Nuclear Cusp Condition Research Articles

Exchange correlation potentials from full configuration interaction in a Slater orbital basis.

Ryabinkin-Kohut-Staroverov (RKS) theory builds a bridge between wave function theory and density functional theory by using quantities from the former to produce accurate exchange-correlation potentials needed by the latter. In this work, the RKS method is developed and tested alongside Slater atomic orbital basis functions for the first time. To evaluate this approach, full configuration interaction computations in the Slater orbital basis are employed to give quality input to RKS, allowing full correlation to be present along with correct nuclei cusps and asymptotic decay of the wavefunction. SlaterRKS is shown to be an efficient algorithm to arrive at exchange-correlation potentials without unphysical artifacts in moderately-sized basis sets. Furthermore, enforcement of the nuclear cusp conditions will be shown to be vital for the success of the Slater-basis RKS method. Examples of weakly and strongly correlated molecular systems will demonstrate the main features of SlaterRKS.

Nuclear cusps and singularities in the nonadditive kinetic potential bifunctional from analytical inversion

The non-additive kinetic potential $v^{\text{NAD}}$ is a key quantity in density-functional theory (DFT) embedding methods, such as frozen density embedding theory and partition DFT. $v^{\text{NAD}}$ is a bi-functional of electron densities $\rho_{\rm B}$ and $\rho_{\rm tot} = \rho_{\rm A} + \rho_{\rm B}$. It can be evaluated using approximate kinetic-energy functionals, but accurate approximations are challenging. The behavior of $v^{\text{NAD}}$ in the vicinity of the nuclei has long been questioned, and singularities were seen in some approximate calculations. In this article, the existence of singularities in $v^{\text{NAD}}$ is analyzed analytically for various choices of $\rho_{\rm B}$ and $\rho_{\rm tot}$, using the nuclear cusp conditions for the density and Kohn-Sham potential. It is shown that no singularities arise from smoothly partitioned ground-state Kohn-Sham densities. We confirm this result by numerical calculations on diatomic test systems HeHe, HeLi$^+$, and H$_2$, using analytical inversion to obtain a numerically exact $v^{\rm NAD}$ for the local density approximation. We examine features of $v^{\rm NAD}$ which can be used for development and testing of approximations to $v^{\rm NAD}[\rho_{\rm B},\rho_{\rm tot}]$ and kinetic-energy functionals.

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.

Finite element calculations for systems with multiple Coulomb centers

The presence of multiple Coulomb centers in molecules or solids poses a challenge when solving the effective Schrödinger equation, required as a crucial ingredient in density functional or Hartree–Fock calculations. This is primarily because Kato’s cusp condition needs to be satisfied close to each nucleus and the matrix elements of the Coulomb potential at the nuclei are rather difficult to evaluate when using global basis functions. A novel method for dealing with these challenges is introduced, rewriting the wavefunction as a product of a function satisfying the nuclear cusp conditions and a smooth function, resulting in a transformed variational principle and a regularized potential. Results of three-dimensional finite element calculations based on this ansatz for the ground state of the molecule H2+ in the Born–Oppenheimer approximation are presented, which were obtained using custom written Python/Fortran code.

Gauge function optimization 2: An accurately solvable model

AbstractThe gauge function is optimized by using the Kennedy–Kobe (KK) theory [Kennedy and Kobe, Phys Rev A, (1984) 30, 51] for the anisotropic harmonic oscillator model in a magnetic field. We extend the KK model to describe various electron density distributions, which resemble to those for the ground and excited states of a H ion. The almost divergence‐free current densities can be obtained by this method. The convergence is the fastest for the h2 model without the node of the wavefunction. When the wavefunction has nodal points, the first‐ and the second‐order derivatives of the gauge function diverge at the nodes. We examine these singularities of the gauge function near the nodes both numerically and analytically. We also investigate the computed divergence‐free current density in detail. Many new stagnation vortices appear after the gauge function optimization. The positions of the vortices are compared with those of the electron density maxima. When the maxima are separated from each other, some correlations are found between them. We also investigate the nuclear cusp condition of current density by inspecting the solutions of the KK equation when the electron density has a cusp. We find that the local mean velocity becomes a C0 continuous function at the nucleus. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

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.

Cusps and derivatives for wave‐functions expanded in Slater orbitals: A density study

AbstractThis 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, N2, and CO2 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. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Two-parameter partially correlated ground-state electron density of some light spherical atoms from Hartree-Fock theory with nonintegral nuclear charge

Partially correlated ground-state electron densities for some spherical light atoms are calculated, into which nonrelativistic ionization potentials represent essential input data. The nuclear cusp condition of Kato is satisfied precisely. The basic theoretical starting point, however, is Hartree-Fock (HF) theory for the N electrons under consideration but with nonintegral nuclear charge Z′ slightly different from the atomic number Z (=N). This HF density is scaled with a parameter λ, near to unity, to preserve normalization. Finally, some tests are performed on the densities for the atoms Ne and Ar, as well as for Be and Mg. © 2007 The American Physical Society.

