Promolecule density

A common reference density, first used by Roux and Daudel (1955), is the superposition of spherical ground-state atoms, centered at the nuclear positions. It is referred to as the promolecule density, or simply the promolecule, as it represents the ensemble of randomly oriented, independent atoms prior to interatomic bonding. It is a hypothetical entity that violates the Pauli exclusion principle. Nevertheless, the promolecule is electrostatically binding if only the electrostatic interactions would exist, the promolecule would be stable (Hirshfeld and Rzotkiewicz 1974). The difference density calculated with the promolecule reference state is commonly called the deformation density, or the standard deformation density. It is the difference between the total density and the density corresponding to the sum of the spherical ground-state atoms located at the positions R... 

Hirshfeld (1984) found the electrostatic charge balance at the F nuclei, based on the experimental deformation density, to be several times more repulsive (i.e., anti-bonding) than that of the promolecule. Very sharp dipolar functions at the exocyclic C, N, and F atoms, oriented along the local bonds, were introduced in a new refinement in which the coefficients of the sharp functions were constrained to satisfy the electrostatic Hellmann-Feynman theorem (chapter 4). The electrostatic imbalance was corrected with negligible changes in the other parameters of the structure. The model deformation maps were virtually unaffected, except for the innermost contour around the nuclear sites. 

Accordingly, each atom is assigned a fraction of the charge density at a point proportional to its investment in the promolecule density at that point. This is the basis of the stockholder concept. We note that Eq. (6.3) can be reformulated as... 

The stockholder recipe partitions the density according to each atom s contribution to the promolecule density. The partitioned fragment distributions... 

The promolecule density shows (3, — 1) critical points along the bond paths, just like the molecule density. But, as the promolecule is hypothetical and violates the exclusion principle, it would be incorrect to infer that the atoms in the promolecule are chemically bonded. In a series of topological analyses, Stewart (1991) has compared the model densities and promolecule densities of urea,... 

The promolecule is the sum over spherical atom densities, so we may write... 

In the derivation of the traceless quadrupole moments from the electrostatic moments, the spherical components are subtracted. Thus, the quadrupole moments can be derived from the second moments, but the opposite is not the case. Spackman (1992) notes that the subtraction introduces an ambiguity in the comparison of quadrupole moments from theory and experiment. The spherical component subtracted is not that of the promolecule, but is based on the distribution itself. It is therefore generally not the same in the two densities being compared. On the other hand, the moments as defined by Eq. (7.1) are based on the total density without the intrusion of a reference state. 

When the electrostatic properties are evaluated by AF summation, the effect of the spherical-atom molecule must be evaluated separately. According to electrostatic theory, on the surface of any spherical charge distribution, the distribution acts as if concentrated at its center. Thus, outside the spherical-atom molecule s density, the potential due to this density is zero. At a point inside the distribution the nuclei are incompletely screened, and the potential will be repulsive, that is, positive. Since the spherical atom potential converges rapidly, it can be evaluated in real space, while the deformation potential A(r) is evaluated in reciprocal space. When the promolecule density, rather than the superposition of rc-modified non-neutral spherical-atom densities advocated by Hansen (1993), is evaluated in direct space, the pertinent expressions are given by (Destro et al. 1989)... 

The term <9>Sphericai-atom crystal (r)> in ET (8-33) is the average potential in the unit cell of the promolecule crystal, equal to 0(0) for the promolecule crystal. Expression (8.33) thus gives the deviation from the average promolecule potential in the crystal. Modification of Eq. (8.33) for the direct space evaluation of the K-modified non-neutral spherical atom densities is straightforward. 

An experiment is performed to evaluate the deformation density in an O—O bond in a molecule. Two different reference states are used to calculate deformation densities. The first is the spherical-atom reference state (the promolecule density) the second is a prepared-atom reference state in which oxygen atoms have the configuration... 

The Voronoi deformation density approach, is based on the partitioning of space into the Voronoi cells of each atom A, that is, the region of space that is closer to that atom than to any other atom (cf. Wigner-Seitz cells in crystals see Chapter 1 of Ref. 202). The VDD charge of an atom A is then calculated as the difference between the (numerical) integral of the electron density p of the real molecule and the superposition of atomic densities SpB of the promolecule in its Voronoi cell (Eq. [42]) ... 

