Hirshfeld electron densities

Fig. 1. The Hirshfeld electron densities (Hh) of bonded hydrogen atoms obtained from the molecular density (H2). The free hydrogen densities (H°) and the resulting electron density of the promolecule (H2) are also shown for comparison. The density values and inter-nuclear distances are in a.u. The zero cusp at nuclear positions is the artifact of the Gaussian basis set used in DFT calculations.

Keywords Hirshfeld Electron density Phosphorous halides Dipole moments... 

This approach, also often called the stockholder scheme, was introduced in 1977 by Hirshfeld [22]. The central idea of the Hirshfeld method originates in x-ray crystallography. It proposes to divide the electron density among the atoms in a molecule, guided by a promolecular density. More precisely, once a molecular geometry is known, a promolecular density p°(r) is composed by simply summing the density of each atom A (denoted p°A(r)) in an isolated state ... 

This share is used as the Hirshfeld operator. The assumption behind the Hirshfeld AIM is that this same weight operator can be used to divide the electron density of the molecule via... 

The properties of a quantum mechanical system such as an AIM are readily calculated from any method as long as they involve an operator acting on the electron density, e.g., for the case of the dipole moment. The problem would seem to become harder for other properties, although the introduction of property densities allows us to generally introduce AIM expectation values [45], The expectation value of a property A for atom a in the Hirshfeld and QCT methods can be written as... 

It is well-known that a superposition of isolated atomic densities looks remarkably much like the total electron density. Such a superposition of atomic densities is best known as a promolecular density, like it has been used by Hirshfeld [30] (see also the chapter on atoms in molecules and population analysis). Carbo-Dorca and coworkers derived a special scheme to obtain approximate electron densities via the so-called atomic shell approximation (ASA) [31-35]. Generally, for a molecule A with atoms N, a promolecular density is defined as... 

Electroneutrality may also be implemented by imposing the requirement that F(000) equal the number of electrons in the unit cell. The equation F(000) = ne can be treated as an observation, with a weight sufficient to keep the crystal practically neutral, but sufficiently small such as not to dominate the least-squares treatment. This slack constraint (Pawley 1972) has been applied in electron density analysis by Hirshfeld (1977). 

Density functional calculations of molecules, using a Hamiltonian including density functionals, frequently reproduce observed properties, such as bond and excitation energies, reaction profiles, and ionization energies (Ziegler 1991). For tetrafluoroterephthalonitrile (l,4-dicyano-2,3,5,6 tetrafluorobenzene), there is excellent agreement between the electron density from a density functional calculation (Delley 1986) and the X-ray diffraction results (Hirshfeld 1992) (see chapter 5). Avery et al. (1984) have proposed the use of experimental densities in crystals as a basis for band structure calculations. 

For a nucleus at R, the peripheral contribution to the potential 0, due to a spherical density component centered at Rj, consists of a point-charge term and a penetration term. The point-charge term is due to the nuclear charge at Rj and the electronic density within the sphere with radius R, — Ry, centered on Rjt which passes through the nucleus i (Fig. 9.1). The penetration terms are due to the electronic charge outside that sphere. They decay exponentially as the distance Rjj = R, — Rj increases (Hirshfeld and Rzotkiewicz 1974). 

The introduction of standard aspherical-atom scattering factors leads to a very significant improvement in Hirshfeld s rigid bond test. The results are a beautiful confirmation of Hirshfeld s (1992) statement that an accurate set of nuclear coordinates (and thermal parameters ) and a detailed map of the electron density can be obtained, via X-ray diffraction, only jointly and simultaneously, never separately or independently . 

Because electron density is a local property, electron density studies of the peptide-like molecules show that the nonspherical part of the deformation density (i.e the P]m parameters of Eq. 8) remain essentially the same for a given atom in the same environment (the peptide residue, a phenyl ring, a methyl group...) [29], The same observation was made for porphyrin ligands [30] and by Brock, Dunitz, and Hirshfeld [37] for naphthalene and anthracene type molecules. All these observations suggest that the multipole parameters are highly transferrable from one atom to a chemically similar atom in different molecules and crystals. A key question is is it possible to determine for each chemical type of a given atom a small set of pseudoatom multipole parameters, and can such parameters be used to calculate electrostatic properties of new molecules To answer this question [29], two accurate but low resolution X-ray data sets (sin 0/Xmax = 0.65 A-1) were... 

Let us begin a survey of this development with a brief reminder of the Hirshfeld (H) partition scheme and its information-theoretic justification. The stockholder division [34] of the molecular electron density p(r). experimental or theoretical, into the AIM densities, p(r) = X< Pa(r)>... 

Therefore (see equation (103)), these Hirshfeld-type -electron density pieces in... 

The substructures presented by the guest molecules in the inorganic waad solvates have been analyzed by estimating the extension and shape of the crystal volume enclosed by the Hirshfeld surfaces (HS) of the guests [17]. The HS of a molecule in a crystal is defined such that for every point on the HS exactly half the electron density is due to the spherically averaged non-interacting atoms comprising the... 

Generally, twinned crystals tend to have a poor effective data to parameter ratio, so they often require restraints in order to obtain a satisfactory refinement (Watkin, 1994). The following restraints can be useful distance restraints for chemically equivalent 1,2- and 1,3-distances, planarity restraints for groups such as phenyl rings, rigid bond ADP restraints (Hirshfeld, 1976 Rollett, 1970 Traeblood and Dunitz, 1983) and similar ADP restraints (Sheldrick, 1997b). Even when restraints are employed, the distribution of the displacement parameters (ORTEPplot) and residual features in a difference electron density map can be less satisfactory than for a normal structure determination. 

Hirshfeld (or stockholder) charges are based on using atomic densities for partitioning the molecular electron density. The promolecular density is defined as the sum of atomic densities placed at the nuclear geometries in the molecule. The actual molecular electron density at each point in space is then partitioned by weighting factors according to the promolecular contributions. 

Bader, R. F. W. (1990) Atoms in molecules, Oxford Science Publications, Oxford. Hirshfeld, F. L. (1992) The role of electron density in X-ray crystallography, in A. Domenicano and I. Hargittai, (eds.). Accurate molecular structures, Oxford Science Publications, New-York. 

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