Intracule Densities Research Articles

Plain Capping for Improved Accuracy of Approximate One- and Two-Electron Densities at Two-Particle Coalescence Points.

The values of the one-electron and intracule densities at two-particle coalescence points that enter the expressions for relativistic corrections to energies of Coulombic systems cannot be efficiently computed with sufficient accuracy from approximate wave functions expressed in terms of cuspless basis functions such as the explicitly correlated Gaussians. A new approach to alleviation of this problem, called plain capping, is proposed. Unlike those offered by the previously published formalisms, such as the expectation value identities and integral transforms, the accuracy improvements effected by the plain capping are attained with negligible computational effort and minimum programming. In the case of the on-top two-electron densities, whose accurate computation is particularly costly, the plain capping constitutes the only viable means of error reduction available at present.

Relativistic effects on electronic pair densities: A perspective from the radial intracule and extracule probability densities.

While the effect of relativity in the electronic density has been widely studied, the effect on the pair probability, intracule, and extracule densities has not been studied before. Thus, in this work, we unveil new insights related to changes in the electronic structure caused by relativistic effects. Our numerical results suggest that the mean inter-electronic distance is reduced (mostly) due to scalar-relativistic effects. As a consequence, an increase in the electron-electron repulsion energy is observed. Preliminary results suggest that this observation is also valid when electronic correlation effects are considered.

Insights from intracules and Coulomb holes

We point out that a typical two-electron distribution function in atoms and molecules, often called the intracule, has a particular sensitivity to the electron–electron repulsion, which leads to the so-called Coulomb correlation. The difference between the intracule densities is computed using the correlated and uncorrelated wave functions, and has been given the name Coulomb hole. The study of intracule densities and Coulomb holes is a subject of considerable current interest. We make use of a three-parameter correlated wave function and its uncorrelated limit to study the properties of intracules and Coulomb holes in He, Li+, Be2+ and Ne8+, providing a transparent physical picture for the interplay between the electron–nucleus attraction and electron–electron repulsion in forming Coulomb holes around electrons in atoms of different Z values.

Electron correlation in Li+, He, H- and the critical nuclear charge system Z C : energies, densities and Coulomb holes.

This paper presents high-accuracy correlation energies, intracule densities and Coulomb hole(s) for the lithium cation, helium, hydride ion and the system with the critical nuclear charge, ZC, for binding two electrons. The fully correlated (FC) wave function and the Hartree–Fock (HF) wave function are both determined using a Laguerre-based wave function. It is found that for the lithium cation and the helium atom a secondary Coulomb hole is present, in agreement with a previous literature finding, confirming a counterintuitive conclusion that electron correlation can act to bring distant electrons closer together. However, no evidence for a tertiary Coulomb hole is found. For the hydride anion and the system just prior to electron detachment only a single Coulomb hole is present and electron correlation decreases the probability of finding the electrons closer together at all radial distances. The emergence of a secondary Coulomb hole is investigated and found to occur between Z = 1.15 and Z = 1.20. The FC and HF energies and intracule densities (in atomic units) used to calculate the correlation energy and Coulomb hole, respectively, are accurate to at least the nano-scale for helium and the cation and at least the micro-scale for the anions.

Electron-Pair Distribution in Chemical Bond Formation.

The chemical formation process has been studied from relaxation holes, Δh(u), resulting from the difference between the radial intracule density and the nonrelaxed counterpart, which is obtained from atomic radial intracule densities and the pair density constructed from the overlap of the atomic densities. Δh(u) plots show that the internal reorganization of electron pairs prior to bond formation and the covalent bond formation from electrons in separate atoms are completely recognizable processes from the shape of the relaxation hole, Δh(u). The magnitude of Δh(u), the shape of Δh(u) ∀ u < Req, and the distance between the minimum and the maximum in Δh(u) provide further information about the nature of the chemical bond formed. A computational affordable approach to calculate the radial intracule density from approximate pair densities has been also suggested, paving the way to study electron-pair distributions in larger systems.

