Topology of electron–electron interactions in atoms and molecules

Jerzy Cioslowski and Guanghua Liu

Department of Chemistry and Supercomputer Computations Research Institute, Florida State University, Tallahassee, Florida 

I. The Hartree–Fock approximation

ABSTRACT

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.

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 http://chem.thu.edu.tw/files/writing_journal/15/43_6624dc9a.pdf

II. The correlation cage

ABSTRACT

The concept of the correlation cage provides new insights into electron–electron interactions in atoms and molecules. The cage constitutes the domain in the space of interelectron distance vectors R within which correlation effects are substantial. Its shape and size are entirely determined by the topological properties of the electron intracule density I(R),$\mathrm{I(}{R}\mathrm{),}$ thus avoiding any references to ill-defined “uncorrelated” quantities. Integration of observables related to I(R)$\mathrm{I(}{R})$ over the correlation cage affords quantitative measures of electron correlation. The number of strongly correlated electron pairs Mcorr[I],$M_{\mathrm{corr}}\mathrm{[I],}$their electron–electron repulsion energy Wcorr[I],$W_{\mathrm{corr}}\mathrm{[I],}$ and the cage volume Vcorr[I]$V_{\mathrm{corr}}\mathrm{[I]}$ that characterizes the spatial extent of electron correlation are functionals of I(R).$\mathrm{I(}{R}\mathrm{).}$ The ratio κ[I]$\mathrm{\kappa [I]}$of I(0)Vcorr[I]$\mathrm{I(}{0}{\mathrm{)V}}_{\mathrm{corr}}\mathrm{[I]}$ and Mcorr[I],$M_{\mathrm{corr}}\mathrm{[I],}$ which measures the strength of short-range correlation effects, is small for systems such as H−${\mathrm{H}}^{\mathrm{-}}$ and closer to one for those with weaker correlation effects.

To download the article click on the following link:

http://chem.thu.edu.tw/files/writing_journal/15/44_e30804bb.pdf

III. Morphology of electron intracule density in two 1Σ+g${}^1{\Sigma }_g^{+}$ states of the hydrogen molecule

ABSTRACT

The differences in electronic structures of two 1Σ+g${}^1{\Sigma }_g^{+}$ states of the hydrogen molecule are vividly reflected in their intracule densities I(r).$\mathrm{I(}{r}\mathrm{).}$ The ground-state wave function of H2${\mathrm{H}}_2$ is associated with two distinct topologies of I(r)$\mathrm{I(}{r})$ (one of which pertains to the united atom limit), whereas no fewer than 11 unequivalent sets of critical entities are found for I(r)$\mathrm{I(}{r})$ of the EF state that involves multiple electronic configurations. These sets and the catastrophes that interrelate them, which arise from conflicts between topological features of I(r)$\mathrm{I(}{r})$pertinent to different configurations, are characterized in detail. The usefulness of topological analysis of I(r)$\mathrm{I(}{r})$ in the detection and characterization of various types of electron correlation is demonstrated.

To download the article click on the following link:

http://chem.thu.edu.tw/files/writing_journal/15/45_09fc51aa.pdf