Kato Cusp Research Articles

Hylleraas- and Kinoshita-type wave functions with correct cusp conditions

The finite terms Hylleraas- and Kinoshita-type variational wave functions are considered for three-body systems. In the Coulombic case local properties of wave functions are restricted by the Kato's cusp conditions. It is showed that Kato's cusp conditions restrict the possible terms in variational calculations. Constraints for the linear expansion coefficients are also derived and a recursion-type solution is given. Two trial functions with correct cusp conditions are determined for the ground state of the He atom. Local and global properties of these wave function are studied through calculations of mean values, local energy, and quantities related to double photoionization.

General coalescence conditions for the exact wave functions. II. Higher-order relations for many-particle systems.

We derived the necessary conditions that must be satisfied by the non-relativistic time-independent exact wave functions for many-particle systems at a two-particle coalescence (or cusp) point. Some simple conditions are known to be Kato's cusp condition (CC) and Rassolov and Chipman's CC. In a previous study, we derived an infinite number of necessary conditions that two-particle wave functions must satisfy at a coalescence point. In the present study, we extend these conditions to many-particle systems. They are called general coalescence conditions (GCCs), and Kato's CC and Rassolov and Chipman's CC are included as special conditions. GCCs can be applied not only to Coulombic systems but also to any system in which the interaction between two particles is represented in a power series of inter-particle distances. We confirmed the correctness of our derivation of the GCCs by applying the exact wave function of a harmonium in electron-electron and electron-nucleus coalescence situations. In addition, we applied the free complement (FC) wave functions of a helium atom to the GCCs to examine the accuracy of the FC wave function in the context of a coalescence situation.

Nuclear cusp conditions and their fulfillment in molecular calculations with SLATER basis sets

A short revision of the main antecedents on the nuclear cusp conditions is followed by a derivation of the general expressions of the conditions which must be accomplished by wave functions and densities. It is stressed that in case of the molecular densities, besides Kato's cusp condition, there are also simple cusp conditions for the three dipole‐type terms of the densities but not for terms of order higher than dipole. The degree of fulfillment of standard atomic and molecular calculations with Slater basis sets is tested, concluding that errors of spherical terms usually affect the third or fourth significant figures, being smaller as nuclear charges increase, and the errors of dipole‐type components are greater than those of the spherical terms. In practice, improvements in the computational method or in the basis set do not guarantee a better fulfillment of the cusp conditions. Hydrogen is a special case which can yield poor results. © 2014 Wiley Periodicals, Inc.

Exact solution of the compressed hydrogen atom

The exact solution to the problem of a hydrogen atom confined in a spherical well (CHA) is discussed; the standard results for the unconfined hydrogen atom (UHA) are recovered as the sphere size becomes large compared to the Bohr radius. The solutions are characterized by a set of three quantum numbers N (= 1, 2, 3,…), L (= 0, 1, 2,…), and M (= − L, − L + 1,…, L − 1, L), and the energy eigenvalues, in contrast to the situation in the UHA, depend on both N and L. All members of a given family n = N + L, however, evolve asymptotically toward the same energy level in the large-sphere limit, recovering the typical n2 degeneracy of the UHA. Besides numerically exact solutions for arbitrary sphere sizes, rigorous analytical approximations are provided for the physically relevant strong- and weak-confinement regimes. A conjecture concerning the ordering of the energy levels is rigorously confirmed. The validity of the virial theorem, Kato's cusp condition, and the role played by the density as an alternative basic variable for the case of the CHA are discussed.

General coalescence conditions for the exact wave functions: Higher-order relations for two-particle systems

We derived the necessary conditions that the non-relativistic time-independent exact wave functions for two-particle systems must satisfy at a coalescence (or cusp) point. Some of such necessary conditions are already known to be Kato's cusp condition (CC) and Rassolov and Chipman's CC. In the present study, we extended and generalized those conditions, calling them generalized coalescence conditions (GCCs). Kato's CC and Rassolov and Chipman's CC were shown to be specific cases included in the GCCs. The GCCs can be applied not only to Coulombic systems but also to any systems where the interaction between two particles is represented in a power series of the inter-particle distance. We confirmed the correctness of our derivation of these GCCs by applying the free complement wave functions of a hydrogen atom in ground and excited states, a harmonic oscillator, and a system with an interacting potential of V = r.

