Intracule Densities Research Articles

Electron-pair densities in position and momentum spaces for multi-determinant wavefunctions

The electronic intracule (relative motion) and extracule (centre-of-mass motion) densities are electron-pair densities which characterize the motion of a pair of electrons in atoms and molecules. A unified method is presented for the evaluation of these electron-pair densities in both position and momentum spaces for wavefunctions expressed as linear combinations of Slater determinants. Detailed expressions are developed for atomic systems where angular integrations can be performed analytically. Interesting relations between atomic intracule and extracule densities and between their moments are discussed. An illustrative application of the results is given for the and states of the helium atom, and the first calculations are reported for the singlet-triplet differences in the extracule densities and in the momentum-space intracule density.

Electron-pair center-of-mass-motion densities of atoms in position and momentum spaces

Spherically averaged center-of-mass-motion (extracule) densities $d(R)$ in position space and $\overline{d}(P)$ in momentum space represent probability densities of finding center-of-mass radii $|{\mathbf{r}}_{j}+{\mathbf{r}}_{k}|/2$ and $|{\mathbf{p}}_{j}+{\mathbf{p}}_{k}|/2$ of any pair of electrons $j$ and $k$ to be $R$ and $P,$ respectively. Theoretical analysis shows that in the Hartree-Fock approximation, the individual spin-orbital-pair component of the extracule density has a definite relation with the corresponding one of the relative-motion (intracule) density, but the total extracule and intracule densities are not related in a simple manner. Using the numerical Hartree-Fock method, the extracule densities $d(R)$ and $\overline{d}(P)$ are constructed and examined systematically for the atoms from He to Xe in their ground state. In position space, the extracule density $d(R)$ is a monotonically decreasing function for all the atoms. In momentum space, however, the extracule densities $\overline{d}(P)$ are found to be classified into two types according to the location of a maximum. These different behaviors of the densities $d(R)$ and $\overline{d}(P)$ are studied in detail based on the contributions of electrons in a pair of atomic subshells and spin orbitals.

Electron-pair relative-motion densities of atoms in position and momentum spaces

Spherically averaged electron-pair relative-motion (intracule) densities $h(u)$ in position space and $\overline{h}(v)$ in momentum space represent probability densities of finding interelectronic position and momentum distances, $|{\mathbf{r}}_{j}\ensuremath{-}{\mathbf{r}}_{k}|$ and $|{\mathbf{p}}_{j}\ensuremath{-}{\mathbf{p}}_{k}|,$ of any pair of electrons $j$ and $k$ to be $u$ and $v,$ respectively. Using the numerical Hartree-Fock method, these electron-pair densities $h(u)$ and $\overline{h}(v)$ are constructed and examined systematically for the atoms from He to Xe in their ground state. In position space, the intracule density $h(u)$ is a monotonically decreasing function for all the atoms. In momentum space, however, the intracule densities $\overline{h}(v)$ are found to be classified into three types according to the number of local maxima and their locations. These different behaviors of the densities $h(u)$ and $\overline{h}(v)$ are studied in detail based on the contributions of electrons in a pair of atomic spinorbitals and subshells.

Second-order quantum similarity measures from intracule and extracule densities

The calculation of quantum similarity measures from second-order density functions contracted to intracule and extracule densities obtained at the Hartree-Fock level is presented and applied to a series of atoms, (He, Li, Be, and Ne), isoelectronic molecules (C2H2, HCN, CNH, CO, and N2), and model hydrogen-transfer processes (H2/H+, H2/Hot, H2/H−). Second-order quantum similarity measures and indices are found to be suitable measures for quantitatively analyzing electron-pair density reorganizations in atoms, molecules, and chemical processes. For the molecular series, a comparative analysis between the topology of pairwise similarity functions as computed from one-electron, intracule, and extracule densities is carried out and the assignment of each particular local similarity maximum to a molecular alignment discussed. In the comparative study of the three hydrogen-transfer reactions considered, second-order quantum similarity indices are found to be more sensitive than first-order indices for analyzing the electron-density reorganization between the reactant complex and the transition state, thus providing additional insights for a better understanding of the mechanistic aspects of each process.

The relevance of the Laplacian of intracule and extracule density distributions for analyzing electron–electron interactions in molecules

A topological analysis of intracule and extracule densities and their Laplacians computed within the Hartree–Fock approximation is presented. The analysis of the density distributions reveals that among all possible electron–electron interactions in atoms and between atoms in molecules only very few are located rigorously as local maxima. In contrast, they are clearly identified as local minima in the topology of Laplacian maps. The conceptually different interpretation of intracule and extracule maps is also discussed in detail. An application example to the C2H2, C2H4, and C2H6 series of molecules is presented.