Salient signature of van der Waals interactions

van der Waals interactions govern the physics of a plethora of molecular structures. It is well known that the leading term in the distance-based London expansion of the van der Waals energy for atomic and molecular dimers decays as $1/{R}^{6}$, where $R$ is the dimer distance. Using perturbation theory, we find the leading term in the distance-based expansion of the intracule pair density at the interatomic distance. Our results unveil a universal $1/{R}^{3}$ decay, which is less prone to numerical errors than the $1/{R}^{6}$ dependency, and it is confirmed numerically in ${\mathrm{H}}_{2}$ and ${\mathrm{He}}_{2}$ molecules. This signature of van der Waals interactions can be directly used in the construction of approximate pair density and energy functionals including vdW corrections.

Quantum effects of nuclear motion in three-particle diatomic ions

A high-accuracy, nonrelativistic wave function is used to study nuclear motion in the ground state of three-particle ${{a}_{1}^{+}{a}_{2}^{+}{a}_{3}^{\ensuremath{-}}}$ electronic and muonic molecular systems without assuming the Born-Oppenheimer approximation. Intracule densities and center-of-mass particle densities show that as the mass ratio ${m}_{{a}_{i}}/{m}_{{a}_{3}}, i=1,2$, becomes smaller, the localization of the like-charged particles (nuclei) ${a}_{1}$ and ${a}_{2}$ decreases. A coordinate system is presented to calculate center-of-mass particle densities for systems where ${a}_{1}\ensuremath{\ne}{a}_{2}$. It is shown that the nuclear motion is strongly correlated and depends on the relative masses of the nuclei ${a}_{1}$ and ${a}_{2}$ rather than just their absolute mass. The heavier particle is always more localized and the lighter the partner mass, the greater the localization. It is shown, for systems with ${m}_{{a}_{1}}&lt;{m}_{{a}_{2}}$, that the ratio of (i) the density maximum and (ii) the FWHM of the radial distribution of each nucleus from the center of mass is directly proportional to the mass ratio of the nuclei: ${m}_{{a}_{1}}/{m}_{{a}_{2}}$ for the former and ${m}_{{a}_{2}}/{m}_{{a}_{1}}$ for the latter, thus quantifying a quantum effect of nuclear correlation.

A simultaneous probability density for the intracule and extracule coordinates

We introduce the intex density X(R,u), which combines both the intracular and extracular coordinates to yield a simultaneous probability density for the position of the center-of-mass radius (R) and relative separation (u) of electron pairs. One of the principle applications of the intex density is to investigate the origin of the recently observed secondary Coulomb hole. The Hartree-Fock (HF) intex densities for the helium atom and heliumlike ions are symmetric functions that may be used to prove the isomorphism 2I(2R)=E(R), where I(u) is the intracule density and E(R) is the extracule density. This is not true of the densities that we have constructed from explicitly correlated wave functions. The difference between these asymmetric functions and their symmetric HF counterparts produces a topologically rich intex correlation hole. From the intex hole distributions (X(exact)(R,u)-X(HF)(R,u)), we conclude that the probability of observing an electron pair with a very large interelectronic separation increases with the inclusion of correlation only when their center-of-mass radius is close to half of their separation.

Anisotropic intracule densities and electron correlation in H2: A quantum Monte Carlo study

We derive efficient quantum Monte Carlo estimators for the anisotropic intracule and extracule densities. These estimators are used in conjunction with an accurate explicitly correlated wave function to investigate the bond-length dependence of electron correlation effects in the ground-state H(2) molecule. It is shown that the localized increase in the magnitude of the correlation energy as the bond is stretched is accompanied by highly anisotropic correlation effects. In addition, we find a small long-range part of the Coulomb hole, which is present even at the equilibrium bond length.

