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Physical Review A

Comment on “Towards a differential equation for the nonrelativistic ground-state electron density of the He-like sequence of atomic ions”

Pablo Serra

Published Jul 18, 2006• 11 citations
Publisher

Abstract

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.
DOI
10.1103/physreva.74.016501

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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.
In the conventional theory of the ground state of heliu m ato m, any solution of the two-electron Schrodinger equation is wrong! That is caused by a strong imbalance of proportions of the attraction versus repulsion operators' acting on a wave function, in the initial differential Schrodinger equation and in its integral algebraic form. In the differential equation, the attraction operators act on the radial part of wave function independently fro m one another obeying the true proportion: one attraction versus one repulsion per electron. But the resulting wave function turns out to satisfy the wrong integral proportion: two attraction energies versus one repulsion energy per electron. The underlying contradiction is un-coincidence of the additivity rules of energies and energy operators: whereas energies of the electrons may be summed the corresponding energy operators referred to different electrons may not. Consequences of this principal contradiction are d iscussed.
The ground-state density nnn of a many-electron system obeys a Schr\"odinger-like differential equation for ${n}^{\frac{1}{2}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, which may be solved by standard Kohn-Sham programs. The exact local effective (nonexternal) potential, ${v}_{\mathrm{eff}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, is displayed explicitly in terms of wave-function expectation values, from which ${v}_{\mathrm{eff}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})>~0$ for all $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$. A derivation for nnn as $|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|\ensuremath{\rightarrow}\ensuremath{\infty}$ implies that this new effective potential tends asymptotically to zero, as does the exact Kohn-Sham potential, with the highest occupied eigenvalue as the exact ionization energy. A new exact expression is also presented for the exchange-correlation hole density ${\ensuremath{\rho}}_{\mathrm{xc}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, {\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}})$ about an electron at $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$, as $|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|\ensuremath{\rightarrow}\ensuremath{\infty}$.
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 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 $\ensuremath{\rho}(r)$ 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.
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).