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Physical Review A

Density matrices in direct and momentum space for a model of the He-like sequence of atomic ions

C. Amovilli, Norman Henry March

Published Oct 18, 2005• 9 citations
Publisher

Abstract

The ground-state electron density $\ensuremath{\rho}(r)$ in a model of the He-like sequence of atomic ions, in which the Coulomb repulsion is replaced by the $s$-wave component, has recently been obtained by Howard and March [Phys. Rev. A 71, 042501 (2005)]. Their result is here generalized to derive the one-particle density matrix yielding the Howard and March $\ensuremath{\rho}(r)$ on the diagonal. Its momentum space counterpart is also presented and the momentum density is studied in detail, with particular emphasis on the asymptotic behavior of momentum distribution. Attention is then focussed on the kinetic energy density and on the spectral decomposition of the ground-state wave function. A natural orbital analysis in both direct and momentum space is also presented. Finally, the relevance for density functional theory of this exactly solvable model is briefly considered.

DOI
10.1103/physreva.72.042504

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In a recent paper [Phys. Rev. A 71, 042501 (2005)], Howard and March presented the exact ground state wave function of the spherical He-like atom, and many physical aspects of this solution were analyzed. We show that this function is not the exact solution of the model.

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Local-density-matrix approximation: Exact asymptotic results for a high-density homogeneous electron gas

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The detailed nature of the correlated first-order density matrix for the model atoms in the title for arbitrary interparticle interaction $u({r}_{12})$ is studied. One representation with contracted information is first explored by constructing the momentum density $\ensuremath{\rho}(\mathbf{p})$ in terms of the wave function of the relative motion, say ${\ensuremath{\Psi}}_{R}({r}_{12})$, which naturally depends on the choice of $u({r}_{12})$. For $u({r}_{12})={e}^{2}∕{r}_{12}$, the so-called Hookean atom, and for the inverse square law $u({r}_{12})=\ensuremath{\lambda}∕{r}_{12}^{2}$, plots are presented of the above density $\ensuremath{\rho}(\mathbf{p})$ in momentum space. The correlated kinetic energy is recovered from averaging ${p}^{2}∕2m$, $m$ denoting the electron mass, with respect to $\ensuremath{\rho}(\mathbf{p})$. The second method developed is in coordinate space and expands the density matrix $\ensuremath{\gamma}({\mathbf{r}}_{1},{\mathbf{r}}_{2})$ in Legendre polynomials, using relative coordinate ${\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}$, center-of-mass coordinate $({\mathbf{r}}_{1}+{\mathbf{r}}_{2})∕2$ and the angle, $\ensuremath{\theta}$ say, between these two vectors. For the Moshinsky atom in which $u({r}_{12})=\frac{1}{2}k{r}_{12}^{2}$ only the $s$ term $(l=0)$ contributes to the Legendre polynomial expansion. The specific example we present of the inverse square law model is shown to be characterized by the low-order terms $(s+d)$ of the Legendre expansion. The Wigner function is finally calculated analytically for both Moshinsky and inverse square law models.