Some model inhomogeneous electron liquid in D dimensions: relation between energy and chemical potential and a spatial generalisation of Kato's nuclear cusp theorem

After a brief survey on the application of the Thomas–Fermi statistical method to bare Coulomb potential satisfying Laplace equation in D dimensions, which then relates energy and chemical potential, we next focus attention on model atomic ions with merely one and two electrons in D dimensions. In particular, for a bare Coulomb field we use the nodeless radial eigenfunctions given by Herschbach in D dimensions to derive the spatial generalization of Kato's nuclear cusp theorem in D dimensions for D > 1. The unbinding of H − as a function of dimensionality D is also briefly referred.

Electron-nucleus cusp correction scheme for the relativistic zeroth-order regular approximation quantum Monte Carlo method

A cusp correction scheme for the relativistic zeroth-order regular approximation (ZORA) quantum Monte Carlo method is proposed by extending the nonrelativistic cusp correction scheme of Ma et al. [J. Chem. Phys. 122, 224322 (2005)]. In this scheme, molecular orbitals that appear in Slater-Jastrow type wave functions are replaced with the exponential-type correction functions within a correction radius. Analysis of the behavior of the ZORA local energy in electron-nucleus collisions reveals that the Kato's cusp condition is not applicable to the ZORA QMC method. The divergence of the electron-nucleus Coulomb potential term in the ZORA local energy is remedied by adding a new logarithmic correction term. This method is shown to be useful for improving the numerical stability of the ZORA-QMC calculations using both Gaussian and Slater basis functions.

Finite-element gauge function optimization for accurate current density calculations

The optimum gauge function is determined for an $\text{H}_{2}{}^{+}$ ion in an ultra strong magnetic field by using the Kennedy-Kobe theory [P. K. Kennedy and D. H. Kobe, Phys. Rev. A 30, 51 (1984)] combined with the finite element method. The divergence-free current density obtained by this method vanishes at the positions of protons. From this observation, we put the question as to whether there exists the nuclear cusp condition of current density or not, which is analogous to Kato's cusp condition of electron density. We will show a few cases where the current density vanishes exactly at the maximum of electron density.

Theoretical study of the (e,3e) double ionization of the K and L shells of beryllium by fast electrons

The production of Be2+(1s2) and hollow Be2+(2s2) in an (e,3e) collision between an electron and neutral Beryllium is studied theoretically. The variations of the fully differential cross sections (FDCS) of the two ionisation cases in terms of incident and ejection energies and detection angles are analysed. Aiming to probe electron-electron correlation, compact analytical wave correlated Jastrow functions satisfying Kato's cusp conditions are used for the four-electron Be initial state as well as for the two-electron Be2+ final states. The two ejected electrons are described by 3C correlated functions.

Second order coalescence conditions of molecular wave functions

Kato's cusp condition gives the exact first order dependence of molecular wave functions on interparticle separation near the coalescence of two charged particles. We derive conditions correct to second order in interparticle separation, which concern second order derivatives of the wave function at the coalescence point. For identical particle coalescence, we give equations correct to third order. In addition to a universal, particle dependent term, a system and state dependent term arises in the higher order conditions, which we interpret as an effect of Coulombic screening. We apply our analysis to the standard orbital-based methods of quantum chemistry and discuss the implications for Jastrow- and R12-type correlation factors.

Correlated analytic ground-state densities for two-electron atomic ions from Be to Ne

Abstract In earlier work, we demonstrated that the three-parameter correlated variational ground-state wave function of Chandrasekhar for two-electron atomic ions led to a closed analytic ground-state electron density ρ ( r ) . Here, we determine the three parameters by imposing (i) Kato's cusp condition at the nucleus, (ii) the correct large-r behavior of ρ ( r ) involving the ionization potential, and (iii) the electrostatic potential created by ρ ( r ) at the nucleus, as given by the 1 / Z expansion of atomic theory. It is then found, from results obtained for Be through Ne, that the electron–electron coalescence condition is increasingly well accounted for as the atomic number Z increases. Finally, the one-body potential of density functional theory is extracted from ρ ( r ) .

Role of the cusp conditions in electron-helium double ionization

We develop two variational wave functions of helium which give accurate energy values and satisfy Kato's cusp conditions adapted to the variational procedure. These wave functions are utilized in calculations for reproducing the electron-helium double-ionization experimental data involving both the small and large momentum transfer. A comparison of the present numerical results for the differential cross sections with experiment indicates a minor role of the cusp conditions.

