChemistryThe Journal of chemical physics
- 2013

The weak-correlation asymptotics of electronic properties of harmonium atoms comprising up to four electrons are investigated and closed-form expressions are derived for the first- and second-order contributions to the Hartree-Fock and correlation energies of eight electronic states, six of which are singly determinantal and two are multi-determinantal.

### Long-range interactions and the sign of natural amplitudes in two-electron systems.

- PhysicsThe Journal of chemical physics
- 2013

It is demonstrated that the amplitudes show an avoided crossing behavior as function of a parameter in the Hamiltonian and this feature is used to show that these amplitudes never become zero, except for special interactions in which infinitely many of them can become zero simultaneously when changing the interaction strength.

### The electron correlation cusp

- Physics
- 1996

The Kais function is an exact solution of the Schrödinger equation for a pair of electrons trapped in a parabolic potential well with r12−1 electron-electron interaction. Partial wave analysis (PWA)…

### Simple approximants for natural orbitals of harmonium.

- MathematicsThe Journal of chemical physics
- 2005

Simple approximations to the natural orbitals (NOs) of harmonium with enforced correct short- and long-range asymptotics yield accurate bounds for the NO occupancies. In particular, expressions…