Visualizing molecular wavefunctions using Monte Carlo methods

AbstractUsing explicitly correlated wavefunctions and variational Monte Carlo we calculate the electron density, the electron density difference, the intracule density, the extracule density, two forms of the kinetic energy density, the Laplacian of the electron density, the Laplacian of the intracule density, and the Laplacian of the extracule density on a dense grid of points for the ground state of the hydrogen molecule at three internuclear distances (0.6, 1.4, 8.0). With these values we construct a contour plot of each function and describe how it can be used to visualize the distribution of electrons in this molecule. We also examine the influence of electron correlation on each expectation value by calculating each function with a Hartree–Fock wavefunction and then comparing these values with our explicitly correlated values. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Correlation holes for the helium dimer

We have investigated the radial electron pair probability distributions (REPPDs) of the helium dimer within the Piris natural orbital functional (PNOF) theory. The analytical formulas to evaluate intracule densities, Fermi, Coulomb, and total correlation holes using our reconstruction functional PNOF-2 [J. Chem. Phys. 126, 214103 (2007)] are derived. The Löwdin's Coulomb holes from PNOF-2 and full configuration interaction calculations are analyzed showing a very similar behavior. New definitions of the Coulomb and Fermi holes based on the cumulant expansion of the two-particle reduced density matrix are presented. The holes are defined in terms of the exact one-particle reduced density matrix and the two-particle cumulant without any reference to the Hartree-Fock state. Through these definitions, we analyze separately the contribution of each component to the total REPPD at several values of the internuclear distance. A straight connection between the Coulomb hole and dispersion interactions is observed.

Intracule densities in the strong-interaction limit of density functional theory

The correlation energy in density functional theory can be expressed exactly in terms of the change in the probability of finding two electrons at a given distance r(12) (intracule density) when the electron-electron interaction is multiplied by a real parameter lambda varying between 0 (Kohn-Sham system) and 1 (physical system). In this process, usually called adiabatic connection, the one-electron density is (ideally) kept fixed by a suitable local one-body potential. While an accurate intracule density of the physical system can only be obtained from expensive wavefunction-based calculations, being able to construct good models starting from Kohn-Sham ingredients would highly improve the accuracy of density functional calculations. To this purpose, we investigate the intracule density in the lambda --> infinity limit of the adiabatic connection. This strong-interaction limit of density functional theory turns out to be, like the opposite non-interacting Kohn-Sham limit, mathematically simple and can be entirely constructed from the knowledge of the one-electron density. We develop here the theoretical framework and, using accurate correlated one-electron densities, we calculate the intracule densities in the strong interaction limit for few atoms. Comparison of our results with the corresponding Kohn-Sham and physical quantities provides useful hints for building approximate intracule densities along the adiabatic connection of density functional theory.

Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density

We construct improved quantum Monte Carlo estimators for the spherically and system-averaged electron pair density (i.e., the probability density of finding two electrons separated by a relative distance u), also known as the spherically averaged electron position intracule density I(u), using the general zero-variance zero-bias principle for observables, introduced by Assaraf and Caffarel. The calculation of I(u) is made vastly more efficient by replacing the average of the local delta-function operator by the average of a smooth nonlocal operator that has several orders of magnitude smaller variance. These new estimators also reduce the systematic error (or bias) of the intracule density due to the approximate trial wave function. Used in combination with the optimization of an increasing number of parameters in trial Jastrow-Slater wave functions, they allow one to obtain well converged correlated intracule densities for atoms and molecules. These ideas can be applied to calculating any pair-correlation function in classical or quantum Monte Carlo calculations.

Non-Born–Oppenheimer treatment of the H2 Hookean molecule

We show that the exact non-Born-Oppenheimer Schrodinger equation for the Hookean diatomic molecule H2 (a two-proton, two-electron system where the electron-proton interaction is harmonic while the proton-proton and electron-electron interactions are Coulombic) can be decoupled into equations describing the relative motion of the electrons, the relative motion of nuclei, the motion of a collective mode representing a three-dimensional harmonic oscillator, and the motion of a free particle expressed as a linear combination of the individual center-of-mass coordinates of the nuclei and electrons. Analytic solutions to the relative motion of electrons can be readily obtained for the given values of the harmonic coupling constant. However, exact analytic solutions to the equation for the relative motion of the nuclei cannot be obtained simultaneously due to the fact that the harmonic constants in these two equations are coupled. For this reason, we present for the relative nuclear motion approximate analytic wave functions, one of them obtained variationally and the other by a series solution where the coefficients are determined recursively. We also explore a variational solution to the Taylor-series expansion of the nuclear interaction potential. Properties of the electronic and nuclear intracule densities are examined at different values of the coupling constant. An interesting result of the present non-Born-Oppenheimer treatment of this harmonic model is the fact that the relative nuclear motion occurs in a highly correlated regime. This leads in a natural way to a spatial localization of the nuclei akin to Wigner electronic crystallization.

