Electron pair density

The theory as presented so far is clearly incomplete. The topology of the density, while recovering the concepts of atoms, bonds and structure, gives no indication of the localized bonded and non-bonded pairs of electrons of the Lewis model of structure and reactivity, a model secondary in importance only to the atomic model. The Lewis model is concerned with the pairing of electrons, information contained in the electron pair density and not in the density itself. Remarkably enough however, the essential information about the spatial pairing of electrons is contained in the Laplacian of the electron density, the sum of the three second derivatives of the density at each point in space, the quantity V2p(r) [44]. 

Note that the bond order index defined by Mayer accounts for the covalent contribution to the bond (this is why of late it is often mentioned as shared electron pair density index, SEDI). As such, the index cannot be expected to produce the integer values corresponding to the Lewis picture if a bond has a significant ionic contribution. The bond order index defined in this way measures the degree of correlation of the fluctuation of electron densities on the two atoms in question [7]. 

Fig. 6. A) Orbital representation B) Electron densities q ) C) Electron pair densities Q ri, ) of the singlet and triplet functions of two electrons in a one-dimensional box. The two axes are n and j-2...

The similarities and differences between methods based on the electron density, electron-pair density, and reduced density matrices have recently been reviewed [5]. This chapter is not intended as a comprehensive review, but as a focused consideration of A-representability constraints that are applicable to diagonal elements of reduced density matrices. Such constraints are useful both to researchers working with the Q-density and to researchers working with g-electron reduced density matrices, and so we shall attempt to review these constraints in a way that is accessible to both audiences. Our focus is on inequalities that arise from the Slater hull because the Slater hull provides an exhaustive list of A-representability conditions for the diagonal elements of the density matrix, Although the Slater hull constraints... 

P. W. Ayers and M. Levy, Generalized density-functional theory conquering the A-representability problem with exact functionals for the electron pair density and the second-order reduced density matrix. J. Chem. Set 117, 507-514 (2005). 

E. R. Davidson, A-representabiUty of the electron-pair density. Chem. Phys. Lett. 246, 209-213... 

The Laplacian of the electron density plays a dominant role throughout the theory.191 In addition, Bader has shown that the topology of the Laplacian recovers the Lewis model of the electron pair, a model that is not evident in the topology of the electron density itself. The Laplacian of the density thus provides a physical valence-shell electron pair repulsion (VSEPR) basis for the model of molecular geometry and for the prediction of the reaction sites and their relative alignment in acid-base reactions. This work is closely tied to earlier studies by Bader of the electron pair density, demonstrating that the spatial localization of electrons is a result of a corresponding localization of the Fermi correlation hole. 

Figure 18. Electron pair density, h(u), its radial derivative, h (u ), and their difference, for H, calculated using a 204 term Hylleraas-type wavefunction, as described in Ref. [71],...

Yrs are two-centre two-electron repulsion energies. For aromatic ions, the onsite electron pair densities, i.e., the diagonal elements of the spinless second order density matrix are lower than those of any classical structure. For radicals, however, the pair densities are increased relative to those in the corresponding classical structures. Thus the resonance energy of the ion exceeds that of the radical by 21 Ec I [33]. FIMO and PMO do not differentiate between the formation of an arylmethyl radical and its carbocation. Empirically, radicals are better described by these methods this has been related to the constant Coulson charge order Q = 1 for arylmethyl radicals as opposed to the variable rt-charge order 1 on arylmethyl cations [16,39]. The actual PPP values of Ec have been correlated to an excellent accuracy to their PMO-o) counterparts [16]. 

The corresponding second-order density matrix yields the electron pair-density upon setting x[ = ri and r2 = t2. 

These qualitative considerations can be put into quantitative terms by probability holes. K electrons did not have charge or spin, the probability of finding an electron at a given position would be independent of the position of a second electron, and the electron pair-density P2 would be given as a simple product of two one-electron densities pi, with a proper normalization factor. 

