Cusp conditions for eigenfunctions ofn-electron systems (original) (raw)
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Non-Isotropic Cusp Conditions and Regularity of the Electron Density of Molecules at the Nuclei
Annales Henri Poincaré, 2007
We investigate regularity properties of molecular oneelectron densities ρ near the nuclei. In particular we derive a representation ρ(x) = e F (x) µ(x) with an explicit function F , only depending on the nuclear charges and the positions of the nuclei, such that µ ∈ C 1,1 (R 3 ), i.e., µ has locally essentially bounded second derivatives. An example constructed using Hydrogenic eigenfunctions shows that this regularity result is sharp. For atomic eigenfunctions which are either even or odd with respect to inversion in the origin, we prove that µ is even C 2,α (R 3 ) for all α ∈ (0, 1). Placing one nucleus at the origin we study ρ in polar coordinates x = rω and investigate ∂ ∂r ρ(r, ω) and ∂ 2 ∂r 2 ρ(r, ω) for fixed ω as r tends to zero. We prove non-isotropic cusp conditions of first and second order, which generalize Kato's classical result.
Cusp conditions for non-Coulombic interactions
Journal of Molecular Structure THEOCHEM
We have derived the conditions that an exact non-relativistic molecular wavefunction must ful®ll at two critical points, namely, at the nucleus±electron cusp (when the position of an electron is coincident with a nucleus) and the electron±electron cusp (when the position of two arbitrary electrons coincide at a point). The interaction potential between the electrons and nuclei has not been restricted to the Coulombic form, but given a general distance dependent form. Using the derived cusp conditions for the wavefunction we have obtained the conditions that rule the one-electron density and the intracule density for the nucleus±electron and the electron±electron cusps, respectively. Finally, the results obtained for the harmonic, Yukawa and Hulthe Ân potentials are illustrated. q
Hylleraas-like functions with the correct cusp conditions: K-shell electrons for the neutral atoms
Journal of Electron Spectroscopy and Related Phenomena, 2007
We present simple correlated wavefunctions for the two K-shell electrons of neutral atoms. A variational method was chosen to calculate the mean energy of the ground state, in which the electrons are subject to a local Hartree potential representing the presence of the outer shell electrons. The functions are constructed in terms of exponential and power series, where special care has been taken in order to fulfill the exact behavior at the electron-electron and electron-nucleus coalescence points (Kato cusp conditions). Global properties, such as the energies and virial coefficients, as well as local properties, such as spatial mean values, are also analyzed.
Physical Review A, 2006
The limit relations for the partial derivatives of the two-electron atomic wave functions at the two-particle coalescence lines have been obtained numerically using accurate correlation function hyperspherical harmonic method wave functions. The asymptotic solutions of the proper two-electron Schrödinger equation have been derived for both electron-nucleus and electron-electron coalescence. It is shown that the solutions for the electron-nucleus coalescence correspond to the ground and singly excited bound states, including triplet ones. The proper solutions at small distances R from the triple coalescence point were presented as the second order expansion on R and ln R. The vanishing of the Fock's logarithmic terms at the electron-nucleus coalescence line was revealed in the frame of this expansion, unlike the case of electron-electron coalescence. On the basis of the obtained boundary solutions the approximate wave function corresponding to both coalescence lines have been proposed in the two-exponential form with no variational parameters. ͯ ץ⌿ ץr 1 ͯ r 1 =0 = − Z⌿͑0,R,R͒ ͑r 2 = r 12 = R͒, ͑2͒ ͯ ץ⌿ ץr 2 ͯ r 2 =0 = − Z⌿͑R,0,R͒ ͑r 1 = r 12 = R͒, ͑3͒ PHYSICAL REVIEW A 73, 012514 ͑2006͒
Criticality of the electron-nucleus cusp condition to local effective potential-energy theories
Physical Review A, 2003
Local(multiplicative) effective potential energy theories of electronic structure comprise the transformation of the Schrödinger equation for interacting fermi systems to model noninteracting fermi or bose systems whereby the equivalent density and energy are obtained. By employing the integrated form of the Kato electron-nucleus cusp condition, we prove that the effective electron -interaction potential energy of these model fermions or bosons is finite at a nucleus. The proof is general and valid for arbitrary system whether it be atomic, molecular, or solid state, and for arbitrary state and symmetry. This then provides justification for all prior work in the literature based on the assumption of finiteness of this potential energy at a nucleus. We further demonstrate the criticality of the electron-nucleus cusp condition to such theories by example of the Hydrogen molecule. We show thereby that both model system effective electron-interaction potential energies, as determined from densities derived from accurate wave functions, will be singular at the nucleus unless the wave function satisfies the electron-nucleus cusp condition.