Hartree–Fock orbitals which obey the nuclear cusp condition

Recently we have constructed a self-consistent field program which uses Slater-type orbital (STO) basis sets. Here, we show how to amend the program so that orbitals are obtained that exactly satisfy the nuclear cusp conditions. Example calculations are presented, and in particular the question of the choice of appropriate STO basis functions is addressed. We examine variational Monte Carlo and the behaviour of the local energy using nuclear cusp-obeying orbitals. We also investigate properties which require the value of the wavefunction at the nucleus; in particular the Fermi contact term is examined.

The molecular physics lecture 2004: (i) Density functional theory, (ii) Quantum Monte Carlo

In this combination of review and new science, we shall look at Density Functional Theory (DFT) and the Quantum Monte Carlo (QMC) method. For DFT, we shall (i) discuss the inadequacy of the Uniform Electron Gas model for chemistry (ii) examine Generalized Gradient Approximation (GGA) exchange functionals (iii) understand that they introduce left–right correlation for molecules (iv) argue that the functionals BLYP and OLYP are parameter-free for molecules (v) look at the ‘remarkable’ performance of these local functionals in the prediction of binding energies (vii) discuss hybrid functionals, and show how they can be improved through the introduction of an Ewald partitioning of 1/r 12, already examined by Hirao, Savin and Gill (viii) show that our new functional CAM-B3LYP offers good energetics as well as good charge transfer excitation energies, already shown by Hirao. For QMC, concentrating on the Diffusion Monte Carlo approach, we shall (i) relate the trial wavefunction to the transcorrelated wavefunction, and look again at the transcorrelated method, (ii) look at the nodal surfaces of the trial wavefunction, demonstrating in particular that the use of finite basis sets in the Hartree–Fock method necessarily introduces ‘spurious’ nodal features which probably cannot be ignored, (iii) argue that, now it is possible to perform Hartree–Fock calculations with slater-type basis sets, it is straightforwardly possible to obtain orbitals which obey the exact nuclear cusp conditions, (iv) therefore argue that these basis sets should be used in Monte Carlo studies.

Recent progress in constructing nonlocal energy density functionals

AbstractBoth model results and some exact relations concerning kinetic, exchange, and correlation energy density functionals will be presented. As to single‐particle kinetic energy, a model of harmonic confinement, motivated by recent experimental activity on Fermion vapors, of an arbitrary number of Fermions will be used to construct exact nonlocal kinetic energy density functionals in both one and three dimensions. Then, some further exact results concerning nuclear cusp conditions for exchange only, and for exchange plus correlation, will be discussed. Finally, in Appendices, the one‐shell harmonic confinement is generalized to include Coulomb repulsion in the two‐electron Hookean atom and some results on the Pauli potentials for many‐Fermion systems are obtained. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

Spatial generalization of the H atom nuclear cusp condition for kinetic energy density for closed shells in a bare Coulomb field

In a recent paper (N.H. March et al., Phys. Rev. A 63 (2000) 012520), it has been shown that there are nuclear cusp conditions for various energy densities of density functional theory, embracing exchange and correlation contributions as well as the full electron–electron interaction energy density. Here, for the single-particle kinetic energy density, an exact nuclear cusp condition for the H atom is shown to have a spatial generalization for an arbitrary number of closed shells in a bare Coulomb field.

Exchange energy density and some approximate exchange potentials obtained from Hartree–Fock theory of the ground state of the Be atom

Hartree–Fock (HF) theory of the ground state of the Be atom is used to calculate first the exchange energy density ϵ x ( r) from the Dirac density matrix. Beyond r=2 a 0, with a 0=ℏ 2/ me 2, ϵ x ( r) rapidly approaches the general asymptotic form − 1 2 e 2ρ(r)/r , with ρ( r) the HF electronic density. The nuclear cusp condition 1 ϵ x(r) ∂ϵ x ∂r r→0=− 2Z a 0 with atomic number Z=4, is also accurately satisfied by the present numerical data. Since a quantum Monte Carlo (QMC) exchange-correlation potential exists for the Be atom, we have compared this with (a) the Slater potential V SL( r)=2 ϵ x ( r)/ ρ( r) and (b) the Harbola–Sahni form. Both have the main features of the QMC exchange-correlation potential, though the magnitude of V SL( r) at r=0 is too large by some 16%. We have also studied how well these two approximate HF exchange potentials fare when inserted into the Levy–Perdew relation between the total exchange energy and the `virial-like' form involving the gradient of the exchange potential.

Nuclear cusp conditions for components of the molecular energy density relevant for density-functional theory

Nuclear cusp conditions are obtained for the full electron-electron interaction energy density as well as for the exchange and correlation energy densities of density-functional theory. Their form is the same as the form of the well known Kato cusp condition for the electron-number density. All these cusp conditions are valid for both the ground and excited states of a molecule or solid.