The ELF has been extensively applied to a large number of systems and has also been used to quantify chemical concepts like the strength of the hydrogen bond [26] and aromaticity [27, 28]. Moreover, other interesting ELF-like scalar fields have been developed, such as the localized orbital locator (LOL) [29], based on the comparison of the local noninteracting kinetic energy density with that of the uniform electron gas, and the electron localizabihty indicator (ELI) [30-32], derived directly from the electron pair density without any reference to the uniform electron gas. 

Molina et joined a lively debate on the nature of multiple bonding in gallynes . They combined a topological analysis of the electron pair density and p and decided that the Ga-Ga bond in bent [HGaGaH] was a result of the sharing of two electron pairs at the HF level. 

Hypervalency. - Dobado et used the electron pair density in... 

Pantano et al used Car-Parrinello MD to compute the influence of thermal fluctuations on the proton hopping properties of a model substrate for enzymatic reactions, benzoylacetone. A plot showing the projection of ELF on the plane defined by the O-H-0 moiety unambiguously demonstrates the presence of electron pair density between the hydrogen and the two H-bond donors, suggesting the presence of a three-centre covalent bond. The character of the bond is o, as evidenced by analysing the Kohn-Sham one-electron wavefimction, in full agreement with AIM. 

The contributions of the correlated and uncorrelated components of the electron-pair density to atomic and molecular intracule I and extracule E densities and their Laplacian functions were analysed at the HF and Cl level. The correlated components of the / and E densities, and their associated Laplacian functions, reveal the short-ranged nature and high isotropy of Fermi and Coulomb correlation in atoms and molecules. In general, it has been found that the uncorrelated / and E have the same topological structure as the parent functions I and E functions. 

Poater et presented new insights in chemical reactivity reactivity drawn from an analysis of electron pairing. They judged that a physically accurate description of the electron pairing in atoms and molecules had to be based on the electron-pair density. Within AIM atomic localization and delocalization indices could be defined that describe the infra- and interatomic distribution of the electron pairs in a molecule. The authors considered these indices as a physically... 

Developing improved approximations to the energy density functional f lp] is very difficult. More than 80 years after the first approximations were proposed, approximating the energy functional r ains a very active area of research. -" This frustratingly slow progress has motivated researchers to consider alternative descriptors that contain more information than the electron density but less information that the electronic wave function. For example, electron pair density (or two-electron distribution function) can also be used as the fundamental descriptor of an electronic system. More generally, k-density (or fe-electron distribution function)... 

Fig. 1. This figure illustrates the geometry of the C—S—S—C system. The third nonbonding orbitals on each sulfur atom (p,) are indicated only to show the orientation of the unshared electron-pair density in each one, assuming no s or d hybridization.

The LDM requires for its determination a quantum chemical calculation since the calculation of the Lis and DIs requires the electron density and the electron pair density contained in the second order density matrix which is inaccessible from experiment [29]. 

Kohout M (2007) Bonding indicators from electron pair density functionals. Faraday Discuss 135 43-54... 

The oft-quoted result of antisynunetrizing the wave function with respect to the permutation of the space-spin coordinates of every pair of electrons is that no two electrons with the same spin can occupy the same point in space. In chemistry, however, one is most interested in the spatial distribution of the electrons. To determine the manner in which the exclusion principle affects the electron distribution and its properties in real space, one must determine how many pairs of electrons, on the average, contribute to the electron density over the region of interest. This information is given by the electron pair density, alternatively called the pair probability function. 

The domain-averaged hole density, which is the starting entity for the evaluation of Fermi orbitals as well as the first step in the calculation of delocalization indices (DI), is computed from the exchange-correlation part pxc of the electron-pair density. At the one-determinantal level of... 

In the following a special class of functions, termed electron localizability indicators [3-8], based on simultaneous evaluation of electron density and electron pair density will be described. This combination is utilized with the aim to analyze the correlation of electronic motion [9]. Apart from the density function point of view, the energy of a molecule can be thought as stemming from two parts - a one-particle terms in wide sense derived from the electron density and a two-particle terms derived from the electron pair density (of course, the full 2-matrix is stiU necessary today). The interplay between the electron density on the one hand and the electron pair density on the other hand could thus elucidate the situation in the molecular system. 

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