Physical Review A•Sep 27, 2007•11 citations
The ground-state electron density \ensuremathρ(r)\ensuremath{\rho}(r)\ensuremathρ(r) in a model of the He-like sequence of atomic ions, in which the Coulomb repulsion is replaced by the sss-wave component, has recently been obtained by Howard and March [Phys. Rev. A 71, 042501 (2005)]. Their result is here generalized to derive the one-particle density matrix yielding the Howard and March \ensuremathρ(r)\ensuremath{\rho}(r)\ensuremathρ(r) on the diagonal. Its momentum space counterpart is also presented and the momentum density is studied in detail, with particular emphasis on the asymptotic behavior of momentum distribution. Attention is then focussed on the kinetic energy density and on the spectral decomposition of the ground-state wave function. A natural orbital analysis in both direct and momentum space is also presented. Finally, the relevance for density functional theory of this exactly solvable model is briefly considered.
We take as a starting point the ground-state electron density in two-electron model atoms in which Coulomb confinement in the He atom is first replaced by harmonic restoring forces. Switching off electron–electron interactions, one readily constructs a third-order differential equation for the ground-state electron density, as in the recent work of March and Ludeña (2004 Phys. Lett. A 330 16). We then switch on two different model interactions, first in the so-called Hookean atom going back to Kestner and Sinanoglu (1962 Phys. Rev. 128 2687), in which e 2 / r 12 is retained as in He, and secondly in the Moshinsky (1968 Am. J. Phys. 36 52) atom in which Kr 2 12 /2 is switched on. Some analyticity properties of the low-order linear homogeneous differential equations which result are next studied. He-like atomic ions are then treated in the limit of large atomic number Z . In this latter case, one identifies both the electron–nuclear cusp, or equivalently Kato's theorem, and the corresponding electron–electron cusp in the ground-state spatial wavefunction Ψ( r 1 , r 2 ). A final comment concerns quantum information and entanglement in relation to the recent work of Amovilli and March (2004 Phys. Rev. A 69 054302).
In a recent paper A. Akbari, N. H. March, and A. Rubio [Phys. Rev. A 76, 032510 (2007)] investigated the one-electron reduced density matrix and the momentum density of several two-electron model atoms, including Hooke's atom. The method used by the authors for deriving an integral form of the momentum density is in fact the most suited for deriving a closed-form expression of the exact reciprocal form factor, in a special case. This function is important inasmuch as it reflects the off-diagonal side of the exact reduced density matrix.
Explicit asymptotic expressions are derived for the reduced one-electron density matrices (the 1-matrices) of strongly correlated two- and three-electron harmonium atoms in the ground and first excited states. These expressions, which are valid at the limit of small confinement strength ω, yield electron densities and kinetic energies in agreement with the published values. In addition, they reveal the ω(5/6) asymptotic scaling of the exchange components of the electron-electron repulsion energies that differs from the ω(2/3) scaling of their Coulomb and correlation counterparts. The natural orbitals of the totally symmetric ground state of the two-electron harmonium atom are found to possess collective occupancies that follow a mixed power/Gaussian dependence on the angular momentum in variance with the simple power-law prediction of Hill's asymptotics. Providing rigorous constraints on energies as functionals of 1-matrices, these results are expected to facilitate development of approximate implementations of the density matrix functional theory and ensure their proper description of strongly correlated systems.
Scaling arguments show that the exchange-correlation energy density of a spin-unpolarized homogeneous electron gas equals $A{\ensuremath{\omega}}^{4∕3}$, where $\ensuremath{\omega}$ is given by a simple functional of the momentum distribution n(k)n(k)n(k) and AAA is a constant. This functional possesses the property of yielding the true n(k)n(k)n(k) (including the discontinuity at the Fermi momentum, which is not reproduced by any of the presently known density matrix functionals) upon minimization of the energy density eee constrained by the electron density $\ensuremath{\rho}$. At the limit of $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\omega}$ coincides with $\ensuremath{\rho}$ and the dependence of $\ensuremath{\omega}$ on n(k)n(k)n(k), which can be readily recovered from the actual values of eee and n(k)n(k)n(k), is given by an explicit analytical formula. Application of this formalism to general Coulombic systems yields a local version of the density matrix theory, in which the exchange-correlation energy is given by a spatial integral of an expression that depends on the amplitudes of natural orbitals at a given point and the respective occupation numbers.
The detailed nature of the correlated first-order density matrix for the model atoms in the title for arbitrary interparticle interaction u(r12)u({r}_{12})u(r12​) is studied. One representation with contracted information is first explored by constructing the momentum density $\ensuremath{\rho}(\mathbf{p})$ in terms of the wave function of the relative motion, say ${\ensuremath{\Psi}}_{R}({r}_{12})$, which naturally depends on the choice of u(r12)u({r}_{12})u(r12​). For u(r12)=e2∕r12u({r}_{12})={e}^{2}∕{r}_{12}u(r12​)=e2∕r12​, the so-called Hookean atom, and for the inverse square law $u({r}_{12})=\ensuremath{\lambda}∕{r}_{12}^{2}$, plots are presented of the above density $\ensuremath{\rho}(\mathbf{p})$ in momentum space. The correlated kinetic energy is recovered from averaging p2∕2m{p}^{2}∕2mp2∕2m, mmm denoting the electron mass, with respect to $\ensuremath{\rho}(\mathbf{p})$. The second method developed is in coordinate space and expands the density matrix $\ensuremath{\gamma}({\mathbf{r}}_{1},{\mathbf{r}}_{2})$ in Legendre polynomials, using relative coordinate ${\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}$, center-of-mass coordinate (r1+r2)∕2({\mathbf{r}}_{1}+{\mathbf{r}}_{2})∕2(r1​+r2​)∕2 and the angle, $\ensuremath{\theta}$ say, between these two vectors. For the Moshinsky atom in which u(r12)=12kr122u({r}_{12})=\frac{1}{2}k{r}_{12}^{2}u(r12​)=21​kr122​ only the sss term (l=0)(l=0)(l=0) contributes to the Legendre polynomial expansion. The specific example we present of the inverse square law model is shown to be characterized by the low-order terms (s+d)(s+d)(s+d) of the Legendre expansion. The Wigner function is finally calculated analytically for both Moshinsky and inverse square law models.