Conditions for accurate description of the electron density of atoms and molecules

AbstractBased on the Kato's cusp condition of the electron density and our recent relations for local strongly decaying properties in an electronic system, necessary conditions for trial electron densities of atomic and molecular systems are derived. These conditions take the form of integral‐differential equations, and their validity is verified numerically. The relevance of these conditions to the Thomas–Fermi problem in the orbital‐less density functional approach is discussed. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

A wave function for three charged particles with arbitrary masses

In this work we present a fully correlated wave function to describe the continuum of the three-body spectra for particles with arbitrary masses interacting with Coulomb potentials. We show that this function can be related with the Φ 2 correlated model introduced few years ago for ion–atom collisions processes. The function includes important properties such as correct asymptotic conditions and Kato's cusp conditions. We discuss the properties of the wave function for one heavy and two light particles, focusing in the case of positron–electron–proton systems. A comparison between the properties of the new model and the C3 one is also performed. This analysis suggests that this function can be a successful alternative choice for the description of positron–atom collisions.

Comment on calculations of excited states by the density functional theory

Valone and Capitani [Phys. Rev. A 23 (1981) 2127] proposed a method for calculating a single selected excited state by finding the ground state energy of the square of a shifted Hamiltonian. Here we prove that this approach can not enable one to calculate excited states on the basis of the Hohenberg and Kohn density functional theory. The justification for the use of the density functional theory for a single excited state on the basis of Kato's cusp theorem is briefly discussed.

Slow antiprotons in an extended electron system: the one-particle aspects of complete screening

A partially linearized response theory of the Thomas–Fermi–Weizsäcker screening is used to study charged particles interacting with jellium. This simple theory can be derived by linearizing contributions involving the Thomas–Fermi kinetic energy-density. The calculations are based on a nonlinear inhomogeneous integrodifferential equation which includes the potential along the lines suggested by the original Thomas–Fermi–Weizsäcker theory. The coefficient of the von Weizsacker term is prescribed according to Kato's cusp condition. This theoretical framework is applied to compute the stopping power of a jellium for slow antiprotons and phase-shifts sums of Friedel and Zwerger for the complete screening in an extended system. Illustrative results are presented and discussed.

Hartree–Fock wave functions with a modified GTO basis for atoms

We have solved the atomic Hartree)Fock equations by using the algebraic approach, expanding the single-particle radial wave function in terms of a . modified Gaussian type orbitals GTOs basis. Several atomic properties such as Kato's cusp condition for the electron density or the correct asymptotic behavior of the electron momentum density distribution are accurately verified. Additionally the energy of the atomic ground state can be obtained by using a smaller number of basis functions than in standard GTO expansions. This study has been performed for several atoms of the first three rows. Q 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 :5 9 )64, 1997

Spatial generalizations of Kato's cusp condition for two-electron atoms with correlations

In this work we study two functions, f(r) and g(r), related to spatial generalizations of Kato's cusp condition for the one-electron density. In particular, we evaluate these functions using accurate correlated densities for low-lying S-states of two-electron atoms. These functions reveal the effects of the electron repulsion and are very sensitive to Coulomb correlations, showing interesting features in the comparison between singlet and triplet states. In addition to this, some relationships between and radial expectation values, previously found from the HF properties of these functions, are extended to more accurate descriptions of these states.

Cusp relations for local strongly decaying properties in electronic systems.

Several expressions are derived that allow one to evaluate any local property at nuclear cusps in atoms and molecules. Applications of the formulas to the Hartree-Fock density and its derivatives for first- and second-row atoms show agreement with Kato's cusp condition. Applications to the exact exchange-only and exchange-correlation energy potential for neutral atoms are given. It is shown that for an atom, the values of ${\mathit{V}}_{\mathrm{x}}$(0) and ${\mathit{V}}_{\mathrm{xc}}$(0) are both close to the negative of the charge of the nucleus.

Single-particle and electron-pair densities at the origin in the ground state of helium-like ions

Here we study the single-particle and the electron-pair densities for the ground state of two-electron atomic systems using the wavefunctions which provide the best values for the energy known up to now. Special attention is paid to the convergence of some properties of the densities when we increase the dimension of the basis used. With these functions we have increased significantly the precision in the determination of the single-particle and the electron-pair densities at the origin and, therefore, of Kato's cusp conditions.