Fukui function from a gradient expansion formula, and estimate of hardness and covalent radius for an atom

The Fukui function for a neutral atom is expressed as its LDA approximation plus a one-parameter gradient correction, and the resultant formula is numerically tested. Expressing hardness as a density functional involving this Fukui function, global hardness values are determined for several atoms. Estimates also are made of the covalent radii of neutral atoms. Calculated Fukui functions exhibit characteristics similar to those reported in the literature. Calculated hardnesses compare favorably with experimental values, and predicted covalent radii are in agreement with existing theoretical values and experimental data. No information other than the electron densities of the neutral species enter in the calculations. An exact nuclear cusp condition on the Fukui function is derived.

Stopping power of an electron gas for antiprotons at intermediate velocities.

The stopping power of antiprotons moving at energies up to 400 keV through a uniform zero-temperature electron gas is calculated within the framework of the kinetic theory. The momentum-transfer cross section required is determined with the aid of a parametric velocity-dependent scattering potential. A nonperturbative constraint, provided by the nuclear-cusp condition, is used to fix the parameter value. A comparison with the result of a quadratic-response-function approach is made. The calculated stopping powers are in good agreement with recent experimental predictions.

Thomas-Fermi-Dirac models of atoms constrained by nuclear cusp and long-range conditions

The traditional Thomas–Fermi–Dirac model of the electronic structure for a neutral atom is deficient in that it predicts an infinite electron density at the nucleus and a sharp cutoff of the electron density at a finite radius. This study was carried out to remedy these faults in the model. Extending an idea used earlier in Thomas–Fermi (TF) theory [Proc. Natl. Acad. Sci. U.S.A. 83, 3577 (1985)], the Thomas–Fermi–Dirac (TFD) energy functional is minimized under constraints ∫ρ(r) dr = N, ∫e−2kr▽2ρ(r)dr < ∞ and ∫(1 − e−k′r)ρ4/3(r)dr < ∞, with k and k′ determined by the nuclear cusp condition and the correct asymptotic behavior. Optimum coordinate scaling also is considered. It is found that the TFD model is substantially improved by constraining the minimization search domain of the energy functional in this way. Energies are given for five noble gas atoms, and Compton profiles for these atoms are calculated. The behavior of electrons in momentum space is improved in both this modified TFD model and in the corresponding modified TF model.

Stopping power of a finite-temperature electron gas for slow unit charges.

The stopping power of protons and antiprotons moving at low velocities through a uniform temperature-dependent electron gas is calculated by making use of the momentum-transfer cross section. The screened scattering potential required is determined with the aid of a parametric dielectric function obtained from the linearized Thomas--Fermi--von Weizsaecker treatment for screening in extended systems. A nonperturbative constraint, provided by the nuclear-cusp condition for the total electron density at the position of the probe charge, is used to fix the parameter value. The temperature-dependent higher-order effect on the stopping power is discussed for metallic densities of the system, showing that first-order approximations are valid only asymptotically.

Appraisal of hermite-gaussian expansion of hydrogen orbitals by calculation of some one-electron properties

Hydrogen orbitals (HOs) are expanded in terms of Hermite-Gaussian (HG) functions thus making polycentric molecular integrals tractable. The quality of the basis sets is examined by calculating various one-electron properties and similarity measures (SM) between approximate HG expansions and exact HOs. It is shown that σ d,< r 2>, < r> and SM may serve as a riddle in eliminating unsatisfactory basis sets. The average distance < r> appears to be rather insensitive in the H atom in contrast to the molecular case. Hermite-Gaussian functions of the fourth degree (HG4-N ) proved very useful in describing HOs. The highest accuracy is obtained for HOs obeying the n- l = 1 condition. It is concluded that HG4 functions are advantageous for HOs of higher quantum numbers n and l, being superior to 1 s GTO functions possessing the equivalent number of nonlinear parameters. It follows that mixed sets employing STOs for inner-shells and HOs for outer valence shell and polarization functions might be useful in molecules involving heavy atoms provided that they are expanded in HG functions. The nuclear cusp condition is briefly discussed and it is shown that implementation of the off-center HG function ( s ) might improve behavior of AOs near the nucleus without considerable increase in computational effort. It should be pointed out that the corresponding hydrogen-like functions are simply obtained by a rescaling procedure.

Compact expressions for the radial electronic density functions for the 2S states of three-electron systems

The radial electronic density function D0(r) has been evaluated in closed form for the 2S states of three-electron systems that are described by Hylleraas-type wave functions. The density function D0(r) can be reduced to the form: D0(r)=𝒥I=17 𝒥K=0gI 𝒜IKrKe−αIr. Numerical values of the expansion coefficients 𝒜IK, summation limits gI and exponents αI are determined for the ground states of the following members of the Li I isoelectronic series: Li, Be+, B2+, C3+, N4+, O5+, F6+, and Ne7+. A discussion is given on the constraints that must be imposed on the choice of the basis set for the Hylleraas wave function, in order that D0(r) be reducible to the aforementioned compact analytic form. Expectation values for several moments 〈rn 〉 are calculated using D0(r). The electron–nuclear cusp condition is evaluated for the wave functions used to determine D0(r) for each member of the Li I sequence examined in this investigation.