Intracule functional models : Part III. The dot intracule and its Fourier transform

Bernard, Yves; Crittenden, Deborah L.; Gill, Peter M. W.

doi:10.1039/b803919d

Cited by 25 publications

(37 citation statements)

References 28 publications

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“…We believe that our results will be useful in the future development of accurate correlation functionals within density-functional theory [42,43] and intracule functional theory [44][45][46][47][48][49].…”

Section: Discussionmentioning

confidence: 99%

Ground state of two electrons on a sphere

Loos

Gill²

2009

Phys. Rev. A

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We have performed a comprehensive study of the singlet ground state of two electrons on the surface of a sphere of radius R. We have used electronic structure models ranging from restricted and unrestricted Hartree-Fock theory to explicitly correlated treatments, the last of which lead to near-exact wavefunctions and energies for any value of R. Møller-Plesset energy corrections (up to fifth-order) are also considered, as well as the asymptotic solution in the large-R regime.

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Section: Discussionmentioning

confidence: 99%

Ground state of two electrons on a sphere

Loos

Gill²

2009

Phys. Rev. A

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“…This kind of relation could be useful in the future development of intracule functional theory [43][44][45][46][47][48].…”

Section: Position Intraculementioning

confidence: 99%

Hooke’s law correlation in two-electron systems

Loos

2010

Phys. Rev. A

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We study the properties of the Hooke's law correlation energy (Ec), defined as the correlation energy when two electrons interact via a harmonic potential in a D-dimensional space. More precisely, we investigate the 1 S ground state properties of two model systems: the Moshinsky atom (in which the electrons move in a quadratic potential) and the spherium model (in which they move on the surface of a sphere). A comparison with their Coulombic counterparts is made, which highlights the main differences of the Ec in both the weakly and strongly correlated limits. Moreover, we show that the Schrödinger equation of the spherium model is exactly solvable for two values of the dimension (D = 1 and 3), and that the exact wave function is based on Mathieu functions.

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“…We believe that our results will be useful in the future development of correlation functionals within densityfunctional theory [33], intracule functional theory [34][35][36][37][38][39], and explicitly correlated methods [5][6][7][8]. They also shed new light on dimension-dependent correlation effects, and may be used as an alternative system for studying quantum dots [40].…”

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confidence: 98%

Two Electrons on a Hypersphere: A Quasiexactly Solvable Model

2009

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We show that the exact wave function for two electrons, interacting through a Coulomb potential but constrained to remain on the surface of a D-sphere (D ≥ 1), is a polynomial in the interelectronic distance u for a countably infinite set of values of the radius R. A selection of these radii, and the associated energies, are reported for ground and excited states on the singlet and triplet manifolds. We conclude that the D = 3 model bears the greatest similarity to normal physical systems. Quantum mechanical models for which it is possible to solve explicitly for a finite portion of the energy spectrum are said to be quasi-exactly solvable [1]. They have ongoing value and are useful both for illuminating more complicated systems and for testing and developing theoretical approaches, such as density functional theory (DFT) [2][3][4] and explicitly correlated methods [5][6][7][8]. One of the most famous two-body models is the Hooke's law atom which consists of a pair of electrons, repelling Coulombically but trapped in a harmonic external potential with force constant k. This system was first considered nearly 50 years ago by Kestner and Sinanoglu [9], solved analytically in 1989 for one particular k value [10], and later for a countably infinite set of k values [11].A related system consists of two electrons trapped on the surface of a sphere of radius R. This has been used by Berry and collaborators [12][13][14][15] to understand both weakly and strongly correlated systems and to suggest an "alternating" version of Hund's rule [16]. Seidl utilized this system to develop new correlation functionals [17] within the adiabatic connection in DFT [18]. We will use the term "spherium" to describe this system. In recent work [19], we examined various schemes and described a method for obtaining near-exact estimates of the 1 S ground state energy of spherium for any given R. Because the corresponding Hartree-Fock (HF) energies are also known exactly [19], this is now one of the most complete theoretical models for understanding electron correlation effects.In this Letter, we consider D-spherium, the generalization in which the two electrons are trapped on a D-sphere of radius R. We adopt the convention that a D-sphere is the surface of a (D + 1)-dimensional ball. (Thus, for example, the Berry system is 2-spherium.) We show that the Schrödinger equation for the 1 S and the 3 P states can be solved exactly for a countably infinite set of R values and that the resulting wave functions are polynomials in the interelectronic distance u = |r 1 − r 2 |. Other spin and angular momentum states can be addressed in the same way using the ansatz derived by Breit [20].The electronic Hamiltonian, in atomic units, iŝand because each electron moves on a D-sphere, it is natural to adopt hyperspherical coordinates [21,22]. For 1 S states, it can be then shown [19] that the wave function S(u) satisfies the Schrödinger equation(2) By introducing the dimensionless variable x = u/2R, this becomes a Heun equation [23] with singular points at x...

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Intracule functional models : Part III. The dot intracule and its Fourier transform

Cited by 25 publications

References 28 publications

Ground state of two electrons on a sphere

Ground state of two electrons on a sphere

Hooke’s law correlation in two-electron systems

Two Electrons on a Hypersphere: A Quasiexactly Solvable Model

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