Deformation density, as originally conceptualized represents the difference between crystallographically observed electron density and calculated densities of the spherical atoms, which consitute the so-called promolecule. The effects of vibrational displacement, represented by ellipsoids in Figure 5.19, and ignored when defining a promolecule by spherical atoms, are most likely... 

The same type of calculations have been performed using experimental X-ray structure factors on crystalline phosphoric acid, 7V-acetyl-a,P-dehydrophenyl-alamine methylamide, and N-acetyl-1 -tryptophan methylamide by Souhassou [60] on urea, 9-methyladenosine, and imidazole by Stewart [32] and on 1-alanine [61] and annulene derivatives [62] by Destro and co-workers. The latter authors collected their X-ray data at 16 K [63]. Stewart [32] showed that the positions of the (3, -1) critical points from the promolecule are very close to those of the multipole electron density, but that large differences appear in comparing the density, the Laplacian maps, and the ellipticities at the critical points. Destro et al. [67] showed that the results obtained may be slightly dependent on the refinement model. 

X-rays are scattered predominantly by electrons rather than atomic nuclei. To determine atomic coordinates, electron densities are therefore assumed to be concentrated spherically around individual nuclei. This assumption ignores all possible effects that chemical bonding may have on electronic density in molecules. Such a hypothetical array of spherical atoms located at the nuclear positions of an actual molecule in a crystal is known as a promolecule. Molecular structures determined by routine crystallographic methods are invariably the structures of promolecules. 

A more realistic outline of a molecular surface can be defined in terms of the outer contours of electron density according to Bader [173]. To avoid excessive computation the densities of large molecules may be built up from previously calculated densities of smaller fragments [213]. The most obvious approach, to approximate molecular density by the sum of atomic densities over the promolecule has also been explored [214]. This approach works well since the deformation density associated with bond formation is small compared to the total density [215]. The total density may therefore be represented by a sum over spherically averaged atomic densities, p(r) = Pa(t)-... 

Using atomic densities calculated from tabulated atomic wave functions, the summation was found [214] to produce results equivalent to the most elaborate molecular Hartree-Fock calculations for a series of small molecules, at a fraction of the computing expense. Surface areas and volumes computed by the two methods were found virtually identical. The promolecule calculation therefore has an obvious advantage in the exploration of surface electron densities, surface areas and molecular volumes of macromolecules for the analysis of molecular recognition. 

The atoms-in-molecules partitioning of electron density (6.3.1) can now be seen in different perspective. The total crystallographically measured electron density is essentially that of the promolecule, which by definition must partition into atomic densities. Calculated densities, on the other hand, can only be obtained after assuming a set of nuclear coordinates. Partitioning into a set of atomic basins therefore simply demonstrates a degree of self-consistency between synthesis and analysis of the density function. 

The difference density distribution is defined in equation 5 Ap(r) = p[molecule]-p[promolecule]... 

Deformation density The difference between the electron density in a molecule, with all its distortions as a result of bonding, and the promolecule density, obtained by forming a molecule with spherical electron density around each atom (free atoms). This map contains effects caused both by the errors in the relative phases of Bragg reflections, experimental errors in the data, and inadequacies in the representations of the scattering factors of free atoms. 

Promolecule density The electron density of spherically symmetrical free atoms with no effects of chemical bonding or other factors that distort the electron density. 

Spackman, M. A. (1999) Hydrogen bond energetics from topological analysis of experimental electron densities recognizing the importance of the promolecule, Chem. Phys. Lett. 301, 425-429. 

The conventional structure factor formalism utilized in standard structure determinations invokes the concept of the promolecule the superposition of isolated (spherical) atomic densities derived, for example, via the Hartree-Fock procedure [45], While this model mimics the dominant topological features of the ED (local maxima at the nuclear positions) reasonably well, it completely neglects density deformations due to bonding. Unfortunately, this omission leads to biases in estimates of the structural [46,47] and thermal parameters [48]. 

Theoretical studies show that weak C H- O HBs have an extremely low signal at the BCP. A direct comparison of pg p with that of the promolecule, and the analysis of the interaction density, suggest that these contacts manifest themselves mainly in the basins of the participating atoms rather than at the BCP. Bond analyses relaying on local topological figures of a model ED extracted from the X-ray diffraction data must confine with these limitations in order to avoid over-interpretations. 

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.

---