Electron correlation studies by means of local-scaling transformations and electron-pair density functions

Electron correlation is one long standing problem of computational electronic structure theory. Even more, with the advent of the density functional theory and, in particular, with its Kohn–Sham implementation, the separation of the non-dynamical and dynamical components of the electron correlation has became an unavoidable requirement towards construction of reliable exchange-correlation functionals. In this paper, we address the analysis of the separation of the non-dynamical and dynamical electron correlation effects from two complementary viewpoints, namely, analysis of the correlation energy components and the analysis of the electron-pair density. The former approach will make use of the local-scaling transformations and the latter will be based on the study of intracule and extracule densities.

Excited states of beryllium atom from explicitly correlated wave functions

A study of the first excited states of beryllium atom starting from explicitly correlated wave functions is carried out. Several properties are obtained and discussed focusing on the analysis of the Hund’s rules in terms of the single-particle and electron pair intracule and extracule densities. A systematic study of the differences on the electronic distributions of the singlet and triplet states is carried out. The trial wave function used to describe the different bound states consists of a generalized Jastrow-type correlation factor times a configuration interaction model wave function. This model wave function has been fixed by using a generalization of the optimized effective potential method to deal with multiconfiguration wave functions. The optimization of the wave function and the calculation of the different quantities is carried out by means of the Variational Monte Carlo method.

Computation and analysis of molecular Hartree—Fock momentum intracules

An efficient general algorithm for the computation of molecular momentum intracule densities from Hartree-Fock wavefunctions using Gaussian basis functions is described. The momentum intracules for a number of systems are examined, and comparison with their position space counterparts discussed.

Comparison of two‐electron densities reconstructed from one‐electron density matrices

AbstractWe consider approximate reconstruction of spinless and spin‐resolved two‐electron charge densities, intracule densities, and extracule densities utilizing only the one‐electron reduced density matrix. Using large basis set, configuration interaction density matrices for the ground state of Be, we compare the two‐electron densities obtained using several proposed reconstruction functionals (“natural orbital functionals”) for the two‐electron density and density matrix. We also analyze the intracule and extracule densities obtained from these reconstructions. Several of the proposed natural orbital functionals are found to produce unphysical negative values for the pair density when two parallel‐spin electrons approach one another. Ancillary results from this study include new, simplified formulas for the intracule and extracule densities, and a mathematical justification for an approximate relation between atomic intracule and extracule densities that has previously been reported on the basis of extensive Hartree–Fock calculations. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

On the extremum property of electron density functions in position and momentum spaces

For statistically uncorrelated wave functions, the electron density ρ( r) is known to be an extremum function of the generalized electron-pair density g( q; a, b), which smoothly connects ρ( r), the electron-pair intracule (relative motion) density h( u), and the electron-pair extracule (center-of-mass motion) density d( R). The present systematic examination of the numerical Hartree–Fock results of moments 〈 r n 〉, 〈 u n 〉, 〈 R n 〉 and density values at the origin ρ(0), h(0), d(0) for 102 neutral atoms and 96 singly charged ions suggests that ρ( r) is local maximum for a small q and is local minimum for a large q of the function g( q; a, b). Analogous results are obtained for the momentum-space counterparts Π( p) and g ̄ (t;a,b) .

A generalized electron-pair density function for atoms

For many-electron atoms, we discuss a generalized electron-pair density function g(q;a,b) that represents the probability density function for the magnitude |ari+brj| of the two-electron vector ari+brj to be q, where a and b are real-valued parameters. It is pointed out that the function g(q;a,b) connects smoothly the single-electron density ρ(r), the electron-pair intracule (relative motion) density h(r12), and the electron-pair extracule (center-of-mass motion) density d(R). Moreover, ρ(r) is found to be a local extremum function of g(q;1,b) with respect to the parameter b. Analogously, the single-electron moments 〈rn〉 are local extrema of the moments 〈qn〉(a,b) associated with g(q;a,b). An illustrative example is given for the helium atom within the Kellner approximation.