Topology of electron–electron interactions in atoms and molecules. I. The Hartree–Fock approximation

Topologies of the electron intracule and extracule densities, I(R) and E(R), are analyzed. These topologies are found to be inherently more complex than those of the one-electron density. The main topological features of I(R) and E(R) are already present in the densities calculated within the Hartree–Fock (HF) approximation. Results of test calculations on several planar systems show that the positions and properties of attractors in I(R) and E(R) are predicted with a surprising fidelity by a naive independent-atom model, making it possible to index distinct types of electron pairs present in atoms and molecules. In general, each pair of atoms in a given molecule has the potential of producing a pair of attractors in I(R). At the HF level of theory, all the atoms collectively furnish a single attractor in I(R) at R=0, but this topological pattern is bound to change upon the inclusion of electron correlation. The attractors in E(R) stem from both individual atoms and atomic pairs. In addition, attractors that are not associated with either of these entities are observed. The plethora of attractors present in I(R) and E(R) give rise to complicated patterns of other critical points. Unusual topological features, such as attractors linked by multiple interaction lines and bifurcations at bond points, are also encountered.

Fast evaluation of electron intracule and extracule densities on large grids of points

A new approach to fast evaluation of the electron intracule and extracule densities on large grids of points is described. Substantial (50- to 100-fold) speed ups over the conventional algorithms are attained through the use of precomputed intermediates in the grid-dependent phase of calculations. These intermediates are evaluated only once in a grid-invariant procedure that employs efficient two-stage integral screening to reduce computational effort. In addition to delivering high performance, the new algorithm facilitates calculations of analytical gradients and Hessians of the intracule and extracule electron densities. For regular grids with shared components of Cartesian coordinates, the present method allows the factorization of the primitive quartet contributions that makes the cost of calculations proportional to the cubic root of the number of grid points.

Electron intracule densities with correct electron coalescence cusps from Hiller–Sucher–Feinberg-type identities

Identities for the electron intracule density I(R) in atoms and molecules are derived within the Hiller–Sucher–Feinberg (HSF) formalism. It is proven that, when applied to arbitrary (exact or approximate) electronic wave functions, these identities produce intracule densities that satisfy a modified condition for the electron coalescence cusp. A corollary of this proof provides a new, simplified derivation of the cusp condition for the exact I(R). An expression for the Hartree–Fock approximation to the HSF electron intracule density that contains only two- and three-electron terms is obtained and its properties are analyzed. A simple scaling of the three-electron contributions in this expression assures integrability of the approximate I(R) and improves its overall accuracy. Numerical tests carried out for the H−, He, Li+, Be2+, Li, and Be systems demonstrate that the application of the scaled HSF-type identity to Hartree–Fock wave functions affords dramatic improvements in the short-range behavior of the electron intracule density.

Bounds for the electron density at the nucleus and for the intracule density at the coalescence point for two-electron atoms

The electron density at the nucleus, π(0), and the interelectronic distance (intracule) density at the coalescence point,h(0), for two electron atoms, which play an important role in the description of the relativistic corrections to the ground state energy, are studied. Lower and upper bounds to these quantities are found analytically. The quality of the bounds is analyzed within the Hartree-Fock framework, for π(0), and by means of wave functions of Hylleraas-type forh(0).

Correlated one- and two-electron densities of low-lying S-states in helium-like atoms

From high-quality, Pekeris-type electronic wavefunctions we calculate the single-particle density rho (r) and the intracule density h(s) of some low-lying S-states of two-electron atoms, obtaining very accurate values for the energies, the cusp-condition ratios and some radial expectation values. The effect of electron correlation on both densities for these states is also Studied by comparison with Hartree-Fock results. Some local important differences appear not only for the ground state, but also for the excited ones. Our results agree with some features previously found, such as the non-monotonic decrease of rho (r) for the excited states. We also find that the Coulomb hole of singlet excited states is strongly dependent on the procedure used for the Hartree-Fock calculations.

Chemical binding and electron correlation effects in x-ray and high energy electron scattering

The total, inelastic and elastic intensities for x-ray and high energy electron scattering from the ten-electron molecules CH4, NH3, H2O, and HF have been calculated with configuration interaction wave functions. The probability density of the interelectronic distance or the radial intracule density is extracted from the intensities. The importance of basis sets, chemical binding, and electron correlation to the scattered intensities and the radial intracule density has been examined. It has been found that scattered intensities predicted with correlated wave functions agree well with experiment. Based on the electron pair distribution, the exchange and correlation holes in molecular systems have been briefly discussed.

Evaluation of electron pair densities and their Laplacians in atomic systems

Analytical formulae for calculating the intracule (relative position) and extracule (center of electron pair mass) pair densities in atomic systems are outlined. This formalism is straightforward and computationally efficient. The application has been demonstrated by computing the pair densities and the Laplacians of these densities for Ne from a near-Hartree-Fock wavefunction constructed from a Slater-type basis.