Chemical Physics Letters, 2012
In very recent work, Amovilli and March have considered the Hookean atom with four electrons. In particular, they demonstrated the cross-over from a triplet P g ground state at large harmonic force constant k to a quintet S u configuration for weak confinement, by means primarily of diffusion quantum Monte Carlo simulation. Here, we focus on such a quintet state for again four spin half fermions but now interacting via a harmonic pair force ÀK r ij . The spatial quintet W is then available, with its eigenvalue. Progress on calculating low-order spinless density matrices is recorded. Finally, this prompts us to examine the Amovilli-March results when extrapolated to small k -0. We display as k ? 0 + approximate results for the four-electron Hookean atom model, including fitted analytic forms for kinetic and Coulomb energy as a function of k as k ? 0 + .
Reports on Mathematical Physics 64 (3), 367-393
In this note we investigate in detail the spectrum of the Schroedinger Hamiltonian with a configuration of three equally spaced one-dimensional point interactions (Dirac distributions), with the external ones having the same negative coupling constant. It will be seen that despite its simplicity, such a toy model exhibits a fairly rich variety of spectral combinations as the two coupling constants and the separation distance are manipulated. By analysing the equation determining the square root of the absolute value of the ground state energy and those determining the same quantity for the two possible excited states, we explicitly calculate the eigenvalues for all possible values of the separation distance and the two coupling constants. As a result of our analysis, we provide the conditions in terms of the three parameters in order to have the emergence of such excited states. Furthermore, we use our findings in order to get the confirmation of the fact that the Hamiltonian with such a configuration of three simple point interactions whose coupling constants undergo a special scaling in terms of the vanishing separation distance, converges in the norm resolvent sense to the Hamiltonian with an attractive δ -interaction centred at the origin, as was shown by Exner and collaborators making the result previously obtained by Cheon et al. mathematically rigorous.
(e,3e)processes on two-electron atoms: Cusp conditions and scaling law
Physical Review A, 2008
We study the double ionization by electron impact of the ground state of heliumlike atoms and propose a scaling law for fully differential ͑e ,3e͒ cross sections. Within the first Born approximation, cross sections are calculated with a three-body Coulomb ͑3C͒ double-continuum wave function and initial states represented by highly accurate wave functions, which satisfy all two-body Kato cusp conditions. We first consider the helium atom in the kinematical and geometrical conditions of the only absolute, high incident energy, experimental data available: our calculations confirm unambiguously that satisfying or not Kato cusp conditions is not a relevant feature of the ground state. Other heliumlike atoms are then considered. Under similar conditions, cross sections for H − are much larger than for helium while the reverse is true for positive ions; a comparison with the rare other theoretical calculations is performed. Finally, within our theoretical framework, we propose an approximate scaling law for ͑e ,3e͒ cross sections for heliumlike positive ions, and confirm it by calculations.
Electron clusters in a quadratic potential
Physics Letters A, 1989
Clusters of electrons in a quadratic external potential are studied. By expanding in inverse powers of the space dimensionality the ground state energy of the N-electron cluster is found in the whole range of variation of the problem's parameter. The rearrangement ofelectron orbitals in the crossover region is investigated.
Cusps and derivatives for wave-functions expanded in Slater orbitals: A density study
International Journal of Quantum Chemistry, 2009
This article recalls basic properties of Slater type Orbital (STO) basis sets, in particular, with respect to satisfying Kato's electron-nucleus cusp condition. It is shown that a suitable starting point is the set of hydrogen-like orbitals or, more flexibly, their generalization as Coulomb Sturmians. A case study is presented on the Hartree-Fock density obtained for a water molecule, N 2 , and CO 2 showing that the nuclear cusp condition may be easily accounted for near the heavier nuclei, but only with difficulties at the hydrogen center. In particular, bond stretching affects the expected cusp condition on hydrogen atoms. Some indications are given on the improvement of STO basis sets.