Electron‐pair distributions and chemical bonding

AbstractThe electron‐pair (intracule and extracule) densities of the first‐row hydrides may be understood on the basis of the different chemical‐bonding schemes in this series. Thus, the LiH pair densities show a strongly ionic nature and may be fairly well described by Li+H−. The CH‐pair densities, on the other hand, may be approximated by the promolecular superposition of a carbon and a hydrogen atom with an accumulation of pair density in the internuclear region signifying covalency. In the case of FH, its pair densities show a predominately ionic structure and are closer to those of F− than to those of the promolecular superposition of a fluorine and a hydrogen atom. The slight deformation of longitudinal pair densities observed in FH is largely due to the presence of the H+. © 1994 John Wiley & Sons, Inc.

Charge and intracule densities in singly excited heliumlike ions

The variation of the charge and intracule densities of singly excited states of two-electron ions with respect to spin multiplicity, nuclear charge, degree of excitation, and angular momentum quantum number is studied systematically. The n 1S, n 3S, n 1P, n 3P, n 1D, and n 3D states with n=3–6 are considered for all the ions from He through Ne8+ using highly accurate explicitly correlated wave functions. Special attention is paid to spin multiplicity differences, that is differences between the densities of a pair of states arising from the same electron configuration of the same ion.

Accurate algebraic densities and intracules for heliumlike ions

AbstractAccurate electron number densities and interelectronic distance (intracule) densities are given in simple algebraic form for the ground states of H−, He, Li+, Be2+, B3+, and Ne8+. The densities are obtained from systematic sequences of increasingly accurate Hylleraas‐type wave functions. The most accurate densities are obtained from 204‐term wave functions that lead to variational energies no more than 16 nano‐Hartrees above the exact the exact ones. The algebraic permit the evaluation of systematic sequences of many expectation values. Some moments of the densities are evaluated in this manner. © 1993 John Wiley & Sons, Inc.

The influence of electron correlation on anisotropic electron intracule and extracule densities

The electron intracule and extracule densities of the neon atom, the hydrogen molecule, and lithium hydride have been studied using both correlated and self-consistent-field (SCF) wave functions. For the first time, the effects of electron correlation on the anisotropic intracule and extracule densities are simultaneously displayed and analyzed.

The Laplacian of the intracule and extracule densities and their relationship to the shell structure of atoms

Numerical evidence of the monotonicity of both the intracule density I(r) and the extracule density E(R) in atoms are presented. However, it is found that their associated Laplacian fields ∇2E(R) and ∇2I(r) show a rich structure. Roots and extrema of these functions have been compared with the corresponding values of the Laplacian of the charge density ∇2ρ(r). Radii corresponding to minima in ∇2I(r) are found to match closely their corresponding values in ∇2ρ(r). Linear relationships between the inverses of the roots of ∇2E(R) and ∇2I(r) and the nuclear charge Z are found. Calculations demonstrate that atomic shell structure is well resolved by the odd numbered zeros of ∇2E(R) and by the radii corresponding to minima in ∇2I(r).

Analysis of the electron pair density for the ground state of carbon dioxide

The centre-of-mass or extracule and relative distance or intracule electron pair densities for the ground state of CO2 have been analysed by means of the radial, longitudinal and transverse reduced distribution functions. Calculations demonstrate that the independent-atom model (IAM) accounts for the chief characteristics of the distributions. The basis set dependence of the pair densities is illustrated by the moments of the distributions. Calculations with the 3-21G, 6-31G and 6-31G* basis sets show that augmenting the number of primitives of the various shells has a more profound effect than the addition of polarization functions. Finally analysis of increment electron-pair densities reveals that the 6-31G* basis favours small and intermediate distances with the respect to both the 3-21G and 6-31G basis set.

The evaluation of electronic extracule and intracule densities and related probability functions in terms of Gaussian basis functions

The evaluation of the basic two-electron integrals involved in the calculation of extracule and intracule densities is described. Expressions are given for the evaluation of the related spherically averaged, longitudinal, and transverse probability functions from wave functions constructed from Gaussian basis sets. All results are expressed in closed analytical forms which are suited to efficient coding. Given that certain pair densities can be related to experimental scattering cross sections, the formulae reported herein will facilitate further comparison between experiment and theory and lead to a more comprehensive understanding of the electronic structures of molecules.

An energy functional of electron-pair density

The electron-pair (or intracule) density is the probability density function for an interelectronic vector and is intimately related to the electron correlation in many-electron systems. Based on the local scaling method, a theory is presented for the direct variational determination of the electron-pair density. Illustrative applications are given for the ground state of the helium atom. Simple electron-pair density functions are reported which compare quantitatively with the exact density.